LMFDB - Embedded newform 15.8.b.a.4.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [15,8,Mod(4,15)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(15, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("15.4");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 15 = 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	15.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$4.68577538226$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 162x^{6} + 7361x^{4} + 87300x^{2} + 160000 $ x^8 + 162x^6 + 7361x^4 + 87300*x^2 + 160000
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{12}\cdot 5^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4.2
Root			$-1.49423i$ of defining polynomial
Character	$\chi$	$=$	15.4
Dual form			15.8.b.a.4.7

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-17.7812i q^{2} -27.0000i q^{3} -188.170 q^{4} +(57.4967 - 273.531i) q^{5} -480.092 q^{6} +1233.23i q^{7} +1069.90i q^{8} -729.000 q^{9} +O(q^{10})$	\(q-17.7812i q^{2} -27.0000i q^{3} -188.170 q^{4} +(57.4967 - 273.531i) q^{5} -480.092 q^{6} +1233.23i q^{7} +1069.90i q^{8} -729.000 q^{9} +(-4863.70 - 1022.36i) q^{10} +1214.92 q^{11} +5080.60i q^{12} -13536.0i q^{13} +21928.3 q^{14} +(-7385.33 - 1552.41i) q^{15} -5061.70 q^{16} -12095.7i q^{17} +12962.5i q^{18} +32874.5 q^{19} +(-10819.2 + 51470.4i) q^{20} +33297.2 q^{21} -21602.7i q^{22} +9426.25i q^{23} +28887.3 q^{24} +(-71513.3 - 31454.2i) q^{25} -240685. q^{26} +19683.0i q^{27} -232058. i q^{28} +79951.8 q^{29} +(-27603.7 + 131320. i) q^{30} +113548. q^{31} +226950. i q^{32} -32802.9i q^{33} -215076. q^{34} +(337327. + 70906.7i) q^{35} +137176. q^{36} +518996. i q^{37} -584547. i q^{38} -365471. q^{39} +(292651. + 61515.8i) q^{40} +604700. q^{41} -592064. i q^{42} -291215. i q^{43} -228612. q^{44} +(-41915.1 + 199404. i) q^{45} +167610. q^{46} -1.09050e6i q^{47} +136666. i q^{48} -697316. q^{49} +(-559293. + 1.27159e6i) q^{50} -326585. q^{51} +2.54707e6i q^{52} +305557. i q^{53} +349987. q^{54} +(69854.0 - 332319. i) q^{55} -1.31944e6 q^{56} -887611. i q^{57} -1.42164e6i q^{58} -144206. q^{59} +(1.38970e6 + 292118. i) q^{60} -80099.7 q^{61} -2.01901e6i q^{62} -899026. i q^{63} +3.38755e6 q^{64} +(-3.70250e6 - 778272. i) q^{65} -583274. q^{66} +3.79572e6i q^{67} +2.27606e6i q^{68} +254509. q^{69} +(1.26081e6 - 5.99807e6i) q^{70} +3.66680e6 q^{71} -779958. i q^{72} +1.67463e6i q^{73} +9.22836e6 q^{74} +(-849264. + 1.93086e6i) q^{75} -6.18601e6 q^{76} +1.49828e6i q^{77} +6.49850e6i q^{78} -3.76132e6 q^{79} +(-291031. + 1.38453e6i) q^{80} +531441. q^{81} -1.07523e7i q^{82} +1.77821e6i q^{83} -6.26556e6 q^{84} +(-3.30856e6 - 695464. i) q^{85} -5.17814e6 q^{86} -2.15870e6i q^{87} +1.29985e6i q^{88} -4.41045e6 q^{89} +(3.54564e6 + 745300. i) q^{90} +1.66930e7 q^{91} -1.77374e6i q^{92} -3.06578e6i q^{93} -1.93904e7 q^{94} +(1.89017e6 - 8.99219e6i) q^{95} +6.12766e6 q^{96} +4.60573e6i q^{97} +1.23991e7i q^{98} -885678. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 666 q^{4} - 444 q^{5} + 486 q^{6} - 5832 q^{9}+O(q^{10}) $ 8 * q - 666 * q^4 - 444 * q^5 + 486 * q^6 - 5832 * q^9	$ 8 q - 666 q^{4} - 444 q^{5} + 486 q^{6} - 5832 q^{9} - 6686 q^{10} + 10752 q^{11} - 13524 q^{14} - 17496 q^{15} + 86530 q^{16} - 30464 q^{19} + 87444 q^{20} + 64152 q^{21} - 110322 q^{24} + 127616 q^{25} - 793524 q^{26} + 240072 q^{29} - 172044 q^{30} + 233728 q^{31} + 184748 q^{34} + 593520 q^{35} + 485514 q^{36} - 454896 q^{39} - 1147102 q^{40} + 507648 q^{41} + 2578572 q^{44} + 323676 q^{45} + 662408 q^{46} - 3267160 q^{49} - 5117736 q^{50} + 264384 q^{51} - 354294 q^{54} + 3525696 q^{55} + 705660 q^{56} + 1091424 q^{59} + 5307606 q^{60} - 6433520 q^{61} - 568594 q^{64} - 2555592 q^{65} - 2382372 q^{66} - 5940864 q^{69} + 12097800 q^{70} - 1381824 q^{71} + 23961276 q^{74} + 7768224 q^{75} + 13115664 q^{76} - 14380160 q^{79} - 31251876 q^{80} + 4251528 q^{81} - 36423756 q^{84} - 1452008 q^{85} - 19837608 q^{86} + 45778896 q^{89} + 4874094 q^{90} + 24075648 q^{91} - 45728896 q^{94} - 25774656 q^{95} + 33586002 q^{96} - 7838208 q^{99}+O(q^{100}) $ 8 * q - 666 * q^4 - 444 * q^5 + 486 * q^6 - 5832 * q^9 - 6686 * q^10 + 10752 * q^11 - 13524 * q^14 - 17496 * q^15 + 86530 * q^16 - 30464 * q^19 + 87444 * q^20 + 64152 * q^21 - 110322 * q^24 + 127616 * q^25 - 793524 * q^26 + 240072 * q^29 - 172044 * q^30 + 233728 * q^31 + 184748 * q^34 + 593520 * q^35 + 485514 * q^36 - 454896 * q^39 - 1147102 * q^40 + 507648 * q^41 + 2578572 * q^44 + 323676 * q^45 + 662408 * q^46 - 3267160 * q^49 - 5117736 * q^50 + 264384 * q^51 - 354294 * q^54 + 3525696 * q^55 + 705660 * q^56 + 1091424 * q^59 + 5307606 * q^60 - 6433520 * q^61 - 568594 * q^64 - 2555592 * q^65 - 2382372 * q^66 - 5940864 * q^69 + 12097800 * q^70 - 1381824 * q^71 + 23961276 * q^74 + 7768224 * q^75 + 13115664 * q^76 - 14380160 * q^79 - 31251876 * q^80 + 4251528 * q^81 - 36423756 * q^84 - 1452008 * q^85 - 19837608 * q^86 + 45778896 * q^89 + 4874094 * q^90 + 24075648 * q^91 - 45728896 * q^94 - 25774656 * q^95 + 33586002 * q^96 - 7838208 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$.

$n$	$7$	$11$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	17.7812i		−	1.57165i	−0.618449	−	0.785825i	$-0.712239\pi$
$2$		−	17.7812i		−	1.57165i	0.618449	−	0.785825i	$-0.287761\pi$
$3$		−	27.0000i		−	0.577350i
$3$		−	27.0000i		−	0.577350i
$4$	−188.170			−1.47008
$5$	57.4967	−	273.531i	0.205706	−	0.978614i
$5$	57.4967	−	273.531i	0.205706	−	0.978614i
$6$	−480.092			−0.907392
$7$			1233.23i			1.35894i	0.733702	+	0.679472i	$0.237791\pi$
$7$			1233.23i			1.35894i	−0.733702	+	0.679472i	$0.762209\pi$
$8$			1069.90i			0.738803i
$9$	−729.000			−0.333333
$10$	−4863.70	−	1022.36i	−1.53804	−	0.323298i
$11$	1214.92			0.275216			0.137608	−	0.990487i	$-0.456059\pi$
$11$	1214.92			0.275216			0.137608	+	0.990487i	$0.456059\pi$
$12$			5080.60i			0.848752i
$13$		−	13536.0i		−	1.70878i	−0.519629	−	0.854392i	$-0.673930\pi$
$13$		−	13536.0i		−	1.70878i	0.519629	−	0.854392i	$-0.326070\pi$
$14$	21928.3			2.13578
$15$	−7385.33	−	1552.41i	−0.565003	−	0.118765i
$16$	−5061.70			−0.308942
$17$		−	12095.7i		−	0.597119i	−0.954391	−	0.298559i	$-0.903494\pi$
$17$		−	12095.7i		−	0.597119i	0.954391	−	0.298559i	$-0.0965061\pi$
$18$			12962.5i			0.523883i
$19$	32874.5			1.09957			0.549783	−	0.835307i	$-0.314710\pi$
$19$	32874.5			1.09957			0.549783	+	0.835307i	$0.314710\pi$
$20$	−10819.2	+	51470.4i	−0.302405	+	1.43864i
$21$	33297.2			0.784586
$22$		−	21602.7i		−	0.432543i
$23$			9426.25i			0.161544i	0.996733	+	0.0807721i	$0.0257386\pi$
$23$			9426.25i			0.161544i	−0.996733	+	0.0807721i	$0.974261\pi$
$24$	28887.3			0.426548
$25$	−71513.3	−	31454.2i	−0.915370	−	0.402614i
$26$	−240685.			−2.68561
$27$			19683.0i			0.192450i
$28$		−	232058.i		−	1.99776i
$29$	79951.8			0.608745			0.304373	−	0.952553i	$-0.401553\pi$
$29$	79951.8			0.608745			0.304373	+	0.952553i	$0.401553\pi$
$30$	−27603.7	+	131320.i	−0.186656	+	0.887986i
$31$	113548.			0.684560			0.342280	−	0.939598i	$-0.388801\pi$
$31$	113548.			0.684560			0.342280	+	0.939598i	$0.388801\pi$
$32$			226950.i			1.22435i
$33$		−	32802.9i		−	0.158896i
$34$	−215076.			−0.938462
$35$	337327.	+	70906.7i	1.32988	+	0.279543i
$36$	137176.			0.490027
$37$			518996.i			1.68445i	0.539127	+	0.842224i	$0.318754\pi$
$37$			518996.i			1.68445i	−0.539127	+	0.842224i	$0.681246\pi$
$38$		−	584547.i		−	1.72813i
$39$	−365471.			−0.986567
$40$	292651.	+	61515.8i	0.723003	+	0.151977i
$41$	604700.			1.37024			0.685119	−	0.728431i	$-0.259750\pi$
$41$	604700.			1.37024			0.685119	+	0.728431i	$0.259750\pi$
$42$		−	592064.i		−	1.23309i
$43$		−	291215.i		−	0.558565i	−0.960209	−	0.279282i	$-0.909903\pi$
$43$		−	291215.i		−	0.558565i	0.960209	−	0.279282i	$-0.0900965\pi$
$44$	−228612.			−0.404590
$45$	−41915.1	+	199404.i	−0.0685688	+	0.326205i
$46$	167610.			0.253891
$47$		−	1.09050e6i		−	1.53209i	−0.642790	−	0.766043i	$-0.722223\pi$
$47$		−	1.09050e6i		−	1.53209i	0.642790	−	0.766043i	$-0.277777\pi$
$48$			136666.i			0.178368i
$49$	−697316.			−0.846727
$50$	−559293.	+	1.27159e6i	−0.632768	+	1.43864i
$51$	−326585.			−0.344747
$52$			2.54707e6i			2.51205i
$53$			305557.i			0.281921i	0.990015	+	0.140960i	$0.0450190\pi$
$53$			305557.i			0.281921i	−0.990015	+	0.140960i	$0.954981\pi$
$54$	349987.			0.302464
$55$	69854.0	−	332319.i	0.0566137	−	0.269330i
$56$	−1.31944e6			−1.00399
$57$		−	887611.i		−	0.634835i
$58$		−	1.42164e6i		−	0.956734i
$59$	−144206.			−0.0914114			−0.0457057	−	0.998955i	$-0.514554\pi$
$59$	−144206.			−0.0914114			−0.0457057	+	0.998955i	$0.514554\pi$
$60$	1.38970e6	+	292118.i	0.830600	+	0.174594i
$61$	−80099.7			−0.0451831			−0.0225916	−	0.999745i	$-0.507192\pi$
$61$	−80099.7			−0.0451831			−0.0225916	+	0.999745i	$0.507192\pi$
$62$		−	2.01901e6i		−	1.07589i
$63$		−	899026.i		−	0.452981i
$64$	3.38755e6			1.61531
$65$	−3.70250e6	−	778272.i	−1.67224	−	0.351508i
$66$	−583274.			−0.249729
$67$			3.79572e6i			1.54181i	0.636948	+	0.770907i	$0.280197\pi$
$67$			3.79572e6i			1.54181i	−0.636948	+	0.770907i	$0.719803\pi$
$68$			2.27606e6i			0.877814i
$69$	254509.			0.0932676
$70$	1.26081e6	−	5.99807e6i	0.439344	−	2.09011i
$71$	3.66680e6			1.21586			0.607929	−	0.793991i	$-0.292000\pi$
$71$	3.66680e6			1.21586			0.607929	+	0.793991i	$0.292000\pi$
$72$		−	779958.i		−	0.246268i
$73$			1.67463e6i			0.503836i	0.967749	+	0.251918i	$0.0810613\pi$
$73$			1.67463e6i			0.503836i	−0.967749	+	0.251918i	$0.918939\pi$
$74$	9.22836e6			2.64736
$75$	−849264.	+	1.93086e6i	−0.232449	+	0.528489i
$76$	−6.18601e6			−1.61645
$77$			1.49828e6i			0.374003i
$78$			6.49850e6i			1.55054i
$79$	−3.76132e6			−0.858312			−0.429156	−	0.903230i	$-0.641189\pi$
$79$	−3.76132e6			−0.858312			−0.429156	+	0.903230i	$0.641189\pi$
$80$	−291031.	+	1.38453e6i	−0.0635513	+	0.302335i
$81$	531441.			0.111111
$82$		−	1.07523e7i		−	2.15353i
$83$			1.77821e6i			0.341358i	0.985327	+	0.170679i	$0.0545961\pi$
$83$			1.77821e6i			0.341358i	−0.985327	+	0.170679i	$0.945404\pi$
$84$	−6.26556e6			−1.15341
$85$	−3.30856e6	−	695464.i	−0.584349	−	0.122831i
$86$	−5.17814e6			−0.877868
$87$		−	2.15870e6i		−	0.351459i
$88$			1.29985e6i			0.203331i
$89$	−4.41045e6			−0.663159			−0.331580	−	0.943427i	$-0.607582\pi$
$89$	−4.41045e6			−0.663159			−0.331580	+	0.943427i	$0.607582\pi$
$90$	3.54564e6	+	745300.i	0.512679	+	0.107766i
$91$	1.66930e7			2.32214
$92$		−	1.77374e6i		−	0.237483i
$93$		−	3.06578e6i		−	0.395231i
$94$	−1.93904e7			−2.40790
$95$	1.89017e6	−	8.99219e6i	0.226188	−	1.07605i
$96$	6.12766e6			0.706880
$97$			4.60573e6i			0.512386i	0.966626	+	0.256193i	$0.0824682\pi$
$97$			4.60573e6i			0.512386i	−0.966626	+	0.256193i	$0.917532\pi$
$98$			1.23991e7i			1.33076i
$99$	−885678.			−0.0917387
$100$	1.34567e7	+	5.91876e6i	1.34567	+	0.591876i
$101$	−1.31837e7			−1.27325			−0.636625	−	0.771174i	$-0.719670\pi$
$101$	−1.31837e7			−1.27325			−0.636625	+	0.771174i	$0.719670\pi$
$102$			5.80706e6i			0.541821i
$103$			3.73897e6i			0.337149i	0.985689	+	0.168574i	$0.0539163\pi$
$103$			3.73897e6i			0.337149i	−0.985689	+	0.168574i	$0.946084\pi$
$104$	1.44821e7			1.26246
$105$	1.91448e6	−	9.10782e6i	0.161394	−	0.767807i
$106$	5.43317e6			0.443080
$107$			3.96146e6i			0.312617i	0.987708	+	0.156308i	$0.0499594\pi$
$107$			3.96146e6i			0.312617i	−0.987708	+	0.156308i	$0.950041\pi$
$108$		−	3.70376e6i		−	0.282917i
$109$	1.46166e7			1.08107			0.540536	−	0.841321i	$-0.318222\pi$
$109$	1.46166e7			1.08107			0.540536	+	0.841321i	$0.318222\pi$
$110$	−5.90902e6	−	1.24209e6i	−0.423293	−	0.0889769i
$111$	1.40129e7			0.972517
$112$		−	6.24225e6i		−	0.419834i
$113$		−	2.16805e7i		−	1.41350i	−0.707465	−	0.706749i	$-0.750161\pi$
$113$		−	2.16805e7i		−	1.41350i	0.707465	−	0.706749i	$-0.249839\pi$
$114$	−1.57828e7			−0.997738
$115$	2.57837e6	+	541978.i	0.158089	+	0.0332307i
$116$	−1.50446e7			−0.894905
$117$			9.86771e6i			0.569595i
$118$			2.56415e6i			0.143667i
$119$	1.49168e7			0.811451
$120$	1.66093e6	−	7.90158e6i	0.0877437	−	0.417426i
$121$	−1.80111e7			−0.924256
$122$			1.42427e6i			0.0710120i
$123$		−	1.63269e7i		−	0.791108i
$124$	−2.13663e7			−1.00636
$125$	−1.27155e7	+	1.77526e7i	−0.582301	+	0.812973i
$126$	−1.59857e7			−0.711928
$127$		−	5.84840e6i		−	0.253352i	−0.991944	−	0.126676i	$-0.959569\pi$
$127$		−	5.84840e6i		−	0.253352i	0.991944	−	0.126676i	$-0.0404308\pi$
$128$		−	3.11850e7i		−	1.31435i
$129$	−7.86279e6			−0.322487
$130$	−1.38386e7	+	6.58348e7i	−0.552447	+	2.62817i
$131$	9.82754e6			0.381940			0.190970	−	0.981596i	$-0.438837\pi$
$131$	9.82754e6			0.381940			0.190970	+	0.981596i	$0.438837\pi$
$132$			6.17253e6i			0.233590i
$133$			4.05419e7i			1.49425i
$134$	6.74923e7			2.42319
$135$	5.38391e6	+	1.13171e6i	0.188334	+	0.0395882i
$136$	1.29412e7			0.441154
$137$			2.59385e7i			0.861833i	0.902392	+	0.430916i	$0.141810\pi$
$137$			2.59385e7i			0.861833i	−0.902392	+	0.430916i	$0.858190\pi$
$138$		−	4.52547e6i		−	0.146584i
$139$	9.06956e6			0.286440			0.143220	−	0.989691i	$-0.454254\pi$
$139$	9.06956e6			0.286440			0.143220	+	0.989691i	$0.454254\pi$
$140$	−6.34749e7	−	1.33425e7i	−1.95503	−	0.410952i
$141$	−2.94435e7			−0.884550
$142$		−	6.52001e7i		−	1.91090i
$143$		−	1.64451e7i		−	0.470285i
$144$	3.68998e6			0.102981
$145$	4.59697e6	−	2.18693e7i	0.125223	−	0.595726i
$146$	2.97769e7			0.791854
$147$			1.88275e7i			0.488858i
$148$		−	9.76597e7i		−	2.47628i
$149$	1.15236e6			0.0285388			0.0142694	−	0.999898i	$-0.495458\pi$
$149$	1.15236e6			0.0285388			0.0142694	+	0.999898i	$0.495458\pi$
$150$	3.43329e7	+	1.51009e7i	0.830599	+	0.365329i
$151$	−1.88225e7			−0.444895			−0.222447	−	0.974945i	$-0.571405\pi$
$151$	−1.88225e7			−0.444895			−0.222447	+	0.974945i	$0.571405\pi$
$152$			3.51725e7i			0.812363i
$153$			8.81779e6i			0.199040i
$154$	2.66412e7			0.587802
$155$	6.52861e6	−	3.10588e7i	0.140818	−	0.669920i
$156$	6.87708e7			1.45033
$157$			1.20969e7i			0.249474i	0.992190	+	0.124737i	$0.0398088\pi$
$157$			1.20969e7i			0.249474i	−0.992190	+	0.124737i	$0.960191\pi$
$158$			6.68807e7i			1.34897i
$159$	8.25004e6			0.162767
$160$	6.20779e7	+	1.30489e7i	1.19817	+	0.251857i
$161$	−1.16247e7			−0.219530
$162$		−	9.44965e6i		−	0.174628i
$163$		−	2.22405e7i		−	0.402242i	−0.979566	−	0.201121i	$-0.935542\pi$
$163$		−	2.22405e7i		−	0.402242i	0.979566	−	0.201121i	$-0.0644585\pi$
$164$	−1.13787e8			−2.01436
$165$	−8.97260e6	−	1.88606e6i	−0.155498	−	0.0326859i
$166$	3.16187e7			0.536495
$167$			8.30044e7i			1.37909i	0.724241	+	0.689547i	$0.242190\pi$
$167$			8.30044e7i			1.37909i	−0.724241	+	0.689547i	$0.757810\pi$
$168$			3.56248e7i			0.579655i
$169$	−1.20474e8			−1.91994
$170$	−1.23662e7	+	5.88300e7i	−0.193048	+	0.918391i
$171$	−2.39655e7			−0.366522
$172$			5.47980e7i			0.821136i
$173$		−	9.00362e7i		−	1.32207i	−0.750353	−	0.661037i	$-0.770117\pi$
$173$		−	9.00362e7i		−	1.32207i	0.750353	−	0.661037i	$-0.229883\pi$
$174$	−3.83842e7			−0.552371
$175$	3.87903e7	−	8.81924e7i	0.547130	−	1.24394i
$176$	−6.14957e6			−0.0850257
$177$			3.89355e6i			0.0527764i
$178$			7.84230e7i			1.04225i
$179$	4.81856e7			0.627960			0.313980	−	0.949430i	$-0.398337\pi$
$179$	4.81856e7			0.627960			0.313980	+	0.949430i	$0.398337\pi$
$180$	7.88718e6	−	3.75219e7i	0.100802	−	0.479547i
$181$	−6.48197e7			−0.812516			−0.406258	−	0.913758i	$-0.633167\pi$
$181$	−6.48197e7			−0.812516			−0.406258	+	0.913758i	$0.633167\pi$
$182$		−	2.96821e8i		−	3.64959i
$183$			2.16269e6i			0.0260865i
$184$	−1.00852e7			−0.119349
$185$	1.41961e8	+	2.98405e7i	1.64842	+	0.346502i
$186$	−5.45133e7			−0.621165
$187$		−	1.46954e7i		−	0.164337i
$188$			2.05200e8i			2.25229i
$189$	−2.42737e7			−0.261529
$190$	−1.59892e8	−	3.36095e7i	−1.69117	−	0.355488i
$191$	−6.74751e7			−0.700691			−0.350346	−	0.936621i	$-0.613936\pi$
$191$	−6.74751e7			−0.700691			−0.350346	+	0.936621i	$0.613936\pi$
$192$		−	9.14638e7i		−	0.932599i
$193$			4.34124e7i			0.434674i	0.976097	+	0.217337i	$0.0697370\pi$
$193$			4.34124e7i			0.434674i	−0.976097	+	0.217337i	$0.930263\pi$
$194$	8.18953e7			0.805291
$195$	−2.10134e7	+	9.99675e7i	−0.202943	+	0.965468i
$196$	1.31214e8			1.24476
$197$		−	1.74800e7i		−	0.162896i	−0.996678	−	0.0814479i	$-0.974046\pi$
$197$		−	1.74800e7i		−	0.162896i	0.996678	−	0.0814479i	$-0.0259544\pi$
$198$			1.57484e7i			0.144181i
$199$	9.50765e7			0.855239			0.427619	−	0.903959i	$-0.359352\pi$
$199$	9.50765e7			0.855239			0.427619	+	0.903959i	$0.359352\pi$
$200$	3.36529e7	−	7.65122e7i	0.297453	−	0.676278i
$201$	1.02484e8			0.890166
$202$			2.34422e8i			2.00110i
$203$			9.85991e7i			0.827250i
$204$	6.14536e7			0.506806
$205$	3.47682e7	−	1.65404e8i	0.281867	−	1.34093i
$206$	6.64833e7			0.529879
$207$		−	6.87174e6i		−	0.0538481i
$208$			6.85149e7i			0.527915i
$209$	3.99399e7			0.302618
$210$	−1.61948e8	−	3.40417e7i	−1.20672	−	0.253655i
$211$	8.57045e7			0.628080			0.314040	−	0.949410i	$-0.398317\pi$
$211$	8.57045e7			0.628080			0.314040	+	0.949410i	$0.398317\pi$
$212$		−	5.74968e7i		−	0.414446i
$213$		−	9.90036e7i		−	0.701976i
$214$	7.04395e7			0.491324
$215$	−7.96562e7	−	1.67439e7i	−0.546619	−	0.114900i
$216$	−2.10589e7			−0.142183
$217$			1.40030e8i			0.930279i
$218$		−	2.59901e8i		−	1.69907i
$219$	4.52151e7			0.290890
$220$	−1.31445e7	+	6.25325e7i	−0.0832268	+	0.395937i
$221$	−1.63727e8			−1.02035
$222$		−	2.49166e8i		−	1.52846i
$223$			2.33644e7i			0.141087i	0.997509	+	0.0705436i	$0.0224734\pi$
$223$			2.33644e7i			0.141087i	−0.997509	+	0.0705436i	$0.977527\pi$
$224$	−2.79882e8			−1.66382
$225$	5.21332e7	+	2.29301e7i	0.305123	+	0.134205i
$226$	−3.85505e8			−2.22152
$227$			1.05529e8i			0.598800i	0.954128	+	0.299400i	$0.0967864\pi$
$227$			1.05529e8i			0.598800i	−0.954128	+	0.299400i	$0.903214\pi$
$228$			1.67022e8i			0.933259i
$229$	3.19979e8			1.76075			0.880375	−	0.474278i	$-0.157291\pi$
$229$	3.19979e8			1.76075			0.880375	+	0.474278i	$0.157291\pi$
$230$	9.63701e6	−	4.58465e7i	0.0522270	−	0.248461i
$231$	4.04535e7			0.215931
$232$			8.55406e7i			0.449743i
$233$			8.00232e7i			0.414448i	0.978293	+	0.207224i	$0.0664429\pi$
$233$			8.00232e7i			0.414448i	−0.978293	+	0.207224i	$0.933557\pi$
$234$	1.75460e8			0.895203
$235$	−2.98285e8	−	6.27001e7i	−1.49932	−	0.315160i
$236$	2.71352e7			0.134382
$237$			1.01556e8i			0.495547i
$238$		−	2.65239e8i		−	1.27532i
$239$	−2.28529e8			−1.08280			−0.541401	−	0.840765i	$-0.682106\pi$
$239$	−2.28529e8			−1.08280			−0.541401	+	0.840765i	$0.682106\pi$
$240$	3.73823e7	+	7.85784e6i	0.174553	+	0.0366913i
$241$	2.19414e8			1.00973			0.504863	−	0.863199i	$-0.331543\pi$
$241$	2.19414e8			1.00973			0.504863	+	0.863199i	$0.331543\pi$
$242$			3.20259e8i			1.45261i
$243$		−	1.43489e7i		−	0.0641500i
$244$	1.50724e7			0.0664229
$245$	−4.00934e7	+	1.90738e8i	−0.174177	+	0.828619i
$246$	−2.90311e8			−1.24334
$247$		−	4.44988e8i		−	1.87892i
$248$			1.21485e8i			0.505756i
$249$	4.80117e7			0.197083
$250$	3.15662e8	+	2.26096e8i	1.27771	+	0.915173i
$251$	3.54594e8			1.41538			0.707690	−	0.706523i	$-0.249737\pi$
$251$	3.54594e8			1.41538			0.707690	+	0.706523i	$0.249737\pi$
$252$			1.69170e8i			0.665919i
$253$			1.14522e7i			0.0444596i
$254$	−1.03991e8			−0.398180
$255$	−1.87775e7	+	8.93310e7i	−0.0709166	+	0.337374i
$256$	−1.20899e8			−0.450386
$257$		−	4.31991e8i		−	1.58748i	−0.608255	−	0.793742i	$-0.708130\pi$
$257$		−	4.31991e8i		−	1.58748i	0.608255	−	0.793742i	$-0.291870\pi$
$258$			1.39810e8i			0.506837i
$259$	−6.40042e8			−2.28907
$260$	6.96701e8	+	1.46448e8i	2.45833	+	0.516745i
$261$	−5.82849e7			−0.202915
$262$		−	1.74745e8i		−	0.600276i
$263$		−	2.94090e8i		−	0.996864i	−0.866929	−	0.498432i	$-0.833909\pi$
$263$		−	2.94090e8i		−	0.996864i	0.866929	−	0.498432i	$-0.166091\pi$
$264$	3.50959e7			0.117393
$265$	8.35793e7	+	1.75685e7i	0.275891	+	0.0579929i
$266$	7.20882e8			2.34844
$267$			1.19082e8i			0.382875i
$268$		−	7.14241e8i		−	2.26659i
$269$	−1.92169e8			−0.601936			−0.300968	−	0.953634i	$-0.597310\pi$
$269$	−1.92169e8			−0.601936			−0.300968	+	0.953634i	$0.597310\pi$
$270$	2.01231e7	−	9.57322e7i	0.0622188	−	0.295995i
$271$	−1.66190e8			−0.507238			−0.253619	−	0.967304i	$-0.581621\pi$
$271$	−1.66190e8			−0.507238			−0.253619	+	0.967304i	$0.581621\pi$
$272$			6.12250e7i			0.184475i
$273$		−	4.50710e8i		−	1.34069i
$274$	4.61217e8			1.35450
$275$	−8.68830e7	−	3.82144e7i	−0.251924	−	0.110806i
$276$	−4.78910e7			−0.137111
$277$			2.17384e8i			0.614538i	0.951623	+	0.307269i	$0.0994152\pi$
$277$			2.17384e8i			0.614538i	−0.951623	+	0.307269i	$0.900585\pi$
$278$		−	1.61267e8i		−	0.450184i
$279$	−8.27762e7			−0.228187
$280$	−7.58632e7	+	3.60907e8i	−0.206528	+	0.982520i
$281$	9.15915e7			0.246254			0.123127	−	0.992391i	$-0.460708\pi$
$281$	9.15915e7			0.246254			0.123127	+	0.992391i	$0.460708\pi$
$282$			5.23540e8i			1.39020i
$283$			6.90362e8i			1.81061i	0.424765	+	0.905303i	$0.360357\pi$
$283$			6.90362e8i			1.81061i	−0.424765	+	0.905303i	$0.639643\pi$
$284$	−6.89984e8			−1.78741
$285$	−2.42789e8	−	5.10347e7i	−0.621258	−	0.130590i
$286$	−2.92414e8			−0.739123
$287$			7.45735e8i			1.86208i
$288$		−	1.65447e8i		−	0.408117i
$289$	2.64032e8			0.643449
$290$	−3.88862e8	−	8.17395e7i	−0.936273	−	0.196806i
$291$	1.24355e8			0.295826
$292$		−	3.15116e8i		−	0.740680i
$293$		−	4.98313e7i		−	0.115735i	−0.998324	−	0.0578677i	$-0.981570\pi$
$293$		−	4.98313e7i		−	0.115735i	0.998324	−	0.0578677i	$-0.0184301\pi$
$294$	3.34776e8			0.768314
$295$	−8.29134e6	+	3.94447e7i	−0.0188039	+	0.0894564i
$296$	−5.55274e8			−1.24448
$297$			2.39133e7i			0.0529654i
$298$		−	2.04903e7i		−	0.0448530i
$299$	1.27593e8			0.276044
$300$	1.59806e8	−	3.63330e8i	0.341720	−	0.776922i
$301$	3.59135e8			0.759058
$302$			3.34686e8i			0.699219i
$303$			3.55961e8i			0.735111i
$304$	−1.66401e8			−0.339702
$305$	−4.60547e6	+	2.19097e7i	−0.00929446	+	0.0442168i
$306$	1.56791e8			0.312821
$307$		−	8.18674e8i		−	1.61483i	−0.589984	−	0.807415i	$-0.700866\pi$
$307$		−	8.18674e8i		−	1.61483i	0.589984	−	0.807415i	$-0.299134\pi$
$308$		−	2.81932e8i		−	0.549815i
$309$	1.00952e8			0.194653
$310$	−5.52261e8	−	1.16086e8i	−1.05288	−	0.221317i
$311$	−8.51281e8			−1.60477			−0.802383	−	0.596810i	$-0.796435\pi$
$311$	−8.51281e8			−1.60477			−0.802383	+	0.596810i	$0.796435\pi$
$312$		−	3.91018e8i		−	0.728879i
$313$			6.91125e8i			1.27395i	0.770886	+	0.636973i	$0.219814\pi$
$313$			6.91125e8i			1.27395i	−0.770886	+	0.636973i	$0.780186\pi$
$314$	2.15098e8			0.392086
$315$	−2.45911e8	−	5.16910e7i	−0.443294	−	0.0931811i
$316$	7.07769e8			1.26179
$317$		−	3.79176e8i		−	0.668550i	−0.942476	−	0.334275i	$-0.891509\pi$
$317$		−	3.79176e8i		−	0.668550i	0.942476	−	0.334275i	$-0.108491\pi$
$318$		−	1.46695e8i		−	0.255813i
$319$	9.71352e7			0.167536
$320$	1.94773e8	−	9.26599e8i	0.332279	−	1.58076i
$321$	1.06960e8			0.180489
$322$			2.06702e8i			0.345023i
$323$		−	3.97641e8i		−	0.656572i
$324$	−1.00001e8			−0.163342
$325$	−4.25763e8	+	9.68000e8i	−0.687981	+	1.56417i
$326$	−3.95462e8			−0.632184
$327$		−	3.94649e8i		−	0.624157i
$328$			6.46969e8i			1.01234i
$329$	1.34484e9			2.08202
$330$	−3.35363e7	+	1.59543e8i	−0.0513708	+	0.244388i
$331$	2.51509e8			0.381203			0.190601	−	0.981668i	$-0.438956\pi$
$331$	2.51509e8			0.381203			0.190601	+	0.981668i	$0.438956\pi$
$332$		−	3.34607e8i		−	0.501824i
$333$		−	3.78348e8i		−	0.561483i
$334$	1.47592e9			2.16745
$335$	1.03825e9	+	2.18241e8i	1.50884	+	0.317161i
$336$	−1.68541e8			−0.242391
$337$			6.36016e7i			0.0905239i	0.998975	+	0.0452620i	$0.0144123\pi$
$337$			6.36016e7i			0.0905239i	−0.998975	+	0.0452620i	$0.985588\pi$
$338$			2.14216e9i			3.01748i
$339$	−5.85374e8			−0.816083
$340$	6.22572e8	+	1.30866e8i	0.859040	+	0.180572i
$341$	1.37951e8			0.188402
$342$			4.26135e8i			0.576044i
$343$			1.55666e8i			0.208289i
$344$	3.11571e8			0.412669
$345$	1.46334e7	−	6.96160e7i	0.0191857	−	0.0912730i
$346$	−1.60095e9			−2.07784
$347$		−	7.64332e8i		−	0.982039i	−0.871148	−	0.491020i	$-0.836624\pi$
$347$		−	7.64332e8i		−	0.982039i	0.871148	−	0.491020i	$-0.163376\pi$
$348$			4.06203e8i			0.516674i
$349$	−8.58029e8			−1.08047			−0.540235	−	0.841514i	$-0.681665\pi$
$349$	−8.58029e8			−1.08047			−0.540235	+	0.841514i	$0.681665\pi$
$350$	−1.56817e9	−	6.89738e8i	−1.95503	−	0.859896i
$351$	2.66428e8			0.328856
$352$			2.75727e8i			0.336961i
$353$		−	8.18349e8i		−	0.990210i	−0.868833	−	0.495105i	$-0.835130\pi$
$353$		−	8.18349e8i		−	0.990210i	0.868833	−	0.495105i	$-0.164870\pi$
$354$	6.92320e7			0.0829459
$355$	2.10829e8	−	1.00298e9i	0.250110	−	1.18986i
$356$	8.29916e8			0.974898
$357$		−	4.02755e8i		−	0.468491i
$358$		−	8.56798e8i		−	0.986933i
$359$	−2.81061e7			−0.0320604			−0.0160302	−	0.999872i	$-0.505103\pi$
$359$	−2.81061e7			−0.0320604			−0.0160302	+	0.999872i	$0.505103\pi$
$360$	−2.13343e8	−	4.48450e7i	−0.241001	−	0.0506589i
$361$	1.86861e8			0.209046
$362$			1.15257e9i			1.27699i
$363$			4.86301e8i			0.533620i
$364$	−3.14112e9			−3.41374
$365$	4.58063e8	+	9.62858e7i	0.493061	+	0.103642i
$366$	3.84552e7			0.0409988
$367$		−	2.88761e8i		−	0.304935i	−0.988308	−	0.152468i	$-0.951278\pi$
$367$		−	2.88761e8i		−	0.304935i	0.988308	−	0.152468i	$-0.0487220\pi$
$368$		−	4.77129e7i		−	0.0499078i
$369$	−4.40826e8			−0.456746
$370$	5.30600e8	−	2.52424e9i	0.544579	−	2.59075i
$371$	−3.76823e8			−0.383114
$372$			5.76890e8i			0.581022i
$373$			4.43383e8i			0.442383i	0.975230	+	0.221191i	$0.0709945\pi$
$373$			4.43383e8i			0.442383i	−0.975230	+	0.221191i	$0.929005\pi$
$374$	−2.61301e8			−0.258280
$375$	4.79319e8	+	3.43318e8i	0.469370	+	0.336192i
$376$	1.16673e9			1.13191
$377$		−	1.08222e9i		−	1.04021i
$378$			4.31615e8i			0.411032i
$379$	7.55920e8			0.713245			0.356622	−	0.934249i	$-0.383928\pi$
$379$	7.55920e8			0.713245			0.356622	+	0.934249i	$0.383928\pi$
$380$	−3.55675e8	+	1.69206e9i	−0.332515	+	1.58188i
$381$	−1.57907e8			−0.146273
$382$			1.19979e9i			1.10124i
$383$			2.04265e9i			1.85779i	0.370338	+	0.928897i	$0.379242\pi$
$383$			2.04265e9i			1.85779i	−0.370338	+	0.928897i	$0.620758\pi$
$384$	−8.41994e8			−0.758839
$385$	4.09826e8	+	8.61461e7i	0.366005	+	0.0769348i
$386$	7.71924e8			0.683154
$387$			2.12295e8i			0.186188i
$388$		−	8.66662e8i		−	0.753249i
$389$	−2.01254e9			−1.73348			−0.866742	−	0.498756i	$-0.833790\pi$
$389$	−2.01254e9			−1.73348			−0.866742	+	0.498756i	$0.833790\pi$
$390$	1.77754e9	+	3.73642e8i	1.51738	+	0.318955i
$391$	1.14017e8			0.0964612
$392$		−	7.46060e8i		−	0.625565i
$393$		−	2.65343e8i		−	0.220513i
$394$	−3.10815e8			−0.256015
$395$	−2.16263e8	+	1.02884e9i	−0.176560	+	0.839956i
$396$	1.66658e8			0.134863
$397$			1.66241e9i			1.33343i	0.745313	+	0.666715i	$0.232300\pi$
$397$			1.66241e9i			1.33343i	−0.745313	+	0.666715i	$0.767700\pi$
$398$		−	1.69057e9i		−	1.34414i
$399$	1.09463e9			0.862705
$400$	3.61979e8	+	1.59212e8i	0.282796	+	0.124384i
$401$	−1.67110e9			−1.29419			−0.647094	−	0.762410i	$-0.724016\pi$
$401$	−1.67110e9			−1.29419			−0.647094	+	0.762410i	$0.724016\pi$
$402$		−	1.82229e9i		−	1.39903i
$403$		−	1.53697e9i		−	1.16977i
$404$	2.48079e9			1.87178
$405$	3.05561e7	−	1.45366e8i	0.0228563	−	0.108735i
$406$	1.75321e9			1.30015
$407$			6.30539e8i			0.463587i
$408$		−	3.49414e8i		−	0.254700i
$409$	1.06939e9			0.772866			0.386433	−	0.922318i	$-0.373707\pi$
$409$	1.06939e9			0.772866			0.386433	+	0.922318i	$0.373707\pi$
$410$	−2.94108e9	−	6.18220e8i	−2.10748	−	0.442996i
$411$	7.00340e8			0.497579
$412$		−	7.03563e8i		−	0.495636i
$413$		−	1.77839e8i		−	0.124223i
$414$	−1.22188e8			−0.0846303
$415$	4.86396e8	+	1.02241e8i	0.334058	+	0.0702195i
$416$	3.07199e9			2.09215
$417$		−	2.44878e8i		−	0.165376i
$418$		−	7.10179e8i		−	0.475610i
$419$	1.33295e9			0.885245			0.442623	−	0.896708i	$-0.354048\pi$
$419$	1.33295e9			0.885245			0.442623	+	0.896708i	$0.354048\pi$
$420$	−3.60249e8	+	1.71382e9i	−0.237263	+	1.12874i
$421$	−2.43203e9			−1.58848			−0.794238	−	0.607606i	$-0.792130\pi$
$421$	−2.43203e9			−1.58848			−0.794238	+	0.607606i	$0.792130\pi$
$422$		−	1.52393e9i		−	0.987122i
$423$			7.94974e8i			0.510695i
$424$	−3.26916e8			−0.208284
$425$	−3.80462e8	+	8.65005e8i	−0.240409	+	0.546585i
$426$	−1.76040e9			−1.10326
$427$		−	9.87814e7i		−	0.0614013i
$428$		−	7.45431e8i		−	0.459572i
$429$	−4.44018e8			−0.271519
$430$	−2.97726e8	+	1.41638e9i	−0.180583	+	0.859093i
$431$	−2.39541e9			−1.44115			−0.720577	−	0.693375i	$-0.756123\pi$
$431$	−2.39541e9			−1.44115			−0.720577	+	0.693375i	$0.756123\pi$
$432$		−	9.96295e7i		−	0.0594559i
$433$		−	2.25008e9i		−	1.33196i	−0.745971	−	0.665979i	$-0.768014\pi$
$433$		−	2.25008e9i		−	1.33196i	0.745971	−	0.665979i	$-0.231986\pi$
$434$	2.48991e9			1.46207
$435$	−5.90471e8	−	1.24118e8i	−0.343943	−	0.0722974i
$436$	−2.75042e9			−1.58926
$437$			3.09883e8i			0.177629i
$438$		−	8.03977e8i		−	0.457177i
$439$	−1.94736e8			−0.109855			−0.0549275	−	0.998490i	$-0.517493\pi$
$439$	−1.94736e8			−0.109855			−0.0549275	+	0.998490i	$0.517493\pi$
$440$	3.55548e8	+	7.47369e7i	0.198982	+	0.0418264i
$441$	5.08344e8			0.282242
$442$			2.91126e9i			1.60363i
$443$			1.65807e9i			0.906128i	0.891478	+	0.453064i	$0.149669\pi$
$443$			1.65807e9i			0.906128i	−0.891478	+	0.453064i	$0.850331\pi$
$444$	−2.63681e9			−1.42968
$445$	−2.53586e8	+	1.20639e9i	−0.136416	+	0.648977i
$446$	4.15447e8			0.221740
$447$		−	3.11137e7i		−	0.0164769i
$448$			4.17763e9i			2.19511i
$449$	1.51217e9			0.788386			0.394193	−	0.919028i	$-0.371024\pi$
$449$	1.51217e9			0.788386			0.394193	+	0.919028i	$0.371024\pi$
$450$	4.07725e8	−	9.26989e8i	0.210923	−	0.479547i
$451$	7.34663e8			0.377112
$452$			4.07963e9i			2.07796i
$453$			5.08207e8i			0.256860i
$454$	1.87643e9			0.941103
$455$	9.59790e8	−	4.56604e9i	0.477679	−	2.27248i
$456$	9.49657e8			0.469018
$457$			1.94052e9i			0.951066i	0.879698	+	0.475533i	$0.157745\pi$
$457$			1.94052e9i			0.951066i	−0.879698	+	0.475533i	$0.842255\pi$
$458$		−	5.68961e9i		−	2.76728i
$459$	2.38080e8			0.114916
$460$	−4.85173e8	−	1.01984e8i	−0.232404	−	0.0488518i
$461$	−3.35145e9			−1.59323			−0.796616	−	0.604485i	$-0.793379\pi$
$461$	−3.35145e9			−1.59323			−0.796616	+	0.604485i	$0.793379\pi$
$462$		−	7.19312e8i		−	0.339368i
$463$		−	2.73755e9i		−	1.28183i	−0.767614	−	0.640913i	$-0.778556\pi$
$463$		−	2.73755e9i		−	1.28183i	0.767614	−	0.640913i	$-0.221444\pi$
$464$	−4.04692e8			−0.188067
$465$	−8.38586e8	−	1.76272e8i	−0.386779	−	0.0813016i
$466$	1.42291e9			0.651367
$467$		−	1.19709e9i		−	0.543898i	−0.962312	−	0.271949i	$-0.912332\pi$
$467$		−	1.19709e9i		−	0.543898i	0.962312	−	0.271949i	$-0.0876683\pi$
$468$		−	1.85681e9i		−	0.837351i
$469$	−4.68099e9			−2.09524
$470$	−1.11488e9	+	5.30386e9i	−0.495321	+	2.35640i
$471$	3.26617e8			0.144034
$472$		−	1.54286e8i		−	0.0675350i
$473$		−	3.53803e8i		−	0.153726i
$474$	1.80578e9			0.778826
$475$	−2.35096e9	−	1.03404e9i	−1.00651	−	0.442701i
$476$	−2.80691e9			−1.19290
$477$		−	2.22751e8i		−	0.0939735i
$478$			4.06352e9i			1.70178i
$479$	4.15303e9			1.72660			0.863299	−	0.504693i	$-0.168394\pi$
$479$	4.15303e9			1.72660			0.863299	+	0.504693i	$0.168394\pi$
$480$	3.52320e8	−	1.67610e9i	0.145410	−	0.691762i
$481$	7.02510e9			2.87836
$482$		−	3.90143e9i		−	1.58694i
$483$			3.13868e8i			0.126745i
$484$	3.38916e9			1.35873
$485$	1.25981e9	+	2.64814e8i	0.501428	+	0.105401i
$486$	−2.55141e8			−0.100821
$487$		−	4.20797e9i		−	1.65090i	−0.564473	−	0.825451i	$-0.690921\pi$
$487$		−	4.20797e9i		−	1.65090i	0.564473	−	0.825451i	$-0.309079\pi$
$488$		−	8.56988e7i		−	0.0333814i
$489$	−6.00493e8			−0.232235
$490$	3.39154e9	+	7.12908e8i	1.30230	+	0.273746i
$491$	2.48815e9			0.948618			0.474309	−	0.880358i	$-0.342698\pi$
$491$	2.48815e9			0.948618			0.474309	+	0.880358i	$0.342698\pi$
$492$			3.07224e9i			1.16299i
$493$		−	9.67076e8i		−	0.363493i
$494$	−7.91241e9			−2.95301
$495$	−5.09235e7	+	2.42260e8i	−0.0188712	+	0.0897768i
$496$	−5.74744e8			−0.211489
$497$			4.52201e9i			1.65228i
$498$		−	8.53705e8i		−	0.309745i
$499$	1.01722e9			0.366489			0.183245	−	0.983067i	$-0.441340\pi$
$499$	1.01722e9			0.366489			0.183245	+	0.983067i	$0.441340\pi$
$500$	2.39268e9	−	3.34051e9i	0.856030	−	1.19514i
$501$	2.24112e9			0.796220
$502$		−	6.30510e9i		−	2.22448i
$503$			4.40347e9i			1.54279i	0.636355	+	0.771396i	$0.280441\pi$
$503$			4.40347e9i			1.54279i	−0.636355	+	0.771396i	$0.719559\pi$
$504$	9.61869e8			0.334664
$505$	−7.58021e8	+	3.60616e9i	−0.261916	+	1.24602i
$506$	2.03633e8			0.0698749
$507$			3.25279e9i			1.10848i
$508$			1.10050e9i			0.372448i
$509$	−3.17397e9			−1.06682			−0.533409	−	0.845857i	$-0.679089\pi$
$509$	−3.17397e9			−1.06682			−0.533409	+	0.845857i	$0.679089\pi$
$510$	1.58841e9	+	3.33887e8i	0.530234	+	0.111456i
$511$	−2.06521e9			−0.684685
$512$		−	1.84194e9i		−	0.606500i
$513$			6.47069e8i			0.211612i
$514$	−7.68132e9			−2.49497
$515$	1.02272e9	+	2.14978e8i	0.329938	+	0.0693536i
$516$	1.47955e9			0.474083
$517$		−	1.32487e9i		−	0.421654i
$518$			1.13807e10i			3.59762i
$519$	−2.43098e9			−0.763300
$520$	8.32675e8	−	3.96131e9i	0.259695	−	1.23546i
$521$	4.37564e9			1.35553			0.677766	−	0.735277i	$-0.262948\pi$
$521$	4.37564e9			1.35553			0.677766	+	0.735277i	$0.262948\pi$
$522$			1.03637e9i			0.318911i
$523$			1.44475e9i			0.441608i	0.975318	+	0.220804i	$0.0708682\pi$
$523$			1.44475e9i			0.441608i	−0.975318	+	0.220804i	$0.929132\pi$
$524$	−1.84925e9			−0.561483
$525$	−2.38119e9	−	1.04734e9i	−0.718187	−	0.315886i
$526$	−5.22928e9			−1.56672
$527$		−	1.37344e9i		−	0.408764i
$528$			1.66038e8i			0.0490896i
$529$	3.31597e9			0.973903
$530$	3.12389e8	−	1.48614e9i	0.0911444	−	0.433604i
$531$	1.05126e8			0.0304705
$532$		−	7.62878e9i		−	2.19667i
$533$		−	8.18519e9i		−	2.34144i
$534$	2.11742e9			0.601746
$535$	1.08358e9	+	2.27771e8i	0.305931	+	0.0643073i
$536$	−4.06104e9			−1.13910
$537$		−	1.30101e9i		−	0.362553i
$538$			3.41699e9i			0.946032i
$539$	−8.47185e8			−0.233033
$540$	−1.01309e9	−	2.12954e8i	−0.276867	−	0.0581979i
$541$	−3.22121e8			−0.0874639			−0.0437320	−	0.999043i	$-0.513925\pi$
$541$	−3.22121e8			−0.0874639			−0.0437320	+	0.999043i	$0.513925\pi$
$542$			2.95505e9i			0.797200i
$543$			1.75013e9i			0.469106i
$544$	2.74513e9			0.731083
$545$	8.40408e8	−	3.99810e9i	0.222383	−	1.05795i
$546$	−8.01416e9			−2.10709
$547$			5.78935e9i			1.51243i	0.654325	+	0.756213i	$0.272953\pi$
$547$			5.78935e9i			1.51243i	−0.654325	+	0.756213i	$0.727047\pi$
$548$		−	4.88086e9i		−	1.26696i
$549$	5.83927e7			0.0150610
$550$	−6.79498e8	+	1.54488e9i	−0.174148	+	0.395937i
$551$	2.62838e9			0.669356
$552$			2.72299e8i			0.0689064i
$553$		−	4.63858e9i		−	1.16640i
$554$	3.86535e9			0.965839
$555$	8.05695e8	−	3.83296e9i	0.200053	−	0.951718i
$556$	−1.70662e9			−0.421091
$557$			6.42196e9i			1.57461i	0.616561	+	0.787307i	$0.288525\pi$
$557$			6.42196e9i			1.57461i	−0.616561	+	0.787307i	$0.711475\pi$
$558$			1.47186e9i			0.358630i
$559$	−3.94187e9			−0.954466
$560$	−1.70745e9	−	3.58909e8i	−0.410856	−	0.0863626i
$561$	−3.96775e8			−0.0948799
$562$		−	1.62861e9i		−	0.387025i
$563$			6.57931e9i			1.55382i	0.629611	+	0.776910i	$0.283214\pi$
$563$			6.57931e9i			1.55382i	−0.629611	+	0.776910i	$0.716786\pi$
$564$	5.54039e9			1.30036
$565$	−5.93029e9	−	1.24656e9i	−1.38327	−	0.290766i
$566$	1.22754e10			2.84564
$567$			6.55390e8i			0.150994i
$568$			3.92312e9i			0.898281i
$569$	1.49620e9			0.340484			0.170242	−	0.985402i	$-0.445545\pi$
$569$	1.49620e9			0.340484			0.170242	+	0.985402i	$0.445545\pi$
$570$	−9.07458e8	+	4.31708e9i	−0.205241	+	0.976400i
$571$	1.46890e8			0.0330192			0.0165096	−	0.999864i	$-0.494745\pi$
$571$	1.46890e8			0.0330192			0.0165096	+	0.999864i	$0.494745\pi$
$572$			3.09449e9i			0.691357i
$573$			1.82183e9i			0.404544i
$574$	1.32600e10			2.92653
$575$	2.96496e8	−	6.74102e8i	0.0650400	−	0.147873i
$576$	−2.46952e9			−0.538436
$577$			1.04552e9i			0.226578i	0.993562	+	0.113289i	$0.0361385\pi$
$577$			1.04552e9i			0.226578i	−0.993562	+	0.113289i	$0.963861\pi$
$578$		−	4.69480e9i		−	1.01128i
$579$	1.17213e9			0.250959
$580$	−8.65013e8	+	4.11515e9i	−0.184088	+	0.875766i
$581$	−2.19295e9			−0.463886
$582$		−	2.21117e9i		−	0.464935i
$583$			3.71228e8i			0.0775891i
$584$	−1.79169e9			−0.372236
$585$	2.69912e9	+	5.67361e8i	0.557413	+	0.117169i
$586$	−8.86060e8			−0.181895
$587$		−	5.62851e9i		−	1.14858i	−0.818653	−	0.574288i	$-0.805279\pi$
$587$		−	5.62851e9i		−	1.14858i	0.818653	−	0.574288i	$-0.194721\pi$
$588$		−	3.54279e9i		−	0.718662i
$589$	3.73282e9			0.752720
$590$	7.01373e8	+	1.47430e8i	0.140594	+	0.0295531i
$591$	−4.71960e8			−0.0940480
$592$		−	2.62700e9i		−	0.520396i
$593$		−	2.78090e9i		−	0.547638i	−0.961781	−	0.273819i	$-0.911713\pi$
$593$		−	2.78090e9i		−	0.547638i	0.961781	−	0.273819i	$-0.0882869\pi$
$594$	4.25207e8			0.0832430
$595$	8.57668e8	−	4.08021e9i	0.166921	−	0.794097i
$596$	−2.16840e8			−0.0419544
$597$		−	2.56707e9i		−	0.493772i
$598$		−	2.26876e9i		−	0.433845i
$599$	−8.60274e9			−1.63547			−0.817735	−	0.575595i	$-0.804771\pi$
$599$	−8.60274e9			−1.63547			−0.817735	+	0.575595i	$0.804771\pi$
$600$	−2.06583e9	−	9.08629e8i	−0.390449	−	0.171734i
$601$	−4.24506e8			−0.0797671			−0.0398835	−	0.999204i	$-0.512699\pi$
$601$	−4.24506e8			−0.0797671			−0.0398835	+	0.999204i	$0.512699\pi$
$602$		−	6.38584e9i		−	1.19297i
$603$		−	2.76708e9i		−	0.513938i
$604$	3.54183e9			0.654032
$605$	−1.03558e9	+	4.92660e9i	−0.190125	+	0.904490i
$606$	6.32940e9			1.15534
$607$			4.40204e9i			0.798902i	0.916754	+	0.399451i	$0.130799\pi$
$607$			4.40204e9i			0.798902i	−0.916754	+	0.399451i	$0.869201\pi$
$608$			7.46088e9i			1.34626i
$609$	2.66218e9			0.477613
$610$	3.89581e8	+	8.18906e7i	0.0694934	+	0.0146076i
$611$	−1.47609e10			−2.61800
$612$		−	1.65925e9i		−	0.292605i
$613$		−	5.74968e7i		−	0.0100817i	−0.999987	−	0.00504083i	$-0.998395\pi$
$613$		−	5.74968e7i		−	0.0100817i	0.999987	−	0.00504083i	$-0.00160455\pi$
$614$	−1.45570e10			−2.53795
$615$	−4.46591e9	−	9.38742e8i	−0.774189	−	0.162736i
$616$	−1.60301e9			−0.276315
$617$		−	5.33595e9i		−	0.914563i	−0.889322	−	0.457281i	$-0.848823\pi$
$617$		−	5.33595e9i		−	0.914563i	0.889322	−	0.457281i	$-0.151177\pi$
$618$		−	1.79505e9i		−	0.305926i
$619$	−3.43964e9			−0.582903			−0.291451	−	0.956586i	$-0.594138\pi$
$619$	−3.43964e9			−0.582903			−0.291451	+	0.956586i	$0.594138\pi$
$620$	−1.22849e9	+	5.84434e9i	−0.207015	+	0.984837i
$621$	−1.85537e8			−0.0310892
$622$			1.51368e10i			2.52213i
$623$		−	5.43911e9i		−	0.901196i
$624$	1.84990e9			0.304792
$625$	4.12478e9	+	4.49879e9i	0.675804	+	0.737082i
$626$	1.22890e10			2.00220
$627$		−	1.07838e9i		−	0.174717i
$628$		−	2.27628e9i		−	0.366748i
$629$	6.27763e9			1.00582
$630$	−9.19127e8	+	4.37259e9i	−0.146448	+	0.696702i
$631$	1.00344e10			1.58998			0.794988	−	0.606625i	$-0.207477\pi$
$631$	1.00344e10			1.58998			0.794988	+	0.606625i	$0.207477\pi$
$632$		−	4.02424e9i		−	0.634124i
$633$		−	2.31402e9i		−	0.362622i
$634$	−6.74220e9			−1.05073
$635$	−1.59972e9	−	3.36264e8i	−0.247934	−	0.0521161i
$636$	−1.55241e9			−0.239281
$637$			9.43884e9i			1.44687i
$638$		−	1.72718e9i		−	0.263309i
$639$	−2.67310e9			−0.405286
$640$	−8.53005e9	−	1.79303e9i	−1.28624	−	0.270370i
$641$	−1.00179e9			−0.150235			−0.0751177	−	0.997175i	$-0.523933\pi$
$641$	−1.00179e9			−0.150235			−0.0751177	+	0.997175i	$0.523933\pi$
$642$		−	1.90187e9i		−	0.283666i
$643$			6.60361e9i			0.979587i	0.871838	+	0.489794i	$0.162928\pi$
$643$			6.60361e9i			0.979587i	−0.871838	+	0.489794i	$0.837072\pi$
$644$	2.18743e9			0.322726
$645$	−4.52084e8	+	2.15072e9i	−0.0663377	+	0.315591i
$646$	−7.07053e9			−1.03190
$647$			6.09411e9i			0.884597i	0.896868	+	0.442299i	$0.145837\pi$
$647$			6.09411e9i			0.884597i	−0.896868	+	0.442299i	$0.854163\pi$
$648$			5.68590e8i			0.0820893i
$649$	−1.75199e8			−0.0251579
$650$	1.72122e10	+	7.57057e9i	2.45833	+	1.08126i
$651$	3.78082e9			0.537097
$652$			4.18500e9i			0.591329i
$653$		−	5.17180e8i		−	0.0726852i	−0.999339	−	0.0363426i	$-0.988429\pi$
$653$		−	5.17180e8i		−	0.0726852i	0.999339	−	0.0363426i	$-0.0115708\pi$
$654$	−7.01733e9			−0.980956
$655$	5.65051e8	−	2.68813e9i	0.0785675	−	0.373772i
$656$	−3.06081e9			−0.423324
$657$		−	1.22081e9i		−	0.167945i
$658$		−	2.39128e10i		−	3.27220i
$659$	−1.84842e9			−0.251594			−0.125797	−	0.992056i	$-0.540149\pi$
$659$	−1.84842e9			−0.251594			−0.125797	+	0.992056i	$0.540149\pi$
$660$	1.68838e9	+	3.54900e8i	0.228595	+	0.0480510i
$661$	−5.39867e9			−0.727079			−0.363539	−	0.931579i	$-0.618432\pi$
$661$	−5.39867e9			−0.727079			−0.363539	+	0.931579i	$0.618432\pi$
$662$		−	4.47213e9i		−	0.599117i
$663$			4.42064e9i			0.589098i
$664$	−1.90251e9			−0.252196
$665$	1.10894e10	+	2.33102e9i	1.46229	+	0.307377i
$666$	−6.72747e9			−0.882454
$667$			7.53646e8i			0.0983393i
$668$		−	1.56190e10i		−	2.02738i
$669$	6.30839e8			0.0814567
$670$	3.88058e9	−	1.84612e10i	0.498466	−	2.37137i
$671$	−9.73148e7			−0.0124351
$672$			7.55682e9i			0.960609i
$673$		−	1.02730e10i		−	1.29911i	−0.760315	−	0.649555i	$-0.774955\pi$
$673$		−	1.02730e10i		−	1.29911i	0.760315	−	0.649555i	$-0.225045\pi$
$674$	1.13091e9			0.142272
$675$	6.19114e8	−	1.40760e9i	0.0774831	−	0.176163i
$676$	2.26696e10			2.82247
$677$			2.95364e9i			0.365845i	0.983127	+	0.182923i	$0.0585558\pi$
$677$			2.95364e9i			0.365845i	−0.983127	+	0.182923i	$0.941444\pi$
$678$			1.04086e10i			1.28260i
$679$	−5.67993e9			−0.696303
$680$	7.44078e8	−	3.53983e9i	0.0907481	−	0.431719i
$681$	2.84928e9			0.345717
$682$		−	2.45294e9i		−	0.296102i
$683$			7.18315e9i			0.862666i	0.902193	+	0.431333i	$0.141957\pi$
$683$			7.18315e9i			0.862666i	−0.902193	+	0.431333i	$0.858043\pi$
$684$	4.50960e9			0.538817
$685$	7.09498e9	+	1.49138e9i	0.843401	+	0.177285i
$686$	2.76793e9			0.327357
$687$		−	8.63944e9i		−	1.01657i
$688$			1.47404e9i			0.172564i
$689$	4.13601e9			0.481741
$690$	−1.23785e9	−	2.60199e8i	−0.143449	−	0.0301533i
$691$	−1.59866e9			−0.184324			−0.0921622	−	0.995744i	$-0.529378\pi$
$691$	−1.59866e9			−0.184324			−0.0921622	+	0.995744i	$0.529378\pi$
$692$			1.69422e10i			1.94356i
$693$		−	1.09225e9i		−	0.124668i
$694$	−1.35907e10			−1.54342
$695$	5.21469e8	−	2.48080e9i	0.0589226	−	0.280315i
$696$	2.30960e9			0.259659
$697$		−	7.31428e9i		−	0.818196i
$698$			1.52568e10i			1.69812i
$699$	2.16063e9			0.239282
$700$	−7.29920e9	+	1.65952e10i	−0.804326	+	1.82869i
$701$	−3.88597e9			−0.426075			−0.213038	−	0.977044i	$-0.568336\pi$
$701$	−3.88597e9			−0.426075			−0.213038	+	0.977044i	$0.568336\pi$
$702$		−	4.73741e9i		−	0.516846i
$703$			1.70617e10i			1.85216i
$704$	4.11561e9			0.444559
$705$	−1.69290e9	+	8.05370e9i	−0.181958	+	0.865633i
$706$	−1.45512e10			−1.55626
$707$		−	1.62586e10i		−	1.73027i
$708$		−	7.32651e8i		−	0.0775856i
$709$	1.30100e10			1.37093			0.685463	−	0.728107i	$-0.259600\pi$
$709$	1.30100e10			1.37093			0.685463	+	0.728107i	$0.259600\pi$
$710$	−1.78342e10	−	3.74879e9i	−1.87004	−	0.393085i
$711$	2.74200e9			0.286104
$712$		−	4.71875e9i		−	0.489944i
$713$			1.07033e9i			0.110587i
$714$	−7.16145e9			−0.736304
$715$	−4.49825e9	−	9.45540e8i	−0.460227	−	0.0967406i
$716$	−9.06711e9			−0.923153
$717$			6.17029e9i			0.625156i
$718$			4.99759e8i			0.0503877i
$719$	8.11116e9			0.813826			0.406913	−	0.913467i	$-0.366605\pi$
$719$	8.11116e9			0.813826			0.406913	+	0.913467i	$0.366605\pi$
$720$	2.12162e8	−	1.00932e9i	0.0211838	−	0.100778i
$721$	−4.61101e9			−0.458166
$722$		−	3.32260e9i		−	0.328548i
$723$		−	5.92417e9i		−	0.582966i
$724$	1.21972e10			1.19446
$725$	−5.71762e9	−	2.51482e9i	−0.557227	−	0.245089i
$726$	8.64700e9			0.838663
$727$		−	9.56697e9i		−	0.923430i	−0.887028	−	0.461715i	$-0.847234\pi$
$727$		−	9.56697e9i		−	0.923430i	0.887028	−	0.461715i	$-0.152766\pi$
$728$			1.78598e10i			1.71561i
$729$	−3.87420e8			−0.0370370
$730$	1.71207e9	−	8.14491e9i	0.162889	−	0.774919i
$731$	−3.52245e9			−0.333530
$732$		−	4.06955e8i		−	0.0383493i
$733$		−	6.19560e9i		−	0.581058i	−0.956866	−	0.290529i	$-0.906169\pi$
$733$		−	6.19560e9i		−	0.581058i	0.956866	−	0.290529i	$-0.0938313\pi$
$734$	−5.13451e9			−0.479251
$735$	5.14991e9	+	1.08252e9i	0.478403	+	0.100561i
$736$	−2.13929e9			−0.197787
$737$			4.61150e9i			0.424332i
$738$			7.83841e9i			0.717845i
$739$	2.06426e10			1.88152			0.940759	−	0.339077i	$-0.110115\pi$
$739$	2.06426e10			1.88152			0.940759	+	0.339077i	$0.110115\pi$
$740$	−2.67129e10	−	5.61511e9i	−2.42332	−	0.509386i
$741$	−1.20147e10			−1.08480
$742$			6.70035e9i			0.602121i
$743$		−	7.98595e9i		−	0.714276i	−0.934052	−	0.357138i	$-0.883753\pi$
$743$		−	7.98595e9i		−	0.714276i	0.934052	−	0.357138i	$-0.116247\pi$
$744$	3.28009e9			0.291998
$745$	6.62568e7	−	3.15206e8i	0.00587061	−	0.0279285i
$746$	7.88387e9			0.695270
$747$		−	1.29632e9i		−	0.113786i
$748$			2.76523e9i			0.241588i
$749$	−4.88540e9			−0.424829
$750$	6.10460e9	−	8.52287e9i	0.528376	−	0.737685i
$751$	−1.60521e10			−1.38291			−0.691453	−	0.722422i	$-0.743029\pi$
$751$	−1.60521e10			−1.38291			−0.691453	+	0.722422i	$0.743029\pi$
$752$			5.51978e9i			0.473325i
$753$		−	9.57403e9i		−	0.817170i
$754$	−1.92432e10			−1.63485
$755$	−1.08223e9	+	5.14853e9i	−0.0915177	+	0.435380i
$756$	4.56759e9			0.384469
$757$		−	2.04385e10i		−	1.71243i	−0.516617	−	0.856217i	$-0.672809\pi$
$757$		−	2.04385e10i		−	1.71243i	0.516617	−	0.856217i	$-0.327191\pi$
$758$		−	1.34411e10i		−	1.12097i
$759$	3.09208e8			0.0256688
$760$	9.62076e9	+	2.02230e9i	0.794990	+	0.167108i
$761$	7.98391e9			0.656703			0.328352	−	0.944556i	$-0.393507\pi$
$761$	7.98391e9			0.656703			0.328352	+	0.944556i	$0.393507\pi$
$762$			2.80777e9i			0.229889i
$763$			1.80257e10i			1.46912i
$764$	1.26968e10			1.03007
$765$	2.41194e9	+	5.06994e8i	0.194783	+	0.0409437i
$766$	3.63207e10			2.91980
$767$			1.95196e9i			0.156202i
$768$			3.26428e9i			0.260030i
$769$	−1.41823e10			−1.12462			−0.562310	−	0.826927i	$-0.690087\pi$
$769$	−1.41823e10			−1.12462			−0.562310	+	0.826927i	$0.690087\pi$
$770$	1.53178e9	−	7.28718e9i	0.120915	−	0.575231i
$771$	−1.16638e10			−0.916534
$772$		−	8.16893e9i		−	0.639006i
$773$			2.50315e9i			0.194921i	0.995239	+	0.0974604i	$0.0310719\pi$
$773$			2.50315e9i			0.194921i	−0.995239	+	0.0974604i	$0.968928\pi$
$774$	3.77486e9			0.292623
$775$	−8.12016e9	−	3.57155e9i	−0.626626	−	0.275614i
$776$	−4.92767e9			−0.378552
$777$			1.72811e10i			1.32160i
$778$			3.57853e10i			2.72443i
$779$	1.98792e10			1.50667
$780$	3.95409e9	−	1.88109e10i	0.298343	−	1.41932i
$781$	4.45488e9			0.334624
$782$		−	2.02736e9i		−	0.151603i
$783$			1.57369e9i			0.117153i
$784$	3.52961e9			0.261589
$785$	3.30888e9	+	6.95533e8i	0.244139	+	0.0513185i
$786$	−4.71812e9			−0.346569
$787$			6.68488e9i			0.488857i	0.969667	+	0.244429i	$0.0786004\pi$
$787$			6.68488e9i			0.488857i	−0.969667	+	0.244429i	$0.921400\pi$
$788$			3.28922e9i			0.239470i
$789$	−7.94044e9			−0.575540
$790$	1.82939e10	+	3.84542e9i	1.32012	+	0.277491i
$791$	2.67371e10			1.92086
$792$		−	9.47588e8i		−	0.0677769i
$793$			1.08423e9i			0.0772082i
$794$	2.95595e10			2.09568
$795$	4.74350e8	−	2.25664e9i	0.0334822	−	0.159286i
$796$	−1.78906e10			−1.25727
$797$			1.69038e9i			0.118271i	0.998250	+	0.0591357i	$0.0188345\pi$
$797$			1.69038e9i			0.118271i	−0.998250	+	0.0591357i	$0.981166\pi$
$798$		−	1.94638e10i		−	1.35587i
$799$	−1.31904e10			−0.914837
$800$	7.13855e9	−	1.62300e10i	0.492941	−	1.12073i
$801$	3.21522e9			0.221053
$802$			2.97142e10i			2.03401i
$803$			2.03455e9i			0.138664i
$804$	−1.92845e10			−1.30862
$805$	−6.68384e8	+	3.17973e9i	−0.0451586	+	0.214835i
$806$	−2.73292e10			−1.83846
$807$			5.18856e9i			0.347528i
$808$		−	1.41053e10i		−	0.940681i
$809$	−2.47159e10			−1.64118			−0.820591	−	0.571516i	$-0.806356\pi$
$809$	−2.47159e10			−1.64118			−0.820591	+	0.571516i	$0.806356\pi$
$810$	−2.58477e9	−	5.43324e8i	−0.170893	−	0.0359220i
$811$	4.01774e9			0.264490			0.132245	−	0.991217i	$-0.457781\pi$
$811$	4.01774e9			0.264490			0.132245	+	0.991217i	$0.457781\pi$
$812$		−	1.85534e10i		−	1.21613i
$813$			4.48712e9i			0.292854i
$814$	1.12117e10			0.728597
$815$	−6.08346e9	−	1.27875e9i	−0.393640	−	0.0827438i
$816$	1.65307e9			0.106507
$817$		−	9.57353e9i		−	0.614179i
$818$		−	1.90150e10i		−	1.21467i
$819$	−1.21692e10			−0.774047
$820$	−6.54235e9	+	3.11241e10i	−0.414367	+	1.97128i
$821$	5.28544e9			0.333334			0.166667	−	0.986013i	$-0.446699\pi$
$821$	5.28544e9			0.333334			0.166667	+	0.986013i	$0.446699\pi$
$822$		−	1.24529e10i		−	0.782020i
$823$			1.27486e10i			0.797194i	0.917126	+	0.398597i	$0.130503\pi$
$823$			1.27486e10i			0.797194i	−0.917126	+	0.398597i	$0.869497\pi$
$824$	−4.00033e9			−0.249087
$825$	−1.03179e9	+	2.34584e9i	−0.0639738	+	0.145449i
$826$	−3.16219e9			−0.195235
$827$		−	7.41064e9i		−	0.455603i	−0.973708	−	0.227802i	$-0.926846\pi$
$827$		−	7.41064e9i		−	0.455603i	0.973708	−	0.227802i	$-0.0731537\pi$
$828$			1.29306e9i			0.0791611i
$829$	−2.85632e10			−1.74127			−0.870636	−	0.491927i	$-0.836293\pi$
$829$	−2.85632e10			−1.74127			−0.870636	+	0.491927i	$0.836293\pi$
$830$	1.81797e9	−	8.64869e9i	0.110360	−	0.525021i
$831$	5.86938e9			0.354804
$832$		−	4.58537e10i		−	2.76021i
$833$			8.43455e9i			0.505597i
$834$	−4.35422e9			−0.259914
$835$	2.27043e10	+	4.77248e9i	1.34960	+	0.283688i
$836$	−7.51551e9			−0.444874
$837$			2.23496e9i			0.131744i
$838$		−	2.37014e10i		−	1.39129i
$839$	−6.28730e9			−0.367534			−0.183767	−	0.982970i	$-0.558829\pi$
$839$	−6.28730e9			−0.367534			−0.183767	+	0.982970i	$0.558829\pi$
$840$	9.74448e9	+	2.04831e9i	0.567258	+	0.119239i
$841$	−1.08576e10			−0.629429
$842$			4.32443e10i			2.49653i
$843$		−	2.47297e9i		−	0.142175i
$844$	−1.61271e10			−0.923329
$845$	−6.92683e9	+	3.29532e10i	−0.394944	+	1.87888i
$846$	1.41356e10			0.802633
$847$		−	2.22119e10i		−	1.25601i
$848$		−	1.54664e9i		−	0.0870970i
$849$	1.86398e10			1.04535
$850$	1.53808e10	+	6.76506e9i	0.859039	+	0.377838i
$851$	−4.89218e9			−0.272113
$852$			1.86296e10i			1.03196i
$853$		−	1.21333e10i		−	0.669354i	−0.942333	−	0.334677i	$-0.891373\pi$
$853$		−	1.21333e10i		−	0.669354i	0.942333	−	0.334677i	$-0.108627\pi$
$854$	−1.75645e9			−0.0965013
$855$	−1.37794e9	+	6.55531e9i	−0.0753960	+	0.358684i
$856$	−4.23838e9			−0.230962
$857$		−	8.42516e8i		−	0.0457241i	−0.999739	−	0.0228621i	$-0.992722\pi$
$857$		−	8.42516e8i		−	0.0457241i	0.999739	−	0.0228621i	$-0.00727786\pi$
$858$			7.89517e9i			0.426733i
$859$	2.86639e9			0.154297			0.0771487	−	0.997020i	$-0.475418\pi$
$859$	2.86639e9			0.154297			0.0771487	+	0.997020i	$0.475418\pi$
$860$	1.49889e10	+	3.15070e9i	0.803575	+	0.168913i
$861$	2.01348e10			1.07507
$862$			4.25933e10i			2.26499i
$863$			8.89034e9i			0.470848i	0.971893	+	0.235424i	$0.0756479\pi$
$863$			8.89034e9i			0.470848i	−0.971893	+	0.235424i	$0.924352\pi$
$864$	−4.46707e9			−0.235627
$865$	−2.46277e10	−	5.17678e9i	−1.29380	−	0.271959i
$866$	−4.00091e10			−2.09337
$867$		−	7.12886e9i		−	0.371495i
$868$		−	2.63496e10i		−	1.36759i
$869$	−4.56971e9			−0.236221
$870$	−2.20697e9	+	1.04993e10i	−0.113626	+	0.540557i
$871$	5.13786e10			2.63463
$872$			1.56384e10i			0.798699i
$873$		−	3.35757e9i		−	0.170795i
$874$	5.51009e9			0.279170
$875$	−2.18930e10	−	1.56811e10i	−1.10478	−	0.791314i
$876$	−8.50814e9			−0.427632
$877$		−	1.47535e10i		−	0.738578i	−0.929315	−	0.369289i	$-0.879601\pi$
$877$		−	1.47535e10i		−	0.738578i	0.929315	−	0.369289i	$-0.120399\pi$
$878$			3.46263e9i			0.172654i
$879$	−1.34545e9			−0.0668198
$880$	−3.53580e8	+	1.68210e9i	−0.0174903	+	0.0832073i
$881$	−1.87087e10			−0.921784			−0.460892	−	0.887456i	$-0.652470\pi$
$881$	−1.87087e10			−0.921784			−0.460892	+	0.887456i	$0.652470\pi$
$882$		−	9.03895e9i		−	0.443586i
$883$			9.38845e9i			0.458914i	0.973319	+	0.229457i	$0.0736951\pi$
$883$			9.38845e9i			0.458914i	−0.973319	+	0.229457i	$0.926305\pi$
$884$	3.08086e10			1.49999
$885$	1.06501e9	+	2.23866e8i	0.0516477	+	0.0108564i
$886$	2.94824e10			1.42412
$887$			8.40643e9i			0.404463i	0.979338	+	0.202232i	$0.0648194\pi$
$887$			8.40643e9i			0.404463i	−0.979338	+	0.202232i	$0.935181\pi$
$888$			1.49924e10i			0.718499i
$889$	7.21243e9			0.344291
$890$	2.14511e10	+	4.50906e9i	1.01996	+	0.214398i
$891$	6.45659e8			0.0305796
$892$		−	4.39649e9i		−	0.207410i
$893$		−	3.58496e10i		−	1.68463i
$894$	−5.53238e8			−0.0258959
$895$	2.77051e9	−	1.31803e10i	0.129175	−	0.614530i
$896$	3.84583e10			1.78613
$897$		−	3.44502e9i		−	0.159374i
$898$		−	2.68882e10i		−	1.23907i
$899$	9.07833e9			0.416723
$900$	−9.80992e9	−	4.31477e9i	−0.448556	−	0.197292i
$901$	3.69594e9			0.168340
$902$		−	1.30632e10i		−	0.592687i
$903$		−	9.69664e9i		−	0.438242i
$904$	2.31960e10			1.04430
$905$	−3.72692e9	+	1.77302e10i	−0.167140	+	0.795139i
$906$	9.03652e9			0.403694
$907$		−	7.97262e9i		−	0.354794i	−0.984139	−	0.177397i	$-0.943232\pi$
$907$		−	7.97262e9i		−	0.354794i	0.984139	−	0.177397i	$-0.0567676\pi$
$908$		−	1.98574e10i		−	0.880284i
$909$	9.61094e9			0.424417
$910$	−8.11896e10	−	1.70662e10i	−3.57154	−	0.750744i
$911$	1.30963e10			0.573897			0.286948	−	0.957946i	$-0.407359\pi$
$911$	1.30963e10			0.573897			0.286948	+	0.957946i	$0.407359\pi$
$912$			4.49282e9i			0.196127i
$913$			2.16039e9i			0.0939472i
$914$	3.45047e10			1.49474
$915$	5.91563e8	+	1.24348e8i	0.0255286	+	0.00536616i
$916$	−6.02106e10			−2.58845
$917$			1.21196e10i			0.519035i
$918$		−	4.23335e9i		−	0.180607i
$919$	2.03307e10			0.864069			0.432034	−	0.901857i	$-0.357796\pi$
$919$	2.03307e10			0.864069			0.432034	+	0.901857i	$0.357796\pi$
$920$	−5.79863e8	+	2.75860e9i	−0.0245509	+	0.116797i
$921$	−2.21042e10			−0.932323
$922$			5.95927e10i			2.50400i
$923$		−	4.96337e10i		−	2.07764i
$924$	−7.61216e9			−0.317436
$925$	1.63246e10	−	3.71151e10i	0.678183	−	1.54189i
$926$	−4.86769e10			−2.01458
$927$		−	2.72571e9i		−	0.112383i
$928$			1.81451e10i			0.745318i
$929$	−1.55355e9			−0.0635726			−0.0317863	−	0.999495i	$-0.510120\pi$
$929$	−1.55355e9			−0.0635726			−0.0317863	+	0.999495i	$0.510120\pi$
$930$	−3.13433e9	+	1.49111e10i	−0.127778	+	0.607880i
$931$	−2.29239e10			−0.931033
$932$		−	1.50580e10i		−	0.609273i
$933$			2.29846e10i			0.926512i
$934$	−2.12857e10			−0.854818
$935$	−4.01964e9	−	8.44935e8i	−0.160822	−	0.0338051i
$936$	−1.05575e10			−0.420818
$937$			2.83219e10i			1.12469i	0.826901	+	0.562347i	$0.190101\pi$
$937$			2.83219e10i			1.12469i	−0.826901	+	0.562347i	$0.809899\pi$
$938$			8.32336e10i			3.29298i
$939$	1.86604e10			0.735513
$940$	5.61285e10	+	1.17983e10i	2.20412	+	0.463310i
$941$	2.10505e10			0.823565			0.411783	−	0.911282i	$-0.364906\pi$
$941$	2.10505e10			0.823565			0.411783	+	0.911282i	$0.364906\pi$
$942$		−	5.80763e9i		−	0.226371i
$943$			5.70005e9i			0.221354i
$944$	7.29926e8			0.0282408
$945$	−1.39566e9	+	6.63960e9i	−0.0537981	+	0.255936i
$946$	−6.29103e9			−0.241603
$947$			2.03722e10i			0.779495i	0.920922	+	0.389747i	$0.127438\pi$
$947$			2.03722e10i			0.779495i	−0.920922	+	0.389747i	$0.872562\pi$
$948$		−	1.91098e10i		−	0.728494i
$949$	2.26677e10			0.860947
$950$	−1.83865e10	+	4.18029e10i	−0.695771	+	1.58188i
$951$	−1.02378e10			−0.385988
$952$			1.59595e10i			0.599503i
$953$			4.66238e10i			1.74495i	0.488659	+	0.872475i	$0.337486\pi$
$953$			4.66238e10i			1.74495i	−0.488659	+	0.872475i	$0.662514\pi$
$954$	−3.96078e9			−0.147693
$955$	−3.87959e9	+	1.84565e10i	−0.144137	+	0.685706i
$956$	4.30024e10			1.59181
$957$		−	2.62265e9i		−	0.0967272i
$958$		−	7.38458e10i		−	2.71361i
$959$	−3.19882e10			−1.17118
$960$	−2.50182e10	−	5.25887e9i	−0.912655	−	0.191842i
$961$	−1.46196e10			−0.531377
$962$		−	1.24915e11i		−	4.52377i
$963$		−	2.88791e9i		−	0.104206i
$964$	−4.12871e10			−1.48438
$965$	1.18746e10	+	2.49607e9i	0.425378	+	0.0894151i
$966$	5.58095e9			0.199199
$967$			2.67331e10i			0.950730i	0.879789	+	0.475365i	$0.157684\pi$
$967$			2.67331e10i			0.950730i	−0.879789	+	0.475365i	$0.842316\pi$
$968$		−	1.92701e10i		−	0.682844i
$969$	−1.07363e10			−0.379072
$970$	4.70871e9	−	2.24009e10i	0.165653	−	0.788069i
$971$	4.54477e10			1.59311			0.796553	−	0.604569i	$-0.206655\pi$
$971$	4.54477e10			1.59311			0.796553	+	0.604569i	$0.206655\pi$
$972$			2.70004e9i			0.0943058i
$973$			1.11849e10i			0.389256i
$974$	−7.48227e10			−2.59464
$975$	2.61360e10	+	1.14956e10i	0.903073	+	0.397206i
$976$	4.05441e8			0.0139590
$977$		−	3.62585e10i		−	1.24388i	−0.783065	−	0.621940i	$-0.786345\pi$
$977$		−	3.62585e10i		−	1.24388i	0.783065	−	0.621940i	$-0.213655\pi$
$978$			1.06775e10i			0.364991i
$979$	−5.35835e9			−0.182512
$980$	7.54439e9	−	3.58912e10i	0.256055	−	1.21814i
$981$	−1.06555e10			−0.360357
$982$		−	4.42423e10i		−	1.49090i
$983$			3.94137e10i			1.32346i	0.749744	+	0.661728i	$0.230177\pi$
$983$			3.94137e10i			1.32346i	−0.749744	+	0.661728i	$0.769823\pi$
$984$	1.74682e10			0.584473
$985$	−4.78132e9	−	1.00504e9i	−0.159412	−	0.0335087i
$986$	−1.71958e10			−0.571284
$987$		−	3.63106e10i		−	1.20205i
$988$			8.37335e10i			2.76217i
$989$	2.74506e9			0.0902329
$990$	4.30767e9	+	9.05481e8i	0.141098	+	0.0296590i
$991$	−3.28617e10			−1.07259			−0.536293	−	0.844032i	$-0.680176\pi$
$991$	−3.28617e10			−1.07259			−0.536293	+	0.844032i	$0.680176\pi$
$992$			2.57697e10i			0.838143i
$993$		−	6.79075e9i		−	0.220088i
$994$	8.04068e10			2.59681
$995$	5.46658e9	−	2.60064e10i	0.175928	−	0.836949i
$996$	−9.03438e9			−0.289728
$997$			4.69907e10i			1.50168i	0.660482	+	0.750842i	$0.270352\pi$
$997$			4.69907e10i			1.50168i	−0.660482	+	0.750842i	$0.729648\pi$
$998$		−	1.80873e10i		−	0.575993i
$999$	−1.02154e10			−0.324172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.8.b.a.4.2	✓	8
3.2	odd	2		45.8.b.d.19.7		8
4.3	odd	2		240.8.f.e.49.7		8
5.2	odd	4		75.8.a.i.1.4		4
5.3	odd	4		75.8.a.j.1.1		4
5.4	even	2	inner	15.8.b.a.4.7	yes	8
15.2	even	4		225.8.a.bb.1.1		4
15.8	even	4		225.8.a.z.1.4		4
15.14	odd	2		45.8.b.d.19.2		8
20.19	odd	2		240.8.f.e.49.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.8.b.a.4.2	✓	8	1.1	even	1	trivial
15.8.b.a.4.7	yes	8	5.4	even	2	inner
45.8.b.d.19.2		8	15.14	odd	2
45.8.b.d.19.7		8	3.2	odd	2
75.8.a.i.1.4		4	5.2	odd	4
75.8.a.j.1.1		4	5.3	odd	4
225.8.a.z.1.4		4	15.8	even	4
225.8.a.bb.1.1		4	15.2	even	4
240.8.f.e.49.3		8	20.19	odd	2
240.8.f.e.49.7		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 15 = 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	15.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-17.7812i q^{2} -27.0000i q^{3} -188.170 q^{4} +(57.4967 - 273.531i) q^{5} -480.092 q^{6} +1233.23i q^{7} +1069.90i q^{8} -729.000 q^{9} +O(q^{10})\)	\(q-17.7812i q^{2} -27.0000i q^{3} -188.170 q^{4} +(57.4967 - 273.531i) q^{5} -480.092 q^{6} +1233.23i q^{7} +1069.90i q^{8} -729.000 q^{9} +(-4863.70 - 1022.36i) q^{10} +1214.92 q^{11} +5080.60i q^{12} -13536.0i q^{13} +21928.3 q^{14} +(-7385.33 - 1552.41i) q^{15} -5061.70 q^{16} -12095.7i q^{17} +12962.5i q^{18} +32874.5 q^{19} +(-10819.2 + 51470.4i) q^{20} +33297.2 q^{21} -21602.7i q^{22} +9426.25i q^{23} +28887.3 q^{24} +(-71513.3 - 31454.2i) q^{25} -240685. q^{26} +19683.0i q^{27} -232058. i q^{28} +79951.8 q^{29} +(-27603.7 + 131320. i) q^{30} +113548. q^{31} +226950. i q^{32} -32802.9i q^{33} -215076. q^{34} +(337327. + 70906.7i) q^{35} +137176. q^{36} +518996. i q^{37} -584547. i q^{38} -365471. q^{39} +(292651. + 61515.8i) q^{40} +604700. q^{41} -592064. i q^{42} -291215. i q^{43} -228612. q^{44} +(-41915.1 + 199404. i) q^{45} +167610. q^{46} -1.09050e6i q^{47} +136666. i q^{48} -697316. q^{49} +(-559293. + 1.27159e6i) q^{50} -326585. q^{51} +2.54707e6i q^{52} +305557. i q^{53} +349987. q^{54} +(69854.0 - 332319. i) q^{55} -1.31944e6 q^{56} -887611. i q^{57} -1.42164e6i q^{58} -144206. q^{59} +(1.38970e6 + 292118. i) q^{60} -80099.7 q^{61} -2.01901e6i q^{62} -899026. i q^{63} +3.38755e6 q^{64} +(-3.70250e6 - 778272. i) q^{65} -583274. q^{66} +3.79572e6i q^{67} +2.27606e6i q^{68} +254509. q^{69} +(1.26081e6 - 5.99807e6i) q^{70} +3.66680e6 q^{71} -779958. i q^{72} +1.67463e6i q^{73} +9.22836e6 q^{74} +(-849264. + 1.93086e6i) q^{75} -6.18601e6 q^{76} +1.49828e6i q^{77} +6.49850e6i q^{78} -3.76132e6 q^{79} +(-291031. + 1.38453e6i) q^{80} +531441. q^{81} -1.07523e7i q^{82} +1.77821e6i q^{83} -6.26556e6 q^{84} +(-3.30856e6 - 695464. i) q^{85} -5.17814e6 q^{86} -2.15870e6i q^{87} +1.29985e6i q^{88} -4.41045e6 q^{89} +(3.54564e6 + 745300. i) q^{90} +1.66930e7 q^{91} -1.77374e6i q^{92} -3.06578e6i q^{93} -1.93904e7 q^{94} +(1.89017e6 - 8.99219e6i) q^{95} +6.12766e6 q^{96} +4.60573e6i q^{97} +1.23991e7i q^{98} -885678. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 666 q^{4} - 444 q^{5} + 486 q^{6} - 5832 q^{9}+O(q^{10}) \) 8 * q - 666 * q^4 - 444 * q^5 + 486 * q^6 - 5832 * q^9	\( 8 q - 666 q^{4} - 444 q^{5} + 486 q^{6} - 5832 q^{9} - 6686 q^{10} + 10752 q^{11} - 13524 q^{14} - 17496 q^{15} + 86530 q^{16} - 30464 q^{19} + 87444 q^{20} + 64152 q^{21} - 110322 q^{24} + 127616 q^{25} - 793524 q^{26} + 240072 q^{29} - 172044 q^{30} + 233728 q^{31} + 184748 q^{34} + 593520 q^{35} + 485514 q^{36} - 454896 q^{39} - 1147102 q^{40} + 507648 q^{41} + 2578572 q^{44} + 323676 q^{45} + 662408 q^{46} - 3267160 q^{49} - 5117736 q^{50} + 264384 q^{51} - 354294 q^{54} + 3525696 q^{55} + 705660 q^{56} + 1091424 q^{59} + 5307606 q^{60} - 6433520 q^{61} - 568594 q^{64} - 2555592 q^{65} - 2382372 q^{66} - 5940864 q^{69} + 12097800 q^{70} - 1381824 q^{71} + 23961276 q^{74} + 7768224 q^{75} + 13115664 q^{76} - 14380160 q^{79} - 31251876 q^{80} + 4251528 q^{81} - 36423756 q^{84} - 1452008 q^{85} - 19837608 q^{86} + 45778896 q^{89} + 4874094 q^{90} + 24075648 q^{91} - 45728896 q^{94} - 25774656 q^{95} + 33586002 q^{96} - 7838208 q^{99}+O(q^{100}) \) 8 * q - 666 * q^4 - 444 * q^5 + 486 * q^6 - 5832 * q^9 - 6686 * q^10 + 10752 * q^11 - 13524 * q^14 - 17496 * q^15 + 86530 * q^16 - 30464 * q^19 + 87444 * q^20 + 64152 * q^21 - 110322 * q^24 + 127616 * q^25 - 793524 * q^26 + 240072 * q^29 - 172044 * q^30 + 233728 * q^31 + 184748 * q^34 + 593520 * q^35 + 485514 * q^36 - 454896 * q^39 - 1147102 * q^40 + 507648 * q^41 + 2578572 * q^44 + 323676 * q^45 + 662408 * q^46 - 3267160 * q^49 - 5117736 * q^50 + 264384 * q^51 - 354294 * q^54 + 3525696 * q^55 + 705660 * q^56 + 1091424 * q^59 + 5307606 * q^60 - 6433520 * q^61 - 568594 * q^64 - 2555592 * q^65 - 2382372 * q^66 - 5940864 * q^69 + 12097800 * q^70 - 1381824 * q^71 + 23961276 * q^74 + 7768224 * q^75 + 13115664 * q^76 - 14380160 * q^79 - 31251876 * q^80 + 4251528 * q^81 - 36423756 * q^84 - 1452008 * q^85 - 19837608 * q^86 + 45778896 * q^89 + 4874094 * q^90 + 24075648 * q^91 - 45728896 * q^94 - 25774656 * q^95 + 33586002 * q^96 - 7838208 * q^99

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