LMFDB - Embedded newform 15.8.b.a.4.8

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			4.8
Root			$-9.60626i$ of defining polynomial
Character	$\chi$	$=$	15.4
Dual form			15.8.b.a.4.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+21.0486i q^{2} -27.0000i q^{3} -315.042 q^{4} +(-238.301 - 146.073i) q^{5} +568.311 q^{6} +923.886i q^{7} -3936.97i q^{8} -729.000 q^{9} +O(q^{10})$	\(q+21.0486i q^{2} -27.0000i q^{3} -315.042 q^{4} +(-238.301 - 146.073i) q^{5} +568.311 q^{6} +923.886i q^{7} -3936.97i q^{8} -729.000 q^{9} +(3074.63 - 5015.90i) q^{10} -3265.52 q^{11} +8506.14i q^{12} +9971.29i q^{13} -19446.5 q^{14} +(-3943.98 + 6434.14i) q^{15} +42542.1 q^{16} -6002.31i q^{17} -15344.4i q^{18} -34306.4 q^{19} +(75075.0 + 46019.2i) q^{20} +24944.9 q^{21} -68734.5i q^{22} +14886.0i q^{23} -106298. q^{24} +(35450.2 + 69619.0i) q^{25} -209881. q^{26} +19683.0i q^{27} -291063. i q^{28} +84126.9 q^{29} +(-135429. - 83015.1i) q^{30} +181823. q^{31} +391519. i q^{32} +88169.0i q^{33} +126340. q^{34} +(134955. - 220163. i) q^{35} +229666. q^{36} +33281.2i q^{37} -722100. i q^{38} +269225. q^{39} +(-575086. + 938185. i) q^{40} -658739. q^{41} +525055. i q^{42} +509770. i q^{43} +1.02878e6 q^{44} +(173722. + 106487. i) q^{45} -313328. q^{46} -139411. i q^{47} -1.14864e6i q^{48} -30022.6 q^{49} +(-1.46538e6 + 746175. i) q^{50} -162062. q^{51} -3.14138e6i q^{52} -848433. i q^{53} -414299. q^{54} +(778178. + 477005. i) q^{55} +3.63731e6 q^{56} +926272. i q^{57} +1.77075e6i q^{58} -1.50633e6 q^{59} +(1.24252e6 - 2.02702e6i) q^{60} -3.47995e6 q^{61} +3.82712e6i q^{62} -673513. i q^{63} -2.79552e6 q^{64} +(1.45654e6 - 2.37617e6i) q^{65} -1.85583e6 q^{66} +1.09706e6i q^{67} +1.89098e6i q^{68} +401921. q^{69} +(4.63412e6 + 2.84061e6i) q^{70} +329871. q^{71} +2.87005e6i q^{72} -2.86250e6i q^{73} -700521. q^{74} +(1.87971e6 - 957154. i) q^{75} +1.08080e7 q^{76} -3.01697e6i q^{77} +5.66680e6i q^{78} +1.04751e6 q^{79} +(-1.01378e7 - 6.21427e6i) q^{80} +531441. q^{81} -1.38655e7i q^{82} +3.76615e6i q^{83} -7.85870e6 q^{84} +(-876778. + 1.43036e6i) q^{85} -1.07299e7 q^{86} -2.27143e6i q^{87} +1.28562e7i q^{88} +9.44386e6 q^{89} +(-2.24141e6 + 3.65659e6i) q^{90} -9.21234e6 q^{91} -4.68970e6i q^{92} -4.90922e6i q^{93} +2.93440e6 q^{94} +(8.17526e6 + 5.01125e6i) q^{95} +1.05710e7 q^{96} +5.39979e6i q^{97} -631932. i q^{98} +2.38056e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			21.0486i			1.86045i	0.366992	+	0.930224i	$0.380387\pi$
$2$			21.0486i			1.86045i	−0.366992	+	0.930224i	$0.619613\pi$
$3$		−	27.0000i		−	0.577350i
$3$		−	27.0000i		−	0.577350i
$4$	−315.042			−2.46127
$5$	−238.301	−	146.073i	−0.852573	−	0.522608i
$5$	−238.301	−	146.073i	−0.852573	−	0.522608i
$6$	568.311			1.07413
$7$			923.886i			1.01806i	0.860747	+	0.509032i	$0.169997\pi$
$7$			923.886i			1.01806i	−0.860747	+	0.509032i	$0.830003\pi$
$8$		−	3936.97i		−	2.71861i
$9$	−729.000			−0.333333
$10$	3074.63	−	5015.90i	0.972285	−	1.58617i
$11$	−3265.52			−0.739738			−0.369869	−	0.929084i	$-0.620597\pi$
$11$	−3265.52			−0.739738			−0.369869	+	0.929084i	$0.620597\pi$
$12$			8506.14i			1.42101i
$13$			9971.29i			1.25878i	0.777090	+	0.629390i	$0.216695\pi$
$13$			9971.29i			1.25878i	−0.777090	+	0.629390i	$0.783305\pi$
$14$	−19446.5			−1.89406
$15$	−3943.98	+	6434.14i	−0.301728	+	0.492233i
$16$	42542.1			2.59656
$17$		−	6002.31i		−	0.296311i	−0.988964	−	0.148155i	$-0.952666\pi$
$17$		−	6002.31i		−	0.296311i	0.988964	−	0.148155i	$-0.0473336\pi$
$18$		−	15344.4i		−	0.620149i
$19$	−34306.4			−1.14746			−0.573730	−	0.819045i	$-0.694504\pi$
$19$	−34306.4			−1.14746			−0.573730	+	0.819045i	$0.694504\pi$
$20$	75075.0	+	46019.2i	2.09841	+	1.28628i
$21$	24944.9			0.587780
$22$		−	68734.5i		−	1.37624i
$23$			14886.0i			0.255111i	0.991831	+	0.127556i	$0.0407131\pi$
$23$			14886.0i			0.255111i	−0.991831	+	0.127556i	$0.959287\pi$
$24$	−106298.			−1.56959
$25$	35450.2	+	69619.0i	0.453762	+	0.891123i
$26$	−209881.			−2.34189
$27$			19683.0i			0.192450i
$28$		−	291063.i		−	2.50573i
$29$	84126.9			0.640534			0.320267	−	0.947327i	$-0.396227\pi$
$29$	84126.9			0.640534			0.320267	+	0.947327i	$0.396227\pi$
$30$	−135429.	−	83015.1i	−0.915774	−	0.561349i
$31$	181823.			1.09618			0.548092	−	0.836418i	$-0.315354\pi$
$31$	181823.			1.09618			0.548092	+	0.836418i	$0.315354\pi$
$32$			391519.i			2.11216i
$33$			88169.0i			0.427088i
$34$	126340.			0.551271
$35$	134955.	−	220163.i	0.532049	−	0.867975i
$36$	229666.			0.820422
$37$			33281.2i			0.108017i	0.998540	+	0.0540086i	$0.0171998\pi$
$37$			33281.2i			0.108017i	−0.998540	+	0.0540086i	$0.982800\pi$
$38$		−	722100.i		−	2.13479i
$39$	269225.			0.726757
$40$	−575086.	+	938185.i	−1.42077	+	2.31781i
$41$	−658739.			−1.49269			−0.746345	−	0.665559i	$-0.768193\pi$
$41$	−658739.			−1.49269			−0.746345	+	0.665559i	$0.768193\pi$
$42$			525055.i			1.09353i
$43$			509770.i			0.977766i	0.872349	+	0.488883i	$0.162596\pi$
$43$			509770.i			0.977766i	−0.872349	+	0.488883i	$0.837404\pi$
$44$	1.02878e6			1.82069
$45$	173722.	+	106487.i	0.284191	+	0.174203i
$46$	−313328.			−0.474621
$47$		−	139411.i		−	0.195864i	−0.995193	−	0.0979319i	$-0.968777\pi$
$47$		−	139411.i		−	0.195864i	0.995193	−	0.0979319i	$-0.0312228\pi$
$48$		−	1.14864e6i		−	1.49913i
$49$	−30022.6			−0.0364554
$50$	−1.46538e6	+	746175.i	−1.65789	+	0.844201i
$51$	−162062.			−0.171075
$52$		−	3.14138e6i		−	3.09819i
$53$		−	848433.i		−	0.782803i	−0.920220	−	0.391401i	$-0.871990\pi$
$53$		−	848433.i		−	0.782803i	0.920220	−	0.391401i	$-0.128010\pi$
$54$	−414299.			−0.358043
$55$	778178.	+	477005.i	0.630681	+	0.386593i
$56$	3.63731e6			2.76772
$57$			926272.i			0.662486i
$58$			1.77075e6i			1.19168i
$59$	−1.50633e6			−0.954857			−0.477428	−	0.878671i	$-0.658431\pi$
$59$	−1.50633e6			−0.954857			−0.477428	+	0.878671i	$0.658431\pi$
$60$	1.24252e6	−	2.02702e6i	0.742632	−	1.21152i
$61$	−3.47995e6			−1.96299			−0.981497	−	0.191476i	$-0.938673\pi$
$61$	−3.47995e6			−1.96299			−0.981497	+	0.191476i	$0.938673\pi$
$62$			3.82712e6i			2.03939i
$63$		−	673513.i		−	0.339355i
$64$	−2.79552e6			−1.33301
$65$	1.45654e6	−	2.37617e6i	0.657848	−	1.07320i
$66$	−1.85583e6			−0.794575
$67$			1.09706e6i			0.445625i	0.974861	+	0.222812i	$0.0715237\pi$
$67$			1.09706e6i			0.445625i	−0.974861	+	0.222812i	$0.928476\pi$
$68$			1.89098e6i			0.729299i
$69$	401921.			0.147289
$70$	4.63412e6	+	2.84061e6i	1.61482	+	0.989849i
$71$	329871.			0.109381			0.0546903	−	0.998503i	$-0.482583\pi$
$71$	329871.			0.109381			0.0546903	+	0.998503i	$0.482583\pi$
$72$			2.87005e6i			0.906203i
$73$		−	2.86250e6i		−	0.861223i	−0.902537	−	0.430611i	$-0.858298\pi$
$73$		−	2.86250e6i		−	0.861223i	0.902537	−	0.430611i	$-0.141702\pi$
$74$	−700521.			−0.200960
$75$	1.87971e6	−	957154.i	0.514490	−	0.261980i
$76$	1.08080e7			2.82420
$77$		−	3.01697e6i		−	0.753101i
$78$			5.66680e6i			1.35209i
$79$	1.04751e6			0.239035			0.119517	−	0.992832i	$-0.461865\pi$
$79$	1.04751e6			0.239035			0.119517	+	0.992832i	$0.461865\pi$
$80$	−1.01378e7	−	6.21427e6i	−2.21376	−	1.35699i
$81$	531441.			0.111111
$82$		−	1.38655e7i		−	2.77707i
$83$			3.76615e6i			0.722977i	0.932377	+	0.361489i	$0.117731\pi$
$83$			3.76615e6i			0.722977i	−0.932377	+	0.361489i	$0.882269\pi$
$84$	−7.85870e6			−1.44668
$85$	−876778.	+	1.43036e6i	−0.154854	+	0.252627i
$86$	−1.07299e7			−1.81908
$87$		−	2.27143e6i		−	0.369812i
$88$			1.28562e7i			2.01106i
$89$	9.44386e6			1.41999			0.709994	−	0.704208i	$-0.248698\pi$
$89$	9.44386e6			1.41999			0.709994	+	0.704208i	$0.248698\pi$
$90$	−2.24141e6	+	3.65659e6i	−0.324095	+	0.528723i
$91$	−9.21234e6			−1.28152
$92$		−	4.68970e6i		−	0.627897i
$93$		−	4.90922e6i		−	0.632882i
$94$	2.93440e6			0.364395
$95$	8.17526e6	+	5.01125e6i	0.978293	+	0.599671i
$96$	1.05710e7			1.21946
$97$			5.39979e6i			0.600726i	0.953825	+	0.300363i	$0.0971078\pi$
$97$			5.39979e6i			0.600726i	−0.953825	+	0.300363i	$0.902892\pi$
$98$		−	631932.i		−	0.0678233i
$99$	2.38056e6			0.246579
$100$	−1.11683e7	−	2.19329e7i	−1.11683	−	2.19329i
$101$	−1.05326e7			−1.01721			−0.508607	−	0.860999i	$-0.669840\pi$
$101$	−1.05326e7			−1.01721			−0.508607	+	0.860999i	$0.669840\pi$
$102$		−	3.41118e6i		−	0.318276i
$103$			1.33641e6i			0.120506i	0.998183	+	0.0602531i	$0.0191908\pi$
$103$			1.33641e6i			0.120506i	−0.998183	+	0.0602531i	$0.980809\pi$
$104$	3.92566e7			3.42213
$105$	−5.94441e6	−	3.64379e6i	−0.501125	−	0.307178i
$106$	1.78583e7			1.45636
$107$			2.47894e7i			1.95625i	0.208026	+	0.978123i	$0.433296\pi$
$107$			2.47894e7i			1.95625i	−0.208026	+	0.978123i	$0.566704\pi$
$108$		−	6.20097e6i		−	0.473671i
$109$	−8.03467e6			−0.594258			−0.297129	−	0.954837i	$-0.596029\pi$
$109$	−8.03467e6			−0.594258			−0.297129	+	0.954837i	$0.596029\pi$
$110$	−1.00403e7	+	1.63795e7i	−0.719236	+	1.17335i
$111$	898592.			0.0623637
$112$			3.93041e7i			2.64347i
$113$			4.51903e6i			0.294626i	0.989090	+	0.147313i	$0.0470624\pi$
$113$			4.51903e6i			0.294626i	−0.989090	+	0.147313i	$0.952938\pi$
$114$	−1.94967e7			−1.23252
$115$	2.17444e6	−	3.54735e6i	0.133323	−	0.217501i
$116$	−2.65035e7			−1.57652
$117$		−	7.26907e6i		−	0.419593i
$118$		−	3.17061e7i		−	1.77646i
$119$	5.54545e6			0.301663
$120$	2.53310e7	+	1.55273e7i	1.33819	+	0.820280i
$121$	−8.82356e6			−0.452788
$122$		−	7.32480e7i		−	3.65205i
$123$			1.77859e7i			0.861805i
$124$	−5.72819e7			−2.69800
$125$	1.72165e6	−	2.17686e7i	0.0788425	−	0.996887i
$126$	1.41765e7			0.631352
$127$			3.42291e7i			1.48280i	0.671063	+	0.741400i	$0.265838\pi$
$127$			3.42291e7i			1.48280i	−0.671063	+	0.741400i	$0.734162\pi$
$128$		−	8.72719e6i		−	0.367824i
$129$	1.37638e7			0.564514
$130$	5.00150e7	+	3.06581e7i	1.99664	+	1.22389i
$131$	3.70918e7			1.44154			0.720772	−	0.693172i	$-0.243787\pi$
$131$	3.70918e7			1.44154			0.720772	+	0.693172i	$0.243787\pi$
$132$		−	2.77769e7i		−	1.05118i
$133$		−	3.16952e7i		−	1.16819i
$134$	−2.30916e7			−0.829061
$135$	2.87516e6	−	4.69049e6i	0.100576	−	0.164078i
$136$	−2.36309e7			−0.805553
$137$		−	1.60427e7i		−	0.533035i	−0.963830	−	0.266518i	$-0.914127\pi$
$137$		−	1.60427e7i		−	0.533035i	0.963830	−	0.266518i	$-0.0858731\pi$
$138$			8.45986e6i			0.274023i
$139$	1.06763e7			0.337184			0.168592	−	0.985686i	$-0.446078\pi$
$139$	1.06763e7			0.337184			0.168592	+	0.985686i	$0.446078\pi$
$140$	−4.25165e7	+	6.93607e7i	−1.30951	+	2.13632i
$141$	−3.76410e6			−0.113082
$142$			6.94332e6i			0.203497i
$143$		−	3.25614e7i		−	0.931167i
$144$	−3.10132e7			−0.865521
$145$	−2.00476e7	−	1.22887e7i	−0.546102	−	0.334748i
$146$	6.02515e7			1.60226
$147$			810609.i			0.0210475i
$148$		−	1.04850e7i		−	0.265859i
$149$	−5.01130e7			−1.24108			−0.620538	−	0.784177i	$-0.713086\pi$
$149$	−5.01130e7			−1.24108			−0.620538	+	0.784177i	$0.713086\pi$
$150$	2.01467e7	+	3.95652e7i	0.487399	+	0.957182i
$151$	6.63039e7			1.56718			0.783591	−	0.621277i	$-0.213386\pi$
$151$	6.63039e7			1.56718			0.783591	+	0.621277i	$0.213386\pi$
$152$			1.35063e8i			3.11949i
$153$			4.37568e6i			0.0987702i
$154$	6.35028e7			1.40110
$155$	−4.33287e7	−	2.65595e7i	−0.934577	−	0.572874i
$156$	−8.48171e7			−1.78874
$157$			2.60565e7i			0.537363i	0.963229	+	0.268682i	$0.0865880\pi$
$157$			2.60565e7i			0.537363i	−0.963229	+	0.268682i	$0.913412\pi$
$158$			2.20485e7i			0.444712i
$159$	−2.29077e7			−0.451951
$160$	5.71904e7	−	9.32995e7i	1.10383	−	1.80077i
$161$	−1.37529e7			−0.259720
$162$			1.11861e7i			0.206716i
$163$		−	1.56423e7i		−	0.282907i	−0.989945	−	0.141454i	$-0.954822\pi$
$163$		−	1.56423e7i		−	0.282907i	0.989945	−	0.141454i	$-0.0451776\pi$
$164$	2.07530e8			3.67391
$165$	1.28791e7	−	2.10108e7i	0.223199	−	0.364124i
$166$	−7.92721e7			−1.34506
$167$			1.10384e7i			0.183400i	0.995787	+	0.0917000i	$0.0292301\pi$
$167$			1.10384e7i			0.183400i	−0.995787	+	0.0917000i	$0.970770\pi$
$168$		−	9.82073e7i		−	1.59794i
$169$	−3.66781e7			−0.584526
$170$	−3.01070e7	−	1.84549e7i	−0.469998	−	0.288098i
$171$	2.50093e7			0.382486
$172$		−	1.60599e8i		−	2.40654i
$173$			6.51735e6i			0.0956995i	0.998855	+	0.0478497i	$0.0152369\pi$
$173$			6.51735e6i			0.0956995i	−0.998855	+	0.0478497i	$0.984763\pi$
$174$	4.78103e7			0.688017
$175$	−6.43200e7	+	3.27519e7i	−0.907221	+	0.461959i
$176$	−1.38922e8			−1.92078
$177$			4.06709e7i			0.551287i
$178$			1.98780e8i			2.64181i
$179$	−5.55709e7			−0.724205			−0.362102	−	0.932138i	$-0.617941\pi$
$179$	−5.55709e7			−0.724205			−0.362102	+	0.932138i	$0.617941\pi$
$180$	−5.47297e7	−	3.35480e7i	−0.699470	−	0.428759i
$181$	3.17736e7			0.398283			0.199142	−	0.979971i	$-0.436185\pi$
$181$	3.17736e7			0.398283			0.199142	+	0.979971i	$0.436185\pi$
$182$		−	1.93906e8i		−	2.38420i
$183$			9.39588e7i			1.13334i
$184$	5.86055e7			0.693548
$185$	4.86149e6	−	7.93095e6i	0.0564506	−	0.0920925i
$186$	1.03332e8			1.17744
$187$			1.96007e7i			0.219192i
$188$			4.39203e7i			0.482073i
$189$	−1.81849e7			−0.195927
$190$	−1.05480e8	+	1.72077e8i	−1.11566	+	1.82006i
$191$	−2.55107e7			−0.264914			−0.132457	−	0.991189i	$-0.542287\pi$
$191$	−2.55107e7			−0.264914			−0.132457	+	0.991189i	$0.542287\pi$
$192$			7.54789e7i			0.769611i
$193$		−	1.82313e8i		−	1.82544i	−0.408584	−	0.912721i	$-0.633977\pi$
$193$		−	1.82313e8i		−	1.82544i	0.408584	−	0.912721i	$-0.366023\pi$
$194$	−1.13658e8			−1.11762
$195$	−6.41567e7	−	3.93266e7i	−0.619613	−	0.379809i
$196$	9.45837e6			0.0897264
$197$		−	5.00315e6i		−	0.0466242i	−0.999728	−	0.0233121i	$-0.992579\pi$
$197$		−	5.00315e6i		−	0.0466242i	0.999728	−	0.0233121i	$-0.00742115\pi$
$198$			5.01074e7i			0.458748i
$199$	1.21656e8			1.09433			0.547164	−	0.837025i	$-0.315707\pi$
$199$	1.21656e8			1.09433			0.547164	+	0.837025i	$0.315707\pi$
$200$	2.74088e8	−	1.39566e8i	2.42261	−	1.23360i
$201$	2.96207e7			0.257282
$202$		−	2.21697e8i		−	1.89248i
$203$			7.77237e7i			0.652105i
$204$	5.10565e7			0.421061
$205$	1.56978e8	+	9.62242e7i	1.27263	+	0.780092i
$206$	−2.81295e7			−0.224195
$207$		−	1.08519e7i		−	0.0850371i
$208$			4.24200e8i			3.26850i
$209$	1.12028e8			0.848819
$210$	7.66965e7	−	1.25121e8i	0.571489	−	0.932317i
$211$	1.39532e8			1.02255			0.511275	−	0.859417i	$-0.329174\pi$
$211$	1.39532e8			1.02255			0.511275	+	0.859417i	$0.329174\pi$
$212$			2.67292e8i			1.92669i
$213$		−	8.90653e6i		−	0.0631509i
$214$	−5.21782e8			−3.63949
$215$	7.44639e7	−	1.21479e8i	0.510988	−	0.833617i
$216$	7.74913e7			0.523197
$217$			1.67984e8i			1.11599i
$218$		−	1.69118e8i		−	1.10559i
$219$	−7.72875e7			−0.497227
$220$	−2.45159e8	−	1.50277e8i	−1.55227	−	0.951508i
$221$	5.98508e7			0.372990
$222$			1.89141e7i			0.116024i
$223$		−	2.25016e8i		−	1.35877i	−0.733782	−	0.679385i	$-0.762247\pi$
$223$		−	2.25016e8i		−	1.35877i	0.733782	−	0.679385i	$-0.237753\pi$
$224$	−3.61719e8			−2.15032
$225$	−2.58432e7	−	5.07522e7i	−0.151254	−	0.297041i
$226$	−9.51191e7			−0.548136
$227$			2.50149e8i			1.41941i	0.704499	+	0.709705i	$0.251172\pi$
$227$			2.50149e8i			1.41941i	−0.704499	+	0.709705i	$0.748828\pi$
$228$		−	2.91815e8i		−	1.63055i
$229$	−1.33900e8			−0.736814			−0.368407	−	0.929665i	$-0.620097\pi$
$229$	−1.33900e8			−0.736814			−0.368407	+	0.929665i	$0.620097\pi$
$230$	7.46666e7	+	4.57689e7i	0.404649	+	0.248041i
$231$	−8.14581e7			−0.434803
$232$		−	3.31205e8i		−	1.74136i
$233$			3.00906e8i			1.55842i	0.626763	+	0.779210i	$0.284380\pi$
$233$			3.00906e8i			1.55842i	−0.626763	+	0.779210i	$0.715620\pi$
$234$	1.53004e8			0.780631
$235$	−2.03642e7	+	3.32218e7i	−0.102360	+	0.166988i
$236$	4.74557e8			2.35016
$237$		−	2.82826e7i		−	0.138007i
$238$			1.16724e8i			0.561229i
$239$	4.73546e7			0.224373			0.112186	−	0.993687i	$-0.464215\pi$
$239$	4.73546e7			0.224373			0.112186	+	0.993687i	$0.464215\pi$
$240$	−1.67785e8	+	2.73722e8i	−0.783456	+	1.27812i
$241$	−1.60378e8			−0.738047			−0.369023	−	0.929420i	$-0.620308\pi$
$241$	−1.60378e8			−0.738047			−0.369023	+	0.929420i	$0.620308\pi$
$242$		−	1.85723e8i		−	0.842388i
$243$		−	1.43489e7i		−	0.0641500i
$244$	1.09633e9			4.83145
$245$	7.15442e6	+	4.38550e6i	0.0310809	+	0.0190519i
$246$	−3.74369e8			−1.60334
$247$		−	3.42079e8i		−	1.44440i
$248$		−	7.15832e8i		−	2.98009i
$249$	1.01686e8			0.417411
$250$	4.58198e8	+	3.62383e7i	1.85466	+	0.146682i
$251$	−3.49310e8			−1.39429			−0.697145	−	0.716930i	$-0.745547\pi$
$251$	−3.49310e8			−1.39429			−0.697145	+	0.716930i	$0.745547\pi$
$252$			2.12185e8i			0.835243i
$253$		−	4.86104e7i		−	0.188715i
$254$	−7.20474e8			−2.75867
$255$	3.86197e7	+	2.36730e7i	0.145854	+	0.0894052i
$256$	−1.74131e8			−0.648689
$257$		−	1.26125e8i		−	0.463484i	−0.972777	−	0.231742i	$-0.925558\pi$
$257$		−	1.26125e8i		−	0.463484i	0.972777	−	0.231742i	$-0.0744425\pi$
$258$			2.89708e8i			1.05025i
$259$	−3.07480e7			−0.109968
$260$	−4.58871e8	+	7.48594e8i	−1.61914	+	2.64143i
$261$	−6.13285e7			−0.213511
$262$			7.80728e8i			2.68192i
$263$			2.64475e8i			0.896479i	0.893914	+	0.448239i	$0.147949\pi$
$263$			2.64475e8i			0.896479i	−0.893914	+	0.448239i	$0.852051\pi$
$264$	3.47118e8			1.16108
$265$	−1.23934e8	+	2.02183e8i	−0.409099	+	0.667396i
$266$	6.67138e8			2.17335
$267$		−	2.54984e8i		−	0.819830i
$268$		−	3.45621e8i		−	1.09680i
$269$	3.76930e8			1.18067			0.590334	−	0.807159i	$-0.298996\pi$
$269$	3.76930e8			1.18067			0.590334	+	0.807159i	$0.298996\pi$
$270$	9.87280e7	+	6.05180e7i	0.305258	+	0.187116i
$271$	−2.26190e8			−0.690368			−0.345184	−	0.938535i	$-0.612184\pi$
$271$	−2.26190e8			−0.690368			−0.345184	+	0.938535i	$0.612184\pi$
$272$		−	2.55351e8i		−	0.769390i
$273$			2.48733e8i			0.739885i
$274$	3.37676e8			0.991685
$275$	−1.15763e8	−	2.27342e8i	−0.335665	−	0.659197i
$276$	−1.26622e8			−0.362516
$277$			5.36580e8i			1.51690i	0.651734	+	0.758448i	$0.274042\pi$
$277$			5.36580e8i			1.51690i	−0.651734	+	0.758448i	$0.725958\pi$
$278$			2.24720e8i			0.627314i
$279$	−1.32549e8			−0.365394
$280$	−8.66776e8	−	5.31314e8i	−2.35968	−	1.44643i
$281$	−2.59895e8			−0.698756			−0.349378	−	0.936982i	$-0.613607\pi$
$281$	−2.59895e8			−0.698756			−0.349378	+	0.936982i	$0.613607\pi$
$282$		−	7.92288e7i		−	0.210383i
$283$		−	6.40732e8i		−	1.68044i	−0.542243	−	0.840222i	$-0.682425\pi$
$283$		−	6.40732e8i		−	1.68044i	0.542243	−	0.840222i	$-0.317575\pi$
$284$	−1.03923e8			−0.269215
$285$	1.35304e8	−	2.20732e8i	0.346220	−	0.564818i
$286$	6.85372e8			1.73239
$287$		−	6.08600e8i		−	1.51965i
$288$		−	2.85417e8i		−	0.704054i
$289$	3.74311e8			0.912200
$290$	2.58660e8	−	4.21973e8i	0.622781	−	1.01599i
$291$	1.45794e8			0.346829
$292$			9.01808e8i			2.11970i
$293$		−	5.02145e8i		−	1.16625i	−0.812381	−	0.583127i	$-0.801829\pi$
$293$		−	5.02145e8i		−	1.16625i	0.812381	−	0.583127i	$-0.198171\pi$
$294$	−1.70622e7			−0.0391578
$295$	3.58961e8	+	2.20035e8i	0.814085	+	0.499016i
$296$	1.31027e8			0.293656
$297$		−	6.42752e7i		−	0.142363i
$298$		−	1.05481e9i		−	2.30896i
$299$	−1.48432e8			−0.321129
$300$	−5.92188e8	+	3.01544e8i	−1.26630	+	0.644802i
$301$	−4.70970e8			−0.995429
$302$			1.39560e9i			2.91566i
$303$			2.84381e8i			0.587289i
$304$	−1.45947e9			−2.97945
$305$	8.29278e8	+	5.08329e8i	1.67360	+	1.02588i
$306$	−9.21019e7			−0.183757
$307$		−	3.39123e8i		−	0.668918i	−0.942410	−	0.334459i	$-0.891446\pi$
$307$		−	3.39123e8i		−	0.668918i	0.942410	−	0.334459i	$-0.108554\pi$
$308$			9.50472e8i			1.85358i
$309$	3.60831e7			0.0695743
$310$	5.59040e8	−	9.12007e8i	1.06580	−	1.73873i
$311$	−3.58567e8			−0.675941			−0.337971	−	0.941157i	$-0.609740\pi$
$311$	−3.58567e8			−0.675941			−0.337971	+	0.941157i	$0.609740\pi$
$312$		−	1.05993e9i		−	1.97577i
$313$			6.27787e8i			1.15720i	0.815612	+	0.578599i	$0.196400\pi$
$313$			6.27787e8i			1.15720i	−0.815612	+	0.578599i	$0.803600\pi$
$314$	−5.48453e8			−0.999737
$315$	−9.83823e7	+	1.60499e8i	−0.177350	+	0.289325i
$316$	−3.30008e8			−0.588329
$317$		−	8.44426e8i		−	1.48886i	−0.667700	−	0.744430i	$-0.732721\pi$
$317$		−	8.44426e8i		−	1.48886i	0.667700	−	0.744430i	$-0.267279\pi$
$318$		−	4.82174e8i		−	0.840832i
$319$	−2.74718e8			−0.473827
$320$	6.66175e8	+	4.08350e8i	1.13648	+	0.696639i
$321$	6.69315e8			1.12944
$322$		−	2.89480e8i		−	0.483195i
$323$			2.05918e8i			0.340004i
$324$	−1.67426e8			−0.273474
$325$	−6.94191e8	+	3.53484e8i	−1.12173	+	0.571186i
$326$	3.29248e8			0.526334
$327$			2.16936e8i			0.343095i
$328$			2.59343e9i			4.05804i
$329$	1.28800e8			0.199402
$330$	4.42247e8	+	2.71087e8i	0.677433	+	0.415251i
$331$	1.15294e9			1.74747			0.873736	−	0.486401i	$-0.161691\pi$
$331$	1.15294e9			1.74747			0.873736	+	0.486401i	$0.161691\pi$
$332$		−	1.18650e9i		−	1.77944i
$333$		−	2.42620e7i		−	0.0360057i
$334$	−2.32343e8			−0.341206
$335$	1.60251e8	−	2.61431e8i	0.232887	−	0.379928i
$336$	1.06121e9			1.52621
$337$			3.43578e8i			0.489014i	0.969648	+	0.244507i	$0.0786261\pi$
$337$			3.43578e8i			0.489014i	−0.969648	+	0.244507i	$0.921374\pi$
$338$		−	7.72022e8i		−	1.08748i
$339$	1.22014e8			0.170102
$340$	2.76222e8	−	4.50623e8i	0.381138	−	0.621781i
$341$	−5.93747e8			−0.810888
$342$			5.26411e8i			0.711596i
$343$			7.33123e8i			0.980951i
$344$	2.00695e9			2.65816
$345$	−9.57784e7	−	5.87099e7i	−0.125574	−	0.0769741i
$346$	−1.37181e8			−0.178044
$347$			3.16194e8i			0.406257i	0.979152	+	0.203128i	$0.0651109\pi$
$347$			3.16194e8i			0.406257i	−0.979152	+	0.203128i	$0.934889\pi$
$348$			7.15595e8i			0.910207i
$349$	−1.05029e9			−1.32257			−0.661287	−	0.750133i	$-0.729989\pi$
$349$	−1.05029e9			−1.32257			−0.661287	+	0.750133i	$0.729989\pi$
$350$	−6.89381e8	−	1.35384e9i	−0.859451	−	1.68784i
$351$	−1.96265e8			−0.242252
$352$		−	1.27851e9i		−	1.56245i
$353$		−	1.06680e8i		−	0.129083i	−0.997915	−	0.0645417i	$-0.979441\pi$
$353$		−	1.06680e8i		−	0.129083i	0.997915	−	0.0645417i	$-0.0205585\pi$
$354$	−8.56064e8			−1.02564
$355$	−7.86088e7	−	4.81854e7i	−0.0932550	−	0.0571632i
$356$	−2.97521e9			−3.49497
$357$		−	1.49727e8i		−	0.174165i
$358$		−	1.16969e9i		−	1.34735i
$359$	−3.25288e8			−0.371054			−0.185527	−	0.982639i	$-0.559399\pi$
$359$	−3.25288e8			−0.371054			−0.185527	+	0.982639i	$0.559399\pi$
$360$	4.19238e8	−	6.83937e8i	0.473589	−	0.772604i
$361$	2.83056e8			0.316663
$362$			6.68790e8i			0.740985i
$363$			2.38236e8i			0.261417i
$364$	2.90227e9			3.15416
$365$	−4.18135e8	+	6.82138e8i	−0.450082	+	0.734255i
$366$	−1.97770e9			−2.10851
$367$			5.10495e8i			0.539089i	0.962988	+	0.269545i	$0.0868732\pi$
$367$			5.10495e8i			0.539089i	−0.962988	+	0.269545i	$0.913127\pi$
$368$			6.33280e8i			0.662413i
$369$	4.80220e8			0.497563
$370$	1.66935e8	+	1.02327e8i	0.171333	+	0.105023i
$371$	7.83856e8			0.796944
$372$			1.54661e9i			1.55769i
$373$		−	1.23881e9i		−	1.23602i	−0.786171	−	0.618009i	$-0.787939\pi$
$373$		−	1.23881e9i		−	1.23602i	0.786171	−	0.618009i	$-0.212061\pi$
$374$	−4.12566e8			−0.407796
$375$	−5.87753e8	−	4.64846e7i	−0.575553	−	0.0455197i
$376$	−5.48856e8			−0.532477
$377$			8.38854e8i			0.806291i
$378$		−	3.82765e8i		−	0.364511i
$379$	2.16554e8			0.204328			0.102164	−	0.994768i	$-0.467423\pi$
$379$	2.16554e8			0.204328			0.102164	+	0.994768i	$0.467423\pi$
$380$	−2.57555e9	−	1.57875e9i	−2.40784	−	1.47595i
$381$	9.24186e8			0.856095
$382$		−	5.36964e8i		−	0.492859i
$383$			4.57458e8i			0.416060i	0.978122	+	0.208030i	$0.0667052\pi$
$383$			4.57458e8i			0.416060i	−0.978122	+	0.208030i	$0.933295\pi$
$384$	−2.35634e8			−0.212363
$385$	−4.40699e8	+	7.18948e8i	−0.393576	+	0.642074i
$386$	3.83743e9			3.39614
$387$		−	3.71623e8i		−	0.325922i
$388$		−	1.70116e9i		−	1.47855i
$389$	3.61833e8			0.311662			0.155831	−	0.987784i	$-0.450194\pi$
$389$	3.61833e8			0.311662			0.155831	+	0.987784i	$0.450194\pi$
$390$	8.27768e8	−	1.35041e9i	0.706614	−	1.15276i
$391$	8.93502e7			0.0755922
$392$			1.18198e8i			0.0991079i
$393$		−	1.00148e9i		−	0.832276i
$394$	1.05309e8			0.0867420
$395$	−2.49622e8	−	1.53013e8i	−0.203795	−	0.124922i
$396$	−7.49978e8			−0.606897
$397$		−	2.41387e9i		−	1.93618i	−0.250598	−	0.968091i	$-0.580627\pi$
$397$		−	2.41387e9i		−	1.93618i	0.250598	−	0.968091i	$-0.419373\pi$
$398$			2.56068e9i			2.03594i
$399$	−8.55770e8			−0.674453
$400$	1.50812e9	+	2.96174e9i	1.17822	+	2.31386i
$401$	2.17916e9			1.68766			0.843829	−	0.536612i	$-0.180296\pi$
$401$	2.17916e9			1.68766			0.843829	+	0.536612i	$0.180296\pi$
$402$			6.23472e8i			0.478659i
$403$			1.81301e9i			1.37985i
$404$	3.31823e9			2.50364
$405$	−1.26643e8	−	7.76294e7i	−0.0947304	−	0.0580675i
$406$	−1.63597e9			−1.21321
$407$		−	1.08680e8i		−	0.0799044i
$408$			6.38034e8i			0.465086i
$409$	−1.27136e9			−0.918830			−0.459415	−	0.888222i	$-0.651941\pi$
$409$	−1.27136e9			−0.918830			−0.459415	+	0.888222i	$0.651941\pi$
$410$	−2.02538e9	+	3.30417e9i	−1.45132	+	2.36766i
$411$	−4.33153e8			−0.307748
$412$		−	4.21025e8i		−	0.296598i
$413$		−	1.39168e9i		−	0.972106i
$414$	2.28416e8			0.158207
$415$	5.50135e8	−	8.97480e8i	0.377834	−	0.616391i
$416$	−3.90395e9			−2.65875
$417$		−	2.88259e8i		−	0.194673i
$418$			2.35803e9i			1.57918i
$419$	8.39512e8			0.557542			0.278771	−	0.960358i	$-0.410073\pi$
$419$	8.39512e8			0.557542			0.278771	+	0.960358i	$0.410073\pi$
$420$	1.87274e9	+	1.14795e9i	1.23340	+	0.756048i
$421$	4.36477e8			0.285085			0.142542	−	0.989789i	$-0.454472\pi$
$421$	4.36477e8			0.285085			0.142542	+	0.989789i	$0.454472\pi$
$422$			2.93694e9i			1.90240i
$423$			1.01631e8i			0.0652880i
$424$	−3.34025e9			−2.12813
$425$	4.17875e8	−	2.12783e8i	0.264049	−	0.134455i
$426$	1.87470e8			0.117489
$427$		−	3.21508e9i		−	1.99846i
$428$		−	7.80972e9i		−	4.81484i
$429$	−8.79159e8			−0.537609
$430$	2.55696e9	+	1.56736e9i	1.55090	+	0.950667i
$431$	−2.57017e9			−1.54629			−0.773147	−	0.634227i	$-0.781318\pi$
$431$	−2.57017e9			−1.54629			−0.773147	+	0.634227i	$0.781318\pi$
$432$			8.37356e8i			0.499709i
$433$			6.59906e8i			0.390638i	0.980740	+	0.195319i	$0.0625742\pi$
$433$			6.59906e8i			0.390638i	−0.980740	+	0.195319i	$0.937426\pi$
$434$	−3.53582e9			−2.07623
$435$	−3.31795e8	+	5.41284e8i	−0.193267	+	0.315292i
$436$	2.53126e9			1.46263
$437$		−	5.10684e8i		−	0.292730i
$438$		−	1.62679e9i		−	0.925065i
$439$	2.75833e9			1.55604			0.778020	−	0.628240i	$-0.216224\pi$
$439$	2.75833e9			1.55604			0.778020	+	0.628240i	$0.216224\pi$
$440$	1.87795e9	−	3.06366e9i	1.05099	−	1.71457i
$441$	2.18865e7			0.0121518
$442$			1.25977e9i			0.693928i
$443$			2.35226e9i			1.28550i	0.766076	+	0.642749i	$0.222206\pi$
$443$			2.35226e9i			1.28550i	−0.766076	+	0.642749i	$0.777794\pi$
$444$	−2.83094e8			−0.153494
$445$	−2.25049e9	−	1.37950e9i	−1.21064	−	0.742097i
$446$	4.73626e9			2.52792
$447$			1.35305e9i			0.716535i
$448$		−	2.58274e9i		−	1.35709i
$449$	−2.21740e9			−1.15606			−0.578031	−	0.816015i	$-0.696179\pi$
$449$	−2.21740e9			−1.15606			−0.578031	+	0.816015i	$0.696179\pi$
$450$	1.06826e9	−	5.43962e8i	0.552629	−	0.281400i
$451$	2.15112e9			1.10420
$452$		−	1.42368e9i		−	0.725152i
$453$		−	1.79020e9i		−	0.904813i
$454$	−5.26527e9			−2.64074
$455$	2.19531e9	+	1.34568e9i	1.09259	+	0.669732i
$456$	3.64670e9			1.80104
$457$			3.45021e9i			1.69098i	0.533992	+	0.845490i	$0.320691\pi$
$457$			3.45021e9i			1.69098i	−0.533992	+	0.845490i	$0.679309\pi$
$458$		−	2.81841e9i		−	1.37080i
$459$	1.18143e8			0.0570250
$460$	−6.85041e8	+	1.11756e9i	−0.328144	+	0.535328i
$461$	−2.84777e9			−1.35379			−0.676896	−	0.736079i	$-0.736675\pi$
$461$	−2.84777e9			−1.35379			−0.676896	+	0.736079i	$0.736675\pi$
$462$		−	1.71458e9i		−	0.808928i
$463$		−	6.54000e8i		−	0.306228i	−0.988209	−	0.153114i	$-0.951070\pi$
$463$		−	6.54000e8i		−	0.306228i	0.988209	−	0.153114i	$-0.0489301\pi$
$464$	3.57894e9			1.66319
$465$	−7.17107e8	+	1.16988e9i	−0.330749	+	0.539578i
$466$	−6.33363e9			−2.89936
$467$			1.92475e9i			0.874510i	0.899338	+	0.437255i	$0.144049\pi$
$467$			1.92475e9i			0.874510i	−0.899338	+	0.437255i	$0.855951\pi$
$468$			2.29006e9i			1.03273i
$469$	−1.01356e9			−0.453675
$470$	−6.99272e8	−	4.28638e8i	−0.310673	−	0.190435i
$471$	7.03527e8			0.310247
$472$			5.93037e9i			2.59588i
$473$		−	1.66467e9i		−	0.723291i
$474$	5.95309e8			0.256755
$475$	−1.21617e9	−	2.38837e9i	−0.520673	−	1.02253i
$476$	−1.74705e9			−0.742474
$477$			6.18508e8i			0.260934i
$478$			9.96747e8i			0.417433i
$479$	1.70920e9			0.710588			0.355294	−	0.934755i	$-0.384381\pi$
$479$	1.70920e9			0.710588			0.355294	+	0.934755i	$0.384381\pi$
$480$	−2.51909e9	−	1.54414e9i	−1.03968	−	0.637298i
$481$	−3.31856e8			−0.135970
$482$		−	3.37572e9i		−	1.37310i
$483$			3.71329e8i			0.149949i
$484$	2.77979e9			1.11443
$485$	7.88766e8	−	1.28678e9i	0.313944	−	0.512163i
$486$	3.02024e8			0.119348
$487$			1.34530e9i			0.527797i	0.964550	+	0.263899i	$0.0850085\pi$
$487$			1.34530e9i			0.527797i	−0.964550	+	0.263899i	$0.914992\pi$
$488$			1.37005e10i			5.33662i
$489$	−4.22342e8			−0.163337
$490$	−9.23084e7	+	1.50590e8i	−0.0354450	+	0.0578243i
$491$	−4.33842e9			−1.65404			−0.827021	−	0.562171i	$-0.809966\pi$
$491$	−4.33842e9			−1.65404			−0.827021	+	0.562171i	$0.809966\pi$
$492$		−	5.60332e9i		−	2.12113i
$493$		−	5.04956e8i		−	0.189797i
$494$	7.20027e9			2.68723
$495$	−5.67292e8	−	3.47737e8i	−0.210227	−	0.128864i
$496$	7.73514e9			2.84631
$497$			3.04764e8i			0.111357i
$498$			2.14035e9i			0.776572i
$499$	1.71564e9			0.618124			0.309062	−	0.951042i	$-0.399985\pi$
$499$	1.71564e9			0.618124			0.309062	+	0.951042i	$0.399985\pi$
$500$	−5.42393e8	+	6.85803e9i	−0.194052	+	2.45360i
$501$	2.98038e8			0.105886
$502$		−	7.35248e9i		−	2.59400i
$503$			1.14006e9i			0.399430i	0.979854	+	0.199715i	$0.0640016\pi$
$503$			1.14006e9i			0.399430i	−0.979854	+	0.199715i	$0.935998\pi$
$504$	−2.65160e9			−0.922573
$505$	2.50995e9	+	1.53854e9i	0.867250	+	0.531605i
$506$	1.02318e9			0.351095
$507$			9.90309e8i			0.337476i
$508$		−	1.07836e10i		−	3.64957i
$509$	1.77349e9			0.596096			0.298048	−	0.954551i	$-0.403665\pi$
$509$	1.77349e9			0.596096			0.298048	+	0.954551i	$0.403665\pi$
$510$	−4.98283e8	+	8.12889e8i	−0.166334	+	0.271354i
$511$	2.64462e9			0.876780
$512$		−	4.78229e9i		−	1.57468i
$513$		−	6.75252e8i		−	0.220829i
$514$	2.65475e9			0.862287
$515$	1.95214e8	−	3.18468e8i	0.0629775	−	0.102740i
$516$	−4.33618e9			−1.38942
$517$			4.55249e8i			0.144888i
$518$		−	6.47202e8i		−	0.204590i
$519$	1.75968e8			0.0552521
$520$	−9.35491e9	−	5.73435e9i	−2.91762	−	1.78843i
$521$	−1.21314e7			−0.00375819			−0.00187910	−	0.999998i	$-0.500598\pi$
$521$	−1.21314e7			−0.00375819			−0.00187910	+	0.999998i	$0.500598\pi$
$522$		−	1.29088e9i		−	0.397227i
$523$			5.94269e9i			1.81647i	0.418466	+	0.908233i	$0.362568\pi$
$523$			5.94269e9i			1.81647i	−0.418466	+	0.908233i	$0.637432\pi$
$524$	−1.16855e10			−3.54802
$525$	8.84302e8	+	1.73664e9i	0.266712	+	0.523784i
$526$	−5.56683e9			−1.66785
$527$		−	1.09136e9i		−	0.324811i
$528$			3.75090e9i			1.10896i
$529$	3.18323e9			0.934918
$530$	−4.25566e9	−	2.60862e9i	−1.24166	−	0.761107i
$531$	1.09811e9			0.318286
$532$			9.98532e9i			2.87522i
$533$		−	6.56847e9i		−	1.87897i
$534$	5.36705e9			1.52525
$535$	3.62108e9	−	5.90736e9i	1.02235	−	1.66784i
$536$	4.31910e9			1.21148
$537$			1.50041e9i			0.418120i
$538$			7.93383e9i			2.19657i
$539$	9.80393e7			0.0269674
$540$	−9.05797e8	+	1.47770e9i	−0.247544	+	0.403839i
$541$	−1.40964e9			−0.382752			−0.191376	−	0.981517i	$-0.561295\pi$
$541$	−1.40964e9			−0.382752			−0.191376	+	0.981517i	$0.561295\pi$
$542$		−	4.76097e9i		−	1.28439i
$543$		−	8.57888e8i		−	0.229949i
$544$	2.35002e9			0.625857
$545$	1.91467e9	+	1.17365e9i	0.506649	+	0.310564i
$546$	−5.23547e9			−1.37652
$547$			1.59514e9i			0.416719i	0.978052	+	0.208359i	$0.0668124\pi$
$547$			1.59514e9i			0.416719i	−0.978052	+	0.208359i	$0.933188\pi$
$548$			5.05413e9i			1.31194i
$549$	2.53689e9			0.654332
$550$	4.78522e9	−	2.43665e9i	1.22640	−	0.624487i
$551$	−2.88609e9			−0.734987
$552$		−	1.58235e9i		−	0.400420i
$553$			9.67775e8i			0.243353i
$554$	−1.12942e10			−2.82210
$555$	−2.14136e8	−	1.31260e8i	−0.0531696	−	0.0325918i
$556$	−3.36347e9			−0.829900
$557$			4.14211e9i			1.01561i	0.861471	+	0.507807i	$0.169544\pi$
$557$			4.14211e9i			1.01561i	−0.861471	+	0.507807i	$0.830456\pi$
$558$		−	2.78997e9i		−	0.679797i
$559$	−5.08307e9			−1.23079
$560$	5.74128e9	−	9.36622e9i	1.38150	−	2.25375i
$561$	5.29218e8			0.126551
$562$		−	5.47042e9i		−	1.30000i
$563$		−	6.84542e9i		−	1.61667i	−0.588724	−	0.808334i	$-0.700370\pi$
$563$		−	6.84542e9i		−	1.61667i	0.588724	−	0.808334i	$-0.299630\pi$
$564$	1.18585e9			0.278325
$565$	6.60110e8	−	1.07689e9i	0.153974	−	0.251190i
$566$	1.34865e10			3.12638
$567$			4.90991e8i			0.113118i
$568$		−	1.29869e9i		−	0.297363i
$569$	4.13520e9			0.941030			0.470515	−	0.882392i	$-0.344068\pi$
$569$	4.13520e9			0.941030			0.470515	+	0.882392i	$0.344068\pi$
$570$	4.64609e9	+	2.84795e9i	1.05081	+	0.644125i
$571$	−6.25937e9			−1.40703			−0.703516	−	0.710679i	$-0.748388\pi$
$571$	−6.25937e9			−1.40703			−0.703516	+	0.710679i	$0.748388\pi$
$572$			1.02582e10i			2.29185i
$573$			6.88789e8i			0.152948i
$574$	1.28101e10			2.82724
$575$	−1.03635e9	+	5.27710e8i	−0.227335	+	0.115760i
$576$	2.03793e9			0.444335
$577$			8.20895e8i			0.177898i	0.996036	+	0.0889492i	$0.0283509\pi$
$577$			8.20895e8i			0.177898i	−0.996036	+	0.0889492i	$0.971649\pi$
$578$			7.87871e9i			1.69710i
$579$	−4.92246e9			−1.05392
$580$	6.31583e9	+	3.87146e9i	1.34410	+	0.823904i
$581$	−3.47950e9			−0.736038
$582$			3.06876e9i			0.645257i
$583$			2.77058e9i			0.579069i
$584$	−1.12696e10			−2.34133
$585$	−1.06182e9	+	1.73223e9i	−0.219283	+	0.357734i
$586$	1.05694e10			2.16975
$587$		−	2.11454e9i		−	0.431502i	−0.976448	−	0.215751i	$-0.930780\pi$
$587$		−	2.11454e9i		−	0.431502i	0.976448	−	0.215751i	$-0.0692200\pi$
$588$		−	2.55376e8i		−	0.0518036i
$589$	−6.23769e9			−1.25783
$590$	−4.63141e9	+	7.55561e9i	−0.928392	+	1.51456i
$591$	−1.35085e8			−0.0269185
$592$			1.41585e9i			0.280473i
$593$			3.04055e9i			0.598771i	0.954132	+	0.299386i	$0.0967817\pi$
$593$			3.04055e9i			0.598771i	−0.954132	+	0.299386i	$0.903218\pi$
$594$	1.35290e9			0.264858
$595$	−1.32149e9	−	8.10043e8i	−0.257190	−	0.157652i
$596$	1.57877e10			3.05462
$597$		−	3.28471e9i		−	0.631810i
$598$		−	3.12429e9i		−	0.597443i
$599$	−3.72414e9			−0.707999			−0.354000	−	0.935246i	$-0.615179\pi$
$599$	−3.72414e9			−0.707999			−0.354000	+	0.935246i	$0.615179\pi$
$600$	−3.76828e9	−	7.40036e9i	−0.712220	−	1.39870i
$601$	3.54365e8			0.0665872			0.0332936	−	0.999446i	$-0.489400\pi$
$601$	3.54365e8			0.0665872			0.0332936	+	0.999446i	$0.489400\pi$
$602$		−	9.91324e9i		−	1.85194i
$603$		−	7.99758e8i		−	0.148542i
$604$	−2.08885e10			−3.85725
$605$	2.10267e9	+	1.28889e9i	0.386035	+	0.236631i
$606$	−5.98582e9			−1.09262
$607$		−	6.16028e9i		−	1.11800i	−0.829169	−	0.558998i	$-0.811186\pi$
$607$		−	6.16028e9i		−	1.11800i	0.829169	−	0.558998i	$-0.188814\pi$
$608$		−	1.34316e10i		−	2.42362i
$609$	2.09854e9			0.376493
$610$	−1.06996e10	+	1.74551e10i	−1.90859	+	3.11364i
$611$	1.39011e9			0.246549
$612$		−	1.37852e9i		−	0.243100i
$613$		−	5.07539e9i		−	0.889933i	−0.895547	−	0.444967i	$-0.853216\pi$
$613$		−	5.07539e9i		−	0.889933i	0.895547	−	0.444967i	$-0.146784\pi$
$614$	7.13806e9			1.24449
$615$	2.59805e9	−	4.23842e9i	0.450386	−	0.734752i
$616$	−1.18777e10			−2.04739
$617$			1.02629e10i			1.75902i	0.475881	+	0.879510i	$0.342129\pi$
$617$			1.02629e10i			1.75902i	−0.475881	+	0.879510i	$0.657871\pi$
$618$			7.59496e8i			0.129439i
$619$	2.67560e8			0.0453423			0.0226712	−	0.999743i	$-0.492783\pi$
$619$	2.67560e8			0.0453423			0.0226712	+	0.999743i	$0.492783\pi$
$620$	1.36504e10	+	8.36736e9i	2.30024	+	1.41000i
$621$	−2.93000e8			−0.0490962
$622$		−	7.54732e9i		−	1.25755i
$623$			8.72505e9i			1.44564i
$624$	1.14534e10			1.88707
$625$	−3.59009e9	+	4.93601e9i	−0.588200	+	0.808715i
$626$	−1.32140e10			−2.15291
$627$		−	3.02476e9i		−	0.490066i
$628$		−	8.20891e9i		−	1.32259i
$629$	1.99764e8			0.0320066
$630$	−3.37828e9	−	2.07081e9i	−0.538274	−	0.329950i
$631$	2.08002e9			0.329582			0.164791	−	0.986328i	$-0.447305\pi$
$631$	2.08002e9			0.329582			0.164791	+	0.986328i	$0.447305\pi$
$632$		−	4.12399e9i		−	0.649843i
$633$		−	3.76735e9i		−	0.590369i
$634$	1.77740e10			2.76995
$635$	4.99996e9	−	8.15685e9i	0.774923	−	1.26420i
$636$	7.21689e9			1.11237
$637$		−	2.99364e8i		−	0.0458893i
$638$		−	5.78242e9i		−	0.881531i
$639$	−2.40476e8			−0.0364602
$640$	−1.27481e9	+	2.07970e9i	−0.192227	+	0.313597i
$641$	1.96047e9			0.294006			0.147003	−	0.989136i	$-0.453037\pi$
$641$	1.96047e9			0.294006			0.147003	+	0.989136i	$0.453037\pi$
$642$			1.40881e10i			2.10126i
$643$			4.42670e9i			0.656662i	0.944563	+	0.328331i	$0.106486\pi$
$643$			4.42670e9i			0.656662i	−0.944563	+	0.328331i	$0.893514\pi$
$644$	4.33275e9			0.639239
$645$	−3.27993e9	−	2.01052e9i	−0.481289	−	0.295019i
$646$	−4.33427e9			−0.632560
$647$		−	6.30245e9i		−	0.914839i	−0.889251	−	0.457420i	$-0.848774\pi$
$647$		−	6.30245e9i		−	0.914839i	0.889251	−	0.457420i	$-0.151226\pi$
$648$		−	2.09227e9i		−	0.302068i
$649$	4.91895e9			0.706344
$650$	−7.44033e9	−	1.46117e10i	−1.06266	−	2.08691i
$651$	4.53556e9			0.644314
$652$			4.92798e9i			0.696310i
$653$		−	8.53328e9i		−	1.19928i	−0.800271	−	0.599639i	$-0.795311\pi$
$653$		−	8.53328e9i		−	1.19928i	0.800271	−	0.599639i	$-0.204689\pi$
$654$	−4.56619e9			−0.638311
$655$	−8.83902e9	−	5.41812e9i	−1.22902	−	0.753362i
$656$	−2.80241e10			−3.87587
$657$			2.08676e9i			0.287074i
$658$			2.71105e9i			0.370977i
$659$	4.84488e9			0.659453			0.329727	−	0.944076i	$-0.393043\pi$
$659$	4.84488e9			0.659453			0.329727	+	0.944076i	$0.393043\pi$
$660$	−4.05747e9	+	6.61929e9i	−0.549353	+	0.896205i
$661$	1.14066e10			1.53621			0.768106	−	0.640323i	$-0.221199\pi$
$661$	1.14066e10			1.53621			0.768106	+	0.640323i	$0.221199\pi$
$662$			2.42678e10i			3.25108i
$663$		−	1.61597e9i		−	0.215346i
$664$	1.48272e10			1.96549
$665$	−4.62982e9	+	7.55301e9i	−0.610504	+	0.995965i
$666$	5.10680e8			0.0669867
$667$			1.25231e9i			0.163407i
$668$		−	3.47757e9i		−	0.451396i
$669$	−6.07543e9			−0.784486
$670$	5.50276e9	+	3.37306e9i	0.706836	+	0.433274i
$671$	1.13639e10			1.45210
$672$			9.76640e9i			1.24149i
$673$		−	2.83977e9i		−	0.359112i	−0.983748	−	0.179556i	$-0.942534\pi$
$673$		−	2.83977e9i		−	0.359112i	0.983748	−	0.179556i	$-0.0574661\pi$
$674$	−7.23183e9			−0.909785
$675$	−1.37031e9	+	6.97765e8i	−0.171497	+	0.0873265i
$676$	1.15552e10			1.43867
$677$			4.43122e9i			0.548862i	0.961607	+	0.274431i	$0.0884895\pi$
$677$			4.43122e9i			0.548862i	−0.961607	+	0.274431i	$0.911511\pi$
$678$			2.56821e9i			0.316466i
$679$	−4.98880e9			−0.611577
$680$	5.63128e9	+	3.45184e9i	0.686793	+	0.420988i
$681$	6.75401e9			0.819496
$682$		−	1.24975e10i		−	1.50862i
$683$			6.08194e9i			0.730415i	0.930926	+	0.365208i	$0.119002\pi$
$683$			6.08194e9i			0.730415i	−0.930926	+	0.365208i	$0.880998\pi$
$684$	−7.87900e9			−0.941401
$685$	−2.34341e9	+	3.82300e9i	−0.278568	+	0.454452i
$686$	−1.54312e10			−1.82501
$687$			3.61531e9i			0.425400i
$688$			2.16867e10i			2.53883i
$689$	8.45998e9			0.985376
$690$	1.23576e9	−	2.01600e9i	0.143206	−	0.233624i
$691$	−6.54644e9			−0.754800			−0.377400	−	0.926050i	$-0.623182\pi$
$691$	−6.54644e9			−0.754800			−0.377400	+	0.926050i	$0.623182\pi$
$692$		−	2.05324e9i		−	0.235542i
$693$			2.19937e9i			0.251034i
$694$	−6.65543e9			−0.755820
$695$	−2.54417e9	−	1.55952e9i	−0.287474	−	0.176215i
$696$	−8.94253e9			−1.00538
$697$			3.95395e9i			0.442300i
$698$		−	2.21071e10i		−	2.46058i
$699$	8.12445e9			0.899755
$700$	2.02635e10	−	1.03182e10i	2.23291	−	1.13700i
$701$	−3.82297e9			−0.419168			−0.209584	−	0.977791i	$-0.567211\pi$
$701$	−3.82297e9			−0.419168			−0.209584	+	0.977791i	$0.567211\pi$
$702$		−	4.13109e9i		−	0.450698i
$703$		−	1.14176e9i		−	0.123945i
$704$	9.12881e9			0.986075
$705$	8.96989e8	+	5.49834e8i	0.0964107	+	0.0590976i
$706$	2.24545e9			0.240153
$707$		−	9.73097e9i		−	1.03559i
$708$		−	1.28130e10i		−	1.35686i
$709$	775984.			8.17694e−5			4.08847e−5	−	1.00000i	$-0.499987\pi$
$709$	775984.			8.17694e−5			4.08847e−5		1.00000i	$0.499987\pi$
$710$	1.01423e9	−	1.65460e9i	0.106349	−	0.173496i
$711$	−7.63631e8			−0.0796783
$712$		−	3.71802e10i		−	3.86039i
$713$			2.70661e9i			0.279649i
$714$	3.15154e9			0.324026
$715$	−4.75636e9	+	7.75944e9i	−0.486635	+	0.793888i
$716$	1.75072e10			1.78246
$717$		−	1.27857e9i		−	0.129542i
$718$		−	6.84684e9i		−	0.690327i
$719$	−6.00502e9			−0.602509			−0.301254	−	0.953544i	$-0.597405\pi$
$719$	−6.00502e9			−0.602509			−0.301254	+	0.953544i	$0.597405\pi$
$720$	7.39049e9	+	4.53020e9i	0.737920	+	0.452328i
$721$	−1.23469e9			−0.122683
$722$			5.95792e9i			0.589134i
$723$			4.33019e9i			0.426112i
$724$	−1.00100e10			−0.980281
$725$	2.98231e9	+	5.85683e9i	0.290650	+	0.570794i
$726$	−5.01453e9			−0.486353
$727$			6.55744e9i			0.632942i	0.948602	+	0.316471i	$0.102498\pi$
$727$			6.55744e9i			0.632942i	−0.948602	+	0.316471i	$0.897502\pi$
$728$			3.62687e10i			3.48395i
$729$	−3.87420e8			−0.0370370
$730$	−1.43580e10	−	8.80114e9i	−1.36604	−	0.837354i
$731$	3.05980e9			0.289723
$732$		−	2.96010e10i		−	2.78944i
$733$			2.19291e9i			0.205663i	0.994699	+	0.102831i	$0.0327902\pi$
$733$			2.19291e9i			0.205663i	−0.994699	+	0.102831i	$0.967210\pi$
$734$	−1.07452e10			−1.00295
$735$	1.18408e8	−	1.93169e8i	0.0109996	−	0.0179446i
$736$	−5.82813e9			−0.538837
$737$		−	3.58248e9i		−	0.329645i
$738$			1.01080e10i			0.925691i
$739$	−1.03179e10			−0.940452			−0.470226	−	0.882546i	$-0.655828\pi$
$739$	−1.03179e10			−0.940452			−0.470226	+	0.882546i	$0.655828\pi$
$740$	−1.53157e9	+	2.49858e9i	−0.138940	+	0.226664i
$741$	−9.23613e9			−0.833924
$742$			1.64990e10i			1.48267i
$743$			1.23675e10i			1.10616i	0.833127	+	0.553082i	$0.186548\pi$
$743$			1.23675e10i			1.10616i	−0.833127	+	0.553082i	$0.813452\pi$
$744$	−1.93275e10			−1.72056
$745$	1.19420e10	+	7.32017e9i	1.05811	+	0.648596i
$746$	2.60752e10			2.29955
$747$		−	2.74553e9i		−	0.240992i
$748$		−	6.17503e9i		−	0.539490i
$749$	−2.29026e10			−1.99159
$750$	9.78434e8	−	1.23714e10i	0.0846871	−	1.07079i
$751$	1.79874e10			1.54963			0.774817	−	0.632185i	$-0.217842\pi$
$751$	1.79874e10			1.54963			0.774817	+	0.632185i	$0.217842\pi$
$752$		−	5.93084e9i		−	0.508573i
$753$			9.43137e9i			0.804994i
$754$	−1.76567e10			−1.50006
$755$	−1.58003e10	−	9.68523e9i	−1.33614	−	0.819022i
$756$	5.72899e9			0.482228
$757$		−	4.81430e9i		−	0.403364i	−0.979451	−	0.201682i	$-0.935359\pi$
$757$		−	4.81430e9i		−	0.403364i	0.979451	−	0.201682i	$-0.0646408\pi$
$758$			4.55815e9i			0.380142i
$759$	−1.31248e9			−0.108955
$760$	1.97291e10	−	3.21857e10i	1.63027	−	2.65960i
$761$	−4.56773e9			−0.375711			−0.187856	−	0.982197i	$-0.560154\pi$
$761$	−4.56773e9			−0.375711			−0.187856	+	0.982197i	$0.560154\pi$
$762$			1.94528e10i			1.59272i
$763$		−	7.42312e9i		−	0.604993i
$764$	8.03694e9			0.652025
$765$	6.39171e8	−	1.04273e9i	0.0516181	−	0.0842088i
$766$	−9.62884e9			−0.774058
$767$		−	1.50201e10i		−	1.20195i
$768$			4.70154e9i			0.374521i
$769$	1.23428e10			0.978752			0.489376	−	0.872073i	$-0.337225\pi$
$769$	1.23428e10			0.978752			0.489376	+	0.872073i	$0.337225\pi$
$770$	−1.51328e10	−	9.27607e9i	−1.19454	−	0.732228i
$771$	−3.40537e9			−0.267592
$772$			5.74364e10i			4.49290i
$773$		−	7.76410e9i		−	0.604592i	−0.953214	−	0.302296i	$-0.902247\pi$
$773$		−	7.76410e9i		−	0.604592i	0.953214	−	0.302296i	$-0.0977532\pi$
$774$	7.82212e9			0.606361
$775$	6.44566e9	+	1.26583e10i	0.497406	+	0.976834i
$776$	2.12588e10			1.63314
$777$			8.30196e8i			0.0634903i
$778$			7.61605e9i			0.579831i
$779$	2.25989e10			1.71280
$780$	2.02120e10	+	1.23895e10i	1.52503	+	0.934810i
$781$	−1.07720e9			−0.0809130
$782$			1.88069e9i			0.140635i
$783$			1.65587e9i			0.123271i
$784$	−1.27722e9			−0.0946587
$785$	3.80617e9	−	6.20931e9i	0.280830	−	0.458142i
$786$	2.10797e10			1.54841
$787$			2.34346e9i			0.171375i	0.996322	+	0.0856873i	$0.0273086\pi$
$787$			2.34346e9i			0.171375i	−0.996322	+	0.0856873i	$0.972691\pi$
$788$			1.57620e9i			0.114755i
$789$	7.14083e9			0.517582
$790$	3.22069e9	−	5.25418e9i	0.232410	−	0.379149i
$791$	−4.17507e9			−0.299948
$792$		−	9.37220e9i		−	0.670353i
$793$		−	3.46996e10i		−	2.47098i
$794$	5.08084e10			3.60217
$795$	5.45894e9	+	3.34620e9i	0.385322	+	0.236193i
$796$	−3.83267e10			−2.69343
$797$		−	1.90723e10i		−	1.33444i	−0.744860	−	0.667221i	$-0.767484\pi$
$797$		−	1.90723e10i		−	1.33444i	0.744860	−	0.667221i	$-0.232516\pi$
$798$		−	1.80127e10i		−	1.25479i
$799$	−8.36788e8			−0.0580366
$800$	−2.72571e10	+	1.38794e10i	−1.88220	+	0.958420i
$801$	−6.88458e9			−0.473329
$802$			4.58683e10i			3.13980i
$803$			9.34755e9i			0.637079i
$804$	−9.33176e9			−0.633238
$805$	3.27734e9	+	2.00894e9i	0.221430	+	0.135732i
$806$	−3.81613e10			−2.56714
$807$		−	1.01771e10i		−	0.681659i
$808$			4.14667e10i			2.76541i
$809$	−2.15987e10			−1.43419			−0.717097	−	0.696973i	$-0.754530\pi$
$809$	−2.15987e10			−1.43419			−0.717097	+	0.696973i	$0.754530\pi$
$810$	1.63399e9	−	2.66566e9i	0.108032	−	0.176241i
$811$	−1.76644e10			−1.16286			−0.581428	−	0.813598i	$-0.697506\pi$
$811$	−1.76644e10			−1.16286			−0.581428	+	0.813598i	$0.697506\pi$
$812$		−	2.44862e10i		−	1.60500i
$813$			6.10713e9i			0.398584i
$814$	2.28756e9			0.148658
$815$	−2.28492e9	+	3.72758e9i	−0.147849	+	0.241199i
$816$	−6.89448e9			−0.444207
$817$		−	1.74884e10i		−	1.12195i
$818$		−	2.67602e10i		−	1.70944i
$819$	6.71579e9			0.427173
$820$	−4.94548e10	−	3.03147e10i	−3.13227	−	1.92001i
$821$	8.00505e9			0.504851			0.252425	−	0.967616i	$-0.418772\pi$
$821$	8.00505e9			0.504851			0.252425	+	0.967616i	$0.418772\pi$
$822$		−	9.11726e9i		−	0.572549i
$823$			9.23334e9i			0.577377i	0.957423	+	0.288688i	$0.0932192\pi$
$823$			9.23334e9i			0.577377i	−0.957423	+	0.288688i	$0.906781\pi$
$824$	5.26140e9			0.327609
$825$	−6.13824e9	+	3.12561e9i	−0.380588	+	0.193796i
$826$	2.92928e10			1.80855
$827$			1.81787e10i			1.11762i	0.829296	+	0.558809i	$0.188742\pi$
$827$			1.81787e10i			1.11762i	−0.829296	+	0.558809i	$0.811258\pi$
$828$			3.41879e9i			0.209299i
$829$	3.08927e10			1.88328			0.941639	−	0.336625i	$-0.109285\pi$
$829$	3.08927e10			1.88328			0.941639	+	0.336625i	$0.109285\pi$
$830$	1.88907e10	+	1.15795e10i	1.14676	+	0.702940i
$831$	1.44877e10			0.875780
$832$		−	2.78749e10i		−	1.67796i
$833$			1.80205e8i			0.0108021i
$834$	6.06744e9			0.362180
$835$	1.61242e9	−	2.63047e9i	0.0958463	−	0.156362i
$836$	−3.52936e10			−2.08917
$837$			3.57882e9i			0.210961i
$838$			1.76705e10i			1.03728i
$839$	−7.74952e9			−0.453010			−0.226505	−	0.974010i	$-0.572730\pi$
$839$	−7.74952e9			−0.453010			−0.226505	+	0.974010i	$0.572730\pi$
$840$	−1.43455e10	+	2.34030e10i	−0.835098	+	1.36236i
$841$	−1.01725e10			−0.589716
$842$			9.18722e9i			0.530386i
$843$			7.01716e9i			0.403427i
$844$	−4.39583e10			−2.51677
$845$	8.74045e9	+	5.35770e9i	0.498351	+	0.305478i
$846$	−2.13918e9			−0.121465
$847$		−	8.15196e9i		−	0.460967i
$848$		−	3.60942e10i		−	2.03260i
$849$	−1.72998e10			−0.970205
$850$	4.47877e9	+	8.79566e9i	0.250146	+	0.491250i
$851$	−4.95422e8			−0.0275564
$852$			2.80593e9i			0.155431i
$853$		−	2.37251e10i		−	1.30884i	−0.756131	−	0.654420i	$-0.772913\pi$
$853$		−	2.37251e10i		−	1.30884i	0.756131	−	0.654420i	$-0.227087\pi$
$854$	6.76728e10			3.71802
$855$	−5.95976e9	−	3.65320e9i	−0.326098	−	0.199890i
$856$	9.75952e10			5.31827
$857$		−	1.52749e10i		−	0.828980i	−0.910054	−	0.414490i	$-0.863960\pi$
$857$		−	1.52749e10i		−	0.828980i	0.910054	−	0.414490i	$-0.136040\pi$
$858$		−	1.85050e10i		−	1.00019i
$859$	1.01749e10			0.547712			0.273856	−	0.961771i	$-0.411701\pi$
$859$	1.01749e10			0.547712			0.273856	+	0.961771i	$0.411701\pi$
$860$	−2.34593e10	+	3.82710e10i	−1.25768	+	2.05175i
$861$	−1.64322e10			−0.877373
$862$		−	5.40985e10i		−	2.87680i
$863$			1.28050e10i			0.678177i	0.940755	+	0.339088i	$0.110119\pi$
$863$			1.28050e10i			0.678177i	−0.940755	+	0.339088i	$0.889881\pi$
$864$	−7.70626e9			−0.406486
$865$	9.52011e8	−	1.55309e9i	0.0500133	−	0.0815908i
$866$	−1.38901e10			−0.726761
$867$		−	1.01064e10i		−	0.526659i
$868$		−	5.29220e10i		−	2.74674i
$869$	−3.42065e9			−0.176823
$870$	−1.13933e10	−	6.98381e9i	−0.586585	−	0.359563i
$871$	−1.09391e10			−0.560943
$872$			3.16322e10i			1.61556i
$873$		−	3.93645e9i		−	0.200242i
$874$	1.07492e10			0.544608
$875$	2.01117e10	+	1.59061e9i	1.01490	+	0.0802667i
$876$	2.43488e10			1.22381
$877$			2.35075e10i			1.17681i	0.808565	+	0.588407i	$0.200245\pi$
$877$			2.35075e10i			1.17681i	−0.808565	+	0.588407i	$0.799755\pi$
$878$			5.80589e10i			2.89493i
$879$	−1.35579e10			−0.673337
$880$	3.31053e10	+	2.02928e10i	1.63760	+	1.00381i
$881$	−1.45948e10			−0.719089			−0.359545	−	0.933128i	$-0.617068\pi$
$881$	−1.45948e10			−0.719089			−0.359545	+	0.933128i	$0.617068\pi$
$882$			4.60678e8i			0.0226078i
$883$			2.28070e10i			1.11482i	0.830236	+	0.557411i	$0.188205\pi$
$883$			2.28070e10i			1.11482i	−0.830236	+	0.557411i	$0.811795\pi$
$884$	−1.88555e10			−0.918027
$885$	5.94094e9	−	9.69194e9i	0.288107	−	0.470012i
$886$	−4.95116e10			−2.39160
$887$			1.84263e10i			0.886553i	0.896385	+	0.443276i	$0.146184\pi$
$887$			1.84263e10i			0.886553i	−0.896385	+	0.443276i	$0.853816\pi$
$888$		−	3.53773e9i		−	0.169543i
$889$	−3.16238e10			−1.50959
$890$	2.90364e10	−	4.73695e10i	1.38063	−	2.25234i
$891$	−1.73543e9			−0.0821931
$892$			7.08894e10i			3.34429i
$893$			4.78268e9i			0.224746i
$894$	−2.84798e10			−1.33308
$895$	1.32426e10	+	8.11742e9i	0.617438	+	0.378475i
$896$	8.06293e9			0.374468
$897$			4.00767e9i			0.185404i
$898$		−	4.66730e10i		−	2.15079i
$899$	1.52962e10			0.702143
$900$	8.14168e9	+	1.59891e10i	0.372276	+	0.731097i
$901$	−5.09256e9			−0.231953
$902$			4.52781e10i			2.05431i
$903$			1.27162e10i			0.574711i
$904$	1.77913e10			0.800972
$905$	−7.57171e9	−	4.64128e9i	−0.339566	−	0.208146i
$906$	3.76812e10			1.68336
$907$			7.44505e9i			0.331316i	0.986183	+	0.165658i	$0.0529747\pi$
$907$			7.44505e9i			0.331316i	−0.986183	+	0.165658i	$0.947025\pi$
$908$		−	7.88073e10i		−	3.49354i
$909$	7.67830e9			0.339072
$910$	−2.83246e10	+	4.62082e10i	−1.24600	+	2.03270i
$911$	−2.92481e8			−0.0128169			−0.00640844	−	0.999979i	$-0.502040\pi$
$911$	−2.92481e8			−0.0128169			−0.00640844	+	0.999979i	$0.502040\pi$
$912$			3.94056e10i			1.72019i
$913$		−	1.22984e10i		−	0.534814i
$914$	−7.26219e10			−3.14598
$915$	1.37249e10	−	2.23905e10i	0.592290	−	0.966251i
$916$	4.21843e10			1.81349
$917$			3.42686e10i			1.46758i
$918$			2.48675e9i			0.106092i
$919$	−4.82234e8			−0.0204953			−0.0102476	−	0.999947i	$-0.503262\pi$
$919$	−4.82234e8			−0.0204953			−0.0102476	+	0.999947i	$0.503262\pi$
$920$	−1.39658e10	−	8.56071e9i	−0.591300	−	0.362453i
$921$	−9.15633e9			−0.386200
$922$		−	5.99415e10i		−	2.51866i
$923$			3.28924e9i			0.137686i
$924$	2.56627e10			1.07017
$925$	−2.31700e9	+	1.17982e9i	−0.0962565	+	0.0490141i
$926$	1.37658e10			0.569721
$927$		−	9.74242e8i		−	0.0401687i
$928$			3.29373e10i			1.35291i
$929$	7.33613e9			0.300201			0.150101	−	0.988671i	$-0.452040\pi$
$929$	7.33613e9			0.300201			0.150101	+	0.988671i	$0.452040\pi$
$930$	−2.46242e10	−	1.50941e10i	−1.00386	−	0.615341i
$931$	1.02997e9			0.0418311
$932$		−	9.47980e10i		−	3.83569i
$933$			9.68131e9i			0.390255i
$934$	−4.05132e10			−1.62698
$935$	2.86313e9	−	4.67087e9i	0.114552	−	0.186877i
$936$	−2.86181e10			−1.14071
$937$		−	1.88156e10i		−	0.747189i	−0.927592	−	0.373595i	$-0.878125\pi$
$937$		−	1.88156e10i		−	0.747189i	0.927592	−	0.373595i	$-0.121875\pi$
$938$		−	2.13340e10i		−	0.844038i
$939$	1.69503e10			0.668108
$940$	6.41559e9	−	1.04663e10i	0.251935	−	0.411003i
$941$	3.63671e10			1.42280			0.711402	−	0.702786i	$-0.248061\pi$
$941$	3.63671e10			1.42280			0.711402	+	0.702786i	$0.248061\pi$
$942$			1.48082e10i			0.577198i
$943$		−	9.80596e9i		−	0.380802i
$944$	−6.40825e10			−2.47935
$945$	4.33348e9	+	2.65632e9i	0.167042	+	0.102393i
$946$	3.50388e10			1.34564
$947$		−	5.58295e9i		−	0.213618i	−0.994280	−	0.106809i	$-0.965937\pi$
$947$		−	5.58295e9i		−	0.213618i	0.994280	−	0.106809i	$-0.0340634\pi$
$948$			8.91022e9i			0.339672i
$949$	2.85428e10			1.08409
$950$	5.02719e10	−	2.55986e10i	1.90236	−	0.968686i
$951$	−2.27995e10			−0.859594
$952$		−	2.18323e10i		−	0.820105i
$953$			2.57190e10i			0.962561i	0.876567	+	0.481281i	$0.159828\pi$
$953$			2.57190e10i			0.962561i	−0.876567	+	0.481281i	$0.840172\pi$
$954$	−1.30187e10			−0.485454
$955$	6.07924e9	+	3.72643e9i	0.225859	+	0.138446i
$956$	−1.49187e10			−0.552241
$957$			7.41739e9i			0.273564i
$958$			3.59762e10i			1.32201i
$959$	1.48216e10			0.542664
$960$	1.10255e10	−	1.79867e10i	0.402205	−	0.656150i
$961$	5.54704e9			0.201618
$962$		−	6.98510e9i		−	0.252965i
$963$		−	1.80715e10i		−	0.652082i
$964$	5.05257e10			1.81653
$965$	−2.66311e10	+	4.34455e10i	−0.953990	+	1.55632i
$966$	−7.81595e9			−0.278973
$967$			1.52250e9i			0.0541457i	0.999633	+	0.0270728i	$0.00861860\pi$
$967$			1.52250e9i			0.0541457i	−0.999633	+	0.0270728i	$0.991381\pi$
$968$			3.47380e10i			1.23095i
$969$	5.55977e9			0.196302
$970$	2.70848e10	+	1.66024e10i	0.952852	+	0.584076i
$971$	3.08348e9			0.108087			0.0540435	−	0.998539i	$-0.482789\pi$
$971$	3.08348e9			0.108087			0.0540435	+	0.998539i	$0.482789\pi$
$972$			4.52051e9i			0.157890i
$973$			9.86365e9i			0.343275i
$974$	−2.83166e10			−0.981939
$975$	9.54406e9	+	1.87432e10i	0.329775	+	0.647629i
$976$	−1.48045e11			−5.09704
$977$		−	4.14054e10i		−	1.42045i	−0.703974	−	0.710226i	$-0.748593\pi$
$977$		−	4.14054e10i		−	1.42045i	0.703974	−	0.710226i	$-0.251407\pi$
$978$		−	8.88969e9i		−	0.303879i
$979$	−3.08391e10			−1.05042
$980$	−2.25394e9	−	1.38162e9i	−0.0764983	−	0.0468917i
$981$	5.85727e9			0.198086
$982$		−	9.13176e10i		−	3.07726i
$983$		−	4.79594e10i		−	1.61041i	−0.592997	−	0.805205i	$-0.702055\pi$
$983$		−	4.79594e10i		−	1.61041i	0.592997	−	0.805205i	$-0.297945\pi$
$984$	7.00227e10			2.34291
$985$	−7.30827e8	+	1.19226e9i	−0.0243662	+	0.0397506i
$986$	1.06286e10			0.353107
$987$		−	3.47760e9i		−	0.115125i
$988$			1.07769e11i			3.55505i
$989$	−7.58842e9			−0.249439
$990$	7.31936e9	−	1.19407e10i	0.239745	−	0.391116i
$991$	−1.52883e10			−0.499000			−0.249500	−	0.968375i	$-0.580266\pi$
$991$	−1.52883e10			−0.499000			−0.249500	+	0.968375i	$0.580266\pi$
$992$			7.11871e10i			2.31532i
$993$		−	3.11295e10i		−	1.00890i
$994$	−6.41484e9			−0.207173
$995$	−2.89908e10	−	1.77707e10i	−0.932995	−	0.571904i
$996$	−3.20354e10			−1.02736
$997$		−	2.89932e8i		−	0.00926538i	−0.999989	−	0.00463269i	$-0.998525\pi$
$997$		−	2.89932e8i		−	0.00926538i	0.999989	−	0.00463269i	$-0.00147464\pi$
$998$			3.61118e10i			1.14999i
$999$	−6.55073e8			−0.0207879

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	15.8.b.a.4.8	yes	8
3.2	odd	2		45.8.b.d.19.1		8
4.3	odd	2		240.8.f.e.49.6		8
5.2	odd	4		75.8.a.i.1.1		4
5.3	odd	4		75.8.a.j.1.4		4
5.4	even	2	inner	15.8.b.a.4.1	✓	8
15.2	even	4		225.8.a.bb.1.4		4
15.8	even	4		225.8.a.z.1.1		4
15.14	odd	2		45.8.b.d.19.8		8
20.19	odd	2		240.8.f.e.49.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.8.b.a.4.1	✓	8	5.4	even	2	inner
15.8.b.a.4.8	yes	8	1.1	even	1	trivial
45.8.b.d.19.1		8	3.2	odd	2
45.8.b.d.19.8		8	15.14	odd	2
75.8.a.i.1.1		4	5.2	odd	4
75.8.a.j.1.4		4	5.3	odd	4
225.8.a.z.1.1		4	15.8	even	4
225.8.a.bb.1.4		4	15.2	even	4
240.8.f.e.49.2		8	20.19	odd	2
240.8.f.e.49.6		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)

\(f(q)\)	\(=\)	\(q+21.0486i q^{2} -27.0000i q^{3} -315.042 q^{4} +(-238.301 - 146.073i) q^{5} +568.311 q^{6} +923.886i q^{7} -3936.97i q^{8} -729.000 q^{9} +O(q^{10})\)	\(q+21.0486i q^{2} -27.0000i q^{3} -315.042 q^{4} +(-238.301 - 146.073i) q^{5} +568.311 q^{6} +923.886i q^{7} -3936.97i q^{8} -729.000 q^{9} +(3074.63 - 5015.90i) q^{10} -3265.52 q^{11} +8506.14i q^{12} +9971.29i q^{13} -19446.5 q^{14} +(-3943.98 + 6434.14i) q^{15} +42542.1 q^{16} -6002.31i q^{17} -15344.4i q^{18} -34306.4 q^{19} +(75075.0 + 46019.2i) q^{20} +24944.9 q^{21} -68734.5i q^{22} +14886.0i q^{23} -106298. q^{24} +(35450.2 + 69619.0i) q^{25} -209881. q^{26} +19683.0i q^{27} -291063. i q^{28} +84126.9 q^{29} +(-135429. - 83015.1i) q^{30} +181823. q^{31} +391519. i q^{32} +88169.0i q^{33} +126340. q^{34} +(134955. - 220163. i) q^{35} +229666. q^{36} +33281.2i q^{37} -722100. i q^{38} +269225. q^{39} +(-575086. + 938185. i) q^{40} -658739. q^{41} +525055. i q^{42} +509770. i q^{43} +1.02878e6 q^{44} +(173722. + 106487. i) q^{45} -313328. q^{46} -139411. i q^{47} -1.14864e6i q^{48} -30022.6 q^{49} +(-1.46538e6 + 746175. i) q^{50} -162062. q^{51} -3.14138e6i q^{52} -848433. i q^{53} -414299. q^{54} +(778178. + 477005. i) q^{55} +3.63731e6 q^{56} +926272. i q^{57} +1.77075e6i q^{58} -1.50633e6 q^{59} +(1.24252e6 - 2.02702e6i) q^{60} -3.47995e6 q^{61} +3.82712e6i q^{62} -673513. i q^{63} -2.79552e6 q^{64} +(1.45654e6 - 2.37617e6i) q^{65} -1.85583e6 q^{66} +1.09706e6i q^{67} +1.89098e6i q^{68} +401921. q^{69} +(4.63412e6 + 2.84061e6i) q^{70} +329871. q^{71} +2.87005e6i q^{72} -2.86250e6i q^{73} -700521. q^{74} +(1.87971e6 - 957154. i) q^{75} +1.08080e7 q^{76} -3.01697e6i q^{77} +5.66680e6i q^{78} +1.04751e6 q^{79} +(-1.01378e7 - 6.21427e6i) q^{80} +531441. q^{81} -1.38655e7i q^{82} +3.76615e6i q^{83} -7.85870e6 q^{84} +(-876778. + 1.43036e6i) q^{85} -1.07299e7 q^{86} -2.27143e6i q^{87} +1.28562e7i q^{88} +9.44386e6 q^{89} +(-2.24141e6 + 3.65659e6i) q^{90} -9.21234e6 q^{91} -4.68970e6i q^{92} -4.90922e6i q^{93} +2.93440e6 q^{94} +(8.17526e6 + 5.01125e6i) q^{95} +1.05710e7 q^{96} +5.39979e6i q^{97} -631932. i q^{98} +2.38056e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 666 q^{4} - 444 q^{5} + 486 q^{6} - 5832 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more