LMFDB - Embedded newform 75.5.f.d.7.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [75,5,Mod(7,75)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(75, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("75.7");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	75.f (of order $4$, degree $2$, 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:	$7.75274723129$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(i, \sqrt{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9 $ x^4 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			7.1
Root			$1.22474 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	75.7
Dual form			75.5.f.d.43.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.550510 + 0.550510i) q^{2} +(3.67423 - 3.67423i) q^{3} -15.3939i q^{4} +4.04541 q^{6} +(-16.7753 - 16.7753i) q^{7} +(17.2827 - 17.2827i) q^{8} -27.0000i q^{9} +O(q^{10})$	\(q+(0.550510 + 0.550510i) q^{2} +(3.67423 - 3.67423i) q^{3} -15.3939i q^{4} +4.04541 q^{6} +(-16.7753 - 16.7753i) q^{7} +(17.2827 - 17.2827i) q^{8} -27.0000i q^{9} -197.060 q^{11} +(-56.5607 - 56.5607i) q^{12} +(120.507 - 120.507i) q^{13} -18.4699i q^{14} -227.273 q^{16} +(152.449 + 152.449i) q^{17} +(14.8638 - 14.8638i) q^{18} -418.242i q^{19} -123.272 q^{21} +(-108.484 - 108.484i) q^{22} +(621.267 - 621.267i) q^{23} -127.001i q^{24} +132.681 q^{26} +(-99.2043 - 99.2043i) q^{27} +(-258.236 + 258.236i) q^{28} +792.756i q^{29} +208.426 q^{31} +(-401.639 - 401.639i) q^{32} +(-724.045 + 724.045i) q^{33} +167.850i q^{34} -415.635 q^{36} +(460.132 + 460.132i) q^{37} +(230.246 - 230.246i) q^{38} -885.545i q^{39} +2436.15 q^{41} +(-67.8627 - 67.8627i) q^{42} +(-2114.72 + 2114.72i) q^{43} +3033.52i q^{44} +684.028 q^{46} +(2910.44 + 2910.44i) q^{47} +(-835.056 + 835.056i) q^{48} -1838.18i q^{49} +1120.27 q^{51} +(-1855.08 - 1855.08i) q^{52} +(1289.93 - 1289.93i) q^{53} -109.226i q^{54} -579.842 q^{56} +(-1536.72 - 1536.72i) q^{57} +(-436.420 + 436.420i) q^{58} +1953.99i q^{59} +1227.13 q^{61} +(114.740 + 114.740i) q^{62} +(-452.932 + 452.932i) q^{63} +3194.16i q^{64} -797.189 q^{66} +(-1165.88 - 1165.88i) q^{67} +(2346.79 - 2346.79i) q^{68} -4565.36i q^{69} -3109.97 q^{71} +(-466.632 - 466.632i) q^{72} +(3457.41 - 3457.41i) q^{73} +506.614i q^{74} -6438.36 q^{76} +(3305.74 + 3305.74i) q^{77} +(487.502 - 487.502i) q^{78} -9879.38i q^{79} -729.000 q^{81} +(1341.13 + 1341.13i) q^{82} +(3296.81 - 3296.81i) q^{83} +1897.64i q^{84} -2328.35 q^{86} +(2912.77 + 2912.77i) q^{87} +(-3405.72 + 3405.72i) q^{88} -6138.84i q^{89} -4043.08 q^{91} +(-9563.71 - 9563.71i) q^{92} +(765.804 - 765.804i) q^{93} +3204.45i q^{94} -2951.43 q^{96} +(-728.123 - 728.123i) q^{97} +(1011.94 - 1011.94i) q^{98} +5320.63i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{2} - 72 q^{6} - 72 q^{7} - 264 q^{8}+O(q^{10}) $ 4 * q + 12 * q^2 - 72 * q^6 - 72 * q^7 - 264 * q^8	$ 4 q + 12 q^{2} - 72 q^{6} - 72 q^{7} - 264 q^{8} - 24 q^{11} - 432 q^{12} + 144 q^{13} - 2320 q^{16} + 600 q^{17} + 324 q^{18} + 36 q^{21} + 1800 q^{22} + 888 q^{23} - 792 q^{26} + 1152 q^{28} + 3244 q^{31} - 6192 q^{32} - 2808 q^{33} + 1512 q^{36} + 2448 q^{37} - 1068 q^{38} + 6864 q^{41} + 1404 q^{42} - 5544 q^{43} - 2496 q^{46} + 5616 q^{47} + 5184 q^{48} + 72 q^{51} - 7920 q^{52} + 1848 q^{53} + 10320 q^{56} - 4104 q^{57} + 5544 q^{58} + 15020 q^{61} + 15636 q^{62} - 1944 q^{63} - 16416 q^{66} - 13320 q^{67} - 8112 q^{68} - 21552 q^{71} + 7128 q^{72} + 19728 q^{73} - 25048 q^{76} - 504 q^{77} + 4860 q^{78} - 2916 q^{81} + 13536 q^{82} + 8592 q^{83} - 18984 q^{86} + 10152 q^{87} - 62064 q^{88} - 4356 q^{91} - 34512 q^{92} - 8856 q^{93} + 33696 q^{96} - 11520 q^{97} + 20136 q^{98}+O(q^{100}) $ 4 * q + 12 * q^2 - 72 * q^6 - 72 * q^7 - 264 * q^8 - 24 * q^11 - 432 * q^12 + 144 * q^13 - 2320 * q^16 + 600 * q^17 + 324 * q^18 + 36 * q^21 + 1800 * q^22 + 888 * q^23 - 792 * q^26 + 1152 * q^28 + 3244 * q^31 - 6192 * q^32 - 2808 * q^33 + 1512 * q^36 + 2448 * q^37 - 1068 * q^38 + 6864 * q^41 + 1404 * q^42 - 5544 * q^43 - 2496 * q^46 + 5616 * q^47 + 5184 * q^48 + 72 * q^51 - 7920 * q^52 + 1848 * q^53 + 10320 * q^56 - 4104 * q^57 + 5544 * q^58 + 15020 * q^61 + 15636 * q^62 - 1944 * q^63 - 16416 * q^66 - 13320 * q^67 - 8112 * q^68 - 21552 * q^71 + 7128 * q^72 + 19728 * q^73 - 25048 * q^76 - 504 * q^77 + 4860 * q^78 - 2916 * q^81 + 13536 * q^82 + 8592 * q^83 - 18984 * q^86 + 10152 * q^87 - 62064 * q^88 - 4356 * q^91 - 34512 * q^92 - 8856 * q^93 + 33696 * q^96 - 11520 * q^97 + 20136 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

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 $

$2$	0.550510	+	0.550510i	0.137628	+	0.137628i	0.772564	−	0.634937i	$-0.218974\pi$
$2$	0.550510	+	0.550510i	0.137628	+	0.137628i	−0.634937	+	0.772564i	$0.718974\pi$
$3$	3.67423	−	3.67423i	0.408248	−	0.408248i
$3$	3.67423	−	3.67423i	0.408248	−	0.408248i
$4$		−	15.3939i		−	0.962117i
$5$		0			0
$5$		0			0
$6$	4.04541			0.112372
$7$	−16.7753	−	16.7753i	−0.342352	−	0.342352i	0.514899	−	0.857251i	$-0.327829\pi$
$7$	−16.7753	−	16.7753i	−0.342352	−	0.342352i	−0.857251	+	0.514899i	$0.827829\pi$
$8$	17.2827	−	17.2827i	0.270041	−	0.270041i
$9$		−	27.0000i		−	0.333333i
$10$		0			0
$11$	−197.060			−1.62860			−0.814298	−	0.580447i	$-0.802878\pi$
$11$	−197.060			−1.62860			−0.814298	+	0.580447i	$0.802878\pi$
$12$	−56.5607	−	56.5607i	−0.392783	−	0.392783i
$13$	120.507	−	120.507i	0.713062	−	0.713062i	−0.254113	−	0.967175i	$-0.581784\pi$
$13$	120.507	−	120.507i	0.713062	−	0.713062i	0.967175	+	0.254113i	$0.0817835\pi$
$14$		−	18.4699i		−	0.0942342i
$15$		0			0
$16$	−227.273			−0.887787
$17$	152.449	+	152.449i	0.527507	+	0.527507i	0.919828	−	0.392321i	$-0.128328\pi$
$17$	152.449	+	152.449i	0.527507	+	0.527507i	−0.392321	+	0.919828i	$0.628328\pi$
$18$	14.8638	−	14.8638i	0.0458759	−	0.0458759i
$19$		−	418.242i		−	1.15856i	−0.815127	−	0.579282i	$-0.803333\pi$
$19$		−	418.242i		−	1.15856i	0.815127	−	0.579282i	$-0.196667\pi$
$20$		0			0
$21$	−123.272			−0.279529
$22$	−108.484	−	108.484i	−0.224140	−	0.224140i
$23$	621.267	−	621.267i	1.17442	−	1.17442i	0.193272	−	0.981145i	$-0.438090\pi$
$23$	621.267	−	621.267i	1.17442	−	1.17442i	0.981145	−	0.193272i	$-0.0619101\pi$
$24$		−	127.001i		−	0.220488i
$25$		0			0
$26$	132.681			0.196274
$27$	−99.2043	−	99.2043i	−0.136083	−	0.136083i
$28$	−258.236	+	258.236i	−0.329383	+	0.329383i
$29$			792.756i			0.942635i	0.881964	+	0.471318i	$0.156221\pi$
$29$			792.756i			0.942635i	−0.881964	+	0.471318i	$0.843779\pi$
$30$		0			0
$31$	208.426			0.216884			0.108442	−	0.994103i	$-0.465414\pi$
$31$	208.426			0.216884			0.108442	+	0.994103i	$0.465414\pi$
$32$	−401.639	−	401.639i	−0.392225	−	0.392225i
$33$	−724.045	+	724.045i	−0.664872	+	0.664872i
$34$			167.850i			0.145199i
$35$		0			0
$36$	−415.635			−0.320706
$37$	460.132	+	460.132i	0.336108	+	0.336108i	0.854900	−	0.518792i	$-0.173618\pi$
$37$	460.132	+	460.132i	0.336108	+	0.336108i	−0.518792	+	0.854900i	$0.673618\pi$
$38$	230.246	−	230.246i	0.159450	−	0.159450i
$39$		−	885.545i		−	0.582212i
$40$		0			0
$41$	2436.15			1.44923			0.724613	−	0.689156i	$-0.242018\pi$
$41$	2436.15			1.44923			0.724613	+	0.689156i	$0.242018\pi$
$42$	−67.8627	−	67.8627i	−0.0384709	−	0.0384709i
$43$	−2114.72	+	2114.72i	−1.14371	+	1.14371i	−0.155946	+	0.987766i	$0.549843\pi$
$43$	−2114.72	+	2114.72i	−1.14371	+	1.14371i	−0.987766	+	0.155946i	$0.950157\pi$
$44$			3033.52i			1.56690i
$45$		0			0
$46$	684.028			0.323264
$47$	2910.44	+	2910.44i	1.31754	+	1.31754i	0.915721	+	0.401815i	$0.131620\pi$
$47$	2910.44	+	2910.44i	1.31754	+	1.31754i	0.401815	+	0.915721i	$0.368380\pi$
$48$	−835.056	+	835.056i	−0.362438	+	0.362438i
$49$		−	1838.18i		−	0.765590i
$50$		0			0
$51$	1120.27			0.430708
$52$	−1855.08	−	1855.08i	−0.686049	−	0.686049i
$53$	1289.93	−	1289.93i	0.459212	−	0.459212i	−0.439185	−	0.898397i	$-0.644733\pi$
$53$	1289.93	−	1289.93i	0.459212	−	0.459212i	0.898397	+	0.439185i	$0.144733\pi$
$54$		−	109.226i		−	0.0374575i
$55$		0			0
$56$	−579.842			−0.184899
$57$	−1536.72	−	1536.72i	−0.472982	−	0.472982i
$58$	−436.420	+	436.420i	−0.129733	+	0.129733i
$59$			1953.99i			0.561331i	0.959806	+	0.280665i	$0.0905551\pi$
$59$			1953.99i			0.561331i	−0.959806	+	0.280665i	$0.909445\pi$
$60$		0			0
$61$	1227.13			0.329784			0.164892	−	0.986312i	$-0.447272\pi$
$61$	1227.13			0.329784			0.164892	+	0.986312i	$0.447272\pi$
$62$	114.740	+	114.740i	0.0298492	+	0.0298492i
$63$	−452.932	+	452.932i	−0.114117	+	0.114117i
$64$			3194.16i			0.779825i
$65$		0			0
$66$	−797.189			−0.183009
$67$	−1165.88	−	1165.88i	−0.259718	−	0.259718i	0.565221	−	0.824939i	$-0.308791\pi$
$67$	−1165.88	−	1165.88i	−0.259718	−	0.259718i	−0.824939	+	0.565221i	$0.808791\pi$
$68$	2346.79	−	2346.79i	0.507524	−	0.507524i
$69$		−	4565.36i		−	0.958908i
$70$		0			0
$71$	−3109.97			−0.616936			−0.308468	−	0.951235i	$-0.599816\pi$
$71$	−3109.97			−0.616936			−0.308468	+	0.951235i	$0.599816\pi$
$72$	−466.632	−	466.632i	−0.0900138	−	0.0900138i
$73$	3457.41	−	3457.41i	0.648791	−	0.648791i	−0.303910	−	0.952701i	$-0.598292\pi$
$73$	3457.41	−	3457.41i	0.648791	−	0.648791i	0.952701	+	0.303910i	$0.0982922\pi$
$74$			506.614i			0.0925154i
$75$		0			0
$76$	−6438.36			−1.11468
$77$	3305.74	+	3305.74i	0.557554	+	0.557554i
$78$	487.502	−	487.502i	0.0801285	−	0.0801285i
$79$		−	9879.38i		−	1.58298i	−0.611182	−	0.791490i	$-0.709306\pi$
$79$		−	9879.38i		−	1.58298i	0.611182	−	0.791490i	$-0.290694\pi$
$80$		0			0
$81$	−729.000			−0.111111
$82$	1341.13	+	1341.13i	0.199454	+	0.199454i
$83$	3296.81	−	3296.81i	0.478562	−	0.478562i	−0.426110	−	0.904671i	$-0.640116\pi$
$83$	3296.81	−	3296.81i	0.478562	−	0.478562i	0.904671	+	0.426110i	$0.140116\pi$
$84$			1897.64i			0.268940i
$85$		0			0
$86$	−2328.35			−0.314813
$87$	2912.77	+	2912.77i	0.384829	+	0.384829i
$88$	−3405.72	+	3405.72i	−0.439789	+	0.439789i
$89$		−	6138.84i		−	0.775008i	−0.921868	−	0.387504i	$-0.873337\pi$
$89$		−	6138.84i		−	0.775008i	0.921868	−	0.387504i	$-0.126663\pi$
$90$		0			0
$91$	−4043.08			−0.488236
$92$	−9563.71	−	9563.71i	−1.12993	−	1.12993i
$93$	765.804	−	765.804i	0.0885425	−	0.0885425i
$94$			3204.45i			0.362658i
$95$		0			0
$96$	−2951.43			−0.320251
$97$	−728.123	−	728.123i	−0.0773858	−	0.0773858i	0.667354	−	0.744740i	$-0.267427\pi$
$97$	−728.123	−	728.123i	−0.0773858	−	0.0773858i	−0.744740	+	0.667354i	$0.767427\pi$
$98$	1011.94	−	1011.94i	0.105366	−	0.105366i
$99$			5320.63i			0.542866i
$100$		0			0
$101$	3284.75			0.322002			0.161001	−	0.986954i	$-0.448528\pi$
$101$	3284.75			0.322002			0.161001	+	0.986954i	$0.448528\pi$
$102$	616.720	+	616.720i	0.0592772	+	0.0592772i
$103$	6786.93	−	6786.93i	0.639733	−	0.639733i	−0.310756	−	0.950490i	$-0.600582\pi$
$103$	6786.93	−	6786.93i	0.639733	−	0.639733i	0.950490	+	0.310756i	$0.100582\pi$
$104$		−	4165.37i		−	0.385112i
$105$		0			0
$106$	1420.24			0.126401
$107$	−1447.22	−	1447.22i	−0.126406	−	0.126406i	0.641073	−	0.767480i	$-0.278489\pi$
$107$	−1447.22	−	1447.22i	−0.126406	−	0.126406i	−0.767480	+	0.641073i	$0.778489\pi$
$108$	−1527.14	+	1527.14i	−0.130928	+	0.130928i
$109$			13927.0i			1.17221i	0.810234	+	0.586106i	$0.199340\pi$
$109$			13927.0i			1.17221i	−0.810234	+	0.586106i	$0.800660\pi$
$110$		0			0
$111$	3381.26			0.274431
$112$	3812.57	+	3812.57i	0.303936	+	0.303936i
$113$	−17279.9	+	17279.9i	−1.35327	+	1.35327i	−0.471287	+	0.881980i	$0.656210\pi$
$113$	−17279.9	+	17279.9i	−1.35327	+	1.35327i	−0.881980	+	0.471287i	$0.843790\pi$
$114$		−	1691.96i		−	0.130191i
$115$		0			0
$116$	12203.6			0.906926
$117$	−3253.70	−	3253.70i	−0.237687	−	0.237687i
$118$	−1075.69	+	1075.69i	−0.0772546	+	0.0772546i
$119$		−	5114.76i		−	0.361186i
$120$		0			0
$121$	24191.7			1.65233
$122$	675.546	+	675.546i	0.0453874	+	0.0453874i
$123$	8950.99	−	8950.99i	0.591644	−	0.591644i
$124$		−	3208.48i		−	0.208668i
$125$		0			0
$126$	−498.687			−0.0314114
$127$	9885.00	+	9885.00i	0.612871	+	0.612871i	0.943693	−	0.330822i	$-0.107326\pi$
$127$	9885.00	+	9885.00i	0.612871	+	0.612871i	−0.330822	+	0.943693i	$0.607326\pi$
$128$	−8184.64	+	8184.64i	−0.499551	+	0.499551i
$129$			15540.0i			0.933837i
$130$		0			0
$131$	−18900.4			−1.10136			−0.550680	−	0.834716i	$-0.685632\pi$
$131$	−18900.4			−1.10136			−0.550680	+	0.834716i	$0.685632\pi$
$132$	11145.9	+	11145.9i	0.639685	+	0.639685i
$133$	−7016.11	+	7016.11i	−0.396637	+	0.396637i
$134$		−	1283.65i		−	0.0714888i
$135$		0			0
$136$	5269.46			0.284897
$137$	1922.12	+	1922.12i	0.102409	+	0.102409i	0.756455	−	0.654046i	$-0.226930\pi$
$137$	1922.12	+	1922.12i	0.102409	+	0.102409i	−0.654046	+	0.756455i	$0.726930\pi$
$138$	2513.28	−	2513.28i	0.131972	−	0.131972i
$139$			12035.8i			0.622939i	0.950256	+	0.311470i	$0.100821\pi$
$139$			12035.8i			0.622939i	−0.950256	+	0.311470i	$0.899179\pi$
$140$		0			0
$141$	21387.3			1.07576
$142$	−1712.07	−	1712.07i	−0.0849074	−	0.0849074i
$143$	−23747.2	+	23747.2i	−1.16129	+	1.16129i
$144$			6136.38i			0.295929i
$145$		0			0
$146$	3806.68			0.178583
$147$	−6753.91	−	6753.91i	−0.312551	−	0.312551i
$148$	7083.21	−	7083.21i	0.323375	−	0.323375i
$149$			2831.35i			0.127533i	0.997965	+	0.0637663i	$0.0203112\pi$
$149$			2831.35i			0.127533i	−0.997965	+	0.0637663i	$0.979689\pi$
$150$		0			0
$151$	−11509.8			−0.504794			−0.252397	−	0.967624i	$-0.581219\pi$
$151$	−11509.8			−0.504794			−0.252397	+	0.967624i	$0.581219\pi$
$152$	−7228.33	−	7228.33i	−0.312860	−	0.312860i
$153$	4116.14	−	4116.14i	0.175836	−	0.175836i
$154$			3639.68i			0.153469i
$155$		0			0
$156$	−13632.0			−0.560157
$157$	−16855.1	−	16855.1i	−0.683806	−	0.683806i	0.277050	−	0.960856i	$-0.410643\pi$
$157$	−16855.1	−	16855.1i	−0.683806	−	0.683806i	−0.960856	+	0.277050i	$0.910643\pi$
$158$	5438.70	−	5438.70i	0.217862	−	0.217862i
$159$		−	9478.99i		−	0.374945i
$160$		0			0
$161$	−20843.8			−0.804129
$162$	−401.322	−	401.322i	−0.0152920	−	0.0152920i
$163$	21091.4	−	21091.4i	0.793836	−	0.793836i	−0.188279	−	0.982116i	$-0.560291\pi$
$163$	21091.4	−	21091.4i	0.793836	−	0.793836i	0.982116	+	0.188279i	$0.0602910\pi$
$164$		−	37501.8i		−	1.39433i
$165$		0			0
$166$	3629.86			0.131727
$167$	−14608.4	−	14608.4i	−0.523804	−	0.523804i	0.394914	−	0.918718i	$-0.370774\pi$
$167$	−14608.4	−	14608.4i	−0.523804	−	0.523804i	−0.918718	+	0.394914i	$0.870774\pi$
$168$	−2130.47	+	2130.47i	−0.0754845	+	0.0754845i
$169$		−	483.065i		−	0.0169134i
$170$		0			0
$171$	−11292.5			−0.386188
$172$	32553.8	+	32553.8i	1.10038	+	1.10038i
$173$	5686.94	−	5686.94i	0.190014	−	0.190014i	−0.605688	−	0.795702i	$-0.707102\pi$
$173$	5686.94	−	5686.94i	0.190014	−	0.190014i	0.795702	+	0.605688i	$0.207102\pi$
$174$			3207.02i			0.105926i
$175$		0			0
$176$	44786.6			1.44585
$177$	7179.42	+	7179.42i	0.229162	+	0.229162i
$178$	3379.49	−	3379.49i	0.106662	−	0.106662i
$179$		−	18806.9i		−	0.586965i	−0.955964	−	0.293482i	$-0.905186\pi$
$179$		−	18806.9i		−	0.586965i	0.955964	−	0.293482i	$-0.0948142\pi$
$180$		0			0
$181$	−35653.8			−1.08830			−0.544150	−	0.838988i	$-0.683148\pi$
$181$	−35653.8			−1.08830			−0.544150	+	0.838988i	$0.683148\pi$
$182$	−2225.76	−	2225.76i	−0.0671948	−	0.0671948i
$183$	4508.75	−	4508.75i	0.134634	−	0.134634i
$184$		−	21474.3i		−	0.634283i
$185$		0			0
$186$	843.166			0.0243718
$187$	−30041.7	−	30041.7i	−0.859096	−	0.859096i
$188$	44802.9	−	44802.9i	1.26762	−	1.26762i
$189$			3328.36i			0.0931765i
$190$		0			0
$191$	−58397.3			−1.60076			−0.800380	−	0.599493i	$-0.795369\pi$
$191$	−58397.3			−1.60076			−0.800380	+	0.599493i	$0.795369\pi$
$192$	11736.1	+	11736.1i	0.318362	+	0.318362i
$193$	−12679.5	+	12679.5i	−0.340398	+	0.340398i	−0.856517	−	0.516119i	$-0.827376\pi$
$193$	−12679.5	+	12679.5i	−0.340398	+	0.340398i	0.516119	+	0.856517i	$0.327376\pi$
$194$		−	801.679i		−	0.0213008i
$195$		0			0
$196$	−28296.7			−0.736587
$197$	20420.8	+	20420.8i	0.526187	+	0.526187i	0.919433	−	0.393246i	$-0.128648\pi$
$197$	20420.8	+	20420.8i	0.526187	+	0.526187i	−0.393246	+	0.919433i	$0.628648\pi$
$198$	−2929.06	+	2929.06i	−0.0747133	+	0.0747133i
$199$			1036.13i			0.0261643i	0.999914	+	0.0130821i	$0.00416429\pi$
$199$			1036.13i			0.0261643i	−0.999914	+	0.0130821i	$0.995836\pi$
$200$		0			0
$201$	−8567.40			−0.212059
$202$	1808.29	+	1808.29i	0.0443164	+	0.0443164i
$203$	13298.7	−	13298.7i	0.322713	−	0.322713i
$204$		−	17245.3i		−	0.414391i
$205$		0			0
$206$	7472.55			0.176090
$207$	−16774.2	−	16774.2i	−0.391472	−	0.391472i
$208$	−27388.1	+	27388.1i	−0.633047	+	0.633047i
$209$			82418.8i			1.88683i
$210$		0			0
$211$	22118.0			0.496800			0.248400	−	0.968658i	$-0.420095\pi$
$211$	22118.0			0.496800			0.248400	+	0.968658i	$0.420095\pi$
$212$	−19857.0	−	19857.0i	−0.441816	−	0.441816i
$213$	−11426.8	+	11426.8i	−0.251863	+	0.251863i
$214$		−	1593.42i		−	0.0347940i
$215$		0			0
$216$	−3429.03			−0.0734960
$217$	−3496.39	−	3496.39i	−0.0742507	−	0.0742507i
$218$	−7666.98	+	7666.98i	−0.161329	+	0.161329i
$219$		−	25406.7i		−	0.529736i
$220$		0			0
$221$	36742.6			0.752290
$222$	1861.42	+	1861.42i	0.0377693	+	0.0377693i
$223$	30120.3	−	30120.3i	0.605688	−	0.605688i	−0.336128	−	0.941816i	$-0.609118\pi$
$223$	30120.3	−	30120.3i	0.605688	−	0.605688i	0.941816	+	0.336128i	$0.109118\pi$
$224$			13475.2i			0.268558i
$225$		0			0
$226$	−19025.5			−0.372494
$227$	48275.2	+	48275.2i	0.936855	+	0.936855i	0.998121	−	0.0612669i	$-0.0195141\pi$
$227$	48275.2	+	48275.2i	0.936855	+	0.936855i	−0.0612669	+	0.998121i	$0.519514\pi$
$228$	−23656.1	+	23656.1i	−0.455064	+	0.455064i
$229$		−	3292.48i		−	0.0627844i	−0.999507	−	0.0313922i	$-0.990006\pi$
$229$		−	3292.48i		−	0.0627844i	0.999507	−	0.0313922i	$-0.00999409\pi$
$230$		0			0
$231$	24292.1			0.455241
$232$	13700.9	+	13700.9i	0.254551	+	0.254551i
$233$	−14693.1	+	14693.1i	−0.270645	+	0.270645i	−0.829360	−	0.558715i	$-0.811295\pi$
$233$	−14693.1	+	14693.1i	−0.270645	+	0.270645i	0.558715	+	0.829360i	$0.311295\pi$
$234$		−	3582.39i		−	0.0654246i
$235$		0			0
$236$	30079.5			0.540066
$237$	−36299.2	−	36299.2i	−0.646249	−	0.646249i
$238$	2815.73	−	2815.73i	0.0497092	−	0.0497092i
$239$			33152.5i			0.580390i	0.956968	+	0.290195i	$0.0937202\pi$
$239$			33152.5i			0.580390i	−0.956968	+	0.290195i	$0.906280\pi$
$240$		0			0
$241$	110527.			1.90299			0.951493	−	0.307671i	$-0.0995497\pi$
$241$	110527.			1.90299			0.951493	+	0.307671i	$0.0995497\pi$
$242$	13317.8	+	13317.8i	0.227406	+	0.227406i
$243$	−2678.52	+	2678.52i	−0.0453609	+	0.0453609i
$244$		−	18890.2i		−	0.317291i
$245$		0			0
$246$	9855.22			0.162853
$247$	−50401.2	−	50401.2i	−0.826128	−	0.826128i
$248$	3602.15	−	3602.15i	0.0585677	−	0.0585677i
$249$		−	24226.5i		−	0.390744i
$250$		0			0
$251$	2118.00			0.0336185			0.0168093	−	0.999859i	$-0.494649\pi$
$251$	2118.00			0.0336185			0.0168093	+	0.999859i	$0.494649\pi$
$252$	6972.38	+	6972.38i	0.109794	+	0.109794i
$253$	−122427.	+	122427.i	−1.91265	+	1.91265i
$254$			10883.6i			0.168696i
$255$		0			0
$256$	42095.2			0.642321
$257$	77692.9	+	77692.9i	1.17629	+	1.17629i	0.980682	+	0.195610i	$0.0626685\pi$
$257$	77692.9	+	77692.9i	1.17629	+	1.17629i	0.195610	+	0.980682i	$0.437331\pi$
$258$	−8554.92	+	8554.92i	−0.128522	+	0.128522i
$259$		−	15437.7i		−	0.230134i
$260$		0			0
$261$	21404.4			0.314212
$262$	−10404.9	−	10404.9i	−0.151578	−	0.151578i
$263$	55670.1	−	55670.1i	0.804842	−	0.804842i	−0.179006	−	0.983848i	$-0.557288\pi$
$263$	55670.1	−	55670.1i	0.804842	−	0.804842i	0.983848	+	0.179006i	$0.0572880\pi$
$264$			25026.8i			0.359086i
$265$		0			0
$266$	−7724.88			−0.109176
$267$	−22555.5	−	22555.5i	−0.316396	−	0.316396i
$268$	−17947.3	+	17947.3i	−0.249880	+	0.249880i
$269$			105854.i			1.46286i	0.681917	+	0.731430i	$0.261147\pi$
$269$			105854.i			1.46286i	−0.681917	+	0.731430i	$0.738853\pi$
$270$		0			0
$271$	−98922.9			−1.34697			−0.673486	−	0.739200i	$-0.735204\pi$
$271$	−98922.9			−1.34697			−0.673486	+	0.739200i	$0.735204\pi$
$272$	−34647.7	−	34647.7i	−0.468314	−	0.468314i
$273$	−14855.2	+	14855.2i	−0.199322	+	0.199322i
$274$			2116.30i			0.0281887i
$275$		0			0
$276$	−70278.6			−0.922582
$277$	63861.7	+	63861.7i	0.832302	+	0.832302i	0.987831	−	0.155529i	$-0.0497083\pi$
$277$	63861.7	+	63861.7i	0.832302	+	0.832302i	−0.155529	+	0.987831i	$0.549708\pi$
$278$	−6625.84	+	6625.84i	−0.0857336	+	0.0857336i
$279$		−	5627.49i		−	0.0722947i
$280$		0			0
$281$	32035.4			0.405712			0.202856	−	0.979209i	$-0.434978\pi$
$281$	32035.4			0.405712			0.202856	+	0.979209i	$0.434978\pi$
$282$	11773.9	+	11773.9i	0.148055	+	0.148055i
$283$	63944.4	−	63944.4i	0.798417	−	0.798417i	−0.184429	−	0.982846i	$-0.559044\pi$
$283$	63944.4	−	63944.4i	0.798417	−	0.798417i	0.982846	+	0.184429i	$0.0590436\pi$
$284$			47874.6i			0.593565i
$285$		0			0
$286$	−26146.2			−0.319651
$287$	−40867.0	−	40867.0i	−0.496146	−	0.496146i
$288$	−10844.2	+	10844.2i	−0.130742	+	0.130742i
$289$		−	37039.3i		−	0.443473i
$290$		0			0
$291$	−5350.59			−0.0631853
$292$	−53222.9	−	53222.9i	−0.624213	−	0.624213i
$293$	−13864.2	+	13864.2i	−0.161495	+	0.161495i	−0.783229	−	0.621734i	$-0.786429\pi$
$293$	−13864.2	+	13864.2i	−0.161495	+	0.161495i	0.621734	+	0.783229i	$0.286429\pi$
$294$		−	7436.19i		−	0.0860312i
$295$		0			0
$296$	15904.6			0.181526
$297$	19549.2	+	19549.2i	0.221624	+	0.221624i
$298$	−1558.69	+	1558.69i	−0.0175520	+	0.0175520i
$299$		−	149734.i		−	1.67486i
$300$		0			0
$301$	70950.0			0.783104
$302$	−6336.26	−	6336.26i	−0.0694735	−	0.0694735i
$303$	12068.9	−	12068.9i	0.131457	−	0.131457i
$304$			95055.3i			1.02856i
$305$		0			0
$306$	4531.95			0.0483997
$307$	94399.9	+	94399.9i	1.00160	+	1.00160i	0.999999	+	0.00160232i	$0.000510033\pi$
$307$	94399.9	+	94399.9i	1.00160	+	1.00160i	0.00160232	+	0.999999i	$0.499490\pi$
$308$	50888.1	−	50888.1i	0.536432	−	0.536432i
$309$		−	49873.5i		−	0.522340i
$310$		0			0
$311$	−88072.6			−0.910584			−0.455292	−	0.890342i	$-0.650465\pi$
$311$	−88072.6			−0.910584			−0.455292	+	0.890342i	$0.650465\pi$
$312$	−15304.6	−	15304.6i	−0.157221	−	0.157221i
$313$	88635.0	−	88635.0i	0.904725	−	0.904725i	−0.0911152	−	0.995840i	$-0.529043\pi$
$313$	88635.0	−	88635.0i	0.904725	−	0.904725i	0.995840	+	0.0911152i	$0.0290431\pi$
$314$		−	18557.8i		−	0.188221i
$315$		0			0
$316$	−152082.			−1.52301
$317$	−40191.1	−	40191.1i	−0.399955	−	0.399955i	0.478262	−	0.878217i	$-0.341267\pi$
$317$	−40191.1	−	40191.1i	−0.399955	−	0.399955i	−0.878217	+	0.478262i	$0.841267\pi$
$318$	5218.28	−	5218.28i	0.0516028	−	0.0516028i
$319$		−	156221.i		−	1.53517i
$320$		0			0
$321$	−10634.9			−0.103210
$322$	−11474.7	−	11474.7i	−0.110670	−	0.110670i
$323$	63760.8	−	63760.8i	0.611151	−	0.611151i
$324$			11222.1i			0.106902i
$325$		0			0
$326$	23222.1			0.218508
$327$	51171.2	+	51171.2i	0.478553	+	0.478553i
$328$	42103.1	−	42103.1i	0.391351	−	0.391351i
$329$		−	97646.6i		−	0.902122i
$330$		0			0
$331$	3414.71			0.0311672			0.0155836	−	0.999879i	$-0.495039\pi$
$331$	3414.71			0.0311672			0.0155836	+	0.999879i	$0.495039\pi$
$332$	−50750.7	−	50750.7i	−0.460432	−	0.460432i
$333$	12423.6	−	12423.6i	0.112036	−	0.112036i
$334$		−	16084.1i		−	0.144180i
$335$		0			0
$336$	28016.6			0.248163
$337$	−123560.	−	123560.i	−1.08797	−	1.08797i	−0.995738	−	0.0922321i	$-0.970600\pi$
$337$	−123560.	−	123560.i	−1.08797	−	1.08797i	−0.0922321	−	0.995738i	$-0.529400\pi$
$338$	265.932	−	265.932i	0.00232776	−	0.00232776i
$339$			126981.i			1.10494i
$340$		0			0
$341$	−41072.4			−0.353217
$342$	−6216.65	−	6216.65i	−0.0531501	−	0.0531501i
$343$	−71113.4	+	71113.4i	−0.604454	+	0.604454i
$344$			73096.0i			0.617699i
$345$		0			0
$346$	6261.44			0.0523025
$347$	28249.8	+	28249.8i	0.234615	+	0.234615i	0.814616	−	0.580001i	$-0.196948\pi$
$347$	28249.8	+	28249.8i	0.234615	+	0.234615i	−0.580001	+	0.814616i	$0.696948\pi$
$348$	44838.9	−	44838.9i	0.370251	−	0.370251i
$349$			105370.i			0.865102i	0.901610	+	0.432551i	$0.142386\pi$
$349$			105370.i			0.865102i	−0.901610	+	0.432551i	$0.857614\pi$
$350$		0			0
$351$	−23909.7			−0.194071
$352$	79147.0	+	79147.0i	0.638777	+	0.638777i
$353$	−121429.	+	121429.i	−0.974482	+	0.974482i	−0.999682	−	0.0252004i	$-0.991978\pi$
$353$	−121429.	+	121429.i	−0.974482	+	0.974482i	0.0252004	+	0.999682i	$0.491978\pi$
$354$			7904.69i			0.0630781i
$355$		0			0
$356$	−94500.5			−0.745649
$357$	−18792.8	−	18792.8i	−0.147454	−	0.147454i
$358$	10353.4	−	10353.4i	0.0807825	−	0.0807825i
$359$		−	200543.i		−	1.55603i	−0.628245	−	0.778016i	$-0.716226\pi$
$359$		−	200543.i		−	1.55603i	0.628245	−	0.778016i	$-0.283774\pi$
$360$		0			0
$361$	−44605.2			−0.342272
$362$	−19627.8	−	19627.8i	−0.149780	−	0.149780i
$363$	88886.1	−	88886.1i	0.674560	−	0.674560i
$364$			62238.7i			0.469741i
$365$		0			0
$366$	4964.23			0.0370586
$367$	−23362.1	−	23362.1i	−0.173452	−	0.173452i	0.615042	−	0.788494i	$-0.289139\pi$
$367$	−23362.1	−	23362.1i	−0.173452	−	0.173452i	−0.788494	+	0.615042i	$0.789139\pi$
$368$	−141197.	+	141197.i	−1.04263	+	1.04263i
$369$		−	65776.0i		−	0.483076i
$370$		0			0
$371$	−43277.7			−0.314425
$372$	−11788.7	−	11788.7i	−0.0851883	−	0.0851883i
$373$	83015.4	−	83015.4i	0.596680	−	0.596680i	−0.342748	−	0.939427i	$-0.611358\pi$
$373$	83015.4	−	83015.4i	0.596680	−	0.596680i	0.939427	+	0.342748i	$0.111358\pi$
$374$		−	33076.6i		−	0.236471i
$375$		0			0
$376$	100600.			0.711578
$377$	95533.0	+	95533.0i	0.672157	+	0.672157i
$378$	−1832.29	+	1832.29i	−0.0128236	+	0.0128236i
$379$			158522.i			1.10360i	0.833977	+	0.551800i	$0.186059\pi$
$379$			158522.i			1.10360i	−0.833977	+	0.551800i	$0.813941\pi$
$380$		0			0
$381$	72639.6			0.500407
$382$	−32148.3	−	32148.3i	−0.220309	−	0.220309i
$383$	−166677.	+	166677.i	−1.13626	+	1.13626i	−0.147148	+	0.989114i	$0.547009\pi$
$383$	−166677.	+	166677.i	−1.13626	+	1.13626i	−0.989114	+	0.147148i	$0.952991\pi$
$384$			60144.6i			0.407882i
$385$		0			0
$386$	−13960.4			−0.0936963
$387$	57097.5	+	57097.5i	0.381237	+	0.381237i
$388$	−11208.6	+	11208.6i	−0.0744542	+	0.0744542i
$389$		−	8918.27i		−	0.0589361i	−0.999566	−	0.0294680i	$-0.990619\pi$
$389$		−	8918.27i		−	0.0589361i	0.999566	−	0.0294680i	$-0.00938133\pi$
$390$		0			0
$391$	189424.			1.23903
$392$	−31768.7	−	31768.7i	−0.206741	−	0.206741i
$393$	−69444.7	+	69444.7i	−0.449629	+	0.449629i
$394$			22483.7i			0.144836i
$395$		0			0
$396$	81905.1			0.522300
$397$	219532.	+	219532.i	1.39289	+	1.39289i	0.818775	+	0.574115i	$0.194654\pi$
$397$	219532.	+	219532.i	1.39289	+	1.39289i	0.574115	+	0.818775i	$0.305346\pi$
$398$	−570.401	+	570.401i	−0.00360092	+	0.00360092i
$399$			51557.7i			0.323853i
$400$		0			0
$401$	278202.			1.73010			0.865050	−	0.501686i	$-0.167287\pi$
$401$	278202.			1.73010			0.865050	+	0.501686i	$0.167287\pi$
$402$	−4716.44	−	4716.44i	−0.0291852	−	0.0291852i
$403$	25116.8	−	25116.8i	0.154652	−	0.154652i
$404$		−	50565.0i		−	0.309804i
$405$		0			0
$406$	14642.1			0.0888284
$407$	−90673.6	−	90673.6i	−0.547384	−	0.547384i
$408$	19361.2	−	19361.2i	0.116309	−	0.116309i
$409$			31311.9i			0.187182i	0.995611	+	0.0935908i	$0.0298346\pi$
$409$			31311.9i			0.187182i	−0.995611	+	0.0935908i	$0.970165\pi$
$410$		0			0
$411$	14124.7			0.0836169
$412$	−104477.	−	104477.i	−0.615498	−	0.615498i
$413$	32778.7	−	32778.7i	0.192173	−	0.192173i
$414$		−	18468.7i		−	0.107755i
$415$		0			0
$416$	−96800.9			−0.559362
$417$	44222.4	+	44222.4i	0.254314	+	0.254314i
$418$	−45372.4	+	45372.4i	−0.259680	+	0.259680i
$419$			83155.1i			0.473653i	0.971552	+	0.236827i	$0.0761074\pi$
$419$			83155.1i			0.473653i	−0.971552	+	0.236827i	$0.923893\pi$
$420$		0			0
$421$	4396.02			0.0248025			0.0124012	−	0.999923i	$-0.496052\pi$
$421$	4396.02			0.0248025			0.0124012	+	0.999923i	$0.496052\pi$
$422$	12176.2	+	12176.2i	0.0683734	+	0.0683734i
$423$	78581.8	−	78581.8i	0.439179	−	0.439179i
$424$		−	44586.7i		−	0.248013i
$425$		0			0
$426$	−12581.1			−0.0693266
$427$	−20585.4	−	20585.4i	−0.112902	−	0.112902i
$428$	−22278.4	+	22278.4i	−0.121618	+	0.121618i
$429$			174506.i			0.948189i
$430$		0			0
$431$	97154.1			0.523006			0.261503	−	0.965203i	$-0.415782\pi$
$431$	97154.1			0.523006			0.261503	+	0.965203i	$0.415782\pi$
$432$	22546.5	+	22546.5i	0.120813	+	0.120813i
$433$	−152799.	+	152799.i	−0.814978	+	0.814978i	−0.985375	−	0.170397i	$-0.945495\pi$
$433$	−152799.	+	152799.i	−0.814978	+	0.814978i	0.170397	+	0.985375i	$0.445495\pi$
$434$		−	3849.60i		−	0.0204379i
$435$		0			0
$436$	214391.			1.12781
$437$	−259840.	−	259840.i	−1.36064	−	1.36064i
$438$	13986.6	−	13986.6i	0.0729062	−	0.0729062i
$439$			30829.0i			0.159967i	0.996796	+	0.0799836i	$0.0254868\pi$
$439$			30829.0i			0.159967i	−0.996796	+	0.0799836i	$0.974513\pi$
$440$		0			0
$441$	−49630.9			−0.255197
$442$	20227.2	+	20227.2i	0.103536	+	0.103536i
$443$	−126571.	+	126571.i	−0.644952	+	0.644952i	−0.951769	−	0.306817i	$-0.900736\pi$
$443$	−126571.	+	126571.i	−0.644952	+	0.644952i	0.306817	+	0.951769i	$0.400736\pi$
$444$		−	52050.7i		−	0.264035i
$445$		0			0
$446$	33163.0			0.166719
$447$	10403.0	+	10403.0i	0.0520650	+	0.0520650i
$448$	53582.9	−	53582.9i	0.266975	−	0.266975i
$449$			135018.i			0.669728i	0.942266	+	0.334864i	$0.108690\pi$
$449$			135018.i			0.669728i	−0.942266	+	0.334864i	$0.891310\pi$
$450$		0			0
$451$	−480068.			−2.36021
$452$	266004.	+	266004.i	1.30200	+	1.30200i
$453$	−42289.7	+	42289.7i	−0.206081	+	0.206081i
$454$			53152.0i			0.257874i
$455$		0			0
$456$	−53117.1			−0.255449
$457$	−6037.20	−	6037.20i	−0.0289070	−	0.0289070i	0.692506	−	0.721413i	$-0.256507\pi$
$457$	−6037.20	−	6037.20i	−0.0289070	−	0.0289070i	−0.721413	+	0.692506i	$0.756507\pi$
$458$	1812.54	−	1812.54i	0.00864087	−	0.00864087i
$459$		−	30247.3i		−	0.143569i
$460$		0			0
$461$	35960.5			0.169209			0.0846046	−	0.996415i	$-0.473037\pi$
$461$	35960.5			0.169209			0.0846046	+	0.996415i	$0.473037\pi$
$462$	13373.0	+	13373.0i	0.0626537	+	0.0626537i
$463$	155969.	−	155969.i	0.727571	−	0.727571i	−0.242565	−	0.970135i	$-0.577989\pi$
$463$	155969.	−	155969.i	0.727571	−	0.727571i	0.970135	+	0.242565i	$0.0779886\pi$
$464$		−	180172.i		−	0.836859i
$465$		0			0
$466$	−16177.4			−0.0744965
$467$	−162294.	−	162294.i	−0.744164	−	0.744164i	0.229212	−	0.973376i	$-0.426385\pi$
$467$	−162294.	−	162294.i	−0.744164	−	0.744164i	−0.973376	+	0.229212i	$0.926385\pi$
$468$	−50087.1	+	50087.1i	−0.228683	+	0.228683i
$469$			39115.7i			0.177830i
$470$		0			0
$471$	−123859.			−0.558325
$472$	33770.2	+	33770.2i	0.151583	+	0.151583i
$473$	416728.	−	416728.i	1.86265	−	1.86265i
$474$		−	39966.1i		−	0.177883i
$475$		0			0
$476$	−78736.0			−0.347504
$477$	−34828.0	−	34828.0i	−0.153071	−	0.153071i
$478$	−18250.8	+	18250.8i	−0.0798776	+	0.0798776i
$479$		−	148721.i		−	0.648190i	−0.946025	−	0.324095i	$-0.894940\pi$
$479$		−	148721.i		−	0.648190i	0.946025	−	0.324095i	$-0.105060\pi$
$480$		0			0
$481$	110899.			0.479331
$482$	60846.4	+	60846.4i	0.261903	+	0.261903i
$483$	−76585.1	+	76585.1i	−0.328284	+	0.328284i
$484$		−	372404.i		−	1.58973i
$485$		0			0
$486$	−2949.10			−0.0124858
$487$	194156.	+	194156.i	0.818639	+	0.818639i	0.985911	−	0.167272i	$-0.0534957\pi$
$487$	194156.	+	194156.i	0.818639	+	0.818639i	−0.167272	+	0.985911i	$0.553496\pi$
$488$	21208.0	−	21208.0i	0.0890554	−	0.0890554i
$489$		−	154990.i		−	0.648165i
$490$		0			0
$491$	−209053.			−0.867148			−0.433574	−	0.901118i	$-0.642748\pi$
$491$	−209053.			−0.867148			−0.433574	+	0.901118i	$0.642748\pi$
$492$	−137790.	−	137790.i	−0.569231	−	0.569231i
$493$	−120855.	+	120855.i	−0.497246	+	0.497246i
$494$		−	55492.8i		−	0.227396i
$495$		0			0
$496$	−47369.6			−0.192547
$497$	52170.6	+	52170.6i	0.211209	+	0.211209i
$498$	13336.9	−	13336.9i	0.0537771	−	0.0537771i
$499$		−	71012.3i		−	0.285189i	−0.989781	−	0.142594i	$-0.954456\pi$
$499$		−	71012.3i		−	0.285189i	0.989781	−	0.142594i	$-0.0455445\pi$
$500$		0			0
$501$	−107349.			−0.427684
$502$	1165.98	+	1165.98i	0.00462683	+	0.00462683i
$503$	31136.6	−	31136.6i	0.123065	−	0.123065i	−0.642892	−	0.765957i	$-0.722266\pi$
$503$	31136.6	−	31136.6i	0.123065	−	0.123065i	0.765957	+	0.642892i	$0.222266\pi$
$504$			15655.7i			0.0616328i
$505$		0			0
$506$	−134795.			−0.526467
$507$	−1774.89	−	1774.89i	−0.00690489	−	0.00690489i
$508$	152169.	−	152169.i	0.589654	−	0.589654i
$509$			223266.i			0.861763i	0.902409	+	0.430881i	$0.141797\pi$
$509$			223266.i			0.861763i	−0.902409	+	0.430881i	$0.858203\pi$
$510$		0			0
$511$	−115998.			−0.444230
$512$	154128.	+	154128.i	0.587952	+	0.587952i
$513$	−41491.4	+	41491.4i	−0.157661	+	0.157661i
$514$			85541.4i			0.323780i
$515$		0			0
$516$	239221.			0.898461
$517$	−573531.	−	573531.i	−2.14573	−	2.14573i
$518$	8498.59	−	8498.59i	0.0316728	−	0.0316728i
$519$		−	41790.3i		−	0.155146i
$520$		0			0
$521$	−88774.7			−0.327050			−0.163525	−	0.986539i	$-0.552286\pi$
$521$	−88774.7			−0.327050			−0.163525	+	0.986539i	$0.552286\pi$
$522$	11783.3	+	11783.3i	0.0432442	+	0.0432442i
$523$	64718.9	−	64718.9i	0.236607	−	0.236607i	−0.578837	−	0.815444i	$-0.696493\pi$
$523$	64718.9	−	64718.9i	0.236607	−	0.236607i	0.815444	+	0.578837i	$0.196493\pi$
$524$			290951.i			1.05964i
$525$		0			0
$526$	61294.0			0.221537
$527$	31774.4	+	31774.4i	0.114408	+	0.114408i
$528$	164556.	−	164556.i	0.590265	−	0.590265i
$529$		−	492104.i		−	1.75851i
$530$		0			0
$531$	52757.8			0.187110
$532$	108005.	+	108005.i	0.381611	+	0.381611i
$533$	293574.	−	293574.i	1.03339	−	1.03339i
$534$		−	24834.1i		−	0.0870895i
$535$		0			0
$536$	−40298.9			−0.140269
$537$	−69101.1	−	69101.1i	−0.239627	−	0.239627i
$538$	−58273.7	+	58273.7i	−0.201330	+	0.201330i
$539$			362232.i			1.24684i
$540$		0			0
$541$	−376988.			−1.28805			−0.644026	−	0.765004i	$-0.722737\pi$
$541$	−376988.			−1.28805			−0.644026	+	0.765004i	$0.722737\pi$
$542$	−54458.1	−	54458.1i	−0.185380	−	0.185380i
$543$	−131000.	+	131000.i	−0.444296	+	0.444296i
$544$		−	122459.i		−	0.413803i
$545$		0			0
$546$	−16355.9			−0.0548643
$547$	−103549.	−	103549.i	−0.346075	−	0.346075i	0.512570	−	0.858645i	$-0.328693\pi$
$547$	−103549.	−	103549.i	−0.346075	−	0.346075i	−0.858645	+	0.512570i	$0.828693\pi$
$548$	29588.9	−	29588.9i	0.0985299	−	0.0985299i
$549$		−	33132.4i		−	0.109928i
$550$		0			0
$551$	331564.			1.09210
$552$	−78901.5	−	78901.5i	−0.258945	−	0.258945i
$553$	−165729.	+	165729.i	−0.541937	+	0.541937i
$554$			70313.0i			0.229095i
$555$		0			0
$556$	185278.			0.599341
$557$	171898.	+	171898.i	0.554065	+	0.554065i	0.927611	−	0.373547i	$-0.121858\pi$
$557$	171898.	+	171898.i	0.554065	+	0.554065i	−0.373547	+	0.927611i	$0.621858\pi$
$558$	3097.99	−	3097.99i	0.00994974	−	0.00994974i
$559$			509680.i			1.63107i
$560$		0			0
$561$	−220761.			−0.701449
$562$	17635.8	+	17635.8i	0.0558372	+	0.0558372i
$563$	−129088.	+	129088.i	−0.407258	+	0.407258i	−0.880781	−	0.473523i	$-0.842982\pi$
$563$	−129088.	+	129088.i	−0.407258	+	0.407258i	0.473523	+	0.880781i	$0.342982\pi$
$564$		−	329233.i		−	1.03501i
$565$		0			0
$566$	70404.1			0.219768
$567$	12229.2	+	12229.2i	0.0380391	+	0.0380391i
$568$	−53748.6	+	53748.6i	−0.166598	+	0.166598i
$569$			259535.i			0.801624i	0.916160	+	0.400812i	$0.131272\pi$
$569$			259535.i			0.801624i	−0.916160	+	0.400812i	$0.868728\pi$
$570$		0			0
$571$	597642.			1.83303			0.916514	−	0.400004i	$-0.130991\pi$
$571$	597642.			1.83303			0.916514	+	0.400004i	$0.130991\pi$
$572$	365562.	+	365562.i	1.11730	+	1.11730i
$573$	−214566.	+	214566.i	−0.653508	+	0.653508i
$574$		−	44995.4i		−	0.136567i
$575$		0			0
$576$	86242.4			0.259942
$577$	335792.	+	335792.i	1.00860	+	1.00860i	0.999963	+	0.00863725i	$0.00274936\pi$
$577$	335792.	+	335792.i	1.00860	+	1.00860i	0.00863725	+	0.999963i	$0.497251\pi$
$578$	20390.5	−	20390.5i	0.0610341	−	0.0610341i
$579$			93174.8i			0.277934i
$580$		0			0
$581$	−110610.			−0.327673
$582$	−2945.56	−	2945.56i	−0.00869603	−	0.00869603i
$583$	−254193.	+	254193.i	−0.747872	+	0.747872i
$584$		−	119506.i		−	0.350401i
$585$		0			0
$586$	−15264.8			−0.0444523
$587$	−189572.	−	189572.i	−0.550171	−	0.550171i	0.376319	−	0.926490i	$-0.377190\pi$
$587$	−189572.	−	189572.i	−0.550171	−	0.550171i	−0.926490	+	0.376319i	$0.877190\pi$
$588$	−103969.	+	103969.i	−0.300711	+	0.300711i
$589$		−	87172.3i		−	0.251274i
$590$		0			0
$591$	150061.			0.429630
$592$	−104576.	−	104576.i	−0.298392	−	0.298392i
$593$	152696.	−	152696.i	0.434228	−	0.434228i	−0.455836	−	0.890064i	$-0.650660\pi$
$593$	152696.	−	152696.i	0.434228	−	0.434228i	0.890064	+	0.455836i	$0.150660\pi$
$594$			21524.1i			0.0610031i
$595$		0			0
$596$	43585.5			0.122701
$597$	3806.99	+	3806.99i	0.0106815	+	0.0106815i
$598$	82430.4	−	82430.4i	0.230507	−	0.230507i
$599$		−	232126.i		−	0.646949i	−0.946237	−	0.323474i	$-0.895149\pi$
$599$		−	232126.i		−	0.646949i	0.946237	−	0.323474i	$-0.104851\pi$
$600$		0			0
$601$	−277987.			−0.769618			−0.384809	−	0.922996i	$-0.625733\pi$
$601$	−277987.			−0.769618			−0.384809	+	0.922996i	$0.625733\pi$
$602$	39058.7	+	39058.7i	0.107777	+	0.107777i
$603$	−31478.6	+	31478.6i	−0.0865728	+	0.0865728i
$604$			177180.i			0.485671i
$605$		0			0
$606$	13288.1			0.0361842
$607$	83749.1	+	83749.1i	0.227302	+	0.227302i	0.811565	−	0.584263i	$-0.198616\pi$
$607$	83749.1	+	83749.1i	0.227302	+	0.227302i	−0.584263	+	0.811565i	$0.698616\pi$
$608$	−167982.	+	167982.i	−0.454418	+	0.454418i
$609$		−	97725.0i		−	0.263494i
$610$		0			0
$611$	701458.			1.87897
$612$	−63363.3	−	63363.3i	−0.169175	−	0.169175i
$613$	−167541.	+	167541.i	−0.445861	+	0.445861i	−0.893976	−	0.448115i	$-0.852096\pi$
$613$	−167541.	+	167541.i	−0.445861	+	0.445861i	0.448115	+	0.893976i	$0.352096\pi$
$614$			103936.i			0.275696i
$615$		0			0
$616$	114264.			0.301125
$617$	98951.6	+	98951.6i	0.259928	+	0.259928i	0.825025	−	0.565097i	$-0.191161\pi$
$617$	98951.6	+	98951.6i	0.259928	+	0.259928i	−0.565097	+	0.825025i	$0.691161\pi$
$618$	27455.9	−	27455.9i	0.0718884	−	0.0718884i
$619$			390730.i			1.01975i	0.860247	+	0.509877i	$0.170309\pi$
$619$			390730.i			1.01975i	−0.860247	+	0.509877i	$0.829691\pi$
$620$		0			0
$621$	−123265.			−0.319636
$622$	−48484.9	−	48484.9i	−0.125321	−	0.125321i
$623$	−102981.	+	102981.i	−0.265326	+	0.265326i
$624$			201261.i			0.516881i
$625$		0			0
$626$	97589.0			0.249030
$627$	302826.	+	302826.i	0.770297	+	0.770297i
$628$	−259466.	+	259466.i	−0.657901	+	0.657901i
$629$			140294.i			0.354598i
$630$		0			0
$631$	47450.1			0.119173			0.0595865	−	0.998223i	$-0.481022\pi$
$631$	47450.1			0.119173			0.0595865	+	0.998223i	$0.481022\pi$
$632$	−170742.	−	170742.i	−0.427470	−	0.427470i
$633$	81266.8	−	81266.8i	0.202818	−	0.202818i
$634$		−	44251.2i		−	0.110090i
$635$		0			0
$636$	−145918.			−0.360741
$637$	−221514.	−	221514.i	−0.545913	−	0.545913i
$638$	86001.1	−	86001.1i	0.211282	−	0.211282i
$639$			83969.3i			0.205645i
$640$		0			0
$641$	461671.			1.12361			0.561807	−	0.827269i	$-0.310106\pi$
$641$	461671.			1.12361			0.561807	+	0.827269i	$0.310106\pi$
$642$	−5854.61	−	5854.61i	−0.0142046	−	0.0142046i
$643$	−47437.8	+	47437.8i	−0.114737	+	0.114737i	−0.762144	−	0.647407i	$-0.775853\pi$
$643$	−47437.8	+	47437.8i	−0.114737	+	0.114737i	0.647407	+	0.762144i	$0.275853\pi$
$644$			320867.i			0.773666i
$645$		0			0
$646$	70201.9			0.168222
$647$	−480099.	−	480099.i	−1.14689	−	1.14689i	−0.987161	−	0.159731i	$-0.948937\pi$
$647$	−480099.	−	480099.i	−1.14689	−	1.14689i	−0.159731	−	0.987161i	$-0.551063\pi$
$648$	−12599.1	+	12599.1i	−0.0300046	+	0.0300046i
$649$		−	385054.i		−	0.914181i
$650$		0			0
$651$	−25693.1			−0.0606254
$652$	−324679.	−	324679.i	−0.763764	−	0.763764i
$653$	259012.	−	259012.i	0.607426	−	0.607426i	−0.334847	−	0.942273i	$-0.608685\pi$
$653$	259012.	−	259012.i	0.607426	−	0.607426i	0.942273	+	0.334847i	$0.108685\pi$
$654$			56340.6i			0.131724i
$655$		0			0
$656$	−553672.			−1.28660
$657$	−93350.0	−	93350.0i	−0.216264	−	0.216264i
$658$	53755.5	−	53755.5i	0.124157	−	0.124157i
$659$		−	347700.i		−	0.800633i	−0.916377	−	0.400316i	$-0.868900\pi$
$659$		−	347700.i		−	0.800633i	0.916377	−	0.400316i	$-0.131100\pi$
$660$		0			0
$661$	−573427.			−1.31243			−0.656214	−	0.754575i	$-0.727843\pi$
$661$	−573427.			−1.31243			−0.656214	+	0.754575i	$0.727843\pi$
$662$	1879.84	+	1879.84i	0.00428947	+	0.00428947i
$663$	135001.	−	135001.i	0.307121	−	0.307121i
$664$		−	113955.i		−	0.258463i
$665$		0			0
$666$	13678.6			0.0308385
$667$	492513.	+	492513.i	1.10705	+	1.10705i
$668$	−224879.	+	224879.i	−0.503961	+	0.503961i
$669$		−	221338.i		−	0.494542i
$670$		0			0
$671$	−241818.			−0.537085
$672$	49511.0	+	49511.0i	0.109639	+	0.109639i
$673$	430119.	−	430119.i	0.949639	−	0.949639i	−0.0491520	−	0.998791i	$-0.515652\pi$
$673$	430119.	−	430119.i	0.949639	−	0.949639i	0.998791	+	0.0491520i	$0.0156519\pi$
$674$		−	136042.i		−	0.299469i
$675$		0			0
$676$	−7436.24			−0.0162727
$677$	232285.	+	232285.i	0.506809	+	0.506809i	0.913545	−	0.406737i	$-0.133333\pi$
$677$	232285.	+	232285.i	0.506809	+	0.506809i	−0.406737	+	0.913545i	$0.633333\pi$
$678$	−69904.1	+	69904.1i	−0.152070	+	0.152070i
$679$			24428.9i			0.0529864i
$680$		0			0
$681$	354749.			0.764939
$682$	−22610.8	−	22610.8i	−0.0486123	−	0.0486123i
$683$	−306372.	+	306372.i	−0.656761	+	0.656761i	−0.954612	−	0.297851i	$-0.903730\pi$
$683$	−306372.	+	306372.i	−0.656761	+	0.656761i	0.297851	+	0.954612i	$0.403730\pi$
$684$			173836.i			0.371558i
$685$		0			0
$686$	−78297.3			−0.166379
$687$	−12097.3	−	12097.3i	−0.0256316	−	0.0256316i
$688$	480620.	−	480620.i	1.01537	−	1.01537i
$689$		−	310892.i		−	0.654893i
$690$		0			0
$691$	568078.			1.18974			0.594870	−	0.803822i	$-0.297204\pi$
$691$	568078.			1.18974			0.594870	+	0.803822i	$0.297204\pi$
$692$	−87544.1	−	87544.1i	−0.182816	−	0.182816i
$693$	89254.8	−	89254.8i	0.185851	−	0.185851i
$694$			31103.6i			0.0645791i
$695$		0			0
$696$	100681.			0.207840
$697$	371390.	+	371390.i	0.764477	+	0.764477i
$698$	−58007.4	+	58007.4i	−0.119062	+	0.119062i
$699$			107972.i			0.220981i
$700$		0			0
$701$	−773879.			−1.57484			−0.787421	−	0.616416i	$-0.788584\pi$
$701$	−773879.			−1.57484			−0.787421	+	0.616416i	$0.788584\pi$
$702$	−13162.5	−	13162.5i	−0.0267095	−	0.0267095i
$703$	192446.	−	192446.i	0.389403	−	0.389403i
$704$		−	629442.i		−	1.27002i
$705$		0			0
$706$	−133696.			−0.268231
$707$	−55102.4	−	55102.4i	−0.110238	−	0.110238i
$708$	110519.	−	110519.i	0.220481	−	0.220481i
$709$			296312.i			0.589463i	0.955580	+	0.294732i	$0.0952302\pi$
$709$			296312.i			0.589463i	−0.955580	+	0.294732i	$0.904770\pi$
$710$		0			0
$711$	−266743.			−0.527660
$712$	−106095.	−	106095.i	−0.209284	−	0.209284i
$713$	129488.	−	129488.i	0.254712	−	0.254712i
$714$		−	20691.3i		−	0.0405874i
$715$		0			0
$716$	−289512.			−0.564729
$717$	121810.	+	121810.i	0.236943	+	0.236943i
$718$	110401.	−	110401.i	0.214153	−	0.214153i
$719$		−	620357.i		−	1.20001i	−0.799997	−	0.600004i	$-0.795166\pi$
$719$		−	620357.i		−	1.20001i	0.799997	−	0.600004i	$-0.204834\pi$
$720$		0			0
$721$	−227705.			−0.438028
$722$	−24555.6	−	24555.6i	−0.0471061	−	0.0471061i
$723$	406103.	−	406103.i	0.776891	−	0.776891i
$724$			548850.i			1.04707i
$725$		0			0
$726$	97865.4			0.185676
$727$	27652.4	+	27652.4i	0.0523195	+	0.0523195i	0.732782	−	0.680463i	$-0.238221\pi$
$727$	27652.4	+	27652.4i	0.0523195	+	0.0523195i	−0.680463	+	0.732782i	$0.738221\pi$
$728$	−69875.2	+	69875.2i	−0.131844	+	0.131844i
$729$			19683.0i			0.0370370i
$730$		0			0
$731$	−644777.			−1.20663
$732$	−69407.2	−	69407.2i	−0.129533	−	0.129533i
$733$	269452.	−	269452.i	0.501503	−	0.501503i	−0.410402	−	0.911905i	$-0.634612\pi$
$733$	269452.	−	269452.i	0.501503	−	0.501503i	0.911905	+	0.410402i	$0.134612\pi$
$734$		−	25722.1i		−	0.0477435i
$735$		0			0
$736$	−499050.			−0.921273
$737$	229748.	+	229748.i	0.422976	+	0.422976i
$738$	36210.4	−	36210.4i	0.0664845	−	0.0664845i
$739$		−	672746.i		−	1.23186i	−0.787800	−	0.615931i	$-0.788780\pi$
$739$		−	672746.i		−	1.23186i	0.787800	−	0.615931i	$-0.211220\pi$
$740$		0			0
$741$	−370372.			−0.674531
$742$	−23824.8	−	23824.8i	−0.0432735	−	0.0432735i
$743$	−413904.	+	413904.i	−0.749759	+	0.749759i	−0.974434	−	0.224675i	$-0.927868\pi$
$743$	−413904.	+	413904.i	−0.749759	+	0.749759i	0.224675	+	0.974434i	$0.427868\pi$
$744$		−	26470.3i		−	0.0478203i
$745$		0			0
$746$	91401.7			0.164239
$747$	−89013.9	−	89013.9i	−0.159521	−	0.159521i
$748$	−462459.	+	462459.i	−0.826551	+	0.826551i
$749$			48555.1i			0.0865509i
$750$		0			0
$751$	320818.			0.568825			0.284413	−	0.958702i	$-0.408201\pi$
$751$	320818.			0.568825			0.284413	+	0.958702i	$0.408201\pi$
$752$	−661465.	−	661465.i	−1.16969	−	1.16969i
$753$	7782.03	−	7782.03i	0.0137247	−	0.0137247i
$754$			105184.i			0.185015i
$755$		0			0
$756$	51236.3			0.0896467
$757$	−435523.	−	435523.i	−0.760011	−	0.760011i	0.216313	−	0.976324i	$-0.430597\pi$
$757$	−435523.	−	435523.i	−0.760011	−	0.760011i	−0.976324	+	0.216313i	$0.930597\pi$
$758$	−87268.0	+	87268.0i	−0.151886	+	0.151886i
$759$			899651.i			1.56167i
$760$		0			0
$761$	479713.			0.828347			0.414174	−	0.910198i	$-0.364071\pi$
$761$	479713.			0.828347			0.414174	+	0.910198i	$0.364071\pi$
$762$	39988.9	+	39988.9i	0.0688699	+	0.0688699i
$763$	233630.	−	233630.i	0.401309	−	0.401309i
$764$			898961.i			1.54012i
$765$		0			0
$766$	−183515.			−0.312762
$767$	235470.	+	235470.i	0.400263	+	0.400263i
$768$	154667.	−	154667.i	0.262226	−	0.262226i
$769$		−	682193.i		−	1.15360i	−0.816886	−	0.576799i	$-0.804301\pi$
$769$		−	682193.i		−	1.15360i	0.816886	−	0.576799i	$-0.195699\pi$
$770$		0			0
$771$	570924.			0.960438
$772$	195186.	+	195186.i	0.327503	+	0.327503i
$773$	−264976.	+	264976.i	−0.443452	+	0.443452i	−0.893171	−	0.449718i	$-0.851524\pi$
$773$	−264976.	+	264976.i	−0.443452	+	0.443452i	0.449718	+	0.893171i	$0.351524\pi$
$774$			62865.5i			0.104938i
$775$		0			0
$776$	−25167.8			−0.0417948
$777$	−56721.6	−	56721.6i	−0.0939520	−	0.0939520i
$778$	4909.60	−	4909.60i	0.00811123	−	0.00811123i
$779$		−	1.01890e6i		−	1.67902i
$780$		0			0
$781$	612852.			1.00474
$782$	104280.	+	104280.i	0.170524	+	0.170524i
$783$	78644.8	−	78644.8i	0.128276	−	0.128276i
$784$			417770.i			0.679681i
$785$		0			0
$786$	−76460.0			−0.123763
$787$	−163669.	−	163669.i	−0.264252	−	0.264252i	0.562527	−	0.826779i	$-0.309829\pi$
$787$	−163669.	−	163669.i	−0.264252	−	0.264252i	−0.826779	+	0.562527i	$0.809829\pi$
$788$	314355.	−	314355.i	0.506253	−	0.506253i
$789$		−	409090.i		−	0.657151i
$790$		0			0
$791$	579748.			0.926588
$792$	91954.5	+	91954.5i	0.146596	+	0.146596i
$793$	147878.	−	147878.i	0.235156	−	0.235156i
$794$			241709.i			0.383400i
$795$		0			0
$796$	15950.1			0.0251731
$797$	−511390.	−	511390.i	−0.805074	−	0.805074i	0.178809	−	0.983884i	$-0.442775\pi$
$797$	−511390.	−	511390.i	−0.805074	−	0.805074i	−0.983884	+	0.178809i	$0.942775\pi$
$798$	−28383.0	+	28383.0i	−0.0445711	+	0.0445711i
$799$			887389.i			1.39002i
$800$		0			0
$801$	−165749.			−0.258336
$802$	153153.	+	153153.i	0.238109	+	0.238109i
$803$	−681317.	+	681317.i	−1.05662	+	1.05662i
$804$			131886.i			0.204026i
$805$		0			0
$806$	27654.1			0.0425687
$807$	388932.	+	388932.i	0.597210	+	0.597210i
$808$	56769.1	−	56769.1i	0.0869540	−	0.0869540i
$809$		−	973697.i		−	1.48774i	−0.668324	−	0.743870i	$-0.732988\pi$
$809$		−	973697.i		−	1.48774i	0.668324	−	0.743870i	$-0.267012\pi$
$810$		0			0
$811$	426534.			0.648504			0.324252	−	0.945971i	$-0.394887\pi$
$811$	426534.			0.648504			0.324252	+	0.945971i	$0.394887\pi$
$812$	−204718.	−	204718.i	−0.310488	−	0.310488i
$813$	−363466.	+	363466.i	−0.549899	+	0.549899i
$814$		−	99833.5i		−	0.150670i
$815$		0			0
$816$	−254608.			−0.382377
$817$	884466.	+	884466.i	1.32506	+	1.32506i
$818$	−17237.5	+	17237.5i	−0.0257614	+	0.0257614i
$819$			109163.i			0.162745i
$820$		0			0
$821$	−263379.			−0.390747			−0.195373	−	0.980729i	$-0.562592\pi$
$821$	−263379.			−0.390747			−0.195373	+	0.980729i	$0.562592\pi$
$822$	7775.77	+	7775.77i	0.0115080	+	0.0115080i
$823$	−704572.	+	704572.i	−1.04022	+	1.04022i	−0.0410646	+	0.999156i	$0.513075\pi$
$823$	−704572.	+	704572.i	−1.04022	+	1.04022i	−0.999156	+	0.0410646i	$0.986925\pi$
$824$		−	234592.i		−	0.345509i
$825$		0			0
$826$	36090.0			0.0528965
$827$	−103507.	−	103507.i	−0.151342	−	0.151342i	0.627375	−	0.778717i	$-0.284129\pi$
$827$	−103507.	−	103507.i	−0.151342	−	0.151342i	−0.778717	+	0.627375i	$0.784129\pi$
$828$	−258220.	+	258220.i	−0.376642	+	0.376642i
$829$			575466.i			0.837357i	0.908134	+	0.418679i	$0.137507\pi$
$829$			575466.i			0.837357i	−0.908134	+	0.418679i	$0.862493\pi$
$830$		0			0
$831$	469286.			0.679572
$832$	384920.	+	384920.i	0.556063	+	0.556063i
$833$	280230.	−	280230.i	0.403854	−	0.403854i
$834$			48689.8i			0.0700012i
$835$		0			0
$836$	1.26875e6			1.81536
$837$	−20676.7	−	20676.7i	−0.0295142	−	0.0295142i
$838$	−45777.7	+	45777.7i	−0.0651878	+	0.0651878i
$839$			1.07622e6i			1.52889i	0.644688	+	0.764446i	$0.276987\pi$
$839$			1.07622e6i			1.52889i	−0.644688	+	0.764446i	$0.723013\pi$
$840$		0			0
$841$	78818.8			0.111439
$842$	2420.05	+	2420.05i	0.00341351	+	0.00341351i
$843$	117706.	−	117706.i	0.165631	−	0.165631i
$844$		−	340482.i		−	0.477980i
$845$		0			0
$846$	86520.1			0.120886
$847$	−405822.	−	405822.i	−0.565678	−	0.565678i
$848$	−293166.	+	293166.i	−0.407683	+	0.407683i
$849$		−	469894.i		−	0.651905i
$850$		0			0
$851$	571729.			0.789462
$852$	175902.	+	175902.i	0.242322	+	0.242322i
$853$	−21249.1	+	21249.1i	−0.0292040	+	0.0292040i	−0.721558	−	0.692354i	$-0.756574\pi$
$853$	−21249.1	+	21249.1i	−0.0292040	+	0.0292040i	0.692354	+	0.721558i	$0.256574\pi$
$854$		−	22664.9i		−	0.0310769i
$855$		0			0
$856$	−50023.8			−0.0682698
$857$	165230.	+	165230.i	0.224971	+	0.224971i	0.810588	−	0.585617i	$-0.199148\pi$
$857$	165230.	+	165230.i	0.224971	+	0.224971i	−0.585617	+	0.810588i	$0.699148\pi$
$858$	−96067.2	+	96067.2i	−0.130497	+	0.130497i
$859$			249324.i			0.337891i	0.985625	+	0.168946i	$0.0540363\pi$
$859$			249324.i			0.337891i	−0.985625	+	0.168946i	$0.945964\pi$
$860$		0			0
$861$	−300310.			−0.405101
$862$	53484.3	+	53484.3i	0.0719800	+	0.0719800i
$863$	−325199.	+	325199.i	−0.436644	+	0.436644i	−0.890881	−	0.454237i	$-0.849912\pi$
$863$	−325199.	+	325199.i	−0.436644	+	0.436644i	0.454237	+	0.890881i	$0.349912\pi$
$864$			79688.6i			0.106750i
$865$		0			0
$866$	−168235.			−0.224327
$867$	−136091.	−	136091.i	−0.181047	−	0.181047i
$868$	−53823.0	+	53823.0i	−0.0714379	+	0.0714379i
$869$			1.94683e6i			2.57804i
$870$		0			0
$871$	−280993.			−0.370390
$872$	240696.	+	240696.i	0.316546	+	0.316546i
$873$	−19659.3	+	19659.3i	−0.0257953	+	0.0257953i
$874$		−	286089.i		−	0.374523i
$875$		0			0
$876$	−391107.			−0.509668
$877$	−578651.	−	578651.i	−0.752346	−	0.752346i	0.222571	−	0.974917i	$-0.428555\pi$
$877$	−578651.	−	578651.i	−0.752346	−	0.752346i	−0.974917	+	0.222571i	$0.928555\pi$
$878$	−16971.7	+	16971.7i	−0.0220159	+	0.0220159i
$879$			101881.i			0.131860i
$880$		0			0
$881$	−1.10283e6			−1.42087			−0.710436	−	0.703762i	$-0.751502\pi$
$881$	−1.10283e6			−1.42087			−0.710436	+	0.703762i	$0.751502\pi$
$882$	−27322.3	−	27322.3i	−0.0351221	−	0.0351221i
$883$	−402039.	+	402039.i	−0.515640	+	0.515640i	−0.916249	−	0.400609i	$-0.868799\pi$
$883$	−402039.	+	402039.i	−0.515640	+	0.515640i	0.400609	+	0.916249i	$0.368799\pi$
$884$		−	565611.i		−	0.723791i
$885$		0			0
$886$	−139357.			−0.177526
$887$	725066.	+	725066.i	0.921575	+	0.921575i	0.997141	−	0.0755662i	$-0.0240764\pi$
$887$	725066.	+	725066.i	0.921575	+	0.921575i	−0.0755662	+	0.997141i	$0.524076\pi$
$888$	58437.2	−	58437.2i	0.0741077	−	0.0741077i
$889$		−	331647.i		−	0.419636i
$890$		0			0
$891$	143657.			0.180955
$892$	−463668.	−	463668.i	−0.582743	−	0.582743i
$893$	1.21727e6	−	1.21727e6i	1.52645	−	1.52645i
$894$			11454.0i			0.0143311i
$895$		0			0
$896$	274599.			0.342045
$897$	−550160.	−	550160.i	−0.683760	−	0.683760i
$898$	−74328.7	+	74328.7i	−0.0921731	+	0.0921731i
$899$			165231.i			0.204442i
$900$		0			0
$901$	393298.			0.484475
$902$	−264282.	−	264282.i	−0.324829	−	0.324829i
$903$	260687.	−	260687.i	0.319701	−	0.319701i
$904$			597284.i			0.730876i
$905$		0			0
$906$	−46561.8			−0.0567249
$907$	118169.	+	118169.i	0.143644	+	0.143644i	0.775272	−	0.631628i	$-0.217613\pi$
$907$	118169.	+	118169.i	0.143644	+	0.143644i	−0.631628	+	0.775272i	$0.717613\pi$
$908$	743142.	−	743142.i	0.901364	−	0.901364i
$909$		−	88688.1i		−	0.107334i
$910$		0			0
$911$	291846.			0.351655			0.175827	−	0.984421i	$-0.443740\pi$
$911$	291846.			0.351655			0.175827	+	0.984421i	$0.443740\pi$
$912$	349255.	+	349255.i	0.419907	+	0.419907i
$913$	−649670.	+	649670.i	−0.779384	+	0.779384i
$914$		−	6647.08i		−	0.00795680i
$915$		0			0
$916$	−50684.0			−0.0604060
$917$	317060.	+	317060.i	0.377053	+	0.377053i
$918$	16651.4	−	16651.4i	0.0197591	−	0.0197591i
$919$		−	487999.i		−	0.577814i	−0.957357	−	0.288907i	$-0.906708\pi$
$919$		−	487999.i		−	0.577814i	0.957357	−	0.288907i	$-0.0932918\pi$
$920$		0			0
$921$	693695.			0.817804
$922$	19796.6	+	19796.6i	0.0232878	+	0.0232878i
$923$	−374775.	+	374775.i	−0.439913	+	0.439913i
$924$		−	373949.i		−	0.437995i
$925$		0			0
$926$	171725.			0.200268
$927$	−183247.	−	183247.i	−0.213244	−	0.213244i
$928$	318402.	−	318402.i	0.369725	−	0.369725i
$929$		−	480347.i		−	0.556575i	−0.960498	−	0.278288i	$-0.910233\pi$
$929$		−	480347.i		−	0.556575i	0.960498	−	0.278288i	$-0.0897669\pi$
$930$		0			0
$931$	−768804.			−0.886986
$932$	226183.	+	226183.i	0.260393	+	0.260393i
$933$	−323599.	+	323599.i	−0.371744	+	0.371744i
$934$		−	178689.i		−	0.204835i
$935$		0			0
$936$	−112465.			−0.128371
$937$	−684823.	−	684823.i	−0.780008	−	0.780008i	0.199824	−	0.979832i	$-0.435963\pi$
$937$	−684823.	−	684823.i	−0.780008	−	0.780008i	−0.979832	+	0.199824i	$0.935963\pi$
$938$	−21533.6	+	21533.6i	−0.0244743	+	0.0244743i
$939$		−	651332.i		−	0.738705i
$940$		0			0
$941$	−1.18932e6			−1.34313			−0.671565	−	0.740946i	$-0.734377\pi$
$941$	−1.18932e6			−1.34313			−0.671565	+	0.740946i	$0.734377\pi$
$942$	−68185.9	−	68185.9i	−0.0768409	−	0.0768409i
$943$	1.51350e6	−	1.51350e6i	1.70200	−	1.70200i
$944$		−	444090.i		−	0.498342i
$945$		0			0
$946$	458826.			0.512703
$947$	242748.	+	242748.i	0.270679	+	0.270679i	0.829374	−	0.558694i	$-0.188698\pi$
$947$	242748.	+	242748.i	0.270679	+	0.270679i	−0.558694	+	0.829374i	$0.688698\pi$
$948$	−558785.	+	558785.i	−0.621767	+	0.621767i
$949$		−	833286.i		−	0.925256i
$950$		0			0
$951$	−295343.			−0.326562
$952$	−88396.6	−	88396.6i	−0.0975352	−	0.0975352i
$953$	851307.	−	851307.i	0.937347	−	0.937347i	−0.0608027	−	0.998150i	$-0.519366\pi$
$953$	851307.	−	851307.i	0.937347	−	0.937347i	0.998150	+	0.0608027i	$0.0193660\pi$
$954$		−	38346.4i		−	0.0421335i
$955$		0			0
$956$	510345.			0.558403
$957$	−573991.	−	573991.i	−0.626732	−	0.626732i
$958$	81872.7	−	81872.7i	0.0892088	−	0.0892088i
$959$		−	64488.2i		−	0.0701202i
$960$		0			0
$961$	−880080.			−0.952961
$962$	61050.8	+	61050.8i	0.0659692	+	0.0659692i
$963$	−39075.1	+	39075.1i	−0.0421354	+	0.0421354i
$964$		−	1.70144e6i		−	1.83090i
$965$		0			0
$966$	−84321.7			−0.0903619
$967$	444643.	+	444643.i	0.475509	+	0.475509i	0.903692	−	0.428183i	$-0.140846\pi$
$967$	444643.	+	444643.i	0.475509	+	0.475509i	−0.428183	+	0.903692i	$0.640846\pi$
$968$	418097.	−	418097.i	0.446197	−	0.446197i
$969$		−	468544.i		−	0.499003i
$970$		0			0
$971$	−1.65301e6			−1.75322			−0.876609	−	0.481204i	$-0.840200\pi$
$971$	−1.65301e6			−1.75322			−0.876609	+	0.481204i	$0.840200\pi$
$972$	41232.8	+	41232.8i	0.0436425	+	0.0436425i
$973$	201904.	−	201904.i	0.213265	−	0.213265i
$974$			213770.i			0.225335i
$975$		0			0
$976$	−278893.			−0.292778
$977$	738999.	+	738999.i	0.774202	+	0.774202i	0.978838	−	0.204636i	$-0.0656010\pi$
$977$	738999.	+	738999.i	0.774202	+	0.774202i	−0.204636	+	0.978838i	$0.565601\pi$
$978$	85323.5	−	85323.5i	0.0892053	−	0.0892053i
$979$			1.20972e6i			1.26218i
$980$		0			0
$981$	376030.			0.390737
$982$	−115086.	−	115086.i	−0.119344	−	0.119344i
$983$	246723.	−	246723.i	0.255330	−	0.255330i	−0.567821	−	0.823152i	$-0.692214\pi$
$983$	246723.	−	246723.i	0.255330	−	0.255330i	0.823152	+	0.567821i	$0.192214\pi$
$984$		−	309394.i		−	0.319537i
$985$		0			0
$986$	−133064.			−0.136870
$987$	−358777.	−	358777.i	−0.368290	−	0.368290i
$988$	−775870.	+	775870.i	−0.794832	+	0.794832i
$989$			2.62761e6i			2.68639i
$990$		0			0
$991$	136401.			0.138890			0.0694449	−	0.997586i	$-0.477877\pi$
$991$	136401.			0.138890			0.0694449	+	0.997586i	$0.477877\pi$
$992$	−83711.8	−	83711.8i	−0.0850674	−	0.0850674i
$993$	12546.5	−	12546.5i	0.0127240	−	0.0127240i
$994$			57440.9i			0.0581365i
$995$		0			0
$996$	−372940.			−0.375941
$997$	−1.13742e6	−	1.13742e6i	−1.14427	−	1.14427i	−0.987660	−	0.156613i	$-0.949943\pi$
$997$	−1.13742e6	−	1.13742e6i	−1.14427	−	1.14427i	−0.156613	−	0.987660i	$-0.550057\pi$
$998$	39093.0	−	39093.0i	0.0392498	−	0.0392498i
$999$		−	91294.1i		−	0.0914770i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	75.5.f.d.7.1	yes	4
3.2	odd	2		225.5.g.d.82.2		4
5.2	odd	4		75.5.f.a.43.2	yes	4
5.3	odd	4	inner	75.5.f.d.43.1	yes	4
5.4	even	2		75.5.f.a.7.2	✓	4
15.2	even	4		225.5.g.l.118.1		4
15.8	even	4		225.5.g.d.118.2		4
15.14	odd	2		225.5.g.l.82.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.5.f.a.7.2	✓	4	5.4	even	2
75.5.f.a.43.2	yes	4	5.2	odd	4
75.5.f.d.7.1	yes	4	1.1	even	1	trivial
75.5.f.d.43.1	yes	4	5.3	odd	4	inner
225.5.g.d.82.2		4	3.2	odd	2
225.5.g.d.118.2		4	15.8	even	4
225.5.g.l.82.1		4	15.14	odd	2
225.5.g.l.118.1		4	15.2	even	4

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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	75.f (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.550510 + 0.550510i) q^{2} +(3.67423 - 3.67423i) q^{3} -15.3939i q^{4} +4.04541 q^{6} +(-16.7753 - 16.7753i) q^{7} +(17.2827 - 17.2827i) q^{8} -27.0000i q^{9} +O(q^{10})\)	\(q+(0.550510 + 0.550510i) q^{2} +(3.67423 - 3.67423i) q^{3} -15.3939i q^{4} +4.04541 q^{6} +(-16.7753 - 16.7753i) q^{7} +(17.2827 - 17.2827i) q^{8} -27.0000i q^{9} -197.060 q^{11} +(-56.5607 - 56.5607i) q^{12} +(120.507 - 120.507i) q^{13} -18.4699i q^{14} -227.273 q^{16} +(152.449 + 152.449i) q^{17} +(14.8638 - 14.8638i) q^{18} -418.242i q^{19} -123.272 q^{21} +(-108.484 - 108.484i) q^{22} +(621.267 - 621.267i) q^{23} -127.001i q^{24} +132.681 q^{26} +(-99.2043 - 99.2043i) q^{27} +(-258.236 + 258.236i) q^{28} +792.756i q^{29} +208.426 q^{31} +(-401.639 - 401.639i) q^{32} +(-724.045 + 724.045i) q^{33} +167.850i q^{34} -415.635 q^{36} +(460.132 + 460.132i) q^{37} +(230.246 - 230.246i) q^{38} -885.545i q^{39} +2436.15 q^{41} +(-67.8627 - 67.8627i) q^{42} +(-2114.72 + 2114.72i) q^{43} +3033.52i q^{44} +684.028 q^{46} +(2910.44 + 2910.44i) q^{47} +(-835.056 + 835.056i) q^{48} -1838.18i q^{49} +1120.27 q^{51} +(-1855.08 - 1855.08i) q^{52} +(1289.93 - 1289.93i) q^{53} -109.226i q^{54} -579.842 q^{56} +(-1536.72 - 1536.72i) q^{57} +(-436.420 + 436.420i) q^{58} +1953.99i q^{59} +1227.13 q^{61} +(114.740 + 114.740i) q^{62} +(-452.932 + 452.932i) q^{63} +3194.16i q^{64} -797.189 q^{66} +(-1165.88 - 1165.88i) q^{67} +(2346.79 - 2346.79i) q^{68} -4565.36i q^{69} -3109.97 q^{71} +(-466.632 - 466.632i) q^{72} +(3457.41 - 3457.41i) q^{73} +506.614i q^{74} -6438.36 q^{76} +(3305.74 + 3305.74i) q^{77} +(487.502 - 487.502i) q^{78} -9879.38i q^{79} -729.000 q^{81} +(1341.13 + 1341.13i) q^{82} +(3296.81 - 3296.81i) q^{83} +1897.64i q^{84} -2328.35 q^{86} +(2912.77 + 2912.77i) q^{87} +(-3405.72 + 3405.72i) q^{88} -6138.84i q^{89} -4043.08 q^{91} +(-9563.71 - 9563.71i) q^{92} +(765.804 - 765.804i) q^{93} +3204.45i q^{94} -2951.43 q^{96} +(-728.123 - 728.123i) q^{97} +(1011.94 - 1011.94i) q^{98} +5320.63i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{2} - 72 q^{6} - 72 q^{7} - 264 q^{8}+O(q^{10}) \) 4 * q + 12 * q^2 - 72 * q^6 - 72 * q^7 - 264 * q^8	\( 4 q + 12 q^{2} - 72 q^{6} - 72 q^{7} - 264 q^{8} - 24 q^{11} - 432 q^{12} + 144 q^{13} - 2320 q^{16} + 600 q^{17} + 324 q^{18} + 36 q^{21} + 1800 q^{22} + 888 q^{23} - 792 q^{26} + 1152 q^{28} + 3244 q^{31} - 6192 q^{32} - 2808 q^{33} + 1512 q^{36} + 2448 q^{37} - 1068 q^{38} + 6864 q^{41} + 1404 q^{42} - 5544 q^{43} - 2496 q^{46} + 5616 q^{47} + 5184 q^{48} + 72 q^{51} - 7920 q^{52} + 1848 q^{53} + 10320 q^{56} - 4104 q^{57} + 5544 q^{58} + 15020 q^{61} + 15636 q^{62} - 1944 q^{63} - 16416 q^{66} - 13320 q^{67} - 8112 q^{68} - 21552 q^{71} + 7128 q^{72} + 19728 q^{73} - 25048 q^{76} - 504 q^{77} + 4860 q^{78} - 2916 q^{81} + 13536 q^{82} + 8592 q^{83} - 18984 q^{86} + 10152 q^{87} - 62064 q^{88} - 4356 q^{91} - 34512 q^{92} - 8856 q^{93} + 33696 q^{96} - 11520 q^{97} + 20136 q^{98}+O(q^{100}) \) 4 * q + 12 * q^2 - 72 * q^6 - 72 * q^7 - 264 * q^8 - 24 * q^11 - 432 * q^12 + 144 * q^13 - 2320 * q^16 + 600 * q^17 + 324 * q^18 + 36 * q^21 + 1800 * q^22 + 888 * q^23 - 792 * q^26 + 1152 * q^28 + 3244 * q^31 - 6192 * q^32 - 2808 * q^33 + 1512 * q^36 + 2448 * q^37 - 1068 * q^38 + 6864 * q^41 + 1404 * q^42 - 5544 * q^43 - 2496 * q^46 + 5616 * q^47 + 5184 * q^48 + 72 * q^51 - 7920 * q^52 + 1848 * q^53 + 10320 * q^56 - 4104 * q^57 + 5544 * q^58 + 15020 * q^61 + 15636 * q^62 - 1944 * q^63 - 16416 * q^66 - 13320 * q^67 - 8112 * q^68 - 21552 * q^71 + 7128 * q^72 + 19728 * q^73 - 25048 * q^76 - 504 * q^77 + 4860 * q^78 - 2916 * q^81 + 13536 * q^82 + 8592 * q^83 - 18984 * q^86 + 10152 * q^87 - 62064 * q^88 - 4356 * q^91 - 34512 * q^92 - 8856 * q^93 + 33696 * q^96 - 11520 * q^97 + 20136 * q^98

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