LMFDB - Embedded newform 177.5.c.a.58.24

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,5,Mod(58,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, base_ring=CyclotomicField(2))

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("177.58");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	177.c (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:	$18.2964834658$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			58.24
Character	$\chi$	$=$	177.58
Dual form			177.5.c.a.58.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.15560i q^{2} -5.19615 q^{3} +11.3534 q^{4} +30.9385 q^{5} -11.2008i q^{6} +73.3307 q^{7} +58.9629i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q+2.15560i q^{2} -5.19615 q^{3} +11.3534 q^{4} +30.9385 q^{5} -11.2008i q^{6} +73.3307 q^{7} +58.9629i q^{8} +27.0000 q^{9} +66.6909i q^{10} -203.451i q^{11} -58.9940 q^{12} -115.092i q^{13} +158.071i q^{14} -160.761 q^{15} +54.5542 q^{16} -149.563 q^{17} +58.2011i q^{18} -97.5337 q^{19} +351.257 q^{20} -381.038 q^{21} +438.557 q^{22} -785.535i q^{23} -306.380i q^{24} +332.189 q^{25} +248.092 q^{26} -140.296 q^{27} +832.553 q^{28} +279.591 q^{29} -346.536i q^{30} +520.224i q^{31} +1061.00i q^{32} +1057.16i q^{33} -322.398i q^{34} +2268.74 q^{35} +306.542 q^{36} +737.478i q^{37} -210.243i q^{38} +598.036i q^{39} +1824.22i q^{40} +540.849 q^{41} -821.364i q^{42} +1171.64i q^{43} -2309.86i q^{44} +835.339 q^{45} +1693.30 q^{46} +1505.45i q^{47} -283.472 q^{48} +2976.40 q^{49} +716.067i q^{50} +777.153 q^{51} -1306.69i q^{52} -3630.24 q^{53} -302.422i q^{54} -6294.45i q^{55} +4323.79i q^{56} +506.800 q^{57} +602.686i q^{58} +(3344.70 - 964.542i) q^{59} -1825.19 q^{60} +5340.44i q^{61} -1121.39 q^{62} +1979.93 q^{63} -1414.23 q^{64} -3560.77i q^{65} -2278.81 q^{66} +1694.69i q^{67} -1698.05 q^{68} +4081.76i q^{69} +4890.49i q^{70} -9066.29 q^{71} +1592.00i q^{72} -876.744i q^{73} -1589.71 q^{74} -1726.11 q^{75} -1107.34 q^{76} -14919.2i q^{77} -1289.12 q^{78} +7895.46 q^{79} +1687.82 q^{80} +729.000 q^{81} +1165.85i q^{82} +3535.87i q^{83} -4326.07 q^{84} -4627.26 q^{85} -2525.58 q^{86} -1452.80 q^{87} +11996.0 q^{88} -11418.1i q^{89} +1800.65i q^{90} -8439.78i q^{91} -8918.49i q^{92} -2703.16i q^{93} -3245.15 q^{94} -3017.55 q^{95} -5513.14i q^{96} +15153.6i q^{97} +6415.91i q^{98} -5493.16i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9}+O(q^{10}) $ 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9	$ 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9} + 360 q^{12} + 144 q^{15} + 3944 q^{16} - 528 q^{17} + 444 q^{19} + 444 q^{20} + 1304 q^{22} + 4880 q^{25} - 1452 q^{26} - 1160 q^{28} - 996 q^{29} + 10320 q^{35} - 8640 q^{36} - 5196 q^{41} - 10476 q^{46} + 576 q^{48} + 5104 q^{49} + 936 q^{51} - 2184 q^{53} - 2520 q^{57} - 11736 q^{59} - 11448 q^{60} + 15240 q^{62} + 2160 q^{63} - 81012 q^{64} + 17352 q^{66} + 29568 q^{68} - 5964 q^{71} + 14376 q^{74} - 2736 q^{75} + 3480 q^{76} + 37692 q^{78} + 19020 q^{79} + 33096 q^{80} + 29160 q^{81} + 25128 q^{84} + 20220 q^{85} - 65880 q^{86} + 1512 q^{87} - 14932 q^{88} - 17864 q^{94} + 11004 q^{95}+O(q^{100}) $ 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9 + 360 * q^12 + 144 * q^15 + 3944 * q^16 - 528 * q^17 + 444 * q^19 + 444 * q^20 + 1304 * q^22 + 4880 * q^25 - 1452 * q^26 - 1160 * q^28 - 996 * q^29 + 10320 * q^35 - 8640 * q^36 - 5196 * q^41 - 10476 * q^46 + 576 * q^48 + 5104 * q^49 + 936 * q^51 - 2184 * q^53 - 2520 * q^57 - 11736 * q^59 - 11448 * q^60 + 15240 * q^62 + 2160 * q^63 - 81012 * q^64 + 17352 * q^66 + 29568 * q^68 - 5964 * q^71 + 14376 * q^74 - 2736 * q^75 + 3480 * q^76 + 37692 * q^78 + 19020 * q^79 + 33096 * q^80 + 29160 * q^81 + 25128 * q^84 + 20220 * q^85 - 65880 * q^86 + 1512 * q^87 - 14932 * q^88 - 17864 * q^94 + 11004 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/177\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$1$

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 $

$2$			2.15560i			0.538899i	0.963014	+	0.269450i	$0.0868418\pi$
$2$			2.15560i			0.538899i	−0.963014	+	0.269450i	$0.913158\pi$
$3$	−5.19615			−0.577350
$3$	−5.19615			−0.577350
$4$	11.3534			0.709588
$5$	30.9385			1.23754			0.618770	−	0.785573i	$-0.287631\pi$
$5$	30.9385			1.23754			0.618770	+	0.785573i	$0.287631\pi$
$6$		−	11.2008i		−	0.311134i
$7$	73.3307			1.49655			0.748273	−	0.663391i	$-0.230883\pi$
$7$	73.3307			1.49655			0.748273	+	0.663391i	$0.230883\pi$
$8$			58.9629i			0.921295i
$9$	27.0000			0.333333
$10$			66.6909i			0.666909i
$11$		−	203.451i		−	1.68141i	−0.541494	−	0.840705i	$-0.682141\pi$
$11$		−	203.451i		−	1.68141i	0.541494	−	0.840705i	$-0.317859\pi$
$12$	−58.9940			−0.409681
$13$		−	115.092i		−	0.681018i	−0.940241	−	0.340509i	$-0.889401\pi$
$13$		−	115.092i		−	0.681018i	0.940241	−	0.340509i	$-0.110599\pi$
$14$			158.071i			0.806487i
$15$	−160.761			−0.714494
$16$	54.5542			0.213102
$17$	−149.563			−0.517520			−0.258760	−	0.965942i	$-0.583314\pi$
$17$	−149.563			−0.517520			−0.258760	+	0.965942i	$0.583314\pi$
$18$			58.2011i			0.179633i
$19$	−97.5337			−0.270177			−0.135088	−	0.990834i	$-0.543132\pi$
$19$	−97.5337			−0.270177			−0.135088	+	0.990834i	$0.543132\pi$
$20$	351.257			0.878143
$21$	−381.038			−0.864031
$22$	438.557			0.906110
$23$		−	785.535i		−	1.48494i	−0.669878	−	0.742472i	$-0.733653\pi$
$23$		−	785.535i		−	1.48494i	0.669878	−	0.742472i	$-0.266347\pi$
$24$		−	306.380i		−	0.531910i
$25$	332.189			0.531503
$26$	248.092			0.367000
$27$	−140.296			−0.192450
$28$	832.553			1.06193
$29$	279.591			0.332451			0.166226	−	0.986088i	$-0.446842\pi$
$29$	279.591			0.332451			0.166226	+	0.986088i	$0.446842\pi$
$30$		−	346.536i		−	0.385040i
$31$			520.224i			0.541336i	0.962673	+	0.270668i	$0.0872445\pi$
$31$			520.224i			0.541336i	−0.962673	+	0.270668i	$0.912755\pi$
$32$			1061.00i			1.03614i
$33$			1057.16i			0.970762i
$34$		−	322.398i		−	0.278891i
$35$	2268.74			1.85203
$36$	306.542			0.236529
$37$			737.478i			0.538699i	0.963043	+	0.269349i	$0.0868086\pi$
$37$			737.478i			0.538699i	−0.963043	+	0.269349i	$0.913191\pi$
$38$		−	210.243i		−	0.145598i
$39$			598.036i			0.393186i
$40$			1824.22i			1.14014i
$41$	540.849			0.321743			0.160871	−	0.986975i	$-0.448570\pi$
$41$	540.849			0.321743			0.160871	+	0.986975i	$0.448570\pi$
$42$		−	821.364i		−	0.465626i
$43$			1171.64i			0.633660i	0.948482	+	0.316830i	$0.102618\pi$
$43$			1171.64i			0.633660i	−0.948482	+	0.316830i	$0.897382\pi$
$44$		−	2309.86i		−	1.19311i
$45$	835.339			0.412513
$46$	1693.30			0.800235
$47$			1505.45i			0.681509i	0.940152	+	0.340755i	$0.110682\pi$
$47$			1505.45i			0.681509i	−0.940152	+	0.340755i	$0.889318\pi$
$48$	−283.472			−0.123035
$49$	2976.40			1.23965
$50$			716.067i			0.286427i
$51$	777.153			0.298790
$52$		−	1306.69i		−	0.483242i
$53$	−3630.24			−1.29236			−0.646179	−	0.763185i	$-0.723634\pi$
$53$	−3630.24			−1.29236			−0.646179	+	0.763185i	$0.723634\pi$
$54$		−	302.422i		−	0.103711i
$55$		−	6294.45i		−	2.08081i
$56$			4323.79i			1.37876i
$57$	506.800			0.155987
$58$			602.686i			0.179158i
$59$	3344.70	−	964.542i	0.960845	−	0.277088i
$59$	3344.70	−	964.542i	0.960845	−	0.277088i
$60$	−1825.19			−0.506996
$61$			5340.44i			1.43522i	0.696447	+	0.717608i	$0.254763\pi$
$61$			5340.44i			1.43522i	−0.696447	+	0.717608i	$0.745237\pi$
$62$	−1121.39			−0.291725
$63$	1979.93			0.498849
$64$	−1414.23			−0.345270
$65$		−	3560.77i		−	0.842786i
$66$	−2278.81			−0.523143
$67$			1694.69i			0.377520i	0.982023	+	0.188760i	$0.0604469\pi$
$67$			1694.69i			0.377520i	−0.982023	+	0.188760i	$0.939553\pi$
$68$	−1698.05			−0.367226
$69$			4081.76i			0.857332i
$70$			4890.49i			0.998059i
$71$	−9066.29			−1.79851			−0.899255	−	0.437425i	$-0.855891\pi$
$71$	−9066.29			−1.79851			−0.899255	+	0.437425i	$0.855891\pi$
$72$			1592.00i			0.307098i
$73$		−	876.744i		−	0.164523i	−0.996611	−	0.0822616i	$-0.973786\pi$
$73$		−	876.744i		−	0.164523i	0.996611	−	0.0822616i	$-0.0262143\pi$
$74$	−1589.71			−0.290304
$75$	−1726.11			−0.306864
$76$	−1107.34			−0.191714
$77$		−	14919.2i		−	2.51631i
$78$	−1289.12			−0.211888
$79$	7895.46			1.26509			0.632547	−	0.774522i	$-0.282009\pi$
$79$	7895.46			1.26509			0.632547	+	0.774522i	$0.282009\pi$
$80$	1687.82			0.263723
$81$	729.000			0.111111
$82$			1165.85i			0.173387i
$83$			3535.87i			0.513264i	0.966509	+	0.256632i	$0.0826128\pi$
$83$			3535.87i			0.513264i	−0.966509	+	0.256632i	$0.917387\pi$
$84$	−4326.07			−0.613106
$85$	−4627.26			−0.640451
$86$	−2525.58			−0.341479
$87$	−1452.80			−0.191941
$88$	11996.0			1.54907
$89$		−	11418.1i		−	1.44150i	−0.693195	−	0.720750i	$-0.743797\pi$
$89$		−	11418.1i		−	1.44150i	0.693195	−	0.720750i	$-0.256203\pi$
$90$			1800.65i			0.222303i
$91$		−	8439.78i		−	1.01917i
$92$		−	8918.49i		−	1.05370i
$93$		−	2703.16i		−	0.312540i
$94$	−3245.15			−0.367265
$95$	−3017.55			−0.334354
$96$		−	5513.14i		−	0.598213i
$97$			15153.6i			1.61054i	0.592909	+	0.805269i	$0.297979\pi$
$97$			15153.6i			1.61054i	−0.592909	+	0.805269i	$0.702021\pi$
$98$			6415.91i			0.668046i
$99$		−	5493.16i		−	0.560470i
$100$	3771.48			0.377148
$101$			10320.2i			1.01168i	0.862626	+	0.505842i	$0.168818\pi$
$101$			10320.2i			1.01168i	−0.862626	+	0.505842i	$0.831182\pi$
$102$			1675.23i			0.161018i
$103$			13724.8i			1.29369i	0.762620	+	0.646847i	$0.223913\pi$
$103$			13724.8i			1.29369i	−0.762620	+	0.646847i	$0.776087\pi$
$104$	6786.16			0.627419
$105$	−11788.7			−1.06927
$106$		−	7825.33i		−	0.696451i
$107$	−9382.60			−0.819513			−0.409756	−	0.912195i	$-0.634386\pi$
$107$	−9382.60			−0.819513			−0.409756	+	0.912195i	$0.634386\pi$
$108$	−1592.84			−0.136560
$109$		−	15647.4i		−	1.31701i	−0.752575	−	0.658507i	$-0.771188\pi$
$109$		−	15647.4i		−	1.31701i	0.752575	−	0.658507i	$-0.228812\pi$
$110$	13568.3			1.12135
$111$		−	3832.05i		−	0.311018i
$112$	4000.50			0.318917
$113$			16451.0i			1.28836i	0.764875	+	0.644179i	$0.222801\pi$
$113$			16451.0i			1.28836i	−0.764875	+	0.644179i	$0.777199\pi$
$114$			1092.46i			0.0840610i
$115$		−	24303.3i		−	1.83768i
$116$	3174.31			0.235903
$117$		−	3107.48i		−	0.227006i
$118$	2079.16	+	7209.82i	0.149322	+	0.517798i
$119$	−10967.6			−0.774492
$120$		−	9478.94i		−	0.658260i
$121$	−26751.1			−1.82714
$122$	−11511.8			−0.773437
$123$	−2810.34			−0.185758
$124$			5906.31i			0.384125i
$125$	−9059.11			−0.579783
$126$			4267.93i			0.268829i
$127$	15007.8			0.930484			0.465242	−	0.885183i	$-0.345967\pi$
$127$	15007.8			0.930484			0.465242	+	0.885183i	$0.345967\pi$
$128$			13927.5i			0.850070i
$129$		−	6088.01i		−	0.365844i
$130$	7675.59			0.454177
$131$		−	521.489i		−	0.0303880i	−0.999885	−	0.0151940i	$-0.995163\pi$
$131$		−	521.489i		−	0.0303880i	0.999885	−	0.0151940i	$-0.00483659\pi$
$132$			12002.4i			0.688841i
$133$	−7152.22			−0.404332
$134$	−3653.07			−0.203445
$135$	−4340.55			−0.238165
$136$		−	8818.68i		−	0.476789i
$137$	−29732.3			−1.58411			−0.792057	−	0.610447i	$-0.790990\pi$
$137$	−29732.3			−1.58411			−0.792057	+	0.610447i	$0.790990\pi$
$138$	−8798.63			−0.462016
$139$	−5711.68			−0.295620			−0.147810	−	0.989016i	$-0.547222\pi$
$139$	−5711.68			−0.295620			−0.147810	+	0.989016i	$0.547222\pi$
$140$	25757.9			1.31418
$141$		−	7822.57i		−	0.393469i
$142$		−	19543.3i		−	0.969216i
$143$	−23415.5			−1.14507
$144$	1472.96			0.0710341
$145$	8650.13			0.411421
$146$	1889.91			0.0886614
$147$	−15465.8			−0.715711
$148$			8372.89i			0.382254i
$149$		−	6283.15i		−	0.283012i	−0.989937	−	0.141506i	$-0.954806\pi$
$149$		−	6283.15i		−	0.283012i	0.989937	−	0.141506i	$-0.0451944\pi$
$150$		−	3720.79i		−	0.165368i
$151$			39803.8i			1.74571i	0.487983	+	0.872853i	$0.337733\pi$
$151$			39803.8i			1.74571i	−0.487983	+	0.872853i	$0.662267\pi$
$152$		−	5750.87i		−	0.248912i
$153$	−4038.21			−0.172507
$154$	32159.7			1.35604
$155$			16094.9i			0.669924i
$156$			6789.74i			0.279000i
$157$		−	2933.82i		−	0.119024i	−0.998228	−	0.0595119i	$-0.981046\pi$
$157$		−	2933.82i		−	0.119024i	0.998228	−	0.0595119i	$-0.0189544\pi$
$158$			17019.4i			0.681758i
$159$	18863.3			0.746144
$160$			32825.8i			1.28226i
$161$		−	57603.8i		−	2.22228i
$162$			1571.43i			0.0598777i
$163$	44939.2			1.69142			0.845708	−	0.533645i	$-0.179178\pi$
$163$	44939.2			1.69142			0.845708	+	0.533645i	$0.179178\pi$
$164$	6140.48			0.228305
$165$			32706.9i			1.20136i
$166$	−7621.92			−0.276597
$167$	5237.32			0.187791			0.0938957	−	0.995582i	$-0.470068\pi$
$167$	5237.32			0.187791			0.0938957	+	0.995582i	$0.470068\pi$
$168$		−	22467.1i		−	0.796028i
$169$	15314.8			0.536215
$170$		−	9974.50i		−	0.345138i
$171$	−2633.41			−0.0900589
$172$			13302.1i			0.449637i
$173$		−	23988.2i		−	0.801503i	−0.916187	−	0.400751i	$-0.868749\pi$
$173$		−	23988.2i		−	0.801503i	0.916187	−	0.400751i	$-0.131251\pi$
$174$		−	3131.65i		−	0.103437i
$175$	24359.7			0.795419
$176$		−	11099.1i		−	0.358312i
$177$	−17379.6	+	5011.91i	−0.554744	+	0.159977i
$178$	24612.9			0.776824
$179$			48379.7i			1.50993i	0.655765	+	0.754965i	$0.272346\pi$
$179$			48379.7i			1.50993i	−0.655765	+	0.754965i	$0.727654\pi$
$180$	9483.94			0.292714
$181$	−9403.26			−0.287026			−0.143513	−	0.989648i	$-0.545840\pi$
$181$	−9403.26			−0.287026			−0.143513	+	0.989648i	$0.545840\pi$
$182$	18192.8			0.549232
$183$		−	27749.7i		−	0.828622i
$184$	46317.4			1.36807
$185$			22816.5i			0.666661i
$186$	5826.93			0.168428
$187$			30428.7i			0.870163i
$188$			17092.0i			0.483590i
$189$	−10288.0			−0.288010
$190$		−	6504.61i		−	0.180183i
$191$		−	26293.8i		−	0.720754i	−0.932807	−	0.360377i	$-0.882648\pi$
$191$		−	26293.8i		−	0.720754i	0.932807	−	0.360377i	$-0.117352\pi$
$192$	7348.54			0.199342
$193$	−35955.0			−0.965260			−0.482630	−	0.875824i	$-0.660318\pi$
$193$	−35955.0			−0.965260			−0.482630	+	0.875824i	$0.660318\pi$
$194$	−32665.0			−0.867918
$195$			18502.3i			0.486583i
$196$	33792.2			0.879639
$197$	16157.0			0.416321			0.208160	−	0.978095i	$-0.433252\pi$
$197$	16157.0			0.416321			0.208160	+	0.978095i	$0.433252\pi$
$198$	11841.0			0.302037
$199$	−8326.75			−0.210266			−0.105133	−	0.994458i	$-0.533527\pi$
$199$	−8326.75			−0.210266			−0.105133	+	0.994458i	$0.533527\pi$
$200$			19586.9i			0.489671i
$201$		−	8805.86i		−	0.217961i
$202$	−22246.2			−0.545195
$203$	20502.6			0.497528
$204$	8823.33			0.212018
$205$	16733.1			0.398169
$206$	−29585.1			−0.697171
$207$		−	21209.4i		−	0.494981i
$208$		−	6278.75i		−	0.145127i
$209$			19843.3i			0.454277i
$210$		−	25411.7i		−	0.576230i
$211$		−	75444.4i		−	1.69458i	−0.531130	−	0.847290i	$-0.678232\pi$
$211$		−	75444.4i		−	1.69458i	0.531130	−	0.847290i	$-0.321768\pi$
$212$	−41215.5			−0.917042
$213$	47109.8			1.03837
$214$		−	20225.1i		−	0.441635i
$215$			36248.7i			0.784179i
$216$		−	8272.27i		−	0.177303i
$217$			38148.4i			0.810134i
$218$	33729.6			0.709738
$219$			4555.70i			0.0949875i
$220$		−	71463.4i		−	1.47652i
$221$			17213.5i			0.352440i
$222$	8260.36			0.167607
$223$	80926.3			1.62735			0.813673	−	0.581322i	$-0.197464\pi$
$223$	80926.3			1.62735			0.813673	+	0.581322i	$0.197464\pi$
$224$			77804.2i			1.55062i
$225$	8969.12			0.177168
$226$	−35461.8			−0.694295
$227$		−	50468.1i		−	0.979411i	−0.871888	−	0.489706i	$-0.837104\pi$
$227$		−	50468.1i		−	0.979411i	0.871888	−	0.489706i	$-0.162896\pi$
$228$	5753.91			0.110686
$229$		−	88590.6i		−	1.68934i	−0.535288	−	0.844669i	$-0.679797\pi$
$229$		−	88590.6i		−	1.68934i	0.535288	−	0.844669i	$-0.320203\pi$
$230$	52388.0			0.990322
$231$			77522.3i			1.45279i
$232$			16485.5i			0.306286i
$233$		−	21367.1i		−	0.393580i	−0.980446	−	0.196790i	$-0.936948\pi$
$233$		−	21367.1i		−	0.393580i	0.980446	−	0.196790i	$-0.0630518\pi$
$234$	6698.48			0.122333
$235$			46576.4i			0.843394i
$236$	37973.7	−	10950.8i	0.681804	−	0.196618i
$237$	−41026.0			−0.730403
$238$		−	23641.7i		−	0.417373i
$239$	27789.3			0.486498			0.243249	−	0.969964i	$-0.421787\pi$
$239$	27789.3			0.486498			0.243249	+	0.969964i	$0.421787\pi$
$240$	−8770.19			−0.152260
$241$	−114284.			−1.96767			−0.983836	−	0.179071i	$-0.942691\pi$
$241$	−114284.			−1.96767			−0.983836	+	0.179071i	$0.942691\pi$
$242$		−	57664.6i		−	0.984643i
$243$	−3788.00			−0.0641500
$244$			60632.1i			1.01841i
$245$	92085.2			1.53411
$246$		−	6057.95i		−	0.100105i
$247$			11225.4i			0.183995i
$248$	−30673.9			−0.498730
$249$		−	18372.9i		−	0.296333i
$250$		−	19527.8i		−	0.312445i
$251$	76189.7			1.20934			0.604671	−	0.796475i	$-0.293305\pi$
$251$	76189.7			1.20934			0.604671	+	0.796475i	$0.293305\pi$
$252$	22478.9			0.353977
$253$	−159818.			−2.49680
$254$			32350.7i			0.501437i
$255$	24043.9			0.369764
$256$	−52649.8			−0.803373
$257$	−9932.87			−0.150386			−0.0751932	−	0.997169i	$-0.523957\pi$
$257$	−9932.87			−0.150386			−0.0751932	+	0.997169i	$0.523957\pi$
$258$	13123.3			0.197153
$259$			54079.8i			0.806187i
$260$		−	40426.9i		−	0.598031i
$261$	7548.97			0.110817
$262$	1124.12			0.0163761
$263$	34983.1			0.505762			0.252881	−	0.967497i	$-0.418622\pi$
$263$	34983.1			0.505762			0.252881	+	0.967497i	$0.418622\pi$
$264$	−62333.2			−0.894359
$265$	−112314.			−1.59934
$266$		−	15417.3i		−	0.217894i
$267$			59330.3i			0.832251i
$268$			19240.5i			0.267884i
$269$		−	61848.8i		−	0.854725i	−0.904080	−	0.427363i	$-0.859443\pi$
$269$		−	61848.8i		−	0.854725i	0.904080	−	0.427363i	$-0.140557\pi$
$270$		−	9356.47i		−	0.128347i
$271$	−91570.5			−1.24686			−0.623429	−	0.781880i	$-0.714261\pi$
$271$	−91570.5			−1.24686			−0.623429	+	0.781880i	$0.714261\pi$
$272$	−8159.30			−0.110285
$273$			43854.4i			0.588421i
$274$		−	64090.7i		−	0.853678i
$275$		−	67584.1i		−	0.893675i
$276$			46341.9i			0.608352i
$277$	−88687.7			−1.15586			−0.577928	−	0.816088i	$-0.696139\pi$
$277$	−88687.7			−1.15586			−0.577928	+	0.816088i	$0.696139\pi$
$278$		−	12312.1i		−	0.159309i
$279$			14046.0i			0.180445i
$280$			133772.i			1.70627i
$281$	−54231.2			−0.686809			−0.343405	−	0.939188i	$-0.611580\pi$
$281$	−54231.2			−0.686809			−0.343405	+	0.939188i	$0.611580\pi$
$282$	16862.3			0.212040
$283$		−	19900.8i		−	0.248483i	−0.992252	−	0.124242i	$-0.960350\pi$
$283$		−	19900.8i		−	0.248483i	0.992252	−	0.124242i	$-0.0396498\pi$
$284$	−102933.			−1.27620
$285$	15679.6			0.193039
$286$		−	50474.5i		−	0.617077i
$287$	39660.9			0.481503
$288$			28647.1i			0.345379i
$289$	−61151.9			−0.732173
$290$			18646.2i			0.221715i
$291$		−	78740.2i		−	0.929845i
$292$		−	9954.03i		−	0.116744i
$293$	88852.6			1.03499			0.517494	−	0.855687i	$-0.326865\pi$
$293$	88852.6			1.03499			0.517494	+	0.855687i	$0.326865\pi$
$294$		−	33338.0i		−	0.385696i
$295$	103480.	−	29841.5i	1.18908	−	0.342907i
$296$	−43483.9			−0.496301
$297$			28543.3i			0.323587i
$298$	13543.9			0.152515
$299$	−90408.8			−1.01127
$300$	−19597.2			−0.217747
$301$			85917.0i			0.948301i
$302$	−85801.0			−0.940760
$303$		−	53625.3i		−	0.584096i
$304$	−5320.88			−0.0575753
$305$			165225.i			1.77614i
$306$		−	8704.74i		−	0.0929636i
$307$	−5886.60			−0.0624579			−0.0312290	−	0.999512i	$-0.509942\pi$
$307$	−5886.60			−0.0624579			−0.0312290	+	0.999512i	$0.509942\pi$
$308$		−	169383.i		−	1.78554i
$309$		−	71316.2i		−	0.746915i
$310$	−34694.2			−0.361022
$311$	−94490.4			−0.976938			−0.488469	−	0.872581i	$-0.662444\pi$
$311$	−94490.4			−0.976938			−0.488469	+	0.872581i	$0.662444\pi$
$312$	−35261.9			−0.362240
$313$		−	90255.1i		−	0.921262i	−0.887592	−	0.460631i	$-0.847623\pi$
$313$		−	90255.1i		−	0.921262i	0.887592	−	0.460631i	$-0.152377\pi$
$314$	6324.13			0.0641418
$315$	61256.0			0.617345
$316$	89640.3			0.897696
$317$	159546.			1.58770			0.793849	−	0.608115i	$-0.208074\pi$
$317$	159546.			1.58770			0.793849	+	0.608115i	$0.208074\pi$
$318$			40661.6i			0.402096i
$319$		−	56883.0i		−	0.558987i
$320$	−43754.1			−0.427286
$321$	48753.4			0.473146
$322$	124171.			1.19759
$323$	14587.5			0.139822
$324$	8276.63			0.0788431
$325$		−	38232.4i		−	0.361963i
$326$			96870.9i			0.911503i
$327$			81306.5i			0.760378i
$328$			31890.1i			0.296420i
$329$			110396.i			1.01991i
$330$	−70502.9			−0.647410
$331$	20831.2			0.190134			0.0950669	−	0.995471i	$-0.469694\pi$
$331$	20831.2			0.190134			0.0950669	+	0.995471i	$0.469694\pi$
$332$			40144.2i			0.364206i
$333$			19911.9i			0.179566i
$334$			11289.5i			0.101201i
$335$			52431.1i			0.467196i
$336$	−20787.2			−0.184127
$337$			189217.i			1.66610i	0.553199	+	0.833049i	$0.313407\pi$
$337$			189217.i			1.66610i	−0.553199	+	0.833049i	$0.686593\pi$
$338$			33012.6i			0.288966i
$339$		−	85482.2i		−	0.743834i
$340$	−52535.1			−0.454456
$341$	105840.			0.910207
$342$		−	5676.57i		−	0.0485326i
$343$	42194.2			0.358645
$344$	−69083.1			−0.583788
$345$			126283.i			1.06098i
$346$	51708.8			0.431929
$347$		−	150636.i		−	1.25104i	−0.780209	−	0.625519i	$-0.784887\pi$
$347$		−	150636.i		−	1.25104i	0.780209	−	0.625519i	$-0.215113\pi$
$348$	−16494.2			−0.136199
$349$		−	80454.6i		−	0.660541i	−0.943886	−	0.330271i	$-0.892860\pi$
$349$		−	80454.6i		−	0.660541i	0.943886	−	0.330271i	$-0.107140\pi$
$350$			52509.7i			0.428651i
$351$			16147.0i			0.131062i
$352$	215862.			1.74217
$353$		−	93087.5i		−	0.747037i	−0.927623	−	0.373518i	$-0.878151\pi$
$353$		−	93087.5i		−	0.747037i	0.927623	−	0.373518i	$-0.121849\pi$
$354$	−10803.6	−	37463.3i	−0.0862112	−	0.298951i
$355$	−280497.			−2.22573
$356$		−	129635.i		−	1.02287i
$357$	56989.2			0.447153
$358$	−104287.			−0.813700
$359$	−36181.0			−0.280732			−0.140366	−	0.990100i	$-0.544828\pi$
$359$	−36181.0			−0.280732			−0.140366	+	0.990100i	$0.544828\pi$
$360$			49254.0i			0.380046i
$361$	−120808.			−0.927005
$362$		−	20269.6i		−	0.154678i
$363$	139003.			1.05490
$364$		−	95820.3i		−	0.723194i
$365$		−	27125.1i		−	0.203604i
$366$	59817.2			0.446544
$367$		−	147497.i		−	1.09509i	−0.836775	−	0.547547i	$-0.815562\pi$
$367$		−	147497.i		−	1.09509i	0.836775	−	0.547547i	$-0.184438\pi$
$368$		−	42854.2i		−	0.316445i
$369$	14602.9			0.107248
$370$	−49183.1			−0.359263
$371$	−266208.			−1.93407
$372$		−	30690.1i		−	0.221775i
$373$	93006.2			0.668489			0.334244	−	0.942486i	$-0.391519\pi$
$373$	93006.2			0.668489			0.334244	+	0.942486i	$0.391519\pi$
$374$	−65592.0			−0.468930
$375$	47072.5			0.334738
$376$	−88765.9			−0.627871
$377$		−	32178.7i		−	0.226405i
$378$		−	22176.8i		−	0.155209i
$379$	−197981.			−1.37831			−0.689153	−	0.724615i	$-0.742017\pi$
$379$	−197981.			−1.37831			−0.689153	+	0.724615i	$0.742017\pi$
$380$	−34259.4			−0.237254
$381$	−77982.7			−0.537215
$382$	56678.9			0.388414
$383$	130011.			0.886307			0.443153	−	0.896446i	$-0.353860\pi$
$383$	130011.			0.886307			0.443153	+	0.896446i	$0.353860\pi$
$384$		−	72369.7i		−	0.490788i
$385$		−	461577.i		−	3.11403i
$386$		−	77504.4i		−	0.520178i
$387$			31634.2i			0.211220i
$388$			172044.i			1.14282i
$389$	−97356.2			−0.643375			−0.321688	−	0.946846i	$-0.604250\pi$
$389$	−97356.2			−0.643375			−0.321688	+	0.946846i	$0.604250\pi$
$390$	−39883.5			−0.262219
$391$			117487.i			0.768487i
$392$			175497.i			1.14208i
$393$			2709.74i			0.0175445i
$394$			34827.9i			0.224355i
$395$	244273.			1.56560
$396$		−	62366.1i		−	0.397703i
$397$			21590.6i			0.136988i	0.997652	+	0.0684942i	$0.0218195\pi$
$397$			21590.6i			0.136988i	−0.997652	+	0.0684942i	$0.978181\pi$
$398$		−	17949.1i		−	0.113312i
$399$	37164.0			0.233441
$400$	18122.3			0.113265
$401$			14919.3i			0.0927809i	0.998923	+	0.0463905i	$0.0147718\pi$
$401$			14919.3i			0.0927809i	−0.998923	+	0.0463905i	$0.985228\pi$
$402$	18981.9			0.117459
$403$	59873.6			0.368659
$404$			117169.i			0.717878i
$405$	22554.2			0.137504
$406$			44195.4i			0.268118i
$407$	150040.			0.905773
$408$			45823.2i			0.275274i
$409$		−	90196.9i		−	0.539194i	−0.962973	−	0.269597i	$-0.913110\pi$
$409$		−	90196.9i		−	0.539194i	0.962973	−	0.269597i	$-0.0868905\pi$
$410$			36069.7i			0.214573i
$411$	154493.			0.914589
$412$			155823.i			0.917989i
$413$	245269.	−	70730.5i	1.43795	−	0.414674i
$414$	45719.0			0.266745
$415$			109395.i			0.635184i
$416$	122113.			0.705627
$417$	29678.7			0.170676
$418$	−42774.1			−0.244810
$419$		−	141820.i		−	0.807808i	−0.914801	−	0.403904i	$-0.867653\pi$
$419$		−	141820.i		−	0.807808i	0.914801	−	0.403904i	$-0.132347\pi$
$420$	−133842.			−0.758742
$421$			328314.i			1.85236i	0.377084	+	0.926179i	$0.376927\pi$
$421$			328314.i			1.85236i	−0.377084	+	0.926179i	$0.623073\pi$
$422$	162628.			0.913208
$423$			40647.2i			0.227170i
$424$		−	214049.i		−	1.19064i
$425$	−49683.3			−0.275063
$426$			101550.i			0.559577i
$427$			391618.i			2.14787i
$428$	−106524.			−0.581516
$429$	121671.			0.661106
$430$	−78137.5			−0.422593
$431$			256724.i			1.38201i	0.722849	+	0.691006i	$0.242832\pi$
$431$			256724.i			1.38201i	−0.722849	+	0.691006i	$0.757168\pi$
$432$	−7653.74			−0.0410116
$433$	63696.1			0.339733			0.169866	−	0.985467i	$-0.445666\pi$
$433$	63696.1			0.339733			0.169866	+	0.985467i	$0.445666\pi$
$434$	−82232.5			−0.436580
$435$	−44947.4			−0.237534
$436$		−	177652.i		−	0.934537i
$437$			76616.2i			0.401197i
$438$	−9820.25			−0.0511887
$439$	301024.			1.56197			0.780985	−	0.624550i	$-0.214717\pi$
$439$	301024.			1.56197			0.780985	+	0.624550i	$0.214717\pi$
$440$	371139.			1.91704
$441$	80362.7			0.413216
$442$	−37105.4			−0.189930
$443$			214549.i			1.09325i	0.837378	+	0.546625i	$0.184088\pi$
$443$			214549.i			1.09325i	−0.837378	+	0.546625i	$0.815912\pi$
$444$		−	43506.8i		−	0.220694i
$445$		−	353260.i		−	1.78391i
$446$			174445.i			0.876976i
$447$			32648.2i			0.163397i
$448$	−103706.			−0.516713
$449$	20938.9			0.103863			0.0519315	−	0.998651i	$-0.483462\pi$
$449$	20938.9			0.103863			0.0519315	+	0.998651i	$0.483462\pi$
$450$			19333.8i			0.0954755i
$451$		−	110036.i		−	0.540981i
$452$			186775.i			0.914203i
$453$		−	206827.i		−	1.00788i
$454$	108789.			0.527804
$455$		−	261114.i		−	1.26127i
$456$			29882.4i			0.143710i
$457$		−	155521.i		−	0.744659i	−0.928101	−	0.372330i	$-0.878559\pi$
$457$		−	155521.i		−	0.744659i	0.928101	−	0.372330i	$-0.121441\pi$
$458$	190966.			0.910383
$459$	20983.1			0.0995967
$460$		−	275925.i		−	1.30399i
$461$	424691.			1.99835			0.999175	−	0.0406056i	$-0.0129287\pi$
$461$	424691.			1.99835			0.999175	+	0.0406056i	$0.0129287\pi$
$462$	−167107.			−0.782907
$463$		−	51304.1i		−	0.239326i	−0.992815	−	0.119663i	$-0.961819\pi$
$463$		−	51304.1i		−	0.239326i	0.992815	−	0.119663i	$-0.0381814\pi$
$464$	15252.9			0.0708461
$465$		−	83631.7i		−	0.386781i
$466$	46058.8			0.212100
$467$		−	18311.1i		−	0.0839614i	−0.999118	−	0.0419807i	$-0.986633\pi$
$467$		−	18311.1i		−	0.0839614i	0.999118	−	0.0419807i	$-0.0133668\pi$
$468$		−	35280.5i		−	0.161081i
$469$			124273.i			0.564976i
$470$	−100400.			−0.454504
$471$			15244.6i			0.0687184i
$472$	56872.2	+	197213.i	0.255279	+	0.885222i
$473$	238370.			1.06544
$474$		−	88435.5i		−	0.393613i
$475$	−32399.7			−0.143600
$476$	−124519.			−0.549570
$477$	−98016.4			−0.430786
$478$			59902.4i			0.262173i
$479$	−117396.			−0.511659			−0.255830	−	0.966722i	$-0.582349\pi$
$479$	−117396.			−0.511659			−0.255830	+	0.966722i	$0.582349\pi$
$480$		−	170568.i		−	0.740313i
$481$	84877.9			0.366863
$482$		−	246351.i		−	1.06038i
$483$			299318.i			1.28304i
$484$	−303716.			−1.29651
$485$			468828.i			1.99310i
$486$		−	8165.39i		−	0.0345704i
$487$	276423.			1.16551			0.582756	−	0.812647i	$-0.301974\pi$
$487$	276423.			1.16551			0.582756	+	0.812647i	$0.301974\pi$
$488$	−314888.			−1.32226
$489$	−233511.			−0.976540
$490$			198499.i			0.826733i
$491$	−95438.0			−0.395875			−0.197938	−	0.980215i	$-0.563424\pi$
$491$	−95438.0			−0.395875			−0.197938	+	0.980215i	$0.563424\pi$
$492$	−31906.9			−0.131812
$493$	−41816.6			−0.172050
$494$	−24197.3			−0.0991548
$495$		−	169950.i		−	0.693603i
$496$			28380.4i			0.115360i
$497$	−664838.			−2.69155
$498$	39604.6			0.159694
$499$	296293.			1.18993			0.594964	−	0.803752i	$-0.297166\pi$
$499$	296293.			1.18993			0.594964	+	0.803752i	$0.297166\pi$
$500$	−102852.			−0.411407
$501$	−27213.9			−0.108421
$502$			164234.i			0.651713i
$503$		−	38668.3i		−	0.152834i	−0.997076	−	0.0764169i	$-0.975652\pi$
$503$		−	38668.3i		−	0.152834i	0.997076	−	0.0764169i	$-0.0243480\pi$
$504$			116742.i			0.459587i
$505$			319291.i			1.25200i
$506$		−	344502.i		−	1.34552i
$507$	−79578.2			−0.309584
$508$	170389.			0.660260
$509$			382145.i			1.47500i	0.675347	+	0.737500i	$0.263994\pi$
$509$			382145.i			1.47500i	−0.675347	+	0.737500i	$0.736006\pi$
$510$			51829.0i			0.199266i
$511$		−	64292.3i		−	0.246217i
$512$			109349.i			0.417133i
$513$	13683.6			0.0519955
$514$		−	21411.3i		−	0.0810431i
$515$			424624.i			1.60100i
$516$		−	69119.6i		−	0.259598i
$517$	306285.			1.14590
$518$	−116574.			−0.434454
$519$			124646.i			0.462748i
$520$	209953.			0.776455
$521$	−193927.			−0.714437			−0.357218	−	0.934021i	$-0.616275\pi$
$521$	−193927.			−0.714437			−0.357218	+	0.934021i	$0.616275\pi$
$522$			16272.5i			0.0597192i
$523$	242685.			0.887238			0.443619	−	0.896216i	$-0.353694\pi$
$523$	242685.			0.887238			0.443619	+	0.896216i	$0.353694\pi$
$524$		−	5920.68i		−	0.0215630i
$525$	−126577.			−0.459235
$526$			75409.4i			0.272555i
$527$		−	77806.3i		−	0.280152i
$528$			57672.5i			0.206872i
$529$	−337224.			−1.20506
$530$		−	242104.i		−	0.861886i
$531$	90306.9	−	26042.6i	0.320282	−	0.0923625i
$532$	−81202.0			−0.286909
$533$		−	62247.5i		−	0.219113i
$534$	−127892.			−0.448499
$535$	−290283.			−1.01418
$536$	−99923.8			−0.347808
$537$		−	251388.i		−	0.871758i
$538$	133321.			0.460611
$539$		−	605550.i		−	2.08436i
$540$	−49280.0			−0.168999
$541$		−	87558.4i		−	0.299160i	−0.988750	−	0.149580i	$-0.952208\pi$
$541$		−	87558.4i		−	0.299160i	0.988750	−	0.149580i	$-0.0477922\pi$
$542$		−	197389.i		−	0.671931i
$543$	48860.8			0.165714
$544$		−	158687.i		−	0.536221i
$545$		−	484108.i		−	1.62986i
$546$	−94532.4			−0.317099
$547$	−169774.			−0.567411			−0.283705	−	0.958912i	$-0.591564\pi$
$547$	−169774.			−0.567411			−0.283705	+	0.958912i	$0.591564\pi$
$548$	−337562.			−1.12407
$549$			144192.i			0.478405i
$550$	145684.			0.481600
$551$	−27269.6			−0.0898205
$552$	−240672.			−0.789856
$553$	578980.			1.89327
$554$		−	191175.i		−	0.622890i
$555$		−	118558.i		−	0.384897i
$556$	−64847.0			−0.209768
$557$	−76223.9			−0.245686			−0.122843	−	0.992426i	$-0.539201\pi$
$557$	−76223.9			−0.245686			−0.122843	+	0.992426i	$0.539201\pi$
$558$	−30277.6			−0.0972418
$559$	134846.			0.431534
$560$	123769.			0.394673
$561$		−	158112.i		−	0.502389i
$562$		−	116900.i		−	0.370121i
$563$			589979.i			1.86131i	0.365892	+	0.930657i	$0.380764\pi$
$563$			589979.i			1.86131i	−0.365892	+	0.930657i	$0.619236\pi$
$564$		−	88812.7i		−	0.279201i
$565$			508970.i			1.59439i
$566$	42898.0			0.133907
$567$	53458.1			0.166283
$568$		−	534575.i		−	1.65696i
$569$		−	312707.i		−	0.965856i	−0.875660	−	0.482928i	$-0.839573\pi$
$569$		−	312707.i		−	0.965856i	0.875660	−	0.482928i	$-0.160427\pi$
$570$			33799.0i			0.104029i
$571$		−	614610.i		−	1.88507i	−0.334110	−	0.942534i	$-0.608436\pi$
$571$		−	614610.i		−	1.88507i	0.334110	−	0.942534i	$-0.391564\pi$
$572$	−265846.			−0.812528
$573$			136627.i			0.416128i
$574$			85492.9i			0.259481i
$575$		−	260946.i		−	0.789252i
$576$	−38184.2			−0.115090
$577$	−170463.			−0.512010			−0.256005	−	0.966675i	$-0.582406\pi$
$577$	−170463.			−0.512010			−0.256005	+	0.966675i	$0.582406\pi$
$578$		−	131819.i		−	0.394568i
$579$	186828.			0.557293
$580$	98208.5			0.291940
$581$			259288.i			0.768123i
$582$	169732.			0.501093
$583$			738574.i			2.17298i
$584$	51695.4			0.151575
$585$		−	96140.8i		−	0.280929i
$586$			191530.i			0.557754i
$587$			315184.i			0.914721i	0.889281	+	0.457360i	$0.151205\pi$
$587$			315184.i			0.914721i	−0.889281	+	0.457360i	$0.848795\pi$
$588$	−175590.			−0.507860
$589$		−	50739.4i		−	0.146256i
$590$	64326.1	+	223061.i	0.184792	+	0.640796i
$591$	−83954.2			−0.240363
$592$			40232.6i			0.114798i
$593$	−93311.9			−0.265355			−0.132678	−	0.991159i	$-0.542357\pi$
$593$	−93311.9			−0.265355			−0.132678	+	0.991159i	$0.542357\pi$
$594$	−61527.9			−0.174381
$595$	−339320.			−0.958464
$596$		−	71335.1i		−	0.200822i
$597$	43267.1			0.121397
$598$		−	194885.i		−	0.544974i
$599$	−70659.2			−0.196932			−0.0984658	−	0.995140i	$-0.531393\pi$
$599$	−70659.2			−0.196932			−0.0984658	+	0.995140i	$0.531393\pi$
$600$		−	101776.i		−	0.282712i
$601$		−	624282.i		−	1.72835i	−0.503190	−	0.864176i	$-0.667841\pi$
$601$		−	624282.i		−	1.72835i	0.503190	−	0.864176i	$-0.332159\pi$
$602$	−185202.			−0.511039
$603$			45756.6i			0.125840i
$604$			451909.i			1.23873i
$605$	−827639.			−2.26115
$606$	115594.			0.314769
$607$	72992.7			0.198108			0.0990540	−	0.995082i	$-0.468418\pi$
$607$	72992.7			0.198108			0.0990540	+	0.995082i	$0.468418\pi$
$608$		−	103484.i		−	0.279940i
$609$	−106535.			−0.287248
$610$	−356158.			−0.957158
$611$	173266.			0.464120
$612$	−45847.4			−0.122409
$613$			446331.i			1.18778i	0.804546	+	0.593890i	$0.202409\pi$
$613$			446331.i			1.18778i	−0.804546	+	0.593890i	$0.797591\pi$
$614$		−	12689.1i		−	0.0336585i
$615$	−86947.5			−0.229883
$616$	879678.			2.31826
$617$	−676732.			−1.77765			−0.888825	−	0.458246i	$-0.848478\pi$
$617$	−676732.			−1.77765			−0.888825	+	0.458246i	$0.848478\pi$
$618$	153729.			0.402512
$619$	217204.			0.566874			0.283437	−	0.958991i	$-0.408525\pi$
$619$	217204.			0.566874			0.283437	+	0.958991i	$0.408525\pi$
$620$			182732.i			0.475370i
$621$			110207.i			0.285777i
$622$		−	203683.i		−	0.526471i
$623$		−	837300.i		−	2.15727i
$624$			32625.4i			0.0837888i
$625$	−487894.			−1.24901
$626$	194554.			0.496467
$627$		−	103109.i		−	0.262277i
$628$		−	33308.8i		−	0.0844578i
$629$		−	110300.i		−	0.278787i
$630$			132043.i			0.332686i
$631$	74809.2			0.187887			0.0939434	−	0.995578i	$-0.470053\pi$
$631$	74809.2			0.187887			0.0939434	+	0.995578i	$0.470053\pi$
$632$			465539.i			1.16553i
$633$			392021.i			0.978366i
$634$			343917.i			0.855609i
$635$	464318.			1.15151
$636$	214162.			0.529454
$637$		−	342559.i		−	0.844223i
$638$	122617.			0.301237
$639$	−244790.			−0.599503
$640$			430897.i			1.05200i
$641$	−461489.			−1.12317			−0.561585	−	0.827419i	$-0.689808\pi$
$641$	−461489.			−1.12317			−0.561585	+	0.827419i	$0.689808\pi$
$642$			105093.i			0.254978i
$643$	−363629.			−0.879501			−0.439751	−	0.898120i	$-0.644933\pi$
$643$	−363629.			−0.879501			−0.439751	+	0.898120i	$0.644933\pi$
$644$		−	654000.i		−	1.57691i
$645$		−	188354.i		−	0.452746i
$646$			31444.7i			0.0753498i
$647$	641950.			1.53353			0.766766	−	0.641926i	$-0.221865\pi$
$647$	641950.			1.53353			0.766766	+	0.641926i	$0.221865\pi$
$648$			42984.0i			0.102366i
$649$	−196237.	−	680481.i	−0.465898	−	1.61557i
$650$	82413.6			0.195062
$651$		−	198225.i		−	0.467731i
$652$	510213.			1.20021
$653$	426122.			0.999327			0.499664	−	0.866220i	$-0.333457\pi$
$653$	426122.			0.999327			0.499664	+	0.866220i	$0.333457\pi$
$654$	−175264.			−0.409767
$655$		−	16134.1i		−	0.0376064i
$656$	29505.6			0.0685641
$657$		−	23672.1i		−	0.0548411i
$658$	−237969.			−0.549628
$659$		−	468334.i		−	1.07841i	−0.842174	−	0.539206i	$-0.818724\pi$
$659$		−	468334.i		−	1.07841i	0.842174	−	0.539206i	$-0.181276\pi$
$660$			371335.i			0.852468i
$661$	485472.			1.11112			0.555561	−	0.831476i	$-0.312504\pi$
$661$	485472.			1.11112			0.555561	+	0.831476i	$0.312504\pi$
$662$			44903.8i			0.102463i
$663$		−	89444.1i		−	0.203481i
$664$	−208485.			−0.472868
$665$	−221279.			−0.500376
$666$	−42922.1			−0.0967681
$667$		−	219629.i		−	0.493671i
$668$	59461.4			0.133255
$669$	−420506.			−0.939549
$670$	−113020.			−0.251772
$671$	1.08651e6			2.41319
$672$		−	404282.i		−	0.895254i
$673$			153856.i			0.339692i	0.985471	+	0.169846i	$0.0543270\pi$
$673$			153856.i			0.339692i	−0.985471	+	0.169846i	$0.945673\pi$
$674$	−407876.			−0.897859
$675$	−46604.9			−0.102288
$676$	173875.			0.380491
$677$	−624029.			−1.36153			−0.680766	−	0.732501i	$-0.738353\pi$
$677$	−624029.			−1.36153			−0.680766	+	0.732501i	$0.738353\pi$
$678$	184265.			0.400852
$679$			1.11122e6i			2.41024i
$680$		−	272837.i		−	0.590044i
$681$			262240.i			0.565463i
$682$			228148.i			0.490510i
$683$			717942.i			1.53903i	0.638627	+	0.769516i	$0.279503\pi$
$683$			717942.i			1.53903i	−0.638627	+	0.769516i	$0.720497\pi$
$684$	−29898.2			−0.0639047
$685$	−919871.			−1.96040
$686$			90953.8i			0.193274i
$687$			460330.i			0.975340i
$688$			63917.7i			0.135034i
$689$			417811.i			0.880120i
$690$	−272216.			−0.571762
$691$		−	647155.i		−	1.35535i	−0.735361	−	0.677676i	$-0.762987\pi$
$691$		−	647155.i		−	1.35535i	0.735361	−	0.677676i	$-0.237013\pi$
$692$		−	272347.i		−	0.568737i
$693$		−	402818.i		−	0.838769i
$694$	324711.			0.674184
$695$	−176711.			−0.365841
$696$		−	85661.3i		−	0.176834i
$697$	−80891.2			−0.166508
$698$	173428.			0.355965
$699$			111027.i			0.227234i
$700$	276565.			0.564419
$701$		−	394820.i		−	0.803457i	−0.915759	−	0.401729i	$-0.868409\pi$
$701$		−	394820.i		−	0.803457i	0.915759	−	0.401729i	$-0.131591\pi$
$702$	−34806.3			−0.0706292
$703$		−	71929.0i		−	0.145544i
$704$			287725.i			0.580541i
$705$		−	242018.i		−	0.486934i
$706$	200659.			0.402578
$707$			756787.i			1.51403i
$708$	−197317.	+	56902.2i	−0.393639	+	0.113517i
$709$	−404978.			−0.805637			−0.402818	−	0.915280i	$-0.631969\pi$
$709$	−404978.			−0.805637			−0.402818	+	0.915280i	$0.631969\pi$
$710$		−	604639.i		−	1.19944i
$711$	213177.			0.421698
$712$	673246.			1.32805
$713$	408654.			0.803853
$714$			122846.i			0.240970i
$715$	−724441.			−1.41707
$716$			549274.i			1.07143i
$717$	−144397.			−0.280880
$718$		−	77991.6i		−	0.151286i
$719$			54443.6i			0.105315i	0.998613	+	0.0526574i	$0.0167691\pi$
$719$			54443.6i			0.105315i	−0.998613	+	0.0526574i	$0.983231\pi$
$720$	45571.3			0.0879075
$721$			1.00645e6i			1.93607i
$722$		−	260414.i		−	0.499562i
$723$	593839.			1.13604
$724$	−106759.			−0.203670
$725$	92877.3			0.176699
$726$			299634.i			0.568484i
$727$	611738.			1.15743			0.578717	−	0.815528i	$-0.303553\pi$
$727$	611738.			1.15743			0.578717	+	0.815528i	$0.303553\pi$
$728$	497634.			0.938961
$729$	19683.0			0.0370370
$730$	58470.9			0.109722
$731$		−	175234.i		−	0.327931i
$732$		−	315054.i		−	0.587980i
$733$	523681.			0.974674			0.487337	−	0.873214i	$-0.337968\pi$
$733$	523681.			0.974674			0.487337	+	0.873214i	$0.337968\pi$
$734$	317944.			0.590146
$735$	−478489.			−0.885721
$736$	833455.			1.53860
$737$	344785.			0.634766
$738$			31478.0i			0.0577956i
$739$			304017.i			0.556684i	0.960482	+	0.278342i	$0.0897848\pi$
$739$			304017.i			0.556684i	−0.960482	+	0.278342i	$0.910215\pi$
$740$			259044.i			0.473054i
$741$		−	58328.7i		−	0.106230i
$742$		−	573837.i		−	1.04227i
$743$	678235.			1.22858			0.614289	−	0.789081i	$-0.289443\pi$
$743$	678235.			1.22858			0.614289	+	0.789081i	$0.289443\pi$
$744$	159386.			0.287942
$745$		−	194391.i		−	0.350238i
$746$			200484.i			0.360248i
$747$			95468.6i			0.171088i
$748$			345469.i			0.617457i
$749$	−688033.			−1.22644
$750$			101469.i			0.180390i
$751$		−	908208.i		−	1.61030i	−0.593074	−	0.805148i	$-0.702086\pi$
$751$		−	908208.i		−	1.61030i	0.593074	−	0.805148i	$-0.297914\pi$
$752$			82128.8i			0.145231i
$753$	−395894.			−0.698214
$754$	69364.4			0.122010
$755$			1.23147e6i			2.16038i
$756$	−116804.			−0.204369
$757$	−761493.			−1.32885			−0.664423	−	0.747357i	$-0.731322\pi$
$757$	−761493.			−1.32885			−0.664423	+	0.747357i	$0.731322\pi$
$758$		−	426768.i		−	0.742768i
$759$	830436.			1.44153
$760$		−	177923.i		−	0.308039i
$761$	631628.			1.09067			0.545334	−	0.838219i	$-0.316403\pi$
$761$	631628.			1.09067			0.545334	+	0.838219i	$0.316403\pi$
$762$		−	168099.i		−	0.289505i
$763$		−	1.14744e6i		−	1.97097i
$764$		−	298525.i		−	0.511438i
$765$	−124936.			−0.213484
$766$			280252.i			0.477630i
$767$	−111011.	−	384948.i	−0.188702	−	0.654352i
$768$	273577.			0.463827
$769$			361776.i			0.611769i	0.952069	+	0.305885i	$0.0989521\pi$
$769$			361776.i			0.611769i	−0.952069	+	0.305885i	$0.901048\pi$
$770$	994973.			1.67815
$771$	51612.7			0.0868256
$772$	−408211.			−0.684937
$773$			300146.i			0.502312i	0.967947	+	0.251156i	$0.0808108\pi$
$773$			300146.i			0.502312i	−0.967947	+	0.251156i	$0.919189\pi$
$774$	−68190.6			−0.113826
$775$			172813.i			0.287722i
$776$	−893498.			−1.48378
$777$		−	281007.i		−	0.465452i
$778$		−	209861.i		−	0.346714i
$779$	−52751.1			−0.0869273
$780$			210064.i			0.345273i
$781$			1.84454e6i			3.02403i
$782$	−253255.			−0.414137
$783$	−39225.6			−0.0639803
$784$	162375.			0.264172
$785$		−	90767.8i		−	0.147297i
$786$	−5841.10			−0.00945474
$787$	−174818.			−0.282253			−0.141126	−	0.989992i	$-0.545072\pi$
$787$	−174818.			−0.282253			−0.141126	+	0.989992i	$0.545072\pi$
$788$	183437.			0.295416
$789$	−181777.			−0.292002
$790$			526555.i			0.843703i
$791$			1.20637e6i			1.92809i
$792$	323893.			0.516358
$793$	614642.			0.977408
$794$	−46540.6			−0.0738229
$795$	583601.			0.923382
$796$	−94537.0			−0.149202
$797$		−	142726.i		−	0.224691i	−0.993669	−	0.112345i	$-0.964164\pi$
$797$		−	142726.i		−	0.224691i	0.993669	−	0.112345i	$-0.0358363\pi$
$798$			80110.7i			0.125801i
$799$		−	225160.i		−	0.352694i
$800$			352454.i			0.550710i
$801$		−	308289.i		−	0.480500i
$802$	−32159.9			−0.0499996
$803$	−178374.			−0.276631
$804$		−	99976.5i		−	0.154663i
$805$		−	1.78218e6i		−	2.75016i
$806$			129063.i			0.198670i
$807$			321376.i			0.493476i
$808$	−608508.			−0.932059
$809$		−	1.05340e6i		−	1.60952i	−0.593600	−	0.804760i	$-0.702294\pi$
$809$		−	1.05340e6i		−	1.60952i	0.593600	−	0.804760i	$-0.297706\pi$
$810$			48617.7i			0.0741010i
$811$			231482.i			0.351945i	0.984395	+	0.175972i	$0.0563070\pi$
$811$			231482.i			0.351945i	−0.984395	+	0.175972i	$0.943693\pi$
$812$	232775.			0.353040
$813$	475814.			0.719874
$814$			323427.i			0.488120i
$815$	1.39035e6			2.09319
$816$	42397.0			0.0636729
$817$		−	114274.i		−	0.171200i
$818$	194428.			0.290571
$819$		−	227874.i		−	0.339725i
$820$	189977.			0.282536
$821$		−	832114.i		−	1.23451i	−0.786761	−	0.617257i	$-0.788244\pi$
$821$		−	832114.i		−	1.23451i	0.786761	−	0.617257i	$-0.211756\pi$
$822$			333025.i			0.492871i
$823$		−	771617.i		−	1.13921i	−0.821920	−	0.569603i	$-0.807097\pi$
$823$		−	771617.i		−	1.13921i	0.821920	−	0.569603i	$-0.192903\pi$
$824$	−809254.			−1.19187
$825$			351177.i			0.515963i
$826$	152467.	+	528702.i	0.223468	+	0.774909i
$827$	−605496.			−0.885320			−0.442660	−	0.896690i	$-0.645965\pi$
$827$	−605496.			−0.885320			−0.442660	+	0.896690i	$0.645965\pi$
$828$		−	240799.i		−	0.351232i
$829$	230768.			0.335789			0.167895	−	0.985805i	$-0.446303\pi$
$829$	230768.			0.335789			0.167895	+	0.985805i	$0.446303\pi$
$830$	−235811.			−0.342300
$831$	460835.			0.667334
$832$			162766.i			0.235135i
$833$	−445159.			−0.641543
$834$			63975.4i			0.0919773i
$835$	162035.			0.232399
$836$			225289.i			0.322350i
$837$		−	72985.4i		−	0.104180i
$838$	305706.			0.435327
$839$		−	321881.i		−	0.457269i	−0.973512	−	0.228634i	$-0.926574\pi$
$839$		−	321881.i		−	0.457269i	0.973512	−	0.228634i	$-0.0734260\pi$
$840$		−	695098.i		−	0.985116i
$841$	−629110.			−0.889476
$842$	−707712.			−0.998234
$843$	281793.			0.396530
$844$		−	856551.i		−	1.20245i
$845$	473817.			0.663587
$846$	−87619.1			−0.122422
$847$	−1.96168e6			−2.73440
$848$	−198045.			−0.275405
$849$			103407.i			0.143462i
$850$		−	107097.i		−	0.148231i
$851$	579315.			0.799937
$852$	534857.			0.736815
$853$	−937941.			−1.28907			−0.644536	−	0.764574i	$-0.722949\pi$
$853$	−937941.			−1.28907			−0.644536	+	0.764574i	$0.722949\pi$
$854$	−844171.			−1.15748
$855$	−81473.7			−0.111451
$856$		−	553225.i		−	0.755013i
$857$			1.06976e6i			1.45655i	0.685285	+	0.728275i	$0.259678\pi$
$857$			1.06976e6i			1.45655i	−0.685285	+	0.728275i	$0.740322\pi$
$858$			262273.i			0.356270i
$859$		−	483593.i		−	0.655381i	−0.944785	−	0.327690i	$-0.893730\pi$
$859$		−	483593.i		−	0.655381i	0.944785	−	0.327690i	$-0.106270\pi$
$860$			411546.i			0.556444i
$861$	−206084.			−0.277996
$862$	−553394.			−0.744766
$863$		−	955910.i		−	1.28350i	−0.766914	−	0.641749i	$-0.778209\pi$
$863$		−	955910.i		−	1.28350i	0.766914	−	0.641749i	$-0.221791\pi$
$864$		−	148855.i		−	0.199404i
$865$		−	742158.i		−	0.991891i
$866$			137303.i			0.183082i
$867$	317754.			0.422720
$868$			433114.i			0.574861i
$869$		−	1.60633e6i		−	2.12714i
$870$		−	96888.5i		−	0.128007i
$871$	195045.			0.257098
$872$	922619.			1.21336
$873$			409146.i			0.536846i
$874$	−165154.			−0.216205
$875$	−664311.			−0.867672
$876$			51722.7i			0.0674020i
$877$	−593572.			−0.771745			−0.385873	−	0.922552i	$-0.626100\pi$
$877$	−593572.			−0.771745			−0.385873	+	0.922552i	$0.626100\pi$
$878$			648887.i			0.841744i
$879$	−461692.			−0.597550
$880$		−	343389.i		−	0.443426i
$881$			194695.i			0.250844i	0.992103	+	0.125422i	$0.0400285\pi$
$881$			194695.i			0.250844i	−0.992103	+	0.125422i	$0.959972\pi$
$882$			173230.i			0.222682i
$883$	−51348.5			−0.0658577			−0.0329288	−	0.999458i	$-0.510483\pi$
$883$	−51348.5			−0.0658577			−0.0329288	+	0.999458i	$0.510483\pi$
$884$			195432.i			0.250087i
$885$	−537698.	+	155061.i	−0.686517	+	0.197977i
$886$	−462481.			−0.589151
$887$		−	15142.4i		−	0.0192463i	−0.999954	−	0.00962314i	$-0.996937\pi$
$887$		−	15142.4i		−	0.0192463i	0.999954	−	0.00962314i	$-0.00306319\pi$
$888$	225949.			0.286539
$889$	1.10053e6			1.39251
$890$	761485.			0.961350
$891$		−	148315.i		−	0.186823i
$892$	918789.			1.15475
$893$		−	146833.i		−	0.184128i
$894$	−70376.3			−0.0880545
$895$			1.49679e6i			1.86860i
$896$			1.02132e6i			1.27217i
$897$	469778.			0.583859
$898$			45135.8i			0.0559717i
$899$			145450.i			0.179968i
$900$	101830.			0.125716
$901$	542950.			0.668821
$902$	237193.			0.291534
$903$		−	446438.i		−	0.547502i
$904$	−970002.			−1.18696
$905$	−290922.			−0.355206
$906$	445835.			0.543148
$907$	−183053.			−0.222517			−0.111258	−	0.993792i	$-0.535488\pi$
$907$	−183053.			−0.222517			−0.111258	+	0.993792i	$0.535488\pi$
$908$		−	572984.i		−	0.694978i
$909$			278645.i			0.337228i
$910$	562857.			0.679696
$911$	−1.37793e6			−1.66032			−0.830159	−	0.557527i	$-0.811750\pi$
$911$	−1.37793e6			−1.66032			−0.830159	+	0.557527i	$0.811750\pi$
$912$	27648.1			0.0332411
$913$	719376.			0.863007
$914$	335241.			0.401296
$915$		−	858534.i		−	1.02545i
$916$		−	1.00580e6i		−	1.19873i
$917$		−	38241.2i		−	0.0454771i
$918$			45231.2i			0.0536726i
$919$			276141.i			0.326964i	0.986546	+	0.163482i	$0.0522726\pi$
$919$			276141.i			0.326964i	−0.986546	+	0.163482i	$0.947727\pi$
$920$	1.43299e6			1.69304
$921$	30587.7			0.0360601
$922$			915463.i			1.07691i
$923$			1.04346e6i			1.22482i
$924$			880142.i			1.03088i
$925$			244983.i			0.286320i
$926$	110591.			0.128973
$927$			370570.i			0.431231i
$928$			296647.i			0.344465i
$929$		−	160509.i		−	0.185981i	−0.995667	−	0.0929904i	$-0.970357\pi$
$929$		−	160509.i		−	0.185981i	0.995667	−	0.0929904i	$-0.0296426\pi$
$930$	180276.			0.208436
$931$	−290299.			−0.334924
$932$		−	242589.i		−	0.279280i
$933$	490987.			0.564036
$934$	39471.3			0.0452467
$935$			941418.i			1.07686i
$936$	183226.			0.209140
$937$		−	90776.5i		−	0.103394i	−0.998663	−	0.0516969i	$-0.983537\pi$
$937$		−	90776.5i		−	0.103394i	0.998663	−	0.0516969i	$-0.0164630\pi$
$938$	−267882.			−0.304465
$939$			468979.i			0.531891i
$940$			528801.i			0.598462i
$941$		−	1.24928e6i		−	1.41085i	−0.708785	−	0.705424i	$-0.750757\pi$
$941$		−	1.24928e6i		−	1.41085i	0.708785	−	0.705424i	$-0.249243\pi$
$942$	−32861.1			−0.0370323
$943$		−	424856.i		−	0.477770i
$944$	182467.	−	52619.8i	0.204758	−	0.0590480i
$945$	−318296.			−0.356424
$946$			513830.i			0.574166i
$947$	620940.			0.692388			0.346194	−	0.938163i	$-0.387474\pi$
$947$	620940.			0.692388			0.346194	+	0.938163i	$0.387474\pi$
$948$	−465785.			−0.518285
$949$	−100906.			−0.112043
$950$		−	69840.7i		−	0.0773858i
$951$	−829026.			−0.916657
$952$		−	646680.i		−	0.713536i
$953$	−1.35262e6			−1.48932			−0.744661	−	0.667443i	$-0.767389\pi$
$953$	−1.35262e6			−1.48932			−0.744661	+	0.667443i	$0.767389\pi$
$954$		−	211284.i		−	0.232150i
$955$		−	813491.i		−	0.891962i
$956$	315503.			0.345213
$957$			295573.i			0.322731i
$958$		−	253058.i		−	0.275733i
$959$	−2.18029e6			−2.37070
$960$	227353.			0.246694
$961$	652888.			0.706956
$962$			182963.i			0.197702i
$963$	−253330.			−0.273171
$964$	−1.29752e6			−1.39624
$965$	−1.11239e6			−1.19455
$966$	−645210.			−0.691427
$967$		−	328428.i		−	0.351226i	−0.984459	−	0.175613i	$-0.943809\pi$
$967$		−	328428.i		−	0.351226i	0.984459	−	0.175613i	$-0.0561908\pi$
$968$		−	1.57732e6i		−	1.68333i
$969$	−75798.7			−0.0807261
$970$	−1.01060e6			−1.07408
$971$	1.82403e6			1.93461			0.967306	−	0.253614i	$-0.0816192\pi$
$971$	1.82403e6			1.93461			0.967306	+	0.253614i	$0.0816192\pi$
$972$	−43006.6			−0.0455201
$973$	−418841.			−0.442409
$974$			595857.i			0.628093i
$975$			198661.i			0.208980i
$976$			291343.i			0.305848i
$977$		−	260498.i		−	0.272907i	−0.990646	−	0.136454i	$-0.956430\pi$
$977$		−	260498.i		−	0.272907i	0.990646	−	0.136454i	$-0.0435705\pi$
$978$		−	503356.i		−	0.526256i
$979$	−2.32302e6			−2.42375
$980$	1.04548e6			1.08859
$981$		−	422481.i		−	0.439005i
$982$		−	205726.i		−	0.213337i
$983$			1.55845e6i			1.61282i	0.591360	+	0.806408i	$0.298591\pi$
$983$			1.55845e6i			1.61282i	−0.591360	+	0.806408i	$0.701409\pi$
$984$		−	165706.i		−	0.171138i
$985$	499873.			0.515213
$986$		−	90139.7i		−	0.0927176i
$987$		−	573634.i		−	0.588845i
$988$			127446.i			0.130561i
$989$	920362.			0.940949
$990$	366344.			0.373782
$991$			1.21576e6i			1.23794i	0.785416	+	0.618969i	$0.212449\pi$
$991$			1.21576e6i			1.23794i	−0.785416	+	0.618969i	$0.787551\pi$
$992$	−551959.			−0.560898
$993$	−108242.			−0.109774
$994$		−	1.43312e6i		−	1.45048i
$995$	−257617.			−0.260213
$996$		−	208595.i		−	0.210274i
$997$	994803.			1.00080			0.500399	−	0.865795i	$-0.333187\pi$
$997$	994803.			1.00080			0.500399	+	0.865795i	$0.333187\pi$
$998$			638689.i			0.641252i
$999$		−	103465.i		−	0.103673i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.5.c.a.58.24	yes	40
3.2	odd	2		531.5.c.d.235.17		40
59.58	odd	2	inner	177.5.c.a.58.17	✓	40
177.176	even	2		531.5.c.d.235.24		40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.5.c.a.58.17	✓	40	59.58	odd	2	inner
177.5.c.a.58.24	yes	40	1.1	even	1	trivial
531.5.c.d.235.17		40	3.2	odd	2
531.5.c.d.235.24		40	177.176	even	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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	177.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.15560i q^{2} -5.19615 q^{3} +11.3534 q^{4} +30.9385 q^{5} -11.2008i q^{6} +73.3307 q^{7} +58.9629i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q+2.15560i q^{2} -5.19615 q^{3} +11.3534 q^{4} +30.9385 q^{5} -11.2008i q^{6} +73.3307 q^{7} +58.9629i q^{8} +27.0000 q^{9} +66.6909i q^{10} -203.451i q^{11} -58.9940 q^{12} -115.092i q^{13} +158.071i q^{14} -160.761 q^{15} +54.5542 q^{16} -149.563 q^{17} +58.2011i q^{18} -97.5337 q^{19} +351.257 q^{20} -381.038 q^{21} +438.557 q^{22} -785.535i q^{23} -306.380i q^{24} +332.189 q^{25} +248.092 q^{26} -140.296 q^{27} +832.553 q^{28} +279.591 q^{29} -346.536i q^{30} +520.224i q^{31} +1061.00i q^{32} +1057.16i q^{33} -322.398i q^{34} +2268.74 q^{35} +306.542 q^{36} +737.478i q^{37} -210.243i q^{38} +598.036i q^{39} +1824.22i q^{40} +540.849 q^{41} -821.364i q^{42} +1171.64i q^{43} -2309.86i q^{44} +835.339 q^{45} +1693.30 q^{46} +1505.45i q^{47} -283.472 q^{48} +2976.40 q^{49} +716.067i q^{50} +777.153 q^{51} -1306.69i q^{52} -3630.24 q^{53} -302.422i q^{54} -6294.45i q^{55} +4323.79i q^{56} +506.800 q^{57} +602.686i q^{58} +(3344.70 - 964.542i) q^{59} -1825.19 q^{60} +5340.44i q^{61} -1121.39 q^{62} +1979.93 q^{63} -1414.23 q^{64} -3560.77i q^{65} -2278.81 q^{66} +1694.69i q^{67} -1698.05 q^{68} +4081.76i q^{69} +4890.49i q^{70} -9066.29 q^{71} +1592.00i q^{72} -876.744i q^{73} -1589.71 q^{74} -1726.11 q^{75} -1107.34 q^{76} -14919.2i q^{77} -1289.12 q^{78} +7895.46 q^{79} +1687.82 q^{80} +729.000 q^{81} +1165.85i q^{82} +3535.87i q^{83} -4326.07 q^{84} -4627.26 q^{85} -2525.58 q^{86} -1452.80 q^{87} +11996.0 q^{88} -11418.1i q^{89} +1800.65i q^{90} -8439.78i q^{91} -8918.49i q^{92} -2703.16i q^{93} -3245.15 q^{94} -3017.55 q^{95} -5513.14i q^{96} +15153.6i q^{97} +6415.91i q^{98} -5493.16i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9}+O(q^{10}) \) 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9	\( 40 q - 320 q^{4} + 80 q^{7} + 1080 q^{9} + 360 q^{12} + 144 q^{15} + 3944 q^{16} - 528 q^{17} + 444 q^{19} + 444 q^{20} + 1304 q^{22} + 4880 q^{25} - 1452 q^{26} - 1160 q^{28} - 996 q^{29} + 10320 q^{35} - 8640 q^{36} - 5196 q^{41} - 10476 q^{46} + 576 q^{48} + 5104 q^{49} + 936 q^{51} - 2184 q^{53} - 2520 q^{57} - 11736 q^{59} - 11448 q^{60} + 15240 q^{62} + 2160 q^{63} - 81012 q^{64} + 17352 q^{66} + 29568 q^{68} - 5964 q^{71} + 14376 q^{74} - 2736 q^{75} + 3480 q^{76} + 37692 q^{78} + 19020 q^{79} + 33096 q^{80} + 29160 q^{81} + 25128 q^{84} + 20220 q^{85} - 65880 q^{86} + 1512 q^{87} - 14932 q^{88} - 17864 q^{94} + 11004 q^{95}+O(q^{100}) \) 40 * q - 320 * q^4 + 80 * q^7 + 1080 * q^9 + 360 * q^12 + 144 * q^15 + 3944 * q^16 - 528 * q^17 + 444 * q^19 + 444 * q^20 + 1304 * q^22 + 4880 * q^25 - 1452 * q^26 - 1160 * q^28 - 996 * q^29 + 10320 * q^35 - 8640 * q^36 - 5196 * q^41 - 10476 * q^46 + 576 * q^48 + 5104 * q^49 + 936 * q^51 - 2184 * q^53 - 2520 * q^57 - 11736 * q^59 - 11448 * q^60 + 15240 * q^62 + 2160 * q^63 - 81012 * q^64 + 17352 * q^66 + 29568 * q^68 - 5964 * q^71 + 14376 * q^74 - 2736 * q^75 + 3480 * q^76 + 37692 * q^78 + 19020 * q^79 + 33096 * q^80 + 29160 * q^81 + 25128 * q^84 + 20220 * q^85 - 65880 * q^86 + 1512 * q^87 - 14932 * q^88 - 17864 * q^94 + 11004 * q^95

Introduction

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