LMFDB - Embedded newform 810.4.e.c.271.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,4,Mod(271,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("810.271");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	810.e (of order $3$, degree $2$, not 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:	$47.7915471046$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			271.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	810.271
Dual form			810.4.e.c.541.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(2.00000 + 3.46410i) q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(2.00000 + 3.46410i) q^{7} +8.00000 q^{8} +10.0000 q^{10} +(-6.00000 - 10.3923i) q^{11} +(29.0000 - 50.2295i) q^{13} +(4.00000 - 6.92820i) q^{14} +(-8.00000 - 13.8564i) q^{16} +66.0000 q^{17} -100.000 q^{19} +(-10.0000 - 17.3205i) q^{20} +(-12.0000 + 20.7846i) q^{22} +(-66.0000 + 114.315i) q^{23} +(-12.5000 - 21.6506i) q^{25} -116.000 q^{26} -16.0000 q^{28} +(45.0000 + 77.9423i) q^{29} +(-76.0000 + 131.636i) q^{31} +(-16.0000 + 27.7128i) q^{32} +(-66.0000 - 114.315i) q^{34} -20.0000 q^{35} -34.0000 q^{37} +(100.000 + 173.205i) q^{38} +(-20.0000 + 34.6410i) q^{40} +(219.000 - 379.319i) q^{41} +(-16.0000 - 27.7128i) q^{43} +48.0000 q^{44} +264.000 q^{46} +(102.000 + 176.669i) q^{47} +(163.500 - 283.190i) q^{49} +(-25.0000 + 43.3013i) q^{50} +(116.000 + 200.918i) q^{52} +222.000 q^{53} +60.0000 q^{55} +(16.0000 + 27.7128i) q^{56} +(90.0000 - 155.885i) q^{58} +(-210.000 + 363.731i) q^{59} +(-451.000 - 781.155i) q^{61} +304.000 q^{62} +64.0000 q^{64} +(145.000 + 251.147i) q^{65} +(512.000 - 886.810i) q^{67} +(-132.000 + 228.631i) q^{68} +(20.0000 + 34.6410i) q^{70} +432.000 q^{71} +362.000 q^{73} +(34.0000 + 58.8897i) q^{74} +(200.000 - 346.410i) q^{76} +(24.0000 - 41.5692i) q^{77} +(80.0000 + 138.564i) q^{79} +80.0000 q^{80} -876.000 q^{82} +(-36.0000 - 62.3538i) q^{83} +(-165.000 + 285.788i) q^{85} +(-32.0000 + 55.4256i) q^{86} +(-48.0000 - 83.1384i) q^{88} +810.000 q^{89} +232.000 q^{91} +(-264.000 - 457.261i) q^{92} +(204.000 - 353.338i) q^{94} +(250.000 - 433.013i) q^{95} +(-553.000 - 957.824i) q^{97} -654.000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 4 q^{4} - 5 q^{5} + 4 q^{7} + 16 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 - 4 * q^4 - 5 * q^5 + 4 * q^7 + 16 * q^8	$ 2 q - 2 q^{2} - 4 q^{4} - 5 q^{5} + 4 q^{7} + 16 q^{8} + 20 q^{10} - 12 q^{11} + 58 q^{13} + 8 q^{14} - 16 q^{16} + 132 q^{17} - 200 q^{19} - 20 q^{20} - 24 q^{22} - 132 q^{23} - 25 q^{25} - 232 q^{26} - 32 q^{28} + 90 q^{29} - 152 q^{31} - 32 q^{32} - 132 q^{34} - 40 q^{35} - 68 q^{37} + 200 q^{38} - 40 q^{40} + 438 q^{41} - 32 q^{43} + 96 q^{44} + 528 q^{46} + 204 q^{47} + 327 q^{49} - 50 q^{50} + 232 q^{52} + 444 q^{53} + 120 q^{55} + 32 q^{56} + 180 q^{58} - 420 q^{59} - 902 q^{61} + 608 q^{62} + 128 q^{64} + 290 q^{65} + 1024 q^{67} - 264 q^{68} + 40 q^{70} + 864 q^{71} + 724 q^{73} + 68 q^{74} + 400 q^{76} + 48 q^{77} + 160 q^{79} + 160 q^{80} - 1752 q^{82} - 72 q^{83} - 330 q^{85} - 64 q^{86} - 96 q^{88} + 1620 q^{89} + 464 q^{91} - 528 q^{92} + 408 q^{94} + 500 q^{95} - 1106 q^{97} - 1308 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 - 4 * q^4 - 5 * q^5 + 4 * q^7 + 16 * q^8 + 20 * q^10 - 12 * q^11 + 58 * q^13 + 8 * q^14 - 16 * q^16 + 132 * q^17 - 200 * q^19 - 20 * q^20 - 24 * q^22 - 132 * q^23 - 25 * q^25 - 232 * q^26 - 32 * q^28 + 90 * q^29 - 152 * q^31 - 32 * q^32 - 132 * q^34 - 40 * q^35 - 68 * q^37 + 200 * q^38 - 40 * q^40 + 438 * q^41 - 32 * q^43 + 96 * q^44 + 528 * q^46 + 204 * q^47 + 327 * q^49 - 50 * q^50 + 232 * q^52 + 444 * q^53 + 120 * q^55 + 32 * q^56 + 180 * q^58 - 420 * q^59 - 902 * q^61 + 608 * q^62 + 128 * q^64 + 290 * q^65 + 1024 * q^67 - 264 * q^68 + 40 * q^70 + 864 * q^71 + 724 * q^73 + 68 * q^74 + 400 * q^76 + 48 * q^77 + 160 * q^79 + 160 * q^80 - 1752 * q^82 - 72 * q^83 - 330 * q^85 - 64 * q^86 - 96 * q^88 + 1620 * q^89 + 464 * q^91 - 528 * q^92 + 408 * q^94 + 500 * q^95 - 1106 * q^97 - 1308 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	−1.00000	−	1.73205i	−0.353553	−	0.612372i
$2$	−1.00000	−	1.73205i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	+	3.46410i	−0.250000	+	0.433013i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$5$	−2.50000	+	4.33013i	−0.223607	+	0.387298i
$6$		0			0
$7$	2.00000	+	3.46410i	0.107990	+	0.187044i	0.914956	−	0.403554i	$-0.132225\pi$
$7$	2.00000	+	3.46410i	0.107990	+	0.187044i	−0.806966	+	0.590598i	$0.798892\pi$
$8$	8.00000			0.353553
$9$		0			0
$10$	10.0000			0.316228
$11$	−6.00000	−	10.3923i	−0.164461	−	0.284854i	0.772003	−	0.635619i	$-0.219255\pi$
$11$	−6.00000	−	10.3923i	−0.164461	−	0.284854i	−0.936464	+	0.350765i	$0.885922\pi$
$12$		0			0
$13$	29.0000	−	50.2295i	0.618704	−	1.07163i	−0.371018	−	0.928626i	$-0.620991\pi$
$13$	29.0000	−	50.2295i	0.618704	−	1.07163i	0.989722	−	0.143001i	$-0.0456753\pi$
$14$	4.00000	−	6.92820i	0.0763604	−	0.132260i
$15$		0			0
$16$	−8.00000	−	13.8564i	−0.125000	−	0.216506i
$17$	66.0000			0.941609			0.470804	−	0.882238i	$-0.343964\pi$
$17$	66.0000			0.941609			0.470804	+	0.882238i	$0.343964\pi$
$18$		0			0
$19$	−100.000			−1.20745			−0.603726	−	0.797192i	$-0.706318\pi$
$19$	−100.000			−1.20745			−0.603726	+	0.797192i	$0.706318\pi$
$20$	−10.0000	−	17.3205i	−0.111803	−	0.193649i
$21$		0			0
$22$	−12.0000	+	20.7846i	−0.116291	+	0.201422i
$23$	−66.0000	+	114.315i	−0.598346	+	1.03637i	0.394720	+	0.918802i	$0.370842\pi$
$23$	−66.0000	+	114.315i	−0.598346	+	1.03637i	−0.993065	+	0.117564i	$0.962492\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$	−116.000			−0.874980
$27$		0			0
$28$	−16.0000			−0.107990
$29$	45.0000	+	77.9423i	0.288148	+	0.499087i	0.973368	−	0.229250i	$-0.0736272\pi$
$29$	45.0000	+	77.9423i	0.288148	+	0.499087i	−0.685220	+	0.728336i	$0.740294\pi$
$30$		0			0
$31$	−76.0000	+	131.636i	−0.440323	+	0.762661i	−0.997713	−	0.0675892i	$-0.978469\pi$
$31$	−76.0000	+	131.636i	−0.440323	+	0.762661i	0.557391	+	0.830250i	$0.311803\pi$
$32$	−16.0000	+	27.7128i	−0.0883883	+	0.153093i
$33$		0			0
$34$	−66.0000	−	114.315i	−0.332909	−	0.576615i
$35$	−20.0000			−0.0965891
$36$		0			0
$37$	−34.0000			−0.151069			−0.0755347	−	0.997143i	$-0.524066\pi$
$37$	−34.0000			−0.151069			−0.0755347	+	0.997143i	$0.524066\pi$
$38$	100.000	+	173.205i	0.426898	+	0.739410i
$39$		0			0
$40$	−20.0000	+	34.6410i	−0.0790569	+	0.136931i
$41$	219.000	−	379.319i	0.834196	−	1.44487i	−0.0604866	−	0.998169i	$-0.519265\pi$
$41$	219.000	−	379.319i	0.834196	−	1.44487i	0.894683	−	0.446702i	$-0.147401\pi$
$42$		0			0
$43$	−16.0000	−	27.7128i	−0.0567437	−	0.0982829i	0.836258	−	0.548336i	$-0.184738\pi$
$43$	−16.0000	−	27.7128i	−0.0567437	−	0.0982829i	−0.893002	+	0.450053i	$0.851405\pi$
$44$	48.0000			0.164461
$45$		0			0
$46$	264.000			0.846189
$47$	102.000	+	176.669i	0.316558	+	0.548295i	0.979767	−	0.200139i	$-0.0641395\pi$
$47$	102.000	+	176.669i	0.316558	+	0.548295i	−0.663209	+	0.748434i	$0.730806\pi$
$48$		0			0
$49$	163.500	−	283.190i	0.476676	−	0.825628i
$50$	−25.0000	+	43.3013i	−0.0707107	+	0.122474i
$51$		0			0
$52$	116.000	+	200.918i	0.309352	+	0.535813i
$53$	222.000			0.575359			0.287680	−	0.957727i	$-0.407116\pi$
$53$	222.000			0.575359			0.287680	+	0.957727i	$0.407116\pi$
$54$		0			0
$55$	60.0000			0.147098
$56$	16.0000	+	27.7128i	0.0381802	+	0.0661300i
$57$		0			0
$58$	90.0000	−	155.885i	0.203751	−	0.352908i
$59$	−210.000	+	363.731i	−0.463384	+	0.802605i	−0.999127	−	0.0417762i	$-0.986698\pi$
$59$	−210.000	+	363.731i	−0.463384	+	0.802605i	0.535743	+	0.844381i	$0.320032\pi$
$60$		0			0
$61$	−451.000	−	781.155i	−0.946633	−	1.63962i	−0.752447	−	0.658652i	$-0.771127\pi$
$61$	−451.000	−	781.155i	−0.946633	−	1.63962i	−0.194186	−	0.980965i	$-0.562206\pi$
$62$	304.000			0.622710
$63$		0			0
$64$	64.0000			0.125000
$65$	145.000	+	251.147i	0.276693	+	0.479246i
$66$		0			0
$67$	512.000	−	886.810i	0.933593	−	1.61703i	0.156470	−	0.987683i	$-0.449989\pi$
$67$	512.000	−	886.810i	0.933593	−	1.61703i	0.777123	−	0.629348i	$-0.216678\pi$
$68$	−132.000	+	228.631i	−0.235402	+	0.407729i
$69$		0			0
$70$	20.0000	+	34.6410i	0.0341494	+	0.0591485i
$71$	432.000			0.722098			0.361049	−	0.932547i	$-0.382419\pi$
$71$	432.000			0.722098			0.361049	+	0.932547i	$0.382419\pi$
$72$		0			0
$73$	362.000			0.580396			0.290198	−	0.956967i	$-0.406279\pi$
$73$	362.000			0.580396			0.290198	+	0.956967i	$0.406279\pi$
$74$	34.0000	+	58.8897i	0.0534111	+	0.0925107i
$75$		0			0
$76$	200.000	−	346.410i	0.301863	−	0.522842i
$77$	24.0000	−	41.5692i	0.0355202	−	0.0615228i
$78$		0			0
$79$	80.0000	+	138.564i	0.113933	+	0.197338i	0.917353	−	0.398075i	$-0.130322\pi$
$79$	80.0000	+	138.564i	0.113933	+	0.197338i	−0.803420	+	0.595413i	$0.796988\pi$
$80$	80.0000			0.111803
$81$		0			0
$82$	−876.000			−1.17973
$83$	−36.0000	−	62.3538i	−0.0476086	−	0.0824605i	0.841239	−	0.540663i	$-0.181827\pi$
$83$	−36.0000	−	62.3538i	−0.0476086	−	0.0824605i	−0.888848	+	0.458203i	$0.848493\pi$
$84$		0			0
$85$	−165.000	+	285.788i	−0.210550	+	0.364684i
$86$	−32.0000	+	55.4256i	−0.0401238	+	0.0694965i
$87$		0			0
$88$	−48.0000	−	83.1384i	−0.0581456	−	0.100711i
$89$	810.000			0.964717			0.482359	−	0.875974i	$-0.339780\pi$
$89$	810.000			0.964717			0.482359	+	0.875974i	$0.339780\pi$
$90$		0			0
$91$	232.000			0.267255
$92$	−264.000	−	457.261i	−0.299173	−	0.518183i
$93$		0			0
$94$	204.000	−	353.338i	0.223840	−	0.387703i
$95$	250.000	−	433.013i	0.269994	−	0.467644i
$96$		0			0
$97$	−553.000	−	957.824i	−0.578852	−	1.00260i	−0.995611	−	0.0935842i	$-0.970168\pi$
$97$	−553.000	−	957.824i	−0.578852	−	1.00260i	0.416759	−	0.909017i	$-0.363166\pi$
$98$	−654.000			−0.674122
$99$		0			0
$100$	100.000			0.100000
$101$	129.000	+	223.435i	0.127089	+	0.220124i	0.922548	−	0.385883i	$-0.126103\pi$
$101$	129.000	+	223.435i	0.127089	+	0.220124i	−0.795459	+	0.606008i	$0.792770\pi$
$102$		0			0
$103$	494.000	−	855.633i	0.472575	−	0.818525i	−0.526932	−	0.849907i	$-0.676658\pi$
$103$	494.000	−	855.633i	0.472575	−	0.818525i	0.999507	+	0.0313828i	$0.00999110\pi$
$104$	232.000	−	401.836i	0.218745	−	0.378877i
$105$		0			0
$106$	−222.000	−	384.515i	−0.203420	−	0.352334i
$107$	−24.0000			−0.0216838			−0.0108419	−	0.999941i	$-0.503451\pi$
$107$	−24.0000			−0.0216838			−0.0108419	+	0.999941i	$0.503451\pi$
$108$		0			0
$109$	950.000			0.834803			0.417401	−	0.908722i	$-0.362941\pi$
$109$	950.000			0.834803			0.417401	+	0.908722i	$0.362941\pi$
$110$	−60.0000	−	103.923i	−0.0520071	−	0.0900789i
$111$		0			0
$112$	32.0000	−	55.4256i	0.0269975	−	0.0467610i
$113$	519.000	−	898.934i	0.432066	−	0.748360i	−0.564985	−	0.825101i	$-0.691118\pi$
$113$	519.000	−	898.934i	0.432066	−	0.748360i	0.997051	+	0.0767413i	$0.0244516\pi$
$114$		0			0
$115$	−330.000	−	571.577i	−0.267588	−	0.463477i
$116$	−360.000			−0.288148
$117$		0			0
$118$	840.000			0.655324
$119$	132.000	+	228.631i	0.101684	+	0.176122i
$120$		0			0
$121$	593.500	−	1027.97i	0.445905	−	0.772331i
$122$	−902.000	+	1562.31i	−0.669371	+	1.15938i
$123$		0			0
$124$	−304.000	−	526.543i	−0.220161	−	0.381331i
$125$	125.000			0.0894427
$126$		0			0
$127$	−124.000			−0.0866395			−0.0433198	−	0.999061i	$-0.513793\pi$
$127$	−124.000			−0.0866395			−0.0433198	+	0.999061i	$0.513793\pi$
$128$	−64.0000	−	110.851i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	290.000	−	502.295i	0.195651	−	0.338878i
$131$	−66.0000	+	114.315i	−0.0440187	+	0.0762426i	−0.887195	−	0.461394i	$-0.847349\pi$
$131$	−66.0000	+	114.315i	−0.0440187	+	0.0762426i	0.843177	+	0.537637i	$0.180683\pi$
$132$		0			0
$133$	−200.000	−	346.410i	−0.130392	−	0.225846i
$134$	−2048.00			−1.32030
$135$		0			0
$136$	528.000			0.332909
$137$	627.000	+	1086.00i	0.391009	+	0.677247i	0.992583	−	0.121570i	$-0.0387928\pi$
$137$	627.000	+	1086.00i	0.391009	+	0.677247i	−0.601574	+	0.798817i	$0.705459\pi$
$138$		0			0
$139$	1430.00	−	2476.83i	0.872597	−	1.51138i	0.0132968	−	0.999912i	$-0.495767\pi$
$139$	1430.00	−	2476.83i	0.872597	−	1.51138i	0.859300	−	0.511471i	$-0.170899\pi$
$140$	40.0000	−	69.2820i	0.0241473	−	0.0418243i
$141$		0			0
$142$	−432.000	−	748.246i	−0.255300	−	0.442193i
$143$	−696.000			−0.407010
$144$		0			0
$145$	−450.000			−0.257727
$146$	−362.000	−	627.002i	−0.205201	−	0.355418i
$147$		0			0
$148$	68.0000	−	117.779i	0.0377673	−	0.0654149i
$149$	−375.000	+	649.519i	−0.206183	+	0.357119i	−0.950509	−	0.310697i	$-0.899437\pi$
$149$	−375.000	+	649.519i	−0.206183	+	0.357119i	0.744326	+	0.667816i	$0.232771\pi$
$150$		0			0
$151$	224.000	+	387.979i	0.120721	+	0.209095i	0.920052	−	0.391796i	$-0.128146\pi$
$151$	224.000	+	387.979i	0.120721	+	0.209095i	−0.799331	+	0.600891i	$0.794813\pi$
$152$	−800.000			−0.426898
$153$		0			0
$154$	−96.0000			−0.0502331
$155$	−380.000	−	658.179i	−0.196918	−	0.341072i
$156$		0			0
$157$	−1123.00	+	1945.09i	−0.570861	+	0.988760i	0.425617	+	0.904903i	$0.360057\pi$
$157$	−1123.00	+	1945.09i	−0.570861	+	0.988760i	−0.996478	+	0.0838566i	$0.973276\pi$
$158$	160.000	−	277.128i	0.0805628	−	0.139539i
$159$		0			0
$160$	−80.0000	−	138.564i	−0.0395285	−	0.0684653i
$161$	−528.000			−0.258461
$162$		0			0
$163$	−568.000			−0.272940			−0.136470	−	0.990644i	$-0.543576\pi$
$163$	−568.000			−0.272940			−0.136470	+	0.990644i	$0.543576\pi$
$164$	876.000	+	1517.28i	0.417098	+	0.722435i
$165$		0			0
$166$	−72.0000	+	124.708i	−0.0336644	+	0.0583084i
$167$	762.000	−	1319.82i	0.353086	−	0.611563i	−0.633703	−	0.773577i	$-0.718466\pi$
$167$	762.000	−	1319.82i	0.353086	−	0.611563i	0.986788	+	0.162014i	$0.0517990\pi$
$168$		0			0
$169$	−583.500	−	1010.65i	−0.265589	−	0.460014i
$170$	660.000			0.297763
$171$		0			0
$172$	128.000			0.0567437
$173$	−1851.00	−	3206.03i	−0.813462	−	1.40896i	−0.910427	−	0.413670i	$-0.864247\pi$
$173$	−1851.00	−	3206.03i	−0.813462	−	1.40896i	0.0969650	−	0.995288i	$-0.469086\pi$
$174$		0			0
$175$	50.0000	−	86.6025i	0.0215980	−	0.0374088i
$176$	−96.0000	+	166.277i	−0.0411152	+	0.0712136i
$177$		0			0
$178$	−810.000	−	1402.96i	−0.341079	−	0.590766i
$179$	3180.00			1.32785			0.663923	−	0.747801i	$-0.268890\pi$
$179$	3180.00			1.32785			0.663923	+	0.747801i	$0.268890\pi$
$180$		0			0
$181$	−2098.00			−0.861564			−0.430782	−	0.902456i	$-0.641762\pi$
$181$	−2098.00			−0.861564			−0.430782	+	0.902456i	$0.641762\pi$
$182$	−232.000	−	401.836i	−0.0944889	−	0.163660i
$183$		0			0
$184$	−528.000	+	914.523i	−0.211547	+	0.366410i
$185$	85.0000	−	147.224i	0.0337801	−	0.0585089i
$186$		0			0
$187$	−396.000	−	685.892i	−0.154858	−	0.268221i
$188$	−816.000			−0.316558
$189$		0			0
$190$	−1000.00			−0.381830
$191$	−2196.00	−	3803.58i	−0.831921	−	1.44093i	−0.896513	−	0.443018i	$-0.853908\pi$
$191$	−2196.00	−	3803.58i	−0.831921	−	1.44093i	0.0645912	−	0.997912i	$-0.479426\pi$
$192$		0			0
$193$	1079.00	−	1868.88i	0.402425	−	0.697021i	−0.591593	−	0.806237i	$-0.701501\pi$
$193$	1079.00	−	1868.88i	0.402425	−	0.697021i	0.994018	+	0.109216i	$0.0348339\pi$
$194$	−1106.00	+	1915.65i	−0.409310	+	0.708946i
$195$		0			0
$196$	654.000	+	1132.76i	0.238338	+	0.412814i
$197$	−1074.00			−0.388423			−0.194212	−	0.980960i	$-0.562215\pi$
$197$	−1074.00			−0.388423			−0.194212	+	0.980960i	$0.562215\pi$
$198$		0			0
$199$	2840.00			1.01167			0.505835	−	0.862630i	$-0.331185\pi$
$199$	2840.00			1.01167			0.505835	+	0.862630i	$0.331185\pi$
$200$	−100.000	−	173.205i	−0.0353553	−	0.0612372i
$201$		0			0
$202$	258.000	−	446.869i	0.0898654	−	0.155651i
$203$	−180.000	+	311.769i	−0.0622341	+	0.107793i
$204$		0			0
$205$	1095.00	+	1896.60i	0.373064	+	0.646166i
$206$	−1976.00			−0.668323
$207$		0			0
$208$	−928.000			−0.309352
$209$	600.000	+	1039.23i	0.198578	+	0.343948i
$210$		0			0
$211$	1334.00	−	2310.56i	0.435243	−	0.753864i	−0.562072	−	0.827088i	$-0.689996\pi$
$211$	1334.00	−	2310.56i	0.435243	−	0.753864i	0.997315	+	0.0732246i	$0.0233290\pi$
$212$	−444.000	+	769.031i	−0.143840	+	0.249138i
$213$		0			0
$214$	24.0000	+	41.5692i	0.00766638	+	0.0132786i
$215$	160.000			0.0507531
$216$		0			0
$217$	−608.000			−0.190202
$218$	−950.000	−	1645.45i	−0.295147	−	0.511210i
$219$		0			0
$220$	−120.000	+	207.846i	−0.0367745	+	0.0636954i
$221$	1914.00	−	3315.15i	0.582577	−	1.00905i
$222$		0			0
$223$	−886.000	−	1534.60i	−0.266058	−	0.460826i	0.701782	−	0.712392i	$-0.252388\pi$
$223$	−886.000	−	1534.60i	−0.266058	−	0.460826i	−0.967840	+	0.251565i	$0.919055\pi$
$224$	−128.000			−0.0381802
$225$		0			0
$226$	−2076.00			−0.611033
$227$	1392.00	+	2411.01i	0.407006	+	0.704954i	0.994553	−	0.104236i	$-0.0332396\pi$
$227$	1392.00	+	2411.01i	0.407006	+	0.704954i	−0.587547	+	0.809190i	$0.699906\pi$
$228$		0			0
$229$	−175.000	+	303.109i	−0.0504992	+	0.0874672i	−0.890170	−	0.455628i	$-0.849415\pi$
$229$	−175.000	+	303.109i	−0.0504992	+	0.0874672i	0.839671	+	0.543096i	$0.182748\pi$
$230$	−660.000	+	1143.15i	−0.189214	+	0.327727i
$231$		0			0
$232$	360.000	+	623.538i	0.101876	+	0.176454i
$233$	1962.00			0.551652			0.275826	−	0.961208i	$-0.411049\pi$
$233$	1962.00			0.551652			0.275826	+	0.961208i	$0.411049\pi$
$234$		0			0
$235$	−1020.00			−0.283138
$236$	−840.000	−	1454.92i	−0.231692	−	0.401303i
$237$		0			0
$238$	264.000	−	457.261i	0.0719016	−	0.124537i
$239$	2160.00	−	3741.23i	0.584597	−	1.01255i	−0.410328	−	0.911938i	$-0.634586\pi$
$239$	2160.00	−	3741.23i	0.584597	−	1.01255i	0.994925	−	0.100614i	$-0.0320809\pi$
$240$		0			0
$241$	239.000	+	413.960i	0.0638811	+	0.110645i	0.896197	−	0.443656i	$-0.146319\pi$
$241$	239.000	+	413.960i	0.0638811	+	0.110645i	−0.832316	+	0.554301i	$0.812985\pi$
$242$	−2374.00			−0.630605
$243$		0			0
$244$	3608.00			0.946633
$245$	817.500	+	1415.95i	0.213176	+	0.369232i
$246$		0			0
$247$	−2900.00	+	5022.95i	−0.747055	+	1.29394i
$248$	−608.000	+	1053.09i	−0.155678	+	0.269641i
$249$		0			0
$250$	−125.000	−	216.506i	−0.0316228	−	0.0547723i
$251$	2652.00			0.666903			0.333452	−	0.942767i	$-0.391787\pi$
$251$	2652.00			0.666903			0.333452	+	0.942767i	$0.391787\pi$
$252$		0			0
$253$	1584.00			0.393617
$254$	124.000	+	214.774i	0.0306317	+	0.0530557i
$255$		0			0
$256$	−128.000	+	221.703i	−0.0312500	+	0.0541266i
$257$	1167.00	−	2021.30i	0.283251	−	0.490605i	−0.688933	−	0.724825i	$-0.741920\pi$
$257$	1167.00	−	2021.30i	0.283251	−	0.490605i	0.972183	+	0.234221i	$0.0752538\pi$
$258$		0			0
$259$	−68.0000	−	117.779i	−0.0163140	−	0.0282566i
$260$	−1160.00			−0.276693
$261$		0			0
$262$	264.000			0.0622518
$263$	1974.00	+	3419.07i	0.462822	+	0.801630i	0.999100	−	0.0424106i	$-0.0135038\pi$
$263$	1974.00	+	3419.07i	0.462822	+	0.801630i	−0.536279	+	0.844041i	$0.680170\pi$
$264$		0			0
$265$	−555.000	+	961.288i	−0.128654	+	0.222836i
$266$	−400.000	+	692.820i	−0.0922014	+	0.159698i
$267$		0			0
$268$	2048.00	+	3547.24i	0.466797	+	0.808516i
$269$	1590.00			0.360387			0.180193	−	0.983631i	$-0.442328\pi$
$269$	1590.00			0.360387			0.180193	+	0.983631i	$0.442328\pi$
$270$		0			0
$271$	4952.00			1.11001			0.555005	−	0.831847i	$-0.312716\pi$
$271$	4952.00			1.11001			0.555005	+	0.831847i	$0.312716\pi$
$272$	−528.000	−	914.523i	−0.117701	−	0.203864i
$273$		0			0
$274$	1254.00	−	2171.99i	0.276485	−	0.478886i
$275$	−150.000	+	259.808i	−0.0328921	+	0.0569709i
$276$		0			0
$277$	−823.000	−	1425.48i	−0.178517	−	0.309201i	0.762856	−	0.646569i	$-0.223797\pi$
$277$	−823.000	−	1425.48i	−0.178517	−	0.309201i	−0.941373	+	0.337368i	$0.890463\pi$
$278$	−5720.00			−1.23404
$279$		0			0
$280$	−160.000			−0.0341494
$281$	579.000	+	1002.86i	0.122919	+	0.212902i	0.920918	−	0.389757i	$-0.127441\pi$
$281$	579.000	+	1002.86i	0.122919	+	0.212902i	−0.797999	+	0.602659i	$0.794108\pi$
$282$		0			0
$283$	−3496.00	+	6055.25i	−0.734331	+	1.27190i	0.220685	+	0.975345i	$0.429171\pi$
$283$	−3496.00	+	6055.25i	−0.734331	+	1.27190i	−0.955016	+	0.296553i	$0.904163\pi$
$284$	−864.000	+	1496.49i	−0.180525	+	0.312678i
$285$		0			0
$286$	696.000	+	1205.51i	0.143900	+	0.249242i
$287$	1752.00			0.360339
$288$		0			0
$289$	−557.000			−0.113373
$290$	450.000	+	779.423i	0.0911204	+	0.157825i
$291$		0			0
$292$	−724.000	+	1254.00i	−0.145099	+	0.251319i
$293$	129.000	−	223.435i	0.0257210	−	0.0445501i	−0.852878	−	0.522110i	$-0.825145\pi$
$293$	129.000	−	223.435i	0.0257210	−	0.0445501i	0.878599	+	0.477560i	$0.158479\pi$
$294$		0			0
$295$	−1050.00	−	1818.65i	−0.207232	−	0.358936i
$296$	−272.000			−0.0534111
$297$		0			0
$298$	1500.00			0.291586
$299$	3828.00	+	6630.29i	0.740398	+	1.28241i
$300$		0			0
$301$	64.0000	−	110.851i	0.0122555	−	0.0212271i
$302$	448.000	−	775.959i	0.0853626	−	0.147852i
$303$		0			0
$304$	800.000	+	1385.64i	0.150931	+	0.261421i
$305$	4510.00			0.846695
$306$		0			0
$307$	−8944.00			−1.66274			−0.831370	−	0.555720i	$-0.812443\pi$
$307$	−8944.00			−1.66274			−0.831370	+	0.555720i	$0.812443\pi$
$308$	96.0000	+	166.277i	0.0177601	+	0.0307614i
$309$		0			0
$310$	−760.000	+	1316.36i	−0.139242	+	0.241175i
$311$	−696.000	+	1205.51i	−0.126902	+	0.219801i	−0.922475	−	0.386057i	$-0.873837\pi$
$311$	−696.000	+	1205.51i	−0.126902	+	0.219801i	0.795573	+	0.605858i	$0.207170\pi$
$312$		0			0
$313$	2939.00	+	5090.50i	0.530742	+	0.919271i	0.999357	+	0.0358688i	$0.0114198\pi$
$313$	2939.00	+	5090.50i	0.530742	+	0.919271i	−0.468615	+	0.883403i	$0.655247\pi$
$314$	4492.00			0.807319
$315$		0			0
$316$	−640.000			−0.113933
$317$	−5163.00	−	8942.58i	−0.914773	−	1.58443i	−0.807234	−	0.590232i	$-0.799036\pi$
$317$	−5163.00	−	8942.58i	−0.914773	−	1.58443i	−0.107539	−	0.994201i	$-0.534297\pi$
$318$		0			0
$319$	540.000	−	935.307i	0.0947780	−	0.164160i
$320$	−160.000	+	277.128i	−0.0279508	+	0.0484123i
$321$		0			0
$322$	528.000	+	914.523i	0.0913798	+	0.158274i
$323$	−6600.00			−1.13695
$324$		0			0
$325$	−1450.00			−0.247482
$326$	568.000	+	983.805i	0.0964988	+	0.167141i
$327$		0			0
$328$	1752.00	−	3034.55i	0.294933	−	0.510839i
$329$	−408.000	+	706.677i	−0.0683701	+	0.118421i
$330$		0			0
$331$	2114.00	+	3661.56i	0.351045	+	0.608028i	0.986433	−	0.164165i	$-0.0524930\pi$
$331$	2114.00	+	3661.56i	0.351045	+	0.608028i	−0.635388	+	0.772193i	$0.719160\pi$
$332$	288.000			0.0476086
$333$		0			0
$334$	−3048.00			−0.499339
$335$	2560.00	+	4434.05i	0.417516	+	0.723158i
$336$		0			0
$337$	−553.000	+	957.824i	−0.0893882	+	0.154825i	−0.907253	−	0.420586i	$-0.861825\pi$
$337$	−553.000	+	957.824i	−0.0893882	+	0.154825i	0.817864	+	0.575411i	$0.195158\pi$
$338$	−1167.00	+	2021.30i	−0.187800	+	0.325279i
$339$		0			0
$340$	−660.000	−	1143.15i	−0.105275	−	0.182342i
$341$	1824.00			0.289663
$342$		0			0
$343$	2680.00			0.421885
$344$	−128.000	−	221.703i	−0.0200619	−	0.0347482i
$345$		0			0
$346$	−3702.00	+	6412.05i	−0.575204	+	0.996283i
$347$	−4668.00	+	8085.21i	−0.722165	+	1.25083i	0.237965	+	0.971274i	$0.423520\pi$
$347$	−4668.00	+	8085.21i	−0.722165	+	1.25083i	−0.960130	+	0.279553i	$0.909814\pi$
$348$		0			0
$349$	5885.00	+	10193.1i	0.902627	+	1.56340i	0.824071	+	0.566486i	$0.191698\pi$
$349$	5885.00	+	10193.1i	0.902627	+	1.56340i	0.0785557	+	0.996910i	$0.474969\pi$
$350$	−200.000			−0.0305441
$351$		0			0
$352$	384.000			0.0581456
$353$	−4161.00	−	7207.06i	−0.627387	−	1.08667i	−0.988074	−	0.153980i	$-0.950791\pi$
$353$	−4161.00	−	7207.06i	−0.627387	−	1.08667i	0.360687	−	0.932687i	$-0.382542\pi$
$354$		0			0
$355$	−1080.00	+	1870.61i	−0.161466	+	0.279667i
$356$	−1620.00	+	2805.92i	−0.241179	+	0.417735i
$357$		0			0
$358$	−3180.00	−	5507.92i	−0.469464	−	0.813136i
$359$	10680.0			1.57011			0.785054	−	0.619427i	$-0.212635\pi$
$359$	10680.0			1.57011			0.785054	+	0.619427i	$0.212635\pi$
$360$		0			0
$361$	3141.00			0.457938
$362$	2098.00	+	3633.84i	0.304609	+	0.527598i
$363$		0			0
$364$	−464.000	+	803.672i	−0.0668138	+	0.115725i
$365$	−905.000	+	1567.51i	−0.129780	+	0.224786i
$366$		0			0
$367$	2942.00	+	5095.69i	0.418450	+	0.724777i	0.995784	−	0.0917316i	$-0.0292402\pi$
$367$	2942.00	+	5095.69i	0.418450	+	0.724777i	−0.577334	+	0.816508i	$0.695907\pi$
$368$	2112.00			0.299173
$369$		0			0
$370$	−340.000			−0.0477723
$371$	444.000	+	769.031i	0.0621330	+	0.107617i
$372$		0			0
$373$	1049.00	−	1816.92i	0.145617	−	0.252216i	−0.783986	−	0.620779i	$-0.786817\pi$
$373$	1049.00	−	1816.92i	0.145617	−	0.252216i	0.929603	+	0.368562i	$0.120150\pi$
$374$	−792.000	+	1371.78i	−0.109501	+	0.189661i
$375$		0			0
$376$	816.000	+	1413.35i	0.111920	+	0.193851i
$377$	5220.00			0.713113
$378$		0			0
$379$	3860.00			0.523153			0.261576	−	0.965183i	$-0.415758\pi$
$379$	3860.00			0.523153			0.261576	+	0.965183i	$0.415758\pi$
$380$	1000.00	+	1732.05i	0.134997	+	0.233822i
$381$		0			0
$382$	−4392.00	+	7607.17i	−0.588257	+	1.01889i
$383$	4794.00	−	8303.45i	0.639587	−	1.10780i	−0.345936	−	0.938258i	$-0.612439\pi$
$383$	4794.00	−	8303.45i	0.639587	−	1.10780i	0.985523	−	0.169540i	$-0.0542281\pi$
$384$		0			0
$385$	120.000	+	207.846i	0.0158851	+	0.0275138i
$386$	−4316.00			−0.569116
$387$		0			0
$388$	4424.00			0.578852
$389$	6705.00	+	11613.4i	0.873925	+	1.51368i	0.857903	+	0.513812i	$0.171767\pi$
$389$	6705.00	+	11613.4i	0.873925	+	1.51368i	0.0160224	+	0.999872i	$0.494900\pi$
$390$		0			0
$391$	−4356.00	+	7544.81i	−0.563408	+	0.975851i
$392$	1308.00	−	2265.52i	0.168531	−	0.291903i
$393$		0			0
$394$	1074.00	+	1860.22i	0.137328	+	0.237860i
$395$	−800.000			−0.101905
$396$		0			0
$397$	−13114.0			−1.65787			−0.828933	−	0.559348i	$-0.811052\pi$
$397$	−13114.0			−1.65787			−0.828933	+	0.559348i	$0.811052\pi$
$398$	−2840.00	−	4919.02i	−0.357679	−	0.619519i
$399$		0			0
$400$	−200.000	+	346.410i	−0.0250000	+	0.0433013i
$401$	2919.00	−	5055.86i	0.363511	−	0.629619i	−0.625025	−	0.780605i	$-0.714911\pi$
$401$	2919.00	−	5055.86i	0.363511	−	0.629619i	0.988536	+	0.150985i	$0.0482446\pi$
$402$		0			0
$403$	4408.00	+	7634.88i	0.544859	+	0.943723i
$404$	−1032.00			−0.127089
$405$		0			0
$406$	720.000			0.0880123
$407$	204.000	+	353.338i	0.0248450	+	0.0430328i
$408$		0			0
$409$	−4765.00	+	8253.22i	−0.576074	+	0.997789i	0.419851	+	0.907593i	$0.362082\pi$
$409$	−4765.00	+	8253.22i	−0.576074	+	0.997789i	−0.995924	+	0.0901954i	$0.971251\pi$
$410$	2190.00	−	3793.19i	0.263796	−	0.456908i
$411$		0			0
$412$	1976.00	+	3422.53i	0.236288	+	0.409262i
$413$	−1680.00			−0.200163
$414$		0			0
$415$	360.000			0.0425824
$416$	928.000	+	1607.34i	0.109372	+	0.189439i
$417$		0			0
$418$	1200.00	−	2078.46i	0.140416	−	0.243208i
$419$	−3630.00	+	6287.34i	−0.423239	+	0.733071i	−0.996254	−	0.0864734i	$-0.972440\pi$
$419$	−3630.00	+	6287.34i	−0.423239	+	0.733071i	0.573015	+	0.819545i	$0.305774\pi$
$420$		0			0
$421$	−6031.00	−	10446.0i	−0.698178	−	1.20928i	−0.969098	−	0.246678i	$-0.920661\pi$
$421$	−6031.00	−	10446.0i	−0.698178	−	1.20928i	0.270920	−	0.962602i	$-0.412672\pi$
$422$	−5336.00			−0.615527
$423$		0			0
$424$	1776.00			0.203420
$425$	−825.000	−	1428.94i	−0.0941609	−	0.163091i
$426$		0			0
$427$	1804.00	−	3124.62i	0.204454	−	0.354124i
$428$	48.0000	−	83.1384i	0.00542095	−	0.00938936i
$429$		0			0
$430$	−160.000	−	277.128i	−0.0179439	−	0.0310798i
$431$	−13608.0			−1.52082			−0.760411	−	0.649442i	$-0.775002\pi$
$431$	−13608.0			−1.52082			−0.760411	+	0.649442i	$0.775002\pi$
$432$		0			0
$433$	−3838.00			−0.425964			−0.212982	−	0.977056i	$-0.568318\pi$
$433$	−3838.00			−0.425964			−0.212982	+	0.977056i	$0.568318\pi$
$434$	608.000	+	1053.09i	0.0672464	+	0.116474i
$435$		0			0
$436$	−1900.00	+	3290.90i	−0.208701	+	0.361480i
$437$	6600.00	−	11431.5i	0.722473	−	1.25136i
$438$		0			0
$439$	−3700.00	−	6408.59i	−0.402258	−	0.696732i	0.591740	−	0.806129i	$-0.298441\pi$
$439$	−3700.00	−	6408.59i	−0.402258	−	0.696732i	−0.993998	+	0.109397i	$0.965108\pi$
$440$	480.000			0.0520071
$441$		0			0
$442$	−7656.00			−0.823889
$443$	−4176.00	−	7233.04i	−0.447873	−	0.775739i	0.550374	−	0.834918i	$-0.314485\pi$
$443$	−4176.00	−	7233.04i	−0.447873	−	0.775739i	−0.998247	+	0.0591792i	$0.981152\pi$
$444$		0			0
$445$	−2025.00	+	3507.40i	−0.215717	+	0.373633i
$446$	−1772.00	+	3069.19i	−0.188131	+	0.325853i
$447$		0			0
$448$	128.000	+	221.703i	0.0134987	+	0.0233805i
$449$	10770.0			1.13200			0.566000	−	0.824405i	$-0.308490\pi$
$449$	10770.0			1.13200			0.566000	+	0.824405i	$0.308490\pi$
$450$		0			0
$451$	−5256.00			−0.548770
$452$	2076.00	+	3595.74i	0.216033	+	0.374180i
$453$		0			0
$454$	2784.00	−	4822.03i	0.287796	−	0.498478i
$455$	−580.000	+	1004.59i	−0.0597600	+	0.103507i
$456$		0			0
$457$	3347.00	+	5797.17i	0.342595	+	0.593392i	0.984914	−	0.173045i	$-0.0553607\pi$
$457$	3347.00	+	5797.17i	0.342595	+	0.593392i	−0.642319	+	0.766438i	$0.722027\pi$
$458$	700.000			0.0714167
$459$		0			0
$460$	2640.00			0.267588
$461$	1509.00	+	2613.66i	0.152454	+	0.264057i	0.932129	−	0.362127i	$-0.117949\pi$
$461$	1509.00	+	2613.66i	0.152454	+	0.264057i	−0.779675	+	0.626184i	$0.784616\pi$
$462$		0			0
$463$	−7246.00	+	12550.4i	−0.727322	+	1.25976i	0.230689	+	0.973028i	$0.425902\pi$
$463$	−7246.00	+	12550.4i	−0.727322	+	1.25976i	−0.958011	+	0.286731i	$0.907431\pi$
$464$	720.000	−	1247.08i	0.0720370	−	0.124772i
$465$		0			0
$466$	−1962.00	−	3398.28i	−0.195038	−	0.337816i
$467$	7776.00			0.770515			0.385257	−	0.922809i	$-0.374113\pi$
$467$	7776.00			0.770515			0.385257	+	0.922809i	$0.374113\pi$
$468$		0			0
$469$	4096.00			0.403274
$470$	1020.00	+	1766.69i	0.100104	+	0.173386i
$471$		0			0
$472$	−1680.00	+	2909.85i	−0.163831	+	0.283764i
$473$	−192.000	+	332.554i	−0.0186642	+	0.0323274i
$474$		0			0
$475$	1250.00	+	2165.06i	0.120745	+	0.209137i
$476$	−1056.00			−0.101684
$477$		0			0
$478$	−8640.00			−0.826746
$479$	6840.00	+	11847.2i	0.652458	+	1.13009i	0.982525	+	0.186133i	$0.0595956\pi$
$479$	6840.00	+	11847.2i	0.652458	+	1.13009i	−0.330066	+	0.943958i	$0.607071\pi$
$480$		0			0
$481$	−986.000	+	1707.80i	−0.0934672	+	0.161890i
$482$	478.000	−	827.920i	0.0451708	−	0.0782380i
$483$		0			0
$484$	2374.00	+	4111.89i	0.222953	+	0.386165i
$485$	5530.00			0.517741
$486$		0			0
$487$	7916.00			0.736567			0.368284	−	0.929714i	$-0.379946\pi$
$487$	7916.00			0.736567			0.368284	+	0.929714i	$0.379946\pi$
$488$	−3608.00	−	6249.24i	−0.334685	−	0.579692i
$489$		0			0
$490$	1635.00	−	2831.90i	0.150738	−	0.261086i
$491$	−6966.00	+	12065.5i	−0.640267	+	1.10898i	0.345106	+	0.938564i	$0.387843\pi$
$491$	−6966.00	+	12065.5i	−0.640267	+	1.10898i	−0.985373	+	0.170411i	$0.945490\pi$
$492$		0			0
$493$	2970.00	+	5144.19i	0.271323	+	0.469945i
$494$	11600.0			1.05650
$495$		0			0
$496$	2432.00			0.220161
$497$	864.000	+	1496.49i	0.0779793	+	0.135064i
$498$		0			0
$499$	4130.00	−	7153.37i	0.370509	−	0.641741i	−0.619135	−	0.785285i	$-0.712516\pi$
$499$	4130.00	−	7153.37i	0.370509	−	0.641741i	0.989644	+	0.143544i	$0.0458498\pi$
$500$	−250.000	+	433.013i	−0.0223607	+	0.0387298i
$501$		0			0
$502$	−2652.00	−	4593.40i	−0.235786	−	0.408393i
$503$	−11148.0			−0.988200			−0.494100	−	0.869405i	$-0.664502\pi$
$503$	−11148.0			−0.988200			−0.494100	+	0.869405i	$0.664502\pi$
$504$		0			0
$505$	−1290.00			−0.113672
$506$	−1584.00	−	2743.57i	−0.139165	−	0.241041i
$507$		0			0
$508$	248.000	−	429.549i	0.0216599	−	0.0375160i
$509$	4845.00	−	8391.79i	0.421907	−	0.730765i	−0.574219	−	0.818702i	$-0.694694\pi$
$509$	4845.00	−	8391.79i	0.421907	−	0.730765i	0.996126	+	0.0879370i	$0.0280274\pi$
$510$		0			0
$511$	724.000	+	1254.00i	0.0626769	+	0.108560i
$512$	512.000			0.0441942
$513$		0			0
$514$	−4668.00			−0.400577
$515$	2470.00	+	4278.17i	0.211342	+	0.366055i
$516$		0			0
$517$	1224.00	−	2120.03i	0.104123	−	0.180346i
$518$	−136.000	+	235.559i	−0.0115357	+	0.0199804i
$519$		0			0
$520$	1160.00	+	2009.18i	0.0978257	+	0.169439i
$521$	−16038.0			−1.34863			−0.674316	−	0.738443i	$-0.735562\pi$
$521$	−16038.0			−1.34863			−0.674316	+	0.738443i	$0.735562\pi$
$522$		0			0
$523$	992.000			0.0829391			0.0414695	−	0.999140i	$-0.486796\pi$
$523$	992.000			0.0829391			0.0414695	+	0.999140i	$0.486796\pi$
$524$	−264.000	−	457.261i	−0.0220093	−	0.0381213i
$525$		0			0
$526$	3948.00	−	6838.14i	0.327264	−	0.566838i
$527$	−5016.00	+	8687.97i	−0.414612	+	0.718129i
$528$		0			0
$529$	−2628.50	−	4552.70i	−0.216035	−	0.374184i
$530$	2220.00			0.181945
$531$		0			0
$532$	1600.00			0.130392
$533$	−12702.0	−	22000.5i	−1.03224	−	1.78789i
$534$		0			0
$535$	60.0000	−	103.923i	0.00484865	−	0.00839810i
$536$	4096.00	−	7094.48i	0.330075	−	0.571707i
$537$		0			0
$538$	−1590.00	−	2753.96i	−0.127416	−	0.220691i
$539$	−3924.00			−0.313578
$540$		0			0
$541$	7142.00			0.567576			0.283788	−	0.958887i	$-0.408409\pi$
$541$	7142.00			0.567576			0.283788	+	0.958887i	$0.408409\pi$
$542$	−4952.00	−	8577.12i	−0.392448	−	0.679739i
$543$		0			0
$544$	−1056.00	+	1829.05i	−0.0832273	+	0.144154i
$545$	−2375.00	+	4113.62i	−0.186668	+	0.323318i
$546$		0			0
$547$	−3808.00	−	6595.65i	−0.297657	−	0.515557i	0.677943	−	0.735115i	$-0.262872\pi$
$547$	−3808.00	−	6595.65i	−0.297657	−	0.515557i	−0.975599	+	0.219558i	$0.929538\pi$
$548$	−5016.00			−0.391009
$549$		0			0
$550$	600.000			0.0465165
$551$	−4500.00	−	7794.23i	−0.347925	−	0.602623i
$552$		0			0
$553$	−320.000	+	554.256i	−0.0246072	+	0.0426209i
$554$	−1646.00	+	2850.96i	−0.126231	+	0.218638i
$555$		0			0
$556$	5720.00	+	9907.33i	0.436299	+	0.755691i
$557$	−10314.0			−0.784593			−0.392296	−	0.919839i	$-0.628319\pi$
$557$	−10314.0			−0.784593			−0.392296	+	0.919839i	$0.628319\pi$
$558$		0			0
$559$	−1856.00			−0.140430
$560$	160.000	+	277.128i	0.0120736	+	0.0209121i
$561$		0			0
$562$	1158.00	−	2005.71i	0.0869169	−	0.150544i
$563$	3564.00	−	6173.03i	0.266793	−	0.462100i	−0.701239	−	0.712927i	$-0.747369\pi$
$563$	3564.00	−	6173.03i	0.266793	−	0.462100i	0.968032	+	0.250827i	$0.0807025\pi$
$564$		0			0
$565$	2595.00	+	4494.67i	0.193226	+	0.334677i
$566$	13984.0			1.03850
$567$		0			0
$568$	3456.00			0.255300
$569$	−1005.00	−	1740.71i	−0.0740453	−	0.128250i	0.826625	−	0.562753i	$-0.190258\pi$
$569$	−1005.00	−	1740.71i	−0.0740453	−	0.128250i	−0.900671	+	0.434502i	$0.856924\pi$
$570$		0			0
$571$	11594.0	−	20081.4i	0.849726	−	1.47177i	−0.0317260	−	0.999497i	$-0.510100\pi$
$571$	11594.0	−	20081.4i	0.849726	−	1.47177i	0.881452	−	0.472273i	$-0.156566\pi$
$572$	1392.00	−	2411.01i	0.101753	−	0.176241i
$573$		0			0
$574$	−1752.00	−	3034.55i	−0.127399	−	0.220662i
$575$	3300.00			0.239338
$576$		0			0
$577$	22466.0			1.62092			0.810461	−	0.585793i	$-0.199217\pi$
$577$	22466.0			1.62092			0.810461	+	0.585793i	$0.199217\pi$
$578$	557.000	+	964.752i	0.0400833	+	0.0694263i
$579$		0			0
$580$	900.000	−	1558.85i	0.0644318	−	0.111599i
$581$	144.000	−	249.415i	0.0102825	−	0.0178098i
$582$		0			0
$583$	−1332.00	−	2307.09i	−0.0946240	−	0.163894i
$584$	2896.00			0.205201
$585$		0			0
$586$	−516.000			−0.0363750
$587$	−11388.0	−	19724.6i	−0.800738	−	1.38692i	−0.919131	−	0.393952i	$-0.871108\pi$
$587$	−11388.0	−	19724.6i	−0.800738	−	1.38692i	0.118393	−	0.992967i	$-0.462226\pi$
$588$		0			0
$589$	7600.00	−	13163.6i	0.531668	−	0.920876i
$590$	−2100.00	+	3637.31i	−0.146535	+	0.253806i
$591$		0			0
$592$	272.000	+	471.118i	0.0188837	+	0.0327075i
$593$	−21198.0			−1.46796			−0.733978	−	0.679174i	$-0.762338\pi$
$593$	−21198.0			−1.46796			−0.733978	+	0.679174i	$0.762338\pi$
$594$		0			0
$595$	−1320.00			−0.0909491
$596$	−1500.00	−	2598.08i	−0.103091	−	0.178559i
$597$		0			0
$598$	7656.00	−	13260.6i	0.523540	−	0.906798i
$599$	−7980.00	+	13821.8i	−0.544330	+	0.942808i	0.454318	+	0.890839i	$0.349883\pi$
$599$	−7980.00	+	13821.8i	−0.544330	+	0.942808i	−0.998649	+	0.0519686i	$0.983450\pi$
$600$		0			0
$601$	−2941.00	−	5093.96i	−0.199610	−	0.345735i	0.748792	−	0.662805i	$-0.230634\pi$
$601$	−2941.00	−	5093.96i	−0.199610	−	0.345735i	−0.948402	+	0.317070i	$0.897301\pi$
$602$	−256.000			−0.0173319
$603$		0			0
$604$	−1792.00			−0.120721
$605$	2967.50	+	5139.86i	0.199415	+	0.345397i
$606$		0			0
$607$	−4258.00	+	7375.07i	−0.284723	+	0.493155i	−0.972542	−	0.232728i	$-0.925235\pi$
$607$	−4258.00	+	7375.07i	−0.284723	+	0.493155i	0.687819	+	0.725882i	$0.258568\pi$
$608$	1600.00	−	2771.28i	0.106725	−	0.184852i
$609$		0			0
$610$	−4510.00	−	7811.55i	−0.299352	−	0.518492i
$611$	11832.0			0.783423
$612$		0			0
$613$	8462.00			0.557548			0.278774	−	0.960357i	$-0.410072\pi$
$613$	8462.00			0.557548			0.278774	+	0.960357i	$0.410072\pi$
$614$	8944.00	+	15491.5i	0.587867	+	1.01822i
$615$		0			0
$616$	192.000	−	332.554i	0.0125583	−	0.0217516i
$617$	5547.00	−	9607.69i	0.361935	−	0.626890i	−0.626344	−	0.779546i	$-0.715450\pi$
$617$	5547.00	−	9607.69i	0.361935	−	0.626890i	0.988279	+	0.152657i	$0.0487829\pi$
$618$		0			0
$619$	−1090.00	−	1887.94i	−0.0707767	−	0.122589i	0.828465	−	0.560041i	$-0.189214\pi$
$619$	−1090.00	−	1887.94i	−0.0707767	−	0.122589i	−0.899242	+	0.437452i	$0.855881\pi$
$620$	3040.00			0.196918
$621$		0			0
$622$	2784.00			0.179467
$623$	1620.00	+	2805.92i	0.104180	+	0.180444i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$	5878.00	−	10181.0i	0.375291	−	0.650023i
$627$		0			0
$628$	−4492.00	−	7780.37i	−0.285430	−	0.494380i
$629$	−2244.00			−0.142248
$630$		0			0
$631$	−26848.0			−1.69382			−0.846911	−	0.531734i	$-0.821541\pi$
$631$	−26848.0			−1.69382			−0.846911	+	0.531734i	$0.821541\pi$
$632$	640.000	+	1108.51i	0.0402814	+	0.0697694i
$633$		0			0
$634$	−10326.0	+	17885.2i	−0.646842	+	1.12036i
$635$	310.000	−	536.936i	0.0193732	−	0.0335553i
$636$		0			0
$637$	−9483.00	−	16425.0i	−0.589843	−	1.02164i
$638$	−2160.00			−0.134036
$639$		0			0
$640$	640.000			0.0395285
$641$	−13161.0	−	22795.5i	−0.810965	−	1.40463i	−0.912190	−	0.409768i	$-0.865610\pi$
$641$	−13161.0	−	22795.5i	−0.810965	−	1.40463i	0.101225	−	0.994864i	$-0.467724\pi$
$642$		0			0
$643$	5084.00	−	8805.75i	0.311809	−	0.540070i	−0.666945	−	0.745107i	$-0.732398\pi$
$643$	5084.00	−	8805.75i	0.311809	−	0.540070i	0.978754	+	0.205038i	$0.0657317\pi$
$644$	1056.00	−	1829.05i	0.0646153	−	0.111917i
$645$		0			0
$646$	6600.00	+	11431.5i	0.401971	+	0.696235i
$647$	−23604.0			−1.43426			−0.717132	−	0.696937i	$-0.754546\pi$
$647$	−23604.0			−1.43426			−0.717132	+	0.696937i	$0.754546\pi$
$648$		0			0
$649$	5040.00			0.304834
$650$	1450.00	+	2511.47i	0.0874980	+	0.151551i
$651$		0			0
$652$	1136.00	−	1967.61i	0.0682350	−	0.118186i
$653$	−8211.00	+	14221.9i	−0.492069	+	0.852289i	−0.999958	−	0.00913349i	$-0.997093\pi$
$653$	−8211.00	+	14221.9i	−0.492069	+	0.852289i	0.507889	+	0.861423i	$0.330426\pi$
$654$		0			0
$655$	−330.000	−	571.577i	−0.0196858	−	0.0340967i
$656$	−7008.00			−0.417098
$657$		0			0
$658$	1632.00			0.0966899
$659$	13050.0	+	22603.3i	0.771405	+	1.33611i	0.936793	+	0.349884i	$0.113779\pi$
$659$	13050.0	+	22603.3i	0.771405	+	1.33611i	−0.165388	+	0.986229i	$0.552888\pi$
$660$		0			0
$661$	1529.00	−	2648.31i	0.0899716	−	0.155835i	−0.817527	−	0.575890i	$-0.804656\pi$
$661$	1529.00	−	2648.31i	0.0899716	−	0.155835i	0.907499	+	0.420055i	$0.137989\pi$
$662$	4228.00	−	7323.11i	0.248226	−	0.429941i
$663$		0			0
$664$	−288.000	−	498.831i	−0.0168322	−	0.0291542i
$665$	2000.00			0.116627
$666$		0			0
$667$	−11880.0			−0.689648
$668$	3048.00	+	5279.29i	0.176543	+	0.305781i
$669$		0			0
$670$	5120.00	−	8868.10i	0.295228	−	0.511350i
$671$	−5412.00	+	9373.86i	−0.311368	+	0.539305i
$672$		0			0
$673$	−5401.00	−	9354.81i	−0.309351	−	0.535812i	0.668870	−	0.743380i	$-0.266778\pi$
$673$	−5401.00	−	9354.81i	−0.309351	−	0.535812i	−0.978221	+	0.207568i	$0.933445\pi$
$674$	2212.00			0.126414
$675$		0			0
$676$	4668.00			0.265589
$677$	5337.00	+	9243.96i	0.302980	+	0.524777i	0.976810	−	0.214110i	$-0.0686851\pi$
$677$	5337.00	+	9243.96i	0.302980	+	0.524777i	−0.673829	+	0.738887i	$0.735352\pi$
$678$		0			0
$679$	2212.00	−	3831.30i	0.125020	−	0.216541i
$680$	−1320.00	+	2286.31i	−0.0744407	+	0.128935i
$681$		0			0
$682$	−1824.00	−	3159.26i	−0.102411	−	0.177382i
$683$	−28608.0			−1.60272			−0.801358	−	0.598185i	$-0.795889\pi$
$683$	−28608.0			−1.60272			−0.801358	+	0.598185i	$0.795889\pi$
$684$		0			0
$685$	−6270.00			−0.349729
$686$	−2680.00	−	4641.90i	−0.149159	−	0.258350i
$687$		0			0
$688$	−256.000	+	443.405i	−0.0141859	+	0.0245707i
$689$	6438.00	−	11150.9i	0.355977	−	0.616571i
$690$		0			0
$691$	1214.00	+	2102.71i	0.0668346	+	0.115761i	0.897506	−	0.441002i	$-0.145377\pi$
$691$	1214.00	+	2102.71i	0.0668346	+	0.115761i	−0.830672	+	0.556763i	$0.812043\pi$
$692$	14808.0			0.813462
$693$		0			0
$694$	18672.0			1.02130
$695$	7150.00	+	12384.2i	0.390237	+	0.675911i
$696$		0			0
$697$	14454.0	−	25035.1i	0.785487	−	1.36050i
$698$	11770.0	−	20386.2i	0.638254	−	1.10549i
$699$		0			0
$700$	200.000	+	346.410i	0.0107990	+	0.0187044i
$701$	−6618.00			−0.356574			−0.178287	−	0.983979i	$-0.557056\pi$
$701$	−6618.00			−0.356574			−0.178287	+	0.983979i	$0.557056\pi$
$702$		0			0
$703$	3400.00			0.182409
$704$	−384.000	−	665.108i	−0.0205576	−	0.0356068i
$705$		0			0
$706$	−8322.00	+	14414.1i	−0.443630	+	0.768389i
$707$	−516.000	+	893.738i	−0.0274486	+	0.0475424i
$708$		0			0
$709$	−10255.0	−	17762.2i	−0.543208	−	0.940864i	−0.998717	−	0.0506331i	$-0.983876\pi$
$709$	−10255.0	−	17762.2i	−0.543208	−	0.940864i	0.455509	−	0.890231i	$-0.349457\pi$
$710$	4320.00			0.228347
$711$		0			0
$712$	6480.00			0.341079
$713$	−10032.0	−	17375.9i	−0.526930	−	0.912670i
$714$		0			0
$715$	1740.00	−	3013.77i	0.0910102	−	0.157634i
$716$	−6360.00	+	11015.8i	−0.331961	+	0.574974i
$717$		0			0
$718$	−10680.0	−	18498.3i	−0.555117	−	0.961491i
$719$	31680.0			1.64321			0.821603	−	0.570061i	$-0.193080\pi$
$719$	31680.0			1.64321			0.821603	+	0.570061i	$0.193080\pi$
$720$		0			0
$721$	3952.00			0.204133
$722$	−3141.00	−	5440.37i	−0.161906	−	0.280429i
$723$		0			0
$724$	4196.00	−	7267.69i	0.215391	−	0.373068i
$725$	1125.00	−	1948.56i	0.0576296	−	0.0998174i
$726$		0			0
$727$	−6598.00	−	11428.1i	−0.336597	−	0.583004i	0.647193	−	0.762326i	$-0.275943\pi$
$727$	−6598.00	−	11428.1i	−0.336597	−	0.583004i	−0.983790	+	0.179322i	$0.942609\pi$
$728$	1856.00			0.0944889
$729$		0			0
$730$	3620.00			0.183537
$731$	−1056.00	−	1829.05i	−0.0534303	−	0.0925440i
$732$		0			0
$733$	−4051.00	+	7016.54i	−0.204130	+	0.353563i	−0.949855	−	0.312690i	$-0.898770\pi$
$733$	−4051.00	+	7016.54i	−0.204130	+	0.353563i	0.745725	+	0.666253i	$0.232103\pi$
$734$	5884.00	−	10191.4i	0.295889	−	0.512494i
$735$		0			0
$736$	−2112.00	−	3658.09i	−0.105774	−	0.183205i
$737$	−12288.0			−0.614158
$738$		0			0
$739$	−12580.0			−0.626201			−0.313101	−	0.949720i	$-0.601368\pi$
$739$	−12580.0			−0.626201			−0.313101	+	0.949720i	$0.601368\pi$
$740$	340.000	+	588.897i	0.0168901	+	0.0292545i
$741$		0			0
$742$	888.000	−	1538.06i	0.0439346	−	0.0760970i
$743$	−14946.0	+	25887.2i	−0.737975	+	1.27821i	0.215430	+	0.976519i	$0.430885\pi$
$743$	−14946.0	+	25887.2i	−0.737975	+	1.27821i	−0.953406	+	0.301691i	$0.902449\pi$
$744$		0			0
$745$	−1875.00	−	3247.60i	−0.0922076	−	0.159708i
$746$	−4196.00			−0.205934
$747$		0			0
$748$	3168.00			0.154858
$749$	−48.0000	−	83.1384i	−0.00234163	−	0.00405582i
$750$		0			0
$751$	20204.0	−	34994.4i	0.981697	−	1.70035i	0.325914	−	0.945399i	$-0.394328\pi$
$751$	20204.0	−	34994.4i	0.981697	−	1.70035i	0.655783	−	0.754950i	$-0.272339\pi$
$752$	1632.00	−	2826.71i	0.0791395	−	0.137074i
$753$		0			0
$754$	−5220.00	−	9041.31i	−0.252124	−	0.436691i
$755$	−2240.00			−0.107976
$756$		0			0
$757$	32366.0			1.55398			0.776990	−	0.629513i	$-0.216746\pi$
$757$	32366.0			1.55398			0.776990	+	0.629513i	$0.216746\pi$
$758$	−3860.00	−	6685.72i	−0.184962	−	0.320364i
$759$		0			0
$760$	2000.00	−	3464.10i	0.0954574	−	0.165337i
$761$	8619.00	−	14928.5i	0.410563	−	0.711116i	−0.584388	−	0.811474i	$-0.698665\pi$
$761$	8619.00	−	14928.5i	0.410563	−	0.711116i	0.994951	+	0.100358i	$0.0319988\pi$
$762$		0			0
$763$	1900.00	+	3290.90i	0.0901502	+	0.156145i
$764$	17568.0			0.831921
$765$		0			0
$766$	−19176.0			−0.904513
$767$	12180.0	+	21096.4i	0.573395	+	0.993150i
$768$		0			0
$769$	−5425.00	+	9396.38i	−0.254396	+	0.440627i	−0.964731	−	0.263237i	$-0.915210\pi$
$769$	−5425.00	+	9396.38i	−0.254396	+	0.440627i	0.710335	+	0.703863i	$0.248543\pi$
$770$	240.000	−	415.692i	0.0112325	−	0.0194552i
$771$		0			0
$772$	4316.00	+	7475.53i	0.201213	+	0.348511i
$773$	9102.00			0.423514			0.211757	−	0.977322i	$-0.432081\pi$
$773$	9102.00			0.423514			0.211757	+	0.977322i	$0.432081\pi$
$774$		0			0
$775$	3800.00			0.176129
$776$	−4424.00	−	7662.59i	−0.204655	−	0.354473i
$777$		0			0
$778$	13410.0	−	23226.8i	0.617959	−	1.07034i
$779$	−21900.0	+	37931.9i	−1.00725	+	1.74461i
$780$		0			0

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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	810.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(2.00000 + 3.46410i) q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+(-1.00000 - 1.73205i) q^{2} +(-2.00000 + 3.46410i) q^{4} +(-2.50000 + 4.33013i) q^{5} +(2.00000 + 3.46410i) q^{7} +8.00000 q^{8} +10.0000 q^{10} +(-6.00000 - 10.3923i) q^{11} +(29.0000 - 50.2295i) q^{13} +(4.00000 - 6.92820i) q^{14} +(-8.00000 - 13.8564i) q^{16} +66.0000 q^{17} -100.000 q^{19} +(-10.0000 - 17.3205i) q^{20} +(-12.0000 + 20.7846i) q^{22} +(-66.0000 + 114.315i) q^{23} +(-12.5000 - 21.6506i) q^{25} -116.000 q^{26} -16.0000 q^{28} +(45.0000 + 77.9423i) q^{29} +(-76.0000 + 131.636i) q^{31} +(-16.0000 + 27.7128i) q^{32} +(-66.0000 - 114.315i) q^{34} -20.0000 q^{35} -34.0000 q^{37} +(100.000 + 173.205i) q^{38} +(-20.0000 + 34.6410i) q^{40} +(219.000 - 379.319i) q^{41} +(-16.0000 - 27.7128i) q^{43} +48.0000 q^{44} +264.000 q^{46} +(102.000 + 176.669i) q^{47} +(163.500 - 283.190i) q^{49} +(-25.0000 + 43.3013i) q^{50} +(116.000 + 200.918i) q^{52} +222.000 q^{53} +60.0000 q^{55} +(16.0000 + 27.7128i) q^{56} +(90.0000 - 155.885i) q^{58} +(-210.000 + 363.731i) q^{59} +(-451.000 - 781.155i) q^{61} +304.000 q^{62} +64.0000 q^{64} +(145.000 + 251.147i) q^{65} +(512.000 - 886.810i) q^{67} +(-132.000 + 228.631i) q^{68} +(20.0000 + 34.6410i) q^{70} +432.000 q^{71} +362.000 q^{73} +(34.0000 + 58.8897i) q^{74} +(200.000 - 346.410i) q^{76} +(24.0000 - 41.5692i) q^{77} +(80.0000 + 138.564i) q^{79} +80.0000 q^{80} -876.000 q^{82} +(-36.0000 - 62.3538i) q^{83} +(-165.000 + 285.788i) q^{85} +(-32.0000 + 55.4256i) q^{86} +(-48.0000 - 83.1384i) q^{88} +810.000 q^{89} +232.000 q^{91} +(-264.000 - 457.261i) q^{92} +(204.000 - 353.338i) q^{94} +(250.000 - 433.013i) q^{95} +(-553.000 - 957.824i) q^{97} -654.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 4 q^{4} - 5 q^{5} + 4 q^{7} + 16 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 - 4 * q^4 - 5 * q^5 + 4 * q^7 + 16 * q^8	\( 2 q - 2 q^{2} - 4 q^{4} - 5 q^{5} + 4 q^{7} + 16 q^{8} + 20 q^{10} - 12 q^{11} + 58 q^{13} + 8 q^{14} - 16 q^{16} + 132 q^{17} - 200 q^{19} - 20 q^{20} - 24 q^{22} - 132 q^{23} - 25 q^{25} - 232 q^{26} - 32 q^{28} + 90 q^{29} - 152 q^{31} - 32 q^{32} - 132 q^{34} - 40 q^{35} - 68 q^{37} + 200 q^{38} - 40 q^{40} + 438 q^{41} - 32 q^{43} + 96 q^{44} + 528 q^{46} + 204 q^{47} + 327 q^{49} - 50 q^{50} + 232 q^{52} + 444 q^{53} + 120 q^{55} + 32 q^{56} + 180 q^{58} - 420 q^{59} - 902 q^{61} + 608 q^{62} + 128 q^{64} + 290 q^{65} + 1024 q^{67} - 264 q^{68} + 40 q^{70} + 864 q^{71} + 724 q^{73} + 68 q^{74} + 400 q^{76} + 48 q^{77} + 160 q^{79} + 160 q^{80} - 1752 q^{82} - 72 q^{83} - 330 q^{85} - 64 q^{86} - 96 q^{88} + 1620 q^{89} + 464 q^{91} - 528 q^{92} + 408 q^{94} + 500 q^{95} - 1106 q^{97} - 1308 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 - 4 * q^4 - 5 * q^5 + 4 * q^7 + 16 * q^8 + 20 * q^10 - 12 * q^11 + 58 * q^13 + 8 * q^14 - 16 * q^16 + 132 * q^17 - 200 * q^19 - 20 * q^20 - 24 * q^22 - 132 * q^23 - 25 * q^25 - 232 * q^26 - 32 * q^28 + 90 * q^29 - 152 * q^31 - 32 * q^32 - 132 * q^34 - 40 * q^35 - 68 * q^37 + 200 * q^38 - 40 * q^40 + 438 * q^41 - 32 * q^43 + 96 * q^44 + 528 * q^46 + 204 * q^47 + 327 * q^49 - 50 * q^50 + 232 * q^52 + 444 * q^53 + 120 * q^55 + 32 * q^56 + 180 * q^58 - 420 * q^59 - 902 * q^61 + 608 * q^62 + 128 * q^64 + 290 * q^65 + 1024 * q^67 - 264 * q^68 + 40 * q^70 + 864 * q^71 + 724 * q^73 + 68 * q^74 + 400 * q^76 + 48 * q^77 + 160 * q^79 + 160 * q^80 - 1752 * q^82 - 72 * q^83 - 330 * q^85 - 64 * q^86 - 96 * q^88 + 1620 * q^89 + 464 * q^91 - 528 * q^92 + 408 * q^94 + 500 * q^95 - 1106 * q^97 - 1308 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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