LMFDB - Embedded newform 475.4.b.c.324.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,4,Mod(324,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("475.324");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 475 = 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	475.b (of order $2$, degree $1$, 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:	$28.0259072527$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	475.324
Dual form			475.4.b.c.324.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{2} +5.00000i q^{3} -1.00000 q^{4} +15.0000 q^{6} +11.0000i q^{7} -21.0000i q^{8} +2.00000 q^{9} +O(q^{10})$	\(q-3.00000i q^{2} +5.00000i q^{3} -1.00000 q^{4} +15.0000 q^{6} +11.0000i q^{7} -21.0000i q^{8} +2.00000 q^{9} -54.0000 q^{11} -5.00000i q^{12} -11.0000i q^{13} +33.0000 q^{14} -71.0000 q^{16} -93.0000i q^{17} -6.00000i q^{18} -19.0000 q^{19} -55.0000 q^{21} +162.000i q^{22} -183.000i q^{23} +105.000 q^{24} -33.0000 q^{26} +145.000i q^{27} -11.0000i q^{28} +249.000 q^{29} +56.0000 q^{31} +45.0000i q^{32} -270.000i q^{33} -279.000 q^{34} -2.00000 q^{36} -250.000i q^{37} +57.0000i q^{38} +55.0000 q^{39} +240.000 q^{41} +165.000i q^{42} +196.000i q^{43} +54.0000 q^{44} -549.000 q^{46} -168.000i q^{47} -355.000i q^{48} +222.000 q^{49} +465.000 q^{51} +11.0000i q^{52} -435.000i q^{53} +435.000 q^{54} +231.000 q^{56} -95.0000i q^{57} -747.000i q^{58} -195.000 q^{59} -358.000 q^{61} -168.000i q^{62} +22.0000i q^{63} -433.000 q^{64} -810.000 q^{66} -961.000i q^{67} +93.0000i q^{68} +915.000 q^{69} -246.000 q^{71} -42.0000i q^{72} -353.000i q^{73} -750.000 q^{74} +19.0000 q^{76} -594.000i q^{77} -165.000i q^{78} +34.0000 q^{79} -671.000 q^{81} -720.000i q^{82} -234.000i q^{83} +55.0000 q^{84} +588.000 q^{86} +1245.00i q^{87} +1134.00i q^{88} +168.000 q^{89} +121.000 q^{91} +183.000i q^{92} +280.000i q^{93} -504.000 q^{94} -225.000 q^{96} +758.000i q^{97} -666.000i q^{98} -108.000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4} + 30 q^{6} + 4 q^{9}+O(q^{10}) $ 2 * q - 2 * q^4 + 30 * q^6 + 4 * q^9	$ 2 q - 2 q^{4} + 30 q^{6} + 4 q^{9} - 108 q^{11} + 66 q^{14} - 142 q^{16} - 38 q^{19} - 110 q^{21} + 210 q^{24} - 66 q^{26} + 498 q^{29} + 112 q^{31} - 558 q^{34} - 4 q^{36} + 110 q^{39} + 480 q^{41} + 108 q^{44} - 1098 q^{46} + 444 q^{49} + 930 q^{51} + 870 q^{54} + 462 q^{56} - 390 q^{59} - 716 q^{61} - 866 q^{64} - 1620 q^{66} + 1830 q^{69} - 492 q^{71} - 1500 q^{74} + 38 q^{76} + 68 q^{79} - 1342 q^{81} + 110 q^{84} + 1176 q^{86} + 336 q^{89} + 242 q^{91} - 1008 q^{94} - 450 q^{96} - 216 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 + 30 * q^6 + 4 * q^9 - 108 * q^11 + 66 * q^14 - 142 * q^16 - 38 * q^19 - 110 * q^21 + 210 * q^24 - 66 * q^26 + 498 * q^29 + 112 * q^31 - 558 * q^34 - 4 * q^36 + 110 * q^39 + 480 * q^41 + 108 * q^44 - 1098 * q^46 + 444 * q^49 + 930 * q^51 + 870 * q^54 + 462 * q^56 - 390 * q^59 - 716 * q^61 - 866 * q^64 - 1620 * q^66 + 1830 * q^69 - 492 * q^71 - 1500 * q^74 + 38 * q^76 + 68 * q^79 - 1342 * q^81 + 110 * q^84 + 1176 * q^86 + 336 * q^89 + 242 * q^91 - 1008 * q^94 - 450 * q^96 - 216 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$.

$n$	$77$	$401$
$\chi(n)$	$-1$	$1$

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 $

$2$	− 3.00000i	− 1.06066i	−0.847791	−	0.530330i	$-0.822068\pi$
$2$	− 3.00000i	− 1.06066i	0.847791	−	0.530330i	$-0.177932\pi$
$3$	5.00000i	0.962250i	0.876652	+	0.481125i	$0.159772\pi$
$3$	5.00000i	0.962250i	−0.876652	+	0.481125i	$0.840228\pi$
$4$	−1.00000	−0.125000
$5$	0	0
$5$	0	0
$6$	15.0000	1.02062
$7$	11.0000i	0.593944i	0.954886	+	0.296972i	$0.0959768\pi$
$7$	11.0000i	0.593944i	−0.954886	+	0.296972i	$0.904023\pi$
$8$	− 21.0000i	− 0.928078i
$9$	2.00000	0.0740741
$10$	0	0
$11$	−54.0000	−1.48015	−0.740073	−	0.672526i	$-0.765209\pi$
$11$	−54.0000	−1.48015	−0.740073	+	0.672526i	$0.765209\pi$
$12$	− 5.00000i	− 0.120281i
$13$	− 11.0000i	− 0.234681i	−0.993092	−	0.117340i	$-0.962563\pi$
$13$	− 11.0000i	− 0.234681i	0.993092	−	0.117340i	$-0.0374369\pi$
$14$	33.0000	0.629973
$15$	0	0
$16$	−71.0000	−1.10938
$17$	− 93.0000i	− 1.32681i	−0.748259	−	0.663406i	$-0.769110\pi$
$17$	− 93.0000i	− 1.32681i	0.748259	−	0.663406i	$-0.230890\pi$
$18$	− 6.00000i	− 0.0785674i
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	−55.0000	−0.571523
$22$	162.000i	1.56993i
$23$	− 183.000i	− 1.65905i	−0.558470	−	0.829525i	$-0.688611\pi$
$23$	− 183.000i	− 1.65905i	0.558470	−	0.829525i	$-0.311389\pi$
$24$	105.000	0.893043
$25$	0	0
$26$	−33.0000	−0.248917
$27$	145.000i	1.03353i
$28$	− 11.0000i	− 0.0742430i
$29$	249.000	1.59442	0.797209	−	0.603703i	$-0.206309\pi$
$29$	249.000	1.59442	0.797209	+	0.603703i	$0.206309\pi$
$30$	0	0
$31$	56.0000	0.324448	0.162224	−	0.986754i	$-0.448133\pi$
$31$	56.0000	0.324448	0.162224	+	0.986754i	$0.448133\pi$
$32$	45.0000i	0.248592i
$33$	− 270.000i	− 1.42427i
$34$	−279.000	−1.40730
$35$	0	0
$36$	−2.00000	−0.00925926
$37$	− 250.000i	− 1.11080i	−0.831582	−	0.555402i	$-0.812564\pi$
$37$	− 250.000i	− 1.11080i	0.831582	−	0.555402i	$-0.187436\pi$
$38$	57.0000i	0.243332i
$39$	55.0000	0.225822
$40$	0	0
$41$	240.000	0.914188	0.457094	−	0.889418i	$-0.348890\pi$
$41$	240.000	0.914188	0.457094	+	0.889418i	$0.348890\pi$
$42$	165.000i	0.606192i
$43$	196.000i	0.695110i	0.937660	+	0.347555i	$0.112988\pi$
$43$	196.000i	0.695110i	−0.937660	+	0.347555i	$0.887012\pi$
$44$	54.0000	0.185018
$45$	0	0
$46$	−549.000	−1.75969
$47$	− 168.000i	− 0.521390i	−0.965421	−	0.260695i	$-0.916048\pi$
$47$	− 168.000i	− 0.521390i	0.965421	−	0.260695i	$-0.0839517\pi$
$48$	− 355.000i	− 1.06750i
$49$	222.000	0.647230
$50$	0	0
$51$	465.000	1.27673
$52$	11.0000i	0.0293351i
$53$	− 435.000i	− 1.12739i	−0.825982	−	0.563697i	$-0.809379\pi$
$53$	− 435.000i	− 1.12739i	0.825982	−	0.563697i	$-0.190621\pi$
$54$	435.000	1.09622
$55$	0	0
$56$	231.000	0.551226
$57$	− 95.0000i	− 0.220755i
$58$	− 747.000i	− 1.69114i
$59$	−195.000	−0.430285	−0.215143	−	0.976583i	$-0.569022\pi$
$59$	−195.000	−0.430285	−0.215143	+	0.976583i	$0.569022\pi$
$60$	0	0
$61$	−358.000	−0.751430	−0.375715	−	0.926735i	$-0.622603\pi$
$61$	−358.000	−0.751430	−0.375715	+	0.926735i	$0.622603\pi$
$62$	− 168.000i	− 0.344129i
$63$	22.0000i	0.0439959i
$64$	−433.000	−0.845703
$65$	0	0
$66$	−810.000	−1.51067
$67$	− 961.000i	− 1.75231i	−0.482029	−	0.876155i	$-0.660100\pi$
$67$	− 961.000i	− 1.75231i	0.482029	−	0.876155i	$-0.339900\pi$
$68$	93.0000i	0.165852i
$69$	915.000	1.59642
$70$	0	0
$71$	−246.000	−0.411195	−0.205597	−	0.978637i	$-0.565914\pi$
$71$	−246.000	−0.411195	−0.205597	+	0.978637i	$0.565914\pi$
$72$	− 42.0000i	− 0.0687465i
$73$	− 353.000i	− 0.565966i	−0.959125	−	0.282983i	$-0.908676\pi$
$73$	− 353.000i	− 0.565966i	0.959125	−	0.282983i	$-0.0913240\pi$
$74$	−750.000	−1.17819
$75$	0	0
$76$	19.0000	0.0286770
$77$	− 594.000i	− 0.879124i
$78$	− 165.000i	− 0.239520i
$79$	34.0000	0.0484215	0.0242108	−	0.999707i	$-0.492293\pi$
$79$	34.0000	0.0484215	0.0242108	+	0.999707i	$0.492293\pi$
$80$	0	0
$81$	−671.000	−0.920439
$82$	− 720.000i	− 0.969643i
$83$	− 234.000i	− 0.309456i	−0.987957	−	0.154728i	$-0.950550\pi$
$83$	− 234.000i	− 0.309456i	0.987957	−	0.154728i	$-0.0494501\pi$
$84$	55.0000	0.0714404
$85$	0	0
$86$	588.000	0.737275
$87$	1245.00i	1.53423i
$88$	1134.00i	1.37369i
$89$	168.000	0.200089	0.100045	−	0.994983i	$-0.468101\pi$
$89$	168.000	0.200089	0.100045	+	0.994983i	$0.468101\pi$
$90$	0	0
$91$	121.000	0.139387
$92$	183.000i	0.207381i
$93$	280.000i	0.312201i
$94$	−504.000	−0.553017
$95$	0	0
$96$	−225.000	−0.239208
$97$	758.000i	0.793435i	0.917941	+	0.396718i	$0.129851\pi$
$97$	758.000i	0.793435i	−0.917941	+	0.396718i	$0.870149\pi$
$98$	− 666.000i	− 0.686491i
$99$	−108.000	−0.109640
$100$	0	0
$101$	−726.000	−0.715245	−0.357622	−	0.933866i	$-0.616412\pi$
$101$	−726.000	−0.715245	−0.357622	+	0.933866i	$0.616412\pi$
$102$	− 1395.00i	− 1.35417i
$103$	− 2.00000i	− 0.00191326i	−1.00000		0.000956630i	$-0.999695\pi$
$103$	− 2.00000i	− 0.00191326i	1.00000		0.000956630i	$-0.000304505\pi$
$104$	−231.000	−0.217802
$105$	0	0
$106$	−1305.00	−1.19578
$107$	1413.00i	1.27663i	0.769773	+	0.638317i	$0.220369\pi$
$107$	1413.00i	1.27663i	−0.769773	+	0.638317i	$0.779631\pi$
$108$	− 145.000i	− 0.129191i
$109$	−389.000	−0.341830	−0.170915	−	0.985286i	$-0.554672\pi$
$109$	−389.000	−0.341830	−0.170915	+	0.985286i	$0.554672\pi$
$110$	0	0
$111$	1250.00	1.06887
$112$	− 781.000i	− 0.658907i
$113$	− 342.000i	− 0.284714i	−0.989815	−	0.142357i	$-0.954532\pi$
$113$	− 342.000i	− 0.284714i	0.989815	−	0.142357i	$-0.0454681\pi$
$114$	−285.000	−0.234146
$115$	0	0
$116$	−249.000	−0.199302
$117$	− 22.0000i	− 0.0173838i
$118$	585.000i	0.456387i
$119$	1023.00	0.788053
$120$	0	0
$121$	1585.00	1.19083
$122$	1074.00i	0.797011i
$123$	1200.00i	0.879678i
$124$	−56.0000	−0.0405560
$125$	0	0
$126$	66.0000	0.0466647
$127$	− 1150.00i	− 0.803512i	−0.915747	−	0.401756i	$-0.868400\pi$
$127$	− 1150.00i	− 0.803512i	0.915747	−	0.401756i	$-0.131600\pi$
$128$	1659.00i	1.14560i
$129$	−980.000	−0.668870
$130$	0	0
$131$	−1452.00	−0.968411	−0.484205	−	0.874954i	$-0.660891\pi$
$131$	−1452.00	−0.968411	−0.484205	+	0.874954i	$0.660891\pi$
$132$	270.000i	0.178034i
$133$	− 209.000i	− 0.136260i
$134$	−2883.00	−1.85861
$135$	0	0
$136$	−1953.00	−1.23139
$137$	− 1689.00i	− 1.05329i	−0.850085	−	0.526646i	$-0.823449\pi$
$137$	− 1689.00i	− 1.05329i	0.850085	−	0.526646i	$-0.176551\pi$
$138$	− 2745.00i	− 1.69326i
$139$	−2144.00	−1.30829	−0.654143	−	0.756371i	$-0.726970\pi$
$139$	−2144.00	−1.30829	−0.654143	+	0.756371i	$0.726970\pi$
$140$	0	0
$141$	840.000	0.501708
$142$	738.000i	0.436138i
$143$	594.000i	0.347362i
$144$	−142.000	−0.0821759
$145$	0	0
$146$	−1059.00	−0.600298
$147$	1110.00i	0.622798i
$148$	250.000i	0.138850i
$149$	3000.00	1.64946	0.824730	−	0.565527i	$-0.191327\pi$
$149$	3000.00	1.64946	0.824730	+	0.565527i	$0.191327\pi$
$150$	0	0
$151$	−1006.00	−0.542166	−0.271083	−	0.962556i	$-0.587382\pi$
$151$	−1006.00	−0.542166	−0.271083	+	0.962556i	$0.587382\pi$
$152$	399.000i	0.212916i
$153$	− 186.000i	− 0.0982824i
$154$	−1782.00	−0.932452
$155$	0	0
$156$	−55.0000	−0.0282277
$157$	2846.00i	1.44672i	0.690469	+	0.723362i	$0.257404\pi$
$157$	2846.00i	1.44672i	−0.690469	+	0.723362i	$0.742596\pi$
$158$	− 102.000i	− 0.0513588i
$159$	2175.00	1.08483
$160$	0	0
$161$	2013.00	0.985383
$162$	2013.00i	0.976273i
$163$	1600.00i	0.768845i	0.923157	+	0.384422i	$0.125599\pi$
$163$	1600.00i	0.768845i	−0.923157	+	0.384422i	$0.874401\pi$
$164$	−240.000	−0.114273
$165$	0	0
$166$	−702.000	−0.328228
$167$	− 2004.00i	− 0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$	− 2004.00i	− 0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$	1155.00i	0.530418i
$169$	2076.00	0.944925
$170$	0	0
$171$	−38.0000	−0.0169938
$172$	− 196.000i	− 0.0868887i
$173$	462.000i	0.203036i	0.994834	+	0.101518i	$0.0323699\pi$
$173$	462.000i	0.203036i	−0.994834	+	0.101518i	$0.967630\pi$
$174$	3735.00	1.62730
$175$	0	0
$176$	3834.00	1.64204
$177$	− 975.000i	− 0.414042i
$178$	− 504.000i	− 0.212227i
$179$	−720.000	−0.300644	−0.150322	−	0.988637i	$-0.548031\pi$
$179$	−720.000	−0.300644	−0.150322	+	0.988637i	$0.548031\pi$
$180$	0	0
$181$	−2338.00	−0.960122	−0.480061	−	0.877235i	$-0.659386\pi$
$181$	−2338.00	−0.960122	−0.480061	+	0.877235i	$0.659386\pi$
$182$	− 363.000i	− 0.147843i
$183$	− 1790.00i	− 0.723063i
$184$	−3843.00	−1.53973
$185$	0	0
$186$	840.000	0.331139
$187$	5022.00i	1.96388i
$188$	168.000i	0.0651737i
$189$	−1595.00	−0.613858
$190$	0	0
$191$	2871.00	1.08763	0.543817	−	0.839204i	$-0.316978\pi$
$191$	2871.00	1.08763	0.543817	+	0.839204i	$0.316978\pi$
$192$	− 2165.00i	− 0.813778i
$193$	− 1658.00i	− 0.618370i	−0.951002	−	0.309185i	$-0.899944\pi$
$193$	− 1658.00i	− 0.618370i	0.951002	−	0.309185i	$-0.100056\pi$
$194$	2274.00	0.841565
$195$	0	0
$196$	−222.000	−0.0809038
$197$	− 4176.00i	− 1.51029i	−0.655556	−	0.755146i	$-0.727566\pi$
$197$	− 4176.00i	− 1.51029i	0.655556	−	0.755146i	$-0.272434\pi$
$198$	324.000i	0.116291i
$199$	241.000	0.0858494	0.0429247	−	0.999078i	$-0.486332\pi$
$199$	241.000	0.0858494	0.0429247	+	0.999078i	$0.486332\pi$
$200$	0	0
$201$	4805.00	1.68616
$202$	2178.00i	0.758631i
$203$	2739.00i	0.946996i
$204$	−465.000	−0.159591
$205$	0	0
$206$	−6.00000	−0.00202932
$207$	− 366.000i	− 0.122893i
$208$	781.000i	0.260349i
$209$	1026.00	0.339569
$210$	0	0
$211$	−745.000	−0.243071	−0.121535	−	0.992587i	$-0.538782\pi$
$211$	−745.000	−0.243071	−0.121535	+	0.992587i	$0.538782\pi$
$212$	435.000i	0.140924i
$213$	− 1230.00i	− 0.395672i
$214$	4239.00	1.35408
$215$	0	0
$216$	3045.00	0.959194
$217$	616.000i	0.192704i
$218$	1167.00i	0.362565i
$219$	1765.00	0.544601
$220$	0	0
$221$	−1023.00	−0.311377
$222$	− 3750.00i	− 1.13371i
$223$	1978.00i	0.593976i	0.954881	+	0.296988i	$0.0959822\pi$
$223$	1978.00i	0.593976i	−0.954881	+	0.296988i	$0.904018\pi$
$224$	−495.000	−0.147650
$225$	0	0
$226$	−1026.00	−0.301985
$227$	5355.00i	1.56574i	0.622183	+	0.782872i	$0.286246\pi$
$227$	5355.00i	1.56574i	−0.622183	+	0.782872i	$0.713754\pi$
$228$	95.0000i	0.0275944i
$229$	6370.00	1.83817	0.919086	−	0.394057i	$-0.128929\pi$
$229$	6370.00	1.83817	0.919086	+	0.394057i	$0.128929\pi$
$230$	0	0
$231$	2970.00	0.845938
$232$	− 5229.00i	− 1.47974i
$233$	2838.00i	0.797955i	0.916961	+	0.398978i	$0.130635\pi$
$233$	2838.00i	0.797955i	−0.916961	+	0.398978i	$0.869365\pi$
$234$	−66.0000	−0.0184383
$235$	0	0
$236$	195.000	0.0537857
$237$	170.000i	0.0465936i
$238$	− 3069.00i	− 0.835856i
$239$	369.000	0.0998687	0.0499344	−	0.998753i	$-0.484099\pi$
$239$	369.000	0.0998687	0.0499344	+	0.998753i	$0.484099\pi$
$240$	0	0
$241$	6608.00	1.76622	0.883109	−	0.469167i	$-0.155446\pi$
$241$	6608.00	1.76622	0.883109	+	0.469167i	$0.155446\pi$
$242$	− 4755.00i	− 1.26307i
$243$	560.000i	0.147835i
$244$	358.000	0.0939287
$245$	0	0
$246$	3600.00	0.933039
$247$	209.000i	0.0538395i
$248$	− 1176.00i	− 0.301113i
$249$	1170.00	0.297774
$250$	0	0
$251$	4674.00	1.17538	0.587690	−	0.809086i	$-0.300038\pi$
$251$	4674.00	1.17538	0.587690	+	0.809086i	$0.300038\pi$
$252$	− 22.0000i	− 0.00549948i
$253$	9882.00i	2.45564i
$254$	−3450.00	−0.852253
$255$	0	0
$256$	1513.00	0.369385
$257$	4512.00i	1.09514i	0.836760	+	0.547570i	$0.184447\pi$
$257$	4512.00i	1.09514i	−0.836760	+	0.547570i	$0.815553\pi$
$258$	2940.00i	0.709443i
$259$	2750.00	0.659756
$260$	0	0
$261$	498.000	0.118105
$262$	4356.00i	1.02715i
$263$	− 3768.00i	− 0.883440i	−0.897153	−	0.441720i	$-0.854368\pi$
$263$	− 3768.00i	− 0.883440i	0.897153	−	0.441720i	$-0.145632\pi$
$264$	−5670.00	−1.32183
$265$	0	0
$266$	−627.000	−0.144526
$267$	840.000i	0.192536i
$268$	961.000i	0.219039i
$269$	−4758.00	−1.07844	−0.539220	−	0.842165i	$-0.681281\pi$
$269$	−4758.00	−1.07844	−0.539220	+	0.842165i	$0.681281\pi$
$270$	0	0
$271$	−2041.00	−0.457498	−0.228749	−	0.973485i	$-0.573463\pi$
$271$	−2041.00	−0.457498	−0.228749	+	0.973485i	$0.573463\pi$
$272$	6603.00i	1.47193i
$273$	605.000i	0.134126i
$274$	−5067.00	−1.11718
$275$	0	0
$276$	−915.000	−0.199553
$277$	1964.00i	0.426012i	0.977051	+	0.213006i	$0.0683254\pi$
$277$	1964.00i	0.426012i	−0.977051	+	0.213006i	$0.931675\pi$
$278$	6432.00i	1.38765i
$279$	112.000	0.0240332
$280$	0	0
$281$	−5496.00	−1.16678	−0.583388	−	0.812194i	$-0.698273\pi$
$281$	−5496.00	−1.16678	−0.583388	+	0.812194i	$0.698273\pi$
$282$	− 2520.00i	− 0.532141i
$283$	− 3098.00i	− 0.650731i	−0.945588	−	0.325366i	$-0.894513\pi$
$283$	− 3098.00i	− 0.650731i	0.945588	−	0.325366i	$-0.105487\pi$
$284$	246.000	0.0513993
$285$	0	0
$286$	1782.00	0.368433
$287$	2640.00i	0.542977i
$288$	90.0000i	0.0184142i
$289$	−3736.00	−0.760432
$290$	0	0
$291$	−3790.00	−0.763484
$292$	353.000i	0.0707458i
$293$	− 117.000i	− 0.0233284i	−0.999932	−	0.0116642i	$-0.996287\pi$
$293$	− 117.000i	− 0.0233284i	0.999932	−	0.0116642i	$-0.00371291\pi$
$294$	3330.00	0.660577
$295$	0	0
$296$	−5250.00	−1.03091
$297$	− 7830.00i	− 1.52977i
$298$	− 9000.00i	− 1.74952i
$299$	−2013.00	−0.389347
$300$	0	0
$301$	−2156.00	−0.412856
$302$	3018.00i	0.575054i
$303$	− 3630.00i	− 0.688244i
$304$	1349.00	0.254508
$305$	0	0
$306$	−558.000	−0.104244
$307$	− 1420.00i	− 0.263986i	−0.991251	−	0.131993i	$-0.957862\pi$
$307$	− 1420.00i	− 0.263986i	0.991251	−	0.131993i	$-0.0421376\pi$
$308$	594.000i	0.109891i
$309$	10.0000	0.00184104
$310$	0	0
$311$	−6561.00	−1.19627	−0.598135	−	0.801395i	$-0.704091\pi$
$311$	−6561.00	−1.19627	−0.598135	+	0.801395i	$0.704091\pi$
$312$	− 1155.00i	− 0.209580i
$313$	1483.00i	0.267809i	0.990994	+	0.133904i	$0.0427515\pi$
$313$	1483.00i	0.267809i	−0.990994	+	0.133904i	$0.957249\pi$
$314$	8538.00	1.53448
$315$	0	0
$316$	−34.0000	−0.00605269
$317$	− 1239.00i	− 0.219524i	−0.993958	−	0.109762i	$-0.964991\pi$
$317$	− 1239.00i	− 0.219524i	0.993958	−	0.109762i	$-0.0350089\pi$
$318$	− 6525.00i	− 1.15064i
$319$	−13446.0	−2.35997
$320$	0	0
$321$	−7065.00	−1.22844
$322$	− 6039.00i	− 1.04516i
$323$	1767.00i	0.304392i
$324$	671.000	0.115055
$325$	0	0
$326$	4800.00	0.815483
$327$	− 1945.00i	− 0.328926i
$328$	− 5040.00i	− 0.848437i
$329$	1848.00	0.309676
$330$	0	0
$331$	−8899.00	−1.47774	−0.738872	−	0.673846i	$-0.764641\pi$
$331$	−8899.00	−1.47774	−0.738872	+	0.673846i	$0.764641\pi$
$332$	234.000i	0.0386820i
$333$	− 500.000i	− 0.0822818i
$334$	−6012.00	−0.984916
$335$	0	0
$336$	3905.00	0.634033
$337$	5816.00i	0.940112i	0.882637	+	0.470056i	$0.155766\pi$
$337$	5816.00i	0.940112i	−0.882637	+	0.470056i	$0.844234\pi$
$338$	− 6228.00i	− 1.00224i
$339$	1710.00	0.273966
$340$	0	0
$341$	−3024.00	−0.480231
$342$	114.000i	0.0180246i
$343$	6215.00i	0.978363i
$344$	4116.00	0.645116
$345$	0	0
$346$	1386.00	0.215352
$347$	− 1578.00i	− 0.244125i	−0.992522	−	0.122063i	$-0.961049\pi$
$347$	− 1578.00i	− 0.244125i	0.992522	−	0.122063i	$-0.0389509\pi$
$348$	− 1245.00i	− 0.191779i
$349$	−1658.00	−0.254300	−0.127150	−	0.991883i	$-0.540583\pi$
$349$	−1658.00	−0.254300	−0.127150	+	0.991883i	$0.540583\pi$
$350$	0	0
$351$	1595.00	0.242549
$352$	− 2430.00i	− 0.367953i
$353$	11367.0i	1.71389i	0.515405	+	0.856947i	$0.327641\pi$
$353$	11367.0i	1.71389i	−0.515405	+	0.856947i	$0.672359\pi$
$354$	−2925.00	−0.439158
$355$	0	0
$356$	−168.000	−0.0250112
$357$	5115.00i	0.758304i
$358$	2160.00i	0.318881i
$359$	−2553.00	−0.375326	−0.187663	−	0.982233i	$-0.560091\pi$
$359$	−2553.00	−0.375326	−0.187663	+	0.982233i	$0.560091\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	7014.00i	1.01836i
$363$	7925.00i	1.14588i
$364$	−121.000	−0.0174234
$365$	0	0
$366$	−5370.00	−0.766925
$367$	− 196.000i	− 0.0278777i	−0.999903	−	0.0139389i	$-0.995563\pi$
$367$	− 196.000i	− 0.0278777i	0.999903	−	0.0139389i	$-0.00443702\pi$
$368$	12993.0i	1.84051i
$369$	480.000	0.0677176
$370$	0	0
$371$	4785.00	0.669609
$372$	− 280.000i	− 0.0390251i
$373$	− 9353.00i	− 1.29834i	−0.760644	−	0.649169i	$-0.775117\pi$
$373$	− 9353.00i	− 1.29834i	0.760644	−	0.649169i	$-0.224883\pi$
$374$	15066.0	2.08301
$375$	0	0
$376$	−3528.00	−0.483890
$377$	− 2739.00i	− 0.374180i
$378$	4785.00i	0.651095i
$379$	−3827.00	−0.518680	−0.259340	−	0.965786i	$-0.583505\pi$
$379$	−3827.00	−0.518680	−0.259340	+	0.965786i	$0.583505\pi$
$380$	0	0
$381$	5750.00	0.773180
$382$	− 8613.00i	− 1.15361i
$383$	− 5694.00i	− 0.759660i	−0.925056	−	0.379830i	$-0.875982\pi$
$383$	− 5694.00i	− 0.759660i	0.925056	−	0.379830i	$-0.124018\pi$
$384$	−8295.00	−1.10235
$385$	0	0
$386$	−4974.00	−0.655881
$387$	392.000i	0.0514896i
$388$	− 758.000i	− 0.0991794i
$389$	−1290.00	−0.168138	−0.0840689	−	0.996460i	$-0.526792\pi$
$389$	−1290.00	−0.168138	−0.0840689	+	0.996460i	$0.526792\pi$
$390$	0	0
$391$	−17019.0	−2.20125
$392$	− 4662.00i	− 0.600680i
$393$	− 7260.00i	− 0.931854i
$394$	−12528.0	−1.60191
$395$	0	0
$396$	108.000	0.0137051
$397$	6536.00i	0.826278i	0.910668	+	0.413139i	$0.135568\pi$
$397$	6536.00i	0.826278i	−0.910668	+	0.413139i	$0.864432\pi$
$398$	− 723.000i	− 0.0910571i
$399$	1045.00	0.131116
$400$	0	0
$401$	2328.00	0.289912	0.144956	−	0.989438i	$-0.453696\pi$
$401$	2328.00	0.289912	0.144956	+	0.989438i	$0.453696\pi$
$402$	− 14415.0i	− 1.78844i
$403$	− 616.000i	− 0.0761418i
$404$	726.000	0.0894056
$405$	0	0
$406$	8217.00	1.00444
$407$	13500.0i	1.64415i
$408$	− 9765.00i	− 1.18490i
$409$	6676.00	0.807107	0.403554	−	0.914956i	$-0.367775\pi$
$409$	6676.00	0.807107	0.403554	+	0.914956i	$0.367775\pi$
$410$	0	0
$411$	8445.00	1.01353
$412$	2.00000i	0 0.000239158i
$413$	− 2145.00i	− 0.255565i
$414$	−1098.00	−0.130347
$415$	0	0
$416$	495.000	0.0583398
$417$	− 10720.0i	− 1.25890i
$418$	− 3078.00i	− 0.360167i
$419$	8136.00	0.948615	0.474307	−	0.880359i	$-0.342699\pi$
$419$	8136.00	0.948615	0.474307	+	0.880359i	$0.342699\pi$
$420$	0	0
$421$	−8665.00	−1.00310	−0.501551	−	0.865128i	$-0.667237\pi$
$421$	−8665.00	−1.00310	−0.501551	+	0.865128i	$0.667237\pi$
$422$	2235.00i	0.257815i
$423$	− 336.000i	− 0.0386215i
$424$	−9135.00	−1.04631
$425$	0	0
$426$	−3690.00	−0.419674
$427$	− 3938.00i	− 0.446307i
$428$	− 1413.00i	− 0.159579i
$429$	−2970.00	−0.334249
$430$	0	0
$431$	750.000	0.0838196	0.0419098	−	0.999121i	$-0.486656\pi$
$431$	750.000	0.0838196	0.0419098	+	0.999121i	$0.486656\pi$
$432$	− 10295.0i	− 1.14657i
$433$	4858.00i	0.539170i	0.962977	+	0.269585i	$0.0868865\pi$
$433$	4858.00i	0.539170i	−0.962977	+	0.269585i	$0.913113\pi$
$434$	1848.00	0.204394
$435$	0	0
$436$	389.000	0.0427287
$437$	3477.00i	0.380612i
$438$	− 5295.00i	− 0.577637i
$439$	−6500.00	−0.706670	−0.353335	−	0.935497i	$-0.614952\pi$
$439$	−6500.00	−0.706670	−0.353335	+	0.935497i	$0.614952\pi$
$440$	0	0
$441$	444.000	0.0479430
$442$	3069.00i	0.330266i
$443$	− 3486.00i	− 0.373871i	−0.982372	−	0.186936i	$-0.940144\pi$
$443$	− 3486.00i	− 0.373871i	0.982372	−	0.186936i	$-0.0598555\pi$
$444$	−1250.00	−0.133609
$445$	0	0
$446$	5934.00	0.630007
$447$	15000.0i	1.58719i
$448$	− 4763.00i	− 0.502300i
$449$	15030.0	1.57975	0.789877	−	0.613265i	$-0.210144\pi$
$449$	15030.0	1.57975	0.789877	+	0.613265i	$0.210144\pi$
$450$	0	0
$451$	−12960.0	−1.35313
$452$	342.000i	0.0355892i
$453$	− 5030.00i	− 0.521700i
$454$	16065.0	1.66072
$455$	0	0
$456$	−1995.00	−0.204878
$457$	− 2959.00i	− 0.302880i	−0.988466	−	0.151440i	$-0.951609\pi$
$457$	− 2959.00i	− 0.302880i	0.988466	−	0.151440i	$-0.0483910\pi$
$458$	− 19110.0i	− 1.94968i
$459$	13485.0	1.37130
$460$	0	0
$461$	−156.000	−0.0157606	−0.00788031	−	0.999969i	$-0.502508\pi$
$461$	−156.000	−0.0157606	−0.00788031	+	0.999969i	$0.502508\pi$
$462$	− 8910.00i	− 0.897253i
$463$	− 4484.00i	− 0.450085i	−0.974349	−	0.225042i	$-0.927748\pi$
$463$	− 4484.00i	− 0.450085i	0.974349	−	0.225042i	$-0.0722520\pi$
$464$	−17679.0	−1.76881
$465$	0	0
$466$	8514.00	0.846359
$467$	8766.00i	0.868613i	0.900765	+	0.434306i	$0.143006\pi$
$467$	8766.00i	0.868613i	−0.900765	+	0.434306i	$0.856994\pi$
$468$	22.0000i	0.00217297i
$469$	10571.0	1.04077
$470$	0	0
$471$	−14230.0	−1.39211
$472$	4095.00i	0.399338i
$473$	− 10584.0i	− 1.02886i
$474$	510.000	0.0494200
$475$	0	0
$476$	−1023.00	−0.0985066
$477$	− 870.000i	− 0.0835106i
$478$	− 1107.00i	− 0.105927i
$479$	18996.0	1.81200	0.906001	−	0.423275i	$-0.139119\pi$
$479$	18996.0	1.81200	0.906001	+	0.423275i	$0.139119\pi$
$480$	0	0
$481$	−2750.00	−0.260684
$482$	− 19824.0i	− 1.87336i
$483$	10065.0i	0.948185i
$484$	−1585.00	−0.148854
$485$	0	0
$486$	1680.00	0.156803
$487$	− 7450.00i	− 0.693207i	−0.938012	−	0.346603i	$-0.887335\pi$
$487$	− 7450.00i	− 0.693207i	0.938012	−	0.346603i	$-0.112665\pi$
$488$	7518.00i	0.697385i
$489$	−8000.00	−0.739821
$490$	0	0
$491$	6180.00	0.568023	0.284012	−	0.958821i	$-0.408335\pi$
$491$	6180.00	0.568023	0.284012	+	0.958821i	$0.408335\pi$
$492$	− 1200.00i	− 0.109960i
$493$	− 23157.0i	− 2.11549i
$494$	627.000	0.0571054
$495$	0	0
$496$	−3976.00	−0.359935
$497$	− 2706.00i	− 0.244227i
$498$	− 3510.00i	− 0.315837i
$499$	−2576.00	−0.231097	−0.115549	−	0.993302i	$-0.536863\pi$
$499$	−2576.00	−0.231097	−0.115549	+	0.993302i	$0.536863\pi$
$500$	0	0
$501$	10020.0	0.893534
$502$	− 14022.0i	− 1.24668i
$503$	10545.0i	0.934748i	0.884060	+	0.467374i	$0.154800\pi$
$503$	10545.0i	0.934748i	−0.884060	+	0.467374i	$0.845200\pi$
$504$	462.000	0.0408316
$505$	0	0
$506$	29646.0	2.60460
$507$	10380.0i	0.909254i
$508$	1150.00i	0.100439i
$509$	14694.0	1.27957	0.639784	−	0.768555i	$-0.279024\pi$
$509$	14694.0	1.27957	0.639784	+	0.768555i	$0.279024\pi$
$510$	0	0
$511$	3883.00	0.336152
$512$	8733.00i	0.753804i
$513$	− 2755.00i	− 0.237108i
$514$	13536.0	1.16157
$515$	0	0
$516$	980.000	0.0836087
$517$	9072.00i	0.771733i
$518$	− 8250.00i	− 0.699776i
$519$	−2310.00	−0.195371
$520$	0	0
$521$	10332.0	0.868816	0.434408	−	0.900716i	$-0.356958\pi$
$521$	10332.0	0.868816	0.434408	+	0.900716i	$0.356958\pi$
$522$	− 1494.00i	− 0.125269i
$523$	− 10937.0i	− 0.914420i	−0.889359	−	0.457210i	$-0.848849\pi$
$523$	− 10937.0i	− 0.914420i	0.889359	−	0.457210i	$-0.151151\pi$
$524$	1452.00	0.121051
$525$	0	0
$526$	−11304.0	−0.937030
$527$	− 5208.00i	− 0.430482i
$528$	19170.0i	1.58005i
$529$	−21322.0	−1.75245
$530$	0	0
$531$	−390.000	−0.0318730
$532$	209.000i	0.0170325i
$533$	− 2640.00i	− 0.214542i
$534$	2520.00	0.204215
$535$	0	0
$536$	−20181.0	−1.62628
$537$	− 3600.00i	− 0.289295i
$538$	14274.0i	1.14386i
$539$	−11988.0	−0.957996
$540$	0	0
$541$	18578.0	1.47640	0.738198	−	0.674584i	$-0.235677\pi$
$541$	18578.0	1.47640	0.738198	+	0.674584i	$0.235677\pi$
$542$	6123.00i	0.485250i
$543$	− 11690.0i	− 0.923878i
$544$	4185.00	0.329835
$545$	0	0
$546$	1815.00	0.142262
$547$	21404.0i	1.67307i	0.547914	+	0.836535i	$0.315422\pi$
$547$	21404.0i	1.67307i	−0.547914	+	0.836535i	$0.684578\pi$
$548$	1689.00i	0.131662i
$549$	−716.000	−0.0556614
$550$	0	0
$551$	−4731.00	−0.365785
$552$	− 19215.0i	− 1.48160i
$553$	374.000i	0.0287597i
$554$	5892.00	0.451854
$555$	0	0
$556$	2144.00	0.163536
$557$	− 3948.00i	− 0.300327i	−0.988661	−	0.150163i	$-0.952020\pi$
$557$	− 3948.00i	− 0.300327i	0.988661	−	0.150163i	$-0.0479800\pi$
$558$	− 336.000i	− 0.0254911i
$559$	2156.00	0.163129
$560$	0	0
$561$	−25110.0	−1.88974
$562$	16488.0i	1.23755i
$563$	− 5724.00i	− 0.428486i	−0.976780	−	0.214243i	$-0.931271\pi$
$563$	− 5724.00i	− 0.428486i	0.976780	−	0.214243i	$-0.0687285\pi$
$564$	−840.000	−0.0627134
$565$	0	0
$566$	−9294.00	−0.690205
$567$	− 7381.00i	− 0.546689i
$568$	5166.00i	0.381621i
$569$	20592.0	1.51716	0.758578	−	0.651582i	$-0.225895\pi$
$569$	20592.0	1.51716	0.758578	+	0.651582i	$0.225895\pi$
$570$	0	0
$571$	20684.0	1.51593	0.757967	−	0.652293i	$-0.226193\pi$
$571$	20684.0	1.51593	0.757967	+	0.652293i	$0.226193\pi$
$572$	− 594.000i	− 0.0434203i
$573$	14355.0i	1.04658i
$574$	7920.00	0.575914
$575$	0	0
$576$	−866.000	−0.0626447
$577$	− 19573.0i	− 1.41219i	−0.708116	−	0.706096i	$-0.750455\pi$
$577$	− 19573.0i	− 1.41219i	0.708116	−	0.706096i	$-0.249545\pi$
$578$	11208.0i	0.806559i
$579$	8290.00	0.595027
$580$	0	0
$581$	2574.00	0.183800
$582$	11370.0i	0.809797i
$583$	23490.0i	1.66871i
$584$	−7413.00	−0.525260
$585$	0	0
$586$	−351.000	−0.0247435
$587$	13524.0i	0.950929i	0.879735	+	0.475464i	$0.157720\pi$
$587$	13524.0i	0.950929i	−0.879735	+	0.475464i	$0.842280\pi$
$588$	− 1110.00i	− 0.0778497i
$589$	−1064.00	−0.0744335
$590$	0	0
$591$	20880.0	1.45328
$592$	17750.0i	1.23230i
$593$	− 8994.00i	− 0.622832i	−0.950274	−	0.311416i	$-0.899197\pi$
$593$	− 8994.00i	− 0.622832i	0.950274	−	0.311416i	$-0.100803\pi$
$594$	−23490.0	−1.62257
$595$	0	0
$596$	−3000.00	−0.206183
$597$	1205.00i	0.0826087i
$598$	6039.00i	0.412965i
$599$	−10128.0	−0.690850	−0.345425	−	0.938446i	$-0.612265\pi$
$599$	−10128.0	−0.690850	−0.345425	+	0.938446i	$0.612265\pi$
$600$	0	0
$601$	−22696.0	−1.54041	−0.770207	−	0.637794i	$-0.779847\pi$
$601$	−22696.0	−1.54041	−0.770207	+	0.637794i	$0.779847\pi$
$602$	6468.00i	0.437900i
$603$	− 1922.00i	− 0.129801i
$604$	1006.00	0.0677708
$605$	0	0
$606$	−10890.0	−0.729993
$607$	− 5182.00i	− 0.346509i	−0.984877	−	0.173254i	$-0.944572\pi$
$607$	− 5182.00i	− 0.346509i	0.984877	−	0.173254i	$-0.0554283\pi$
$608$	− 855.000i	− 0.0570310i
$609$	−13695.0	−0.911247
$610$	0	0
$611$	−1848.00	−0.122360
$612$	186.000i	0.0122853i
$613$	− 10082.0i	− 0.664287i	−0.943229	−	0.332144i	$-0.892228\pi$
$613$	− 10082.0i	− 0.664287i	0.943229	−	0.332144i	$-0.107772\pi$
$614$	−4260.00	−0.279999
$615$	0	0
$616$	−12474.0	−0.815896
$617$	− 12174.0i	− 0.794338i	−0.917745	−	0.397169i	$-0.869993\pi$
$617$	− 12174.0i	− 0.794338i	0.917745	−	0.397169i	$-0.130007\pi$
$618$	− 30.0000i	− 0.00195271i
$619$	−7490.00	−0.486347	−0.243173	−	0.969983i	$-0.578188\pi$
$619$	−7490.00	−0.486347	−0.243173	+	0.969983i	$0.578188\pi$
$620$	0	0
$621$	26535.0	1.71467
$622$	19683.0i	1.26884i
$623$	1848.00i	0.118842i
$624$	−3905.00	−0.250521
$625$	0	0
$626$	4449.00	0.284054
$627$	5130.00i	0.326750i
$628$	− 2846.00i	− 0.180840i
$629$	−23250.0	−1.47383
$630$	0	0
$631$	11072.0	0.698525	0.349263	−	0.937025i	$-0.386432\pi$
$631$	11072.0	0.698525	0.349263	+	0.937025i	$0.386432\pi$
$632$	− 714.000i	− 0.0449389i
$633$	− 3725.00i	− 0.233895i
$634$	−3717.00	−0.232841
$635$	0	0
$636$	−2175.00	−0.135604
$637$	− 2442.00i	− 0.151893i
$638$	40338.0i	2.50313i
$639$	−492.000	−0.0304589
$640$	0	0
$641$	−18894.0	−1.16422	−0.582112	−	0.813108i	$-0.697774\pi$
$641$	−18894.0	−1.16422	−0.582112	+	0.813108i	$0.697774\pi$
$642$	21195.0i	1.30296i
$643$	19834.0i	1.21645i	0.793765	+	0.608224i	$0.208118\pi$
$643$	19834.0i	1.21645i	−0.793765	+	0.608224i	$0.791882\pi$
$644$	−2013.00	−0.123173
$645$	0	0
$646$	5301.00	0.322856
$647$	3375.00i	0.205077i	0.994729	+	0.102539i	$0.0326965\pi$
$647$	3375.00i	0.205077i	−0.994729	+	0.102539i	$0.967303\pi$
$648$	14091.0i	0.854239i
$649$	10530.0	0.636885
$650$	0	0
$651$	−3080.00	−0.185430
$652$	− 1600.00i	− 0.0961056i
$653$	24948.0i	1.49509i	0.664214	+	0.747543i	$0.268766\pi$
$653$	24948.0i	1.49509i	−0.664214	+	0.747543i	$0.731234\pi$
$654$	−5835.00	−0.348879
$655$	0	0
$656$	−17040.0	−1.01418
$657$	− 706.000i	− 0.0419234i
$658$	− 5544.00i	− 0.328461i
$659$	9879.00	0.583962	0.291981	−	0.956424i	$-0.405686\pi$
$659$	9879.00	0.583962	0.291981	+	0.956424i	$0.405686\pi$
$660$	0	0
$661$	−14155.0	−0.832928	−0.416464	−	0.909152i	$-0.636731\pi$
$661$	−14155.0	−0.832928	−0.416464	+	0.909152i	$0.636731\pi$
$662$	26697.0i	1.56738i
$663$	− 5115.00i	− 0.299623i
$664$	−4914.00	−0.287199
$665$	0	0
$666$	−1500.00	−0.0872730
$667$	− 45567.0i	− 2.64522i
$668$	2004.00i	0.116073i
$669$	−9890.00	−0.571554
$670$	0	0
$671$	19332.0	1.11223
$672$	− 2475.00i	− 0.142076i
$673$	− 8948.00i	− 0.512511i	−0.966609	−	0.256256i	$-0.917511\pi$
$673$	− 8948.00i	− 0.512511i	0.966609	−	0.256256i	$-0.0824889\pi$
$674$	17448.0	0.997139
$675$	0	0
$676$	−2076.00	−0.118116
$677$	− 11511.0i	− 0.653477i	−0.945115	−	0.326738i	$-0.894050\pi$
$677$	− 11511.0i	− 0.653477i	0.945115	−	0.326738i	$-0.105950\pi$
$678$	− 5130.00i	− 0.290585i
$679$	−8338.00	−0.471256
$680$	0	0
$681$	−26775.0	−1.50664
$682$	9072.00i	0.509362i
$683$	10476.0i	0.586900i	0.955974	+	0.293450i	$0.0948035\pi$
$683$	10476.0i	0.586900i	−0.955974	+	0.293450i	$0.905197\pi$
$684$	38.0000	0.00212422
$685$	0	0
$686$	18645.0	1.03771
$687$	31850.0i	1.76878i
$688$	− 13916.0i	− 0.771137i
$689$	−4785.00	−0.264578
$690$	0	0
$691$	30098.0	1.65699	0.828496	−	0.559995i	$-0.189197\pi$
$691$	30098.0	1.65699	0.828496	+	0.559995i	$0.189197\pi$
$692$	− 462.000i	− 0.0253795i
$693$	− 1188.00i	− 0.0651203i
$694$	−4734.00	−0.258934
$695$	0	0
$696$	26145.0	1.42388
$697$	− 22320.0i	− 1.21296i
$698$	4974.00i	0.269726i
$699$	−14190.0	−0.767833
$700$	0	0
$701$	−14700.0	−0.792028	−0.396014	−	0.918245i	$-0.629607\pi$
$701$	−14700.0	−0.792028	−0.396014	+	0.918245i	$0.629607\pi$
$702$	− 4785.00i	− 0.257262i
$703$	4750.00i	0.254836i
$704$	23382.0	1.25176
$705$	0	0
$706$	34101.0	1.81786
$707$	− 7986.00i	− 0.424815i
$708$	975.000i	0.0517553i
$709$	−31178.0	−1.65150	−0.825751	−	0.564035i	$-0.809248\pi$
$709$	−31178.0	−1.65150	−0.825751	+	0.564035i	$0.809248\pi$
$710$	0	0
$711$	68.0000	0.00358678
$712$	− 3528.00i	− 0.185699i
$713$	− 10248.0i	− 0.538276i
$714$	15345.0	0.804303
$715$	0	0
$716$	720.000	0.0375805
$717$	1845.00i	0.0960987i
$718$	7659.00i	0.398094i
$719$	33285.0	1.72645	0.863227	−	0.504815i	$-0.168439\pi$
$719$	33285.0	1.72645	0.863227	+	0.504815i	$0.168439\pi$
$720$	0	0
$721$	22.0000	0.00113637
$722$	− 1083.00i	− 0.0558242i
$723$	33040.0i	1.69954i
$724$	2338.00	0.120015
$725$	0	0
$726$	23775.0	1.21539
$727$	− 34729.0i	− 1.77170i	−0.463970	−	0.885851i	$-0.653575\pi$
$727$	− 34729.0i	− 1.77170i	0.463970	−	0.885851i	$-0.346425\pi$
$728$	− 2541.00i	− 0.129362i
$729$	−20917.0	−1.06269
$730$	0	0
$731$	18228.0	0.922280
$732$	1790.00i	0.0903829i
$733$	− 4196.00i	− 0.211436i	−0.994396	−	0.105718i	$-0.966286\pi$
$733$	− 4196.00i	− 0.211436i	0.994396	−	0.105718i	$-0.0337141\pi$
$734$	−588.000	−0.0295688
$735$	0	0
$736$	8235.00	0.412427
$737$	51894.0i	2.59368i
$738$	− 1440.00i	− 0.0718254i
$739$	10744.0	0.534810	0.267405	−	0.963584i	$-0.413834\pi$
$739$	10744.0	0.534810	0.267405	+	0.963584i	$0.413834\pi$
$740$	0	0
$741$	−1045.00	−0.0518071
$742$	− 14355.0i	− 0.710227i
$743$	2208.00i	0.109022i	0.998513	+	0.0545112i	$0.0173601\pi$
$743$	2208.00i	0.109022i	−0.998513	+	0.0545112i	$0.982640\pi$
$744$	5880.00	0.289746
$745$	0	0
$746$	−28059.0	−1.37710
$747$	− 468.000i	− 0.0229227i
$748$	− 5022.00i	− 0.245485i
$749$	−15543.0	−0.758249
$750$	0	0
$751$	13160.0	0.639434	0.319717	−	0.947513i	$-0.396412\pi$
$751$	13160.0	0.639434	0.319717	+	0.947513i	$0.396412\pi$
$752$	11928.0i	0.578417i
$753$	23370.0i	1.13101i
$754$	−8217.00	−0.396877
$755$	0	0
$756$	1595.00	0.0767323
$757$	758.000i	0.0363936i	0.999834	+	0.0181968i	$0.00579255\pi$
$757$	758.000i	0.0363936i	−0.999834	+	0.0181968i	$0.994207\pi$
$758$	11481.0i	0.550143i
$759$	−49410.0	−2.36294
$760$	0	0
$761$	4851.00	0.231076	0.115538	−	0.993303i	$-0.463141\pi$
$761$	4851.00	0.231076	0.115538	+	0.993303i	$0.463141\pi$
$762$	− 17250.0i	− 0.820081i
$763$	− 4279.00i	− 0.203028i
$764$	−2871.00	−0.135954
$765$	0	0
$766$	−17082.0	−0.805741
$767$	2145.00i	0.100980i
$768$	7565.00i	0.355441i
$769$	33091.0	1.55175	0.775873	−	0.630890i	$-0.217310\pi$
$769$	33091.0	1.55175	0.775873	+	0.630890i	$0.217310\pi$
$770$	0	0
$771$	−22560.0	−1.05380
$772$	1658.00i	0.0772963i
$773$	− 42357.0i	− 1.97086i	−0.170079	−	0.985430i	$-0.554402\pi$
$773$	− 42357.0i	− 1.97086i	0.170079	−	0.985430i	$-0.445598\pi$
$774$	1176.00	0.0546130
$775$	0	0
$776$	15918.0	0.736370
$777$	13750.0i	0.634850i
$778$	3870.00i	0.178337i
$779$	−4560.00	−0.209729
$780$	0	0
$781$	13284.0	0.608629
$782$	51057.0i	2.33478i
$783$	36105.0i	1.64788i
$784$	−15762.0	−0.718021
$785$	0	0
$786$	−21780.0	−0.988380
$787$	− 39877.0i	− 1.80618i	−0.429454	−	0.903089i	$-0.641294\pi$
$787$	− 39877.0i	− 1.80618i	0.429454	−	0.903089i	$-0.358706\pi$
$788$	4176.00i	0.188787i
$789$	18840.0	0.850091
$790$	0	0
$791$	3762.00	0.169104
$792$	2268.00i	0.101755i
$793$	3938.00i	0.176346i
$794$	19608.0	0.876400
$795$	0	0
$796$	−241.000	−0.0107312
$797$	− 30033.0i	− 1.33478i	−0.744706	−	0.667392i	$-0.767410\pi$
$797$	− 30033.0i	− 1.33478i	0.744706	−	0.667392i	$-0.232590\pi$
$798$	− 3135.00i	− 0.139070i
$799$	−15624.0	−0.691786
$800$	0	0
$801$	336.000	0.0148214
$802$	− 6984.00i	− 0.307498i
$803$	19062.0i	0.837713i
$804$	−4805.00	−0.210770
$805$	0	0
$806$	−1848.00	−0.0807606
$807$	− 23790.0i	− 1.03773i
$808$	15246.0i	0.663802i
$809$	−585.000	−0.0254234	−0.0127117	−	0.999919i	$-0.504046\pi$
$809$	−585.000	−0.0254234	−0.0127117	+	0.999919i	$0.504046\pi$
$810$	0	0
$811$	28361.0	1.22798	0.613989	−	0.789315i	$-0.289564\pi$
$811$	28361.0	1.22798	0.613989	+	0.789315i	$0.289564\pi$
$812$	− 2739.00i	− 0.118374i
$813$	− 10205.0i	− 0.440228i
$814$	40500.0	1.74389
$815$	0	0
$816$	−33015.0	−1.41637
$817$	− 3724.00i	− 0.159469i
$818$	− 20028.0i	− 0.856067i
$819$	242.000	0.0103250
$820$	0	0
$821$	25068.0	1.06563	0.532813	−	0.846233i	$-0.321135\pi$
$821$	25068.0	1.06563	0.532813	+	0.846233i	$0.321135\pi$
$822$	− 25335.0i	− 1.07501i
$823$	− 10901.0i	− 0.461707i	−0.972989	−	0.230854i	$-0.925848\pi$
$823$	− 10901.0i	− 0.461707i	0.972989	−	0.230854i	$-0.0741518\pi$
$824$	−42.0000	−0.00177565
$825$	0	0
$826$	−6435.00	−0.271068
$827$	12027.0i	0.505707i	0.967505	+	0.252854i	$0.0813691\pi$
$827$	12027.0i	0.505707i	−0.967505	+	0.252854i	$0.918631\pi$
$828$	366.000i	0.0153616i
$829$	19339.0	0.810219	0.405109	−	0.914268i	$-0.367233\pi$
$829$	19339.0	0.810219	0.405109	+	0.914268i	$0.367233\pi$
$830$	0	0
$831$	−9820.00	−0.409930
$832$	4763.00i	0.198470i
$833$	− 20646.0i	− 0.858753i
$834$	−32160.0	−1.33526
$835$	0	0
$836$	−1026.00	−0.0424461
$837$	8120.00i	0.335326i
$838$	− 24408.0i	− 1.00616i
$839$	13188.0	0.542670	0.271335	−	0.962485i	$-0.412535\pi$
$839$	13188.0	0.542670	0.271335	+	0.962485i	$0.412535\pi$
$840$	0	0
$841$	37612.0	1.54217
$842$	25995.0i	1.06395i
$843$	− 27480.0i	− 1.12273i
$844$	745.000	0.0303838
$845$	0	0
$846$	−1008.00	−0.0409642
$847$	17435.0i	0.707289i
$848$	30885.0i	1.25070i
$849$	15490.0	0.626167
$850$	0	0
$851$	−45750.0	−1.84288
$852$	1230.00i	0.0494590i
$853$	4678.00i	0.187775i	0.995583	+	0.0938873i	$0.0299293\pi$
$853$	4678.00i	0.187775i	−0.995583	+	0.0938873i	$0.970071\pi$
$854$	−11814.0	−0.473380
$855$	0	0
$856$	29673.0	1.18482
$857$	15252.0i	0.607933i	0.952683	+	0.303966i	$0.0983111\pi$
$857$	15252.0i	0.607933i	−0.952683	+	0.303966i	$0.901689\pi$
$858$	8910.00i	0.354525i
$859$	610.000	0.0242293	0.0121146	−	0.999927i	$-0.496144\pi$
$859$	610.000	0.0242293	0.0121146	+	0.999927i	$0.496144\pi$
$860$	0	0
$861$	−13200.0	−0.522479
$862$	− 2250.00i	− 0.0889041i
$863$	− 774.000i	− 0.0305299i	−0.999883	−	0.0152649i	$-0.995141\pi$
$863$	− 774.000i	− 0.0305299i	0.999883	−	0.0152649i	$-0.00485917\pi$
$864$	−6525.00	−0.256927
$865$	0	0
$866$	14574.0	0.571876
$867$	− 18680.0i	− 0.731726i
$868$	− 616.000i	− 0.0240880i
$869$	−1836.00	−0.0716709
$870$	0	0
$871$	−10571.0	−0.411234
$872$	8169.00i	0.317245i
$873$	1516.00i	0.0587730i
$874$	10431.0	0.403700
$875$	0	0
$876$	−1765.00	−0.0680751
$877$	− 31039.0i	− 1.19511i	−0.801827	−	0.597556i	$-0.796139\pi$
$877$	− 31039.0i	− 1.19511i	0.801827	−	0.597556i	$-0.203861\pi$
$878$	19500.0i	0.749537i
$879$	585.000	0.0224477
$880$	0	0
$881$	33678.0	1.28790	0.643950	−	0.765067i	$-0.277294\pi$
$881$	33678.0	1.28790	0.643950	+	0.765067i	$0.277294\pi$
$882$	− 1332.00i	− 0.0508512i
$883$	42982.0i	1.63812i	0.573708	+	0.819060i	$0.305504\pi$
$883$	42982.0i	1.63812i	−0.573708	+	0.819060i	$0.694496\pi$
$884$	1023.00	0.0389222
$885$	0	0
$886$	−10458.0	−0.396550
$887$	4494.00i	0.170117i	0.996376	+	0.0850585i	$0.0271077\pi$
$887$	4494.00i	0.170117i	−0.996376	+	0.0850585i	$0.972892\pi$
$888$	− 26250.0i	− 0.991996i
$889$	12650.0	0.477241
$890$	0	0
$891$	36234.0	1.36238
$892$	− 1978.00i	− 0.0742470i
$893$	3192.00i	0.119615i
$894$	45000.0	1.68347
$895$	0	0
$896$	−18249.0	−0.680420
$897$	− 10065.0i	− 0.374649i
$898$	− 45090.0i	− 1.67558i
$899$	13944.0	0.517306
$900$	0	0
$901$	−40455.0	−1.49584
$902$	38880.0i	1.43521i
$903$	− 10780.0i	− 0.397271i
$904$	−7182.00	−0.264236
$905$	0	0
$906$	−15090.0	−0.553346
$907$	− 23839.0i	− 0.872724i	−0.899771	−	0.436362i	$-0.856267\pi$
$907$	− 23839.0i	− 0.872724i	0.899771	−	0.436362i	$-0.143733\pi$
$908$	− 5355.00i	− 0.195718i
$909$	−1452.00	−0.0529811
$910$	0	0
$911$	−10332.0	−0.375757	−0.187878	−	0.982192i	$-0.560161\pi$
$911$	−10332.0	−0.375757	−0.187878	+	0.982192i	$0.560161\pi$
$912$	6745.00i	0.244901i
$913$	12636.0i	0.458040i
$914$	−8877.00	−0.321253
$915$	0	0
$916$	−6370.00	−0.229772
$917$	− 15972.0i	− 0.575182i
$918$	− 40455.0i	− 1.45448i
$919$	14371.0	0.515838	0.257919	−	0.966166i	$-0.416963\pi$
$919$	14371.0	0.515838	0.257919	+	0.966166i	$0.416963\pi$
$920$	0	0
$921$	7100.00	0.254021
$922$	468.000i	0.0167167i
$923$	2706.00i	0.0964995i
$924$	−2970.00	−0.105742
$925$	0	0
$926$	−13452.0	−0.477387
$927$	− 4.00000i	0 0.000141723i
$928$	11205.0i	0.396360i
$929$	−26889.0	−0.949623	−0.474811	−	0.880088i	$-0.657484\pi$
$929$	−26889.0	−0.949623	−0.474811	+	0.880088i	$0.657484\pi$
$930$	0	0
$931$	−4218.00	−0.148485
$932$	− 2838.00i	− 0.0997444i
$933$	− 32805.0i	− 1.15111i
$934$	26298.0	0.921303
$935$	0	0
$936$	−462.000	−0.0161335
$937$	785.000i	0.0273691i	0.999906	+	0.0136845i	$0.00435606\pi$
$937$	785.000i	0.0273691i	−0.999906	+	0.0136845i	$0.995644\pi$
$938$	− 31713.0i	− 1.10391i
$939$	−7415.00	−0.257699
$940$	0	0
$941$	−18141.0	−0.628459	−0.314229	−	0.949347i	$-0.601746\pi$
$941$	−18141.0	−0.628459	−0.314229	+	0.949347i	$0.601746\pi$
$942$	42690.0i	1.47656i
$943$	− 43920.0i	− 1.51668i
$944$	13845.0	0.477348
$945$	0	0
$946$	−31752.0	−1.09128
$947$	23100.0i	0.792660i	0.918108	+	0.396330i	$0.129716\pi$
$947$	23100.0i	0.792660i	−0.918108	+	0.396330i	$0.870284\pi$
$948$	− 170.000i	− 0.00582420i
$949$	−3883.00	−0.132821
$950$	0	0
$951$	6195.00	0.211237
$952$	− 21483.0i	− 0.731374i
$953$	− 45690.0i	− 1.55304i	−0.630094	−	0.776519i	$-0.716984\pi$
$953$	− 45690.0i	− 1.55304i	0.630094	−	0.776519i	$-0.283016\pi$
$954$	−2610.00	−0.0885764
$955$	0	0
$956$	−369.000	−0.0124836
$957$	− 67230.0i	− 2.27089i
$958$	− 56988.0i	− 1.92192i
$959$	18579.0	0.625597
$960$	0	0
$961$	−26655.0	−0.894733
$962$	8250.00i	0.276498i
$963$	2826.00i	0.0945655i
$964$	−6608.00	−0.220777
$965$	0	0
$966$	30195.0	1.00570
$967$	21584.0i	0.717781i	0.933380	+	0.358891i	$0.116845\pi$
$967$	21584.0i	0.717781i	−0.933380	+	0.358891i	$0.883155\pi$
$968$	− 33285.0i	− 1.10519i
$969$	−8835.00	−0.292901
$970$	0	0
$971$	−50556.0	−1.67087	−0.835437	−	0.549586i	$-0.814786\pi$
$971$	−50556.0	−1.67087	−0.835437	+	0.549586i	$0.814786\pi$
$972$	− 560.000i	− 0.0184794i
$973$	− 23584.0i	− 0.777049i
$974$	−22350.0	−0.735257
$975$	0	0
$976$	25418.0	0.833617
$977$	8568.00i	0.280568i	0.990111	+	0.140284i	$0.0448015\pi$
$977$	8568.00i	0.280568i	−0.990111	+	0.140284i	$0.955198\pi$
$978$	24000.0i	0.784699i
$979$	−9072.00	−0.296162
$980$	0	0
$981$	−778.000	−0.0253207
$982$	− 18540.0i	− 0.602480i
$983$	− 29706.0i	− 0.963860i	−0.876210	−	0.481930i	$-0.839936\pi$
$983$	− 29706.0i	− 0.963860i	0.876210	−	0.481930i	$-0.160064\pi$
$984$	25200.0	0.816409
$985$	0	0
$986$	−69471.0	−2.24382
$987$	9240.00i	0.297986i
$988$	− 209.000i	− 0.00672993i
$989$	35868.0	1.15322
$990$	0	0
$991$	30512.0	0.978048	0.489024	−	0.872270i	$-0.337353\pi$
$991$	30512.0	0.978048	0.489024	+	0.872270i	$0.337353\pi$
$992$	2520.00i	0.0806553i
$993$	− 44495.0i	− 1.42196i
$994$	−8118.00	−0.259042
$995$	0	0
$996$	−1170.00	−0.0372218
$997$	47756.0i	1.51700i	0.651674	+	0.758499i	$0.274067\pi$
$997$	47756.0i	1.51700i	−0.651674	+	0.758499i	$0.725933\pi$
$998$	7728.00i	0.245116i
$999$	36250.0	1.14805

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	475.4.b.c.324.1		2
5.2	odd	4		475.4.a.e.1.1		1
5.3	odd	4		19.4.a.a.1.1	✓	1
5.4	even	2	inner	475.4.b.c.324.2		2
15.8	even	4		171.4.a.d.1.1		1
20.3	even	4		304.4.a.b.1.1		1
35.13	even	4		931.4.a.a.1.1		1
40.3	even	4		1216.4.a.a.1.1		1
40.13	odd	4		1216.4.a.f.1.1		1
55.43	even	4		2299.4.a.b.1.1		1
95.18	even	4		361.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.4.a.a.1.1	✓	1	5.3	odd	4
171.4.a.d.1.1		1	15.8	even	4
304.4.a.b.1.1		1	20.3	even	4
361.4.a.b.1.1		1	95.18	even	4
475.4.a.e.1.1		1	5.2	odd	4
475.4.b.c.324.1		2	1.1	even	1	trivial
475.4.b.c.324.2		2	5.4	even	2	inner
931.4.a.a.1.1		1	35.13	even	4
1216.4.a.a.1.1		1	40.3	even	4
1216.4.a.f.1.1		1	40.13	odd	4
2299.4.a.b.1.1		1	55.43	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 475 = 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	475.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{2} +5.00000i q^{3} -1.00000 q^{4} +15.0000 q^{6} +11.0000i q^{7} -21.0000i q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q-3.00000i q^{2} +5.00000i q^{3} -1.00000 q^{4} +15.0000 q^{6} +11.0000i q^{7} -21.0000i q^{8} +2.00000 q^{9} -54.0000 q^{11} -5.00000i q^{12} -11.0000i q^{13} +33.0000 q^{14} -71.0000 q^{16} -93.0000i q^{17} -6.00000i q^{18} -19.0000 q^{19} -55.0000 q^{21} +162.000i q^{22} -183.000i q^{23} +105.000 q^{24} -33.0000 q^{26} +145.000i q^{27} -11.0000i q^{28} +249.000 q^{29} +56.0000 q^{31} +45.0000i q^{32} -270.000i q^{33} -279.000 q^{34} -2.00000 q^{36} -250.000i q^{37} +57.0000i q^{38} +55.0000 q^{39} +240.000 q^{41} +165.000i q^{42} +196.000i q^{43} +54.0000 q^{44} -549.000 q^{46} -168.000i q^{47} -355.000i q^{48} +222.000 q^{49} +465.000 q^{51} +11.0000i q^{52} -435.000i q^{53} +435.000 q^{54} +231.000 q^{56} -95.0000i q^{57} -747.000i q^{58} -195.000 q^{59} -358.000 q^{61} -168.000i q^{62} +22.0000i q^{63} -433.000 q^{64} -810.000 q^{66} -961.000i q^{67} +93.0000i q^{68} +915.000 q^{69} -246.000 q^{71} -42.0000i q^{72} -353.000i q^{73} -750.000 q^{74} +19.0000 q^{76} -594.000i q^{77} -165.000i q^{78} +34.0000 q^{79} -671.000 q^{81} -720.000i q^{82} -234.000i q^{83} +55.0000 q^{84} +588.000 q^{86} +1245.00i q^{87} +1134.00i q^{88} +168.000 q^{89} +121.000 q^{91} +183.000i q^{92} +280.000i q^{93} -504.000 q^{94} -225.000 q^{96} +758.000i q^{97} -666.000i q^{98} -108.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4} + 30 q^{6} + 4 q^{9}+O(q^{10}) \) 2 * q - 2 * q^4 + 30 * q^6 + 4 * q^9	\( 2 q - 2 q^{4} + 30 q^{6} + 4 q^{9} - 108 q^{11} + 66 q^{14} - 142 q^{16} - 38 q^{19} - 110 q^{21} + 210 q^{24} - 66 q^{26} + 498 q^{29} + 112 q^{31} - 558 q^{34} - 4 q^{36} + 110 q^{39} + 480 q^{41} + 108 q^{44} - 1098 q^{46} + 444 q^{49} + 930 q^{51} + 870 q^{54} + 462 q^{56} - 390 q^{59} - 716 q^{61} - 866 q^{64} - 1620 q^{66} + 1830 q^{69} - 492 q^{71} - 1500 q^{74} + 38 q^{76} + 68 q^{79} - 1342 q^{81} + 110 q^{84} + 1176 q^{86} + 336 q^{89} + 242 q^{91} - 1008 q^{94} - 450 q^{96} - 216 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 + 30 * q^6 + 4 * q^9 - 108 * q^11 + 66 * q^14 - 142 * q^16 - 38 * q^19 - 110 * q^21 + 210 * q^24 - 66 * q^26 + 498 * q^29 + 112 * q^31 - 558 * q^34 - 4 * q^36 + 110 * q^39 + 480 * q^41 + 108 * q^44 - 1098 * q^46 + 444 * q^49 + 930 * q^51 + 870 * q^54 + 462 * q^56 - 390 * q^59 - 716 * q^61 - 866 * q^64 - 1620 * q^66 + 1830 * q^69 - 492 * q^71 - 1500 * q^74 + 38 * q^76 + 68 * q^79 - 1342 * q^81 + 110 * q^84 + 1176 * q^86 + 336 * q^89 + 242 * q^91 - 1008 * q^94 - 450 * q^96 - 216 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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