LMFDB - Embedded newform 975.2.c.k.274.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(274,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("975.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.c (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:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.6
Root			$0.403032 + 0.403032i$ of defining polynomial
Character	$\chi$	$=$	975.274
Dual form			975.2.c.k.274.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.67513i q^{2} -1.00000i q^{3} -5.15633 q^{4} +2.67513 q^{6} -2.28726i q^{7} -8.44358i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q+2.67513i q^{2} -1.00000i q^{3} -5.15633 q^{4} +2.67513 q^{6} -2.28726i q^{7} -8.44358i q^{8} -1.00000 q^{9} -0.130933 q^{11} +5.15633i q^{12} +1.00000i q^{13} +6.11871 q^{14} +12.2750 q^{16} +7.96239i q^{17} -2.67513i q^{18} +5.11871 q^{19} -2.28726 q^{21} -0.350262i q^{22} +5.50659i q^{23} -8.44358 q^{24} -2.67513 q^{26} +1.00000i q^{27} +11.7938i q^{28} +3.00000 q^{29} -7.79384 q^{31} +15.9502i q^{32} +0.130933i q^{33} -21.3004 q^{34} +5.15633 q^{36} +4.80606i q^{37} +13.6932i q^{38} +1.00000 q^{39} +11.4314 q^{41} -6.11871i q^{42} +2.93207i q^{43} +0.675131 q^{44} -14.7308 q^{46} +2.67513i q^{47} -12.2750i q^{48} +1.76845 q^{49} +7.96239 q^{51} -5.15633i q^{52} +8.50659i q^{53} -2.67513 q^{54} -19.3127 q^{56} -5.11871i q^{57} +8.02539i q^{58} +10.4133 q^{59} +5.18664 q^{61} -20.8496i q^{62} +2.28726i q^{63} -18.1187 q^{64} -0.350262 q^{66} -6.05571i q^{67} -41.0567i q^{68} +5.50659 q^{69} -15.0435 q^{71} +8.44358i q^{72} +0.932071i q^{73} -12.8568 q^{74} -26.3938 q^{76} +0.299477i q^{77} +2.67513i q^{78} -8.85685 q^{79} +1.00000 q^{81} +30.5804i q^{82} -6.80114i q^{83} +11.7938 q^{84} -7.84367 q^{86} -3.00000i q^{87} +1.10554i q^{88} +8.12601 q^{89} +2.28726 q^{91} -28.3938i q^{92} +7.79384i q^{93} -7.15633 q^{94} +15.9502 q^{96} +18.9624i q^{97} +4.73084i q^{98} +0.130933 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 10 q^{4} + 6 q^{6} - 6 q^{9}+O(q^{10}) $ 6 * q - 10 * q^4 + 6 * q^6 - 6 * q^9	$ 6 q - 10 q^{4} + 6 q^{6} - 6 q^{9} - 10 q^{11} - 6 q^{14} + 10 q^{16} - 12 q^{19} - 2 q^{21} - 18 q^{24} - 6 q^{26} + 18 q^{29} + 6 q^{31} - 34 q^{34} + 10 q^{36} + 6 q^{39} - 16 q^{41} - 6 q^{44} - 44 q^{46} - 12 q^{49} + 26 q^{51} - 6 q^{54} - 74 q^{56} + 34 q^{59} + 6 q^{61} - 66 q^{64} + 18 q^{66} - 8 q^{69} - 4 q^{71} - 16 q^{74} - 52 q^{76} + 8 q^{79} + 6 q^{81} + 18 q^{84} - 68 q^{86} + 32 q^{89} + 2 q^{91} - 22 q^{94} + 22 q^{96} + 10 q^{99}+O(q^{100}) $ 6 * q - 10 * q^4 + 6 * q^6 - 6 * q^9 - 10 * q^11 - 6 * q^14 + 10 * q^16 - 12 * q^19 - 2 * q^21 - 18 * q^24 - 6 * q^26 + 18 * q^29 + 6 * q^31 - 34 * q^34 + 10 * q^36 + 6 * q^39 - 16 * q^41 - 6 * q^44 - 44 * q^46 - 12 * q^49 + 26 * q^51 - 6 * q^54 - 74 * q^56 + 34 * q^59 + 6 * q^61 - 66 * q^64 + 18 * q^66 - 8 * q^69 - 4 * q^71 - 16 * q^74 - 52 * q^76 + 8 * q^79 + 6 * q^81 + 18 * q^84 - 68 * q^86 + 32 * q^89 + 2 * q^91 - 22 * q^94 + 22 * q^96 + 10 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/975\mathbb{Z}\right)^\times$.

$n$	$301$	$326$	$352$
$\chi(n)$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.67513i	1.89160i	0.324745	+	0.945802i	$0.394721\pi$
$2$	2.67513i	1.89160i	−0.324745	+	0.945802i	$0.605279\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	−5.15633	−2.57816
$5$	0	0
$5$	0	0
$6$	2.67513	1.09212
$7$	− 2.28726i	− 0.864502i	−0.901753	−	0.432251i	$-0.857719\pi$
$7$	− 2.28726i	− 0.864502i	0.901753	−	0.432251i	$-0.142281\pi$
$8$	− 8.44358i	− 2.98526i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−0.130933	−0.0394777	−0.0197388	−	0.999805i	$-0.506283\pi$
$11$	−0.130933	−0.0394777	−0.0197388	+	0.999805i	$0.506283\pi$
$12$	5.15633i	1.48850i
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	6.11871	1.63530
$15$	0	0
$16$	12.2750	3.06876
$17$	7.96239i	1.93116i	0.260101	+	0.965581i	$0.416244\pi$
$17$	7.96239i	1.93116i	−0.260101	+	0.965581i	$0.583756\pi$
$18$	− 2.67513i	− 0.630534i
$19$	5.11871	1.17431	0.587157	−	0.809473i	$-0.300247\pi$
$19$	5.11871	1.17431	0.587157	+	0.809473i	$0.300247\pi$
$20$	0	0
$21$	−2.28726	−0.499121
$22$	− 0.350262i	− 0.0746761i
$23$	5.50659i	1.14820i	0.818784	+	0.574101i	$0.194648\pi$
$23$	5.50659i	1.14820i	−0.818784	+	0.574101i	$0.805352\pi$
$24$	−8.44358	−1.72354
$25$	0	0
$26$	−2.67513	−0.524636
$27$	1.00000i	0.192450i
$28$	11.7938i	2.22883i
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−7.79384	−1.39982	−0.699908	−	0.714233i	$-0.746776\pi$
$31$	−7.79384	−1.39982	−0.699908	+	0.714233i	$0.746776\pi$
$32$	15.9502i	2.81962i
$33$	0.130933i	0.0227924i
$34$	−21.3004	−3.65299
$35$	0	0
$36$	5.15633	0.859388
$37$	4.80606i	0.790112i	0.918657	+	0.395056i	$0.129275\pi$
$37$	4.80606i	0.790112i	−0.918657	+	0.395056i	$0.870725\pi$
$38$	13.6932i	2.22134i
$39$	1.00000	0.160128
$40$	0	0
$41$	11.4314	1.78528	0.892640	−	0.450771i	$-0.148851\pi$
$41$	11.4314	1.78528	0.892640	+	0.450771i	$0.148851\pi$
$42$	− 6.11871i	− 0.944138i
$43$	2.93207i	0.447137i	0.974688	+	0.223568i	$0.0717706\pi$
$43$	2.93207i	0.447137i	−0.974688	+	0.223568i	$0.928229\pi$
$44$	0.675131	0.101780
$45$	0	0
$46$	−14.7308	−2.17194
$47$	2.67513i	0.390208i	0.980783	+	0.195104i	$0.0625044\pi$
$47$	2.67513i	0.390208i	−0.980783	+	0.195104i	$0.937496\pi$
$48$	− 12.2750i	− 1.77175i
$49$	1.76845	0.252636
$50$	0	0
$51$	7.96239	1.11496
$52$	− 5.15633i	− 0.715054i
$53$	8.50659i	1.16847i	0.811585	+	0.584235i	$0.198605\pi$
$53$	8.50659i	1.16847i	−0.811585	+	0.584235i	$0.801395\pi$
$54$	−2.67513	−0.364039
$55$	0	0
$56$	−19.3127	−2.58076
$57$	− 5.11871i	− 0.677990i
$58$	8.02539i	1.05379i
$59$	10.4133	1.35569	0.677846	−	0.735204i	$-0.262914\pi$
$59$	10.4133	1.35569	0.677846	+	0.735204i	$0.262914\pi$
$60$	0	0
$61$	5.18664	0.664082	0.332041	−	0.943265i	$-0.392263\pi$
$61$	5.18664	0.664082	0.332041	+	0.943265i	$0.392263\pi$
$62$	− 20.8496i	− 2.64790i
$63$	2.28726i	0.288167i
$64$	−18.1187	−2.26484
$65$	0	0
$66$	−0.350262	−0.0431142
$67$	− 6.05571i	− 0.739823i	−0.929067	−	0.369911i	$-0.879388\pi$
$67$	− 6.05571i	− 0.739823i	0.929067	−	0.369911i	$-0.120612\pi$
$68$	− 41.0567i	− 4.97885i
$69$	5.50659	0.662915
$70$	0	0
$71$	−15.0435	−1.78533	−0.892667	−	0.450717i	$-0.851168\pi$
$71$	−15.0435	−1.78533	−0.892667	+	0.450717i	$0.851168\pi$
$72$	8.44358i	0.995086i
$73$	0.932071i	0.109091i	0.998511	+	0.0545454i	$0.0173709\pi$
$73$	0.932071i	0.109091i	−0.998511	+	0.0545454i	$0.982629\pi$
$74$	−12.8568	−1.49458
$75$	0	0
$76$	−26.3938	−3.02757
$77$	0.299477i	0.0341285i
$78$	2.67513i	0.302899i
$79$	−8.85685	−0.996473	−0.498237	−	0.867041i	$-0.666019\pi$
$79$	−8.85685	−0.996473	−0.498237	+	0.867041i	$0.666019\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	30.5804i	3.37704i
$83$	− 6.80114i	− 0.746522i	−0.927726	−	0.373261i	$-0.878240\pi$
$83$	− 6.80114i	− 0.746522i	0.927726	−	0.373261i	$-0.121760\pi$
$84$	11.7938	1.28681
$85$	0	0
$86$	−7.84367	−0.845805
$87$	− 3.00000i	− 0.321634i
$88$	1.10554i	0.117851i
$89$	8.12601	0.861355	0.430678	−	0.902506i	$-0.358275\pi$
$89$	8.12601	0.861355	0.430678	+	0.902506i	$0.358275\pi$
$90$	0	0
$91$	2.28726	0.239770
$92$	− 28.3938i	− 2.96025i
$93$	7.79384i	0.808184i
$94$	−7.15633	−0.738119
$95$	0	0
$96$	15.9502	1.62791
$97$	18.9624i	1.92534i	0.270680	+	0.962669i	$0.412752\pi$
$97$	18.9624i	1.92534i	−0.270680	+	0.962669i	$0.587248\pi$
$98$	4.73084i	0.477887i
$99$	0.130933	0.0131592
$100$	0	0
$101$	−3.54420	−0.352661	−0.176330	−	0.984331i	$-0.556423\pi$
$101$	−3.54420	−0.352661	−0.176330	+	0.984331i	$0.556423\pi$
$102$	21.3004i	2.10906i
$103$	− 14.0811i	− 1.38745i	−0.720239	−	0.693726i	$-0.755968\pi$
$103$	− 14.0811i	− 1.38745i	0.720239	−	0.693726i	$-0.244032\pi$
$104$	8.44358	0.827961
$105$	0	0
$106$	−22.7562	−2.21028
$107$	− 2.64974i	− 0.256160i	−0.991764	−	0.128080i	$-0.959119\pi$
$107$	− 2.64974i	− 0.256160i	0.991764	−	0.128080i	$-0.0408814\pi$
$108$	− 5.15633i	− 0.496168i
$109$	9.31994	0.892689	0.446344	−	0.894861i	$-0.352726\pi$
$109$	9.31994	0.892689	0.446344	+	0.894861i	$0.352726\pi$
$110$	0	0
$111$	4.80606	0.456171
$112$	− 28.0762i	− 2.65295i
$113$	− 13.3503i	− 1.25589i	−0.778259	−	0.627943i	$-0.783897\pi$
$113$	− 13.3503i	− 1.25589i	0.778259	−	0.627943i	$-0.216103\pi$
$114$	13.6932	1.28249
$115$	0	0
$116$	−15.4690	−1.43626
$117$	− 1.00000i	− 0.0924500i
$118$	27.8568i	2.56443i
$119$	18.2120	1.66949
$120$	0	0
$121$	−10.9829	−0.998442
$122$	13.8749i	1.25618i
$123$	− 11.4314i	− 1.03073i
$124$	40.1876	3.60895
$125$	0	0
$126$	−6.11871	−0.545098
$127$	8.73084i	0.774737i	0.921925	+	0.387368i	$0.126616\pi$
$127$	8.73084i	0.774737i	−0.921925	+	0.387368i	$0.873384\pi$
$128$	− 16.5696i	− 1.46456i
$129$	2.93207	0.258154
$130$	0	0
$131$	19.0435	1.66384	0.831919	−	0.554897i	$-0.187243\pi$
$131$	19.0435	1.66384	0.831919	+	0.554897i	$0.187243\pi$
$132$	− 0.675131i	− 0.0587626i
$133$	− 11.7078i	− 1.01520i
$134$	16.1998	1.39945
$135$	0	0
$136$	67.2311	5.76502
$137$	7.66291i	0.654687i	0.944905	+	0.327343i	$0.106153\pi$
$137$	7.66291i	0.654687i	−0.944905	+	0.327343i	$0.893847\pi$
$138$	14.7308i	1.25397i
$139$	−7.61213	−0.645652	−0.322826	−	0.946458i	$-0.604633\pi$
$139$	−7.61213	−0.645652	−0.322826	+	0.946458i	$0.604633\pi$
$140$	0	0
$141$	2.67513	0.225287
$142$	− 40.2433i	− 3.37714i
$143$	− 0.130933i	− 0.0109491i
$144$	−12.2750	−1.02292
$145$	0	0
$146$	−2.49341	−0.206356
$147$	− 1.76845i	− 0.145859i
$148$	− 24.7816i	− 2.03704i
$149$	−0.887166	−0.0726795	−0.0363397	−	0.999339i	$-0.511570\pi$
$149$	−0.887166	−0.0726795	−0.0363397	+	0.999339i	$0.511570\pi$
$150$	0	0
$151$	2.93700	0.239009	0.119505	−	0.992834i	$-0.461869\pi$
$151$	2.93700	0.239009	0.119505	+	0.992834i	$0.461869\pi$
$152$	− 43.2203i	− 3.50563i
$153$	− 7.96239i	− 0.643721i
$154$	−0.801139	−0.0645576
$155$	0	0
$156$	−5.15633	−0.412836
$157$	15.7816i	1.25951i	0.776793	+	0.629755i	$0.216845\pi$
$157$	15.7816i	1.25951i	−0.776793	+	0.629755i	$0.783155\pi$
$158$	− 23.6932i	− 1.88493i
$159$	8.50659	0.674616
$160$	0	0
$161$	12.5950	0.992624
$162$	2.67513i	0.210178i
$163$	17.4314i	1.36533i	0.730732	+	0.682665i	$0.239179\pi$
$163$	17.4314i	1.36533i	−0.730732	+	0.682665i	$0.760821\pi$
$164$	−58.9438	−4.60274
$165$	0	0
$166$	18.1939	1.41212
$167$	− 1.91890i	− 0.148489i	−0.997240	−	0.0742444i	$-0.976346\pi$
$167$	− 1.91890i	− 0.148489i	0.997240	−	0.0742444i	$-0.0236545\pi$
$168$	19.3127i	1.49000i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−5.11871	−0.391438
$172$	− 15.1187i	− 1.15279i
$173$	7.93207i	0.603064i	0.953456	+	0.301532i	$0.0974980\pi$
$173$	7.93207i	0.603064i	−0.953456	+	0.301532i	$0.902502\pi$
$174$	8.02539	0.608403
$175$	0	0
$176$	−1.60720	−0.121147
$177$	− 10.4133i	− 0.782709i
$178$	21.7381i	1.62934i
$179$	−14.9175	−1.11499	−0.557493	−	0.830182i	$-0.688236\pi$
$179$	−14.9175	−1.11499	−0.557493	+	0.830182i	$0.688236\pi$
$180$	0	0
$181$	−23.5804	−1.75272	−0.876358	−	0.481659i	$-0.840034\pi$
$181$	−23.5804	−1.75272	−0.876358	+	0.481659i	$0.840034\pi$
$182$	6.11871i	0.453549i
$183$	− 5.18664i	− 0.383408i
$184$	46.4953	3.42768
$185$	0	0
$186$	−20.8496	−1.52876
$187$	− 1.04254i	− 0.0762378i
$188$	− 13.7938i	− 1.00602i
$189$	2.28726	0.166374
$190$	0	0
$191$	8.38787	0.606925	0.303463	−	0.952843i	$-0.401857\pi$
$191$	8.38787	0.606925	0.303463	+	0.952843i	$0.401857\pi$
$192$	18.1187i	1.30761i
$193$	− 18.1563i	− 1.30692i	−0.756961	−	0.653460i	$-0.773317\pi$
$193$	− 18.1563i	− 1.30692i	0.756961	−	0.653460i	$-0.226683\pi$
$194$	−50.7269	−3.64198
$195$	0	0
$196$	−9.11871	−0.651337
$197$	− 1.19394i	− 0.0850645i	−0.999095	−	0.0425322i	$-0.986457\pi$
$197$	− 1.19394i	− 0.0850645i	0.999095	−	0.0425322i	$-0.0135425\pi$
$198$	0.350262i	0.0248920i
$199$	−0.700523	−0.0496588	−0.0248294	−	0.999692i	$-0.507904\pi$
$199$	−0.700523	−0.0496588	−0.0248294	+	0.999692i	$0.507904\pi$
$200$	0	0
$201$	−6.05571	−0.427137
$202$	− 9.48119i	− 0.667095i
$203$	− 6.86177i	− 0.481602i
$204$	−41.0567	−2.87454
$205$	0	0
$206$	37.6688	2.62451
$207$	− 5.50659i	− 0.382734i
$208$	12.2750i	0.851121i
$209$	−0.670206	−0.0463591
$210$	0	0
$211$	16.9829	1.16915	0.584574	−	0.811340i	$-0.301262\pi$
$211$	16.9829	1.16915	0.584574	+	0.811340i	$0.301262\pi$
$212$	− 43.8627i	− 3.01250i
$213$	15.0435i	1.03076i
$214$	7.08840	0.484553
$215$	0	0
$216$	8.44358	0.574513
$217$	17.8265i	1.21014i
$218$	24.9321i	1.68861i
$219$	0.932071	0.0629836
$220$	0	0
$221$	−7.96239	−0.535608
$222$	12.8568i	0.862895i
$223$	0.806063i	0.0539780i	0.999636	+	0.0269890i	$0.00859191\pi$
$223$	0.806063i	0.0539780i	−0.999636	+	0.0269890i	$0.991408\pi$
$224$	36.4821	2.43757
$225$	0	0
$226$	35.7137	2.37564
$227$	7.68243i	0.509900i	0.966954	+	0.254950i	$0.0820591\pi$
$227$	7.68243i	0.509900i	−0.966954	+	0.254950i	$0.917941\pi$
$228$	26.3938i	1.74797i
$229$	4.44851	0.293966	0.146983	−	0.989139i	$-0.453044\pi$
$229$	4.44851	0.293966	0.146983	+	0.989139i	$0.453044\pi$
$230$	0	0
$231$	0.299477	0.0197041
$232$	− 25.3307i	− 1.66305i
$233$	10.1260i	0.663377i	0.943389	+	0.331688i	$0.107618\pi$
$233$	10.1260i	0.663377i	−0.943389	+	0.331688i	$0.892382\pi$
$234$	2.67513	0.174879
$235$	0	0
$236$	−53.6942	−3.49519
$237$	8.85685i	0.575314i
$238$	48.7196i	3.15802i
$239$	19.4363	1.25723	0.628615	−	0.777717i	$-0.283622\pi$
$239$	19.4363	1.25723	0.628615	+	0.777717i	$0.283622\pi$
$240$	0	0
$241$	−6.46898	−0.416703	−0.208352	−	0.978054i	$-0.566810\pi$
$241$	−6.46898	−0.416703	−0.208352	+	0.978054i	$0.566810\pi$
$242$	− 29.3806i	− 1.88866i
$243$	− 1.00000i	− 0.0641500i
$244$	−26.7440	−1.71211
$245$	0	0
$246$	30.5804	1.94973
$247$	5.11871i	0.325696i
$248$	65.8080i	4.17881i
$249$	−6.80114	−0.431005
$250$	0	0
$251$	−17.7685	−1.12153	−0.560767	−	0.827973i	$-0.689494\pi$
$251$	−17.7685	−1.12153	−0.560767	+	0.827973i	$0.689494\pi$
$252$	− 11.7938i	− 0.742942i
$253$	− 0.720992i	− 0.0453283i
$254$	−23.3561	−1.46549
$255$	0	0
$256$	8.08840	0.505525
$257$	− 4.81924i	− 0.300616i	−0.988639	−	0.150308i	$-0.951974\pi$
$257$	− 4.81924i	− 0.300616i	0.988639	−	0.150308i	$-0.0480265\pi$
$258$	7.84367i	0.488326i
$259$	10.9927	0.683054
$260$	0	0
$261$	−3.00000	−0.185695
$262$	50.9438i	3.14732i
$263$	1.30536i	0.0804917i	0.999190	+	0.0402459i	$0.0128141\pi$
$263$	1.30536i	0.0804917i	−0.999190	+	0.0402459i	$0.987186\pi$
$264$	1.10554	0.0680413
$265$	0	0
$266$	31.3199	1.92035
$267$	− 8.12601i	− 0.497304i
$268$	31.2252i	1.90738i
$269$	8.79877	0.536470	0.268235	−	0.963353i	$-0.413560\pi$
$269$	8.79877	0.536470	0.268235	+	0.963353i	$0.413560\pi$
$270$	0	0
$271$	17.4264	1.05858	0.529290	−	0.848441i	$-0.322458\pi$
$271$	17.4264	1.05858	0.529290	+	0.848441i	$0.322458\pi$
$272$	97.7386i	5.92627i
$273$	− 2.28726i	− 0.138431i
$274$	−20.4993	−1.23841
$275$	0	0
$276$	−28.3938	−1.70910
$277$	11.7381i	0.705276i	0.935760	+	0.352638i	$0.114715\pi$
$277$	11.7381i	0.705276i	−0.935760	+	0.352638i	$0.885285\pi$
$278$	− 20.3634i	− 1.22132i
$279$	7.79384	0.466605
$280$	0	0
$281$	−14.5647	−0.868855	−0.434428	−	0.900707i	$-0.643049\pi$
$281$	−14.5647	−0.868855	−0.434428	+	0.900707i	$0.643049\pi$
$282$	7.15633i	0.426153i
$283$	− 26.1016i	− 1.55158i	−0.630993	−	0.775789i	$-0.717352\pi$
$283$	− 26.1016i	− 1.55158i	0.630993	−	0.775789i	$-0.282648\pi$
$284$	77.5691	4.60288
$285$	0	0
$286$	0.350262	0.0207114
$287$	− 26.1465i	− 1.54338i
$288$	− 15.9502i	− 0.939873i
$289$	−46.3996	−2.72939
$290$	0	0
$291$	18.9624	1.11159
$292$	− 4.80606i	− 0.281254i
$293$	− 9.07381i	− 0.530098i	−0.964235	−	0.265049i	$-0.914612\pi$
$293$	− 9.07381i	− 0.530098i	0.964235	−	0.265049i	$-0.0853881\pi$
$294$	4.73084	0.275908
$295$	0	0
$296$	40.5804	2.35869
$297$	− 0.130933i	− 0.00759748i
$298$	− 2.37328i	− 0.137481i
$299$	−5.50659	−0.318454
$300$	0	0
$301$	6.70640	0.386551
$302$	7.85685i	0.452111i
$303$	3.54420i	0.203609i
$304$	62.8324	3.60369
$305$	0	0
$306$	21.3004	1.21766
$307$	− 11.3806i	− 0.649524i	−0.945796	−	0.324762i	$-0.894716\pi$
$307$	− 11.3806i	− 0.649524i	0.945796	−	0.324762i	$-0.105284\pi$
$308$	− 1.54420i	− 0.0879889i
$309$	−14.0811	−0.801046
$310$	0	0
$311$	9.22425	0.523059	0.261530	−	0.965195i	$-0.415773\pi$
$311$	9.22425	0.523059	0.261530	+	0.965195i	$0.415773\pi$
$312$	− 8.44358i	− 0.478024i
$313$	− 18.7685i	− 1.06086i	−0.847730	−	0.530428i	$-0.822031\pi$
$313$	− 18.7685i	− 1.06086i	0.847730	−	0.530428i	$-0.177969\pi$
$314$	−42.2179	−2.38249
$315$	0	0
$316$	45.6688	2.56907
$317$	10.5442i	0.592221i	0.955154	+	0.296111i	$0.0956897\pi$
$317$	10.5442i	0.592221i	−0.955154	+	0.296111i	$0.904310\pi$
$318$	22.7562i	1.27611i
$319$	−0.392798	−0.0219924
$320$	0	0
$321$	−2.64974	−0.147894
$322$	33.6932i	1.87765i
$323$	40.7572i	2.26779i
$324$	−5.15633	−0.286463
$325$	0	0
$326$	−46.6312	−2.58266
$327$	− 9.31994i	− 0.515394i
$328$	− 96.5217i	− 5.32952i
$329$	6.11871	0.337336
$330$	0	0
$331$	4.95651	0.272434	0.136217	−	0.990679i	$-0.456506\pi$
$331$	4.95651	0.272434	0.136217	+	0.990679i	$0.456506\pi$
$332$	35.0689i	1.92466i
$333$	− 4.80606i	− 0.263371i
$334$	5.13330	0.280882
$335$	0	0
$336$	−28.0762	−1.53168
$337$	− 32.6082i	− 1.77628i	−0.459573	−	0.888140i	$-0.651998\pi$
$337$	− 32.6082i	− 1.77628i	0.459573	−	0.888140i	$-0.348002\pi$
$338$	− 2.67513i	− 0.145508i
$339$	−13.3503	−0.725087
$340$	0	0
$341$	1.02047	0.0552614
$342$	− 13.6932i	− 0.740445i
$343$	− 20.0557i	− 1.08291i
$344$	24.7572	1.33482
$345$	0	0
$346$	−21.2193	−1.14076
$347$	− 22.1622i	− 1.18973i	−0.803826	−	0.594865i	$-0.797206\pi$
$347$	− 22.1622i	− 1.18973i	0.803826	−	0.594865i	$-0.202794\pi$
$348$	15.4690i	0.829224i
$349$	4.34297	0.232474	0.116237	−	0.993222i	$-0.462917\pi$
$349$	4.34297	0.232474	0.116237	+	0.993222i	$0.462917\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	− 2.08840i	− 0.111312i
$353$	6.96827i	0.370883i	0.982655	+	0.185442i	$0.0593716\pi$
$353$	6.96827i	0.370883i	−0.982655	+	0.185442i	$0.940628\pi$
$354$	27.8568	1.48057
$355$	0	0
$356$	−41.9003	−2.22071
$357$	− 18.2120i	− 0.963883i
$358$	− 39.9062i	− 2.10911i
$359$	15.0122	0.792315	0.396157	−	0.918183i	$-0.370343\pi$
$359$	15.0122	0.792315	0.396157	+	0.918183i	$0.370343\pi$
$360$	0	0
$361$	7.20123	0.379012
$362$	− 63.0806i	− 3.31544i
$363$	10.9829i	0.576450i
$364$	−11.7938	−0.618165
$365$	0	0
$366$	13.8749	0.725255
$367$	27.0738i	1.41324i	0.707593	+	0.706621i	$0.249781\pi$
$367$	27.0738i	1.41324i	−0.707593	+	0.706621i	$0.750219\pi$
$368$	67.5936i	3.52356i
$369$	−11.4314	−0.595093
$370$	0	0
$371$	19.4568	1.01014
$372$	− 40.1876i	− 2.08363i
$373$	− 3.49341i	− 0.180882i	−0.995902	−	0.0904410i	$-0.971172\pi$
$373$	− 3.49341i	− 0.180882i	0.995902	−	0.0904410i	$-0.0288277\pi$
$374$	2.78892	0.144212
$375$	0	0
$376$	22.5877	1.16487
$377$	3.00000i	0.154508i
$378$	6.11871i	0.314713i
$379$	−26.9937	−1.38657	−0.693286	−	0.720663i	$-0.743838\pi$
$379$	−26.9937	−1.38657	−0.693286	+	0.720663i	$0.743838\pi$
$380$	0	0
$381$	8.73084	0.447295
$382$	22.4387i	1.14806i
$383$	− 5.52118i	− 0.282119i	−0.990001	−	0.141059i	$-0.954949\pi$
$383$	− 5.52118i	− 0.282119i	0.990001	−	0.141059i	$-0.0450509\pi$
$384$	−16.5696	−0.845563
$385$	0	0
$386$	48.5705	2.47218
$387$	− 2.93207i	− 0.149046i
$388$	− 97.7762i	− 4.96384i
$389$	15.9003	0.806179	0.403090	−	0.915161i	$-0.367936\pi$
$389$	15.9003	0.806179	0.403090	+	0.915161i	$0.367936\pi$
$390$	0	0
$391$	−43.8456	−2.21737
$392$	− 14.9321i	− 0.754183i
$393$	− 19.0435i	− 0.960617i
$394$	3.19394	0.160908
$395$	0	0
$396$	−0.675131	−0.0339266
$397$	− 17.1998i	− 0.863234i	−0.902057	−	0.431617i	$-0.857943\pi$
$397$	− 17.1998i	− 0.863234i	0.902057	−	0.431617i	$-0.142057\pi$
$398$	− 1.87399i	− 0.0939347i
$399$	−11.7078	−0.586124
$400$	0	0
$401$	19.0435	0.950987	0.475493	−	0.879719i	$-0.342270\pi$
$401$	19.0435	0.950987	0.475493	+	0.879719i	$0.342270\pi$
$402$	− 16.1998i	− 0.807973i
$403$	− 7.79384i	− 0.388239i
$404$	18.2750	0.909217
$405$	0	0
$406$	18.3561	0.911000
$407$	− 0.629270i	− 0.0311918i
$408$	− 67.2311i	− 3.32843i
$409$	−12.4182	−0.614040	−0.307020	−	0.951703i	$-0.599332\pi$
$409$	−12.4182	−0.614040	−0.307020	+	0.951703i	$0.599332\pi$
$410$	0	0
$411$	7.66291	0.377984
$412$	72.6067i	3.57708i
$413$	− 23.8178i	− 1.17200i
$414$	14.7308	0.723981
$415$	0	0
$416$	−15.9502	−0.782021
$417$	7.61213i	0.372767i
$418$	− 1.79289i	− 0.0876931i
$419$	−6.14174	−0.300043	−0.150022	−	0.988683i	$-0.547934\pi$
$419$	−6.14174	−0.300043	−0.150022	+	0.988683i	$0.547934\pi$
$420$	0	0
$421$	−17.6121	−0.858363	−0.429181	−	0.903218i	$-0.641198\pi$
$421$	−17.6121	−0.858363	−0.429181	+	0.903218i	$0.641198\pi$
$422$	45.4314i	2.21156i
$423$	− 2.67513i	− 0.130069i
$424$	71.8261	3.48818
$425$	0	0
$426$	−40.2433	−1.94979
$427$	− 11.8632i	− 0.574100i
$428$	13.6629i	0.660422i
$429$	−0.130933	−0.00632148
$430$	0	0
$431$	−3.65703	−0.176153	−0.0880765	−	0.996114i	$-0.528072\pi$
$431$	−3.65703	−0.176153	−0.0880765	+	0.996114i	$0.528072\pi$
$432$	12.2750i	0.590583i
$433$	− 9.28963i	− 0.446431i	−0.974769	−	0.223216i	$-0.928345\pi$
$433$	− 9.28963i	− 0.446431i	0.974769	−	0.223216i	$-0.0716554\pi$
$434$	−47.6883	−2.28911
$435$	0	0
$436$	−48.0567	−2.30150
$437$	28.1866i	1.34835i
$438$	2.49341i	0.119140i
$439$	−27.0581	−1.29141	−0.645706	−	0.763586i	$-0.723437\pi$
$439$	−27.0581	−1.29141	−0.645706	+	0.763586i	$0.723437\pi$
$440$	0	0
$441$	−1.76845	−0.0842120
$442$	− 21.3004i	− 1.01316i
$443$	− 7.09428i	− 0.337059i	−0.985697	−	0.168530i	$-0.946098\pi$
$443$	− 7.09428i	− 0.337059i	0.985697	−	0.168530i	$-0.0539019\pi$
$444$	−24.7816	−1.17608
$445$	0	0
$446$	−2.15633	−0.102105
$447$	0.887166i	0.0419615i
$448$	41.4422i	1.95796i
$449$	14.7612	0.696622	0.348311	−	0.937379i	$-0.386755\pi$
$449$	14.7612	0.696622	0.348311	+	0.937379i	$0.386755\pi$
$450$	0	0
$451$	−1.49674	−0.0704786
$452$	68.8383i	3.23788i
$453$	− 2.93700i	− 0.137992i
$454$	−20.5515	−0.964529
$455$	0	0
$456$	−43.2203	−2.02398
$457$	2.28233i	0.106763i	0.998574	+	0.0533815i	$0.0169999\pi$
$457$	2.28233i	0.106763i	−0.998574	+	0.0533815i	$0.983000\pi$
$458$	11.9003i	0.556066i
$459$	−7.96239	−0.371652
$460$	0	0
$461$	−10.5647	−0.492046	−0.246023	−	0.969264i	$-0.579124\pi$
$461$	−10.5647	−0.492046	−0.246023	+	0.969264i	$0.579124\pi$
$462$	0.801139i	0.0372724i
$463$	6.18172i	0.287289i	0.989629	+	0.143644i	$0.0458821\pi$
$463$	6.18172i	0.287289i	−0.989629	+	0.143644i	$0.954118\pi$
$464$	36.8251	1.70956
$465$	0	0
$466$	−27.0884	−1.25485
$467$	6.32250i	0.292570i	0.989242	+	0.146285i	$0.0467317\pi$
$467$	6.32250i	0.292570i	−0.989242	+	0.146285i	$0.953268\pi$
$468$	5.15633i	0.238351i
$469$	−13.8510	−0.639578
$470$	0	0
$471$	15.7816	0.727179
$472$	− 87.9253i	− 4.04709i
$473$	− 0.383904i	− 0.0176519i
$474$	−23.6932	−1.08827
$475$	0	0
$476$	−93.9072	−4.30423
$477$	− 8.50659i	− 0.389490i
$478$	51.9946i	2.37818i
$479$	−3.33075	−0.152186	−0.0760929	−	0.997101i	$-0.524245\pi$
$479$	−3.33075	−0.152186	−0.0760929	+	0.997101i	$0.524245\pi$
$480$	0	0
$481$	−4.80606	−0.219138
$482$	− 17.3054i	− 0.788237i
$483$	− 12.5950i	− 0.573092i
$484$	56.6312	2.57414
$485$	0	0
$486$	2.67513	0.121346
$487$	8.44358i	0.382615i	0.981530	+	0.191308i	$0.0612728\pi$
$487$	8.44358i	0.382615i	−0.981530	+	0.191308i	$0.938727\pi$
$488$	− 43.7938i	− 1.98245i
$489$	17.4314	0.788274
$490$	0	0
$491$	−23.3112	−1.05202	−0.526011	−	0.850478i	$-0.676313\pi$
$491$	−23.3112	−1.05202	−0.526011	+	0.850478i	$0.676313\pi$
$492$	58.9438i	2.65739i
$493$	23.8872i	1.07582i
$494$	−13.6932	−0.616088
$495$	0	0
$496$	−95.6697	−4.29570
$497$	34.4083i	1.54343i
$498$	− 18.1939i	− 0.815290i
$499$	31.8651	1.42648	0.713239	−	0.700921i	$-0.247227\pi$
$499$	31.8651	1.42648	0.713239	+	0.700921i	$0.247227\pi$
$500$	0	0
$501$	−1.91890	−0.0857300
$502$	− 47.5329i	− 2.12150i
$503$	9.43136i	0.420524i	0.977645	+	0.210262i	$0.0674317\pi$
$503$	9.43136i	0.420524i	−0.977645	+	0.210262i	$0.932568\pi$
$504$	19.3127	0.860254
$505$	0	0
$506$	1.92875	0.0857433
$507$	1.00000i	0.0444116i
$508$	− 45.0191i	− 1.99740i
$509$	−37.4821	−1.66137	−0.830684	−	0.556745i	$-0.812050\pi$
$509$	−37.4821	−1.66137	−0.830684	+	0.556745i	$0.812050\pi$
$510$	0	0
$511$	2.13189	0.0943092
$512$	− 11.5017i	− 0.508306i
$513$	5.11871i	0.225997i
$514$	12.8921	0.568646
$515$	0	0
$516$	−15.1187	−0.665564
$517$	− 0.350262i	− 0.0154045i
$518$	29.4069i	1.29207i
$519$	7.93207	0.348179
$520$	0	0
$521$	15.9248	0.697677	0.348839	−	0.937183i	$-0.386576\pi$
$521$	15.9248	0.697677	0.348839	+	0.937183i	$0.386576\pi$
$522$	− 8.02539i	− 0.351262i
$523$	− 2.27645i	− 0.0995424i	−0.998761	−	0.0497712i	$-0.984151\pi$
$523$	− 2.27645i	− 0.0995424i	0.998761	−	0.0497712i	$-0.0158492\pi$
$524$	−98.1944	−4.28964
$525$	0	0
$526$	−3.49200	−0.152258
$527$	− 62.0576i	− 2.70327i
$528$	1.60720i	0.0699445i
$529$	−7.32250	−0.318370
$530$	0	0
$531$	−10.4133	−0.451897
$532$	60.3693i	2.61734i
$533$	11.4314i	0.495147i
$534$	21.7381	0.940701
$535$	0	0
$536$	−51.1319	−2.20856
$537$	14.9175i	0.643737i
$538$	23.5379i	1.01479i
$539$	−0.231548	−0.00997348
$540$	0	0
$541$	19.3317	0.831135	0.415567	−	0.909562i	$-0.363583\pi$
$541$	19.3317	0.831135	0.415567	+	0.909562i	$0.363583\pi$
$542$	46.6180i	2.00241i
$543$	23.5804i	1.01193i
$544$	−127.001	−5.44514
$545$	0	0
$546$	6.11871	0.261857
$547$	− 40.9438i	− 1.75063i	−0.483552	−	0.875316i	$-0.660653\pi$
$547$	− 40.9438i	− 1.75063i	0.483552	−	0.875316i	$-0.339347\pi$
$548$	− 39.5125i	− 1.68789i
$549$	−5.18664	−0.221361
$550$	0	0
$551$	15.3561	0.654194
$552$	− 46.4953i	− 1.97897i
$553$	20.2579i	0.861453i
$554$	−31.4010	−1.33410
$555$	0	0
$556$	39.2506	1.66460
$557$	− 36.4953i	− 1.54636i	−0.634189	−	0.773178i	$-0.718666\pi$
$557$	− 36.4953i	− 1.54636i	0.634189	−	0.773178i	$-0.281334\pi$
$558$	20.8496i	0.882632i
$559$	−2.93207	−0.124013
$560$	0	0
$561$	−1.04254	−0.0440159
$562$	− 38.9624i	− 1.64353i
$563$	− 1.04349i	− 0.0439779i	−0.999758	−	0.0219890i	$-0.993000\pi$
$563$	− 1.04349i	− 0.0439779i	0.999758	−	0.0219890i	$-0.00699986\pi$
$564$	−13.7938	−0.580826
$565$	0	0
$566$	69.8251	2.93497
$567$	− 2.28726i	− 0.0960558i
$568$	127.021i	5.32968i
$569$	39.4807	1.65512	0.827559	−	0.561378i	$-0.189729\pi$
$569$	39.4807	1.65512	0.827559	+	0.561378i	$0.189729\pi$
$570$	0	0
$571$	35.4920	1.48529	0.742647	−	0.669683i	$-0.233570\pi$
$571$	35.4920	1.48529	0.742647	+	0.669683i	$0.233570\pi$
$572$	0.675131i	0.0282286i
$573$	− 8.38787i	− 0.350408i
$574$	69.9452	2.91946
$575$	0	0
$576$	18.1187	0.754946
$577$	− 36.0625i	− 1.50130i	−0.660698	−	0.750652i	$-0.729740\pi$
$577$	− 36.0625i	− 1.50130i	0.660698	−	0.750652i	$-0.270260\pi$
$578$	− 124.125i	− 5.16292i
$579$	−18.1563	−0.754551
$580$	0	0
$581$	−15.5560	−0.645370
$582$	50.7269i	2.10270i
$583$	− 1.11379i	− 0.0461284i
$584$	7.87002	0.325664
$585$	0	0
$586$	24.2736	1.00273
$587$	− 22.4944i	− 0.928442i	−0.885719	−	0.464221i	$-0.846334\pi$
$587$	− 22.4944i	− 0.928442i	0.885719	−	0.464221i	$-0.153666\pi$
$588$	9.11871i	0.376049i
$589$	−39.8945	−1.64382
$590$	0	0
$591$	−1.19394	−0.0491120
$592$	58.9946i	2.42466i
$593$	28.8627i	1.18525i	0.805478	+	0.592625i	$0.201908\pi$
$593$	28.8627i	1.18525i	−0.805478	+	0.592625i	$0.798092\pi$
$594$	0.350262	0.0143714
$595$	0	0
$596$	4.57452	0.187379
$597$	0.700523i	0.0286705i
$598$	− 14.7308i	− 0.602389i
$599$	−2.62927	−0.107429	−0.0537145	−	0.998556i	$-0.517106\pi$
$599$	−2.62927	−0.107429	−0.0537145	+	0.998556i	$0.517106\pi$
$600$	0	0
$601$	−31.6048	−1.28919	−0.644594	−	0.764525i	$-0.722974\pi$
$601$	−31.6048	−1.28919	−0.644594	+	0.764525i	$0.722974\pi$
$602$	17.9405i	0.731200i
$603$	6.05571i	0.246608i
$604$	−15.1441	−0.616205
$605$	0	0
$606$	−9.48119	−0.385147
$607$	13.9149i	0.564790i	0.959298	+	0.282395i	$0.0911288\pi$
$607$	13.9149i	0.564790i	−0.959298	+	0.282395i	$0.908871\pi$
$608$	81.6444i	3.31112i
$609$	−6.86177	−0.278053
$610$	0	0
$611$	−2.67513	−0.108224
$612$	41.0567i	1.65962i
$613$	− 25.4471i	− 1.02780i	−0.857851	−	0.513899i	$-0.828201\pi$
$613$	− 25.4471i	− 1.02780i	0.857851	−	0.513899i	$-0.171799\pi$
$614$	30.4445	1.22864
$615$	0	0
$616$	2.52865	0.101882
$617$	30.6497i	1.23391i	0.786998	+	0.616956i	$0.211634\pi$
$617$	30.6497i	1.23391i	−0.786998	+	0.616956i	$0.788366\pi$
$618$	− 37.6688i	− 1.51526i
$619$	7.99412	0.321311	0.160655	−	0.987011i	$-0.448639\pi$
$619$	7.99412	0.321311	0.160655	+	0.987011i	$0.448639\pi$
$620$	0	0
$621$	−5.50659	−0.220972
$622$	24.6761i	0.989421i
$623$	− 18.5863i	− 0.744643i
$624$	12.2750	0.491395
$625$	0	0
$626$	50.2081	2.00672
$627$	0.670206i	0.0267655i
$628$	− 81.3752i	− 3.24722i
$629$	−38.2677	−1.52583
$630$	0	0
$631$	41.3463	1.64597	0.822985	−	0.568063i	$-0.192307\pi$
$631$	41.3463	1.64597	0.822985	+	0.568063i	$0.192307\pi$
$632$	74.7835i	2.97473i
$633$	− 16.9829i	− 0.675008i
$634$	−28.2071	−1.12025
$635$	0	0
$636$	−43.8627	−1.73927
$637$	1.76845i	0.0700686i
$638$	− 1.05079i	− 0.0416010i
$639$	15.0435	0.595111
$640$	0	0
$641$	3.26187	0.128836	0.0644180	−	0.997923i	$-0.479481\pi$
$641$	3.26187	0.128836	0.0644180	+	0.997923i	$0.479481\pi$
$642$	− 7.08840i	− 0.279757i
$643$	6.82065i	0.268980i	0.990915	+	0.134490i	$0.0429397\pi$
$643$	6.82065i	0.268980i	−0.990915	+	0.134490i	$0.957060\pi$
$644$	−64.9438	−2.55915
$645$	0	0
$646$	−109.031	−4.28976
$647$	− 30.9478i	− 1.21668i	−0.793675	−	0.608342i	$-0.791835\pi$
$647$	− 30.9478i	− 1.21668i	0.793675	−	0.608342i	$-0.208165\pi$
$648$	− 8.44358i	− 0.331695i
$649$	−1.36344	−0.0535195
$650$	0	0
$651$	17.8265	0.698677
$652$	− 89.8818i	− 3.52004i
$653$	− 16.9003i	− 0.661361i	−0.943743	−	0.330681i	$-0.892722\pi$
$653$	− 16.9003i	− 0.661361i	0.943743	−	0.330681i	$-0.107278\pi$
$654$	24.9321	0.974921
$655$	0	0
$656$	140.320	5.47859
$657$	− 0.932071i	− 0.0363636i
$658$	16.3684i	0.638105i
$659$	−4.79621	−0.186834	−0.0934170	−	0.995627i	$-0.529779\pi$
$659$	−4.79621	−0.186834	−0.0934170	+	0.995627i	$0.529779\pi$
$660$	0	0
$661$	−0.105540	−0.00410503	−0.00205251	−	0.999998i	$-0.500653\pi$
$661$	−0.105540	−0.00410503	−0.00205251	+	0.999998i	$0.500653\pi$
$662$	13.2593i	0.515338i
$663$	7.96239i	0.309234i
$664$	−57.4260	−2.22856
$665$	0	0
$666$	12.8568	0.498193
$667$	16.5198i	0.639648i
$668$	9.89446i	0.382828i
$669$	0.806063	0.0311642
$670$	0	0
$671$	−0.679100	−0.0262164
$672$	− 36.4821i	− 1.40733i
$673$	− 3.86670i	− 0.149050i	−0.997219	−	0.0745251i	$-0.976256\pi$
$673$	− 3.86670i	− 0.149050i	0.997219	−	0.0745251i	$-0.0237441\pi$
$674$	87.2311	3.36002
$675$	0	0
$676$	5.15633	0.198320
$677$	30.1622i	1.15923i	0.814891	+	0.579614i	$0.196797\pi$
$677$	30.1622i	1.15923i	−0.814891	+	0.579614i	$0.803203\pi$
$678$	− 35.7137i	− 1.37158i
$679$	43.3719	1.66446
$680$	0	0
$681$	7.68243	0.294391
$682$	2.72989i	0.104533i
$683$	17.9805i	0.688004i	0.938969	+	0.344002i	$0.111783\pi$
$683$	17.9805i	0.688004i	−0.938969	+	0.344002i	$0.888217\pi$
$684$	26.3938	1.00919
$685$	0	0
$686$	53.6516	2.04843
$687$	− 4.44851i	− 0.169721i
$688$	35.9913i	1.37216i
$689$	−8.50659	−0.324075
$690$	0	0
$691$	31.4363	1.19589	0.597946	−	0.801536i	$-0.295984\pi$
$691$	31.4363	1.19589	0.597946	+	0.801536i	$0.295984\pi$
$692$	− 40.9003i	− 1.55480i
$693$	− 0.299477i	− 0.0113762i
$694$	59.2868	2.25050
$695$	0	0
$696$	−25.3307	−0.960160
$697$	91.0210i	3.44766i
$698$	11.6180i	0.439748i
$699$	10.1260	0.383001
$700$	0	0
$701$	5.32724	0.201207	0.100604	−	0.994927i	$-0.467923\pi$
$701$	5.32724	0.201207	0.100604	+	0.994927i	$0.467923\pi$
$702$	− 2.67513i	− 0.100966i
$703$	24.6009i	0.927839i
$704$	2.37233	0.0894105
$705$	0	0
$706$	−18.6410	−0.701564
$707$	8.10650i	0.304876i
$708$	53.6942i	2.01795i
$709$	−27.8046	−1.04423	−0.522113	−	0.852876i	$-0.674856\pi$
$709$	−27.8046	−1.04423	−0.522113	+	0.852876i	$0.674856\pi$
$710$	0	0
$711$	8.85685	0.332158
$712$	− 68.6126i	− 2.57137i
$713$	− 42.9175i	− 1.60727i
$714$	48.7196	1.82328
$715$	0	0
$716$	76.9194	2.87461
$717$	− 19.4363i	− 0.725862i
$718$	40.1596i	1.49874i
$719$	0.992706	0.0370217	0.0185108	−	0.999829i	$-0.494107\pi$
$719$	0.992706	0.0370217	0.0185108	+	0.999829i	$0.494107\pi$
$720$	0	0
$721$	−32.2071	−1.19946
$722$	19.2642i	0.716941i
$723$	6.46898i	0.240584i
$724$	121.588	4.51879
$725$	0	0
$726$	−29.3806	−1.09042
$727$	− 40.0468i	− 1.48525i	−0.669705	−	0.742627i	$-0.733580\pi$
$727$	− 40.0468i	− 1.48525i	0.669705	−	0.742627i	$-0.266420\pi$
$728$	− 19.3127i	− 0.715774i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−23.3463	−0.863494
$732$	26.7440i	0.988487i
$733$	30.8627i	1.13994i	0.821665	+	0.569970i	$0.193045\pi$
$733$	30.8627i	1.13994i	−0.821665	+	0.569970i	$0.806955\pi$
$734$	−72.4260	−2.67329
$735$	0	0
$736$	−87.8310	−3.23749
$737$	0.792890i	0.0292065i
$738$	− 30.5804i	− 1.12568i
$739$	−34.1309	−1.25553	−0.627763	−	0.778404i	$-0.716029\pi$
$739$	−34.1309	−1.25553	−0.627763	+	0.778404i	$0.716029\pi$
$740$	0	0
$741$	5.11871	0.188041
$742$	52.0494i	1.91079i
$743$	14.1916i	0.520638i	0.965523	+	0.260319i	$0.0838277\pi$
$743$	14.1916i	0.520638i	−0.965523	+	0.260319i	$0.916172\pi$
$744$	65.8080	2.41264
$745$	0	0
$746$	9.34534	0.342157
$747$	6.80114i	0.248841i
$748$	5.37565i	0.196553i
$749$	−6.06063	−0.221451
$750$	0	0
$751$	13.4050	0.489156	0.244578	−	0.969630i	$-0.421351\pi$
$751$	13.4050	0.489156	0.244578	+	0.969630i	$0.421351\pi$
$752$	32.8373i	1.19745i
$753$	17.7685i	0.647518i
$754$	−8.02539	−0.292268
$755$	0	0
$756$	−11.7938	−0.428938
$757$	5.76116i	0.209393i	0.994504	+	0.104696i	$0.0333871\pi$
$757$	5.76116i	0.209393i	−0.994504	+	0.104696i	$0.966613\pi$
$758$	− 72.2116i	− 2.62284i
$759$	−0.720992	−0.0261703
$760$	0	0
$761$	−9.05334	−0.328183	−0.164092	−	0.986445i	$-0.552469\pi$
$761$	−9.05334	−0.328183	−0.164092	+	0.986445i	$0.552469\pi$
$762$	23.3561i	0.846104i
$763$	− 21.3171i	− 0.771731i
$764$	−43.2506	−1.56475
$765$	0	0
$766$	14.7699	0.533657
$767$	10.4133i	0.376001i
$768$	− 8.08840i	− 0.291865i
$769$	38.8383	1.40054	0.700272	−	0.713876i	$-0.253062\pi$
$769$	38.8383	1.40054	0.700272	+	0.713876i	$0.253062\pi$
$770$	0	0
$771$	−4.81924	−0.173561
$772$	93.6199i	3.36945i
$773$	23.7381i	0.853801i	0.904299	+	0.426901i	$0.140395\pi$
$773$	23.7381i	0.853801i	−0.904299	+	0.426901i	$0.859605\pi$
$774$	7.84367	0.281935
$775$	0	0
$776$	160.111	5.74763
$777$	− 10.9927i	− 0.394361i
$778$	42.5355i	1.52497i
$779$	58.5139	2.09648
$780$	0	0
$781$	1.96968	0.0704808
$782$	− 117.293i	− 4.19438i
$783$	3.00000i	0.107211i
$784$	21.7078	0.775279
$785$	0	0
$786$	50.9438	1.81711
$787$	19.1803i	0.683704i	0.939754	+	0.341852i	$0.111054\pi$
$787$	19.1803i	0.683704i	−0.939754	+	0.341852i	$0.888946\pi$
$788$	6.15633i	0.219310i
$789$	1.30536	0.0464719
$790$	0	0
$791$	−30.5355	−1.08572
$792$	− 1.10554i	− 0.0392837i
$793$	5.18664i	0.184183i
$794$	46.0118	1.63290
$795$	0	0
$796$	3.61213	0.128028
$797$	51.8627i	1.83707i	0.395337	+	0.918536i	$0.370628\pi$
$797$	51.8627i	1.83707i	−0.395337	+	0.918536i	$0.629372\pi$
$798$	− 31.3199i	− 1.10871i
$799$	−21.3004	−0.753555
$800$	0	0
$801$	−8.12601	−0.287118
$802$	50.9438i	1.79889i
$803$	− 0.122039i	− 0.00430665i
$804$	31.2252	1.10123
$805$	0	0
$806$	20.8496	0.734394
$807$	− 8.79877i	− 0.309731i
$808$	29.9257i	1.05278i
$809$	18.1866	0.639408	0.319704	−	0.947517i	$-0.396416\pi$
$809$	18.1866	0.639408	0.319704	+	0.947517i	$0.396416\pi$
$810$	0	0
$811$	4.71133	0.165437	0.0827185	−	0.996573i	$-0.473640\pi$
$811$	4.71133	0.165437	0.0827185	+	0.996573i	$0.473640\pi$
$812$	35.3815i	1.24165i
$813$	− 17.4264i	− 0.611172i
$814$	1.68338	0.0590024
$815$	0	0
$816$	97.7386	3.42154
$817$	15.0084i	0.525079i
$818$	− 33.2203i	− 1.16152i
$819$	−2.28726	−0.0799233
$820$	0	0
$821$	−33.8046	−1.17979	−0.589895	−	0.807480i	$-0.700831\pi$
$821$	−33.8046	−1.17979	−0.589895	+	0.807480i	$0.700831\pi$
$822$	20.4993i	0.714995i
$823$	2.82653i	0.0985267i	0.998786	+	0.0492633i	$0.0156874\pi$
$823$	2.82653i	0.0985267i	−0.998786	+	0.0492633i	$0.984313\pi$
$824$	−118.895	−4.14190
$825$	0	0
$826$	63.7158	2.21696
$827$	39.9560i	1.38941i	0.719296	+	0.694704i	$0.244465\pi$
$827$	39.9560i	1.38941i	−0.719296	+	0.694704i	$0.755535\pi$
$828$	28.3938i	0.986751i
$829$	−1.30280	−0.0452482	−0.0226241	−	0.999744i	$-0.507202\pi$
$829$	−1.30280	−0.0452482	−0.0226241	+	0.999744i	$0.507202\pi$
$830$	0	0
$831$	11.7381	0.407191
$832$	− 18.1187i	− 0.628153i
$833$	14.0811i	0.487881i
$834$	−20.3634	−0.705128
$835$	0	0
$836$	3.45580	0.119521
$837$	− 7.79384i	− 0.269395i
$838$	− 16.4299i	− 0.567563i
$839$	17.9551	0.619879	0.309939	−	0.950756i	$-0.399691\pi$
$839$	17.9551	0.619879	0.309939	+	0.950756i	$0.399691\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	− 47.1147i	− 1.62368i
$843$	14.5647i	0.501634i
$844$	−87.5691	−3.01425
$845$	0	0
$846$	7.15633	0.246040
$847$	25.1206i	0.863155i
$848$	104.419i	3.58575i
$849$	−26.1016	−0.895804
$850$	0	0
$851$	−26.4650	−0.907209
$852$	− 77.5691i	− 2.65747i
$853$	43.3258i	1.48345i	0.670705	+	0.741724i	$0.265992\pi$
$853$	43.3258i	1.48345i	−0.670705	+	0.741724i	$0.734008\pi$
$854$	31.7356	1.08597
$855$	0	0
$856$	−22.3733	−0.764703
$857$	− 15.5223i	− 0.530232i	−0.964216	−	0.265116i	$-0.914590\pi$
$857$	− 15.5223i	− 0.530232i	0.964216	−	0.265116i	$-0.0854103\pi$
$858$	− 0.350262i	− 0.0119577i
$859$	−14.0957	−0.480939	−0.240469	−	0.970657i	$-0.577301\pi$
$859$	−14.0957	−0.480939	−0.240469	+	0.970657i	$0.577301\pi$
$860$	0	0
$861$	−26.1465	−0.891070
$862$	− 9.78304i	− 0.333212i
$863$	46.6105i	1.58664i	0.608804	+	0.793320i	$0.291649\pi$
$863$	46.6105i	1.58664i	−0.608804	+	0.793320i	$0.708351\pi$
$864$	−15.9502	−0.542636
$865$	0	0
$866$	24.8510	0.844470
$867$	46.3996i	1.57581i
$868$	− 91.9194i	− 3.11995i
$869$	1.15965	0.0393384
$870$	0	0
$871$	6.05571	0.205190
$872$	− 78.6937i	− 2.66491i
$873$	− 18.9624i	− 0.641780i
$874$	−75.4030	−2.55054
$875$	0	0
$876$	−4.80606	−0.162382
$877$	− 33.3357i	− 1.12567i	−0.826571	−	0.562833i	$-0.809711\pi$
$877$	− 33.3357i	− 1.12567i	0.826571	−	0.562833i	$-0.190289\pi$
$878$	− 72.3839i	− 2.44284i
$879$	−9.07381	−0.306052
$880$	0	0
$881$	22.1793	0.747241	0.373621	−	0.927582i	$-0.378116\pi$
$881$	22.1793	0.747241	0.373621	+	0.927582i	$0.378116\pi$
$882$	− 4.73084i	− 0.159296i
$883$	27.4109i	0.922450i	0.887283	+	0.461225i	$0.152590\pi$
$883$	27.4109i	0.922450i	−0.887283	+	0.461225i	$0.847410\pi$
$884$	41.0567	1.38089
$885$	0	0
$886$	18.9781	0.637582
$887$	1.33170i	0.0447142i	0.999750	+	0.0223571i	$0.00711708\pi$
$887$	1.33170i	0.0447142i	−0.999750	+	0.0223571i	$0.992883\pi$
$888$	− 40.5804i	− 1.36179i
$889$	19.9697	0.669762
$890$	0	0
$891$	−0.130933	−0.00438641
$892$	− 4.15633i	− 0.139164i
$893$	13.6932i	0.458226i
$894$	−2.37328	−0.0793745
$895$	0	0
$896$	−37.8989	−1.26611
$897$	5.50659i	0.183860i
$898$	39.4880i	1.31773i
$899$	−23.3815	−0.779818
$900$	0	0
$901$	−67.7328	−2.25651
$902$	− 4.00397i	− 0.133318i
$903$	− 6.70640i	− 0.223175i
$904$	−112.724	−3.74915
$905$	0	0
$906$	7.85685	0.261026
$907$	− 31.1754i	− 1.03516i	−0.855634	−	0.517581i	$-0.826833\pi$
$907$	− 31.1754i	− 1.03516i	0.855634	−	0.517581i	$-0.173167\pi$
$908$	− 39.6131i	− 1.31461i
$909$	3.54420	0.117554
$910$	0	0
$911$	9.56722	0.316976	0.158488	−	0.987361i	$-0.449338\pi$
$911$	9.56722	0.316976	0.158488	+	0.987361i	$0.449338\pi$
$912$	− 62.8324i	− 2.08059i
$913$	0.890491i	0.0294709i
$914$	−6.10554	−0.201953
$915$	0	0
$916$	−22.9380	−0.757891
$917$	− 43.5574i	− 1.43839i
$918$	− 21.3004i	− 0.703019i
$919$	35.8397	1.18224	0.591121	−	0.806583i	$-0.298685\pi$
$919$	35.8397	1.18224	0.591121	+	0.806583i	$0.298685\pi$
$920$	0	0
$921$	−11.3806	−0.375003
$922$	− 28.2619i	− 0.930755i
$923$	− 15.0435i	− 0.495163i
$924$	−1.54420	−0.0508004
$925$	0	0
$926$	−16.5369	−0.543436
$927$	14.0811i	0.462484i
$928$	47.8505i	1.57077i
$929$	−19.7137	−0.646785	−0.323393	−	0.946265i	$-0.604823\pi$
$929$	−19.7137	−0.646785	−0.323393	+	0.946265i	$0.604823\pi$
$930$	0	0
$931$	9.05220	0.296674
$932$	− 52.2130i	− 1.71029i
$933$	− 9.22425i	− 0.301989i
$934$	−16.9135	−0.553427
$935$	0	0
$936$	−8.44358	−0.275987
$937$	− 57.2736i	− 1.87105i	−0.353263	−	0.935524i	$-0.614928\pi$
$937$	− 57.2736i	− 1.87105i	0.353263	−	0.935524i	$-0.385072\pi$
$938$	− 37.0532i	− 1.20983i
$939$	−18.7685	−0.612485
$940$	0	0
$941$	−25.2447	−0.822954	−0.411477	−	0.911420i	$-0.634987\pi$
$941$	−25.2447	−0.822954	−0.411477	+	0.911420i	$0.634987\pi$
$942$	42.2179i	1.37553i
$943$	62.9478i	2.04986i
$944$	127.823	4.16029
$945$	0	0
$946$	1.02699	0.0333904
$947$	− 5.04254i	− 0.163860i	−0.996638	−	0.0819302i	$-0.973892\pi$
$947$	− 5.04254i	− 0.163860i	0.996638	−	0.0819302i	$-0.0261085\pi$
$948$	− 45.6688i	− 1.48325i
$949$	−0.932071	−0.0302563
$950$	0	0
$951$	10.5442	0.341919
$952$	− 153.775i	− 4.98387i
$953$	15.4426i	0.500236i	0.968215	+	0.250118i	$0.0804694\pi$
$953$	15.4426i	0.500236i	−0.968215	+	0.250118i	$0.919531\pi$
$954$	22.7562	0.736760
$955$	0	0
$956$	−100.220	−3.24134
$957$	0.392798i	0.0126973i
$958$	− 8.91019i	− 0.287875i
$959$	17.5271	0.565978
$960$	0	0
$961$	29.7440	0.959484
$962$	− 12.8568i	− 0.414521i
$963$	2.64974i	0.0853866i
$964$	33.3561	1.07433
$965$	0	0
$966$	33.6932	1.08406
$967$	− 30.2979i	− 0.974314i	−0.873314	−	0.487157i	$-0.838034\pi$
$967$	− 30.2979i	− 0.974314i	0.873314	−	0.487157i	$-0.161966\pi$
$968$	92.7347i	2.98060i
$969$	40.7572	1.30931
$970$	0	0
$971$	−40.7367	−1.30730	−0.653652	−	0.756795i	$-0.726764\pi$
$971$	−40.7367	−1.30730	−0.653652	+	0.756795i	$0.726764\pi$
$972$	5.15633i	0.165389i
$973$	17.4109i	0.558168i
$974$	−22.5877	−0.723756
$975$	0	0
$976$	63.6662	2.03791
$977$	53.1608i	1.70076i	0.526166	+	0.850382i	$0.323629\pi$
$977$	53.1608i	1.70076i	−0.526166	+	0.850382i	$0.676371\pi$
$978$	46.6312i	1.49110i
$979$	−1.06396	−0.0340043
$980$	0	0
$981$	−9.31994	−0.297563
$982$	− 62.3606i	− 1.99001i
$983$	− 11.1831i	− 0.356687i	−0.983968	−	0.178343i	$-0.942926\pi$
$983$	− 11.1831i	− 0.356687i	0.983968	−	0.178343i	$-0.0570737\pi$
$984$	−96.5217	−3.07700
$985$	0	0
$986$	−63.9013	−2.03503
$987$	− 6.11871i	− 0.194761i
$988$	− 26.3938i	− 0.839697i
$989$	−16.1457	−0.513404
$990$	0	0
$991$	−57.9208	−1.83992	−0.919958	−	0.392018i	$-0.871777\pi$
$991$	−57.9208	−1.83992	−0.919958	+	0.392018i	$0.871777\pi$
$992$	− 124.313i	− 3.94695i
$993$	− 4.95651i	− 0.157290i
$994$	−92.0468	−2.91955
$995$	0	0
$996$	35.0689	1.11120
$997$	− 27.4241i	− 0.868529i	−0.900785	−	0.434265i	$-0.857008\pi$
$997$	− 27.4241i	− 0.868529i	0.900785	−	0.434265i	$-0.142992\pi$
$998$	85.2433i	2.69833i
$999$	−4.80606	−0.152057

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.c.k.274.6		6
3.2	odd	2		2925.2.c.y.2224.1		6
5.2	odd	4		975.2.a.m.1.1	✓	3
5.3	odd	4		975.2.a.q.1.3	yes	3
5.4	even	2	inner	975.2.c.k.274.1		6
15.2	even	4		2925.2.a.bk.1.3		3
15.8	even	4		2925.2.a.be.1.1		3
15.14	odd	2		2925.2.c.y.2224.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
975.2.a.m.1.1	✓	3	5.2	odd	4
975.2.a.q.1.3	yes	3	5.3	odd	4
975.2.c.k.274.1		6	5.4	even	2	inner
975.2.c.k.274.6		6	1.1	even	1	trivial
2925.2.a.be.1.1		3	15.8	even	4
2925.2.a.bk.1.3		3	15.2	even	4
2925.2.c.y.2224.1		6	3.2	odd	2
2925.2.c.y.2224.6		6	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.67513i q^{2} -1.00000i q^{3} -5.15633 q^{4} +2.67513 q^{6} -2.28726i q^{7} -8.44358i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+2.67513i q^{2} -1.00000i q^{3} -5.15633 q^{4} +2.67513 q^{6} -2.28726i q^{7} -8.44358i q^{8} -1.00000 q^{9} -0.130933 q^{11} +5.15633i q^{12} +1.00000i q^{13} +6.11871 q^{14} +12.2750 q^{16} +7.96239i q^{17} -2.67513i q^{18} +5.11871 q^{19} -2.28726 q^{21} -0.350262i q^{22} +5.50659i q^{23} -8.44358 q^{24} -2.67513 q^{26} +1.00000i q^{27} +11.7938i q^{28} +3.00000 q^{29} -7.79384 q^{31} +15.9502i q^{32} +0.130933i q^{33} -21.3004 q^{34} +5.15633 q^{36} +4.80606i q^{37} +13.6932i q^{38} +1.00000 q^{39} +11.4314 q^{41} -6.11871i q^{42} +2.93207i q^{43} +0.675131 q^{44} -14.7308 q^{46} +2.67513i q^{47} -12.2750i q^{48} +1.76845 q^{49} +7.96239 q^{51} -5.15633i q^{52} +8.50659i q^{53} -2.67513 q^{54} -19.3127 q^{56} -5.11871i q^{57} +8.02539i q^{58} +10.4133 q^{59} +5.18664 q^{61} -20.8496i q^{62} +2.28726i q^{63} -18.1187 q^{64} -0.350262 q^{66} -6.05571i q^{67} -41.0567i q^{68} +5.50659 q^{69} -15.0435 q^{71} +8.44358i q^{72} +0.932071i q^{73} -12.8568 q^{74} -26.3938 q^{76} +0.299477i q^{77} +2.67513i q^{78} -8.85685 q^{79} +1.00000 q^{81} +30.5804i q^{82} -6.80114i q^{83} +11.7938 q^{84} -7.84367 q^{86} -3.00000i q^{87} +1.10554i q^{88} +8.12601 q^{89} +2.28726 q^{91} -28.3938i q^{92} +7.79384i q^{93} -7.15633 q^{94} +15.9502 q^{96} +18.9624i q^{97} +4.73084i q^{98} +0.130933 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 10 q^{4} + 6 q^{6} - 6 q^{9}+O(q^{10}) \) 6 * q - 10 * q^4 + 6 * q^6 - 6 * q^9	\( 6 q - 10 q^{4} + 6 q^{6} - 6 q^{9} - 10 q^{11} - 6 q^{14} + 10 q^{16} - 12 q^{19} - 2 q^{21} - 18 q^{24} - 6 q^{26} + 18 q^{29} + 6 q^{31} - 34 q^{34} + 10 q^{36} + 6 q^{39} - 16 q^{41} - 6 q^{44} - 44 q^{46} - 12 q^{49} + 26 q^{51} - 6 q^{54} - 74 q^{56} + 34 q^{59} + 6 q^{61} - 66 q^{64} + 18 q^{66} - 8 q^{69} - 4 q^{71} - 16 q^{74} - 52 q^{76} + 8 q^{79} + 6 q^{81} + 18 q^{84} - 68 q^{86} + 32 q^{89} + 2 q^{91} - 22 q^{94} + 22 q^{96} + 10 q^{99}+O(q^{100}) \) 6 * q - 10 * q^4 + 6 * q^6 - 6 * q^9 - 10 * q^11 - 6 * q^14 + 10 * q^16 - 12 * q^19 - 2 * q^21 - 18 * q^24 - 6 * q^26 + 18 * q^29 + 6 * q^31 - 34 * q^34 + 10 * q^36 + 6 * q^39 - 16 * q^41 - 6 * q^44 - 44 * q^46 - 12 * q^49 + 26 * q^51 - 6 * q^54 - 74 * q^56 + 34 * q^59 + 6 * q^61 - 66 * q^64 + 18 * q^66 - 8 * q^69 - 4 * q^71 - 16 * q^74 - 52 * q^76 + 8 * q^79 + 6 * q^81 + 18 * q^84 - 68 * q^86 + 32 * q^89 + 2 * q^91 - 22 * q^94 + 22 * q^96 + 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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