LMFDB - Embedded newform 1075.2.b.g.474.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1075,2,Mod(474,1075)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1075, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1075.474");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1075 = 5^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1075.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:	$8.58391821729$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.6594624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 9x^{4} + 18x^{2} + 1 $ x^6 + 9x^4 + 18x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 215)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			474.4
Root			$-0.239123i$ of defining polynomial
Character	$\chi$	$=$	1075.474
Dual form			1075.2.b.g.474.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.760877i q^{2} +0.239123i q^{3} +1.42107 q^{4} -0.181943 q^{6} -1.94282i q^{7} +2.60301i q^{8} +2.94282 q^{9} +O(q^{10})$	\(q+0.760877i q^{2} +0.239123i q^{3} +1.42107 q^{4} -0.181943 q^{6} -1.94282i q^{7} +2.60301i q^{8} +2.94282 q^{9} +5.66019 q^{11} +0.339810i q^{12} -0.478247i q^{13} +1.47825 q^{14} +0.861564 q^{16} -3.52175i q^{17} +2.23912i q^{18} -7.88564 q^{19} +0.464574 q^{21} +4.30671i q^{22} -7.88564i q^{23} -0.622440 q^{24} +0.363887 q^{26} +1.42107i q^{27} -2.76088i q^{28} -0.478247 q^{29} +1.57893 q^{31} +5.86156i q^{32} +1.35348i q^{33} +2.67962 q^{34} +4.18194 q^{36} -0.339810i q^{37} -6.00000i q^{38} +0.114360 q^{39} -0.603010 q^{41} +0.353483i q^{42} -1.00000i q^{43} +8.04351 q^{44} +6.00000 q^{46} +12.8421i q^{47} +0.206020i q^{48} +3.22545 q^{49} +0.842133 q^{51} -0.679620i q^{52} +2.47825i q^{53} -1.08126 q^{54} +5.05718 q^{56} -1.88564i q^{57} -0.363887i q^{58} +6.96690 q^{59} +8.24953 q^{61} +1.20137i q^{62} -5.71737i q^{63} -2.73680 q^{64} -1.02983 q^{66} +7.88564i q^{67} -5.00465i q^{68} +1.88564 q^{69} -13.2060 q^{71} +7.66019i q^{72} +6.60301i q^{73} +0.258554 q^{74} -11.2060 q^{76} -10.9967i q^{77} +0.0870138i q^{78} -3.06758 q^{79} +8.48865 q^{81} -0.458816i q^{82} -4.56526i q^{83} +0.660190 q^{84} +0.760877 q^{86} -0.114360i q^{87} +14.7335i q^{88} -14.2495 q^{89} -0.929147 q^{91} -11.2060i q^{92} +0.377560i q^{93} -9.77128 q^{94} -1.40164 q^{96} +9.77128i q^{97} +2.45417i q^{98} +16.6569 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 8 q^{4} + 16 q^{6}+O(q^{10}) $ 6 * q - 8 * q^4 + 16 * q^6	$ 6 q - 8 q^{4} + 16 q^{6} + 18 q^{11} + 10 q^{14} + 20 q^{16} - 12 q^{19} - 16 q^{21} - 18 q^{24} - 32 q^{26} - 4 q^{29} + 26 q^{31} + 48 q^{34} + 8 q^{36} + 36 q^{39} + 30 q^{41} + 46 q^{44} + 36 q^{46} - 28 q^{51} + 26 q^{54} + 48 q^{56} - 26 q^{59} - 20 q^{61} - 66 q^{64} + 16 q^{66} - 24 q^{69} - 12 q^{71} + 50 q^{74} + 34 q^{79} - 18 q^{81} - 12 q^{84} + 4 q^{86} - 16 q^{89} + 32 q^{91} + 12 q^{94} + 56 q^{96} - 6 q^{99}+O(q^{100}) $ 6 * q - 8 * q^4 + 16 * q^6 + 18 * q^11 + 10 * q^14 + 20 * q^16 - 12 * q^19 - 16 * q^21 - 18 * q^24 - 32 * q^26 - 4 * q^29 + 26 * q^31 + 48 * q^34 + 8 * q^36 + 36 * q^39 + 30 * q^41 + 46 * q^44 + 36 * q^46 - 28 * q^51 + 26 * q^54 + 48 * q^56 - 26 * q^59 - 20 * q^61 - 66 * q^64 + 16 * q^66 - 24 * q^69 - 12 * q^71 + 50 * q^74 + 34 * q^79 - 18 * q^81 - 12 * q^84 + 4 * q^86 - 16 * q^89 + 32 * q^91 + 12 * q^94 + 56 * q^96 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1075\mathbb{Z}\right)^\times$.

$n$	$302$	$476$
$\chi(n)$	$-1$	$1$

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 $

$2$	0.760877i	0.538021i	0.963137	+	0.269011i	$0.0866966\pi$
$2$	0.760877i	0.538021i	−0.963137	+	0.269011i	$0.913303\pi$
$3$	0.239123i	0.138058i	0.997615	+	0.0690289i	$0.0219901\pi$
$3$	0.239123i	0.138058i	−0.997615	+	0.0690289i	$0.978010\pi$
$4$	1.42107	0.710533
$5$	0	0
$5$	0	0
$6$	−0.181943	−0.0742781
$7$	− 1.94282i	− 0.734317i	−0.930158	−	0.367158i	$-0.880331\pi$
$7$	− 1.94282i	− 0.734317i	0.930158	−	0.367158i	$-0.119669\pi$
$8$	2.60301i	0.920303i
$9$	2.94282	0.980940
$10$	0	0
$11$	5.66019	1.70661	0.853306	−	0.521411i	$-0.174594\pi$
$11$	5.66019	1.70661	0.853306	+	0.521411i	$0.174594\pi$
$12$	0.339810i	0.0980947i
$13$	− 0.478247i	− 0.132642i	−0.997798	−	0.0663209i	$-0.978874\pi$
$13$	− 0.478247i	− 0.132642i	0.997798	−	0.0663209i	$-0.0211261\pi$
$14$	1.47825	0.395078
$15$	0	0
$16$	0.861564	0.215391
$17$	− 3.52175i	− 0.854151i	−0.904216	−	0.427075i	$-0.859544\pi$
$17$	− 3.52175i	− 0.854151i	0.904216	−	0.427075i	$-0.140456\pi$
$18$	2.23912i	0.527766i
$19$	−7.88564	−1.80909	−0.904545	−	0.426378i	$-0.859789\pi$
$19$	−7.88564	−1.80909	−0.904545	+	0.426378i	$0.859789\pi$
$20$	0	0
$21$	0.464574	0.101378
$22$	4.30671i	0.918193i
$23$	− 7.88564i	− 1.64427i	−0.569293	−	0.822135i	$-0.692783\pi$
$23$	− 7.88564i	− 1.64427i	0.569293	−	0.822135i	$-0.307217\pi$
$24$	−0.622440	−0.127055
$25$	0	0
$26$	0.363887	0.0713640
$27$	1.42107i	0.273484i
$28$	− 2.76088i	− 0.521757i
$29$	−0.478247	−0.0888082	−0.0444041	−	0.999014i	$-0.514139\pi$
$29$	−0.478247	−0.0888082	−0.0444041	+	0.999014i	$0.514139\pi$
$30$	0	0
$31$	1.57893	0.283585	0.141792	−	0.989896i	$-0.454713\pi$
$31$	1.57893	0.283585	0.141792	+	0.989896i	$0.454713\pi$
$32$	5.86156i	1.03619i
$33$	1.35348i	0.235611i
$34$	2.67962	0.459551
$35$	0	0
$36$	4.18194	0.696991
$37$	− 0.339810i	− 0.0558644i	−0.999610	−	0.0279322i	$-0.991108\pi$
$37$	− 0.339810i	− 0.0558644i	0.999610	−	0.0279322i	$-0.00889226\pi$
$38$	− 6.00000i	− 0.973329i
$39$	0.114360	0.0183122
$40$	0	0
$41$	−0.603010	−0.0941743	−0.0470872	−	0.998891i	$-0.514994\pi$
$41$	−0.603010	−0.0941743	−0.0470872	+	0.998891i	$0.514994\pi$
$42$	0.353483i	0.0545436i
$43$	− 1.00000i	− 0.152499i
$43$	− 1.00000i	− 0.152499i
$44$	8.04351	1.21260
$45$	0	0
$46$	6.00000	0.884652
$47$	12.8421i	1.87322i	0.350377	+	0.936609i	$0.386054\pi$
$47$	12.8421i	1.87322i	−0.350377	+	0.936609i	$0.613946\pi$
$48$	0.206020i	0.0297364i
$49$	3.22545	0.460779
$50$	0	0
$51$	0.842133	0.117922
$52$	− 0.679620i	− 0.0942464i
$53$	2.47825i	0.340413i	0.985408	+	0.170207i	$0.0544435\pi$
$53$	2.47825i	0.340413i	−0.985408	+	0.170207i	$0.945556\pi$
$54$	−1.08126	−0.147140
$55$	0	0
$56$	5.05718	0.675794
$57$	− 1.88564i	− 0.249759i
$58$	− 0.363887i	− 0.0477807i
$59$	6.96690	0.907013	0.453506	−	0.891253i	$-0.350173\pi$
$59$	6.96690	0.907013	0.453506	+	0.891253i	$0.350173\pi$
$60$	0	0
$61$	8.24953	1.05624	0.528122	−	0.849169i	$-0.322896\pi$
$61$	8.24953	1.05624	0.528122	+	0.849169i	$0.322896\pi$
$62$	1.20137i	0.152575i
$63$	− 5.71737i	− 0.720321i
$64$	−2.73680	−0.342100
$65$	0	0
$66$	−1.02983	−0.126764
$67$	7.88564i	0.963384i	0.876341	+	0.481692i	$0.159978\pi$
$67$	7.88564i	0.963384i	−0.876341	+	0.481692i	$0.840022\pi$
$68$	− 5.00465i	− 0.606903i
$69$	1.88564	0.227004
$70$	0	0
$71$	−13.2060	−1.56727	−0.783633	−	0.621224i	$-0.786636\pi$
$71$	−13.2060	−1.56727	−0.783633	+	0.621224i	$0.786636\pi$
$72$	7.66019i	0.902762i
$73$	6.60301i	0.772824i	0.922326	+	0.386412i	$0.126286\pi$
$73$	6.60301i	0.772824i	−0.922326	+	0.386412i	$0.873714\pi$
$74$	0.258554	0.0300562
$75$	0	0
$76$	−11.2060	−1.28542
$77$	− 10.9967i	− 1.25319i
$78$	0.0870138i	0.00985237i
$79$	−3.06758	−0.345130	−0.172565	−	0.984998i	$-0.555206\pi$
$79$	−3.06758	−0.345130	−0.172565	+	0.984998i	$0.555206\pi$
$80$	0	0
$81$	8.48865	0.943183
$82$	− 0.458816i	− 0.0506678i
$83$	− 4.56526i	− 0.501102i	−0.968103	−	0.250551i	$-0.919388\pi$
$83$	− 4.56526i	− 0.501102i	0.968103	−	0.250551i	$-0.0806118\pi$
$84$	0.660190	0.0720326
$85$	0	0
$86$	0.760877	0.0820474
$87$	− 0.114360i	− 0.0122607i
$88$	14.7335i	1.57060i
$89$	−14.2495	−1.51045	−0.755223	−	0.655467i	$-0.772472\pi$
$89$	−14.2495	−1.51045	−0.755223	+	0.655467i	$0.772472\pi$
$90$	0	0
$91$	−0.929147	−0.0974011
$92$	− 11.2060i	− 1.16831i
$93$	0.377560i	0.0391511i
$94$	−9.77128	−1.00783
$95$	0	0
$96$	−1.40164	−0.143054
$97$	9.77128i	0.992123i	0.868287	+	0.496062i	$0.165221\pi$
$97$	9.77128i	0.992123i	−0.868287	+	0.496062i	$0.834779\pi$
$98$	2.45417i	0.247909i
$99$	16.6569	1.67408
$100$	0	0
$101$	−6.67059	−0.663749	−0.331874	−	0.943324i	$-0.607681\pi$
$101$	−6.67059	−0.663749	−0.331874	+	0.943324i	$0.607681\pi$
$102$	0.640760i	0.0634447i
$103$	13.7713i	1.35692i	0.734635	+	0.678462i	$0.237353\pi$
$103$	13.7713i	1.35692i	−0.734635	+	0.678462i	$0.762647\pi$
$104$	1.24488	0.122071
$105$	0	0
$106$	−1.88564	−0.183150
$107$	− 19.5699i	− 1.89189i	−0.324321	−	0.945947i	$-0.605136\pi$
$107$	− 19.5699i	− 1.89189i	0.324321	−	0.945947i	$-0.394864\pi$
$108$	2.01943i	0.194320i
$109$	−6.05718	−0.580173	−0.290086	−	0.957000i	$-0.593684\pi$
$109$	−6.05718	−0.580173	−0.290086	+	0.957000i	$0.593684\pi$
$110$	0	0
$111$	0.0812565	0.00771253
$112$	− 1.67386i	− 0.158165i
$113$	− 11.3776i	− 1.07031i	−0.844754	−	0.535155i	$-0.820253\pi$
$113$	− 11.3776i	− 1.07031i	0.844754	−	0.535155i	$-0.179747\pi$
$114$	1.43474	0.134376
$115$	0	0
$116$	−0.679620	−0.0631012
$117$	− 1.40739i	− 0.130114i
$118$	5.30095i	0.487992i
$119$	−6.84213	−0.627217
$120$	0	0
$121$	21.0377	1.91252
$122$	6.27687i	0.568281i
$123$	− 0.144194i	− 0.0130015i
$124$	2.24377	0.201496
$125$	0	0
$126$	4.35021	0.387548
$127$	16.2495i	1.44191i	0.692981	+	0.720956i	$0.256297\pi$
$127$	16.2495i	1.44191i	−0.692981	+	0.720956i	$0.743703\pi$
$128$	9.64076i	0.852131i
$129$	0.239123	0.0210536
$130$	0	0
$131$	−7.52175	−0.657179	−0.328589	−	0.944473i	$-0.606573\pi$
$131$	−7.52175	−0.657179	−0.328589	+	0.944473i	$0.606573\pi$
$132$	1.92339i	0.167410i
$133$	15.3204i	1.32845i
$134$	−6.00000	−0.518321
$135$	0	0
$136$	9.16716	0.786077
$137$	11.8090i	1.00891i	0.863437	+	0.504457i	$0.168307\pi$
$137$	11.8090i	1.00891i	−0.863437	+	0.504457i	$0.831693\pi$
$138$	1.43474i	0.122133i
$139$	−9.53216	−0.808507	−0.404254	−	0.914647i	$-0.632469\pi$
$139$	−9.53216	−0.808507	−0.404254	+	0.914647i	$0.632469\pi$
$140$	0	0
$141$	−3.07085	−0.258612
$142$	− 10.0482i	− 0.843222i
$143$	− 2.70697i	− 0.226368i
$144$	2.53543	0.211286
$145$	0	0
$146$	−5.02408	−0.415796
$147$	0.771280i	0.0636141i
$148$	− 0.482893i	− 0.0396935i
$149$	6.92915	0.567658	0.283829	−	0.958875i	$-0.408395\pi$
$149$	6.92915	0.567658	0.283829	+	0.958875i	$0.408395\pi$
$150$	0	0
$151$	12.6134	1.02647	0.513233	−	0.858250i	$-0.328448\pi$
$151$	12.6134	1.02647	0.513233	+	0.858250i	$0.328448\pi$
$152$	− 20.5264i	− 1.66491i
$153$	− 10.3639i	− 0.837871i
$154$	8.36716	0.674245
$155$	0	0
$156$	0.162513	0.0130115
$157$	2.12476i	0.169575i	0.996399	+	0.0847873i	$0.0270211\pi$
$157$	2.12476i	0.169575i	−0.996399	+	0.0847873i	$0.972979\pi$
$158$	− 2.33405i	− 0.185687i
$159$	−0.592606	−0.0469967
$160$	0	0
$161$	−15.3204	−1.20742
$162$	6.45882i	0.507453i
$163$	− 17.9097i	− 1.40280i	−0.712769	−	0.701399i	$-0.752559\pi$
$163$	− 17.9097i	− 1.40280i	0.712769	−	0.701399i	$-0.247441\pi$
$164$	−0.856917	−0.0669140
$165$	0	0
$166$	3.47360	0.269604
$167$	5.20602i	0.402854i	0.979504	+	0.201427i	$0.0645579\pi$
$167$	5.20602i	0.402854i	−0.979504	+	0.201427i	$0.935442\pi$
$168$	1.20929i	0.0932987i
$169$	12.7713	0.982406
$170$	0	0
$171$	−23.2060	−1.77461
$172$	− 1.42107i	− 0.108355i
$173$	− 16.4782i	− 1.25282i	−0.779495	−	0.626409i	$-0.784524\pi$
$173$	− 16.4782i	− 1.25282i	0.779495	−	0.626409i	$-0.215476\pi$
$174$	0.0870138	0.00659650
$175$	0	0
$176$	4.87661	0.367589
$177$	1.66595i	0.125220i
$178$	− 10.8421i	− 0.812652i
$179$	8.24953	0.616599	0.308299	−	0.951289i	$-0.400240\pi$
$179$	8.24953	0.616599	0.308299	+	0.951289i	$0.400240\pi$
$180$	0	0
$181$	−15.1683	−1.12745	−0.563724	−	0.825963i	$-0.690632\pi$
$181$	−15.1683	−1.12745	−0.563724	+	0.825963i	$0.690632\pi$
$182$	− 0.706966i	− 0.0524038i
$183$	1.97265i	0.145823i
$184$	20.5264	1.51323
$185$	0	0
$186$	−0.287276	−0.0210641
$187$	− 19.9338i	− 1.45770i
$188$	18.2495i	1.33098i
$189$	2.76088	0.200824
$190$	0	0
$191$	−17.6843	−1.27959	−0.639794	−	0.768546i	$-0.720980\pi$
$191$	−17.6843	−1.27959	−0.639794	+	0.768546i	$0.720980\pi$
$192$	− 0.654433i	− 0.0472296i
$193$	18.1352i	1.30540i	0.757617	+	0.652699i	$0.226363\pi$
$193$	18.1352i	1.30540i	−0.757617	+	0.652699i	$0.773637\pi$
$194$	−7.43474	−0.533783
$195$	0	0
$196$	4.58358	0.327399
$197$	− 4.72777i	− 0.336840i	−0.985715	−	0.168420i	$-0.946134\pi$
$197$	− 4.72777i	− 0.336840i	0.985715	−	0.168420i	$-0.0538665\pi$
$198$	12.6739i	0.900692i
$199$	−2.56526	−0.181846	−0.0909232	−	0.995858i	$-0.528982\pi$
$199$	−2.56526	−0.181846	−0.0909232	+	0.995858i	$0.528982\pi$
$200$	0	0
$201$	−1.88564	−0.133003
$202$	− 5.07550i	− 0.357111i
$203$	0.929147i	0.0652133i
$204$	1.19673	0.0837877
$205$	0	0
$206$	−10.4782	−0.730054
$207$	− 23.2060i	− 1.61293i
$208$	− 0.412040i	− 0.0285698i
$209$	−44.6342	−3.08741
$210$	0	0
$211$	−25.4074	−1.74912	−0.874559	−	0.484920i	$-0.838849\pi$
$211$	−25.4074	−1.74912	−0.874559	+	0.484920i	$0.838849\pi$
$212$	3.52175i	0.241875i
$213$	− 3.15787i	− 0.216373i
$214$	14.8903	1.01788
$215$	0	0
$216$	−3.69905	−0.251689
$217$	− 3.06758i	− 0.208241i
$218$	− 4.60877i	− 0.312145i
$219$	−1.57893	−0.106694
$220$	0	0
$221$	−1.68427	−0.113296
$222$	0.0618262i	0.00414950i
$223$	4.00000i	0.267860i	0.990991	+	0.133930i	$0.0427597\pi$
$223$	4.00000i	0.267860i	−0.990991	+	0.133930i	$0.957240\pi$
$224$	11.3880	0.760890
$225$	0	0
$226$	8.65692	0.575850
$227$	− 4.87197i	− 0.323364i	−0.986843	−	0.161682i	$-0.948308\pi$
$227$	− 4.87197i	− 0.323364i	0.986843	−	0.161682i	$-0.0516919\pi$
$228$	− 2.67962i	− 0.177462i
$229$	−13.5458	−0.895134	−0.447567	−	0.894250i	$-0.647709\pi$
$229$	−13.5458	−0.895134	−0.447567	+	0.894250i	$0.647709\pi$
$230$	0	0
$231$	2.62957	0.173013
$232$	− 1.24488i	− 0.0817304i
$233$	− 11.3308i	− 0.742304i	−0.928572	−	0.371152i	$-0.878963\pi$
$233$	− 11.3308i	− 0.742304i	0.928572	−	0.371152i	$-0.121037\pi$
$234$	1.07085	0.0700039
$235$	0	0
$236$	9.90042	0.644463
$237$	− 0.733531i	− 0.0476479i
$238$	− 5.20602i	− 0.337456i
$239$	−9.35021	−0.604815	−0.302408	−	0.953179i	$-0.597790\pi$
$239$	−9.35021	−0.604815	−0.302408	+	0.953179i	$0.597790\pi$
$240$	0	0
$241$	2.22872	0.143564	0.0717822	−	0.997420i	$-0.477131\pi$
$241$	2.22872	0.143564	0.0717822	+	0.997420i	$0.477131\pi$
$242$	16.0071i	1.02898i
$243$	6.29303i	0.403698i
$244$	11.7231	0.750496
$245$	0	0
$246$	0.109714	0.00699509
$247$	3.77128i	0.239961i
$248$	4.10998i	0.260984i
$249$	1.09166	0.0691811
$250$	0	0
$251$	−1.68427	−0.106310	−0.0531550	−	0.998586i	$-0.516928\pi$
$251$	−1.68427	−0.106310	−0.0531550	+	0.998586i	$0.516928\pi$
$252$	− 8.12476i	− 0.511812i
$253$	− 44.6342i	− 2.80613i
$254$	−12.3639	−0.775779
$255$	0	0
$256$	−12.8090	−0.800564
$257$	− 2.70370i	− 0.168652i	−0.996438	−	0.0843260i	$-0.973126\pi$
$257$	− 2.70370i	− 0.168652i	0.996438	−	0.0843260i	$-0.0268737\pi$
$258$	0.181943i	0.0113273i
$259$	−0.660190	−0.0410222
$260$	0	0
$261$	−1.40739	−0.0871155
$262$	− 5.72313i	− 0.353576i
$263$	− 3.54583i	− 0.218645i	−0.994006	−	0.109323i	$-0.965132\pi$
$263$	− 3.54583i	− 0.218645i	0.994006	−	0.109323i	$-0.0348682\pi$
$264$	−3.52313	−0.216834
$265$	0	0
$266$	−11.6569	−0.714732
$267$	− 3.40739i	− 0.208529i
$268$	11.2060i	0.684517i
$269$	−28.1866	−1.71857	−0.859283	−	0.511500i	$-0.829090\pi$
$269$	−28.1866	−1.71857	−0.859283	+	0.511500i	$0.829090\pi$
$270$	0	0
$271$	−22.4887	−1.36609	−0.683044	−	0.730377i	$-0.739344\pi$
$271$	−22.4887	−1.36609	−0.683044	+	0.730377i	$0.739344\pi$
$272$	− 3.03421i	− 0.183976i
$273$	− 0.222181i	− 0.0134470i
$274$	−8.98522	−0.542817
$275$	0	0
$276$	2.67962	0.161294
$277$	− 5.57893i	− 0.335206i	−0.985855	−	0.167603i	$-0.946397\pi$
$277$	− 5.57893i	− 0.335206i	0.985855	−	0.167603i	$-0.0536026\pi$
$278$	− 7.25280i	− 0.434994i
$279$	4.64652	0.278180
$280$	0	0
$281$	28.6375	1.70837	0.854185	−	0.519970i	$-0.174057\pi$
$281$	28.6375	1.70837	0.854185	+	0.519970i	$0.174057\pi$
$282$	− 2.33654i	− 0.139139i
$283$	− 13.1190i	− 0.779844i	−0.920848	−	0.389922i	$-0.872502\pi$
$283$	− 13.1190i	− 0.779844i	0.920848	−	0.389922i	$-0.127498\pi$
$284$	−18.7666	−1.11359
$285$	0	0
$286$	2.05967	0.121791
$287$	1.17154i	0.0691538i
$288$	17.2495i	1.01644i
$289$	4.59725	0.270427
$290$	0	0
$291$	−2.33654	−0.136970
$292$	9.38332i	0.549117i
$293$	− 11.6361i	− 0.679789i	−0.940464	−	0.339894i	$-0.889609\pi$
$293$	− 11.6361i	− 0.679789i	0.940464	−	0.339894i	$-0.110391\pi$
$294$	−0.586849	−0.0342257
$295$	0	0
$296$	0.884529	0.0514122
$297$	8.04351i	0.466732i
$298$	5.27223i	0.305412i
$299$	−3.77128	−0.218099
$300$	0	0
$301$	−1.94282	−0.111982
$302$	9.59725i	0.552260i
$303$	− 1.59509i	− 0.0916358i
$304$	−6.79398	−0.389661
$305$	0	0
$306$	7.88564	0.450792
$307$	− 7.09166i	− 0.404742i	−0.979309	−	0.202371i	$-0.935135\pi$
$307$	− 7.09166i	− 0.404742i	0.979309	−	0.202371i	$-0.0648648\pi$
$308$	− 15.6271i	− 0.890436i
$309$	−3.29303	−0.187334
$310$	0	0
$311$	5.53216	0.313700	0.156850	−	0.987622i	$-0.449866\pi$
$311$	5.53216	0.313700	0.156850	+	0.987622i	$0.449866\pi$
$312$	0.297680i	0.0168528i
$313$	19.1488i	1.08236i	0.840908	+	0.541178i	$0.182022\pi$
$313$	19.1488i	1.08236i	−0.840908	+	0.541178i	$0.817978\pi$
$314$	−1.61668	−0.0912347
$315$	0	0
$316$	−4.35924	−0.245226
$317$	12.7004i	0.713327i	0.934233	+	0.356664i	$0.116086\pi$
$317$	12.7004i	0.713327i	−0.934233	+	0.356664i	$0.883914\pi$
$318$	− 0.450900i	− 0.0252852i
$319$	−2.70697	−0.151561
$320$	0	0
$321$	4.67962	0.261191
$322$	− 11.6569i	− 0.649615i
$323$	27.7713i	1.54524i
$324$	12.0629	0.670163
$325$	0	0
$326$	13.6271	0.754735
$327$	− 1.44841i	− 0.0800974i
$328$	− 1.56964i	− 0.0866689i
$329$	24.9500	1.37554
$330$	0	0
$331$	5.49441	0.302000	0.151000	−	0.988534i	$-0.451751\pi$
$331$	5.49441	0.302000	0.151000	+	0.988534i	$0.451751\pi$
$332$	− 6.48754i	− 0.356050i
$333$	− 1.00000i	− 0.0547997i
$334$	−3.96114	−0.216744
$335$	0	0
$336$	0.400260	0.0218360
$337$	− 22.4782i	− 1.22447i	−0.790677	−	0.612234i	$-0.790271\pi$
$337$	− 22.4782i	− 1.22447i	0.790677	−	0.612234i	$-0.209729\pi$
$338$	9.71737i	0.528555i
$339$	2.72064	0.147765
$340$	0	0
$341$	8.93706	0.483969
$342$	− 17.6569i	− 0.954777i
$343$	− 19.8662i	− 1.07267i
$344$	2.60301	0.140345
$345$	0	0
$346$	12.5379	0.674042
$347$	− 15.8227i	− 0.849407i	−0.905332	−	0.424704i	$-0.860378\pi$
$347$	− 15.8227i	− 0.849407i	0.905332	−	0.424704i	$-0.139622\pi$
$348$	− 0.162513i	− 0.00871161i
$349$	21.7713	1.16539	0.582695	−	0.812691i	$-0.301998\pi$
$349$	21.7713	1.16539	0.582695	+	0.812691i	$0.301998\pi$
$350$	0	0
$351$	0.679620	0.0362754
$352$	33.1776i	1.76837i
$353$	24.1352i	1.28459i	0.766459	+	0.642293i	$0.222017\pi$
$353$	24.1352i	1.28459i	−0.766459	+	0.642293i	$0.777983\pi$
$354$	−1.26758	−0.0673711
$355$	0	0
$356$	−20.2495	−1.07322
$357$	− 1.63611i	− 0.0865923i
$358$	6.27687i	0.331743i
$359$	−27.7713	−1.46571	−0.732856	−	0.680384i	$-0.761813\pi$
$359$	−27.7713	−1.46571	−0.732856	+	0.680384i	$0.761813\pi$
$360$	0	0
$361$	43.1833	2.27281
$362$	− 11.5412i	− 0.606591i
$363$	5.03062i	0.264039i
$364$	−1.32038	−0.0692067
$365$	0	0
$366$	−1.50095	−0.0784557
$367$	8.36389i	0.436591i	0.975883	+	0.218296i	$0.0700497\pi$
$367$	8.36389i	0.436591i	−0.975883	+	0.218296i	$0.929950\pi$
$368$	− 6.79398i	− 0.354161i
$369$	−1.77455	−0.0923794
$370$	0	0
$371$	4.81479	0.249971
$372$	0.536538i	0.0278182i
$373$	18.3067i	0.947885i	0.880556	+	0.473943i	$0.157170\pi$
$373$	18.3067i	0.947885i	−0.880556	+	0.473943i	$0.842830\pi$
$374$	15.1672	0.784275
$375$	0	0
$376$	−33.4282	−1.72393
$377$	0.228720i	0.0117797i
$378$	2.10069i	0.108048i
$379$	−1.88891	−0.0970268	−0.0485134	−	0.998823i	$-0.515448\pi$
$379$	−1.88891	−0.0970268	−0.0485134	+	0.998823i	$0.515448\pi$
$380$	0	0
$381$	−3.88564	−0.199067
$382$	− 13.4555i	− 0.688446i
$383$	− 33.8687i	− 1.73061i	−0.501246	−	0.865305i	$-0.667125\pi$
$383$	− 33.8687i	− 1.73061i	0.501246	−	0.865305i	$-0.332875\pi$
$384$	−2.30533	−0.117643
$385$	0	0
$386$	−13.7986	−0.702332
$387$	− 2.94282i	− 0.149592i
$388$	13.8856i	0.704937i
$389$	7.13052	0.361532	0.180766	−	0.983526i	$-0.442142\pi$
$389$	7.13052	0.361532	0.180766	+	0.983526i	$0.442142\pi$
$390$	0	0
$391$	−27.7713	−1.40445
$392$	8.39588i	0.424056i
$393$	− 1.79863i	− 0.0907287i
$394$	3.59725	0.181227
$395$	0	0
$396$	23.6706	1.18949
$397$	− 1.56991i	− 0.0787914i	−0.999224	−	0.0393957i	$-0.987457\pi$
$397$	− 1.56991i	− 0.0787914i	0.999224	−	0.0393957i	$-0.0125433\pi$
$398$	− 1.95185i	− 0.0978372i
$399$	−3.66346	−0.183402
$400$	0	0
$401$	6.89931	0.344535	0.172268	−	0.985050i	$-0.444891\pi$
$401$	6.89931	0.344535	0.172268	+	0.985050i	$0.444891\pi$
$402$	− 1.43474i	− 0.0715583i
$403$	− 0.755119i	− 0.0376152i
$404$	−9.47936	−0.471616
$405$	0	0
$406$	−0.706966	−0.0350861
$407$	− 1.92339i	− 0.0953389i
$408$	2.19208i	0.108524i
$409$	24.8421	1.22836	0.614182	−	0.789164i	$-0.289486\pi$
$409$	24.8421	1.22836	0.614182	+	0.789164i	$0.289486\pi$
$410$	0	0
$411$	−2.82381	−0.139288
$412$	19.5699i	0.964140i
$413$	− 13.5354i	− 0.666035i
$414$	17.6569	0.867790
$415$	0	0
$416$	2.80327	0.137442
$417$	− 2.27936i	− 0.111621i
$418$	− 33.9611i	− 1.66109i
$419$	34.6134	1.69098	0.845488	−	0.533994i	$-0.179310\pi$
$419$	34.6134	1.69098	0.845488	+	0.533994i	$0.179310\pi$
$420$	0	0
$421$	14.4509	0.704294	0.352147	−	0.935945i	$-0.385452\pi$
$421$	14.4509	0.704294	0.352147	+	0.935945i	$0.385452\pi$
$422$	− 19.3319i	− 0.941062i
$423$	37.7921i	1.83751i
$424$	−6.45090	−0.313283
$425$	0	0
$426$	2.40275	0.116413
$427$	− 16.0273i	− 0.775618i
$428$	− 27.8101i	− 1.34425i
$429$	0.647299	0.0312519
$430$	0	0
$431$	−14.2599	−0.686877	−0.343438	−	0.939175i	$-0.611592\pi$
$431$	−14.2599	−0.686877	−0.343438	+	0.939175i	$0.611592\pi$
$432$	1.22434i	0.0589060i
$433$	22.4750i	1.08008i	0.841640	+	0.540039i	$0.181591\pi$
$433$	22.4750i	1.08008i	−0.841640	+	0.540039i	$0.818409\pi$
$434$	2.33405	0.112038
$435$	0	0
$436$	−8.60766	−0.412232
$437$	62.1833i	2.97463i
$438$	− 1.20137i	− 0.0574039i
$439$	−30.8389	−1.47186	−0.735929	−	0.677058i	$-0.763254\pi$
$439$	−30.8389	−1.47186	−0.735929	+	0.677058i	$0.763254\pi$
$440$	0	0
$441$	9.49192	0.451996
$442$	− 1.28152i	− 0.0609556i
$443$	27.9338i	1.32717i	0.748099	+	0.663587i	$0.230967\pi$
$443$	27.9338i	1.32717i	−0.748099	+	0.663587i	$0.769033\pi$
$444$	0.115471	0.00548001
$445$	0	0
$446$	−3.04351	−0.144114
$447$	1.65692i	0.0783696i
$448$	5.31711i	0.251210i
$449$	−8.20137	−0.387047	−0.193523	−	0.981096i	$-0.561992\pi$
$449$	−8.20137	−0.387047	−0.193523	+	0.981096i	$0.561992\pi$
$450$	0	0
$451$	−3.41315	−0.160719
$452$	− 16.1683i	− 0.760491i
$453$	3.01616i	0.141712i
$454$	3.70697	0.173977
$455$	0	0
$456$	4.90834	0.229854
$457$	10.1248i	0.473616i	0.971556	+	0.236808i	$0.0761013\pi$
$457$	10.1248i	0.473616i	−0.971556	+	0.236808i	$0.923899\pi$
$458$	− 10.3067i	− 0.481601i
$459$	5.00465	0.233597
$460$	0	0
$461$	−16.3606	−0.761990	−0.380995	−	0.924577i	$-0.624418\pi$
$461$	−16.3606	−0.761990	−0.380995	+	0.924577i	$0.624418\pi$
$462$	2.00078i	0.0930848i
$463$	0.738177i	0.0343060i	0.999853	+	0.0171530i	$0.00546024\pi$
$463$	0.738177i	0.0343060i	−0.999853	+	0.0171530i	$0.994540\pi$
$464$	−0.412040	−0.0191285
$465$	0	0
$466$	8.62133	0.399375
$467$	11.5426i	0.534126i	0.963679	+	0.267063i	$0.0860532\pi$
$467$	11.5426i	0.534126i	−0.963679	+	0.267063i	$0.913947\pi$
$468$	− 2.00000i	− 0.0924500i
$469$	15.3204	0.707429
$470$	0	0
$471$	−0.508080	−0.0234111
$472$	18.1349i	0.834726i
$473$	− 5.66019i	− 0.260256i
$474$	0.558126	0.0256356
$475$	0	0
$476$	−9.72313	−0.445659
$477$	7.29303i	0.333925i
$478$	− 7.11436i	− 0.325403i
$479$	−11.3776	−0.519854	−0.259927	−	0.965628i	$-0.583698\pi$
$479$	−11.3776	−0.519854	−0.259927	+	0.965628i	$0.583698\pi$
$480$	0	0
$481$	−0.162513	−0.00740996
$482$	1.69578i	0.0772407i
$483$	− 3.66346i	− 0.166693i
$484$	29.8960	1.35891
$485$	0	0
$486$	−4.78822	−0.217198
$487$	25.0917i	1.13701i	0.822679	+	0.568506i	$0.192478\pi$
$487$	25.0917i	1.13701i	−0.822679	+	0.568506i	$0.807522\pi$
$488$	21.4736i	0.972064i
$489$	4.28263	0.193667
$490$	0	0
$491$	31.3685	1.41564	0.707821	−	0.706392i	$-0.249678\pi$
$491$	31.3685	1.41564	0.707821	+	0.706392i	$0.249678\pi$
$492$	− 0.204909i	− 0.00923801i
$493$	1.68427i	0.0758555i
$494$	−2.86948	−0.129104
$495$	0	0
$496$	1.36035	0.0610816
$497$	25.6569i	1.15087i
$498$	0.830619i	0.0372209i
$499$	−18.4991	−0.828131	−0.414066	−	0.910247i	$-0.635892\pi$
$499$	−18.4991	−0.828131	−0.414066	+	0.910247i	$0.635892\pi$
$500$	0	0
$501$	−1.24488	−0.0556172
$502$	− 1.28152i	− 0.0571970i
$503$	− 36.8539i	− 1.64323i	−0.570039	−	0.821617i	$-0.693072\pi$
$503$	− 36.8539i	− 1.64323i	0.570039	−	0.821617i	$-0.306928\pi$
$504$	14.8824	0.662913
$505$	0	0
$506$	33.9611	1.50976
$507$	3.05391i	0.135629i
$508$	23.0917i	1.02453i
$509$	4.98633	0.221015	0.110508	−	0.993875i	$-0.464752\pi$
$509$	4.98633	0.221015	0.110508	+	0.993875i	$0.464752\pi$
$510$	0	0
$511$	12.8285	0.567498
$512$	9.53543i	0.421410i
$513$	− 11.2060i	− 0.494758i
$514$	2.05718	0.0907383
$515$	0	0
$516$	0.339810	0.0149593
$517$	72.6889i	3.19685i
$518$	− 0.502323i	− 0.0220708i
$519$	3.94033	0.172961
$520$	0	0
$521$	6.11436	0.267875	0.133937	−	0.990990i	$-0.457238\pi$
$521$	6.11436	0.267875	0.133937	+	0.990990i	$0.457238\pi$
$522$	− 1.07085i	− 0.0468700i
$523$	16.9532i	0.741313i	0.928770	+	0.370656i	$0.120867\pi$
$523$	16.9532i	0.741313i	−0.928770	+	0.370656i	$0.879133\pi$
$524$	−10.6889	−0.466947
$525$	0	0
$526$	2.69794	0.117636
$527$	− 5.56061i	− 0.242224i
$528$	1.16611i	0.0507485i
$529$	−39.1833	−1.70362
$530$	0	0
$531$	20.5023	0.889725
$532$	21.7713i	0.943905i
$533$	0.288387i	0.0124914i
$534$	2.59261	0.112193
$535$	0	0
$536$	−20.5264	−0.886605
$537$	1.97265i	0.0851263i
$538$	− 21.4465i	− 0.924625i
$539$	18.2567	0.786370
$540$	0	0
$541$	−29.0884	−1.25061	−0.625304	−	0.780381i	$-0.715025\pi$
$541$	−29.0884	−1.25061	−0.625304	+	0.780381i	$0.715025\pi$
$542$	− 17.1111i	− 0.734984i
$543$	− 3.62709i	− 0.155653i
$544$	20.6430	0.885061
$545$	0	0
$546$	0.169052	0.00723476
$547$	− 20.0000i	− 0.855138i	−0.903983	−	0.427569i	$-0.859370\pi$
$547$	− 20.0000i	− 0.855138i	0.903983	−	0.427569i	$-0.140630\pi$
$548$	16.7814i	0.716867i
$549$	24.2769	1.03611
$550$	0	0
$551$	3.77128	0.160662
$552$	4.90834i	0.208913i
$553$	5.95976i	0.253435i
$554$	4.24488	0.180348
$555$	0	0
$556$	−13.5458	−0.574471
$557$	− 36.0963i	− 1.52945i	−0.644357	−	0.764725i	$-0.722875\pi$
$557$	− 36.0963i	− 1.52945i	0.644357	−	0.764725i	$-0.277125\pi$
$558$	3.53543i	0.149667i
$559$	−0.478247	−0.0202277
$560$	0	0
$561$	4.76663	0.201247
$562$	21.7896i	0.919139i
$563$	− 7.43474i	− 0.313337i	−0.987651	−	0.156668i	$-0.949925\pi$
$563$	− 7.43474i	− 0.313337i	0.987651	−	0.156668i	$-0.0500754\pi$
$564$	−4.36389	−0.183753
$565$	0	0
$566$	9.98195	0.419573
$567$	− 16.4919i	− 0.692596i
$568$	− 34.3754i	− 1.44236i
$569$	5.16035	0.216333	0.108167	−	0.994133i	$-0.465502\pi$
$569$	5.16035	0.216333	0.108167	+	0.994133i	$0.465502\pi$
$570$	0	0
$571$	17.5426	0.734133	0.367067	−	0.930195i	$-0.380362\pi$
$571$	17.5426	0.734133	0.367067	+	0.930195i	$0.380362\pi$
$572$	− 3.84678i	− 0.160842i
$573$	− 4.22872i	− 0.176657i
$574$	−0.891397	−0.0372062
$575$	0	0
$576$	−8.05391	−0.335580
$577$	− 19.9130i	− 0.828988i	−0.910052	−	0.414494i	$-0.863959\pi$
$577$	− 19.9130i	− 0.828988i	0.910052	−	0.414494i	$-0.136041\pi$
$578$	3.49794i	0.145495i
$579$	−4.33654	−0.180220
$580$	0	0
$581$	−8.86948	−0.367968
$582$	− 1.77782i	− 0.0736930i
$583$	14.0273i	0.580953i
$584$	−17.1877	−0.711232
$585$	0	0
$586$	8.85365	0.365741
$587$	6.89931i	0.284765i	0.989812	+	0.142383i	$0.0454763\pi$
$587$	6.89931i	0.284765i	−0.989812	+	0.142383i	$0.954524\pi$
$588$	1.09604i	0.0452000i
$589$	−12.4509	−0.513030
$590$	0	0
$591$	1.13052	0.0465034
$592$	− 0.292768i	− 0.0120327i
$593$	1.74720i	0.0717491i	0.999356	+	0.0358745i	$0.0114217\pi$
$593$	1.74720i	0.0717491i	−0.999356	+	0.0358745i	$0.988578\pi$
$594$	−6.12012	−0.251111
$595$	0	0
$596$	9.84678	0.403340
$597$	− 0.613413i	− 0.0251053i
$598$	− 2.86948i	− 0.117342i
$599$	−23.2085	−0.948274	−0.474137	−	0.880451i	$-0.657240\pi$
$599$	−23.2085	−0.948274	−0.474137	+	0.880451i	$0.657240\pi$
$600$	0	0
$601$	−8.41204	−0.343134	−0.171567	−	0.985172i	$-0.554883\pi$
$601$	−8.41204	−0.343134	−0.171567	+	0.985172i	$0.554883\pi$
$602$	− 1.47825i	− 0.0602488i
$603$	23.2060i	0.945022i
$604$	17.9245	0.729338
$605$	0	0
$606$	1.21367	0.0493020
$607$	− 24.9877i	− 1.01422i	−0.861881	−	0.507110i	$-0.830714\pi$
$607$	− 24.9877i	− 1.01422i	0.861881	−	0.507110i	$-0.169286\pi$
$608$	− 46.2222i	− 1.87456i
$609$	−0.222181	−0.00900322
$610$	0	0
$611$	6.14171	0.248467
$612$	− 14.7278i	− 0.595335i
$613$	16.0755i	0.649283i	0.945837	+	0.324642i	$0.105244\pi$
$613$	16.0755i	0.649283i	−0.945837	+	0.324642i	$0.894756\pi$
$614$	5.39588	0.217760
$615$	0	0
$616$	28.6246	1.15332
$617$	15.2722i	0.614837i	0.951574	+	0.307418i	$0.0994651\pi$
$617$	15.2722i	0.614837i	−0.951574	+	0.307418i	$0.900535\pi$
$618$	− 2.50559i	− 0.100790i
$619$	−12.8512	−0.516532	−0.258266	−	0.966074i	$-0.583151\pi$
$619$	−12.8512	−0.516532	−0.258266	+	0.966074i	$0.583151\pi$
$620$	0	0
$621$	11.2060	0.449682
$622$	4.20929i	0.168777i
$623$	27.6843i	1.10915i
$624$	0.0985283	0.00394429
$625$	0	0
$626$	−14.5699	−0.582331
$627$	− 10.6731i	− 0.426242i
$628$	3.01943i	0.120488i
$629$	−1.19673	−0.0477166
$630$	0	0
$631$	−0.842133	−0.0335248	−0.0167624	−	0.999860i	$-0.505336\pi$
$631$	−0.842133	−0.0335248	−0.0167624	+	0.999860i	$0.505336\pi$
$632$	− 7.98495i	− 0.317624i
$633$	− 6.07550i	− 0.241479i
$634$	−9.66346	−0.383785
$635$	0	0
$636$	−0.842133	−0.0333928
$637$	− 1.54256i	− 0.0611185i
$638$	− 2.05967i	− 0.0815430i
$639$	−38.8629	−1.53739
$640$	0	0
$641$	18.1833	0.718198	0.359099	−	0.933299i	$-0.383084\pi$
$641$	18.1833	0.718198	0.359099	+	0.933299i	$0.383084\pi$
$642$	3.56061i	0.140526i
$643$	− 49.3893i	− 1.94773i	−0.227137	−	0.973863i	$-0.572937\pi$
$643$	− 49.3893i	− 1.94773i	0.227137	−	0.973863i	$-0.427063\pi$
$644$	−21.7713	−0.857909
$645$	0	0
$646$	−21.1305	−0.831369
$647$	7.07334i	0.278082i	0.990287	+	0.139041i	$0.0444019\pi$
$647$	7.07334i	0.278082i	−0.990287	+	0.139041i	$0.955598\pi$
$648$	22.0960i	0.868014i
$649$	39.4340	1.54792
$650$	0	0
$651$	0.733531	0.0287493
$652$	− 25.4509i	− 0.996734i
$653$	6.01832i	0.235515i	0.993042	+	0.117758i	$0.0375706\pi$
$653$	6.01832i	0.235515i	−0.993042	+	0.117758i	$0.962429\pi$
$654$	1.10206	0.0430941
$655$	0	0
$656$	−0.519531	−0.0202843
$657$	19.4315i	0.758094i
$658$	18.9838i	0.740067i
$659$	−0.401636	−0.0156455	−0.00782276	−	0.999969i	$-0.502490\pi$
$659$	−0.401636	−0.0156455	−0.00782276	+	0.999969i	$0.502490\pi$
$660$	0	0
$661$	36.8662	1.43393	0.716965	−	0.697109i	$-0.245531\pi$
$661$	36.8662	1.43393	0.716965	+	0.697109i	$0.245531\pi$
$662$	4.18057i	0.162482i
$663$	− 0.402747i	− 0.0156414i
$664$	11.8834	0.461166
$665$	0	0
$666$	0.760877	0.0294834
$667$	3.77128i	0.146025i
$668$	7.39810i	0.286241i
$669$	−0.956493	−0.0369802
$670$	0	0
$671$	46.6939	1.80260
$672$	2.72313i	0.105047i
$673$	48.3333i	1.86311i	0.363600	+	0.931555i	$0.381548\pi$
$673$	48.3333i	1.86311i	−0.363600	+	0.931555i	$0.618452\pi$
$674$	17.1032	0.658790
$675$	0	0
$676$	18.1488	0.698032
$677$	31.4796i	1.20986i	0.796279	+	0.604930i	$0.206799\pi$
$677$	31.4796i	1.20986i	−0.796279	+	0.604930i	$0.793201\pi$
$678$	2.07007i	0.0795006i
$679$	18.9838	0.728533
$680$	0	0
$681$	1.16500	0.0446429
$682$	6.80000i	0.260386i
$683$	− 7.88564i	− 0.301736i	−0.988554	−	0.150868i	$-0.951793\pi$
$683$	− 7.88564i	− 0.301736i	0.988554	−	0.150868i	$-0.0482068\pi$
$684$	−32.9773	−1.26092
$685$	0	0
$686$	15.1157	0.577122
$687$	− 3.23912i	− 0.123580i
$688$	− 0.861564i	− 0.0328468i
$689$	1.18521	0.0451530
$690$	0	0
$691$	−30.7759	−1.17077	−0.585386	−	0.810755i	$-0.699057\pi$
$691$	−30.7759	−1.17077	−0.585386	+	0.810755i	$0.699057\pi$
$692$	− 23.4167i	− 0.890169i
$693$	− 32.3614i	− 1.22931i
$694$	12.0391	0.456999
$695$	0	0
$696$	0.297680	0.0112835
$697$	2.12365i	0.0804391i
$698$	16.5653i	0.627004i
$699$	2.70945	0.102481
$700$	0	0
$701$	−14.7310	−0.556384	−0.278192	−	0.960526i	$-0.589735\pi$
$701$	−14.7310	−0.556384	−0.278192	+	0.960526i	$0.589735\pi$
$702$	0.517107i	0.0195170i
$703$	2.67962i	0.101064i
$704$	−15.4908	−0.583832
$705$	0	0
$706$	−18.3639	−0.691134
$707$	12.9598i	0.487402i
$708$	2.36742i	0.0889732i
$709$	24.6238	0.924767	0.462383	−	0.886680i	$-0.346994\pi$
$709$	24.6238	0.924767	0.462383	+	0.886680i	$0.346994\pi$
$710$	0	0
$711$	−9.02735	−0.338552
$712$	− 37.0917i	− 1.39007i
$713$	− 12.4509i	− 0.466290i
$714$	1.24488	0.0465885
$715$	0	0
$716$	11.7231	0.438114
$717$	− 2.23585i	− 0.0834995i
$718$	− 21.1305i	− 0.788584i
$719$	41.4315	1.54513	0.772567	−	0.634934i	$-0.218973\pi$
$719$	41.4315	1.54513	0.772567	+	0.634934i	$0.218973\pi$
$720$	0	0
$721$	26.7551	0.996413
$722$	32.8572i	1.22282i
$723$	0.532939i	0.0198202i
$724$	−21.5551	−0.801090
$725$	0	0
$726$	−3.82768	−0.142058
$727$	− 41.4588i	− 1.53762i	−0.639476	−	0.768811i	$-0.720849\pi$
$727$	− 41.4588i	− 1.53762i	0.639476	−	0.768811i	$-0.279151\pi$
$728$	− 2.41858i	− 0.0896385i
$729$	23.9611	0.887450
$730$	0	0
$731$	−3.52175	−0.130257
$732$	2.80327i	0.103612i
$733$	24.4120i	0.901679i	0.892605	+	0.450840i	$0.148875\pi$
$733$	24.4120i	0.901679i	−0.892605	+	0.450840i	$0.851125\pi$
$734$	−6.36389	−0.234895
$735$	0	0
$736$	46.2222	1.70377
$737$	44.6342i	1.64412i
$738$	− 1.35021i	− 0.0497021i
$739$	0.613413	0.0225648	0.0112824	−	0.999936i	$-0.496409\pi$
$739$	0.613413	0.0225648	0.0112824	+	0.999936i	$0.496409\pi$
$740$	0	0
$741$	−0.901801	−0.0331285
$742$	3.66346i	0.134490i
$743$	− 38.1182i	− 1.39842i	−0.714915	−	0.699211i	$-0.753535\pi$
$743$	− 38.1182i	− 1.39842i	0.714915	−	0.699211i	$-0.246465\pi$
$744$	−0.982792	−0.0360309
$745$	0	0
$746$	−13.9291	−0.509982
$747$	− 13.4347i	− 0.491551i
$748$	− 28.3272i	− 1.03575i
$749$	−38.0208	−1.38925
$750$	0	0
$751$	−8.11436	−0.296097	−0.148049	−	0.988980i	$-0.547299\pi$
$751$	−8.11436	−0.296097	−0.148049	+	0.988980i	$0.547299\pi$
$752$	11.0643i	0.403474i
$753$	− 0.402747i	− 0.0146769i
$754$	−0.174028	−0.00633771
$755$	0	0
$756$	3.92339	0.142692
$757$	− 32.6044i	− 1.18503i	−0.805561	−	0.592513i	$-0.798136\pi$
$757$	− 32.6044i	− 1.18503i	0.805561	−	0.592513i	$-0.201864\pi$
$758$	− 1.43723i	− 0.0522024i
$759$	10.6731	0.387408
$760$	0	0
$761$	36.0000	1.30500	0.652499	−	0.757789i	$-0.273720\pi$
$761$	36.0000	1.30500	0.652499	+	0.757789i	$0.273720\pi$
$762$	− 2.95649i	− 0.107102i
$763$	11.7680i	0.426031i
$764$	−25.1305	−0.909190
$765$	0	0
$766$	25.7699	0.931104
$767$	− 3.33189i	− 0.120308i
$768$	− 3.06294i	− 0.110524i
$769$	−13.7141	−0.494543	−0.247272	−	0.968946i	$-0.579534\pi$
$769$	−13.7141	−0.494543	−0.247272	+	0.968946i	$0.579534\pi$
$770$	0	0
$771$	0.646517	0.0232837
$772$	25.7713i	0.927529i
$773$	− 32.2794i	− 1.16101i	−0.814257	−	0.580504i	$-0.802856\pi$
$773$	− 32.2794i	− 1.16101i	0.814257	−	0.580504i	$-0.197144\pi$
$774$	2.23912	0.0804836
$775$	0	0
$776$	−25.4347	−0.913054
$777$	− 0.157867i	− 0.00566344i
$778$	5.42545i	0.194512i
$779$	4.75512	0.170370
$780$	0	0
$781$	−74.7486	−2.67471
$782$	− 21.1305i	− 0.755626i
$783$	− 0.679620i	− 0.0242876i
$784$	2.77893	0.0992475
$785$	0	0
$786$	1.36853	0.0488140
$787$	43.6777i	1.55694i	0.627680	+	0.778471i	$0.284005\pi$
$787$	43.6777i	1.55694i	−0.627680	+	0.778471i	$0.715995\pi$
$788$	− 6.71848i	− 0.239336i
$789$	0.847890	0.0301857
$790$	0	0
$791$	−22.1046	−0.785947
$792$	43.3581i	1.54066i
$793$	− 3.94531i	− 0.140102i
$794$	1.19451	0.0423914
$795$	0	0
$796$	−3.64541	−0.129208
$797$	4.18057i	0.148083i	0.997255	+	0.0740416i	$0.0235898\pi$
$797$	4.18057i	0.148083i	−0.997255	+	0.0740416i	$0.976410\pi$
$798$	− 2.78744i	− 0.0986743i
$799$	45.2268	1.60001
$800$	0	0
$801$	−41.9338	−1.48166
$802$	5.24953i	0.185367i
$803$	37.3743i	1.31891i
$804$	−2.67962	−0.0945029
$805$	0	0
$806$	0.574553	0.0202378
$807$	− 6.74007i	− 0.237262i
$808$	− 17.3636i	− 0.610850i
$809$	21.0058	0.738523	0.369262	−	0.929326i	$-0.379611\pi$
$809$	21.0058	0.738523	0.369262	+	0.929326i	$0.379611\pi$
$810$	0	0
$811$	−3.69578	−0.129776	−0.0648882	−	0.997893i	$-0.520669\pi$
$811$	−3.69578	−0.129776	−0.0648882	+	0.997893i	$0.520669\pi$
$812$	1.32038i	0.0463362i
$813$	− 5.37756i	− 0.188599i
$814$	1.46346	0.0512943
$815$	0	0
$816$	0.725551	0.0253994
$817$	7.88564i	0.275884i
$818$	18.9018i	0.660886i
$819$	−2.73431	−0.0955446
$820$	0	0
$821$	31.6673	1.10520	0.552599	−	0.833448i	$-0.313636\pi$
$821$	31.6673	1.10520	0.552599	+	0.833448i	$0.313636\pi$
$822$	− 2.14857i	− 0.0749401i
$823$	40.8903i	1.42535i	0.701497	+	0.712673i	$0.252516\pi$
$823$	40.8903i	1.42535i	−0.701497	+	0.712673i	$0.747484\pi$
$824$	−35.8468	−1.24878
$825$	0	0
$826$	10.2988	0.358341
$827$	4.07550i	0.141719i	0.997486	+	0.0708595i	$0.0225742\pi$
$827$	4.07550i	0.141719i	−0.997486	+	0.0708595i	$0.977426\pi$
$828$	− 32.9773i	− 1.14604i
$829$	23.4074	0.812972	0.406486	−	0.913657i	$-0.366754\pi$
$829$	23.4074	0.812972	0.406486	+	0.913657i	$0.366754\pi$
$830$	0	0
$831$	1.33405	0.0462778
$832$	1.30887i	0.0453767i
$833$	− 11.3592i	− 0.393574i
$834$	1.73431	0.0600543
$835$	0	0
$836$	−63.4282	−2.19371
$837$	2.24377i	0.0775560i
$838$	26.3365i	0.909781i
$839$	6.45090	0.222710	0.111355	−	0.993781i	$-0.464481\pi$
$839$	6.45090	0.222710	0.111355	+	0.993781i	$0.464481\pi$
$840$	0	0
$841$	−28.7713	−0.992113
$842$	10.9954i	0.378925i
$843$	6.84789i	0.235854i
$844$	−36.1056	−1.24281
$845$	0	0
$846$	−28.7551	−0.988621
$847$	− 40.8726i	− 1.40440i
$848$	2.13517i	0.0733219i
$849$	3.13706	0.107664
$850$	0	0
$851$	−2.67962	−0.0918562
$852$	− 4.48754i	− 0.153741i
$853$	− 9.82597i	− 0.336435i	−0.985750	−	0.168217i	$-0.946199\pi$
$853$	− 9.82597i	− 0.336435i	0.985750	−	0.168217i	$-0.0538011\pi$
$854$	12.1948	0.417299
$855$	0	0
$856$	50.9407	1.74112
$857$	− 42.0093i	− 1.43501i	−0.696553	−	0.717505i	$-0.745284\pi$
$857$	− 42.0093i	− 1.43501i	0.696553	−	0.717505i	$-0.254716\pi$
$858$	0.492514i	0.0168142i
$859$	23.0643	0.786944	0.393472	−	0.919337i	$-0.371274\pi$
$859$	23.0643	0.786944	0.393472	+	0.919337i	$0.371274\pi$
$860$	0	0
$861$	−0.280142	−0.00954723
$862$	− 10.8500i	− 0.369554i
$863$	42.3822i	1.44271i	0.692567	+	0.721354i	$0.256480\pi$
$863$	42.3822i	1.44271i	−0.692567	+	0.721354i	$0.743520\pi$
$864$	−8.32967	−0.283381
$865$	0	0
$866$	−17.1007	−0.581105
$867$	1.09931i	0.0373345i
$868$	− 4.35924i	− 0.147962i
$869$	−17.3631	−0.589003
$870$	0	0
$871$	3.77128	0.127785
$872$	− 15.7669i	− 0.533935i
$873$	28.7551i	0.973213i
$874$	−47.3138	−1.60041
$875$	0	0
$876$	−2.24377	−0.0758099
$877$	− 16.0755i	− 0.542831i	−0.962462	−	0.271416i	$-0.912508\pi$
$877$	− 16.0755i	− 0.542831i	0.962462	−	0.271416i	$-0.0874918\pi$
$878$	− 23.4646i	− 0.791891i
$879$	2.78247	0.0938502
$880$	0	0
$881$	2.23121	0.0751713	0.0375856	−	0.999293i	$-0.488033\pi$
$881$	2.23121	0.0751713	0.0375856	+	0.999293i	$0.488033\pi$
$882$	7.22218i	0.243183i
$883$	52.2977i	1.75996i	0.475013	+	0.879979i	$0.342443\pi$
$883$	52.2977i	1.75996i	−0.475013	+	0.879979i	$0.657557\pi$
$884$	−2.39345	−0.0805006
$885$	0	0
$886$	−21.2542	−0.714048
$887$	− 32.4991i	− 1.09121i	−0.838042	−	0.545606i	$-0.816300\pi$
$887$	− 32.4991i	− 1.09121i	0.838042	−	0.545606i	$-0.183700\pi$
$888$	0.211511i	0.00709786i
$889$	31.5699	1.05882
$890$	0	0
$891$	48.0474	1.60965
$892$	5.68427i	0.190323i
$893$	− 101.268i	− 3.38882i
$894$	−1.26071	−0.0421645
$895$	0	0
$896$	18.7303	0.625734
$897$	− 0.901801i	− 0.0301103i
$898$	− 6.24023i	− 0.208239i
$899$	−0.755119	−0.0251846
$900$	0	0
$901$	8.72777	0.290764
$902$	− 2.59699i	− 0.0864702i
$903$	− 0.464574i	− 0.0154600i
$904$	29.6159	0.985010
$905$	0	0
$906$	−2.29493	−0.0762438
$907$	− 7.69578i	− 0.255534i	−0.991804	−	0.127767i	$-0.959219\pi$
$907$	− 7.69578i	− 0.255534i	0.991804	−	0.127767i	$-0.0407810\pi$
$908$	− 6.92339i	− 0.229761i
$909$	−19.6304	−0.651098
$910$	0	0
$911$	−31.3412	−1.03838	−0.519190	−	0.854659i	$-0.673766\pi$
$911$	−31.3412	−1.03838	−0.519190	+	0.854659i	$0.673766\pi$
$912$	− 1.62460i	− 0.0537958i
$913$	− 25.8402i	− 0.855187i
$914$	−7.70370	−0.254816
$915$	0	0
$916$	−19.2495	−0.636022
$917$	14.6134i	0.482577i
$918$	3.80792i	0.125680i
$919$	16.9784	0.560066	0.280033	−	0.959990i	$-0.409655\pi$
$919$	16.9784	0.560066	0.280033	+	0.959990i	$0.409655\pi$
$920$	0	0
$921$	1.69578	0.0558779
$922$	− 12.4484i	− 0.409967i
$923$	6.31573i	0.207885i
$924$	3.73680	0.122932
$925$	0	0
$926$	−0.561662	−0.0184573
$927$	40.5264i	1.33106i
$928$	− 2.80327i	− 0.0920219i
$929$	50.3847	1.65307	0.826534	−	0.562887i	$-0.190309\pi$
$929$	50.3847	1.65307	0.826534	+	0.562887i	$0.190309\pi$
$930$	0	0
$931$	−25.4347	−0.833590
$932$	− 16.1018i	− 0.527432i
$933$	1.32287i	0.0433087i
$934$	−8.78247	−0.287371
$935$	0	0
$936$	3.66346	0.119744
$937$	− 27.1488i	− 0.886914i	−0.896296	−	0.443457i	$-0.853752\pi$
$937$	− 27.1488i	− 0.886914i	0.896296	−	0.443457i	$-0.146248\pi$
$938$	11.6569i	0.380612i
$939$	−4.57893	−0.149428
$940$	0	0
$941$	−0.640760	−0.0208882	−0.0104441	−	0.999945i	$-0.503325\pi$
$941$	−0.640760	−0.0208882	−0.0104441	+	0.999945i	$0.503325\pi$
$942$	− 0.386587i	− 0.0125957i
$943$	4.75512i	0.154848i
$944$	6.00242	0.195362
$945$	0	0
$946$	4.30671	0.140023
$947$	− 30.5587i	− 0.993025i	−0.868030	−	0.496513i	$-0.834614\pi$
$947$	− 30.5587i	− 0.993025i	0.868030	−	0.496513i	$-0.165386\pi$
$948$	− 1.04240i	− 0.0338554i
$949$	3.15787	0.102509
$950$	0	0
$951$	−3.03697	−0.0984804
$952$	− 17.8101i	− 0.577230i
$953$	20.4484i	0.662389i	0.943562	+	0.331195i	$0.107452\pi$
$953$	20.4484i	0.662389i	−0.943562	+	0.331195i	$0.892548\pi$
$954$	−5.54910	−0.179659
$955$	0	0
$956$	−13.2873	−0.429741
$957$	− 0.647299i	− 0.0209242i
$958$	− 8.65692i	− 0.279692i
$959$	22.9428	0.740862
$960$	0	0
$961$	−28.5070	−0.919580
$962$	− 0.123652i	− 0.00398671i
$963$	− 57.5907i	− 1.85583i
$964$	3.16716	0.102007
$965$	0	0
$966$	2.78744	0.0896844
$967$	− 13.6181i	− 0.437927i	−0.975733	−	0.218964i	$-0.929732\pi$
$967$	− 13.6181i	− 0.437927i	0.975733	−	0.218964i	$-0.0702676\pi$
$968$	54.7615i	1.76010i
$969$	−6.64076	−0.213332
$970$	0	0
$971$	17.5789	0.564135	0.282067	−	0.959395i	$-0.408980\pi$
$971$	17.5789	0.564135	0.282067	+	0.959395i	$0.408980\pi$
$972$	8.94282i	0.286841i
$973$	18.5193i	0.593700i
$974$	−19.0917	−0.611736
$975$	0	0
$976$	7.10749	0.227505
$977$	− 32.7004i	− 1.04618i	−0.852278	−	0.523090i	$-0.824779\pi$
$977$	− 32.7004i	− 1.04618i	0.852278	−	0.523090i	$-0.175221\pi$
$978$	3.25855i	0.104197i
$979$	−80.6550	−2.57775
$980$	0	0
$981$	−17.8252	−0.569115
$982$	23.8676i	0.761645i
$983$	− 29.1819i	− 0.930759i	−0.885111	−	0.465380i	$-0.845918\pi$
$983$	− 29.1819i	− 0.930759i	0.885111	−	0.465380i	$-0.154082\pi$
$984$	0.375338	0.0119653
$985$	0	0
$986$	−1.28152	−0.0408119
$987$	5.96611i	0.189904i
$988$	5.35924i	0.170500i
$989$	−7.88564	−0.250749
$990$	0	0
$991$	−3.58796	−0.113975	−0.0569877	−	0.998375i	$-0.518150\pi$
$991$	−3.58796	−0.113975	−0.0569877	+	0.998375i	$0.518150\pi$
$992$	9.25502i	0.293847i
$993$	1.31384i	0.0416935i
$994$	−19.5218	−0.619192
$995$	0	0
$996$	1.55132	0.0491555
$997$	31.8889i	1.00993i	0.863139	+	0.504966i	$0.168495\pi$
$997$	31.8889i	1.00993i	−0.863139	+	0.504966i	$0.831505\pi$
$998$	− 14.0755i	− 0.445552i
$999$	0.482893	0.0152781

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1075.2.b.g.474.4		6
5.2	odd	4		215.2.a.b.1.2	✓	3
5.3	odd	4		1075.2.a.l.1.2		3
5.4	even	2	inner	1075.2.b.g.474.3		6
15.2	even	4		1935.2.a.r.1.2		3
15.8	even	4		9675.2.a.bp.1.2		3
20.7	even	4		3440.2.a.l.1.2		3
215.42	even	4		9245.2.a.h.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
215.2.a.b.1.2	✓	3	5.2	odd	4
1075.2.a.l.1.2		3	5.3	odd	4
1075.2.b.g.474.3		6	5.4	even	2	inner
1075.2.b.g.474.4		6	1.1	even	1	trivial
1935.2.a.r.1.2		3	15.2	even	4
3440.2.a.l.1.2		3	20.7	even	4
9245.2.a.h.1.2		3	215.42	even	4
9675.2.a.bp.1.2		3	15.8	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:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1075.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.760877i q^{2} +0.239123i q^{3} +1.42107 q^{4} -0.181943 q^{6} -1.94282i q^{7} +2.60301i q^{8} +2.94282 q^{9} +O(q^{10})\)	\(q+0.760877i q^{2} +0.239123i q^{3} +1.42107 q^{4} -0.181943 q^{6} -1.94282i q^{7} +2.60301i q^{8} +2.94282 q^{9} +5.66019 q^{11} +0.339810i q^{12} -0.478247i q^{13} +1.47825 q^{14} +0.861564 q^{16} -3.52175i q^{17} +2.23912i q^{18} -7.88564 q^{19} +0.464574 q^{21} +4.30671i q^{22} -7.88564i q^{23} -0.622440 q^{24} +0.363887 q^{26} +1.42107i q^{27} -2.76088i q^{28} -0.478247 q^{29} +1.57893 q^{31} +5.86156i q^{32} +1.35348i q^{33} +2.67962 q^{34} +4.18194 q^{36} -0.339810i q^{37} -6.00000i q^{38} +0.114360 q^{39} -0.603010 q^{41} +0.353483i q^{42} -1.00000i q^{43} +8.04351 q^{44} +6.00000 q^{46} +12.8421i q^{47} +0.206020i q^{48} +3.22545 q^{49} +0.842133 q^{51} -0.679620i q^{52} +2.47825i q^{53} -1.08126 q^{54} +5.05718 q^{56} -1.88564i q^{57} -0.363887i q^{58} +6.96690 q^{59} +8.24953 q^{61} +1.20137i q^{62} -5.71737i q^{63} -2.73680 q^{64} -1.02983 q^{66} +7.88564i q^{67} -5.00465i q^{68} +1.88564 q^{69} -13.2060 q^{71} +7.66019i q^{72} +6.60301i q^{73} +0.258554 q^{74} -11.2060 q^{76} -10.9967i q^{77} +0.0870138i q^{78} -3.06758 q^{79} +8.48865 q^{81} -0.458816i q^{82} -4.56526i q^{83} +0.660190 q^{84} +0.760877 q^{86} -0.114360i q^{87} +14.7335i q^{88} -14.2495 q^{89} -0.929147 q^{91} -11.2060i q^{92} +0.377560i q^{93} -9.77128 q^{94} -1.40164 q^{96} +9.77128i q^{97} +2.45417i q^{98} +16.6569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 8 q^{4} + 16 q^{6}+O(q^{10}) \) 6 * q - 8 * q^4 + 16 * q^6	\( 6 q - 8 q^{4} + 16 q^{6} + 18 q^{11} + 10 q^{14} + 20 q^{16} - 12 q^{19} - 16 q^{21} - 18 q^{24} - 32 q^{26} - 4 q^{29} + 26 q^{31} + 48 q^{34} + 8 q^{36} + 36 q^{39} + 30 q^{41} + 46 q^{44} + 36 q^{46} - 28 q^{51} + 26 q^{54} + 48 q^{56} - 26 q^{59} - 20 q^{61} - 66 q^{64} + 16 q^{66} - 24 q^{69} - 12 q^{71} + 50 q^{74} + 34 q^{79} - 18 q^{81} - 12 q^{84} + 4 q^{86} - 16 q^{89} + 32 q^{91} + 12 q^{94} + 56 q^{96} - 6 q^{99}+O(q^{100}) \) 6 * q - 8 * q^4 + 16 * q^6 + 18 * q^11 + 10 * q^14 + 20 * q^16 - 12 * q^19 - 16 * q^21 - 18 * q^24 - 32 * q^26 - 4 * q^29 + 26 * q^31 + 48 * q^34 + 8 * q^36 + 36 * q^39 + 30 * q^41 + 46 * q^44 + 36 * q^46 - 28 * q^51 + 26 * q^54 + 48 * q^56 - 26 * q^59 - 20 * q^61 - 66 * q^64 + 16 * q^66 - 24 * q^69 - 12 * q^71 + 50 * q^74 + 34 * q^79 - 18 * q^81 - 12 * q^84 + 4 * q^86 - 16 * q^89 + 32 * q^91 + 12 * q^94 + 56 * q^96 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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