LMFDB - Embedded newform 7381.2.a.r.1.35

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7381,2,Mod(1,7381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7381, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7381 = 11^{2} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7381.a (trivial)

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:	yes
Analytic conductor:	$58.9375817319$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.35
Character	$\chi$	$=$	7381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.43787 q^{2} +1.95628 q^{3} +0.0674703 q^{4} -2.13246 q^{5} +2.81287 q^{6} -3.51697 q^{7} -2.77873 q^{8} +0.827019 q^{9} +O(q^{10})$	\(q+1.43787 q^{2} +1.95628 q^{3} +0.0674703 q^{4} -2.13246 q^{5} +2.81287 q^{6} -3.51697 q^{7} -2.77873 q^{8} +0.827019 q^{9} -3.06620 q^{10} +0.131991 q^{12} -4.22369 q^{13} -5.05695 q^{14} -4.17169 q^{15} -4.13039 q^{16} +6.93135 q^{17} +1.18915 q^{18} -4.37933 q^{19} -0.143878 q^{20} -6.88017 q^{21} +3.18253 q^{23} -5.43596 q^{24} -0.452605 q^{25} -6.07312 q^{26} -4.25095 q^{27} -0.237291 q^{28} +8.09032 q^{29} -5.99834 q^{30} +2.88982 q^{31} -0.381509 q^{32} +9.96638 q^{34} +7.49981 q^{35} +0.0557992 q^{36} +8.14521 q^{37} -6.29690 q^{38} -8.26272 q^{39} +5.92553 q^{40} +4.27055 q^{41} -9.89280 q^{42} +7.99180 q^{43} -1.76359 q^{45} +4.57606 q^{46} -13.1513 q^{47} -8.08018 q^{48} +5.36911 q^{49} -0.650787 q^{50} +13.5596 q^{51} -0.284974 q^{52} -3.87987 q^{53} -6.11232 q^{54} +9.77271 q^{56} -8.56718 q^{57} +11.6328 q^{58} -7.58701 q^{59} -0.281465 q^{60} +1.00000 q^{61} +4.15518 q^{62} -2.90860 q^{63} +7.71222 q^{64} +9.00687 q^{65} +5.08244 q^{67} +0.467661 q^{68} +6.22590 q^{69} +10.7838 q^{70} -2.96532 q^{71} -2.29806 q^{72} -7.70755 q^{73} +11.7117 q^{74} -0.885420 q^{75} -0.295475 q^{76} -11.8807 q^{78} +4.13051 q^{79} +8.80790 q^{80} -10.7971 q^{81} +6.14050 q^{82} +9.72969 q^{83} -0.464208 q^{84} -14.7808 q^{85} +11.4912 q^{86} +15.8269 q^{87} +1.48882 q^{89} -2.53581 q^{90} +14.8546 q^{91} +0.214726 q^{92} +5.65328 q^{93} -18.9099 q^{94} +9.33875 q^{95} -0.746337 q^{96} -1.99118 q^{97} +7.72008 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 10 q^{2} - 2 q^{3} + 50 q^{4} - 2 q^{5} + 12 q^{6} + 8 q^{7} + 30 q^{8} + 48 q^{9}+O(q^{10}) $ 50 * q + 10 * q^2 - 2 * q^3 + 50 * q^4 - 2 * q^5 + 12 * q^6 + 8 * q^7 + 30 * q^8 + 48 * q^9	\( 50 q + 10 q^{2} - 2 q^{3} + 50 q^{4} - 2 q^{5} + 12 q^{6} + 8 q^{7} + 30 q^{8} + 48 q^{9} + 12 q^{10} - 14 q^{12} + 8 q^{13} - 2 q^{14} - 16 q^{15} + 42 q^{16} + 22 q^{17} + 32 q^{19} - 8 q^{20} + 24 q^{21} + 12 q^{23} - 8 q^{24} + 40 q^{25} + 10 q^{26} - 8 q^{27} + 72 q^{28} + 56 q^{29} + 24 q^{30} + 10 q^{31} + 70 q^{32} - 32 q^{34} + 70 q^{35} + 34 q^{36} - 8 q^{37} - 14 q^{38} + 96 q^{39} - 54 q^{40} + 56 q^{41} - 8 q^{42} + 44 q^{43} - 24 q^{45} - 4 q^{46} - 4 q^{47} - 28 q^{48} + 38 q^{49} + 120 q^{50} + 76 q^{51} + 24 q^{52} + 4 q^{53} + 48 q^{54} - 18 q^{56} + 8 q^{57} + 28 q^{58} + 12 q^{59} - 60 q^{60} + 50 q^{61} + 8 q^{62} + 30 q^{63} + 10 q^{64} + 64 q^{65} + 18 q^{67} - 22 q^{68} - 8 q^{69} + 34 q^{70} + 12 q^{71} + 104 q^{72} - 16 q^{73} + 84 q^{74} - 26 q^{75} + 64 q^{76} + 40 q^{78} + 78 q^{79} - 36 q^{80} + 34 q^{81} + 54 q^{82} + 68 q^{83} - 78 q^{84} - 4 q^{85} + 36 q^{86} + 48 q^{87} + 26 q^{89} - 20 q^{90} + 32 q^{92} + 22 q^{93} + 156 q^{94} + 100 q^{95} - 4 q^{96} - 14 q^{97} + 70 q^{98}+O(q^{100}) \) 50 * q + 10 * q^2 - 2 * q^3 + 50 * q^4 - 2 * q^5 + 12 * q^6 + 8 * q^7 + 30 * q^8 + 48 * q^9 + 12 * q^10 - 14 * q^12 + 8 * q^13 - 2 * q^14 - 16 * q^15 + 42 * q^16 + 22 * q^17 + 32 * q^19 - 8 * q^20 + 24 * q^21 + 12 * q^23 - 8 * q^24 + 40 * q^25 + 10 * q^26 - 8 * q^27 + 72 * q^28 + 56 * q^29 + 24 * q^30 + 10 * q^31 + 70 * q^32 - 32 * q^34 + 70 * q^35 + 34 * q^36 - 8 * q^37 - 14 * q^38 + 96 * q^39 - 54 * q^40 + 56 * q^41 - 8 * q^42 + 44 * q^43 - 24 * q^45 - 4 * q^46 - 4 * q^47 - 28 * q^48 + 38 * q^49 + 120 * q^50 + 76 * q^51 + 24 * q^52 + 4 * q^53 + 48 * q^54 - 18 * q^56 + 8 * q^57 + 28 * q^58 + 12 * q^59 - 60 * q^60 + 50 * q^61 + 8 * q^62 + 30 * q^63 + 10 * q^64 + 64 * q^65 + 18 * q^67 - 22 * q^68 - 8 * q^69 + 34 * q^70 + 12 * q^71 + 104 * q^72 - 16 * q^73 + 84 * q^74 - 26 * q^75 + 64 * q^76 + 40 * q^78 + 78 * q^79 - 36 * q^80 + 34 * q^81 + 54 * q^82 + 68 * q^83 - 78 * q^84 - 4 * q^85 + 36 * q^86 + 48 * q^87 + 26 * q^89 - 20 * q^90 + 32 * q^92 + 22 * q^93 + 156 * q^94 + 100 * q^95 - 4 * q^96 - 14 * q^97 + 70 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.43787	1.01673	0.508364	−	0.861142i	$-0.330251\pi$
$2$	1.43787	1.01673	0.508364	+	0.861142i	$0.330251\pi$
$3$	1.95628	1.12946	0.564728	−	0.825277i	$-0.308981\pi$
$3$	1.95628	1.12946	0.564728	+	0.825277i	$0.308981\pi$
$4$	0.0674703	0.0337352
$5$	−2.13246	−0.953666	−0.476833	−	0.878994i	$-0.658215\pi$
$5$	−2.13246	−0.953666	−0.476833	+	0.878994i	$0.658215\pi$
$6$	2.81287	1.14835
$7$	−3.51697	−1.32929	−0.664646	−	0.747159i	$-0.731417\pi$
$7$	−3.51697	−1.32929	−0.664646	+	0.747159i	$0.731417\pi$
$8$	−2.77873	−0.982428
$9$	0.827019	0.275673
$10$	−3.06620	−0.969619
$11$	0	0
$11$	0	0
$12$	0.131991	0.0381024
$13$	−4.22369	−1.17144	−0.585721	−	0.810513i	$-0.699189\pi$
$13$	−4.22369	−1.17144	−0.585721	+	0.810513i	$0.699189\pi$
$14$	−5.05695	−1.35153
$15$	−4.17169	−1.07712
$16$	−4.13039	−1.03260
$17$	6.93135	1.68110	0.840550	−	0.541734i	$-0.182232\pi$
$17$	6.93135	1.68110	0.840550	+	0.541734i	$0.182232\pi$
$18$	1.18915	0.280284
$19$	−4.37933	−1.00469	−0.502343	−	0.864668i	$-0.667529\pi$
$19$	−4.37933	−1.00469	−0.502343	+	0.864668i	$0.667529\pi$
$20$	−0.143878	−0.0321721
$21$	−6.88017	−1.50138
$22$	0	0
$23$	3.18253	0.663603	0.331801	−	0.943349i	$-0.392344\pi$
$23$	3.18253	0.663603	0.331801	+	0.943349i	$0.392344\pi$
$24$	−5.43596	−1.10961
$25$	−0.452605	−0.0905210
$26$	−6.07312	−1.19104
$27$	−4.25095	−0.818096
$28$	−0.237291	−0.0448439
$29$	8.09032	1.50234	0.751168	−	0.660112i	$-0.229491\pi$
$29$	8.09032	1.50234	0.751168	+	0.660112i	$0.229491\pi$
$30$	−5.99834	−1.09514
$31$	2.88982	0.519026	0.259513	−	0.965740i	$-0.416438\pi$
$31$	2.88982	0.519026	0.259513	+	0.965740i	$0.416438\pi$
$32$	−0.381509	−0.0674419
$33$	0	0
$34$	9.96638	1.70922
$35$	7.49981	1.26770
$36$	0.0557992	0.00929987
$37$	8.14521	1.33906	0.669532	−	0.742783i	$-0.266495\pi$
$37$	8.14521	1.33906	0.669532	+	0.742783i	$0.266495\pi$
$38$	−6.29690	−1.02149
$39$	−8.26272	−1.32309
$40$	5.92553	0.936908
$41$	4.27055	0.666948	0.333474	−	0.942759i	$-0.391779\pi$
$41$	4.27055	0.666948	0.333474	+	0.942759i	$0.391779\pi$
$42$	−9.89280	−1.52649
$43$	7.99180	1.21874	0.609369	−	0.792886i	$-0.291423\pi$
$43$	7.99180	1.21874	0.609369	+	0.792886i	$0.291423\pi$
$44$	0	0
$45$	−1.76359	−0.262900
$46$	4.57606	0.674703
$47$	−13.1513	−1.91832	−0.959160	−	0.282864i	$-0.908715\pi$
$47$	−13.1513	−1.91832	−0.959160	+	0.282864i	$0.908715\pi$
$48$	−8.08018	−1.16627
$49$	5.36911	0.767015
$50$	−0.650787	−0.0920352
$51$	13.5596	1.89873
$52$	−0.284974	−0.0395188
$53$	−3.87987	−0.532941	−0.266470	−	0.963843i	$-0.585857\pi$
$53$	−3.87987	−0.532941	−0.266470	+	0.963843i	$0.585857\pi$
$54$	−6.11232	−0.831781
$55$	0	0
$56$	9.77271	1.30593
$57$	−8.56718	−1.13475
$58$	11.6328	1.52747
$59$	−7.58701	−0.987745	−0.493873	−	0.869534i	$-0.664419\pi$
$59$	−7.58701	−0.987745	−0.493873	+	0.869534i	$0.664419\pi$
$60$	−0.281465	−0.0363370
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	4.15518	0.527708
$63$	−2.90860	−0.366450
$64$	7.71222	0.964027
$65$	9.00687	1.11716
$66$	0	0
$67$	5.08244	0.620919	0.310459	−	0.950587i	$-0.399517\pi$
$67$	5.08244	0.620919	0.310459	+	0.950587i	$0.399517\pi$
$68$	0.467661	0.0567122
$69$	6.22590	0.749511
$70$	10.7838	1.28891
$71$	−2.96532	−0.351919	−0.175959	−	0.984397i	$-0.556303\pi$
$71$	−2.96532	−0.351919	−0.175959	+	0.984397i	$0.556303\pi$
$72$	−2.29806	−0.270829
$73$	−7.70755	−0.902101	−0.451050	−	0.892498i	$-0.648951\pi$
$73$	−7.70755	−0.902101	−0.451050	+	0.892498i	$0.648951\pi$
$74$	11.7117	1.36146
$75$	−0.885420	−0.102240
$76$	−0.295475	−0.0338933
$77$	0	0
$78$	−11.8807	−1.34523
$79$	4.13051	0.464719	0.232359	−	0.972630i	$-0.425355\pi$
$79$	4.13051	0.464719	0.232359	+	0.972630i	$0.425355\pi$
$80$	8.80790	0.984753
$81$	−10.7971	−1.19968
$82$	6.14050	0.678104
$83$	9.72969	1.06797	0.533986	−	0.845493i	$-0.320693\pi$
$83$	9.72969	1.06797	0.533986	+	0.845493i	$0.320693\pi$
$84$	−0.464208	−0.0506492
$85$	−14.7808	−1.60321
$86$	11.4912	1.23913
$87$	15.8269	1.69682
$88$	0	0
$89$	1.48882	0.157814	0.0789071	−	0.996882i	$-0.474857\pi$
$89$	1.48882	0.157814	0.0789071	+	0.996882i	$0.474857\pi$
$90$	−2.53581	−0.267298
$91$	14.8546	1.55719
$92$	0.214726	0.0223868
$93$	5.65328	0.586218
$94$	−18.9099	−1.95041
$95$	9.33875	0.958136
$96$	−0.746337	−0.0761727
$97$	−1.99118	−0.202174	−0.101087	−	0.994878i	$-0.532232\pi$
$97$	−1.99118	−0.202174	−0.101087	+	0.994878i	$0.532232\pi$
$98$	7.72008	0.779846
$99$	0	0
$100$	−0.0305374	−0.00305374
$101$	−2.16067	−0.214995	−0.107497	−	0.994205i	$-0.534284\pi$
$101$	−2.16067	−0.214995	−0.107497	+	0.994205i	$0.534284\pi$
$102$	19.4970	1.93049
$103$	−4.67013	−0.460162	−0.230081	−	0.973172i	$-0.573899\pi$
$103$	−4.67013	−0.460162	−0.230081	+	0.973172i	$0.573899\pi$
$104$	11.7365	1.15086
$105$	14.6717	1.43181
$106$	−5.57874	−0.541856
$107$	11.1372	1.07668	0.538338	−	0.842729i	$-0.319052\pi$
$107$	11.1372	1.07668	0.538338	+	0.842729i	$0.319052\pi$
$108$	−0.286813	−0.0275986
$109$	15.2969	1.46518	0.732591	−	0.680669i	$-0.238311\pi$
$109$	15.2969	1.46518	0.732591	+	0.680669i	$0.238311\pi$
$110$	0	0
$111$	15.9343	1.51241
$112$	14.5265	1.37262
$113$	17.7051	1.66556	0.832779	−	0.553606i	$-0.186749\pi$
$113$	17.7051	1.66556	0.832779	+	0.553606i	$0.186749\pi$
$114$	−12.3185	−1.15373
$115$	−6.78662	−0.632856
$116$	0.545857	0.0506815
$117$	−3.49307	−0.322935
$118$	−10.9091	−1.00427
$119$	−24.3774	−2.23467
$120$	11.5920	1.05820
$121$	0	0
$122$	1.43787	0.130179
$123$	8.35438	0.753289
$124$	0.194977	0.0175094
$125$	11.6275	1.03999
$126$	−4.18219	−0.372579
$127$	−5.02501	−0.445898	−0.222949	−	0.974830i	$-0.571568\pi$
$127$	−5.02501	−0.445898	−0.222949	+	0.974830i	$0.571568\pi$
$128$	11.8522	1.04759
$129$	15.6342	1.37651
$130$	12.9507	1.13585
$131$	−6.15503	−0.537767	−0.268884	−	0.963173i	$-0.586655\pi$
$131$	−6.15503	−0.537767	−0.268884	+	0.963173i	$0.586655\pi$
$132$	0	0
$133$	15.4020	1.33552
$134$	7.30788	0.631305
$135$	9.06500	0.780191
$136$	−19.2603	−1.65156
$137$	7.02715	0.600370	0.300185	−	0.953881i	$-0.402952\pi$
$137$	7.02715	0.600370	0.300185	+	0.953881i	$0.402952\pi$
$138$	8.95204	0.762048
$139$	14.9893	1.27137	0.635687	−	0.771947i	$-0.280717\pi$
$139$	14.9893	1.27137	0.635687	+	0.771947i	$0.280717\pi$
$140$	0.506015	0.0427661
$141$	−25.7277	−2.16666
$142$	−4.26374	−0.357805
$143$	0	0
$144$	−3.41591	−0.284659
$145$	−17.2523	−1.43273
$146$	−11.0825	−0.917191
$147$	10.5035	0.866311
$148$	0.549560	0.0451735
$149$	18.3484	1.50316	0.751578	−	0.659644i	$-0.229293\pi$
$149$	18.3484	1.50316	0.751578	+	0.659644i	$0.229293\pi$
$150$	−1.27312	−0.103950
$151$	16.1766	1.31644	0.658218	−	0.752827i	$-0.271310\pi$
$151$	16.1766	1.31644	0.658218	+	0.752827i	$0.271310\pi$
$152$	12.1690	0.987032
$153$	5.73236	0.463434
$154$	0	0
$155$	−6.16242	−0.494978
$156$	−0.557488	−0.0446348
$157$	−6.13046	−0.489264	−0.244632	−	0.969616i	$-0.578667\pi$
$157$	−6.13046	−0.489264	−0.244632	+	0.969616i	$0.578667\pi$
$158$	5.93914	0.472493
$159$	−7.59009	−0.601934
$160$	0.813553	0.0643170
$161$	−11.1929	−0.882121
$162$	−15.5248	−1.21975
$163$	9.50411	0.744420	0.372210	−	0.928149i	$-0.378600\pi$
$163$	9.50411	0.744420	0.372210	+	0.928149i	$0.378600\pi$
$164$	0.288135	0.0224996
$165$	0	0
$166$	13.9900	1.08584
$167$	3.22809	0.249797	0.124898	−	0.992170i	$-0.460140\pi$
$167$	3.22809	0.249797	0.124898	+	0.992170i	$0.460140\pi$
$168$	19.1181	1.47500
$169$	4.83959	0.372277
$170$	−21.2529	−1.63003
$171$	−3.62179	−0.276965
$172$	0.539210	0.0411144
$173$	−13.4276	−1.02088	−0.510440	−	0.859913i	$-0.670518\pi$
$173$	−13.4276	−1.02088	−0.510440	+	0.859913i	$0.670518\pi$
$174$	22.7570	1.72521
$175$	1.59180	0.120329
$176$	0	0
$177$	−14.8423	−1.11562
$178$	2.14072	0.160454
$179$	12.2486	0.915500	0.457750	−	0.889081i	$-0.348655\pi$
$179$	12.2486	0.915500	0.457750	+	0.889081i	$0.348655\pi$
$180$	−0.118990	−0.00886897
$181$	−19.0869	−1.41872	−0.709359	−	0.704847i	$-0.751016\pi$
$181$	−19.0869	−1.41872	−0.709359	+	0.704847i	$0.751016\pi$
$182$	21.3590	1.58324
$183$	1.95628	0.144612
$184$	−8.84337	−0.651942
$185$	−17.3693	−1.27702
$186$	8.12868	0.596024
$187$	0	0
$188$	−0.887325	−0.0647148
$189$	14.9505	1.08749
$190$	13.4279	0.974163
$191$	16.0262	1.15962	0.579808	−	0.814753i	$-0.303128\pi$
$191$	16.0262	1.15962	0.579808	+	0.814753i	$0.303128\pi$
$192$	15.0872	1.08883
$193$	−4.96332	−0.357268	−0.178634	−	0.983916i	$-0.557168\pi$
$193$	−4.96332	−0.357268	−0.178634	+	0.983916i	$0.557168\pi$
$194$	−2.86306	−0.205555
$195$	17.6199	1.26179
$196$	0.362255	0.0258754
$197$	1.76251	0.125574	0.0627870	−	0.998027i	$-0.480001\pi$
$197$	1.76251	0.125574	0.0627870	+	0.998027i	$0.480001\pi$
$198$	0	0
$199$	20.2158	1.43306	0.716531	−	0.697555i	$-0.245729\pi$
$199$	20.2158	1.43306	0.716531	+	0.697555i	$0.245729\pi$
$200$	1.25767	0.0889303
$201$	9.94265	0.701301
$202$	−3.10676	−0.218591
$203$	−28.4535	−1.99704
$204$	0.914874	0.0640540
$205$	−9.10679	−0.636045
$206$	−6.71504	−0.467859
$207$	2.63201	0.182937
$208$	17.4455	1.20963
$209$	0	0
$210$	21.0960	1.45576
$211$	0.341442	0.0235059	0.0117529	−	0.999931i	$-0.496259\pi$
$211$	0.341442	0.0235059	0.0117529	+	0.999931i	$0.496259\pi$
$212$	−0.261776	−0.0179788
$213$	−5.80098	−0.397477
$214$	16.0139	1.09469
$215$	−17.0422	−1.16227
$216$	11.8122	0.803721
$217$	−10.1634	−0.689937
$218$	21.9950	1.48969
$219$	−15.0781	−1.01888
$220$	0	0
$221$	−29.2759	−1.96931
$222$	22.9114	1.53771
$223$	−18.5892	−1.24482	−0.622411	−	0.782691i	$-0.713847\pi$
$223$	−18.5892	−1.24482	−0.622411	+	0.782691i	$0.713847\pi$
$224$	1.34176	0.0896499
$225$	−0.374313	−0.0249542
$226$	25.4577	1.69342
$227$	−29.1175	−1.93260	−0.966298	−	0.257428i	$-0.917125\pi$
$227$	−29.1175	−1.93260	−0.966298	+	0.257428i	$0.917125\pi$
$228$	−0.578030	−0.0382810
$229$	−10.1945	−0.673671	−0.336835	−	0.941564i	$-0.609357\pi$
$229$	−10.1945	−0.673671	−0.336835	+	0.941564i	$0.609357\pi$
$230$	−9.75828	−0.643442
$231$	0	0
$232$	−22.4808	−1.47594
$233$	−24.1880	−1.58461	−0.792305	−	0.610125i	$-0.791119\pi$
$233$	−24.1880	−1.58461	−0.792305	+	0.610125i	$0.791119\pi$
$234$	−5.02259	−0.328337
$235$	28.0447	1.82944
$236$	−0.511898	−0.0333217
$237$	8.08043	0.524880
$238$	−35.0515	−2.27205
$239$	−17.6815	−1.14372	−0.571859	−	0.820352i	$-0.693778\pi$
$239$	−17.6815	−1.14372	−0.571859	+	0.820352i	$0.693778\pi$
$240$	17.2307	1.11224
$241$	4.46326	0.287504	0.143752	−	0.989614i	$-0.454083\pi$
$241$	4.46326	0.287504	0.143752	+	0.989614i	$0.454083\pi$
$242$	0	0
$243$	−8.36925	−0.536888
$244$	0.0674703	0.00431935
$245$	−11.4494	−0.731476
$246$	12.0125	0.765890
$247$	18.4969	1.17693
$248$	−8.03001	−0.509906
$249$	19.0340	1.20623
$250$	16.7188	1.05739
$251$	7.90143	0.498734	0.249367	−	0.968409i	$-0.419777\pi$
$251$	7.90143	0.498734	0.249367	+	0.968409i	$0.419777\pi$
$252$	−0.196244	−0.0123622
$253$	0	0
$254$	−7.22531	−0.453357
$255$	−28.9154	−1.81075
$256$	1.61747	0.101092
$257$	−10.8673	−0.677882	−0.338941	−	0.940808i	$-0.610069\pi$
$257$	−10.8673	−0.677882	−0.338941	+	0.940808i	$0.610069\pi$
$258$	22.4799	1.39954
$259$	−28.6465	−1.78001
$260$	0.607696	0.0376877
$261$	6.69085	0.414153
$262$	−8.85013	−0.546763
$263$	31.3601	1.93374	0.966872	−	0.255262i	$-0.0821616\pi$
$263$	31.3601	1.93374	0.966872	+	0.255262i	$0.0821616\pi$
$264$	0	0
$265$	8.27367	0.508247
$266$	22.1460	1.35786
$267$	2.91254	0.178244
$268$	0.342914	0.0209468
$269$	1.91455	0.116732	0.0583661	−	0.998295i	$-0.481411\pi$
$269$	1.91455	0.116732	0.0583661	+	0.998295i	$0.481411\pi$
$270$	13.0343	0.793241
$271$	19.7941	1.20241	0.601203	−	0.799097i	$-0.294688\pi$
$271$	19.7941	1.20241	0.601203	+	0.799097i	$0.294688\pi$
$272$	−28.6292	−1.73590
$273$	29.0598	1.75878
$274$	10.1041	0.610412
$275$	0	0
$276$	0.420064	0.0252849
$277$	−13.3630	−0.802903	−0.401451	−	0.915880i	$-0.631494\pi$
$277$	−13.3630	−0.802903	−0.401451	+	0.915880i	$0.631494\pi$
$278$	21.5526	1.29264
$279$	2.38993	0.143081
$280$	−20.8399	−1.24542
$281$	18.5558	1.10695	0.553473	−	0.832867i	$-0.313302\pi$
$281$	18.5558	1.10695	0.553473	+	0.832867i	$0.313302\pi$
$282$	−36.9930	−2.20290
$283$	10.8347	0.644058	0.322029	−	0.946730i	$-0.395635\pi$
$283$	10.8347	0.644058	0.322029	+	0.946730i	$0.395635\pi$
$284$	−0.200071	−0.0118720
$285$	18.2692	1.08217
$286$	0	0
$287$	−15.0194	−0.886568
$288$	−0.315515	−0.0185919
$289$	31.0436	1.82610
$290$	−24.8066	−1.45669
$291$	−3.89530	−0.228346
$292$	−0.520031	−0.0304325
$293$	−27.2028	−1.58921	−0.794603	−	0.607129i	$-0.792321\pi$
$293$	−27.2028	−1.58921	−0.794603	+	0.607129i	$0.792321\pi$
$294$	15.1026	0.880802
$295$	16.1790	0.941979
$296$	−22.6333	−1.31553
$297$	0	0
$298$	26.3826	1.52830
$299$	−13.4420	−0.777372
$300$	−0.0597396	−0.00344907
$301$	−28.1070	−1.62006
$302$	23.2599	1.33846
$303$	−4.22687	−0.242827
$304$	18.0883	1.03744
$305$	−2.13246	−0.122104
$306$	8.24238	0.471186
$307$	23.8449	1.36090	0.680449	−	0.732795i	$-0.261785\pi$
$307$	23.8449	1.36090	0.680449	+	0.732795i	$0.261785\pi$
$308$	0	0
$309$	−9.13607	−0.519733
$310$	−8.86076	−0.503258
$311$	−7.49912	−0.425236	−0.212618	−	0.977135i	$-0.568199\pi$
$311$	−7.49912	−0.425236	−0.212618	+	0.977135i	$0.568199\pi$
$312$	22.9598	1.29984
$313$	32.6206	1.84383	0.921914	−	0.387395i	$-0.126625\pi$
$313$	32.6206	1.84383	0.921914	+	0.387395i	$0.126625\pi$
$314$	−8.81481	−0.497449
$315$	6.20249	0.349470
$316$	0.278687	0.0156774
$317$	−32.2879	−1.81347	−0.906735	−	0.421700i	$-0.861433\pi$
$317$	−32.2879	−1.81347	−0.906735	+	0.421700i	$0.861433\pi$
$318$	−10.9136	−0.612002
$319$	0	0
$320$	−16.4460	−0.919360
$321$	21.7875	1.21606
$322$	−16.0939	−0.896877
$323$	−30.3547	−1.68898
$324$	−0.728484	−0.0404713
$325$	1.91166	0.106040
$326$	13.6657	0.756872
$327$	29.9251	1.65486
$328$	−11.8667	−0.655228
$329$	46.2529	2.55001
$330$	0	0
$331$	17.3099	0.951438	0.475719	−	0.879597i	$-0.342188\pi$
$331$	17.3099	0.951438	0.475719	+	0.879597i	$0.342188\pi$
$332$	0.656466	0.0360282
$333$	6.73624	0.369144
$334$	4.64157	0.253975
$335$	−10.8381	−0.592149
$336$	28.4178	1.55032
$337$	19.0306	1.03666	0.518331	−	0.855180i	$-0.326554\pi$
$337$	19.0306	1.03666	0.518331	+	0.855180i	$0.326554\pi$
$338$	6.95871	0.378504
$339$	34.6361	1.88117
$340$	−0.997269	−0.0540845
$341$	0	0
$342$	−5.20766	−0.281598
$343$	5.73581	0.309705
$344$	−22.2070	−1.19732
$345$	−13.2765	−0.714783
$346$	−19.3071	−1.03796
$347$	6.83943	0.367160	0.183580	−	0.983005i	$-0.441231\pi$
$347$	6.83943	0.367160	0.183580	+	0.983005i	$0.441231\pi$
$348$	1.06785	0.0572426
$349$	−26.0678	−1.39538	−0.697690	−	0.716400i	$-0.745789\pi$
$349$	−26.0678	−1.39538	−0.697690	+	0.716400i	$0.745789\pi$
$350$	2.28880	0.122342
$351$	17.9547	0.958352
$352$	0	0
$353$	−8.58988	−0.457193	−0.228597	−	0.973521i	$-0.573414\pi$
$353$	−8.58988	−0.457193	−0.228597	+	0.973521i	$0.573414\pi$
$354$	−21.3413	−1.13428
$355$	6.32343	0.335613
$356$	0.100451	0.00532389
$357$	−47.6889	−2.52396
$358$	17.6118	0.930815
$359$	−22.6157	−1.19361	−0.596805	−	0.802387i	$-0.703563\pi$
$359$	−22.6157	−1.19361	−0.596805	+	0.802387i	$0.703563\pi$
$360$	4.90052	0.258280
$361$	0.178509	0.00939522
$362$	−27.4445	−1.44245
$363$	0	0
$364$	1.00225	0.0525320
$365$	16.4361	0.860303
$366$	2.81287	0.147031
$367$	−18.5955	−0.970676	−0.485338	−	0.874327i	$-0.661303\pi$
$367$	−18.5955	−0.970676	−0.485338	+	0.874327i	$0.661303\pi$
$368$	−13.1451	−0.685234
$369$	3.53182	0.183859
$370$	−24.9749	−1.29838
$371$	13.6454	0.708433
$372$	0.381429	0.0197762
$373$	29.4776	1.52629	0.763147	−	0.646225i	$-0.223653\pi$
$373$	29.4776	1.52629	0.763147	+	0.646225i	$0.223653\pi$
$374$	0	0
$375$	22.7466	1.17463
$376$	36.5440	1.88461
$377$	−34.1711	−1.75990
$378$	21.4969	1.10568
$379$	−3.18579	−0.163643	−0.0818214	−	0.996647i	$-0.526074\pi$
$379$	−3.18579	−0.163643	−0.0818214	+	0.996647i	$0.526074\pi$
$380$	0.630089	0.0323229
$381$	−9.83031	−0.503622
$382$	23.0436	1.17901
$383$	−27.5668	−1.40860	−0.704299	−	0.709904i	$-0.748738\pi$
$383$	−27.5668	−1.40860	−0.704299	+	0.709904i	$0.748738\pi$
$384$	23.1862	1.18321
$385$	0	0
$386$	−7.13662	−0.363244
$387$	6.60937	0.335973
$388$	−0.134345	−0.00682036
$389$	32.7270	1.65932	0.829662	−	0.558266i	$-0.188533\pi$
$389$	32.7270	1.65932	0.829662	+	0.558266i	$0.188533\pi$
$390$	25.3352	1.28290
$391$	22.0592	1.11558
$392$	−14.9193	−0.753537
$393$	−12.0409	−0.607385
$394$	2.53427	0.127674
$395$	−8.80816	−0.443187
$396$	0	0
$397$	−34.2908	−1.72100	−0.860502	−	0.509447i	$-0.829850\pi$
$397$	−34.2908	−1.72100	−0.860502	+	0.509447i	$0.829850\pi$
$398$	29.0677	1.45703
$399$	30.1305	1.50841
$400$	1.86943	0.0934717
$401$	−20.2505	−1.01126	−0.505630	−	0.862750i	$-0.668740\pi$
$401$	−20.2505	−1.01126	−0.505630	+	0.862750i	$0.668740\pi$
$402$	14.2962	0.713032
$403$	−12.2057	−0.608009
$404$	−0.145781	−0.00725289
$405$	23.0244	1.14409
$406$	−40.9124	−2.03045
$407$	0	0
$408$	−37.6785	−1.86537
$409$	−1.71957	−0.0850272	−0.0425136	−	0.999096i	$-0.513537\pi$
$409$	−1.71957	−0.0850272	−0.0425136	+	0.999096i	$0.513537\pi$
$410$	−13.0944	−0.646685
$411$	13.7470	0.678092
$412$	−0.315095	−0.0155236
$413$	26.6833	1.31300
$414$	3.78449	0.185997
$415$	−20.7482	−1.01849
$416$	1.61138	0.0790042
$417$	29.3232	1.43596
$418$	0	0
$419$	6.08960	0.297497	0.148748	−	0.988875i	$-0.452476\pi$
$419$	6.08960	0.297497	0.148748	+	0.988875i	$0.452476\pi$
$420$	0.989905	0.0483024
$421$	−39.2682	−1.91382	−0.956908	−	0.290390i	$-0.906215\pi$
$421$	−39.2682	−1.91382	−0.956908	+	0.290390i	$0.906215\pi$
$422$	0.490950	0.0238991
$423$	−10.8764	−0.528829
$424$	10.7811	0.523576
$425$	−3.13716	−0.152175
$426$	−8.34106	−0.404126
$427$	−3.51697	−0.170198
$428$	0.751432	0.0363218
$429$	0	0
$430$	−24.5045	−1.18171
$431$	−9.76798	−0.470507	−0.235253	−	0.971934i	$-0.575592\pi$
$431$	−9.76798	−0.470507	−0.235253	+	0.971934i	$0.575592\pi$
$432$	17.5581	0.844764
$433$	4.00348	0.192395	0.0961974	−	0.995362i	$-0.469332\pi$
$433$	4.00348	0.192395	0.0961974	+	0.995362i	$0.469332\pi$
$434$	−14.6137	−0.701478
$435$	−33.7503	−1.61820
$436$	1.03209	0.0494281
$437$	−13.9373	−0.666713
$438$	−21.6804	−1.03593
$439$	17.2178	0.821760	0.410880	−	0.911689i	$-0.365221\pi$
$439$	17.2178	0.821760	0.410880	+	0.911689i	$0.365221\pi$
$440$	0	0
$441$	4.44035	0.211445
$442$	−42.0950	−2.00225
$443$	−22.9465	−1.09022	−0.545111	−	0.838364i	$-0.683513\pi$
$443$	−22.9465	−1.09022	−0.545111	+	0.838364i	$0.683513\pi$
$444$	1.07509	0.0510216
$445$	−3.17484	−0.150502
$446$	−26.7288	−1.26564
$447$	35.8945	1.69775
$448$	−27.1237	−1.28147
$449$	−12.6010	−0.594680	−0.297340	−	0.954772i	$-0.596099\pi$
$449$	−12.6010	−0.594680	−0.297340	+	0.954772i	$0.596099\pi$
$450$	−0.538213	−0.0253716
$451$	0	0
$452$	1.19457	0.0561878
$453$	31.6460	1.48686
$454$	−41.8672	−1.96492
$455$	−31.6769	−1.48504
$456$	23.8058	1.11481
$457$	25.2361	1.18049	0.590247	−	0.807222i	$-0.299030\pi$
$457$	25.2361	1.18049	0.590247	+	0.807222i	$0.299030\pi$
$458$	−14.6583	−0.684940
$459$	−29.4648	−1.37530
$460$	−0.457895	−0.0213495
$461$	−37.4458	−1.74402	−0.872012	−	0.489484i	$-0.837185\pi$
$461$	−37.4458	−1.74402	−0.872012	+	0.489484i	$0.837185\pi$
$462$	0	0
$463$	29.4121	1.36690	0.683448	−	0.730000i	$-0.260480\pi$
$463$	29.4121	1.36690	0.683448	+	0.730000i	$0.260480\pi$
$464$	−33.4162	−1.55131
$465$	−12.0554	−0.559056
$466$	−34.7792	−1.61112
$467$	−7.43261	−0.343940	−0.171970	−	0.985102i	$-0.555013\pi$
$467$	−7.43261	−0.343940	−0.171970	+	0.985102i	$0.555013\pi$
$468$	−0.235679	−0.0108943
$469$	−17.8748	−0.825382
$470$	40.3247	1.86004
$471$	−11.9929	−0.552603
$472$	21.0822	0.970389
$473$	0	0
$474$	11.6186	0.533660
$475$	1.98210	0.0909452
$476$	−1.64475	−0.0753870
$477$	−3.20872	−0.146917
$478$	−25.4236	−1.16285
$479$	−4.67273	−0.213502	−0.106751	−	0.994286i	$-0.534045\pi$
$479$	−4.67273	−0.213502	−0.106751	+	0.994286i	$0.534045\pi$
$480$	1.59153	0.0726433
$481$	−34.4029	−1.56864
$482$	6.41758	0.292313
$483$	−21.8963	−0.996318
$484$	0	0
$485$	4.24611	0.192806
$486$	−12.0339	−0.545868
$487$	0.00374509	0.000169706 0	8.48531e−5	−	1.00000i	$-0.499973\pi$
$487$	0.00374509	0.000169706 0	8.48531e−5		1.00000i	$0.499973\pi$
$488$	−2.77873	−0.125787
$489$	18.5927	0.840790
$490$	−16.4628	−0.743712
$491$	25.1283	1.13402	0.567012	−	0.823709i	$-0.308099\pi$
$491$	25.1283	1.13402	0.567012	+	0.823709i	$0.308099\pi$
$492$	0.563673	0.0254123
$493$	56.0769	2.52558
$494$	26.5962	1.19662
$495$	0	0
$496$	−11.9361	−0.535945
$497$	10.4289	0.467802
$498$	27.3684	1.22641
$499$	33.2637	1.48909	0.744543	−	0.667575i	$-0.232668\pi$
$499$	33.2637	1.48909	0.744543	+	0.667575i	$0.232668\pi$
$500$	0.784510	0.0350843
$501$	6.31503	0.282135
$502$	11.3612	0.507077
$503$	27.9025	1.24411	0.622055	−	0.782973i	$-0.286298\pi$
$503$	27.9025	1.24411	0.622055	+	0.782973i	$0.286298\pi$
$504$	8.08221	0.360010
$505$	4.60755	0.205033
$506$	0	0
$507$	9.46759	0.420470
$508$	−0.339039	−0.0150424
$509$	−1.59842	−0.0708488	−0.0354244	−	0.999372i	$-0.511278\pi$
$509$	−1.59842	−0.0708488	−0.0354244	+	0.999372i	$0.511278\pi$
$510$	−41.5766	−1.84104
$511$	27.1073	1.19915
$512$	−21.3787	−0.944812
$513$	18.6163	0.821930
$514$	−15.6257	−0.689222
$515$	9.95888	0.438841
$516$	1.05484	0.0464369
$517$	0	0
$518$	−41.1899	−1.80978
$519$	−26.2681	−1.15304
$520$	−25.0276	−1.09753
$521$	28.8814	1.26532	0.632660	−	0.774430i	$-0.281963\pi$
$521$	28.8814	1.26532	0.632660	+	0.774430i	$0.281963\pi$
$522$	9.62057	0.421081
$523$	−24.3339	−1.06405	−0.532025	−	0.846729i	$-0.678569\pi$
$523$	−24.3339	−1.06405	−0.532025	+	0.846729i	$0.678569\pi$
$524$	−0.415282	−0.0181417
$525$	3.11400	0.135906
$526$	45.0917	1.96609
$527$	20.0303	0.872535
$528$	0	0
$529$	−12.8715	−0.559631
$530$	11.8965	0.516749
$531$	−6.27460	−0.272295
$532$	1.03918	0.0450540
$533$	−18.0375	−0.781291
$534$	4.18785	0.181226
$535$	−23.7497	−1.02679
$536$	−14.1227	−0.610008
$537$	23.9616	1.03402
$538$	2.75287	0.118685
$539$	0	0
$540$	0.611618	0.0263199
$541$	21.0998	0.907152	0.453576	−	0.891218i	$-0.350148\pi$
$541$	21.0998	0.907152	0.453576	+	0.891218i	$0.350148\pi$
$542$	28.4613	1.22252
$543$	−37.3393	−1.60238
$544$	−2.64437	−0.113376
$545$	−32.6202	−1.39729
$546$	41.7842	1.78820
$547$	32.5329	1.39101	0.695503	−	0.718523i	$-0.255182\pi$
$547$	32.5329	1.39101	0.695503	+	0.718523i	$0.255182\pi$
$548$	0.474124	0.0202536
$549$	0.827019	0.0352963
$550$	0	0
$551$	−35.4302	−1.50938
$552$	−17.3001	−0.736341
$553$	−14.5269	−0.617747
$554$	−19.2142	−0.816333
$555$	−33.9792	−1.44234
$556$	1.01133	0.0428900
$557$	27.0768	1.14728	0.573640	−	0.819108i	$-0.305531\pi$
$557$	27.0768	1.14728	0.573640	+	0.819108i	$0.305531\pi$
$558$	3.43641	0.145475
$559$	−33.7549	−1.42768
$560$	−30.9771	−1.30902
$561$	0	0
$562$	26.6808	1.12546
$563$	10.1001	0.425670	0.212835	−	0.977088i	$-0.431730\pi$
$563$	10.1001	0.425670	0.212835	+	0.977088i	$0.431730\pi$
$564$	−1.73585	−0.0730926
$565$	−37.7555	−1.58839
$566$	15.5789	0.654832
$567$	37.9731	1.59472
$568$	8.23981	0.345735
$569$	20.1036	0.842787	0.421394	−	0.906878i	$-0.361541\pi$
$569$	20.1036	0.842787	0.421394	+	0.906878i	$0.361541\pi$
$570$	26.2687	1.10028
$571$	−31.6928	−1.32630	−0.663151	−	0.748485i	$-0.730781\pi$
$571$	−31.6928	−1.32630	−0.663151	+	0.748485i	$0.730781\pi$
$572$	0	0
$573$	31.3517	1.30974
$574$	−21.5960	−0.901398
$575$	−1.44043	−0.0600700
$576$	6.37815	0.265756
$577$	−5.27576	−0.219633	−0.109816	−	0.993952i	$-0.535026\pi$
$577$	−5.27576	−0.219633	−0.109816	+	0.993952i	$0.535026\pi$
$578$	44.6367	1.85664
$579$	−9.70964	−0.403519
$580$	−1.16402	−0.0483333
$581$	−34.2191	−1.41965
$582$	−5.60093	−0.232166
$583$	0	0
$584$	21.4172	0.886249
$585$	7.44885	0.307972
$586$	−39.1141	−1.61579
$587$	12.1321	0.500744	0.250372	−	0.968150i	$-0.419447\pi$
$587$	12.1321	0.500744	0.250372	+	0.968150i	$0.419447\pi$
$588$	0.708672	0.0292251
$589$	−12.6555	−0.521459
$590$	23.2633	0.957736
$591$	3.44797	0.141830
$592$	−33.6429	−1.38271
$593$	2.37969	0.0977220	0.0488610	−	0.998806i	$-0.484441\pi$
$593$	2.37969	0.0977220	0.0488610	+	0.998806i	$0.484441\pi$
$594$	0	0
$595$	51.9838	2.13113
$596$	1.23797	0.0507092
$597$	39.5478	1.61858
$598$	−19.3279	−0.790376
$599$	18.6565	0.762283	0.381141	−	0.924517i	$-0.375531\pi$
$599$	18.6565	0.762283	0.381141	+	0.924517i	$0.375531\pi$
$600$	2.46034	0.100443
$601$	2.28876	0.0933606	0.0466803	−	0.998910i	$-0.485136\pi$
$601$	2.28876	0.0933606	0.0466803	+	0.998910i	$0.485136\pi$
$602$	−40.4142	−1.64716
$603$	4.20327	0.171170
$604$	1.09144	0.0444102
$605$	0	0
$606$	−6.07769	−0.246889
$607$	24.9768	1.01378	0.506888	−	0.862012i	$-0.330796\pi$
$607$	24.9768	1.01378	0.506888	+	0.862012i	$0.330796\pi$
$608$	1.67075	0.0677579
$609$	−55.6628	−2.25557
$610$	−3.06620	−0.124147
$611$	55.5472	2.24720
$612$	0.386764	0.0156340
$613$	−8.37270	−0.338170	−0.169085	−	0.985601i	$-0.554081\pi$
$613$	−8.37270	−0.338170	−0.169085	+	0.985601i	$0.554081\pi$
$614$	34.2858	1.38366
$615$	−17.8154	−0.718386
$616$	0	0
$617$	16.8174	0.677043	0.338521	−	0.940959i	$-0.390073\pi$
$617$	16.8174	0.677043	0.338521	+	0.940959i	$0.390073\pi$
$618$	−13.1365	−0.528427
$619$	4.98028	0.200175	0.100087	−	0.994979i	$-0.468088\pi$
$619$	4.98028	0.200175	0.100087	+	0.994979i	$0.468088\pi$
$620$	−0.415781	−0.0166982
$621$	−13.5288	−0.542891
$622$	−10.7828	−0.432349
$623$	−5.23613	−0.209781
$624$	34.1282	1.36622
$625$	−22.5321	−0.901285
$626$	46.9042	1.87467
$627$	0	0
$628$	−0.413625	−0.0165054
$629$	56.4573	2.25110
$630$	8.91837	0.355316
$631$	−24.2902	−0.966977	−0.483488	−	0.875351i	$-0.660630\pi$
$631$	−24.2902	−0.966977	−0.483488	+	0.875351i	$0.660630\pi$
$632$	−11.4776	−0.456553
$633$	0.667956	0.0265489
$634$	−46.4258	−1.84381
$635$	10.7156	0.425238
$636$	−0.512106	−0.0203063
$637$	−22.6775	−0.898514
$638$	0	0
$639$	−2.45237	−0.0970144
$640$	−25.2743	−0.999056
$641$	−2.01263	−0.0794941	−0.0397470	−	0.999210i	$-0.512655\pi$
$641$	−2.01263	−0.0794941	−0.0397470	+	0.999210i	$0.512655\pi$
$642$	31.3276	1.23640
$643$	−2.30689	−0.0909749	−0.0454874	−	0.998965i	$-0.514484\pi$
$643$	−2.30689	−0.0909749	−0.0454874	+	0.998965i	$0.514484\pi$
$644$	−0.755186	−0.0297585
$645$	−33.3393	−1.31273
$646$	−43.6461	−1.71723
$647$	16.4021	0.644833	0.322416	−	0.946598i	$-0.395505\pi$
$647$	16.4021	0.644833	0.322416	+	0.946598i	$0.395505\pi$
$648$	30.0022	1.17860
$649$	0	0
$650$	2.74873	0.107814
$651$	−19.8824	−0.779254
$652$	0.641246	0.0251131
$653$	0.298643	0.0116868	0.00584340	−	0.999983i	$-0.498140\pi$
$653$	0.298643	0.0116868	0.00584340	+	0.999983i	$0.498140\pi$
$654$	43.0283	1.68254
$655$	13.1254	0.512851
$656$	−17.6390	−0.688688
$657$	−6.37429	−0.248685
$658$	66.5057	2.59266
$659$	−8.71936	−0.339658	−0.169829	−	0.985474i	$-0.554322\pi$
$659$	−8.71936	−0.339658	−0.169829	+	0.985474i	$0.554322\pi$
$660$	0	0
$661$	9.84044	0.382749	0.191374	−	0.981517i	$-0.438706\pi$
$661$	9.84044	0.382749	0.191374	+	0.981517i	$0.438706\pi$
$662$	24.8894	0.967353
$663$	−57.2718	−2.22425
$664$	−27.0362	−1.04921
$665$	−32.8441	−1.27364
$666$	9.68583	0.375318
$667$	25.7477	0.996954
$668$	0.217800	0.00842694
$669$	−36.3655	−1.40597
$670$	−15.5838	−0.602054
$671$	0	0
$672$	2.62485	0.101256
$673$	20.6920	0.797619	0.398810	−	0.917034i	$-0.369423\pi$
$673$	20.6920	0.797619	0.398810	+	0.917034i	$0.369423\pi$
$674$	27.3635	1.05400
$675$	1.92400	0.0740549
$676$	0.326529	0.0125588
$677$	11.6812	0.448944	0.224472	−	0.974481i	$-0.427934\pi$
$677$	11.6812	0.448944	0.224472	+	0.974481i	$0.427934\pi$
$678$	49.8022	1.91264
$679$	7.00292	0.268748
$680$	41.0719	1.57504
$681$	−56.9619	−2.18278
$682$	0	0
$683$	22.5849	0.864188	0.432094	−	0.901828i	$-0.357775\pi$
$683$	22.5849	0.864188	0.432094	+	0.901828i	$0.357775\pi$
$684$	−0.244363	−0.00934346
$685$	−14.9851	−0.572552
$686$	8.24735	0.314885
$687$	−19.9432	−0.760882
$688$	−33.0093	−1.25847
$689$	16.3874	0.624309
$690$	−19.0899	−0.726740
$691$	7.88744	0.300052	0.150026	−	0.988682i	$-0.452064\pi$
$691$	7.88744	0.300052	0.150026	+	0.988682i	$0.452064\pi$
$692$	−0.905964	−0.0344396
$693$	0	0
$694$	9.83422	0.373302
$695$	−31.9641	−1.21247
$696$	−43.9787	−1.66701
$697$	29.6007	1.12121
$698$	−37.4822	−1.41872
$699$	−47.3185	−1.78975
$700$	0.107399	0.00405931
$701$	−27.0438	−1.02143	−0.510715	−	0.859750i	$-0.670619\pi$
$701$	−27.0438	−1.02143	−0.510715	+	0.859750i	$0.670619\pi$
$702$	25.8166	0.974383
$703$	−35.6705	−1.34534
$704$	0	0
$705$	54.8633	2.06627
$706$	−12.3511	−0.464841
$707$	7.59902	0.285791
$708$	−1.00141	−0.0376355
$709$	−24.6330	−0.925113	−0.462557	−	0.886590i	$-0.653068\pi$
$709$	−24.6330	−0.925113	−0.462557	+	0.886590i	$0.653068\pi$
$710$	9.09227	0.341227
$711$	3.41601	0.128110
$712$	−4.13701	−0.155041
$713$	9.19692	0.344427
$714$	−68.5704	−2.56618
$715$	0	0
$716$	0.826415	0.0308846
$717$	−34.5898	−1.29178
$718$	−32.5184	−1.21358
$719$	20.3846	0.760217	0.380109	−	0.924942i	$-0.375887\pi$
$719$	20.3846	0.760217	0.380109	+	0.924942i	$0.375887\pi$
$720$	7.28429	0.271470
$721$	16.4247	0.611689
$722$	0.256673	0.00955238
$723$	8.73137	0.324723
$724$	−1.28780	−0.0478607
$725$	−3.66172	−0.135993
$726$	0	0
$727$	22.7796	0.844847	0.422424	−	0.906399i	$-0.361179\pi$
$727$	22.7796	0.844847	0.422424	+	0.906399i	$0.361179\pi$
$728$	−41.2769	−1.52983
$729$	16.0187	0.593286
$730$	23.6329	0.874694
$731$	55.3940	2.04882
$732$	0.131991	0.00487851
$733$	−15.2059	−0.561642	−0.280821	−	0.959760i	$-0.590607\pi$
$733$	−15.2059	−0.561642	−0.280821	+	0.959760i	$0.590607\pi$
$734$	−26.7379	−0.986913
$735$	−22.3982	−0.826171
$736$	−1.21416	−0.0447546
$737$	0	0
$738$	5.07830	0.186935
$739$	16.1291	0.593319	0.296660	−	0.954983i	$-0.404127\pi$
$739$	16.1291	0.593319	0.296660	+	0.954983i	$0.404127\pi$
$740$	−1.17192	−0.0430805
$741$	36.1851	1.32929
$742$	19.6203	0.720284
$743$	17.6105	0.646067	0.323034	−	0.946387i	$-0.395297\pi$
$743$	17.6105	0.646067	0.323034	+	0.946387i	$0.395297\pi$
$744$	−15.7089	−0.575917
$745$	−39.1272	−1.43351
$746$	42.3850	1.55182
$747$	8.04664	0.294411
$748$	0	0
$749$	−39.1693	−1.43122
$750$	32.7066	1.19428
$751$	21.2710	0.776190	0.388095	−	0.921619i	$-0.373133\pi$
$751$	21.2710	0.776190	0.388095	+	0.921619i	$0.373133\pi$
$752$	54.3201	1.98085
$753$	15.4574	0.563299
$754$	−49.1335	−1.78934
$755$	−34.4961	−1.25544
$756$	1.00871	0.0366866
$757$	39.2881	1.42795	0.713974	−	0.700172i	$-0.246893\pi$
$757$	39.2881	1.42795	0.713974	+	0.700172i	$0.246893\pi$
$758$	−4.58075	−0.166380
$759$	0	0
$760$	−25.9498	−0.941299
$761$	36.9632	1.33992	0.669958	−	0.742399i	$-0.266312\pi$
$761$	36.9632	1.33992	0.669958	+	0.742399i	$0.266312\pi$
$762$	−14.1347	−0.512047
$763$	−53.7990	−1.94765
$764$	1.08129	0.0391198
$765$	−12.2240	−0.441961
$766$	−39.6375	−1.43216
$767$	32.0452	1.15709
$768$	3.16421	0.114179
$769$	42.3005	1.52540	0.762698	−	0.646754i	$-0.223874\pi$
$769$	42.3005	1.52540	0.762698	+	0.646754i	$0.223874\pi$
$770$	0	0
$771$	−21.2594	−0.765639
$772$	−0.334877	−0.0120525
$773$	12.4295	0.447059	0.223530	−	0.974697i	$-0.428242\pi$
$773$	12.4295	0.447059	0.223530	+	0.974697i	$0.428242\pi$
$774$	9.50342	0.341593
$775$	−1.30794	−0.0469828
$776$	5.53294	0.198621
$777$	−56.0404	−2.01044
$778$	47.0572	1.68708
$779$	−18.7021	−0.670074
$780$	1.18882	0.0425667
$781$	0	0
$782$	31.7183	1.13424
$783$	−34.3916	−1.22905
$784$	−22.1765	−0.792018
$785$	13.0730	0.466595
$786$	−17.3133	−0.617545
$787$	−10.4534	−0.372623	−0.186311	−	0.982491i	$-0.559653\pi$
$787$	−10.4534	−0.372623	−0.186311	+	0.982491i	$0.559653\pi$
$788$	0.118917	0.00423626
$789$	61.3490	2.18408
$790$	−12.6650	−0.450600
$791$	−62.2684	−2.21401
$792$	0	0
$793$	−4.22369	−0.149988
$794$	−49.3057	−1.74979
$795$	16.1856	0.574044
$796$	1.36397	0.0483446
$797$	−46.2887	−1.63963	−0.819815	−	0.572629i	$-0.805924\pi$
$797$	−46.2887	−1.63963	−0.819815	+	0.572629i	$0.805924\pi$
$798$	43.3238	1.53365
$799$	−91.1566	−3.22489
$800$	0.172673	0.00610490
$801$	1.23128	0.0435051
$802$	−29.1175	−1.02818
$803$	0	0
$804$	0.670834	0.0236585
$805$	23.8684	0.841249
$806$	−17.5502	−0.618180
$807$	3.74539	0.131844
$808$	6.00391	0.211217
$809$	−36.7955	−1.29366	−0.646830	−	0.762634i	$-0.723906\pi$
$809$	−36.7955	−1.29366	−0.646830	+	0.762634i	$0.723906\pi$
$810$	33.1061	1.16323
$811$	8.73575	0.306754	0.153377	−	0.988168i	$-0.450985\pi$
$811$	8.73575	0.306754	0.153377	+	0.988168i	$0.450985\pi$
$812$	−1.91976	−0.0673705
$813$	38.7227	1.35807
$814$	0	0
$815$	−20.2672	−0.709928
$816$	−56.0066	−1.96062
$817$	−34.9987	−1.22445
$818$	−2.47252	−0.0864495
$819$	12.2850	0.429274
$820$	−0.614438	−0.0214571
$821$	−41.4737	−1.44744	−0.723721	−	0.690092i	$-0.757570\pi$
$821$	−41.4737	−1.44744	−0.723721	+	0.690092i	$0.757570\pi$
$822$	19.7665	0.689435
$823$	−34.0282	−1.18615	−0.593075	−	0.805147i	$-0.702086\pi$
$823$	−34.0282	−1.18615	−0.593075	+	0.805147i	$0.702086\pi$
$824$	12.9770	0.452076
$825$	0	0
$826$	38.3672	1.33496
$827$	52.0180	1.80884	0.904422	−	0.426639i	$-0.140303\pi$
$827$	52.0180	1.80884	0.904422	+	0.426639i	$0.140303\pi$
$828$	0.177583	0.00617142
$829$	−22.4678	−0.780338	−0.390169	−	0.920743i	$-0.627583\pi$
$829$	−22.4678	−0.780338	−0.390169	+	0.920743i	$0.627583\pi$
$830$	−29.8332	−1.03553
$831$	−26.1417	−0.906844
$832$	−32.5740	−1.12930
$833$	37.2152	1.28943
$834$	42.1629	1.45998
$835$	−6.88377	−0.238223
$836$	0	0
$837$	−12.2845	−0.424613
$838$	8.75606	0.302473
$839$	27.5894	0.952490	0.476245	−	0.879313i	$-0.341998\pi$
$839$	27.5894	0.952490	0.476245	+	0.879313i	$0.341998\pi$
$840$	−40.7687	−1.40665
$841$	36.4533	1.25701
$842$	−56.4626	−1.94583
$843$	36.3002	1.25025
$844$	0.0230372	0.000792974 0
$845$	−10.3203	−0.355028
$846$	−15.6389	−0.537675
$847$	0	0
$848$	16.0254	0.550313
$849$	21.1957	0.727436
$850$	−4.51083	−0.154720
$851$	25.9223	0.888607
$852$	−0.391394	−0.0134089
$853$	10.1059	0.346020	0.173010	−	0.984920i	$-0.444651\pi$
$853$	10.1059	0.346020	0.173010	+	0.984920i	$0.444651\pi$
$854$	−5.05695	−0.173045
$855$	7.72332	0.264132
$856$	−30.9473	−1.05776
$857$	−40.7260	−1.39117	−0.695586	−	0.718443i	$-0.744855\pi$
$857$	−40.7260	−1.39117	−0.695586	+	0.718443i	$0.744855\pi$
$858$	0	0
$859$	−12.1909	−0.415948	−0.207974	−	0.978134i	$-0.566687\pi$
$859$	−12.1909	−0.415948	−0.207974	+	0.978134i	$0.566687\pi$
$860$	−1.14984	−0.0392094
$861$	−29.3821	−1.00134
$862$	−14.0451	−0.478377
$863$	35.5364	1.20967	0.604836	−	0.796350i	$-0.293239\pi$
$863$	35.5364	1.20967	0.604836	+	0.796350i	$0.293239\pi$
$864$	1.62178	0.0551739
$865$	28.6338	0.973579
$866$	5.75648	0.195613
$867$	60.7299	2.06250
$868$	−0.685728	−0.0232751
$869$	0	0
$870$	−48.5285	−1.64527
$871$	−21.4667	−0.727370
$872$	−42.5060	−1.43944
$873$	−1.64674	−0.0557338
$874$	−20.0401	−0.677865
$875$	−40.8935	−1.38245
$876$	−1.01733	−0.0343722
$877$	53.6382	1.81123	0.905617	−	0.424097i	$-0.139408\pi$
$877$	53.6382	1.81123	0.905617	+	0.424097i	$0.139408\pi$
$878$	24.7569	0.835506
$879$	−53.2163	−1.79494
$880$	0	0
$881$	24.0000	0.808580	0.404290	−	0.914631i	$-0.367519\pi$
$881$	24.0000	0.808580	0.404290	+	0.914631i	$0.367519\pi$
$882$	6.38465	0.214982
$883$	15.7500	0.530028	0.265014	−	0.964245i	$-0.414623\pi$
$883$	15.7500	0.530028	0.265014	+	0.964245i	$0.414623\pi$
$884$	−1.97526	−0.0664350
$885$	31.6506	1.06392
$886$	−32.9941	−1.10846
$887$	−42.8840	−1.43991	−0.719953	−	0.694023i	$-0.755836\pi$
$887$	−42.8840	−1.43991	−0.719953	+	0.694023i	$0.755836\pi$
$888$	−44.2770	−1.48584
$889$	17.6728	0.592728
$890$	−4.56501	−0.153020
$891$	0	0
$892$	−1.25422	−0.0419943
$893$	57.5940	1.92731
$894$	51.6116	1.72615
$895$	−26.1196	−0.873082
$896$	−41.6838	−1.39256
$897$	−26.2963	−0.878008
$898$	−18.1186	−0.604627
$899$	23.3795	0.779751
$900$	−0.0252550	−0.000841833 0
$901$	−26.8927	−0.895926
$902$	0	0
$903$	−54.9850	−1.82979
$904$	−49.1977	−1.63629
$905$	40.7021	1.35298
$906$	45.5028	1.51173
$907$	−6.28293	−0.208621	−0.104311	−	0.994545i	$-0.533264\pi$
$907$	−6.28293	−0.208621	−0.104311	+	0.994545i	$0.533264\pi$
$908$	−1.96457	−0.0651964
$909$	−1.78692	−0.0592682
$910$	−45.5473	−1.50988
$911$	−51.9322	−1.72059	−0.860295	−	0.509797i	$-0.829720\pi$
$911$	−51.9322	−1.72059	−0.860295	+	0.509797i	$0.829720\pi$
$912$	35.3858	1.17174
$913$	0	0
$914$	36.2862	1.20024
$915$	−4.17169	−0.137912
$916$	−0.687825	−0.0227264
$917$	21.6471	0.714850
$918$	−42.3666	−1.39831
$919$	13.6031	0.448725	0.224363	−	0.974506i	$-0.427970\pi$
$919$	13.6031	0.448725	0.224363	+	0.974506i	$0.427970\pi$
$920$	18.8582	0.621735
$921$	46.6472	1.53708
$922$	−53.8422	−1.77320
$923$	12.5246	0.412252
$924$	0	0
$925$	−3.68656	−0.121213
$926$	42.2907	1.38976
$927$	−3.86229	−0.126854
$928$	−3.08653	−0.101320
$929$	15.6355	0.512986	0.256493	−	0.966546i	$-0.417433\pi$
$929$	15.6355	0.512986	0.256493	+	0.966546i	$0.417433\pi$
$930$	−17.3341	−0.568408
$931$	−23.5131	−0.770610
$932$	−1.63197	−0.0534571
$933$	−14.6704	−0.480286
$934$	−10.6871	−0.349693
$935$	0	0
$936$	9.70630	0.317260
$937$	21.3924	0.698861	0.349430	−	0.936962i	$-0.386375\pi$
$937$	21.3924	0.698861	0.349430	+	0.936962i	$0.386375\pi$
$938$	−25.7016	−0.839188
$939$	63.8150	2.08252
$940$	1.89219	0.0617164
$941$	6.38421	0.208119	0.104060	−	0.994571i	$-0.466817\pi$
$941$	6.38421	0.208119	0.104060	+	0.994571i	$0.466817\pi$
$942$	−17.2442	−0.561847
$943$	13.5911	0.442588
$944$	31.3373	1.01994
$945$	−31.8814	−1.03710
$946$	0	0
$947$	−29.7062	−0.965323	−0.482662	−	0.875807i	$-0.660330\pi$
$947$	−29.7062	−0.965323	−0.482662	+	0.875807i	$0.660330\pi$
$948$	0.545189	0.0177069
$949$	32.5544	1.05676
$950$	2.85001	0.0924665
$951$	−63.1641	−2.04824
$952$	67.7381	2.19540
$953$	41.5947	1.34739	0.673693	−	0.739011i	$-0.264707\pi$
$953$	41.5947	1.34739	0.673693	+	0.739011i	$0.264707\pi$
$954$	−4.61373	−0.149375
$955$	−34.1753	−1.10589
$956$	−1.19297	−0.0385835
$957$	0	0
$958$	−6.71877	−0.217074
$959$	−24.7143	−0.798066
$960$	−32.1729	−1.03838
$961$	−22.6490	−0.730612
$962$	−49.4669	−1.59488
$963$	9.21069	0.296810
$964$	0.301137	0.00969899
$965$	10.5841	0.340714
$966$	−31.4841	−1.01298
$967$	18.7810	0.603957	0.301978	−	0.953315i	$-0.402353\pi$
$967$	18.7810	0.603957	0.301978	+	0.953315i	$0.402353\pi$
$968$	0	0
$969$	−59.3821	−1.90763
$970$	6.10536	0.196031
$971$	−22.9370	−0.736084	−0.368042	−	0.929809i	$-0.619972\pi$
$971$	−22.9370	−0.736084	−0.368042	+	0.929809i	$0.619972\pi$
$972$	−0.564676	−0.0181120
$973$	−52.7169	−1.69003
$974$	0.00538495	0.000172545 0
$975$	3.73974	0.119768
$976$	−4.13039	−0.132211
$977$	20.4952	0.655700	0.327850	−	0.944730i	$-0.393676\pi$
$977$	20.4952	0.655700	0.327850	+	0.944730i	$0.393676\pi$
$978$	26.7338	0.854854
$979$	0	0
$980$	−0.772496	−0.0246765
$981$	12.6509	0.403911
$982$	36.1312	1.15299
$983$	27.4965	0.877001	0.438500	−	0.898731i	$-0.355510\pi$
$983$	27.4965	0.877001	0.438500	+	0.898731i	$0.355510\pi$
$984$	−23.2145	−0.740052
$985$	−3.75849	−0.119756
$986$	80.6312	2.56782
$987$	90.4835	2.88012
$988$	1.24799	0.0397040
$989$	25.4341	0.808758
$990$	0	0
$991$	−24.7381	−0.785833	−0.392916	−	0.919574i	$-0.628534\pi$
$991$	−24.7381	−0.785833	−0.392916	+	0.919574i	$0.628534\pi$
$992$	−1.10249	−0.0350041
$993$	33.8629	1.07461
$994$	14.9955	0.475628
$995$	−43.1095	−1.36666
$996$	1.28423	0.0406923
$997$	−23.6738	−0.749755	−0.374878	−	0.927074i	$-0.622315\pi$
$997$	−23.6738	−0.749755	−0.374878	+	0.927074i	$0.622315\pi$
$998$	47.8288	1.51399
$999$	−34.6249	−1.09548

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7381.2.a.r.1.35	yes	50
11.10	odd	2		7381.2.a.q.1.16	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7381.2.a.q.1.16	✓	50	11.10	odd	2
7381.2.a.r.1.35	yes	50	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7381 = 11^{2} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7381.a (trivial)

\(f(q)\)	\(=\)	\(q+1.43787 q^{2} +1.95628 q^{3} +0.0674703 q^{4} -2.13246 q^{5} +2.81287 q^{6} -3.51697 q^{7} -2.77873 q^{8} +0.827019 q^{9} +O(q^{10})\)	\(q+1.43787 q^{2} +1.95628 q^{3} +0.0674703 q^{4} -2.13246 q^{5} +2.81287 q^{6} -3.51697 q^{7} -2.77873 q^{8} +0.827019 q^{9} -3.06620 q^{10} +0.131991 q^{12} -4.22369 q^{13} -5.05695 q^{14} -4.17169 q^{15} -4.13039 q^{16} +6.93135 q^{17} +1.18915 q^{18} -4.37933 q^{19} -0.143878 q^{20} -6.88017 q^{21} +3.18253 q^{23} -5.43596 q^{24} -0.452605 q^{25} -6.07312 q^{26} -4.25095 q^{27} -0.237291 q^{28} +8.09032 q^{29} -5.99834 q^{30} +2.88982 q^{31} -0.381509 q^{32} +9.96638 q^{34} +7.49981 q^{35} +0.0557992 q^{36} +8.14521 q^{37} -6.29690 q^{38} -8.26272 q^{39} +5.92553 q^{40} +4.27055 q^{41} -9.89280 q^{42} +7.99180 q^{43} -1.76359 q^{45} +4.57606 q^{46} -13.1513 q^{47} -8.08018 q^{48} +5.36911 q^{49} -0.650787 q^{50} +13.5596 q^{51} -0.284974 q^{52} -3.87987 q^{53} -6.11232 q^{54} +9.77271 q^{56} -8.56718 q^{57} +11.6328 q^{58} -7.58701 q^{59} -0.281465 q^{60} +1.00000 q^{61} +4.15518 q^{62} -2.90860 q^{63} +7.71222 q^{64} +9.00687 q^{65} +5.08244 q^{67} +0.467661 q^{68} +6.22590 q^{69} +10.7838 q^{70} -2.96532 q^{71} -2.29806 q^{72} -7.70755 q^{73} +11.7117 q^{74} -0.885420 q^{75} -0.295475 q^{76} -11.8807 q^{78} +4.13051 q^{79} +8.80790 q^{80} -10.7971 q^{81} +6.14050 q^{82} +9.72969 q^{83} -0.464208 q^{84} -14.7808 q^{85} +11.4912 q^{86} +15.8269 q^{87} +1.48882 q^{89} -2.53581 q^{90} +14.8546 q^{91} +0.214726 q^{92} +5.65328 q^{93} -18.9099 q^{94} +9.33875 q^{95} -0.746337 q^{96} -1.99118 q^{97} +7.72008 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 10 q^{2} - 2 q^{3} + 50 q^{4} - 2 q^{5} + 12 q^{6} + 8 q^{7} + 30 q^{8} + 48 q^{9}+O(q^{10}) \) 50 * q + 10 * q^2 - 2 * q^3 + 50 * q^4 - 2 * q^5 + 12 * q^6 + 8 * q^7 + 30 * q^8 + 48 * q^9	\( 50 q + 10 q^{2} - 2 q^{3} + 50 q^{4} - 2 q^{5} + 12 q^{6} + 8 q^{7} + 30 q^{8} + 48 q^{9} + 12 q^{10} - 14 q^{12} + 8 q^{13} - 2 q^{14} - 16 q^{15} + 42 q^{16} + 22 q^{17} + 32 q^{19} - 8 q^{20} + 24 q^{21} + 12 q^{23} - 8 q^{24} + 40 q^{25} + 10 q^{26} - 8 q^{27} + 72 q^{28} + 56 q^{29} + 24 q^{30} + 10 q^{31} + 70 q^{32} - 32 q^{34} + 70 q^{35} + 34 q^{36} - 8 q^{37} - 14 q^{38} + 96 q^{39} - 54 q^{40} + 56 q^{41} - 8 q^{42} + 44 q^{43} - 24 q^{45} - 4 q^{46} - 4 q^{47} - 28 q^{48} + 38 q^{49} + 120 q^{50} + 76 q^{51} + 24 q^{52} + 4 q^{53} + 48 q^{54} - 18 q^{56} + 8 q^{57} + 28 q^{58} + 12 q^{59} - 60 q^{60} + 50 q^{61} + 8 q^{62} + 30 q^{63} + 10 q^{64} + 64 q^{65} + 18 q^{67} - 22 q^{68} - 8 q^{69} + 34 q^{70} + 12 q^{71} + 104 q^{72} - 16 q^{73} + 84 q^{74} - 26 q^{75} + 64 q^{76} + 40 q^{78} + 78 q^{79} - 36 q^{80} + 34 q^{81} + 54 q^{82} + 68 q^{83} - 78 q^{84} - 4 q^{85} + 36 q^{86} + 48 q^{87} + 26 q^{89} - 20 q^{90} + 32 q^{92} + 22 q^{93} + 156 q^{94} + 100 q^{95} - 4 q^{96} - 14 q^{97} + 70 q^{98}+O(q^{100}) \) 50 * q + 10 * q^2 - 2 * q^3 + 50 * q^4 - 2 * q^5 + 12 * q^6 + 8 * q^7 + 30 * q^8 + 48 * q^9 + 12 * q^10 - 14 * q^12 + 8 * q^13 - 2 * q^14 - 16 * q^15 + 42 * q^16 + 22 * q^17 + 32 * q^19 - 8 * q^20 + 24 * q^21 + 12 * q^23 - 8 * q^24 + 40 * q^25 + 10 * q^26 - 8 * q^27 + 72 * q^28 + 56 * q^29 + 24 * q^30 + 10 * q^31 + 70 * q^32 - 32 * q^34 + 70 * q^35 + 34 * q^36 - 8 * q^37 - 14 * q^38 + 96 * q^39 - 54 * q^40 + 56 * q^41 - 8 * q^42 + 44 * q^43 - 24 * q^45 - 4 * q^46 - 4 * q^47 - 28 * q^48 + 38 * q^49 + 120 * q^50 + 76 * q^51 + 24 * q^52 + 4 * q^53 + 48 * q^54 - 18 * q^56 + 8 * q^57 + 28 * q^58 + 12 * q^59 - 60 * q^60 + 50 * q^61 + 8 * q^62 + 30 * q^63 + 10 * q^64 + 64 * q^65 + 18 * q^67 - 22 * q^68 - 8 * q^69 + 34 * q^70 + 12 * q^71 + 104 * q^72 - 16 * q^73 + 84 * q^74 - 26 * q^75 + 64 * q^76 + 40 * q^78 + 78 * q^79 - 36 * q^80 + 34 * q^81 + 54 * q^82 + 68 * q^83 - 78 * q^84 - 4 * q^85 + 36 * q^86 + 48 * q^87 + 26 * q^89 - 20 * q^90 + 32 * q^92 + 22 * q^93 + 156 * q^94 + 100 * q^95 - 4 * q^96 - 14 * q^97 + 70 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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