LMFDB - Embedded newform 7381.2.a.u.1.22

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:	$64$
Twist minimal:	no (minimal twist has level 671)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.22
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)

$f(q)$	$=$	$q-1.24293 q^{2} -2.95954 q^{3} -0.455124 q^{4} +2.31409 q^{5} +3.67850 q^{6} -2.50119 q^{7} +3.05155 q^{8} +5.75889 q^{9} +O(q^{10})$	\(q-1.24293 q^{2} -2.95954 q^{3} -0.455124 q^{4} +2.31409 q^{5} +3.67850 q^{6} -2.50119 q^{7} +3.05155 q^{8} +5.75889 q^{9} -2.87626 q^{10} +1.34696 q^{12} -4.51421 q^{13} +3.10881 q^{14} -6.84866 q^{15} -2.88261 q^{16} -1.92814 q^{17} -7.15790 q^{18} -7.30384 q^{19} -1.05320 q^{20} +7.40239 q^{21} +4.71783 q^{23} -9.03119 q^{24} +0.355028 q^{25} +5.61085 q^{26} -8.16505 q^{27} +1.13835 q^{28} -5.82388 q^{29} +8.51240 q^{30} -1.06977 q^{31} -2.52021 q^{32} +2.39655 q^{34} -5.78800 q^{35} -2.62101 q^{36} -2.79133 q^{37} +9.07816 q^{38} +13.3600 q^{39} +7.06157 q^{40} -8.69094 q^{41} -9.20065 q^{42} -4.66444 q^{43} +13.3266 q^{45} -5.86393 q^{46} -1.70551 q^{47} +8.53122 q^{48} -0.744029 q^{49} -0.441275 q^{50} +5.70642 q^{51} +2.05453 q^{52} -12.0575 q^{53} +10.1486 q^{54} -7.63251 q^{56} +21.6160 q^{57} +7.23868 q^{58} +6.98634 q^{59} +3.11699 q^{60} -1.00000 q^{61} +1.32965 q^{62} -14.4041 q^{63} +8.89767 q^{64} -10.4463 q^{65} -5.99579 q^{67} +0.877545 q^{68} -13.9626 q^{69} +7.19408 q^{70} +13.7942 q^{71} +17.5735 q^{72} -6.94426 q^{73} +3.46943 q^{74} -1.05072 q^{75} +3.32415 q^{76} -16.6056 q^{78} +10.3483 q^{79} -6.67064 q^{80} +6.88815 q^{81} +10.8022 q^{82} +1.00082 q^{83} -3.36901 q^{84} -4.46191 q^{85} +5.79757 q^{86} +17.2360 q^{87} -6.01134 q^{89} -16.5640 q^{90} +11.2909 q^{91} -2.14720 q^{92} +3.16604 q^{93} +2.11983 q^{94} -16.9018 q^{95} +7.45866 q^{96} -5.25239 q^{97} +0.924776 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 64 q - 5 q^{2} + 19 q^{3} + 73 q^{4} + 21 q^{5} + q^{6} - 6 q^{7} - 3 q^{8} + 81 q^{9}+O(q^{10}) $ 64 * q - 5 * q^2 + 19 * q^3 + 73 * q^4 + 21 * q^5 + q^6 - 6 * q^7 - 3 * q^8 + 81 * q^9	\( 64 q - 5 q^{2} + 19 q^{3} + 73 q^{4} + 21 q^{5} + q^{6} - 6 q^{7} - 3 q^{8} + 81 q^{9} + 4 q^{10} + 41 q^{12} + q^{13} + 41 q^{14} + 28 q^{15} + 107 q^{16} - 13 q^{17} - 38 q^{18} + q^{19} + 65 q^{20} - q^{21} + 52 q^{23} - 3 q^{24} + 87 q^{25} + 37 q^{26} + 88 q^{27} + q^{28} - 19 q^{29} + 19 q^{30} + 45 q^{31} + 24 q^{32} - 23 q^{34} - 10 q^{35} + 146 q^{36} + 26 q^{37} - 4 q^{38} + 6 q^{39} - 84 q^{40} + 12 q^{41} + 28 q^{42} - 5 q^{43} + 71 q^{45} + 5 q^{46} + 63 q^{47} + 85 q^{48} + 78 q^{49} - 14 q^{50} - 22 q^{51} + 24 q^{52} + 86 q^{53} + 114 q^{54} + 119 q^{56} + 29 q^{57} + q^{58} + 119 q^{59} - 22 q^{60} - 64 q^{61} - 13 q^{62} + 28 q^{63} + 135 q^{64} + 30 q^{65} + 2 q^{67} - 59 q^{68} + 63 q^{69} - 2 q^{70} + 126 q^{71} - 48 q^{72} - 8 q^{73} + 27 q^{74} + 107 q^{75} + 82 q^{76} - 13 q^{78} + 5 q^{79} + 119 q^{80} + 96 q^{81} - 27 q^{82} - 14 q^{83} - 182 q^{84} + 52 q^{85} + 60 q^{86} + 8 q^{87} + 59 q^{89} + 45 q^{90} + 83 q^{91} + 111 q^{92} - 7 q^{93} - 21 q^{94} + 26 q^{95} + 86 q^{96} - 39 q^{97} + 57 q^{98}+O(q^{100}) \) 64 * q - 5 * q^2 + 19 * q^3 + 73 * q^4 + 21 * q^5 + q^6 - 6 * q^7 - 3 * q^8 + 81 * q^9 + 4 * q^10 + 41 * q^12 + q^13 + 41 * q^14 + 28 * q^15 + 107 * q^16 - 13 * q^17 - 38 * q^18 + q^19 + 65 * q^20 - q^21 + 52 * q^23 - 3 * q^24 + 87 * q^25 + 37 * q^26 + 88 * q^27 + q^28 - 19 * q^29 + 19 * q^30 + 45 * q^31 + 24 * q^32 - 23 * q^34 - 10 * q^35 + 146 * q^36 + 26 * q^37 - 4 * q^38 + 6 * q^39 - 84 * q^40 + 12 * q^41 + 28 * q^42 - 5 * q^43 + 71 * q^45 + 5 * q^46 + 63 * q^47 + 85 * q^48 + 78 * q^49 - 14 * q^50 - 22 * q^51 + 24 * q^52 + 86 * q^53 + 114 * q^54 + 119 * q^56 + 29 * q^57 + q^58 + 119 * q^59 - 22 * q^60 - 64 * q^61 - 13 * q^62 + 28 * q^63 + 135 * q^64 + 30 * q^65 + 2 * q^67 - 59 * q^68 + 63 * q^69 - 2 * q^70 + 126 * q^71 - 48 * q^72 - 8 * q^73 + 27 * q^74 + 107 * q^75 + 82 * q^76 - 13 * q^78 + 5 * q^79 + 119 * q^80 + 96 * q^81 - 27 * q^82 - 14 * q^83 - 182 * q^84 + 52 * q^85 + 60 * q^86 + 8 * q^87 + 59 * q^89 + 45 * q^90 + 83 * q^91 + 111 * q^92 - 7 * q^93 - 21 * q^94 + 26 * q^95 + 86 * q^96 - 39 * q^97 + 57 * 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 $

$2$	−1.24293	−0.878885	−0.439442	−	0.898271i	$-0.644824\pi$
$2$	−1.24293	−0.878885	−0.439442	+	0.898271i	$0.644824\pi$
$3$	−2.95954	−1.70869	−0.854346	−	0.519704i	$-0.826042\pi$
$3$	−2.95954	−1.70869	−0.854346	+	0.519704i	$0.826042\pi$
$4$	−0.455124	−0.227562
$5$	2.31409	1.03489	0.517447	−	0.855715i	$-0.326882\pi$
$5$	2.31409	1.03489	0.517447	+	0.855715i	$0.326882\pi$
$6$	3.67850	1.50174
$7$	−2.50119	−0.945362	−0.472681	−	0.881233i	$-0.656714\pi$
$7$	−2.50119	−0.945362	−0.472681	+	0.881233i	$0.656714\pi$
$8$	3.05155	1.07889
$9$	5.75889	1.91963
$10$	−2.87626	−0.909552
$11$	0	0
$11$	0	0
$12$	1.34696	0.388834
$13$	−4.51421	−1.25202	−0.626009	−	0.779816i	$-0.715313\pi$
$13$	−4.51421	−1.25202	−0.626009	+	0.779816i	$0.715313\pi$
$14$	3.10881	0.830864
$15$	−6.84866	−1.76832
$16$	−2.88261	−0.720653
$17$	−1.92814	−0.467644	−0.233822	−	0.972279i	$-0.575123\pi$
$17$	−1.92814	−0.467644	−0.233822	+	0.972279i	$0.575123\pi$
$18$	−7.15790	−1.68713
$19$	−7.30384	−1.67562	−0.837808	−	0.545965i	$-0.816163\pi$
$19$	−7.30384	−1.67562	−0.837808	+	0.545965i	$0.816163\pi$
$20$	−1.05320	−0.235503
$21$	7.40239	1.61533
$22$	0	0
$23$	4.71783	0.983735	0.491867	−	0.870670i	$-0.336314\pi$
$23$	4.71783	0.983735	0.491867	+	0.870670i	$0.336314\pi$
$24$	−9.03119	−1.84348
$25$	0.355028	0.0710056
$26$	5.61085	1.10038
$27$	−8.16505	−1.57137
$28$	1.13835	0.215129
$29$	−5.82388	−1.08147	−0.540734	−	0.841194i	$-0.681853\pi$
$29$	−5.82388	−1.08147	−0.540734	+	0.841194i	$0.681853\pi$
$30$	8.51240	1.55415
$31$	−1.06977	−0.192137	−0.0960685	−	0.995375i	$-0.530627\pi$
$31$	−1.06977	−0.192137	−0.0960685	+	0.995375i	$0.530627\pi$
$32$	−2.52021	−0.445514
$33$	0	0
$34$	2.39655	0.411005
$35$	−5.78800	−0.978350
$36$	−2.62101	−0.436835
$37$	−2.79133	−0.458892	−0.229446	−	0.973321i	$-0.573691\pi$
$37$	−2.79133	−0.458892	−0.229446	+	0.973321i	$0.573691\pi$
$38$	9.07816	1.47267
$39$	13.3600	2.13931
$40$	7.06157	1.11653
$41$	−8.69094	−1.35730	−0.678648	−	0.734463i	$-0.737434\pi$
$41$	−8.69094	−1.35730	−0.678648	+	0.734463i	$0.737434\pi$
$42$	−9.20065	−1.41969
$43$	−4.66444	−0.711320	−0.355660	−	0.934615i	$-0.615744\pi$
$43$	−4.66444	−0.711320	−0.355660	+	0.934615i	$0.615744\pi$
$44$	0	0
$45$	13.3266	1.98661
$46$	−5.86393	−0.864589
$47$	−1.70551	−0.248775	−0.124387	−	0.992234i	$-0.539697\pi$
$47$	−1.70551	−0.248775	−0.124387	+	0.992234i	$0.539697\pi$
$48$	8.53122	1.23138
$49$	−0.744029	−0.106290
$50$	−0.441275	−0.0624058
$51$	5.70642	0.799059
$52$	2.05453	0.284912
$53$	−12.0575	−1.65622	−0.828110	−	0.560566i	$-0.810584\pi$
$53$	−12.0575	−1.65622	−0.828110	+	0.560566i	$0.810584\pi$
$54$	10.1486	1.38105
$55$	0	0
$56$	−7.63251	−1.01994
$57$	21.6160	2.86311
$58$	7.23868	0.950485
$59$	6.98634	0.909544	0.454772	−	0.890608i	$-0.349721\pi$
$59$	6.98634	0.909544	0.454772	+	0.890608i	$0.349721\pi$
$60$	3.11699	0.402402
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	1.32965	0.168866
$63$	−14.4041	−1.81475
$64$	8.89767	1.11221
$65$	−10.4463	−1.29571
$66$	0	0
$67$	−5.99579	−0.732502	−0.366251	−	0.930516i	$-0.619359\pi$
$67$	−5.99579	−0.732502	−0.366251	+	0.930516i	$0.619359\pi$
$68$	0.877545	0.106418
$69$	−13.9626	−1.68090
$70$	7.19408	0.859857
$71$	13.7942	1.63707	0.818535	−	0.574457i	$-0.194787\pi$
$71$	13.7942	1.63707	0.818535	+	0.574457i	$0.194787\pi$
$72$	17.5735	2.07106
$73$	−6.94426	−0.812764	−0.406382	−	0.913703i	$-0.633210\pi$
$73$	−6.94426	−0.812764	−0.406382	+	0.913703i	$0.633210\pi$
$74$	3.46943	0.403313
$75$	−1.05072	−0.121327
$76$	3.32415	0.381306
$77$	0	0
$78$	−16.6056	−1.88021
$79$	10.3483	1.16428	0.582140	−	0.813089i	$-0.302216\pi$
$79$	10.3483	1.16428	0.582140	+	0.813089i	$0.302216\pi$
$80$	−6.67064	−0.745800
$81$	6.88815	0.765350
$82$	10.8022	1.19291
$83$	1.00082	0.109855	0.0549274	−	0.998490i	$-0.482507\pi$
$83$	1.00082	0.109855	0.0549274	+	0.998490i	$0.482507\pi$
$84$	−3.36901	−0.367589
$85$	−4.46191	−0.483962
$86$	5.79757	0.625168
$87$	17.2360	1.84789
$88$	0	0
$89$	−6.01134	−0.637201	−0.318601	−	0.947889i	$-0.603213\pi$
$89$	−6.01134	−0.637201	−0.318601	+	0.947889i	$0.603213\pi$
$90$	−16.5640	−1.74600
$91$	11.2909	1.18361
$92$	−2.14720	−0.223861
$93$	3.16604	0.328303
$94$	2.11983	0.218644
$95$	−16.9018	−1.73408
$96$	7.45866	0.761247
$97$	−5.25239	−0.533299	−0.266650	−	0.963794i	$-0.585917\pi$
$97$	−5.25239	−0.533299	−0.266650	+	0.963794i	$0.585917\pi$
$98$	0.924776	0.0934165
$99$	0	0
$100$	−0.161582	−0.0161582
$101$	5.33552	0.530904	0.265452	−	0.964124i	$-0.414479\pi$
$101$	5.33552	0.530904	0.265452	+	0.964124i	$0.414479\pi$
$102$	−7.09269	−0.702281
$103$	10.0740	0.992625	0.496312	−	0.868144i	$-0.334687\pi$
$103$	10.0740	0.992625	0.496312	+	0.868144i	$0.334687\pi$
$104$	−13.7753	−1.35078
$105$	17.1298	1.67170
$106$	14.9866	1.45563
$107$	15.4932	1.49778	0.748890	−	0.662694i	$-0.230587\pi$
$107$	15.4932	1.49778	0.748890	+	0.662694i	$0.230587\pi$
$108$	3.71611	0.357583
$109$	2.14492	0.205447	0.102723	−	0.994710i	$-0.467244\pi$
$109$	2.14492	0.205447	0.102723	+	0.994710i	$0.467244\pi$
$110$	0	0
$111$	8.26106	0.784105
$112$	7.20998	0.681279
$113$	−12.8805	−1.21169	−0.605847	−	0.795581i	$-0.707166\pi$
$113$	−12.8805	−1.21169	−0.605847	+	0.795581i	$0.707166\pi$
$114$	−26.8672	−2.51634
$115$	10.9175	1.01806
$116$	2.65059	0.246101
$117$	−25.9969	−2.40341
$118$	−8.68353	−0.799384
$119$	4.82266	0.442093
$120$	−20.8990	−1.90781
$121$	0	0
$122$	1.24293	0.112530
$123$	25.7212	2.31920
$124$	0.486880	0.0437231
$125$	−10.7489	−0.961411
$126$	17.9033	1.59495
$127$	−14.3800	−1.27602	−0.638008	−	0.770030i	$-0.720241\pi$
$127$	−14.3800	−1.27602	−0.638008	+	0.770030i	$0.720241\pi$
$128$	−6.01877	−0.531989
$129$	13.8046	1.21543
$130$	12.9840	1.13878
$131$	−15.7183	−1.37331	−0.686656	−	0.726982i	$-0.740922\pi$
$131$	−15.7183	−1.37331	−0.686656	+	0.726982i	$0.740922\pi$
$132$	0	0
$133$	18.2683	1.58406
$134$	7.45234	0.643785
$135$	−18.8947	−1.62620
$136$	−5.88382	−0.504534
$137$	−23.2193	−1.98376	−0.991878	−	0.127197i	$-0.959402\pi$
$137$	−23.2193	−1.98376	−0.991878	+	0.127197i	$0.959402\pi$
$138$	17.3545	1.47732
$139$	−0.543994	−0.0461410	−0.0230705	−	0.999734i	$-0.507344\pi$
$139$	−0.543994	−0.0461410	−0.0230705	+	0.999734i	$0.507344\pi$
$140$	2.63426	0.222635
$141$	5.04754	0.425079
$142$	−17.1452	−1.43880
$143$	0	0
$144$	−16.6007	−1.38339
$145$	−13.4770	−1.11920
$146$	8.63123	0.714326
$147$	2.20198	0.181617
$148$	1.27040	0.104426
$149$	10.5862	0.867259	0.433629	−	0.901091i	$-0.357233\pi$
$149$	10.5862	0.867259	0.433629	+	0.901091i	$0.357233\pi$
$150$	1.30597	0.106632
$151$	−11.4731	−0.933670	−0.466835	−	0.884344i	$-0.654606\pi$
$151$	−11.4731	−0.933670	−0.466835	+	0.884344i	$0.654606\pi$
$152$	−22.2880	−1.80780
$153$	−11.1040	−0.897703
$154$	0	0
$155$	−2.47556	−0.198841
$156$	−6.08046	−0.486826
$157$	5.66928	0.452458	0.226229	−	0.974074i	$-0.427360\pi$
$157$	5.66928	0.452458	0.226229	+	0.974074i	$0.427360\pi$
$158$	−12.8623	−1.02327
$159$	35.6846	2.82997
$160$	−5.83200	−0.461060
$161$	−11.8002	−0.929986
$162$	−8.56149	−0.672654
$163$	−10.5033	−0.822678	−0.411339	−	0.911482i	$-0.634939\pi$
$163$	−10.5033	−0.822678	−0.411339	+	0.911482i	$0.634939\pi$
$164$	3.95545	0.308869
$165$	0	0
$166$	−1.24396	−0.0965497
$167$	−17.6331	−1.36449	−0.682244	−	0.731125i	$-0.738996\pi$
$167$	−17.6331	−1.36449	−0.682244	+	0.731125i	$0.738996\pi$
$168$	22.5887	1.74276
$169$	7.37813	0.567548
$170$	5.54584	0.425346
$171$	−42.0620	−3.21656
$172$	2.12290	0.161870
$173$	12.0719	0.917811	0.458906	−	0.888485i	$-0.348242\pi$
$173$	12.0719	0.917811	0.458906	+	0.888485i	$0.348242\pi$
$174$	−21.4232	−1.62409
$175$	−0.887994	−0.0671261
$176$	0	0
$177$	−20.6764	−1.55413
$178$	7.47168	0.560026
$179$	−8.60947	−0.643502	−0.321751	−	0.946824i	$-0.604271\pi$
$179$	−8.60947	−0.643502	−0.321751	+	0.946824i	$0.604271\pi$
$180$	−6.06526	−0.452078
$181$	2.44325	0.181606	0.0908028	−	0.995869i	$-0.471057\pi$
$181$	2.44325	0.181606	0.0908028	+	0.995869i	$0.471057\pi$
$182$	−14.0338	−1.04026
$183$	2.95954	0.218776
$184$	14.3967	1.06134
$185$	−6.45940	−0.474905
$186$	−3.93517	−0.288540
$187$	0	0
$188$	0.776220	0.0566116
$189$	20.4224	1.48551
$190$	21.0077	1.52406
$191$	−8.24107	−0.596303	−0.298151	−	0.954519i	$-0.596370\pi$
$191$	−8.24107	−0.596303	−0.298151	+	0.954519i	$0.596370\pi$
$192$	−26.3330	−1.90042
$193$	26.1249	1.88051	0.940256	−	0.340469i	$-0.110586\pi$
$193$	26.1249	1.88051	0.940256	+	0.340469i	$0.110586\pi$
$194$	6.52836	0.468709
$195$	30.9163	2.21396
$196$	0.338625	0.0241875
$197$	−18.9201	−1.34800	−0.674002	−	0.738729i	$-0.735426\pi$
$197$	−18.9201	−1.34800	−0.674002	+	0.738729i	$0.735426\pi$
$198$	0	0
$199$	−19.8047	−1.40392	−0.701960	−	0.712216i	$-0.747692\pi$
$199$	−19.8047	−1.40392	−0.701960	+	0.712216i	$0.747692\pi$
$200$	1.08339	0.0766069
$201$	17.7448	1.25162
$202$	−6.63168	−0.466604
$203$	14.5667	1.02238
$204$	−2.59713	−0.181836
$205$	−20.1116	−1.40466
$206$	−12.5213	−0.872402
$207$	27.1694	1.88841
$208$	13.0127	0.902271
$209$	0	0
$210$	−21.2912	−1.46923
$211$	0.139660	0.00961457	0.00480729	−	0.999988i	$-0.498470\pi$
$211$	0.139660	0.00961457	0.00480729	+	0.999988i	$0.498470\pi$
$212$	5.48764	0.376893
$213$	−40.8245	−2.79725
$214$	−19.2569	−1.31638
$215$	−10.7939	−0.736141
$216$	−24.9161	−1.69532
$217$	2.67571	0.181639
$218$	−2.66599	−0.180564
$219$	20.5518	1.38876
$220$	0	0
$221$	8.70405	0.585498
$222$	−10.2679	−0.689138
$223$	5.11414	0.342468	0.171234	−	0.985230i	$-0.445225\pi$
$223$	5.11414	0.342468	0.171234	+	0.985230i	$0.445225\pi$
$224$	6.30353	0.421172
$225$	2.04457	0.136305
$226$	16.0095	1.06494
$227$	−29.4015	−1.95144	−0.975722	−	0.219014i	$-0.929716\pi$
$227$	−29.4015	−1.95144	−0.975722	+	0.219014i	$0.929716\pi$
$228$	−9.83797	−0.651535
$229$	21.4868	1.41989	0.709945	−	0.704257i	$-0.248720\pi$
$229$	21.4868	1.41989	0.709945	+	0.704257i	$0.248720\pi$
$230$	−13.5697	−0.894758
$231$	0	0
$232$	−17.7718	−1.16678
$233$	−11.7244	−0.768089	−0.384044	−	0.923315i	$-0.625469\pi$
$233$	−11.7244	−0.768089	−0.384044	+	0.923315i	$0.625469\pi$
$234$	32.3123	2.11232
$235$	−3.94671	−0.257455
$236$	−3.17965	−0.206978
$237$	−30.6264	−1.98940
$238$	−5.99423	−0.388548
$239$	−20.4575	−1.32328	−0.661642	−	0.749820i	$-0.730140\pi$
$239$	−20.4575	−1.32328	−0.661642	+	0.749820i	$0.730140\pi$
$240$	19.7420	1.27434
$241$	−30.6700	−1.97563	−0.987814	−	0.155639i	$-0.950256\pi$
$241$	−30.6700	−1.97563	−0.987814	+	0.155639i	$0.950256\pi$
$242$	0	0
$243$	4.10940	0.263618
$244$	0.455124	0.0291363
$245$	−1.72175	−0.109999
$246$	−31.9697	−2.03831
$247$	32.9711	2.09790
$248$	−3.26446	−0.207294
$249$	−2.96198	−0.187708
$250$	13.3601	0.844969
$251$	0.0140463	0.000886593 0	0.000443296	−	1.00000i	$-0.499859\pi$
$251$	0.0140463	0.000886593 0	0.000443296		1.00000i	$0.499859\pi$
$252$	6.55565	0.412967
$253$	0	0
$254$	17.8733	1.12147
$255$	13.2052	0.826942
$256$	−10.3144	−0.644652
$257$	−13.5896	−0.847698	−0.423849	−	0.905733i	$-0.639321\pi$
$257$	−13.5896	−0.847698	−0.423849	+	0.905733i	$0.639321\pi$
$258$	−17.1582	−1.06822
$259$	6.98166	0.433819
$260$	4.75437	0.294853
$261$	−33.5391	−2.07602
$262$	19.5367	1.20698
$263$	−21.0396	−1.29735	−0.648677	−	0.761064i	$-0.724677\pi$
$263$	−21.0396	−1.29735	−0.648677	+	0.761064i	$0.724677\pi$
$264$	0	0
$265$	−27.9021	−1.71401
$266$	−22.7062	−1.39221
$267$	17.7908	1.08878
$268$	2.72883	0.166690
$269$	−1.08198	−0.0659697	−0.0329848	−	0.999456i	$-0.510501\pi$
$269$	−1.08198	−0.0659697	−0.0329848	+	0.999456i	$0.510501\pi$
$270$	23.4848	1.42924
$271$	17.6757	1.07372	0.536861	−	0.843671i	$-0.319610\pi$
$271$	17.6757	1.07372	0.536861	+	0.843671i	$0.319610\pi$
$272$	5.55809	0.337009
$273$	−33.4160	−2.02243
$274$	28.8599	1.74349
$275$	0	0
$276$	6.35472	0.382509
$277$	6.29646	0.378318	0.189159	−	0.981947i	$-0.439424\pi$
$277$	6.29646	0.378318	0.189159	+	0.981947i	$0.439424\pi$
$278$	0.676147	0.0405526
$279$	−6.16071	−0.368832
$280$	−17.6624	−1.05553
$281$	−3.69422	−0.220379	−0.110189	−	0.993911i	$-0.535146\pi$
$281$	−3.69422	−0.220379	−0.110189	+	0.993911i	$0.535146\pi$
$282$	−6.27374	−0.373595
$283$	7.63839	0.454055	0.227028	−	0.973888i	$-0.427099\pi$
$283$	7.63839	0.454055	0.227028	+	0.973888i	$0.427099\pi$
$284$	−6.27807	−0.372535
$285$	50.0215	2.96302
$286$	0	0
$287$	21.7377	1.28314
$288$	−14.5136	−0.855222
$289$	−13.2823	−0.781309
$290$	16.7510	0.983651
$291$	15.5447	0.911245
$292$	3.16050	0.184954
$293$	−0.312368	−0.0182487	−0.00912437	−	0.999958i	$-0.502904\pi$
$293$	−0.312368	−0.0182487	−0.00912437	+	0.999958i	$0.502904\pi$
$294$	−2.73691	−0.159620
$295$	16.1670	0.941281
$296$	−8.51788	−0.495092
$297$	0	0
$298$	−13.1580	−0.762220
$299$	−21.2973	−1.23165
$300$	0.478208	0.0276094
$301$	11.6667	0.672456
$302$	14.2603	0.820589
$303$	−15.7907	−0.907152
$304$	21.0541	1.20754
$305$	−2.31409	−0.132505
$306$	13.8015	0.788977
$307$	10.8387	0.618600	0.309300	−	0.950965i	$-0.399905\pi$
$307$	10.8387	0.618600	0.309300	+	0.950965i	$0.399905\pi$
$308$	0	0
$309$	−29.8145	−1.69609
$310$	3.07694	0.174759
$311$	−2.71939	−0.154202	−0.0771012	−	0.997023i	$-0.524566\pi$
$311$	−2.71939	−0.154202	−0.0771012	+	0.997023i	$0.524566\pi$
$312$	40.7687	2.30807
$313$	−19.6878	−1.11282	−0.556409	−	0.830909i	$-0.687821\pi$
$313$	−19.6878	−1.11282	−0.556409	+	0.830909i	$0.687821\pi$
$314$	−7.04652	−0.397658
$315$	−33.3324	−1.87807
$316$	−4.70978	−0.264946
$317$	11.7072	0.657543	0.328771	−	0.944410i	$-0.393365\pi$
$317$	11.7072	0.657543	0.328771	+	0.944410i	$0.393365\pi$
$318$	−44.3534	−2.48722
$319$	0	0
$320$	20.5900	1.15102
$321$	−45.8526	−2.55925
$322$	14.6668	0.817350
$323$	14.0829	0.783591
$324$	−3.13496	−0.174165
$325$	−1.60267	−0.0889003
$326$	13.0548	0.723039
$327$	−6.34800	−0.351045
$328$	−26.5208	−1.46437
$329$	4.26582	0.235182
$330$	0	0
$331$	0.827883	0.0455046	0.0227523	−	0.999741i	$-0.492757\pi$
$331$	0.827883	0.0455046	0.0227523	+	0.999741i	$0.492757\pi$
$332$	−0.455500	−0.0249988
$333$	−16.0750	−0.880903
$334$	21.9167	1.19923
$335$	−13.8748	−0.758062
$336$	−21.3382	−1.16410
$337$	−6.15940	−0.335524	−0.167762	−	0.985828i	$-0.553654\pi$
$337$	−6.15940	−0.335524	−0.167762	+	0.985828i	$0.553654\pi$
$338$	−9.17050	−0.498809
$339$	38.1203	2.07041
$340$	2.03072	0.110131
$341$	0	0
$342$	52.2801	2.82699
$343$	19.3693	1.04584
$344$	−14.2338	−0.767433
$345$	−32.3108	−1.73955
$346$	−15.0046	−0.806650
$347$	21.8424	1.17256	0.586281	−	0.810108i	$-0.300591\pi$
$347$	21.8424	1.17256	0.586281	+	0.810108i	$0.300591\pi$
$348$	−7.84453	−0.420511
$349$	−14.2068	−0.760474	−0.380237	−	0.924889i	$-0.624157\pi$
$349$	−14.2068	−0.760474	−0.380237	+	0.924889i	$0.624157\pi$
$350$	1.10372	0.0589961
$351$	36.8588	1.96738
$352$	0	0
$353$	−11.6717	−0.621220	−0.310610	−	0.950537i	$-0.600533\pi$
$353$	−11.6717	−0.621220	−0.310610	+	0.950537i	$0.600533\pi$
$354$	25.6993	1.36590
$355$	31.9211	1.69419
$356$	2.73591	0.145003
$357$	−14.2729	−0.755401
$358$	10.7010	0.565564
$359$	13.7460	0.725484	0.362742	−	0.931890i	$-0.381841\pi$
$359$	13.7460	0.725484	0.362742	+	0.931890i	$0.381841\pi$
$360$	40.6668	2.14333
$361$	34.3461	1.80769
$362$	−3.03679	−0.159610
$363$	0	0
$364$	−5.13877	−0.269345
$365$	−16.0697	−0.841125
$366$	−3.67850	−0.192279
$367$	22.1819	1.15788	0.578942	−	0.815369i	$-0.303466\pi$
$367$	22.1819	1.15788	0.578942	+	0.815369i	$0.303466\pi$
$368$	−13.5997	−0.708932
$369$	−50.0502	−2.60551
$370$	8.02859	0.417386
$371$	30.1580	1.56573
$372$	−1.44094	−0.0747093
$373$	−3.43076	−0.177638	−0.0888189	−	0.996048i	$-0.528309\pi$
$373$	−3.43076	−0.177638	−0.0888189	+	0.996048i	$0.528309\pi$
$374$	0	0
$375$	31.8118	1.64276
$376$	−5.20445	−0.268399
$377$	26.2902	1.35402
$378$	−25.3836	−1.30559
$379$	26.4492	1.35860	0.679301	−	0.733860i	$-0.262283\pi$
$379$	26.4492	1.35860	0.679301	+	0.733860i	$0.262283\pi$
$380$	7.69240	0.394612
$381$	42.5581	2.18032
$382$	10.2431	0.524081
$383$	30.0484	1.53540	0.767700	−	0.640810i	$-0.221401\pi$
$383$	30.0484	1.53540	0.767700	+	0.640810i	$0.221401\pi$
$384$	17.8128	0.909006
$385$	0	0
$386$	−32.4714	−1.65275
$387$	−26.8620	−1.36547
$388$	2.39049	0.121359
$389$	17.4915	0.886855	0.443427	−	0.896310i	$-0.353762\pi$
$389$	17.4915	0.886855	0.443427	+	0.896310i	$0.353762\pi$
$390$	−38.4268	−1.94582
$391$	−9.09665	−0.460037
$392$	−2.27044	−0.114675
$393$	46.5189	2.34657
$394$	23.5164	1.18474
$395$	23.9470	1.20491
$396$	0	0
$397$	−2.09107	−0.104948	−0.0524739	−	0.998622i	$-0.516711\pi$
$397$	−2.09107	−0.104948	−0.0524739	+	0.998622i	$0.516711\pi$
$398$	24.6159	1.23388
$399$	−54.0659	−2.70668
$400$	−1.02341	−0.0511705
$401$	−26.6197	−1.32932	−0.664662	−	0.747144i	$-0.731425\pi$
$401$	−26.6197	−1.32932	−0.664662	+	0.747144i	$0.731425\pi$
$402$	−22.0555	−1.10003
$403$	4.82919	0.240559
$404$	−2.42832	−0.120814
$405$	15.9398	0.792056
$406$	−18.1053	−0.898553
$407$	0	0
$408$	17.4134	0.862093
$409$	2.96929	0.146822	0.0734109	−	0.997302i	$-0.476612\pi$
$409$	2.96929	0.146822	0.0734109	+	0.997302i	$0.476612\pi$
$410$	24.9974	1.23453
$411$	68.7184	3.38963
$412$	−4.58494	−0.225884
$413$	−17.4742	−0.859849
$414$	−33.7697	−1.65969
$415$	2.31600	0.113688
$416$	11.3768	0.557791
$417$	1.60997	0.0788407
$418$	0	0
$419$	19.3538	0.945493	0.472747	−	0.881198i	$-0.343263\pi$
$419$	19.3538	0.945493	0.472747	+	0.881198i	$0.343263\pi$
$420$	−7.79619	−0.380415
$421$	24.5977	1.19882	0.599408	−	0.800443i	$-0.295403\pi$
$421$	24.5977	1.19882	0.599408	+	0.800443i	$0.295403\pi$
$422$	−0.173587	−0.00845010
$423$	−9.82186	−0.477555
$424$	−36.7939	−1.78687
$425$	−0.684545	−0.0332053
$426$	50.7420	2.45846
$427$	2.50119	0.121041
$428$	−7.05131	−0.340838
$429$	0	0
$430$	13.4161	0.646983
$431$	34.6610	1.66956	0.834781	−	0.550582i	$-0.185594\pi$
$431$	34.6610	1.66956	0.834781	+	0.550582i	$0.185594\pi$
$432$	23.5367	1.13241
$433$	21.3682	1.02689	0.513445	−	0.858123i	$-0.328369\pi$
$433$	21.3682	1.02689	0.513445	+	0.858123i	$0.328369\pi$
$434$	−3.32572	−0.159640
$435$	39.8858	1.91238
$436$	−0.976207	−0.0467518
$437$	−34.4582	−1.64836
$438$	−25.5445	−1.22056
$439$	7.55472	0.360567	0.180283	−	0.983615i	$-0.442298\pi$
$439$	7.55472	0.360567	0.180283	+	0.983615i	$0.442298\pi$
$440$	0	0
$441$	−4.28478	−0.204037
$442$	−10.8185	−0.514585
$443$	41.4280	1.96830	0.984152	−	0.177329i	$-0.0567455\pi$
$443$	41.4280	1.96830	0.984152	+	0.177329i	$0.0567455\pi$
$444$	−3.75981	−0.178433
$445$	−13.9108	−0.659436
$446$	−6.35652	−0.300990
$447$	−31.3304	−1.48188
$448$	−22.2548	−1.05144
$449$	−2.77735	−0.131071	−0.0655356	−	0.997850i	$-0.520876\pi$
$449$	−2.77735	−0.131071	−0.0655356	+	0.997850i	$0.520876\pi$
$450$	−2.54126	−0.119796
$451$	0	0
$452$	5.86222	0.275735
$453$	33.9552	1.59536
$454$	36.5440	1.71509
$455$	26.1283	1.22491
$456$	65.9623	3.08897
$457$	−17.3713	−0.812597	−0.406299	−	0.913740i	$-0.633181\pi$
$457$	−17.3713	−0.812597	−0.406299	+	0.913740i	$0.633181\pi$
$458$	−26.7066	−1.24792
$459$	15.7434	0.734839
$460$	−4.96881	−0.231672
$461$	10.6475	0.495906	0.247953	−	0.968772i	$-0.420242\pi$
$461$	10.6475	0.495906	0.247953	+	0.968772i	$0.420242\pi$
$462$	0	0
$463$	−17.4918	−0.812912	−0.406456	−	0.913670i	$-0.633236\pi$
$463$	−17.4918	−0.812912	−0.406456	+	0.913670i	$0.633236\pi$
$464$	16.7880	0.779363
$465$	7.32651	0.339759
$466$	14.5726	0.675061
$467$	17.4564	0.807787	0.403893	−	0.914806i	$-0.367657\pi$
$467$	17.4564	0.807787	0.403893	+	0.914806i	$0.367657\pi$
$468$	11.8318	0.546925
$469$	14.9966	0.692480
$470$	4.90549	0.226273
$471$	−16.7785	−0.773112
$472$	21.3191	0.981293
$473$	0	0
$474$	38.0664	1.74845
$475$	−2.59307	−0.118978
$476$	−2.19491	−0.100604
$477$	−69.4376	−3.17933
$478$	25.4272	1.16301
$479$	−3.91645	−0.178947	−0.0894736	−	0.995989i	$-0.528518\pi$
$479$	−3.91645	−0.178947	−0.0894736	+	0.995989i	$0.528518\pi$
$480$	17.2600	0.787809
$481$	12.6007	0.574541
$482$	38.1207	1.73635
$483$	34.9232	1.58906
$484$	0	0
$485$	−12.1545	−0.551908
$486$	−5.10769	−0.231690
$487$	−36.8283	−1.66885	−0.834426	−	0.551121i	$-0.814200\pi$
$487$	−36.8283	−1.66885	−0.834426	+	0.551121i	$0.814200\pi$
$488$	−3.05155	−0.138137
$489$	31.0848	1.40570
$490$	2.14002	0.0966762
$491$	−38.6443	−1.74399	−0.871996	−	0.489512i	$-0.837175\pi$
$491$	−38.6443	−1.74399	−0.871996	+	0.489512i	$0.837175\pi$
$492$	−11.7063	−0.527762
$493$	11.2293	0.505741
$494$	−40.9808	−1.84381
$495$	0	0
$496$	3.08374	0.138464
$497$	−34.5020	−1.54762
$498$	3.68154	0.164974
$499$	−2.43966	−0.109214	−0.0546070	−	0.998508i	$-0.517391\pi$
$499$	−2.43966	−0.109214	−0.0546070	+	0.998508i	$0.517391\pi$
$500$	4.89208	0.218781
$501$	52.1858	2.33149
$502$	−0.0174585	−0.000779213 0
$503$	−21.2141	−0.945891	−0.472946	−	0.881092i	$-0.656809\pi$
$503$	−21.2141	−0.945891	−0.472946	+	0.881092i	$0.656809\pi$
$504$	−43.9548	−1.95790
$505$	12.3469	0.549430
$506$	0	0
$507$	−21.8359	−0.969765
$508$	6.54467	0.290373
$509$	−24.0143	−1.06441	−0.532207	−	0.846614i	$-0.678637\pi$
$509$	−24.0143	−1.06441	−0.532207	+	0.846614i	$0.678637\pi$
$510$	−16.4131	−0.726786
$511$	17.3689	0.768357
$512$	24.8577	1.09856
$513$	59.6362	2.63300
$514$	16.8910	0.745028
$515$	23.3123	1.02726
$516$	−6.28281	−0.276585
$517$	0	0
$518$	−8.67772	−0.381277
$519$	−35.7274	−1.56826
$520$	−31.8774	−1.39792
$521$	28.2610	1.23814	0.619069	−	0.785337i	$-0.287510\pi$
$521$	28.2610	1.23814	0.619069	+	0.785337i	$0.287510\pi$
$522$	41.6867	1.82458
$523$	−37.8934	−1.65696	−0.828481	−	0.560018i	$-0.810794\pi$
$523$	−37.8934	−1.65696	−0.828481	+	0.560018i	$0.810794\pi$
$524$	7.15377	0.312514
$525$	2.62806	0.114698
$526$	26.1507	1.14022
$527$	2.06268	0.0898516
$528$	0	0
$529$	−0.742119	−0.0322661
$530$	34.6803	1.50642
$531$	40.2336	1.74599
$532$	−8.31435	−0.360473
$533$	39.2327	1.69936
$534$	−22.1128	−0.956912
$535$	35.8526	1.55004
$536$	−18.2964	−0.790285
$537$	25.4801	1.09955
$538$	1.34483	0.0579797
$539$	0	0
$540$	8.59943	0.370061
$541$	8.06387	0.346693	0.173346	−	0.984861i	$-0.444542\pi$
$541$	8.06387	0.346693	0.173346	+	0.984861i	$0.444542\pi$
$542$	−21.9696	−0.943677
$543$	−7.23091	−0.310308
$544$	4.85932	0.208342
$545$	4.96356	0.212615
$546$	41.5337	1.77748
$547$	5.07863	0.217147	0.108573	−	0.994088i	$-0.465372\pi$
$547$	5.07863	0.217147	0.108573	+	0.994088i	$0.465372\pi$
$548$	10.5676	0.451427
$549$	−5.75889	−0.245783
$550$	0	0
$551$	42.5367	1.81212
$552$	−42.6076	−1.81350
$553$	−25.8832	−1.10067
$554$	−7.82606	−0.332497
$555$	19.1169	0.811466
$556$	0.247585	0.0104999
$557$	3.56510	0.151058	0.0755291	−	0.997144i	$-0.475935\pi$
$557$	3.56510	0.151058	0.0755291	+	0.997144i	$0.475935\pi$
$558$	7.65733	0.324161
$559$	21.0563	0.890586
$560$	16.6846	0.705051
$561$	0	0
$562$	4.59166	0.193687
$563$	42.7990	1.80376	0.901882	−	0.431982i	$-0.142185\pi$
$563$	42.7990	1.80376	0.901882	+	0.431982i	$0.142185\pi$
$564$	−2.29725	−0.0967319
$565$	−29.8066	−1.25397
$566$	−9.49399	−0.399062
$567$	−17.2286	−0.723533
$568$	42.0936	1.76621
$569$	0.0889222	0.00372781	0.00186391	−	0.999998i	$-0.499407\pi$
$569$	0.0889222	0.00372781	0.00186391	+	0.999998i	$0.499407\pi$
$570$	−62.1732	−2.60415
$571$	34.8764	1.45953	0.729765	−	0.683698i	$-0.239629\pi$
$571$	34.8764	1.45953	0.729765	+	0.683698i	$0.239629\pi$
$572$	0	0
$573$	24.3898	1.01890
$574$	−27.0185	−1.12773
$575$	1.67496	0.0698507
$576$	51.2407	2.13503
$577$	6.31046	0.262708	0.131354	−	0.991336i	$-0.458068\pi$
$577$	6.31046	0.262708	0.131354	+	0.991336i	$0.458068\pi$
$578$	16.5089	0.686681
$579$	−77.3177	−3.21322
$580$	6.13371	0.254688
$581$	−2.50326	−0.103853
$582$	−19.3209	−0.800879
$583$	0	0
$584$	−21.1907	−0.876879
$585$	−60.1592	−2.48728
$586$	0.388252	0.0160385
$587$	−44.6729	−1.84385	−0.921923	−	0.387373i	$-0.873383\pi$
$587$	−44.6729	−1.84385	−0.921923	+	0.387373i	$0.873383\pi$
$588$	−1.00218	−0.0413290
$589$	7.81345	0.321948
$590$	−20.0945	−0.827278
$591$	55.9950	2.30332
$592$	8.04633	0.330702
$593$	−43.6270	−1.79155	−0.895774	−	0.444510i	$-0.853378\pi$
$593$	−43.6270	−1.79155	−0.895774	+	0.444510i	$0.853378\pi$
$594$	0	0
$595$	11.1601	0.457519
$596$	−4.81805	−0.197355
$597$	58.6130	2.39887
$598$	26.4710	1.08248
$599$	47.0379	1.92192	0.960959	−	0.276690i	$-0.0892376\pi$
$599$	47.0379	1.92192	0.960959	+	0.276690i	$0.0892376\pi$
$600$	−3.20633	−0.130898
$601$	0.791913	0.0323028	0.0161514	−	0.999870i	$-0.494859\pi$
$601$	0.791913	0.0323028	0.0161514	+	0.999870i	$0.494859\pi$
$602$	−14.5009	−0.591011
$603$	−34.5291	−1.40613
$604$	5.22170	0.212468
$605$	0	0
$606$	19.6267	0.797282
$607$	−27.7316	−1.12559	−0.562796	−	0.826596i	$-0.690275\pi$
$607$	−27.7316	−1.12559	−0.562796	+	0.826596i	$0.690275\pi$
$608$	18.4072	0.746510
$609$	−43.1106	−1.74693
$610$	2.87626	0.116456
$611$	7.69905	0.311470
$612$	5.05368	0.204283
$613$	−2.86929	−0.115890	−0.0579448	−	0.998320i	$-0.518455\pi$
$613$	−2.86929	−0.115890	−0.0579448	+	0.998320i	$0.518455\pi$
$614$	−13.4718	−0.543678
$615$	59.5213	2.40013
$616$	0	0
$617$	−7.55145	−0.304010	−0.152005	−	0.988380i	$-0.548573\pi$
$617$	−7.55145	−0.304010	−0.152005	+	0.988380i	$0.548573\pi$
$618$	37.0574	1.49067
$619$	−29.2243	−1.17462	−0.587312	−	0.809361i	$-0.699814\pi$
$619$	−29.2243	−1.17462	−0.587312	+	0.809361i	$0.699814\pi$
$620$	1.12668	0.0452487
$621$	−38.5213	−1.54581
$622$	3.38001	0.135526
$623$	15.0355	0.602386
$624$	−38.5117	−1.54170
$625$	−26.6491	−1.06596
$626$	24.4705	0.978038
$627$	0	0
$628$	−2.58023	−0.102962
$629$	5.38209	0.214598
$630$	41.4299	1.65061
$631$	−0.297910	−0.0118596	−0.00592981	−	0.999982i	$-0.501888\pi$
$631$	−0.297910	−0.0118596	−0.00592981	+	0.999982i	$0.501888\pi$
$632$	31.5785	1.25612
$633$	−0.413329	−0.0164283
$634$	−14.5513	−0.577904
$635$	−33.2766	−1.32054
$636$	−16.2409	−0.643994
$637$	3.35870	0.133077
$638$	0	0
$639$	79.4392	3.14257
$640$	−13.9280	−0.550552
$641$	−26.8206	−1.05935	−0.529676	−	0.848200i	$-0.677686\pi$
$641$	−26.8206	−1.05935	−0.529676	+	0.848200i	$0.677686\pi$
$642$	56.9916	2.24928
$643$	−47.2258	−1.86240	−0.931202	−	0.364504i	$-0.881239\pi$
$643$	−47.2258	−1.86240	−0.931202	+	0.364504i	$0.881239\pi$
$644$	5.37055	0.211629
$645$	31.9451	1.25784
$646$	−17.5040	−0.688686
$647$	−30.4636	−1.19765	−0.598823	−	0.800881i	$-0.704365\pi$
$647$	−30.4636	−1.19765	−0.598823	+	0.800881i	$0.704365\pi$
$648$	21.0195	0.825724
$649$	0	0
$650$	1.99201	0.0781331
$651$	−7.91888	−0.310365
$652$	4.78028	0.187210
$653$	−13.6627	−0.534664	−0.267332	−	0.963605i	$-0.586142\pi$
$653$	−13.6627	−0.534664	−0.267332	+	0.963605i	$0.586142\pi$
$654$	7.89012	0.308528
$655$	−36.3736	−1.42123
$656$	25.0526	0.978140
$657$	−39.9912	−1.56021
$658$	−5.30211	−0.206698
$659$	11.2672	0.438908	0.219454	−	0.975623i	$-0.429572\pi$
$659$	11.2672	0.438908	0.219454	+	0.975623i	$0.429572\pi$
$660$	0	0
$661$	21.8482	0.849796	0.424898	−	0.905241i	$-0.360310\pi$
$661$	21.8482	0.849796	0.424898	+	0.905241i	$0.360310\pi$
$662$	−1.02900	−0.0399933
$663$	−25.7600	−1.00044
$664$	3.05407	0.118521
$665$	42.2746	1.63934
$666$	19.9801	0.774212
$667$	−27.4760	−1.06388
$668$	8.02523	0.310506
$669$	−15.1355	−0.585173
$670$	17.2454	0.666249
$671$	0	0
$672$	−18.6556	−0.719654
$673$	−5.32110	−0.205113	−0.102557	−	0.994727i	$-0.532702\pi$
$673$	−5.32110	−0.205113	−0.102557	+	0.994727i	$0.532702\pi$
$674$	7.65570	0.294887
$675$	−2.89882	−0.111576
$676$	−3.35796	−0.129152
$677$	−6.51268	−0.250303	−0.125151	−	0.992138i	$-0.539942\pi$
$677$	−6.51268	−0.250303	−0.125151	+	0.992138i	$0.539942\pi$
$678$	−47.3809	−1.81965
$679$	13.1372	0.504161
$680$	−13.6157	−0.522139
$681$	87.0149	3.33442
$682$	0	0
$683$	−34.3410	−1.31402	−0.657012	−	0.753880i	$-0.728180\pi$
$683$	−34.3410	−1.31402	−0.657012	+	0.753880i	$0.728180\pi$
$684$	19.1434	0.731967
$685$	−53.7315	−2.05298
$686$	−24.0747	−0.919177
$687$	−63.5912	−2.42615
$688$	13.4458	0.512615
$689$	54.4299	2.07362
$690$	40.1600	1.52887
$691$	28.6070	1.08826	0.544131	−	0.839001i	$-0.316860\pi$
$691$	28.6070	1.08826	0.544131	+	0.839001i	$0.316860\pi$
$692$	−5.49422	−0.208859
$693$	0	0
$694$	−27.1486	−1.03055
$695$	−1.25885	−0.0477510
$696$	52.5965	1.99367
$697$	16.7574	0.634731
$698$	17.6581	0.668368
$699$	34.6988	1.31243
$700$	0.404148	0.0152753
$701$	42.2172	1.59452	0.797261	−	0.603635i	$-0.206282\pi$
$701$	42.2172	1.59452	0.797261	+	0.603635i	$0.206282\pi$
$702$	−45.8129	−1.72910
$703$	20.3874	0.768927
$704$	0	0
$705$	11.6805	0.439912
$706$	14.5071	0.545980
$707$	−13.3452	−0.501897
$708$	9.41031	0.353661
$709$	34.8628	1.30930	0.654651	−	0.755931i	$-0.272816\pi$
$709$	34.8628	1.30930	0.654651	+	0.755931i	$0.272816\pi$
$710$	−39.6756	−1.48900
$711$	59.5950	2.23499
$712$	−18.3439	−0.687467
$713$	−5.04700	−0.189012
$714$	17.7402	0.663910
$715$	0	0
$716$	3.91838	0.146437
$717$	60.5447	2.26109
$718$	−17.0853	−0.637617
$719$	−19.5101	−0.727603	−0.363801	−	0.931477i	$-0.618521\pi$
$719$	−19.5101	−0.727603	−0.363801	+	0.931477i	$0.618521\pi$
$720$	−38.4155	−1.43166
$721$	−25.1971	−0.938390
$722$	−42.6898	−1.58875
$723$	90.7691	3.37574
$724$	−1.11198	−0.0413265
$725$	−2.06764	−0.0767903
$726$	0	0
$727$	−39.5594	−1.46718	−0.733589	−	0.679593i	$-0.762156\pi$
$727$	−39.5594	−1.46718	−0.733589	+	0.679593i	$0.762156\pi$
$728$	34.4548	1.27698
$729$	−32.8264	−1.21579
$730$	19.9735	0.739251
$731$	8.99371	0.332644
$732$	−1.34696	−0.0497850
$733$	15.8434	0.585191	0.292596	−	0.956236i	$-0.405481\pi$
$733$	15.8434	0.585191	0.292596	+	0.956236i	$0.405481\pi$
$734$	−27.5705	−1.01765
$735$	5.09560	0.187954
$736$	−11.8899	−0.438268
$737$	0	0
$738$	62.2089	2.28994
$739$	34.0846	1.25382	0.626912	−	0.779090i	$-0.284319\pi$
$739$	34.0846	1.25382	0.626912	+	0.779090i	$0.284319\pi$
$740$	2.93983	0.108070
$741$	−97.5793	−3.58467
$742$	−37.4843	−1.37609
$743$	6.20359	0.227588	0.113794	−	0.993504i	$-0.463700\pi$
$743$	6.20359	0.227588	0.113794	+	0.993504i	$0.463700\pi$
$744$	9.66132	0.354201
$745$	24.4975	0.897521
$746$	4.26419	0.156123
$747$	5.76364	0.210881
$748$	0	0
$749$	−38.7514	−1.41594
$750$	−39.5399	−1.44379
$751$	−29.3860	−1.07231	−0.536154	−	0.844120i	$-0.680123\pi$
$751$	−29.3860	−1.07231	−0.536154	+	0.844120i	$0.680123\pi$
$752$	4.91633	0.179280
$753$	−0.0415705	−0.00151491
$754$	−32.6769	−1.19002
$755$	−26.5499	−0.966250
$756$	−9.29472	−0.338046
$757$	36.0789	1.31131	0.655656	−	0.755060i	$-0.272392\pi$
$757$	36.0789	1.31131	0.655656	+	0.755060i	$0.272392\pi$
$758$	−32.8745	−1.19405
$759$	0	0
$760$	−51.5765	−1.87088
$761$	−20.5796	−0.746010	−0.373005	−	0.927829i	$-0.621673\pi$
$761$	−20.5796	−0.746010	−0.373005	+	0.927829i	$0.621673\pi$
$762$	−52.8968	−1.91625
$763$	−5.36487	−0.194221
$764$	3.75071	0.135696
$765$	−25.6956	−0.929027
$766$	−37.3480	−1.34944
$767$	−31.5378	−1.13876
$768$	30.5260	1.10151
$769$	6.64509	0.239628	0.119814	−	0.992796i	$-0.461770\pi$
$769$	6.64509	0.239628	0.119814	+	0.992796i	$0.461770\pi$
$770$	0	0
$771$	40.2191	1.44845
$772$	−11.8901	−0.427933
$773$	−20.4417	−0.735236	−0.367618	−	0.929977i	$-0.619827\pi$
$773$	−20.4417	−0.735236	−0.367618	+	0.929977i	$0.619827\pi$
$774$	33.3876	1.20009
$775$	−0.379800	−0.0136428
$776$	−16.0279	−0.575369
$777$	−20.6625	−0.741264
$778$	−21.7407	−0.779443
$779$	63.4772	2.27431
$780$	−14.0708	−0.503814
$781$	0	0
$782$	11.3065	0.404320
$783$	47.5523	1.69938
$784$	2.14475	0.0765981
$785$	13.1192	0.468246
$786$	−57.8198	−2.06236
$787$	−23.6116	−0.841664	−0.420832	−	0.907139i	$-0.638262\pi$
$787$	−23.6116	−0.841664	−0.420832	+	0.907139i	$0.638262\pi$
$788$	8.61101	0.306755
$789$	62.2675	2.21678
$790$	−29.7645	−1.05897
$791$	32.2166	1.14549
$792$	0	0
$793$	4.51421	0.160304
$794$	2.59905	0.0922369
$795$	82.5774	2.92872
$796$	9.01361	0.319479
$797$	29.1891	1.03393	0.516966	−	0.856006i	$-0.327061\pi$
$797$	29.1891	1.03393	0.516966	+	0.856006i	$0.327061\pi$
$798$	67.2001	2.37886
$799$	3.28847	0.116338
$800$	−0.894745	−0.0316340
$801$	−34.6187	−1.22319
$802$	33.0864	1.16832
$803$	0	0
$804$	−8.07608	−0.284821
$805$	−27.3068	−0.962437
$806$	−6.00234	−0.211423
$807$	3.20217	0.112722
$808$	16.2816	0.572785
$809$	−21.4278	−0.753363	−0.376682	−	0.926343i	$-0.622935\pi$
$809$	−21.4278	−0.753363	−0.376682	+	0.926343i	$0.622935\pi$
$810$	−19.8121	−0.696125
$811$	−28.4369	−0.998555	−0.499277	−	0.866442i	$-0.666401\pi$
$811$	−28.4369	−0.998555	−0.499277	+	0.866442i	$0.666401\pi$
$812$	−6.62963	−0.232655
$813$	−52.3119	−1.83466
$814$	0	0
$815$	−24.3055	−0.851385
$816$	−16.4494	−0.575845
$817$	34.0683	1.19190
$818$	−3.69062	−0.129039
$819$	65.0232	2.27209
$820$	9.15329	0.319647
$821$	15.7922	0.551152	0.275576	−	0.961279i	$-0.411131\pi$
$821$	15.7922	0.551152	0.275576	+	0.961279i	$0.411131\pi$
$822$	−85.4122	−2.97909
$823$	−21.6123	−0.753358	−0.376679	−	0.926344i	$-0.622934\pi$
$823$	−21.6123	−0.753358	−0.376679	+	0.926344i	$0.622934\pi$
$824$	30.7414	1.07093
$825$	0	0
$826$	21.7192	0.755708
$827$	12.3501	0.429456	0.214728	−	0.976674i	$-0.431114\pi$
$827$	12.3501	0.429456	0.214728	+	0.976674i	$0.431114\pi$
$828$	−12.3655	−0.429730
$829$	−21.4390	−0.744606	−0.372303	−	0.928111i	$-0.621432\pi$
$829$	−21.4390	−0.744606	−0.372303	+	0.928111i	$0.621432\pi$
$830$	−2.87863	−0.0999187
$831$	−18.6346	−0.646428
$832$	−40.1660	−1.39251
$833$	1.43459	0.0497058
$834$	−2.00108	−0.0692919
$835$	−40.8046	−1.41210
$836$	0	0
$837$	8.73475	0.301917
$838$	−24.0554	−0.830979
$839$	54.4560	1.88003	0.940015	−	0.341132i	$-0.110810\pi$
$839$	54.4560	1.88003	0.940015	+	0.341132i	$0.110810\pi$
$840$	52.2725	1.80357
$841$	4.91757	0.169571
$842$	−30.5732	−1.05362
$843$	10.9332	0.376559
$844$	−0.0635625	−0.00218791
$845$	17.0737	0.587352
$846$	12.2079	0.419716
$847$	0	0
$848$	34.7570	1.19356
$849$	−22.6061	−0.775841
$850$	0.850842	0.0291837
$851$	−13.1690	−0.451428
$852$	18.5802	0.636548
$853$	−3.47442	−0.118962	−0.0594809	−	0.998229i	$-0.518945\pi$
$853$	−3.47442	−0.118962	−0.0594809	+	0.998229i	$0.518945\pi$
$854$	−3.10881	−0.106381
$855$	−97.3354	−3.32880
$856$	47.2781	1.61593
$857$	1.72960	0.0590819	0.0295409	−	0.999564i	$-0.490595\pi$
$857$	1.72960	0.0590819	0.0295409	+	0.999564i	$0.490595\pi$
$858$	0	0
$859$	−9.16126	−0.312578	−0.156289	−	0.987711i	$-0.549953\pi$
$859$	−9.16126	−0.312578	−0.156289	+	0.987711i	$0.549953\pi$
$860$	4.91259	0.167518
$861$	−64.3337	−2.19249
$862$	−43.0812	−1.46735
$863$	27.8430	0.947786	0.473893	−	0.880582i	$-0.342848\pi$
$863$	27.8430	0.947786	0.473893	+	0.880582i	$0.342848\pi$
$864$	20.5776	0.700065
$865$	27.9356	0.949838
$866$	−26.5592	−0.902517
$867$	39.3094	1.33502
$868$	−1.21778	−0.0413342
$869$	0	0
$870$	−49.5752	−1.68076
$871$	27.0663	0.917105
$872$	6.54534	0.221653
$873$	−30.2479	−1.02374
$874$	42.8292	1.44872
$875$	26.8851	0.908882
$876$	−9.35363	−0.316030
$877$	42.3113	1.42875	0.714376	−	0.699762i	$-0.246711\pi$
$877$	42.3113	1.42875	0.714376	+	0.699762i	$0.246711\pi$
$878$	−9.38999	−0.316897
$879$	0.924467	0.0311815
$880$	0	0
$881$	−52.3095	−1.76235	−0.881177	−	0.472787i	$-0.843248\pi$
$881$	−52.3095	−1.76235	−0.881177	+	0.472787i	$0.843248\pi$
$882$	5.32568	0.179325
$883$	−46.8485	−1.57658	−0.788289	−	0.615305i	$-0.789033\pi$
$883$	−46.8485	−1.57658	−0.788289	+	0.615305i	$0.789033\pi$
$884$	−3.96142	−0.133237
$885$	−47.8470	−1.60836
$886$	−51.4921	−1.72991
$887$	10.9421	0.367398	0.183699	−	0.982983i	$-0.441193\pi$
$887$	10.9421	0.367398	0.183699	+	0.982983i	$0.441193\pi$
$888$	25.2090	0.845960
$889$	35.9671	1.20630
$890$	17.2902	0.579568
$891$	0	0
$892$	−2.32757	−0.0779327
$893$	12.4568	0.416850
$894$	38.9415	1.30240
$895$	−19.9231	−0.665957
$896$	15.0541	0.502923
$897$	63.0302	2.10452
$898$	3.45205	0.115196
$899$	6.23023	0.207790
$900$	−0.930532	−0.0310177
$901$	23.2485	0.774520
$902$	0	0
$903$	−34.5280	−1.14902
$904$	−39.3054	−1.30728
$905$	5.65392	0.187943
$906$	−42.2040	−1.40213
$907$	−12.0909	−0.401472	−0.200736	−	0.979645i	$-0.564333\pi$
$907$	−12.0909	−0.401472	−0.200736	+	0.979645i	$0.564333\pi$
$908$	13.3813	0.444074
$909$	30.7267	1.01914
$910$	−32.4756	−1.07656
$911$	26.7829	0.887358	0.443679	−	0.896186i	$-0.353673\pi$
$911$	26.7829	0.887358	0.443679	+	0.896186i	$0.353673\pi$
$912$	−62.3106	−2.06331
$913$	0	0
$914$	21.5914	0.714179
$915$	6.84866	0.226410
$916$	−9.77917	−0.323113
$917$	39.3145	1.29828
$918$	−19.5679	−0.645838
$919$	23.1167	0.762550	0.381275	−	0.924462i	$-0.375485\pi$
$919$	23.1167	0.762550	0.381275	+	0.924462i	$0.375485\pi$
$920$	33.3152	1.09837
$921$	−32.0777	−1.05700
$922$	−13.2342	−0.435844
$923$	−62.2699	−2.04964
$924$	0	0
$925$	−0.991001	−0.0325839
$926$	21.7411	0.714456
$927$	58.0153	1.90547
$928$	14.6774	0.481809
$929$	−21.0391	−0.690270	−0.345135	−	0.938553i	$-0.612167\pi$
$929$	−21.0391	−0.690270	−0.345135	+	0.938553i	$0.612167\pi$
$930$	−9.10634	−0.298609
$931$	5.43427	0.178101
$932$	5.33604	0.174788
$933$	8.04815	0.263485
$934$	−21.6971	−0.709951
$935$	0	0
$936$	−79.3307	−2.59300
$937$	−35.7549	−1.16806	−0.584030	−	0.811732i	$-0.698525\pi$
$937$	−35.7549	−1.16806	−0.584030	+	0.811732i	$0.698525\pi$
$938$	−18.6398	−0.608610
$939$	58.2667	1.90146
$940$	1.79624	0.0585870
$941$	49.9018	1.62675	0.813377	−	0.581737i	$-0.197627\pi$
$941$	49.9018	1.62675	0.813377	+	0.581737i	$0.197627\pi$
$942$	20.8545	0.679476
$943$	−41.0023	−1.33522
$944$	−20.1389	−0.655466
$945$	47.2593	1.53734
$946$	0	0
$947$	40.4078	1.31308	0.656538	−	0.754293i	$-0.272020\pi$
$947$	40.4078	1.31308	0.656538	+	0.754293i	$0.272020\pi$
$948$	13.9388	0.452711
$949$	31.3479	1.01759
$950$	3.22300	0.104568
$951$	−34.6480	−1.12354
$952$	14.7166	0.476967
$953$	0.0405872	0.00131475	0.000657374	−	1.00000i	$-0.499791\pi$
$953$	0.0405872	0.00131475	0.000657374		1.00000i	$0.499791\pi$
$954$	86.3061	2.79426
$955$	−19.0706	−0.617110
$956$	9.31069	0.301129
$957$	0	0
$958$	4.86788	0.157274
$959$	58.0759	1.87537
$960$	−60.9371	−1.96674
$961$	−29.8556	−0.963083
$962$	−15.6617	−0.504955
$963$	89.2234	2.87518
$964$	13.9587	0.449578
$965$	60.4555	1.94613
$966$	−43.4071	−1.39660
$967$	−15.6015	−0.501710	−0.250855	−	0.968025i	$-0.580712\pi$
$967$	−15.6015	−0.501710	−0.250855	+	0.968025i	$0.580712\pi$
$968$	0	0
$969$	−41.6788	−1.33892
$970$	15.1072	0.485064
$971$	−25.7623	−0.826750	−0.413375	−	0.910561i	$-0.635650\pi$
$971$	−25.7623	−0.826750	−0.413375	+	0.910561i	$0.635650\pi$
$972$	−1.87029	−0.0599894
$973$	1.36063	0.0436199
$974$	45.7751	1.46673
$975$	4.74318	0.151903
$976$	2.88261	0.0922702
$977$	17.7278	0.567163	0.283582	−	0.958948i	$-0.408477\pi$
$977$	17.7278	0.567163	0.283582	+	0.958948i	$0.408477\pi$
$978$	−38.6363	−1.23545
$979$	0	0
$980$	0.783611	0.0250315
$981$	12.3524	0.394381
$982$	48.0322	1.53277
$983$	16.1630	0.515521	0.257761	−	0.966209i	$-0.417015\pi$
$983$	16.1630	0.515521	0.257761	+	0.966209i	$0.417015\pi$
$984$	78.4895	2.50215
$985$	−43.7830	−1.39504
$986$	−13.9572	−0.444488
$987$	−12.6249	−0.401854
$988$	−15.0059	−0.477402
$989$	−22.0060	−0.699751
$990$	0	0
$991$	40.5102	1.28685	0.643425	−	0.765509i	$-0.277513\pi$
$991$	40.5102	1.28685	0.643425	+	0.765509i	$0.277513\pi$
$992$	2.69605	0.0855997
$993$	−2.45016	−0.0777533
$994$	42.8835	1.36018
$995$	−45.8300	−1.45291
$996$	1.34807	0.0427152
$997$	−10.7828	−0.341495	−0.170747	−	0.985315i	$-0.554618\pi$
$997$	−10.7828	−0.341495	−0.170747	+	0.985315i	$0.554618\pi$
$998$	3.03232	0.0959865
$999$	22.7914	0.721087

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7381.2.a.u.1.22		64
11.7	odd	10		671.2.j.c.489.23	yes	128
11.8	odd	10		671.2.j.c.306.23	✓	128
11.10	odd	2		7381.2.a.v.1.43		64

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
671.2.j.c.306.23	✓	128	11.8	odd	10
671.2.j.c.489.23	yes	128	11.7	odd	10
7381.2.a.u.1.22		64	1.1	even	1	trivial
7381.2.a.v.1.43		64	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.24293 q^{2} -2.95954 q^{3} -0.455124 q^{4} +2.31409 q^{5} +3.67850 q^{6} -2.50119 q^{7} +3.05155 q^{8} +5.75889 q^{9} +O(q^{10})\)	\(q-1.24293 q^{2} -2.95954 q^{3} -0.455124 q^{4} +2.31409 q^{5} +3.67850 q^{6} -2.50119 q^{7} +3.05155 q^{8} +5.75889 q^{9} -2.87626 q^{10} +1.34696 q^{12} -4.51421 q^{13} +3.10881 q^{14} -6.84866 q^{15} -2.88261 q^{16} -1.92814 q^{17} -7.15790 q^{18} -7.30384 q^{19} -1.05320 q^{20} +7.40239 q^{21} +4.71783 q^{23} -9.03119 q^{24} +0.355028 q^{25} +5.61085 q^{26} -8.16505 q^{27} +1.13835 q^{28} -5.82388 q^{29} +8.51240 q^{30} -1.06977 q^{31} -2.52021 q^{32} +2.39655 q^{34} -5.78800 q^{35} -2.62101 q^{36} -2.79133 q^{37} +9.07816 q^{38} +13.3600 q^{39} +7.06157 q^{40} -8.69094 q^{41} -9.20065 q^{42} -4.66444 q^{43} +13.3266 q^{45} -5.86393 q^{46} -1.70551 q^{47} +8.53122 q^{48} -0.744029 q^{49} -0.441275 q^{50} +5.70642 q^{51} +2.05453 q^{52} -12.0575 q^{53} +10.1486 q^{54} -7.63251 q^{56} +21.6160 q^{57} +7.23868 q^{58} +6.98634 q^{59} +3.11699 q^{60} -1.00000 q^{61} +1.32965 q^{62} -14.4041 q^{63} +8.89767 q^{64} -10.4463 q^{65} -5.99579 q^{67} +0.877545 q^{68} -13.9626 q^{69} +7.19408 q^{70} +13.7942 q^{71} +17.5735 q^{72} -6.94426 q^{73} +3.46943 q^{74} -1.05072 q^{75} +3.32415 q^{76} -16.6056 q^{78} +10.3483 q^{79} -6.67064 q^{80} +6.88815 q^{81} +10.8022 q^{82} +1.00082 q^{83} -3.36901 q^{84} -4.46191 q^{85} +5.79757 q^{86} +17.2360 q^{87} -6.01134 q^{89} -16.5640 q^{90} +11.2909 q^{91} -2.14720 q^{92} +3.16604 q^{93} +2.11983 q^{94} -16.9018 q^{95} +7.45866 q^{96} -5.25239 q^{97} +0.924776 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 64 q - 5 q^{2} + 19 q^{3} + 73 q^{4} + 21 q^{5} + q^{6} - 6 q^{7} - 3 q^{8} + 81 q^{9}+O(q^{10}) \) 64 * q - 5 * q^2 + 19 * q^3 + 73 * q^4 + 21 * q^5 + q^6 - 6 * q^7 - 3 * q^8 + 81 * q^9	\( 64 q - 5 q^{2} + 19 q^{3} + 73 q^{4} + 21 q^{5} + q^{6} - 6 q^{7} - 3 q^{8} + 81 q^{9} + 4 q^{10} + 41 q^{12} + q^{13} + 41 q^{14} + 28 q^{15} + 107 q^{16} - 13 q^{17} - 38 q^{18} + q^{19} + 65 q^{20} - q^{21} + 52 q^{23} - 3 q^{24} + 87 q^{25} + 37 q^{26} + 88 q^{27} + q^{28} - 19 q^{29} + 19 q^{30} + 45 q^{31} + 24 q^{32} - 23 q^{34} - 10 q^{35} + 146 q^{36} + 26 q^{37} - 4 q^{38} + 6 q^{39} - 84 q^{40} + 12 q^{41} + 28 q^{42} - 5 q^{43} + 71 q^{45} + 5 q^{46} + 63 q^{47} + 85 q^{48} + 78 q^{49} - 14 q^{50} - 22 q^{51} + 24 q^{52} + 86 q^{53} + 114 q^{54} + 119 q^{56} + 29 q^{57} + q^{58} + 119 q^{59} - 22 q^{60} - 64 q^{61} - 13 q^{62} + 28 q^{63} + 135 q^{64} + 30 q^{65} + 2 q^{67} - 59 q^{68} + 63 q^{69} - 2 q^{70} + 126 q^{71} - 48 q^{72} - 8 q^{73} + 27 q^{74} + 107 q^{75} + 82 q^{76} - 13 q^{78} + 5 q^{79} + 119 q^{80} + 96 q^{81} - 27 q^{82} - 14 q^{83} - 182 q^{84} + 52 q^{85} + 60 q^{86} + 8 q^{87} + 59 q^{89} + 45 q^{90} + 83 q^{91} + 111 q^{92} - 7 q^{93} - 21 q^{94} + 26 q^{95} + 86 q^{96} - 39 q^{97} + 57 q^{98}+O(q^{100}) \) 64 * q - 5 * q^2 + 19 * q^3 + 73 * q^4 + 21 * q^5 + q^6 - 6 * q^7 - 3 * q^8 + 81 * q^9 + 4 * q^10 + 41 * q^12 + q^13 + 41 * q^14 + 28 * q^15 + 107 * q^16 - 13 * q^17 - 38 * q^18 + q^19 + 65 * q^20 - q^21 + 52 * q^23 - 3 * q^24 + 87 * q^25 + 37 * q^26 + 88 * q^27 + q^28 - 19 * q^29 + 19 * q^30 + 45 * q^31 + 24 * q^32 - 23 * q^34 - 10 * q^35 + 146 * q^36 + 26 * q^37 - 4 * q^38 + 6 * q^39 - 84 * q^40 + 12 * q^41 + 28 * q^42 - 5 * q^43 + 71 * q^45 + 5 * q^46 + 63 * q^47 + 85 * q^48 + 78 * q^49 - 14 * q^50 - 22 * q^51 + 24 * q^52 + 86 * q^53 + 114 * q^54 + 119 * q^56 + 29 * q^57 + q^58 + 119 * q^59 - 22 * q^60 - 64 * q^61 - 13 * q^62 + 28 * q^63 + 135 * q^64 + 30 * q^65 + 2 * q^67 - 59 * q^68 + 63 * q^69 - 2 * q^70 + 126 * q^71 - 48 * q^72 - 8 * q^73 + 27 * q^74 + 107 * q^75 + 82 * q^76 - 13 * q^78 + 5 * q^79 + 119 * q^80 + 96 * q^81 - 27 * q^82 - 14 * q^83 - 182 * q^84 + 52 * q^85 + 60 * q^86 + 8 * q^87 + 59 * q^89 + 45 * q^90 + 83 * q^91 + 111 * q^92 - 7 * q^93 - 21 * q^94 + 26 * q^95 + 86 * q^96 - 39 * q^97 + 57 * q^98

Introduction

L-functions

Modular forms

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Database

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