LMFDB - Embedded newform 1681.2.a.m.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1681,2,Mod(1,1681)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1681, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1681.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1681 = 41^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1681.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:	$13.4228525798$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 41)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	1681.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.47960 q^{2} -2.68766 q^{3} +0.189211 q^{4} -0.774365 q^{5} -3.97665 q^{6} -4.67709 q^{7} -2.67924 q^{8} +4.22349 q^{9} +O(q^{10})$	\(q+1.47960 q^{2} -2.68766 q^{3} +0.189211 q^{4} -0.774365 q^{5} -3.97665 q^{6} -4.67709 q^{7} -2.67924 q^{8} +4.22349 q^{9} -1.14575 q^{10} -0.545050 q^{11} -0.508535 q^{12} -0.907468 q^{13} -6.92022 q^{14} +2.08123 q^{15} -4.34262 q^{16} -5.99197 q^{17} +6.24907 q^{18} +4.38157 q^{19} -0.146519 q^{20} +12.5704 q^{21} -0.806456 q^{22} +1.04686 q^{23} +7.20087 q^{24} -4.40036 q^{25} -1.34269 q^{26} -3.28832 q^{27} -0.884959 q^{28} -1.23385 q^{29} +3.07938 q^{30} +1.52019 q^{31} -1.06686 q^{32} +1.46491 q^{33} -8.86571 q^{34} +3.62178 q^{35} +0.799132 q^{36} -7.25720 q^{37} +6.48297 q^{38} +2.43896 q^{39} +2.07471 q^{40} +18.5992 q^{42} +6.22101 q^{43} -0.103130 q^{44} -3.27052 q^{45} +1.54893 q^{46} +3.97043 q^{47} +11.6715 q^{48} +14.8752 q^{49} -6.51076 q^{50} +16.1043 q^{51} -0.171703 q^{52} +5.35331 q^{53} -4.86540 q^{54} +0.422068 q^{55} +12.5311 q^{56} -11.7762 q^{57} -1.82560 q^{58} +8.87496 q^{59} +0.393792 q^{60} +1.38786 q^{61} +2.24927 q^{62} -19.7537 q^{63} +7.10673 q^{64} +0.702711 q^{65} +2.16747 q^{66} +5.63556 q^{67} -1.13375 q^{68} -2.81360 q^{69} +5.35878 q^{70} +1.08971 q^{71} -11.3157 q^{72} -5.61291 q^{73} -10.7377 q^{74} +11.8266 q^{75} +0.829043 q^{76} +2.54925 q^{77} +3.60868 q^{78} -12.1959 q^{79} +3.36278 q^{80} -3.83260 q^{81} -13.1576 q^{83} +2.37846 q^{84} +4.63997 q^{85} +9.20460 q^{86} +3.31616 q^{87} +1.46032 q^{88} +1.95076 q^{89} -4.83906 q^{90} +4.24431 q^{91} +0.198078 q^{92} -4.08575 q^{93} +5.87464 q^{94} -3.39294 q^{95} +2.86734 q^{96} -1.94271 q^{97} +22.0093 q^{98} -2.30202 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 8 q^{2} + 20 q^{4} + 12 q^{5} + 24 q^{8} + 24 q^{9}+O(q^{10}) $ 24 * q + 8 * q^2 + 20 * q^4 + 12 * q^5 + 24 * q^8 + 24 * q^9	$ 24 q + 8 q^{2} + 20 q^{4} + 12 q^{5} + 24 q^{8} + 24 q^{9} - 4 q^{10} + 20 q^{16} + 20 q^{18} + 40 q^{20} + 56 q^{21} + 48 q^{23} + 8 q^{25} + 32 q^{31} + 36 q^{32} + 24 q^{33} + 108 q^{36} + 60 q^{39} + 44 q^{40} - 40 q^{42} + 36 q^{43} - 36 q^{45} + 36 q^{46} + 32 q^{49} - 60 q^{50} + 36 q^{51} - 60 q^{57} + 116 q^{59} + 40 q^{61} + 48 q^{62} + 60 q^{64} + 16 q^{66} + 4 q^{72} + 16 q^{73} + 24 q^{74} - 16 q^{77} - 60 q^{78} + 84 q^{80} - 28 q^{81} + 80 q^{83} - 24 q^{84} - 24 q^{86} + 76 q^{87} + 56 q^{90} + 40 q^{91} - 40 q^{92} + 36 q^{98}+O(q^{100}) $ 24 * q + 8 * q^2 + 20 * q^4 + 12 * q^5 + 24 * q^8 + 24 * q^9 - 4 * q^10 + 20 * q^16 + 20 * q^18 + 40 * q^20 + 56 * q^21 + 48 * q^23 + 8 * q^25 + 32 * q^31 + 36 * q^32 + 24 * q^33 + 108 * q^36 + 60 * q^39 + 44 * q^40 - 40 * q^42 + 36 * q^43 - 36 * q^45 + 36 * q^46 + 32 * q^49 - 60 * q^50 + 36 * q^51 - 60 * q^57 + 116 * q^59 + 40 * q^61 + 48 * q^62 + 60 * q^64 + 16 * q^66 + 4 * q^72 + 16 * q^73 + 24 * q^74 - 16 * q^77 - 60 * q^78 + 84 * q^80 - 28 * q^81 + 80 * q^83 - 24 * q^84 - 24 * q^86 + 76 * q^87 + 56 * q^90 + 40 * q^91 - 40 * q^92 + 36 * 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 $

$2$	1.47960	1.04623	0.523117	−	0.852261i	$-0.324769\pi$
$2$	1.47960	1.04623	0.523117	+	0.852261i	$0.324769\pi$
$3$	−2.68766	−1.55172	−0.775859	−	0.630906i	$-0.782683\pi$
$3$	−2.68766	−1.55172	−0.775859	+	0.630906i	$0.782683\pi$
$4$	0.189211	0.0946056
$5$	−0.774365	−0.346307	−0.173153	−	0.984895i	$-0.555396\pi$
$5$	−0.774365	−0.346307	−0.173153	+	0.984895i	$0.555396\pi$
$6$	−3.97665	−1.62346
$7$	−4.67709	−1.76777	−0.883887	−	0.467700i	$-0.845083\pi$
$7$	−4.67709	−1.76777	−0.883887	+	0.467700i	$0.845083\pi$
$8$	−2.67924	−0.947254
$9$	4.22349	1.40783
$10$	−1.14575	−0.362318
$11$	−0.545050	−0.164339	−0.0821694	−	0.996618i	$-0.526185\pi$
$11$	−0.545050	−0.164339	−0.0821694	+	0.996618i	$0.526185\pi$
$12$	−0.508535	−0.146801
$13$	−0.907468	−0.251686	−0.125843	−	0.992050i	$-0.540164\pi$
$13$	−0.907468	−0.251686	−0.125843	+	0.992050i	$0.540164\pi$
$14$	−6.92022	−1.84951
$15$	2.08123	0.537371
$16$	−4.34262	−1.08566
$17$	−5.99197	−1.45327	−0.726633	−	0.687026i	$-0.758916\pi$
$17$	−5.99197	−1.45327	−0.726633	+	0.687026i	$0.758916\pi$
$18$	6.24907	1.47292
$19$	4.38157	1.00520	0.502601	−	0.864519i	$-0.332377\pi$
$19$	4.38157	1.00520	0.502601	+	0.864519i	$0.332377\pi$
$20$	−0.146519	−0.0327626
$21$	12.5704	2.74309
$22$	−0.806456	−0.171937
$23$	1.04686	0.218286	0.109143	−	0.994026i	$-0.465189\pi$
$23$	1.04686	0.218286	0.109143	+	0.994026i	$0.465189\pi$
$24$	7.20087	1.46987
$25$	−4.40036	−0.880072
$26$	−1.34269	−0.263323
$27$	−3.28832	−0.632838
$28$	−0.884959	−0.167241
$29$	−1.23385	−0.229120	−0.114560	−	0.993416i	$-0.536546\pi$
$29$	−1.23385	−0.229120	−0.114560	+	0.993416i	$0.536546\pi$
$30$	3.07938	0.562215
$31$	1.52019	0.273035	0.136517	−	0.990638i	$-0.456409\pi$
$31$	1.52019	0.273035	0.136517	+	0.990638i	$0.456409\pi$
$32$	−1.06686	−0.188595
$33$	1.46491	0.255008
$34$	−8.86571	−1.52046
$35$	3.62178	0.612192
$36$	0.799132	0.133189
$37$	−7.25720	−1.19308	−0.596539	−	0.802584i	$-0.703458\pi$
$37$	−7.25720	−1.19308	−0.596539	+	0.802584i	$0.703458\pi$
$38$	6.48297	1.05168
$39$	2.43896	0.390546
$40$	2.07471	0.328041
$41$	0	0
$41$	0	0
$42$	18.5992	2.86991
$43$	6.22101	0.948696	0.474348	−	0.880338i	$-0.342684\pi$
$43$	6.22101	0.948696	0.474348	+	0.880338i	$0.342684\pi$
$44$	−0.103130	−0.0155474
$45$	−3.27052	−0.487541
$46$	1.54893	0.228378
$47$	3.97043	0.579146	0.289573	−	0.957156i	$-0.406487\pi$
$47$	3.97043	0.579146	0.289573	+	0.957156i	$0.406487\pi$
$48$	11.6715	1.68463
$49$	14.8752	2.12503
$50$	−6.51076	−0.920761
$51$	16.1043	2.25506
$52$	−0.171703	−0.0238109
$53$	5.35331	0.735334	0.367667	−	0.929957i	$-0.380157\pi$
$53$	5.35331	0.735334	0.367667	+	0.929957i	$0.380157\pi$
$54$	−4.86540	−0.662096
$55$	0.422068	0.0569117
$56$	12.5311	1.67453
$57$	−11.7762	−1.55979
$58$	−1.82560	−0.239713
$59$	8.87496	1.15542	0.577711	−	0.816242i	$-0.303946\pi$
$59$	8.87496	1.15542	0.577711	+	0.816242i	$0.303946\pi$
$60$	0.393792	0.0508383
$61$	1.38786	0.177698	0.0888489	−	0.996045i	$-0.471681\pi$
$61$	1.38786	0.177698	0.0888489	+	0.996045i	$0.471681\pi$
$62$	2.24927	0.285658
$63$	−19.7537	−2.48873
$64$	7.10673	0.888341
$65$	0.702711	0.0871606
$66$	2.16747	0.266798
$67$	5.63556	0.688493	0.344247	−	0.938879i	$-0.388134\pi$
$67$	5.63556	0.688493	0.344247	+	0.938879i	$0.388134\pi$
$68$	−1.13375	−0.137487
$69$	−2.81360	−0.338718
$70$	5.35878	0.640496
$71$	1.08971	0.129325	0.0646623	−	0.997907i	$-0.479403\pi$
$71$	1.08971	0.129325	0.0646623	+	0.997907i	$0.479403\pi$
$72$	−11.3157	−1.33357
$73$	−5.61291	−0.656941	−0.328471	−	0.944514i	$-0.606533\pi$
$73$	−5.61291	−0.656941	−0.328471	+	0.944514i	$0.606533\pi$
$74$	−10.7377	−1.24824
$75$	11.8266	1.36562
$76$	0.829043	0.0950977
$77$	2.54925	0.290514
$78$	3.60868	0.408603
$79$	−12.1959	−1.37214	−0.686072	−	0.727534i	$-0.740666\pi$
$79$	−12.1959	−1.37214	−0.686072	+	0.727534i	$0.740666\pi$
$80$	3.36278	0.375970
$81$	−3.83260	−0.425844
$82$	0	0
$83$	−13.1576	−1.44423	−0.722116	−	0.691772i	$-0.756830\pi$
$83$	−13.1576	−1.44423	−0.722116	+	0.691772i	$0.756830\pi$
$84$	2.37846	0.259512
$85$	4.63997	0.503276
$86$	9.20460	0.992558
$87$	3.31616	0.355530
$88$	1.46032	0.155671
$89$	1.95076	0.206781	0.103390	−	0.994641i	$-0.467031\pi$
$89$	1.95076	0.206781	0.103390	+	0.994641i	$0.467031\pi$
$90$	−4.83906	−0.510082
$91$	4.24431	0.444925
$92$	0.198078	0.0206510
$93$	−4.08575	−0.423673
$94$	5.87464	0.605922
$95$	−3.39294	−0.348108
$96$	2.86734	0.292647
$97$	−1.94271	−0.197252	−0.0986262	−	0.995125i	$-0.531445\pi$
$97$	−1.94271	−0.197252	−0.0986262	+	0.995125i	$0.531445\pi$
$98$	22.0093	2.22328
$99$	−2.30202	−0.231361
$100$	−0.832597	−0.0832597
$101$	8.41293	0.837117	0.418559	−	0.908190i	$-0.362535\pi$
$101$	8.41293	0.837117	0.418559	+	0.908190i	$0.362535\pi$
$102$	23.8280	2.35932
$103$	−4.29663	−0.423359	−0.211680	−	0.977339i	$-0.567893\pi$
$103$	−4.29663	−0.423359	−0.211680	+	0.977339i	$0.567893\pi$
$104$	2.43132	0.238411
$105$	−9.73409	−0.949950
$106$	7.92076	0.769332
$107$	−12.3843	−1.19723	−0.598617	−	0.801035i	$-0.704283\pi$
$107$	−12.3843	−1.19723	−0.598617	+	0.801035i	$0.704283\pi$
$108$	−0.622188	−0.0598700
$109$	−1.73022	−0.165725	−0.0828624	−	0.996561i	$-0.526406\pi$
$109$	−1.73022	−0.165725	−0.0828624	+	0.996561i	$0.526406\pi$
$110$	0.624491	0.0595429
$111$	19.5049	1.85132
$112$	20.3108	1.91919
$113$	1.16784	0.109861	0.0549305	−	0.998490i	$-0.482506\pi$
$113$	1.16784	0.109861	0.0549305	+	0.998490i	$0.482506\pi$
$114$	−17.4240	−1.63191
$115$	−0.810653	−0.0755937
$116$	−0.233458	−0.0216761
$117$	−3.83268	−0.354331
$118$	13.1314	1.20884
$119$	28.0250	2.56905
$120$	−5.57611	−0.509027
$121$	−10.7029	−0.972993
$122$	2.05348	0.185913
$123$	0	0
$124$	0.287638	0.0258306
$125$	7.27931	0.651081
$126$	−29.2275	−2.60379
$127$	9.99403	0.886827	0.443413	−	0.896317i	$-0.353767\pi$
$127$	9.99403	0.886827	0.443413	+	0.896317i	$0.353767\pi$
$128$	12.6488	1.11801
$129$	−16.7199	−1.47211
$130$	1.03973	0.0911904
$131$	0.814910	0.0711990	0.0355995	−	0.999366i	$-0.488666\pi$
$131$	0.814910	0.0711990	0.0355995	+	0.999366i	$0.488666\pi$
$132$	0.277177	0.0241252
$133$	−20.4930	−1.77697
$134$	8.33836	0.720325
$135$	2.54636	0.219156
$136$	16.0539	1.37661
$137$	−4.30393	−0.367710	−0.183855	−	0.982953i	$-0.558858\pi$
$137$	−4.30393	−0.367710	−0.183855	+	0.982953i	$0.558858\pi$
$138$	−4.16300	−0.354378
$139$	5.02414	0.426142	0.213071	−	0.977037i	$-0.431653\pi$
$139$	5.02414	0.426142	0.213071	+	0.977037i	$0.431653\pi$
$140$	0.685281	0.0579168
$141$	−10.6711	−0.898672
$142$	1.61233	0.135304
$143$	0.494616	0.0413618
$144$	−18.3410	−1.52842
$145$	0.955451	0.0793459
$146$	−8.30485	−0.687314
$147$	−39.9794	−3.29745
$148$	−1.37315	−0.112872
$149$	−7.92556	−0.649287	−0.324644	−	0.945836i	$-0.605244\pi$
$149$	−7.92556	−0.649287	−0.324644	+	0.945836i	$0.605244\pi$
$150$	17.4987	1.42876
$151$	20.6732	1.68236	0.841182	−	0.540752i	$-0.181860\pi$
$151$	20.6732	1.68236	0.841182	+	0.540752i	$0.181860\pi$
$152$	−11.7393	−0.952182
$153$	−25.3070	−2.04595
$154$	3.77187	0.303946
$155$	−1.17718	−0.0945537
$156$	0.461479	0.0369479
$157$	12.0702	0.963309	0.481654	−	0.876361i	$-0.340036\pi$
$157$	12.0702	0.963309	0.481654	+	0.876361i	$0.340036\pi$
$158$	−18.0450	−1.43558
$159$	−14.3879	−1.14103
$160$	0.826136	0.0653118
$161$	−4.89626	−0.385880
$162$	−5.67070	−0.445533
$163$	2.00089	0.156722	0.0783609	−	0.996925i	$-0.475031\pi$
$163$	2.00089	0.156722	0.0783609	+	0.996925i	$0.475031\pi$
$164$	0	0
$165$	−1.13437	−0.0883109
$166$	−19.4679	−1.51100
$167$	19.1232	1.47980	0.739899	−	0.672718i	$-0.234873\pi$
$167$	19.1232	1.47980	0.739899	+	0.672718i	$0.234873\pi$
$168$	−33.6792	−2.59840
$169$	−12.1765	−0.936654
$170$	6.86530	0.526544
$171$	18.5055	1.41515
$172$	1.17709	0.0897520
$173$	−6.00388	−0.456467	−0.228233	−	0.973606i	$-0.573295\pi$
$173$	−6.00388	−0.456467	−0.228233	+	0.973606i	$0.573295\pi$
$174$	4.90659	0.371968
$175$	20.5809	1.55577
$176$	2.36695	0.178415
$177$	−23.8528	−1.79289
$178$	2.88635	0.216341
$179$	−9.15427	−0.684222	−0.342111	−	0.939660i	$-0.611142\pi$
$179$	−9.15427	−0.684222	−0.342111	+	0.939660i	$0.611142\pi$
$180$	−0.618820	−0.0461241
$181$	−19.0537	−1.41625	−0.708125	−	0.706087i	$-0.750459\pi$
$181$	−19.0537	−1.41625	−0.708125	+	0.706087i	$0.750459\pi$
$182$	6.27987	0.465495
$183$	−3.73010	−0.275737
$184$	−2.80479	−0.206772
$185$	5.61973	0.413171
$186$	−6.04527	−0.443261
$187$	3.26593	0.238828
$188$	0.751250	0.0547905
$189$	15.3798	1.11871
$190$	−5.02019	−0.364203
$191$	−11.2658	−0.815166	−0.407583	−	0.913168i	$-0.633628\pi$
$191$	−11.2658	−0.815166	−0.407583	+	0.913168i	$0.633628\pi$
$192$	−19.1004	−1.37845
$193$	11.6925	0.841647	0.420823	−	0.907143i	$-0.361741\pi$
$193$	11.6925	0.841647	0.420823	+	0.907143i	$0.361741\pi$
$194$	−2.87443	−0.206372
$195$	−1.88865	−0.135249
$196$	2.81456	0.201040
$197$	−10.1058	−0.720008	−0.360004	−	0.932951i	$-0.617225\pi$
$197$	−10.1058	−0.720008	−0.360004	+	0.932951i	$0.617225\pi$
$198$	−3.40606	−0.242058
$199$	8.41556	0.596563	0.298281	−	0.954478i	$-0.403587\pi$
$199$	8.41556	0.596563	0.298281	+	0.954478i	$0.403587\pi$
$200$	11.7896	0.833652
$201$	−15.1464	−1.06835
$202$	12.4478	0.875821
$203$	5.77083	0.405033
$204$	3.04712	0.213341
$205$	0	0
$206$	−6.35728	−0.442933
$207$	4.42141	0.307309
$208$	3.94079	0.273245
$209$	−2.38818	−0.165194
$210$	−14.4025	−0.993870
$211$	12.9839	0.893851	0.446925	−	0.894571i	$-0.352519\pi$
$211$	12.9839	0.893851	0.446925	+	0.894571i	$0.352519\pi$
$212$	1.01291	0.0695668
$213$	−2.92876	−0.200675
$214$	−18.3238	−1.25259
$215$	−4.81734	−0.328540
$216$	8.81020	0.599458
$217$	−7.11008	−0.482664
$218$	−2.56003	−0.173387
$219$	15.0856	1.01939
$220$	0.0798601	0.00538416
$221$	5.43752	0.365767
$222$	28.8594	1.93691
$223$	−17.4981	−1.17176	−0.585881	−	0.810397i	$-0.699251\pi$
$223$	−17.4981	−1.17176	−0.585881	+	0.810397i	$0.699251\pi$
$224$	4.98978	0.333394
$225$	−18.5849	−1.23899
$226$	1.72793	0.114940
$227$	27.2403	1.80800	0.904002	−	0.427529i	$-0.140616\pi$
$227$	27.2403	1.80800	0.904002	+	0.427529i	$0.140616\pi$
$228$	−2.22818	−0.147565
$229$	22.6347	1.49574	0.747870	−	0.663845i	$-0.231077\pi$
$229$	22.6347	1.49574	0.747870	+	0.663845i	$0.231077\pi$
$230$	−1.19944	−0.0790888
$231$	−6.85151	−0.450796
$232$	3.30578	0.217035
$233$	−24.2104	−1.58607	−0.793036	−	0.609174i	$-0.791501\pi$
$233$	−24.2104	−1.58607	−0.793036	+	0.609174i	$0.791501\pi$
$234$	−5.67083	−0.370714
$235$	−3.07456	−0.200562
$236$	1.67924	0.109309
$237$	32.7783	2.12918
$238$	41.4657	2.68782
$239$	−22.2811	−1.44125	−0.720623	−	0.693327i	$-0.756144\pi$
$239$	−22.2811	−1.44125	−0.720623	+	0.693327i	$0.756144\pi$
$240$	−9.03798	−0.583399
$241$	12.0839	0.778394	0.389197	−	0.921155i	$-0.372753\pi$
$241$	12.0839	0.778394	0.389197	+	0.921155i	$0.372753\pi$
$242$	−15.8360	−1.01798
$243$	20.1657	1.29363
$244$	0.262599	0.0168112
$245$	−11.5188	−0.735911
$246$	0	0
$247$	−3.97613	−0.252995
$248$	−4.07296	−0.258633
$249$	35.3630	2.24104
$250$	10.7705	0.681184
$251$	7.94589	0.501540	0.250770	−	0.968047i	$-0.419316\pi$
$251$	7.94589	0.501540	0.250770	+	0.968047i	$0.419316\pi$
$252$	−3.73762	−0.235448
$253$	−0.570592	−0.0358728
$254$	14.7872	0.927828
$255$	−12.4706	−0.780942
$256$	4.50171	0.281357
$257$	−23.5058	−1.46625	−0.733125	−	0.680094i	$-0.761939\pi$
$257$	−23.5058	−1.46625	−0.733125	+	0.680094i	$0.761939\pi$
$258$	−24.7388	−1.54017
$259$	33.9426	2.10909
$260$	0.132961	0.00824589
$261$	−5.21115	−0.322562
$262$	1.20574	0.0744909
$263$	21.3787	1.31827	0.659134	−	0.752026i	$-0.270923\pi$
$263$	21.3787	1.31827	0.659134	+	0.752026i	$0.270923\pi$
$264$	−3.92484	−0.241557
$265$	−4.14542	−0.254651
$266$	−30.3214	−1.85913
$267$	−5.24298	−0.320865
$268$	1.06631	0.0651353
$269$	27.0347	1.64834	0.824169	−	0.566344i	$-0.191643\pi$
$269$	27.0347	1.64834	0.824169	+	0.566344i	$0.191643\pi$
$270$	3.76759	0.229288
$271$	11.9376	0.725158	0.362579	−	0.931953i	$-0.381896\pi$
$271$	11.9376	0.725158	0.362579	+	0.931953i	$0.381896\pi$
$272$	26.0209	1.57775
$273$	−11.4072	−0.690398
$274$	−6.36809	−0.384710
$275$	2.39842	0.144630
$276$	−0.532365	−0.0320446
$277$	−14.6668	−0.881242	−0.440621	−	0.897693i	$-0.645242\pi$
$277$	−14.6668	−0.881242	−0.440621	+	0.897693i	$0.645242\pi$
$278$	7.43371	0.445844
$279$	6.42052	0.384386
$280$	−9.70361	−0.579902
$281$	10.6965	0.638098	0.319049	−	0.947738i	$-0.396636\pi$
$281$	10.6965	0.638098	0.319049	+	0.947738i	$0.396636\pi$
$282$	−15.7890	−0.940221
$283$	−19.0733	−1.13379	−0.566894	−	0.823791i	$-0.691855\pi$
$283$	−19.0733	−1.13379	−0.566894	+	0.823791i	$0.691855\pi$
$284$	0.206185	0.0122348
$285$	9.11905	0.540166
$286$	0.731832	0.0432742
$287$	0	0
$288$	−4.50586	−0.265510
$289$	18.9037	1.11198
$290$	1.41368	0.0830143
$291$	5.22134	0.306080
$292$	−1.06203	−0.0621504
$293$	−25.9387	−1.51535	−0.757677	−	0.652629i	$-0.773666\pi$
$293$	−25.9387	−1.51535	−0.757677	+	0.652629i	$0.773666\pi$
$294$	−59.1535	−3.44990
$295$	−6.87246	−0.400130
$296$	19.4438	1.13015
$297$	1.79230	0.104000
$298$	−11.7266	−0.679306
$299$	−0.949992	−0.0549395
$300$	2.23773	0.129196
$301$	−29.0963	−1.67708
$302$	30.5881	1.76015
$303$	−22.6110	−1.29897
$304$	−19.0275	−1.09130
$305$	−1.07471	−0.0615379
$306$	−37.4442	−2.14054
$307$	−8.65704	−0.494084	−0.247042	−	0.969005i	$-0.579459\pi$
$307$	−8.65704	−0.494084	−0.247042	+	0.969005i	$0.579459\pi$
$308$	0.482347	0.0274843
$309$	11.5479	0.656935
$310$	−1.74176	−0.0989253
$311$	8.07904	0.458120	0.229060	−	0.973412i	$-0.426435\pi$
$311$	8.07904	0.458120	0.229060	+	0.973412i	$0.426435\pi$
$312$	−6.53456	−0.369947
$313$	3.45127	0.195077	0.0975387	−	0.995232i	$-0.468903\pi$
$313$	3.45127	0.195077	0.0975387	+	0.995232i	$0.468903\pi$
$314$	17.8591	1.00785
$315$	15.2965	0.861863
$316$	−2.30760	−0.129813
$317$	−27.9163	−1.56794	−0.783968	−	0.620801i	$-0.786807\pi$
$317$	−27.9163	−1.56794	−0.783968	+	0.620801i	$0.786807\pi$
$318$	−21.2883	−1.19379
$319$	0.672510	0.0376534
$320$	−5.50320	−0.307638
$321$	33.2847	1.85777
$322$	−7.24451	−0.403720
$323$	−26.2542	−1.46083
$324$	−0.725171	−0.0402873
$325$	3.99318	0.221502
$326$	2.96051	0.163968
$327$	4.65023	0.257158
$328$	0	0
$329$	−18.5701	−1.02380
$330$	−1.67842	−0.0923938
$331$	−22.5781	−1.24100	−0.620501	−	0.784205i	$-0.713071\pi$
$331$	−22.5781	−1.24100	−0.620501	+	0.784205i	$0.713071\pi$
$332$	−2.48956	−0.136632
$333$	−30.6507	−1.67965
$334$	28.2947	1.54822
$335$	−4.36398	−0.238430
$336$	−54.5885	−2.97805
$337$	24.7909	1.35045	0.675223	−	0.737614i	$-0.264048\pi$
$337$	24.7909	1.35045	0.675223	+	0.737614i	$0.264048\pi$
$338$	−18.0163	−0.979959
$339$	−3.13875	−0.170473
$340$	0.877935	0.0476127
$341$	−0.828581	−0.0448702
$342$	27.3808	1.48058
$343$	−36.8330	−1.98880
$344$	−16.6676	−0.898656
$345$	2.17875	0.117300
$346$	−8.88334	−0.477571
$347$	−3.59498	−0.192989	−0.0964943	−	0.995334i	$-0.530763\pi$
$347$	−3.59498	−0.192989	−0.0964943	+	0.995334i	$0.530763\pi$
$348$	0.627456	0.0336351
$349$	0.286874	0.0153560	0.00767800	−	0.999971i	$-0.497556\pi$
$349$	0.286874	0.0153560	0.00767800	+	0.999971i	$0.497556\pi$
$350$	30.4514	1.62770
$351$	2.98405	0.159277
$352$	0.581490	0.0309935
$353$	23.5120	1.25142	0.625708	−	0.780057i	$-0.284810\pi$
$353$	23.5120	1.25142	0.625708	+	0.780057i	$0.284810\pi$
$354$	−35.2926	−1.87578
$355$	−0.843832	−0.0447860
$356$	0.369107	0.0195626
$357$	−75.3215	−3.98644
$358$	−13.5446	−0.715857
$359$	34.3543	1.81315	0.906575	−	0.422045i	$-0.138687\pi$
$359$	34.3543	1.81315	0.906575	+	0.422045i	$0.138687\pi$
$360$	8.76252	0.461825
$361$	0.198176	0.0104303
$362$	−28.1918	−1.48173
$363$	28.7658	1.50981
$364$	0.803071	0.0420924
$365$	4.34644	0.227503
$366$	−5.51905	−0.288485
$367$	26.0833	1.36154	0.680769	−	0.732498i	$-0.261646\pi$
$367$	26.0833	1.36154	0.680769	+	0.732498i	$0.261646\pi$
$368$	−4.54612	−0.236983
$369$	0	0
$370$	8.31494	0.432273
$371$	−25.0380	−1.29991
$372$	−0.773070	−0.0400818
$373$	10.0988	0.522894	0.261447	−	0.965218i	$-0.415800\pi$
$373$	10.0988	0.522894	0.261447	+	0.965218i	$0.415800\pi$
$374$	4.83226	0.249870
$375$	−19.5643	−1.01030
$376$	−10.6377	−0.548599
$377$	1.11968	0.0576664
$378$	22.7559	1.17044
$379$	−13.2339	−0.679778	−0.339889	−	0.940466i	$-0.610389\pi$
$379$	−13.2339	−0.679778	−0.339889	+	0.940466i	$0.610389\pi$
$380$	−0.641982	−0.0329330
$381$	−26.8605	−1.37611
$382$	−16.6689	−0.852855
$383$	6.87869	0.351485	0.175742	−	0.984436i	$-0.443767\pi$
$383$	6.87869	0.351485	0.175742	+	0.984436i	$0.443767\pi$
$384$	−33.9956	−1.73483
$385$	−1.97405	−0.100607
$386$	17.3002	0.880559
$387$	26.2744	1.33560
$388$	−0.367583	−0.0186612
$389$	30.5169	1.54727	0.773633	−	0.633634i	$-0.218437\pi$
$389$	30.5169	1.54727	0.773633	+	0.633634i	$0.218437\pi$
$390$	−2.79444	−0.141502
$391$	−6.27276	−0.317227
$392$	−39.8542	−2.01294
$393$	−2.19020	−0.110481
$394$	−14.9525	−0.753296
$395$	9.44406	0.475182
$396$	−0.435567	−0.0218881
$397$	24.0343	1.20625	0.603124	−	0.797647i	$-0.293922\pi$
$397$	24.0343	1.20625	0.603124	+	0.797647i	$0.293922\pi$
$398$	12.4516	0.624145
$399$	55.0782	2.75736
$400$	19.1091	0.955455
$401$	−4.76630	−0.238018	−0.119009	−	0.992893i	$-0.537972\pi$
$401$	−4.76630	−0.238018	−0.119009	+	0.992893i	$0.537972\pi$
$402$	−22.4106	−1.11774
$403$	−1.37953	−0.0687190
$404$	1.59182	0.0791960
$405$	2.96783	0.147473
$406$	8.53851	0.423759
$407$	3.95554	0.196069
$408$	−43.1474	−2.13612
$409$	37.1120	1.83507	0.917534	−	0.397657i	$-0.130176\pi$
$409$	37.1120	1.83507	0.917534	+	0.397657i	$0.130176\pi$
$410$	0	0
$411$	11.5675	0.570582
$412$	−0.812971	−0.0400522
$413$	−41.5090	−2.04252
$414$	6.54191	0.321517
$415$	10.1888	0.500147
$416$	0.968137	0.0474668
$417$	−13.5031	−0.661252
$418$	−3.53354	−0.172831
$419$	11.2273	0.548491	0.274246	−	0.961660i	$-0.411572\pi$
$419$	11.2273	0.548491	0.274246	+	0.961660i	$0.411572\pi$
$420$	−1.84180	−0.0898706
$421$	−16.9472	−0.825956	−0.412978	−	0.910741i	$-0.635511\pi$
$421$	−16.9472	−0.825956	−0.412978	+	0.910741i	$0.635511\pi$
$422$	19.2110	0.935177
$423$	16.7691	0.815340
$424$	−14.3428	−0.696549
$425$	26.3668	1.27898
$426$	−4.33339	−0.209953
$427$	−6.49117	−0.314130
$428$	−2.34325	−0.113265
$429$	−1.32936	−0.0641819
$430$	−7.12772	−0.343729
$431$	20.3505	0.980248	0.490124	−	0.871653i	$-0.336952\pi$
$431$	20.3505	0.980248	0.490124	+	0.871653i	$0.336952\pi$
$432$	14.2799	0.687044
$433$	−21.1989	−1.01875	−0.509377	−	0.860543i	$-0.670124\pi$
$433$	−21.1989	−1.01875	−0.509377	+	0.860543i	$0.670124\pi$
$434$	−10.5201	−0.504979
$435$	−2.56792	−0.123122
$436$	−0.327377	−0.0156785
$437$	4.58690	0.219421
$438$	22.3206	1.06652
$439$	−20.6673	−0.986397	−0.493198	−	0.869917i	$-0.664172\pi$
$439$	−20.6673	−0.986397	−0.493198	+	0.869917i	$0.664172\pi$
$440$	−1.13082	−0.0539098
$441$	62.8253	2.99168
$442$	8.04534	0.382678
$443$	21.7029	1.03114	0.515568	−	0.856849i	$-0.327581\pi$
$443$	21.7029	1.03114	0.515568	+	0.856849i	$0.327581\pi$
$444$	3.69054	0.175145
$445$	−1.51060	−0.0716095
$446$	−25.8902	−1.22594
$447$	21.3012	1.00751
$448$	−33.2388	−1.57039
$449$	−5.55296	−0.262061	−0.131030	−	0.991378i	$-0.541829\pi$
$449$	−5.55296	−0.262061	−0.131030	+	0.991378i	$0.541829\pi$
$450$	−27.4981	−1.29628
$451$	0	0
$452$	0.220968	0.0103935
$453$	−55.5625	−2.61055
$454$	40.3047	1.89159
$455$	−3.28665	−0.154080
$456$	31.5511	1.47752
$457$	38.7172	1.81111	0.905557	−	0.424224i	$-0.139453\pi$
$457$	38.7172	1.81111	0.905557	+	0.424224i	$0.139453\pi$
$458$	33.4902	1.56489
$459$	19.7035	0.919682
$460$	−0.153385	−0.00715160
$461$	5.09040	0.237084	0.118542	−	0.992949i	$-0.462178\pi$
$461$	5.09040	0.237084	0.118542	+	0.992949i	$0.462178\pi$
$462$	−10.1375	−0.471638
$463$	−2.88369	−0.134017	−0.0670083	−	0.997752i	$-0.521345\pi$
$463$	−2.88369	−0.134017	−0.0670083	+	0.997752i	$0.521345\pi$
$464$	5.35814	0.248746
$465$	3.16387	0.146721
$466$	−35.8216	−1.65940
$467$	36.8563	1.70551	0.852753	−	0.522315i	$-0.174931\pi$
$467$	36.8563	1.70551	0.852753	+	0.522315i	$0.174931\pi$
$468$	−0.725186	−0.0335218
$469$	−26.3580	−1.21710
$470$	−4.54911	−0.209835
$471$	−32.4406	−1.49478
$472$	−23.7781	−1.09448
$473$	−3.39077	−0.155908
$474$	48.4987	2.22762
$475$	−19.2805	−0.884649
$476$	5.30265	0.243046
$477$	22.6097	1.03523
$478$	−32.9671	−1.50788
$479$	31.9019	1.45763	0.728817	−	0.684708i	$-0.240070\pi$
$479$	31.9019	1.45763	0.728817	+	0.684708i	$0.240070\pi$
$480$	−2.22037	−0.101346
$481$	6.58568	0.300281
$482$	17.8793	0.814382
$483$	13.1595	0.598777
$484$	−2.02511	−0.0920506
$485$	1.50437	0.0683099
$486$	29.8371	1.35344
$487$	−17.9320	−0.812578	−0.406289	−	0.913745i	$-0.633177\pi$
$487$	−17.9320	−0.812578	−0.406289	+	0.913745i	$0.633177\pi$
$488$	−3.71842	−0.168325
$489$	−5.37770	−0.243188
$490$	−17.0433	−0.769936
$491$	−25.0172	−1.12901	−0.564505	−	0.825430i	$-0.690933\pi$
$491$	−25.0172	−1.12901	−0.564505	+	0.825430i	$0.690933\pi$
$492$	0	0
$493$	7.39319	0.332973
$494$	−5.88308	−0.264692
$495$	1.78260	0.0801219
$496$	−6.60162	−0.296421
$497$	−5.09667	−0.228617
$498$	52.3231	2.34465
$499$	−25.8218	−1.15594	−0.577970	−	0.816058i	$-0.696155\pi$
$499$	−25.8218	−1.15594	−0.577970	+	0.816058i	$0.696155\pi$
$500$	1.37733	0.0615960
$501$	−51.3966	−2.29623
$502$	11.7567	0.524728
$503$	−18.4128	−0.820985	−0.410493	−	0.911864i	$-0.634643\pi$
$503$	−18.4128	−0.820985	−0.410493	+	0.911864i	$0.634643\pi$
$504$	52.9248	2.35746
$505$	−6.51468	−0.289899
$506$	−0.844247	−0.0375313
$507$	32.7262	1.45342
$508$	1.89098	0.0838988
$509$	18.5528	0.822338	0.411169	−	0.911559i	$-0.365121\pi$
$509$	18.5528	0.822338	0.411169	+	0.911559i	$0.365121\pi$
$510$	−18.4516	−0.817048
$511$	26.2521	1.16132
$512$	−18.6369	−0.823642
$513$	−14.4080	−0.636130
$514$	−34.7791	−1.53404
$515$	3.32716	0.146612
$516$	−3.16360	−0.139270
$517$	−2.16408	−0.0951762
$518$	50.2214	2.20660
$519$	16.1364	0.708308
$520$	−1.88273	−0.0825633
$521$	8.23340	0.360712	0.180356	−	0.983601i	$-0.442275\pi$
$521$	8.23340	0.360712	0.180356	+	0.983601i	$0.442275\pi$
$522$	−7.71041	−0.337476
$523$	−34.2863	−1.49924	−0.749618	−	0.661870i	$-0.769763\pi$
$523$	−34.2863	−1.49924	−0.749618	+	0.661870i	$0.769763\pi$
$524$	0.154190	0.00673583
$525$	−55.3143	−2.41411
$526$	31.6319	1.37922
$527$	−9.10894	−0.396792
$528$	−6.36154	−0.276850
$529$	−21.9041	−0.952351
$530$	−6.13356	−0.266425
$531$	37.4833	1.62664
$532$	−3.87751	−0.168111
$533$	0	0
$534$	−7.75751	−0.335700
$535$	9.58996	0.414610
$536$	−15.0990	−0.652178
$537$	24.6035	1.06172
$538$	40.0006	1.72455
$539$	−8.10773	−0.349225
$540$	0.481800	0.0207334
$541$	−0.834839	−0.0358925	−0.0179463	−	0.999839i	$-0.505713\pi$
$541$	−0.834839	−0.0358925	−0.0179463	+	0.999839i	$0.505713\pi$
$542$	17.6629	0.758685
$543$	51.2097	2.19762
$544$	6.39257	0.274079
$545$	1.33982	0.0573916
$546$	−16.8781	−0.722318
$547$	31.9375	1.36555	0.682775	−	0.730629i	$-0.260773\pi$
$547$	31.9375	1.36555	0.682775	+	0.730629i	$0.260773\pi$
$548$	−0.814353	−0.0347874
$549$	5.86163	0.250168
$550$	3.54869	0.151317
$551$	−5.40620	−0.230312
$552$	7.53831	0.320852
$553$	57.0412	2.42564
$554$	−21.7010	−0.921985
$555$	−15.1039	−0.641124
$556$	0.950624	0.0403154
$557$	27.8329	1.17932	0.589660	−	0.807652i	$-0.299262\pi$
$557$	27.8329	1.17932	0.589660	+	0.807652i	$0.299262\pi$
$558$	9.49979	0.402158
$559$	−5.64537	−0.238774
$560$	−15.7280	−0.664630
$561$	−8.77768	−0.370594
$562$	15.8265	0.667600
$563$	29.5266	1.24440	0.622200	−	0.782859i	$-0.286239\pi$
$563$	29.5266	1.24440	0.622200	+	0.782859i	$0.286239\pi$
$564$	−2.01910	−0.0850194
$565$	−0.904334	−0.0380456
$566$	−28.2208	−1.18621
$567$	17.9254	0.752797
$568$	−2.91959	−0.122503
$569$	28.7842	1.20670	0.603348	−	0.797478i	$-0.293833\pi$
$569$	28.7842	1.20670	0.603348	+	0.797478i	$0.293833\pi$
$570$	13.4925	0.565140
$571$	27.3640	1.14515	0.572574	−	0.819853i	$-0.305945\pi$
$571$	27.3640	1.14515	0.572574	+	0.819853i	$0.305945\pi$
$572$	0.0935868	0.00391306
$573$	30.2786	1.26491
$574$	0	0
$575$	−4.60656	−0.192107
$576$	30.0152	1.25063
$577$	9.05413	0.376929	0.188464	−	0.982080i	$-0.439649\pi$
$577$	9.05413	0.376929	0.188464	+	0.982080i	$0.439649\pi$
$578$	27.9699	1.16339
$579$	−31.4255	−1.30600
$580$	0.180782	0.00750657
$581$	61.5392	2.55308
$582$	7.72548	0.320232
$583$	−2.91783	−0.120844
$584$	15.0383	0.622291
$585$	2.96790	0.122707
$586$	−38.3788	−1.58542
$587$	−5.18703	−0.214091	−0.107046	−	0.994254i	$-0.534139\pi$
$587$	−5.18703	−0.214091	−0.107046	+	0.994254i	$0.534139\pi$
$588$	−7.56455	−0.311957
$589$	6.66083	0.274455
$590$	−10.1685	−0.418630
$591$	27.1609	1.11725
$592$	31.5153	1.29527
$593$	−23.4369	−0.962437	−0.481219	−	0.876601i	$-0.659806\pi$
$593$	−23.4369	−0.962437	−0.481219	+	0.876601i	$0.659806\pi$
$594$	2.65189	0.108808
$595$	−21.7016	−0.889678
$596$	−1.49961	−0.0614262
$597$	−22.6181	−0.925698
$598$	−1.40561	−0.0574795
$599$	30.6974	1.25426	0.627130	−	0.778914i	$-0.284229\pi$
$599$	30.6974	1.25426	0.627130	+	0.778914i	$0.284229\pi$
$600$	−31.6864	−1.29359
$601$	−7.38136	−0.301092	−0.150546	−	0.988603i	$-0.548103\pi$
$601$	−7.38136	−0.301092	−0.150546	+	0.988603i	$0.548103\pi$
$602$	−43.0508	−1.75462
$603$	23.8017	0.969281
$604$	3.91161	0.159161
$605$	8.28797	0.336954
$606$	−33.4553	−1.35903
$607$	31.3718	1.27334	0.636671	−	0.771136i	$-0.280311\pi$
$607$	31.3718	1.27334	0.636671	+	0.771136i	$0.280311\pi$
$608$	−4.67451	−0.189576
$609$	−15.5100	−0.628497
$610$	−1.59014	−0.0643831
$611$	−3.60303	−0.145763
$612$	−4.78838	−0.193559
$613$	17.6297	0.712059	0.356029	−	0.934475i	$-0.384130\pi$
$613$	17.6297	0.712059	0.356029	+	0.934475i	$0.384130\pi$
$614$	−12.8089	−0.516927
$615$	0	0
$616$	−6.83006	−0.275191
$617$	8.04598	0.323919	0.161959	−	0.986797i	$-0.448219\pi$
$617$	8.04598	0.323919	0.161959	+	0.986797i	$0.448219\pi$
$618$	17.0862	0.687307
$619$	−6.35587	−0.255464	−0.127732	−	0.991809i	$-0.540770\pi$
$619$	−6.35587	−0.255464	−0.127732	+	0.991809i	$0.540770\pi$
$620$	−0.222737	−0.00894531
$621$	−3.44241	−0.138139
$622$	11.9537	0.479301
$623$	−9.12391	−0.365542
$624$	−10.5915	−0.423999
$625$	16.3649	0.654598
$626$	5.10649	0.204097
$627$	6.41860	0.256334
$628$	2.28382	0.0911345
$629$	43.4849	1.73386
$630$	22.6327	0.901710
$631$	4.18327	0.166533	0.0832667	−	0.996527i	$-0.473465\pi$
$631$	4.18327	0.166533	0.0832667	+	0.996527i	$0.473465\pi$
$632$	32.6757	1.29977
$633$	−34.8963	−1.38700
$634$	−41.3049	−1.64043
$635$	−7.73903	−0.307114
$636$	−2.72235	−0.107948
$637$	−13.4988	−0.534840
$638$	0.995045	0.0393942
$639$	4.60237	0.182067
$640$	−9.79480	−0.387173
$641$	−19.6147	−0.774734	−0.387367	−	0.921926i	$-0.626615\pi$
$641$	−19.6147	−0.774734	−0.387367	+	0.921926i	$0.626615\pi$
$642$	49.2480	1.94366
$643$	11.1171	0.438416	0.219208	−	0.975678i	$-0.429653\pi$
$643$	11.1171	0.438416	0.219208	+	0.975678i	$0.429653\pi$
$644$	−0.926429	−0.0365064
$645$	12.9473	0.509801
$646$	−38.8457	−1.52837
$647$	10.4368	0.410313	0.205157	−	0.978729i	$-0.434230\pi$
$647$	10.4368	0.410313	0.205157	+	0.978729i	$0.434230\pi$
$648$	10.2684	0.403383
$649$	−4.83730	−0.189881
$650$	5.90831	0.231743
$651$	19.1094	0.748958
$652$	0.378591	0.0148268
$653$	−26.9884	−1.05614	−0.528069	−	0.849202i	$-0.677084\pi$
$653$	−26.9884	−1.05614	−0.528069	+	0.849202i	$0.677084\pi$
$654$	6.88047	0.269048
$655$	−0.631038	−0.0246567
$656$	0	0
$657$	−23.7061	−0.924862
$658$	−27.4762	−1.07113
$659$	−15.3561	−0.598190	−0.299095	−	0.954223i	$-0.596685\pi$
$659$	−15.3561	−0.598190	−0.299095	+	0.954223i	$0.596685\pi$
$660$	−0.214636	−0.00835471
$661$	0.339382	0.0132004	0.00660021	−	0.999978i	$-0.497899\pi$
$661$	0.339382	0.0132004	0.00660021	+	0.999978i	$0.497899\pi$
$662$	−33.4065	−1.29838
$663$	−14.6142	−0.567567
$664$	35.2523	1.36806
$665$	15.8691	0.615377
$666$	−45.3508	−1.75731
$667$	−1.29167	−0.0500136
$668$	3.61833	0.139997
$669$	47.0290	1.81824
$670$	−6.45694	−0.249453
$671$	−0.756456	−0.0292026
$672$	−13.4108	−0.517334
$673$	−12.5261	−0.482847	−0.241424	−	0.970420i	$-0.577614\pi$
$673$	−12.5261	−0.482847	−0.241424	+	0.970420i	$0.577614\pi$
$674$	36.6805	1.41288
$675$	14.4698	0.556943
$676$	−2.30393	−0.0886128
$677$	29.2489	1.12413	0.562063	−	0.827094i	$-0.310008\pi$
$677$	29.2489	1.12413	0.562063	+	0.827094i	$0.310008\pi$
$678$	−4.64409	−0.178355
$679$	9.08624	0.348698
$680$	−12.4316	−0.476730
$681$	−73.2126	−2.80551
$682$	−1.22597	−0.0469447
$683$	−23.6545	−0.905115	−0.452558	−	0.891735i	$-0.649488\pi$
$683$	−23.6545	−0.905115	−0.452558	+	0.891735i	$0.649488\pi$
$684$	3.50146	0.133881
$685$	3.33282	0.127340
$686$	−54.4981	−2.08075
$687$	−60.8342	−2.32097
$688$	−27.0155	−1.02996
$689$	−4.85796	−0.185074
$690$	3.22368	0.122723
$691$	−16.7116	−0.635739	−0.317870	−	0.948134i	$-0.602967\pi$
$691$	−16.7116	−0.635739	−0.317870	+	0.948134i	$0.602967\pi$
$692$	−1.13600	−0.0431843
$693$	10.7667	0.408995
$694$	−5.31912	−0.201911
$695$	−3.89052	−0.147576
$696$	−8.88480	−0.336777
$697$	0	0
$698$	0.424458	0.0160660
$699$	65.0691	2.46114
$700$	3.89414	0.147184
$701$	−35.8707	−1.35482	−0.677409	−	0.735607i	$-0.736897\pi$
$701$	−35.8707	−1.35482	−0.677409	+	0.735607i	$0.736897\pi$
$702$	4.41519	0.166641
$703$	−31.7980	−1.19928
$704$	−3.87352	−0.145989
$705$	8.26336	0.311216
$706$	34.7883	1.30927
$707$	−39.3480	−1.47984
$708$	−4.51323	−0.169617
$709$	18.5313	0.695958	0.347979	−	0.937502i	$-0.386868\pi$
$709$	18.5313	0.695958	0.347979	+	0.937502i	$0.386868\pi$
$710$	−1.24853	−0.0468566
$711$	−51.5092	−1.93174
$712$	−5.22657	−0.195874
$713$	1.59143	0.0595995
$714$	−111.446	−4.17075
$715$	−0.383013	−0.0143239
$716$	−1.73209	−0.0647313
$717$	59.8840	2.23641
$718$	50.8305	1.89698
$719$	29.8646	1.11376	0.556882	−	0.830592i	$-0.311998\pi$
$719$	29.8646	1.11376	0.556882	+	0.830592i	$0.311998\pi$
$720$	14.2027	0.529302
$721$	20.0957	0.748404
$722$	0.293222	0.0109126
$723$	−32.4774	−1.20785
$724$	−3.60517	−0.133985
$725$	5.42938	0.201642
$726$	42.5618	1.57962
$727$	−5.10520	−0.189342	−0.0946708	−	0.995509i	$-0.530180\pi$
$727$	−5.10520	−0.189342	−0.0946708	+	0.995509i	$0.530180\pi$
$728$	−11.3715	−0.421457
$729$	−42.7006	−1.58150
$730$	6.43099	0.238022
$731$	−37.2761	−1.37871
$732$	−0.705777	−0.0260863
$733$	19.6414	0.725471	0.362735	−	0.931892i	$-0.381843\pi$
$733$	19.6414	0.725471	0.362735	+	0.931892i	$0.381843\pi$
$734$	38.5928	1.42449
$735$	30.9587	1.14193
$736$	−1.11685	−0.0411676
$737$	−3.07166	−0.113146
$738$	0	0
$739$	21.4364	0.788552	0.394276	−	0.918992i	$-0.370995\pi$
$739$	21.4364	0.788552	0.394276	+	0.918992i	$0.370995\pi$
$740$	1.06332	0.0390883
$741$	10.6865	0.392578
$742$	−37.0461	−1.36001
$743$	26.6988	0.979485	0.489742	−	0.871867i	$-0.337091\pi$
$743$	26.6988	0.979485	0.489742	+	0.871867i	$0.337091\pi$
$744$	10.9467	0.401326
$745$	6.13728	0.224852
$746$	14.9421	0.547069
$747$	−55.5709	−2.03323
$748$	0.617950	0.0225945
$749$	57.9225	2.11644
$750$	−28.9473	−1.05701
$751$	−18.3027	−0.667876	−0.333938	−	0.942595i	$-0.608378\pi$
$751$	−18.3027	−0.667876	−0.333938	+	0.942595i	$0.608378\pi$
$752$	−17.2421	−0.628753
$753$	−21.3558	−0.778249
$754$	1.65667	0.0603325
$755$	−16.0086	−0.582614
$756$	2.91003	0.105837
$757$	17.8721	0.649571	0.324785	−	0.945788i	$-0.394708\pi$
$757$	17.8721	0.649571	0.324785	+	0.945788i	$0.394708\pi$
$758$	−19.5808	−0.711207
$759$	1.53355	0.0556645
$760$	9.09049	0.329747
$761$	−6.49099	−0.235298	−0.117649	−	0.993055i	$-0.537536\pi$
$761$	−6.49099	−0.235298	−0.117649	+	0.993055i	$0.537536\pi$
$762$	−39.7428	−1.43973
$763$	8.09239	0.292964
$764$	−2.13162	−0.0771193
$765$	19.5969	0.708527
$766$	10.1777	0.367735
$767$	−8.05374	−0.290804
$768$	−12.0990	−0.436587
$769$	14.8889	0.536907	0.268454	−	0.963293i	$-0.413487\pi$
$769$	14.8889	0.536907	0.268454	+	0.963293i	$0.413487\pi$
$770$	−2.92080	−0.105258
$771$	63.1754	2.27521
$772$	2.21236	0.0796245
$773$	−33.2262	−1.19506	−0.597532	−	0.801845i	$-0.703852\pi$
$773$	−33.2262	−1.19506	−0.597532	+	0.801845i	$0.703852\pi$
$774$	38.8755	1.39735
$775$	−6.68939	−0.240290
$776$	5.20499	0.186848
$777$	−91.2261	−3.27272
$778$	45.1527	1.61880
$779$	0	0
$780$	−0.357353	−0.0127953
$781$	−0.593946	−0.0212531
$782$	−9.28116	−0.331894
$783$	4.05730	0.144996
$784$	−64.5974	−2.30705
$785$	−9.34676	−0.333600
$786$	−3.24061	−0.115589
$787$	−51.6698	−1.84183	−0.920914	−	0.389765i	$-0.872556\pi$
$787$	−51.6698	−1.84183	−0.920914	+	0.389765i	$0.872556\pi$
$788$	−1.91213	−0.0681168
$789$	−57.4586	−2.04558
$790$	13.9734	0.497152
$791$	−5.46209	−0.194210
$792$	6.16765	0.219158
$793$	−1.25944	−0.0447241
$794$	35.5611	1.26202
$795$	11.1415	0.395147
$796$	1.59232	0.0564382
$797$	14.3536	0.508431	0.254216	−	0.967148i	$-0.418183\pi$
$797$	14.3536	0.508431	0.254216	+	0.967148i	$0.418183\pi$
$798$	81.4936	2.88484
$799$	−23.7907	−0.841654
$800$	4.69455	0.165977
$801$	8.23904	0.291112
$802$	−7.05221	−0.249022
$803$	3.05932	0.107961
$804$	−2.86588	−0.101072
$805$	3.79150	0.133633
$806$	−2.04114	−0.0718962
$807$	−72.6601	−2.55776
$808$	−22.5402	−0.792963
$809$	−44.1719	−1.55300	−0.776501	−	0.630116i	$-0.783007\pi$
$809$	−44.1719	−1.55300	−0.776501	+	0.630116i	$0.783007\pi$
$810$	4.39120	0.154291
$811$	27.7113	0.973074	0.486537	−	0.873660i	$-0.338260\pi$
$811$	27.7113	0.973074	0.486537	+	0.873660i	$0.338260\pi$
$812$	1.09191	0.0383184
$813$	−32.0842	−1.12524
$814$	5.85261	0.205134
$815$	−1.54942	−0.0542738
$816$	−69.9351	−2.44822
$817$	27.2578	0.953630
$818$	54.9108	1.91991
$819$	17.9258	0.626378
$820$	0	0
$821$	11.1582	0.389424	0.194712	−	0.980860i	$-0.437623\pi$
$821$	11.1582	0.389424	0.194712	+	0.980860i	$0.437623\pi$
$822$	17.1152	0.596962
$823$	−40.3927	−1.40800	−0.704000	−	0.710200i	$-0.748604\pi$
$823$	−40.3927	−1.40800	−0.704000	+	0.710200i	$0.748604\pi$
$824$	11.5117	0.401029
$825$	−6.44612	−0.224425
$826$	−61.4167	−2.13696
$827$	2.11653	0.0735990	0.0367995	−	0.999323i	$-0.488284\pi$
$827$	2.11653	0.0735990	0.0367995	+	0.999323i	$0.488284\pi$
$828$	0.836580	0.0290732
$829$	−22.5376	−0.782765	−0.391382	−	0.920228i	$-0.628003\pi$
$829$	−22.5376	−0.782765	−0.391382	+	0.920228i	$0.628003\pi$
$830$	15.0753	0.523271
$831$	39.4193	1.36744
$832$	−6.44912	−0.223583
$833$	−89.1317	−3.08823
$834$	−19.9792	−0.691824
$835$	−14.8083	−0.512464
$836$	−0.451870	−0.0156283
$837$	−4.99888	−0.172787
$838$	16.6119	0.573850
$839$	30.3508	1.04783	0.523913	−	0.851772i	$-0.324472\pi$
$839$	30.3508	1.04783	0.523913	+	0.851772i	$0.324472\pi$
$840$	26.0800	0.899844
$841$	−27.4776	−0.947504
$842$	−25.0750	−0.864143
$843$	−28.7484	−0.990149
$844$	2.45671	0.0845633
$845$	9.42906	0.324370
$846$	24.8115	0.853036
$847$	50.0585	1.72003
$848$	−23.2474	−0.798320
$849$	51.2624	1.75932
$850$	39.0123	1.33811
$851$	−7.59728	−0.260431
$852$	−0.554154	−0.0189850
$853$	−18.6542	−0.638708	−0.319354	−	0.947635i	$-0.603466\pi$
$853$	−18.6542	−0.638708	−0.319354	+	0.947635i	$0.603466\pi$
$854$	−9.60432	−0.328653
$855$	−14.3300	−0.490077
$856$	33.1805	1.13409
$857$	48.1230	1.64385	0.821926	−	0.569595i	$-0.192900\pi$
$857$	48.1230	1.64385	0.821926	+	0.569595i	$0.192900\pi$
$858$	−1.96691	−0.0671493
$859$	−17.9691	−0.613098	−0.306549	−	0.951855i	$-0.599174\pi$
$859$	−17.9691	−0.613098	−0.306549	+	0.951855i	$0.599174\pi$
$860$	−0.911495	−0.0310817
$861$	0	0
$862$	30.1105	1.02557
$863$	33.8047	1.15073	0.575363	−	0.817898i	$-0.304861\pi$
$863$	33.8047	1.15073	0.575363	+	0.817898i	$0.304861\pi$
$864$	3.50817	0.119350
$865$	4.64920	0.158078
$866$	−31.3659	−1.06586
$867$	−50.8066	−1.72548
$868$	−1.34531	−0.0456627
$869$	6.64737	0.225496
$870$	−3.79949	−0.128815
$871$	−5.11409	−0.173284
$872$	4.63567	0.156983
$873$	−8.20502	−0.277698
$874$	6.78676	0.229566
$875$	−34.0460	−1.15097
$876$	2.85436	0.0964399
$877$	−27.0528	−0.913507	−0.456753	−	0.889593i	$-0.650988\pi$
$877$	−27.0528	−0.913507	−0.456753	+	0.889593i	$0.650988\pi$
$878$	−30.5793	−1.03200
$879$	69.7143	2.35140
$880$	−1.83288	−0.0617864
$881$	−16.7259	−0.563510	−0.281755	−	0.959486i	$-0.590916\pi$
$881$	−16.7259	−0.563510	−0.281755	+	0.959486i	$0.590916\pi$
$882$	92.9561	3.13000
$883$	−25.5414	−0.859536	−0.429768	−	0.902939i	$-0.641405\pi$
$883$	−25.5414	−0.859536	−0.429768	+	0.902939i	$0.641405\pi$
$884$	1.02884	0.0346036
$885$	18.4708	0.620889
$886$	32.1116	1.07881
$887$	−40.0127	−1.34349	−0.671747	−	0.740780i	$-0.734456\pi$
$887$	−40.0127	−1.34349	−0.671747	+	0.740780i	$0.734456\pi$
$888$	−52.2582	−1.75367
$889$	−46.7430	−1.56771
$890$	−2.23509	−0.0749203
$891$	2.08896	0.0699827
$892$	−3.31085	−0.110855
$893$	17.3967	0.582159
$894$	31.5172	1.05409
$895$	7.08875	0.236951
$896$	−59.1597	−1.97639
$897$	2.55325	0.0852506
$898$	−8.21615	−0.274177
$899$	−1.87569	−0.0625577
$900$	−3.51647	−0.117216
$901$	−32.0769	−1.06864
$902$	0	0
$903$	78.2007	2.60236
$904$	−3.12892	−0.104066
$905$	14.7545	0.490457
$906$	−82.2102	−2.73125
$907$	−24.3470	−0.808428	−0.404214	−	0.914664i	$-0.632455\pi$
$907$	−24.3470	−0.808428	−0.404214	+	0.914664i	$0.632455\pi$
$908$	5.15418	0.171047
$909$	35.5319	1.17852
$910$	−4.86292	−0.161204
$911$	24.0894	0.798118	0.399059	−	0.916925i	$-0.369337\pi$
$911$	24.0894	0.798118	0.399059	+	0.916925i	$0.369337\pi$
$912$	51.1394	1.69339
$913$	7.17154	0.237343
$914$	57.2859	1.89485
$915$	2.88846	0.0954895
$916$	4.28273	0.141505
$917$	−3.81141	−0.125864
$918$	29.1533	0.962202
$919$	−7.19903	−0.237474	−0.118737	−	0.992926i	$-0.537885\pi$
$919$	−7.19903	−0.237474	−0.118737	+	0.992926i	$0.537885\pi$
$920$	2.17193	0.0716065
$921$	23.2672	0.766679
$922$	7.53175	0.248045
$923$	−0.988875	−0.0325492
$924$	−1.29638	−0.0426479
$925$	31.9343	1.04999
$926$	−4.26671	−0.140213
$927$	−18.1468	−0.596018
$928$	1.31634	0.0432110
$929$	−6.89462	−0.226205	−0.113102	−	0.993583i	$-0.536079\pi$
$929$	−6.89462	−0.226205	−0.113102	+	0.993583i	$0.536079\pi$
$930$	4.68125	0.153504
$931$	65.1768	2.13608
$932$	−4.58087	−0.150051
$933$	−21.7137	−0.710873
$934$	54.5325	1.78436
$935$	−2.52902	−0.0827078
$936$	10.2687	0.335642
$937$	57.9860	1.89432	0.947160	−	0.320762i	$-0.103939\pi$
$937$	57.9860	1.89432	0.947160	+	0.320762i	$0.103939\pi$
$938$	−38.9993	−1.27337
$939$	−9.27583	−0.302705
$940$	−0.581742	−0.0189743
$941$	40.7464	1.32829	0.664147	−	0.747602i	$-0.268795\pi$
$941$	40.7464	1.32829	0.664147	+	0.747602i	$0.268795\pi$
$942$	−47.9991	−1.56389
$943$	0	0
$944$	−38.5406	−1.25439
$945$	−11.9096	−0.387418
$946$	−5.01697	−0.163116
$947$	−28.0328	−0.910945	−0.455472	−	0.890250i	$-0.650530\pi$
$947$	−28.0328	−0.910945	−0.455472	+	0.890250i	$0.650530\pi$
$948$	6.20203	0.201432
$949$	5.09353	0.165343
$950$	−28.5274	−0.925550
$951$	75.0294	2.43300
$952$	−75.0857	−2.43354
$953$	38.5353	1.24828	0.624140	−	0.781312i	$-0.285449\pi$
$953$	38.5353	1.24828	0.624140	+	0.781312i	$0.285449\pi$
$954$	33.4532	1.08309
$955$	8.72386	0.282298
$956$	−4.21584	−0.136350
$957$	−1.80748	−0.0584274
$958$	47.2020	1.52503
$959$	20.1299	0.650028
$960$	14.7907	0.477368
$961$	−28.6890	−0.925452
$962$	9.74416	0.314164
$963$	−52.3049	−1.68550
$964$	2.28641	0.0736404
$965$	−9.05429	−0.291468
$966$	19.4707	0.626461
$967$	−42.2286	−1.35798	−0.678990	−	0.734147i	$-0.737582\pi$
$967$	−42.2286	−1.35798	−0.678990	+	0.734147i	$0.737582\pi$
$968$	28.6757	0.921672
$969$	70.5624	2.26679
$970$	2.22586	0.0714681
$971$	2.08564	0.0669315	0.0334657	−	0.999440i	$-0.489346\pi$
$971$	2.08564	0.0669315	0.0334657	+	0.999440i	$0.489346\pi$
$972$	3.81557	0.122385
$973$	−23.4984	−0.753323
$974$	−26.5322	−0.850146
$975$	−10.7323	−0.343709
$976$	−6.02697	−0.192918
$977$	29.8162	0.953904	0.476952	−	0.878929i	$-0.341742\pi$
$977$	29.8162	0.953904	0.476952	+	0.878929i	$0.341742\pi$
$978$	−7.95684	−0.254432
$979$	−1.06326	−0.0339821
$980$	−2.17949	−0.0696214
$981$	−7.30756	−0.233312
$982$	−37.0154	−1.18121
$983$	36.2484	1.15615	0.578073	−	0.815985i	$-0.303805\pi$
$983$	36.2484	1.15615	0.578073	+	0.815985i	$0.303805\pi$
$984$	0	0
$985$	7.82557	0.249343
$986$	10.9390	0.348367
$987$	49.9099	1.58865
$988$	−0.752330	−0.0239348
$989$	6.51253	0.207087
$990$	2.63753	0.0838263
$991$	−13.5186	−0.429433	−0.214717	−	0.976676i	$-0.568883\pi$
$991$	−13.5186	−0.429433	−0.214717	+	0.976676i	$0.568883\pi$
$992$	−1.62183	−0.0514930
$993$	60.6821	1.92569
$994$	−7.54102	−0.239187
$995$	−6.51672	−0.206594
$996$	6.69109	0.212015
$997$	27.8577	0.882262	0.441131	−	0.897443i	$-0.354577\pi$
$997$	27.8577	0.882262	0.441131	+	0.897443i	$0.354577\pi$
$998$	−38.2058	−1.20938
$999$	23.8640	0.755024

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1681.2.a.m.1.17		24
41.12	odd	40		41.2.g.a.21.2	yes	24
41.24	odd	40		41.2.g.a.2.2	✓	24
41.40	even	2	inner	1681.2.a.m.1.18		24
123.53	even	40		369.2.u.a.226.2		24
123.65	even	40		369.2.u.a.289.2		24
164.135	even	40		656.2.bs.d.513.3		24
164.147	even	40		656.2.bs.d.289.3		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.2.g.a.2.2	✓	24	41.24	odd	40
41.2.g.a.21.2	yes	24	41.12	odd	40
369.2.u.a.226.2		24	123.53	even	40
369.2.u.a.289.2		24	123.65	even	40
656.2.bs.d.289.3		24	164.147	even	40
656.2.bs.d.513.3		24	164.135	even	40
1681.2.a.m.1.17		24	1.1	even	1	trivial
1681.2.a.m.1.18		24	41.40	even	2	inner

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:	\( N \)	\(=\)	\( 1681 = 41^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1681.a (trivial)

\(f(q)\)	\(=\)	\(q+1.47960 q^{2} -2.68766 q^{3} +0.189211 q^{4} -0.774365 q^{5} -3.97665 q^{6} -4.67709 q^{7} -2.67924 q^{8} +4.22349 q^{9} +O(q^{10})\)	\(q+1.47960 q^{2} -2.68766 q^{3} +0.189211 q^{4} -0.774365 q^{5} -3.97665 q^{6} -4.67709 q^{7} -2.67924 q^{8} +4.22349 q^{9} -1.14575 q^{10} -0.545050 q^{11} -0.508535 q^{12} -0.907468 q^{13} -6.92022 q^{14} +2.08123 q^{15} -4.34262 q^{16} -5.99197 q^{17} +6.24907 q^{18} +4.38157 q^{19} -0.146519 q^{20} +12.5704 q^{21} -0.806456 q^{22} +1.04686 q^{23} +7.20087 q^{24} -4.40036 q^{25} -1.34269 q^{26} -3.28832 q^{27} -0.884959 q^{28} -1.23385 q^{29} +3.07938 q^{30} +1.52019 q^{31} -1.06686 q^{32} +1.46491 q^{33} -8.86571 q^{34} +3.62178 q^{35} +0.799132 q^{36} -7.25720 q^{37} +6.48297 q^{38} +2.43896 q^{39} +2.07471 q^{40} +18.5992 q^{42} +6.22101 q^{43} -0.103130 q^{44} -3.27052 q^{45} +1.54893 q^{46} +3.97043 q^{47} +11.6715 q^{48} +14.8752 q^{49} -6.51076 q^{50} +16.1043 q^{51} -0.171703 q^{52} +5.35331 q^{53} -4.86540 q^{54} +0.422068 q^{55} +12.5311 q^{56} -11.7762 q^{57} -1.82560 q^{58} +8.87496 q^{59} +0.393792 q^{60} +1.38786 q^{61} +2.24927 q^{62} -19.7537 q^{63} +7.10673 q^{64} +0.702711 q^{65} +2.16747 q^{66} +5.63556 q^{67} -1.13375 q^{68} -2.81360 q^{69} +5.35878 q^{70} +1.08971 q^{71} -11.3157 q^{72} -5.61291 q^{73} -10.7377 q^{74} +11.8266 q^{75} +0.829043 q^{76} +2.54925 q^{77} +3.60868 q^{78} -12.1959 q^{79} +3.36278 q^{80} -3.83260 q^{81} -13.1576 q^{83} +2.37846 q^{84} +4.63997 q^{85} +9.20460 q^{86} +3.31616 q^{87} +1.46032 q^{88} +1.95076 q^{89} -4.83906 q^{90} +4.24431 q^{91} +0.198078 q^{92} -4.08575 q^{93} +5.87464 q^{94} -3.39294 q^{95} +2.86734 q^{96} -1.94271 q^{97} +22.0093 q^{98} -2.30202 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8 q^{2} + 20 q^{4} + 12 q^{5} + 24 q^{8} + 24 q^{9}+O(q^{10}) \) 24 * q + 8 * q^2 + 20 * q^4 + 12 * q^5 + 24 * q^8 + 24 * q^9	\( 24 q + 8 q^{2} + 20 q^{4} + 12 q^{5} + 24 q^{8} + 24 q^{9} - 4 q^{10} + 20 q^{16} + 20 q^{18} + 40 q^{20} + 56 q^{21} + 48 q^{23} + 8 q^{25} + 32 q^{31} + 36 q^{32} + 24 q^{33} + 108 q^{36} + 60 q^{39} + 44 q^{40} - 40 q^{42} + 36 q^{43} - 36 q^{45} + 36 q^{46} + 32 q^{49} - 60 q^{50} + 36 q^{51} - 60 q^{57} + 116 q^{59} + 40 q^{61} + 48 q^{62} + 60 q^{64} + 16 q^{66} + 4 q^{72} + 16 q^{73} + 24 q^{74} - 16 q^{77} - 60 q^{78} + 84 q^{80} - 28 q^{81} + 80 q^{83} - 24 q^{84} - 24 q^{86} + 76 q^{87} + 56 q^{90} + 40 q^{91} - 40 q^{92} + 36 q^{98}+O(q^{100}) \) 24 * q + 8 * q^2 + 20 * q^4 + 12 * q^5 + 24 * q^8 + 24 * q^9 - 4 * q^10 + 20 * q^16 + 20 * q^18 + 40 * q^20 + 56 * q^21 + 48 * q^23 + 8 * q^25 + 32 * q^31 + 36 * q^32 + 24 * q^33 + 108 * q^36 + 60 * q^39 + 44 * q^40 - 40 * q^42 + 36 * q^43 - 36 * q^45 + 36 * q^46 + 32 * q^49 - 60 * q^50 + 36 * q^51 - 60 * q^57 + 116 * q^59 + 40 * q^61 + 48 * q^62 + 60 * q^64 + 16 * q^66 + 4 * q^72 + 16 * q^73 + 24 * q^74 - 16 * q^77 - 60 * q^78 + 84 * q^80 - 28 * q^81 + 80 * q^83 - 24 * q^84 - 24 * q^86 + 76 * q^87 + 56 * q^90 + 40 * q^91 - 40 * q^92 + 36 * q^98

Introduction

L-functions

Modular forms

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