LMFDB - Embedded newform 1681.2.a.m.1.9

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			1.9
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)

$f(q)$	$=$	$q-0.734595 q^{2} -0.0612231 q^{3} -1.46037 q^{4} -0.718161 q^{5} +0.0449742 q^{6} -3.13283 q^{7} +2.54197 q^{8} -2.99625 q^{9} +O(q^{10})$	\(q-0.734595 q^{2} -0.0612231 q^{3} -1.46037 q^{4} -0.718161 q^{5} +0.0449742 q^{6} -3.13283 q^{7} +2.54197 q^{8} -2.99625 q^{9} +0.527558 q^{10} -3.10797 q^{11} +0.0894084 q^{12} +2.01078 q^{13} +2.30136 q^{14} +0.0439680 q^{15} +1.05342 q^{16} +1.94609 q^{17} +2.20103 q^{18} -4.39059 q^{19} +1.04878 q^{20} +0.191802 q^{21} +2.28310 q^{22} -0.892338 q^{23} -0.155627 q^{24} -4.48425 q^{25} -1.47711 q^{26} +0.367109 q^{27} +4.57509 q^{28} -9.15088 q^{29} -0.0322987 q^{30} -6.98046 q^{31} -5.85778 q^{32} +0.190280 q^{33} -1.42959 q^{34} +2.24988 q^{35} +4.37563 q^{36} +4.31757 q^{37} +3.22531 q^{38} -0.123106 q^{39} -1.82554 q^{40} -0.140897 q^{42} +10.8556 q^{43} +4.53878 q^{44} +2.15179 q^{45} +0.655507 q^{46} +11.5493 q^{47} -0.0644935 q^{48} +2.81463 q^{49} +3.29411 q^{50} -0.119146 q^{51} -2.93648 q^{52} -2.97712 q^{53} -0.269677 q^{54} +2.23202 q^{55} -7.96357 q^{56} +0.268806 q^{57} +6.72220 q^{58} +4.44997 q^{59} -0.0642096 q^{60} +11.9422 q^{61} +5.12781 q^{62} +9.38675 q^{63} +2.19626 q^{64} -1.44406 q^{65} -0.139778 q^{66} +8.10052 q^{67} -2.84201 q^{68} +0.0546317 q^{69} -1.65275 q^{70} -1.00042 q^{71} -7.61639 q^{72} -6.49752 q^{73} -3.17167 q^{74} +0.274539 q^{75} +6.41188 q^{76} +9.73674 q^{77} +0.0904333 q^{78} -6.26974 q^{79} -0.756523 q^{80} +8.96628 q^{81} +7.83744 q^{83} -0.280101 q^{84} -1.39761 q^{85} -7.97450 q^{86} +0.560246 q^{87} -7.90037 q^{88} -14.1184 q^{89} -1.58070 q^{90} -6.29943 q^{91} +1.30314 q^{92} +0.427365 q^{93} -8.48403 q^{94} +3.15315 q^{95} +0.358632 q^{96} -6.41956 q^{97} -2.06762 q^{98} +9.31226 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.734595	−0.519437	−0.259719	−	0.965684i	$-0.583630\pi$
$2$	−0.734595	−0.519437	−0.259719	+	0.965684i	$0.583630\pi$
$3$	−0.0612231	−0.0353472	−0.0176736	−	0.999844i	$-0.505626\pi$
$3$	−0.0612231	−0.0353472	−0.0176736	+	0.999844i	$0.505626\pi$
$4$	−1.46037	−0.730185
$5$	−0.718161	−0.321171	−0.160586	−	0.987022i	$-0.551338\pi$
$5$	−0.718161	−0.321171	−0.160586	+	0.987022i	$0.551338\pi$
$6$	0.0449742	0.0183607
$7$	−3.13283	−1.18410	−0.592049	−	0.805902i	$-0.701681\pi$
$7$	−3.13283	−1.18410	−0.592049	+	0.805902i	$0.701681\pi$
$8$	2.54197	0.898723
$9$	−2.99625	−0.998751
$10$	0.527558	0.166828
$11$	−3.10797	−0.937088	−0.468544	−	0.883440i	$-0.655221\pi$
$11$	−3.10797	−0.937088	−0.468544	+	0.883440i	$0.655221\pi$
$12$	0.0894084	0.0258100
$13$	2.01078	0.557690	0.278845	−	0.960336i	$-0.410048\pi$
$13$	2.01078	0.557690	0.278845	+	0.960336i	$0.410048\pi$
$14$	2.30136	0.615065
$15$	0.0439680	0.0113525
$16$	1.05342	0.263355
$17$	1.94609	0.471996	0.235998	−	0.971754i	$-0.424164\pi$
$17$	1.94609	0.471996	0.235998	+	0.971754i	$0.424164\pi$
$18$	2.20103	0.518788
$19$	−4.39059	−1.00727	−0.503635	−	0.863916i	$-0.668004\pi$
$19$	−4.39059	−1.00727	−0.503635	+	0.863916i	$0.668004\pi$
$20$	1.04878	0.234514
$21$	0.191802	0.0418546
$22$	2.28310	0.486759
$23$	−0.892338	−0.186065	−0.0930326	−	0.995663i	$-0.529656\pi$
$23$	−0.892338	−0.186065	−0.0930326	+	0.995663i	$0.529656\pi$
$24$	−0.155627	−0.0317673
$25$	−4.48425	−0.896849
$26$	−1.47711	−0.289685
$27$	0.367109	0.0706502
$28$	4.57509	0.864611
$29$	−9.15088	−1.69928	−0.849638	−	0.527366i	$-0.823180\pi$
$29$	−9.15088	−1.69928	−0.849638	+	0.527366i	$0.823180\pi$
$30$	−0.0322987	−0.00589691
$31$	−6.98046	−1.25373	−0.626864	−	0.779129i	$-0.715662\pi$
$31$	−6.98046	−1.25373	−0.626864	+	0.779129i	$0.715662\pi$
$32$	−5.85778	−1.03552
$33$	0.190280	0.0331234
$34$	−1.42959	−0.245172
$35$	2.24988	0.380299
$36$	4.37563	0.729272
$37$	4.31757	0.709805	0.354902	−	0.934903i	$-0.384514\pi$
$37$	4.31757	0.709805	0.354902	+	0.934903i	$0.384514\pi$
$38$	3.22531	0.523214
$39$	−0.123106	−0.0197128
$40$	−1.82554	−0.288644
$41$	0	0
$41$	0	0
$42$	−0.140897	−0.0217408
$43$	10.8556	1.65547	0.827734	−	0.561121i	$-0.189630\pi$
$43$	10.8556	1.65547	0.827734	+	0.561121i	$0.189630\pi$
$44$	4.53878	0.684247
$45$	2.15179	0.320770
$46$	0.655507	0.0966493
$47$	11.5493	1.68463	0.842316	−	0.538984i	$-0.181192\pi$
$47$	11.5493	1.68463	0.842316	+	0.538984i	$0.181192\pi$
$48$	−0.0644935	−0.00930884
$49$	2.81463	0.402090
$50$	3.29411	0.465857
$51$	−0.119146	−0.0166837
$52$	−2.93648	−0.407217
$53$	−2.97712	−0.408939	−0.204469	−	0.978873i	$-0.565547\pi$
$53$	−2.97712	−0.408939	−0.204469	+	0.978873i	$0.565547\pi$
$54$	−0.269677	−0.0366984
$55$	2.23202	0.300966
$56$	−7.96357	−1.06418
$57$	0.268806	0.0356042
$58$	6.72220	0.882668
$59$	4.44997	0.579337	0.289669	−	0.957127i	$-0.406455\pi$
$59$	4.44997	0.579337	0.289669	+	0.957127i	$0.406455\pi$
$60$	−0.0642096	−0.00828942
$61$	11.9422	1.52904	0.764519	−	0.644601i	$-0.222977\pi$
$61$	11.9422	1.52904	0.764519	+	0.644601i	$0.222977\pi$
$62$	5.12781	0.651233
$63$	9.38675	1.18262
$64$	2.19626	0.274533
$65$	−1.44406	−0.179114
$66$	−0.139778	−0.0172055
$67$	8.10052	0.989636	0.494818	−	0.868997i	$-0.335235\pi$
$67$	8.10052	0.989636	0.494818	+	0.868997i	$0.335235\pi$
$68$	−2.84201	−0.344644
$69$	0.0546317	0.00657688
$70$	−1.65275	−0.197541
$71$	−1.00042	−0.118728	−0.0593639	−	0.998236i	$-0.518907\pi$
$71$	−1.00042	−0.118728	−0.0593639	+	0.998236i	$0.518907\pi$
$72$	−7.61639	−0.897600
$73$	−6.49752	−0.760477	−0.380238	−	0.924889i	$-0.624158\pi$
$73$	−6.49752	−0.760477	−0.380238	+	0.924889i	$0.624158\pi$
$74$	−3.17167	−0.368699
$75$	0.274539	0.0317011
$76$	6.41188	0.735494
$77$	9.73674	1.10960
$78$	0.0904333	0.0102396
$79$	−6.26974	−0.705400	−0.352700	−	0.935736i	$-0.614736\pi$
$79$	−6.26974	−0.705400	−0.352700	+	0.935736i	$0.614736\pi$
$80$	−0.756523	−0.0845819
$81$	8.96628	0.996253
$82$	0	0
$83$	7.83744	0.860271	0.430136	−	0.902764i	$-0.358466\pi$
$83$	7.83744	0.860271	0.430136	+	0.902764i	$0.358466\pi$
$84$	−0.280101	−0.0305616
$85$	−1.39761	−0.151592
$86$	−7.97450	−0.859912
$87$	0.560246	0.0600646
$88$	−7.90037	−0.842182
$89$	−14.1184	−1.49655	−0.748276	−	0.663388i	$-0.769118\pi$
$89$	−14.1184	−1.49655	−0.748276	+	0.663388i	$0.769118\pi$
$90$	−1.58070	−0.166620
$91$	−6.29943	−0.660360
$92$	1.30314	0.135862
$93$	0.427365	0.0443157
$94$	−8.48403	−0.875061
$95$	3.15315	0.323506
$96$	0.358632	0.0366027
$97$	−6.41956	−0.651808	−0.325904	−	0.945403i	$-0.605669\pi$
$97$	−6.41956	−0.651808	−0.325904	+	0.945403i	$0.605669\pi$
$98$	−2.06762	−0.208861
$99$	9.31226	0.935917
$100$	6.54865	0.654865
$101$	−12.0325	−1.19727	−0.598637	−	0.801020i	$-0.704291\pi$
$101$	−12.0325	−1.19727	−0.598637	+	0.801020i	$0.704291\pi$
$102$	0.0875239	0.00866615
$103$	15.3757	1.51501	0.757504	−	0.652830i	$-0.226419\pi$
$103$	15.3757	1.51501	0.757504	+	0.652830i	$0.226419\pi$
$104$	5.11135	0.501209
$105$	−0.137744	−0.0134425
$106$	2.18698	0.212418
$107$	1.11557	0.107846	0.0539231	−	0.998545i	$-0.482827\pi$
$107$	1.11557	0.107846	0.0539231	+	0.998545i	$0.482827\pi$
$108$	−0.536115	−0.0515877
$109$	5.92066	0.567096	0.283548	−	0.958958i	$-0.408488\pi$
$109$	5.92066	0.567096	0.283548	+	0.958958i	$0.408488\pi$
$110$	−1.63963	−0.156333
$111$	−0.264335	−0.0250896
$112$	−3.30018	−0.311838
$113$	15.4477	1.45320	0.726600	−	0.687060i	$-0.241099\pi$
$113$	15.4477	1.45320	0.726600	+	0.687060i	$0.241099\pi$
$114$	−0.197463	−0.0184941
$115$	0.640842	0.0597588
$116$	13.3637	1.24079
$117$	−6.02480	−0.556993
$118$	−3.26893	−0.300929
$119$	−6.09677	−0.558890
$120$	0.111766	0.0102027
$121$	−1.34053	−0.121866
$122$	−8.77266	−0.794240
$123$	0	0
$124$	10.1940	0.915452
$125$	6.81121	0.609213
$126$	−6.89547	−0.614297
$127$	4.33588	0.384747	0.192374	−	0.981322i	$-0.438381\pi$
$127$	4.33588	0.384747	0.192374	+	0.981322i	$0.438381\pi$
$128$	10.1022	0.892916
$129$	−0.664615	−0.0585161
$130$	1.06080	0.0930385
$131$	16.5744	1.44811	0.724055	−	0.689742i	$-0.242276\pi$
$131$	16.5744	1.44811	0.724055	+	0.689742i	$0.242276\pi$
$132$	−0.277878	−0.0241862
$133$	13.7550	1.19271
$134$	−5.95060	−0.514054
$135$	−0.263643	−0.0226908
$136$	4.94691	0.424194
$137$	0.235971	0.0201604	0.0100802	−	0.999949i	$-0.496791\pi$
$137$	0.235971	0.0201604	0.0100802	+	0.999949i	$0.496791\pi$
$138$	−0.0401322	−0.00341628
$139$	−2.35539	−0.199782	−0.0998909	−	0.994998i	$-0.531849\pi$
$139$	−2.35539	−0.199782	−0.0998909	+	0.994998i	$0.531849\pi$
$140$	−3.28565	−0.277688
$141$	−0.707081	−0.0595470
$142$	0.734903	0.0616717
$143$	−6.24944	−0.522605
$144$	−3.15631	−0.263025
$145$	6.57180	0.545759
$146$	4.77305	0.395020
$147$	−0.172321	−0.0142128
$148$	−6.30525	−0.518288
$149$	−8.39135	−0.687446	−0.343723	−	0.939071i	$-0.611688\pi$
$149$	−8.39135	−0.687446	−0.343723	+	0.939071i	$0.611688\pi$
$150$	−0.201675	−0.0164667
$151$	3.62231	0.294779	0.147390	−	0.989079i	$-0.452913\pi$
$151$	3.62231	0.294779	0.147390	+	0.989079i	$0.452913\pi$
$152$	−11.1608	−0.905257
$153$	−5.83097	−0.471406
$154$	−7.15257	−0.576370
$155$	5.01309	0.402661
$156$	0.179781	0.0143940
$157$	−13.7802	−1.09978	−0.549890	−	0.835237i	$-0.685330\pi$
$157$	−13.7802	−1.09978	−0.549890	+	0.835237i	$0.685330\pi$
$158$	4.60572	0.366411
$159$	0.182268	0.0144548
$160$	4.20683	0.332579
$161$	2.79554	0.220320
$162$	−6.58659	−0.517491
$163$	0.923374	0.0723243	0.0361621	−	0.999346i	$-0.488487\pi$
$163$	0.923374	0.0723243	0.0361621	+	0.999346i	$0.488487\pi$
$164$	0	0
$165$	−0.136651	−0.0106383
$166$	−5.75735	−0.446857
$167$	17.0021	1.31566	0.657832	−	0.753164i	$-0.271474\pi$
$167$	17.0021	1.31566	0.657832	+	0.753164i	$0.271474\pi$
$168$	0.487555	0.0376156
$169$	−8.95676	−0.688982
$170$	1.02667	0.0787423
$171$	13.1553	1.00601
$172$	−15.8532	−1.20880
$173$	12.1428	0.923198	0.461599	−	0.887089i	$-0.347276\pi$
$173$	12.1428	0.923198	0.461599	+	0.887089i	$0.347276\pi$
$174$	−0.411554	−0.0311998
$175$	14.0484	1.06196
$176$	−3.27399	−0.246786
$177$	−0.272441	−0.0204779
$178$	10.3713	0.777365
$179$	2.13864	0.159849	0.0799247	−	0.996801i	$-0.474532\pi$
$179$	2.13864	0.159849	0.0799247	+	0.996801i	$0.474532\pi$
$180$	−3.14241	−0.234221
$181$	−5.61782	−0.417569	−0.208785	−	0.977962i	$-0.566951\pi$
$181$	−5.61782	−0.417569	−0.208785	+	0.977962i	$0.566951\pi$
$182$	4.62754	0.343016
$183$	−0.731137	−0.0540472
$184$	−2.26830	−0.167221
$185$	−3.10071	−0.227969
$186$	−0.313941	−0.0230192
$187$	−6.04839	−0.442302
$188$	−16.8662	−1.23009
$189$	−1.15009	−0.0836568
$190$	−2.31629	−0.168041
$191$	13.1210	0.949401	0.474701	−	0.880147i	$-0.342556\pi$
$191$	13.1210	0.949401	0.474701	+	0.880147i	$0.342556\pi$
$192$	−0.134462	−0.00970396
$193$	16.5769	1.19323	0.596614	−	0.802528i	$-0.296512\pi$
$193$	16.5769	1.19323	0.596614	+	0.802528i	$0.296512\pi$
$194$	4.71578	0.338573
$195$	0.0884100	0.00633117
$196$	−4.11040	−0.293600
$197$	−14.4810	−1.03173	−0.515864	−	0.856671i	$-0.672529\pi$
$197$	−14.4810	−1.03173	−0.515864	+	0.856671i	$0.672529\pi$
$198$	−6.84074	−0.486150
$199$	−18.0390	−1.27875	−0.639375	−	0.768895i	$-0.720807\pi$
$199$	−18.0390	−1.27875	−0.639375	+	0.768895i	$0.720807\pi$
$200$	−11.3988	−0.806019
$201$	−0.495939	−0.0349808
$202$	8.83899	0.621909
$203$	28.6682	2.01211
$204$	0.173997	0.0121822
$205$	0	0
$206$	−11.2949	−0.786952
$207$	2.67367	0.185833
$208$	2.11819	0.146870
$209$	13.6458	0.943901
$210$	0.101186	0.00698253
$211$	13.5016	0.929492	0.464746	−	0.885444i	$-0.346146\pi$
$211$	13.5016	0.929492	0.464746	+	0.885444i	$0.346146\pi$
$212$	4.34769	0.298601
$213$	0.0612487	0.00419670
$214$	−0.819492	−0.0560193
$215$	−7.79609	−0.531689
$216$	0.933181	0.0634949
$217$	21.8686	1.48454
$218$	−4.34929	−0.294571
$219$	0.397798	0.0268807
$220$	−3.25958	−0.219761
$221$	3.91316	0.263227
$222$	0.194179	0.0130325
$223$	−7.58982	−0.508252	−0.254126	−	0.967171i	$-0.581788\pi$
$223$	−7.58982	−0.508252	−0.254126	+	0.967171i	$0.581788\pi$
$224$	18.3514	1.22616
$225$	13.4359	0.895729
$226$	−11.3478	−0.754847
$227$	−21.7141	−1.44121	−0.720606	−	0.693345i	$-0.756136\pi$
$227$	−21.7141	−1.44121	−0.720606	+	0.693345i	$0.756136\pi$
$228$	−0.392556	−0.0259976
$229$	2.96197	0.195733	0.0978663	−	0.995200i	$-0.468798\pi$
$229$	2.96197	0.195733	0.0978663	+	0.995200i	$0.468798\pi$
$230$	−0.470760	−0.0310410
$231$	−0.596114	−0.0392214
$232$	−23.2613	−1.52718
$233$	−17.4010	−1.13998	−0.569988	−	0.821653i	$-0.693052\pi$
$233$	−17.4010	−1.13998	−0.569988	+	0.821653i	$0.693052\pi$
$234$	4.42579	0.289323
$235$	−8.29422	−0.541055
$236$	−6.49860	−0.423023
$237$	0.383853	0.0249339
$238$	4.47866	0.290308
$239$	5.94031	0.384247	0.192123	−	0.981371i	$-0.438463\pi$
$239$	5.94031	0.384247	0.192123	+	0.981371i	$0.438463\pi$
$240$	0.0463167	0.00298973
$241$	−8.78248	−0.565729	−0.282865	−	0.959160i	$-0.591285\pi$
$241$	−8.78248	−0.565729	−0.282865	+	0.959160i	$0.591285\pi$
$242$	0.984748	0.0633020
$243$	−1.65027	−0.105865
$244$	−17.4400	−1.11648
$245$	−2.02136	−0.129140
$246$	0	0
$247$	−8.82851	−0.561745
$248$	−17.7441	−1.12675
$249$	−0.479833	−0.0304082
$250$	−5.00349	−0.316448
$251$	−5.12696	−0.323611	−0.161805	−	0.986823i	$-0.551732\pi$
$251$	−5.12696	−0.323611	−0.161805	+	0.986823i	$0.551732\pi$
$252$	−13.7081	−0.863531
$253$	2.77336	0.174360
$254$	−3.18512	−0.199852
$255$	0.0855657	0.00535833
$256$	−11.8136	−0.738347
$257$	1.09929	0.0685719	0.0342860	−	0.999412i	$-0.489084\pi$
$257$	1.09929	0.0685719	0.0342860	+	0.999412i	$0.489084\pi$
$258$	0.488224	0.0303955
$259$	−13.5262	−0.840479
$260$	2.10887	0.130786
$261$	27.4184	1.69715
$262$	−12.1755	−0.752203
$263$	−12.7631	−0.787006	−0.393503	−	0.919323i	$-0.628737\pi$
$263$	−12.7631	−0.787006	−0.393503	+	0.919323i	$0.628737\pi$
$264$	0.483685	0.0297688
$265$	2.13805	0.131339
$266$	−10.1043	−0.619537
$267$	0.864375	0.0528989
$268$	−11.8297	−0.722617
$269$	−0.552080	−0.0336609	−0.0168304	−	0.999858i	$-0.505358\pi$
$269$	−0.552080	−0.0336609	−0.0168304	+	0.999858i	$0.505358\pi$
$270$	0.193671	0.0117865
$271$	2.76027	0.167675	0.0838373	−	0.996479i	$-0.473282\pi$
$271$	2.76027	0.167675	0.0838373	+	0.996479i	$0.473282\pi$
$272$	2.05005	0.124302
$273$	0.385671	0.0233419
$274$	−0.173343	−0.0104721
$275$	13.9369	0.840426
$276$	−0.0797825	−0.00480234
$277$	21.2618	1.27750	0.638748	−	0.769416i	$-0.279452\pi$
$277$	21.2618	1.27750	0.638748	+	0.769416i	$0.279452\pi$
$278$	1.73026	0.103774
$279$	20.9152	1.25216
$280$	5.71912	0.341783
$281$	1.79245	0.106928	0.0534642	−	0.998570i	$-0.482974\pi$
$281$	1.79245	0.106928	0.0534642	+	0.998570i	$0.482974\pi$
$282$	0.519419	0.0309309
$283$	3.71847	0.221040	0.110520	−	0.993874i	$-0.464748\pi$
$283$	3.71847	0.221040	0.110520	+	0.993874i	$0.464748\pi$
$284$	1.46098	0.0866933
$285$	−0.193046	−0.0114350
$286$	4.59081	0.271460
$287$	0	0
$288$	17.5514	1.03423
$289$	−13.2127	−0.777220
$290$	−4.82762	−0.283487
$291$	0.393026	0.0230396
$292$	9.48878	0.555289
$293$	−13.9029	−0.812213	−0.406107	−	0.913826i	$-0.633114\pi$
$293$	−13.9029	−0.812213	−0.406107	+	0.913826i	$0.633114\pi$
$294$	0.126586	0.00738264
$295$	−3.19580	−0.186066
$296$	10.9751	0.637917
$297$	−1.14096	−0.0662054
$298$	6.16424	0.357085
$299$	−1.79429	−0.103767
$300$	−0.400929	−0.0231477
$301$	−34.0089	−1.96024
$302$	−2.66093	−0.153119
$303$	0.736665	0.0423203
$304$	−4.62513	−0.265269
$305$	−8.57640	−0.491083
$306$	4.28341	0.244866
$307$	−21.8696	−1.24817	−0.624083	−	0.781358i	$-0.714527\pi$
$307$	−21.8696	−1.24817	−0.624083	+	0.781358i	$0.714527\pi$
$308$	−14.2192	−0.810216
$309$	−0.941346	−0.0535513
$310$	−3.68259	−0.209157
$311$	15.9497	0.904426	0.452213	−	0.891910i	$-0.350635\pi$
$311$	15.9497	0.904426	0.452213	+	0.891910i	$0.350635\pi$
$312$	−0.312933	−0.0177163
$313$	−0.821288	−0.0464220	−0.0232110	−	0.999731i	$-0.507389\pi$
$313$	−0.821288	−0.0464220	−0.0232110	+	0.999731i	$0.507389\pi$
$314$	10.1229	0.571267
$315$	−6.74120	−0.379823
$316$	9.15613	0.515073
$317$	28.2922	1.58905	0.794523	−	0.607234i	$-0.207721\pi$
$317$	28.2922	1.58905	0.794523	+	0.607234i	$0.207721\pi$
$318$	−0.133894	−0.00750838
$319$	28.4407	1.59237
$320$	−1.57727	−0.0881720
$321$	−0.0682986	−0.00381206
$322$	−2.05359	−0.114442
$323$	−8.54448	−0.475428
$324$	−13.0941	−0.727449
$325$	−9.01683	−0.500164
$326$	−0.678307	−0.0375679
$327$	−0.362481	−0.0200453
$328$	0	0
$329$	−36.1819	−1.99477
$330$	0.100383	0.00552592
$331$	−30.7547	−1.69043	−0.845216	−	0.534425i	$-0.820528\pi$
$331$	−30.7547	−1.69043	−0.845216	+	0.534425i	$0.820528\pi$
$332$	−11.4456	−0.628157
$333$	−12.9365	−0.708918
$334$	−12.4897	−0.683406
$335$	−5.81747	−0.317843
$336$	0.202047	0.0110226
$337$	−26.0552	−1.41932	−0.709658	−	0.704546i	$-0.751151\pi$
$337$	−26.0552	−1.41932	−0.709658	+	0.704546i	$0.751151\pi$
$338$	6.57960	0.357883
$339$	−0.945758	−0.0513665
$340$	2.04102	0.110690
$341$	21.6950	1.17485
$342$	−9.66383	−0.522560
$343$	13.1121	0.707984
$344$	27.5947	1.48781
$345$	−0.0392343	−0.00211231
$346$	−8.92003	−0.479544
$347$	6.48814	0.348302	0.174151	−	0.984719i	$-0.444282\pi$
$347$	6.48814	0.348302	0.174151	+	0.984719i	$0.444282\pi$
$348$	−0.818166	−0.0438583
$349$	−7.55915	−0.404632	−0.202316	−	0.979320i	$-0.564847\pi$
$349$	−7.55915	−0.404632	−0.202316	+	0.979320i	$0.564847\pi$
$350$	−10.3199	−0.551621
$351$	0.738176	0.0394009
$352$	18.2058	0.970372
$353$	37.2015	1.98004	0.990019	−	0.140931i	$-0.0450097\pi$
$353$	37.2015	1.98004	0.990019	+	0.140931i	$0.0450097\pi$
$354$	0.200134	0.0106370
$355$	0.718461	0.0381320
$356$	20.6181	1.09276
$357$	0.373263	0.0197552
$358$	−1.57103	−0.0830318
$359$	21.7538	1.14812	0.574060	−	0.818813i	$-0.305367\pi$
$359$	21.7538	1.14812	0.574060	+	0.818813i	$0.305367\pi$
$360$	5.46979	0.288283
$361$	0.277286	0.0145940
$362$	4.12683	0.216901
$363$	0.0820715	0.00430763
$364$	9.19950	0.482185
$365$	4.66626	0.244243
$366$	0.537090	0.0280741
$367$	−17.4952	−0.913242	−0.456621	−	0.889661i	$-0.650940\pi$
$367$	−17.4952	−0.913242	−0.456621	+	0.889661i	$0.650940\pi$
$368$	−0.940005	−0.0490011
$369$	0	0
$370$	2.27777	0.118416
$371$	9.32681	0.484224
$372$	−0.624111	−0.0323587
$373$	7.16524	0.371002	0.185501	−	0.982644i	$-0.440609\pi$
$373$	7.16524	0.371002	0.185501	+	0.982644i	$0.440609\pi$
$374$	4.44312	0.229748
$375$	−0.417004	−0.0215340
$376$	29.3579	1.51402
$377$	−18.4004	−0.947669
$378$	0.844852	0.0434545
$379$	−7.89135	−0.405351	−0.202676	−	0.979246i	$-0.564964\pi$
$379$	−7.89135	−0.405351	−0.202676	+	0.979246i	$0.564964\pi$
$380$	−4.60476	−0.236219
$381$	−0.265456	−0.0135997
$382$	−9.63861	−0.493154
$383$	6.15635	0.314575	0.157287	−	0.987553i	$-0.449725\pi$
$383$	6.15635	0.314575	0.157287	+	0.987553i	$0.449725\pi$
$384$	−0.618488	−0.0315621
$385$	−6.99255	−0.356373
$386$	−12.1773	−0.619807
$387$	−32.5262	−1.65340
$388$	9.37494	0.475940
$389$	−22.2173	−1.12646	−0.563231	−	0.826299i	$-0.690442\pi$
$389$	−22.2173	−1.12646	−0.563231	+	0.826299i	$0.690442\pi$
$390$	−0.0649456	−0.00328865
$391$	−1.73657	−0.0878221
$392$	7.15472	0.361368
$393$	−1.01473	−0.0511866
$394$	10.6377	0.535918
$395$	4.50268	0.226554
$396$	−13.5993	−0.683392
$397$	6.52941	0.327702	0.163851	−	0.986485i	$-0.447608\pi$
$397$	6.52941	0.327702	0.163851	+	0.986485i	$0.447608\pi$
$398$	13.2514	0.664231
$399$	−0.842123	−0.0421589
$400$	−4.72378	−0.236189
$401$	21.3099	1.06416	0.532082	−	0.846693i	$-0.321410\pi$
$401$	21.3099	1.06416	0.532082	+	0.846693i	$0.321410\pi$
$402$	0.364315	0.0181704
$403$	−14.0362	−0.699191
$404$	17.5718	0.874232
$405$	−6.43923	−0.319968
$406$	−21.0595	−1.04517
$407$	−13.4189	−0.665149
$408$	−0.302865	−0.0149940
$409$	−3.89625	−0.192657	−0.0963287	−	0.995350i	$-0.530710\pi$
$409$	−3.89625	−0.192657	−0.0963287	+	0.995350i	$0.530710\pi$
$410$	0	0
$411$	−0.0144469	−0.000712613 0
$412$	−22.4541	−1.10624
$413$	−13.9410	−0.685992
$414$	−1.96406	−0.0965285
$415$	−5.62854	−0.276294
$416$	−11.7787	−0.577499
$417$	0.144204	0.00706172
$418$	−10.0242	−0.490298
$419$	14.8236	0.724179	0.362089	−	0.932143i	$-0.382063\pi$
$419$	14.8236	0.724179	0.362089	+	0.932143i	$0.382063\pi$
$420$	0.201158	0.00981549
$421$	11.1278	0.542334	0.271167	−	0.962532i	$-0.412590\pi$
$421$	11.1278	0.542334	0.271167	+	0.962532i	$0.412590\pi$
$422$	−9.91825	−0.482813
$423$	−34.6045	−1.68253
$424$	−7.56775	−0.367522
$425$	−8.72674	−0.423309
$426$	−0.0449931	−0.00217992
$427$	−37.4128	−1.81053
$428$	−1.62914	−0.0787476
$429$	0.382610	0.0184726
$430$	5.72697	0.276179
$431$	−26.4706	−1.27504	−0.637521	−	0.770433i	$-0.720040\pi$
$431$	−26.4706	−1.27504	−0.637521	+	0.770433i	$0.720040\pi$
$432$	0.386719	0.0186060
$433$	−0.467593	−0.0224711	−0.0112355	−	0.999937i	$-0.503576\pi$
$433$	−0.467593	−0.0224711	−0.0112355	+	0.999937i	$0.503576\pi$
$434$	−16.0646	−0.771124
$435$	−0.402346	−0.0192910
$436$	−8.64635	−0.414085
$437$	3.91789	0.187418
$438$	−0.292221	−0.0139628
$439$	−19.5402	−0.932605	−0.466303	−	0.884625i	$-0.654414\pi$
$439$	−19.5402	−0.932605	−0.466303	+	0.884625i	$0.654414\pi$
$440$	5.67373	0.270485
$441$	−8.43335	−0.401588
$442$	−2.87459	−0.136730
$443$	7.44868	0.353898	0.176949	−	0.984220i	$-0.443377\pi$
$443$	7.44868	0.353898	0.176949	+	0.984220i	$0.443377\pi$
$444$	0.386027	0.0183200
$445$	10.1393	0.480649
$446$	5.57545	0.264005
$447$	0.513744	0.0242993
$448$	−6.88052	−0.325074
$449$	25.5421	1.20541	0.602704	−	0.797965i	$-0.294090\pi$
$449$	25.5421	1.20541	0.602704	+	0.797965i	$0.294090\pi$
$450$	−9.86997	−0.465275
$451$	0	0
$452$	−22.5594	−1.06110
$453$	−0.221769	−0.0104196
$454$	15.9510	0.748620
$455$	4.52401	0.212089
$456$	0.683296	0.0319983
$457$	35.3076	1.65162	0.825811	−	0.563948i	$-0.190718\pi$
$457$	35.3076	1.65162	0.825811	+	0.563948i	$0.190718\pi$
$458$	−2.17585	−0.101671
$459$	0.714427	0.0333466
$460$	−0.935866	−0.0436350
$461$	−28.4899	−1.32691	−0.663453	−	0.748218i	$-0.730910\pi$
$461$	−28.4899	−1.32691	−0.663453	+	0.748218i	$0.730910\pi$
$462$	0.437902	0.0203731
$463$	12.8381	0.596638	0.298319	−	0.954466i	$-0.403574\pi$
$463$	12.8381	0.596638	0.298319	+	0.954466i	$0.403574\pi$
$464$	−9.63971	−0.447512
$465$	−0.306917	−0.0142329
$466$	12.7827	0.592146
$467$	29.4622	1.36335	0.681674	−	0.731656i	$-0.261252\pi$
$467$	29.4622	1.36335	0.681674	+	0.731656i	$0.261252\pi$
$468$	8.79844	0.406708
$469$	−25.3776	−1.17183
$470$	6.09290	0.281044
$471$	0.843667	0.0388741
$472$	11.3117	0.520663
$473$	−33.7390	−1.55132
$474$	−0.281977	−0.0129516
$475$	19.6885	0.903370
$476$	8.90354	0.408093
$477$	8.92019	0.408428
$478$	−4.36373	−0.199592
$479$	−26.7173	−1.22075	−0.610373	−	0.792114i	$-0.708980\pi$
$479$	−26.7173	−1.22075	−0.610373	+	0.792114i	$0.708980\pi$
$480$	−0.257555	−0.0117557
$481$	8.68169	0.395851
$482$	6.45157	0.293861
$483$	−0.171152	−0.00778768
$484$	1.95767	0.0889850
$485$	4.61028	0.209342
$486$	1.21228	0.0549902
$487$	−11.6064	−0.525938	−0.262969	−	0.964804i	$-0.584702\pi$
$487$	−11.6064	−0.525938	−0.262969	+	0.964804i	$0.584702\pi$
$488$	30.3567	1.37418
$489$	−0.0565319	−0.00255646
$490$	1.48488	0.0670801
$491$	26.8852	1.21331	0.606656	−	0.794964i	$-0.292511\pi$
$491$	26.8852	1.21331	0.606656	+	0.794964i	$0.292511\pi$
$492$	0	0
$493$	−17.8084	−0.802052
$494$	6.48538	0.291791
$495$	−6.68770	−0.300590
$496$	−7.35334	−0.330175
$497$	3.13414	0.140586
$498$	0.352483	0.0157951
$499$	12.8573	0.575572	0.287786	−	0.957695i	$-0.407081\pi$
$499$	12.8573	0.575572	0.287786	+	0.957695i	$0.407081\pi$
$500$	−9.94689	−0.444838
$501$	−1.04092	−0.0465050
$502$	3.76624	0.168096
$503$	−12.0787	−0.538565	−0.269282	−	0.963061i	$-0.586786\pi$
$503$	−12.0787	−0.538565	−0.269282	+	0.963061i	$0.586786\pi$
$504$	23.8609	1.06285
$505$	8.64124	0.384530
$506$	−2.03730	−0.0905689
$507$	0.548361	0.0243536
$508$	−6.33199	−0.280936
$509$	29.0868	1.28925	0.644624	−	0.764499i	$-0.277014\pi$
$509$	29.0868	1.28925	0.644624	+	0.764499i	$0.277014\pi$
$510$	−0.0628562	−0.00278332
$511$	20.3556	0.900480
$512$	−11.5262	−0.509391
$513$	−1.61183	−0.0711639
$514$	−0.807535	−0.0356188
$515$	−11.0422	−0.486577
$516$	0.970584	0.0427276
$517$	−35.8947	−1.57865
$518$	9.93630	0.436576
$519$	−0.743419	−0.0326325
$520$	−3.67077	−0.160974
$521$	21.7063	0.950972	0.475486	−	0.879723i	$-0.342272\pi$
$521$	21.7063	0.950972	0.475486	+	0.879723i	$0.342272\pi$
$522$	−20.1414	−0.881565
$523$	−15.7771	−0.689885	−0.344942	−	0.938624i	$-0.612102\pi$
$523$	−15.7771	−0.689885	−0.344942	+	0.938624i	$0.612102\pi$
$524$	−24.2047	−1.05739
$525$	−0.860086	−0.0375372
$526$	9.37571	0.408801
$527$	−13.5846	−0.591754
$528$	0.200444	0.00872320
$529$	−22.2037	−0.965380
$530$	−1.57060	−0.0682226
$531$	−13.3332	−0.578613
$532$	−20.0874	−0.870897
$533$	0	0
$534$	−0.634966	−0.0274777
$535$	−0.801158	−0.0346371
$536$	20.5913	0.889408
$537$	−0.130934	−0.00565023
$538$	0.405555	0.0174847
$539$	−8.74779	−0.376794
$540$	0.385017	0.0165685
$541$	−4.22799	−0.181775	−0.0908877	−	0.995861i	$-0.528970\pi$
$541$	−4.22799	−0.181775	−0.0908877	+	0.995861i	$0.528970\pi$
$542$	−2.02768	−0.0870965
$543$	0.343941	0.0147599
$544$	−11.3998	−0.488761
$545$	−4.25199	−0.182135
$546$	−0.283312	−0.0121246
$547$	2.38523	0.101985	0.0509925	−	0.998699i	$-0.483762\pi$
$547$	2.38523	0.101985	0.0509925	+	0.998699i	$0.483762\pi$
$548$	−0.344605	−0.0147208
$549$	−35.7817	−1.52713
$550$	−10.2380	−0.436549
$551$	40.1778	1.71163
$552$	0.138872	0.00591080
$553$	19.6420	0.835264
$554$	−15.6188	−0.663580
$555$	0.189835	0.00805805
$556$	3.43974	0.145878
$557$	−7.73176	−0.327605	−0.163803	−	0.986493i	$-0.552376\pi$
$557$	−7.73176	−0.327605	−0.163803	+	0.986493i	$0.552376\pi$
$558$	−15.3642	−0.650419
$559$	21.8283	0.923238
$560$	2.37006	0.100153
$561$	0.370301	0.0156341
$562$	−1.31672	−0.0555426
$563$	29.8025	1.25603	0.628013	−	0.778203i	$-0.283869\pi$
$563$	29.8025	1.25603	0.628013	+	0.778203i	$0.283869\pi$
$564$	1.03260	0.0434803
$565$	−11.0940	−0.466726
$566$	−2.73157	−0.114817
$567$	−28.0898	−1.17966
$568$	−2.54304	−0.106703
$569$	8.84184	0.370669	0.185335	−	0.982675i	$-0.440663\pi$
$569$	8.84184	0.370669	0.185335	+	0.982675i	$0.440663\pi$
$570$	0.141810	0.00593979
$571$	22.3396	0.934885	0.467442	−	0.884024i	$-0.345176\pi$
$571$	22.3396	0.934885	0.467442	+	0.884024i	$0.345176\pi$
$572$	9.12649	0.381598
$573$	−0.803308	−0.0335587
$574$	0	0
$575$	4.00146	0.166872
$576$	−6.58055	−0.274190
$577$	−12.0489	−0.501604	−0.250802	−	0.968038i	$-0.580694\pi$
$577$	−12.0489	−0.501604	−0.250802	+	0.968038i	$0.580694\pi$
$578$	9.70602	0.403717
$579$	−1.01489	−0.0421773
$580$	−9.59726	−0.398505
$581$	−24.5534	−1.01865
$582$	−0.288715	−0.0119676
$583$	9.25279	0.383211
$584$	−16.5165	−0.683458
$585$	4.32678	0.178890
$586$	10.2130	0.421894
$587$	25.8033	1.06502	0.532508	−	0.846425i	$-0.321250\pi$
$587$	25.8033	1.06502	0.532508	+	0.846425i	$0.321250\pi$
$588$	0.251652	0.0103779
$589$	30.6483	1.26284
$590$	2.34762	0.0966499
$591$	0.886571	0.0364686
$592$	4.54821	0.186930
$593$	−20.8287	−0.855333	−0.427667	−	0.903937i	$-0.640664\pi$
$593$	−20.8287	−0.855333	−0.427667	+	0.903937i	$0.640664\pi$
$594$	0.838147	0.0343896
$595$	4.37846	0.179499
$596$	12.2545	0.501962
$597$	1.10440	0.0452002
$598$	1.31808	0.0539003
$599$	18.3686	0.750522	0.375261	−	0.926919i	$-0.377553\pi$
$599$	18.3686	0.750522	0.375261	+	0.926919i	$0.377553\pi$
$600$	0.697872	0.0284905
$601$	−10.7755	−0.439543	−0.219772	−	0.975551i	$-0.570531\pi$
$601$	−10.7755	−0.439543	−0.219772	+	0.975551i	$0.570531\pi$
$602$	24.9828	1.01822
$603$	−24.2712	−0.988399
$604$	−5.28991	−0.215243
$605$	0.962716	0.0391400
$606$	−0.541151	−0.0219827
$607$	−13.9057	−0.564415	−0.282208	−	0.959353i	$-0.591067\pi$
$607$	−13.9057	−0.564415	−0.282208	+	0.959353i	$0.591067\pi$
$608$	25.7191	1.04305
$609$	−1.75516	−0.0711225
$610$	6.30018	0.255087
$611$	23.2230	0.939502
$612$	8.51538	0.344214
$613$	−14.9295	−0.602995	−0.301498	−	0.953467i	$-0.597487\pi$
$613$	−14.9295	−0.602995	−0.301498	+	0.953467i	$0.597487\pi$
$614$	16.0653	0.648344
$615$	0	0
$616$	24.7505	0.997227
$617$	−8.40580	−0.338405	−0.169202	−	0.985581i	$-0.554119\pi$
$617$	−8.40580	−0.338405	−0.169202	+	0.985581i	$0.554119\pi$
$618$	0.691508	0.0278165
$619$	−27.2430	−1.09499	−0.547495	−	0.836809i	$-0.684418\pi$
$619$	−27.2430	−1.09499	−0.547495	+	0.836809i	$0.684418\pi$
$620$	−7.32096	−0.294017
$621$	−0.327585	−0.0131456
$622$	−11.7166	−0.469793
$623$	44.2307	1.77207
$624$	−0.129682	−0.00519145
$625$	17.5297	0.701187
$626$	0.603315	0.0241133
$627$	−0.835440	−0.0333642
$628$	20.1242	0.803043
$629$	8.40238	0.335025
$630$	4.95205	0.197294
$631$	3.20106	0.127432	0.0637160	−	0.997968i	$-0.479705\pi$
$631$	3.20106	0.127432	0.0637160	+	0.997968i	$0.479705\pi$
$632$	−15.9375	−0.633959
$633$	−0.826613	−0.0328549
$634$	−20.7833	−0.825410
$635$	−3.11386	−0.123570
$636$	−0.266179	−0.0105547
$637$	5.65961	0.224242
$638$	−20.8924	−0.827137
$639$	2.99751	0.118580
$640$	−7.25500	−0.286779
$641$	−43.1828	−1.70562	−0.852809	−	0.522223i	$-0.825103\pi$
$641$	−43.1828	−1.70562	−0.852809	+	0.522223i	$0.825103\pi$
$642$	0.0501719	0.00198013
$643$	−4.00507	−0.157945	−0.0789723	−	0.996877i	$-0.525164\pi$
$643$	−4.00507	−0.157945	−0.0789723	+	0.996877i	$0.525164\pi$
$644$	−4.08253	−0.160874
$645$	0.477301	0.0187937
$646$	6.27674	0.246955
$647$	16.9523	0.666465	0.333232	−	0.942845i	$-0.391861\pi$
$647$	16.9523	0.666465	0.333232	+	0.942845i	$0.391861\pi$
$648$	22.7920	0.895355
$649$	−13.8304	−0.542890
$650$	6.62372	0.259804
$651$	−1.33886	−0.0524742
$652$	−1.34847	−0.0528101
$653$	17.4583	0.683195	0.341598	−	0.939846i	$-0.389032\pi$
$653$	17.4583	0.683195	0.341598	+	0.939846i	$0.389032\pi$
$654$	0.266277	0.0104123
$655$	−11.9031	−0.465091
$656$	0	0
$657$	19.4682	0.759527
$658$	26.5790	1.03616
$659$	27.4685	1.07002	0.535011	−	0.844845i	$-0.320308\pi$
$659$	27.4685	1.07002	0.535011	+	0.844845i	$0.320308\pi$
$660$	0.199561	0.00776792
$661$	−7.52676	−0.292757	−0.146379	−	0.989229i	$-0.546762\pi$
$661$	−7.52676	−0.292757	−0.146379	+	0.989229i	$0.546762\pi$
$662$	22.5923	0.878073
$663$	−0.239576	−0.00930435
$664$	19.9226	0.773145
$665$	−9.87829	−0.383063
$666$	9.50312	0.368238
$667$	8.16568	0.316176
$668$	−24.8294	−0.960678
$669$	0.464673	0.0179653
$670$	4.27349	0.165099
$671$	−37.1159	−1.43284
$672$	−1.12353	−0.0433412
$673$	42.5265	1.63927	0.819637	−	0.572883i	$-0.194175\pi$
$673$	42.5265	1.63927	0.819637	+	0.572883i	$0.194175\pi$
$674$	19.1400	0.737246
$675$	−1.64621	−0.0633626
$676$	13.0802	0.503084
$677$	46.2980	1.77938	0.889688	−	0.456569i	$-0.150922\pi$
$677$	46.2980	1.77938	0.889688	+	0.456569i	$0.150922\pi$
$678$	0.694750	0.0266817
$679$	20.1114	0.771805
$680$	−3.55267	−0.136239
$681$	1.32940	0.0509428
$682$	−15.9371	−0.610262
$683$	15.3985	0.589208	0.294604	−	0.955619i	$-0.404812\pi$
$683$	15.3985	0.589208	0.294604	+	0.955619i	$0.404812\pi$
$684$	−19.2116	−0.734575
$685$	−0.169465	−0.00647493
$686$	−9.63205	−0.367753
$687$	−0.181341	−0.00691859
$688$	11.4355	0.435975
$689$	−5.98633	−0.228061
$690$	0.0288214	0.00109721
$691$	−42.8858	−1.63145	−0.815726	−	0.578438i	$-0.803662\pi$
$691$	−42.8858	−1.63145	−0.815726	+	0.578438i	$0.803662\pi$
$692$	−17.7329	−0.674105
$693$	−29.1737	−1.10822
$694$	−4.76616	−0.180921
$695$	1.69155	0.0641642
$696$	1.42413	0.0539815
$697$	0	0
$698$	5.55292	0.210181
$699$	1.06534	0.0402949
$700$	−20.5158	−0.775426
$701$	29.2453	1.10458	0.552289	−	0.833653i	$-0.313754\pi$
$701$	29.2453	1.10458	0.552289	+	0.833653i	$0.313754\pi$
$702$	−0.542261	−0.0204663
$703$	−18.9567	−0.714965
$704$	−6.82591	−0.257261
$705$	0.507798	0.0191248
$706$	−27.3281	−1.02851
$707$	37.6957	1.41769
$708$	0.397865	0.0149527
$709$	−8.48265	−0.318572	−0.159286	−	0.987232i	$-0.550919\pi$
$709$	−8.48265	−0.318572	−0.159286	+	0.987232i	$0.550919\pi$
$710$	−0.527778	−0.0198072
$711$	18.7857	0.704519
$712$	−35.8887	−1.34498
$713$	6.22893	0.233275
$714$	−0.274198	−0.0102616
$715$	4.48810	0.167846
$716$	−3.12320	−0.116720
$717$	−0.363684	−0.0135820
$718$	−15.9802	−0.596377
$719$	38.5164	1.43642	0.718210	−	0.695827i	$-0.244962\pi$
$719$	38.5164	1.43642	0.718210	+	0.695827i	$0.244962\pi$
$720$	2.26673	0.0844762
$721$	−48.1693	−1.79392
$722$	−0.203693	−0.00758066
$723$	0.537691	0.0199969
$724$	8.20409	0.304903
$725$	41.0348	1.52399
$726$	−0.0602893	−0.00223755
$727$	−51.7389	−1.91889	−0.959444	−	0.281899i	$-0.909036\pi$
$727$	−51.7389	−1.91889	−0.959444	+	0.281899i	$0.909036\pi$
$728$	−16.0130	−0.593481
$729$	−26.7978	−0.992511
$730$	−3.42781	−0.126869
$731$	21.1260	0.781374
$732$	1.06773	0.0394644
$733$	−0.628345	−0.0232084	−0.0116042	−	0.999933i	$-0.503694\pi$
$733$	−0.628345	−0.0232084	−0.0116042	+	0.999933i	$0.503694\pi$
$734$	12.8519	0.474372
$735$	0.123754	0.00456473
$736$	5.22712	0.192674
$737$	−25.1762	−0.927376
$738$	0	0
$739$	40.6971	1.49707	0.748533	−	0.663097i	$-0.230758\pi$
$739$	40.6971	1.49707	0.748533	+	0.663097i	$0.230758\pi$
$740$	4.52818	0.166459
$741$	0.540509	0.0198561
$742$	−6.85143	−0.251524
$743$	16.0543	0.588976	0.294488	−	0.955655i	$-0.404851\pi$
$743$	16.0543	0.588976	0.294488	+	0.955655i	$0.404851\pi$
$744$	1.08635	0.0398275
$745$	6.02633	0.220788
$746$	−5.26355	−0.192712
$747$	−23.4830	−0.859196
$748$	8.83288	0.322962
$749$	−3.49489	−0.127700
$750$	0.306329	0.0111856
$751$	−29.4138	−1.07332	−0.536662	−	0.843797i	$-0.680315\pi$
$751$	−29.4138	−1.07332	−0.536662	+	0.843797i	$0.680315\pi$
$752$	12.1662	0.443655
$753$	0.313888	0.0114387
$754$	13.5169	0.492255
$755$	−2.60140	−0.0946746
$756$	1.67956	0.0610849
$757$	−36.6271	−1.33124	−0.665618	−	0.746292i	$-0.731832\pi$
$757$	−36.6271	−1.33124	−0.665618	+	0.746292i	$0.731832\pi$
$758$	5.79695	0.210555
$759$	−0.169794	−0.00616312
$760$	8.01522	0.290742
$761$	26.5837	0.963659	0.481830	−	0.876265i	$-0.339972\pi$
$761$	26.5837	0.963659	0.481830	+	0.876265i	$0.339972\pi$
$762$	0.195003	0.00706421
$763$	−18.5484	−0.671498
$764$	−19.1615	−0.693238
$765$	4.18758	0.151402
$766$	−4.52243	−0.163402
$767$	8.94792	0.323091
$768$	0.723262	0.0260985
$769$	−22.0920	−0.796657	−0.398329	−	0.917243i	$-0.630410\pi$
$769$	−22.0920	−0.796657	−0.398329	+	0.917243i	$0.630410\pi$
$770$	5.13669	0.185114
$771$	−0.0673020	−0.00242382
$772$	−24.2083	−0.871277
$773$	34.5161	1.24146	0.620729	−	0.784025i	$-0.286837\pi$
$773$	34.5161	1.24146	0.620729	+	0.784025i	$0.286837\pi$
$774$	23.8936	0.858838
$775$	31.3021	1.12440
$776$	−16.3184	−0.585795
$777$	0.828118	0.0297086
$778$	16.3207	0.585127
$779$	0	0
$780$	−0.129111	−0.00462293
$781$	3.10927	0.111258
$782$	1.27568	0.0456181
$783$	−3.35937	−0.120054
$784$	2.96498	0.105892
$785$	9.89640	0.353218
$786$	0.745420	0.0265882
$787$	−8.16106	−0.290910	−0.145455	−	0.989365i	$-0.546465\pi$
$787$	−8.16106	−0.290910	−0.145455	+	0.989365i	$0.546465\pi$
$788$	21.1476	0.753351
$789$	0.781397	0.0278185
$790$	−3.30765	−0.117681
$791$	−48.3951	−1.72073
$792$	23.6715	0.841130
$793$	24.0131	0.852729
$794$	−4.79647	−0.170221
$795$	−0.130898	−0.00464247
$796$	26.3436	0.933724
$797$	46.2100	1.63684	0.818420	−	0.574620i	$-0.194850\pi$
$797$	46.2100	1.63684	0.818420	+	0.574620i	$0.194850\pi$
$798$	0.618620	0.0218989
$799$	22.4759	0.795139
$800$	26.2677	0.928704
$801$	42.3024	1.49468
$802$	−15.6541	−0.552767
$803$	20.1941	0.712634
$804$	0.724254	0.0255425
$805$	−2.00765	−0.0707604
$806$	10.3109	0.363186
$807$	0.0338000	0.00118982
$808$	−30.5862	−1.07602
$809$	0.363099	0.0127659	0.00638295	−	0.999980i	$-0.497968\pi$
$809$	0.363099	0.0127659	0.00638295	+	0.999980i	$0.497968\pi$
$810$	4.73023	0.166203
$811$	−22.6805	−0.796419	−0.398209	−	0.917295i	$-0.630368\pi$
$811$	−22.6805	−0.796419	−0.398209	+	0.917295i	$0.630368\pi$
$812$	−41.8661	−1.46921
$813$	−0.168992	−0.00592683
$814$	9.85745	0.345503
$815$	−0.663131	−0.0232285
$816$	−0.125510	−0.00439374
$817$	−47.6626	−1.66750
$818$	2.86217	0.100073
$819$	18.8747	0.659535
$820$	0	0
$821$	−12.5246	−0.437111	−0.218556	−	0.975824i	$-0.570135\pi$
$821$	−12.5246	−0.437111	−0.218556	+	0.975824i	$0.570135\pi$
$822$	0.0106126	0.000370158 0
$823$	30.8082	1.07390	0.536952	−	0.843613i	$-0.319575\pi$
$823$	30.8082	1.07390	0.536952	+	0.843613i	$0.319575\pi$
$824$	39.0845	1.36157
$825$	−0.853260	−0.0297067
$826$	10.2410	0.356330
$827$	−12.6832	−0.441039	−0.220520	−	0.975383i	$-0.570775\pi$
$827$	−12.6832	−0.441039	−0.220520	+	0.975383i	$0.570775\pi$
$828$	−3.90454	−0.135692
$829$	47.8117	1.66057	0.830284	−	0.557341i	$-0.188178\pi$
$829$	47.8117	1.66057	0.830284	+	0.557341i	$0.188178\pi$
$830$	4.13470	0.143518
$831$	−1.30171	−0.0451559
$832$	4.41620	0.153104
$833$	5.47753	0.189785
$834$	−0.105932	−0.00366812
$835$	−12.2103	−0.422554
$836$	−19.9279	−0.689222
$837$	−2.56259	−0.0885761
$838$	−10.8893	−0.376166
$839$	5.97006	0.206109	0.103055	−	0.994676i	$-0.467138\pi$
$839$	5.97006	0.206109	0.103055	+	0.994676i	$0.467138\pi$
$840$	−0.350142	−0.0120811
$841$	54.7387	1.88754
$842$	−8.17440	−0.281708
$843$	−0.109739	−0.00377962
$844$	−19.7174	−0.678701
$845$	6.43240	0.221281
$846$	25.4203	0.873967
$847$	4.19966	0.144302
$848$	−3.13615	−0.107696
$849$	−0.227656	−0.00781315
$850$	6.41063	0.219883
$851$	−3.85273	−0.132070
$852$	−0.0894458	−0.00306436
$853$	−5.96549	−0.204255	−0.102127	−	0.994771i	$-0.532565\pi$
$853$	−5.96549	−0.204255	−0.102127	+	0.994771i	$0.532565\pi$
$854$	27.4833	0.940458
$855$	−9.44763	−0.323102
$856$	2.83575	0.0969238
$857$	−8.64635	−0.295354	−0.147677	−	0.989036i	$-0.547180\pi$
$857$	−8.64635	−0.295354	−0.147677	+	0.989036i	$0.547180\pi$
$858$	−0.281064	−0.00959536
$859$	36.2717	1.23757	0.618787	−	0.785559i	$-0.287624\pi$
$859$	36.2717	1.23757	0.618787	+	0.785559i	$0.287624\pi$
$860$	11.3852	0.388231
$861$	0	0
$862$	19.4452	0.662305
$863$	34.3370	1.16885	0.584423	−	0.811449i	$-0.301321\pi$
$863$	34.3370	1.16885	0.584423	+	0.811449i	$0.301321\pi$
$864$	−2.15044	−0.0731596
$865$	−8.72046	−0.296505
$866$	0.343492	0.0116723
$867$	0.808925	0.0274725
$868$	−31.9362	−1.08399
$869$	19.4861	0.661022
$870$	0.295562	0.0100205
$871$	16.2884	0.551910
$872$	15.0502	0.509663
$873$	19.2346	0.650994
$874$	−2.87806	−0.0973520
$875$	−21.3384	−0.721369
$876$	−0.580932	−0.0196279
$877$	−44.9675	−1.51844	−0.759222	−	0.650832i	$-0.774420\pi$
$877$	−44.9675	−1.51844	−0.759222	+	0.650832i	$0.774420\pi$
$878$	14.3542	0.484430
$879$	0.851176	0.0287095
$880$	2.35125	0.0792607
$881$	28.0364	0.944569	0.472285	−	0.881446i	$-0.343429\pi$
$881$	28.0364	0.944569	0.472285	+	0.881446i	$0.343429\pi$
$882$	6.19510	0.208600
$883$	38.2278	1.28647	0.643234	−	0.765669i	$-0.277592\pi$
$883$	38.2278	1.28647	0.643234	+	0.765669i	$0.277592\pi$
$884$	−5.71466	−0.192205
$885$	0.195657	0.00657692
$886$	−5.47177	−0.183828
$887$	15.9010	0.533902	0.266951	−	0.963710i	$-0.413984\pi$
$887$	15.9010	0.533902	0.266951	+	0.963710i	$0.413984\pi$
$888$	−0.671933	−0.0225486
$889$	−13.5836	−0.455579
$890$	−7.44829	−0.249667
$891$	−27.8669	−0.933577
$892$	11.0839	0.371118
$893$	−50.7080	−1.69688
$894$	−0.377394	−0.0126220
$895$	−1.53589	−0.0513390
$896$	−31.6485	−1.05730
$897$	0.109852	0.00366786
$898$	−18.7631	−0.626134
$899$	63.8773	2.13043
$900$	−19.6214	−0.654047
$901$	−5.79374	−0.193017
$902$	0	0
$903$	2.08213	0.0692889
$904$	39.2677	1.30602
$905$	4.03450	0.134111
$906$	0.162911	0.00541234
$907$	43.1182	1.43172	0.715858	−	0.698246i	$-0.246036\pi$
$907$	43.1182	1.43172	0.715858	+	0.698246i	$0.246036\pi$
$908$	31.7105	1.05235
$909$	36.0523	1.19578
$910$	−3.32331	−0.110167
$911$	26.3499	0.873010	0.436505	−	0.899702i	$-0.356216\pi$
$911$	26.3499	0.873010	0.436505	+	0.899702i	$0.356216\pi$
$912$	0.283165	0.00937652
$913$	−24.3585	−0.806150
$914$	−25.9368	−0.857914
$915$	0.525074	0.0173584
$916$	−4.32557	−0.142921
$917$	−51.9247	−1.71471
$918$	−0.524815	−0.0173215
$919$	30.3070	0.999736	0.499868	−	0.866102i	$-0.333382\pi$
$919$	30.3070	0.999736	0.499868	+	0.866102i	$0.333382\pi$
$920$	1.62900	0.0537066
$921$	1.33893	0.0441192
$922$	20.9285	0.689244
$923$	−2.01162	−0.0662133
$924$	0.870546	0.0286389
$925$	−19.3611	−0.636588
$926$	−9.43083	−0.309916
$927$	−46.0693	−1.51312
$928$	53.6039	1.75963
$929$	−20.3568	−0.667886	−0.333943	−	0.942593i	$-0.608379\pi$
$929$	−20.3568	−0.667886	−0.333943	+	0.942593i	$0.608379\pi$
$930$	0.225460	0.00739312
$931$	−12.3579	−0.405014
$932$	25.4119	0.832393
$933$	−0.976492	−0.0319689
$934$	−21.6428	−0.708174
$935$	4.34371	0.142055
$936$	−15.3149	−0.500582
$937$	37.3624	1.22058	0.610288	−	0.792179i	$-0.291054\pi$
$937$	37.3624	1.22058	0.610288	+	0.792179i	$0.291054\pi$
$938$	18.6422	0.608691
$939$	0.0502818	0.00164089
$940$	12.1126	0.395070
$941$	−24.5271	−0.799560	−0.399780	−	0.916611i	$-0.630914\pi$
$941$	−24.5271	−0.799560	−0.399780	+	0.916611i	$0.630914\pi$
$942$	−0.619754	−0.0201927
$943$	0	0
$944$	4.68768	0.152571
$945$	0.825950	0.0268682
$946$	24.7845	0.805813
$947$	−57.2372	−1.85996	−0.929979	−	0.367612i	$-0.880175\pi$
$947$	−57.2372	−1.85996	−0.929979	+	0.367612i	$0.880175\pi$
$948$	−0.560567	−0.0182064
$949$	−13.0651	−0.424110
$950$	−14.4631	−0.469244
$951$	−1.73214	−0.0561683
$952$	−15.4978	−0.502287
$953$	−25.8126	−0.836152	−0.418076	−	0.908412i	$-0.637295\pi$
$953$	−25.8126	−0.836152	−0.418076	+	0.908412i	$0.637295\pi$
$954$	−6.55273	−0.212153
$955$	−9.42297	−0.304920
$956$	−8.67505	−0.280571
$957$	−1.74123	−0.0562858
$958$	19.6264	0.634101
$959$	−0.739258	−0.0238719
$960$	0.0965653	0.00311663
$961$	17.7268	0.571831
$962$	−6.37753	−0.205620
$963$	−3.34253	−0.107711
$964$	12.8257	0.413087
$965$	−11.9048	−0.383231
$966$	0.125727	0.00404521
$967$	50.2189	1.61493	0.807466	−	0.589915i	$-0.200839\pi$
$967$	50.2189	1.61493	0.807466	+	0.589915i	$0.200839\pi$
$968$	−3.40759	−0.109524
$969$	0.523120	0.0168050
$970$	−3.38669	−0.108740
$971$	−24.5844	−0.788950	−0.394475	−	0.918907i	$-0.629074\pi$
$971$	−24.5844	−0.788950	−0.394475	+	0.918907i	$0.629074\pi$
$972$	2.41001	0.0773010
$973$	7.37905	0.236561
$974$	8.52604	0.273192
$975$	0.552038	0.0176794
$976$	12.5801	0.402679
$977$	36.8420	1.17868	0.589341	−	0.807885i	$-0.299388\pi$
$977$	36.8420	1.17868	0.589341	+	0.807885i	$0.299388\pi$
$978$	0.0415280	0.00132792
$979$	43.8797	1.40240
$980$	2.95193	0.0942959
$981$	−17.7398	−0.566388
$982$	−19.7497	−0.630240
$983$	−24.1327	−0.769713	−0.384857	−	0.922976i	$-0.625749\pi$
$983$	−24.1327	−0.769713	−0.384857	+	0.922976i	$0.625749\pi$
$984$	0	0
$985$	10.3997	0.331361
$986$	13.0820	0.416616
$987$	2.21517	0.0705095
$988$	12.8929	0.410177
$989$	−9.68689	−0.308025
$990$	4.91275	0.156137
$991$	−6.95394	−0.220899	−0.110450	−	0.993882i	$-0.535229\pi$
$991$	−6.95394	−0.220899	−0.110450	+	0.993882i	$0.535229\pi$
$992$	40.8900	1.29826
$993$	1.88290	0.0597520
$994$	−2.30233	−0.0730254
$995$	12.9549	0.410698
$996$	0.700733	0.0222036
$997$	32.7607	1.03754	0.518772	−	0.854913i	$-0.326390\pi$
$997$	32.7607	1.03754	0.518772	+	0.854913i	$0.326390\pi$
$998$	−9.44492	−0.298974
$999$	1.58502	0.0501478

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1681.2.a.m.1.9		24
41.17	odd	40		41.2.g.a.2.3	✓	24
41.29	odd	40		41.2.g.a.21.3	yes	24
41.40	even	2	inner	1681.2.a.m.1.10		24
123.17	even	40		369.2.u.a.289.1		24
123.29	even	40		369.2.u.a.226.1		24
164.99	even	40		656.2.bs.d.289.2		24
164.111	even	40		656.2.bs.d.513.2		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
41.2.g.a.2.3	✓	24	41.17	odd	40
41.2.g.a.21.3	yes	24	41.29	odd	40
369.2.u.a.226.1		24	123.29	even	40
369.2.u.a.289.1		24	123.17	even	40
656.2.bs.d.289.2		24	164.99	even	40
656.2.bs.d.513.2		24	164.111	even	40
1681.2.a.m.1.9		24	1.1	even	1	trivial
1681.2.a.m.1.10		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)

\(f(q)\)	\(=\)	\(q-0.734595 q^{2} -0.0612231 q^{3} -1.46037 q^{4} -0.718161 q^{5} +0.0449742 q^{6} -3.13283 q^{7} +2.54197 q^{8} -2.99625 q^{9} +O(q^{10})\)	\(q-0.734595 q^{2} -0.0612231 q^{3} -1.46037 q^{4} -0.718161 q^{5} +0.0449742 q^{6} -3.13283 q^{7} +2.54197 q^{8} -2.99625 q^{9} +0.527558 q^{10} -3.10797 q^{11} +0.0894084 q^{12} +2.01078 q^{13} +2.30136 q^{14} +0.0439680 q^{15} +1.05342 q^{16} +1.94609 q^{17} +2.20103 q^{18} -4.39059 q^{19} +1.04878 q^{20} +0.191802 q^{21} +2.28310 q^{22} -0.892338 q^{23} -0.155627 q^{24} -4.48425 q^{25} -1.47711 q^{26} +0.367109 q^{27} +4.57509 q^{28} -9.15088 q^{29} -0.0322987 q^{30} -6.98046 q^{31} -5.85778 q^{32} +0.190280 q^{33} -1.42959 q^{34} +2.24988 q^{35} +4.37563 q^{36} +4.31757 q^{37} +3.22531 q^{38} -0.123106 q^{39} -1.82554 q^{40} -0.140897 q^{42} +10.8556 q^{43} +4.53878 q^{44} +2.15179 q^{45} +0.655507 q^{46} +11.5493 q^{47} -0.0644935 q^{48} +2.81463 q^{49} +3.29411 q^{50} -0.119146 q^{51} -2.93648 q^{52} -2.97712 q^{53} -0.269677 q^{54} +2.23202 q^{55} -7.96357 q^{56} +0.268806 q^{57} +6.72220 q^{58} +4.44997 q^{59} -0.0642096 q^{60} +11.9422 q^{61} +5.12781 q^{62} +9.38675 q^{63} +2.19626 q^{64} -1.44406 q^{65} -0.139778 q^{66} +8.10052 q^{67} -2.84201 q^{68} +0.0546317 q^{69} -1.65275 q^{70} -1.00042 q^{71} -7.61639 q^{72} -6.49752 q^{73} -3.17167 q^{74} +0.274539 q^{75} +6.41188 q^{76} +9.73674 q^{77} +0.0904333 q^{78} -6.26974 q^{79} -0.756523 q^{80} +8.96628 q^{81} +7.83744 q^{83} -0.280101 q^{84} -1.39761 q^{85} -7.97450 q^{86} +0.560246 q^{87} -7.90037 q^{88} -14.1184 q^{89} -1.58070 q^{90} -6.29943 q^{91} +1.30314 q^{92} +0.427365 q^{93} -8.48403 q^{94} +3.15315 q^{95} +0.358632 q^{96} -6.41956 q^{97} -2.06762 q^{98} +9.31226 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 8 q^{2} + 20 q^{4} + 12 q^{5} + 24 q^{8} + 24 q^{9}+O(q^{10}) \) 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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