LMFDB - Embedded newform 4022.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4022,2,Mod(1,4022)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4022, 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("4022.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4022 = 2 \cdot 2011 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4022.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:	$32.1158316930$
Analytic rank:	$0$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	4022.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.20664 q^{3} +1.00000 q^{4} -0.511117 q^{5} -3.20664 q^{6} +0.304730 q^{7} +1.00000 q^{8} +7.28252 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.20664 q^{3} +1.00000 q^{4} -0.511117 q^{5} -3.20664 q^{6} +0.304730 q^{7} +1.00000 q^{8} +7.28252 q^{9} -0.511117 q^{10} -5.47625 q^{11} -3.20664 q^{12} -4.55294 q^{13} +0.304730 q^{14} +1.63897 q^{15} +1.00000 q^{16} -0.0995111 q^{17} +7.28252 q^{18} -2.49354 q^{19} -0.511117 q^{20} -0.977157 q^{21} -5.47625 q^{22} +3.77686 q^{23} -3.20664 q^{24} -4.73876 q^{25} -4.55294 q^{26} -13.7325 q^{27} +0.304730 q^{28} -2.52025 q^{29} +1.63897 q^{30} +8.08601 q^{31} +1.00000 q^{32} +17.5604 q^{33} -0.0995111 q^{34} -0.155752 q^{35} +7.28252 q^{36} +2.81456 q^{37} -2.49354 q^{38} +14.5996 q^{39} -0.511117 q^{40} +0.268079 q^{41} -0.977157 q^{42} +2.78357 q^{43} -5.47625 q^{44} -3.72222 q^{45} +3.77686 q^{46} +6.14783 q^{47} -3.20664 q^{48} -6.90714 q^{49} -4.73876 q^{50} +0.319096 q^{51} -4.55294 q^{52} -13.7699 q^{53} -13.7325 q^{54} +2.79901 q^{55} +0.304730 q^{56} +7.99587 q^{57} -2.52025 q^{58} +3.86826 q^{59} +1.63897 q^{60} -3.61577 q^{61} +8.08601 q^{62} +2.21920 q^{63} +1.00000 q^{64} +2.32708 q^{65} +17.5604 q^{66} -6.70163 q^{67} -0.0995111 q^{68} -12.1110 q^{69} -0.155752 q^{70} +5.87209 q^{71} +7.28252 q^{72} -8.47756 q^{73} +2.81456 q^{74} +15.1955 q^{75} -2.49354 q^{76} -1.66878 q^{77} +14.5996 q^{78} -1.83982 q^{79} -0.511117 q^{80} +22.1875 q^{81} +0.268079 q^{82} +15.1734 q^{83} -0.977157 q^{84} +0.0508618 q^{85} +2.78357 q^{86} +8.08152 q^{87} -5.47625 q^{88} -10.3228 q^{89} -3.72222 q^{90} -1.38741 q^{91} +3.77686 q^{92} -25.9289 q^{93} +6.14783 q^{94} +1.27449 q^{95} -3.20664 q^{96} +19.2804 q^{97} -6.90714 q^{98} -39.8809 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q + 50 q^{2} + 18 q^{3} + 50 q^{4} + 11 q^{5} + 18 q^{6} + 30 q^{7} + 50 q^{8} + 66 q^{9}+O(q^{10}) $ 50 * q + 50 * q^2 + 18 * q^3 + 50 * q^4 + 11 * q^5 + 18 * q^6 + 30 * q^7 + 50 * q^8 + 66 * q^9	\( 50 q + 50 q^{2} + 18 q^{3} + 50 q^{4} + 11 q^{5} + 18 q^{6} + 30 q^{7} + 50 q^{8} + 66 q^{9} + 11 q^{10} + 21 q^{11} + 18 q^{12} + 26 q^{13} + 30 q^{14} + 17 q^{15} + 50 q^{16} + 24 q^{17} + 66 q^{18} + 39 q^{19} + 11 q^{20} - q^{21} + 21 q^{22} + 28 q^{23} + 18 q^{24} + 79 q^{25} + 26 q^{26} + 66 q^{27} + 30 q^{28} - 5 q^{29} + 17 q^{30} + 60 q^{31} + 50 q^{32} + 37 q^{33} + 24 q^{34} + 38 q^{35} + 66 q^{36} + 35 q^{37} + 39 q^{38} + 37 q^{39} + 11 q^{40} + 42 q^{41} - q^{42} + 44 q^{43} + 21 q^{44} + 31 q^{45} + 28 q^{46} + 60 q^{47} + 18 q^{48} + 92 q^{49} + 79 q^{50} + 26 q^{51} + 26 q^{52} - 2 q^{53} + 66 q^{54} + 33 q^{55} + 30 q^{56} + 15 q^{57} - 5 q^{58} + 65 q^{59} + 17 q^{60} + 15 q^{61} + 60 q^{62} + 56 q^{63} + 50 q^{64} + 6 q^{65} + 37 q^{66} + 48 q^{67} + 24 q^{68} - 9 q^{69} + 38 q^{70} + 34 q^{71} + 66 q^{72} + 91 q^{73} + 35 q^{74} + 54 q^{75} + 39 q^{76} - 6 q^{77} + 37 q^{78} + 29 q^{79} + 11 q^{80} + 66 q^{81} + 42 q^{82} + 43 q^{83} - q^{84} + 44 q^{86} + 32 q^{87} + 21 q^{88} + 38 q^{89} + 31 q^{90} + 55 q^{91} + 28 q^{92} - 15 q^{93} + 60 q^{94} + 9 q^{95} + 18 q^{96} + 80 q^{97} + 92 q^{98} + 20 q^{99}+O(q^{100}) \) 50 * q + 50 * q^2 + 18 * q^3 + 50 * q^4 + 11 * q^5 + 18 * q^6 + 30 * q^7 + 50 * q^8 + 66 * q^9 + 11 * q^10 + 21 * q^11 + 18 * q^12 + 26 * q^13 + 30 * q^14 + 17 * q^15 + 50 * q^16 + 24 * q^17 + 66 * q^18 + 39 * q^19 + 11 * q^20 - q^21 + 21 * q^22 + 28 * q^23 + 18 * q^24 + 79 * q^25 + 26 * q^26 + 66 * q^27 + 30 * q^28 - 5 * q^29 + 17 * q^30 + 60 * q^31 + 50 * q^32 + 37 * q^33 + 24 * q^34 + 38 * q^35 + 66 * q^36 + 35 * q^37 + 39 * q^38 + 37 * q^39 + 11 * q^40 + 42 * q^41 - q^42 + 44 * q^43 + 21 * q^44 + 31 * q^45 + 28 * q^46 + 60 * q^47 + 18 * q^48 + 92 * q^49 + 79 * q^50 + 26 * q^51 + 26 * q^52 - 2 * q^53 + 66 * q^54 + 33 * q^55 + 30 * q^56 + 15 * q^57 - 5 * q^58 + 65 * q^59 + 17 * q^60 + 15 * q^61 + 60 * q^62 + 56 * q^63 + 50 * q^64 + 6 * q^65 + 37 * q^66 + 48 * q^67 + 24 * q^68 - 9 * q^69 + 38 * q^70 + 34 * q^71 + 66 * q^72 + 91 * q^73 + 35 * q^74 + 54 * q^75 + 39 * q^76 - 6 * q^77 + 37 * q^78 + 29 * q^79 + 11 * q^80 + 66 * q^81 + 42 * q^82 + 43 * q^83 - q^84 + 44 * q^86 + 32 * q^87 + 21 * q^88 + 38 * q^89 + 31 * q^90 + 55 * q^91 + 28 * q^92 - 15 * q^93 + 60 * q^94 + 9 * q^95 + 18 * q^96 + 80 * q^97 + 92 * q^98 + 20 * q^99

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−3.20664	−1.85135	−0.925676	−	0.378317i	$-0.876503\pi$
$3$	−3.20664	−1.85135	−0.925676	+	0.378317i	$0.876503\pi$
$4$	1.00000	0.500000
$5$	−0.511117	−0.228578	−0.114289	−	0.993448i	$-0.536459\pi$
$5$	−0.511117	−0.228578	−0.114289	+	0.993448i	$0.536459\pi$
$6$	−3.20664	−1.30910
$7$	0.304730	0.115177	0.0575885	−	0.998340i	$-0.481659\pi$
$7$	0.304730	0.115177	0.0575885	+	0.998340i	$0.481659\pi$
$8$	1.00000	0.353553
$9$	7.28252	2.42751
$10$	−0.511117	−0.161629
$11$	−5.47625	−1.65115	−0.825576	−	0.564290i	$-0.809150\pi$
$11$	−5.47625	−1.65115	−0.825576	+	0.564290i	$0.809150\pi$
$12$	−3.20664	−0.925676
$13$	−4.55294	−1.26276	−0.631379	−	0.775475i	$-0.717511\pi$
$13$	−4.55294	−1.26276	−0.631379	+	0.775475i	$0.717511\pi$
$14$	0.304730	0.0814424
$15$	1.63897	0.423179
$16$	1.00000	0.250000
$17$	−0.0995111	−0.0241350	−0.0120675	−	0.999927i	$-0.503841\pi$
$17$	−0.0995111	−0.0241350	−0.0120675	+	0.999927i	$0.503841\pi$
$18$	7.28252	1.71651
$19$	−2.49354	−0.572057	−0.286029	−	0.958221i	$-0.592335\pi$
$19$	−2.49354	−0.572057	−0.286029	+	0.958221i	$0.592335\pi$
$20$	−0.511117	−0.114289
$21$	−0.977157	−0.213233
$22$	−5.47625	−1.16754
$23$	3.77686	0.787530	0.393765	−	0.919211i	$-0.371172\pi$
$23$	3.77686	0.787530	0.393765	+	0.919211i	$0.371172\pi$
$24$	−3.20664	−0.654552
$25$	−4.73876	−0.947752
$26$	−4.55294	−0.892904
$27$	−13.7325	−2.64282
$28$	0.304730	0.0575885
$29$	−2.52025	−0.467998	−0.233999	−	0.972237i	$-0.575181\pi$
$29$	−2.52025	−0.467998	−0.233999	+	0.972237i	$0.575181\pi$
$30$	1.63897	0.299233
$31$	8.08601	1.45229	0.726146	−	0.687541i	$-0.241310\pi$
$31$	8.08601	1.45229	0.726146	+	0.687541i	$0.241310\pi$
$32$	1.00000	0.176777
$33$	17.5604	3.05687
$34$	−0.0995111	−0.0170660
$35$	−0.155752	−0.0263270
$36$	7.28252	1.21375
$37$	2.81456	0.462711	0.231356	−	0.972869i	$-0.425684\pi$
$37$	2.81456	0.462711	0.231356	+	0.972869i	$0.425684\pi$
$38$	−2.49354	−0.404506
$39$	14.5996	2.33781
$40$	−0.511117	−0.0808147
$41$	0.268079	0.0418670	0.0209335	−	0.999781i	$-0.493336\pi$
$41$	0.268079	0.0418670	0.0209335	+	0.999781i	$0.493336\pi$
$42$	−0.977157	−0.150779
$43$	2.78357	0.424491	0.212245	−	0.977216i	$-0.431922\pi$
$43$	2.78357	0.424491	0.212245	+	0.977216i	$0.431922\pi$
$44$	−5.47625	−0.825576
$45$	−3.72222	−0.554875
$46$	3.77686	0.556867
$47$	6.14783	0.896753	0.448377	−	0.893845i	$-0.352002\pi$
$47$	6.14783	0.896753	0.448377	+	0.893845i	$0.352002\pi$
$48$	−3.20664	−0.462838
$49$	−6.90714	−0.986734
$50$	−4.73876	−0.670162
$51$	0.319096	0.0446824
$52$	−4.55294	−0.631379
$53$	−13.7699	−1.89144	−0.945720	−	0.324983i	$-0.894641\pi$
$53$	−13.7699	−1.89144	−0.945720	+	0.324983i	$0.894641\pi$
$54$	−13.7325	−1.86875
$55$	2.79901	0.377418
$56$	0.304730	0.0407212
$57$	7.99587	1.05908
$58$	−2.52025	−0.330925
$59$	3.86826	0.503604	0.251802	−	0.967779i	$-0.418977\pi$
$59$	3.86826	0.503604	0.251802	+	0.967779i	$0.418977\pi$
$60$	1.63897	0.211590
$61$	−3.61577	−0.462952	−0.231476	−	0.972841i	$-0.574356\pi$
$61$	−3.61577	−0.462952	−0.231476	+	0.972841i	$0.574356\pi$
$62$	8.08601	1.02692
$63$	2.21920	0.279593
$64$	1.00000	0.125000
$65$	2.32708	0.288639
$66$	17.5604	2.16153
$67$	−6.70163	−0.818735	−0.409367	−	0.912370i	$-0.634251\pi$
$67$	−6.70163	−0.818735	−0.409367	+	0.912370i	$0.634251\pi$
$68$	−0.0995111	−0.0120675
$69$	−12.1110	−1.45799
$70$	−0.155752	−0.0186160
$71$	5.87209	0.696889	0.348445	−	0.937329i	$-0.386710\pi$
$71$	5.87209	0.696889	0.348445	+	0.937329i	$0.386710\pi$
$72$	7.28252	0.858253
$73$	−8.47756	−0.992223	−0.496112	−	0.868259i	$-0.665239\pi$
$73$	−8.47756	−0.992223	−0.496112	+	0.868259i	$0.665239\pi$
$74$	2.81456	0.327186
$75$	15.1955	1.75462
$76$	−2.49354	−0.286029
$77$	−1.66878	−0.190175
$78$	14.5996	1.65308
$79$	−1.83982	−0.206996	−0.103498	−	0.994630i	$-0.533004\pi$
$79$	−1.83982	−0.206996	−0.103498	+	0.994630i	$0.533004\pi$
$80$	−0.511117	−0.0571446
$81$	22.1875	2.46528
$82$	0.268079	0.0296044
$83$	15.1734	1.66550	0.832749	−	0.553650i	$-0.186766\pi$
$83$	15.1734	1.66550	0.832749	+	0.553650i	$0.186766\pi$
$84$	−0.977157	−0.106617
$85$	0.0508618	0.00551674
$86$	2.78357	0.300160
$87$	8.08152	0.866429
$88$	−5.47625	−0.583771
$89$	−10.3228	−1.09421	−0.547105	−	0.837064i	$-0.684270\pi$
$89$	−10.3228	−1.09421	−0.547105	+	0.837064i	$0.684270\pi$
$90$	−3.72222	−0.392356
$91$	−1.38741	−0.145441
$92$	3.77686	0.393765
$93$	−25.9289	−2.68870
$94$	6.14783	0.634100
$95$	1.27449	0.130760
$96$	−3.20664	−0.327276
$97$	19.2804	1.95763	0.978814	−	0.204752i	$-0.0656386\pi$
$97$	19.2804	1.95763	0.978814	+	0.204752i	$0.0656386\pi$
$98$	−6.90714	−0.697726
$99$	−39.8809	−4.00818
$100$	−4.73876	−0.473876
$101$	6.81151	0.677770	0.338885	−	0.940828i	$-0.389950\pi$
$101$	6.81151	0.677770	0.338885	+	0.940828i	$0.389950\pi$
$102$	0.319096	0.0315952
$103$	18.5771	1.83046	0.915230	−	0.402932i	$-0.132009\pi$
$103$	18.5771	1.83046	0.915230	+	0.402932i	$0.132009\pi$
$104$	−4.55294	−0.446452
$105$	0.499441	0.0487405
$106$	−13.7699	−1.33745
$107$	−18.7182	−1.80955	−0.904777	−	0.425886i	$-0.859963\pi$
$107$	−18.7182	−1.80955	−0.904777	+	0.425886i	$0.859963\pi$
$108$	−13.7325	−1.32141
$109$	−10.8747	−1.04161	−0.520804	−	0.853677i	$-0.674368\pi$
$109$	−10.8747	−1.04161	−0.520804	+	0.853677i	$0.674368\pi$
$110$	2.79901	0.266875
$111$	−9.02527	−0.856641
$112$	0.304730	0.0287942
$113$	−16.2825	−1.53173	−0.765864	−	0.643003i	$-0.777688\pi$
$113$	−16.2825	−1.53173	−0.765864	+	0.643003i	$0.777688\pi$
$114$	7.99587	0.748882
$115$	−1.93042	−0.180012
$116$	−2.52025	−0.233999
$117$	−33.1568	−3.06535
$118$	3.86826	0.356102
$119$	−0.0303240	−0.00277979
$120$	1.63897	0.149616
$121$	18.9894	1.72630
$122$	−3.61577	−0.327357
$123$	−0.859633	−0.0775105
$124$	8.08601	0.726146
$125$	4.97765	0.445214
$126$	2.21920	0.197702
$127$	17.6871	1.56947	0.784736	−	0.619830i	$-0.212798\pi$
$127$	17.6871	1.56947	0.784736	+	0.619830i	$0.212798\pi$
$128$	1.00000	0.0883883
$129$	−8.92591	−0.785882
$130$	2.32708	0.204099
$131$	20.2685	1.77087	0.885433	−	0.464767i	$-0.153862\pi$
$131$	20.2685	1.77087	0.885433	+	0.464767i	$0.153862\pi$
$132$	17.5604	1.52843
$133$	−0.759855	−0.0658878
$134$	−6.70163	−0.578933
$135$	7.01890	0.604091
$136$	−0.0995111	−0.00853300
$137$	9.35924	0.799614	0.399807	−	0.916599i	$-0.369077\pi$
$137$	9.35924	0.799614	0.399807	+	0.916599i	$0.369077\pi$
$138$	−12.1110	−1.03096
$139$	11.0069	0.933593	0.466797	−	0.884365i	$-0.345408\pi$
$139$	11.0069	0.933593	0.466797	+	0.884365i	$0.345408\pi$
$140$	−0.155752	−0.0131635
$141$	−19.7139	−1.66021
$142$	5.87209	0.492775
$143$	24.9330	2.08500
$144$	7.28252	0.606876
$145$	1.28814	0.106974
$146$	−8.47756	−0.701608
$147$	22.1487	1.82679
$148$	2.81456	0.231356
$149$	12.7245	1.04243	0.521215	−	0.853426i	$-0.325479\pi$
$149$	12.7245	1.04243	0.521215	+	0.853426i	$0.325479\pi$
$150$	15.1955	1.24071
$151$	−15.0182	−1.22216	−0.611081	−	0.791568i	$-0.709265\pi$
$151$	−15.0182	−1.22216	−0.611081	+	0.791568i	$0.709265\pi$
$152$	−2.49354	−0.202253
$153$	−0.724691	−0.0585878
$154$	−1.66878	−0.134474
$155$	−4.13290	−0.331962
$156$	14.5996	1.16890
$157$	8.06414	0.643589	0.321794	−	0.946810i	$-0.395714\pi$
$157$	8.06414	0.643589	0.321794	+	0.946810i	$0.395714\pi$
$158$	−1.83982	−0.146368
$159$	44.1550	3.50172
$160$	−0.511117	−0.0404073
$161$	1.15092	0.0907052
$162$	22.1875	1.74321
$163$	22.3647	1.75174	0.875871	−	0.482546i	$-0.160288\pi$
$163$	22.3647	1.75174	0.875871	+	0.482546i	$0.160288\pi$
$164$	0.268079	0.0209335
$165$	−8.97539	−0.698734
$166$	15.1734	1.17769
$167$	8.52996	0.660068	0.330034	−	0.943969i	$-0.392940\pi$
$167$	8.52996	0.660068	0.330034	+	0.943969i	$0.392940\pi$
$168$	−0.977157	−0.0753893
$169$	7.72923	0.594556
$170$	0.0508618	0.00390092
$171$	−18.1592	−1.38867
$172$	2.78357	0.212245
$173$	19.5571	1.48690	0.743450	−	0.668791i	$-0.233188\pi$
$173$	19.5571	1.48690	0.743450	+	0.668791i	$0.233188\pi$
$174$	8.08152	0.612658
$175$	−1.44404	−0.109159
$176$	−5.47625	−0.412788
$177$	−12.4041	−0.932349
$178$	−10.3228	−0.773723
$179$	−4.59652	−0.343560	−0.171780	−	0.985135i	$-0.554952\pi$
$179$	−4.59652	−0.343560	−0.171780	+	0.985135i	$0.554952\pi$
$180$	−3.72222	−0.277438
$181$	8.27870	0.615351	0.307676	−	0.951491i	$-0.400449\pi$
$181$	8.27870	0.615351	0.307676	+	0.951491i	$0.400449\pi$
$182$	−1.38741	−0.102842
$183$	11.5945	0.857088
$184$	3.77686	0.278434
$185$	−1.43857	−0.105766
$186$	−25.9289	−1.90120
$187$	0.544948	0.0398505
$188$	6.14783	0.448377
$189$	−4.18469	−0.304391
$190$	1.27449	0.0924612
$191$	−1.59309	−0.115272	−0.0576359	−	0.998338i	$-0.518356\pi$
$191$	−1.59309	−0.115272	−0.0576359	+	0.998338i	$0.518356\pi$
$192$	−3.20664	−0.231419
$193$	0.0427242	0.00307536	0.00153768	−	0.999999i	$-0.499511\pi$
$193$	0.0427242	0.00307536	0.00153768	+	0.999999i	$0.499511\pi$
$194$	19.2804	1.38425
$195$	−7.46211	−0.534373
$196$	−6.90714	−0.493367
$197$	−4.36537	−0.311019	−0.155510	−	0.987834i	$-0.549702\pi$
$197$	−4.36537	−0.311019	−0.155510	+	0.987834i	$0.549702\pi$
$198$	−39.8809	−2.83421
$199$	7.47568	0.529937	0.264969	−	0.964257i	$-0.414638\pi$
$199$	7.47568	0.529937	0.264969	+	0.964257i	$0.414638\pi$
$200$	−4.73876	−0.335081
$201$	21.4897	1.51577
$202$	6.81151	0.479256
$203$	−0.767994	−0.0539026
$204$	0.319096	0.0223412
$205$	−0.137020	−0.00956988
$206$	18.5771	1.29433
$207$	27.5050	1.91173
$208$	−4.55294	−0.315689
$209$	13.6553	0.944554
$210$	0.499441	0.0344647
$211$	−2.28119	−0.157043	−0.0785217	−	0.996912i	$-0.525020\pi$
$211$	−2.28119	−0.157043	−0.0785217	+	0.996912i	$0.525020\pi$
$212$	−13.7699	−0.945720
$213$	−18.8297	−1.29019
$214$	−18.7182	−1.27955
$215$	−1.42273	−0.0970295
$216$	−13.7325	−0.934376
$217$	2.46405	0.167270
$218$	−10.8747	−0.736527
$219$	27.1844	1.83695
$220$	2.79901	0.188709
$221$	0.453068	0.0304766
$222$	−9.02527	−0.605737
$223$	7.72993	0.517634	0.258817	−	0.965926i	$-0.416667\pi$
$223$	7.72993	0.517634	0.258817	+	0.965926i	$0.416667\pi$
$224$	0.304730	0.0203606
$225$	−34.5101	−2.30067
$226$	−16.2825	−1.08310
$227$	−10.8012	−0.716900	−0.358450	−	0.933549i	$-0.616695\pi$
$227$	−10.8012	−0.716900	−0.358450	+	0.933549i	$0.616695\pi$
$228$	7.99587	0.529540
$229$	24.7410	1.63493	0.817467	−	0.575975i	$-0.195377\pi$
$229$	24.7410	1.63493	0.817467	+	0.575975i	$0.195377\pi$
$230$	−1.93042	−0.127288
$231$	5.35116	0.352080
$232$	−2.52025	−0.165462
$233$	22.1006	1.44786	0.723930	−	0.689873i	$-0.242334\pi$
$233$	22.1006	1.44786	0.723930	+	0.689873i	$0.242334\pi$
$234$	−33.1568	−2.16753
$235$	−3.14226	−0.204978
$236$	3.86826	0.251802
$237$	5.89963	0.383222
$238$	−0.0303240	−0.00196561
$239$	−0.970013	−0.0627449	−0.0313725	−	0.999508i	$-0.509988\pi$
$239$	−0.970013	−0.0627449	−0.0313725	+	0.999508i	$0.509988\pi$
$240$	1.63897	0.105795
$241$	−10.0867	−0.649738	−0.324869	−	0.945759i	$-0.605320\pi$
$241$	−10.0867	−0.649738	−0.324869	+	0.945759i	$0.605320\pi$
$242$	18.9894	1.22068
$243$	−29.9498	−1.92128
$244$	−3.61577	−0.231476
$245$	3.53036	0.225546
$246$	−0.859633	−0.0548082
$247$	11.3529	0.722369
$248$	8.08601	0.513462
$249$	−48.6556	−3.08342
$250$	4.97765	0.314814
$251$	3.88688	0.245338	0.122669	−	0.992448i	$-0.460855\pi$
$251$	3.88688	0.245338	0.122669	+	0.992448i	$0.460855\pi$
$252$	2.21920	0.139796
$253$	−20.6830	−1.30033
$254$	17.6871	1.10978
$255$	−0.163095	−0.0102134
$256$	1.00000	0.0625000
$257$	16.9818	1.05930	0.529648	−	0.848218i	$-0.322324\pi$
$257$	16.9818	1.05930	0.529648	+	0.848218i	$0.322324\pi$
$258$	−8.92591	−0.555703
$259$	0.857680	0.0532936
$260$	2.32708	0.144320
$261$	−18.3537	−1.13607
$262$	20.2685	1.25219
$263$	−6.28942	−0.387822	−0.193911	−	0.981019i	$-0.562117\pi$
$263$	−6.28942	−0.387822	−0.193911	+	0.981019i	$0.562117\pi$
$264$	17.5604	1.08077
$265$	7.03802	0.432342
$266$	−0.759855	−0.0465897
$267$	33.1013	2.02577
$268$	−6.70163	−0.409367
$269$	2.87948	0.175565	0.0877825	−	0.996140i	$-0.472022\pi$
$269$	2.87948	0.175565	0.0877825	+	0.996140i	$0.472022\pi$
$270$	7.01890	0.427157
$271$	−13.1347	−0.797878	−0.398939	−	0.916977i	$-0.630622\pi$
$271$	−13.1347	−0.797878	−0.398939	+	0.916977i	$0.630622\pi$
$272$	−0.0995111	−0.00603374
$273$	4.44893	0.269262
$274$	9.35924	0.565412
$275$	25.9506	1.56488
$276$	−12.1110	−0.728997
$277$	16.4892	0.990739	0.495370	−	0.868682i	$-0.335033\pi$
$277$	16.4892	0.990739	0.495370	+	0.868682i	$0.335033\pi$
$278$	11.0069	0.660150
$279$	58.8865	3.52544
$280$	−0.155752	−0.00930799
$281$	−26.4941	−1.58051	−0.790253	−	0.612781i	$-0.790051\pi$
$281$	−26.4941	−1.58051	−0.790253	+	0.612781i	$0.790051\pi$
$282$	−19.7139	−1.17394
$283$	−23.2728	−1.38342	−0.691711	−	0.722174i	$-0.743143\pi$
$283$	−23.2728	−1.38342	−0.691711	+	0.722174i	$0.743143\pi$
$284$	5.87209	0.348445
$285$	−4.08683	−0.242083
$286$	24.9330	1.47432
$287$	0.0816917	0.00482211
$288$	7.28252	0.429126
$289$	−16.9901	−0.999418
$290$	1.28814	0.0756422
$291$	−61.8252	−3.62426
$292$	−8.47756	−0.496112
$293$	−3.40579	−0.198969	−0.0994843	−	0.995039i	$-0.531719\pi$
$293$	−3.40579	−0.198969	−0.0994843	+	0.995039i	$0.531719\pi$
$294$	22.1487	1.29174
$295$	−1.97713	−0.115113
$296$	2.81456	0.163593
$297$	75.2025	4.36369
$298$	12.7245	0.737109
$299$	−17.1958	−0.994459
$300$	15.1955	0.877311
$301$	0.848237	0.0488916
$302$	−15.0182	−0.864198
$303$	−21.8420	−1.25479
$304$	−2.49354	−0.143014
$305$	1.84808	0.105821
$306$	−0.724691	−0.0414278
$307$	8.52120	0.486330	0.243165	−	0.969985i	$-0.421814\pi$
$307$	8.52120	0.486330	0.243165	+	0.969985i	$0.421814\pi$
$308$	−1.66878	−0.0950873
$309$	−59.5701	−3.38883
$310$	−4.13290	−0.234733
$311$	−19.2623	−1.09227	−0.546133	−	0.837698i	$-0.683901\pi$
$311$	−19.2623	−1.09227	−0.546133	+	0.837698i	$0.683901\pi$
$312$	14.5996	0.826540
$313$	−6.17973	−0.349299	−0.174650	−	0.984631i	$-0.555879\pi$
$313$	−6.17973	−0.349299	−0.174650	+	0.984631i	$0.555879\pi$
$314$	8.06414	0.455086
$315$	−1.13427	−0.0639089
$316$	−1.83982	−0.103498
$317$	32.9797	1.85232	0.926161	−	0.377128i	$-0.123088\pi$
$317$	32.9797	1.85232	0.926161	+	0.377128i	$0.123088\pi$
$318$	44.1550	2.47609
$319$	13.8015	0.772736
$320$	−0.511117	−0.0285723
$321$	60.0224	3.35012
$322$	1.15092	0.0641383
$323$	0.248135	0.0138066
$324$	22.1875	1.23264
$325$	21.5753	1.19678
$326$	22.3647	1.23867
$327$	34.8712	1.92838
$328$	0.268079	0.0148022
$329$	1.87343	0.103285
$330$	−8.97539	−0.494079
$331$	6.51620	0.358163	0.179081	−	0.983834i	$-0.442687\pi$
$331$	6.51620	0.358163	0.179081	+	0.983834i	$0.442687\pi$
$332$	15.1734	0.832749
$333$	20.4971	1.12323
$334$	8.52996	0.466739
$335$	3.42532	0.187145
$336$	−0.977157	−0.0533083
$337$	4.93373	0.268757	0.134379	−	0.990930i	$-0.457096\pi$
$337$	4.93373	0.268757	0.134379	+	0.990930i	$0.457096\pi$
$338$	7.72923	0.420415
$339$	52.2120	2.83577
$340$	0.0508618	0.00275837
$341$	−44.2811	−2.39795
$342$	−18.1592	−0.981939
$343$	−4.23792	−0.228826
$344$	2.78357	0.150080
$345$	6.19014	0.333266
$346$	19.5571	1.05140
$347$	−3.51442	−0.188664	−0.0943319	−	0.995541i	$-0.530071\pi$
$347$	−3.51442	−0.188664	−0.0943319	+	0.995541i	$0.530071\pi$
$348$	8.08152	0.433215
$349$	27.4961	1.47183	0.735915	−	0.677074i	$-0.236752\pi$
$349$	27.4961	1.47183	0.735915	+	0.677074i	$0.236752\pi$
$350$	−1.44404	−0.0771872
$351$	62.5231	3.33723
$352$	−5.47625	−0.291885
$353$	−17.7729	−0.945958	−0.472979	−	0.881074i	$-0.656821\pi$
$353$	−17.7729	−0.945958	−0.472979	+	0.881074i	$0.656821\pi$
$354$	−12.4041	−0.659271
$355$	−3.00133	−0.159294
$356$	−10.3228	−0.547105
$357$	0.0972379	0.00514638
$358$	−4.59652	−0.242933
$359$	29.0846	1.53503	0.767513	−	0.641033i	$-0.221494\pi$
$359$	29.0846	1.53503	0.767513	+	0.641033i	$0.221494\pi$
$360$	−3.72222	−0.196178
$361$	−12.7823	−0.672751
$362$	8.27870	0.435119
$363$	−60.8920	−3.19600
$364$	−1.38741	−0.0727203
$365$	4.33302	0.226801
$366$	11.5945	0.606053
$367$	−24.3797	−1.27261	−0.636305	−	0.771438i	$-0.719538\pi$
$367$	−24.3797	−1.27261	−0.636305	+	0.771438i	$0.719538\pi$
$368$	3.77686	0.196882
$369$	1.95229	0.101632
$370$	−1.43857	−0.0747877
$371$	−4.19609	−0.217850
$372$	−25.9289	−1.34435
$373$	18.3450	0.949867	0.474933	−	0.880022i	$-0.342472\pi$
$373$	18.3450	0.949867	0.474933	+	0.880022i	$0.342472\pi$
$374$	0.544948	0.0281786
$375$	−15.9615	−0.824248
$376$	6.14783	0.317050
$377$	11.4745	0.590968
$378$	−4.18469	−0.215237
$379$	17.7293	0.910694	0.455347	−	0.890314i	$-0.349515\pi$
$379$	17.7293	0.910694	0.455347	+	0.890314i	$0.349515\pi$
$380$	1.27449	0.0653800
$381$	−56.7160	−2.90565
$382$	−1.59309	−0.0815094
$383$	−31.8343	−1.62666	−0.813329	−	0.581804i	$-0.802347\pi$
$383$	−31.8343	−1.62666	−0.813329	+	0.581804i	$0.802347\pi$
$384$	−3.20664	−0.163638
$385$	0.852940	0.0434698
$386$	0.0427242	0.00217460
$387$	20.2714	1.03045
$388$	19.2804	0.978814
$389$	24.3207	1.23311	0.616554	−	0.787312i	$-0.288528\pi$
$389$	24.3207	1.23311	0.616554	+	0.787312i	$0.288528\pi$
$390$	−7.46211	−0.377859
$391$	−0.375839	−0.0190070
$392$	−6.90714	−0.348863
$393$	−64.9937	−3.27850
$394$	−4.36537	−0.219924
$395$	0.940363	0.0473148
$396$	−39.8809	−2.00409
$397$	31.1635	1.56405	0.782026	−	0.623246i	$-0.214187\pi$
$397$	31.1635	1.56405	0.782026	+	0.623246i	$0.214187\pi$
$398$	7.47568	0.374722
$399$	2.43658	0.121982
$400$	−4.73876	−0.236938
$401$	−13.4381	−0.671066	−0.335533	−	0.942028i	$-0.608916\pi$
$401$	−13.4381	−0.671066	−0.335533	+	0.942028i	$0.608916\pi$
$402$	21.4897	1.07181
$403$	−36.8151	−1.83389
$404$	6.81151	0.338885
$405$	−11.3404	−0.563509
$406$	−0.767994	−0.0381149
$407$	−15.4133	−0.764007
$408$	0.319096	0.0157976
$409$	5.06988	0.250689	0.125345	−	0.992113i	$-0.459996\pi$
$409$	5.06988	0.250689	0.125345	+	0.992113i	$0.459996\pi$
$410$	−0.137020	−0.00676693
$411$	−30.0117	−1.48037
$412$	18.5771	0.915230
$413$	1.17877	0.0580036
$414$	27.5050	1.35180
$415$	−7.75539	−0.380697
$416$	−4.55294	−0.223226
$417$	−35.2951	−1.72841
$418$	13.6553	0.667900
$419$	−24.5114	−1.19746	−0.598731	−	0.800950i	$-0.704328\pi$
$419$	−24.5114	−1.19746	−0.598731	+	0.800950i	$0.704328\pi$
$420$	0.499441	0.0243702
$421$	9.13325	0.445127	0.222564	−	0.974918i	$-0.428558\pi$
$421$	9.13325	0.445127	0.222564	+	0.974918i	$0.428558\pi$
$422$	−2.28119	−0.111046
$423$	44.7717	2.17687
$424$	−13.7699	−0.668725
$425$	0.471559	0.0228740
$426$	−18.8297	−0.912300
$427$	−1.10183	−0.0533214
$428$	−18.7182	−0.904777
$429$	−79.9512	−3.86008
$430$	−1.42273	−0.0686102
$431$	−19.6253	−0.945319	−0.472659	−	0.881245i	$-0.656706\pi$
$431$	−19.6253	−0.945319	−0.472659	+	0.881245i	$0.656706\pi$
$432$	−13.7325	−0.660704
$433$	0.982708	0.0472259	0.0236130	−	0.999721i	$-0.492483\pi$
$433$	0.982708	0.0472259	0.0236130	+	0.999721i	$0.492483\pi$
$434$	2.46405	0.118278
$435$	−4.13060	−0.198047
$436$	−10.8747	−0.520804
$437$	−9.41775	−0.450512
$438$	27.1844	1.29892
$439$	29.2197	1.39458	0.697290	−	0.716789i	$-0.254389\pi$
$439$	29.2197	1.39458	0.697290	+	0.716789i	$0.254389\pi$
$440$	2.79901	0.133437
$441$	−50.3014	−2.39530
$442$	0.453068	0.0215502
$443$	−24.1131	−1.14565	−0.572824	−	0.819678i	$-0.694152\pi$
$443$	−24.1131	−1.14565	−0.572824	+	0.819678i	$0.694152\pi$
$444$	−9.02527	−0.428321
$445$	5.27613	0.250113
$446$	7.72993	0.366023
$447$	−40.8028	−1.92990
$448$	0.304730	0.0143971
$449$	−15.0064	−0.708195	−0.354098	−	0.935208i	$-0.615212\pi$
$449$	−15.0064	−0.708195	−0.354098	+	0.935208i	$0.615212\pi$
$450$	−34.5101	−1.62682
$451$	−1.46807	−0.0691287
$452$	−16.2825	−0.765864
$453$	48.1578	2.26265
$454$	−10.8012	−0.506925
$455$	0.709131	0.0332446
$456$	7.99587	0.374441
$457$	−3.84754	−0.179980	−0.0899902	−	0.995943i	$-0.528684\pi$
$457$	−3.84754	−0.179980	−0.0899902	+	0.995943i	$0.528684\pi$
$458$	24.7410	1.15607
$459$	1.36653	0.0637843
$460$	−1.93042	−0.0900061
$461$	−7.37485	−0.343481	−0.171741	−	0.985142i	$-0.554939\pi$
$461$	−7.37485	−0.343481	−0.171741	+	0.985142i	$0.554939\pi$
$462$	5.35116	0.248958
$463$	5.34371	0.248343	0.124172	−	0.992261i	$-0.460373\pi$
$463$	5.34371	0.248343	0.124172	+	0.992261i	$0.460373\pi$
$464$	−2.52025	−0.117000
$465$	13.2527	0.614579
$466$	22.1006	1.02379
$467$	14.6205	0.676554	0.338277	−	0.941047i	$-0.390156\pi$
$467$	14.6205	0.676554	0.338277	+	0.941047i	$0.390156\pi$
$468$	−33.1568	−1.53268
$469$	−2.04219	−0.0942994
$470$	−3.14226	−0.144942
$471$	−25.8588	−1.19151
$472$	3.86826	0.178051
$473$	−15.2436	−0.700899
$474$	5.89963	0.270979
$475$	11.8163	0.542168
$476$	−0.0303240	−0.00138990
$477$	−100.279	−4.59148
$478$	−0.970013	−0.0443674
$479$	11.2440	0.513752	0.256876	−	0.966444i	$-0.417307\pi$
$479$	11.2440	0.513752	0.256876	+	0.966444i	$0.417307\pi$
$480$	1.63897	0.0748082
$481$	−12.8145	−0.584292
$482$	−10.0867	−0.459434
$483$	−3.69058	−0.167927
$484$	18.9894	0.863152
$485$	−9.85454	−0.447472
$486$	−29.9498	−1.35855
$487$	35.3399	1.60140	0.800702	−	0.599063i	$-0.204460\pi$
$487$	35.3399	1.60140	0.800702	+	0.599063i	$0.204460\pi$
$488$	−3.61577	−0.163678
$489$	−71.7156	−3.24309
$490$	3.53036	0.159485
$491$	−10.4301	−0.470704	−0.235352	−	0.971910i	$-0.575624\pi$
$491$	−10.4301	−0.470704	−0.235352	+	0.971910i	$0.575624\pi$
$492$	−0.859633	−0.0387552
$493$	0.250792	0.0112951
$494$	11.3529	0.510792
$495$	20.3838	0.916184
$496$	8.08601	0.363073
$497$	1.78940	0.0802655
$498$	−48.6556	−2.18031
$499$	21.1325	0.946019	0.473009	−	0.881057i	$-0.343168\pi$
$499$	21.1325	0.946019	0.473009	+	0.881057i	$0.343168\pi$
$500$	4.97765	0.222607
$501$	−27.3525	−1.22202
$502$	3.88688	0.173480
$503$	2.44173	0.108871	0.0544357	−	0.998517i	$-0.482664\pi$
$503$	2.44173	0.108871	0.0544357	+	0.998517i	$0.482664\pi$
$504$	2.21920	0.0988509
$505$	−3.48148	−0.154924
$506$	−20.6830	−0.919473
$507$	−24.7848	−1.10073
$508$	17.6871	0.784736
$509$	−39.4508	−1.74863	−0.874313	−	0.485362i	$-0.838688\pi$
$509$	−39.4508	−1.74863	−0.874313	+	0.485362i	$0.838688\pi$
$510$	−0.163095	−0.00722198
$511$	−2.58336	−0.114281
$512$	1.00000	0.0441942
$513$	34.2425	1.51184
$514$	16.9818	0.749035
$515$	−9.49509	−0.418404
$516$	−8.92591	−0.392941
$517$	−33.6671	−1.48068
$518$	0.857680	0.0376843
$519$	−62.7126	−2.75278
$520$	2.32708	0.102049
$521$	2.34235	0.102620	0.0513101	−	0.998683i	$-0.483660\pi$
$521$	2.34235	0.102620	0.0513101	+	0.998683i	$0.483660\pi$
$522$	−18.3537	−0.803321
$523$	5.68680	0.248666	0.124333	−	0.992241i	$-0.460321\pi$
$523$	5.68680	0.248666	0.124333	+	0.992241i	$0.460321\pi$
$524$	20.2685	0.885433
$525$	4.63051	0.202092
$526$	−6.28942	−0.274232
$527$	−0.804648	−0.0350510
$528$	17.5604	0.764216
$529$	−8.73534	−0.379797
$530$	7.03802	0.305712
$531$	28.1707	1.22250
$532$	−0.759855	−0.0329439
$533$	−1.22055	−0.0528678
$534$	33.1013	1.43243
$535$	9.56718	0.413625
$536$	−6.70163	−0.289466
$537$	14.7394	0.636050
$538$	2.87948	0.124143
$539$	37.8253	1.62925
$540$	7.01890	0.302045
$541$	−19.0328	−0.818283	−0.409142	−	0.912471i	$-0.634172\pi$
$541$	−19.0328	−0.818283	−0.409142	+	0.912471i	$0.634172\pi$
$542$	−13.1347	−0.564185
$543$	−26.5468	−1.13923
$544$	−0.0995111	−0.00426650
$545$	5.55824	0.238089
$546$	4.44893	0.190397
$547$	−10.6452	−0.455154	−0.227577	−	0.973760i	$-0.573080\pi$
$547$	−10.6452	−0.455154	−0.227577	+	0.973760i	$0.573080\pi$
$548$	9.35924	0.399807
$549$	−26.3319	−1.12382
$550$	25.9506	1.10654
$551$	6.28434	0.267722
$552$	−12.1110	−0.515479
$553$	−0.560647	−0.0238412
$554$	16.4892	0.700559
$555$	4.61297	0.195810
$556$	11.0069	0.466797
$557$	7.13259	0.302218	0.151109	−	0.988517i	$-0.451716\pi$
$557$	7.13259	0.302218	0.151109	+	0.988517i	$0.451716\pi$
$558$	58.8865	2.49287
$559$	−12.6734	−0.536029
$560$	−0.155752	−0.00658174
$561$	−1.74745	−0.0737774
$562$	−26.4941	−1.11759
$563$	−32.2295	−1.35831	−0.679156	−	0.733994i	$-0.737654\pi$
$563$	−32.2295	−1.35831	−0.679156	+	0.733994i	$0.737654\pi$
$564$	−19.7139	−0.830103
$565$	8.32226	0.350120
$566$	−23.2728	−0.978228
$567$	6.76119	0.283943
$568$	5.87209	0.246387
$569$	−10.8564	−0.455123	−0.227561	−	0.973764i	$-0.573075\pi$
$569$	−10.8564	−0.455123	−0.227561	+	0.973764i	$0.573075\pi$
$570$	−4.08683	−0.171178
$571$	39.8787	1.66887	0.834437	−	0.551104i	$-0.185793\pi$
$571$	39.8787	1.66887	0.834437	+	0.551104i	$0.185793\pi$
$572$	24.9330	1.04250
$573$	5.10845	0.213409
$574$	0.0816917	0.00340975
$575$	−17.8976	−0.746383
$576$	7.28252	0.303438
$577$	28.2121	1.17448	0.587242	−	0.809411i	$-0.300214\pi$
$577$	28.2121	1.17448	0.587242	+	0.809411i	$0.300214\pi$
$578$	−16.9901	−0.706695
$579$	−0.137001	−0.00569357
$580$	1.28814	0.0534871
$581$	4.62379	0.191827
$582$	−61.8252	−2.56274
$583$	75.4074	3.12306
$584$	−8.47756	−0.350804
$585$	16.9470	0.700673
$586$	−3.40579	−0.140692
$587$	−36.9756	−1.52615	−0.763073	−	0.646312i	$-0.776311\pi$
$587$	−36.9756	−1.52615	−0.763073	+	0.646312i	$0.776311\pi$
$588$	22.1487	0.913396
$589$	−20.1628	−0.830794
$590$	−1.97713	−0.0813973
$591$	13.9981	0.575806
$592$	2.81456	0.115678
$593$	−15.1550	−0.622341	−0.311171	−	0.950354i	$-0.600721\pi$
$593$	−15.1550	−0.622341	−0.311171	+	0.950354i	$0.600721\pi$
$594$	75.2025	3.08560
$595$	0.0154991	0.000635401 0
$596$	12.7245	0.521215
$597$	−23.9718	−0.981100
$598$	−17.1958	−0.703188
$599$	38.8289	1.58651	0.793253	−	0.608892i	$-0.208386\pi$
$599$	38.8289	1.58651	0.793253	+	0.608892i	$0.208386\pi$
$600$	15.1955	0.620353
$601$	2.33090	0.0950793	0.0475397	−	0.998869i	$-0.484862\pi$
$601$	2.33090	0.0950793	0.0475397	+	0.998869i	$0.484862\pi$
$602$	0.848237	0.0345715
$603$	−48.8048	−1.98748
$604$	−15.0182	−0.611081
$605$	−9.70578	−0.394596
$606$	−21.8420	−0.887272
$607$	4.27595	0.173556	0.0867778	−	0.996228i	$-0.472343\pi$
$607$	4.27595	0.173556	0.0867778	+	0.996228i	$0.472343\pi$
$608$	−2.49354	−0.101126
$609$	2.46268	0.0997927
$610$	1.84808	0.0748267
$611$	−27.9907	−1.13238
$612$	−0.724691	−0.0292939
$613$	−21.3854	−0.863750	−0.431875	−	0.901934i	$-0.642148\pi$
$613$	−21.3854	−0.863750	−0.431875	+	0.901934i	$0.642148\pi$
$614$	8.52120	0.343887
$615$	0.439373	0.0177172
$616$	−1.66878	−0.0672369
$617$	−3.68897	−0.148512	−0.0742562	−	0.997239i	$-0.523658\pi$
$617$	−3.68897	−0.148512	−0.0742562	+	0.997239i	$0.523658\pi$
$618$	−59.5701	−2.39626
$619$	−24.4936	−0.984481	−0.492240	−	0.870459i	$-0.663822\pi$
$619$	−24.4936	−0.984481	−0.492240	+	0.870459i	$0.663822\pi$
$620$	−4.13290	−0.165981
$621$	−51.8656	−2.08130
$622$	−19.2623	−0.772349
$623$	−3.14565	−0.126028
$624$	14.5996	0.584452
$625$	21.1496	0.845986
$626$	−6.17973	−0.246992
$627$	−43.7874	−1.74870
$628$	8.06414	0.321794
$629$	−0.280080	−0.0111675
$630$	−1.13427	−0.0451904
$631$	−34.9026	−1.38945	−0.694725	−	0.719276i	$-0.744474\pi$
$631$	−34.9026	−1.38945	−0.694725	+	0.719276i	$0.744474\pi$
$632$	−1.83982	−0.0731841
$633$	7.31493	0.290743
$634$	32.9797	1.30979
$635$	−9.04016	−0.358748
$636$	44.1550	1.75086
$637$	31.4478	1.24601
$638$	13.8015	0.546407
$639$	42.7636	1.69170
$640$	−0.511117	−0.0202037
$641$	49.9465	1.97277	0.986384	−	0.164458i	$-0.0525873\pi$
$641$	49.9465	1.97277	0.986384	+	0.164458i	$0.0525873\pi$
$642$	60.0224	2.36889
$643$	32.5626	1.28414	0.642071	−	0.766645i	$-0.278075\pi$
$643$	32.5626	1.28414	0.642071	+	0.766645i	$0.278075\pi$
$644$	1.15092	0.0453526
$645$	4.56218	0.179636
$646$	0.248135	0.00976273
$647$	36.2287	1.42430	0.712149	−	0.702029i	$-0.247722\pi$
$647$	36.2287	1.42430	0.712149	+	0.702029i	$0.247722\pi$
$648$	22.1875	0.871607
$649$	−21.1836	−0.831528
$650$	21.5753	0.846252
$651$	−7.90130	−0.309676
$652$	22.3647	0.875871
$653$	10.8788	0.425720	0.212860	−	0.977083i	$-0.431722\pi$
$653$	10.8788	0.425720	0.212860	+	0.977083i	$0.431722\pi$
$654$	34.8712	1.36357
$655$	−10.3596	−0.404782
$656$	0.268079	0.0104667
$657$	−61.7380	−2.40863
$658$	1.87343	0.0730337
$659$	−21.6668	−0.844019	−0.422009	−	0.906591i	$-0.638675\pi$
$659$	−21.6668	−0.844019	−0.422009	+	0.906591i	$0.638675\pi$
$660$	−8.97539	−0.349367
$661$	−33.1721	−1.29025	−0.645123	−	0.764079i	$-0.723194\pi$
$661$	−33.1721	−1.29025	−0.645123	+	0.764079i	$0.723194\pi$
$662$	6.51620	0.253259
$663$	−1.45282	−0.0564230
$664$	15.1734	0.588843
$665$	0.388375	0.0150605
$666$	20.4971	0.794246
$667$	−9.51862	−0.368562
$668$	8.52996	0.330034
$669$	−24.7871	−0.958324
$670$	3.42532	0.132332
$671$	19.8009	0.764405
$672$	−0.977157	−0.0376946
$673$	35.8542	1.38208	0.691040	−	0.722817i	$-0.257153\pi$
$673$	35.8542	1.38208	0.691040	+	0.722817i	$0.257153\pi$
$674$	4.93373	0.190040
$675$	65.0749	2.50473
$676$	7.72923	0.297278
$677$	11.3533	0.436343	0.218172	−	0.975910i	$-0.429991\pi$
$677$	11.3533	0.436343	0.218172	+	0.975910i	$0.429991\pi$
$678$	52.2120	2.00519
$679$	5.87531	0.225474
$680$	0.0508618	0.00195046
$681$	34.6355	1.32724
$682$	−44.2811	−1.69561
$683$	7.32674	0.280350	0.140175	−	0.990127i	$-0.455234\pi$
$683$	7.32674	0.280350	0.140175	+	0.990127i	$0.455234\pi$
$684$	−18.1592	−0.694336
$685$	−4.78366	−0.182774
$686$	−4.23792	−0.161804
$687$	−79.3355	−3.02684
$688$	2.78357	0.106123
$689$	62.6934	2.38843
$690$	6.19014	0.235655
$691$	−12.2374	−0.465531	−0.232765	−	0.972533i	$-0.574777\pi$
$691$	−12.2374	−0.465531	−0.232765	+	0.972533i	$0.574777\pi$
$692$	19.5571	0.743450
$693$	−12.1529	−0.461650
$694$	−3.51442	−0.133405
$695$	−5.62581	−0.213399
$696$	8.08152	0.306329
$697$	−0.0266769	−0.00101046
$698$	27.4961	1.04074
$699$	−70.8687	−2.68050
$700$	−1.44404	−0.0545796
$701$	42.2082	1.59418	0.797090	−	0.603860i	$-0.206371\pi$
$701$	42.2082	1.59418	0.797090	+	0.603860i	$0.206371\pi$
$702$	62.5231	2.35978
$703$	−7.01822	−0.264697
$704$	−5.47625	−0.206394
$705$	10.0761	0.379487
$706$	−17.7729	−0.668894
$707$	2.07567	0.0780635
$708$	−12.4041	−0.466175
$709$	38.6075	1.44993	0.724967	−	0.688783i	$-0.241855\pi$
$709$	38.6075	1.44993	0.724967	+	0.688783i	$0.241855\pi$
$710$	−3.00133	−0.112638
$711$	−13.3985	−0.502484
$712$	−10.3228	−0.386862
$713$	30.5397	1.14372
$714$	0.0972379	0.00363904
$715$	−12.7437	−0.476587
$716$	−4.59652	−0.171780
$717$	3.11048	0.116163
$718$	29.0846	1.08543
$719$	−45.1955	−1.68551	−0.842753	−	0.538300i	$-0.819067\pi$
$719$	−45.1955	−1.68551	−0.842753	+	0.538300i	$0.819067\pi$
$720$	−3.72222	−0.138719
$721$	5.66100	0.210827
$722$	−12.7823	−0.475706
$723$	32.3442	1.20289
$724$	8.27870	0.307676
$725$	11.9428	0.443546
$726$	−60.8920	−2.25991
$727$	9.45896	0.350813	0.175407	−	0.984496i	$-0.443876\pi$
$727$	9.45896	0.350813	0.175407	+	0.984496i	$0.443876\pi$
$728$	−1.38741	−0.0514210
$729$	29.4757	1.09169
$730$	4.33302	0.160372
$731$	−0.276996	−0.0102451
$732$	11.5945	0.428544
$733$	−33.7298	−1.24584	−0.622920	−	0.782286i	$-0.714054\pi$
$733$	−33.7298	−1.24584	−0.622920	+	0.782286i	$0.714054\pi$
$734$	−24.3797	−0.899871
$735$	−11.3206	−0.417565
$736$	3.77686	0.139217
$737$	36.6998	1.35186
$738$	1.95229	0.0718649
$739$	−42.4834	−1.56278	−0.781389	−	0.624045i	$-0.785488\pi$
$739$	−42.4834	−1.56278	−0.781389	+	0.624045i	$0.785488\pi$
$740$	−1.43857	−0.0528829
$741$	−36.4047	−1.33736
$742$	−4.19609	−0.154043
$743$	−20.6902	−0.759051	−0.379526	−	0.925181i	$-0.623913\pi$
$743$	−20.6902	−0.759051	−0.379526	+	0.925181i	$0.623913\pi$
$744$	−25.9289	−0.950600
$745$	−6.50369	−0.238277
$746$	18.3450	0.671657
$747$	110.501	4.04301
$748$	0.544948	0.0199253
$749$	−5.70398	−0.208419
$750$	−15.9615	−0.582831
$751$	−21.3708	−0.779833	−0.389917	−	0.920850i	$-0.627496\pi$
$751$	−21.3708	−0.779833	−0.389917	+	0.920850i	$0.627496\pi$
$752$	6.14783	0.224188
$753$	−12.4638	−0.454207
$754$	11.4745	0.417877
$755$	7.67604	0.279360
$756$	−4.18469	−0.152196
$757$	−53.3919	−1.94056	−0.970281	−	0.241979i	$-0.922203\pi$
$757$	−53.3919	−1.94056	−0.970281	+	0.241979i	$0.922203\pi$
$758$	17.7293	0.643958
$759$	66.3230	2.40737
$760$	1.27449	0.0462306
$761$	0.402994	0.0146085	0.00730425	−	0.999973i	$-0.497675\pi$
$761$	0.402994	0.0146085	0.00730425	+	0.999973i	$0.497675\pi$
$762$	−56.7160	−2.05460
$763$	−3.31384	−0.119969
$764$	−1.59309	−0.0576359
$765$	0.370402	0.0133919
$766$	−31.8343	−1.15022
$767$	−17.6119	−0.635930
$768$	−3.20664	−0.115710
$769$	−49.0623	−1.76923	−0.884615	−	0.466322i	$-0.845579\pi$
$769$	−49.0623	−1.76923	−0.884615	+	0.466322i	$0.845579\pi$
$770$	0.852940	0.0307378
$771$	−54.4544	−1.96113
$772$	0.0427242	0.00153768
$773$	−39.5587	−1.42283	−0.711413	−	0.702774i	$-0.751944\pi$
$773$	−39.5587	−1.42283	−0.711413	+	0.702774i	$0.751944\pi$
$774$	20.2714	0.728641
$775$	−38.3177	−1.37641
$776$	19.2804	0.692126
$777$	−2.75027	−0.0986653
$778$	24.3207	0.871940
$779$	−0.668466	−0.0239503
$780$	−7.46211	−0.267186
$781$	−32.1571	−1.15067
$782$	−0.375839	−0.0134400
$783$	34.6092	1.23683
$784$	−6.90714	−0.246684
$785$	−4.12172	−0.147110
$786$	−64.9937	−2.31825
$787$	25.5776	0.911743	0.455872	−	0.890046i	$-0.349328\pi$
$787$	25.5776	0.911743	0.455872	+	0.890046i	$0.349328\pi$
$788$	−4.36537	−0.155510
$789$	20.1679	0.717996
$790$	0.940363	0.0334566
$791$	−4.96176	−0.176420
$792$	−39.8809	−1.41711
$793$	16.4624	0.584596
$794$	31.1635	1.10595
$795$	−22.5684	−0.800418
$796$	7.47568	0.264969
$797$	−30.6975	−1.08736	−0.543681	−	0.839292i	$-0.682970\pi$
$797$	−30.6975	−1.08736	−0.543681	+	0.839292i	$0.682970\pi$
$798$	2.43658	0.0862540
$799$	−0.611777	−0.0216431
$800$	−4.73876	−0.167540
$801$	−75.1756	−2.65620
$802$	−13.4381	−0.474516
$803$	46.4253	1.63831
$804$	21.4897	0.757883
$805$	−0.588255	−0.0207333
$806$	−36.8151	−1.29676
$807$	−9.23344	−0.325033
$808$	6.81151	0.239628
$809$	−22.4927	−0.790801	−0.395401	−	0.918509i	$-0.629394\pi$
$809$	−22.4927	−0.790801	−0.395401	+	0.918509i	$0.629394\pi$
$810$	−11.3404	−0.398461
$811$	−13.5816	−0.476914	−0.238457	−	0.971153i	$-0.576642\pi$
$811$	−13.5816	−0.476914	−0.238457	+	0.971153i	$0.576642\pi$
$812$	−0.767994	−0.0269513
$813$	42.1183	1.47715
$814$	−15.4133	−0.540234
$815$	−11.4310	−0.400410
$816$	0.319096	0.0111706
$817$	−6.94095	−0.242833
$818$	5.06988	0.177264
$819$	−10.1039	−0.353058
$820$	−0.137020	−0.00478494
$821$	36.8701	1.28677	0.643387	−	0.765541i	$-0.277529\pi$
$821$	36.8701	1.28677	0.643387	+	0.765541i	$0.277529\pi$
$822$	−30.0117	−1.04678
$823$	17.0440	0.594115	0.297058	−	0.954860i	$-0.403995\pi$
$823$	17.0440	0.594115	0.297058	+	0.954860i	$0.403995\pi$
$824$	18.5771	0.647165
$825$	−83.2143	−2.89715
$826$	1.17877	0.0410148
$827$	26.3442	0.916077	0.458038	−	0.888932i	$-0.348552\pi$
$827$	26.3442	0.916077	0.458038	+	0.888932i	$0.348552\pi$
$828$	27.5050	0.955866
$829$	50.8842	1.76728	0.883640	−	0.468167i	$-0.155085\pi$
$829$	50.8842	1.76728	0.883640	+	0.468167i	$0.155085\pi$
$830$	−7.75539	−0.269193
$831$	−52.8748	−1.83421
$832$	−4.55294	−0.157845
$833$	0.687337	0.0238148
$834$	−35.2951	−1.22217
$835$	−4.35981	−0.150877
$836$	13.6553	0.472277
$837$	−111.041	−3.83814
$838$	−24.5114	−0.846734
$839$	−35.5410	−1.22701	−0.613506	−	0.789690i	$-0.710241\pi$
$839$	−35.5410	−1.22701	−0.613506	+	0.789690i	$0.710241\pi$
$840$	0.499441	0.0172324
$841$	−22.6484	−0.780978
$842$	9.13325	0.314752
$843$	84.9569	2.92607
$844$	−2.28119	−0.0785217
$845$	−3.95054	−0.135903
$846$	44.7717	1.53928
$847$	5.78662	0.198831
$848$	−13.7699	−0.472860
$849$	74.6273	2.56120
$850$	0.471559	0.0161743
$851$	10.6302	0.364399
$852$	−18.8297	−0.645094
$853$	32.1776	1.10174	0.550870	−	0.834591i	$-0.314296\pi$
$853$	32.1776	1.10174	0.550870	+	0.834591i	$0.314296\pi$
$854$	−1.10183	−0.0377039
$855$	9.28150	0.317421
$856$	−18.7182	−0.639774
$857$	4.58563	0.156642	0.0783210	−	0.996928i	$-0.475044\pi$
$857$	4.58563	0.156642	0.0783210	+	0.996928i	$0.475044\pi$
$858$	−79.9512	−2.72949
$859$	39.8992	1.36134	0.680672	−	0.732588i	$-0.261688\pi$
$859$	39.8992	1.36134	0.680672	+	0.732588i	$0.261688\pi$
$860$	−1.42273	−0.0485147
$861$	−0.261956	−0.00892742
$862$	−19.6253	−0.668441
$863$	−12.1240	−0.412704	−0.206352	−	0.978478i	$-0.566159\pi$
$863$	−12.1240	−0.412704	−0.206352	+	0.978478i	$0.566159\pi$
$864$	−13.7325	−0.467188
$865$	−9.99598	−0.339873
$866$	0.982708	0.0333938
$867$	54.4811	1.85027
$868$	2.46405	0.0836352
$869$	10.0753	0.341782
$870$	−4.13060	−0.140040
$871$	30.5121	1.03386
$872$	−10.8747	−0.368264
$873$	140.410	4.75215
$874$	−9.41775	−0.318560
$875$	1.51684	0.0512784
$876$	27.1844	0.918477
$877$	6.22184	0.210097	0.105048	−	0.994467i	$-0.466500\pi$
$877$	6.22184	0.210097	0.105048	+	0.994467i	$0.466500\pi$
$878$	29.2197	0.986118
$879$	10.9211	0.368361
$880$	2.79901	0.0943545
$881$	23.0617	0.776967	0.388483	−	0.921456i	$-0.372999\pi$
$881$	23.0617	0.776967	0.388483	+	0.921456i	$0.372999\pi$
$882$	−50.3014	−1.69373
$883$	5.93444	0.199710	0.0998549	−	0.995002i	$-0.468162\pi$
$883$	5.93444	0.199710	0.0998549	+	0.995002i	$0.468162\pi$
$884$	0.453068	0.0152383
$885$	6.33995	0.213115
$886$	−24.1131	−0.810096
$887$	−14.3768	−0.482727	−0.241363	−	0.970435i	$-0.577595\pi$
$887$	−14.3768	−0.482727	−0.241363	+	0.970435i	$0.577595\pi$
$888$	−9.02527	−0.302868
$889$	5.38977	0.180767
$890$	5.27613	0.176856
$891$	−121.504	−4.07055
$892$	7.72993	0.258817
$893$	−15.3299	−0.512994
$894$	−40.8028	−1.36465
$895$	2.34936	0.0785304
$896$	0.304730	0.0101803
$897$	55.1407	1.84109
$898$	−15.0064	−0.500770
$899$	−20.3788	−0.679669
$900$	−34.5101	−1.15034
$901$	1.37026	0.0456499
$902$	−1.46807	−0.0488814
$903$	−2.71999	−0.0905155
$904$	−16.2825	−0.541548
$905$	−4.23138	−0.140656
$906$	48.1578	1.59994
$907$	32.6023	1.08254	0.541271	−	0.840848i	$-0.317943\pi$
$907$	32.6023	1.08254	0.541271	+	0.840848i	$0.317943\pi$
$908$	−10.8012	−0.358450
$909$	49.6049	1.64529
$910$	0.709131	0.0235075
$911$	16.3942	0.543163	0.271582	−	0.962415i	$-0.412453\pi$
$911$	16.3942	0.543163	0.271582	+	0.962415i	$0.412453\pi$
$912$	7.99587	0.264770
$913$	−83.0935	−2.74999
$914$	−3.84754	−0.127265
$915$	−5.92613	−0.195912
$916$	24.7410	0.817467
$917$	6.17641	0.203963
$918$	1.36653	0.0451023
$919$	26.3788	0.870157	0.435079	−	0.900392i	$-0.356721\pi$
$919$	26.3788	0.870157	0.435079	+	0.900392i	$0.356721\pi$
$920$	−1.93042	−0.0636440
$921$	−27.3244	−0.900369
$922$	−7.37485	−0.242878
$923$	−26.7353	−0.880002
$924$	5.35116	0.176040
$925$	−13.3375	−0.438535
$926$	5.34371	0.175605
$927$	135.288	4.44345
$928$	−2.52025	−0.0827312
$929$	−50.5399	−1.65816	−0.829080	−	0.559131i	$-0.811135\pi$
$929$	−50.5399	−1.65816	−0.829080	+	0.559131i	$0.811135\pi$
$930$	13.2527	0.434573
$931$	17.2232	0.564468
$932$	22.1006	0.723930
$933$	61.7673	2.02217
$934$	14.6205	0.478396
$935$	−0.278532	−0.00910897
$936$	−33.1568	−1.08377
$937$	−3.39658	−0.110961	−0.0554807	−	0.998460i	$-0.517669\pi$
$937$	−3.39658	−0.110961	−0.0554807	+	0.998460i	$0.517669\pi$
$938$	−2.04219	−0.0666797
$939$	19.8162	0.646676
$940$	−3.14226	−0.102489
$941$	6.26321	0.204175	0.102087	−	0.994775i	$-0.467448\pi$
$941$	6.26321	0.204175	0.102087	+	0.994775i	$0.467448\pi$
$942$	−25.8588	−0.842524
$943$	1.01250	0.0329715
$944$	3.86826	0.125901
$945$	2.13887	0.0695773
$946$	−15.2436	−0.495611
$947$	6.89093	0.223925	0.111963	−	0.993712i	$-0.464286\pi$
$947$	6.89093	0.223925	0.111963	+	0.993712i	$0.464286\pi$
$948$	5.89963	0.191611
$949$	38.5978	1.25294
$950$	11.8163	0.383371
$951$	−105.754	−3.42930
$952$	−0.0303240	−0.000982805 0
$953$	36.5014	1.18240	0.591198	−	0.806527i	$-0.298655\pi$
$953$	36.5014	1.18240	0.591198	+	0.806527i	$0.298655\pi$
$954$	−100.279	−3.24667
$955$	0.814254	0.0263486
$956$	−0.970013	−0.0313725
$957$	−44.2564	−1.43061
$958$	11.2440	0.363278
$959$	2.85204	0.0920970
$960$	1.63897	0.0528974
$961$	34.3836	1.10915
$962$	−12.8145	−0.413157
$963$	−136.315	−4.39270
$964$	−10.0867	−0.324869
$965$	−0.0218371	−0.000702960 0
$966$	−3.69058	−0.118743
$967$	−16.3233	−0.524921	−0.262460	−	0.964943i	$-0.584534\pi$
$967$	−16.3233	−0.524921	−0.262460	+	0.964943i	$0.584534\pi$
$968$	18.9894	0.610341
$969$	−0.795678	−0.0255609
$970$	−9.85454	−0.316410
$971$	−19.7362	−0.633364	−0.316682	−	0.948532i	$-0.602569\pi$
$971$	−19.7362	−0.633364	−0.316682	+	0.948532i	$0.602569\pi$
$972$	−29.9498	−0.960641
$973$	3.35413	0.107528
$974$	35.3399	1.13236
$975$	−69.1840	−2.21566
$976$	−3.61577	−0.115738
$977$	−20.5083	−0.656119	−0.328060	−	0.944657i	$-0.606395\pi$
$977$	−20.5083	−0.656119	−0.328060	+	0.944657i	$0.606395\pi$
$978$	−71.7156	−2.29321
$979$	56.5300	1.80671
$980$	3.53036	0.112773
$981$	−79.1952	−2.52851
$982$	−10.4301	−0.332838
$983$	−10.3417	−0.329848	−0.164924	−	0.986306i	$-0.552738\pi$
$983$	−10.3417	−0.329848	−0.164924	+	0.986306i	$0.552738\pi$
$984$	−0.859633	−0.0274041
$985$	2.23121	0.0710923
$986$	0.250792	0.00798686
$987$	−6.00739	−0.191217
$988$	11.3529	0.361185
$989$	10.5132	0.334299
$990$	20.3838	0.647840
$991$	−41.2466	−1.31024	−0.655120	−	0.755525i	$-0.727382\pi$
$991$	−41.2466	−1.31024	−0.655120	+	0.755525i	$0.727382\pi$
$992$	8.08601	0.256731
$993$	−20.8951	−0.663086
$994$	1.78940	0.0567563
$995$	−3.82095	−0.121132
$996$	−48.6556	−1.54171
$997$	38.7599	1.22754	0.613770	−	0.789485i	$-0.289652\pi$
$997$	38.7599	1.22754	0.613770	+	0.789485i	$0.289652\pi$
$998$	21.1325	0.668936
$999$	−38.6509	−1.22286

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4022.2.a.f.1.1	✓	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4022.2.a.f.1.1	✓	50	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4022 = 2 \cdot 2011 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4022.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.20664 q^{3} +1.00000 q^{4} -0.511117 q^{5} -3.20664 q^{6} +0.304730 q^{7} +1.00000 q^{8} +7.28252 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.20664 q^{3} +1.00000 q^{4} -0.511117 q^{5} -3.20664 q^{6} +0.304730 q^{7} +1.00000 q^{8} +7.28252 q^{9} -0.511117 q^{10} -5.47625 q^{11} -3.20664 q^{12} -4.55294 q^{13} +0.304730 q^{14} +1.63897 q^{15} +1.00000 q^{16} -0.0995111 q^{17} +7.28252 q^{18} -2.49354 q^{19} -0.511117 q^{20} -0.977157 q^{21} -5.47625 q^{22} +3.77686 q^{23} -3.20664 q^{24} -4.73876 q^{25} -4.55294 q^{26} -13.7325 q^{27} +0.304730 q^{28} -2.52025 q^{29} +1.63897 q^{30} +8.08601 q^{31} +1.00000 q^{32} +17.5604 q^{33} -0.0995111 q^{34} -0.155752 q^{35} +7.28252 q^{36} +2.81456 q^{37} -2.49354 q^{38} +14.5996 q^{39} -0.511117 q^{40} +0.268079 q^{41} -0.977157 q^{42} +2.78357 q^{43} -5.47625 q^{44} -3.72222 q^{45} +3.77686 q^{46} +6.14783 q^{47} -3.20664 q^{48} -6.90714 q^{49} -4.73876 q^{50} +0.319096 q^{51} -4.55294 q^{52} -13.7699 q^{53} -13.7325 q^{54} +2.79901 q^{55} +0.304730 q^{56} +7.99587 q^{57} -2.52025 q^{58} +3.86826 q^{59} +1.63897 q^{60} -3.61577 q^{61} +8.08601 q^{62} +2.21920 q^{63} +1.00000 q^{64} +2.32708 q^{65} +17.5604 q^{66} -6.70163 q^{67} -0.0995111 q^{68} -12.1110 q^{69} -0.155752 q^{70} +5.87209 q^{71} +7.28252 q^{72} -8.47756 q^{73} +2.81456 q^{74} +15.1955 q^{75} -2.49354 q^{76} -1.66878 q^{77} +14.5996 q^{78} -1.83982 q^{79} -0.511117 q^{80} +22.1875 q^{81} +0.268079 q^{82} +15.1734 q^{83} -0.977157 q^{84} +0.0508618 q^{85} +2.78357 q^{86} +8.08152 q^{87} -5.47625 q^{88} -10.3228 q^{89} -3.72222 q^{90} -1.38741 q^{91} +3.77686 q^{92} -25.9289 q^{93} +6.14783 q^{94} +1.27449 q^{95} -3.20664 q^{96} +19.2804 q^{97} -6.90714 q^{98} -39.8809 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q + 50 q^{2} + 18 q^{3} + 50 q^{4} + 11 q^{5} + 18 q^{6} + 30 q^{7} + 50 q^{8} + 66 q^{9}+O(q^{10}) \) 50 * q + 50 * q^2 + 18 * q^3 + 50 * q^4 + 11 * q^5 + 18 * q^6 + 30 * q^7 + 50 * q^8 + 66 * q^9	\( 50 q + 50 q^{2} + 18 q^{3} + 50 q^{4} + 11 q^{5} + 18 q^{6} + 30 q^{7} + 50 q^{8} + 66 q^{9} + 11 q^{10} + 21 q^{11} + 18 q^{12} + 26 q^{13} + 30 q^{14} + 17 q^{15} + 50 q^{16} + 24 q^{17} + 66 q^{18} + 39 q^{19} + 11 q^{20} - q^{21} + 21 q^{22} + 28 q^{23} + 18 q^{24} + 79 q^{25} + 26 q^{26} + 66 q^{27} + 30 q^{28} - 5 q^{29} + 17 q^{30} + 60 q^{31} + 50 q^{32} + 37 q^{33} + 24 q^{34} + 38 q^{35} + 66 q^{36} + 35 q^{37} + 39 q^{38} + 37 q^{39} + 11 q^{40} + 42 q^{41} - q^{42} + 44 q^{43} + 21 q^{44} + 31 q^{45} + 28 q^{46} + 60 q^{47} + 18 q^{48} + 92 q^{49} + 79 q^{50} + 26 q^{51} + 26 q^{52} - 2 q^{53} + 66 q^{54} + 33 q^{55} + 30 q^{56} + 15 q^{57} - 5 q^{58} + 65 q^{59} + 17 q^{60} + 15 q^{61} + 60 q^{62} + 56 q^{63} + 50 q^{64} + 6 q^{65} + 37 q^{66} + 48 q^{67} + 24 q^{68} - 9 q^{69} + 38 q^{70} + 34 q^{71} + 66 q^{72} + 91 q^{73} + 35 q^{74} + 54 q^{75} + 39 q^{76} - 6 q^{77} + 37 q^{78} + 29 q^{79} + 11 q^{80} + 66 q^{81} + 42 q^{82} + 43 q^{83} - q^{84} + 44 q^{86} + 32 q^{87} + 21 q^{88} + 38 q^{89} + 31 q^{90} + 55 q^{91} + 28 q^{92} - 15 q^{93} + 60 q^{94} + 9 q^{95} + 18 q^{96} + 80 q^{97} + 92 q^{98} + 20 q^{99}+O(q^{100}) \) 50 * q + 50 * q^2 + 18 * q^3 + 50 * q^4 + 11 * q^5 + 18 * q^6 + 30 * q^7 + 50 * q^8 + 66 * q^9 + 11 * q^10 + 21 * q^11 + 18 * q^12 + 26 * q^13 + 30 * q^14 + 17 * q^15 + 50 * q^16 + 24 * q^17 + 66 * q^18 + 39 * q^19 + 11 * q^20 - q^21 + 21 * q^22 + 28 * q^23 + 18 * q^24 + 79 * q^25 + 26 * q^26 + 66 * q^27 + 30 * q^28 - 5 * q^29 + 17 * q^30 + 60 * q^31 + 50 * q^32 + 37 * q^33 + 24 * q^34 + 38 * q^35 + 66 * q^36 + 35 * q^37 + 39 * q^38 + 37 * q^39 + 11 * q^40 + 42 * q^41 - q^42 + 44 * q^43 + 21 * q^44 + 31 * q^45 + 28 * q^46 + 60 * q^47 + 18 * q^48 + 92 * q^49 + 79 * q^50 + 26 * q^51 + 26 * q^52 - 2 * q^53 + 66 * q^54 + 33 * q^55 + 30 * q^56 + 15 * q^57 - 5 * q^58 + 65 * q^59 + 17 * q^60 + 15 * q^61 + 60 * q^62 + 56 * q^63 + 50 * q^64 + 6 * q^65 + 37 * q^66 + 48 * q^67 + 24 * q^68 - 9 * q^69 + 38 * q^70 + 34 * q^71 + 66 * q^72 + 91 * q^73 + 35 * q^74 + 54 * q^75 + 39 * q^76 - 6 * q^77 + 37 * q^78 + 29 * q^79 + 11 * q^80 + 66 * q^81 + 42 * q^82 + 43 * q^83 - q^84 + 44 * q^86 + 32 * q^87 + 21 * q^88 + 38 * q^89 + 31 * q^90 + 55 * q^91 + 28 * q^92 - 15 * q^93 + 60 * q^94 + 9 * q^95 + 18 * q^96 + 80 * q^97 + 92 * q^98 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more