LMFDB - Embedded newform 61.2.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [61,2,Mod(1,61)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(61, 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("61.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	61.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:	$0.487087452330$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	61.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.17009 q^{2} -1.70928 q^{3} +2.70928 q^{4} -1.63090 q^{5} -3.70928 q^{6} -0.460811 q^{7} +1.53919 q^{8} -0.0783777 q^{9} +O(q^{10})$	\(q+2.17009 q^{2} -1.70928 q^{3} +2.70928 q^{4} -1.63090 q^{5} -3.70928 q^{6} -0.460811 q^{7} +1.53919 q^{8} -0.0783777 q^{9} -3.53919 q^{10} +6.17009 q^{11} -4.63090 q^{12} -4.07838 q^{13} -1.00000 q^{14} +2.78765 q^{15} -2.07838 q^{16} +0.630898 q^{17} -0.170086 q^{18} +7.12783 q^{19} -4.41855 q^{20} +0.787653 q^{21} +13.3896 q^{22} -0.170086 q^{23} -2.63090 q^{24} -2.34017 q^{25} -8.85043 q^{26} +5.26180 q^{27} -1.24846 q^{28} +2.63090 q^{29} +6.04945 q^{30} -10.3896 q^{31} -7.58864 q^{32} -10.5464 q^{33} +1.36910 q^{34} +0.751536 q^{35} -0.212347 q^{36} +5.12783 q^{37} +15.4680 q^{38} +6.97107 q^{39} -2.51026 q^{40} +3.15676 q^{41} +1.70928 q^{42} -3.36910 q^{43} +16.7165 q^{44} +0.127826 q^{45} -0.369102 q^{46} +0.183417 q^{47} +3.55252 q^{48} -6.78765 q^{49} -5.07838 q^{50} -1.07838 q^{51} -11.0494 q^{52} -4.34017 q^{53} +11.4186 q^{54} -10.0628 q^{55} -0.709275 q^{56} -12.1834 q^{57} +5.70928 q^{58} +1.78047 q^{59} +7.55252 q^{60} +1.00000 q^{61} -22.5464 q^{62} +0.0361173 q^{63} -12.3112 q^{64} +6.65142 q^{65} -22.8865 q^{66} -8.55971 q^{67} +1.70928 q^{68} +0.290725 q^{69} +1.63090 q^{70} +14.3896 q^{71} -0.120638 q^{72} -0.552520 q^{73} +11.1278 q^{74} +4.00000 q^{75} +19.3112 q^{76} -2.84324 q^{77} +15.1278 q^{78} -7.00719 q^{79} +3.38962 q^{80} -8.75872 q^{81} +6.85043 q^{82} +6.83710 q^{83} +2.13397 q^{84} -1.02893 q^{85} -7.31124 q^{86} -4.49693 q^{87} +9.49693 q^{88} +4.49693 q^{89} +0.277394 q^{90} +1.87936 q^{91} -0.460811 q^{92} +17.7587 q^{93} +0.398032 q^{94} -11.6248 q^{95} +12.9711 q^{96} +8.52359 q^{97} -14.7298 q^{98} -0.483597 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 2 q^{3} + q^{4} - q^{5} - 4 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + q^2 + 2 * q^3 + q^4 - q^5 - 4 * q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + q^{2} + 2 q^{3} + q^{4} - q^{5} - 4 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} - 9 q^{10} + 13 q^{11} - 10 q^{12} - 9 q^{13} - 3 q^{14} - 2 q^{15} - 3 q^{16} - 2 q^{17} + 5 q^{18} + q^{20} - 8 q^{21} + 11 q^{22} + 5 q^{23} - 4 q^{24} + 4 q^{25} + q^{26} + 8 q^{27} + 5 q^{28} + 4 q^{29} - 2 q^{31} - 3 q^{32} + 4 q^{33} + 8 q^{34} + 11 q^{35} - 11 q^{36} - 6 q^{37} + 14 q^{38} + 6 q^{39} + 9 q^{40} + 3 q^{41} - 2 q^{42} - 14 q^{43} + 9 q^{44} - 21 q^{45} - 5 q^{46} - 4 q^{47} + 10 q^{48} - 10 q^{49} - 12 q^{50} - 15 q^{52} - 2 q^{53} + 20 q^{54} - 13 q^{55} + 5 q^{56} - 32 q^{57} + 10 q^{58} + 29 q^{59} + 22 q^{60} + 3 q^{61} - 32 q^{62} - 19 q^{63} - 11 q^{64} - 17 q^{65} - 22 q^{66} + 9 q^{67} - 2 q^{68} + 8 q^{69} + q^{70} + 14 q^{71} - 13 q^{72} - q^{73} + 12 q^{74} + 12 q^{75} + 32 q^{76} - 15 q^{77} + 24 q^{78} + 13 q^{79} - 19 q^{80} - q^{81} - 7 q^{82} - 8 q^{83} + 20 q^{84} - 18 q^{85} + 4 q^{86} + 4 q^{87} + 11 q^{88} - 4 q^{89} + 7 q^{90} - 7 q^{91} - 3 q^{92} + 28 q^{93} + 20 q^{94} + 4 q^{95} + 24 q^{96} + 10 q^{97} - 4 q^{98} + 17 q^{99}+O(q^{100}) \) 3 * q + q^2 + 2 * q^3 + q^4 - q^5 - 4 * q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9 - 9 * q^10 + 13 * q^11 - 10 * q^12 - 9 * q^13 - 3 * q^14 - 2 * q^15 - 3 * q^16 - 2 * q^17 + 5 * q^18 + q^20 - 8 * q^21 + 11 * q^22 + 5 * q^23 - 4 * q^24 + 4 * q^25 + q^26 + 8 * q^27 + 5 * q^28 + 4 * q^29 - 2 * q^31 - 3 * q^32 + 4 * q^33 + 8 * q^34 + 11 * q^35 - 11 * q^36 - 6 * q^37 + 14 * q^38 + 6 * q^39 + 9 * q^40 + 3 * q^41 - 2 * q^42 - 14 * q^43 + 9 * q^44 - 21 * q^45 - 5 * q^46 - 4 * q^47 + 10 * q^48 - 10 * q^49 - 12 * q^50 - 15 * q^52 - 2 * q^53 + 20 * q^54 - 13 * q^55 + 5 * q^56 - 32 * q^57 + 10 * q^58 + 29 * q^59 + 22 * q^60 + 3 * q^61 - 32 * q^62 - 19 * q^63 - 11 * q^64 - 17 * q^65 - 22 * q^66 + 9 * q^67 - 2 * q^68 + 8 * q^69 + q^70 + 14 * q^71 - 13 * q^72 - q^73 + 12 * q^74 + 12 * q^75 + 32 * q^76 - 15 * q^77 + 24 * q^78 + 13 * q^79 - 19 * q^80 - q^81 - 7 * q^82 - 8 * q^83 + 20 * q^84 - 18 * q^85 + 4 * q^86 + 4 * q^87 + 11 * q^88 - 4 * q^89 + 7 * q^90 - 7 * q^91 - 3 * q^92 + 28 * q^93 + 20 * q^94 + 4 * q^95 + 24 * q^96 + 10 * q^97 - 4 * q^98 + 17 * 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 $

$2$	2.17009	1.53448	0.767241	−	0.641358i	$-0.221629\pi$
$2$	2.17009	1.53448	0.767241	+	0.641358i	$0.221629\pi$
$3$	−1.70928	−0.986851	−0.493425	−	0.869788i	$-0.664255\pi$
$3$	−1.70928	−0.986851	−0.493425	+	0.869788i	$0.664255\pi$
$4$	2.70928	1.35464
$5$	−1.63090	−0.729360	−0.364680	−	0.931133i	$-0.618822\pi$
$5$	−1.63090	−0.729360	−0.364680	+	0.931133i	$0.618822\pi$
$6$	−3.70928	−1.51431
$7$	−0.460811	−0.174170	−0.0870851	−	0.996201i	$-0.527755\pi$
$7$	−0.460811	−0.174170	−0.0870851	+	0.996201i	$0.527755\pi$
$8$	1.53919	0.544185
$9$	−0.0783777	−0.0261259
$10$	−3.53919	−1.11919
$11$	6.17009	1.86035	0.930176	−	0.367115i	$-0.119654\pi$
$11$	6.17009	1.86035	0.930176	+	0.367115i	$0.119654\pi$
$12$	−4.63090	−1.33682
$13$	−4.07838	−1.13114	−0.565569	−	0.824701i	$-0.691343\pi$
$13$	−4.07838	−1.13114	−0.565569	+	0.824701i	$0.691343\pi$
$14$	−1.00000	−0.267261
$15$	2.78765	0.719769
$16$	−2.07838	−0.519594
$17$	0.630898	0.153015	0.0765076	−	0.997069i	$-0.475623\pi$
$17$	0.630898	0.153015	0.0765076	+	0.997069i	$0.475623\pi$
$18$	−0.170086	−0.0400898
$19$	7.12783	1.63524	0.817618	−	0.575761i	$-0.195294\pi$
$19$	7.12783	1.63524	0.817618	+	0.575761i	$0.195294\pi$
$20$	−4.41855	−0.988018
$21$	0.787653	0.171880
$22$	13.3896	2.85468
$23$	−0.170086	−0.0354655	−0.0177327	−	0.999843i	$-0.505645\pi$
$23$	−0.170086	−0.0354655	−0.0177327	+	0.999843i	$0.505645\pi$
$24$	−2.63090	−0.537030
$25$	−2.34017	−0.468035
$26$	−8.85043	−1.73571
$27$	5.26180	1.01263
$28$	−1.24846	−0.235938
$29$	2.63090	0.488545	0.244273	−	0.969707i	$-0.421451\pi$
$29$	2.63090	0.488545	0.244273	+	0.969707i	$0.421451\pi$
$30$	6.04945	1.10447
$31$	−10.3896	−1.86603	−0.933016	−	0.359836i	$-0.882833\pi$
$31$	−10.3896	−1.86603	−0.933016	+	0.359836i	$0.882833\pi$
$32$	−7.58864	−1.34149
$33$	−10.5464	−1.83589
$34$	1.36910	0.234799
$35$	0.751536	0.127033
$36$	−0.212347	−0.0353911
$37$	5.12783	0.843009	0.421505	−	0.906826i	$-0.361502\pi$
$37$	5.12783	0.843009	0.421505	+	0.906826i	$0.361502\pi$
$38$	15.4680	2.50924
$39$	6.97107	1.11626
$40$	−2.51026	−0.396907
$41$	3.15676	0.493002	0.246501	−	0.969142i	$-0.420719\pi$
$41$	3.15676	0.493002	0.246501	+	0.969142i	$0.420719\pi$
$42$	1.70928	0.263747
$43$	−3.36910	−0.513783	−0.256892	−	0.966440i	$-0.582698\pi$
$43$	−3.36910	−0.513783	−0.256892	+	0.966440i	$0.582698\pi$
$44$	16.7165	2.52010
$45$	0.127826	0.0190552
$46$	−0.369102	−0.0544212
$47$	0.183417	0.0267542	0.0133771	−	0.999911i	$-0.495742\pi$
$47$	0.183417	0.0267542	0.0133771	+	0.999911i	$0.495742\pi$
$48$	3.55252	0.512762
$49$	−6.78765	−0.969665
$50$	−5.07838	−0.718191
$51$	−1.07838	−0.151003
$52$	−11.0494	−1.53228
$53$	−4.34017	−0.596169	−0.298084	−	0.954540i	$-0.596348\pi$
$53$	−4.34017	−0.596169	−0.298084	+	0.954540i	$0.596348\pi$
$54$	11.4186	1.55387
$55$	−10.0628	−1.35686
$56$	−0.709275	−0.0947809
$57$	−12.1834	−1.61373
$58$	5.70928	0.749665
$59$	1.78047	0.231797	0.115898	−	0.993261i	$-0.463025\pi$
$59$	1.78047	0.231797	0.115898	+	0.993261i	$0.463025\pi$
$60$	7.55252	0.975026
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	−22.5464	−2.86339
$63$	0.0361173	0.00455036
$64$	−12.3112	−1.53891
$65$	6.65142	0.825007
$66$	−22.8865	−2.81714
$67$	−8.55971	−1.04573	−0.522867	−	0.852414i	$-0.675138\pi$
$67$	−8.55971	−1.04573	−0.522867	+	0.852414i	$0.675138\pi$
$68$	1.70928	0.207280
$69$	0.290725	0.0349991
$70$	1.63090	0.194930
$71$	14.3896	1.70773	0.853867	−	0.520491i	$-0.174251\pi$
$71$	14.3896	1.70773	0.853867	+	0.520491i	$0.174251\pi$
$72$	−0.120638	−0.0142173
$73$	−0.552520	−0.0646676	−0.0323338	−	0.999477i	$-0.510294\pi$
$73$	−0.552520	−0.0646676	−0.0323338	+	0.999477i	$0.510294\pi$
$74$	11.1278	1.29358
$75$	4.00000	0.461880
$76$	19.3112	2.21515
$77$	−2.84324	−0.324018
$78$	15.1278	1.71289
$79$	−7.00719	−0.788370	−0.394185	−	0.919031i	$-0.628973\pi$
$79$	−7.00719	−0.788370	−0.394185	+	0.919031i	$0.628973\pi$
$80$	3.38962	0.378971
$81$	−8.75872	−0.973192
$82$	6.85043	0.756504
$83$	6.83710	0.750469	0.375235	−	0.926930i	$-0.377562\pi$
$83$	6.83710	0.750469	0.375235	+	0.926930i	$0.377562\pi$
$84$	2.13397	0.232835
$85$	−1.02893	−0.111603
$86$	−7.31124	−0.788392
$87$	−4.49693	−0.482121
$88$	9.49693	1.01238
$89$	4.49693	0.476673	0.238337	−	0.971183i	$-0.423398\pi$
$89$	4.49693	0.476673	0.238337	+	0.971183i	$0.423398\pi$
$90$	0.277394	0.0292399
$91$	1.87936	0.197011
$92$	−0.460811	−0.0480429
$93$	17.7587	1.84149
$94$	0.398032	0.0410538
$95$	−11.6248	−1.19267
$96$	12.9711	1.32385
$97$	8.52359	0.865439	0.432720	−	0.901528i	$-0.357554\pi$
$97$	8.52359	0.865439	0.432720	+	0.901528i	$0.357554\pi$
$98$	−14.7298	−1.48793
$99$	−0.483597	−0.0486034
$100$	−6.34017	−0.634017
$101$	−6.78765	−0.675397	−0.337698	−	0.941254i	$-0.609648\pi$
$101$	−6.78765	−0.675397	−0.337698	+	0.941254i	$0.609648\pi$
$102$	−2.34017	−0.231712
$103$	11.1278	1.09646	0.548229	−	0.836328i	$-0.315302\pi$
$103$	11.1278	1.09646	0.548229	+	0.836328i	$0.315302\pi$
$104$	−6.27739	−0.615549
$105$	−1.28458	−0.125362
$106$	−9.41855	−0.914811
$107$	15.5753	1.50572	0.752861	−	0.658180i	$-0.228673\pi$
$107$	15.5753	1.50572	0.752861	+	0.658180i	$0.228673\pi$
$108$	14.2557	1.37175
$109$	11.6803	1.11877	0.559387	−	0.828907i	$-0.311037\pi$
$109$	11.6803	1.11877	0.559387	+	0.828907i	$0.311037\pi$
$110$	−21.8371	−2.08209
$111$	−8.76487	−0.831924
$112$	0.957740	0.0904979
$113$	−12.9421	−1.21749	−0.608747	−	0.793364i	$-0.708328\pi$
$113$	−12.9421	−1.21749	−0.608747	+	0.793364i	$0.708328\pi$
$114$	−26.4391	−2.47625
$115$	0.277394	0.0258671
$116$	7.12783	0.661802
$117$	0.319654	0.0295520
$118$	3.86376	0.355688
$119$	−0.290725	−0.0266507
$120$	4.29072	0.391688
$121$	27.0700	2.46091
$122$	2.17009	0.196470
$123$	−5.39576	−0.486520
$124$	−28.1483	−2.52780
$125$	11.9711	1.07073
$126$	0.0783777	0.00698244
$127$	−1.02893	−0.0913027	−0.0456514	−	0.998957i	$-0.514536\pi$
$127$	−1.02893	−0.0913027	−0.0456514	+	0.998957i	$0.514536\pi$
$128$	−11.5392	−1.01993
$129$	5.75872	0.507027
$130$	14.4341	1.26596
$131$	2.44748	0.213837	0.106919	−	0.994268i	$-0.465902\pi$
$131$	2.44748	0.213837	0.106919	+	0.994268i	$0.465902\pi$
$132$	−28.5730	−2.48696
$133$	−3.28458	−0.284809
$134$	−18.5753	−1.60466
$135$	−8.58145	−0.738574
$136$	0.971071	0.0832686
$137$	−16.0700	−1.37295	−0.686475	−	0.727153i	$-0.740843\pi$
$137$	−16.0700	−1.37295	−0.686475	+	0.727153i	$0.740843\pi$
$138$	0.630898	0.0537056
$139$	−1.53919	−0.130552	−0.0652761	−	0.997867i	$-0.520793\pi$
$139$	−1.53919	−0.130552	−0.0652761	+	0.997867i	$0.520793\pi$
$140$	2.03612	0.172083
$141$	−0.313511	−0.0264024
$142$	31.2267	2.62049
$143$	−25.1639	−2.10431
$144$	0.162899	0.0135749
$145$	−4.29072	−0.356325
$146$	−1.19902	−0.0992313
$147$	11.6020	0.956914
$148$	13.8927	1.14197
$149$	1.92162	0.157425	0.0787127	−	0.996897i	$-0.474919\pi$
$149$	1.92162	0.157425	0.0787127	+	0.996897i	$0.474919\pi$
$150$	8.68035	0.708747
$151$	9.32457	0.758823	0.379412	−	0.925228i	$-0.376126\pi$
$151$	9.32457	0.758823	0.379412	+	0.925228i	$0.376126\pi$
$152$	10.9711	0.889871
$153$	−0.0494483	−0.00399766
$154$	−6.17009	−0.497200
$155$	16.9444	1.36101
$156$	18.8865	1.51213
$157$	−13.4947	−1.07699	−0.538496	−	0.842628i	$-0.681007\pi$
$157$	−13.4947	−1.07699	−0.538496	+	0.842628i	$0.681007\pi$
$158$	−15.2062	−1.20974
$159$	7.41855	0.588329
$160$	12.3763	0.978432
$161$	0.0783777	0.00617703
$162$	−19.0072	−1.49335
$163$	−10.7031	−0.838334	−0.419167	−	0.907909i	$-0.637678\pi$
$163$	−10.7031	−0.838334	−0.419167	+	0.907909i	$0.637678\pi$
$164$	8.55252	0.667840
$165$	17.2001	1.33902
$166$	14.8371	1.15158
$167$	4.88655	0.378133	0.189066	−	0.981964i	$-0.439454\pi$
$167$	4.88655	0.378133	0.189066	+	0.981964i	$0.439454\pi$
$168$	1.21235	0.0935346
$169$	3.63317	0.279474
$170$	−2.23287	−0.171253
$171$	−0.558663	−0.0427220
$172$	−9.12783	−0.695990
$173$	−0.738205	−0.0561247	−0.0280623	−	0.999606i	$-0.508934\pi$
$173$	−0.738205	−0.0561247	−0.0280623	+	0.999606i	$0.508934\pi$
$174$	−9.75872	−0.739807
$175$	1.07838	0.0815177
$176$	−12.8238	−0.966628
$177$	−3.04331	−0.228749
$178$	9.75872	0.731447
$179$	1.90110	0.142095	0.0710476	−	0.997473i	$-0.477366\pi$
$179$	1.90110	0.142095	0.0710476	+	0.997473i	$0.477366\pi$
$180$	0.346316	0.0258129
$181$	−21.3607	−1.58773	−0.793864	−	0.608096i	$-0.791934\pi$
$181$	−21.3607	−1.58773	−0.793864	+	0.608096i	$0.791934\pi$
$182$	4.07838	0.302309
$183$	−1.70928	−0.126353
$184$	−0.261795	−0.0192998
$185$	−8.36296	−0.614857
$186$	38.5380	2.82574
$187$	3.89269	0.284662
$188$	0.496928	0.0362422
$189$	−2.42469	−0.176371
$190$	−25.2267	−1.83014
$191$	2.88550	0.208788	0.104394	−	0.994536i	$-0.466710\pi$
$191$	2.88550	0.208788	0.104394	+	0.994536i	$0.466710\pi$
$192$	21.0433	1.51867
$193$	0.680346	0.0489724	0.0244862	−	0.999700i	$-0.492205\pi$
$193$	0.680346	0.0489724	0.0244862	+	0.999700i	$0.492205\pi$
$194$	18.4969	1.32800
$195$	−11.3691	−0.814158
$196$	−18.3896	−1.31354
$197$	−19.3896	−1.38145	−0.690727	−	0.723116i	$-0.742709\pi$
$197$	−19.3896	−1.38145	−0.690727	+	0.723116i	$0.742709\pi$
$198$	−1.04945	−0.0745810
$199$	−21.4186	−1.51832	−0.759160	−	0.650904i	$-0.774390\pi$
$199$	−21.4186	−1.51832	−0.759160	+	0.650904i	$0.774390\pi$
$200$	−3.60197	−0.254698
$201$	14.6309	1.03198
$202$	−14.7298	−1.03638
$203$	−1.21235	−0.0850901
$204$	−2.92162	−0.204554
$205$	−5.14834	−0.359576
$206$	24.1483	1.68249
$207$	0.0133310	0.000926568 0
$208$	8.47641	0.587733
$209$	43.9793	3.04211
$210$	−2.78765	−0.192366
$211$	5.39576	0.371460	0.185730	−	0.982601i	$-0.440535\pi$
$211$	5.39576	0.371460	0.185730	+	0.982601i	$0.440535\pi$
$212$	−11.7587	−0.807592
$213$	−24.5958	−1.68528
$214$	33.7998	2.31050
$215$	5.49466	0.374733
$216$	8.09890	0.551060
$217$	4.78765	0.325007
$218$	25.3474	1.71674
$219$	0.944409	0.0638172
$220$	−27.2628	−1.83806
$221$	−2.57304	−0.173081
$222$	−19.0205	−1.27657
$223$	12.4257	0.832089	0.416045	−	0.909344i	$-0.363416\pi$
$223$	12.4257	0.832089	0.416045	+	0.909344i	$0.363416\pi$
$224$	3.49693	0.233648
$225$	0.183417	0.0122278
$226$	−28.0856	−1.86822
$227$	−22.3679	−1.48461	−0.742304	−	0.670063i	$-0.766267\pi$
$227$	−22.3679	−1.48461	−0.742304	+	0.670063i	$0.766267\pi$
$228$	−33.0082	−2.18602
$229$	−0.185685	−0.0122704	−0.00613520	−	0.999981i	$-0.501953\pi$
$229$	−0.185685	−0.0122704	−0.00613520	+	0.999981i	$0.501953\pi$
$230$	0.601968	0.0396926
$231$	4.85989	0.319757
$232$	4.04945	0.265859
$233$	−17.8660	−1.17044	−0.585221	−	0.810874i	$-0.698992\pi$
$233$	−17.8660	−1.17044	−0.585221	+	0.810874i	$0.698992\pi$
$234$	0.693677	0.0453471
$235$	−0.299135	−0.0195134
$236$	4.82377	0.314001
$237$	11.9772	0.778004
$238$	−0.630898	−0.0408950
$239$	−22.3896	−1.44826	−0.724132	−	0.689661i	$-0.757759\pi$
$239$	−22.3896	−1.44826	−0.724132	+	0.689661i	$0.757759\pi$
$240$	−5.79380	−0.373988
$241$	20.0433	1.29110	0.645551	−	0.763717i	$-0.276628\pi$
$241$	20.0433	1.29110	0.645551	+	0.763717i	$0.276628\pi$
$242$	58.7442	3.77622
$243$	−0.814315	−0.0522383
$244$	2.70928	0.173444
$245$	11.0700	0.707234
$246$	−11.7093	−0.746556
$247$	−29.0700	−1.84968
$248$	−15.9916	−1.01547
$249$	−11.6865	−0.740601
$250$	25.9783	1.64301
$251$	7.05559	0.445345	0.222672	−	0.974893i	$-0.428522\pi$
$251$	7.05559	0.445345	0.222672	+	0.974893i	$0.428522\pi$
$252$	0.0978518	0.00616408
$253$	−1.04945	−0.0659783
$254$	−2.23287	−0.140102
$255$	1.75872	0.110136
$256$	−0.418551	−0.0261594
$257$	3.26180	0.203465	0.101733	−	0.994812i	$-0.467561\pi$
$257$	3.26180	0.203465	0.101733	+	0.994812i	$0.467561\pi$
$258$	12.4969	0.778025
$259$	−2.36296	−0.146827
$260$	18.0205	1.11759
$261$	−0.206204	−0.0127637
$262$	5.31124	0.328130
$263$	−20.5730	−1.26859	−0.634294	−	0.773092i	$-0.718709\pi$
$263$	−20.5730	−1.26859	−0.634294	+	0.773092i	$0.718709\pi$
$264$	−16.2329	−0.999064
$265$	7.07838	0.434821
$266$	−7.12783	−0.437035
$267$	−7.68649	−0.470405
$268$	−23.1906	−1.41659
$269$	−3.97334	−0.242259	−0.121129	−	0.992637i	$-0.538652\pi$
$269$	−3.97334	−0.242259	−0.121129	+	0.992637i	$0.538652\pi$
$270$	−18.6225	−1.13333
$271$	−4.99773	−0.303591	−0.151795	−	0.988412i	$-0.548505\pi$
$271$	−4.99773	−0.303591	−0.151795	+	0.988412i	$0.548505\pi$
$272$	−1.31124	−0.0795058
$273$	−3.21235	−0.194420
$274$	−34.8732	−2.10677
$275$	−14.4391	−0.870709
$276$	0.787653	0.0474111
$277$	17.9649	1.07941	0.539704	−	0.841855i	$-0.318536\pi$
$277$	17.9649	1.07941	0.539704	+	0.841855i	$0.318536\pi$
$278$	−3.34017	−0.200330
$279$	0.814315	0.0487518
$280$	1.15676	0.0691294
$281$	15.6020	0.930735	0.465368	−	0.885117i	$-0.345922\pi$
$281$	15.6020	0.930735	0.465368	+	0.885117i	$0.345922\pi$
$282$	−0.680346	−0.0405140
$283$	24.7298	1.47003	0.735017	−	0.678049i	$-0.237174\pi$
$283$	24.7298	1.47003	0.735017	+	0.678049i	$0.237174\pi$
$284$	38.9854	2.31336
$285$	19.8699	1.17699
$286$	−54.6079	−3.22903
$287$	−1.45467	−0.0858663
$288$	0.594780	0.0350478
$289$	−16.6020	−0.976586
$290$	−9.31124	−0.546775
$291$	−14.5692	−0.854059
$292$	−1.49693	−0.0876011
$293$	24.1711	1.41209	0.706046	−	0.708166i	$-0.250477\pi$
$293$	24.1711	1.41209	0.706046	+	0.708166i	$0.250477\pi$
$294$	25.1773	1.46837
$295$	−2.90376	−0.169063
$296$	7.89269	0.458753
$297$	32.4657	1.88385
$298$	4.17009	0.241567
$299$	0.693677	0.0401164
$300$	10.8371	0.625680
$301$	1.55252	0.0894858
$302$	20.2351	1.16440
$303$	11.6020	0.666516
$304$	−14.8143	−0.849659
$305$	−1.63090	−0.0933849
$306$	−0.107307	−0.00613434
$307$	19.5041	1.11316	0.556579	−	0.830794i	$-0.312114\pi$
$307$	19.5041	1.11316	0.556579	+	0.830794i	$0.312114\pi$
$308$	−7.70313	−0.438927
$309$	−19.0205	−1.08204
$310$	36.7708	2.08844
$311$	8.35350	0.473684	0.236842	−	0.971548i	$-0.423888\pi$
$311$	8.35350	0.473684	0.236842	+	0.971548i	$0.423888\pi$
$312$	10.7298	0.607455
$313$	2.09890	0.118637	0.0593183	−	0.998239i	$-0.481107\pi$
$313$	2.09890	0.118637	0.0593183	+	0.998239i	$0.481107\pi$
$314$	−29.2846	−1.65262
$315$	−0.0589037	−0.00331885
$316$	−18.9844	−1.06796
$317$	27.7587	1.55909	0.779543	−	0.626349i	$-0.215452\pi$
$317$	27.7587	1.55909	0.779543	+	0.626349i	$0.215452\pi$
$318$	16.0989	0.902781
$319$	16.2329	0.908866
$320$	20.0784	1.12242
$321$	−26.6225	−1.48592
$322$	0.170086	0.00947855
$323$	4.49693	0.250216
$324$	−23.7298	−1.31832
$325$	9.54411	0.529412
$326$	−23.2267	−1.28641
$327$	−19.9649	−1.10406
$328$	4.85884	0.268285
$329$	−0.0845208	−0.00465978
$330$	37.3256	2.05471
$331$	−1.19902	−0.0659039	−0.0329519	−	0.999457i	$-0.510491\pi$
$331$	−1.19902	−0.0659039	−0.0329519	+	0.999457i	$0.510491\pi$
$332$	18.5236	1.01661
$333$	−0.401907	−0.0220244
$334$	10.6042	0.580238
$335$	13.9600	0.762717
$336$	−1.63704	−0.0893079
$337$	20.9672	1.14216	0.571078	−	0.820896i	$-0.306525\pi$
$337$	20.9672	1.14216	0.571078	+	0.820896i	$0.306525\pi$
$338$	7.88428	0.428848
$339$	22.1217	1.20148
$340$	−2.78765	−0.151182
$341$	−64.1049	−3.47147
$342$	−1.21235	−0.0655562
$343$	6.35350	0.343057
$344$	−5.18568	−0.279593
$345$	−0.474142	−0.0255270
$346$	−1.60197	−0.0861223
$347$	20.3896	1.09457	0.547286	−	0.836946i	$-0.315661\pi$
$347$	20.3896	1.09457	0.547286	+	0.836946i	$0.315661\pi$
$348$	−12.1834	−0.653100
$349$	−7.65142	−0.409571	−0.204785	−	0.978807i	$-0.565650\pi$
$349$	−7.65142	−0.409571	−0.204785	+	0.978807i	$0.565650\pi$
$350$	2.34017	0.125088
$351$	−21.4596	−1.14543
$352$	−46.8225	−2.49565
$353$	−19.6765	−1.04727	−0.523636	−	0.851942i	$-0.675425\pi$
$353$	−19.6765	−1.04727	−0.523636	+	0.851942i	$0.675425\pi$
$354$	−6.60424	−0.351011
$355$	−23.4680	−1.24555
$356$	12.1834	0.645720
$357$	0.496928	0.0263002
$358$	4.12556	0.218043
$359$	20.8599	1.10094	0.550471	−	0.834854i	$-0.314448\pi$
$359$	20.8599	1.10094	0.550471	+	0.834854i	$0.314448\pi$
$360$	0.196748	0.0103696
$361$	31.8059	1.67399
$362$	−46.3545	−2.43634
$363$	−46.2700	−2.42855
$364$	5.09171	0.266878
$365$	0.901103	0.0471659
$366$	−3.70928	−0.193887
$367$	−5.31965	−0.277684	−0.138842	−	0.990315i	$-0.544338\pi$
$367$	−5.31965	−0.277684	−0.138842	+	0.990315i	$0.544338\pi$
$368$	0.353504	0.0184277
$369$	−0.247419	−0.0128801
$370$	−18.1483	−0.943488
$371$	2.00000	0.103835
$372$	48.1133	2.49456
$373$	−20.0905	−1.04025	−0.520123	−	0.854091i	$-0.674114\pi$
$373$	−20.0905	−1.04025	−0.520123	+	0.854091i	$0.674114\pi$
$374$	8.44748	0.436809
$375$	−20.4619	−1.05665
$376$	0.282314	0.0145592
$377$	−10.7298	−0.552613
$378$	−5.26180	−0.270638
$379$	−26.4040	−1.35628	−0.678141	−	0.734932i	$-0.737214\pi$
$379$	−26.4040	−1.35628	−0.678141	+	0.734932i	$0.737214\pi$
$380$	−31.4947	−1.61564
$381$	1.75872	0.0901021
$382$	6.26180	0.320381
$383$	−10.9350	−0.558750	−0.279375	−	0.960182i	$-0.590127\pi$
$383$	−10.9350	−0.558750	−0.279375	+	0.960182i	$0.590127\pi$
$384$	19.7237	1.00652
$385$	4.63704	0.236325
$386$	1.47641	0.0751473
$387$	0.264063	0.0134231
$388$	23.0928	1.17236
$389$	−5.50307	−0.279017	−0.139508	−	0.990221i	$-0.544552\pi$
$389$	−5.50307	−0.279017	−0.139508	+	0.990221i	$0.544552\pi$
$390$	−24.6719	−1.24931
$391$	−0.107307	−0.00542676
$392$	−10.4475	−0.527677
$393$	−4.18342	−0.211025
$394$	−42.0772	−2.11982
$395$	11.4280	0.575005
$396$	−1.31020	−0.0658400
$397$	14.7565	0.740605	0.370303	−	0.928911i	$-0.379254\pi$
$397$	14.7565	0.740605	0.370303	+	0.928911i	$0.379254\pi$
$398$	−46.4801	−2.32984
$399$	5.61425	0.281064
$400$	4.86376	0.243188
$401$	−17.7587	−0.886828	−0.443414	−	0.896317i	$-0.646233\pi$
$401$	−17.7587	−0.886828	−0.443414	+	0.896317i	$0.646233\pi$
$402$	31.7503	1.58356
$403$	42.3728	2.11074
$404$	−18.3896	−0.914918
$405$	14.2846	0.709807
$406$	−2.63090	−0.130569
$407$	31.6391	1.56829
$408$	−1.65983	−0.0821737
$409$	−16.4391	−0.812860	−0.406430	−	0.913682i	$-0.633226\pi$
$409$	−16.4391	−0.812860	−0.406430	+	0.913682i	$0.633226\pi$
$410$	−11.1724	−0.551763
$411$	27.4680	1.35490
$412$	30.1483	1.48530
$413$	−0.820458	−0.0403721
$414$	0.0289294	0.00142180
$415$	−11.1506	−0.547362
$416$	30.9493	1.51742
$417$	2.63090	0.128836
$418$	95.4389	4.66807
$419$	32.0183	1.56419	0.782097	−	0.623157i	$-0.214150\pi$
$419$	32.0183	1.56419	0.782097	+	0.623157i	$0.214150\pi$
$420$	−3.48029	−0.169821
$421$	−0.630898	−0.0307481	−0.0153740	−	0.999882i	$-0.504894\pi$
$421$	−0.630898	−0.0307481	−0.0153740	+	0.999882i	$0.504894\pi$
$422$	11.7093	0.569999
$423$	−0.0143758	−0.000698978 0
$424$	−6.68035	−0.324426
$425$	−1.47641	−0.0716164
$426$	−53.3751	−2.58603
$427$	−0.460811	−0.0223002
$428$	42.1978	2.03971
$429$	43.0121	2.07664
$430$	11.9239	0.575021
$431$	−16.6225	−0.800677	−0.400339	−	0.916367i	$-0.631107\pi$
$431$	−16.6225	−0.800677	−0.400339	+	0.916367i	$0.631107\pi$
$432$	−10.9360	−0.526158
$433$	−20.4969	−0.985020	−0.492510	−	0.870307i	$-0.663920\pi$
$433$	−20.4969	−0.985020	−0.492510	+	0.870307i	$0.663920\pi$
$434$	10.3896	0.498718
$435$	7.33403	0.351640
$436$	31.6453	1.51553
$437$	−1.21235	−0.0579944
$438$	2.04945	0.0979264
$439$	−2.34858	−0.112092	−0.0560459	−	0.998428i	$-0.517849\pi$
$439$	−2.34858	−0.112092	−0.0560459	+	0.998428i	$0.517849\pi$
$440$	−15.4885	−0.738386
$441$	0.532001	0.0253334
$442$	−5.58372	−0.265590
$443$	16.5958	0.788491	0.394246	−	0.919005i	$-0.371006\pi$
$443$	16.5958	0.788491	0.394246	+	0.919005i	$0.371006\pi$
$444$	−23.7464	−1.12696
$445$	−7.33403	−0.347666
$446$	26.9649	1.27683
$447$	−3.28458	−0.155355
$448$	5.67316	0.268032
$449$	−20.4680	−0.965945	−0.482972	−	0.875636i	$-0.660443\pi$
$449$	−20.4680	−0.965945	−0.482972	+	0.875636i	$0.660443\pi$
$450$	0.398032	0.0187634
$451$	19.4775	0.917158
$452$	−35.0638	−1.64926
$453$	−15.9383	−0.748845
$454$	−48.5402	−2.27811
$455$	−3.06505	−0.143692
$456$	−18.7526	−0.878170
$457$	23.9506	1.12036	0.560180	−	0.828371i	$-0.310732\pi$
$457$	23.9506	1.12036	0.560180	+	0.828371i	$0.310732\pi$
$458$	−0.402952	−0.0188287
$459$	3.31965	0.154948
$460$	0.751536	0.0350405
$461$	−26.5113	−1.23475	−0.617377	−	0.786667i	$-0.711805\pi$
$461$	−26.5113	−1.23475	−0.617377	+	0.786667i	$0.711805\pi$
$462$	10.5464	0.490662
$463$	34.0410	1.58202	0.791011	−	0.611802i	$-0.209555\pi$
$463$	34.0410	1.58202	0.791011	+	0.611802i	$0.209555\pi$
$464$	−5.46800	−0.253845
$465$	−28.9627	−1.34311
$466$	−38.7708	−1.79602
$467$	4.55971	0.210998	0.105499	−	0.994419i	$-0.466356\pi$
$467$	4.55971	0.210998	0.105499	+	0.994419i	$0.466356\pi$
$468$	0.866031	0.0400323
$469$	3.94441	0.182136
$470$	−0.649149	−0.0299430
$471$	23.0661	1.06283
$472$	2.74047	0.126140
$473$	−20.7877	−0.955817
$474$	25.9916	1.19383
$475$	−16.6803	−0.765347
$476$	−0.787653	−0.0361020
$477$	0.340173	0.0155755
$478$	−48.5874	−2.22234
$479$	−0.653684	−0.0298676	−0.0149338	−	0.999888i	$-0.504754\pi$
$479$	−0.653684	−0.0298676	−0.0149338	+	0.999888i	$0.504754\pi$
$480$	−21.1545	−0.965566
$481$	−20.9132	−0.953560
$482$	43.4957	1.98118
$483$	−0.133969	−0.00609581
$484$	73.3400	3.33364
$485$	−13.9011	−0.631217
$486$	−1.76713	−0.0801588
$487$	−0.0227863	−0.00103255	−0.000516274	−	1.00000i	$-0.500164\pi$
$487$	−0.0227863	−0.00103255	−0.000516274		1.00000i	$0.500164\pi$
$488$	1.53919	0.0696758
$489$	18.2946	0.827310
$490$	24.0228	1.08524
$491$	−2.94441	−0.132879	−0.0664396	−	0.997790i	$-0.521164\pi$
$491$	−2.94441	−0.132879	−0.0664396	+	0.997790i	$0.521164\pi$
$492$	−14.6186	−0.659058
$493$	1.65983	0.0747548
$494$	−63.0843	−2.83830
$495$	0.788698	0.0354493
$496$	21.5936	0.969579
$497$	−6.63090	−0.297436
$498$	−25.3607	−1.13644
$499$	23.0878	1.03355	0.516777	−	0.856120i	$-0.327132\pi$
$499$	23.0878	1.03355	0.516777	+	0.856120i	$0.327132\pi$
$500$	32.4329	1.45044
$501$	−8.35246	−0.373160
$502$	15.3112	0.683374
$503$	5.62475	0.250795	0.125398	−	0.992107i	$-0.459979\pi$
$503$	5.62475	0.250795	0.125398	+	0.992107i	$0.459979\pi$
$504$	0.0555914	0.00247624
$505$	11.0700	0.492607
$506$	−2.27739	−0.101242
$507$	−6.21008	−0.275799
$508$	−2.78765	−0.123682
$509$	−19.5525	−0.866650	−0.433325	−	0.901238i	$-0.642660\pi$
$509$	−19.5525	−0.866650	−0.433325	+	0.901238i	$0.642660\pi$
$510$	3.81658	0.169001
$511$	0.254607	0.0112632
$512$	22.1701	0.979789
$513$	37.5052	1.65589
$514$	7.07838	0.312214
$515$	−18.1483	−0.799712
$516$	15.6020	0.686838
$517$	1.13170	0.0497722
$518$	−5.12783	−0.225304
$519$	1.26180	0.0553867
$520$	10.2378	0.448957
$521$	16.7442	0.733575	0.366788	−	0.930305i	$-0.380458\pi$
$521$	16.7442	0.733575	0.366788	+	0.930305i	$0.380458\pi$
$522$	−0.447480	−0.0195857
$523$	−1.69982	−0.0743279	−0.0371640	−	0.999309i	$-0.511832\pi$
$523$	−1.69982	−0.0743279	−0.0371640	+	0.999309i	$0.511832\pi$
$524$	6.63090	0.289672
$525$	−1.84324	−0.0804458
$526$	−44.6453	−1.94663
$527$	−6.55479	−0.285531
$528$	21.9194	0.953917
$529$	−22.9711	−0.998742
$530$	15.3607	0.667226
$531$	−0.139549	−0.00605590
$532$	−8.89884	−0.385813
$533$	−12.8744	−0.557654
$534$	−16.6803	−0.721829
$535$	−25.4017	−1.09821
$536$	−13.1750	−0.569074
$537$	−3.24951	−0.140227
$538$	−8.62249	−0.371742
$539$	−41.8804	−1.80392
$540$	−23.2495	−1.00050
$541$	13.3074	0.572128	0.286064	−	0.958210i	$-0.407653\pi$
$541$	13.3074	0.572128	0.286064	+	0.958210i	$0.407653\pi$
$542$	−10.8455	−0.465855
$543$	36.5113	1.56685
$544$	−4.78765	−0.205269
$545$	−19.0494	−0.815989
$546$	−6.97107	−0.298334
$547$	−24.2690	−1.03767	−0.518833	−	0.854875i	$-0.673634\pi$
$547$	−24.2690	−1.03767	−0.518833	+	0.854875i	$0.673634\pi$
$548$	−43.5380	−1.85985
$549$	−0.0783777	−0.00334508
$550$	−31.3340	−1.33609
$551$	18.7526	0.798887
$552$	0.447480	0.0190460
$553$	3.22899	0.137311
$554$	38.9854	1.65633
$555$	14.2946	0.606772
$556$	−4.17009	−0.176851
$557$	1.36069	0.0576544	0.0288272	−	0.999584i	$-0.490823\pi$
$557$	1.36069	0.0576544	0.0288272	+	0.999584i	$0.490823\pi$
$558$	1.76713	0.0748088
$559$	13.7405	0.581160
$560$	−1.56198	−0.0660055
$561$	−6.65368	−0.280919
$562$	33.8576	1.42820
$563$	14.4885	0.610618	0.305309	−	0.952253i	$-0.401240\pi$
$563$	14.4885	0.610618	0.305309	+	0.952253i	$0.401240\pi$
$564$	−0.849388	−0.0357657
$565$	21.1073	0.887991
$566$	53.6658	2.25574
$567$	4.03612	0.169501
$568$	22.1483	0.929324
$569$	−3.78765	−0.158787	−0.0793933	−	0.996843i	$-0.525298\pi$
$569$	−3.78765	−0.158787	−0.0793933	+	0.996843i	$0.525298\pi$
$570$	43.1194	1.80607
$571$	−3.87444	−0.162140	−0.0810702	−	0.996708i	$-0.525834\pi$
$571$	−3.87444	−0.162140	−0.0810702	+	0.996708i	$0.525834\pi$
$572$	−68.1761	−2.85058
$573$	−4.93212	−0.206042
$574$	−3.15676	−0.131760
$575$	0.398032	0.0165991
$576$	0.964928	0.0402053
$577$	−23.9071	−0.995264	−0.497632	−	0.867388i	$-0.665797\pi$
$577$	−23.9071	−0.995264	−0.497632	+	0.867388i	$0.665797\pi$
$578$	−36.0277	−1.49856
$579$	−1.16290	−0.0483284
$580$	−11.6248	−0.482692
$581$	−3.15061	−0.130709
$582$	−31.6163	−1.31054
$583$	−26.7792	−1.10908
$584$	−0.850432	−0.0351911
$585$	−0.521323	−0.0215541
$586$	52.4534	2.16683
$587$	12.3318	0.508986	0.254493	−	0.967075i	$-0.418091\pi$
$587$	12.3318	0.508986	0.254493	+	0.967075i	$0.418091\pi$
$588$	31.4329	1.29627
$589$	−74.0554	−3.05140
$590$	−6.30140	−0.259425
$591$	33.1422	1.36329
$592$	−10.6576	−0.438023
$593$	11.1278	0.456965	0.228483	−	0.973548i	$-0.426624\pi$
$593$	11.1278	0.456965	0.228483	+	0.973548i	$0.426624\pi$
$594$	70.4534	2.89074
$595$	0.474142	0.0194379
$596$	5.20620	0.213254
$597$	36.6102	1.49836
$598$	1.50534	0.0615579
$599$	23.0878	0.943343	0.471672	−	0.881774i	$-0.343651\pi$
$599$	23.0878	0.943343	0.471672	+	0.881774i	$0.343651\pi$
$600$	6.15676	0.251348
$601$	−10.3340	−0.421534	−0.210767	−	0.977536i	$-0.567596\pi$
$601$	−10.3340	−0.421534	−0.210767	+	0.977536i	$0.567596\pi$
$602$	3.36910	0.137314
$603$	0.670891	0.0273208
$604$	25.2628	1.02793
$605$	−44.1483	−1.79489
$606$	25.1773	1.02276
$607$	−24.8371	−1.00811	−0.504053	−	0.863672i	$-0.668159\pi$
$607$	−24.8371	−1.00811	−0.504053	+	0.863672i	$0.668159\pi$
$608$	−54.0905	−2.19366
$609$	2.07223	0.0839712
$610$	−3.53919	−0.143298
$611$	−0.748046	−0.0302627
$612$	−0.133969	−0.00541538
$613$	−23.9877	−0.968855	−0.484427	−	0.874832i	$-0.660972\pi$
$613$	−23.9877	−0.968855	−0.484427	+	0.874832i	$0.660972\pi$
$614$	42.3256	1.70812
$615$	8.79994	0.354848
$616$	−4.37629	−0.176326
$617$	21.6020	0.869662	0.434831	−	0.900512i	$-0.356808\pi$
$617$	21.6020	0.869662	0.434831	+	0.900512i	$0.356808\pi$
$618$	−41.2762	−1.66037
$619$	−46.2388	−1.85850	−0.929248	−	0.369457i	$-0.879544\pi$
$619$	−46.2388	−1.85850	−0.929248	+	0.369457i	$0.879544\pi$
$620$	45.9071	1.84367
$621$	−0.894960	−0.0359135
$622$	18.1278	0.726860
$623$	−2.07223	−0.0830223
$624$	−14.4885	−0.580005
$625$	−7.82273	−0.312909
$626$	4.55479	0.182046
$627$	−75.1727	−3.00211
$628$	−36.5608	−1.45893
$629$	3.23513	0.128993
$630$	−0.127826	−0.00509271
$631$	33.2628	1.32417	0.662086	−	0.749428i	$-0.269671\pi$
$631$	33.2628	1.32417	0.662086	+	0.749428i	$0.269671\pi$
$632$	−10.7854	−0.429020
$633$	−9.22285	−0.366575
$634$	60.2388	2.39239
$635$	1.67808	0.0665925
$636$	20.0989	0.796973
$637$	27.6826	1.09683
$638$	35.2267	1.39464
$639$	−1.12783	−0.0446161
$640$	18.8192	0.743896
$641$	33.7009	1.33110	0.665552	−	0.746351i	$-0.268196\pi$
$641$	33.7009	1.33110	0.665552	+	0.746351i	$0.268196\pi$
$642$	−57.7731	−2.28012
$643$	−6.97107	−0.274912	−0.137456	−	0.990508i	$-0.543893\pi$
$643$	−6.97107	−0.274912	−0.137456	+	0.990508i	$0.543893\pi$
$644$	0.212347	0.00836764
$645$	−9.39189	−0.369805
$646$	9.75872	0.383952
$647$	−16.0049	−0.629218	−0.314609	−	0.949221i	$-0.601873\pi$
$647$	−16.0049	−0.629218	−0.314609	+	0.949221i	$0.601873\pi$
$648$	−13.4813	−0.529597
$649$	10.9856	0.431223
$650$	20.7115	0.812374
$651$	−8.18342	−0.320733
$652$	−28.9977	−1.13564
$653$	43.2762	1.69353	0.846764	−	0.531969i	$-0.178548\pi$
$653$	43.2762	1.69353	0.846764	+	0.531969i	$0.178548\pi$
$654$	−43.3256	−1.69417
$655$	−3.99159	−0.155964
$656$	−6.56093	−0.256161
$657$	0.0433053	0.00168950
$658$	−0.183417	−0.00715036
$659$	−16.9276	−0.659405	−0.329703	−	0.944085i	$-0.606948\pi$
$659$	−16.9276	−0.659405	−0.329703	+	0.944085i	$0.606948\pi$
$660$	46.5997	1.81389
$661$	15.6781	0.609807	0.304903	−	0.952383i	$-0.401376\pi$
$661$	15.6781	0.609807	0.304903	+	0.952383i	$0.401376\pi$
$662$	−2.60197	−0.101128
$663$	4.39803	0.170805
$664$	10.5236	0.408395
$665$	5.35682	0.207728
$666$	−0.872174	−0.0337961
$667$	−0.447480	−0.0173265
$668$	13.2390	0.512233
$669$	−21.2390	−0.821148
$670$	30.2944	1.17038
$671$	6.17009	0.238194
$672$	−5.97721	−0.230576
$673$	−50.6330	−1.95176	−0.975879	−	0.218312i	$-0.929945\pi$
$673$	−50.6330	−1.95176	−0.975879	+	0.218312i	$0.929945\pi$
$674$	45.5006	1.75262
$675$	−12.3135	−0.473947
$676$	9.84324	0.378586
$677$	38.1711	1.46704	0.733518	−	0.679670i	$-0.237877\pi$
$677$	38.1711	1.46704	0.733518	+	0.679670i	$0.237877\pi$
$678$	48.0060	1.84366
$679$	−3.92777	−0.150734
$680$	−1.58372	−0.0607328
$681$	38.2329	1.46509
$682$	−139.113	−5.32692
$683$	36.6309	1.40164	0.700821	−	0.713337i	$-0.252817\pi$
$683$	36.6309	1.40164	0.700821	+	0.713337i	$0.252817\pi$
$684$	−1.51357	−0.0578729
$685$	26.2085	1.00137
$686$	13.7877	0.526415
$687$	0.317387	0.0121091
$688$	7.00227	0.266959
$689$	17.7009	0.674349
$690$	−1.02893	−0.0391707
$691$	−17.4764	−0.664834	−0.332417	−	0.943133i	$-0.607864\pi$
$691$	−17.4764	−0.664834	−0.332417	+	0.943133i	$0.607864\pi$
$692$	−2.00000	−0.0760286
$693$	0.222847	0.00846526
$694$	44.2472	1.67960
$695$	2.51026	0.0952196
$696$	−6.92162	−0.262363
$697$	1.99159	0.0754368
$698$	−16.6042	−0.628480
$699$	30.5380	1.15505
$700$	2.92162	0.110427
$701$	−9.83096	−0.371310	−0.185655	−	0.982615i	$-0.559441\pi$
$701$	−9.83096	−0.371310	−0.185655	+	0.982615i	$0.559441\pi$
$702$	−46.5692	−1.75764
$703$	36.5503	1.37852
$704$	−75.9614	−2.86290
$705$	0.511304	0.0192568
$706$	−42.6996	−1.60702
$707$	3.12783	0.117634
$708$	−8.24515	−0.309872
$709$	35.9214	1.34906	0.674529	−	0.738248i	$-0.264347\pi$
$709$	35.9214	1.34906	0.674529	+	0.738248i	$0.264347\pi$
$710$	−50.9276	−1.91128
$711$	0.549208	0.0205969
$712$	6.92162	0.259399
$713$	1.76713	0.0661797
$714$	1.07838	0.0403573
$715$	41.0398	1.53480
$716$	5.15061	0.192487
$717$	38.2700	1.42922
$718$	45.2678	1.68938
$719$	−20.3773	−0.759946	−0.379973	−	0.924998i	$-0.624067\pi$
$719$	−20.3773	−0.759946	−0.379973	+	0.924998i	$0.624067\pi$
$720$	−0.265671	−0.00990097
$721$	−5.12783	−0.190970
$722$	69.0216	2.56872
$723$	−34.2595	−1.27413
$724$	−57.8720	−2.15080
$725$	−6.15676	−0.228656
$726$	−100.410	−3.72656
$727$	30.0722	1.11532	0.557659	−	0.830070i	$-0.311700\pi$
$727$	30.0722	1.11532	0.557659	+	0.830070i	$0.311700\pi$
$728$	2.89269	0.107210
$729$	27.6681	1.02474
$730$	1.95547	0.0723753
$731$	−2.12556	−0.0786166
$732$	−4.63090	−0.171163
$733$	1.44360	0.0533207	0.0266604	−	0.999645i	$-0.491513\pi$
$733$	1.44360	0.0533207	0.0266604	+	0.999645i	$0.491513\pi$
$734$	−11.5441	−0.426101
$735$	−18.9216	−0.697935
$736$	1.29072	0.0475767
$737$	−52.8141	−1.94543
$738$	−0.536921	−0.0197644
$739$	−16.1122	−0.592698	−0.296349	−	0.955080i	$-0.595769\pi$
$739$	−16.1122	−0.592698	−0.296349	+	0.955080i	$0.595769\pi$
$740$	−22.6576	−0.832908
$741$	49.6886	1.82536
$742$	4.34017	0.159333
$743$	37.1100	1.36143	0.680716	−	0.732547i	$-0.261669\pi$
$743$	37.1100	1.36143	0.680716	+	0.732547i	$0.261669\pi$
$744$	27.3340	1.00211
$745$	−3.13397	−0.114820
$746$	−43.5981	−1.59624
$747$	−0.535877	−0.0196067
$748$	10.5464	0.385614
$749$	−7.17727	−0.262252
$750$	−44.4040	−1.62140
$751$	15.6430	0.570821	0.285411	−	0.958405i	$-0.407870\pi$
$751$	15.6430	0.570821	0.285411	+	0.958405i	$0.407870\pi$
$752$	−0.381211	−0.0139013
$753$	−12.0599	−0.439489
$754$	−23.2846	−0.847974
$755$	−15.2074	−0.553455
$756$	−6.56916	−0.238918
$757$	−41.3728	−1.50372	−0.751860	−	0.659323i	$-0.770843\pi$
$757$	−41.3728	−1.50372	−0.751860	+	0.659323i	$0.770843\pi$
$758$	−57.2990	−2.08119
$759$	1.79380	0.0651107
$760$	−17.8927	−0.649036
$761$	−39.4947	−1.43168	−0.715840	−	0.698264i	$-0.753956\pi$
$761$	−39.4947	−1.43168	−0.715840	+	0.698264i	$0.753956\pi$
$762$	3.81658	0.138260
$763$	−5.38243	−0.194857
$764$	7.81763	0.282832
$765$	0.0806452	0.00291573
$766$	−23.7298	−0.857392
$767$	−7.26141	−0.262194
$768$	0.715418	0.0258154
$769$	−34.7031	−1.25143	−0.625713	−	0.780053i	$-0.715192\pi$
$769$	−34.7031	−1.25143	−0.625713	+	0.780053i	$0.715192\pi$
$770$	10.0628	0.362637
$771$	−5.57531	−0.200790
$772$	1.84324	0.0663398
$773$	3.34632	0.120359	0.0601793	−	0.998188i	$-0.480833\pi$
$773$	3.34632	0.120359	0.0601793	+	0.998188i	$0.480833\pi$
$774$	0.573039	0.0205975
$775$	24.3135	0.873367
$776$	13.1194	0.470960
$777$	4.03895	0.144896
$778$	−11.9421	−0.428147
$779$	22.5008	0.806175
$780$	−30.8020	−1.10289
$781$	88.7852	3.17698
$782$	−0.232866	−0.00832726
$783$	13.8432	0.494717
$784$	14.1073	0.503832
$785$	22.0084	0.785514
$786$	−9.07838	−0.323815
$787$	−32.0950	−1.14406	−0.572032	−	0.820231i	$-0.693845\pi$
$787$	−32.0950	−1.14406	−0.572032	+	0.820231i	$0.693845\pi$
$788$	−52.5318	−1.87137
$789$	35.1650	1.25191
$790$	24.7998	0.882336
$791$	5.96388	0.212051
$792$	−0.744348	−0.0264492
$793$	−4.07838	−0.144827
$794$	32.0228	1.13645
$795$	−12.0989	−0.429104
$796$	−58.0288	−2.05677
$797$	38.3979	1.36012	0.680061	−	0.733156i	$-0.261953\pi$
$797$	38.3979	1.36012	0.680061	+	0.733156i	$0.261953\pi$
$798$	12.1834	0.431288
$799$	0.115718	0.00409380
$800$	17.7587	0.627866
$801$	−0.352459	−0.0124535
$802$	−38.5380	−1.36082
$803$	−3.40910	−0.120304
$804$	39.6391	1.39796
$805$	−0.127826	−0.00450528
$806$	91.9526	3.23889
$807$	6.79153	0.239073
$808$	−10.4475	−0.367541
$809$	−22.0577	−0.775507	−0.387753	−	0.921763i	$-0.626749\pi$
$809$	−22.0577	−0.775507	−0.387753	+	0.921763i	$0.626749\pi$
$810$	30.9988	1.08919
$811$	−31.1100	−1.09242	−0.546209	−	0.837649i	$-0.683930\pi$
$811$	−31.1100	−1.09242	−0.546209	+	0.837649i	$0.683930\pi$
$812$	−3.28458	−0.115266
$813$	8.54250	0.299599
$814$	68.6596	2.40652
$815$	17.4557	0.611447
$816$	2.24128	0.0784604
$817$	−24.0144	−0.840157
$818$	−35.6742	−1.24732
$819$	−0.147300	−0.00514708
$820$	−13.9483	−0.487095
$821$	−35.4641	−1.23771	−0.618853	−	0.785507i	$-0.712402\pi$
$821$	−35.4641	−1.23771	−0.618853	+	0.785507i	$0.712402\pi$
$822$	59.6079	2.07907
$823$	−4.89884	−0.170763	−0.0853813	−	0.996348i	$-0.527211\pi$
$823$	−4.89884	−0.170763	−0.0853813	+	0.996348i	$0.527211\pi$
$824$	17.1278	0.596676
$825$	24.6803	0.859259
$826$	−1.78047	−0.0619503
$827$	13.9506	0.485108	0.242554	−	0.970138i	$-0.422015\pi$
$827$	13.9506	0.485108	0.242554	+	0.970138i	$0.422015\pi$
$828$	0.0361173	0.00125516
$829$	42.4863	1.47561	0.737804	−	0.675015i	$-0.235863\pi$
$829$	42.4863	1.47561	0.737804	+	0.675015i	$0.235863\pi$
$830$	−24.1978	−0.839918
$831$	−30.7070	−1.06521
$832$	50.2099	1.74072
$833$	−4.28231	−0.148373
$834$	5.70928	0.197696
$835$	−7.96946	−0.275795
$836$	119.152	4.12096
$837$	−54.6681	−1.88960
$838$	69.4824	2.40023
$839$	−11.9421	−0.412288	−0.206144	−	0.978522i	$-0.566092\pi$
$839$	−11.9421	−0.412288	−0.206144	+	0.978522i	$0.566092\pi$
$840$	−1.97721	−0.0682204
$841$	−22.0784	−0.761323
$842$	−1.36910	−0.0471824
$843$	−26.6681	−0.918497
$844$	14.6186	0.503193
$845$	−5.92532	−0.203837
$846$	−0.0311968	−0.00107257
$847$	−12.4741	−0.428617
$848$	9.02052	0.309766
$849$	−42.2700	−1.45070
$850$	−3.20394	−0.109894
$851$	−0.872174	−0.0298977
$852$	−66.6369	−2.28294
$853$	16.2800	0.557418	0.278709	−	0.960376i	$-0.410093\pi$
$853$	16.2800	0.557418	0.278709	+	0.960376i	$0.410093\pi$
$854$	−1.00000	−0.0342193
$855$	0.911122	0.0311597
$856$	23.9733	0.819392
$857$	26.2085	0.895264	0.447632	−	0.894218i	$-0.352267\pi$
$857$	26.2085	0.895264	0.447632	+	0.894218i	$0.352267\pi$
$858$	93.3400	3.18657
$859$	−0.0350725	−0.00119666	−0.000598329	−	1.00000i	$-0.500190\pi$
$859$	−0.0350725	−0.00119666	−0.000598329		1.00000i	$0.500190\pi$
$860$	14.8865	0.507627
$861$	2.48643	0.0847372
$862$	−36.0722	−1.22863
$863$	10.8515	0.369389	0.184694	−	0.982796i	$-0.440871\pi$
$863$	10.8515	0.369389	0.184694	+	0.982796i	$0.440871\pi$
$864$	−39.9299	−1.35844
$865$	1.20394	0.0409351
$866$	−44.4801	−1.51150
$867$	28.3773	0.963745
$868$	12.9711	0.440267
$869$	−43.2350	−1.46665
$870$	15.9155	0.539585
$871$	34.9097	1.18287
$872$	17.9783	0.608821
$873$	−0.668060	−0.0226104
$874$	−2.63090	−0.0889914
$875$	−5.51640	−0.186488
$876$	2.55866	0.0864492
$877$	0.665970	0.0224882	0.0112441	−	0.999937i	$-0.496421\pi$
$877$	0.665970	0.0224882	0.0112441	+	0.999937i	$0.496421\pi$
$878$	−5.09663	−0.172003
$879$	−41.3151	−1.39352
$880$	20.9143	0.705019
$881$	14.4101	0.485490	0.242745	−	0.970090i	$-0.421952\pi$
$881$	14.4101	0.485490	0.242745	+	0.970090i	$0.421952\pi$
$882$	1.15449	0.0388736
$883$	57.8937	1.94828	0.974140	−	0.225947i	$-0.0725475\pi$
$883$	57.8937	1.94828	0.974140	+	0.225947i	$0.0725475\pi$
$884$	−6.97107	−0.234462
$885$	4.96332	0.166840
$886$	36.0144	1.20993
$887$	−38.4040	−1.28948	−0.644740	−	0.764402i	$-0.723034\pi$
$887$	−38.4040	−1.28948	−0.644740	+	0.764402i	$0.723034\pi$
$888$	−13.4908	−0.452721
$889$	0.474142	0.0159022
$890$	−15.9155	−0.533488
$891$	−54.0421	−1.81048
$892$	33.6647	1.12718
$893$	1.30737	0.0437494
$894$	−7.12783	−0.238390
$895$	−3.10050	−0.103638
$896$	5.31739	0.177641
$897$	−1.18568	−0.0395889
$898$	−44.4173	−1.48223
$899$	−27.3340	−0.911641
$900$	0.496928	0.0165643
$901$	−2.73820	−0.0912228
$902$	42.2678	1.40736
$903$	−2.65368	−0.0883091
$904$	−19.9204	−0.662543
$905$	34.8371	1.15802
$906$	−34.5874	−1.14909
$907$	−42.6407	−1.41586	−0.707931	−	0.706281i	$-0.750371\pi$
$907$	−42.6407	−1.41586	−0.707931	+	0.706281i	$0.750371\pi$
$908$	−60.6007	−2.01111
$909$	0.532001	0.0176454
$910$	−6.65142	−0.220492
$911$	18.6042	0.616386	0.308193	−	0.951324i	$-0.400276\pi$
$911$	18.6042	0.616386	0.308193	+	0.951324i	$0.400276\pi$
$912$	25.3217	0.838487
$913$	42.1855	1.39614
$914$	51.9748	1.71917
$915$	2.78765	0.0921570
$916$	−0.503072	−0.0166220
$917$	−1.12783	−0.0372441
$918$	7.20394	0.237765
$919$	33.2306	1.09618	0.548088	−	0.836421i	$-0.315356\pi$
$919$	33.2306	1.09618	0.548088	+	0.836421i	$0.315356\pi$
$920$	0.426961	0.0140765
$921$	−33.3379	−1.09852
$922$	−57.5318	−1.89471
$923$	−58.6863	−1.93168
$924$	13.1668	0.433155
$925$	−12.0000	−0.394558
$926$	73.8720	2.42758
$927$	−0.872174	−0.0286460
$928$	−19.9649	−0.655381
$929$	13.0989	0.429761	0.214880	−	0.976640i	$-0.431064\pi$
$929$	13.0989	0.429761	0.214880	+	0.976640i	$0.431064\pi$
$930$	−62.8515	−2.06098
$931$	−48.3812	−1.58563
$932$	−48.4040	−1.58553
$933$	−14.2784	−0.467455
$934$	9.89496	0.323773
$935$	−6.34858	−0.207621
$936$	0.492008	0.0160818
$937$	−21.0144	−0.686510	−0.343255	−	0.939242i	$-0.611529\pi$
$937$	−21.0144	−0.686510	−0.343255	+	0.939242i	$0.611529\pi$
$938$	8.55971	0.279484
$939$	−3.58759	−0.117077
$940$	−0.810439	−0.0264336
$941$	−37.3523	−1.21765	−0.608825	−	0.793305i	$-0.708359\pi$
$941$	−37.3523	−1.21765	−0.608825	+	0.793305i	$0.708359\pi$
$942$	50.0554	1.63089
$943$	−0.536921	−0.0174846
$944$	−3.70048	−0.120440
$945$	3.95443	0.128638
$946$	−45.1110	−1.46669
$947$	−32.2729	−1.04873	−0.524363	−	0.851495i	$-0.675697\pi$
$947$	−32.2729	−1.04873	−0.524363	+	0.851495i	$0.675697\pi$
$948$	32.4496	1.05391
$949$	2.25338	0.0731480
$950$	−36.1978	−1.17441
$951$	−47.4473	−1.53858
$952$	−0.447480	−0.0145029
$953$	51.3400	1.66307	0.831533	−	0.555476i	$-0.187464\pi$
$953$	51.3400	1.66307	0.831533	+	0.555476i	$0.187464\pi$
$954$	0.738205	0.0239003
$955$	−4.70596	−0.152281
$956$	−60.6596	−1.96187
$957$	−27.7464	−0.896915
$958$	−1.41855	−0.0458313
$959$	7.40522	0.239127
$960$	−34.3195	−1.10766
$961$	76.9442	2.48207
$962$	−45.3835	−1.46322
$963$	−1.22076	−0.0393384
$964$	54.3028	1.74898
$965$	−1.10957	−0.0357185
$966$	−0.290725	−0.00935391
$967$	−0.746615	−0.0240095	−0.0120048	−	0.999928i	$-0.503821\pi$
$967$	−0.746615	−0.0240095	−0.0120048	+	0.999928i	$0.503821\pi$
$968$	41.6658	1.33919
$969$	−7.68649	−0.246926
$970$	−30.1666	−0.968591
$971$	58.1276	1.86541	0.932703	−	0.360647i	$-0.117444\pi$
$971$	58.1276	1.86541	0.932703	+	0.360647i	$0.117444\pi$
$972$	−2.20620	−0.0707640
$973$	0.709275	0.0227383
$974$	−0.0494483	−0.00158443
$975$	−16.3135	−0.522450
$976$	−2.07838	−0.0665273
$977$	6.86376	0.219591	0.109796	−	0.993954i	$-0.464980\pi$
$977$	6.86376	0.219591	0.109796	+	0.993954i	$0.464980\pi$
$978$	39.7009	1.26949
$979$	27.7464	0.886780
$980$	29.9916	0.958046
$981$	−0.915479	−0.0292290
$982$	−6.38962	−0.203901
$983$	−27.8660	−0.888788	−0.444394	−	0.895831i	$-0.646581\pi$
$983$	−27.8660	−0.888788	−0.444394	+	0.895831i	$0.646581\pi$
$984$	−8.30510	−0.264757
$985$	31.6225	1.00758
$986$	3.60197	0.114710
$987$	0.144469	0.00459851
$988$	−78.7585	−2.50564
$989$	0.573039	0.0182216
$990$	1.71154	0.0543964
$991$	−20.0312	−0.636312	−0.318156	−	0.948038i	$-0.603064\pi$
$991$	−20.0312	−0.636312	−0.318156	+	0.948038i	$0.603064\pi$
$992$	78.8431	2.50327
$993$	2.04945	0.0650373
$994$	−14.3896	−0.456411
$995$	34.9315	1.10740
$996$	−31.6619	−1.00325
$997$	0.500804	0.0158606	0.00793031	−	0.999969i	$-0.497476\pi$
$997$	0.500804	0.0158606	0.00793031	+	0.999969i	$0.497476\pi$
$998$	50.1026	1.58597
$999$	26.9816	0.853659

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.2.a.b.1.3	✓	3
3.2	odd	2		549.2.a.g.1.1		3
4.3	odd	2		976.2.a.f.1.3		3
5.2	odd	4		1525.2.b.b.1099.6		6
5.3	odd	4		1525.2.b.b.1099.1		6
5.4	even	2		1525.2.a.d.1.1		3
7.6	odd	2		2989.2.a.i.1.3		3
8.3	odd	2		3904.2.a.w.1.1		3
8.5	even	2		3904.2.a.r.1.3		3
11.10	odd	2		7381.2.a.f.1.1		3
12.11	even	2		8784.2.a.bn.1.2		3
61.60	even	2		3721.2.a.c.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.2.a.b.1.3	✓	3	1.1	even	1	trivial
549.2.a.g.1.1		3	3.2	odd	2
976.2.a.f.1.3		3	4.3	odd	2
1525.2.a.d.1.1		3	5.4	even	2
1525.2.b.b.1099.1		6	5.3	odd	4
1525.2.b.b.1099.6		6	5.2	odd	4
2989.2.a.i.1.3		3	7.6	odd	2
3721.2.a.c.1.1		3	61.60	even	2
3904.2.a.r.1.3		3	8.5	even	2
3904.2.a.w.1.1		3	8.3	odd	2
7381.2.a.f.1.1		3	11.10	odd	2
8784.2.a.bn.1.2		3	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	61.a (trivial)

\(f(q)\)	\(=\)	\(q+2.17009 q^{2} -1.70928 q^{3} +2.70928 q^{4} -1.63090 q^{5} -3.70928 q^{6} -0.460811 q^{7} +1.53919 q^{8} -0.0783777 q^{9} +O(q^{10})\)	\(q+2.17009 q^{2} -1.70928 q^{3} +2.70928 q^{4} -1.63090 q^{5} -3.70928 q^{6} -0.460811 q^{7} +1.53919 q^{8} -0.0783777 q^{9} -3.53919 q^{10} +6.17009 q^{11} -4.63090 q^{12} -4.07838 q^{13} -1.00000 q^{14} +2.78765 q^{15} -2.07838 q^{16} +0.630898 q^{17} -0.170086 q^{18} +7.12783 q^{19} -4.41855 q^{20} +0.787653 q^{21} +13.3896 q^{22} -0.170086 q^{23} -2.63090 q^{24} -2.34017 q^{25} -8.85043 q^{26} +5.26180 q^{27} -1.24846 q^{28} +2.63090 q^{29} +6.04945 q^{30} -10.3896 q^{31} -7.58864 q^{32} -10.5464 q^{33} +1.36910 q^{34} +0.751536 q^{35} -0.212347 q^{36} +5.12783 q^{37} +15.4680 q^{38} +6.97107 q^{39} -2.51026 q^{40} +3.15676 q^{41} +1.70928 q^{42} -3.36910 q^{43} +16.7165 q^{44} +0.127826 q^{45} -0.369102 q^{46} +0.183417 q^{47} +3.55252 q^{48} -6.78765 q^{49} -5.07838 q^{50} -1.07838 q^{51} -11.0494 q^{52} -4.34017 q^{53} +11.4186 q^{54} -10.0628 q^{55} -0.709275 q^{56} -12.1834 q^{57} +5.70928 q^{58} +1.78047 q^{59} +7.55252 q^{60} +1.00000 q^{61} -22.5464 q^{62} +0.0361173 q^{63} -12.3112 q^{64} +6.65142 q^{65} -22.8865 q^{66} -8.55971 q^{67} +1.70928 q^{68} +0.290725 q^{69} +1.63090 q^{70} +14.3896 q^{71} -0.120638 q^{72} -0.552520 q^{73} +11.1278 q^{74} +4.00000 q^{75} +19.3112 q^{76} -2.84324 q^{77} +15.1278 q^{78} -7.00719 q^{79} +3.38962 q^{80} -8.75872 q^{81} +6.85043 q^{82} +6.83710 q^{83} +2.13397 q^{84} -1.02893 q^{85} -7.31124 q^{86} -4.49693 q^{87} +9.49693 q^{88} +4.49693 q^{89} +0.277394 q^{90} +1.87936 q^{91} -0.460811 q^{92} +17.7587 q^{93} +0.398032 q^{94} -11.6248 q^{95} +12.9711 q^{96} +8.52359 q^{97} -14.7298 q^{98} -0.483597 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 2 q^{3} + q^{4} - q^{5} - 4 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + q^2 + 2 * q^3 + q^4 - q^5 - 4 * q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + q^{2} + 2 q^{3} + q^{4} - q^{5} - 4 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} - 9 q^{10} + 13 q^{11} - 10 q^{12} - 9 q^{13} - 3 q^{14} - 2 q^{15} - 3 q^{16} - 2 q^{17} + 5 q^{18} + q^{20} - 8 q^{21} + 11 q^{22} + 5 q^{23} - 4 q^{24} + 4 q^{25} + q^{26} + 8 q^{27} + 5 q^{28} + 4 q^{29} - 2 q^{31} - 3 q^{32} + 4 q^{33} + 8 q^{34} + 11 q^{35} - 11 q^{36} - 6 q^{37} + 14 q^{38} + 6 q^{39} + 9 q^{40} + 3 q^{41} - 2 q^{42} - 14 q^{43} + 9 q^{44} - 21 q^{45} - 5 q^{46} - 4 q^{47} + 10 q^{48} - 10 q^{49} - 12 q^{50} - 15 q^{52} - 2 q^{53} + 20 q^{54} - 13 q^{55} + 5 q^{56} - 32 q^{57} + 10 q^{58} + 29 q^{59} + 22 q^{60} + 3 q^{61} - 32 q^{62} - 19 q^{63} - 11 q^{64} - 17 q^{65} - 22 q^{66} + 9 q^{67} - 2 q^{68} + 8 q^{69} + q^{70} + 14 q^{71} - 13 q^{72} - q^{73} + 12 q^{74} + 12 q^{75} + 32 q^{76} - 15 q^{77} + 24 q^{78} + 13 q^{79} - 19 q^{80} - q^{81} - 7 q^{82} - 8 q^{83} + 20 q^{84} - 18 q^{85} + 4 q^{86} + 4 q^{87} + 11 q^{88} - 4 q^{89} + 7 q^{90} - 7 q^{91} - 3 q^{92} + 28 q^{93} + 20 q^{94} + 4 q^{95} + 24 q^{96} + 10 q^{97} - 4 q^{98} + 17 q^{99}+O(q^{100}) \) 3 * q + q^2 + 2 * q^3 + q^4 - q^5 - 4 * q^6 - 3 * q^7 + 3 * q^8 + 3 * q^9 - 9 * q^10 + 13 * q^11 - 10 * q^12 - 9 * q^13 - 3 * q^14 - 2 * q^15 - 3 * q^16 - 2 * q^17 + 5 * q^18 + q^20 - 8 * q^21 + 11 * q^22 + 5 * q^23 - 4 * q^24 + 4 * q^25 + q^26 + 8 * q^27 + 5 * q^28 + 4 * q^29 - 2 * q^31 - 3 * q^32 + 4 * q^33 + 8 * q^34 + 11 * q^35 - 11 * q^36 - 6 * q^37 + 14 * q^38 + 6 * q^39 + 9 * q^40 + 3 * q^41 - 2 * q^42 - 14 * q^43 + 9 * q^44 - 21 * q^45 - 5 * q^46 - 4 * q^47 + 10 * q^48 - 10 * q^49 - 12 * q^50 - 15 * q^52 - 2 * q^53 + 20 * q^54 - 13 * q^55 + 5 * q^56 - 32 * q^57 + 10 * q^58 + 29 * q^59 + 22 * q^60 + 3 * q^61 - 32 * q^62 - 19 * q^63 - 11 * q^64 - 17 * q^65 - 22 * q^66 + 9 * q^67 - 2 * q^68 + 8 * q^69 + q^70 + 14 * q^71 - 13 * q^72 - q^73 + 12 * q^74 + 12 * q^75 + 32 * q^76 - 15 * q^77 + 24 * q^78 + 13 * q^79 - 19 * q^80 - q^81 - 7 * q^82 - 8 * q^83 + 20 * q^84 - 18 * q^85 + 4 * q^86 + 4 * q^87 + 11 * q^88 - 4 * q^89 + 7 * q^90 - 7 * q^91 - 3 * q^92 + 28 * q^93 + 20 * q^94 + 4 * q^95 + 24 * q^96 + 10 * q^97 - 4 * q^98 + 17 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more