LMFDB - Embedded newform 7406.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7406,2,Mod(1,7406)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7406, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7406.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7406 = 2 \cdot 7 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7406.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:	$59.1372077370$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.334904$ of defining polynomial
Character	$\chi$	$=$	7406.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.74912 q^{3} +1.00000 q^{4} -2.80853 q^{5} -1.74912 q^{6} +1.00000 q^{7} +1.00000 q^{8} +0.0594122 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.74912 q^{3} +1.00000 q^{4} -2.80853 q^{5} -1.74912 q^{6} +1.00000 q^{7} +1.00000 q^{8} +0.0594122 q^{9} -2.80853 q^{10} -0.585786 q^{11} -1.74912 q^{12} +1.14343 q^{13} +1.00000 q^{14} +4.91245 q^{15} +1.00000 q^{16} +0.808530 q^{17} +0.0594122 q^{18} +1.49824 q^{19} -2.80853 q^{20} -1.74912 q^{21} -0.585786 q^{22} -1.74912 q^{24} +2.88784 q^{25} +1.14343 q^{26} +5.14343 q^{27} +1.00000 q^{28} -3.21803 q^{29} +4.91245 q^{30} -1.86794 q^{31} +1.00000 q^{32} +1.02461 q^{33} +0.808530 q^{34} -2.80853 q^{35} +0.0594122 q^{36} -2.36147 q^{37} +1.49824 q^{38} -2.00000 q^{39} -2.80853 q^{40} +0.550984 q^{41} -1.74912 q^{42} -12.6135 q^{43} -0.585786 q^{44} -0.166861 q^{45} +6.85499 q^{47} -1.74912 q^{48} +1.00000 q^{49} +2.88784 q^{50} -1.41421 q^{51} +1.14343 q^{52} +9.19932 q^{53} +5.14343 q^{54} +1.64520 q^{55} +1.00000 q^{56} -2.62059 q^{57} -3.21803 q^{58} +3.86794 q^{59} +4.91245 q^{60} +7.96362 q^{61} -1.86794 q^{62} +0.0594122 q^{63} +1.00000 q^{64} -3.21137 q^{65} +1.02461 q^{66} -12.3615 q^{67} +0.808530 q^{68} -2.80853 q^{70} +11.2739 q^{71} +0.0594122 q^{72} +6.20951 q^{73} -2.36147 q^{74} -5.05117 q^{75} +1.49824 q^{76} -0.585786 q^{77} -2.00000 q^{78} +13.2524 q^{79} -2.80853 q^{80} -9.17471 q^{81} +0.550984 q^{82} +2.85657 q^{83} -1.74912 q^{84} -2.27078 q^{85} -12.6135 q^{86} +5.62872 q^{87} -0.585786 q^{88} -12.0691 q^{89} -0.166861 q^{90} +1.14343 q^{91} +3.26725 q^{93} +6.85499 q^{94} -4.20784 q^{95} -1.74912 q^{96} +0.955493 q^{97} +1.00000 q^{98} -0.0348029 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} + 4 q^{8} - 4 q^{10} - 8 q^{11} - 4 q^{13} + 4 q^{14} + 4 q^{16} - 4 q^{17} - 8 q^{19} - 4 q^{20} - 8 q^{22} - 4 q^{26} + 12 q^{27} + 4 q^{28} - 4 q^{29} + 4 q^{32} - 4 q^{33} - 4 q^{34} - 4 q^{35} + 8 q^{37} - 8 q^{38} - 8 q^{39} - 4 q^{40} - 8 q^{43} - 8 q^{44} - 16 q^{45} + 4 q^{49} - 4 q^{52} + 12 q^{54} + 12 q^{55} + 4 q^{56} - 24 q^{57} - 4 q^{58} + 8 q^{59} - 12 q^{61} + 4 q^{64} - 16 q^{65} - 4 q^{66} - 32 q^{67} - 4 q^{68} - 4 q^{70} + 8 q^{71} + 4 q^{73} + 8 q^{74} + 4 q^{75} - 8 q^{76} - 8 q^{77} - 8 q^{78} - 16 q^{79} - 4 q^{80} - 8 q^{81} + 20 q^{83} - 12 q^{85} - 8 q^{86} - 20 q^{87} - 8 q^{88} - 28 q^{89} - 16 q^{90} - 4 q^{91} - 12 q^{93} + 8 q^{95} + 16 q^{97} + 4 q^{98} - 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 + 4 * q^8 - 4 * q^10 - 8 * q^11 - 4 * q^13 + 4 * q^14 + 4 * q^16 - 4 * q^17 - 8 * q^19 - 4 * q^20 - 8 * q^22 - 4 * q^26 + 12 * q^27 + 4 * q^28 - 4 * q^29 + 4 * q^32 - 4 * q^33 - 4 * q^34 - 4 * q^35 + 8 * q^37 - 8 * q^38 - 8 * q^39 - 4 * q^40 - 8 * q^43 - 8 * q^44 - 16 * q^45 + 4 * q^49 - 4 * q^52 + 12 * q^54 + 12 * q^55 + 4 * q^56 - 24 * q^57 - 4 * q^58 + 8 * q^59 - 12 * q^61 + 4 * q^64 - 16 * q^65 - 4 * q^66 - 32 * q^67 - 4 * q^68 - 4 * q^70 + 8 * q^71 + 4 * q^73 + 8 * q^74 + 4 * q^75 - 8 * q^76 - 8 * q^77 - 8 * q^78 - 16 * q^79 - 4 * q^80 - 8 * q^81 + 20 * q^83 - 12 * q^85 - 8 * q^86 - 20 * q^87 - 8 * q^88 - 28 * q^89 - 16 * q^90 - 4 * q^91 - 12 * q^93 + 8 * q^95 + 16 * q^97 + 4 * q^98 - 8 * 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
$3$	−1.74912	−1.00985	−0.504927	−	0.863162i	$-0.668480\pi$
$3$	−1.74912	−1.00985	−0.504927	+	0.863162i	$0.668480\pi$
$4$	1.00000	0.500000
$5$	−2.80853	−1.25601	−0.628006	−	0.778208i	$-0.716129\pi$
$5$	−2.80853	−1.25601	−0.628006	+	0.778208i	$0.716129\pi$
$6$	−1.74912	−0.714074
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0.0594122	0.0198041
$10$	−2.80853	−0.888135
$11$	−0.585786	−0.176621	−0.0883106	−	0.996093i	$-0.528147\pi$
$11$	−0.585786	−0.176621	−0.0883106	+	0.996093i	$0.528147\pi$
$12$	−1.74912	−0.504927
$13$	1.14343	0.317131	0.158566	−	0.987348i	$-0.449313\pi$
$13$	1.14343	0.317131	0.158566	+	0.987348i	$0.449313\pi$
$14$	1.00000	0.267261
$15$	4.91245	1.26839
$16$	1.00000	0.250000
$17$	0.808530	0.196097	0.0980486	−	0.995182i	$-0.468740\pi$
$17$	0.808530	0.196097	0.0980486	+	0.995182i	$0.468740\pi$
$18$	0.0594122	0.0140036
$19$	1.49824	0.343719	0.171859	−	0.985121i	$-0.445023\pi$
$19$	1.49824	0.343719	0.171859	+	0.985121i	$0.445023\pi$
$20$	−2.80853	−0.628006
$21$	−1.74912	−0.381689
$22$	−0.585786	−0.124890
$23$	0	0
$23$	0	0
$24$	−1.74912	−0.357037
$25$	2.88784	0.577568
$26$	1.14343	0.224246
$27$	5.14343	0.989854
$28$	1.00000	0.188982
$29$	−3.21803	−0.597573	−0.298787	−	0.954320i	$-0.596582\pi$
$29$	−3.21803	−0.597573	−0.298787	+	0.954320i	$0.596582\pi$
$30$	4.91245	0.896886
$31$	−1.86794	−0.335492	−0.167746	−	0.985830i	$-0.553649\pi$
$31$	−1.86794	−0.335492	−0.167746	+	0.985830i	$0.553649\pi$
$32$	1.00000	0.176777
$33$	1.02461	0.178362
$34$	0.808530	0.138662
$35$	−2.80853	−0.474728
$36$	0.0594122	0.00990203
$37$	−2.36147	−0.388222	−0.194111	−	0.980980i	$-0.562182\pi$
$37$	−2.36147	−0.388222	−0.194111	+	0.980980i	$0.562182\pi$
$38$	1.49824	0.243046
$39$	−2.00000	−0.320256
$40$	−2.80853	−0.444068
$41$	0.550984	0.0860492	0.0430246	−	0.999074i	$-0.486301\pi$
$41$	0.550984	0.0860492	0.0430246	+	0.999074i	$0.486301\pi$
$42$	−1.74912	−0.269895
$43$	−12.6135	−1.92355	−0.961773	−	0.273849i	$-0.911703\pi$
$43$	−12.6135	−1.92355	−0.961773	+	0.273849i	$0.911703\pi$
$44$	−0.585786	−0.0883106
$45$	−0.166861	−0.0248742
$46$	0	0
$47$	6.85499	0.999903	0.499951	−	0.866053i	$-0.333351\pi$
$47$	6.85499	0.999903	0.499951	+	0.866053i	$0.333351\pi$
$48$	−1.74912	−0.252463
$49$	1.00000	0.142857
$50$	2.88784	0.408402
$51$	−1.41421	−0.198030
$52$	1.14343	0.158566
$53$	9.19932	1.26362	0.631812	−	0.775122i	$-0.282312\pi$
$53$	9.19932	1.26362	0.631812	+	0.775122i	$0.282312\pi$
$54$	5.14343	0.699933
$55$	1.64520	0.221839
$56$	1.00000	0.133631
$57$	−2.62059	−0.347106
$58$	−3.21803	−0.422548
$59$	3.86794	0.503563	0.251782	−	0.967784i	$-0.418984\pi$
$59$	3.86794	0.503563	0.251782	+	0.967784i	$0.418984\pi$
$60$	4.91245	0.634194
$61$	7.96362	1.01964	0.509818	−	0.860282i	$-0.329713\pi$
$61$	7.96362	1.01964	0.509818	+	0.860282i	$0.329713\pi$
$62$	−1.86794	−0.237229
$63$	0.0594122	0.00748523
$64$	1.00000	0.125000
$65$	−3.21137	−0.398321
$66$	1.02461	0.126121
$67$	−12.3615	−1.51019	−0.755097	−	0.655613i	$-0.772410\pi$
$67$	−12.3615	−1.51019	−0.755097	+	0.655613i	$0.772410\pi$
$68$	0.808530	0.0980486
$69$	0	0
$70$	−2.80853	−0.335684
$71$	11.2739	1.33797	0.668984	−	0.743277i	$-0.266730\pi$
$71$	11.2739	1.33797	0.668984	+	0.743277i	$0.266730\pi$
$72$	0.0594122	0.00700179
$73$	6.20951	0.726768	0.363384	−	0.931639i	$-0.381621\pi$
$73$	6.20951	0.726768	0.363384	+	0.931639i	$0.381621\pi$
$74$	−2.36147	−0.274515
$75$	−5.05117	−0.583259
$76$	1.49824	0.171859
$77$	−0.585786	−0.0667566
$78$	−2.00000	−0.226455
$79$	13.2524	1.49102	0.745508	−	0.666497i	$-0.232207\pi$
$79$	13.2524	1.49102	0.745508	+	0.666497i	$0.232207\pi$
$80$	−2.80853	−0.314003
$81$	−9.17471	−1.01941
$82$	0.550984	0.0608460
$83$	2.85657	0.313549	0.156774	−	0.987634i	$-0.449890\pi$
$83$	2.85657	0.313549	0.156774	+	0.987634i	$0.449890\pi$
$84$	−1.74912	−0.190844
$85$	−2.27078	−0.246301
$86$	−12.6135	−1.36015
$87$	5.62872	0.603462
$88$	−0.585786	−0.0624450
$89$	−12.0691	−1.27932	−0.639662	−	0.768656i	$-0.720926\pi$
$89$	−12.0691	−1.27932	−0.639662	+	0.768656i	$0.720926\pi$
$90$	−0.166861	−0.0175887
$91$	1.14343	0.119864
$92$	0	0
$93$	3.26725	0.338798
$94$	6.85499	0.707038
$95$	−4.20784	−0.431715
$96$	−1.74912	−0.178519
$97$	0.955493	0.0970156	0.0485078	−	0.998823i	$-0.484553\pi$
$97$	0.955493	0.0970156	0.0485078	+	0.998823i	$0.484553\pi$
$98$	1.00000	0.101015
$99$	−0.0348029	−0.00349782
$100$	2.88784	0.288784
$101$	10.1305	1.00802	0.504010	−	0.863698i	$-0.331857\pi$
$101$	10.1305	1.00802	0.504010	+	0.863698i	$0.331857\pi$
$102$	−1.41421	−0.140028
$103$	1.44902	0.142776	0.0713879	−	0.997449i	$-0.477257\pi$
$103$	1.44902	0.142776	0.0713879	+	0.997449i	$0.477257\pi$
$104$	1.14343	0.112123
$105$	4.91245	0.479406
$106$	9.19932	0.893517
$107$	−9.15509	−0.885056	−0.442528	−	0.896755i	$-0.645918\pi$
$107$	−9.15509	−0.885056	−0.442528	+	0.896755i	$0.645918\pi$
$108$	5.14343	0.494927
$109$	−15.1899	−1.45493	−0.727464	−	0.686146i	$-0.759301\pi$
$109$	−15.1899	−1.45493	−0.727464	+	0.686146i	$0.759301\pi$
$110$	1.64520	0.156864
$111$	4.13048	0.392048
$112$	1.00000	0.0944911
$113$	−4.96873	−0.467419	−0.233709	−	0.972307i	$-0.575086\pi$
$113$	−4.96873	−0.467419	−0.233709	+	0.972307i	$0.575086\pi$
$114$	−2.62059	−0.245441
$115$	0	0
$116$	−3.21803	−0.298787
$117$	0.0679339	0.00628049
$118$	3.86794	0.356073
$119$	0.808530	0.0741178
$120$	4.91245	0.448443
$121$	−10.6569	−0.968805
$122$	7.96362	0.720992
$123$	−0.963735	−0.0868971
$124$	−1.86794	−0.167746
$125$	5.93207	0.530580
$126$	0.0594122	0.00529286
$127$	14.5573	1.29175	0.645874	−	0.763444i	$-0.276493\pi$
$127$	14.5573	1.29175	0.645874	+	0.763444i	$0.276493\pi$
$128$	1.00000	0.0883883
$129$	22.0625	1.94250
$130$	−3.21137	−0.281656
$131$	−16.1172	−1.40817	−0.704085	−	0.710116i	$-0.748643\pi$
$131$	−16.1172	−1.40817	−0.704085	+	0.710116i	$0.748643\pi$
$132$	1.02461	0.0891808
$133$	1.49824	0.129913
$134$	−12.3615	−1.06787
$135$	−14.4455	−1.24327
$136$	0.808530	0.0693309
$137$	−5.70108	−0.487076	−0.243538	−	0.969891i	$-0.578308\pi$
$137$	−5.70108	−0.487076	−0.243538	+	0.969891i	$0.578308\pi$
$138$	0	0
$139$	17.0230	1.44387	0.721937	−	0.691958i	$-0.243252\pi$
$139$	17.0230	1.44387	0.721937	+	0.691958i	$0.243252\pi$
$140$	−2.80853	−0.237364
$141$	−11.9902	−1.00976
$142$	11.2739	0.946086
$143$	−0.669808	−0.0560122
$144$	0.0594122	0.00495102
$145$	9.03794	0.750560
$146$	6.20951	0.513903
$147$	−1.74912	−0.144265
$148$	−2.36147	−0.194111
$149$	−2.22931	−0.182632	−0.0913162	−	0.995822i	$-0.529107\pi$
$149$	−2.22931	−0.182632	−0.0913162	+	0.995822i	$0.529107\pi$
$150$	−5.05117	−0.412426
$151$	−6.78157	−0.551877	−0.275938	−	0.961175i	$-0.588989\pi$
$151$	−6.78157	−0.551877	−0.275938	+	0.961175i	$0.588989\pi$
$152$	1.49824	0.121523
$153$	0.0480365	0.00388352
$154$	−0.585786	−0.0472040
$155$	5.24617	0.421383
$156$	−2.00000	−0.160128
$157$	−16.7406	−1.33604	−0.668022	−	0.744141i	$-0.732859\pi$
$157$	−16.7406	−1.33604	−0.668022	+	0.744141i	$0.732859\pi$
$158$	13.2524	1.05431
$159$	−16.0907	−1.27607
$160$	−2.80853	−0.222034
$161$	0	0
$162$	−9.17471	−0.720833
$163$	3.15509	0.247126	0.123563	−	0.992337i	$-0.460568\pi$
$163$	3.15509	0.247126	0.123563	+	0.992337i	$0.460568\pi$
$164$	0.550984	0.0430246
$165$	−2.87765	−0.224024
$166$	2.85657	0.221713
$167$	−4.77596	−0.369575	−0.184787	−	0.982779i	$-0.559160\pi$
$167$	−4.77596	−0.369575	−0.184787	+	0.982779i	$0.559160\pi$
$168$	−1.74912	−0.134947
$169$	−11.6926	−0.899428
$170$	−2.27078	−0.174161
$171$	0.0890134	0.00680703
$172$	−12.6135	−0.961773
$173$	−13.5045	−1.02673	−0.513364	−	0.858171i	$-0.671601\pi$
$173$	−13.5045	−1.02673	−0.513364	+	0.858171i	$0.671601\pi$
$174$	5.62872	0.426712
$175$	2.88784	0.218300
$176$	−0.585786	−0.0441553
$177$	−6.76549	−0.508525
$178$	−12.0691	−0.904618
$179$	−21.4155	−1.60067	−0.800334	−	0.599554i	$-0.795345\pi$
$179$	−21.4155	−1.60067	−0.800334	+	0.599554i	$0.795345\pi$
$180$	−0.166861	−0.0124371
$181$	−15.9425	−1.18500	−0.592500	−	0.805571i	$-0.701859\pi$
$181$	−15.9425	−1.18500	−0.592500	+	0.805571i	$0.701859\pi$
$182$	1.14343	0.0847569
$183$	−13.9293	−1.02968
$184$	0	0
$185$	6.63224	0.487612
$186$	3.26725	0.239566
$187$	−0.473626	−0.0346349
$188$	6.85499	0.499951
$189$	5.14343	0.374130
$190$	−4.20784	−0.305269
$191$	−11.2435	−0.813554	−0.406777	−	0.913527i	$-0.633347\pi$
$191$	−11.2435	−0.813554	−0.406777	+	0.913527i	$0.633347\pi$
$192$	−1.74912	−0.126232
$193$	−6.70960	−0.482968	−0.241484	−	0.970405i	$-0.577634\pi$
$193$	−6.70960	−0.482968	−0.241484	+	0.970405i	$0.577634\pi$
$194$	0.955493	0.0686004
$195$	5.61706	0.402246
$196$	1.00000	0.0714286
$197$	−7.23412	−0.515410	−0.257705	−	0.966224i	$-0.582966\pi$
$197$	−7.23412	−0.515410	−0.257705	+	0.966224i	$0.582966\pi$
$198$	−0.0348029	−0.00247333
$199$	−25.1386	−1.78203	−0.891014	−	0.453975i	$-0.850005\pi$
$199$	−25.1386	−1.78203	−0.891014	+	0.453975i	$0.850005\pi$
$200$	2.88784	0.204201
$201$	21.6217	1.52507
$202$	10.1305	0.712778
$203$	−3.21803	−0.225862
$204$	−1.41421	−0.0990148
$205$	−1.54745	−0.108079
$206$	1.44902	0.100958
$207$	0	0
$208$	1.14343	0.0792829
$209$	−0.877646	−0.0607080
$210$	4.91245	0.338991
$211$	−12.6041	−0.867702	−0.433851	−	0.900985i	$-0.642846\pi$
$211$	−12.6041	−0.867702	−0.433851	+	0.900985i	$0.642846\pi$
$212$	9.19932	0.631812
$213$	−19.7194	−1.35115
$214$	−9.15509	−0.625829
$215$	35.4255	2.41600
$216$	5.14343	0.349966
$217$	−1.86794	−0.126804
$218$	−15.1899	−1.02879
$219$	−10.8612	−0.733929
$220$	1.64520	0.110919
$221$	0.924500	0.0621886
$222$	4.13048	0.277220
$223$	4.91902	0.329402	0.164701	−	0.986344i	$-0.447334\pi$
$223$	4.91902	0.329402	0.164701	+	0.986344i	$0.447334\pi$
$224$	1.00000	0.0668153
$225$	0.171573	0.0114382
$226$	−4.96873	−0.330515
$227$	−18.6698	−1.23916	−0.619579	−	0.784934i	$-0.712697\pi$
$227$	−18.6698	−1.23916	−0.619579	+	0.784934i	$0.712697\pi$
$228$	−2.62059	−0.173553
$229$	21.8394	1.44319	0.721594	−	0.692316i	$-0.243410\pi$
$229$	21.8394	1.44319	0.721594	+	0.692316i	$0.243410\pi$
$230$	0	0
$231$	1.02461	0.0674143
$232$	−3.21803	−0.211274
$233$	2.95392	0.193517	0.0967587	−	0.995308i	$-0.469152\pi$
$233$	2.95392	0.193517	0.0967587	+	0.995308i	$0.469152\pi$
$234$	0.0679339	0.00444098
$235$	−19.2524	−1.25589
$236$	3.86794	0.251782
$237$	−23.1801	−1.50571
$238$	0.808530	0.0524092
$239$	4.95668	0.320621	0.160310	−	0.987067i	$-0.448751\pi$
$239$	4.95668	0.320621	0.160310	+	0.987067i	$0.448751\pi$
$240$	4.91245	0.317097
$241$	−24.4717	−1.57636	−0.788179	−	0.615446i	$-0.788976\pi$
$241$	−24.4717	−1.57636	−0.788179	+	0.615446i	$0.788976\pi$
$242$	−10.6569	−0.685049
$243$	0.617339	0.0396023
$244$	7.96362	0.509818
$245$	−2.80853	−0.179430
$246$	−0.963735	−0.0614455
$247$	1.71313	0.109004
$248$	−1.86794	−0.118614
$249$	−4.99647	−0.316638
$250$	5.93207	0.375177
$251$	3.83822	0.242267	0.121133	−	0.992636i	$-0.461347\pi$
$251$	3.83822	0.242267	0.121133	+	0.992636i	$0.461347\pi$
$252$	0.0594122	0.00374262
$253$	0	0
$254$	14.5573	0.913403
$255$	3.97186	0.248728
$256$	1.00000	0.0625000
$257$	−17.3270	−1.08083	−0.540415	−	0.841398i	$-0.681733\pi$
$257$	−17.3270	−1.08083	−0.540415	+	0.841398i	$0.681733\pi$
$258$	22.0625	1.37355
$259$	−2.36147	−0.146734
$260$	−3.21137	−0.199161
$261$	−0.191190	−0.0118344
$262$	−16.1172	−0.995727
$263$	0.180095	0.0111051	0.00555255	−	0.999985i	$-0.498233\pi$
$263$	0.180095	0.0111051	0.00555255	+	0.999985i	$0.498233\pi$
$264$	1.02461	0.0630603
$265$	−25.8366	−1.58713
$266$	1.49824	0.0918627
$267$	21.1103	1.29193
$268$	−12.3615	−0.755097
$269$	5.56394	0.339239	0.169620	−	0.985510i	$-0.445746\pi$
$269$	5.56394	0.339239	0.169620	+	0.985510i	$0.445746\pi$
$270$	−14.4455	−0.879124
$271$	22.1968	1.34836	0.674181	−	0.738566i	$-0.264497\pi$
$271$	22.1968	1.34836	0.674181	+	0.738566i	$0.264497\pi$
$272$	0.808530	0.0490243
$273$	−2.00000	−0.121046
$274$	−5.70108	−0.344415
$275$	−1.69166	−0.102011
$276$	0	0
$277$	12.4825	0.750002	0.375001	−	0.927024i	$-0.377642\pi$
$277$	12.4825	0.750002	0.375001	+	0.927024i	$0.377642\pi$
$278$	17.0230	1.02097
$279$	−0.110979	−0.00664411
$280$	−2.80853	−0.167842
$281$	0.893601	0.0533078	0.0266539	−	0.999645i	$-0.491515\pi$
$281$	0.893601	0.0533078	0.0266539	+	0.999645i	$0.491515\pi$
$282$	−11.9902	−0.714005
$283$	−17.4420	−1.03682	−0.518408	−	0.855133i	$-0.673475\pi$
$283$	−17.4420	−1.03682	−0.518408	+	0.855133i	$0.673475\pi$
$284$	11.2739	0.668984
$285$	7.36000	0.435969
$286$	−0.669808	−0.0396066
$287$	0.550984	0.0325235
$288$	0.0594122	0.00350090
$289$	−16.3463	−0.961546
$290$	9.03794	0.530726
$291$	−1.67127	−0.0979716
$292$	6.20951	0.363384
$293$	30.4319	1.77785	0.888924	−	0.458055i	$-0.151454\pi$
$293$	30.4319	1.77785	0.888924	+	0.458055i	$0.151454\pi$
$294$	−1.74912	−0.102011
$295$	−10.8632	−0.632482
$296$	−2.36147	−0.137257
$297$	−3.01295	−0.174829
$298$	−2.22931	−0.129141
$299$	0	0
$300$	−5.05117	−0.291629
$301$	−12.6135	−0.727032
$302$	−6.78157	−0.390236
$303$	−17.7194	−1.01795
$304$	1.49824	0.0859297
$305$	−22.3661	−1.28068
$306$	0.0480365	0.00274607
$307$	0.0850711	0.00485526	0.00242763	−	0.999997i	$-0.499227\pi$
$307$	0.0850711	0.00485526	0.00242763	+	0.999997i	$0.499227\pi$
$308$	−0.585786	−0.0333783
$309$	−2.53450	−0.144183
$310$	5.24617	0.297962
$311$	31.9610	1.81234	0.906171	−	0.422912i	$-0.138992\pi$
$311$	31.9610	1.81234	0.906171	+	0.422912i	$0.138992\pi$
$312$	−2.00000	−0.113228
$313$	−24.7889	−1.40115	−0.700575	−	0.713578i	$-0.747073\pi$
$313$	−24.7889	−1.40115	−0.700575	+	0.713578i	$0.747073\pi$
$314$	−16.7406	−0.944726
$315$	−0.166861	−0.00940155
$316$	13.2524	0.745508
$317$	−10.1216	−0.568485	−0.284242	−	0.958752i	$-0.591742\pi$
$317$	−10.1216	−0.568485	−0.284242	+	0.958752i	$0.591742\pi$
$318$	−16.0907	−0.902321
$319$	1.88508	0.105544
$320$	−2.80853	−0.157002
$321$	16.0133	0.893777
$322$	0	0
$323$	1.21137	0.0674023
$324$	−9.17471	−0.509706
$325$	3.30205	0.183165
$326$	3.15509	0.174744
$327$	26.5689	1.46926
$328$	0.550984	0.0304230
$329$	6.85499	0.377928
$330$	−2.87765	−0.158409
$331$	−27.5215	−1.51272	−0.756360	−	0.654155i	$-0.773024\pi$
$331$	−27.5215	−1.51272	−0.756360	+	0.654155i	$0.773024\pi$
$332$	2.85657	0.156774
$333$	−0.140300	−0.00768838
$334$	−4.77596	−0.261329
$335$	34.7175	1.89682
$336$	−1.74912	−0.0954222
$337$	24.8397	1.35310	0.676552	−	0.736395i	$-0.263473\pi$
$337$	24.8397	1.35310	0.676552	+	0.736395i	$0.263473\pi$
$338$	−11.6926	−0.635991
$339$	8.69089	0.472024
$340$	−2.27078	−0.123150
$341$	1.09422	0.0592551
$342$	0.0890134	0.00481330
$343$	1.00000	0.0539949
$344$	−12.6135	−0.680076
$345$	0	0
$346$	−13.5045	−0.726007
$347$	19.5045	1.04706	0.523528	−	0.852008i	$-0.324615\pi$
$347$	19.5045	1.04706	0.523528	+	0.852008i	$0.324615\pi$
$348$	5.62872	0.301731
$349$	−23.7725	−1.27251	−0.636257	−	0.771477i	$-0.719518\pi$
$349$	−23.7725	−1.27251	−0.636257	+	0.771477i	$0.719518\pi$
$350$	2.88784	0.154362
$351$	5.88118	0.313914
$352$	−0.585786	−0.0312225
$353$	18.7511	0.998019	0.499009	−	0.866597i	$-0.333697\pi$
$353$	18.7511	0.998019	0.499009	+	0.866597i	$0.333697\pi$
$354$	−6.76549	−0.359581
$355$	−31.6631	−1.68050
$356$	−12.0691	−0.639662
$357$	−1.41421	−0.0748481
$358$	−21.4155	−1.13184
$359$	−32.8789	−1.73528	−0.867642	−	0.497190i	$-0.834365\pi$
$359$	−32.8789	−1.73528	−0.867642	+	0.497190i	$0.834365\pi$
$360$	−0.166861	−0.00879434
$361$	−16.7553	−0.881857
$362$	−15.9425	−0.837921
$363$	18.6401	0.978351
$364$	1.14343	0.0599322
$365$	−17.4396	−0.912830
$366$	−13.9293	−0.728096
$367$	−9.11306	−0.475698	−0.237849	−	0.971302i	$-0.576442\pi$
$367$	−9.11306	−0.475698	−0.237849	+	0.971302i	$0.576442\pi$
$368$	0	0
$369$	0.0327351	0.00170412
$370$	6.63224	0.344794
$371$	9.19932	0.477605
$372$	3.26725	0.169399
$373$	3.35794	0.173867	0.0869336	−	0.996214i	$-0.472293\pi$
$373$	3.35794	0.173867	0.0869336	+	0.996214i	$0.472293\pi$
$374$	−0.473626	−0.0244906
$375$	−10.3759	−0.535808
$376$	6.85499	0.353519
$377$	−3.67961	−0.189509
$378$	5.14343	0.264550
$379$	−9.32756	−0.479125	−0.239562	−	0.970881i	$-0.577004\pi$
$379$	−9.32756	−0.479125	−0.239562	+	0.970881i	$0.577004\pi$
$380$	−4.20784	−0.215858
$381$	−25.4623	−1.30448
$382$	−11.2435	−0.575270
$383$	−7.00757	−0.358070	−0.179035	−	0.983843i	$-0.557298\pi$
$383$	−7.00757	−0.358070	−0.179035	+	0.983843i	$0.557298\pi$
$384$	−1.74912	−0.0892593
$385$	1.64520	0.0838471
$386$	−6.70960	−0.341510
$387$	−0.749397	−0.0380940
$388$	0.955493	0.0485078
$389$	−26.6483	−1.35112	−0.675562	−	0.737303i	$-0.736099\pi$
$389$	−26.6483	−1.35112	−0.675562	+	0.737303i	$0.736099\pi$
$390$	5.61706	0.284431
$391$	0	0
$392$	1.00000	0.0505076
$393$	28.1910	1.42205
$394$	−7.23412	−0.364450
$395$	−37.2199	−1.87273
$396$	−0.0348029	−0.00174891
$397$	−28.5854	−1.43466	−0.717330	−	0.696734i	$-0.754636\pi$
$397$	−28.5854	−1.43466	−0.717330	+	0.696734i	$0.754636\pi$
$398$	−25.1386	−1.26008
$399$	−2.62059	−0.131194
$400$	2.88784	0.144392
$401$	21.2462	1.06098	0.530492	−	0.847690i	$-0.322007\pi$
$401$	21.2462	1.06098	0.530492	+	0.847690i	$0.322007\pi$
$402$	21.6217	1.07839
$403$	−2.13587	−0.106395
$404$	10.1305	0.504010
$405$	25.7674	1.28039
$406$	−3.21803	−0.159708
$407$	1.38331	0.0685683
$408$	−1.41421	−0.0700140
$409$	14.8893	0.736229	0.368114	−	0.929781i	$-0.380004\pi$
$409$	14.8893	0.736229	0.368114	+	0.929781i	$0.380004\pi$
$410$	−1.54745	−0.0764233
$411$	9.97186	0.491876
$412$	1.44902	0.0713879
$413$	3.86794	0.190329
$414$	0	0
$415$	−8.02275	−0.393821
$416$	1.14343	0.0560615
$417$	−29.7753	−1.45810
$418$	−0.877646	−0.0429271
$419$	6.64334	0.324548	0.162274	−	0.986746i	$-0.448117\pi$
$419$	6.64334	0.324548	0.162274	+	0.986746i	$0.448117\pi$
$420$	4.91245	0.239703
$421$	−14.5523	−0.709234	−0.354617	−	0.935012i	$-0.615389\pi$
$421$	−14.5523	−0.709234	−0.354617	+	0.935012i	$0.615389\pi$
$422$	−12.6041	−0.613558
$423$	0.407270	0.0198021
$424$	9.19932	0.446758
$425$	2.33490	0.113259
$426$	−19.7194	−0.955408
$427$	7.96362	0.385387
$428$	−9.15509	−0.442528
$429$	1.17157	0.0565641
$430$	35.4255	1.70837
$431$	−20.5119	−0.988025	−0.494013	−	0.869455i	$-0.664470\pi$
$431$	−20.5119	−0.988025	−0.494013	+	0.869455i	$0.664470\pi$
$432$	5.14343	0.247464
$433$	2.48410	0.119378	0.0596891	−	0.998217i	$-0.480989\pi$
$433$	2.48410	0.119378	0.0596891	+	0.998217i	$0.480989\pi$
$434$	−1.86794	−0.0896641
$435$	−15.8084	−0.757955
$436$	−15.1899	−0.727464
$437$	0	0
$438$	−10.8612	−0.518966
$439$	11.0941	0.529494	0.264747	−	0.964318i	$-0.414712\pi$
$439$	11.0941	0.529494	0.264747	+	0.964318i	$0.414712\pi$
$440$	1.64520	0.0784318
$441$	0.0594122	0.00282915
$442$	0.924500	0.0439740
$443$	23.0723	1.09620	0.548100	−	0.836413i	$-0.315351\pi$
$443$	23.0723	1.09620	0.548100	+	0.836413i	$0.315351\pi$
$444$	4.13048	0.196024
$445$	33.8965	1.60685
$446$	4.91902	0.232922
$447$	3.89933	0.184432
$448$	1.00000	0.0472456
$449$	18.4545	0.870922	0.435461	−	0.900208i	$-0.356585\pi$
$449$	18.4545	0.870922	0.435461	+	0.900208i	$0.356585\pi$
$450$	0.171573	0.00808802
$451$	−0.322759	−0.0151981
$452$	−4.96873	−0.233709
$453$	11.8618	0.557314
$454$	−18.6698	−0.876218
$455$	−3.21137	−0.150551
$456$	−2.62059	−0.122720
$457$	8.95851	0.419061	0.209531	−	0.977802i	$-0.432806\pi$
$457$	8.95851	0.419061	0.209531	+	0.977802i	$0.432806\pi$
$458$	21.8394	1.02049
$459$	4.15862	0.194108
$460$	0	0
$461$	35.4388	1.65055	0.825275	−	0.564731i	$-0.191020\pi$
$461$	35.4388	1.65055	0.825275	+	0.564731i	$0.191020\pi$
$462$	1.02461	0.0476691
$463$	−8.12509	−0.377605	−0.188803	−	0.982015i	$-0.560461\pi$
$463$	−8.12509	−0.377605	−0.188803	+	0.982015i	$0.560461\pi$
$464$	−3.21803	−0.149393
$465$	−9.17617	−0.425535
$466$	2.95392	0.136837
$467$	−5.33019	−0.246652	−0.123326	−	0.992366i	$-0.539356\pi$
$467$	−5.33019	−0.246652	−0.123326	+	0.992366i	$0.539356\pi$
$468$	0.0679339	0.00314025
$469$	−12.3615	−0.570799
$470$	−19.2524	−0.888049
$471$	29.2813	1.34921
$472$	3.86794	0.178036
$473$	7.38883	0.339739
$474$	−23.1801	−1.06470
$475$	4.32666	0.198521
$476$	0.808530	0.0370589
$477$	0.546552	0.0250249
$478$	4.95668	0.226713
$479$	−5.99058	−0.273716	−0.136858	−	0.990591i	$-0.543700\pi$
$479$	−5.99058	−0.273716	−0.136858	+	0.990591i	$0.543700\pi$
$480$	4.91245	0.224222
$481$	−2.70018	−0.123118
$482$	−24.4717	−1.11465
$483$	0	0
$484$	−10.6569	−0.484402
$485$	−2.68353	−0.121853
$486$	0.617339	0.0280031
$487$	−39.3503	−1.78313	−0.891567	−	0.452888i	$-0.850394\pi$
$487$	−39.3503	−1.78313	−0.891567	+	0.452888i	$0.850394\pi$
$488$	7.96362	0.360496
$489$	−5.51862	−0.249561
$490$	−2.80853	−0.126876
$491$	−2.21726	−0.100064	−0.0500318	−	0.998748i	$-0.515932\pi$
$491$	−2.21726	−0.100064	−0.0500318	+	0.998748i	$0.515932\pi$
$492$	−0.963735	−0.0434485
$493$	−2.60187	−0.117183
$494$	1.71313	0.0770775
$495$	0.0977449	0.00439330
$496$	−1.86794	−0.0838731
$497$	11.2739	0.505704
$498$	−4.99647	−0.223897
$499$	−6.81510	−0.305086	−0.152543	−	0.988297i	$-0.548746\pi$
$499$	−6.81510	−0.305086	−0.152543	+	0.988297i	$0.548746\pi$
$500$	5.93207	0.265290
$501$	8.35371	0.373216
$502$	3.83822	0.171308
$503$	2.21726	0.0988628	0.0494314	−	0.998778i	$-0.484259\pi$
$503$	2.21726	0.0988628	0.0494314	+	0.998778i	$0.484259\pi$
$504$	0.0594122	0.00264643
$505$	−28.4518	−1.26609
$506$	0	0
$507$	20.4517	0.908290
$508$	14.5573	0.645874
$509$	38.7557	1.71781	0.858907	−	0.512131i	$-0.171144\pi$
$509$	38.7557	1.71781	0.858907	+	0.512131i	$0.171144\pi$
$510$	3.97186	0.175877
$511$	6.20951	0.274693
$512$	1.00000	0.0441942
$513$	7.70607	0.340231
$514$	−17.3270	−0.764263
$515$	−4.06961	−0.179328
$516$	22.0625	0.971249
$517$	−4.01556	−0.176604
$518$	−2.36147	−0.103757
$519$	23.6210	1.03685
$520$	−3.21137	−0.140828
$521$	−8.79428	−0.385284	−0.192642	−	0.981269i	$-0.561706\pi$
$521$	−8.79428	−0.385284	−0.192642	+	0.981269i	$0.561706\pi$
$522$	−0.191190	−0.00836817
$523$	−34.0181	−1.48751	−0.743753	−	0.668455i	$-0.766956\pi$
$523$	−34.0181	−1.48751	−0.743753	+	0.668455i	$0.766956\pi$
$524$	−16.1172	−0.704085
$525$	−5.05117	−0.220451
$526$	0.180095	0.00785250
$527$	−1.51029	−0.0657891
$528$	1.02461	0.0445904
$529$	0	0
$530$	−25.8366	−1.12227
$531$	0.229803	0.00997260
$532$	1.49824	0.0649567
$533$	0.630013	0.0272889
$534$	21.1103	0.913532
$535$	25.7123	1.11164
$536$	−12.3615	−0.533934
$537$	37.4582	1.61644
$538$	5.56394	0.239878
$539$	−0.585786	−0.0252316
$540$	−14.4455	−0.621635
$541$	−20.6265	−0.886801	−0.443400	−	0.896324i	$-0.646228\pi$
$541$	−20.6265	−0.886801	−0.443400	+	0.896324i	$0.646228\pi$
$542$	22.1968	0.953436
$543$	27.8854	1.19668
$544$	0.808530	0.0346654
$545$	42.6613	1.82741
$546$	−2.00000	−0.0855921
$547$	34.0202	1.45460	0.727298	−	0.686321i	$-0.240776\pi$
$547$	34.0202	1.45460	0.727298	+	0.686321i	$0.240776\pi$
$548$	−5.70108	−0.243538
$549$	0.473136	0.0201930
$550$	−1.69166	−0.0721325
$551$	−4.82137	−0.205397
$552$	0	0
$553$	13.2524	0.563551
$554$	12.4825	0.530332
$555$	−11.6006	−0.492417
$556$	17.0230	0.721937
$557$	−45.6626	−1.93479	−0.967393	−	0.253280i	$-0.918491\pi$
$557$	−45.6626	−1.93479	−0.967393	+	0.253280i	$0.918491\pi$
$558$	−0.110979	−0.00469810
$559$	−14.4227	−0.610017
$560$	−2.80853	−0.118682
$561$	0.828427	0.0349762
$562$	0.893601	0.0376943
$563$	−20.6041	−0.868360	−0.434180	−	0.900826i	$-0.642962\pi$
$563$	−20.6041	−0.868360	−0.434180	+	0.900826i	$0.642962\pi$
$564$	−11.9902	−0.504878
$565$	13.9548	0.587084
$566$	−17.4420	−0.733140
$567$	−9.17471	−0.385301
$568$	11.2739	0.473043
$569$	39.6809	1.66351	0.831755	−	0.555144i	$-0.187337\pi$
$569$	39.6809	1.66351	0.831755	+	0.555144i	$0.187337\pi$
$570$	7.36000	0.308277
$571$	24.1086	1.00891	0.504457	−	0.863437i	$-0.331693\pi$
$571$	24.1086	1.00891	0.504457	+	0.863437i	$0.331693\pi$
$572$	−0.669808	−0.0280061
$573$	19.6663	0.821571
$574$	0.550984	0.0229976
$575$	0	0
$576$	0.0594122	0.00247551
$577$	−14.2284	−0.592334	−0.296167	−	0.955136i	$-0.595709\pi$
$577$	−14.2284	−0.592334	−0.296167	+	0.955136i	$0.595709\pi$
$578$	−16.3463	−0.679916
$579$	11.7359	0.487727
$580$	9.03794	0.375280
$581$	2.85657	0.118510
$582$	−1.67127	−0.0692764
$583$	−5.38883	−0.223183
$584$	6.20951	0.256951
$585$	−0.190794	−0.00788838
$586$	30.4319	1.25713
$587$	−33.5968	−1.38669	−0.693343	−	0.720607i	$-0.743863\pi$
$587$	−33.5968	−1.38669	−0.693343	+	0.720607i	$0.743863\pi$
$588$	−1.74912	−0.0721324
$589$	−2.79862	−0.115315
$590$	−10.8632	−0.447232
$591$	12.6533	0.520488
$592$	−2.36147	−0.0970556
$593$	−14.9839	−0.615315	−0.307657	−	0.951497i	$-0.599545\pi$
$593$	−14.9839	−0.615315	−0.307657	+	0.951497i	$0.599545\pi$
$594$	−3.01295	−0.123623
$595$	−2.27078	−0.0930929
$596$	−2.22931	−0.0913162
$597$	43.9704	1.79959
$598$	0	0
$599$	5.72019	0.233721	0.116860	−	0.993148i	$-0.462717\pi$
$599$	5.72019	0.233721	0.116860	+	0.993148i	$0.462717\pi$
$600$	−5.05117	−0.206213
$601$	19.5162	0.796081	0.398040	−	0.917368i	$-0.369690\pi$
$601$	19.5162	0.796081	0.398040	+	0.917368i	$0.369690\pi$
$602$	−12.6135	−0.514089
$603$	−0.734422	−0.0299080
$604$	−6.78157	−0.275938
$605$	29.9301	1.21683
$606$	−17.7194	−0.719801
$607$	−36.9940	−1.50154	−0.750769	−	0.660564i	$-0.770317\pi$
$607$	−36.9940	−1.50154	−0.750769	+	0.660564i	$0.770317\pi$
$608$	1.49824	0.0607615
$609$	5.62872	0.228087
$610$	−22.3661	−0.905575
$611$	7.83822	0.317101
$612$	0.0480365	0.00194176
$613$	26.3411	1.06391	0.531953	−	0.846774i	$-0.321458\pi$
$613$	26.3411	1.06391	0.531953	+	0.846774i	$0.321458\pi$
$614$	0.0850711	0.00343319
$615$	2.70668	0.109144
$616$	−0.585786	−0.0236020
$617$	−25.0305	−1.00769	−0.503845	−	0.863794i	$-0.668082\pi$
$617$	−25.0305	−1.00769	−0.503845	+	0.863794i	$0.668082\pi$
$618$	−2.53450	−0.101953
$619$	0.758689	0.0304943	0.0152471	−	0.999884i	$-0.495146\pi$
$619$	0.758689	0.0304943	0.0152471	+	0.999884i	$0.495146\pi$
$620$	5.24617	0.210691
$621$	0	0
$622$	31.9610	1.28152
$623$	−12.0691	−0.483539
$624$	−2.00000	−0.0800641
$625$	−31.0996	−1.24398
$626$	−24.7889	−0.990763
$627$	1.53511	0.0613062
$628$	−16.7406	−0.668022
$629$	−1.90931	−0.0761294
$630$	−0.166861	−0.00664790
$631$	7.53931	0.300135	0.150068	−	0.988676i	$-0.452051\pi$
$631$	7.53931	0.300135	0.150068	+	0.988676i	$0.452051\pi$
$632$	13.2524	0.527154
$633$	22.0461	0.876252
$634$	−10.1216	−0.401979
$635$	−40.8845	−1.62245
$636$	−16.0907	−0.638037
$637$	1.14343	0.0453045
$638$	1.88508	0.0746310
$639$	0.669808	0.0264972
$640$	−2.80853	−0.111017
$641$	37.9589	1.49929	0.749643	−	0.661843i	$-0.230225\pi$
$641$	37.9589	1.49929	0.749643	+	0.661843i	$0.230225\pi$
$642$	16.0133	0.631996
$643$	−7.52861	−0.296899	−0.148450	−	0.988920i	$-0.547428\pi$
$643$	−7.52861	−0.296899	−0.148450	+	0.988920i	$0.547428\pi$
$644$	0	0
$645$	−61.9633	−2.43980
$646$	1.21137	0.0476606
$647$	−16.9550	−0.666568	−0.333284	−	0.942826i	$-0.608157\pi$
$647$	−16.9550	−0.666568	−0.333284	+	0.942826i	$0.608157\pi$
$648$	−9.17471	−0.360417
$649$	−2.26579	−0.0889400
$650$	3.30205	0.129517
$651$	3.26725	0.128054
$652$	3.15509	0.123563
$653$	−50.6988	−1.98400	−0.991999	−	0.126244i	$-0.959708\pi$
$653$	−50.6988	−1.98400	−0.991999	+	0.126244i	$0.959708\pi$
$654$	26.5689	1.03893
$655$	45.2658	1.76868
$656$	0.550984	0.0215123
$657$	0.368921	0.0143930
$658$	6.85499	0.267235
$659$	36.5612	1.42422	0.712110	−	0.702068i	$-0.247740\pi$
$659$	36.5612	1.42422	0.712110	+	0.702068i	$0.247740\pi$
$660$	−2.87765	−0.112012
$661$	−35.3626	−1.37545	−0.687723	−	0.725973i	$-0.741390\pi$
$661$	−35.3626	−1.37545	−0.687723	+	0.725973i	$0.741390\pi$
$662$	−27.5215	−1.06966
$663$	−1.61706	−0.0628014
$664$	2.85657	0.110856
$665$	−4.20784	−0.163173
$666$	−0.140300	−0.00543651
$667$	0	0
$668$	−4.77596	−0.184787
$669$	−8.60394	−0.332648
$670$	34.7175	1.34126
$671$	−4.66498	−0.180090
$672$	−1.74912	−0.0674737
$673$	31.6494	1.21999	0.609997	−	0.792403i	$-0.291170\pi$
$673$	31.6494	1.21999	0.609997	+	0.792403i	$0.291170\pi$
$674$	24.8397	0.956789
$675$	14.8534	0.571708
$676$	−11.6926	−0.449714
$677$	−10.4395	−0.401222	−0.200611	−	0.979671i	$-0.564293\pi$
$677$	−10.4395	−0.401222	−0.200611	+	0.979671i	$0.564293\pi$
$678$	8.69089	0.333772
$679$	0.955493	0.0366685
$680$	−2.27078	−0.0870804
$681$	32.6557	1.25137
$682$	1.09422	0.0418997
$683$	−11.0779	−0.423883	−0.211941	−	0.977282i	$-0.567979\pi$
$683$	−11.0779	−0.423883	−0.211941	+	0.977282i	$0.567979\pi$
$684$	0.0890134	0.00340351
$685$	16.0117	0.611774
$686$	1.00000	0.0381802
$687$	−38.1997	−1.45741
$688$	−12.6135	−0.480886
$689$	10.5188	0.400735
$690$	0	0
$691$	−30.3775	−1.15561	−0.577807	−	0.816174i	$-0.696091\pi$
$691$	−30.3775	−1.15561	−0.577807	+	0.816174i	$0.696091\pi$
$692$	−13.5045	−0.513364
$693$	−0.0348029	−0.00132205
$694$	19.5045	0.740381
$695$	−47.8097	−1.81352
$696$	5.62872	0.213356
$697$	0.445487	0.0168740
$698$	−23.7725	−0.899803
$699$	−5.16674	−0.195424
$700$	2.88784	0.109150
$701$	8.95224	0.338122	0.169061	−	0.985606i	$-0.445927\pi$
$701$	8.95224	0.338122	0.169061	+	0.985606i	$0.445927\pi$
$702$	5.88118	0.221971
$703$	−3.53803	−0.133439
$704$	−0.585786	−0.0220777
$705$	33.6748	1.26827
$706$	18.7511	0.705706
$707$	10.1305	0.380996
$708$	−6.76549	−0.254263
$709$	−4.77478	−0.179320	−0.0896602	−	0.995972i	$-0.528578\pi$
$709$	−4.77478	−0.179320	−0.0896602	+	0.995972i	$0.528578\pi$
$710$	−31.6631	−1.18830
$711$	0.787356	0.0295282
$712$	−12.0691	−0.452309
$713$	0	0
$714$	−1.41421	−0.0529256
$715$	1.88118	0.0703520
$716$	−21.4155	−0.800334
$717$	−8.66981	−0.323780
$718$	−32.8789	−1.22703
$719$	−37.6162	−1.40285	−0.701423	−	0.712745i	$-0.747452\pi$
$719$	−37.6162	−1.40285	−0.701423	+	0.712745i	$0.747452\pi$
$720$	−0.166861	−0.00621854
$721$	1.44902	0.0539642
$722$	−16.7553	−0.623567
$723$	42.8038	1.59189
$724$	−15.9425	−0.592500
$725$	−9.29316	−0.345139
$726$	18.6401	0.691799
$727$	−7.34904	−0.272561	−0.136280	−	0.990670i	$-0.543515\pi$
$727$	−7.34904	−0.272561	−0.136280	+	0.990670i	$0.543515\pi$
$728$	1.14343	0.0423785
$729$	26.4443	0.979419
$730$	−17.4396	−0.645468
$731$	−10.1984	−0.377202
$732$	−13.9293	−0.514842
$733$	−6.10891	−0.225638	−0.112819	−	0.993616i	$-0.535988\pi$
$733$	−6.10891	−0.225638	−0.112819	+	0.993616i	$0.535988\pi$
$734$	−9.11306	−0.336369
$735$	4.91245	0.181198
$736$	0	0
$737$	7.24118	0.266732
$738$	0.0327351	0.00120500
$739$	−37.1177	−1.36540	−0.682698	−	0.730701i	$-0.739193\pi$
$739$	−37.1177	−1.36540	−0.682698	+	0.730701i	$0.739193\pi$
$740$	6.63224	0.243806
$741$	−2.99647	−0.110078
$742$	9.19932	0.337718
$743$	8.31851	0.305177	0.152588	−	0.988290i	$-0.451239\pi$
$743$	8.31851	0.305177	0.152588	+	0.988290i	$0.451239\pi$
$744$	3.26725	0.119783
$745$	6.26109	0.229389
$746$	3.35794	0.122943
$747$	0.169715	0.00620954
$748$	−0.473626	−0.0173175
$749$	−9.15509	−0.334520
$750$	−10.3759	−0.378874
$751$	−5.71106	−0.208400	−0.104200	−	0.994556i	$-0.533228\pi$
$751$	−5.71106	−0.208400	−0.104200	+	0.994556i	$0.533228\pi$
$752$	6.85499	0.249976
$753$	−6.71351	−0.244654
$754$	−3.67961	−0.134003
$755$	19.0462	0.693164
$756$	5.14343	0.187065
$757$	−10.1340	−0.368326	−0.184163	−	0.982896i	$-0.558957\pi$
$757$	−10.1340	−0.368326	−0.184163	+	0.982896i	$0.558957\pi$
$758$	−9.32756	−0.338792
$759$	0	0
$760$	−4.20784	−0.152634
$761$	29.5264	1.07033	0.535165	−	0.844747i	$-0.320249\pi$
$761$	29.5264	1.07033	0.535165	+	0.844747i	$0.320249\pi$
$762$	−25.4623	−0.922403
$763$	−15.1899	−0.549911
$764$	−11.2435	−0.406777
$765$	−0.134912	−0.00487775
$766$	−7.00757	−0.253194
$767$	4.42274	0.159696
$768$	−1.74912	−0.0631158
$769$	−15.2977	−0.551650	−0.275825	−	0.961208i	$-0.588951\pi$
$769$	−15.2977	−0.551650	−0.275825	+	0.961208i	$0.588951\pi$
$770$	1.64520	0.0592888
$771$	30.3070	1.09148
$772$	−6.70960	−0.241484
$773$	40.4998	1.45668	0.728338	−	0.685218i	$-0.240293\pi$
$773$	40.4998	1.45668	0.728338	+	0.685218i	$0.240293\pi$
$774$	−0.749397	−0.0269365
$775$	−5.39432	−0.193770
$776$	0.955493	0.0343002
$777$	4.13048	0.148180
$778$	−26.6483	−0.955389
$779$	0.825503	0.0295767
$780$	5.61706	0.201123
$781$	−6.60411	−0.236313
$782$	0	0
$783$	−16.5517	−0.591511
$784$	1.00000	0.0357143
$785$	47.0165	1.67809
$786$	28.1910	1.00554
$787$	0.206354	0.00735571	0.00367785	−	0.999993i	$-0.498829\pi$
$787$	0.206354	0.00735571	0.00367785	+	0.999993i	$0.498829\pi$
$788$	−7.23412	−0.257705
$789$	−0.315007	−0.0112145
$790$	−37.2199	−1.32422
$791$	−4.96873	−0.176668
$792$	−0.0348029	−0.00123667
$793$	9.10587	0.323359
$794$	−28.5854	−1.01446
$795$	45.1912	1.60277
$796$	−25.1386	−0.891014
$797$	33.9355	1.20206	0.601028	−	0.799228i	$-0.294758\pi$
$797$	33.9355	1.20206	0.601028	+	0.799228i	$0.294758\pi$
$798$	−2.62059	−0.0927679
$799$	5.54246	0.196078
$800$	2.88784	0.102101
$801$	−0.717053	−0.0253358
$802$	21.2462	0.750228
$803$	−3.63745	−0.128363
$804$	21.6217	0.762537
$805$	0	0
$806$	−2.13587	−0.0752327
$807$	−9.73198	−0.342582
$808$	10.1305	0.356389
$809$	40.8276	1.43542	0.717712	−	0.696340i	$-0.245190\pi$
$809$	40.8276	1.43542	0.717712	+	0.696340i	$0.245190\pi$
$810$	25.7674	0.905375
$811$	47.5372	1.66926	0.834628	−	0.550813i	$-0.185682\pi$
$811$	47.5372	1.66926	0.834628	+	0.550813i	$0.185682\pi$
$812$	−3.21803	−0.112931
$813$	−38.8249	−1.36165
$814$	1.38331	0.0484851
$815$	−8.86116	−0.310393
$816$	−1.41421	−0.0495074
$817$	−18.8980	−0.661158
$818$	14.8893	0.520592
$819$	0.0679339	0.00237380
$820$	−1.54745	−0.0540394
$821$	3.46747	0.121015	0.0605077	−	0.998168i	$-0.480728\pi$
$821$	3.46747	0.121015	0.0605077	+	0.998168i	$0.480728\pi$
$822$	9.97186	0.347809
$823$	29.3660	1.02364	0.511818	−	0.859094i	$-0.328972\pi$
$823$	29.3660	1.02364	0.511818	+	0.859094i	$0.328972\pi$
$824$	1.44902	0.0504789
$825$	2.95891	0.103016
$826$	3.86794	0.134583
$827$	−42.6337	−1.48252	−0.741259	−	0.671219i	$-0.765771\pi$
$827$	−42.6337	−1.48252	−0.741259	+	0.671219i	$0.765771\pi$
$828$	0	0
$829$	19.6828	0.683611	0.341805	−	0.939771i	$-0.388962\pi$
$829$	19.6828	0.683611	0.341805	+	0.939771i	$0.388962\pi$
$830$	−8.02275	−0.278474
$831$	−21.8334	−0.757392
$832$	1.14343	0.0396414
$833$	0.808530	0.0280139
$834$	−29.7753	−1.03103
$835$	13.4134	0.464191
$836$	−0.877646	−0.0303540
$837$	−9.60764	−0.332088
$838$	6.64334	0.229490
$839$	−20.1438	−0.695441	−0.347721	−	0.937598i	$-0.613044\pi$
$839$	−20.1438	−0.695441	−0.347721	+	0.937598i	$0.613044\pi$
$840$	4.91245	0.169496
$841$	−18.6443	−0.642906
$842$	−14.5523	−0.501504
$843$	−1.56301	−0.0538330
$844$	−12.6041	−0.433851
$845$	32.8389	1.12969
$846$	0.407270	0.0140022
$847$	−10.6569	−0.366174
$848$	9.19932	0.315906
$849$	30.5080	1.04703
$850$	2.33490	0.0800865
$851$	0	0
$852$	−19.7194	−0.675576
$853$	−12.5223	−0.428755	−0.214377	−	0.976751i	$-0.568772\pi$
$853$	−12.5223	−0.428755	−0.214377	+	0.976751i	$0.568772\pi$
$854$	7.96362	0.272509
$855$	−0.249997	−0.00854971
$856$	−9.15509	−0.312915
$857$	14.9263	0.509873	0.254936	−	0.966958i	$-0.417945\pi$
$857$	14.9263	0.509873	0.254936	+	0.966958i	$0.417945\pi$
$858$	1.17157	0.0399968
$859$	−5.17223	−0.176474	−0.0882370	−	0.996100i	$-0.528123\pi$
$859$	−5.17223	−0.176474	−0.0882370	+	0.996100i	$0.528123\pi$
$860$	35.4255	1.20800
$861$	−0.963735	−0.0328440
$862$	−20.5119	−0.698639
$863$	28.1890	0.959564	0.479782	−	0.877388i	$-0.340716\pi$
$863$	28.1890	0.959564	0.479782	+	0.877388i	$0.340716\pi$
$864$	5.14343	0.174983
$865$	37.9278	1.28958
$866$	2.48410	0.0844131
$867$	28.5916	0.971020
$868$	−1.86794	−0.0634021
$869$	−7.76310	−0.263345
$870$	−15.8084	−0.535955
$871$	−14.1345	−0.478930
$872$	−15.1899	−0.514395
$873$	0.0567680	0.00192130
$874$	0	0
$875$	5.93207	0.200540
$876$	−10.8612	−0.366965
$877$	6.39587	0.215973	0.107987	−	0.994152i	$-0.465560\pi$
$877$	6.39587	0.215973	0.107987	+	0.994152i	$0.465560\pi$
$878$	11.0941	0.374408
$879$	−53.2289	−1.79537
$880$	1.64520	0.0554596
$881$	6.33267	0.213353	0.106677	−	0.994294i	$-0.465979\pi$
$881$	6.33267	0.213353	0.106677	+	0.994294i	$0.465979\pi$
$882$	0.0594122	0.00200051
$883$	8.23611	0.277167	0.138584	−	0.990351i	$-0.455745\pi$
$883$	8.23611	0.277167	0.138584	+	0.990351i	$0.455745\pi$
$884$	0.924500	0.0310943
$885$	19.0011	0.638714
$886$	23.0723	0.775131
$887$	−4.48686	−0.150654	−0.0753270	−	0.997159i	$-0.524000\pi$
$887$	−4.48686	−0.150654	−0.0753270	+	0.997159i	$0.524000\pi$
$888$	4.13048	0.138610
$889$	14.5573	0.488235
$890$	33.8965	1.13621
$891$	5.37442	0.180050
$892$	4.91902	0.164701
$893$	10.2704	0.343685
$894$	3.89933	0.130413
$895$	60.1460	2.01046
$896$	1.00000	0.0334077
$897$	0	0
$898$	18.4545	0.615835
$899$	6.01110	0.200481
$900$	0.171573	0.00571910
$901$	7.43792	0.247793
$902$	−0.322759	−0.0107467
$903$	22.0625	0.734196
$904$	−4.96873	−0.165257
$905$	44.7751	1.48837
$906$	11.8618	0.394081
$907$	5.73273	0.190352	0.0951761	−	0.995460i	$-0.469659\pi$
$907$	5.73273	0.190352	0.0951761	+	0.995460i	$0.469659\pi$
$908$	−18.6698	−0.619579
$909$	0.601874	0.0199629
$910$	−3.21137	−0.106456
$911$	−15.1148	−0.500775	−0.250387	−	0.968146i	$-0.580558\pi$
$911$	−15.1148	−0.500775	−0.250387	+	0.968146i	$0.580558\pi$
$912$	−2.62059	−0.0867764
$913$	−1.67334	−0.0553794
$914$	8.95851	0.296321
$915$	39.1209	1.29330
$916$	21.8394	0.721594
$917$	−16.1172	−0.532238
$918$	4.15862	0.137255
$919$	47.0765	1.55291	0.776456	−	0.630172i	$-0.217015\pi$
$919$	47.0765	1.55291	0.776456	+	0.630172i	$0.217015\pi$
$920$	0	0
$921$	−0.148799	−0.00490310
$922$	35.4388	1.16711
$923$	12.8910	0.424312
$924$	1.02461	0.0337072
$925$	−6.81953	−0.224225
$926$	−8.12509	−0.267007
$927$	0.0860892	0.00282754
$928$	−3.21803	−0.105637
$929$	48.7501	1.59944	0.799720	−	0.600373i	$-0.204981\pi$
$929$	48.7501	1.59944	0.799720	+	0.600373i	$0.204981\pi$
$930$	−9.17617	−0.300898
$931$	1.49824	0.0491027
$932$	2.95392	0.0967587
$933$	−55.9035	−1.83020
$934$	−5.33019	−0.174409
$935$	1.33019	0.0435019
$936$	0.0679339	0.00222049
$937$	20.3012	0.663213	0.331606	−	0.943418i	$-0.392409\pi$
$937$	20.3012	0.663213	0.331606	+	0.943418i	$0.392409\pi$
$938$	−12.3615	−0.403616
$939$	43.3587	1.41496
$940$	−19.2524	−0.627945
$941$	15.5790	0.507861	0.253931	−	0.967222i	$-0.418277\pi$
$941$	15.5790	0.507861	0.253931	+	0.967222i	$0.418277\pi$
$942$	29.2813	0.954035
$943$	0	0
$944$	3.86794	0.125891
$945$	−14.4455	−0.469912
$946$	7.38883	0.240232
$947$	−42.1490	−1.36966	−0.684829	−	0.728704i	$-0.740123\pi$
$947$	−42.1490	−1.36966	−0.684829	+	0.728704i	$0.740123\pi$
$948$	−23.1801	−0.752854
$949$	7.10016	0.230481
$950$	4.32666	0.140375
$951$	17.7038	0.574086
$952$	0.808530	0.0262046
$953$	38.0581	1.23282	0.616412	−	0.787424i	$-0.288586\pi$
$953$	38.0581	1.23282	0.616412	+	0.787424i	$0.288586\pi$
$954$	0.546552	0.0176953
$955$	31.5778	1.02183
$956$	4.95668	0.160310
$957$	−3.29722	−0.106584
$958$	−5.99058	−0.193547
$959$	−5.70108	−0.184098
$960$	4.91245	0.158549
$961$	−27.5108	−0.887445
$962$	−2.70018	−0.0870573
$963$	−0.543924	−0.0175277
$964$	−24.4717	−0.788179
$965$	18.8441	0.606614
$966$	0	0
$967$	−15.2841	−0.491504	−0.245752	−	0.969333i	$-0.579035\pi$
$967$	−15.2841	−0.491504	−0.245752	+	0.969333i	$0.579035\pi$
$968$	−10.6569	−0.342524
$969$	−2.11882	−0.0680664
$970$	−2.68353	−0.0861630
$971$	35.7942	1.14869	0.574345	−	0.818614i	$-0.305257\pi$
$971$	35.7942	1.14869	0.574345	+	0.818614i	$0.305257\pi$
$972$	0.617339	0.0198012
$973$	17.0230	0.545733
$974$	−39.3503	−1.26087
$975$	−5.77568	−0.184970
$976$	7.96362	0.254909
$977$	24.3091	0.777717	0.388858	−	0.921298i	$-0.372870\pi$
$977$	24.3091	0.777717	0.388858	+	0.921298i	$0.372870\pi$
$978$	−5.51862	−0.176466
$979$	7.06992	0.225956
$980$	−2.80853	−0.0897152
$981$	−0.902465	−0.0288135
$982$	−2.21726	−0.0707557
$983$	13.8171	0.440698	0.220349	−	0.975421i	$-0.429280\pi$
$983$	13.8171	0.440698	0.220349	+	0.975421i	$0.429280\pi$
$984$	−0.963735	−0.0307228
$985$	20.3172	0.647361
$986$	−2.60187	−0.0828606
$987$	−11.9902	−0.381652
$988$	1.71313	0.0545020
$989$	0	0
$990$	0.0977449	0.00310654
$991$	−48.1460	−1.52941	−0.764705	−	0.644381i	$-0.777115\pi$
$991$	−48.1460	−1.52941	−0.764705	+	0.644381i	$0.777115\pi$
$992$	−1.86794	−0.0593072
$993$	48.1384	1.52763
$994$	11.2739	0.357587
$995$	70.6025	2.23825
$996$	−4.99647	−0.158319
$997$	−20.2267	−0.640586	−0.320293	−	0.947319i	$-0.603781\pi$
$997$	−20.2267	−0.640586	−0.320293	+	0.947319i	$0.603781\pi$
$998$	−6.81510	−0.215728
$999$	−12.1460	−0.384284

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7406.2.a.bb.1.1	✓	4
23.22	odd	2		7406.2.a.bc.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7406.2.a.bb.1.1	✓	4	1.1	even	1	trivial
7406.2.a.bc.1.1	yes	4	23.22	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7406 = 2 \cdot 7 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7406.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.74912 q^{3} +1.00000 q^{4} -2.80853 q^{5} -1.74912 q^{6} +1.00000 q^{7} +1.00000 q^{8} +0.0594122 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.74912 q^{3} +1.00000 q^{4} -2.80853 q^{5} -1.74912 q^{6} +1.00000 q^{7} +1.00000 q^{8} +0.0594122 q^{9} -2.80853 q^{10} -0.585786 q^{11} -1.74912 q^{12} +1.14343 q^{13} +1.00000 q^{14} +4.91245 q^{15} +1.00000 q^{16} +0.808530 q^{17} +0.0594122 q^{18} +1.49824 q^{19} -2.80853 q^{20} -1.74912 q^{21} -0.585786 q^{22} -1.74912 q^{24} +2.88784 q^{25} +1.14343 q^{26} +5.14343 q^{27} +1.00000 q^{28} -3.21803 q^{29} +4.91245 q^{30} -1.86794 q^{31} +1.00000 q^{32} +1.02461 q^{33} +0.808530 q^{34} -2.80853 q^{35} +0.0594122 q^{36} -2.36147 q^{37} +1.49824 q^{38} -2.00000 q^{39} -2.80853 q^{40} +0.550984 q^{41} -1.74912 q^{42} -12.6135 q^{43} -0.585786 q^{44} -0.166861 q^{45} +6.85499 q^{47} -1.74912 q^{48} +1.00000 q^{49} +2.88784 q^{50} -1.41421 q^{51} +1.14343 q^{52} +9.19932 q^{53} +5.14343 q^{54} +1.64520 q^{55} +1.00000 q^{56} -2.62059 q^{57} -3.21803 q^{58} +3.86794 q^{59} +4.91245 q^{60} +7.96362 q^{61} -1.86794 q^{62} +0.0594122 q^{63} +1.00000 q^{64} -3.21137 q^{65} +1.02461 q^{66} -12.3615 q^{67} +0.808530 q^{68} -2.80853 q^{70} +11.2739 q^{71} +0.0594122 q^{72} +6.20951 q^{73} -2.36147 q^{74} -5.05117 q^{75} +1.49824 q^{76} -0.585786 q^{77} -2.00000 q^{78} +13.2524 q^{79} -2.80853 q^{80} -9.17471 q^{81} +0.550984 q^{82} +2.85657 q^{83} -1.74912 q^{84} -2.27078 q^{85} -12.6135 q^{86} +5.62872 q^{87} -0.585786 q^{88} -12.0691 q^{89} -0.166861 q^{90} +1.14343 q^{91} +3.26725 q^{93} +6.85499 q^{94} -4.20784 q^{95} -1.74912 q^{96} +0.955493 q^{97} +1.00000 q^{98} -0.0348029 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} - 4 q^{5} + 4 q^{7} + 4 q^{8} - 4 q^{10} - 8 q^{11} - 4 q^{13} + 4 q^{14} + 4 q^{16} - 4 q^{17} - 8 q^{19} - 4 q^{20} - 8 q^{22} - 4 q^{26} + 12 q^{27} + 4 q^{28} - 4 q^{29} + 4 q^{32} - 4 q^{33} - 4 q^{34} - 4 q^{35} + 8 q^{37} - 8 q^{38} - 8 q^{39} - 4 q^{40} - 8 q^{43} - 8 q^{44} - 16 q^{45} + 4 q^{49} - 4 q^{52} + 12 q^{54} + 12 q^{55} + 4 q^{56} - 24 q^{57} - 4 q^{58} + 8 q^{59} - 12 q^{61} + 4 q^{64} - 16 q^{65} - 4 q^{66} - 32 q^{67} - 4 q^{68} - 4 q^{70} + 8 q^{71} + 4 q^{73} + 8 q^{74} + 4 q^{75} - 8 q^{76} - 8 q^{77} - 8 q^{78} - 16 q^{79} - 4 q^{80} - 8 q^{81} + 20 q^{83} - 12 q^{85} - 8 q^{86} - 20 q^{87} - 8 q^{88} - 28 q^{89} - 16 q^{90} - 4 q^{91} - 12 q^{93} + 8 q^{95} + 16 q^{97} + 4 q^{98} - 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 - 4 * q^5 + 4 * q^7 + 4 * q^8 - 4 * q^10 - 8 * q^11 - 4 * q^13 + 4 * q^14 + 4 * q^16 - 4 * q^17 - 8 * q^19 - 4 * q^20 - 8 * q^22 - 4 * q^26 + 12 * q^27 + 4 * q^28 - 4 * q^29 + 4 * q^32 - 4 * q^33 - 4 * q^34 - 4 * q^35 + 8 * q^37 - 8 * q^38 - 8 * q^39 - 4 * q^40 - 8 * q^43 - 8 * q^44 - 16 * q^45 + 4 * q^49 - 4 * q^52 + 12 * q^54 + 12 * q^55 + 4 * q^56 - 24 * q^57 - 4 * q^58 + 8 * q^59 - 12 * q^61 + 4 * q^64 - 16 * q^65 - 4 * q^66 - 32 * q^67 - 4 * q^68 - 4 * q^70 + 8 * q^71 + 4 * q^73 + 8 * q^74 + 4 * q^75 - 8 * q^76 - 8 * q^77 - 8 * q^78 - 16 * q^79 - 4 * q^80 - 8 * q^81 + 20 * q^83 - 12 * q^85 - 8 * q^86 - 20 * q^87 - 8 * q^88 - 28 * q^89 - 16 * q^90 - 4 * q^91 - 12 * q^93 + 8 * q^95 + 16 * q^97 + 4 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more