LMFDB - Embedded newform 8034.2.a.t.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8034,2,Mod(1,8034)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8034, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8034.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8034 = 2 \cdot 3 \cdot 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8034.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:	$64.1518129839$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 4 x^{10} - 24 x^{9} + 88 x^{8} + 220 x^{7} - 637 x^{6} - 977 x^{5} + 1739 x^{4} + 1872 x^{3} + \cdots - 162 $ x^11 - 4x^10 - 24x^9 + 88x^8 + 220x^7 - 637x^6 - 977x^5 + 1739x^4 + 1872x^3 - 1494x^2 - 1161x - 162
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$3.85844$ of defining polynomial
Character	$\chi$	$=$	8034.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.85844 q^{5} -1.00000 q^{6} +3.88029 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.85844 q^{5} -1.00000 q^{6} +3.88029 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.85844 q^{10} -5.32451 q^{11} +1.00000 q^{12} -1.00000 q^{13} -3.88029 q^{14} +3.85844 q^{15} +1.00000 q^{16} +0.277336 q^{17} -1.00000 q^{18} +3.65879 q^{19} +3.85844 q^{20} +3.88029 q^{21} +5.32451 q^{22} +0.824024 q^{23} -1.00000 q^{24} +9.88752 q^{25} +1.00000 q^{26} +1.00000 q^{27} +3.88029 q^{28} +10.0368 q^{29} -3.85844 q^{30} -4.97252 q^{31} -1.00000 q^{32} -5.32451 q^{33} -0.277336 q^{34} +14.9719 q^{35} +1.00000 q^{36} +2.46936 q^{37} -3.65879 q^{38} -1.00000 q^{39} -3.85844 q^{40} +5.80382 q^{41} -3.88029 q^{42} -4.68269 q^{43} -5.32451 q^{44} +3.85844 q^{45} -0.824024 q^{46} +4.44885 q^{47} +1.00000 q^{48} +8.05668 q^{49} -9.88752 q^{50} +0.277336 q^{51} -1.00000 q^{52} -10.2989 q^{53} -1.00000 q^{54} -20.5443 q^{55} -3.88029 q^{56} +3.65879 q^{57} -10.0368 q^{58} +10.0204 q^{59} +3.85844 q^{60} +2.38408 q^{61} +4.97252 q^{62} +3.88029 q^{63} +1.00000 q^{64} -3.85844 q^{65} +5.32451 q^{66} -7.40243 q^{67} +0.277336 q^{68} +0.824024 q^{69} -14.9719 q^{70} +14.1804 q^{71} -1.00000 q^{72} -14.8181 q^{73} -2.46936 q^{74} +9.88752 q^{75} +3.65879 q^{76} -20.6607 q^{77} +1.00000 q^{78} -1.14178 q^{79} +3.85844 q^{80} +1.00000 q^{81} -5.80382 q^{82} +6.90807 q^{83} +3.88029 q^{84} +1.07008 q^{85} +4.68269 q^{86} +10.0368 q^{87} +5.32451 q^{88} +8.07997 q^{89} -3.85844 q^{90} -3.88029 q^{91} +0.824024 q^{92} -4.97252 q^{93} -4.44885 q^{94} +14.1172 q^{95} -1.00000 q^{96} -11.1288 q^{97} -8.05668 q^{98} -5.32451 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 11 q^{2} + 11 q^{3} + 11 q^{4} + 4 q^{5} - 11 q^{6} + 4 q^{7} - 11 q^{8} + 11 q^{9}+O(q^{10}) $ 11 * q - 11 * q^2 + 11 * q^3 + 11 * q^4 + 4 * q^5 - 11 * q^6 + 4 * q^7 - 11 * q^8 + 11 * q^9	\( 11 q - 11 q^{2} + 11 q^{3} + 11 q^{4} + 4 q^{5} - 11 q^{6} + 4 q^{7} - 11 q^{8} + 11 q^{9} - 4 q^{10} + 5 q^{11} + 11 q^{12} - 11 q^{13} - 4 q^{14} + 4 q^{15} + 11 q^{16} + 8 q^{17} - 11 q^{18} - 2 q^{19} + 4 q^{20} + 4 q^{21} - 5 q^{22} + 3 q^{23} - 11 q^{24} + 9 q^{25} + 11 q^{26} + 11 q^{27} + 4 q^{28} + 7 q^{29} - 4 q^{30} + 20 q^{31} - 11 q^{32} + 5 q^{33} - 8 q^{34} + 9 q^{35} + 11 q^{36} + q^{37} + 2 q^{38} - 11 q^{39} - 4 q^{40} + 37 q^{41} - 4 q^{42} - 16 q^{43} + 5 q^{44} + 4 q^{45} - 3 q^{46} + 28 q^{47} + 11 q^{48} + 17 q^{49} - 9 q^{50} + 8 q^{51} - 11 q^{52} - 5 q^{53} - 11 q^{54} - 28 q^{55} - 4 q^{56} - 2 q^{57} - 7 q^{58} + 31 q^{59} + 4 q^{60} + 8 q^{61} - 20 q^{62} + 4 q^{63} + 11 q^{64} - 4 q^{65} - 5 q^{66} - 22 q^{67} + 8 q^{68} + 3 q^{69} - 9 q^{70} + 42 q^{71} - 11 q^{72} - 4 q^{73} - q^{74} + 9 q^{75} - 2 q^{76} - 21 q^{77} + 11 q^{78} + 33 q^{79} + 4 q^{80} + 11 q^{81} - 37 q^{82} + 18 q^{83} + 4 q^{84} + 17 q^{85} + 16 q^{86} + 7 q^{87} - 5 q^{88} + 67 q^{89} - 4 q^{90} - 4 q^{91} + 3 q^{92} + 20 q^{93} - 28 q^{94} + 32 q^{95} - 11 q^{96} - 15 q^{97} - 17 q^{98} + 5 q^{99}+O(q^{100}) \) 11 * q - 11 * q^2 + 11 * q^3 + 11 * q^4 + 4 * q^5 - 11 * q^6 + 4 * q^7 - 11 * q^8 + 11 * q^9 - 4 * q^10 + 5 * q^11 + 11 * q^12 - 11 * q^13 - 4 * q^14 + 4 * q^15 + 11 * q^16 + 8 * q^17 - 11 * q^18 - 2 * q^19 + 4 * q^20 + 4 * q^21 - 5 * q^22 + 3 * q^23 - 11 * q^24 + 9 * q^25 + 11 * q^26 + 11 * q^27 + 4 * q^28 + 7 * q^29 - 4 * q^30 + 20 * q^31 - 11 * q^32 + 5 * q^33 - 8 * q^34 + 9 * q^35 + 11 * q^36 + q^37 + 2 * q^38 - 11 * q^39 - 4 * q^40 + 37 * q^41 - 4 * q^42 - 16 * q^43 + 5 * q^44 + 4 * q^45 - 3 * q^46 + 28 * q^47 + 11 * q^48 + 17 * q^49 - 9 * q^50 + 8 * q^51 - 11 * q^52 - 5 * q^53 - 11 * q^54 - 28 * q^55 - 4 * q^56 - 2 * q^57 - 7 * q^58 + 31 * q^59 + 4 * q^60 + 8 * q^61 - 20 * q^62 + 4 * q^63 + 11 * q^64 - 4 * q^65 - 5 * q^66 - 22 * q^67 + 8 * q^68 + 3 * q^69 - 9 * q^70 + 42 * q^71 - 11 * q^72 - 4 * q^73 - q^74 + 9 * q^75 - 2 * q^76 - 21 * q^77 + 11 * q^78 + 33 * q^79 + 4 * q^80 + 11 * q^81 - 37 * q^82 + 18 * q^83 + 4 * q^84 + 17 * q^85 + 16 * q^86 + 7 * q^87 - 5 * q^88 + 67 * q^89 - 4 * q^90 - 4 * q^91 + 3 * q^92 + 20 * q^93 - 28 * q^94 + 32 * q^95 - 11 * q^96 - 15 * q^97 - 17 * q^98 + 5 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	3.85844	1.72554	0.862772	−	0.505593i	$-0.168726\pi$
$5$	3.85844	1.72554	0.862772	+	0.505593i	$0.168726\pi$
$6$	−1.00000	−0.408248
$7$	3.88029	1.46661	0.733307	−	0.679898i	$-0.237976\pi$
$7$	3.88029	1.46661	0.733307	+	0.679898i	$0.237976\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−3.85844	−1.22014
$11$	−5.32451	−1.60540	−0.802700	−	0.596383i	$-0.796604\pi$
$11$	−5.32451	−1.60540	−0.802700	+	0.596383i	$0.796604\pi$
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−3.88029	−1.03705
$15$	3.85844	0.996244
$16$	1.00000	0.250000
$17$	0.277336	0.0672640	0.0336320	−	0.999434i	$-0.489293\pi$
$17$	0.277336	0.0672640	0.0336320	+	0.999434i	$0.489293\pi$
$18$	−1.00000	−0.235702
$19$	3.65879	0.839384	0.419692	−	0.907667i	$-0.362138\pi$
$19$	3.65879	0.839384	0.419692	+	0.907667i	$0.362138\pi$
$20$	3.85844	0.862772
$21$	3.88029	0.846750
$22$	5.32451	1.13519
$23$	0.824024	0.171821	0.0859105	−	0.996303i	$-0.472620\pi$
$23$	0.824024	0.171821	0.0859105	+	0.996303i	$0.472620\pi$
$24$	−1.00000	−0.204124
$25$	9.88752	1.97750
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	3.88029	0.733307
$29$	10.0368	1.86379	0.931896	−	0.362725i	$-0.118153\pi$
$29$	10.0368	1.86379	0.931896	+	0.362725i	$0.118153\pi$
$30$	−3.85844	−0.704451
$31$	−4.97252	−0.893092	−0.446546	−	0.894761i	$-0.647346\pi$
$31$	−4.97252	−0.893092	−0.446546	+	0.894761i	$0.647346\pi$
$32$	−1.00000	−0.176777
$33$	−5.32451	−0.926878
$34$	−0.277336	−0.0475628
$35$	14.9719	2.53071
$36$	1.00000	0.166667
$37$	2.46936	0.405960	0.202980	−	0.979183i	$-0.434937\pi$
$37$	2.46936	0.405960	0.202980	+	0.979183i	$0.434937\pi$
$38$	−3.65879	−0.593534
$39$	−1.00000	−0.160128
$40$	−3.85844	−0.610072
$41$	5.80382	0.906405	0.453202	−	0.891408i	$-0.350281\pi$
$41$	5.80382	0.906405	0.453202	+	0.891408i	$0.350281\pi$
$42$	−3.88029	−0.598742
$43$	−4.68269	−0.714103	−0.357051	−	0.934085i	$-0.616218\pi$
$43$	−4.68269	−0.714103	−0.357051	+	0.934085i	$0.616218\pi$
$44$	−5.32451	−0.802700
$45$	3.85844	0.575182
$46$	−0.824024	−0.121496
$47$	4.44885	0.648931	0.324465	−	0.945898i	$-0.394816\pi$
$47$	4.44885	0.648931	0.324465	+	0.945898i	$0.394816\pi$
$48$	1.00000	0.144338
$49$	8.05668	1.15095
$50$	−9.88752	−1.39831
$51$	0.277336	0.0388349
$52$	−1.00000	−0.138675
$53$	−10.2989	−1.41467	−0.707333	−	0.706880i	$-0.750102\pi$
$53$	−10.2989	−1.41467	−0.707333	+	0.706880i	$0.750102\pi$
$54$	−1.00000	−0.136083
$55$	−20.5443	−2.77019
$56$	−3.88029	−0.518526
$57$	3.65879	0.484618
$58$	−10.0368	−1.31790
$59$	10.0204	1.30455	0.652273	−	0.757984i	$-0.273816\pi$
$59$	10.0204	1.30455	0.652273	+	0.757984i	$0.273816\pi$
$60$	3.85844	0.498122
$61$	2.38408	0.305251	0.152625	−	0.988284i	$-0.451227\pi$
$61$	2.38408	0.305251	0.152625	+	0.988284i	$0.451227\pi$
$62$	4.97252	0.631511
$63$	3.88029	0.488871
$64$	1.00000	0.125000
$65$	−3.85844	−0.478580
$66$	5.32451	0.655402
$67$	−7.40243	−0.904351	−0.452175	−	0.891929i	$-0.649352\pi$
$67$	−7.40243	−0.904351	−0.452175	+	0.891929i	$0.649352\pi$
$68$	0.277336	0.0336320
$69$	0.824024	0.0992009
$70$	−14.9719	−1.78948
$71$	14.1804	1.68290	0.841452	−	0.540332i	$-0.181701\pi$
$71$	14.1804	1.68290	0.841452	+	0.540332i	$0.181701\pi$
$72$	−1.00000	−0.117851
$73$	−14.8181	−1.73433	−0.867163	−	0.498025i	$-0.834059\pi$
$73$	−14.8181	−1.73433	−0.867163	+	0.498025i	$0.834059\pi$
$74$	−2.46936	−0.287057
$75$	9.88752	1.14171
$76$	3.65879	0.419692
$77$	−20.6607	−2.35450
$78$	1.00000	0.113228
$79$	−1.14178	−0.128461	−0.0642303	−	0.997935i	$-0.520459\pi$
$79$	−1.14178	−0.128461	−0.0642303	+	0.997935i	$0.520459\pi$
$80$	3.85844	0.431386
$81$	1.00000	0.111111
$82$	−5.80382	−0.640925
$83$	6.90807	0.758259	0.379129	−	0.925344i	$-0.376223\pi$
$83$	6.90807	0.758259	0.379129	+	0.925344i	$0.376223\pi$
$84$	3.88029	0.423375
$85$	1.07008	0.116067
$86$	4.68269	0.504947
$87$	10.0368	1.07606
$88$	5.32451	0.567595
$89$	8.07997	0.856475	0.428238	−	0.903666i	$-0.359135\pi$
$89$	8.07997	0.856475	0.428238	+	0.903666i	$0.359135\pi$
$90$	−3.85844	−0.406715
$91$	−3.88029	−0.406765
$92$	0.824024	0.0859105
$93$	−4.97252	−0.515627
$94$	−4.44885	−0.458863
$95$	14.1172	1.44839
$96$	−1.00000	−0.102062
$97$	−11.1288	−1.12996	−0.564978	−	0.825106i	$-0.691115\pi$
$97$	−11.1288	−1.12996	−0.564978	+	0.825106i	$0.691115\pi$
$98$	−8.05668	−0.813848
$99$	−5.32451	−0.535133
$100$	9.88752	0.988752
$101$	2.34120	0.232958	0.116479	−	0.993193i	$-0.462839\pi$
$101$	2.34120	0.232958	0.116479	+	0.993193i	$0.462839\pi$
$102$	−0.277336	−0.0274604
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	1.00000	0.0980581
$105$	14.9719	1.46110
$106$	10.2989	1.00032
$107$	−5.54334	−0.535895	−0.267947	−	0.963434i	$-0.586345\pi$
$107$	−5.54334	−0.535895	−0.267947	+	0.963434i	$0.586345\pi$
$108$	1.00000	0.0962250
$109$	−19.1559	−1.83480	−0.917399	−	0.397969i	$-0.869715\pi$
$109$	−19.1559	−1.83480	−0.917399	+	0.397969i	$0.869715\pi$
$110$	20.5443	1.95882
$111$	2.46936	0.234381
$112$	3.88029	0.366653
$113$	−16.5498	−1.55688	−0.778438	−	0.627721i	$-0.783988\pi$
$113$	−16.5498	−1.55688	−0.778438	+	0.627721i	$0.783988\pi$
$114$	−3.65879	−0.342677
$115$	3.17944	0.296485
$116$	10.0368	0.931896
$117$	−1.00000	−0.0924500
$118$	−10.0204	−0.922453
$119$	1.07615	0.0986502
$120$	−3.85844	−0.352225
$121$	17.3504	1.57731
$122$	−2.38408	−0.215845
$123$	5.80382	0.523313
$124$	−4.97252	−0.446546
$125$	18.8582	1.68673
$126$	−3.88029	−0.345684
$127$	17.6642	1.56745	0.783724	−	0.621109i	$-0.213318\pi$
$127$	17.6642	1.56745	0.783724	+	0.621109i	$0.213318\pi$
$128$	−1.00000	−0.0883883
$129$	−4.68269	−0.412287
$130$	3.85844	0.338407
$131$	21.2850	1.85968	0.929839	−	0.367968i	$-0.119946\pi$
$131$	21.2850	1.85968	0.929839	+	0.367968i	$0.119946\pi$
$132$	−5.32451	−0.463439
$133$	14.1972	1.23105
$134$	7.40243	0.639473
$135$	3.85844	0.332081
$136$	−0.277336	−0.0237814
$137$	−8.67169	−0.740872	−0.370436	−	0.928858i	$-0.620792\pi$
$137$	−8.67169	−0.740872	−0.370436	+	0.928858i	$0.620792\pi$
$138$	−0.824024	−0.0701456
$139$	−13.8301	−1.17305	−0.586525	−	0.809931i	$-0.699505\pi$
$139$	−13.8301	−1.17305	−0.586525	+	0.809931i	$0.699505\pi$
$140$	14.9719	1.26535
$141$	4.44885	0.374660
$142$	−14.1804	−1.18999
$143$	5.32451	0.445258
$144$	1.00000	0.0833333
$145$	38.7265	3.21606
$146$	14.8181	1.22635
$147$	8.05668	0.664504
$148$	2.46936	0.202980
$149$	−5.91958	−0.484951	−0.242475	−	0.970158i	$-0.577959\pi$
$149$	−5.91958	−0.484951	−0.242475	+	0.970158i	$0.577959\pi$
$150$	−9.88752	−0.807313
$151$	21.1663	1.72249	0.861246	−	0.508188i	$-0.169685\pi$
$151$	21.1663	1.72249	0.861246	+	0.508188i	$0.169685\pi$
$152$	−3.65879	−0.296767
$153$	0.277336	0.0224213
$154$	20.6607	1.66488
$155$	−19.1862	−1.54107
$156$	−1.00000	−0.0800641
$157$	−2.43450	−0.194294	−0.0971472	−	0.995270i	$-0.530972\pi$
$157$	−2.43450	−0.194294	−0.0971472	+	0.995270i	$0.530972\pi$
$158$	1.14178	0.0908353
$159$	−10.2989	−0.816758
$160$	−3.85844	−0.305036
$161$	3.19746	0.251995
$162$	−1.00000	−0.0785674
$163$	13.5810	1.06374	0.531872	−	0.846824i	$-0.321489\pi$
$163$	13.5810	1.06374	0.531872	+	0.846824i	$0.321489\pi$
$164$	5.80382	0.453202
$165$	−20.5443	−1.59937
$166$	−6.90807	−0.536170
$167$	15.9891	1.23728	0.618638	−	0.785676i	$-0.287685\pi$
$167$	15.9891	1.23728	0.618638	+	0.785676i	$0.287685\pi$
$168$	−3.88029	−0.299371
$169$	1.00000	0.0769231
$170$	−1.07008	−0.0820717
$171$	3.65879	0.279795
$172$	−4.68269	−0.357051
$173$	−15.7820	−1.19989	−0.599943	−	0.800043i	$-0.704810\pi$
$173$	−15.7820	−1.19989	−0.599943	+	0.800043i	$0.704810\pi$
$174$	−10.0368	−0.760890
$175$	38.3665	2.90023
$176$	−5.32451	−0.401350
$177$	10.0204	0.753180
$178$	−8.07997	−0.605619
$179$	23.3585	1.74590	0.872948	−	0.487814i	$-0.162205\pi$
$179$	23.3585	1.74590	0.872948	+	0.487814i	$0.162205\pi$
$180$	3.85844	0.287591
$181$	9.70526	0.721386	0.360693	−	0.932685i	$-0.382540\pi$
$181$	9.70526	0.721386	0.360693	+	0.932685i	$0.382540\pi$
$182$	3.88029	0.287627
$183$	2.38408	0.176237
$184$	−0.824024	−0.0607479
$185$	9.52786	0.700502
$186$	4.97252	0.364603
$187$	−1.47668	−0.107986
$188$	4.44885	0.324465
$189$	3.88029	0.282250
$190$	−14.1172	−1.02417
$191$	14.7455	1.06695	0.533474	−	0.845816i	$-0.320886\pi$
$191$	14.7455	1.06695	0.533474	+	0.845816i	$0.320886\pi$
$192$	1.00000	0.0721688
$193$	2.18383	0.157196	0.0785979	−	0.996906i	$-0.474956\pi$
$193$	2.18383	0.157196	0.0785979	+	0.996906i	$0.474956\pi$
$194$	11.1288	0.798999
$195$	−3.85844	−0.276308
$196$	8.05668	0.575477
$197$	−3.77989	−0.269306	−0.134653	−	0.990893i	$-0.542992\pi$
$197$	−3.77989	−0.269306	−0.134653	+	0.990893i	$0.542992\pi$
$198$	5.32451	0.378396
$199$	−5.73499	−0.406543	−0.203271	−	0.979122i	$-0.565157\pi$
$199$	−5.73499	−0.406543	−0.203271	+	0.979122i	$0.565157\pi$
$200$	−9.88752	−0.699153
$201$	−7.40243	−0.522127
$202$	−2.34120	−0.164726
$203$	38.9459	2.73346
$204$	0.277336	0.0194174
$205$	22.3937	1.56404
$206$	1.00000	0.0696733
$207$	0.824024	0.0572736
$208$	−1.00000	−0.0693375
$209$	−19.4813	−1.34755
$210$	−14.9719	−1.03316
$211$	−27.3332	−1.88169	−0.940847	−	0.338831i	$-0.889969\pi$
$211$	−27.3332	−1.88169	−0.940847	+	0.338831i	$0.889969\pi$
$212$	−10.2989	−0.707333
$213$	14.1804	0.971625
$214$	5.54334	0.378935
$215$	−18.0678	−1.23222
$216$	−1.00000	−0.0680414
$217$	−19.2949	−1.30982
$218$	19.1559	1.29740
$219$	−14.8181	−1.00131
$220$	−20.5443	−1.38509
$221$	−0.277336	−0.0186557
$222$	−2.46936	−0.165733
$223$	9.43754	0.631985	0.315992	−	0.948762i	$-0.397663\pi$
$223$	9.43754	0.631985	0.315992	+	0.948762i	$0.397663\pi$
$224$	−3.88029	−0.259263
$225$	9.88752	0.659168
$226$	16.5498	1.10088
$227$	−17.8183	−1.18264	−0.591320	−	0.806437i	$-0.701393\pi$
$227$	−17.8183	−1.18264	−0.591320	+	0.806437i	$0.701393\pi$
$228$	3.65879	0.242309
$229$	8.92659	0.589886	0.294943	−	0.955515i	$-0.404699\pi$
$229$	8.92659	0.589886	0.294943	+	0.955515i	$0.404699\pi$
$230$	−3.17944	−0.209646
$231$	−20.6607	−1.35937
$232$	−10.0368	−0.658950
$233$	−8.18269	−0.536066	−0.268033	−	0.963410i	$-0.586374\pi$
$233$	−8.18269	−0.536066	−0.268033	+	0.963410i	$0.586374\pi$
$234$	1.00000	0.0653720
$235$	17.1656	1.11976
$236$	10.0204	0.652273
$237$	−1.14178	−0.0741667
$238$	−1.07615	−0.0697562
$239$	−7.84265	−0.507299	−0.253650	−	0.967296i	$-0.581631\pi$
$239$	−7.84265	−0.507299	−0.253650	+	0.967296i	$0.581631\pi$
$240$	3.85844	0.249061
$241$	−4.67358	−0.301052	−0.150526	−	0.988606i	$-0.548097\pi$
$241$	−4.67358	−0.301052	−0.150526	+	0.988606i	$0.548097\pi$
$242$	−17.3504	−1.11533
$243$	1.00000	0.0641500
$244$	2.38408	0.152625
$245$	31.0862	1.98602
$246$	−5.80382	−0.370038
$247$	−3.65879	−0.232803
$248$	4.97252	0.315756
$249$	6.90807	0.437781
$250$	−18.8582	−1.19270
$251$	5.65760	0.357104	0.178552	−	0.983930i	$-0.442859\pi$
$251$	5.65760	0.357104	0.178552	+	0.983930i	$0.442859\pi$
$252$	3.88029	0.244436
$253$	−4.38753	−0.275841
$254$	−17.6642	−1.10835
$255$	1.07008	0.0670113
$256$	1.00000	0.0625000
$257$	−6.88357	−0.429385	−0.214693	−	0.976682i	$-0.568875\pi$
$257$	−6.88357	−0.429385	−0.214693	+	0.976682i	$0.568875\pi$
$258$	4.68269	0.291531
$259$	9.58184	0.595386
$260$	−3.85844	−0.239290
$261$	10.0368	0.621264
$262$	−21.2850	−1.31499
$263$	17.0875	1.05366	0.526831	−	0.849970i	$-0.323380\pi$
$263$	17.0875	1.05366	0.526831	+	0.849970i	$0.323380\pi$
$264$	5.32451	0.327701
$265$	−39.7377	−2.44107
$266$	−14.1972	−0.870485
$267$	8.07997	0.494486
$268$	−7.40243	−0.452175
$269$	9.67753	0.590050	0.295025	−	0.955490i	$-0.404672\pi$
$269$	9.67753	0.590050	0.295025	+	0.955490i	$0.404672\pi$
$270$	−3.85844	−0.234817
$271$	−8.84507	−0.537300	−0.268650	−	0.963238i	$-0.586577\pi$
$271$	−8.84507	−0.537300	−0.268650	+	0.963238i	$0.586577\pi$
$272$	0.277336	0.0168160
$273$	−3.88029	−0.234846
$274$	8.67169	0.523876
$275$	−52.6462	−3.17469
$276$	0.824024	0.0496004
$277$	11.7447	0.705669	0.352835	−	0.935686i	$-0.385218\pi$
$277$	11.7447	0.705669	0.352835	+	0.935686i	$0.385218\pi$
$278$	13.8301	0.829472
$279$	−4.97252	−0.297697
$280$	−14.9719	−0.894740
$281$	5.95584	0.355296	0.177648	−	0.984094i	$-0.443151\pi$
$281$	5.95584	0.355296	0.177648	+	0.984094i	$0.443151\pi$
$282$	−4.44885	−0.264925
$283$	10.6198	0.631282	0.315641	−	0.948879i	$-0.397780\pi$
$283$	10.6198	0.631282	0.315641	+	0.948879i	$0.397780\pi$
$284$	14.1804	0.841452
$285$	14.1172	0.836231
$286$	−5.32451	−0.314845
$287$	22.5205	1.32935
$288$	−1.00000	−0.0589256
$289$	−16.9231	−0.995476
$290$	−38.7265	−2.27410
$291$	−11.1288	−0.652380
$292$	−14.8181	−0.867163
$293$	−11.1321	−0.650342	−0.325171	−	0.945655i	$-0.605422\pi$
$293$	−11.1321	−0.650342	−0.325171	+	0.945655i	$0.605422\pi$
$294$	−8.05668	−0.469875
$295$	38.6631	2.25105
$296$	−2.46936	−0.143529
$297$	−5.32451	−0.308959
$298$	5.91958	0.342912
$299$	−0.824024	−0.0476546
$300$	9.88752	0.570856
$301$	−18.1702	−1.04731
$302$	−21.1663	−1.21799
$303$	2.34120	0.134498
$304$	3.65879	0.209846
$305$	9.19883	0.526724
$306$	−0.277336	−0.0158543
$307$	8.56114	0.488610	0.244305	−	0.969698i	$-0.421440\pi$
$307$	8.56114	0.488610	0.244305	+	0.969698i	$0.421440\pi$
$308$	−20.6607	−1.17725
$309$	−1.00000	−0.0568880
$310$	19.1862	1.08970
$311$	−6.64717	−0.376926	−0.188463	−	0.982080i	$-0.560351\pi$
$311$	−6.64717	−0.376926	−0.188463	+	0.982080i	$0.560351\pi$
$312$	1.00000	0.0566139
$313$	4.58305	0.259049	0.129525	−	0.991576i	$-0.458655\pi$
$313$	4.58305	0.259049	0.129525	+	0.991576i	$0.458655\pi$
$314$	2.43450	0.137387
$315$	14.9719	0.843569
$316$	−1.14178	−0.0642303
$317$	−3.18049	−0.178634	−0.0893169	−	0.996003i	$-0.528468\pi$
$317$	−3.18049	−0.178634	−0.0893169	+	0.996003i	$0.528468\pi$
$318$	10.2989	0.577535
$319$	−53.4412	−2.99213
$320$	3.85844	0.215693
$321$	−5.54334	−0.309399
$322$	−3.19746	−0.178187
$323$	1.01472	0.0564603
$324$	1.00000	0.0555556
$325$	−9.88752	−0.548461
$326$	−13.5810	−0.752181
$327$	−19.1559	−1.05932
$328$	−5.80382	−0.320462
$329$	17.2628	0.951730
$330$	20.5443	1.13093
$331$	−12.6377	−0.694633	−0.347316	−	0.937748i	$-0.612907\pi$
$331$	−12.6377	−0.694633	−0.347316	+	0.937748i	$0.612907\pi$
$332$	6.90807	0.379129
$333$	2.46936	0.135320
$334$	−15.9891	−0.874887
$335$	−28.5618	−1.56050
$336$	3.88029	0.211687
$337$	19.7606	1.07643	0.538215	−	0.842808i	$-0.319099\pi$
$337$	19.7606	1.07643	0.538215	+	0.842808i	$0.319099\pi$
$338$	−1.00000	−0.0543928
$339$	−16.5498	−0.898863
$340$	1.07008	0.0580335
$341$	26.4763	1.43377
$342$	−3.65879	−0.197845
$343$	4.10023	0.221391
$344$	4.68269	0.252473
$345$	3.17944	0.171176
$346$	15.7820	0.848447
$347$	−4.04555	−0.217177	−0.108588	−	0.994087i	$-0.534633\pi$
$347$	−4.04555	−0.217177	−0.108588	+	0.994087i	$0.534633\pi$
$348$	10.0368	0.538031
$349$	8.62227	0.461540	0.230770	−	0.973008i	$-0.425876\pi$
$349$	8.62227	0.461540	0.230770	+	0.973008i	$0.425876\pi$
$350$	−38.3665	−2.05078
$351$	−1.00000	−0.0533761
$352$	5.32451	0.283797
$353$	6.26141	0.333261	0.166631	−	0.986019i	$-0.446711\pi$
$353$	6.26141	0.333261	0.166631	+	0.986019i	$0.446711\pi$
$354$	−10.0204	−0.532579
$355$	54.7142	2.90393
$356$	8.07997	0.428238
$357$	1.07615	0.0569557
$358$	−23.3585	−1.23453
$359$	−17.9502	−0.947373	−0.473687	−	0.880694i	$-0.657077\pi$
$359$	−17.9502	−0.947373	−0.473687	+	0.880694i	$0.657077\pi$
$360$	−3.85844	−0.203357
$361$	−5.61326	−0.295435
$362$	−9.70526	−0.510097
$363$	17.3504	0.910660
$364$	−3.88029	−0.203383
$365$	−57.1746	−2.99266
$366$	−2.38408	−0.124618
$367$	−21.9556	−1.14608	−0.573038	−	0.819529i	$-0.694235\pi$
$367$	−21.9556	−1.14608	−0.573038	+	0.819529i	$0.694235\pi$
$368$	0.824024	0.0429552
$369$	5.80382	0.302135
$370$	−9.52786	−0.495330
$371$	−39.9628	−2.07477
$372$	−4.97252	−0.257813
$373$	−27.5391	−1.42592	−0.712960	−	0.701204i	$-0.752646\pi$
$373$	−27.5391	−1.42592	−0.712960	+	0.701204i	$0.752646\pi$
$374$	1.47668	0.0763573
$375$	18.8582	0.973833
$376$	−4.44885	−0.229432
$377$	−10.0368	−0.516923
$378$	−3.88029	−0.199581
$379$	−30.3231	−1.55759	−0.778797	−	0.627276i	$-0.784170\pi$
$379$	−30.3231	−1.55759	−0.778797	+	0.627276i	$0.784170\pi$
$380$	14.1172	0.724197
$381$	17.6642	0.904967
$382$	−14.7455	−0.754446
$383$	19.7901	1.01123	0.505613	−	0.862760i	$-0.331266\pi$
$383$	19.7901	1.01123	0.505613	+	0.862760i	$0.331266\pi$
$384$	−1.00000	−0.0510310
$385$	−79.7178	−4.06280
$386$	−2.18383	−0.111154
$387$	−4.68269	−0.238034
$388$	−11.1288	−0.564978
$389$	−15.2114	−0.771246	−0.385623	−	0.922656i	$-0.626013\pi$
$389$	−15.2114	−0.771246	−0.385623	+	0.922656i	$0.626013\pi$
$390$	3.85844	0.195379
$391$	0.228532	0.0115574
$392$	−8.05668	−0.406924
$393$	21.2850	1.07369
$394$	3.77989	0.190428
$395$	−4.40549	−0.221664
$396$	−5.32451	−0.267567
$397$	13.0316	0.654036	0.327018	−	0.945018i	$-0.393956\pi$
$397$	13.0316	0.654036	0.327018	+	0.945018i	$0.393956\pi$
$398$	5.73499	0.287469
$399$	14.1972	0.710748
$400$	9.88752	0.494376
$401$	−11.7498	−0.586759	−0.293380	−	0.955996i	$-0.594780\pi$
$401$	−11.7498	−0.586759	−0.293380	+	0.955996i	$0.594780\pi$
$402$	7.40243	0.369200
$403$	4.97252	0.247699
$404$	2.34120	0.116479
$405$	3.85844	0.191727
$406$	−38.9459	−1.93285
$407$	−13.1481	−0.651728
$408$	−0.277336	−0.0137302
$409$	−0.0245352	−0.00121319	−0.000606595	−	1.00000i	$-0.500193\pi$
$409$	−0.0245352	−0.00121319	−0.000606595		1.00000i	$0.500193\pi$
$410$	−22.3937	−1.10594
$411$	−8.67169	−0.427743
$412$	−1.00000	−0.0492665
$413$	38.8821	1.91326
$414$	−0.824024	−0.0404986
$415$	26.6543	1.30841
$416$	1.00000	0.0490290
$417$	−13.8301	−0.677261
$418$	19.4813	0.952860
$419$	−14.7320	−0.719703	−0.359852	−	0.933010i	$-0.617173\pi$
$419$	−14.7320	−0.719703	−0.359852	+	0.933010i	$0.617173\pi$
$420$	14.9719	0.730552
$421$	13.4256	0.654323	0.327161	−	0.944969i	$-0.393908\pi$
$421$	13.4256	0.654323	0.327161	+	0.944969i	$0.393908\pi$
$422$	27.3332	1.33056
$423$	4.44885	0.216310
$424$	10.2989	0.500160
$425$	2.74217	0.133015
$426$	−14.1804	−0.687043
$427$	9.25094	0.447685
$428$	−5.54334	−0.267947
$429$	5.32451	0.257070
$430$	18.0678	0.871309
$431$	−2.64155	−0.127239	−0.0636194	−	0.997974i	$-0.520264\pi$
$431$	−2.64155	−0.127239	−0.0636194	+	0.997974i	$0.520264\pi$
$432$	1.00000	0.0481125
$433$	8.88156	0.426820	0.213410	−	0.976963i	$-0.431543\pi$
$433$	8.88156	0.426820	0.213410	+	0.976963i	$0.431543\pi$
$434$	19.2949	0.926183
$435$	38.7265	1.85679
$436$	−19.1559	−0.917399
$437$	3.01493	0.144224
$438$	14.8181	0.708035
$439$	−22.9953	−1.09751	−0.548754	−	0.835984i	$-0.684897\pi$
$439$	−22.9953	−1.09751	−0.548754	+	0.835984i	$0.684897\pi$
$440$	20.5443	0.979410
$441$	8.05668	0.383651
$442$	0.277336	0.0131915
$443$	0.196660	0.00934361	0.00467181	−	0.999989i	$-0.498513\pi$
$443$	0.196660	0.00934361	0.00467181	+	0.999989i	$0.498513\pi$
$444$	2.46936	0.117191
$445$	31.1760	1.47789
$446$	−9.43754	−0.446881
$447$	−5.91958	−0.279987
$448$	3.88029	0.183327
$449$	2.74048	0.129331	0.0646657	−	0.997907i	$-0.479402\pi$
$449$	2.74048	0.129331	0.0646657	+	0.997907i	$0.479402\pi$
$450$	−9.88752	−0.466102
$451$	−30.9025	−1.45514
$452$	−16.5498	−0.778438
$453$	21.1663	0.994481
$454$	17.8183	0.836253
$455$	−14.9719	−0.701892
$456$	−3.65879	−0.171338
$457$	3.59423	0.168131	0.0840655	−	0.996460i	$-0.473210\pi$
$457$	3.59423	0.168131	0.0840655	+	0.996460i	$0.473210\pi$
$458$	−8.92659	−0.417112
$459$	0.277336	0.0129450
$460$	3.17944	0.148242
$461$	−19.5112	−0.908727	−0.454364	−	0.890816i	$-0.650133\pi$
$461$	−19.5112	−0.908727	−0.454364	+	0.890816i	$0.650133\pi$
$462$	20.6607	0.961221
$463$	−35.6296	−1.65585	−0.827924	−	0.560841i	$-0.810478\pi$
$463$	−35.6296	−1.65585	−0.827924	+	0.560841i	$0.810478\pi$
$464$	10.0368	0.465948
$465$	−19.1862	−0.889737
$466$	8.18269	0.379056
$467$	−3.06220	−0.141702	−0.0708508	−	0.997487i	$-0.522571\pi$
$467$	−3.06220	−0.141702	−0.0708508	+	0.997487i	$0.522571\pi$
$468$	−1.00000	−0.0462250
$469$	−28.7236	−1.32633
$470$	−17.1656	−0.791789
$471$	−2.43450	−0.112176
$472$	−10.0204	−0.461227
$473$	24.9330	1.14642
$474$	1.14178	0.0524438
$475$	36.1764	1.65989
$476$	1.07615	0.0493251
$477$	−10.2989	−0.471555
$478$	7.84265	0.358715
$479$	18.0189	0.823305	0.411653	−	0.911341i	$-0.364952\pi$
$479$	18.0189	0.823305	0.411653	+	0.911341i	$0.364952\pi$
$480$	−3.85844	−0.176113
$481$	−2.46936	−0.112593
$482$	4.67358	0.212876
$483$	3.19746	0.145489
$484$	17.3504	0.788655
$485$	−42.9396	−1.94979
$486$	−1.00000	−0.0453609
$487$	−27.4929	−1.24582	−0.622911	−	0.782293i	$-0.714050\pi$
$487$	−27.4929	−1.24582	−0.622911	+	0.782293i	$0.714050\pi$
$488$	−2.38408	−0.107922
$489$	13.5810	0.614153
$490$	−31.0862	−1.40433
$491$	21.4115	0.966287	0.483144	−	0.875541i	$-0.339495\pi$
$491$	21.4115	0.966287	0.483144	+	0.875541i	$0.339495\pi$
$492$	5.80382	0.261657
$493$	2.78358	0.125366
$494$	3.65879	0.164617
$495$	−20.5443	−0.923397
$496$	−4.97252	−0.223273
$497$	55.0241	2.46817
$498$	−6.90807	−0.309558
$499$	40.2409	1.80143	0.900715	−	0.434410i	$-0.143043\pi$
$499$	40.2409	1.80143	0.900715	+	0.434410i	$0.143043\pi$
$500$	18.8582	0.843364
$501$	15.9891	0.714342
$502$	−5.65760	−0.252511
$503$	−41.1931	−1.83671	−0.918354	−	0.395759i	$-0.870482\pi$
$503$	−41.1931	−1.83671	−0.918354	+	0.395759i	$0.870482\pi$
$504$	−3.88029	−0.172842
$505$	9.03336	0.401979
$506$	4.38753	0.195049
$507$	1.00000	0.0444116
$508$	17.6642	0.783724
$509$	5.40010	0.239355	0.119678	−	0.992813i	$-0.461814\pi$
$509$	5.40010	0.239355	0.119678	+	0.992813i	$0.461814\pi$
$510$	−1.07008	−0.0473841
$511$	−57.4985	−2.54358
$512$	−1.00000	−0.0441942
$513$	3.65879	0.161539
$514$	6.88357	0.303621
$515$	−3.85844	−0.170023
$516$	−4.68269	−0.206144
$517$	−23.6879	−1.04179
$518$	−9.58184	−0.421002
$519$	−15.7820	−0.692754
$520$	3.85844	0.169204
$521$	31.8154	1.39386	0.696929	−	0.717140i	$-0.254549\pi$
$521$	31.8154	1.39386	0.696929	+	0.717140i	$0.254549\pi$
$522$	−10.0368	−0.439300
$523$	34.5666	1.51149	0.755747	−	0.654864i	$-0.227274\pi$
$523$	34.5666	1.51149	0.755747	+	0.654864i	$0.227274\pi$
$524$	21.2850	0.929839
$525$	38.3665	1.67445
$526$	−17.0875	−0.745051
$527$	−1.37906	−0.0600729
$528$	−5.32451	−0.231720
$529$	−22.3210	−0.970478
$530$	39.7377	1.72610
$531$	10.0204	0.434849
$532$	14.1972	0.615526
$533$	−5.80382	−0.251391
$534$	−8.07997	−0.349654
$535$	−21.3886	−0.924710
$536$	7.40243	0.319736
$537$	23.3585	1.00799
$538$	−9.67753	−0.417228
$539$	−42.8979	−1.84774
$540$	3.85844	0.166041
$541$	8.83781	0.379967	0.189984	−	0.981787i	$-0.439157\pi$
$541$	8.83781	0.379967	0.189984	+	0.981787i	$0.439157\pi$
$542$	8.84507	0.379928
$543$	9.70526	0.416492
$544$	−0.277336	−0.0118907
$545$	−73.9116	−3.16603
$546$	3.88029	0.166061
$547$	8.34188	0.356673	0.178336	−	0.983970i	$-0.442928\pi$
$547$	8.34188	0.356673	0.178336	+	0.983970i	$0.442928\pi$
$548$	−8.67169	−0.370436
$549$	2.38408	0.101750
$550$	52.6462	2.24484
$551$	36.7226	1.56444
$552$	−0.824024	−0.0350728
$553$	−4.43045	−0.188402
$554$	−11.7447	−0.498983
$555$	9.52786	0.404435
$556$	−13.8301	−0.586525
$557$	18.8303	0.797864	0.398932	−	0.916981i	$-0.369381\pi$
$557$	18.8303	0.797864	0.398932	+	0.916981i	$0.369381\pi$
$558$	4.97252	0.210504
$559$	4.68269	0.198057
$560$	14.9719	0.632677
$561$	−1.47668	−0.0623455
$562$	−5.95584	−0.251232
$563$	−12.9685	−0.546556	−0.273278	−	0.961935i	$-0.588108\pi$
$563$	−12.9685	−0.546556	−0.273278	+	0.961935i	$0.588108\pi$
$564$	4.44885	0.187330
$565$	−63.8564	−2.68646
$566$	−10.6198	−0.446384
$567$	3.88029	0.162957
$568$	−14.1804	−0.594996
$569$	27.4941	1.15261	0.576307	−	0.817233i	$-0.304493\pi$
$569$	27.4941	1.15261	0.576307	+	0.817233i	$0.304493\pi$
$570$	−14.1172	−0.591304
$571$	31.1754	1.30465	0.652326	−	0.757939i	$-0.273794\pi$
$571$	31.1754	1.30465	0.652326	+	0.757939i	$0.273794\pi$
$572$	5.32451	0.222629
$573$	14.7455	0.616003
$574$	−22.5205	−0.939989
$575$	8.14756	0.339777
$576$	1.00000	0.0416667
$577$	−16.3596	−0.681058	−0.340529	−	0.940234i	$-0.610606\pi$
$577$	−16.3596	−0.681058	−0.340529	+	0.940234i	$0.610606\pi$
$578$	16.9231	0.703908
$579$	2.18383	0.0907570
$580$	38.7265	1.60803
$581$	26.8053	1.11207
$582$	11.1288	0.461302
$583$	54.8367	2.27111
$584$	14.8181	0.613177
$585$	−3.85844	−0.159527
$586$	11.1321	0.459861
$587$	20.5766	0.849287	0.424643	−	0.905361i	$-0.360399\pi$
$587$	20.5766	0.849287	0.424643	+	0.905361i	$0.360399\pi$
$588$	8.05668	0.332252
$589$	−18.1934	−0.749647
$590$	−38.6631	−1.59173
$591$	−3.77989	−0.155484
$592$	2.46936	0.101490
$593$	32.6991	1.34279	0.671395	−	0.741099i	$-0.265695\pi$
$593$	32.6991	1.34279	0.671395	+	0.741099i	$0.265695\pi$
$594$	5.32451	0.218467
$595$	4.15224	0.170225
$596$	−5.91958	−0.242475
$597$	−5.73499	−0.234718
$598$	0.824024	0.0336969
$599$	−32.4528	−1.32599	−0.662994	−	0.748625i	$-0.730714\pi$
$599$	−32.4528	−1.32599	−0.662994	+	0.748625i	$0.730714\pi$
$600$	−9.88752	−0.403656
$601$	19.5184	0.796172	0.398086	−	0.917348i	$-0.369675\pi$
$601$	19.5184	0.796172	0.398086	+	0.917348i	$0.369675\pi$
$602$	18.1702	0.740562
$603$	−7.40243	−0.301450
$604$	21.1663	0.861246
$605$	66.9454	2.72172
$606$	−2.34120	−0.0951047
$607$	27.3114	1.10854	0.554268	−	0.832338i	$-0.312998\pi$
$607$	27.3114	1.10854	0.554268	+	0.832338i	$0.312998\pi$
$608$	−3.65879	−0.148383
$609$	38.9459	1.57817
$610$	−9.19883	−0.372450
$611$	−4.44885	−0.179981
$612$	0.277336	0.0112107
$613$	33.8820	1.36848	0.684241	−	0.729256i	$-0.260134\pi$
$613$	33.8820	1.36848	0.684241	+	0.729256i	$0.260134\pi$
$614$	−8.56114	−0.345500
$615$	22.3937	0.903000
$616$	20.6607	0.832442
$617$	−13.1208	−0.528225	−0.264112	−	0.964492i	$-0.585079\pi$
$617$	−13.1208	−0.528225	−0.264112	+	0.964492i	$0.585079\pi$
$618$	1.00000	0.0402259
$619$	−8.89296	−0.357438	−0.178719	−	0.983900i	$-0.557195\pi$
$619$	−8.89296	−0.357438	−0.178719	+	0.983900i	$0.557195\pi$
$620$	−19.1862	−0.770535
$621$	0.824024	0.0330670
$622$	6.64717	0.266527
$623$	31.3527	1.25612
$624$	−1.00000	−0.0400320
$625$	23.3255	0.933019
$626$	−4.58305	−0.183175
$627$	−19.4813	−0.778007
$628$	−2.43450	−0.0971472
$629$	0.684843	0.0273065
$630$	−14.9719	−0.596493
$631$	−11.0790	−0.441049	−0.220525	−	0.975381i	$-0.570777\pi$
$631$	−11.0790	−0.441049	−0.220525	+	0.975381i	$0.570777\pi$
$632$	1.14178	0.0454177
$633$	−27.3332	−1.08640
$634$	3.18049	0.126313
$635$	68.1564	2.70470
$636$	−10.2989	−0.408379
$637$	−8.05668	−0.319217
$638$	53.4412	2.11576
$639$	14.1804	0.560968
$640$	−3.85844	−0.152518
$641$	25.1822	0.994636	0.497318	−	0.867568i	$-0.334318\pi$
$641$	25.1822	0.994636	0.497318	+	0.867568i	$0.334318\pi$
$642$	5.54334	0.218778
$643$	−23.9057	−0.942748	−0.471374	−	0.881933i	$-0.656242\pi$
$643$	−23.9057	−0.942748	−0.471374	+	0.881933i	$0.656242\pi$
$644$	3.19746	0.125997
$645$	−18.0678	−0.711420
$646$	−1.01472	−0.0399234
$647$	19.7210	0.775313	0.387657	−	0.921804i	$-0.373285\pi$
$647$	19.7210	0.775313	0.387657	+	0.921804i	$0.373285\pi$
$648$	−1.00000	−0.0392837
$649$	−53.3538	−2.09432
$650$	9.88752	0.387821
$651$	−19.2949	−0.756225
$652$	13.5810	0.531872
$653$	−33.8860	−1.32606	−0.663031	−	0.748592i	$-0.730730\pi$
$653$	−33.8860	−1.32606	−0.663031	+	0.748592i	$0.730730\pi$
$654$	19.1559	0.749053
$655$	82.1267	3.20896
$656$	5.80382	0.226601
$657$	−14.8181	−0.578109
$658$	−17.2628	−0.672975
$659$	19.6589	0.765801	0.382900	−	0.923790i	$-0.374925\pi$
$659$	19.6589	0.765801	0.382900	+	0.923790i	$0.374925\pi$
$660$	−20.5443	−0.799685
$661$	12.9317	0.502983	0.251492	−	0.967859i	$-0.419079\pi$
$661$	12.9317	0.502983	0.251492	+	0.967859i	$0.419079\pi$
$662$	12.6377	0.491179
$663$	−0.277336	−0.0107709
$664$	−6.90807	−0.268085
$665$	54.7789	2.12423
$666$	−2.46936	−0.0956857
$667$	8.27059	0.320239
$668$	15.9891	0.618638
$669$	9.43754	0.364876
$670$	28.5618	1.10344
$671$	−12.6941	−0.490049
$672$	−3.88029	−0.149686
$673$	−26.1032	−1.00621	−0.503103	−	0.864227i	$-0.667808\pi$
$673$	−26.1032	−1.00621	−0.503103	+	0.864227i	$0.667808\pi$
$674$	−19.7606	−0.761150
$675$	9.88752	0.380571
$676$	1.00000	0.0384615
$677$	−31.8840	−1.22540	−0.612702	−	0.790314i	$-0.709917\pi$
$677$	−31.8840	−1.22540	−0.612702	+	0.790314i	$0.709917\pi$
$678$	16.5498	0.635592
$679$	−43.1829	−1.65721
$680$	−1.07008	−0.0410359
$681$	−17.8183	−0.682797
$682$	−26.4763	−1.01383
$683$	5.50929	0.210807	0.105404	−	0.994430i	$-0.466387\pi$
$683$	5.50929	0.210807	0.105404	+	0.994430i	$0.466387\pi$
$684$	3.65879	0.139897
$685$	−33.4592	−1.27841
$686$	−4.10023	−0.156547
$687$	8.92659	0.340571
$688$	−4.68269	−0.178526
$689$	10.2989	0.392358
$690$	−3.17944	−0.121039
$691$	34.5212	1.31325	0.656625	−	0.754217i	$-0.271983\pi$
$691$	34.5212	1.31325	0.656625	+	0.754217i	$0.271983\pi$
$692$	−15.7820	−0.599943
$693$	−20.6607	−0.784834
$694$	4.04555	0.153567
$695$	−53.3624	−2.02415
$696$	−10.0368	−0.380445
$697$	1.60961	0.0609684
$698$	−8.62227	−0.326358
$699$	−8.18269	−0.309498
$700$	38.3665	1.45012
$701$	−25.7039	−0.970823	−0.485411	−	0.874286i	$-0.661330\pi$
$701$	−25.7039	−0.970823	−0.485411	+	0.874286i	$0.661330\pi$
$702$	1.00000	0.0377426
$703$	9.03486	0.340756
$704$	−5.32451	−0.200675
$705$	17.1656	0.646493
$706$	−6.26141	−0.235651
$707$	9.08454	0.341659
$708$	10.0204	0.376590
$709$	−40.1891	−1.50933	−0.754667	−	0.656108i	$-0.772201\pi$
$709$	−40.1891	−1.50933	−0.754667	+	0.656108i	$0.772201\pi$
$710$	−54.7142	−2.05339
$711$	−1.14178	−0.0428202
$712$	−8.07997	−0.302810
$713$	−4.09748	−0.153452
$714$	−1.07615	−0.0402738
$715$	20.5443	0.768312
$716$	23.3585	0.872948
$717$	−7.84265	−0.292889
$718$	17.9502	0.669894
$719$	38.8496	1.44885	0.724423	−	0.689356i	$-0.242106\pi$
$719$	38.8496	1.44885	0.724423	+	0.689356i	$0.242106\pi$
$720$	3.85844	0.143795
$721$	−3.88029	−0.144510
$722$	5.61326	0.208904
$723$	−4.67358	−0.173812
$724$	9.70526	0.360693
$725$	99.2394	3.68566
$726$	−17.3504	−0.643934
$727$	−25.9987	−0.964238	−0.482119	−	0.876106i	$-0.660133\pi$
$727$	−25.9987	−0.964238	−0.482119	+	0.876106i	$0.660133\pi$
$728$	3.88029	0.143813
$729$	1.00000	0.0370370
$730$	57.1746	2.11613
$731$	−1.29868	−0.0480334
$732$	2.38408	0.0881183
$733$	−2.74844	−0.101516	−0.0507579	−	0.998711i	$-0.516164\pi$
$733$	−2.74844	−0.101516	−0.0507579	+	0.998711i	$0.516164\pi$
$734$	21.9556	0.810398
$735$	31.0862	1.14663
$736$	−0.824024	−0.0303739
$737$	39.4143	1.45185
$738$	−5.80382	−0.213642
$739$	−51.9138	−1.90968	−0.954841	−	0.297118i	$-0.903974\pi$
$739$	−51.9138	−1.90968	−0.954841	+	0.297118i	$0.903974\pi$
$740$	9.52786	0.350251
$741$	−3.65879	−0.134409
$742$	39.9628	1.46708
$743$	3.26212	0.119675	0.0598377	−	0.998208i	$-0.480942\pi$
$743$	3.26212	0.119675	0.0598377	+	0.998208i	$0.480942\pi$
$744$	4.97252	0.182302
$745$	−22.8403	−0.836805
$746$	27.5391	1.00828
$747$	6.90807	0.252753
$748$	−1.47668	−0.0539928
$749$	−21.5098	−0.785950
$750$	−18.8582	−0.688604
$751$	18.4495	0.673230	0.336615	−	0.941642i	$-0.390718\pi$
$751$	18.4495	0.673230	0.336615	+	0.941642i	$0.390718\pi$
$752$	4.44885	0.162233
$753$	5.65760	0.206174
$754$	10.0368	0.365520
$755$	81.6689	2.97224
$756$	3.88029	0.141125
$757$	−1.61678	−0.0587627	−0.0293814	−	0.999568i	$-0.509354\pi$
$757$	−1.61678	−0.0587627	−0.0293814	+	0.999568i	$0.509354\pi$
$758$	30.3231	1.10139
$759$	−4.38753	−0.159257
$760$	−14.1172	−0.512085
$761$	12.0570	0.437067	0.218534	−	0.975829i	$-0.429873\pi$
$761$	12.0570	0.437067	0.218534	+	0.975829i	$0.429873\pi$
$762$	−17.6642	−0.639908
$763$	−74.3303	−2.69094
$764$	14.7455	0.533474
$765$	1.07008	0.0386890
$766$	−19.7901	−0.715045
$767$	−10.0204	−0.361816
$768$	1.00000	0.0360844
$769$	−51.0158	−1.83968	−0.919838	−	0.392299i	$-0.871680\pi$
$769$	−51.0158	−1.83968	−0.919838	+	0.392299i	$0.871680\pi$
$770$	79.7178	2.87283
$771$	−6.88357	−0.247906
$772$	2.18383	0.0785979
$773$	−31.2100	−1.12254	−0.561272	−	0.827631i	$-0.689688\pi$
$773$	−31.2100	−1.12254	−0.561272	+	0.827631i	$0.689688\pi$
$774$	4.68269	0.168316
$775$	−49.1659	−1.76609
$776$	11.1288	0.399500
$777$	9.58184	0.343747
$778$	15.2114	0.545353
$779$	21.2350	0.760821
$780$	−3.85844	−0.138154
$781$	−75.5037	−2.70173
$782$	−0.228532	−0.00817229
$783$	10.0368	0.358687
$784$	8.05668	0.287739
$785$	−9.39337	−0.335264
$786$	−21.2850	−0.759210
$787$	27.7816	0.990307	0.495154	−	0.868805i	$-0.335112\pi$
$787$	27.7816	0.990307	0.495154	+	0.868805i	$0.335112\pi$
$788$	−3.77989	−0.134653
$789$	17.0875	0.608332
$790$	4.40549	0.156740
$791$	−64.2182	−2.28334
$792$	5.32451	0.189198
$793$	−2.38408	−0.0846613
$794$	−13.0316	−0.462473
$795$	−39.7377	−1.40935
$796$	−5.73499	−0.203271
$797$	−21.7841	−0.771632	−0.385816	−	0.922576i	$-0.626080\pi$
$797$	−21.7841	−0.771632	−0.385816	+	0.922576i	$0.626080\pi$
$798$	−14.1972	−0.502575
$799$	1.23383	0.0436497
$800$	−9.88752	−0.349577
$801$	8.07997	0.285492
$802$	11.7498	0.414901
$803$	78.8990	2.78429
$804$	−7.40243	−0.261064
$805$	12.3372	0.434828
$806$	−4.97252	−0.175150
$807$	9.67753	0.340665
$808$	−2.34120	−0.0823631
$809$	−13.5223	−0.475420	−0.237710	−	0.971336i	$-0.576397\pi$
$809$	−13.5223	−0.475420	−0.237710	+	0.971336i	$0.576397\pi$
$810$	−3.85844	−0.135572
$811$	−18.9459	−0.665281	−0.332641	−	0.943054i	$-0.607940\pi$
$811$	−18.9459	−0.665281	−0.332641	+	0.943054i	$0.607940\pi$
$812$	38.9459	1.36673
$813$	−8.84507	−0.310210
$814$	13.1481	0.460842
$815$	52.4014	1.83554
$816$	0.277336	0.00970872
$817$	−17.1330	−0.599406
$818$	0.0245352	0.000857855 0
$819$	−3.88029	−0.135588
$820$	22.3937	0.782021
$821$	−4.92907	−0.172026	−0.0860128	−	0.996294i	$-0.527413\pi$
$821$	−4.92907	−0.172026	−0.0860128	+	0.996294i	$0.527413\pi$
$822$	8.67169	0.302460
$823$	36.6840	1.27872	0.639362	−	0.768906i	$-0.279199\pi$
$823$	36.6840	1.27872	0.639362	+	0.768906i	$0.279199\pi$
$824$	1.00000	0.0348367
$825$	−52.6462	−1.83291
$826$	−38.8821	−1.35288
$827$	−34.0653	−1.18457	−0.592283	−	0.805730i	$-0.701773\pi$
$827$	−34.0653	−1.18457	−0.592283	+	0.805730i	$0.701773\pi$
$828$	0.824024	0.0286368
$829$	−50.9163	−1.76839	−0.884197	−	0.467114i	$-0.845294\pi$
$829$	−50.9163	−1.76839	−0.884197	+	0.467114i	$0.845294\pi$
$830$	−26.6543	−0.925185
$831$	11.7447	0.407418
$832$	−1.00000	−0.0346688
$833$	2.23441	0.0774177
$834$	13.8301	0.478896
$835$	61.6931	2.13498
$836$	−19.4813	−0.673773
$837$	−4.97252	−0.171876
$838$	14.7320	0.508907
$839$	−50.8167	−1.75439	−0.877194	−	0.480136i	$-0.840587\pi$
$839$	−50.8167	−1.75439	−0.877194	+	0.480136i	$0.840587\pi$
$840$	−14.9719	−0.516578
$841$	71.7380	2.47372
$842$	−13.4256	−0.462676
$843$	5.95584	0.205130
$844$	−27.3332	−0.940847
$845$	3.85844	0.132734
$846$	−4.44885	−0.152954
$847$	67.3247	2.31330
$848$	−10.2989	−0.353666
$849$	10.6198	0.364471
$850$	−2.74217	−0.0940557
$851$	2.03481	0.0697524
$852$	14.1804	0.485813
$853$	3.10533	0.106324	0.0531622	−	0.998586i	$-0.483070\pi$
$853$	3.10533	0.106324	0.0531622	+	0.998586i	$0.483070\pi$
$854$	−9.25094	−0.316561
$855$	14.1172	0.482798
$856$	5.54334	0.189467
$857$	0.968802	0.0330937	0.0165468	−	0.999863i	$-0.494733\pi$
$857$	0.968802	0.0330937	0.0165468	+	0.999863i	$0.494733\pi$
$858$	−5.32451	−0.181776
$859$	−7.73637	−0.263961	−0.131981	−	0.991252i	$-0.542134\pi$
$859$	−7.73637	−0.263961	−0.131981	+	0.991252i	$0.542134\pi$
$860$	−18.0678	−0.616108
$861$	22.5205	0.767498
$862$	2.64155	0.0899715
$863$	17.5195	0.596370	0.298185	−	0.954508i	$-0.403619\pi$
$863$	17.5195	0.596370	0.298185	+	0.954508i	$0.403619\pi$
$864$	−1.00000	−0.0340207
$865$	−60.8939	−2.07046
$866$	−8.88156	−0.301808
$867$	−16.9231	−0.574738
$868$	−19.2949	−0.654910
$869$	6.07943	0.206231
$870$	−38.7265	−1.31295
$871$	7.40243	0.250822
$872$	19.1559	0.648699
$873$	−11.1288	−0.376652
$874$	−3.01493	−0.101982
$875$	73.1753	2.47378
$876$	−14.8181	−0.500657
$877$	−47.6563	−1.60924	−0.804620	−	0.593790i	$-0.797631\pi$
$877$	−47.6563	−1.60924	−0.804620	+	0.593790i	$0.797631\pi$
$878$	22.9953	0.776055
$879$	−11.1321	−0.375475
$880$	−20.5443	−0.692547
$881$	18.6784	0.629292	0.314646	−	0.949209i	$-0.398114\pi$
$881$	18.6784	0.629292	0.314646	+	0.949209i	$0.398114\pi$
$882$	−8.05668	−0.271283
$883$	−39.3267	−1.32345	−0.661725	−	0.749747i	$-0.730175\pi$
$883$	−39.3267	−1.32345	−0.661725	+	0.749747i	$0.730175\pi$
$884$	−0.277336	−0.00932783
$885$	38.6631	1.29965
$886$	−0.196660	−0.00660693
$887$	−40.0048	−1.34323	−0.671615	−	0.740901i	$-0.734399\pi$
$887$	−40.0048	−1.34323	−0.671615	+	0.740901i	$0.734399\pi$
$888$	−2.46936	−0.0828663
$889$	68.5425	2.29884
$890$	−31.1760	−1.04502
$891$	−5.32451	−0.178378
$892$	9.43754	0.315992
$893$	16.2774	0.544702
$894$	5.91958	0.197980
$895$	90.1272	3.01262
$896$	−3.88029	−0.129632
$897$	−0.824024	−0.0275134
$898$	−2.74048	−0.0914511
$899$	−49.9084	−1.66454
$900$	9.88752	0.329584
$901$	−2.85627	−0.0951560
$902$	30.9025	1.02894
$903$	−18.1702	−0.604666
$904$	16.5498	0.550439
$905$	37.4471	1.24478
$906$	−21.1663	−0.703204
$907$	18.2889	0.607273	0.303636	−	0.952788i	$-0.401799\pi$
$907$	18.2889	0.607273	0.303636	+	0.952788i	$0.401799\pi$
$908$	−17.8183	−0.591320
$909$	2.34120	0.0776527
$910$	14.9719	0.496312
$911$	9.54766	0.316328	0.158164	−	0.987413i	$-0.449443\pi$
$911$	9.54766	0.316328	0.158164	+	0.987413i	$0.449443\pi$
$912$	3.65879	0.121155
$913$	−36.7821	−1.21731
$914$	−3.59423	−0.118887
$915$	9.19883	0.304104
$916$	8.92659	0.294943
$917$	82.5920	2.72743
$918$	−0.277336	−0.00915347
$919$	29.6461	0.977934	0.488967	−	0.872302i	$-0.337374\pi$
$919$	29.6461	0.977934	0.488967	+	0.872302i	$0.337374\pi$
$920$	−3.17944	−0.104823
$921$	8.56114	0.282099
$922$	19.5112	0.642567
$923$	−14.1804	−0.466754
$924$	−20.6607	−0.679686
$925$	24.4158	0.802788
$926$	35.6296	1.17086
$927$	−1.00000	−0.0328443
$928$	−10.0368	−0.329475
$929$	−17.4659	−0.573037	−0.286518	−	0.958075i	$-0.592498\pi$
$929$	−17.4659	−0.573037	−0.286518	+	0.958075i	$0.592498\pi$
$930$	19.1862	0.629139
$931$	29.4777	0.966092
$932$	−8.18269	−0.268033
$933$	−6.64717	−0.217619
$934$	3.06220	0.100198
$935$	−5.69768	−0.186334
$936$	1.00000	0.0326860
$937$	−51.4584	−1.68107	−0.840537	−	0.541754i	$-0.817760\pi$
$937$	−51.4584	−1.68107	−0.840537	+	0.541754i	$0.817760\pi$
$938$	28.7236	0.937859
$939$	4.58305	0.149562
$940$	17.1656	0.559879
$941$	32.3417	1.05431	0.527155	−	0.849769i	$-0.323259\pi$
$941$	32.3417	1.05431	0.527155	+	0.849769i	$0.323259\pi$
$942$	2.43450	0.0793203
$943$	4.78249	0.155739
$944$	10.0204	0.326136
$945$	14.9719	0.487035
$946$	−24.9330	−0.810642
$947$	−22.7376	−0.738873	−0.369437	−	0.929256i	$-0.620449\pi$
$947$	−22.7376	−0.738873	−0.369437	+	0.929256i	$0.620449\pi$
$948$	−1.14178	−0.0370834
$949$	14.8181	0.481015
$950$	−36.1764	−1.17372
$951$	−3.18049	−0.103134
$952$	−1.07615	−0.0348781
$953$	20.7926	0.673539	0.336770	−	0.941587i	$-0.390666\pi$
$953$	20.7926	0.673539	0.336770	+	0.941587i	$0.390666\pi$
$954$	10.2989	0.333440
$955$	56.8946	1.84107
$956$	−7.84265	−0.253650
$957$	−53.4412	−1.72751
$958$	−18.0189	−0.582165
$959$	−33.6487	−1.08657
$960$	3.85844	0.124530
$961$	−6.27400	−0.202387
$962$	2.46936	0.0796153
$963$	−5.54334	−0.178632
$964$	−4.67358	−0.150526
$965$	8.42618	0.271248
$966$	−3.19746	−0.102876
$967$	59.0769	1.89978	0.949892	−	0.312577i	$-0.101192\pi$
$967$	59.0769	1.89978	0.949892	+	0.312577i	$0.101192\pi$
$968$	−17.3504	−0.557663
$969$	1.01472	0.0325974
$970$	42.9396	1.37871
$971$	33.9019	1.08796	0.543981	−	0.839097i	$-0.316916\pi$
$971$	33.9019	1.08796	0.543981	+	0.839097i	$0.316916\pi$
$972$	1.00000	0.0320750
$973$	−53.6647	−1.72041
$974$	27.4929	0.880929
$975$	−9.88752	−0.316654
$976$	2.38408	0.0763126
$977$	20.0312	0.640855	0.320427	−	0.947273i	$-0.396173\pi$
$977$	20.0312	0.640855	0.320427	+	0.947273i	$0.396173\pi$
$978$	−13.5810	−0.434272
$979$	−43.0219	−1.37499
$980$	31.0862	0.993011
$981$	−19.1559	−0.611599
$982$	−21.4115	−0.683268
$983$	−13.1532	−0.419521	−0.209761	−	0.977753i	$-0.567269\pi$
$983$	−13.1532	−0.419521	−0.209761	+	0.977753i	$0.567269\pi$
$984$	−5.80382	−0.185019
$985$	−14.5845	−0.464699
$986$	−2.78358	−0.0886472
$987$	17.2628	0.549482
$988$	−3.65879	−0.116402
$989$	−3.85865	−0.122698
$990$	20.5443	0.652940
$991$	37.9295	1.20487	0.602436	−	0.798168i	$-0.294197\pi$
$991$	37.9295	1.20487	0.602436	+	0.798168i	$0.294197\pi$
$992$	4.97252	0.157878
$993$	−12.6377	−0.401046
$994$	−55.0241	−1.74526
$995$	−22.1281	−0.701508
$996$	6.90807	0.218891
$997$	−1.45145	−0.0459680	−0.0229840	−	0.999736i	$-0.507317\pi$
$997$	−1.45145	−0.0459680	−0.0229840	+	0.999736i	$0.507317\pi$
$998$	−40.2409	−1.27380
$999$	2.46936	0.0781271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8034.2.a.t.1.11	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8034.2.a.t.1.11	✓	11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8034.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.85844 q^{5} -1.00000 q^{6} +3.88029 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.85844 q^{5} -1.00000 q^{6} +3.88029 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.85844 q^{10} -5.32451 q^{11} +1.00000 q^{12} -1.00000 q^{13} -3.88029 q^{14} +3.85844 q^{15} +1.00000 q^{16} +0.277336 q^{17} -1.00000 q^{18} +3.65879 q^{19} +3.85844 q^{20} +3.88029 q^{21} +5.32451 q^{22} +0.824024 q^{23} -1.00000 q^{24} +9.88752 q^{25} +1.00000 q^{26} +1.00000 q^{27} +3.88029 q^{28} +10.0368 q^{29} -3.85844 q^{30} -4.97252 q^{31} -1.00000 q^{32} -5.32451 q^{33} -0.277336 q^{34} +14.9719 q^{35} +1.00000 q^{36} +2.46936 q^{37} -3.65879 q^{38} -1.00000 q^{39} -3.85844 q^{40} +5.80382 q^{41} -3.88029 q^{42} -4.68269 q^{43} -5.32451 q^{44} +3.85844 q^{45} -0.824024 q^{46} +4.44885 q^{47} +1.00000 q^{48} +8.05668 q^{49} -9.88752 q^{50} +0.277336 q^{51} -1.00000 q^{52} -10.2989 q^{53} -1.00000 q^{54} -20.5443 q^{55} -3.88029 q^{56} +3.65879 q^{57} -10.0368 q^{58} +10.0204 q^{59} +3.85844 q^{60} +2.38408 q^{61} +4.97252 q^{62} +3.88029 q^{63} +1.00000 q^{64} -3.85844 q^{65} +5.32451 q^{66} -7.40243 q^{67} +0.277336 q^{68} +0.824024 q^{69} -14.9719 q^{70} +14.1804 q^{71} -1.00000 q^{72} -14.8181 q^{73} -2.46936 q^{74} +9.88752 q^{75} +3.65879 q^{76} -20.6607 q^{77} +1.00000 q^{78} -1.14178 q^{79} +3.85844 q^{80} +1.00000 q^{81} -5.80382 q^{82} +6.90807 q^{83} +3.88029 q^{84} +1.07008 q^{85} +4.68269 q^{86} +10.0368 q^{87} +5.32451 q^{88} +8.07997 q^{89} -3.85844 q^{90} -3.88029 q^{91} +0.824024 q^{92} -4.97252 q^{93} -4.44885 q^{94} +14.1172 q^{95} -1.00000 q^{96} -11.1288 q^{97} -8.05668 q^{98} -5.32451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 11 q^{2} + 11 q^{3} + 11 q^{4} + 4 q^{5} - 11 q^{6} + 4 q^{7} - 11 q^{8} + 11 q^{9}+O(q^{10}) \) 11 * q - 11 * q^2 + 11 * q^3 + 11 * q^4 + 4 * q^5 - 11 * q^6 + 4 * q^7 - 11 * q^8 + 11 * q^9	\( 11 q - 11 q^{2} + 11 q^{3} + 11 q^{4} + 4 q^{5} - 11 q^{6} + 4 q^{7} - 11 q^{8} + 11 q^{9} - 4 q^{10} + 5 q^{11} + 11 q^{12} - 11 q^{13} - 4 q^{14} + 4 q^{15} + 11 q^{16} + 8 q^{17} - 11 q^{18} - 2 q^{19} + 4 q^{20} + 4 q^{21} - 5 q^{22} + 3 q^{23} - 11 q^{24} + 9 q^{25} + 11 q^{26} + 11 q^{27} + 4 q^{28} + 7 q^{29} - 4 q^{30} + 20 q^{31} - 11 q^{32} + 5 q^{33} - 8 q^{34} + 9 q^{35} + 11 q^{36} + q^{37} + 2 q^{38} - 11 q^{39} - 4 q^{40} + 37 q^{41} - 4 q^{42} - 16 q^{43} + 5 q^{44} + 4 q^{45} - 3 q^{46} + 28 q^{47} + 11 q^{48} + 17 q^{49} - 9 q^{50} + 8 q^{51} - 11 q^{52} - 5 q^{53} - 11 q^{54} - 28 q^{55} - 4 q^{56} - 2 q^{57} - 7 q^{58} + 31 q^{59} + 4 q^{60} + 8 q^{61} - 20 q^{62} + 4 q^{63} + 11 q^{64} - 4 q^{65} - 5 q^{66} - 22 q^{67} + 8 q^{68} + 3 q^{69} - 9 q^{70} + 42 q^{71} - 11 q^{72} - 4 q^{73} - q^{74} + 9 q^{75} - 2 q^{76} - 21 q^{77} + 11 q^{78} + 33 q^{79} + 4 q^{80} + 11 q^{81} - 37 q^{82} + 18 q^{83} + 4 q^{84} + 17 q^{85} + 16 q^{86} + 7 q^{87} - 5 q^{88} + 67 q^{89} - 4 q^{90} - 4 q^{91} + 3 q^{92} + 20 q^{93} - 28 q^{94} + 32 q^{95} - 11 q^{96} - 15 q^{97} - 17 q^{98} + 5 q^{99}+O(q^{100}) \) 11 * q - 11 * q^2 + 11 * q^3 + 11 * q^4 + 4 * q^5 - 11 * q^6 + 4 * q^7 - 11 * q^8 + 11 * q^9 - 4 * q^10 + 5 * q^11 + 11 * q^12 - 11 * q^13 - 4 * q^14 + 4 * q^15 + 11 * q^16 + 8 * q^17 - 11 * q^18 - 2 * q^19 + 4 * q^20 + 4 * q^21 - 5 * q^22 + 3 * q^23 - 11 * q^24 + 9 * q^25 + 11 * q^26 + 11 * q^27 + 4 * q^28 + 7 * q^29 - 4 * q^30 + 20 * q^31 - 11 * q^32 + 5 * q^33 - 8 * q^34 + 9 * q^35 + 11 * q^36 + q^37 + 2 * q^38 - 11 * q^39 - 4 * q^40 + 37 * q^41 - 4 * q^42 - 16 * q^43 + 5 * q^44 + 4 * q^45 - 3 * q^46 + 28 * q^47 + 11 * q^48 + 17 * q^49 - 9 * q^50 + 8 * q^51 - 11 * q^52 - 5 * q^53 - 11 * q^54 - 28 * q^55 - 4 * q^56 - 2 * q^57 - 7 * q^58 + 31 * q^59 + 4 * q^60 + 8 * q^61 - 20 * q^62 + 4 * q^63 + 11 * q^64 - 4 * q^65 - 5 * q^66 - 22 * q^67 + 8 * q^68 + 3 * q^69 - 9 * q^70 + 42 * q^71 - 11 * q^72 - 4 * q^73 - q^74 + 9 * q^75 - 2 * q^76 - 21 * q^77 + 11 * q^78 + 33 * q^79 + 4 * q^80 + 11 * q^81 - 37 * q^82 + 18 * q^83 + 4 * q^84 + 17 * q^85 + 16 * q^86 + 7 * q^87 - 5 * q^88 + 67 * q^89 - 4 * q^90 - 4 * q^91 + 3 * q^92 + 20 * q^93 - 28 * q^94 + 32 * q^95 - 11 * q^96 - 15 * q^97 - 17 * q^98 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more