LMFDB - Embedded newform 2166.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2166,2,Mod(1,2166)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2166, 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("2166.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2166 = 2 \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2166.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:	$17.2955970778$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	2166.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.18479 q^{5} -1.00000 q^{6} -1.46791 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.18479 q^{5} -1.00000 q^{6} -1.46791 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.18479 q^{10} +1.18479 q^{11} -1.00000 q^{12} -2.57398 q^{13} -1.46791 q^{14} +1.18479 q^{15} +1.00000 q^{16} +2.22668 q^{17} +1.00000 q^{18} -1.18479 q^{20} +1.46791 q^{21} +1.18479 q^{22} +1.18479 q^{23} -1.00000 q^{24} -3.59627 q^{25} -2.57398 q^{26} -1.00000 q^{27} -1.46791 q^{28} -9.41147 q^{29} +1.18479 q^{30} +7.92902 q^{31} +1.00000 q^{32} -1.18479 q^{33} +2.22668 q^{34} +1.73917 q^{35} +1.00000 q^{36} +0.0641778 q^{37} +2.57398 q^{39} -1.18479 q^{40} -9.68004 q^{41} +1.46791 q^{42} -7.24897 q^{43} +1.18479 q^{44} -1.18479 q^{45} +1.18479 q^{46} -4.86484 q^{47} -1.00000 q^{48} -4.84524 q^{49} -3.59627 q^{50} -2.22668 q^{51} -2.57398 q^{52} +1.45336 q^{53} -1.00000 q^{54} -1.40373 q^{55} -1.46791 q^{56} -9.41147 q^{58} -1.40373 q^{59} +1.18479 q^{60} +7.80066 q^{61} +7.92902 q^{62} -1.46791 q^{63} +1.00000 q^{64} +3.04963 q^{65} -1.18479 q^{66} -8.71688 q^{67} +2.22668 q^{68} -1.18479 q^{69} +1.73917 q^{70} -14.6382 q^{71} +1.00000 q^{72} +6.53983 q^{73} +0.0641778 q^{74} +3.59627 q^{75} -1.73917 q^{77} +2.57398 q^{78} -12.9017 q^{79} -1.18479 q^{80} +1.00000 q^{81} -9.68004 q^{82} +3.89899 q^{83} +1.46791 q^{84} -2.63816 q^{85} -7.24897 q^{86} +9.41147 q^{87} +1.18479 q^{88} -12.4115 q^{89} -1.18479 q^{90} +3.77837 q^{91} +1.18479 q^{92} -7.92902 q^{93} -4.86484 q^{94} -1.00000 q^{96} -12.9145 q^{97} -4.84524 q^{98} +1.18479 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} - 9 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 - 9 * q^7 + 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} - 9 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{12} - 9 q^{14} + 3 q^{16} + 3 q^{18} + 9 q^{21} - 3 q^{24} + 3 q^{25} - 3 q^{27} - 9 q^{28} - 18 q^{29} - 9 q^{31} + 3 q^{32} - 9 q^{35} + 3 q^{36} - 9 q^{37} - 9 q^{41} + 9 q^{42} - 9 q^{43} + 9 q^{47} - 3 q^{48} + 12 q^{49} + 3 q^{50} - 9 q^{53} - 3 q^{54} - 18 q^{55} - 9 q^{56} - 18 q^{58} - 18 q^{59} + 9 q^{61} - 9 q^{62} - 9 q^{63} + 3 q^{64} - 18 q^{65} - 18 q^{67} - 9 q^{70} - 27 q^{71} + 3 q^{72} - 9 q^{73} - 9 q^{74} - 3 q^{75} + 9 q^{77} - 27 q^{79} + 3 q^{81} - 9 q^{82} + 9 q^{83} + 9 q^{84} + 9 q^{85} - 9 q^{86} + 18 q^{87} - 27 q^{89} + 3 q^{91} + 9 q^{93} + 9 q^{94} - 3 q^{96} + 12 q^{97} + 12 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 - 9 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^12 - 9 * q^14 + 3 * q^16 + 3 * q^18 + 9 * q^21 - 3 * q^24 + 3 * q^25 - 3 * q^27 - 9 * q^28 - 18 * q^29 - 9 * q^31 + 3 * q^32 - 9 * q^35 + 3 * q^36 - 9 * q^37 - 9 * q^41 + 9 * q^42 - 9 * q^43 + 9 * q^47 - 3 * q^48 + 12 * q^49 + 3 * q^50 - 9 * q^53 - 3 * q^54 - 18 * q^55 - 9 * q^56 - 18 * q^58 - 18 * q^59 + 9 * q^61 - 9 * q^62 - 9 * q^63 + 3 * q^64 - 18 * q^65 - 18 * q^67 - 9 * q^70 - 27 * q^71 + 3 * q^72 - 9 * q^73 - 9 * q^74 - 3 * q^75 + 9 * q^77 - 27 * q^79 + 3 * q^81 - 9 * q^82 + 9 * q^83 + 9 * q^84 + 9 * q^85 - 9 * q^86 + 18 * q^87 - 27 * q^89 + 3 * q^91 + 9 * q^93 + 9 * q^94 - 3 * q^96 + 12 * q^97 + 12 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−1.18479	−0.529855	−0.264928	−	0.964268i	$-0.585348\pi$
$5$	−1.18479	−0.529855	−0.264928	+	0.964268i	$0.585348\pi$
$6$	−1.00000	−0.408248
$7$	−1.46791	−0.554818	−0.277409	−	0.960752i	$-0.589476\pi$
$7$	−1.46791	−0.554818	−0.277409	+	0.960752i	$0.589476\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.18479	−0.374664
$11$	1.18479	0.357228	0.178614	−	0.983919i	$-0.442839\pi$
$11$	1.18479	0.357228	0.178614	+	0.983919i	$0.442839\pi$
$12$	−1.00000	−0.288675
$13$	−2.57398	−0.713893	−0.356947	−	0.934125i	$-0.616182\pi$
$13$	−2.57398	−0.713893	−0.356947	+	0.934125i	$0.616182\pi$
$14$	−1.46791	−0.392316
$15$	1.18479	0.305912
$16$	1.00000	0.250000
$17$	2.22668	0.540050	0.270025	−	0.962853i	$-0.412968\pi$
$17$	2.22668	0.540050	0.270025	+	0.962853i	$0.412968\pi$
$18$	1.00000	0.235702
$19$	0	0
$19$	0	0
$20$	−1.18479	−0.264928
$21$	1.46791	0.320324
$22$	1.18479	0.252599
$23$	1.18479	0.247046	0.123523	−	0.992342i	$-0.460581\pi$
$23$	1.18479	0.247046	0.123523	+	0.992342i	$0.460581\pi$
$24$	−1.00000	−0.204124
$25$	−3.59627	−0.719253
$26$	−2.57398	−0.504799
$27$	−1.00000	−0.192450
$28$	−1.46791	−0.277409
$29$	−9.41147	−1.74767	−0.873833	−	0.486225i	$-0.838373\pi$
$29$	−9.41147	−1.74767	−0.873833	+	0.486225i	$0.838373\pi$
$30$	1.18479	0.216313
$31$	7.92902	1.42409	0.712047	−	0.702132i	$-0.247768\pi$
$31$	7.92902	1.42409	0.712047	+	0.702132i	$0.247768\pi$
$32$	1.00000	0.176777
$33$	−1.18479	−0.206246
$34$	2.22668	0.381873
$35$	1.73917	0.293973
$36$	1.00000	0.166667
$37$	0.0641778	0.0105508	0.00527538	−	0.999986i	$-0.498321\pi$
$37$	0.0641778	0.0105508	0.00527538	+	0.999986i	$0.498321\pi$
$38$	0	0
$39$	2.57398	0.412166
$40$	−1.18479	−0.187332
$41$	−9.68004	−1.51177	−0.755885	−	0.654705i	$-0.772793\pi$
$41$	−9.68004	−1.51177	−0.755885	+	0.654705i	$0.772793\pi$
$42$	1.46791	0.226504
$43$	−7.24897	−1.10546	−0.552729	−	0.833361i	$-0.686413\pi$
$43$	−7.24897	−1.10546	−0.552729	+	0.833361i	$0.686413\pi$
$44$	1.18479	0.178614
$45$	−1.18479	−0.176618
$46$	1.18479	0.174688
$47$	−4.86484	−0.709609	−0.354805	−	0.934940i	$-0.615453\pi$
$47$	−4.86484	−0.709609	−0.354805	+	0.934940i	$0.615453\pi$
$48$	−1.00000	−0.144338
$49$	−4.84524	−0.692177
$50$	−3.59627	−0.508589
$51$	−2.22668	−0.311798
$52$	−2.57398	−0.356947
$53$	1.45336	0.199635	0.0998174	−	0.995006i	$-0.468174\pi$
$53$	1.45336	0.199635	0.0998174	+	0.995006i	$0.468174\pi$
$54$	−1.00000	−0.136083
$55$	−1.40373	−0.189279
$56$	−1.46791	−0.196158
$57$	0	0
$58$	−9.41147	−1.23579
$59$	−1.40373	−0.182751	−0.0913753	−	0.995817i	$-0.529126\pi$
$59$	−1.40373	−0.182751	−0.0913753	+	0.995817i	$0.529126\pi$
$60$	1.18479	0.152956
$61$	7.80066	0.998772	0.499386	−	0.866380i	$-0.333559\pi$
$61$	7.80066	0.998772	0.499386	+	0.866380i	$0.333559\pi$
$62$	7.92902	1.00699
$63$	−1.46791	−0.184939
$64$	1.00000	0.125000
$65$	3.04963	0.378260
$66$	−1.18479	−0.145838
$67$	−8.71688	−1.06494	−0.532468	−	0.846450i	$-0.678735\pi$
$67$	−8.71688	−1.06494	−0.532468	+	0.846450i	$0.678735\pi$
$68$	2.22668	0.270025
$69$	−1.18479	−0.142632
$70$	1.73917	0.207871
$71$	−14.6382	−1.73723	−0.868615	−	0.495488i	$-0.834989\pi$
$71$	−14.6382	−1.73723	−0.868615	+	0.495488i	$0.834989\pi$
$72$	1.00000	0.117851
$73$	6.53983	0.765429	0.382715	−	0.923867i	$-0.374989\pi$
$73$	6.53983	0.765429	0.382715	+	0.923867i	$0.374989\pi$
$74$	0.0641778	0.00746051
$75$	3.59627	0.415261
$76$	0	0
$77$	−1.73917	−0.198197
$78$	2.57398	0.291446
$79$	−12.9017	−1.45155	−0.725776	−	0.687931i	$-0.758519\pi$
$79$	−12.9017	−1.45155	−0.725776	+	0.687931i	$0.758519\pi$
$80$	−1.18479	−0.132464
$81$	1.00000	0.111111
$82$	−9.68004	−1.06898
$83$	3.89899	0.427969	0.213985	−	0.976837i	$-0.431356\pi$
$83$	3.89899	0.427969	0.213985	+	0.976837i	$0.431356\pi$
$84$	1.46791	0.160162
$85$	−2.63816	−0.286148
$86$	−7.24897	−0.781677
$87$	9.41147	1.00902
$88$	1.18479	0.126299
$89$	−12.4115	−1.31561	−0.657807	−	0.753187i	$-0.728516\pi$
$89$	−12.4115	−1.31561	−0.657807	+	0.753187i	$0.728516\pi$
$90$	−1.18479	−0.124888
$91$	3.77837	0.396081
$92$	1.18479	0.123523
$93$	−7.92902	−0.822201
$94$	−4.86484	−0.501770
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−12.9145	−1.31127	−0.655633	−	0.755080i	$-0.727598\pi$
$97$	−12.9145	−1.31127	−0.655633	+	0.755080i	$0.727598\pi$
$98$	−4.84524	−0.489443
$99$	1.18479	0.119076
$100$	−3.59627	−0.359627
$101$	13.7392	1.36710	0.683549	−	0.729904i	$-0.260435\pi$
$101$	13.7392	1.36710	0.683549	+	0.729904i	$0.260435\pi$
$102$	−2.22668	−0.220474
$103$	−17.1361	−1.68847	−0.844235	−	0.535973i	$-0.819945\pi$
$103$	−17.1361	−1.68847	−0.844235	+	0.535973i	$0.819945\pi$
$104$	−2.57398	−0.252399
$105$	−1.73917	−0.169726
$106$	1.45336	0.141163
$107$	10.5030	1.01536	0.507681	−	0.861545i	$-0.330503\pi$
$107$	10.5030	1.01536	0.507681	+	0.861545i	$0.330503\pi$
$108$	−1.00000	−0.0962250
$109$	8.04189	0.770273	0.385137	−	0.922860i	$-0.374154\pi$
$109$	8.04189	0.770273	0.385137	+	0.922860i	$0.374154\pi$
$110$	−1.40373	−0.133841
$111$	−0.0641778	−0.00609148
$112$	−1.46791	−0.138705
$113$	19.1411	1.80065	0.900324	−	0.435220i	$-0.143330\pi$
$113$	19.1411	1.80065	0.900324	+	0.435220i	$0.143330\pi$
$114$	0	0
$115$	−1.40373	−0.130899
$116$	−9.41147	−0.873833
$117$	−2.57398	−0.237964
$118$	−1.40373	−0.129224
$119$	−3.26857	−0.299629
$120$	1.18479	0.108156
$121$	−9.59627	−0.872388
$122$	7.80066	0.706239
$123$	9.68004	0.872820
$124$	7.92902	0.712047
$125$	10.1848	0.910956
$126$	−1.46791	−0.130772
$127$	−7.27631	−0.645668	−0.322834	−	0.946456i	$-0.604636\pi$
$127$	−7.27631	−0.645668	−0.322834	+	0.946456i	$0.604636\pi$
$128$	1.00000	0.0883883
$129$	7.24897	0.638236
$130$	3.04963	0.267470
$131$	−6.89899	−0.602767	−0.301384	−	0.953503i	$-0.597448\pi$
$131$	−6.89899	−0.602767	−0.301384	+	0.953503i	$0.597448\pi$
$132$	−1.18479	−0.103123
$133$	0	0
$134$	−8.71688	−0.753024
$135$	1.18479	0.101971
$136$	2.22668	0.190936
$137$	15.8229	1.35185	0.675923	−	0.736972i	$-0.263745\pi$
$137$	15.8229	1.35185	0.675923	+	0.736972i	$0.263745\pi$
$138$	−1.18479	−0.100856
$139$	−15.4757	−1.31263	−0.656314	−	0.754488i	$-0.727885\pi$
$139$	−15.4757	−1.31263	−0.656314	+	0.754488i	$0.727885\pi$
$140$	1.73917	0.146987
$141$	4.86484	0.409693
$142$	−14.6382	−1.22841
$143$	−3.04963	−0.255023
$144$	1.00000	0.0833333
$145$	11.1506	0.926011
$146$	6.53983	0.541240
$147$	4.84524	0.399628
$148$	0.0641778	0.00527538
$149$	2.22668	0.182417	0.0912084	−	0.995832i	$-0.470927\pi$
$149$	2.22668	0.182417	0.0912084	+	0.995832i	$0.470927\pi$
$150$	3.59627	0.293634
$151$	−6.59627	−0.536797	−0.268398	−	0.963308i	$-0.586494\pi$
$151$	−6.59627	−0.536797	−0.268398	+	0.963308i	$0.586494\pi$
$152$	0	0
$153$	2.22668	0.180017
$154$	−1.73917	−0.140146
$155$	−9.39424	−0.754563
$156$	2.57398	0.206083
$157$	9.25402	0.738551	0.369276	−	0.929320i	$-0.379606\pi$
$157$	9.25402	0.738551	0.369276	+	0.929320i	$0.379606\pi$
$158$	−12.9017	−1.02640
$159$	−1.45336	−0.115259
$160$	−1.18479	−0.0936661
$161$	−1.73917	−0.137066
$162$	1.00000	0.0785674
$163$	−3.14022	−0.245961	−0.122980	−	0.992409i	$-0.539245\pi$
$163$	−3.14022	−0.245961	−0.122980	+	0.992409i	$0.539245\pi$
$164$	−9.68004	−0.755885
$165$	1.40373	0.109280
$166$	3.89899	0.302620
$167$	−15.8726	−1.22826	−0.614128	−	0.789206i	$-0.710492\pi$
$167$	−15.8726	−1.22826	−0.614128	+	0.789206i	$0.710492\pi$
$168$	1.46791	0.113252
$169$	−6.37464	−0.490357
$170$	−2.63816	−0.202337
$171$	0	0
$172$	−7.24897	−0.552729
$173$	−3.46110	−0.263143	−0.131571	−	0.991307i	$-0.542002\pi$
$173$	−3.46110	−0.263143	−0.131571	+	0.991307i	$0.542002\pi$
$174$	9.41147	0.713482
$175$	5.27900	0.399055
$176$	1.18479	0.0893071
$177$	1.40373	0.105511
$178$	−12.4115	−0.930279
$179$	0.630415	0.0471194	0.0235597	−	0.999722i	$-0.492500\pi$
$179$	0.630415	0.0471194	0.0235597	+	0.999722i	$0.492500\pi$
$180$	−1.18479	−0.0883092
$181$	11.7888	0.876255	0.438127	−	0.898913i	$-0.355642\pi$
$181$	11.7888	0.876255	0.438127	+	0.898913i	$0.355642\pi$
$182$	3.77837	0.280071
$183$	−7.80066	−0.576641
$184$	1.18479	0.0873441
$185$	−0.0760373	−0.00559038
$186$	−7.92902	−0.581384
$187$	2.63816	0.192921
$188$	−4.86484	−0.354805
$189$	1.46791	0.106775
$190$	0	0
$191$	11.4192	0.826265	0.413133	−	0.910671i	$-0.364435\pi$
$191$	11.4192	0.826265	0.413133	+	0.910671i	$0.364435\pi$
$192$	−1.00000	−0.0721688
$193$	−9.49020	−0.683120	−0.341560	−	0.939860i	$-0.610955\pi$
$193$	−9.49020	−0.683120	−0.341560	+	0.939860i	$0.610955\pi$
$194$	−12.9145	−0.927205
$195$	−3.04963	−0.218389
$196$	−4.84524	−0.346088
$197$	7.13516	0.508359	0.254180	−	0.967157i	$-0.418195\pi$
$197$	7.13516	0.508359	0.254180	+	0.967157i	$0.418195\pi$
$198$	1.18479	0.0841995
$199$	−17.7520	−1.25840	−0.629202	−	0.777242i	$-0.716618\pi$
$199$	−17.7520	−1.25840	−0.629202	+	0.777242i	$0.716618\pi$
$200$	−3.59627	−0.254294
$201$	8.71688	0.614841
$202$	13.7392	0.966685
$203$	13.8152	0.969637
$204$	−2.22668	−0.155899
$205$	11.4688	0.801019
$206$	−17.1361	−1.19393
$207$	1.18479	0.0823488
$208$	−2.57398	−0.178473
$209$	0	0
$210$	−1.73917	−0.120014
$211$	−2.16250	−0.148873	−0.0744365	−	0.997226i	$-0.523716\pi$
$211$	−2.16250	−0.148873	−0.0744365	+	0.997226i	$0.523716\pi$
$212$	1.45336	0.0998174
$213$	14.6382	1.00299
$214$	10.5030	0.717970
$215$	8.58853	0.585733
$216$	−1.00000	−0.0680414
$217$	−11.6391	−0.790113
$218$	8.04189	0.544665
$219$	−6.53983	−0.441921
$220$	−1.40373	−0.0946397
$221$	−5.73143	−0.385538
$222$	−0.0641778	−0.00430733
$223$	28.2550	1.89209	0.946046	−	0.324033i	$-0.105039\pi$
$223$	28.2550	1.89209	0.946046	+	0.324033i	$0.105039\pi$
$224$	−1.46791	−0.0980789
$225$	−3.59627	−0.239751
$226$	19.1411	1.27325
$227$	21.0232	1.39536	0.697680	−	0.716409i	$-0.254216\pi$
$227$	21.0232	1.39536	0.697680	+	0.716409i	$0.254216\pi$
$228$	0	0
$229$	7.96080	0.526064	0.263032	−	0.964787i	$-0.415277\pi$
$229$	7.96080	0.526064	0.263032	+	0.964787i	$0.415277\pi$
$230$	−1.40373	−0.0925594
$231$	1.73917	0.114429
$232$	−9.41147	−0.617894
$233$	14.6878	0.962229	0.481114	−	0.876658i	$-0.340232\pi$
$233$	14.6878	0.962229	0.481114	+	0.876658i	$0.340232\pi$
$234$	−2.57398	−0.168266
$235$	5.76382	0.375990
$236$	−1.40373	−0.0913753
$237$	12.9017	0.838054
$238$	−3.26857	−0.211870
$239$	15.7297	1.01747	0.508734	−	0.860924i	$-0.330114\pi$
$239$	15.7297	1.01747	0.508734	+	0.860924i	$0.330114\pi$
$240$	1.18479	0.0764780
$241$	−9.85204	−0.634626	−0.317313	−	0.948321i	$-0.602781\pi$
$241$	−9.85204	−0.634626	−0.317313	+	0.948321i	$0.602781\pi$
$242$	−9.59627	−0.616871
$243$	−1.00000	−0.0641500
$244$	7.80066	0.499386
$245$	5.74060	0.366754
$246$	9.68004	0.617177
$247$	0	0
$248$	7.92902	0.503493
$249$	−3.89899	−0.247088
$250$	10.1848	0.644143
$251$	4.62267	0.291781	0.145890	−	0.989301i	$-0.453395\pi$
$251$	4.62267	0.291781	0.145890	+	0.989301i	$0.453395\pi$
$252$	−1.46791	−0.0924697
$253$	1.40373	0.0882520
$254$	−7.27631	−0.456556
$255$	2.63816	0.165208
$256$	1.00000	0.0625000
$257$	3.01724	0.188210	0.0941050	−	0.995562i	$-0.470001\pi$
$257$	3.01724	0.188210	0.0941050	+	0.995562i	$0.470001\pi$
$258$	7.24897	0.451301
$259$	−0.0942073	−0.00585375
$260$	3.04963	0.189130
$261$	−9.41147	−0.582556
$262$	−6.89899	−0.426221
$263$	−5.00774	−0.308791	−0.154395	−	0.988009i	$-0.549343\pi$
$263$	−5.00774	−0.308791	−0.154395	+	0.988009i	$0.549343\pi$
$264$	−1.18479	−0.0729189
$265$	−1.72193	−0.105778
$266$	0	0
$267$	12.4115	0.759570
$268$	−8.71688	−0.532468
$269$	24.5868	1.49908	0.749541	−	0.661958i	$-0.230274\pi$
$269$	24.5868	1.49908	0.749541	+	0.661958i	$0.230274\pi$
$270$	1.18479	0.0721042
$271$	−1.68004	−0.102055	−0.0510277	−	0.998697i	$-0.516250\pi$
$271$	−1.68004	−0.102055	−0.0510277	+	0.998697i	$0.516250\pi$
$272$	2.22668	0.135012
$273$	−3.77837	−0.228677
$274$	15.8229	0.955899
$275$	−4.26083	−0.256938
$276$	−1.18479	−0.0713161
$277$	−29.9813	−1.80140	−0.900702	−	0.434438i	$-0.856947\pi$
$277$	−29.9813	−1.80140	−0.900702	+	0.434438i	$0.856947\pi$
$278$	−15.4757	−0.928168
$279$	7.92902	0.474698
$280$	1.73917	0.103935
$281$	−17.6117	−1.05063	−0.525314	−	0.850908i	$-0.676052\pi$
$281$	−17.6117	−1.05063	−0.525314	+	0.850908i	$0.676052\pi$
$282$	4.86484	0.289697
$283$	15.9145	0.946017	0.473008	−	0.881058i	$-0.343168\pi$
$283$	15.9145	0.946017	0.473008	+	0.881058i	$0.343168\pi$
$284$	−14.6382	−0.868615
$285$	0	0
$286$	−3.04963	−0.180328
$287$	14.2094	0.838757
$288$	1.00000	0.0589256
$289$	−12.0419	−0.708346
$290$	11.1506	0.654788
$291$	12.9145	0.757059
$292$	6.53983	0.382715
$293$	−0.268571	−0.0156901	−0.00784503	−	0.999969i	$-0.502497\pi$
$293$	−0.268571	−0.0156901	−0.00784503	+	0.999969i	$0.502497\pi$
$294$	4.84524	0.282580
$295$	1.66313	0.0968313
$296$	0.0641778	0.00373026
$297$	−1.18479	−0.0687486
$298$	2.22668	0.128988
$299$	−3.04963	−0.176365
$300$	3.59627	0.207631
$301$	10.6408	0.613328
$302$	−6.59627	−0.379572
$303$	−13.7392	−0.789295
$304$	0	0
$305$	−9.24216	−0.529205
$306$	2.22668	0.127291
$307$	0.571290	0.0326052	0.0163026	−	0.999867i	$-0.494810\pi$
$307$	0.571290	0.0326052	0.0163026	+	0.999867i	$0.494810\pi$
$308$	−1.73917	−0.0990984
$309$	17.1361	0.974838
$310$	−9.39424	−0.533557
$311$	14.3523	0.813847	0.406924	−	0.913462i	$-0.366602\pi$
$311$	14.3523	0.813847	0.406924	+	0.913462i	$0.366602\pi$
$312$	2.57398	0.145723
$313$	−2.40373	−0.135867	−0.0679335	−	0.997690i	$-0.521641\pi$
$313$	−2.40373	−0.135867	−0.0679335	+	0.997690i	$0.521641\pi$
$314$	9.25402	0.522235
$315$	1.73917	0.0979911
$316$	−12.9017	−0.725776
$317$	−14.0574	−0.789541	−0.394770	−	0.918780i	$-0.629176\pi$
$317$	−14.0574	−0.789541	−0.394770	+	0.918780i	$0.629176\pi$
$318$	−1.45336	−0.0815006
$319$	−11.1506	−0.624316
$320$	−1.18479	−0.0662319
$321$	−10.5030	−0.586220
$322$	−1.73917	−0.0969202
$323$	0	0
$324$	1.00000	0.0555556
$325$	9.25671	0.513470
$326$	−3.14022	−0.173920
$327$	−8.04189	−0.444717
$328$	−9.68004	−0.534491
$329$	7.14115	0.393704
$330$	1.40373	0.0772730
$331$	−26.8425	−1.47540	−0.737700	−	0.675129i	$-0.764088\pi$
$331$	−26.8425	−1.47540	−0.737700	+	0.675129i	$0.764088\pi$
$332$	3.89899	0.213985
$333$	0.0641778	0.00351692
$334$	−15.8726	−0.868509
$335$	10.3277	0.564262
$336$	1.46791	0.0800811
$337$	20.9213	1.13965	0.569827	−	0.821765i	$-0.307010\pi$
$337$	20.9213	1.13965	0.569827	+	0.821765i	$0.307010\pi$
$338$	−6.37464	−0.346735
$339$	−19.1411	−1.03960
$340$	−2.63816	−0.143074
$341$	9.39424	0.508727
$342$	0	0
$343$	17.3878	0.938851
$344$	−7.24897	−0.390838
$345$	1.40373	0.0755745
$346$	−3.46110	−0.186070
$347$	−33.6364	−1.80570	−0.902848	−	0.429959i	$-0.858528\pi$
$347$	−33.6364	−1.80570	−0.902848	+	0.429959i	$0.858528\pi$
$348$	9.41147	0.504508
$349$	27.1516	1.45339	0.726695	−	0.686960i	$-0.241055\pi$
$349$	27.1516	1.45339	0.726695	+	0.686960i	$0.241055\pi$
$350$	5.27900	0.282174
$351$	2.57398	0.137389
$352$	1.18479	0.0631497
$353$	−16.7452	−0.891255	−0.445627	−	0.895219i	$-0.647019\pi$
$353$	−16.7452	−0.891255	−0.445627	+	0.895219i	$0.647019\pi$
$354$	1.40373	0.0746076
$355$	17.3432	0.920480
$356$	−12.4115	−0.657807
$357$	3.26857	0.172991
$358$	0.630415	0.0333185
$359$	−4.73917	−0.250124	−0.125062	−	0.992149i	$-0.539913\pi$
$359$	−4.73917	−0.250124	−0.125062	+	0.992149i	$0.539913\pi$
$360$	−1.18479	−0.0624440
$361$	0	0
$362$	11.7888	0.619606
$363$	9.59627	0.503673
$364$	3.77837	0.198040
$365$	−7.74834	−0.405567
$366$	−7.80066	−0.407747
$367$	8.79292	0.458987	0.229493	−	0.973310i	$-0.426293\pi$
$367$	8.79292	0.458987	0.229493	+	0.973310i	$0.426293\pi$
$368$	1.18479	0.0617616
$369$	−9.68004	−0.503923
$370$	−0.0760373	−0.00395299
$371$	−2.13341	−0.110761
$372$	−7.92902	−0.411100
$373$	9.44562	0.489076	0.244538	−	0.969640i	$-0.421364\pi$
$373$	9.44562	0.489076	0.244538	+	0.969640i	$0.421364\pi$
$374$	2.63816	0.136416
$375$	−10.1848	−0.525940
$376$	−4.86484	−0.250885
$377$	24.2249	1.24765
$378$	1.46791	0.0755012
$379$	−36.9077	−1.89582	−0.947910	−	0.318540i	$-0.896808\pi$
$379$	−36.9077	−1.89582	−0.947910	+	0.318540i	$0.896808\pi$
$380$	0	0
$381$	7.27631	0.372777
$382$	11.4192	0.584258
$383$	29.4688	1.50579	0.752894	−	0.658142i	$-0.228657\pi$
$383$	29.4688	1.50579	0.752894	+	0.658142i	$0.228657\pi$
$384$	−1.00000	−0.0510310
$385$	2.06056	0.105016
$386$	−9.49020	−0.483038
$387$	−7.24897	−0.368486
$388$	−12.9145	−0.655633
$389$	17.6878	0.896806	0.448403	−	0.893831i	$-0.351993\pi$
$389$	17.6878	0.896806	0.448403	+	0.893831i	$0.351993\pi$
$390$	−3.04963	−0.154424
$391$	2.63816	0.133417
$392$	−4.84524	−0.244721
$393$	6.89899	0.348008
$394$	7.13516	0.359464
$395$	15.2858	0.769112
$396$	1.18479	0.0595381
$397$	12.5895	0.631847	0.315923	−	0.948785i	$-0.397686\pi$
$397$	12.5895	0.631847	0.315923	+	0.948785i	$0.397686\pi$
$398$	−17.7520	−0.889826
$399$	0	0
$400$	−3.59627	−0.179813
$401$	24.0729	1.20214	0.601070	−	0.799196i	$-0.294741\pi$
$401$	24.0729	1.20214	0.601070	+	0.799196i	$0.294741\pi$
$402$	8.71688	0.434759
$403$	−20.4091	−1.01665
$404$	13.7392	0.683549
$405$	−1.18479	−0.0588728
$406$	13.8152	0.685637
$407$	0.0760373	0.00376903
$408$	−2.22668	−0.110237
$409$	35.3601	1.74844	0.874222	−	0.485526i	$-0.161372\pi$
$409$	35.3601	1.74844	0.874222	+	0.485526i	$0.161372\pi$
$410$	11.4688	0.566406
$411$	−15.8229	−0.780488
$412$	−17.1361	−0.844235
$413$	2.06056	0.101393
$414$	1.18479	0.0582294
$415$	−4.61949	−0.226762
$416$	−2.57398	−0.126200
$417$	15.4757	0.757846
$418$	0	0
$419$	4.57903	0.223700	0.111850	−	0.993725i	$-0.464322\pi$
$419$	4.57903	0.223700	0.111850	+	0.993725i	$0.464322\pi$
$420$	−1.73917	−0.0848628
$421$	−0.278066	−0.0135521	−0.00677606	−	0.999977i	$-0.502157\pi$
$421$	−0.278066	−0.0135521	−0.00677606	+	0.999977i	$0.502157\pi$
$422$	−2.16250	−0.105269
$423$	−4.86484	−0.236536
$424$	1.45336	0.0705816
$425$	−8.00774	−0.388432
$426$	14.6382	0.709221
$427$	−11.4507	−0.554137
$428$	10.5030	0.507681
$429$	3.04963	0.147238
$430$	8.58853	0.414175
$431$	11.9828	0.577189	0.288595	−	0.957451i	$-0.406812\pi$
$431$	11.9828	0.577189	0.288595	+	0.957451i	$0.406812\pi$
$432$	−1.00000	−0.0481125
$433$	29.1480	1.40076	0.700381	−	0.713769i	$-0.253014\pi$
$433$	29.1480	1.40076	0.700381	+	0.713769i	$0.253014\pi$
$434$	−11.6391	−0.558694
$435$	−11.1506	−0.534632
$436$	8.04189	0.385137
$437$	0	0
$438$	−6.53983	−0.312485
$439$	−6.29767	−0.300571	−0.150286	−	0.988643i	$-0.548019\pi$
$439$	−6.29767	−0.300571	−0.150286	+	0.988643i	$0.548019\pi$
$440$	−1.40373	−0.0669204
$441$	−4.84524	−0.230726
$442$	−5.73143	−0.272616
$443$	20.8803	0.992054	0.496027	−	0.868307i	$-0.334792\pi$
$443$	20.8803	0.992054	0.496027	+	0.868307i	$0.334792\pi$
$444$	−0.0641778	−0.00304574
$445$	14.7050	0.697085
$446$	28.2550	1.33791
$447$	−2.22668	−0.105318
$448$	−1.46791	−0.0693523
$449$	−41.3756	−1.95263	−0.976317	−	0.216345i	$-0.930586\pi$
$449$	−41.3756	−1.95263	−0.976317	+	0.216345i	$0.930586\pi$
$450$	−3.59627	−0.169530
$451$	−11.4688	−0.540047
$452$	19.1411	0.900324
$453$	6.59627	0.309920
$454$	21.0232	0.986669
$455$	−4.47659	−0.209866
$456$	0	0
$457$	−27.6245	−1.29222	−0.646111	−	0.763244i	$-0.723606\pi$
$457$	−27.6245	−1.29222	−0.646111	+	0.763244i	$0.723606\pi$
$458$	7.96080	0.371984
$459$	−2.22668	−0.103933
$460$	−1.40373	−0.0654494
$461$	−18.5107	−0.862131	−0.431065	−	0.902321i	$-0.641862\pi$
$461$	−18.5107	−0.862131	−0.431065	+	0.902321i	$0.641862\pi$
$462$	1.73917	0.0809135
$463$	−3.61856	−0.168169	−0.0840843	−	0.996459i	$-0.526796\pi$
$463$	−3.61856	−0.168169	−0.0840843	+	0.996459i	$0.526796\pi$
$464$	−9.41147	−0.436917
$465$	9.39424	0.435647
$466$	14.6878	0.680399
$467$	−14.4020	−0.666444	−0.333222	−	0.942848i	$-0.608136\pi$
$467$	−14.4020	−0.666444	−0.333222	+	0.942848i	$0.608136\pi$
$468$	−2.57398	−0.118982
$469$	12.7956	0.590846
$470$	5.76382	0.265865
$471$	−9.25402	−0.426403
$472$	−1.40373	−0.0646121
$473$	−8.58853	−0.394901
$474$	12.9017	0.592594
$475$	0	0
$476$	−3.26857	−0.149815
$477$	1.45336	0.0665449
$478$	15.7297	0.719459
$479$	−31.8557	−1.45552	−0.727761	−	0.685831i	$-0.759439\pi$
$479$	−31.8557	−1.45552	−0.727761	+	0.685831i	$0.759439\pi$
$480$	1.18479	0.0540781
$481$	−0.165192	−0.00753211
$482$	−9.85204	−0.448748
$483$	1.73917	0.0791350
$484$	−9.59627	−0.436194
$485$	15.3010	0.694781
$486$	−1.00000	−0.0453609
$487$	19.7469	0.894818	0.447409	−	0.894329i	$-0.352347\pi$
$487$	19.7469	0.894818	0.447409	+	0.894329i	$0.352347\pi$
$488$	7.80066	0.353119
$489$	3.14022	0.142005
$490$	5.74060	0.259334
$491$	−29.1830	−1.31701	−0.658506	−	0.752575i	$-0.728811\pi$
$491$	−29.1830	−1.31701	−0.658506	+	0.752575i	$0.728811\pi$
$492$	9.68004	0.436410
$493$	−20.9564	−0.943827
$494$	0	0
$495$	−1.40373	−0.0630931
$496$	7.92902	0.356023
$497$	21.4875	0.963847
$498$	−3.89899	−0.174718
$499$	−5.38413	−0.241027	−0.120513	−	0.992712i	$-0.538454\pi$
$499$	−5.38413	−0.241027	−0.120513	+	0.992712i	$0.538454\pi$
$500$	10.1848	0.455478
$501$	15.8726	0.709134
$502$	4.62267	0.206320
$503$	24.5868	1.09627	0.548135	−	0.836390i	$-0.315338\pi$
$503$	24.5868	1.09627	0.548135	+	0.836390i	$0.315338\pi$
$504$	−1.46791	−0.0653860
$505$	−16.2781	−0.724364
$506$	1.40373	0.0624036
$507$	6.37464	0.283108
$508$	−7.27631	−0.322834
$509$	4.75641	0.210824	0.105412	−	0.994429i	$-0.466384\pi$
$509$	4.75641	0.210824	0.105412	+	0.994429i	$0.466384\pi$
$510$	2.63816	0.116819
$511$	−9.59989	−0.424674
$512$	1.00000	0.0441942
$513$	0	0
$514$	3.01724	0.133085
$515$	20.3027	0.894645
$516$	7.24897	0.319118
$517$	−5.76382	−0.253493
$518$	−0.0942073	−0.00413923
$519$	3.46110	0.151926
$520$	3.04963	0.133735
$521$	29.6290	1.29807	0.649035	−	0.760759i	$-0.275173\pi$
$521$	29.6290	1.29807	0.649035	+	0.760759i	$0.275173\pi$
$522$	−9.41147	−0.411929
$523$	10.7365	0.469474	0.234737	−	0.972059i	$-0.424577\pi$
$523$	10.7365	0.469474	0.234737	+	0.972059i	$0.424577\pi$
$524$	−6.89899	−0.301384
$525$	−5.27900	−0.230394
$526$	−5.00774	−0.218348
$527$	17.6554	0.769081
$528$	−1.18479	−0.0515615
$529$	−21.5963	−0.938968
$530$	−1.72193	−0.0747960
$531$	−1.40373	−0.0609168
$532$	0	0
$533$	24.9162	1.07924
$534$	12.4115	0.537097
$535$	−12.4439	−0.537995
$536$	−8.71688	−0.376512
$537$	−0.630415	−0.0272044
$538$	24.5868	1.06001
$539$	−5.74060	−0.247265
$540$	1.18479	0.0509854
$541$	29.6064	1.27288	0.636439	−	0.771327i	$-0.280407\pi$
$541$	29.6064	1.27288	0.636439	+	0.771327i	$0.280407\pi$
$542$	−1.68004	−0.0721641
$543$	−11.7888	−0.505906
$544$	2.22668	0.0954682
$545$	−9.52797	−0.408133
$546$	−3.77837	−0.161699
$547$	−13.3628	−0.571351	−0.285676	−	0.958326i	$-0.592218\pi$
$547$	−13.3628	−0.571351	−0.285676	+	0.958326i	$0.592218\pi$
$548$	15.8229	0.675923
$549$	7.80066	0.332924
$550$	−4.26083	−0.181682
$551$	0	0
$552$	−1.18479	−0.0504281
$553$	18.9385	0.805347
$554$	−29.9813	−1.27378
$555$	0.0760373	0.00322761
$556$	−15.4757	−0.656314
$557$	−3.39424	−0.143818	−0.0719092	−	0.997411i	$-0.522909\pi$
$557$	−3.39424	−0.143818	−0.0719092	+	0.997411i	$0.522909\pi$
$558$	7.92902	0.335662
$559$	18.6587	0.789178
$560$	1.73917	0.0734934
$561$	−2.63816	−0.111383
$562$	−17.6117	−0.742907
$563$	41.5449	1.75091	0.875454	−	0.483301i	$-0.160562\pi$
$563$	41.5449	1.75091	0.875454	+	0.483301i	$0.160562\pi$
$564$	4.86484	0.204847
$565$	−22.6783	−0.954083
$566$	15.9145	0.668935
$567$	−1.46791	−0.0616465
$568$	−14.6382	−0.614203
$569$	−2.17705	−0.0912668	−0.0456334	−	0.998958i	$-0.514531\pi$
$569$	−2.17705	−0.0912668	−0.0456334	+	0.998958i	$0.514531\pi$
$570$	0	0
$571$	−2.50980	−0.105032	−0.0525159	−	0.998620i	$-0.516724\pi$
$571$	−2.50980	−0.105032	−0.0525159	+	0.998620i	$0.516724\pi$
$572$	−3.04963	−0.127511
$573$	−11.4192	−0.477045
$574$	14.2094	0.593091
$575$	−4.26083	−0.177689
$576$	1.00000	0.0416667
$577$	11.5107	0.479198	0.239599	−	0.970872i	$-0.422984\pi$
$577$	11.5107	0.479198	0.239599	+	0.970872i	$0.422984\pi$
$578$	−12.0419	−0.500877
$579$	9.49020	0.394399
$580$	11.1506	0.463005
$581$	−5.72336	−0.237445
$582$	12.9145	0.535322
$583$	1.72193	0.0713152
$584$	6.53983	0.270620
$585$	3.04963	0.126087
$586$	−0.268571	−0.0110946
$587$	16.6955	0.689098	0.344549	−	0.938768i	$-0.388032\pi$
$587$	16.6955	0.689098	0.344549	+	0.938768i	$0.388032\pi$
$588$	4.84524	0.199814
$589$	0	0
$590$	1.66313	0.0684701
$591$	−7.13516	−0.293501
$592$	0.0641778	0.00263769
$593$	44.5853	1.83090	0.915450	−	0.402431i	$-0.131835\pi$
$593$	44.5853	1.83090	0.915450	+	0.402431i	$0.131835\pi$
$594$	−1.18479	−0.0486126
$595$	3.87258	0.158760
$596$	2.22668	0.0912084
$597$	17.7520	0.726539
$598$	−3.04963	−0.124709
$599$	8.51249	0.347811	0.173905	−	0.984762i	$-0.444361\pi$
$599$	8.51249	0.347811	0.173905	+	0.984762i	$0.444361\pi$
$600$	3.59627	0.146817
$601$	−33.5226	−1.36742	−0.683708	−	0.729756i	$-0.739634\pi$
$601$	−33.5226	−1.36742	−0.683708	+	0.729756i	$0.739634\pi$
$602$	10.6408	0.433688
$603$	−8.71688	−0.354979
$604$	−6.59627	−0.268398
$605$	11.3696	0.462239
$606$	−13.7392	−0.558116
$607$	−1.89393	−0.0768724	−0.0384362	−	0.999261i	$-0.512238\pi$
$607$	−1.89393	−0.0768724	−0.0384362	+	0.999261i	$0.512238\pi$
$608$	0	0
$609$	−13.8152	−0.559820
$610$	−9.24216	−0.374204
$611$	12.5220	0.506585
$612$	2.22668	0.0900083
$613$	8.49558	0.343133	0.171567	−	0.985173i	$-0.445117\pi$
$613$	8.49558	0.343133	0.171567	+	0.985173i	$0.445117\pi$
$614$	0.571290	0.0230554
$615$	−11.4688	−0.462468
$616$	−1.73917	−0.0700732
$617$	−28.1676	−1.13398	−0.566992	−	0.823724i	$-0.691893\pi$
$617$	−28.1676	−1.13398	−0.566992	+	0.823724i	$0.691893\pi$
$618$	17.1361	0.689315
$619$	−22.8966	−0.920293	−0.460146	−	0.887843i	$-0.652203\pi$
$619$	−22.8966	−0.920293	−0.460146	+	0.887843i	$0.652203\pi$
$620$	−9.39424	−0.377282
$621$	−1.18479	−0.0475441
$622$	14.3523	0.575477
$623$	18.2189	0.729926
$624$	2.57398	0.103042
$625$	5.91447	0.236579
$626$	−2.40373	−0.0960725
$627$	0	0
$628$	9.25402	0.369276
$629$	0.142903	0.00569793
$630$	1.73917	0.0692902
$631$	−32.4894	−1.29338	−0.646691	−	0.762752i	$-0.723848\pi$
$631$	−32.4894	−1.29338	−0.646691	+	0.762752i	$0.723848\pi$
$632$	−12.9017	−0.513201
$633$	2.16250	0.0859518
$634$	−14.0574	−0.558290
$635$	8.62092	0.342111
$636$	−1.45336	−0.0576296
$637$	12.4715	0.494140
$638$	−11.1506	−0.441458
$639$	−14.6382	−0.579076
$640$	−1.18479	−0.0468330
$641$	−7.40373	−0.292430	−0.146215	−	0.989253i	$-0.546709\pi$
$641$	−7.40373	−0.292430	−0.146215	+	0.989253i	$0.546709\pi$
$642$	−10.5030	−0.414520
$643$	−42.5485	−1.67795	−0.838975	−	0.544171i	$-0.816844\pi$
$643$	−42.5485	−1.67795	−0.838975	+	0.544171i	$0.816844\pi$
$644$	−1.73917	−0.0685329
$645$	−8.58853	−0.338173
$646$	0	0
$647$	−41.2063	−1.61999	−0.809993	−	0.586440i	$-0.800529\pi$
$647$	−41.2063	−1.61999	−0.809993	+	0.586440i	$0.800529\pi$
$648$	1.00000	0.0392837
$649$	−1.66313	−0.0652837
$650$	9.25671	0.363078
$651$	11.6391	0.456172
$652$	−3.14022	−0.122980
$653$	−26.6705	−1.04370	−0.521850	−	0.853037i	$-0.674758\pi$
$653$	−26.6705	−1.04370	−0.521850	+	0.853037i	$0.674758\pi$
$654$	−8.04189	−0.314463
$655$	8.17387	0.319379
$656$	−9.68004	−0.377942
$657$	6.53983	0.255143
$658$	7.14115	0.278391
$659$	12.5868	0.490311	0.245155	−	0.969484i	$-0.421161\pi$
$659$	12.5868	0.490311	0.245155	+	0.969484i	$0.421161\pi$
$660$	1.40373	0.0546402
$661$	−8.29498	−0.322637	−0.161319	−	0.986902i	$-0.551575\pi$
$661$	−8.29498	−0.322637	−0.161319	+	0.986902i	$0.551575\pi$
$662$	−26.8425	−1.04327
$663$	5.73143	0.222590
$664$	3.89899	0.151310
$665$	0	0
$666$	0.0641778	0.00248684
$667$	−11.1506	−0.431755
$668$	−15.8726	−0.614128
$669$	−28.2550	−1.09240
$670$	10.3277	0.398994
$671$	9.24216	0.356790
$672$	1.46791	0.0566259
$673$	−1.62536	−0.0626531	−0.0313266	−	0.999509i	$-0.509973\pi$
$673$	−1.62536	−0.0626531	−0.0313266	+	0.999509i	$0.509973\pi$
$674$	20.9213	0.805857
$675$	3.59627	0.138420
$676$	−6.37464	−0.245178
$677$	13.2841	0.510548	0.255274	−	0.966869i	$-0.417834\pi$
$677$	13.2841	0.510548	0.255274	+	0.966869i	$0.417834\pi$
$678$	−19.1411	−0.735111
$679$	18.9573	0.727514
$680$	−2.63816	−0.101169
$681$	−21.0232	−0.805612
$682$	9.39424	0.359724
$683$	−2.90673	−0.111223	−0.0556114	−	0.998452i	$-0.517711\pi$
$683$	−2.90673	−0.111223	−0.0556114	+	0.998452i	$0.517711\pi$
$684$	0	0
$685$	−18.7469	−0.716283
$686$	17.3878	0.663868
$687$	−7.96080	−0.303723
$688$	−7.24897	−0.276364
$689$	−3.74092	−0.142518
$690$	1.40373	0.0534392
$691$	−14.3601	−0.546284	−0.273142	−	0.961974i	$-0.588063\pi$
$691$	−14.3601	−0.546284	−0.273142	+	0.961974i	$0.588063\pi$
$692$	−3.46110	−0.131571
$693$	−1.73917	−0.0660656
$694$	−33.6364	−1.27682
$695$	18.3354	0.695503
$696$	9.41147	0.356741
$697$	−21.5544	−0.816430
$698$	27.1516	1.02770
$699$	−14.6878	−0.555543
$700$	5.27900	0.199527
$701$	48.6536	1.83762	0.918811	−	0.394697i	$-0.129150\pi$
$701$	48.6536	1.83762	0.918811	+	0.394697i	$0.129150\pi$
$702$	2.57398	0.0971485
$703$	0	0
$704$	1.18479	0.0446535
$705$	−5.76382	−0.217078
$706$	−16.7452	−0.630212
$707$	−20.1679	−0.758491
$708$	1.40373	0.0527555
$709$	−9.74422	−0.365952	−0.182976	−	0.983117i	$-0.558573\pi$
$709$	−9.74422	−0.365952	−0.182976	+	0.983117i	$0.558573\pi$
$710$	17.3432	0.650878
$711$	−12.9017	−0.483851
$712$	−12.4115	−0.465140
$713$	9.39424	0.351817
$714$	3.26857	0.122323
$715$	3.61318	0.135125
$716$	0.630415	0.0235597
$717$	−15.7297	−0.587435
$718$	−4.73917	−0.176864
$719$	4.84760	0.180785	0.0903925	−	0.995906i	$-0.471188\pi$
$719$	4.84760	0.180785	0.0903925	+	0.995906i	$0.471188\pi$
$720$	−1.18479	−0.0441546
$721$	25.1543	0.936794
$722$	0	0
$723$	9.85204	0.366401
$724$	11.7888	0.438127
$725$	33.8462	1.25702
$726$	9.59627	0.356151
$727$	−14.1916	−0.526337	−0.263169	−	0.964750i	$-0.584768\pi$
$727$	−14.1916	−0.526337	−0.263169	+	0.964750i	$0.584768\pi$
$728$	3.77837	0.140036
$729$	1.00000	0.0370370
$730$	−7.74834	−0.286779
$731$	−16.1411	−0.597002
$732$	−7.80066	−0.288321
$733$	10.6382	0.392929	0.196465	−	0.980511i	$-0.437054\pi$
$733$	10.6382	0.392929	0.196465	+	0.980511i	$0.437054\pi$
$734$	8.79292	0.324553
$735$	−5.74060	−0.211745
$736$	1.18479	0.0436720
$737$	−10.3277	−0.380426
$738$	−9.68004	−0.356327
$739$	−27.6013	−1.01533	−0.507665	−	0.861554i	$-0.669491\pi$
$739$	−27.6013	−1.01533	−0.507665	+	0.861554i	$0.669491\pi$
$740$	−0.0760373	−0.00279519
$741$	0	0
$742$	−2.13341	−0.0783199
$743$	−1.21120	−0.0444346	−0.0222173	−	0.999753i	$-0.507073\pi$
$743$	−1.21120	−0.0444346	−0.0222173	+	0.999753i	$0.507073\pi$
$744$	−7.92902	−0.290692
$745$	−2.63816	−0.0966545
$746$	9.44562	0.345829
$747$	3.89899	0.142656
$748$	2.63816	0.0964605
$749$	−15.4175	−0.563342
$750$	−10.1848	−0.371896
$751$	−49.0506	−1.78988	−0.894940	−	0.446186i	$-0.852782\pi$
$751$	−49.0506	−1.78988	−0.894940	+	0.446186i	$0.852782\pi$
$752$	−4.86484	−0.177402
$753$	−4.62267	−0.168460
$754$	24.2249	0.882220
$755$	7.81521	0.284425
$756$	1.46791	0.0533874
$757$	36.0755	1.31119	0.655594	−	0.755114i	$-0.272418\pi$
$757$	36.0755	1.31119	0.655594	+	0.755114i	$0.272418\pi$
$758$	−36.9077	−1.34055
$759$	−1.40373	−0.0509523
$760$	0	0
$761$	4.72193	0.171170	0.0855850	−	0.996331i	$-0.472724\pi$
$761$	4.72193	0.171170	0.0855850	+	0.996331i	$0.472724\pi$
$762$	7.27631	0.263593
$763$	−11.8048	−0.427362
$764$	11.4192	0.413133
$765$	−2.63816	−0.0953827
$766$	29.4688	1.06475
$767$	3.61318	0.130464
$768$	−1.00000	−0.0360844
$769$	21.6979	0.782446	0.391223	−	0.920296i	$-0.372052\pi$
$769$	21.6979	0.782446	0.391223	+	0.920296i	$0.372052\pi$
$770$	2.06056	0.0742573
$771$	−3.01724	−0.108663
$772$	−9.49020	−0.341560
$773$	−44.0898	−1.58580	−0.792899	−	0.609353i	$-0.791429\pi$
$773$	−44.0898	−1.58580	−0.792899	+	0.609353i	$0.791429\pi$
$774$	−7.24897	−0.260559
$775$	−28.5149	−1.02428
$776$	−12.9145	−0.463602
$777$	0.0942073	0.00337967
$778$	17.6878	0.634138
$779$	0	0
$780$	−3.04963	−0.109194
$781$	−17.3432	−0.620588
$782$	2.63816	0.0943403
$783$	9.41147	0.336339
$784$	−4.84524	−0.173044
$785$	−10.9641	−0.391325
$786$	6.89899	0.246079
$787$	20.1388	0.717870	0.358935	−	0.933363i	$-0.383140\pi$
$787$	20.1388	0.717870	0.358935	+	0.933363i	$0.383140\pi$
$788$	7.13516	0.254180
$789$	5.00774	0.178280
$790$	15.2858	0.543845
$791$	−28.0975	−0.999032
$792$	1.18479	0.0420998
$793$	−20.0787	−0.713016
$794$	12.5895	0.446783
$795$	1.72193	0.0610707
$796$	−17.7520	−0.629202
$797$	−12.0729	−0.427642	−0.213821	−	0.976873i	$-0.568591\pi$
$797$	−12.0729	−0.427642	−0.213821	+	0.976873i	$0.568591\pi$
$798$	0	0
$799$	−10.8324	−0.383224
$800$	−3.59627	−0.127147
$801$	−12.4115	−0.438538
$802$	24.0729	0.850042
$803$	7.74834	0.273433
$804$	8.71688	0.307421
$805$	2.06056	0.0726250
$806$	−20.4091	−0.718880
$807$	−24.5868	−0.865495
$808$	13.7392	0.483342
$809$	−38.9736	−1.37024	−0.685119	−	0.728431i	$-0.740250\pi$
$809$	−38.9736	−1.37024	−0.685119	+	0.728431i	$0.740250\pi$
$810$	−1.18479	−0.0416294
$811$	21.4020	0.751525	0.375763	−	0.926716i	$-0.377381\pi$
$811$	21.4020	0.751525	0.375763	+	0.926716i	$0.377381\pi$
$812$	13.8152	0.484819
$813$	1.68004	0.0589217
$814$	0.0760373	0.00266511
$815$	3.72050	0.130324
$816$	−2.22668	−0.0779494
$817$	0	0
$818$	35.3601	1.23634
$819$	3.77837	0.132027
$820$	11.4688	0.400509
$821$	48.8958	1.70648	0.853238	−	0.521522i	$-0.174636\pi$
$821$	48.8958	1.70648	0.853238	+	0.521522i	$0.174636\pi$
$822$	−15.8229	−0.551889
$823$	−29.9581	−1.04427	−0.522137	−	0.852862i	$-0.674865\pi$
$823$	−29.9581	−1.04427	−0.522137	+	0.852862i	$0.674865\pi$
$824$	−17.1361	−0.596964
$825$	4.26083	0.148343
$826$	2.06056	0.0716959
$827$	23.1566	0.805235	0.402617	−	0.915368i	$-0.368100\pi$
$827$	23.1566	0.805235	0.402617	+	0.915368i	$0.368100\pi$
$828$	1.18479	0.0411744
$829$	−30.8580	−1.07174	−0.535872	−	0.844299i	$-0.680017\pi$
$829$	−30.8580	−1.07174	−0.535872	+	0.844299i	$0.680017\pi$
$830$	−4.61949	−0.160345
$831$	29.9813	1.04004
$832$	−2.57398	−0.0892366
$833$	−10.7888	−0.373810
$834$	15.4757	0.535878
$835$	18.8057	0.650798
$836$	0	0
$837$	−7.92902	−0.274067
$838$	4.57903	0.158180
$839$	−11.2703	−0.389095	−0.194547	−	0.980893i	$-0.562324\pi$
$839$	−11.2703	−0.389095	−0.194547	+	0.980893i	$0.562324\pi$
$840$	−1.73917	−0.0600071
$841$	59.5758	2.05434
$842$	−0.278066	−0.00958279
$843$	17.6117	0.606581
$844$	−2.16250	−0.0744365
$845$	7.55262	0.259818
$846$	−4.86484	−0.167257
$847$	14.0865	0.484017
$848$	1.45336	0.0499087
$849$	−15.9145	−0.546183
$850$	−8.00774	−0.274663
$851$	0.0760373	0.00260653
$852$	14.6382	0.501495
$853$	−19.6168	−0.671666	−0.335833	−	0.941921i	$-0.609018\pi$
$853$	−19.6168	−0.671666	−0.335833	+	0.941921i	$0.609018\pi$
$854$	−11.4507	−0.391834
$855$	0	0
$856$	10.5030	0.358985
$857$	−23.1679	−0.791400	−0.395700	−	0.918380i	$-0.629498\pi$
$857$	−23.1679	−0.791400	−0.395700	+	0.918380i	$0.629498\pi$
$858$	3.04963	0.104113
$859$	2.42190	0.0826343	0.0413171	−	0.999146i	$-0.486845\pi$
$859$	2.42190	0.0826343	0.0413171	+	0.999146i	$0.486845\pi$
$860$	8.58853	0.292866
$861$	−14.2094	−0.484257
$862$	11.9828	0.408135
$863$	16.6723	0.567532	0.283766	−	0.958894i	$-0.408416\pi$
$863$	16.6723	0.567532	0.283766	+	0.958894i	$0.408416\pi$
$864$	−1.00000	−0.0340207
$865$	4.10069	0.139428
$866$	29.1480	0.990488
$867$	12.0419	0.408964
$868$	−11.6391	−0.395056
$869$	−15.2858	−0.518535
$870$	−11.1506	−0.378042
$871$	22.4371	0.760251
$872$	8.04189	0.272333
$873$	−12.9145	−0.437088
$874$	0	0
$875$	−14.9504	−0.505415
$876$	−6.53983	−0.220960
$877$	38.1976	1.28984	0.644920	−	0.764250i	$-0.276891\pi$
$877$	38.1976	1.28984	0.644920	+	0.764250i	$0.276891\pi$
$878$	−6.29767	−0.212536
$879$	0.268571	0.00905866
$880$	−1.40373	−0.0473198
$881$	−23.5185	−0.792358	−0.396179	−	0.918173i	$-0.629664\pi$
$881$	−23.5185	−0.792358	−0.396179	+	0.918173i	$0.629664\pi$
$882$	−4.84524	−0.163148
$883$	−18.4148	−0.619706	−0.309853	−	0.950784i	$-0.600280\pi$
$883$	−18.4148	−0.619706	−0.309853	+	0.950784i	$0.600280\pi$
$884$	−5.73143	−0.192769
$885$	−1.66313	−0.0559056
$886$	20.8803	0.701488
$887$	27.5452	0.924878	0.462439	−	0.886651i	$-0.346975\pi$
$887$	27.5452	0.924878	0.462439	+	0.886651i	$0.346975\pi$
$888$	−0.0641778	−0.00215367
$889$	10.6810	0.358229
$890$	14.7050	0.492913
$891$	1.18479	0.0396920
$892$	28.2550	0.946046
$893$	0	0
$894$	−2.22668	−0.0744714
$895$	−0.746911	−0.0249665
$896$	−1.46791	−0.0490395
$897$	3.04963	0.101824
$898$	−41.3756	−1.38072
$899$	−74.6237	−2.48884
$900$	−3.59627	−0.119876
$901$	3.23618	0.107813
$902$	−11.4688	−0.381871
$903$	−10.6408	−0.354105
$904$	19.1411	0.636625
$905$	−13.9673	−0.464288
$906$	6.59627	0.219146
$907$	−20.8289	−0.691613	−0.345807	−	0.938306i	$-0.612395\pi$
$907$	−20.8289	−0.691613	−0.345807	+	0.938306i	$0.612395\pi$
$908$	21.0232	0.697680
$909$	13.7392	0.455700
$910$	−4.47659	−0.148397
$911$	−13.7300	−0.454895	−0.227448	−	0.973790i	$-0.573038\pi$
$911$	−13.7300	−0.454895	−0.227448	+	0.973790i	$0.573038\pi$
$912$	0	0
$913$	4.61949	0.152883
$914$	−27.6245	−0.913739
$915$	9.24216	0.305536
$916$	7.96080	0.263032
$917$	10.1271	0.334426
$918$	−2.22668	−0.0734914
$919$	−44.4489	−1.46623	−0.733117	−	0.680102i	$-0.761935\pi$
$919$	−44.4489	−1.46623	−0.733117	+	0.680102i	$0.761935\pi$
$920$	−1.40373	−0.0462797
$921$	−0.571290	−0.0188246
$922$	−18.5107	−0.609619
$923$	37.6783	1.24020
$924$	1.73917	0.0572145
$925$	−0.230800	−0.00758867
$926$	−3.61856	−0.118913
$927$	−17.1361	−0.562823
$928$	−9.41147	−0.308947
$929$	9.71452	0.318723	0.159361	−	0.987220i	$-0.449056\pi$
$929$	9.71452	0.318723	0.159361	+	0.987220i	$0.449056\pi$
$930$	9.39424	0.308049
$931$	0	0
$932$	14.6878	0.481114
$933$	−14.3523	−0.469875
$934$	−14.4020	−0.471247
$935$	−3.12567	−0.102220
$936$	−2.57398	−0.0841331
$937$	−11.4489	−0.374020	−0.187010	−	0.982358i	$-0.559880\pi$
$937$	−11.4489	−0.374020	−0.187010	+	0.982358i	$0.559880\pi$
$938$	12.7956	0.417791
$939$	2.40373	0.0784429
$940$	5.76382	0.187995
$941$	−45.0729	−1.46933	−0.734666	−	0.678428i	$-0.762661\pi$
$941$	−45.0729	−1.46933	−0.734666	+	0.678428i	$0.762661\pi$
$942$	−9.25402	−0.301512
$943$	−11.4688	−0.373477
$944$	−1.40373	−0.0456876
$945$	−1.73917	−0.0565752
$946$	−8.58853	−0.279237
$947$	44.1130	1.43348	0.716740	−	0.697341i	$-0.245634\pi$
$947$	44.1130	1.43348	0.716740	+	0.697341i	$0.245634\pi$
$948$	12.9017	0.419027
$949$	−16.8334	−0.546435
$950$	0	0
$951$	14.0574	0.455841
$952$	−3.26857	−0.105935
$953$	−1.83244	−0.0593587	−0.0296793	−	0.999559i	$-0.509449\pi$
$953$	−1.83244	−0.0593587	−0.0296793	+	0.999559i	$0.509449\pi$
$954$	1.45336	0.0470544
$955$	−13.5294	−0.437801
$956$	15.7297	0.508734
$957$	11.1506	0.360449
$958$	−31.8557	−1.02921
$959$	−23.2267	−0.750029
$960$	1.18479	0.0382390
$961$	31.8693	1.02804
$962$	−0.165192	−0.00532601
$963$	10.5030	0.338454
$964$	−9.85204	−0.317313
$965$	11.2439	0.361955
$966$	1.73917	0.0559569
$967$	16.4884	0.530233	0.265116	−	0.964216i	$-0.414590\pi$
$967$	16.4884	0.530233	0.265116	+	0.964216i	$0.414590\pi$
$968$	−9.59627	−0.308436
$969$	0	0
$970$	15.3010	0.491284
$971$	−44.5681	−1.43026	−0.715129	−	0.698992i	$-0.753632\pi$
$971$	−44.5681	−1.43026	−0.715129	+	0.698992i	$0.753632\pi$
$972$	−1.00000	−0.0320750
$973$	22.7169	0.728270
$974$	19.7469	0.632732
$975$	−9.25671	−0.296452
$976$	7.80066	0.249693
$977$	−31.4097	−1.00489	−0.502443	−	0.864610i	$-0.667565\pi$
$977$	−31.4097	−1.00489	−0.502443	+	0.864610i	$0.667565\pi$
$978$	3.14022	0.100413
$979$	−14.7050	−0.469975
$980$	5.74060	0.183377
$981$	8.04189	0.256758
$982$	−29.1830	−0.931268
$983$	29.0574	0.926786	0.463393	−	0.886153i	$-0.346632\pi$
$983$	29.0574	0.926786	0.463393	+	0.886153i	$0.346632\pi$
$984$	9.68004	0.308589
$985$	−8.45369	−0.269357
$986$	−20.9564	−0.667386
$987$	−7.14115	−0.227305
$988$	0	0
$989$	−8.58853	−0.273099
$990$	−1.40373	−0.0446136
$991$	−3.95130	−0.125517	−0.0627587	−	0.998029i	$-0.519990\pi$
$991$	−3.95130	−0.125517	−0.0627587	+	0.998029i	$0.519990\pi$
$992$	7.92902	0.251746
$993$	26.8425	0.851823
$994$	21.4875	0.681542
$995$	21.0324	0.666772
$996$	−3.89899	−0.123544
$997$	−47.0515	−1.49014	−0.745068	−	0.666989i	$-0.767583\pi$
$997$	−47.0515	−1.49014	−0.745068	+	0.666989i	$0.767583\pi$
$998$	−5.38413	−0.170432
$999$	−0.0641778	−0.00203049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2166.2.a.s.1.2		3
3.2	odd	2		6498.2.a.bm.1.2		3
19.2	odd	18		114.2.i.a.61.1	yes	6
19.10	odd	18		114.2.i.a.43.1	✓	6
19.18	odd	2		2166.2.a.q.1.2		3
57.2	even	18		342.2.u.e.289.1		6
57.29	even	18		342.2.u.e.271.1		6
57.56	even	2		6498.2.a.br.1.2		3
76.59	even	18		912.2.bo.a.289.1		6
76.67	even	18		912.2.bo.a.385.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.a.43.1	✓	6	19.10	odd	18
114.2.i.a.61.1	yes	6	19.2	odd	18
342.2.u.e.271.1		6	57.29	even	18
342.2.u.e.289.1		6	57.2	even	18
912.2.bo.a.289.1		6	76.59	even	18
912.2.bo.a.385.1		6	76.67	even	18
2166.2.a.q.1.2		3	19.18	odd	2
2166.2.a.s.1.2		3	1.1	even	1	trivial
6498.2.a.bm.1.2		3	3.2	odd	2
6498.2.a.br.1.2		3	57.56	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2166.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.18479 q^{5} -1.00000 q^{6} -1.46791 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.18479 q^{5} -1.00000 q^{6} -1.46791 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.18479 q^{10} +1.18479 q^{11} -1.00000 q^{12} -2.57398 q^{13} -1.46791 q^{14} +1.18479 q^{15} +1.00000 q^{16} +2.22668 q^{17} +1.00000 q^{18} -1.18479 q^{20} +1.46791 q^{21} +1.18479 q^{22} +1.18479 q^{23} -1.00000 q^{24} -3.59627 q^{25} -2.57398 q^{26} -1.00000 q^{27} -1.46791 q^{28} -9.41147 q^{29} +1.18479 q^{30} +7.92902 q^{31} +1.00000 q^{32} -1.18479 q^{33} +2.22668 q^{34} +1.73917 q^{35} +1.00000 q^{36} +0.0641778 q^{37} +2.57398 q^{39} -1.18479 q^{40} -9.68004 q^{41} +1.46791 q^{42} -7.24897 q^{43} +1.18479 q^{44} -1.18479 q^{45} +1.18479 q^{46} -4.86484 q^{47} -1.00000 q^{48} -4.84524 q^{49} -3.59627 q^{50} -2.22668 q^{51} -2.57398 q^{52} +1.45336 q^{53} -1.00000 q^{54} -1.40373 q^{55} -1.46791 q^{56} -9.41147 q^{58} -1.40373 q^{59} +1.18479 q^{60} +7.80066 q^{61} +7.92902 q^{62} -1.46791 q^{63} +1.00000 q^{64} +3.04963 q^{65} -1.18479 q^{66} -8.71688 q^{67} +2.22668 q^{68} -1.18479 q^{69} +1.73917 q^{70} -14.6382 q^{71} +1.00000 q^{72} +6.53983 q^{73} +0.0641778 q^{74} +3.59627 q^{75} -1.73917 q^{77} +2.57398 q^{78} -12.9017 q^{79} -1.18479 q^{80} +1.00000 q^{81} -9.68004 q^{82} +3.89899 q^{83} +1.46791 q^{84} -2.63816 q^{85} -7.24897 q^{86} +9.41147 q^{87} +1.18479 q^{88} -12.4115 q^{89} -1.18479 q^{90} +3.77837 q^{91} +1.18479 q^{92} -7.92902 q^{93} -4.86484 q^{94} -1.00000 q^{96} -12.9145 q^{97} -4.84524 q^{98} +1.18479 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} - 9 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 - 9 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} - 9 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{12} - 9 q^{14} + 3 q^{16} + 3 q^{18} + 9 q^{21} - 3 q^{24} + 3 q^{25} - 3 q^{27} - 9 q^{28} - 18 q^{29} - 9 q^{31} + 3 q^{32} - 9 q^{35} + 3 q^{36} - 9 q^{37} - 9 q^{41} + 9 q^{42} - 9 q^{43} + 9 q^{47} - 3 q^{48} + 12 q^{49} + 3 q^{50} - 9 q^{53} - 3 q^{54} - 18 q^{55} - 9 q^{56} - 18 q^{58} - 18 q^{59} + 9 q^{61} - 9 q^{62} - 9 q^{63} + 3 q^{64} - 18 q^{65} - 18 q^{67} - 9 q^{70} - 27 q^{71} + 3 q^{72} - 9 q^{73} - 9 q^{74} - 3 q^{75} + 9 q^{77} - 27 q^{79} + 3 q^{81} - 9 q^{82} + 9 q^{83} + 9 q^{84} + 9 q^{85} - 9 q^{86} + 18 q^{87} - 27 q^{89} + 3 q^{91} + 9 q^{93} + 9 q^{94} - 3 q^{96} + 12 q^{97} + 12 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 - 3 * q^3 + 3 * q^4 - 3 * q^6 - 9 * q^7 + 3 * q^8 + 3 * q^9 - 3 * q^12 - 9 * q^14 + 3 * q^16 + 3 * q^18 + 9 * q^21 - 3 * q^24 + 3 * q^25 - 3 * q^27 - 9 * q^28 - 18 * q^29 - 9 * q^31 + 3 * q^32 - 9 * q^35 + 3 * q^36 - 9 * q^37 - 9 * q^41 + 9 * q^42 - 9 * q^43 + 9 * q^47 - 3 * q^48 + 12 * q^49 + 3 * q^50 - 9 * q^53 - 3 * q^54 - 18 * q^55 - 9 * q^56 - 18 * q^58 - 18 * q^59 + 9 * q^61 - 9 * q^62 - 9 * q^63 + 3 * q^64 - 18 * q^65 - 18 * q^67 - 9 * q^70 - 27 * q^71 + 3 * q^72 - 9 * q^73 - 9 * q^74 - 3 * q^75 + 9 * q^77 - 27 * q^79 + 3 * q^81 - 9 * q^82 + 9 * q^83 + 9 * q^84 + 9 * q^85 - 9 * q^86 + 18 * q^87 - 27 * q^89 + 3 * q^91 + 9 * q^93 + 9 * q^94 - 3 * q^96 + 12 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more