LMFDB - Embedded newform 8670.2.a.bu.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8670,2,Mod(1,8670)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8670, 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("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8670.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:	$69.2302985525$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.765367$ of defining polynomial
Character	$\chi$	$=$	8670.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.08239 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.08239 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.41421 q^{11} -1.00000 q^{12} -3.84776 q^{13} +1.08239 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +2.82843 q^{19} +1.00000 q^{20} +1.08239 q^{21} -1.41421 q^{22} -2.98067 q^{23} +1.00000 q^{24} +1.00000 q^{25} +3.84776 q^{26} -1.00000 q^{27} -1.08239 q^{28} -8.64047 q^{29} +1.00000 q^{30} +3.44993 q^{31} -1.00000 q^{32} -1.41421 q^{33} -1.08239 q^{35} +1.00000 q^{36} +7.39104 q^{37} -2.82843 q^{38} +3.84776 q^{39} -1.00000 q^{40} +9.83938 q^{41} -1.08239 q^{42} -4.27836 q^{43} +1.41421 q^{44} +1.00000 q^{45} +2.98067 q^{46} +9.39104 q^{47} -1.00000 q^{48} -5.82843 q^{49} -1.00000 q^{50} -3.84776 q^{52} -3.29769 q^{53} +1.00000 q^{54} +1.41421 q^{55} +1.08239 q^{56} -2.82843 q^{57} +8.64047 q^{58} -13.1792 q^{59} -1.00000 q^{60} +2.46927 q^{61} -3.44993 q^{62} -1.08239 q^{63} +1.00000 q^{64} -3.84776 q^{65} +1.41421 q^{66} +4.95974 q^{67} +2.98067 q^{69} +1.08239 q^{70} +3.24718 q^{71} -1.00000 q^{72} -8.54168 q^{73} -7.39104 q^{74} -1.00000 q^{75} +2.82843 q^{76} -1.53073 q^{77} -3.84776 q^{78} -0.328878 q^{79} +1.00000 q^{80} +1.00000 q^{81} -9.83938 q^{82} -0.537522 q^{83} +1.08239 q^{84} +4.27836 q^{86} +8.64047 q^{87} -1.41421 q^{88} +17.5718 q^{89} -1.00000 q^{90} +4.16478 q^{91} -2.98067 q^{92} -3.44993 q^{93} -9.39104 q^{94} +2.82843 q^{95} +1.00000 q^{96} +0.253965 q^{97} +5.82843 q^{98} +1.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^8 + 4 * q^9	$ 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{9} - 4 q^{10} - 4 q^{12} - 8 q^{13} - 4 q^{15} + 4 q^{16} - 4 q^{18} + 4 q^{20} - 8 q^{23} + 4 q^{24} + 4 q^{25} + 8 q^{26} - 4 q^{27} - 8 q^{29} + 4 q^{30} + 16 q^{31} - 4 q^{32} + 4 q^{36} + 8 q^{39} - 4 q^{40} + 8 q^{41} - 8 q^{43} + 4 q^{45} + 8 q^{46} + 8 q^{47} - 4 q^{48} - 12 q^{49} - 4 q^{50} - 8 q^{52} - 8 q^{53} + 4 q^{54} + 8 q^{58} - 4 q^{60} + 16 q^{61} - 16 q^{62} + 4 q^{64} - 8 q^{65} + 8 q^{67} + 8 q^{69} - 4 q^{72} - 8 q^{73} - 4 q^{75} - 8 q^{78} + 4 q^{80} + 4 q^{81} - 8 q^{82} - 16 q^{83} + 8 q^{86} + 8 q^{87} - 8 q^{89} - 4 q^{90} + 8 q^{91} - 8 q^{92} - 16 q^{93} - 8 q^{94} + 4 q^{96} + 8 q^{97} + 12 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^8 + 4 * q^9 - 4 * q^10 - 4 * q^12 - 8 * q^13 - 4 * q^15 + 4 * q^16 - 4 * q^18 + 4 * q^20 - 8 * q^23 + 4 * q^24 + 4 * q^25 + 8 * q^26 - 4 * q^27 - 8 * q^29 + 4 * q^30 + 16 * q^31 - 4 * q^32 + 4 * q^36 + 8 * q^39 - 4 * q^40 + 8 * q^41 - 8 * q^43 + 4 * q^45 + 8 * q^46 + 8 * q^47 - 4 * q^48 - 12 * q^49 - 4 * q^50 - 8 * q^52 - 8 * q^53 + 4 * q^54 + 8 * q^58 - 4 * q^60 + 16 * q^61 - 16 * q^62 + 4 * q^64 - 8 * q^65 + 8 * q^67 + 8 * q^69 - 4 * q^72 - 8 * q^73 - 4 * q^75 - 8 * q^78 + 4 * q^80 + 4 * q^81 - 8 * q^82 - 16 * q^83 + 8 * q^86 + 8 * q^87 - 8 * q^89 - 4 * q^90 + 8 * q^91 - 8 * q^92 - 16 * q^93 - 8 * q^94 + 4 * q^96 + 8 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.00000	0.408248
$7$	−1.08239	−0.409106	−0.204553	−	0.978856i	$-0.565574\pi$
$7$	−1.08239	−0.409106	−0.204553	+	0.978856i	$0.565574\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	1.41421	0.426401	0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421	0.426401	0.213201	+	0.977008i	$0.431611\pi$
$12$	−1.00000	−0.288675
$13$	−3.84776	−1.06718	−0.533588	−	0.845744i	$-0.679157\pi$
$13$	−3.84776	−1.06718	−0.533588	+	0.845744i	$0.679157\pi$
$14$	1.08239	0.289281
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−1.00000	−0.235702
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	1.00000	0.223607
$21$	1.08239	0.236197
$22$	−1.41421	−0.301511
$23$	−2.98067	−0.621512	−0.310756	−	0.950490i	$-0.600582\pi$
$23$	−2.98067	−0.621512	−0.310756	+	0.950490i	$0.600582\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	3.84776	0.754608
$27$	−1.00000	−0.192450
$28$	−1.08239	−0.204553
$29$	−8.64047	−1.60449	−0.802247	−	0.596992i	$-0.796362\pi$
$29$	−8.64047	−1.60449	−0.802247	+	0.596992i	$0.796362\pi$
$30$	1.00000	0.182574
$31$	3.44993	0.619626	0.309813	−	0.950797i	$-0.399733\pi$
$31$	3.44993	0.619626	0.309813	+	0.950797i	$0.399733\pi$
$32$	−1.00000	−0.176777
$33$	−1.41421	−0.246183
$34$	0	0
$35$	−1.08239	−0.182958
$36$	1.00000	0.166667
$37$	7.39104	1.21508	0.607539	−	0.794290i	$-0.292157\pi$
$37$	7.39104	1.21508	0.607539	+	0.794290i	$0.292157\pi$
$38$	−2.82843	−0.458831
$39$	3.84776	0.616135
$40$	−1.00000	−0.158114
$41$	9.83938	1.53665	0.768326	−	0.640058i	$-0.221090\pi$
$41$	9.83938	1.53665	0.768326	+	0.640058i	$0.221090\pi$
$42$	−1.08239	−0.167017
$43$	−4.27836	−0.652444	−0.326222	−	0.945293i	$-0.605776\pi$
$43$	−4.27836	−0.652444	−0.326222	+	0.945293i	$0.605776\pi$
$44$	1.41421	0.213201
$45$	1.00000	0.149071
$46$	2.98067	0.439476
$47$	9.39104	1.36982	0.684912	−	0.728626i	$-0.259841\pi$
$47$	9.39104	1.36982	0.684912	+	0.728626i	$0.259841\pi$
$48$	−1.00000	−0.144338
$49$	−5.82843	−0.832632
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−3.84776	−0.533588
$53$	−3.29769	−0.452973	−0.226487	−	0.974014i	$-0.572724\pi$
$53$	−3.29769	−0.452973	−0.226487	+	0.974014i	$0.572724\pi$
$54$	1.00000	0.136083
$55$	1.41421	0.190693
$56$	1.08239	0.144641
$57$	−2.82843	−0.374634
$58$	8.64047	1.13455
$59$	−13.1792	−1.71579	−0.857893	−	0.513828i	$-0.828227\pi$
$59$	−13.1792	−1.71579	−0.857893	+	0.513828i	$0.828227\pi$
$60$	−1.00000	−0.129099
$61$	2.46927	0.316157	0.158079	−	0.987427i	$-0.449470\pi$
$61$	2.46927	0.316157	0.158079	+	0.987427i	$0.449470\pi$
$62$	−3.44993	−0.438142
$63$	−1.08239	−0.136369
$64$	1.00000	0.125000
$65$	−3.84776	−0.477256
$66$	1.41421	0.174078
$67$	4.95974	0.605929	0.302965	−	0.953002i	$-0.402024\pi$
$67$	4.95974	0.605929	0.302965	+	0.953002i	$0.402024\pi$
$68$	0	0
$69$	2.98067	0.358830
$70$	1.08239	0.129371
$71$	3.24718	0.385369	0.192684	−	0.981261i	$-0.438281\pi$
$71$	3.24718	0.385369	0.192684	+	0.981261i	$0.438281\pi$
$72$	−1.00000	−0.117851
$73$	−8.54168	−0.999729	−0.499864	−	0.866104i	$-0.666617\pi$
$73$	−8.54168	−0.999729	−0.499864	+	0.866104i	$0.666617\pi$
$74$	−7.39104	−0.859191
$75$	−1.00000	−0.115470
$76$	2.82843	0.324443
$77$	−1.53073	−0.174443
$78$	−3.84776	−0.435673
$79$	−0.328878	−0.0370017	−0.0185008	−	0.999829i	$-0.505889\pi$
$79$	−0.328878	−0.0370017	−0.0185008	+	0.999829i	$0.505889\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−9.83938	−1.08658
$83$	−0.537522	−0.0590007	−0.0295004	−	0.999565i	$-0.509392\pi$
$83$	−0.537522	−0.0590007	−0.0295004	+	0.999565i	$0.509392\pi$
$84$	1.08239	0.118099
$85$	0	0
$86$	4.27836	0.461348
$87$	8.64047	0.926355
$88$	−1.41421	−0.150756
$89$	17.5718	1.86261	0.931305	−	0.364239i	$-0.118671\pi$
$89$	17.5718	1.86261	0.931305	+	0.364239i	$0.118671\pi$
$90$	−1.00000	−0.105409
$91$	4.16478	0.436588
$92$	−2.98067	−0.310756
$93$	−3.44993	−0.357742
$94$	−9.39104	−0.968611
$95$	2.82843	0.290191
$96$	1.00000	0.102062
$97$	0.253965	0.0257862	0.0128931	−	0.999917i	$-0.495896\pi$
$97$	0.253965	0.0257862	0.0128931	+	0.999917i	$0.495896\pi$
$98$	5.82843	0.588760
$99$	1.41421	0.142134
$100$	1.00000	0.100000
$101$	13.0353	1.29707	0.648533	−	0.761187i	$-0.275383\pi$
$101$	13.0353	1.29707	0.648533	+	0.761187i	$0.275383\pi$
$102$	0	0
$103$	−7.92856	−0.781224	−0.390612	−	0.920555i	$-0.627737\pi$
$103$	−7.92856	−0.781224	−0.390612	+	0.920555i	$0.627737\pi$
$104$	3.84776	0.377304
$105$	1.08239	0.105631
$106$	3.29769	0.320300
$107$	−13.1161	−1.26799	−0.633993	−	0.773339i	$-0.718585\pi$
$107$	−13.1161	−1.26799	−0.633993	+	0.773339i	$0.718585\pi$
$108$	−1.00000	−0.0962250
$109$	−1.10332	−0.105679	−0.0528393	−	0.998603i	$-0.516827\pi$
$109$	−1.10332	−0.105679	−0.0528393	+	0.998603i	$0.516827\pi$
$110$	−1.41421	−0.134840
$111$	−7.39104	−0.701526
$112$	−1.08239	−0.102276
$113$	−4.87547	−0.458646	−0.229323	−	0.973350i	$-0.573651\pi$
$113$	−4.87547	−0.458646	−0.229323	+	0.973350i	$0.573651\pi$
$114$	2.82843	0.264906
$115$	−2.98067	−0.277949
$116$	−8.64047	−0.802247
$117$	−3.84776	−0.355725
$118$	13.1792	1.21324
$119$	0	0
$120$	1.00000	0.0912871
$121$	−9.00000	−0.818182
$122$	−2.46927	−0.223557
$123$	−9.83938	−0.887187
$124$	3.44993	0.309813
$125$	1.00000	0.0894427
$126$	1.08239	0.0964272
$127$	−12.3205	−1.09327	−0.546634	−	0.837372i	$-0.684091\pi$
$127$	−12.3205	−1.09327	−0.546634	+	0.837372i	$0.684091\pi$
$128$	−1.00000	−0.0883883
$129$	4.27836	0.376689
$130$	3.84776	0.337471
$131$	3.96722	0.346618	0.173309	−	0.984868i	$-0.444554\pi$
$131$	3.96722	0.346618	0.173309	+	0.984868i	$0.444554\pi$
$132$	−1.41421	−0.123091
$133$	−3.06147	−0.265463
$134$	−4.95974	−0.428457
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−18.0019	−1.53801	−0.769003	−	0.639245i	$-0.779247\pi$
$137$	−18.0019	−1.53801	−0.769003	+	0.639245i	$0.779247\pi$
$138$	−2.98067	−0.253731
$139$	16.7274	1.41880	0.709400	−	0.704807i	$-0.248966\pi$
$139$	16.7274	1.41880	0.709400	+	0.704807i	$0.248966\pi$
$140$	−1.08239	−0.0914788
$141$	−9.39104	−0.790868
$142$	−3.24718	−0.272497
$143$	−5.44155	−0.455046
$144$	1.00000	0.0833333
$145$	−8.64047	−0.717552
$146$	8.54168	0.706915
$147$	5.82843	0.480721
$148$	7.39104	0.607539
$149$	−7.95295	−0.651531	−0.325766	−	0.945451i	$-0.605622\pi$
$149$	−7.95295	−0.651531	−0.325766	+	0.945451i	$0.605622\pi$
$150$	1.00000	0.0816497
$151$	−13.4006	−1.09053	−0.545264	−	0.838264i	$-0.683571\pi$
$151$	−13.4006	−1.09053	−0.545264	+	0.838264i	$0.683571\pi$
$152$	−2.82843	−0.229416
$153$	0	0
$154$	1.53073	0.123350
$155$	3.44993	0.277105
$156$	3.84776	0.308067
$157$	6.64168	0.530064	0.265032	−	0.964240i	$-0.414617\pi$
$157$	6.64168	0.530064	0.265032	+	0.964240i	$0.414617\pi$
$158$	0.328878	0.0261641
$159$	3.29769	0.261524
$160$	−1.00000	−0.0790569
$161$	3.22625	0.254264
$162$	−1.00000	−0.0785674
$163$	−7.04148	−0.551531	−0.275765	−	0.961225i	$-0.588931\pi$
$163$	−7.04148	−0.551531	−0.275765	+	0.961225i	$0.588931\pi$
$164$	9.83938	0.768326
$165$	−1.41421	−0.110096
$166$	0.537522	0.0417198
$167$	−0.474462	−0.0367150	−0.0183575	−	0.999831i	$-0.505844\pi$
$167$	−0.474462	−0.0367150	−0.0183575	+	0.999831i	$0.505844\pi$
$168$	−1.08239	−0.0835084
$169$	1.80525	0.138865
$170$	0	0
$171$	2.82843	0.216295
$172$	−4.27836	−0.326222
$173$	16.4936	1.25399	0.626993	−	0.779025i	$-0.284286\pi$
$173$	16.4936	1.25399	0.626993	+	0.779025i	$0.284286\pi$
$174$	−8.64047	−0.655032
$175$	−1.08239	−0.0818212
$176$	1.41421	0.106600
$177$	13.1792	0.990610
$178$	−17.5718	−1.31706
$179$	1.23880	0.0925919	0.0462960	−	0.998928i	$-0.485258\pi$
$179$	1.23880	0.0925919	0.0462960	+	0.998928i	$0.485258\pi$
$180$	1.00000	0.0745356
$181$	8.85802	0.658411	0.329206	−	0.944258i	$-0.393219\pi$
$181$	8.85802	0.658411	0.329206	+	0.944258i	$0.393219\pi$
$182$	−4.16478	−0.308714
$183$	−2.46927	−0.182533
$184$	2.98067	0.219738
$185$	7.39104	0.543400
$186$	3.44993	0.252961
$187$	0	0
$188$	9.39104	0.684912
$189$	1.08239	0.0787324
$190$	−2.82843	−0.205196
$191$	−15.7889	−1.14244	−0.571221	−	0.820796i	$-0.693530\pi$
$191$	−15.7889	−1.14244	−0.571221	+	0.820796i	$0.693530\pi$
$192$	−1.00000	−0.0721688
$193$	−1.33145	−0.0958397	−0.0479198	−	0.998851i	$-0.515259\pi$
$193$	−1.33145	−0.0958397	−0.0479198	+	0.998851i	$0.515259\pi$
$194$	−0.253965	−0.0182336
$195$	3.84776	0.275544
$196$	−5.82843	−0.416316
$197$	−8.86709	−0.631754	−0.315877	−	0.948800i	$-0.602299\pi$
$197$	−8.86709	−0.631754	−0.315877	+	0.948800i	$0.602299\pi$
$198$	−1.41421	−0.100504
$199$	15.6603	1.11013	0.555066	−	0.831807i	$-0.312693\pi$
$199$	15.6603	1.11013	0.555066	+	0.831807i	$0.312693\pi$
$200$	−1.00000	−0.0707107
$201$	−4.95974	−0.349833
$202$	−13.0353	−0.917164
$203$	9.35237	0.656408
$204$	0	0
$205$	9.83938	0.687212
$206$	7.92856	0.552409
$207$	−2.98067	−0.207171
$208$	−3.84776	−0.266794
$209$	4.00000	0.276686
$210$	−1.08239	−0.0746922
$211$	23.7502	1.63503	0.817515	−	0.575907i	$-0.195351\pi$
$211$	23.7502	1.63503	0.817515	+	0.575907i	$0.195351\pi$
$212$	−3.29769	−0.226487
$213$	−3.24718	−0.222493
$214$	13.1161	0.896601
$215$	−4.27836	−0.291782
$216$	1.00000	0.0680414
$217$	−3.73418	−0.253493
$218$	1.10332	0.0747261
$219$	8.54168	0.577194
$220$	1.41421	0.0953463
$221$	0	0
$222$	7.39104	0.496054
$223$	0.0714415	0.00478408	0.00239204	−	0.999997i	$-0.499239\pi$
$223$	0.0714415	0.00478408	0.00239204	+	0.999997i	$0.499239\pi$
$224$	1.08239	0.0723204
$225$	1.00000	0.0666667
$226$	4.87547	0.324312
$227$	−18.8799	−1.25310	−0.626552	−	0.779379i	$-0.715534\pi$
$227$	−18.8799	−1.25310	−0.626552	+	0.779379i	$0.715534\pi$
$228$	−2.82843	−0.187317
$229$	−14.7183	−0.972614	−0.486307	−	0.873788i	$-0.661656\pi$
$229$	−14.7183	−0.972614	−0.486307	+	0.873788i	$0.661656\pi$
$230$	2.98067	0.196539
$231$	1.53073	0.100715
$232$	8.64047	0.567274
$233$	−4.38078	−0.286994	−0.143497	−	0.989651i	$-0.545835\pi$
$233$	−4.38078	−0.286994	−0.143497	+	0.989651i	$0.545835\pi$
$234$	3.84776	0.251536
$235$	9.39104	0.612604
$236$	−13.1792	−0.857893
$237$	0.328878	0.0213629
$238$	0	0
$239$	15.6401	1.01167	0.505837	−	0.862629i	$-0.331184\pi$
$239$	15.6401	1.01167	0.505837	+	0.862629i	$0.331184\pi$
$240$	−1.00000	−0.0645497
$241$	−3.64847	−0.235019	−0.117509	−	0.993072i	$-0.537491\pi$
$241$	−3.64847	−0.235019	−0.117509	+	0.993072i	$0.537491\pi$
$242$	9.00000	0.578542
$243$	−1.00000	−0.0641500
$244$	2.46927	0.158079
$245$	−5.82843	−0.372365
$246$	9.83938	0.627336
$247$	−10.8831	−0.692475
$248$	−3.44993	−0.219071
$249$	0.537522	0.0340641
$250$	−1.00000	−0.0632456
$251$	9.89059	0.624288	0.312144	−	0.950035i	$-0.398953\pi$
$251$	9.89059	0.624288	0.312144	+	0.950035i	$0.398953\pi$
$252$	−1.08239	−0.0681843
$253$	−4.21530	−0.265014
$254$	12.3205	0.773057
$255$	0	0
$256$	1.00000	0.0625000
$257$	−20.7338	−1.29334	−0.646670	−	0.762770i	$-0.723839\pi$
$257$	−20.7338	−1.29334	−0.646670	+	0.762770i	$0.723839\pi$
$258$	−4.27836	−0.266359
$259$	−8.00000	−0.497096
$260$	−3.84776	−0.238628
$261$	−8.64047	−0.534831
$262$	−3.96722	−0.245096
$263$	−21.3839	−1.31859	−0.659293	−	0.751886i	$-0.729144\pi$
$263$	−21.3839	−1.31859	−0.659293	+	0.751886i	$0.729144\pi$
$264$	1.41421	0.0870388
$265$	−3.29769	−0.202576
$266$	3.06147	0.187711
$267$	−17.5718	−1.07538
$268$	4.95974	0.302965
$269$	−18.7949	−1.14595	−0.572973	−	0.819574i	$-0.694210\pi$
$269$	−18.7949	−1.14595	−0.572973	+	0.819574i	$0.694210\pi$
$270$	1.00000	0.0608581
$271$	−20.3192	−1.23430	−0.617152	−	0.786844i	$-0.711714\pi$
$271$	−20.3192	−1.23430	−0.617152	+	0.786844i	$0.711714\pi$
$272$	0	0
$273$	−4.16478	−0.252064
$274$	18.0019	1.08754
$275$	1.41421	0.0852803
$276$	2.98067	0.179415
$277$	5.60859	0.336988	0.168494	−	0.985703i	$-0.446110\pi$
$277$	5.60859	0.336988	0.168494	+	0.985703i	$0.446110\pi$
$278$	−16.7274	−1.00324
$279$	3.44993	0.206542
$280$	1.08239	0.0646853
$281$	−5.85123	−0.349055	−0.174528	−	0.984652i	$-0.555840\pi$
$281$	−5.85123	−0.349055	−0.174528	+	0.984652i	$0.555840\pi$
$282$	9.39104	0.559228
$283$	24.5792	1.46108	0.730539	−	0.682870i	$-0.239269\pi$
$283$	24.5792	1.46108	0.730539	+	0.682870i	$0.239269\pi$
$284$	3.24718	0.192684
$285$	−2.82843	−0.167542
$286$	5.44155	0.321766
$287$	−10.6501	−0.628654
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	8.64047	0.507386
$291$	−0.253965	−0.0148877
$292$	−8.54168	−0.499864
$293$	−15.9317	−0.930742	−0.465371	−	0.885116i	$-0.654079\pi$
$293$	−15.9317	−0.930742	−0.465371	+	0.885116i	$0.654079\pi$
$294$	−5.82843	−0.339921
$295$	−13.1792	−0.767323
$296$	−7.39104	−0.429595
$297$	−1.41421	−0.0820610
$298$	7.95295	0.460702
$299$	11.4689	0.663263
$300$	−1.00000	−0.0577350
$301$	4.63087	0.266919
$302$	13.4006	0.771120
$303$	−13.0353	−0.748861
$304$	2.82843	0.162221
$305$	2.46927	0.141390
$306$	0	0
$307$	25.3632	1.44755	0.723777	−	0.690034i	$-0.242404\pi$
$307$	25.3632	1.44755	0.723777	+	0.690034i	$0.242404\pi$
$308$	−1.53073	−0.0872216
$309$	7.92856	0.451040
$310$	−3.44993	−0.195943
$311$	−20.3975	−1.15664	−0.578318	−	0.815812i	$-0.696291\pi$
$311$	−20.3975	−1.15664	−0.578318	+	0.815812i	$0.696291\pi$
$312$	−3.84776	−0.217836
$313$	−10.8267	−0.611961	−0.305981	−	0.952038i	$-0.598984\pi$
$313$	−10.8267	−0.611961	−0.305981	+	0.952038i	$0.598984\pi$
$314$	−6.64168	−0.374812
$315$	−1.08239	−0.0609859
$316$	−0.328878	−0.0185008
$317$	−21.3942	−1.20162	−0.600810	−	0.799392i	$-0.705155\pi$
$317$	−21.3942	−1.20162	−0.600810	+	0.799392i	$0.705155\pi$
$318$	−3.29769	−0.184925
$319$	−12.2195	−0.684159
$320$	1.00000	0.0559017
$321$	13.1161	0.732072
$322$	−3.22625	−0.179792
$323$	0	0
$324$	1.00000	0.0555556
$325$	−3.84776	−0.213435
$326$	7.04148	0.389991
$327$	1.10332	0.0610136
$328$	−9.83938	−0.543289
$329$	−10.1648	−0.560403
$330$	1.41421	0.0778499
$331$	18.7402	1.03006	0.515028	−	0.857173i	$-0.327782\pi$
$331$	18.7402	1.03006	0.515028	+	0.857173i	$0.327782\pi$
$332$	−0.537522	−0.0295004
$333$	7.39104	0.405026
$334$	0.474462	0.0259614
$335$	4.95974	0.270980
$336$	1.08239	0.0590493
$337$	11.4061	0.621328	0.310664	−	0.950520i	$-0.399449\pi$
$337$	11.4061	0.621328	0.310664	+	0.950520i	$0.399449\pi$
$338$	−1.80525	−0.0981926
$339$	4.87547	0.264799
$340$	0	0
$341$	4.87894	0.264210
$342$	−2.82843	−0.152944
$343$	13.8854	0.749741
$344$	4.27836	0.230674
$345$	2.98067	0.160474
$346$	−16.4936	−0.886702
$347$	21.2777	1.14225	0.571125	−	0.820863i	$-0.306507\pi$
$347$	21.2777	1.14225	0.571125	+	0.820863i	$0.306507\pi$
$348$	8.64047	0.463178
$349$	16.0411	0.858661	0.429330	−	0.903148i	$-0.358750\pi$
$349$	16.0411	0.858661	0.429330	+	0.903148i	$0.358750\pi$
$350$	1.08239	0.0578563
$351$	3.84776	0.205378
$352$	−1.41421	−0.0753778
$353$	−35.1353	−1.87007	−0.935033	−	0.354561i	$-0.884630\pi$
$353$	−35.1353	−1.87007	−0.935033	+	0.354561i	$0.884630\pi$
$354$	−13.1792	−0.700467
$355$	3.24718	0.172342
$356$	17.5718	0.931305
$357$	0	0
$358$	−1.23880	−0.0654724
$359$	−6.84519	−0.361275	−0.180638	−	0.983550i	$-0.557816\pi$
$359$	−6.84519	−0.361275	−0.180638	+	0.983550i	$0.557816\pi$
$360$	−1.00000	−0.0527046
$361$	−11.0000	−0.578947
$362$	−8.85802	−0.465567
$363$	9.00000	0.472377
$364$	4.16478	0.218294
$365$	−8.54168	−0.447092
$366$	2.46927	0.129071
$367$	10.7915	0.563311	0.281656	−	0.959516i	$-0.409116\pi$
$367$	10.7915	0.563311	0.281656	+	0.959516i	$0.409116\pi$
$368$	−2.98067	−0.155378
$369$	9.83938	0.512218
$370$	−7.39104	−0.384242
$371$	3.56940	0.185314
$372$	−3.44993	−0.178871
$373$	30.5657	1.58263	0.791317	−	0.611406i	$-0.209396\pi$
$373$	30.5657	1.58263	0.791317	+	0.611406i	$0.209396\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	−9.39104	−0.484306
$377$	33.2464	1.71228
$378$	−1.08239	−0.0556722
$379$	−36.5643	−1.87818	−0.939091	−	0.343669i	$-0.888330\pi$
$379$	−36.5643	−1.87818	−0.939091	+	0.343669i	$0.888330\pi$
$380$	2.82843	0.145095
$381$	12.3205	0.631198
$382$	15.7889	0.807828
$383$	12.1158	0.619087	0.309544	−	0.950885i	$-0.399824\pi$
$383$	12.1158	0.619087	0.309544	+	0.950885i	$0.399824\pi$
$384$	1.00000	0.0510310
$385$	−1.53073	−0.0780134
$386$	1.33145	0.0677689
$387$	−4.27836	−0.217481
$388$	0.253965	0.0128931
$389$	−26.5242	−1.34483	−0.672416	−	0.740173i	$-0.734743\pi$
$389$	−26.5242	−1.34483	−0.672416	+	0.740173i	$0.734743\pi$
$390$	−3.84776	−0.194839
$391$	0	0
$392$	5.82843	0.294380
$393$	−3.96722	−0.200120
$394$	8.86709	0.446718
$395$	−0.328878	−0.0165477
$396$	1.41421	0.0710669
$397$	9.97364	0.500562	0.250281	−	0.968173i	$-0.419477\pi$
$397$	9.97364	0.500562	0.250281	+	0.968173i	$0.419477\pi$
$398$	−15.6603	−0.784981
$399$	3.06147	0.153265
$400$	1.00000	0.0500000
$401$	−6.68138	−0.333652	−0.166826	−	0.985986i	$-0.553352\pi$
$401$	−6.68138	−0.333652	−0.166826	+	0.985986i	$0.553352\pi$
$402$	4.95974	0.247370
$403$	−13.2745	−0.661251
$404$	13.0353	0.648533
$405$	1.00000	0.0496904
$406$	−9.35237	−0.464150
$407$	10.4525	0.518111
$408$	0	0
$409$	1.48756	0.0735553	0.0367777	−	0.999323i	$-0.488291\pi$
$409$	1.48756	0.0735553	0.0367777	+	0.999323i	$0.488291\pi$
$410$	−9.83938	−0.485932
$411$	18.0019	0.887969
$412$	−7.92856	−0.390612
$413$	14.2651	0.701938
$414$	2.98067	0.146492
$415$	−0.537522	−0.0263859
$416$	3.84776	0.188652
$417$	−16.7274	−0.819144
$418$	−4.00000	−0.195646
$419$	−22.6261	−1.10536	−0.552680	−	0.833394i	$-0.686395\pi$
$419$	−22.6261	−1.10536	−0.552680	+	0.833394i	$0.686395\pi$
$420$	1.08239	0.0528153
$421$	−26.8469	−1.30844	−0.654218	−	0.756306i	$-0.727002\pi$
$421$	−26.8469	−1.30844	−0.654218	+	0.756306i	$0.727002\pi$
$422$	−23.7502	−1.15614
$423$	9.39104	0.456608
$424$	3.29769	0.160150
$425$	0	0
$426$	3.24718	0.157326
$427$	−2.67271	−0.129342
$428$	−13.1161	−0.633993
$429$	5.44155	0.262721
$430$	4.27836	0.206321
$431$	11.5837	0.557967	0.278983	−	0.960296i	$-0.410003\pi$
$431$	11.5837	0.557967	0.278983	+	0.960296i	$0.410003\pi$
$432$	−1.00000	−0.0481125
$433$	16.6971	0.802409	0.401205	−	0.915988i	$-0.368592\pi$
$433$	16.6971	0.802409	0.401205	+	0.915988i	$0.368592\pi$
$434$	3.73418	0.179246
$435$	8.64047	0.414279
$436$	−1.10332	−0.0528393
$437$	−8.43060	−0.403290
$438$	−8.54168	−0.408137
$439$	7.64603	0.364925	0.182463	−	0.983213i	$-0.441593\pi$
$439$	7.64603	0.364925	0.182463	+	0.983213i	$0.441593\pi$
$440$	−1.41421	−0.0674200
$441$	−5.82843	−0.277544
$442$	0	0
$443$	−15.4265	−0.732936	−0.366468	−	0.930431i	$-0.619433\pi$
$443$	−15.4265	−0.732936	−0.366468	+	0.930431i	$0.619433\pi$
$444$	−7.39104	−0.350763
$445$	17.5718	0.832985
$446$	−0.0714415	−0.00338285
$447$	7.95295	0.376162
$448$	−1.08239	−0.0511382
$449$	4.26229	0.201150	0.100575	−	0.994929i	$-0.467932\pi$
$449$	4.26229	0.201150	0.100575	+	0.994929i	$0.467932\pi$
$450$	−1.00000	−0.0471405
$451$	13.9150	0.655231
$452$	−4.87547	−0.229323
$453$	13.4006	0.629617
$454$	18.8799	0.886079
$455$	4.16478	0.195248
$456$	2.82843	0.132453
$457$	−9.90215	−0.463203	−0.231602	−	0.972811i	$-0.574397\pi$
$457$	−9.90215	−0.463203	−0.231602	+	0.972811i	$0.574397\pi$
$458$	14.7183	0.687742
$459$	0	0
$460$	−2.98067	−0.138974
$461$	30.7765	1.43340	0.716702	−	0.697380i	$-0.245651\pi$
$461$	30.7765	1.43340	0.716702	+	0.697380i	$0.245651\pi$
$462$	−1.53073	−0.0712162
$463$	18.6378	0.866172	0.433086	−	0.901353i	$-0.357425\pi$
$463$	18.6378	0.866172	0.433086	+	0.901353i	$0.357425\pi$
$464$	−8.64047	−0.401124
$465$	−3.44993	−0.159987
$466$	4.38078	0.202936
$467$	−15.2886	−0.707473	−0.353737	−	0.935345i	$-0.615089\pi$
$467$	−15.2886	−0.707473	−0.353737	+	0.935345i	$0.615089\pi$
$468$	−3.84776	−0.177863
$469$	−5.36839	−0.247889
$470$	−9.39104	−0.433176
$471$	−6.64168	−0.306033
$472$	13.1792	0.606622
$473$	−6.05052	−0.278203
$474$	−0.328878	−0.0151059
$475$	2.82843	0.129777
$476$	0	0
$477$	−3.29769	−0.150991
$478$	−15.6401	−0.715361
$479$	25.0385	1.14404	0.572019	−	0.820241i	$-0.306161\pi$
$479$	25.0385	1.14404	0.572019	+	0.820241i	$0.306161\pi$
$480$	1.00000	0.0456435
$481$	−28.4389	−1.29670
$482$	3.64847	0.166183
$483$	−3.22625	−0.146800
$484$	−9.00000	−0.409091
$485$	0.253965	0.0115320
$486$	1.00000	0.0453609
$487$	15.4994	0.702346	0.351173	−	0.936311i	$-0.385783\pi$
$487$	15.4994	0.702346	0.351173	+	0.936311i	$0.385783\pi$
$488$	−2.46927	−0.111778
$489$	7.04148	0.318427
$490$	5.82843	0.263301
$491$	−28.2731	−1.27595	−0.637974	−	0.770058i	$-0.720227\pi$
$491$	−28.2731	−1.27595	−0.637974	+	0.770058i	$0.720227\pi$
$492$	−9.83938	−0.443593
$493$	0	0
$494$	10.8831	0.489654
$495$	1.41421	0.0635642
$496$	3.44993	0.154907
$497$	−3.51472	−0.157657
$498$	−0.537522	−0.0240869
$499$	36.9410	1.65370	0.826852	−	0.562419i	$-0.190129\pi$
$499$	36.9410	1.65370	0.826852	+	0.562419i	$0.190129\pi$
$500$	1.00000	0.0447214
$501$	0.474462	0.0211974
$502$	−9.89059	−0.441438
$503$	7.63490	0.340423	0.170212	−	0.985408i	$-0.445555\pi$
$503$	7.63490	0.340423	0.170212	+	0.985408i	$0.445555\pi$
$504$	1.08239	0.0482136
$505$	13.0353	0.580065
$506$	4.21530	0.187393
$507$	−1.80525	−0.0801740
$508$	−12.3205	−0.546634
$509$	−40.3578	−1.78883	−0.894414	−	0.447241i	$-0.852407\pi$
$509$	−40.3578	−1.78883	−0.894414	+	0.447241i	$0.852407\pi$
$510$	0	0
$511$	9.24545	0.408995
$512$	−1.00000	−0.0441942
$513$	−2.82843	−0.124878
$514$	20.7338	0.914529
$515$	−7.92856	−0.349374
$516$	4.27836	0.188344
$517$	13.2809	0.584095
$518$	8.00000	0.351500
$519$	−16.4936	−0.723989
$520$	3.84776	0.168735
$521$	−28.5441	−1.25054	−0.625270	−	0.780408i	$-0.715011\pi$
$521$	−28.5441	−1.25054	−0.625270	+	0.780408i	$0.715011\pi$
$522$	8.64047	0.378183
$523$	−38.1253	−1.66710	−0.833552	−	0.552441i	$-0.813696\pi$
$523$	−38.1253	−1.66710	−0.833552	+	0.552441i	$0.813696\pi$
$524$	3.96722	0.173309
$525$	1.08239	0.0472395
$526$	21.3839	0.932381
$527$	0	0
$528$	−1.41421	−0.0615457
$529$	−14.1156	−0.613723
$530$	3.29769	0.143243
$531$	−13.1792	−0.571929
$532$	−3.06147	−0.132731
$533$	−37.8596	−1.63988
$534$	17.5718	0.760408
$535$	−13.1161	−0.567060
$536$	−4.95974	−0.214228
$537$	−1.23880	−0.0534580
$538$	18.7949	0.810306
$539$	−8.24264	−0.355036
$540$	−1.00000	−0.0430331
$541$	−5.80562	−0.249603	−0.124802	−	0.992182i	$-0.539829\pi$
$541$	−5.80562	−0.249603	−0.124802	+	0.992182i	$0.539829\pi$
$542$	20.3192	0.872785
$543$	−8.85802	−0.380134
$544$	0	0
$545$	−1.10332	−0.0472609
$546$	4.16478	0.178236
$547$	15.2508	0.652078	0.326039	−	0.945356i	$-0.394286\pi$
$547$	15.2508	0.652078	0.326039	+	0.945356i	$0.394286\pi$
$548$	−18.0019	−0.769003
$549$	2.46927	0.105386
$550$	−1.41421	−0.0603023
$551$	−24.4389	−1.04113
$552$	−2.98067	−0.126866
$553$	0.355975	0.0151376
$554$	−5.60859	−0.238286
$555$	−7.39104	−0.313732
$556$	16.7274	0.709400
$557$	4.27759	0.181247	0.0906237	−	0.995885i	$-0.471114\pi$
$557$	4.27759	0.181247	0.0906237	+	0.995885i	$0.471114\pi$
$558$	−3.44993	−0.146047
$559$	16.4621	0.696273
$560$	−1.08239	−0.0457394
$561$	0	0
$562$	5.85123	0.246819
$563$	5.86725	0.247275	0.123637	−	0.992327i	$-0.460544\pi$
$563$	5.86725	0.247275	0.123637	+	0.992327i	$0.460544\pi$
$564$	−9.39104	−0.395434
$565$	−4.87547	−0.205113
$566$	−24.5792	−1.03314
$567$	−1.08239	−0.0454562
$568$	−3.24718	−0.136249
$569$	−28.4971	−1.19466	−0.597329	−	0.801996i	$-0.703771\pi$
$569$	−28.4971	−1.19466	−0.597329	+	0.801996i	$0.703771\pi$
$570$	2.82843	0.118470
$571$	47.0266	1.96800	0.984001	−	0.178160i	$-0.0570146\pi$
$571$	47.0266	1.96800	0.984001	+	0.178160i	$0.0570146\pi$
$572$	−5.44155	−0.227523
$573$	15.7889	0.659589
$574$	10.6501	0.444525
$575$	−2.98067	−0.124302
$576$	1.00000	0.0416667
$577$	−31.8177	−1.32459	−0.662294	−	0.749244i	$-0.730417\pi$
$577$	−31.8177	−1.32459	−0.662294	+	0.749244i	$0.730417\pi$
$578$	0	0
$579$	1.33145	0.0553331
$580$	−8.64047	−0.358776
$581$	0.581810	0.0241375
$582$	0.253965	0.0105272
$583$	−4.66364	−0.193148
$584$	8.54168	0.353457
$585$	−3.84776	−0.159085
$586$	15.9317	0.658134
$587$	−3.95815	−0.163370	−0.0816852	−	0.996658i	$-0.526030\pi$
$587$	−3.95815	−0.163370	−0.0816852	+	0.996658i	$0.526030\pi$
$588$	5.82843	0.240360
$589$	9.75789	0.402067
$590$	13.1792	0.542579
$591$	8.86709	0.364743
$592$	7.39104	0.303770
$593$	−22.9977	−0.944403	−0.472202	−	0.881491i	$-0.656541\pi$
$593$	−22.9977	−0.944403	−0.472202	+	0.881491i	$0.656541\pi$
$594$	1.41421	0.0580259
$595$	0	0
$596$	−7.95295	−0.325766
$597$	−15.6603	−0.640934
$598$	−11.4689	−0.468998
$599$	−34.7762	−1.42092	−0.710458	−	0.703739i	$-0.751512\pi$
$599$	−34.7762	−1.42092	−0.710458	+	0.703739i	$0.751512\pi$
$600$	1.00000	0.0408248
$601$	23.3155	0.951057	0.475529	−	0.879700i	$-0.342257\pi$
$601$	23.3155	0.951057	0.475529	+	0.879700i	$0.342257\pi$
$602$	−4.63087	−0.188740
$603$	4.95974	0.201976
$604$	−13.4006	−0.545264
$605$	−9.00000	−0.365902
$606$	13.0353	0.529525
$607$	−4.81339	−0.195369	−0.0976847	−	0.995217i	$-0.531144\pi$
$607$	−4.81339	−0.195369	−0.0976847	+	0.995217i	$0.531144\pi$
$608$	−2.82843	−0.114708
$609$	−9.35237	−0.378977
$610$	−2.46927	−0.0999777
$611$	−36.1344	−1.46184
$612$	0	0
$613$	36.9458	1.49223	0.746114	−	0.665818i	$-0.231917\pi$
$613$	36.9458	1.49223	0.746114	+	0.665818i	$0.231917\pi$
$614$	−25.3632	−1.02358
$615$	−9.83938	−0.396762
$616$	1.53073	0.0616750
$617$	−31.2457	−1.25791	−0.628953	−	0.777443i	$-0.716516\pi$
$617$	−31.2457	−1.25791	−0.628953	+	0.777443i	$0.716516\pi$
$618$	−7.92856	−0.318933
$619$	3.74882	0.150678	0.0753388	−	0.997158i	$-0.475996\pi$
$619$	3.74882	0.150678	0.0753388	+	0.997158i	$0.475996\pi$
$620$	3.44993	0.138553
$621$	2.98067	0.119610
$622$	20.3975	0.817865
$623$	−19.0196	−0.762005
$624$	3.84776	0.154034
$625$	1.00000	0.0400000
$626$	10.8267	0.432722
$627$	−4.00000	−0.159745
$628$	6.64168	0.265032
$629$	0	0
$630$	1.08239	0.0431235
$631$	15.0156	0.597763	0.298882	−	0.954290i	$-0.403386\pi$
$631$	15.0156	0.597763	0.298882	+	0.954290i	$0.403386\pi$
$632$	0.328878	0.0130821
$633$	−23.7502	−0.943986
$634$	21.3942	0.849673
$635$	−12.3205	−0.488924
$636$	3.29769	0.130762
$637$	22.4264	0.888566
$638$	12.2195	0.483773
$639$	3.24718	0.128456
$640$	−1.00000	−0.0395285
$641$	−29.5767	−1.16821	−0.584106	−	0.811678i	$-0.698555\pi$
$641$	−29.5767	−1.16821	−0.584106	+	0.811678i	$0.698555\pi$
$642$	−13.1161	−0.517653
$643$	−36.5554	−1.44161	−0.720803	−	0.693140i	$-0.756227\pi$
$643$	−36.5554	−1.44161	−0.720803	+	0.693140i	$0.756227\pi$
$644$	3.22625	0.127132
$645$	4.27836	0.168460
$646$	0	0
$647$	−6.72548	−0.264406	−0.132203	−	0.991223i	$-0.542205\pi$
$647$	−6.72548	−0.264406	−0.132203	+	0.991223i	$0.542205\pi$
$648$	−1.00000	−0.0392837
$649$	−18.6382	−0.731614
$650$	3.84776	0.150922
$651$	3.73418	0.146354
$652$	−7.04148	−0.275765
$653$	−25.5113	−0.998333	−0.499167	−	0.866506i	$-0.666360\pi$
$653$	−25.5113	−0.998333	−0.499167	+	0.866506i	$0.666360\pi$
$654$	−1.10332	−0.0431431
$655$	3.96722	0.155012
$656$	9.83938	0.384163
$657$	−8.54168	−0.333243
$658$	10.1648	0.396265
$659$	−4.92599	−0.191889	−0.0959446	−	0.995387i	$-0.530587\pi$
$659$	−4.92599	−0.191889	−0.0959446	+	0.995387i	$0.530587\pi$
$660$	−1.41421	−0.0550482
$661$	−39.6903	−1.54377	−0.771886	−	0.635761i	$-0.780687\pi$
$661$	−39.6903	−1.54377	−0.771886	+	0.635761i	$0.780687\pi$
$662$	−18.7402	−0.728359
$663$	0	0
$664$	0.537522	0.0208599
$665$	−3.06147	−0.118719
$666$	−7.39104	−0.286397
$667$	25.7544	0.997213
$668$	−0.474462	−0.0183575
$669$	−0.0714415	−0.00276209
$670$	−4.95974	−0.191612
$671$	3.49207	0.134810
$672$	−1.08239	−0.0417542
$673$	−29.2517	−1.12757	−0.563785	−	0.825922i	$-0.690655\pi$
$673$	−29.2517	−1.12757	−0.563785	+	0.825922i	$0.690655\pi$
$674$	−11.4061	−0.439346
$675$	−1.00000	−0.0384900
$676$	1.80525	0.0694327
$677$	−28.5960	−1.09903	−0.549517	−	0.835482i	$-0.685188\pi$
$677$	−28.5960	−1.09903	−0.549517	+	0.835482i	$0.685188\pi$
$678$	−4.87547	−0.187241
$679$	−0.274890	−0.0105493
$680$	0	0
$681$	18.8799	0.723480
$682$	−4.87894	−0.186824
$683$	−39.0151	−1.49287	−0.746436	−	0.665457i	$-0.768237\pi$
$683$	−39.0151	−1.49287	−0.746436	+	0.665457i	$0.768237\pi$
$684$	2.82843	0.108148
$685$	−18.0019	−0.687818
$686$	−13.8854	−0.530147
$687$	14.7183	0.561539
$688$	−4.27836	−0.163111
$689$	12.6887	0.483402
$690$	−2.98067	−0.113472
$691$	−46.7736	−1.77935	−0.889676	−	0.456592i	$-0.849070\pi$
$691$	−46.7736	−1.77935	−0.889676	+	0.456592i	$0.849070\pi$
$692$	16.4936	0.626993
$693$	−1.53073	−0.0581478
$694$	−21.2777	−0.807692
$695$	16.7274	0.634506
$696$	−8.64047	−0.327516
$697$	0	0
$698$	−16.0411	−0.607165
$699$	4.38078	0.165696
$700$	−1.08239	−0.0409106
$701$	17.2629	0.652011	0.326005	−	0.945368i	$-0.394297\pi$
$701$	17.2629	0.652011	0.326005	+	0.945368i	$0.394297\pi$
$702$	−3.84776	−0.145224
$703$	20.9050	0.788447
$704$	1.41421	0.0533002
$705$	−9.39104	−0.353687
$706$	35.1353	1.32234
$707$	−14.1094	−0.530637
$708$	13.1792	0.495305
$709$	9.99093	0.375217	0.187609	−	0.982244i	$-0.439926\pi$
$709$	9.99093	0.375217	0.187609	+	0.982244i	$0.439926\pi$
$710$	−3.24718	−0.121864
$711$	−0.328878	−0.0123339
$712$	−17.5718	−0.658532
$713$	−10.2831	−0.385105
$714$	0	0
$715$	−5.44155	−0.203503
$716$	1.23880	0.0462960
$717$	−15.6401	−0.584090
$718$	6.84519	0.255460
$719$	26.4407	0.986070	0.493035	−	0.870010i	$-0.335887\pi$
$719$	26.4407	0.986070	0.493035	+	0.870010i	$0.335887\pi$
$720$	1.00000	0.0372678
$721$	8.58181	0.319603
$722$	11.0000	0.409378
$723$	3.64847	0.135688
$724$	8.85802	0.329206
$725$	−8.64047	−0.320899
$726$	−9.00000	−0.334021
$727$	−32.9914	−1.22358	−0.611792	−	0.791019i	$-0.709551\pi$
$727$	−32.9914	−1.22358	−0.611792	+	0.791019i	$0.709551\pi$
$728$	−4.16478	−0.154357
$729$	1.00000	0.0370370
$730$	8.54168	0.316142
$731$	0	0
$732$	−2.46927	−0.0912667
$733$	−14.5678	−0.538074	−0.269037	−	0.963130i	$-0.586705\pi$
$733$	−14.5678	−0.538074	−0.269037	+	0.963130i	$0.586705\pi$
$734$	−10.7915	−0.398321
$735$	5.82843	0.214985
$736$	2.98067	0.109869
$737$	7.01414	0.258369
$738$	−9.83938	−0.362193
$739$	−42.8096	−1.57478	−0.787388	−	0.616457i	$-0.788567\pi$
$739$	−42.8096	−1.57478	−0.787388	+	0.616457i	$0.788567\pi$
$740$	7.39104	0.271700
$741$	10.8831	0.399801
$742$	−3.56940	−0.131037
$743$	−27.9665	−1.02599	−0.512996	−	0.858391i	$-0.671465\pi$
$743$	−27.9665	−1.02599	−0.512996	+	0.858391i	$0.671465\pi$
$744$	3.44993	0.126481
$745$	−7.95295	−0.291374
$746$	−30.5657	−1.11909
$747$	−0.537522	−0.0196669
$748$	0	0
$749$	14.1968	0.518740
$750$	1.00000	0.0365148
$751$	16.6702	0.608305	0.304152	−	0.952623i	$-0.401627\pi$
$751$	16.6702	0.608305	0.304152	+	0.952623i	$0.401627\pi$
$752$	9.39104	0.342456
$753$	−9.89059	−0.360433
$754$	−33.2464	−1.21076
$755$	−13.4006	−0.487699
$756$	1.08239	0.0393662
$757$	8.65664	0.314631	0.157316	−	0.987548i	$-0.449716\pi$
$757$	8.65664	0.314631	0.157316	+	0.987548i	$0.449716\pi$
$758$	36.5643	1.32808
$759$	4.21530	0.153006
$760$	−2.82843	−0.102598
$761$	24.1950	0.877068	0.438534	−	0.898714i	$-0.355498\pi$
$761$	24.1950	0.877068	0.438534	+	0.898714i	$0.355498\pi$
$762$	−12.3205	−0.446325
$763$	1.19422	0.0432337
$764$	−15.7889	−0.571221
$765$	0	0
$766$	−12.1158	−0.437761
$767$	50.7104	1.83105
$768$	−1.00000	−0.0360844
$769$	−30.3808	−1.09556	−0.547780	−	0.836623i	$-0.684527\pi$
$769$	−30.3808	−1.09556	−0.547780	+	0.836623i	$0.684527\pi$
$770$	1.53073	0.0551638
$771$	20.7338	0.746710
$772$	−1.33145	−0.0479198
$773$	−31.8564	−1.14579	−0.572897	−	0.819627i	$-0.694180\pi$
$773$	−31.8564	−1.14579	−0.572897	+	0.819627i	$0.694180\pi$
$774$	4.27836	0.153783
$775$	3.44993	0.123925
$776$	−0.253965	−0.00911681
$777$	8.00000	0.286998
$778$	26.5242	0.950940
$779$	27.8300	0.997112
$780$	3.84776	0.137772
$781$	4.59220	0.164322
$782$	0	0
$783$	8.64047	0.308785
$784$	−5.82843	−0.208158
$785$	6.64168	0.237052
$786$	3.96722	0.141506
$787$	13.5777	0.483994	0.241997	−	0.970277i	$-0.422198\pi$
$787$	13.5777	0.483994	0.241997	+	0.970277i	$0.422198\pi$
$788$	−8.86709	−0.315877
$789$	21.3839	0.761286
$790$	0.328878	0.0117010
$791$	5.27717	0.187635
$792$	−1.41421	−0.0502519
$793$	−9.50114	−0.337395
$794$	−9.97364	−0.353951
$795$	3.29769	0.116957
$796$	15.6603	0.555066
$797$	−13.4639	−0.476914	−0.238457	−	0.971153i	$-0.576642\pi$
$797$	−13.4639	−0.476914	−0.238457	+	0.971153i	$0.576642\pi$
$798$	−3.06147	−0.108375
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	17.5718	0.620870
$802$	6.68138	0.235928
$803$	−12.0798	−0.426286
$804$	−4.95974	−0.174917
$805$	3.22625	0.113710
$806$	13.2745	0.467575
$807$	18.7949	0.661612
$808$	−13.0353	−0.458582
$809$	29.1235	1.02393	0.511964	−	0.859007i	$-0.328918\pi$
$809$	29.1235	1.02393	0.511964	+	0.859007i	$0.328918\pi$
$810$	−1.00000	−0.0351364
$811$	−27.6273	−0.970124	−0.485062	−	0.874480i	$-0.661203\pi$
$811$	−27.6273	−0.970124	−0.485062	+	0.874480i	$0.661203\pi$
$812$	9.35237	0.328204
$813$	20.3192	0.712626
$814$	−10.4525	−0.366360
$815$	−7.04148	−0.246652
$816$	0	0
$817$	−12.1010	−0.423362
$818$	−1.48756	−0.0520115
$819$	4.16478	0.145529
$820$	9.83938	0.343606
$821$	−15.8222	−0.552197	−0.276099	−	0.961129i	$-0.589042\pi$
$821$	−15.8222	−0.552197	−0.276099	+	0.961129i	$0.589042\pi$
$822$	−18.0019	−0.627889
$823$	−45.0552	−1.57053	−0.785263	−	0.619162i	$-0.787472\pi$
$823$	−45.0552	−1.57053	−0.785263	+	0.619162i	$0.787472\pi$
$824$	7.92856	0.276204
$825$	−1.41421	−0.0492366
$826$	−14.2651	−0.496345
$827$	37.1986	1.29352	0.646761	−	0.762693i	$-0.276123\pi$
$827$	37.1986	1.29352	0.646761	+	0.762693i	$0.276123\pi$
$828$	−2.98067	−0.103585
$829$	−52.0539	−1.80791	−0.903954	−	0.427630i	$-0.859349\pi$
$829$	−52.0539	−1.80791	−0.903954	+	0.427630i	$0.859349\pi$
$830$	0.537522	0.0186577
$831$	−5.60859	−0.194560
$832$	−3.84776	−0.133397
$833$	0	0
$834$	16.7274	0.579222
$835$	−0.474462	−0.0164194
$836$	4.00000	0.138343
$837$	−3.44993	−0.119247
$838$	22.6261	0.781607
$839$	29.8445	1.03035	0.515174	−	0.857086i	$-0.327727\pi$
$839$	29.8445	1.03035	0.515174	+	0.857086i	$0.327727\pi$
$840$	−1.08239	−0.0373461
$841$	45.6576	1.57440
$842$	26.8469	0.925205
$843$	5.85123	0.201527
$844$	23.7502	0.817515
$845$	1.80525	0.0621025
$846$	−9.39104	−0.322870
$847$	9.74153	0.334723
$848$	−3.29769	−0.113243
$849$	−24.5792	−0.843554
$850$	0	0
$851$	−22.0302	−0.755186
$852$	−3.24718	−0.111246
$853$	−43.2383	−1.48045	−0.740225	−	0.672359i	$-0.765281\pi$
$853$	−43.2383	−1.48045	−0.740225	+	0.672359i	$0.765281\pi$
$854$	2.67271	0.0914584
$855$	2.82843	0.0967302
$856$	13.1161	0.448301
$857$	−24.9681	−0.852893	−0.426446	−	0.904513i	$-0.640235\pi$
$857$	−24.9681	−0.852893	−0.426446	+	0.904513i	$0.640235\pi$
$858$	−5.44155	−0.185772
$859$	−24.7577	−0.844723	−0.422361	−	0.906427i	$-0.638799\pi$
$859$	−24.7577	−0.844723	−0.422361	+	0.906427i	$0.638799\pi$
$860$	−4.27836	−0.145891
$861$	10.6501	0.362953
$862$	−11.5837	−0.394542
$863$	40.6749	1.38459	0.692295	−	0.721614i	$-0.256600\pi$
$863$	40.6749	1.38459	0.692295	+	0.721614i	$0.256600\pi$
$864$	1.00000	0.0340207
$865$	16.4936	0.560799
$866$	−16.6971	−0.567389
$867$	0	0
$868$	−3.73418	−0.126746
$869$	−0.465104	−0.0157776
$870$	−8.64047	−0.292939
$871$	−19.0839	−0.646633
$872$	1.10332	0.0373630
$873$	0.253965	0.00859542
$874$	8.43060	0.285169
$875$	−1.08239	−0.0365915
$876$	8.54168	0.288597
$877$	−45.8310	−1.54760	−0.773802	−	0.633428i	$-0.781647\pi$
$877$	−45.8310	−1.54760	−0.773802	+	0.633428i	$0.781647\pi$
$878$	−7.64603	−0.258041
$879$	15.9317	0.537364
$880$	1.41421	0.0476731
$881$	36.8865	1.24274	0.621369	−	0.783518i	$-0.286577\pi$
$881$	36.8865	1.24274	0.621369	+	0.783518i	$0.286577\pi$
$882$	5.82843	0.196253
$883$	38.2494	1.28719	0.643597	−	0.765364i	$-0.277441\pi$
$883$	38.2494	1.28719	0.643597	+	0.765364i	$0.277441\pi$
$884$	0	0
$885$	13.1792	0.443014
$886$	15.4265	0.518264
$887$	41.2499	1.38504	0.692518	−	0.721401i	$-0.256501\pi$
$887$	41.2499	1.38504	0.692518	+	0.721401i	$0.256501\pi$
$888$	7.39104	0.248027
$889$	13.3356	0.447262
$890$	−17.5718	−0.589009
$891$	1.41421	0.0473779
$892$	0.0714415	0.00239204
$893$	26.5619	0.888859
$894$	−7.95295	−0.265987
$895$	1.23880	0.0414084
$896$	1.08239	0.0361602
$897$	−11.4689	−0.382935
$898$	−4.26229	−0.142234
$899$	−29.8090	−0.994187
$900$	1.00000	0.0333333
$901$	0	0
$902$	−13.9150	−0.463318
$903$	−4.63087	−0.154106
$904$	4.87547	0.162156
$905$	8.85802	0.294451
$906$	−13.4006	−0.445206
$907$	43.2447	1.43592	0.717958	−	0.696086i	$-0.245077\pi$
$907$	43.2447	1.43592	0.717958	+	0.696086i	$0.245077\pi$
$908$	−18.8799	−0.626552
$909$	13.0353	0.432355
$910$	−4.16478	−0.138061
$911$	−1.50605	−0.0498977	−0.0249489	−	0.999689i	$-0.507942\pi$
$911$	−1.50605	−0.0498977	−0.0249489	+	0.999689i	$0.507942\pi$
$912$	−2.82843	−0.0936586
$913$	−0.760171	−0.0251580
$914$	9.90215	0.327534
$915$	−2.46927	−0.0816314
$916$	−14.7183	−0.486307
$917$	−4.29409	−0.141803
$918$	0	0
$919$	−14.2426	−0.469821	−0.234911	−	0.972017i	$-0.575480\pi$
$919$	−14.2426	−0.469821	−0.234911	+	0.972017i	$0.575480\pi$
$920$	2.98067	0.0982697
$921$	−25.3632	−0.835746
$922$	−30.7765	−1.01357
$923$	−12.4944	−0.411257
$924$	1.53073	0.0503574
$925$	7.39104	0.243016
$926$	−18.6378	−0.612476
$927$	−7.92856	−0.260408
$928$	8.64047	0.283637
$929$	−35.0789	−1.15090	−0.575451	−	0.817836i	$-0.695174\pi$
$929$	−35.0789	−1.15090	−0.575451	+	0.817836i	$0.695174\pi$
$930$	3.44993	0.113128
$931$	−16.4853	−0.540283
$932$	−4.38078	−0.143497
$933$	20.3975	0.667784
$934$	15.2886	0.500259
$935$	0	0
$936$	3.84776	0.125768
$937$	−33.4881	−1.09401	−0.547005	−	0.837129i	$-0.684232\pi$
$937$	−33.4881	−1.09401	−0.547005	+	0.837129i	$0.684232\pi$
$938$	5.36839	0.175284
$939$	10.8267	0.353316
$940$	9.39104	0.306302
$941$	−31.5699	−1.02915	−0.514575	−	0.857445i	$-0.672050\pi$
$941$	−31.5699	−1.02915	−0.514575	+	0.857445i	$0.672050\pi$
$942$	6.64168	0.216398
$943$	−29.3279	−0.955048
$944$	−13.1792	−0.428947
$945$	1.08239	0.0352102
$946$	6.05052	0.196719
$947$	5.41474	0.175955	0.0879777	−	0.996122i	$-0.471960\pi$
$947$	5.41474	0.175955	0.0879777	+	0.996122i	$0.471960\pi$
$948$	0.328878	0.0106815
$949$	32.8663	1.06689
$950$	−2.82843	−0.0917663
$951$	21.3942	0.693755
$952$	0	0
$953$	−41.3447	−1.33929	−0.669643	−	0.742683i	$-0.733553\pi$
$953$	−41.3447	−1.33929	−0.669643	+	0.742683i	$0.733553\pi$
$954$	3.29769	0.106767
$955$	−15.7889	−0.510916
$956$	15.6401	0.505837
$957$	12.2195	0.394999
$958$	−25.0385	−0.808956
$959$	19.4851	0.629208
$960$	−1.00000	−0.0322749
$961$	−19.0980	−0.616063
$962$	28.4389	0.916908
$963$	−13.1161	−0.422662
$964$	−3.64847	−0.117509
$965$	−1.33145	−0.0428608
$966$	3.22625	0.103803
$967$	−15.5004	−0.498459	−0.249230	−	0.968444i	$-0.580177\pi$
$967$	−15.5004	−0.498459	−0.249230	+	0.968444i	$0.580177\pi$
$968$	9.00000	0.289271
$969$	0	0
$970$	−0.253965	−0.00815433
$971$	14.9735	0.480524	0.240262	−	0.970708i	$-0.422767\pi$
$971$	14.9735	0.480524	0.240262	+	0.970708i	$0.422767\pi$
$972$	−1.00000	−0.0320750
$973$	−18.1056	−0.580439
$974$	−15.4994	−0.496633
$975$	3.84776	0.123227
$976$	2.46927	0.0790393
$977$	−34.3333	−1.09842	−0.549210	−	0.835684i	$-0.685071\pi$
$977$	−34.3333	−1.09842	−0.549210	+	0.835684i	$0.685071\pi$
$978$	−7.04148	−0.225162
$979$	24.8503	0.794220
$980$	−5.82843	−0.186182
$981$	−1.10332	−0.0352262
$982$	28.2731	0.902231
$983$	−20.1177	−0.641656	−0.320828	−	0.947137i	$-0.603961\pi$
$983$	−20.1177	−0.641656	−0.320828	+	0.947137i	$0.603961\pi$
$984$	9.83938	0.313668
$985$	−8.86709	−0.282529
$986$	0	0
$987$	10.1648	0.323549
$988$	−10.8831	−0.346238
$989$	12.7524	0.405502
$990$	−1.41421	−0.0449467
$991$	10.0748	0.320038	0.160019	−	0.987114i	$-0.448845\pi$
$991$	10.0748	0.320038	0.160019	+	0.987114i	$0.448845\pi$
$992$	−3.44993	−0.109536
$993$	−18.7402	−0.594703
$994$	3.51472	0.111480
$995$	15.6603	0.496466
$996$	0.537522	0.0170320
$997$	23.1772	0.734029	0.367015	−	0.930215i	$-0.380380\pi$
$997$	23.1772	0.734029	0.367015	+	0.930215i	$0.380380\pi$
$998$	−36.9410	−1.16935
$999$	−7.39104	−0.233842

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.bu.1.2		4
17.11	odd	16		510.2.u.b.121.2	✓	8
17.14	odd	16		510.2.u.b.451.2	yes	8
17.16	even	2		8670.2.a.bv.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.u.b.121.2	✓	8	17.11	odd	16
510.2.u.b.451.2	yes	8	17.14	odd	16
8670.2.a.bu.1.2		4	1.1	even	1	trivial
8670.2.a.bv.1.3		4	17.16	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:	\( N \)	\(=\)	\( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8670.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.08239 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -1.08239 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.41421 q^{11} -1.00000 q^{12} -3.84776 q^{13} +1.08239 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +2.82843 q^{19} +1.00000 q^{20} +1.08239 q^{21} -1.41421 q^{22} -2.98067 q^{23} +1.00000 q^{24} +1.00000 q^{25} +3.84776 q^{26} -1.00000 q^{27} -1.08239 q^{28} -8.64047 q^{29} +1.00000 q^{30} +3.44993 q^{31} -1.00000 q^{32} -1.41421 q^{33} -1.08239 q^{35} +1.00000 q^{36} +7.39104 q^{37} -2.82843 q^{38} +3.84776 q^{39} -1.00000 q^{40} +9.83938 q^{41} -1.08239 q^{42} -4.27836 q^{43} +1.41421 q^{44} +1.00000 q^{45} +2.98067 q^{46} +9.39104 q^{47} -1.00000 q^{48} -5.82843 q^{49} -1.00000 q^{50} -3.84776 q^{52} -3.29769 q^{53} +1.00000 q^{54} +1.41421 q^{55} +1.08239 q^{56} -2.82843 q^{57} +8.64047 q^{58} -13.1792 q^{59} -1.00000 q^{60} +2.46927 q^{61} -3.44993 q^{62} -1.08239 q^{63} +1.00000 q^{64} -3.84776 q^{65} +1.41421 q^{66} +4.95974 q^{67} +2.98067 q^{69} +1.08239 q^{70} +3.24718 q^{71} -1.00000 q^{72} -8.54168 q^{73} -7.39104 q^{74} -1.00000 q^{75} +2.82843 q^{76} -1.53073 q^{77} -3.84776 q^{78} -0.328878 q^{79} +1.00000 q^{80} +1.00000 q^{81} -9.83938 q^{82} -0.537522 q^{83} +1.08239 q^{84} +4.27836 q^{86} +8.64047 q^{87} -1.41421 q^{88} +17.5718 q^{89} -1.00000 q^{90} +4.16478 q^{91} -2.98067 q^{92} -3.44993 q^{93} -9.39104 q^{94} +2.82843 q^{95} +1.00000 q^{96} +0.253965 q^{97} +5.82843 q^{98} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^8 + 4 * q^9	\( 4 q - 4 q^{2} - 4 q^{3} + 4 q^{4} + 4 q^{5} + 4 q^{6} - 4 q^{8} + 4 q^{9} - 4 q^{10} - 4 q^{12} - 8 q^{13} - 4 q^{15} + 4 q^{16} - 4 q^{18} + 4 q^{20} - 8 q^{23} + 4 q^{24} + 4 q^{25} + 8 q^{26} - 4 q^{27} - 8 q^{29} + 4 q^{30} + 16 q^{31} - 4 q^{32} + 4 q^{36} + 8 q^{39} - 4 q^{40} + 8 q^{41} - 8 q^{43} + 4 q^{45} + 8 q^{46} + 8 q^{47} - 4 q^{48} - 12 q^{49} - 4 q^{50} - 8 q^{52} - 8 q^{53} + 4 q^{54} + 8 q^{58} - 4 q^{60} + 16 q^{61} - 16 q^{62} + 4 q^{64} - 8 q^{65} + 8 q^{67} + 8 q^{69} - 4 q^{72} - 8 q^{73} - 4 q^{75} - 8 q^{78} + 4 q^{80} + 4 q^{81} - 8 q^{82} - 16 q^{83} + 8 q^{86} + 8 q^{87} - 8 q^{89} - 4 q^{90} + 8 q^{91} - 8 q^{92} - 16 q^{93} - 8 q^{94} + 4 q^{96} + 8 q^{97} + 12 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 - 4 * q^3 + 4 * q^4 + 4 * q^5 + 4 * q^6 - 4 * q^8 + 4 * q^9 - 4 * q^10 - 4 * q^12 - 8 * q^13 - 4 * q^15 + 4 * q^16 - 4 * q^18 + 4 * q^20 - 8 * q^23 + 4 * q^24 + 4 * q^25 + 8 * q^26 - 4 * q^27 - 8 * q^29 + 4 * q^30 + 16 * q^31 - 4 * q^32 + 4 * q^36 + 8 * q^39 - 4 * q^40 + 8 * q^41 - 8 * q^43 + 4 * q^45 + 8 * q^46 + 8 * q^47 - 4 * q^48 - 12 * q^49 - 4 * q^50 - 8 * q^52 - 8 * q^53 + 4 * q^54 + 8 * q^58 - 4 * q^60 + 16 * q^61 - 16 * q^62 + 4 * q^64 - 8 * q^65 + 8 * q^67 + 8 * q^69 - 4 * q^72 - 8 * q^73 - 4 * q^75 - 8 * q^78 + 4 * q^80 + 4 * q^81 - 8 * q^82 - 16 * q^83 + 8 * q^86 + 8 * q^87 - 8 * q^89 - 4 * q^90 + 8 * q^91 - 8 * q^92 - 16 * q^93 - 8 * q^94 + 4 * q^96 + 8 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more