LMFDB - Embedded newform 8670.2.a.bv.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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.08239 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.08239 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.41421 q^{11} +1.00000 q^{12} -0.152241 q^{13} +1.08239 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +2.82843 q^{19} -1.00000 q^{20} -1.08239 q^{21} +1.41421 q^{22} +6.67619 q^{23} -1.00000 q^{24} +1.00000 q^{25} +0.152241 q^{26} +1.00000 q^{27} -1.08239 q^{28} -1.81204 q^{29} +1.00000 q^{30} -10.2069 q^{31} -1.00000 q^{32} -1.41421 q^{33} +1.08239 q^{35} +1.00000 q^{36} +7.39104 q^{37} -2.82843 q^{38} -0.152241 q^{39} +1.00000 q^{40} +5.83938 q^{41} +1.08239 q^{42} -11.0353 q^{43} -1.41421 q^{44} -1.00000 q^{45} -6.67619 q^{46} -5.39104 q^{47} +1.00000 q^{48} -5.82843 q^{49} -1.00000 q^{50} -0.152241 q^{52} -6.35916 q^{53} -1.00000 q^{54} +1.41421 q^{55} +1.08239 q^{56} +2.82843 q^{57} +1.81204 q^{58} +1.86550 q^{59} -1.00000 q^{60} -5.53073 q^{61} +10.2069 q^{62} -1.08239 q^{63} +1.00000 q^{64} +0.152241 q^{65} +1.41421 q^{66} +4.69711 q^{67} +6.67619 q^{69} -1.08239 q^{70} +3.24718 q^{71} -1.00000 q^{72} -10.1985 q^{73} -7.39104 q^{74} +1.00000 q^{75} +2.82843 q^{76} +1.53073 q^{77} +0.152241 q^{78} +16.6417 q^{79} -1.00000 q^{80} +1.00000 q^{81} -5.83938 q^{82} -1.80562 q^{83} -1.08239 q^{84} +11.0353 q^{86} -1.81204 q^{87} +1.41421 q^{88} -4.60127 q^{89} +1.00000 q^{90} +0.164784 q^{91} +6.67619 q^{92} -10.2069 q^{93} +5.39104 q^{94} -2.82843 q^{95} -1.00000 q^{96} +1.91082 q^{97} +5.82843 q^{98} -1.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	1.00000	0.288675
$13$	−0.152241	−0.0422240	−0.0211120	−	0.999777i	$-0.506721\pi$
$13$	−0.152241	−0.0422240	−0.0211120	+	0.999777i	$0.506721\pi$
$14$	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
$23$	6.67619	1.39208	0.696041	−	0.718002i	$-0.254943\pi$
$23$	6.67619	1.39208	0.696041	+	0.718002i	$0.254943\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	0.152241	0.0298569
$27$	1.00000	0.192450
$28$	−1.08239	−0.204553
$29$	−1.81204	−0.336487	−0.168244	−	0.985745i	$-0.553809\pi$
$29$	−1.81204	−0.336487	−0.168244	+	0.985745i	$0.553809\pi$
$30$	1.00000	0.182574
$31$	−10.2069	−1.83322	−0.916608	−	0.399786i	$-0.869084\pi$
$31$	−10.2069	−1.83322	−0.916608	+	0.399786i	$0.869084\pi$
$32$	−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
$39$	−0.152241	−0.0243781
$40$	1.00000	0.158114
$41$	5.83938	0.911958	0.455979	−	0.889991i	$-0.349289\pi$
$41$	5.83938	0.911958	0.455979	+	0.889991i	$0.349289\pi$
$42$	1.08239	0.167017
$43$	−11.0353	−1.68287	−0.841437	−	0.540355i	$-0.818290\pi$
$43$	−11.0353	−1.68287	−0.841437	+	0.540355i	$0.818290\pi$
$44$	−1.41421	−0.213201
$45$	−1.00000	−0.149071
$46$	−6.67619	−0.984350
$47$	−5.39104	−0.786363	−0.393182	−	0.919461i	$-0.628626\pi$
$47$	−5.39104	−0.786363	−0.393182	+	0.919461i	$0.628626\pi$
$48$	1.00000	0.144338
$49$	−5.82843	−0.832632
$50$	−1.00000	−0.141421
$51$	0	0
$52$	−0.152241	−0.0211120
$53$	−6.35916	−0.873498	−0.436749	−	0.899583i	$-0.643870\pi$
$53$	−6.35916	−0.873498	−0.436749	+	0.899583i	$0.643870\pi$
$54$	−1.00000	−0.136083
$55$	1.41421	0.190693
$56$	1.08239	0.144641
$57$	2.82843	0.374634
$58$	1.81204	0.237932
$59$	1.86550	0.242867	0.121434	−	0.992600i	$-0.461251\pi$
$59$	1.86550	0.242867	0.121434	+	0.992600i	$0.461251\pi$
$60$	−1.00000	−0.129099
$61$	−5.53073	−0.708138	−0.354069	−	0.935219i	$-0.615202\pi$
$61$	−5.53073	−0.708138	−0.354069	+	0.935219i	$0.615202\pi$
$62$	10.2069	1.29628
$63$	−1.08239	−0.136369
$64$	1.00000	0.125000
$65$	0.152241	0.0188832
$66$	1.41421	0.174078
$67$	4.69711	0.573843	0.286922	−	0.957954i	$-0.407368\pi$
$67$	4.69711	0.573843	0.286922	+	0.957954i	$0.407368\pi$
$68$	0	0
$69$	6.67619	0.803718
$70$	−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
$73$	−10.1985	−1.19365	−0.596824	−	0.802372i	$-0.703571\pi$
$73$	−10.1985	−1.19365	−0.596824	+	0.802372i	$0.703571\pi$
$74$	−7.39104	−0.859191
$75$	1.00000	0.115470
$76$	2.82843	0.324443
$77$	1.53073	0.174443
$78$	0.152241	0.0172379
$79$	16.6417	1.87234	0.936168	−	0.351553i	$-0.114346\pi$
$79$	16.6417	1.87234	0.936168	+	0.351553i	$0.114346\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−5.83938	−0.644851
$83$	−1.80562	−0.198193	−0.0990965	−	0.995078i	$-0.531595\pi$
$83$	−1.80562	−0.198193	−0.0990965	+	0.995078i	$0.531595\pi$
$84$	−1.08239	−0.118099
$85$	0	0
$86$	11.0353	1.18997
$87$	−1.81204	−0.194271
$88$	1.41421	0.150756
$89$	−4.60127	−0.487734	−0.243867	−	0.969809i	$-0.578416\pi$
$89$	−4.60127	−0.487734	−0.243867	+	0.969809i	$0.578416\pi$
$90$	1.00000	0.105409
$91$	0.164784	0.0172741
$92$	6.67619	0.696041
$93$	−10.2069	−1.05841
$94$	5.39104	0.556043
$95$	−2.82843	−0.290191
$96$	−1.00000	−0.102062
$97$	1.91082	0.194014	0.0970072	−	0.995284i	$-0.469073\pi$
$97$	1.91082	0.194014	0.0970072	+	0.995284i	$0.469073\pi$
$98$	5.82843	0.588760
$99$	−1.41421	−0.142134
$100$	1.00000	0.100000
$101$	6.27836	0.624720	0.312360	−	0.949964i	$-0.398880\pi$
$101$	6.27836	0.624720	0.312360	+	0.949964i	$0.398880\pi$
$102$	0	0
$103$	5.58541	0.550347	0.275174	−	0.961395i	$-0.411265\pi$
$103$	5.58541	0.550347	0.275174	+	0.961395i	$0.411265\pi$
$104$	0.152241	0.0149285
$105$	1.08239	0.105631
$106$	6.35916	0.617656
$107$	−3.45929	−0.334422	−0.167211	−	0.985921i	$-0.553476\pi$
$107$	−3.45929	−0.334422	−0.167211	+	0.985921i	$0.553476\pi$
$108$	1.00000	0.0962250
$109$	2.89668	0.277452	0.138726	−	0.990331i	$-0.455699\pi$
$109$	2.89668	0.277452	0.138726	+	0.990331i	$0.455699\pi$
$110$	−1.41421	−0.134840
$111$	7.39104	0.701526
$112$	−1.08239	−0.102276
$113$	9.46767	0.890644	0.445322	−	0.895371i	$-0.353089\pi$
$113$	9.46767	0.890644	0.445322	+	0.895371i	$0.353089\pi$
$114$	−2.82843	−0.264906
$115$	−6.67619	−0.622558
$116$	−1.81204	−0.168244
$117$	−0.152241	−0.0140747
$118$	−1.86550	−0.171733
$119$	0	0
$120$	1.00000	0.0912871
$121$	−9.00000	−0.818182
$122$	5.53073	0.500729
$123$	5.83938	0.526519
$124$	−10.2069	−0.916608
$125$	−1.00000	−0.0894427
$126$	1.08239	0.0964272
$127$	−16.6501	−1.47745	−0.738727	−	0.674005i	$-0.764573\pi$
$127$	−16.6501	−1.47745	−0.738727	+	0.674005i	$0.764573\pi$
$128$	−1.00000	−0.0883883
$129$	−11.0353	−0.971608
$130$	−0.152241	−0.0133524
$131$	16.9378	1.47986	0.739931	−	0.672683i	$-0.234858\pi$
$131$	16.9378	1.47986	0.739931	+	0.672683i	$0.234858\pi$
$132$	−1.41421	−0.123091
$133$	−3.06147	−0.265463
$134$	−4.69711	−0.405769
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	12.8303	1.09617	0.548085	−	0.836423i	$-0.315357\pi$
$137$	12.8303	1.09617	0.548085	+	0.836423i	$0.315357\pi$
$138$	−6.67619	−0.568315
$139$	2.38425	0.202229	0.101115	−	0.994875i	$-0.467759\pi$
$139$	2.38425	0.202229	0.101115	+	0.994875i	$0.467759\pi$
$140$	1.08239	0.0914788
$141$	−5.39104	−0.454007
$142$	−3.24718	−0.272497
$143$	0.215301	0.0180044
$144$	1.00000	0.0833333
$145$	1.81204	0.150482
$146$	10.1985	0.844037
$147$	−5.82843	−0.480721
$148$	7.39104	0.607539
$149$	−3.36075	−0.275324	−0.137662	−	0.990479i	$-0.543959\pi$
$149$	−3.36075	−0.275324	−0.137662	+	0.990479i	$0.543959\pi$
$150$	−1.00000	−0.0816497
$151$	−4.74150	−0.385858	−0.192929	−	0.981213i	$-0.561799\pi$
$151$	−4.74150	−0.385858	−0.192929	+	0.981213i	$0.561799\pi$
$152$	−2.82843	−0.229416
$153$	0	0
$154$	−1.53073	−0.123350
$155$	10.2069	0.819840
$156$	−0.152241	−0.0121890
$157$	−9.67112	−0.771840	−0.385920	−	0.922532i	$-0.626116\pi$
$157$	−9.67112	−0.771840	−0.385920	+	0.922532i	$0.626116\pi$
$158$	−16.6417	−1.32394
$159$	−6.35916	−0.504314
$160$	1.00000	0.0790569
$161$	−7.22625	−0.569508
$162$	−1.00000	−0.0785674
$163$	1.44381	0.113088	0.0565438	−	0.998400i	$-0.481992\pi$
$163$	1.44381	0.113088	0.0565438	+	0.998400i	$0.481992\pi$
$164$	5.83938	0.455979
$165$	1.41421	0.110096
$166$	1.80562	0.140144
$167$	0.211830	0.0163919	0.00819593	−	0.999966i	$-0.497391\pi$
$167$	0.211830	0.0163919	0.00819593	+	0.999966i	$0.497391\pi$
$168$	1.08239	0.0835084
$169$	−12.9768	−0.998217
$170$	0	0
$171$	2.82843	0.216295
$172$	−11.0353	−0.841437
$173$	23.5230	1.78842	0.894212	−	0.447644i	$-0.147737\pi$
$173$	23.5230	1.78842	0.894212	+	0.447644i	$0.147737\pi$
$174$	1.81204	0.137370
$175$	−1.08239	−0.0818212
$176$	−1.41421	−0.106600
$177$	1.86550	0.140219
$178$	4.60127	0.344880
$179$	−17.2388	−1.28849	−0.644244	−	0.764820i	$-0.722828\pi$
$179$	−17.2388	−1.28849	−0.644244	+	0.764820i	$0.722828\pi$
$180$	−1.00000	−0.0745356
$181$	−14.4557	−1.07448	−0.537241	−	0.843428i	$-0.680534\pi$
$181$	−14.4557	−1.07448	−0.537241	+	0.843428i	$0.680534\pi$
$182$	−0.164784	−0.0122146
$183$	−5.53073	−0.408844
$184$	−6.67619	−0.492175
$185$	−7.39104	−0.543400
$186$	10.2069	0.748408
$187$	0	0
$188$	−5.39104	−0.393182
$189$	−1.08239	−0.0787324
$190$	2.82843	0.205196
$191$	9.44572	0.683468	0.341734	−	0.939797i	$-0.388986\pi$
$191$	9.44572	0.683468	0.341734	+	0.939797i	$0.388986\pi$
$192$	1.00000	0.0721688
$193$	−10.0177	−0.721093	−0.360546	−	0.932741i	$-0.617410\pi$
$193$	−10.0177	−0.721093	−0.360546	+	0.932741i	$0.617410\pi$
$194$	−1.91082	−0.137189
$195$	0.152241	0.0109022
$196$	−5.82843	−0.416316
$197$	1.47605	0.105165	0.0525823	−	0.998617i	$-0.483255\pi$
$197$	1.47605	0.105165	0.0525823	+	0.998617i	$0.483255\pi$
$198$	1.41421	0.100504
$199$	−20.6239	−1.46199	−0.730996	−	0.682381i	$-0.760944\pi$
$199$	−20.6239	−1.46199	−0.730996	+	0.682381i	$0.760944\pi$
$200$	−1.00000	−0.0707107
$201$	4.69711	0.331309
$202$	−6.27836	−0.441744
$203$	1.96134	0.137659
$204$	0	0
$205$	−5.83938	−0.407840
$206$	−5.58541	−0.389154
$207$	6.67619	0.464027
$208$	−0.152241	−0.0105560
$209$	−4.00000	−0.276686
$210$	−1.08239	−0.0746922
$211$	−5.90666	−0.406631	−0.203316	−	0.979113i	$-0.565172\pi$
$211$	−5.90666	−0.406631	−0.203316	+	0.979113i	$0.565172\pi$
$212$	−6.35916	−0.436749
$213$	3.24718	0.222493
$214$	3.45929	0.236472
$215$	11.0353	0.752604
$216$	−1.00000	−0.0680414
$217$	11.0479	0.749980
$218$	−2.89668	−0.196188
$219$	−10.1985	−0.689153
$220$	1.41421	0.0953463
$221$	0	0
$222$	−7.39104	−0.496054
$223$	13.5854	0.909747	0.454873	−	0.890556i	$-0.349685\pi$
$223$	13.5854	0.909747	0.454873	+	0.890556i	$0.349685\pi$
$224$	1.08239	0.0723204
$225$	1.00000	0.0666667
$226$	−9.46767	−0.629780
$227$	−6.87992	−0.456636	−0.228318	−	0.973587i	$-0.573323\pi$
$227$	−6.87992	−0.456636	−0.228318	+	0.973587i	$0.573323\pi$
$228$	2.82843	0.187317
$229$	−8.59539	−0.567999	−0.284000	−	0.958824i	$-0.591661\pi$
$229$	−8.59539	−0.567999	−0.284000	+	0.958824i	$0.591661\pi$
$230$	6.67619	0.440215
$231$	1.53073	0.100715
$232$	1.81204	0.118966
$233$	−19.6945	−1.29023	−0.645114	−	0.764086i	$-0.723190\pi$
$233$	−19.6945	−1.29023	−0.645114	+	0.764086i	$0.723190\pi$
$234$	0.152241	0.00995230
$235$	5.39104	0.351672
$236$	1.86550	0.121434
$237$	16.6417	1.08099
$238$	0	0
$239$	−8.32638	−0.538589	−0.269294	−	0.963058i	$-0.586790\pi$
$239$	−8.32638	−0.538589	−0.269294	+	0.963058i	$0.586790\pi$
$240$	−1.00000	−0.0645497
$241$	−8.33476	−0.536889	−0.268445	−	0.963295i	$-0.586510\pi$
$241$	−8.33476	−0.536889	−0.268445	+	0.963295i	$0.586510\pi$
$242$	9.00000	0.578542
$243$	1.00000	0.0641500
$244$	−5.53073	−0.354069
$245$	5.82843	0.372365
$246$	−5.83938	−0.372305
$247$	−0.430602	−0.0273986
$248$	10.2069	0.648140
$249$	−1.80562	−0.114427
$250$	1.00000	0.0632456
$251$	−21.2043	−1.33840	−0.669202	−	0.743081i	$-0.733364\pi$
$251$	−21.2043	−1.33840	−0.669202	+	0.743081i	$0.733364\pi$
$252$	−1.08239	−0.0681843
$253$	−9.44155	−0.593585
$254$	16.6501	1.04472
$255$	0	0
$256$	1.00000	0.0625000
$257$	7.56224	0.471719	0.235860	−	0.971787i	$-0.424209\pi$
$257$	7.56224	0.471719	0.235860	+	0.971787i	$0.424209\pi$
$258$	11.0353	0.687031
$259$	−8.00000	−0.497096
$260$	0.152241	0.00944158
$261$	−1.81204	−0.112162
$262$	−16.9378	−1.04642
$263$	11.2417	0.693195	0.346598	−	0.938014i	$-0.387337\pi$
$263$	11.2417	0.693195	0.346598	+	0.938014i	$0.387337\pi$
$264$	1.41421	0.0870388
$265$	6.35916	0.390640
$266$	3.06147	0.187711
$267$	−4.60127	−0.281593
$268$	4.69711	0.286922
$269$	−29.1380	−1.77658	−0.888289	−	0.459285i	$-0.848106\pi$
$269$	−29.1380	−1.77658	−0.888289	+	0.459285i	$0.848106\pi$
$270$	1.00000	0.0608581
$271$	21.4908	1.30547	0.652736	−	0.757585i	$-0.273621\pi$
$271$	21.4908	1.30547	0.652736	+	0.757585i	$0.273621\pi$
$272$	0	0
$273$	0.164784	0.00997321
$274$	−12.8303	−0.775109
$275$	−1.41421	−0.0852803
$276$	6.67619	0.401859
$277$	−6.87669	−0.413180	−0.206590	−	0.978428i	$-0.566237\pi$
$277$	−6.87669	−0.413180	−0.206590	+	0.978428i	$0.566237\pi$
$278$	−2.38425	−0.142998
$279$	−10.2069	−0.611072
$280$	−1.08239	−0.0646853
$281$	−7.11933	−0.424704	−0.212352	−	0.977193i	$-0.568112\pi$
$281$	−7.11933	−0.424704	−0.212352	+	0.977193i	$0.568112\pi$
$282$	5.39104	0.321032
$283$	−25.8473	−1.53646	−0.768230	−	0.640174i	$-0.778862\pi$
$283$	−25.8473	−1.53646	−0.768230	+	0.640174i	$0.778862\pi$
$284$	3.24718	0.192684
$285$	−2.82843	−0.167542
$286$	−0.215301	−0.0127310
$287$	−6.32050	−0.373087
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	−1.81204	−0.106407
$291$	1.91082	0.112014
$292$	−10.1985	−0.596824
$293$	−17.7251	−1.03551	−0.517756	−	0.855528i	$-0.673232\pi$
$293$	−17.7251	−1.03551	−0.517756	+	0.855528i	$0.673232\pi$
$294$	5.82843	0.339921
$295$	−1.86550	−0.108614
$296$	−7.39104	−0.429595
$297$	−1.41421	−0.0820610
$298$	3.36075	0.194683
$299$	−1.01639	−0.0587793
$300$	1.00000	0.0577350
$301$	11.9446	0.688474
$302$	4.74150	0.272843
$303$	6.27836	0.360682
$304$	2.82843	0.162221
$305$	5.53073	0.316689
$306$	0	0
$307$	18.2348	1.04071	0.520357	−	0.853949i	$-0.325799\pi$
$307$	18.2348	1.04071	0.520357	+	0.853949i	$0.325799\pi$
$308$	1.53073	0.0872216
$309$	5.58541	0.317743
$310$	−10.2069	−0.579714
$311$	−31.7112	−1.79818	−0.899088	−	0.437767i	$-0.855769\pi$
$311$	−31.7112	−1.79818	−0.899088	+	0.437767i	$0.855769\pi$
$312$	0.152241	0.00861895
$313$	19.1144	1.08041	0.540206	−	0.841533i	$-0.318346\pi$
$313$	19.1144	1.08041	0.540206	+	0.841533i	$0.318346\pi$
$314$	9.67112	0.545773
$315$	1.08239	0.0609859
$316$	16.6417	0.936168
$317$	21.9195	1.23112	0.615561	−	0.788090i	$-0.288930\pi$
$317$	21.9195	1.23112	0.615561	+	0.788090i	$0.288930\pi$
$318$	6.35916	0.356604
$319$	2.56261	0.143479
$320$	−1.00000	−0.0559017
$321$	−3.45929	−0.193079
$322$	7.22625	0.402703
$323$	0	0
$324$	1.00000	0.0555556
$325$	−0.152241	−0.00844481
$326$	−1.44381	−0.0799650
$327$	2.89668	0.160187
$328$	−5.83938	−0.322426
$329$	5.83522	0.321706
$330$	−1.41421	−0.0778499
$331$	−18.7402	−1.03006	−0.515028	−	0.857173i	$-0.672218\pi$
$331$	−18.7402	−1.03006	−0.515028	+	0.857173i	$0.672218\pi$
$332$	−1.80562	−0.0990965
$333$	7.39104	0.405026
$334$	−0.211830	−0.0115908
$335$	−4.69711	−0.256631
$336$	−1.08239	−0.0590493
$337$	−28.8782	−1.57310	−0.786548	−	0.617529i	$-0.788134\pi$
$337$	−28.8782	−1.57310	−0.786548	+	0.617529i	$0.788134\pi$
$338$	12.9768	0.705846
$339$	9.46767	0.514213
$340$	0	0
$341$	14.4348	0.781686
$342$	−2.82843	−0.152944
$343$	13.8854	0.749741
$344$	11.0353	0.594986
$345$	−6.67619	−0.359434
$346$	−23.5230	−1.26461
$347$	14.9346	0.801731	0.400866	−	0.916137i	$-0.368709\pi$
$347$	14.9346	0.801731	0.400866	+	0.916137i	$0.368709\pi$
$348$	−1.81204	−0.0971354
$349$	−3.07054	−0.164362	−0.0821811	−	0.996617i	$-0.526189\pi$
$349$	−3.07054	−0.164362	−0.0821811	+	0.996617i	$0.526189\pi$
$350$	1.08239	0.0578563
$351$	−0.152241	−0.00812602
$352$	1.41421	0.0753778
$353$	−30.8058	−1.63963	−0.819813	−	0.572631i	$-0.805923\pi$
$353$	−30.8058	−1.63963	−0.819813	+	0.572631i	$0.805923\pi$
$354$	−1.86550	−0.0991501
$355$	−3.24718	−0.172342
$356$	−4.60127	−0.243867
$357$	0	0
$358$	17.2388	0.911099
$359$	−30.8117	−1.62618	−0.813089	−	0.582140i	$-0.802216\pi$
$359$	−30.8117	−1.62618	−0.813089	+	0.582140i	$0.802216\pi$
$360$	1.00000	0.0527046
$361$	−11.0000	−0.578947
$362$	14.4557	0.759774
$363$	−9.00000	−0.472377
$364$	0.164784	0.00863705
$365$	10.1985	0.533816
$366$	5.53073	0.289096
$367$	−9.89480	−0.516505	−0.258252	−	0.966078i	$-0.583147\pi$
$367$	−9.89480	−0.516505	−0.258252	+	0.966078i	$0.583147\pi$
$368$	6.67619	0.348020
$369$	5.83938	0.303986
$370$	7.39104	0.384242
$371$	6.88311	0.357353
$372$	−10.2069	−0.529204
$373$	−20.9089	−1.08262	−0.541310	−	0.840823i	$-0.682072\pi$
$373$	−20.9089	−1.08262	−0.541310	+	0.840823i	$0.682072\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	5.39104	0.278021
$377$	0.275866	0.0142078
$378$	1.08239	0.0556722
$379$	−16.9663	−0.871501	−0.435751	−	0.900067i	$-0.643517\pi$
$379$	−16.9663	−0.871501	−0.435751	+	0.900067i	$0.643517\pi$
$380$	−2.82843	−0.145095
$381$	−16.6501	−0.853009
$382$	−9.44572	−0.483285
$383$	−17.9736	−0.918410	−0.459205	−	0.888330i	$-0.651866\pi$
$383$	−17.9736	−0.918410	−0.459205	+	0.888330i	$0.651866\pi$
$384$	−1.00000	−0.0510310
$385$	−1.53073	−0.0780134
$386$	10.0177	0.509889
$387$	−11.0353	−0.560958
$388$	1.91082	0.0970072
$389$	−8.78948	−0.445644	−0.222822	−	0.974859i	$-0.571527\pi$
$389$	−8.78948	−0.445644	−0.222822	+	0.974859i	$0.571527\pi$
$390$	−0.152241	−0.00770902
$391$	0	0
$392$	5.82843	0.294380
$393$	16.9378	0.854398
$394$	−1.47605	−0.0743626
$395$	−16.6417	−0.837334
$396$	−1.41421	−0.0710669
$397$	20.1158	1.00958	0.504791	−	0.863242i	$-0.331570\pi$
$397$	20.1158	1.00958	0.504791	+	0.863242i	$0.331570\pi$
$398$	20.6239	1.03378
$399$	−3.06147	−0.153265
$400$	1.00000	0.0500000
$401$	−0.338236	−0.0168907	−0.00844535	−	0.999964i	$-0.502688\pi$
$401$	−0.338236	−0.0168907	−0.00844535	+	0.999964i	$0.502688\pi$
$402$	−4.69711	−0.234271
$403$	1.55391	0.0774058
$404$	6.27836	0.312360
$405$	−1.00000	−0.0496904
$406$	−1.96134	−0.0973395
$407$	−10.4525	−0.518111
$408$	0	0
$409$	−15.8307	−0.782778	−0.391389	−	0.920225i	$-0.628005\pi$
$409$	−15.8307	−0.782778	−0.391389	+	0.920225i	$0.628005\pi$
$410$	5.83938	0.288386
$411$	12.8303	0.632874
$412$	5.58541	0.275174
$413$	−2.01920	−0.0993584
$414$	−6.67619	−0.328117
$415$	1.80562	0.0886346
$416$	0.152241	0.00746423
$417$	2.38425	0.116757
$418$	4.00000	0.195646
$419$	−23.5134	−1.14871	−0.574353	−	0.818607i	$-0.694746\pi$
$419$	−23.5134	−1.14871	−0.574353	+	0.818607i	$0.694746\pi$
$420$	1.08239	0.0528153
$421$	−12.0648	−0.588003	−0.294001	−	0.955805i	$-0.594987\pi$
$421$	−12.0648	−0.588003	−0.294001	+	0.955805i	$0.594987\pi$
$422$	5.90666	0.287532
$423$	−5.39104	−0.262121
$424$	6.35916	0.308828
$425$	0	0
$426$	−3.24718	−0.157326
$427$	5.98642	0.289703
$428$	−3.45929	−0.167211
$429$	0.215301	0.0103948
$430$	−11.0353	−0.532172
$431$	−6.35744	−0.306227	−0.153113	−	0.988209i	$-0.548930\pi$
$431$	−6.35744	−0.306227	−0.153113	+	0.988209i	$0.548930\pi$
$432$	1.00000	0.0481125
$433$	−35.0402	−1.68392	−0.841962	−	0.539536i	$-0.818600\pi$
$433$	−35.0402	−1.68392	−0.841962	+	0.539536i	$0.818600\pi$
$434$	−11.0479	−0.530316
$435$	1.81204	0.0868806
$436$	2.89668	0.138726
$437$	18.8831	0.903301
$438$	10.1985	0.487305
$439$	−7.38340	−0.352391	−0.176195	−	0.984355i	$-0.556379\pi$
$439$	−7.38340	−0.352391	−0.176195	+	0.984355i	$0.556379\pi$
$440$	−1.41421	−0.0674200
$441$	−5.82843	−0.277544
$442$	0	0
$443$	22.0539	1.04781	0.523907	−	0.851776i	$-0.324474\pi$
$443$	22.0539	1.04781	0.523907	+	0.851776i	$0.324474\pi$
$444$	7.39104	0.350763
$445$	4.60127	0.218121
$446$	−13.5854	−0.643288
$447$	−3.36075	−0.158958
$448$	−1.08239	−0.0511382
$449$	37.9191	1.78952	0.894758	−	0.446552i	$-0.147348\pi$
$449$	37.9191	1.78952	0.894758	+	0.446552i	$0.147348\pi$
$450$	−1.00000	−0.0471405
$451$	−8.25813	−0.388860
$452$	9.46767	0.445322
$453$	−4.74150	−0.222775
$454$	6.87992	0.322891
$455$	−0.164784	−0.00772521
$456$	−2.82843	−0.132453
$457$	−6.09785	−0.285245	−0.142623	−	0.989777i	$-0.545554\pi$
$457$	−6.09785	−0.285245	−0.142623	+	0.989777i	$0.545554\pi$
$458$	8.59539	0.401636
$459$	0	0
$460$	−6.67619	−0.311279
$461$	19.1647	0.892587	0.446293	−	0.894887i	$-0.352744\pi$
$461$	19.1647	0.892587	0.446293	+	0.894887i	$0.352744\pi$
$462$	−1.53073	−0.0712162
$463$	11.9896	0.557204	0.278602	−	0.960407i	$-0.410129\pi$
$463$	11.9896	0.557204	0.278602	+	0.960407i	$0.410129\pi$
$464$	−1.81204	−0.0841218
$465$	10.2069	0.473335
$466$	19.6945	0.912329
$467$	−31.3388	−1.45019	−0.725093	−	0.688651i	$-0.758203\pi$
$467$	−31.3388	−1.45019	−0.725093	+	0.688651i	$0.758203\pi$
$468$	−0.152241	−0.00703734
$469$	−5.08412	−0.234763
$470$	−5.39104	−0.248670
$471$	−9.67112	−0.445622
$472$	−1.86550	−0.0858665
$473$	15.6063	0.717580
$474$	−16.6417	−0.764378
$475$	2.82843	0.129777
$476$	0	0
$477$	−6.35916	−0.291166
$478$	8.32638	0.380840
$479$	18.4111	0.841223	0.420611	−	0.907241i	$-0.361816\pi$
$479$	18.4111	0.841223	0.420611	+	0.907241i	$0.361816\pi$
$480$	1.00000	0.0456435
$481$	−1.12522	−0.0513055
$482$	8.33476	0.379638
$483$	−7.22625	−0.328806
$484$	−9.00000	−0.409091
$485$	−1.91082	−0.0867658
$486$	−1.00000	−0.0453609
$487$	−15.1280	−0.685515	−0.342758	−	0.939424i	$-0.611361\pi$
$487$	−15.1280	−0.685515	−0.342758	+	0.939424i	$0.611361\pi$
$488$	5.53073	0.250365
$489$	1.44381	0.0652911
$490$	−5.82843	−0.263301
$491$	13.6457	0.615821	0.307911	−	0.951415i	$-0.400370\pi$
$491$	13.6457	0.615821	0.307911	+	0.951415i	$0.400370\pi$
$492$	5.83938	0.263259
$493$	0	0
$494$	0.430602	0.0193737
$495$	1.41421	0.0635642
$496$	−10.2069	−0.458304
$497$	−3.51472	−0.157657
$498$	1.80562	0.0809119
$499$	−31.3433	−1.40312	−0.701559	−	0.712611i	$-0.747513\pi$
$499$	−31.3433	−1.40312	−0.701559	+	0.712611i	$0.747513\pi$
$500$	−1.00000	−0.0447214
$501$	0.211830	0.00946385
$502$	21.2043	0.946394
$503$	13.0075	0.579975	0.289987	−	0.957030i	$-0.406349\pi$
$503$	13.0075	0.579975	0.289987	+	0.957030i	$0.406349\pi$
$504$	1.08239	0.0482136
$505$	−6.27836	−0.279383
$506$	9.44155	0.419728
$507$	−12.9768	−0.576321
$508$	−16.6501	−0.738727
$509$	21.0441	0.932762	0.466381	−	0.884584i	$-0.345558\pi$
$509$	21.0441	0.932762	0.466381	+	0.884584i	$0.345558\pi$
$510$	0	0
$511$	11.0388	0.488329
$512$	−1.00000	−0.0441942
$513$	2.82843	0.124878
$514$	−7.56224	−0.333556
$515$	−5.58541	−0.246123
$516$	−11.0353	−0.485804
$517$	7.62408	0.335307
$518$	8.00000	0.351500
$519$	23.5230	1.03255
$520$	−0.152241	−0.00667621
$521$	−1.91671	−0.0839724	−0.0419862	−	0.999118i	$-0.513369\pi$
$521$	−1.91671	−0.0839724	−0.0419862	+	0.999118i	$0.513369\pi$
$522$	1.81204	0.0793108
$523$	37.8410	1.65467	0.827337	−	0.561706i	$-0.189855\pi$
$523$	37.8410	1.65467	0.827337	+	0.561706i	$0.189855\pi$
$524$	16.9378	0.739931
$525$	−1.08239	−0.0472395
$526$	−11.2417	−0.490163
$527$	0	0
$528$	−1.41421	−0.0615457
$529$	21.5715	0.937890
$530$	−6.35916	−0.276224
$531$	1.86550	0.0809557
$532$	−3.06147	−0.132731
$533$	−0.888992	−0.0385065
$534$	4.60127	0.199117
$535$	3.45929	0.149558
$536$	−4.69711	−0.202884
$537$	−17.2388	−0.743909
$538$	29.1380	1.25623
$539$	8.24264	0.355036
$540$	−1.00000	−0.0430331
$541$	4.53752	0.195083	0.0975417	−	0.995231i	$-0.468902\pi$
$541$	4.53752	0.195083	0.0975417	+	0.995231i	$0.468902\pi$
$542$	−21.4908	−0.923109
$543$	−14.4557	−0.620353
$544$	0	0
$545$	−2.89668	−0.124080
$546$	−0.164784	−0.00705212
$547$	30.3635	1.29825	0.649125	−	0.760682i	$-0.275135\pi$
$547$	30.3635	1.29825	0.649125	+	0.760682i	$0.275135\pi$
$548$	12.8303	0.548085
$549$	−5.53073	−0.236046
$550$	1.41421	0.0603023
$551$	−5.12522	−0.218342
$552$	−6.67619	−0.284157
$553$	−18.0128	−0.765983
$554$	6.87669	0.292163
$555$	−7.39104	−0.313732
$556$	2.38425	0.101115
$557$	−37.5324	−1.59030	−0.795150	−	0.606413i	$-0.792608\pi$
$557$	−37.5324	−1.59030	−0.795150	+	0.606413i	$0.792608\pi$
$558$	10.2069	0.432093
$559$	1.68003	0.0710578
$560$	1.08239	0.0457394
$561$	0	0
$562$	7.11933	0.300311
$563$	4.07388	0.171694	0.0858468	−	0.996308i	$-0.472640\pi$
$563$	4.07388	0.171694	0.0858468	+	0.996308i	$0.472640\pi$
$564$	−5.39104	−0.227004
$565$	−9.46767	−0.398308
$566$	25.8473	1.08644
$567$	−1.08239	−0.0454562
$568$	−3.24718	−0.136249
$569$	28.0950	1.17781	0.588903	−	0.808204i	$-0.299560\pi$
$569$	28.0950	1.17781	0.588903	+	0.808204i	$0.299560\pi$
$570$	2.82843	0.118470
$571$	13.3698	0.559507	0.279754	−	0.960072i	$-0.409747\pi$
$571$	13.3698	0.559507	0.279754	+	0.960072i	$0.409747\pi$
$572$	0.215301	0.00900220
$573$	9.44572	0.394600
$574$	6.32050	0.263812
$575$	6.67619	0.278416
$576$	1.00000	0.0416667
$577$	14.8471	0.618095	0.309047	−	0.951047i	$-0.399990\pi$
$577$	14.8471	0.618095	0.309047	+	0.951047i	$0.399990\pi$
$578$	0	0
$579$	−10.0177	−0.416323
$580$	1.81204	0.0752408
$581$	1.95439	0.0810819
$582$	−1.91082	−0.0792060
$583$	8.99321	0.372461
$584$	10.1985	0.422019
$585$	0.152241	0.00629439
$586$	17.7251	0.732218
$587$	3.95815	0.163370	0.0816852	−	0.996658i	$-0.473970\pi$
$587$	3.95815	0.163370	0.0816852	+	0.996658i	$0.473970\pi$
$588$	−5.82843	−0.240360
$589$	−28.8695	−1.18955
$590$	1.86550	0.0768013
$591$	1.47605	0.0607168
$592$	7.39104	0.303770
$593$	−40.3160	−1.65558	−0.827790	−	0.561039i	$-0.810402\pi$
$593$	−40.3160	−1.65558	−0.827790	+	0.561039i	$0.810402\pi$
$594$	1.41421	0.0580259
$595$	0	0
$596$	−3.36075	−0.137662
$597$	−20.6239	−0.844082
$598$	1.01639	0.0415632
$599$	−33.5081	−1.36910	−0.684552	−	0.728964i	$-0.740002\pi$
$599$	−33.5081	−1.36910	−0.684552	+	0.728964i	$0.740002\pi$
$600$	−1.00000	−0.0408248
$601$	−36.5668	−1.49159	−0.745795	−	0.666175i	$-0.767930\pi$
$601$	−36.5668	−1.49159	−0.745795	+	0.666175i	$0.767930\pi$
$602$	−11.9446	−0.486824
$603$	4.69711	0.191281
$604$	−4.74150	−0.192929
$605$	9.00000	0.365902
$606$	−6.27836	−0.255041
$607$	−27.4408	−1.11379	−0.556894	−	0.830584i	$-0.688007\pi$
$607$	−27.4408	−1.11379	−0.556894	+	0.830584i	$0.688007\pi$
$608$	−2.82843	−0.114708
$609$	1.96134	0.0794774
$610$	−5.53073	−0.223933
$611$	0.820736	0.0332034
$612$	0	0
$613$	−13.6321	−0.550596	−0.275298	−	0.961359i	$-0.588777\pi$
$613$	−13.6321	−0.550596	−0.275298	+	0.961359i	$0.588777\pi$
$614$	−18.2348	−0.735896
$615$	−5.83938	−0.235466
$616$	−1.53073	−0.0616750
$617$	22.6954	0.913682	0.456841	−	0.889548i	$-0.348981\pi$
$617$	22.6954	0.913682	0.456841	+	0.889548i	$0.348981\pi$
$618$	−5.58541	−0.224678
$619$	−35.8492	−1.44090	−0.720450	−	0.693507i	$-0.756065\pi$
$619$	−35.8492	−1.44090	−0.720450	+	0.693507i	$0.756065\pi$
$620$	10.2069	0.409920
$621$	6.67619	0.267906
$622$	31.7112	1.27150
$623$	4.98038	0.199535
$624$	−0.152241	−0.00609451
$625$	1.00000	0.0400000
$626$	−19.1144	−0.763966
$627$	−4.00000	−0.159745
$628$	−9.67112	−0.385920
$629$	0	0
$630$	−1.08239	−0.0431235
$631$	−1.55980	−0.0620945	−0.0310473	−	0.999518i	$-0.509884\pi$
$631$	−1.55980	−0.0620945	−0.0310473	+	0.999518i	$0.509884\pi$
$632$	−16.6417	−0.661971
$633$	−5.90666	−0.234769
$634$	−21.9195	−0.870534
$635$	16.6501	0.660738
$636$	−6.35916	−0.252157
$637$	0.887325	0.0351571
$638$	−2.56261	−0.101455
$639$	3.24718	0.128456
$640$	1.00000	0.0395285
$641$	14.4233	0.569684	0.284842	−	0.958574i	$-0.408059\pi$
$641$	14.4233	0.569684	0.284842	+	0.958574i	$0.408059\pi$
$642$	3.45929	0.136527
$643$	3.92983	0.154978	0.0774888	−	0.996993i	$-0.475310\pi$
$643$	3.92983	0.154978	0.0774888	+	0.996993i	$0.475310\pi$
$644$	−7.22625	−0.284754
$645$	11.0353	0.434516
$646$	0	0
$647$	−18.4461	−0.725191	−0.362595	−	0.931947i	$-0.618109\pi$
$647$	−18.4461	−0.725191	−0.362595	+	0.931947i	$0.618109\pi$
$648$	−1.00000	−0.0392837
$649$	−2.63821	−0.103559
$650$	0.152241	0.00597138
$651$	11.0479	0.433001
$652$	1.44381	0.0565438
$653$	42.0867	1.64698	0.823490	−	0.567331i	$-0.192024\pi$
$653$	42.0867	1.64698	0.823490	+	0.567331i	$0.192024\pi$
$654$	−2.89668	−0.113269
$655$	−16.9378	−0.661814
$656$	5.83938	0.227989
$657$	−10.1985	−0.397883
$658$	−5.83522	−0.227480
$659$	−19.0740	−0.743018	−0.371509	−	0.928429i	$-0.621160\pi$
$659$	−19.0740	−0.743018	−0.371509	+	0.928429i	$0.621160\pi$
$660$	1.41421	0.0550482
$661$	16.3766	0.636974	0.318487	−	0.947927i	$-0.396825\pi$
$661$	16.3766	0.636974	0.318487	+	0.947927i	$0.396825\pi$
$662$	18.7402	0.728359
$663$	0	0
$664$	1.80562	0.0700718
$665$	3.06147	0.118719
$666$	−7.39104	−0.286397
$667$	−12.0975	−0.468417
$668$	0.211830	0.00819593
$669$	13.5854	0.525242
$670$	4.69711	0.181465
$671$	7.82164	0.301951
$672$	1.08239	0.0417542
$673$	44.4052	1.71169	0.855847	−	0.517229i	$-0.173037\pi$
$673$	44.4052	1.71169	0.855847	+	0.517229i	$0.173037\pi$
$674$	28.8782	1.11235
$675$	1.00000	0.0384900
$676$	−12.9768	−0.499109
$677$	−42.2529	−1.62391	−0.811955	−	0.583720i	$-0.801597\pi$
$677$	−42.2529	−1.62391	−0.811955	+	0.583720i	$0.801597\pi$
$678$	−9.46767	−0.363604
$679$	−2.06826	−0.0793724
$680$	0	0
$681$	−6.87992	−0.263639
$682$	−14.4348	−0.552736
$683$	3.32803	0.127344	0.0636718	−	0.997971i	$-0.479719\pi$
$683$	3.32803	0.127344	0.0636718	+	0.997971i	$0.479719\pi$
$684$	2.82843	0.108148
$685$	−12.8303	−0.490222
$686$	−13.8854	−0.530147
$687$	−8.59539	−0.327934
$688$	−11.0353	−0.420719
$689$	0.968125	0.0368826
$690$	6.67619	0.254158
$691$	−46.7736	−1.77935	−0.889676	−	0.456592i	$-0.849070\pi$
$691$	−46.7736	−1.77935	−0.889676	+	0.456592i	$0.849070\pi$
$692$	23.5230	0.894212
$693$	1.53073	0.0581478
$694$	−14.9346	−0.566910
$695$	−2.38425	−0.0904397
$696$	1.81204	0.0686851
$697$	0	0
$698$	3.07054	0.116222
$699$	−19.6945	−0.744914
$700$	−1.08239	−0.0409106
$701$	18.0508	0.681769	0.340885	−	0.940105i	$-0.389273\pi$
$701$	18.0508	0.681769	0.340885	+	0.940105i	$0.389273\pi$
$702$	0.152241	0.00574596
$703$	20.9050	0.788447
$704$	−1.41421	−0.0533002
$705$	5.39104	0.203038
$706$	30.8058	1.15939
$707$	−6.79565	−0.255577
$708$	1.86550	0.0701097
$709$	−22.9796	−0.863018	−0.431509	−	0.902109i	$-0.642019\pi$
$709$	−22.9796	−0.863018	−0.431509	+	0.902109i	$0.642019\pi$
$710$	3.24718	0.121864
$711$	16.6417	0.624112
$712$	4.60127	0.172440
$713$	−68.1433	−2.55199
$714$	0	0
$715$	−0.215301	−0.00805181
$716$	−17.2388	−0.644244
$717$	−8.32638	−0.310954
$718$	30.8117	1.14988
$719$	11.4112	0.425566	0.212783	−	0.977099i	$-0.431747\pi$
$719$	11.4112	0.425566	0.212783	+	0.977099i	$0.431747\pi$
$720$	−1.00000	−0.0372678
$721$	−6.04561	−0.225150
$722$	11.0000	0.409378
$723$	−8.33476	−0.309973
$724$	−14.4557	−0.537241
$725$	−1.81204	−0.0672974
$726$	9.00000	0.334021
$727$	36.5894	1.35703	0.678513	−	0.734589i	$-0.262625\pi$
$727$	36.5894	1.35703	0.678513	+	0.734589i	$0.262625\pi$
$728$	−0.164784	−0.00610732
$729$	1.00000	0.0370370
$730$	−10.1985	−0.377465
$731$	0	0
$732$	−5.53073	−0.204422
$733$	3.53837	0.130693	0.0653463	−	0.997863i	$-0.479185\pi$
$733$	3.53837	0.130693	0.0653463	+	0.997863i	$0.479185\pi$
$734$	9.89480	0.365224
$735$	5.82843	0.214985
$736$	−6.67619	−0.246087
$737$	−6.64272	−0.244688
$738$	−5.83938	−0.214950
$739$	14.5253	0.534323	0.267161	−	0.963652i	$-0.413914\pi$
$739$	14.5253	0.534323	0.267161	+	0.963652i	$0.413914\pi$
$740$	−7.39104	−0.271700
$741$	−0.430602	−0.0158186
$742$	−6.88311	−0.252687
$743$	32.0335	1.17519	0.587597	−	0.809154i	$-0.300074\pi$
$743$	32.0335	1.17519	0.587597	+	0.809154i	$0.300074\pi$
$744$	10.2069	0.374204
$745$	3.36075	0.123128
$746$	20.9089	0.765529
$747$	−1.80562	−0.0660643
$748$	0	0
$749$	3.74431	0.136814
$750$	1.00000	0.0365148
$751$	32.2682	1.17748	0.588742	−	0.808321i	$-0.299624\pi$
$751$	32.2682	1.17748	0.588742	+	0.808321i	$0.299624\pi$
$752$	−5.39104	−0.196591
$753$	−21.2043	−0.772727
$754$	−0.275866	−0.0100465
$755$	4.74150	0.172561
$756$	−1.08239	−0.0393662
$757$	−48.9409	−1.77879	−0.889394	−	0.457142i	$-0.848873\pi$
$757$	−48.9409	−1.77879	−0.889394	+	0.457142i	$0.848873\pi$
$758$	16.9663	0.616244
$759$	−9.44155	−0.342707
$760$	2.82843	0.102598
$761$	−51.5087	−1.86719	−0.933595	−	0.358331i	$-0.883346\pi$
$761$	−51.5087	−1.86719	−0.933595	+	0.358331i	$0.883346\pi$
$762$	16.6501	0.603168
$763$	−3.13535	−0.113507
$764$	9.44572	0.341734
$765$	0	0
$766$	17.9736	0.649414
$767$	−0.284005	−0.0102548
$768$	1.00000	0.0360844
$769$	−27.8446	−1.00410	−0.502051	−	0.864838i	$-0.667421\pi$
$769$	−27.8446	−1.00410	−0.502051	+	0.864838i	$0.667421\pi$
$770$	1.53073	0.0551638
$771$	7.56224	0.272347
$772$	−10.0177	−0.360546
$773$	22.1995	0.798461	0.399231	−	0.916851i	$-0.369277\pi$
$773$	22.1995	0.798461	0.399231	+	0.916851i	$0.369277\pi$
$774$	11.0353	0.396657
$775$	−10.2069	−0.366643
$776$	−1.91082	−0.0685944
$777$	−8.00000	−0.286998
$778$	8.78948	0.315118
$779$	16.5163	0.591756
$780$	0.152241	0.00545110
$781$	−4.59220	−0.164322
$782$	0	0
$783$	−1.81204	−0.0647570
$784$	−5.82843	−0.208158
$785$	9.67112	0.345177
$786$	−16.9378	−0.604151
$787$	36.8914	1.31504	0.657519	−	0.753438i	$-0.271606\pi$
$787$	36.8914	1.31504	0.657519	+	0.753438i	$0.271606\pi$
$788$	1.47605	0.0525823
$789$	11.2417	0.400216
$790$	16.6417	0.592085
$791$	−10.2477	−0.364368
$792$	1.41421	0.0502519
$793$	0.842004	0.0299004
$794$	−20.1158	−0.713882
$795$	6.35916	0.225536
$796$	−20.6239	−0.730996
$797$	37.7481	1.33711	0.668554	−	0.743664i	$-0.266914\pi$
$797$	37.7481	1.33711	0.668554	+	0.743664i	$0.266914\pi$
$798$	3.06147	0.108375
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−4.60127	−0.162578
$802$	0.338236	0.0119435
$803$	14.4229	0.508973
$804$	4.69711	0.165654
$805$	7.22625	0.254692
$806$	−1.55391	−0.0547342
$807$	−29.1380	−1.02571
$808$	−6.27836	−0.220872
$809$	−7.84707	−0.275888	−0.137944	−	0.990440i	$-0.544049\pi$
$809$	−7.84707	−0.275888	−0.137944	+	0.990440i	$0.544049\pi$
$810$	1.00000	0.0351364
$811$	22.0296	0.773564	0.386782	−	0.922171i	$-0.373587\pi$
$811$	22.0296	0.773564	0.386782	+	0.922171i	$0.373587\pi$
$812$	1.96134	0.0688294
$813$	21.4908	0.753715
$814$	10.4525	0.366360
$815$	−1.44381	−0.0505743
$816$	0	0
$817$	−31.2127	−1.09199
$818$	15.8307	0.553508
$819$	0.164784	0.00575803
$820$	−5.83938	−0.203920
$821$	30.6042	1.06810	0.534048	−	0.845454i	$-0.320670\pi$
$821$	30.6042	1.06810	0.534048	+	0.845454i	$0.320670\pi$
$822$	−12.8303	−0.447509
$823$	25.5722	0.891390	0.445695	−	0.895185i	$-0.352957\pi$
$823$	25.5722	0.891390	0.445695	+	0.895185i	$0.352957\pi$
$824$	−5.58541	−0.194577
$825$	−1.41421	−0.0492366
$826$	2.01920	0.0702570
$827$	49.4829	1.72069	0.860344	−	0.509713i	$-0.170249\pi$
$827$	49.4829	1.72069	0.860344	+	0.509713i	$0.170249\pi$
$828$	6.67619	0.232014
$829$	−14.5735	−0.506158	−0.253079	−	0.967446i	$-0.581443\pi$
$829$	−14.5735	−0.506158	−0.253079	+	0.967446i	$0.581443\pi$
$830$	−1.80562	−0.0626741
$831$	−6.87669	−0.238550
$832$	−0.152241	−0.00527800
$833$	0	0
$834$	−2.38425	−0.0825597
$835$	−0.211830	−0.00733066
$836$	−4.00000	−0.138343
$837$	−10.2069	−0.352803
$838$	23.5134	0.812258
$839$	41.1582	1.42094	0.710470	−	0.703728i	$-0.248482\pi$
$839$	41.1582	1.42094	0.710470	+	0.703728i	$0.248482\pi$
$840$	−1.08239	−0.0373461
$841$	−25.7165	−0.886776
$842$	12.0648	0.415781
$843$	−7.11933	−0.245203
$844$	−5.90666	−0.203316
$845$	12.9768	0.446416
$846$	5.39104	0.185348
$847$	9.74153	0.334723
$848$	−6.35916	−0.218374
$849$	−25.8473	−0.887075
$850$	0	0
$851$	49.3439	1.69149
$852$	3.24718	0.111246
$853$	27.1881	0.930903	0.465452	−	0.885073i	$-0.345892\pi$
$853$	27.1881	0.930903	0.465452	+	0.885073i	$0.345892\pi$
$854$	−5.98642	−0.204851
$855$	−2.82843	−0.0967302
$856$	3.45929	0.118236
$857$	−15.3112	−0.523021	−0.261511	−	0.965201i	$-0.584221\pi$
$857$	−15.3112	−0.523021	−0.261511	+	0.965201i	$0.584221\pi$
$858$	−0.215301	−0.00735026
$859$	−38.2717	−1.30581	−0.652907	−	0.757438i	$-0.726451\pi$
$859$	−38.2717	−1.30581	−0.652907	+	0.757438i	$0.726451\pi$
$860$	11.0353	0.376302
$861$	−6.32050	−0.215402
$862$	6.35744	0.216535
$863$	12.3789	0.421382	0.210691	−	0.977553i	$-0.432429\pi$
$863$	12.3789	0.421382	0.210691	+	0.977553i	$0.432429\pi$
$864$	−1.00000	−0.0340207
$865$	−23.5230	−0.799807
$866$	35.0402	1.19071
$867$	0	0
$868$	11.0479	0.374990
$869$	−23.5349	−0.798367
$870$	−1.81204	−0.0614339
$871$	−0.715093	−0.0242300
$872$	−2.89668	−0.0980941
$873$	1.91082	0.0646714
$874$	−18.8831	−0.638731
$875$	1.08239	0.0365915
$876$	−10.1985	−0.344577
$877$	39.7081	1.34085	0.670423	−	0.741979i	$-0.266112\pi$
$877$	39.7081	1.34085	0.670423	+	0.741979i	$0.266112\pi$
$878$	7.38340	0.249178
$879$	−17.7251	−0.597853
$880$	1.41421	0.0476731
$881$	20.6022	0.694107	0.347054	−	0.937845i	$-0.387182\pi$
$881$	20.6022	0.694107	0.347054	+	0.937845i	$0.387182\pi$
$882$	5.82843	0.196253
$883$	−37.5631	−1.26410	−0.632050	−	0.774928i	$-0.717786\pi$
$883$	−37.5631	−1.26410	−0.632050	+	0.774928i	$0.717786\pi$
$884$	0	0
$885$	−1.86550	−0.0627080
$886$	−22.0539	−0.740916
$887$	−12.6912	−0.426130	−0.213065	−	0.977038i	$-0.568345\pi$
$887$	−12.6912	−0.426130	−0.213065	+	0.977038i	$0.568345\pi$
$888$	−7.39104	−0.248027
$889$	18.0219	0.604435
$890$	−4.60127	−0.154235
$891$	−1.41421	−0.0473779
$892$	13.5854	0.454873
$893$	−15.2482	−0.510260
$894$	3.36075	0.112400
$895$	17.2388	0.576229
$896$	1.08239	0.0361602
$897$	−1.01639	−0.0339362
$898$	−37.9191	−1.26538
$899$	18.4953	0.616854
$900$	1.00000	0.0333333
$901$	0	0
$902$	8.25813	0.274966
$903$	11.9446	0.397491
$904$	−9.46767	−0.314890
$905$	14.4557	0.480523
$906$	4.74150	0.157526
$907$	−18.0101	−0.598017	−0.299008	−	0.954251i	$-0.596656\pi$
$907$	−18.0101	−0.598017	−0.299008	+	0.954251i	$0.596656\pi$
$908$	−6.87992	−0.228318
$909$	6.27836	0.208240
$910$	0.164784	0.00546255
$911$	−0.133468	−0.00442200	−0.00221100	−	0.999998i	$-0.500704\pi$
$911$	−0.133468	−0.00442200	−0.00221100	+	0.999998i	$0.500704\pi$
$912$	2.82843	0.0936586
$913$	2.55354	0.0845098
$914$	6.09785	0.201699
$915$	5.53073	0.182840
$916$	−8.59539	−0.284000
$917$	−18.3333	−0.605420
$918$	0	0
$919$	−14.2426	−0.469821	−0.234911	−	0.972017i	$-0.575480\pi$
$919$	−14.2426	−0.469821	−0.234911	+	0.972017i	$0.575480\pi$
$920$	6.67619	0.220107
$921$	18.2348	0.600857
$922$	−19.1647	−0.631154
$923$	−0.494353	−0.0162718
$924$	1.53073	0.0503574
$925$	7.39104	0.243016
$926$	−11.9896	−0.394003
$927$	5.58541	0.183449
$928$	1.81204	0.0594831
$929$	49.4896	1.62370	0.811851	−	0.583865i	$-0.198460\pi$
$929$	49.4896	1.62370	0.811851	+	0.583865i	$0.198460\pi$
$930$	−10.2069	−0.334698
$931$	−16.4853	−0.540283
$932$	−19.6945	−0.645114
$933$	−31.7112	−1.03818
$934$	31.3388	1.02544
$935$	0	0
$936$	0.152241	0.00497615
$937$	4.51757	0.147583	0.0737914	−	0.997274i	$-0.476490\pi$
$937$	4.51757	0.147583	0.0737914	+	0.997274i	$0.476490\pi$
$938$	5.08412	0.166002
$939$	19.1144	0.623776
$940$	5.39104	0.175836
$941$	40.2291	1.31143	0.655715	−	0.755009i	$-0.272367\pi$
$941$	40.2291	1.31143	0.655715	+	0.755009i	$0.272367\pi$
$942$	9.67112	0.315102
$943$	38.9848	1.26952
$944$	1.86550	0.0607168
$945$	1.08239	0.0352102
$946$	−15.6063	−0.507406
$947$	−24.5264	−0.797000	−0.398500	−	0.917168i	$-0.630469\pi$
$947$	−24.5264	−0.797000	−0.398500	+	0.917168i	$0.630469\pi$
$948$	16.6417	0.540497
$949$	1.55264	0.0504007
$950$	−2.82843	−0.0917663
$951$	21.9195	0.710788
$952$	0	0
$953$	3.00154	0.0972293	0.0486146	−	0.998818i	$-0.484519\pi$
$953$	3.00154	0.0972293	0.0486146	+	0.998818i	$0.484519\pi$
$954$	6.35916	0.205885
$955$	−9.44572	−0.305656
$956$	−8.32638	−0.269294
$957$	2.56261	0.0828374
$958$	−18.4111	−0.594834
$959$	−13.8875	−0.448449
$960$	−1.00000	−0.0322749
$961$	73.1812	2.36068
$962$	1.12522	0.0362785
$963$	−3.45929	−0.111474
$964$	−8.33476	−0.268445
$965$	10.0177	0.322482
$966$	7.22625	0.232501
$967$	20.1867	0.649160	0.324580	−	0.945858i	$-0.394777\pi$
$967$	20.1867	0.649160	0.324580	+	0.945858i	$0.394777\pi$
$968$	9.00000	0.289271
$969$	0	0
$970$	1.91082	0.0613527
$971$	−38.9735	−1.25072	−0.625360	−	0.780336i	$-0.715048\pi$
$971$	−38.9735	−1.25072	−0.625360	+	0.780336i	$0.715048\pi$
$972$	1.00000	0.0320750
$973$	−2.58069	−0.0827332
$974$	15.1280	0.484732
$975$	−0.152241	−0.00487561
$976$	−5.53073	−0.177034
$977$	−20.2941	−0.649265	−0.324633	−	0.945840i	$-0.605241\pi$
$977$	−20.2941	−0.649265	−0.324633	+	0.945840i	$0.605241\pi$
$978$	−1.44381	−0.0461678
$979$	6.50718	0.207970
$980$	5.82843	0.186182
$981$	2.89668	0.0924840
$982$	−13.6457	−0.435452
$983$	11.1960	0.357096	0.178548	−	0.983931i	$-0.442860\pi$
$983$	11.1960	0.357096	0.178548	+	0.983931i	$0.442860\pi$
$984$	−5.83938	−0.186153
$985$	−1.47605	−0.0470310
$986$	0	0
$987$	5.83522	0.185737
$988$	−0.430602	−0.0136993
$989$	−73.6740	−2.34270
$990$	−1.41421	−0.0449467
$991$	44.9865	1.42904	0.714522	−	0.699613i	$-0.246644\pi$
$991$	44.9865	1.42904	0.714522	+	0.699613i	$0.246644\pi$
$992$	10.2069	0.324070
$993$	−18.7402	−0.594703
$994$	3.51472	0.111480
$995$	20.6239	0.653823
$996$	−1.80562	−0.0572134
$997$	−26.7639	−0.847622	−0.423811	−	0.905751i	$-0.639308\pi$
$997$	−26.7639	−0.847622	−0.423811	+	0.905751i	$0.639308\pi$
$998$	31.3433	0.992155
$999$	7.39104	0.233842

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.bv.1.2		4
17.11	odd	16		510.2.u.b.121.1	✓	8
17.14	odd	16		510.2.u.b.451.1	yes	8
17.16	even	2		8670.2.a.bu.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.u.b.121.1	✓	8	17.11	odd	16
510.2.u.b.451.1	yes	8	17.14	odd	16
8670.2.a.bu.1.3		4	17.16	even	2
8670.2.a.bv.1.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8670.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.08239 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.08239 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -1.41421 q^{11} +1.00000 q^{12} -0.152241 q^{13} +1.08239 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +2.82843 q^{19} -1.00000 q^{20} -1.08239 q^{21} +1.41421 q^{22} +6.67619 q^{23} -1.00000 q^{24} +1.00000 q^{25} +0.152241 q^{26} +1.00000 q^{27} -1.08239 q^{28} -1.81204 q^{29} +1.00000 q^{30} -10.2069 q^{31} -1.00000 q^{32} -1.41421 q^{33} +1.08239 q^{35} +1.00000 q^{36} +7.39104 q^{37} -2.82843 q^{38} -0.152241 q^{39} +1.00000 q^{40} +5.83938 q^{41} +1.08239 q^{42} -11.0353 q^{43} -1.41421 q^{44} -1.00000 q^{45} -6.67619 q^{46} -5.39104 q^{47} +1.00000 q^{48} -5.82843 q^{49} -1.00000 q^{50} -0.152241 q^{52} -6.35916 q^{53} -1.00000 q^{54} +1.41421 q^{55} +1.08239 q^{56} +2.82843 q^{57} +1.81204 q^{58} +1.86550 q^{59} -1.00000 q^{60} -5.53073 q^{61} +10.2069 q^{62} -1.08239 q^{63} +1.00000 q^{64} +0.152241 q^{65} +1.41421 q^{66} +4.69711 q^{67} +6.67619 q^{69} -1.08239 q^{70} +3.24718 q^{71} -1.00000 q^{72} -10.1985 q^{73} -7.39104 q^{74} +1.00000 q^{75} +2.82843 q^{76} +1.53073 q^{77} +0.152241 q^{78} +16.6417 q^{79} -1.00000 q^{80} +1.00000 q^{81} -5.83938 q^{82} -1.80562 q^{83} -1.08239 q^{84} +11.0353 q^{86} -1.81204 q^{87} +1.41421 q^{88} -4.60127 q^{89} +1.00000 q^{90} +0.164784 q^{91} +6.67619 q^{92} -10.2069 q^{93} +5.39104 q^{94} -2.82843 q^{95} -1.00000 q^{96} +1.91082 q^{97} +5.82843 q^{98} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} - 4 q^{5} - 4 q^{6} - 4 q^{8} + 4 q^{9}+O(q^{10}) \) 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