LMFDB - Embedded newform 8470.2.a.cw.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.19898000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 7x^{3} + 24x^{2} - 15x - 5 $ x^6 - x^5 - 10x^4 + 7x^3 + 24x^2 - 15x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.935683$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.935683 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.935683 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.12450 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.935683 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.935683 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.12450 q^{9} -1.00000 q^{10} -0.935683 q^{12} -2.51397 q^{13} -1.00000 q^{14} -0.935683 q^{15} +1.00000 q^{16} +4.68602 q^{17} +2.12450 q^{18} +4.05554 q^{19} +1.00000 q^{20} -0.935683 q^{21} -3.46945 q^{23} +0.935683 q^{24} +1.00000 q^{25} +2.51397 q^{26} +4.79491 q^{27} +1.00000 q^{28} -7.46463 q^{29} +0.935683 q^{30} +1.54591 q^{31} -1.00000 q^{32} -4.68602 q^{34} +1.00000 q^{35} -2.12450 q^{36} -4.92622 q^{37} -4.05554 q^{38} +2.35228 q^{39} -1.00000 q^{40} +1.65079 q^{41} +0.935683 q^{42} -7.97092 q^{43} -2.12450 q^{45} +3.46945 q^{46} +1.77448 q^{47} -0.935683 q^{48} +1.00000 q^{49} -1.00000 q^{50} -4.38463 q^{51} -2.51397 q^{52} +7.14297 q^{53} -4.79491 q^{54} -1.00000 q^{56} -3.79470 q^{57} +7.46463 q^{58} -0.426837 q^{59} -0.935683 q^{60} +6.15201 q^{61} -1.54591 q^{62} -2.12450 q^{63} +1.00000 q^{64} -2.51397 q^{65} -5.99409 q^{67} +4.68602 q^{68} +3.24631 q^{69} -1.00000 q^{70} +6.53922 q^{71} +2.12450 q^{72} -0.855372 q^{73} +4.92622 q^{74} -0.935683 q^{75} +4.05554 q^{76} -2.35228 q^{78} -9.86178 q^{79} +1.00000 q^{80} +1.88697 q^{81} -1.65079 q^{82} -8.15674 q^{83} -0.935683 q^{84} +4.68602 q^{85} +7.97092 q^{86} +6.98453 q^{87} +1.00048 q^{89} +2.12450 q^{90} -2.51397 q^{91} -3.46945 q^{92} -1.44649 q^{93} -1.77448 q^{94} +4.05554 q^{95} +0.935683 q^{96} +11.3338 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) $ 6 * q - 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 + q^6 + 6 * q^7 - 6 * q^8 + 3 * q^9	$ 6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} + 3 q^{9} - 6 q^{10} - q^{12} - 9 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} - 9 q^{17} - 3 q^{18} - 12 q^{19} + 6 q^{20} - q^{21} + 4 q^{23} + q^{24} + 6 q^{25} + 9 q^{26} - 4 q^{27} + 6 q^{28} - 15 q^{29} + q^{30} + 8 q^{31} - 6 q^{32} + 9 q^{34} + 6 q^{35} + 3 q^{36} - 4 q^{37} + 12 q^{38} + 19 q^{39} - 6 q^{40} - 4 q^{41} + q^{42} - 30 q^{43} + 3 q^{45} - 4 q^{46} - 7 q^{47} - q^{48} + 6 q^{49} - 6 q^{50} - 16 q^{51} - 9 q^{52} - 6 q^{53} + 4 q^{54} - 6 q^{56} + 14 q^{57} + 15 q^{58} + 4 q^{59} - q^{60} + 14 q^{61} - 8 q^{62} + 3 q^{63} + 6 q^{64} - 9 q^{65} + 18 q^{67} - 9 q^{68} - 10 q^{69} - 6 q^{70} + 23 q^{71} - 3 q^{72} - 23 q^{73} + 4 q^{74} - q^{75} - 12 q^{76} - 19 q^{78} - 21 q^{79} + 6 q^{80} - 18 q^{81} + 4 q^{82} - 25 q^{83} - q^{84} - 9 q^{85} + 30 q^{86} - 14 q^{87} - 18 q^{89} - 3 q^{90} - 9 q^{91} + 4 q^{92} - 24 q^{93} + 7 q^{94} - 12 q^{95} + q^{96} + 7 q^{97} - 6 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 + q^6 + 6 * q^7 - 6 * q^8 + 3 * q^9 - 6 * q^10 - q^12 - 9 * q^13 - 6 * q^14 - q^15 + 6 * q^16 - 9 * q^17 - 3 * q^18 - 12 * q^19 + 6 * q^20 - q^21 + 4 * q^23 + q^24 + 6 * q^25 + 9 * q^26 - 4 * q^27 + 6 * q^28 - 15 * q^29 + q^30 + 8 * q^31 - 6 * q^32 + 9 * q^34 + 6 * q^35 + 3 * q^36 - 4 * q^37 + 12 * q^38 + 19 * q^39 - 6 * q^40 - 4 * q^41 + q^42 - 30 * q^43 + 3 * q^45 - 4 * q^46 - 7 * q^47 - q^48 + 6 * q^49 - 6 * q^50 - 16 * q^51 - 9 * q^52 - 6 * q^53 + 4 * q^54 - 6 * q^56 + 14 * q^57 + 15 * q^58 + 4 * q^59 - q^60 + 14 * q^61 - 8 * q^62 + 3 * q^63 + 6 * q^64 - 9 * q^65 + 18 * q^67 - 9 * q^68 - 10 * q^69 - 6 * q^70 + 23 * q^71 - 3 * q^72 - 23 * q^73 + 4 * q^74 - q^75 - 12 * q^76 - 19 * q^78 - 21 * q^79 + 6 * q^80 - 18 * q^81 + 4 * q^82 - 25 * q^83 - q^84 - 9 * q^85 + 30 * q^86 - 14 * q^87 - 18 * q^89 - 3 * q^90 - 9 * q^91 + 4 * q^92 - 24 * q^93 + 7 * q^94 - 12 * q^95 + q^96 + 7 * q^97 - 6 * 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
$3$	−0.935683	−0.540217	−0.270109	−	0.962830i	$-0.587060\pi$
$3$	−0.935683	−0.540217	−0.270109	+	0.962830i	$0.587060\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.935683	0.381991
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	−2.12450	−0.708165
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	−0.935683	−0.270109
$13$	−2.51397	−0.697249	−0.348625	−	0.937262i	$-0.613351\pi$
$13$	−2.51397	−0.697249	−0.348625	+	0.937262i	$0.613351\pi$
$14$	−1.00000	−0.267261
$15$	−0.935683	−0.241592
$16$	1.00000	0.250000
$17$	4.68602	1.13653	0.568263	−	0.822847i	$-0.307616\pi$
$17$	4.68602	1.13653	0.568263	+	0.822847i	$0.307616\pi$
$18$	2.12450	0.500749
$19$	4.05554	0.930405	0.465203	−	0.885204i	$-0.345982\pi$
$19$	4.05554	0.930405	0.465203	+	0.885204i	$0.345982\pi$
$20$	1.00000	0.223607
$21$	−0.935683	−0.204183
$22$	0	0
$23$	−3.46945	−0.723431	−0.361716	−	0.932289i	$-0.617809\pi$
$23$	−3.46945	−0.723431	−0.361716	+	0.932289i	$0.617809\pi$
$24$	0.935683	0.190996
$25$	1.00000	0.200000
$26$	2.51397	0.493030
$27$	4.79491	0.922780
$28$	1.00000	0.188982
$29$	−7.46463	−1.38615	−0.693074	−	0.720867i	$-0.743744\pi$
$29$	−7.46463	−1.38615	−0.693074	+	0.720867i	$0.743744\pi$
$30$	0.935683	0.170832
$31$	1.54591	0.277654	0.138827	−	0.990317i	$-0.455667\pi$
$31$	1.54591	0.277654	0.138827	+	0.990317i	$0.455667\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−4.68602	−0.803645
$35$	1.00000	0.169031
$36$	−2.12450	−0.354083
$37$	−4.92622	−0.809866	−0.404933	−	0.914346i	$-0.632705\pi$
$37$	−4.92622	−0.809866	−0.404933	+	0.914346i	$0.632705\pi$
$38$	−4.05554	−0.657896
$39$	2.35228	0.376666
$40$	−1.00000	−0.158114
$41$	1.65079	0.257810	0.128905	−	0.991657i	$-0.458854\pi$
$41$	1.65079	0.257810	0.128905	+	0.991657i	$0.458854\pi$
$42$	0.935683	0.144379
$43$	−7.97092	−1.21555	−0.607777	−	0.794108i	$-0.707939\pi$
$43$	−7.97092	−1.21555	−0.607777	+	0.794108i	$0.707939\pi$
$44$	0	0
$45$	−2.12450	−0.316701
$46$	3.46945	0.511543
$47$	1.77448	0.258834	0.129417	−	0.991590i	$-0.458689\pi$
$47$	1.77448	0.258834	0.129417	+	0.991590i	$0.458689\pi$
$48$	−0.935683	−0.135054
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−4.38463	−0.613971
$52$	−2.51397	−0.348625
$53$	7.14297	0.981162	0.490581	−	0.871395i	$-0.336784\pi$
$53$	7.14297	0.981162	0.490581	+	0.871395i	$0.336784\pi$
$54$	−4.79491	−0.652504
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−3.79470	−0.502621
$58$	7.46463	0.980154
$59$	−0.426837	−0.0555694	−0.0277847	−	0.999614i	$-0.508845\pi$
$59$	−0.426837	−0.0555694	−0.0277847	+	0.999614i	$0.508845\pi$
$60$	−0.935683	−0.120796
$61$	6.15201	0.787684	0.393842	−	0.919178i	$-0.371146\pi$
$61$	6.15201	0.787684	0.393842	+	0.919178i	$0.371146\pi$
$62$	−1.54591	−0.196331
$63$	−2.12450	−0.267661
$64$	1.00000	0.125000
$65$	−2.51397	−0.311819
$66$	0	0
$67$	−5.99409	−0.732295	−0.366147	−	0.930557i	$-0.619323\pi$
$67$	−5.99409	−0.732295	−0.366147	+	0.930557i	$0.619323\pi$
$68$	4.68602	0.568263
$69$	3.24631	0.390810
$70$	−1.00000	−0.119523
$71$	6.53922	0.776063	0.388031	−	0.921646i	$-0.373155\pi$
$71$	6.53922	0.776063	0.388031	+	0.921646i	$0.373155\pi$
$72$	2.12450	0.250374
$73$	−0.855372	−0.100114	−0.0500568	−	0.998746i	$-0.515940\pi$
$73$	−0.855372	−0.100114	−0.0500568	+	0.998746i	$0.515940\pi$
$74$	4.92622	0.572662
$75$	−0.935683	−0.108043
$76$	4.05554	0.465203
$77$	0	0
$78$	−2.35228	−0.266343
$79$	−9.86178	−1.10954	−0.554768	−	0.832005i	$-0.687193\pi$
$79$	−9.86178	−1.10954	−0.554768	+	0.832005i	$0.687193\pi$
$80$	1.00000	0.111803
$81$	1.88697	0.209664
$82$	−1.65079	−0.182300
$83$	−8.15674	−0.895319	−0.447659	−	0.894204i	$-0.647742\pi$
$83$	−8.15674	−0.895319	−0.447659	+	0.894204i	$0.647742\pi$
$84$	−0.935683	−0.102091
$85$	4.68602	0.508270
$86$	7.97092	0.859526
$87$	6.98453	0.748820
$88$	0	0
$89$	1.00048	0.106051	0.0530255	−	0.998593i	$-0.483114\pi$
$89$	1.00048	0.106051	0.0530255	+	0.998593i	$0.483114\pi$
$90$	2.12450	0.223942
$91$	−2.51397	−0.263535
$92$	−3.46945	−0.361716
$93$	−1.44649	−0.149994
$94$	−1.77448	−0.183023
$95$	4.05554	0.416090
$96$	0.935683	0.0954978
$97$	11.3338	1.15078	0.575388	−	0.817880i	$-0.304851\pi$
$97$	11.3338	1.15078	0.575388	+	0.817880i	$0.304851\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	3.16660	0.315088	0.157544	−	0.987512i	$-0.449642\pi$
$101$	3.16660	0.315088	0.157544	+	0.987512i	$0.449642\pi$
$102$	4.38463	0.434143
$103$	15.0475	1.48267	0.741335	−	0.671135i	$-0.234193\pi$
$103$	15.0475	1.48267	0.741335	+	0.671135i	$0.234193\pi$
$104$	2.51397	0.246515
$105$	−0.935683	−0.0913134
$106$	−7.14297	−0.693787
$107$	−14.4377	−1.39574	−0.697872	−	0.716222i	$-0.745870\pi$
$107$	−14.4377	−1.39574	−0.697872	+	0.716222i	$0.745870\pi$
$108$	4.79491	0.461390
$109$	−12.1869	−1.16729	−0.583646	−	0.812008i	$-0.698374\pi$
$109$	−12.1869	−1.16729	−0.583646	+	0.812008i	$0.698374\pi$
$110$	0	0
$111$	4.60938	0.437503
$112$	1.00000	0.0944911
$113$	−11.7294	−1.10341	−0.551706	−	0.834039i	$-0.686023\pi$
$113$	−11.7294	−1.10341	−0.551706	+	0.834039i	$0.686023\pi$
$114$	3.79470	0.355407
$115$	−3.46945	−0.323528
$116$	−7.46463	−0.693074
$117$	5.34092	0.493768
$118$	0.426837	0.0392935
$119$	4.68602	0.429567
$120$	0.935683	0.0854158
$121$	0	0
$122$	−6.15201	−0.556977
$123$	−1.54462	−0.139274
$124$	1.54591	0.138827
$125$	1.00000	0.0894427
$126$	2.12450	0.189265
$127$	3.68316	0.326828	0.163414	−	0.986558i	$-0.447749\pi$
$127$	3.68316	0.326828	0.163414	+	0.986558i	$0.447749\pi$
$128$	−1.00000	−0.0883883
$129$	7.45826	0.656663
$130$	2.51397	0.220490
$131$	8.57781	0.749447	0.374724	−	0.927137i	$-0.377738\pi$
$131$	8.57781	0.749447	0.374724	+	0.927137i	$0.377738\pi$
$132$	0	0
$133$	4.05554	0.351660
$134$	5.99409	0.517811
$135$	4.79491	0.412680
$136$	−4.68602	−0.401823
$137$	1.63086	0.139334	0.0696671	−	0.997570i	$-0.477806\pi$
$137$	1.63086	0.139334	0.0696671	+	0.997570i	$0.477806\pi$
$138$	−3.24631	−0.276344
$139$	−23.1799	−1.96609	−0.983046	−	0.183360i	$-0.941303\pi$
$139$	−23.1799	−1.96609	−0.983046	+	0.183360i	$0.941303\pi$
$140$	1.00000	0.0845154
$141$	−1.66035	−0.139827
$142$	−6.53922	−0.548759
$143$	0	0
$144$	−2.12450	−0.177041
$145$	−7.46463	−0.619904
$146$	0.855372	0.0707911
$147$	−0.935683	−0.0771739
$148$	−4.92622	−0.404933
$149$	−1.43386	−0.117466	−0.0587332	−	0.998274i	$-0.518706\pi$
$149$	−1.43386	−0.117466	−0.0587332	+	0.998274i	$0.518706\pi$
$150$	0.935683	0.0763982
$151$	17.7952	1.44815	0.724075	−	0.689722i	$-0.242267\pi$
$151$	17.7952	1.44815	0.724075	+	0.689722i	$0.242267\pi$
$152$	−4.05554	−0.328948
$153$	−9.95543	−0.804849
$154$	0	0
$155$	1.54591	0.124171
$156$	2.35228	0.188333
$157$	−7.29144	−0.581920	−0.290960	−	0.956735i	$-0.593975\pi$
$157$	−7.29144	−0.581920	−0.290960	+	0.956735i	$0.593975\pi$
$158$	9.86178	0.784561
$159$	−6.68356	−0.530041
$160$	−1.00000	−0.0790569
$161$	−3.46945	−0.273431
$162$	−1.88697	−0.148255
$163$	9.41304	0.737286	0.368643	−	0.929571i	$-0.379822\pi$
$163$	9.41304	0.737286	0.368643	+	0.929571i	$0.379822\pi$
$164$	1.65079	0.128905
$165$	0	0
$166$	8.15674	0.633086
$167$	−0.289918	−0.0224345	−0.0112173	−	0.999937i	$-0.503571\pi$
$167$	−0.289918	−0.0224345	−0.0112173	+	0.999937i	$0.503571\pi$
$168$	0.935683	0.0721895
$169$	−6.67997	−0.513844
$170$	−4.68602	−0.359401
$171$	−8.61598	−0.658881
$172$	−7.97092	−0.607777
$173$	−10.4188	−0.792126	−0.396063	−	0.918223i	$-0.629624\pi$
$173$	−10.4188	−0.792126	−0.396063	+	0.918223i	$0.629624\pi$
$174$	−6.98453	−0.529496
$175$	1.00000	0.0755929
$176$	0	0
$177$	0.399384	0.0300196
$178$	−1.00048	−0.0749893
$179$	−14.9115	−1.11454	−0.557270	−	0.830331i	$-0.688151\pi$
$179$	−14.9115	−1.11454	−0.557270	+	0.830331i	$0.688151\pi$
$180$	−2.12450	−0.158351
$181$	4.37330	0.325065	0.162533	−	0.986703i	$-0.448034\pi$
$181$	4.37330	0.325065	0.162533	+	0.986703i	$0.448034\pi$
$182$	2.51397	0.186348
$183$	−5.75633	−0.425520
$184$	3.46945	0.255772
$185$	−4.92622	−0.362183
$186$	1.44649	0.106062
$187$	0	0
$188$	1.77448	0.129417
$189$	4.79491	0.348778
$190$	−4.05554	−0.294220
$191$	−10.6197	−0.768416	−0.384208	−	0.923246i	$-0.625526\pi$
$191$	−10.6197	−0.768416	−0.384208	+	0.923246i	$0.625526\pi$
$192$	−0.935683	−0.0675271
$193$	7.98145	0.574517	0.287259	−	0.957853i	$-0.407256\pi$
$193$	7.98145	0.574517	0.287259	+	0.957853i	$0.407256\pi$
$194$	−11.3338	−0.813722
$195$	2.35228	0.168450
$196$	1.00000	0.0714286
$197$	−7.28621	−0.519121	−0.259560	−	0.965727i	$-0.583578\pi$
$197$	−7.28621	−0.519121	−0.259560	+	0.965727i	$0.583578\pi$
$198$	0	0
$199$	14.1853	1.00557	0.502783	−	0.864413i	$-0.332309\pi$
$199$	14.1853	1.00557	0.502783	+	0.864413i	$0.332309\pi$
$200$	−1.00000	−0.0707107
$201$	5.60857	0.395598
$202$	−3.16660	−0.222801
$203$	−7.46463	−0.523914
$204$	−4.38463	−0.306985
$205$	1.65079	0.115296
$206$	−15.0475	−1.04841
$207$	7.37084	0.512309
$208$	−2.51397	−0.174312
$209$	0	0
$210$	0.935683	0.0645683
$211$	−23.2770	−1.60246	−0.801229	−	0.598358i	$-0.795820\pi$
$211$	−23.2770	−1.60246	−0.801229	+	0.598358i	$0.795820\pi$
$212$	7.14297	0.490581
$213$	−6.11864	−0.419242
$214$	14.4377	0.986940
$215$	−7.97092	−0.543612
$216$	−4.79491	−0.326252
$217$	1.54591	0.104943
$218$	12.1869	0.825400
$219$	0.800357	0.0540831
$220$	0	0
$221$	−11.7805	−0.792442
$222$	−4.60938	−0.309362
$223$	−8.09748	−0.542248	−0.271124	−	0.962544i	$-0.587395\pi$
$223$	−8.09748	−0.542248	−0.271124	+	0.962544i	$0.587395\pi$
$224$	−1.00000	−0.0668153
$225$	−2.12450	−0.141633
$226$	11.7294	0.780230
$227$	−18.8354	−1.25015	−0.625074	−	0.780565i	$-0.714931\pi$
$227$	−18.8354	−1.25015	−0.625074	+	0.780565i	$0.714931\pi$
$228$	−3.79470	−0.251310
$229$	10.8721	0.718451	0.359225	−	0.933251i	$-0.383041\pi$
$229$	10.8721	0.718451	0.359225	+	0.933251i	$0.383041\pi$
$230$	3.46945	0.228769
$231$	0	0
$232$	7.46463	0.490077
$233$	−8.97639	−0.588063	−0.294031	−	0.955796i	$-0.594997\pi$
$233$	−8.97639	−0.588063	−0.294031	+	0.955796i	$0.594997\pi$
$234$	−5.34092	−0.349147
$235$	1.77448	0.115754
$236$	−0.426837	−0.0277847
$237$	9.22750	0.599391
$238$	−4.68602	−0.303749
$239$	−9.78125	−0.632696	−0.316348	−	0.948643i	$-0.602457\pi$
$239$	−9.78125	−0.632696	−0.316348	+	0.948643i	$0.602457\pi$
$240$	−0.935683	−0.0603981
$241$	23.3855	1.50639	0.753195	−	0.657797i	$-0.228512\pi$
$241$	23.3855	1.50639	0.753195	+	0.657797i	$0.228512\pi$
$242$	0	0
$243$	−16.1503	−1.03604
$244$	6.15201	0.393842
$245$	1.00000	0.0638877
$246$	1.54462	0.0984813
$247$	−10.1955	−0.648724
$248$	−1.54591	−0.0981656
$249$	7.63213	0.483666
$250$	−1.00000	−0.0632456
$251$	26.0939	1.64703	0.823517	−	0.567291i	$-0.192009\pi$
$251$	26.0939	1.64703	0.823517	+	0.567291i	$0.192009\pi$
$252$	−2.12450	−0.133831
$253$	0	0
$254$	−3.68316	−0.231102
$255$	−4.38463	−0.274576
$256$	1.00000	0.0625000
$257$	5.51935	0.344287	0.172144	−	0.985072i	$-0.444931\pi$
$257$	5.51935	0.344287	0.172144	+	0.985072i	$0.444931\pi$
$258$	−7.45826	−0.464331
$259$	−4.92622	−0.306101
$260$	−2.51397	−0.155910
$261$	15.8586	0.981622
$262$	−8.57781	−0.529939
$263$	−27.1823	−1.67613	−0.838067	−	0.545568i	$-0.816314\pi$
$263$	−27.1823	−1.67613	−0.838067	+	0.545568i	$0.816314\pi$
$264$	0	0
$265$	7.14297	0.438789
$266$	−4.05554	−0.248661
$267$	−0.936135	−0.0572905
$268$	−5.99409	−0.366147
$269$	−8.14480	−0.496597	−0.248299	−	0.968684i	$-0.579871\pi$
$269$	−8.14480	−0.496597	−0.248299	+	0.968684i	$0.579871\pi$
$270$	−4.79491	−0.291809
$271$	−6.30590	−0.383056	−0.191528	−	0.981487i	$-0.561344\pi$
$271$	−6.30590	−0.383056	−0.191528	+	0.981487i	$0.561344\pi$
$272$	4.68602	0.284132
$273$	2.35228	0.142366
$274$	−1.63086	−0.0985241
$275$	0	0
$276$	3.24631	0.195405
$277$	−26.1396	−1.57057	−0.785287	−	0.619132i	$-0.787485\pi$
$277$	−26.1396	−1.57057	−0.785287	+	0.619132i	$0.787485\pi$
$278$	23.1799	1.39024
$279$	−3.28429	−0.196625
$280$	−1.00000	−0.0597614
$281$	14.7356	0.879055	0.439527	−	0.898229i	$-0.355146\pi$
$281$	14.7356	0.879055	0.439527	+	0.898229i	$0.355146\pi$
$282$	1.66035	0.0988723
$283$	21.5060	1.27840	0.639199	−	0.769041i	$-0.279266\pi$
$283$	21.5060	1.27840	0.639199	+	0.769041i	$0.279266\pi$
$284$	6.53922	0.388031
$285$	−3.79470	−0.224779
$286$	0	0
$287$	1.65079	0.0974432
$288$	2.12450	0.125187
$289$	4.95876	0.291692
$290$	7.46463	0.438338
$291$	−10.6049	−0.621669
$292$	−0.855372	−0.0500568
$293$	−6.11604	−0.357303	−0.178651	−	0.983912i	$-0.557173\pi$
$293$	−6.11604	−0.357303	−0.178651	+	0.983912i	$0.557173\pi$
$294$	0.935683	0.0545702
$295$	−0.426837	−0.0248514
$296$	4.92622	0.286331
$297$	0	0
$298$	1.43386	0.0830613
$299$	8.72209	0.504412
$300$	−0.935683	−0.0540217
$301$	−7.97092	−0.459436
$302$	−17.7952	−1.02400
$303$	−2.96293	−0.170216
$304$	4.05554	0.232601
$305$	6.15201	0.352263
$306$	9.95543	0.569114
$307$	24.8721	1.41952	0.709762	−	0.704441i	$-0.248802\pi$
$307$	24.8721	1.41952	0.709762	+	0.704441i	$0.248802\pi$
$308$	0	0
$309$	−14.0797	−0.800963
$310$	−1.54591	−0.0878020
$311$	3.37946	0.191632	0.0958158	−	0.995399i	$-0.469454\pi$
$311$	3.37946	0.191632	0.0958158	+	0.995399i	$0.469454\pi$
$312$	−2.35228	−0.133172
$313$	−20.7401	−1.17230	−0.586150	−	0.810203i	$-0.699357\pi$
$313$	−20.7401	−1.17230	−0.586150	+	0.810203i	$0.699357\pi$
$314$	7.29144	0.411480
$315$	−2.12450	−0.119702
$316$	−9.86178	−0.554768
$317$	7.46200	0.419108	0.209554	−	0.977797i	$-0.432799\pi$
$317$	7.46200	0.419108	0.209554	+	0.977797i	$0.432799\pi$
$318$	6.68356	0.374795
$319$	0	0
$320$	1.00000	0.0559017
$321$	13.5091	0.754005
$322$	3.46945	0.193345
$323$	19.0043	1.05743
$324$	1.88697	0.104832
$325$	−2.51397	−0.139450
$326$	−9.41304	−0.521340
$327$	11.4031	0.630591
$328$	−1.65079	−0.0911498
$329$	1.77448	0.0978301
$330$	0	0
$331$	−8.26702	−0.454396	−0.227198	−	0.973849i	$-0.572956\pi$
$331$	−8.26702	−0.454396	−0.227198	+	0.973849i	$0.572956\pi$
$332$	−8.15674	−0.447659
$333$	10.4657	0.573519
$334$	0.289918	0.0158636
$335$	−5.99409	−0.327492
$336$	−0.935683	−0.0510457
$337$	0.309107	0.0168381	0.00841906	−	0.999965i	$-0.497320\pi$
$337$	0.309107	0.0168381	0.00841906	+	0.999965i	$0.497320\pi$
$338$	6.67997	0.363342
$339$	10.9750	0.596082
$340$	4.68602	0.254135
$341$	0	0
$342$	8.61598	0.465899
$343$	1.00000	0.0539949
$344$	7.97092	0.429763
$345$	3.24631	0.174775
$346$	10.4188	0.560118
$347$	−6.12313	−0.328707	−0.164353	−	0.986402i	$-0.552554\pi$
$347$	−6.12313	−0.328707	−0.164353	+	0.986402i	$0.552554\pi$
$348$	6.98453	0.374410
$349$	19.7838	1.05900	0.529501	−	0.848309i	$-0.322379\pi$
$349$	19.7838	1.05900	0.529501	+	0.848309i	$0.322379\pi$
$350$	−1.00000	−0.0534522
$351$	−12.0542	−0.643408
$352$	0	0
$353$	−30.8256	−1.64068	−0.820340	−	0.571876i	$-0.806216\pi$
$353$	−30.8256	−1.64068	−0.820340	+	0.571876i	$0.806216\pi$
$354$	−0.399384	−0.0212270
$355$	6.53922	0.347066
$356$	1.00048	0.0530255
$357$	−4.38463	−0.232059
$358$	14.9115	0.788099
$359$	25.2431	1.33228	0.666140	−	0.745827i	$-0.267945\pi$
$359$	25.2431	1.33228	0.666140	+	0.745827i	$0.267945\pi$
$360$	2.12450	0.111971
$361$	−2.55258	−0.134347
$362$	−4.37330	−0.229856
$363$	0	0
$364$	−2.51397	−0.131768
$365$	−0.855372	−0.0447722
$366$	5.75633	0.300888
$367$	−8.87025	−0.463023	−0.231512	−	0.972832i	$-0.574367\pi$
$367$	−8.87025	−0.463023	−0.231512	+	0.972832i	$0.574367\pi$
$368$	−3.46945	−0.180858
$369$	−3.50710	−0.182572
$370$	4.92622	0.256102
$371$	7.14297	0.370845
$372$	−1.44649	−0.0749968
$373$	−23.0860	−1.19535	−0.597675	−	0.801739i	$-0.703909\pi$
$373$	−23.0860	−1.19535	−0.597675	+	0.801739i	$0.703909\pi$
$374$	0	0
$375$	−0.935683	−0.0483185
$376$	−1.77448	−0.0915116
$377$	18.7658	0.966490
$378$	−4.79491	−0.246623
$379$	31.6315	1.62480	0.812399	−	0.583101i	$-0.198161\pi$
$379$	31.6315	1.62480	0.812399	+	0.583101i	$0.198161\pi$
$380$	4.05554	0.208045
$381$	−3.44627	−0.176558
$382$	10.6197	0.543352
$383$	1.84985	0.0945228	0.0472614	−	0.998883i	$-0.484951\pi$
$383$	1.84985	0.0945228	0.0472614	+	0.998883i	$0.484951\pi$
$384$	0.935683	0.0477489
$385$	0	0
$386$	−7.98145	−0.406245
$387$	16.9342	0.860813
$388$	11.3338	0.575388
$389$	15.6624	0.794116	0.397058	−	0.917793i	$-0.370031\pi$
$389$	15.6624	0.794116	0.397058	+	0.917793i	$0.370031\pi$
$390$	−2.35228	−0.119112
$391$	−16.2579	−0.822198
$392$	−1.00000	−0.0505076
$393$	−8.02612	−0.404864
$394$	7.28621	0.367074
$395$	−9.86178	−0.496200
$396$	0	0
$397$	−14.8207	−0.743827	−0.371914	−	0.928267i	$-0.621298\pi$
$397$	−14.8207	−0.743827	−0.371914	+	0.928267i	$0.621298\pi$
$398$	−14.1853	−0.711043
$399$	−3.79470	−0.189973
$400$	1.00000	0.0500000
$401$	14.4322	0.720710	0.360355	−	0.932815i	$-0.382656\pi$
$401$	14.4322	0.720710	0.360355	+	0.932815i	$0.382656\pi$
$402$	−5.60857	−0.279730
$403$	−3.88638	−0.193594
$404$	3.16660	0.157544
$405$	1.88697	0.0937645
$406$	7.46463	0.370463
$407$	0	0
$408$	4.38463	0.217071
$409$	−38.0339	−1.88066	−0.940328	−	0.340270i	$-0.889482\pi$
$409$	−38.0339	−1.88066	−0.940328	+	0.340270i	$0.889482\pi$
$410$	−1.65079	−0.0815268
$411$	−1.52597	−0.0752707
$412$	15.0475	0.741335
$413$	−0.426837	−0.0210033
$414$	−7.37084	−0.362257
$415$	−8.15674	−0.400399
$416$	2.51397	0.123257
$417$	21.6890	1.06212
$418$	0	0
$419$	24.2138	1.18292	0.591460	−	0.806334i	$-0.298552\pi$
$419$	24.2138	1.18292	0.591460	+	0.806334i	$0.298552\pi$
$420$	−0.935683	−0.0456567
$421$	38.6678	1.88455	0.942276	−	0.334837i	$-0.108681\pi$
$421$	38.6678	1.88455	0.942276	+	0.334837i	$0.108681\pi$
$422$	23.2770	1.13311
$423$	−3.76987	−0.183297
$424$	−7.14297	−0.346893
$425$	4.68602	0.227305
$426$	6.11864	0.296449
$427$	6.15201	0.297717
$428$	−14.4377	−0.697872
$429$	0	0
$430$	7.97092	0.384392
$431$	−23.6029	−1.13691	−0.568455	−	0.822714i	$-0.692459\pi$
$431$	−23.6029	−1.13691	−0.568455	+	0.822714i	$0.692459\pi$
$432$	4.79491	0.230695
$433$	−30.6532	−1.47310	−0.736550	−	0.676384i	$-0.763546\pi$
$433$	−30.6532	−1.47310	−0.736550	+	0.676384i	$0.763546\pi$
$434$	−1.54591	−0.0742062
$435$	6.98453	0.334883
$436$	−12.1869	−0.583646
$437$	−14.0705	−0.673084
$438$	−0.800357	−0.0382426
$439$	23.4152	1.11755	0.558774	−	0.829320i	$-0.311272\pi$
$439$	23.4152	1.11755	0.558774	+	0.829320i	$0.311272\pi$
$440$	0	0
$441$	−2.12450	−0.101166
$442$	11.7805	0.560341
$443$	0.833237	0.0395883	0.0197941	−	0.999804i	$-0.493699\pi$
$443$	0.833237	0.0395883	0.0197941	+	0.999804i	$0.493699\pi$
$444$	4.60938	0.218752
$445$	1.00048	0.0474274
$446$	8.09748	0.383427
$447$	1.34164	0.0634573
$448$	1.00000	0.0472456
$449$	−5.98030	−0.282228	−0.141114	−	0.989993i	$-0.545068\pi$
$449$	−5.98030	−0.282228	−0.141114	+	0.989993i	$0.545068\pi$
$450$	2.12450	0.100150
$451$	0	0
$452$	−11.7294	−0.551706
$453$	−16.6506	−0.782315
$454$	18.8354	0.883989
$455$	−2.51397	−0.117857
$456$	3.79470	0.177703
$457$	−41.9762	−1.96356	−0.981782	−	0.190010i	$-0.939148\pi$
$457$	−41.9762	−1.96356	−0.981782	+	0.190010i	$0.939148\pi$
$458$	−10.8721	−0.508021
$459$	22.4690	1.04876
$460$	−3.46945	−0.161764
$461$	−32.1474	−1.49725	−0.748626	−	0.662992i	$-0.769286\pi$
$461$	−32.1474	−1.49725	−0.748626	+	0.662992i	$0.769286\pi$
$462$	0	0
$463$	−8.14625	−0.378588	−0.189294	−	0.981920i	$-0.560620\pi$
$463$	−8.14625	−0.378588	−0.189294	+	0.981920i	$0.560620\pi$
$464$	−7.46463	−0.346537
$465$	−1.44649	−0.0670792
$466$	8.97639	0.415823
$467$	7.67994	0.355385	0.177693	−	0.984086i	$-0.443137\pi$
$467$	7.67994	0.355385	0.177693	+	0.984086i	$0.443137\pi$
$468$	5.34092	0.246884
$469$	−5.99409	−0.276781
$470$	−1.77448	−0.0818505
$471$	6.82248	0.314363
$472$	0.426837	0.0196468
$473$	0	0
$474$	−9.22750	−0.423833
$475$	4.05554	0.186081
$476$	4.68602	0.214783
$477$	−15.1752	−0.694825
$478$	9.78125	0.447384
$479$	−22.7225	−1.03822	−0.519109	−	0.854708i	$-0.673736\pi$
$479$	−22.7225	−1.03822	−0.519109	+	0.854708i	$0.673736\pi$
$480$	0.935683	0.0427079
$481$	12.3844	0.564678
$482$	−23.3855	−1.06518
$483$	3.24631	0.147712
$484$	0	0
$485$	11.3338	0.514643
$486$	16.1503	0.732594
$487$	33.4458	1.51557	0.757786	−	0.652503i	$-0.226281\pi$
$487$	33.4458	1.51557	0.757786	+	0.652503i	$0.226281\pi$
$488$	−6.15201	−0.278488
$489$	−8.80763	−0.398295
$490$	−1.00000	−0.0451754
$491$	6.10991	0.275736	0.137868	−	0.990451i	$-0.455975\pi$
$491$	6.10991	0.275736	0.137868	+	0.990451i	$0.455975\pi$
$492$	−1.54462	−0.0696368
$493$	−34.9794	−1.57539
$494$	10.1955	0.458717
$495$	0	0
$496$	1.54591	0.0694136
$497$	6.53922	0.293324
$498$	−7.63213	−0.342004
$499$	21.1193	0.945430	0.472715	−	0.881215i	$-0.343274\pi$
$499$	21.1193	0.945430	0.472715	+	0.881215i	$0.343274\pi$
$500$	1.00000	0.0447214
$501$	0.271271	0.0121195
$502$	−26.0939	−1.16463
$503$	−25.5325	−1.13844	−0.569219	−	0.822186i	$-0.692754\pi$
$503$	−25.5325	−1.13844	−0.569219	+	0.822186i	$0.692754\pi$
$504$	2.12450	0.0946326
$505$	3.16660	0.140912
$506$	0	0
$507$	6.25033	0.277587
$508$	3.68316	0.163414
$509$	−22.0002	−0.975144	−0.487572	−	0.873083i	$-0.662117\pi$
$509$	−22.0002	−0.975144	−0.487572	+	0.873083i	$0.662117\pi$
$510$	4.38463	0.194155
$511$	−0.855372	−0.0378394
$512$	−1.00000	−0.0441942
$513$	19.4459	0.858559
$514$	−5.51935	−0.243448
$515$	15.0475	0.663070
$516$	7.45826	0.328332
$517$	0	0
$518$	4.92622	0.216446
$519$	9.74870	0.427920
$520$	2.51397	0.110245
$521$	−27.5451	−1.20677	−0.603387	−	0.797449i	$-0.706182\pi$
$521$	−27.5451	−1.20677	−0.603387	+	0.797449i	$0.706182\pi$
$522$	−15.8586	−0.694111
$523$	2.58863	0.113193	0.0565963	−	0.998397i	$-0.481975\pi$
$523$	2.58863	0.113193	0.0565963	+	0.998397i	$0.481975\pi$
$524$	8.57781	0.374724
$525$	−0.935683	−0.0408366
$526$	27.1823	1.18521
$527$	7.24418	0.315561
$528$	0	0
$529$	−10.9629	−0.476647
$530$	−7.14297	−0.310271
$531$	0.906814	0.0393524
$532$	4.05554	0.175830
$533$	−4.15004	−0.179758
$534$	0.936135	0.0405105
$535$	−14.4377	−0.624196
$536$	5.99409	0.258905
$537$	13.9525	0.602094
$538$	8.14480	0.351147
$539$	0	0
$540$	4.79491	0.206340
$541$	16.6493	0.715809	0.357905	−	0.933758i	$-0.383491\pi$
$541$	16.6493	0.715809	0.357905	+	0.933758i	$0.383491\pi$
$542$	6.30590	0.270861
$543$	−4.09203	−0.175606
$544$	−4.68602	−0.200911
$545$	−12.1869	−0.522029
$546$	−2.35228	−0.100668
$547$	−32.2258	−1.37788	−0.688938	−	0.724821i	$-0.741923\pi$
$547$	−32.2258	−1.37788	−0.688938	+	0.724821i	$0.741923\pi$
$548$	1.63086	0.0696671
$549$	−13.0699	−0.557811
$550$	0	0
$551$	−30.2731	−1.28968
$552$	−3.24631	−0.138172
$553$	−9.86178	−0.419366
$554$	26.1396	1.11056
$555$	4.60938	0.195657
$556$	−23.1799	−0.983046
$557$	−43.4129	−1.83946	−0.919732	−	0.392548i	$-0.871594\pi$
$557$	−43.4129	−1.83946	−0.919732	+	0.392548i	$0.871594\pi$
$558$	3.28429	0.139035
$559$	20.0386	0.847544
$560$	1.00000	0.0422577
$561$	0	0
$562$	−14.7356	−0.621586
$563$	−12.0403	−0.507437	−0.253718	−	0.967278i	$-0.581654\pi$
$563$	−12.0403	−0.507437	−0.253718	+	0.967278i	$0.581654\pi$
$564$	−1.66035	−0.0699133
$565$	−11.7294	−0.493461
$566$	−21.5060	−0.903964
$567$	1.88697	0.0792455
$568$	−6.53922	−0.274380
$569$	−26.3405	−1.10425	−0.552126	−	0.833761i	$-0.686183\pi$
$569$	−26.3405	−1.10425	−0.552126	+	0.833761i	$0.686183\pi$
$570$	3.79470	0.158943
$571$	22.2032	0.929176	0.464588	−	0.885527i	$-0.346203\pi$
$571$	22.2032	0.929176	0.464588	+	0.885527i	$0.346203\pi$
$572$	0	0
$573$	9.93670	0.415112
$574$	−1.65079	−0.0689027
$575$	−3.46945	−0.144686
$576$	−2.12450	−0.0885207
$577$	33.9425	1.41304	0.706522	−	0.707691i	$-0.250263\pi$
$577$	33.9425	1.41304	0.706522	+	0.707691i	$0.250263\pi$
$578$	−4.95876	−0.206257
$579$	−7.46811	−0.310364
$580$	−7.46463	−0.309952
$581$	−8.15674	−0.338399
$582$	10.6049	0.439587
$583$	0	0
$584$	0.855372	0.0353955
$585$	5.34092	0.220820
$586$	6.11604	0.252651
$587$	−10.6550	−0.439779	−0.219889	−	0.975525i	$-0.570570\pi$
$587$	−10.6550	−0.439779	−0.219889	+	0.975525i	$0.570570\pi$
$588$	−0.935683	−0.0385869
$589$	6.26952	0.258331
$590$	0.426837	0.0175726
$591$	6.81758	0.280438
$592$	−4.92622	−0.202466
$593$	44.1567	1.81330	0.906649	−	0.421886i	$-0.138632\pi$
$593$	44.1567	1.81330	0.906649	+	0.421886i	$0.138632\pi$
$594$	0	0
$595$	4.68602	0.192108
$596$	−1.43386	−0.0587332
$597$	−13.2729	−0.543224
$598$	−8.72209	−0.356673
$599$	25.7621	1.05261	0.526306	−	0.850295i	$-0.323577\pi$
$599$	25.7621	1.05261	0.526306	+	0.850295i	$0.323577\pi$
$600$	0.935683	0.0381991
$601$	48.8171	1.99129	0.995645	−	0.0932309i	$-0.0297195\pi$
$601$	48.8171	1.99129	0.995645	+	0.0932309i	$0.0297195\pi$
$602$	7.97092	0.324870
$603$	12.7344	0.518586
$604$	17.7952	0.724075
$605$	0	0
$606$	2.96293	0.120361
$607$	−23.4113	−0.950237	−0.475118	−	0.879922i	$-0.657595\pi$
$607$	−23.4113	−0.950237	−0.475118	+	0.879922i	$0.657595\pi$
$608$	−4.05554	−0.164474
$609$	6.98453	0.283028
$610$	−6.15201	−0.249088
$611$	−4.46098	−0.180472
$612$	−9.95543	−0.402424
$613$	−34.3936	−1.38914	−0.694572	−	0.719424i	$-0.744406\pi$
$613$	−34.3936	−1.38914	−0.694572	+	0.719424i	$0.744406\pi$
$614$	−24.8721	−1.00376
$615$	−1.54462	−0.0622851
$616$	0	0
$617$	−33.9149	−1.36536	−0.682681	−	0.730717i	$-0.739186\pi$
$617$	−33.9149	−1.36536	−0.682681	+	0.730717i	$0.739186\pi$
$618$	14.0797	0.566367
$619$	−46.5283	−1.87013	−0.935066	−	0.354474i	$-0.884660\pi$
$619$	−46.5283	−1.87013	−0.935066	+	0.354474i	$0.884660\pi$
$620$	1.54591	0.0620854
$621$	−16.6357	−0.667568
$622$	−3.37946	−0.135504
$623$	1.00048	0.0400835
$624$	2.35228	0.0941665
$625$	1.00000	0.0400000
$626$	20.7401	0.828941
$627$	0	0
$628$	−7.29144	−0.290960
$629$	−23.0844	−0.920434
$630$	2.12450	0.0846420
$631$	−49.1136	−1.95518	−0.977591	−	0.210514i	$-0.932486\pi$
$631$	−49.1136	−1.95518	−0.977591	+	0.210514i	$0.932486\pi$
$632$	9.86178	0.392281
$633$	21.7799	0.865675
$634$	−7.46200	−0.296354
$635$	3.68316	0.146162
$636$	−6.68356	−0.265020
$637$	−2.51397	−0.0996070
$638$	0	0
$639$	−13.8926	−0.549581
$640$	−1.00000	−0.0395285
$641$	−10.5819	−0.417959	−0.208979	−	0.977920i	$-0.567014\pi$
$641$	−10.5819	−0.417959	−0.208979	+	0.977920i	$0.567014\pi$
$642$	−13.5091	−0.533162
$643$	−27.2101	−1.07306	−0.536531	−	0.843881i	$-0.680266\pi$
$643$	−27.2101	−1.07306	−0.536531	+	0.843881i	$0.680266\pi$
$644$	−3.46945	−0.136716
$645$	7.45826	0.293669
$646$	−19.0043	−0.747716
$647$	−32.6756	−1.28461	−0.642304	−	0.766450i	$-0.722021\pi$
$647$	−32.6756	−1.28461	−0.642304	+	0.766450i	$0.722021\pi$
$648$	−1.88697	−0.0741273
$649$	0	0
$650$	2.51397	0.0986059
$651$	−1.44649	−0.0566923
$652$	9.41304	0.368643
$653$	−35.8727	−1.40381	−0.701904	−	0.712272i	$-0.747666\pi$
$653$	−35.8727	−1.40381	−0.701904	+	0.712272i	$0.747666\pi$
$654$	−11.4031	−0.445895
$655$	8.57781	0.335163
$656$	1.65079	0.0644526
$657$	1.81723	0.0708971
$658$	−1.77448	−0.0691763
$659$	−2.91875	−0.113698	−0.0568492	−	0.998383i	$-0.518105\pi$
$659$	−2.91875	−0.113698	−0.0568492	+	0.998383i	$0.518105\pi$
$660$	0	0
$661$	−45.9224	−1.78617	−0.893087	−	0.449883i	$-0.851466\pi$
$661$	−45.9224	−1.78617	−0.893087	+	0.449883i	$0.851466\pi$
$662$	8.26702	0.321307
$663$	11.0228	0.428091
$664$	8.15674	0.316543
$665$	4.05554	0.157267
$666$	−10.4657	−0.405539
$667$	25.8982	1.00278
$668$	−0.289918	−0.0112173
$669$	7.57668	0.292931
$670$	5.99409	0.231572
$671$	0	0
$672$	0.935683	0.0360948
$673$	−17.1172	−0.659818	−0.329909	−	0.944013i	$-0.607018\pi$
$673$	−17.1172	−0.659818	−0.329909	+	0.944013i	$0.607018\pi$
$674$	−0.309107	−0.0119063
$675$	4.79491	0.184556
$676$	−6.67997	−0.256922
$677$	−0.309569	−0.0118977	−0.00594886	−	0.999982i	$-0.501894\pi$
$677$	−0.309569	−0.0118977	−0.00594886	+	0.999982i	$0.501894\pi$
$678$	−10.9750	−0.421494
$679$	11.3338	0.434953
$680$	−4.68602	−0.179701
$681$	17.6240	0.675352
$682$	0	0
$683$	44.6187	1.70729	0.853644	−	0.520856i	$-0.174387\pi$
$683$	44.6187	1.70729	0.853644	+	0.520856i	$0.174387\pi$
$684$	−8.61598	−0.329440
$685$	1.63086	0.0623121
$686$	−1.00000	−0.0381802
$687$	−10.1729	−0.388119
$688$	−7.97092	−0.303888
$689$	−17.9572	−0.684115
$690$	−3.24631	−0.123585
$691$	1.84640	0.0702406	0.0351203	−	0.999383i	$-0.488819\pi$
$691$	1.84640	0.0702406	0.0351203	+	0.999383i	$0.488819\pi$
$692$	−10.4188	−0.396063
$693$	0	0
$694$	6.12313	0.232431
$695$	−23.1799	−0.879263
$696$	−6.98453	−0.264748
$697$	7.73564	0.293008
$698$	−19.7838	−0.748828
$699$	8.39906	0.317682
$700$	1.00000	0.0377964
$701$	−13.9594	−0.527240	−0.263620	−	0.964627i	$-0.584917\pi$
$701$	−13.9594	−0.527240	−0.263620	+	0.964627i	$0.584917\pi$
$702$	12.0542	0.454958
$703$	−19.9785	−0.753503
$704$	0	0
$705$	−1.66035	−0.0625323
$706$	30.8256	1.16014
$707$	3.16660	0.119092
$708$	0.399384	0.0150098
$709$	−12.2455	−0.459891	−0.229945	−	0.973204i	$-0.573855\pi$
$709$	−12.2455	−0.459891	−0.229945	+	0.973204i	$0.573855\pi$
$710$	−6.53922	−0.245413
$711$	20.9513	0.785736
$712$	−1.00048	−0.0374947
$713$	−5.36348	−0.200864
$714$	4.38463	0.164091
$715$	0	0
$716$	−14.9115	−0.557270
$717$	9.15215	0.341793
$718$	−25.2431	−0.942064
$719$	−11.5631	−0.431230	−0.215615	−	0.976478i	$-0.569176\pi$
$719$	−11.5631	−0.431230	−0.215615	+	0.976478i	$0.569176\pi$
$720$	−2.12450	−0.0791753
$721$	15.0475	0.560396
$722$	2.55258	0.0949973
$723$	−21.8814	−0.813778
$724$	4.37330	0.162533
$725$	−7.46463	−0.277229
$726$	0	0
$727$	−11.2457	−0.417080	−0.208540	−	0.978014i	$-0.566871\pi$
$727$	−11.2457	−0.417080	−0.208540	+	0.978014i	$0.566871\pi$
$728$	2.51397	0.0931738
$729$	9.45068	0.350025
$730$	0.855372	0.0316587
$731$	−37.3519	−1.38151
$732$	−5.75633	−0.212760
$733$	−13.3226	−0.492081	−0.246041	−	0.969260i	$-0.579130\pi$
$733$	−13.3226	−0.492081	−0.246041	+	0.969260i	$0.579130\pi$
$734$	8.87025	0.327407
$735$	−0.935683	−0.0345132
$736$	3.46945	0.127886
$737$	0	0
$738$	3.50710	0.129098
$739$	−42.5012	−1.56343	−0.781716	−	0.623635i	$-0.785655\pi$
$739$	−42.5012	−1.56343	−0.781716	+	0.623635i	$0.785655\pi$
$740$	−4.92622	−0.181092
$741$	9.53976	0.350452
$742$	−7.14297	−0.262227
$743$	22.0875	0.810311	0.405156	−	0.914248i	$-0.367217\pi$
$743$	22.0875	0.810311	0.405156	+	0.914248i	$0.367217\pi$
$744$	1.44649	0.0530308
$745$	−1.43386	−0.0525326
$746$	23.0860	0.845240
$747$	17.3290	0.634034
$748$	0	0
$749$	−14.4377	−0.527542
$750$	0.935683	0.0341663
$751$	20.2698	0.739657	0.369829	−	0.929100i	$-0.379416\pi$
$751$	20.2698	0.739657	0.369829	+	0.929100i	$0.379416\pi$
$752$	1.77448	0.0647085
$753$	−24.4157	−0.889756
$754$	−18.7658	−0.683412
$755$	17.7952	0.647632
$756$	4.79491	0.174389
$757$	−5.51054	−0.200284	−0.100142	−	0.994973i	$-0.531930\pi$
$757$	−5.51054	−0.200284	−0.100142	+	0.994973i	$0.531930\pi$
$758$	−31.6315	−1.14891
$759$	0	0
$760$	−4.05554	−0.147110
$761$	33.9696	1.23140	0.615699	−	0.787981i	$-0.288874\pi$
$761$	33.9696	1.23140	0.615699	+	0.787981i	$0.288874\pi$
$762$	3.44627	0.124845
$763$	−12.1869	−0.441195
$764$	−10.6197	−0.384208
$765$	−9.95543	−0.359939
$766$	−1.84985	−0.0668377
$767$	1.07305	0.0387458
$768$	−0.935683	−0.0337636
$769$	0.669422	0.0241400	0.0120700	−	0.999927i	$-0.496158\pi$
$769$	0.669422	0.0241400	0.0120700	+	0.999927i	$0.496158\pi$
$770$	0	0
$771$	−5.16436	−0.185990
$772$	7.98145	0.287259
$773$	−32.7311	−1.17725	−0.588627	−	0.808405i	$-0.700331\pi$
$773$	−32.7311	−1.17725	−0.588627	+	0.808405i	$0.700331\pi$
$774$	−16.9342	−0.608687
$775$	1.54591	0.0555309
$776$	−11.3338	−0.406861
$777$	4.60938	0.165361
$778$	−15.6624	−0.561525
$779$	6.69486	0.239868
$780$	2.35228	0.0842251
$781$	0	0
$782$	16.2579	0.581382
$783$	−35.7922	−1.27911
$784$	1.00000	0.0357143
$785$	−7.29144	−0.260243
$786$	8.02612	0.286282
$787$	−16.9929	−0.605732	−0.302866	−	0.953033i	$-0.597944\pi$
$787$	−16.9929	−0.605732	−0.302866	+	0.953033i	$0.597944\pi$
$788$	−7.28621	−0.259560
$789$	25.4340	0.905476
$790$	9.86178	0.350866
$791$	−11.7294	−0.417050
$792$	0	0
$793$	−15.4659	−0.549212
$794$	14.8207	0.525965
$795$	−6.68356	−0.237041
$796$	14.1853	0.502783
$797$	46.4576	1.64561	0.822806	−	0.568322i	$-0.192407\pi$
$797$	46.4576	1.64561	0.822806	+	0.568322i	$0.192407\pi$
$798$	3.79470	0.134331
$799$	8.31523	0.294172
$800$	−1.00000	−0.0353553
$801$	−2.12552	−0.0751016
$802$	−14.4322	−0.509619
$803$	0	0
$804$	5.60857	0.197799
$805$	−3.46945	−0.122282
$806$	3.88638	0.136892
$807$	7.62096	0.268270
$808$	−3.16660	−0.111401
$809$	2.84705	0.100097	0.0500484	−	0.998747i	$-0.484062\pi$
$809$	2.84705	0.100097	0.0500484	+	0.998747i	$0.484062\pi$
$810$	−1.88697	−0.0663015
$811$	1.46624	0.0514866	0.0257433	−	0.999669i	$-0.491805\pi$
$811$	1.46624	0.0514866	0.0257433	+	0.999669i	$0.491805\pi$
$812$	−7.46463	−0.261957
$813$	5.90032	0.206933
$814$	0	0
$815$	9.41304	0.329724
$816$	−4.38463	−0.153493
$817$	−32.3264	−1.13096
$818$	38.0339	1.32982
$819$	5.34092	0.186627
$820$	1.65079	0.0576482
$821$	−19.6491	−0.685757	−0.342879	−	0.939380i	$-0.611402\pi$
$821$	−19.6491	−0.685757	−0.342879	+	0.939380i	$0.611402\pi$
$822$	1.52597	0.0532244
$823$	−17.0752	−0.595203	−0.297601	−	0.954690i	$-0.596187\pi$
$823$	−17.0752	−0.595203	−0.297601	+	0.954690i	$0.596187\pi$
$824$	−15.0475	−0.524203
$825$	0	0
$826$	0.426837	0.0148516
$827$	15.9823	0.555759	0.277880	−	0.960616i	$-0.410368\pi$
$827$	15.9823	0.555759	0.277880	+	0.960616i	$0.410368\pi$
$828$	7.37084	0.256154
$829$	17.6458	0.612863	0.306432	−	0.951893i	$-0.400865\pi$
$829$	17.6458	0.612863	0.306432	+	0.951893i	$0.400865\pi$
$830$	8.15674	0.283125
$831$	24.4584	0.848451
$832$	−2.51397	−0.0871561
$833$	4.68602	0.162361
$834$	−21.6890	−0.751030
$835$	−0.289918	−0.0100330
$836$	0	0
$837$	7.41251	0.256214
$838$	−24.2138	−0.836451
$839$	7.96223	0.274887	0.137443	−	0.990510i	$-0.456111\pi$
$839$	7.96223	0.274887	0.137443	+	0.990510i	$0.456111\pi$
$840$	0.935683	0.0322841
$841$	26.7207	0.921404
$842$	−38.6678	−1.33258
$843$	−13.7879	−0.474880
$844$	−23.2770	−0.801229
$845$	−6.67997	−0.229798
$846$	3.76987	0.129611
$847$	0	0
$848$	7.14297	0.245291
$849$	−20.1228	−0.690613
$850$	−4.68602	−0.160729
$851$	17.0913	0.585882
$852$	−6.11864	−0.209621
$853$	−39.3256	−1.34648	−0.673242	−	0.739422i	$-0.735099\pi$
$853$	−39.3256	−1.34648	−0.673242	+	0.739422i	$0.735099\pi$
$854$	−6.15201	−0.210517
$855$	−8.61598	−0.294660
$856$	14.4377	0.493470
$857$	−42.5288	−1.45276	−0.726379	−	0.687295i	$-0.758798\pi$
$857$	−42.5288	−1.45276	−0.726379	+	0.687295i	$0.758798\pi$
$858$	0	0
$859$	36.8680	1.25792	0.628960	−	0.777438i	$-0.283481\pi$
$859$	36.8680	1.25792	0.628960	+	0.777438i	$0.283481\pi$
$860$	−7.97092	−0.271806
$861$	−1.54462	−0.0526405
$862$	23.6029	0.803917
$863$	25.0975	0.854327	0.427164	−	0.904174i	$-0.359513\pi$
$863$	25.0975	0.854327	0.427164	+	0.904174i	$0.359513\pi$
$864$	−4.79491	−0.163126
$865$	−10.4188	−0.354250
$866$	30.6532	1.04164
$867$	−4.63983	−0.157577
$868$	1.54591	0.0524717
$869$	0	0
$870$	−6.98453	−0.236798
$871$	15.0690	0.510592
$872$	12.1869	0.412700
$873$	−24.0787	−0.814940
$874$	14.0705	0.475942
$875$	1.00000	0.0338062
$876$	0.800357	0.0270416
$877$	−54.6433	−1.84517	−0.922586	−	0.385792i	$-0.873928\pi$
$877$	−54.6433	−1.84517	−0.922586	+	0.385792i	$0.873928\pi$
$878$	−23.4152	−0.790226
$879$	5.72268	0.193021
$880$	0	0
$881$	−49.1584	−1.65619	−0.828094	−	0.560590i	$-0.810575\pi$
$881$	−49.1584	−1.65619	−0.828094	+	0.560590i	$0.810575\pi$
$882$	2.12450	0.0715355
$883$	−26.9252	−0.906105	−0.453053	−	0.891484i	$-0.649665\pi$
$883$	−26.9252	−0.906105	−0.453053	+	0.891484i	$0.649665\pi$
$884$	−11.7805	−0.396221
$885$	0.399384	0.0134252
$886$	−0.833237	−0.0279931
$887$	31.8126	1.06816	0.534082	−	0.845433i	$-0.320658\pi$
$887$	31.8126	1.06816	0.534082	+	0.845433i	$0.320658\pi$
$888$	−4.60938	−0.154681
$889$	3.68316	0.123529
$890$	−1.00048	−0.0335363
$891$	0	0
$892$	−8.09748	−0.271124
$893$	7.19646	0.240820
$894$	−1.34164	−0.0448711
$895$	−14.9115	−0.498438
$896$	−1.00000	−0.0334077
$897$	−8.16112	−0.272492
$898$	5.98030	0.199565
$899$	−11.5397	−0.384870
$900$	−2.12450	−0.0708165
$901$	33.4721	1.11512
$902$	0	0
$903$	7.45826	0.248195
$904$	11.7294	0.390115
$905$	4.37330	0.145374
$906$	16.6506	0.553180
$907$	27.8459	0.924609	0.462304	−	0.886721i	$-0.347023\pi$
$907$	27.8459	0.924609	0.462304	+	0.886721i	$0.347023\pi$
$908$	−18.8354	−0.625074
$909$	−6.72743	−0.223135
$910$	2.51397	0.0833372
$911$	−54.9974	−1.82214	−0.911072	−	0.412247i	$-0.864744\pi$
$911$	−54.9974	−1.82214	−0.911072	+	0.412247i	$0.864744\pi$
$912$	−3.79470	−0.125655
$913$	0	0
$914$	41.9762	1.38845
$915$	−5.75633	−0.190298
$916$	10.8721	0.359225
$917$	8.57781	0.283264
$918$	−22.4690	−0.741588
$919$	16.9015	0.557529	0.278764	−	0.960360i	$-0.410075\pi$
$919$	16.9015	0.557529	0.278764	+	0.960360i	$0.410075\pi$
$920$	3.46945	0.114384
$921$	−23.2724	−0.766851
$922$	32.1474	1.05872
$923$	−16.4394	−0.541109
$924$	0	0
$925$	−4.92622	−0.161973
$926$	8.14625	0.267702
$927$	−31.9683	−1.04998
$928$	7.46463	0.245039
$929$	−16.9182	−0.555069	−0.277535	−	0.960716i	$-0.589517\pi$
$929$	−16.9182	−0.555069	−0.277535	+	0.960716i	$0.589517\pi$
$930$	1.44649	0.0474321
$931$	4.05554	0.132915
$932$	−8.97639	−0.294031
$933$	−3.16210	−0.103523
$934$	−7.67994	−0.251295
$935$	0	0
$936$	−5.34092	−0.174573
$937$	−5.70923	−0.186512	−0.0932562	−	0.995642i	$-0.529728\pi$
$937$	−5.70923	−0.186512	−0.0932562	+	0.995642i	$0.529728\pi$
$938$	5.99409	0.195714
$939$	19.4062	0.633297
$940$	1.77448	0.0578770
$941$	−34.2699	−1.11717	−0.558584	−	0.829448i	$-0.688655\pi$
$941$	−34.2699	−1.11717	−0.558584	+	0.829448i	$0.688655\pi$
$942$	−6.82248	−0.222288
$943$	−5.72735	−0.186508
$944$	−0.426837	−0.0138924
$945$	4.79491	0.155978
$946$	0	0
$947$	−29.4856	−0.958154	−0.479077	−	0.877773i	$-0.659029\pi$
$947$	−29.4856	−0.958154	−0.479077	+	0.877773i	$0.659029\pi$
$948$	9.22750	0.299695
$949$	2.15038	0.0698042
$950$	−4.05554	−0.131579
$951$	−6.98207	−0.226409
$952$	−4.68602	−0.151875
$953$	−43.8435	−1.42023	−0.710116	−	0.704085i	$-0.751357\pi$
$953$	−43.8435	−1.42023	−0.710116	+	0.704085i	$0.751357\pi$
$954$	15.1752	0.491316
$955$	−10.6197	−0.343646
$956$	−9.78125	−0.316348
$957$	0	0
$958$	22.7225	0.734131
$959$	1.63086	0.0526633
$960$	−0.935683	−0.0301991
$961$	−28.6102	−0.922908
$962$	−12.3844	−0.399288
$963$	30.6728	0.988418
$964$	23.3855	0.753195
$965$	7.98145	0.256932
$966$	−3.24631	−0.104448
$967$	−43.1164	−1.38653	−0.693265	−	0.720683i	$-0.743828\pi$
$967$	−43.1164	−1.38653	−0.693265	+	0.720683i	$0.743828\pi$
$968$	0	0
$969$	−17.7820	−0.571242
$970$	−11.3338	−0.363908
$971$	−55.6910	−1.78721	−0.893604	−	0.448856i	$-0.851831\pi$
$971$	−55.6910	−1.78721	−0.893604	+	0.448856i	$0.851831\pi$
$972$	−16.1503	−0.518022
$973$	−23.1799	−0.743113
$974$	−33.4458	−1.07167
$975$	2.35228	0.0753332
$976$	6.15201	0.196921
$977$	−9.63540	−0.308264	−0.154132	−	0.988050i	$-0.549258\pi$
$977$	−9.63540	−0.308264	−0.154132	+	0.988050i	$0.549258\pi$
$978$	8.80763	0.281637
$979$	0	0
$980$	1.00000	0.0319438
$981$	25.8910	0.826636
$982$	−6.10991	−0.194975
$983$	54.5716	1.74056	0.870281	−	0.492555i	$-0.163937\pi$
$983$	54.5716	1.74056	0.870281	+	0.492555i	$0.163937\pi$
$984$	1.54462	0.0492407
$985$	−7.28621	−0.232158
$986$	34.9794	1.11397
$987$	−1.66035	−0.0528495
$988$	−10.1955	−0.324362
$989$	27.6547	0.879370
$990$	0	0
$991$	−50.0492	−1.58986	−0.794932	−	0.606699i	$-0.792494\pi$
$991$	−50.0492	−1.58986	−0.794932	+	0.606699i	$0.792494\pi$
$992$	−1.54591	−0.0490828
$993$	7.73531	0.245473
$994$	−6.53922	−0.207411
$995$	14.1853	0.449703
$996$	7.63213	0.241833
$997$	22.8919	0.724994	0.362497	−	0.931985i	$-0.381924\pi$
$997$	22.8919	0.724994	0.362497	+	0.931985i	$0.381924\pi$
$998$	−21.1193	−0.668520
$999$	−23.6208	−0.747328

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.cw.1.3		6
11.3	even	5		770.2.n.j.141.2	yes	12
11.4	even	5		770.2.n.j.71.2	✓	12
11.10	odd	2		8470.2.a.dc.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.j.71.2	✓	12	11.4	even	5
770.2.n.j.141.2	yes	12	11.3	even	5
8470.2.a.cw.1.3		6	1.1	even	1	trivial
8470.2.a.dc.1.3		6	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.935683 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.935683 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.12450 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.935683 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.935683 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.12450 q^{9} -1.00000 q^{10} -0.935683 q^{12} -2.51397 q^{13} -1.00000 q^{14} -0.935683 q^{15} +1.00000 q^{16} +4.68602 q^{17} +2.12450 q^{18} +4.05554 q^{19} +1.00000 q^{20} -0.935683 q^{21} -3.46945 q^{23} +0.935683 q^{24} +1.00000 q^{25} +2.51397 q^{26} +4.79491 q^{27} +1.00000 q^{28} -7.46463 q^{29} +0.935683 q^{30} +1.54591 q^{31} -1.00000 q^{32} -4.68602 q^{34} +1.00000 q^{35} -2.12450 q^{36} -4.92622 q^{37} -4.05554 q^{38} +2.35228 q^{39} -1.00000 q^{40} +1.65079 q^{41} +0.935683 q^{42} -7.97092 q^{43} -2.12450 q^{45} +3.46945 q^{46} +1.77448 q^{47} -0.935683 q^{48} +1.00000 q^{49} -1.00000 q^{50} -4.38463 q^{51} -2.51397 q^{52} +7.14297 q^{53} -4.79491 q^{54} -1.00000 q^{56} -3.79470 q^{57} +7.46463 q^{58} -0.426837 q^{59} -0.935683 q^{60} +6.15201 q^{61} -1.54591 q^{62} -2.12450 q^{63} +1.00000 q^{64} -2.51397 q^{65} -5.99409 q^{67} +4.68602 q^{68} +3.24631 q^{69} -1.00000 q^{70} +6.53922 q^{71} +2.12450 q^{72} -0.855372 q^{73} +4.92622 q^{74} -0.935683 q^{75} +4.05554 q^{76} -2.35228 q^{78} -9.86178 q^{79} +1.00000 q^{80} +1.88697 q^{81} -1.65079 q^{82} -8.15674 q^{83} -0.935683 q^{84} +4.68602 q^{85} +7.97092 q^{86} +6.98453 q^{87} +1.00048 q^{89} +2.12450 q^{90} -2.51397 q^{91} -3.46945 q^{92} -1.44649 q^{93} -1.77448 q^{94} +4.05554 q^{95} +0.935683 q^{96} +11.3338 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) \) 6 * q - 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 + q^6 + 6 * q^7 - 6 * q^8 + 3 * q^9	\( 6 q - 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} - 6 q^{8} + 3 q^{9} - 6 q^{10} - q^{12} - 9 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} - 9 q^{17} - 3 q^{18} - 12 q^{19} + 6 q^{20} - q^{21} + 4 q^{23} + q^{24} + 6 q^{25} + 9 q^{26} - 4 q^{27} + 6 q^{28} - 15 q^{29} + q^{30} + 8 q^{31} - 6 q^{32} + 9 q^{34} + 6 q^{35} + 3 q^{36} - 4 q^{37} + 12 q^{38} + 19 q^{39} - 6 q^{40} - 4 q^{41} + q^{42} - 30 q^{43} + 3 q^{45} - 4 q^{46} - 7 q^{47} - q^{48} + 6 q^{49} - 6 q^{50} - 16 q^{51} - 9 q^{52} - 6 q^{53} + 4 q^{54} - 6 q^{56} + 14 q^{57} + 15 q^{58} + 4 q^{59} - q^{60} + 14 q^{61} - 8 q^{62} + 3 q^{63} + 6 q^{64} - 9 q^{65} + 18 q^{67} - 9 q^{68} - 10 q^{69} - 6 q^{70} + 23 q^{71} - 3 q^{72} - 23 q^{73} + 4 q^{74} - q^{75} - 12 q^{76} - 19 q^{78} - 21 q^{79} + 6 q^{80} - 18 q^{81} + 4 q^{82} - 25 q^{83} - q^{84} - 9 q^{85} + 30 q^{86} - 14 q^{87} - 18 q^{89} - 3 q^{90} - 9 q^{91} + 4 q^{92} - 24 q^{93} + 7 q^{94} - 12 q^{95} + q^{96} + 7 q^{97} - 6 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 + q^6 + 6 * q^7 - 6 * q^8 + 3 * q^9 - 6 * q^10 - q^12 - 9 * q^13 - 6 * q^14 - q^15 + 6 * q^16 - 9 * q^17 - 3 * q^18 - 12 * q^19 + 6 * q^20 - q^21 + 4 * q^23 + q^24 + 6 * q^25 + 9 * q^26 - 4 * q^27 + 6 * q^28 - 15 * q^29 + q^30 + 8 * q^31 - 6 * q^32 + 9 * q^34 + 6 * q^35 + 3 * q^36 - 4 * q^37 + 12 * q^38 + 19 * q^39 - 6 * q^40 - 4 * q^41 + q^42 - 30 * q^43 + 3 * q^45 - 4 * q^46 - 7 * q^47 - q^48 + 6 * q^49 - 6 * q^50 - 16 * q^51 - 9 * q^52 - 6 * q^53 + 4 * q^54 - 6 * q^56 + 14 * q^57 + 15 * q^58 + 4 * q^59 - q^60 + 14 * q^61 - 8 * q^62 + 3 * q^63 + 6 * q^64 - 9 * q^65 + 18 * q^67 - 9 * q^68 - 10 * q^69 - 6 * q^70 + 23 * q^71 - 3 * q^72 - 23 * q^73 + 4 * q^74 - q^75 - 12 * q^76 - 19 * q^78 - 21 * q^79 + 6 * q^80 - 18 * q^81 + 4 * q^82 - 25 * q^83 - q^84 - 9 * q^85 + 30 * q^86 - 14 * q^87 - 18 * q^89 - 3 * q^90 - 9 * q^91 + 4 * q^92 - 24 * q^93 + 7 * q^94 - 12 * q^95 + q^96 + 7 * q^97 - 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more