LMFDB - Embedded newform 8470.2.a.ct.1.1

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.5225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + x + 11 $ x^4 - x^3 - 8*x^2 + x + 11
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			1.1
Root			$1.26498$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.04678 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.04678 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.18929 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.04678 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.04678 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.18929 q^{9} -1.00000 q^{10} -2.04678 q^{12} +3.50105 q^{13} +1.00000 q^{14} +2.04678 q^{15} +1.00000 q^{16} -4.28284 q^{17} +1.18929 q^{18} +4.44661 q^{19} -1.00000 q^{20} -2.04678 q^{21} -2.00000 q^{23} -2.04678 q^{24} +1.00000 q^{25} +3.50105 q^{26} +3.70611 q^{27} +1.00000 q^{28} +1.30628 q^{29} +2.04678 q^{30} -5.70820 q^{31} +1.00000 q^{32} -4.28284 q^{34} -1.00000 q^{35} +1.18929 q^{36} -0.327530 q^{37} +4.44661 q^{38} -7.16586 q^{39} -1.00000 q^{40} -3.87535 q^{41} -2.04678 q^{42} -2.63928 q^{43} -1.18929 q^{45} -2.00000 q^{46} +12.2258 q^{47} -2.04678 q^{48} +1.00000 q^{49} +1.00000 q^{50} +8.76602 q^{51} +3.50105 q^{52} +1.09146 q^{53} +3.70611 q^{54} +1.00000 q^{56} -9.10121 q^{57} +1.30628 q^{58} +10.0613 q^{59} +2.04678 q^{60} +13.0021 q^{61} -5.70820 q^{62} +1.18929 q^{63} +1.00000 q^{64} -3.50105 q^{65} -8.44661 q^{67} -4.28284 q^{68} +4.09355 q^{69} -1.00000 q^{70} -5.87963 q^{71} +1.18929 q^{72} -7.44187 q^{73} -0.327530 q^{74} -2.04678 q^{75} +4.44661 q^{76} -7.16586 q^{78} +0.498955 q^{79} -1.00000 q^{80} -11.1535 q^{81} -3.87535 q^{82} -12.6125 q^{83} -2.04678 q^{84} +4.28284 q^{85} -2.63928 q^{86} -2.67366 q^{87} +1.39645 q^{89} -1.18929 q^{90} +3.50105 q^{91} -2.00000 q^{92} +11.6834 q^{93} +12.2258 q^{94} -4.44661 q^{95} -2.04678 q^{96} -9.79837 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} + 4 q^{7} + 4 q^{8} + 6 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 + 4 * q^7 + 4 * q^8 + 6 * q^9	$ 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} + 4 q^{7} + 4 q^{8} + 6 q^{9} - 4 q^{10} + 2 q^{12} + q^{13} + 4 q^{14} - 2 q^{15} + 4 q^{16} + 2 q^{17} + 6 q^{18} - 3 q^{19} - 4 q^{20} + 2 q^{21} - 8 q^{23} + 2 q^{24} + 4 q^{25} + q^{26} + 14 q^{27} + 4 q^{28} + 15 q^{29} - 2 q^{30} + 4 q^{31} + 4 q^{32} + 2 q^{34} - 4 q^{35} + 6 q^{36} + 2 q^{37} - 3 q^{38} - q^{39} - 4 q^{40} + 11 q^{41} + 2 q^{42} + 7 q^{43} - 6 q^{45} - 8 q^{46} + 5 q^{47} + 2 q^{48} + 4 q^{49} + 4 q^{50} + 18 q^{51} + q^{52} + 10 q^{53} + 14 q^{54} + 4 q^{56} - 34 q^{57} + 15 q^{58} + 13 q^{59} - 2 q^{60} + 26 q^{61} + 4 q^{62} + 6 q^{63} + 4 q^{64} - q^{65} - 13 q^{67} + 2 q^{68} - 4 q^{69} - 4 q^{70} - 13 q^{71} + 6 q^{72} - 18 q^{73} + 2 q^{74} + 2 q^{75} - 3 q^{76} - q^{78} + 15 q^{79} - 4 q^{80} - 8 q^{81} + 11 q^{82} - 2 q^{83} + 2 q^{84} - 2 q^{85} + 7 q^{86} + 26 q^{87} - 7 q^{89} - 6 q^{90} + q^{91} - 8 q^{92} + 2 q^{93} + 5 q^{94} + 3 q^{95} + 2 q^{96} + 10 q^{97} + 4 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 + 4 * q^7 + 4 * q^8 + 6 * q^9 - 4 * q^10 + 2 * q^12 + q^13 + 4 * q^14 - 2 * q^15 + 4 * q^16 + 2 * q^17 + 6 * q^18 - 3 * q^19 - 4 * q^20 + 2 * q^21 - 8 * q^23 + 2 * q^24 + 4 * q^25 + q^26 + 14 * q^27 + 4 * q^28 + 15 * q^29 - 2 * q^30 + 4 * q^31 + 4 * q^32 + 2 * q^34 - 4 * q^35 + 6 * q^36 + 2 * q^37 - 3 * q^38 - q^39 - 4 * q^40 + 11 * q^41 + 2 * q^42 + 7 * q^43 - 6 * q^45 - 8 * q^46 + 5 * q^47 + 2 * q^48 + 4 * q^49 + 4 * q^50 + 18 * q^51 + q^52 + 10 * q^53 + 14 * q^54 + 4 * q^56 - 34 * q^57 + 15 * q^58 + 13 * q^59 - 2 * q^60 + 26 * q^61 + 4 * q^62 + 6 * q^63 + 4 * q^64 - q^65 - 13 * q^67 + 2 * q^68 - 4 * q^69 - 4 * q^70 - 13 * q^71 + 6 * q^72 - 18 * q^73 + 2 * q^74 + 2 * q^75 - 3 * q^76 - q^78 + 15 * q^79 - 4 * q^80 - 8 * q^81 + 11 * q^82 - 2 * q^83 + 2 * q^84 - 2 * q^85 + 7 * q^86 + 26 * q^87 - 7 * q^89 - 6 * q^90 + q^91 - 8 * q^92 + 2 * q^93 + 5 * q^94 + 3 * q^95 + 2 * q^96 + 10 * q^97 + 4 * 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
$3$	−2.04678	−1.18171	−0.590853	−	0.806779i	$-0.701209\pi$
$3$	−2.04678	−1.18171	−0.590853	+	0.806779i	$0.701209\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−2.04678	−0.835593
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	1.18929	0.396431
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	−2.04678	−0.590853
$13$	3.50105	0.971015	0.485508	−	0.874232i	$-0.338635\pi$
$13$	3.50105	0.971015	0.485508	+	0.874232i	$0.338635\pi$
$14$	1.00000	0.267261
$15$	2.04678	0.528475
$16$	1.00000	0.250000
$17$	−4.28284	−1.03874	−0.519371	−	0.854549i	$-0.673834\pi$
$17$	−4.28284	−1.03874	−0.519371	+	0.854549i	$0.673834\pi$
$18$	1.18929	0.280319
$19$	4.44661	1.02012	0.510061	−	0.860138i	$-0.329623\pi$
$19$	4.44661	1.02012	0.510061	+	0.860138i	$0.329623\pi$
$20$	−1.00000	−0.223607
$21$	−2.04678	−0.446643
$22$	0	0
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	−2.04678	−0.417796
$25$	1.00000	0.200000
$26$	3.50105	0.686611
$27$	3.70611	0.713242
$28$	1.00000	0.188982
$29$	1.30628	0.242570	0.121285	−	0.992618i	$-0.461298\pi$
$29$	1.30628	0.242570	0.121285	+	0.992618i	$0.461298\pi$
$30$	2.04678	0.373688
$31$	−5.70820	−1.02522	−0.512612	−	0.858620i	$-0.671322\pi$
$31$	−5.70820	−1.02522	−0.512612	+	0.858620i	$0.671322\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.28284	−0.734502
$35$	−1.00000	−0.169031
$36$	1.18929	0.198215
$37$	−0.327530	−0.0538456	−0.0269228	−	0.999638i	$-0.508571\pi$
$37$	−0.327530	−0.0538456	−0.0269228	+	0.999638i	$0.508571\pi$
$38$	4.44661	0.721335
$39$	−7.16586	−1.14746
$40$	−1.00000	−0.158114
$41$	−3.87535	−0.605228	−0.302614	−	0.953113i	$-0.597859\pi$
$41$	−3.87535	−0.605228	−0.302614	+	0.953113i	$0.597859\pi$
$42$	−2.04678	−0.315824
$43$	−2.63928	−0.402487	−0.201243	−	0.979541i	$-0.564498\pi$
$43$	−2.63928	−0.402487	−0.201243	+	0.979541i	$0.564498\pi$
$44$	0	0
$45$	−1.18929	−0.177289
$46$	−2.00000	−0.294884
$47$	12.2258	1.78331	0.891655	−	0.452715i	$-0.149544\pi$
$47$	12.2258	1.78331	0.891655	+	0.452715i	$0.149544\pi$
$48$	−2.04678	−0.295427
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	8.76602	1.22749
$52$	3.50105	0.485508
$53$	1.09146	0.149924	0.0749619	−	0.997186i	$-0.476116\pi$
$53$	1.09146	0.149924	0.0749619	+	0.997186i	$0.476116\pi$
$54$	3.70611	0.504338
$55$	0	0
$56$	1.00000	0.133631
$57$	−9.10121	−1.20549
$58$	1.30628	0.171523
$59$	10.0613	1.30986	0.654932	−	0.755688i	$-0.272697\pi$
$59$	10.0613	1.30986	0.654932	+	0.755688i	$0.272697\pi$
$60$	2.04678	0.264238
$61$	13.0021	1.66475	0.832374	−	0.554215i	$-0.186981\pi$
$61$	13.0021	1.66475	0.832374	+	0.554215i	$0.186981\pi$
$62$	−5.70820	−0.724943
$63$	1.18929	0.149837
$64$	1.00000	0.125000
$65$	−3.50105	−0.434251
$66$	0	0
$67$	−8.44661	−1.03192	−0.515959	−	0.856613i	$-0.672564\pi$
$67$	−8.44661	−1.03192	−0.515959	+	0.856613i	$0.672564\pi$
$68$	−4.28284	−0.519371
$69$	4.09355	0.492806
$70$	−1.00000	−0.119523
$71$	−5.87963	−0.697784	−0.348892	−	0.937163i	$-0.613442\pi$
$71$	−5.87963	−0.697784	−0.348892	+	0.937163i	$0.613442\pi$
$72$	1.18929	0.140159
$73$	−7.44187	−0.871006	−0.435503	−	0.900187i	$-0.643429\pi$
$73$	−7.44187	−0.871006	−0.435503	+	0.900187i	$0.643429\pi$
$74$	−0.327530	−0.0380746
$75$	−2.04678	−0.236341
$76$	4.44661	0.510061
$77$	0	0
$78$	−7.16586	−0.811373
$79$	0.498955	0.0561368	0.0280684	−	0.999606i	$-0.491064\pi$
$79$	0.498955	0.0561368	0.0280684	+	0.999606i	$0.491064\pi$
$80$	−1.00000	−0.111803
$81$	−11.1535	−1.23927
$82$	−3.87535	−0.427961
$83$	−12.6125	−1.38440	−0.692199	−	0.721707i	$-0.743358\pi$
$83$	−12.6125	−1.38440	−0.692199	+	0.721707i	$0.743358\pi$
$84$	−2.04678	−0.223322
$85$	4.28284	0.464540
$86$	−2.63928	−0.284601
$87$	−2.67366	−0.286647
$88$	0	0
$89$	1.39645	0.148023	0.0740117	−	0.997257i	$-0.476420\pi$
$89$	1.39645	0.148023	0.0740117	+	0.997257i	$0.476420\pi$
$90$	−1.18929	−0.125362
$91$	3.50105	0.367009
$92$	−2.00000	−0.208514
$93$	11.6834	1.21151
$94$	12.2258	1.26099
$95$	−4.44661	−0.456212
$96$	−2.04678	−0.208898
$97$	−9.79837	−0.994874	−0.497437	−	0.867500i	$-0.665725\pi$
$97$	−9.79837	−0.994874	−0.497437	+	0.867500i	$0.665725\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	17.3874	1.73011	0.865057	−	0.501673i	$-0.167282\pi$
$101$	17.3874	1.73011	0.865057	+	0.501673i	$0.167282\pi$
$102$	8.76602	0.867966
$103$	6.86988	0.676909	0.338455	−	0.940983i	$-0.390096\pi$
$103$	6.86988	0.676909	0.338455	+	0.940983i	$0.390096\pi$
$104$	3.50105	0.343306
$105$	2.04678	0.199745
$106$	1.09146	0.106012
$107$	1.75025	0.169203	0.0846013	−	0.996415i	$-0.473038\pi$
$107$	1.75025	0.169203	0.0846013	+	0.996415i	$0.473038\pi$
$108$	3.70611	0.356621
$109$	−5.88436	−0.563620	−0.281810	−	0.959470i	$-0.590935\pi$
$109$	−5.88436	−0.563620	−0.281810	+	0.959470i	$0.590935\pi$
$110$	0	0
$111$	0.670380	0.0636297
$112$	1.00000	0.0944911
$113$	17.1279	1.61126	0.805630	−	0.592419i	$-0.201827\pi$
$113$	17.1279	1.61126	0.805630	+	0.592419i	$0.201827\pi$
$114$	−9.10121	−0.852407
$115$	2.00000	0.186501
$116$	1.30628	0.121285
$117$	4.16376	0.384940
$118$	10.0613	0.926214
$119$	−4.28284	−0.392608
$120$	2.04678	0.186844
$121$	0	0
$122$	13.0021	1.17715
$123$	7.93198	0.715202
$124$	−5.70820	−0.512612
$125$	−1.00000	−0.0894427
$126$	1.18929	0.105951
$127$	−9.94218	−0.882226	−0.441113	−	0.897452i	$-0.645416\pi$
$127$	−9.94218	−0.882226	−0.441113	+	0.897452i	$0.645416\pi$
$128$	1.00000	0.0883883
$129$	5.40202	0.475621
$130$	−3.50105	−0.307062
$131$	−7.65112	−0.668482	−0.334241	−	0.942488i	$-0.608480\pi$
$131$	−7.65112	−0.668482	−0.334241	+	0.942488i	$0.608480\pi$
$132$	0	0
$133$	4.44661	0.385570
$134$	−8.44661	−0.729676
$135$	−3.70611	−0.318971
$136$	−4.28284	−0.367251
$137$	2.16038	0.184574	0.0922870	−	0.995732i	$-0.470582\pi$
$137$	2.16038	0.184574	0.0922870	+	0.995732i	$0.470582\pi$
$138$	4.09355	0.348466
$139$	−21.1314	−1.79234	−0.896170	−	0.443711i	$-0.853662\pi$
$139$	−21.1314	−1.79234	−0.896170	+	0.443711i	$0.853662\pi$
$140$	−1.00000	−0.0845154
$141$	−25.0234	−2.10735
$142$	−5.87963	−0.493408
$143$	0	0
$144$	1.18929	0.0991077
$145$	−1.30628	−0.108481
$146$	−7.44187	−0.615894
$147$	−2.04678	−0.168815
$148$	−0.327530	−0.0269228
$149$	16.0089	1.31150	0.655751	−	0.754977i	$-0.272352\pi$
$149$	16.0089	1.31150	0.655751	+	0.754977i	$0.272352\pi$
$150$	−2.04678	−0.167119
$151$	−12.9386	−1.05293	−0.526466	−	0.850196i	$-0.676483\pi$
$151$	−12.9386	−1.05293	−0.526466	+	0.850196i	$0.676483\pi$
$152$	4.44661	0.360668
$153$	−5.09355	−0.411789
$154$	0	0
$155$	5.70820	0.458494
$156$	−7.16586	−0.573728
$157$	18.4955	1.47610	0.738050	−	0.674746i	$-0.235747\pi$
$157$	18.4955	1.47610	0.738050	+	0.674746i	$0.235747\pi$
$158$	0.498955	0.0396947
$159$	−2.23398	−0.177166
$160$	−1.00000	−0.0790569
$161$	−2.00000	−0.157622
$162$	−11.1535	−0.876299
$163$	6.52922	0.511408	0.255704	−	0.966755i	$-0.417693\pi$
$163$	6.52922	0.511408	0.255704	+	0.966755i	$0.417693\pi$
$164$	−3.87535	−0.302614
$165$	0	0
$166$	−12.6125	−0.978917
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	−2.04678	−0.157912
$169$	−0.742683	−0.0571295
$170$	4.28284	0.328479
$171$	5.28832	0.404408
$172$	−2.63928	−0.201243
$173$	7.54354	0.573525	0.286762	−	0.958002i	$-0.407421\pi$
$173$	7.54354	0.573525	0.286762	+	0.958002i	$0.407421\pi$
$174$	−2.67366	−0.202690
$175$	1.00000	0.0755929
$176$	0	0
$177$	−20.5931	−1.54788
$178$	1.39645	0.104668
$179$	6.66819	0.498404	0.249202	−	0.968452i	$-0.419832\pi$
$179$	6.66819	0.498404	0.249202	+	0.968452i	$0.419832\pi$
$180$	−1.18929	−0.0886446
$181$	−3.26233	−0.242487	−0.121244	−	0.992623i	$-0.538688\pi$
$181$	−3.26233	−0.242487	−0.121244	+	0.992623i	$0.538688\pi$
$182$	3.50105	0.259515
$183$	−26.6124	−1.96724
$184$	−2.00000	−0.147442
$185$	0.327530	0.0240805
$186$	11.6834	0.856670
$187$	0	0
$188$	12.2258	0.891655
$189$	3.70611	0.269580
$190$	−4.44661	−0.322591
$191$	−13.7035	−0.991548	−0.495774	−	0.868452i	$-0.665116\pi$
$191$	−13.7035	−0.991548	−0.495774	+	0.868452i	$0.665116\pi$
$192$	−2.04678	−0.147713
$193$	−11.1467	−0.802357	−0.401178	−	0.916000i	$-0.631399\pi$
$193$	−11.1467	−0.802357	−0.401178	+	0.916000i	$0.631399\pi$
$194$	−9.79837	−0.703482
$195$	7.16586	0.513158
$196$	1.00000	0.0714286
$197$	3.51254	0.250258	0.125129	−	0.992140i	$-0.460065\pi$
$197$	3.51254	0.250258	0.125129	+	0.992140i	$0.460065\pi$
$198$	0	0
$199$	19.5167	1.38350	0.691752	−	0.722135i	$-0.256839\pi$
$199$	19.5167	1.38350	0.691752	+	0.722135i	$0.256839\pi$
$200$	1.00000	0.0707107
$201$	17.2883	1.21942
$202$	17.3874	1.22338
$203$	1.30628	0.0916829
$204$	8.76602	0.613744
$205$	3.87535	0.270666
$206$	6.86988	0.478647
$207$	−2.37858	−0.165323
$208$	3.50105	0.242754
$209$	0	0
$210$	2.04678	0.141241
$211$	14.3495	0.987861	0.493931	−	0.869501i	$-0.335560\pi$
$211$	14.3495	0.987861	0.493931	+	0.869501i	$0.335560\pi$
$212$	1.09146	0.0749619
$213$	12.0343	0.824576
$214$	1.75025	0.119644
$215$	2.63928	0.179998
$216$	3.70611	0.252169
$217$	−5.70820	−0.387498
$218$	−5.88436	−0.398539
$219$	15.2319	1.02927
$220$	0	0
$221$	−14.9944	−1.00863
$222$	0.670380	0.0449930
$223$	13.2981	0.890504	0.445252	−	0.895405i	$-0.353114\pi$
$223$	13.2981	0.890504	0.445252	+	0.895405i	$0.353114\pi$
$224$	1.00000	0.0668153
$225$	1.18929	0.0792861
$226$	17.1279	1.13933
$227$	18.9140	1.25537	0.627683	−	0.778469i	$-0.284003\pi$
$227$	18.9140	1.25537	0.627683	+	0.778469i	$0.284003\pi$
$228$	−9.10121	−0.602743
$229$	3.89113	0.257133	0.128566	−	0.991701i	$-0.458962\pi$
$229$	3.89113	0.257133	0.128566	+	0.991701i	$0.458962\pi$
$230$	2.00000	0.131876
$231$	0	0
$232$	1.30628	0.0857615
$233$	−6.92431	−0.453627	−0.226813	−	0.973938i	$-0.572831\pi$
$233$	−6.92431	−0.453627	−0.226813	+	0.973938i	$0.572831\pi$
$234$	4.16376	0.272194
$235$	−12.2258	−0.797521
$236$	10.0613	0.654932
$237$	−1.02125	−0.0663372
$238$	−4.28284	−0.277616
$239$	26.6477	1.72370	0.861850	−	0.507164i	$-0.169306\pi$
$239$	26.6477	1.72370	0.861850	+	0.507164i	$0.169306\pi$
$240$	2.04678	0.132119
$241$	0.0928140	0.00597868	0.00298934	−	0.999996i	$-0.499048\pi$
$241$	0.0928140	0.00597868	0.00298934	+	0.999996i	$0.499048\pi$
$242$	0	0
$243$	11.7103	0.751216
$244$	13.0021	0.832374
$245$	−1.00000	−0.0638877
$246$	7.93198	0.505724
$247$	15.5678	0.990554
$248$	−5.70820	−0.362471
$249$	25.8149	1.63595
$250$	−1.00000	−0.0632456
$251$	21.0063	1.32590	0.662952	−	0.748662i	$-0.269303\pi$
$251$	21.0063	1.32590	0.662952	+	0.748662i	$0.269303\pi$
$252$	1.18929	0.0749184
$253$	0	0
$254$	−9.94218	−0.623828
$255$	−8.76602	−0.548950
$256$	1.00000	0.0625000
$257$	−4.23060	−0.263897	−0.131949	−	0.991257i	$-0.542123\pi$
$257$	−4.23060	−0.263897	−0.131949	+	0.991257i	$0.542123\pi$
$258$	5.40202	0.336315
$259$	−0.327530	−0.0203517
$260$	−3.50105	−0.217126
$261$	1.55355	0.0961623
$262$	−7.65112	−0.472688
$263$	16.4789	1.01613	0.508066	−	0.861318i	$-0.330361\pi$
$263$	16.4789	1.01613	0.508066	+	0.861318i	$0.330361\pi$
$264$	0	0
$265$	−1.09146	−0.0670480
$266$	4.44661	0.272639
$267$	−2.85822	−0.174920
$268$	−8.44661	−0.515959
$269$	20.2848	1.23679	0.618394	−	0.785868i	$-0.287784\pi$
$269$	20.2848	1.23679	0.618394	+	0.785868i	$0.287784\pi$
$270$	−3.70611	−0.225547
$271$	13.8443	0.840979	0.420489	−	0.907298i	$-0.361858\pi$
$271$	13.8443	0.840979	0.420489	+	0.907298i	$0.361858\pi$
$272$	−4.28284	−0.259686
$273$	−7.16586	−0.433697
$274$	2.16038	0.130513
$275$	0	0
$276$	4.09355	0.246403
$277$	−8.50369	−0.510937	−0.255469	−	0.966817i	$-0.582230\pi$
$277$	−8.50369	−0.510937	−0.255469	+	0.966817i	$0.582230\pi$
$278$	−21.1314	−1.26738
$279$	−6.78872	−0.406430
$280$	−1.00000	−0.0597614
$281$	2.06245	0.123036	0.0615179	−	0.998106i	$-0.480406\pi$
$281$	2.06245	0.123036	0.0615179	+	0.998106i	$0.480406\pi$
$282$	−25.0234	−1.49012
$283$	27.3214	1.62409	0.812044	−	0.583596i	$-0.198355\pi$
$283$	27.3214	1.62409	0.812044	+	0.583596i	$0.198355\pi$
$284$	−5.87963	−0.348892
$285$	9.10121	0.539109
$286$	0	0
$287$	−3.87535	−0.228755
$288$	1.18929	0.0700797
$289$	1.34275	0.0789854
$290$	−1.30628	−0.0767075
$291$	20.0551	1.17565
$292$	−7.44187	−0.435503
$293$	19.0599	1.11349	0.556746	−	0.830683i	$-0.312050\pi$
$293$	19.0599	1.11349	0.556746	+	0.830683i	$0.312050\pi$
$294$	−2.04678	−0.119370
$295$	−10.0613	−0.585789
$296$	−0.327530	−0.0190373
$297$	0	0
$298$	16.0089	0.927372
$299$	−7.00209	−0.404941
$300$	−2.04678	−0.118171
$301$	−2.63928	−0.152126
$302$	−12.9386	−0.744535
$303$	−35.5882	−2.04449
$304$	4.44661	0.255031
$305$	−13.0021	−0.744498
$306$	−5.09355	−0.291179
$307$	−27.8834	−1.59139	−0.795694	−	0.605699i	$-0.792894\pi$
$307$	−27.8834	−1.59139	−0.795694	+	0.605699i	$0.792894\pi$
$308$	0	0
$309$	−14.0611	−0.799908
$310$	5.70820	0.324204
$311$	21.0336	1.19271	0.596354	−	0.802721i	$-0.296615\pi$
$311$	21.0336	1.19271	0.596354	+	0.802721i	$0.296615\pi$
$312$	−7.16586	−0.405687
$313$	3.94821	0.223166	0.111583	−	0.993755i	$-0.464408\pi$
$313$	3.94821	0.223166	0.111583	+	0.993755i	$0.464408\pi$
$314$	18.4955	1.04376
$315$	−1.18929	−0.0670090
$316$	0.498955	0.0280684
$317$	−28.1225	−1.57952	−0.789759	−	0.613417i	$-0.789794\pi$
$317$	−28.1225	−1.57952	−0.789759	+	0.613417i	$0.789794\pi$
$318$	−2.23398	−0.125275
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	−3.58236	−0.199948
$322$	−2.00000	−0.111456
$323$	−19.0441	−1.05964
$324$	−11.1535	−0.619637
$325$	3.50105	0.194203
$326$	6.52922	0.361620
$327$	12.0440	0.666033
$328$	−3.87535	−0.213980
$329$	12.2258	0.674028
$330$	0	0
$331$	30.2738	1.66400	0.831999	−	0.554777i	$-0.187196\pi$
$331$	30.2738	1.66400	0.831999	+	0.554777i	$0.187196\pi$
$332$	−12.6125	−0.692199
$333$	−0.389529	−0.0213460
$334$	12.0000	0.656611
$335$	8.44661	0.461488
$336$	−2.04678	−0.111661
$337$	−32.0764	−1.74731	−0.873656	−	0.486544i	$-0.838257\pi$
$337$	−32.0764	−1.74731	−0.873656	+	0.486544i	$0.838257\pi$
$338$	−0.742683	−0.0403966
$339$	−35.0570	−1.90404
$340$	4.28284	0.232270
$341$	0	0
$342$	5.28832	0.285959
$343$	1.00000	0.0539949
$344$	−2.63928	−0.142301
$345$	−4.09355	−0.220389
$346$	7.54354	0.405543
$347$	16.6585	0.894275	0.447138	−	0.894465i	$-0.352443\pi$
$347$	16.6585	0.894275	0.447138	+	0.894465i	$0.352443\pi$
$348$	−2.67366	−0.143323
$349$	−5.44944	−0.291702	−0.145851	−	0.989307i	$-0.546592\pi$
$349$	−5.44944	−0.291702	−0.145851	+	0.989307i	$0.546592\pi$
$350$	1.00000	0.0534522
$351$	12.9753	0.692569
$352$	0	0
$353$	12.9302	0.688207	0.344104	−	0.938932i	$-0.388183\pi$
$353$	12.9302	0.688207	0.344104	+	0.938932i	$0.388183\pi$
$354$	−20.5931	−1.09451
$355$	5.87963	0.312058
$356$	1.39645	0.0740117
$357$	8.76602	0.463947
$358$	6.66819	0.352425
$359$	30.5347	1.61156	0.805780	−	0.592216i	$-0.201747\pi$
$359$	30.5347	1.61156	0.805780	+	0.592216i	$0.201747\pi$
$360$	−1.18929	−0.0626812
$361$	0.772330	0.0406490
$362$	−3.26233	−0.171464
$363$	0	0
$364$	3.50105	0.183505
$365$	7.44187	0.389526
$366$	−26.6124	−1.39105
$367$	27.8437	1.45343	0.726715	−	0.686939i	$-0.241046\pi$
$367$	27.8437	1.45343	0.726715	+	0.686939i	$0.241046\pi$
$368$	−2.00000	−0.104257
$369$	−4.60892	−0.239931
$370$	0.327530	0.0170275
$371$	1.09146	0.0566659
$372$	11.6834	0.605757
$373$	−14.5146	−0.751539	−0.375770	−	0.926713i	$-0.622622\pi$
$373$	−14.5146	−0.751539	−0.375770	+	0.926713i	$0.622622\pi$
$374$	0	0
$375$	2.04678	0.105695
$376$	12.2258	0.630496
$377$	4.57335	0.235539
$378$	3.70611	0.190622
$379$	−11.9208	−0.612329	−0.306165	−	0.951979i	$-0.599046\pi$
$379$	−11.9208	−0.612329	−0.306165	+	0.951979i	$0.599046\pi$
$380$	−4.44661	−0.228106
$381$	20.3494	1.04253
$382$	−13.7035	−0.701131
$383$	31.7926	1.62453	0.812264	−	0.583290i	$-0.198235\pi$
$383$	31.7926	1.62453	0.812264	+	0.583290i	$0.198235\pi$
$384$	−2.04678	−0.104449
$385$	0	0
$386$	−11.1467	−0.567352
$387$	−3.13888	−0.159558
$388$	−9.79837	−0.497437
$389$	3.03309	0.153784	0.0768919	−	0.997039i	$-0.475500\pi$
$389$	3.03309	0.153784	0.0768919	+	0.997039i	$0.475500\pi$
$390$	7.16586	0.362857
$391$	8.56569	0.433185
$392$	1.00000	0.0505076
$393$	15.6601	0.789949
$394$	3.51254	0.176959
$395$	−0.498955	−0.0251051
$396$	0	0
$397$	−28.8927	−1.45008	−0.725041	−	0.688706i	$-0.758179\pi$
$397$	−28.8927	−1.45008	−0.725041	+	0.688706i	$0.758179\pi$
$398$	19.5167	0.978285
$399$	−9.10121	−0.455631
$400$	1.00000	0.0500000
$401$	36.2856	1.81202	0.906009	−	0.423258i	$-0.139114\pi$
$401$	36.2856	1.81202	0.906009	+	0.423258i	$0.139114\pi$
$402$	17.2883	0.862263
$403$	−19.9847	−0.995508
$404$	17.3874	0.865057
$405$	11.1535	0.554220
$406$	1.30628	0.0648296
$407$	0	0
$408$	8.76602	0.433983
$409$	−5.94636	−0.294029	−0.147014	−	0.989134i	$-0.546966\pi$
$409$	−5.94636	−0.294029	−0.147014	+	0.989134i	$0.546966\pi$
$410$	3.87535	0.191390
$411$	−4.42182	−0.218112
$412$	6.86988	0.338455
$413$	10.0613	0.495082
$414$	−2.37858	−0.116901
$415$	12.6125	0.619122
$416$	3.50105	0.171653
$417$	43.2512	2.11802
$418$	0	0
$419$	7.10047	0.346881	0.173440	−	0.984844i	$-0.444512\pi$
$419$	7.10047	0.346881	0.173440	+	0.984844i	$0.444512\pi$
$420$	2.04678	0.0998724
$421$	−21.7460	−1.05983	−0.529917	−	0.848050i	$-0.677777\pi$
$421$	−21.7460	−1.05983	−0.529917	+	0.848050i	$0.677777\pi$
$422$	14.3495	0.698524
$423$	14.5400	0.706959
$424$	1.09146	0.0530061
$425$	−4.28284	−0.207748
$426$	12.0343	0.583063
$427$	13.0021	0.629215
$428$	1.75025	0.0846013
$429$	0	0
$430$	2.63928	0.127278
$431$	18.7940	0.905277	0.452638	−	0.891694i	$-0.350483\pi$
$431$	18.7940	0.905277	0.452638	+	0.891694i	$0.350483\pi$
$432$	3.70611	0.178310
$433$	−32.4488	−1.55939	−0.779694	−	0.626161i	$-0.784625\pi$
$433$	−32.4488	−1.55939	−0.779694	+	0.626161i	$0.784625\pi$
$434$	−5.70820	−0.274003
$435$	2.67366	0.128192
$436$	−5.88436	−0.281810
$437$	−8.89322	−0.425420
$438$	15.2319	0.727806
$439$	17.1535	0.818690	0.409345	−	0.912380i	$-0.365757\pi$
$439$	17.1535	0.818690	0.409345	+	0.912380i	$0.365757\pi$
$440$	0	0
$441$	1.18929	0.0566330
$442$	−14.9944	−0.713212
$443$	4.23861	0.201383	0.100691	−	0.994918i	$-0.467895\pi$
$443$	4.23861	0.201383	0.100691	+	0.994918i	$0.467895\pi$
$444$	0.670380	0.0318148
$445$	−1.39645	−0.0661981
$446$	13.2981	0.629682
$447$	−32.7667	−1.54981
$448$	1.00000	0.0472456
$449$	13.1589	0.621008	0.310504	−	0.950572i	$-0.399502\pi$
$449$	13.1589	0.621008	0.310504	+	0.950572i	$0.399502\pi$
$450$	1.18929	0.0560638
$451$	0	0
$452$	17.1279	0.805630
$453$	26.4825	1.24426
$454$	18.9140	0.887678
$455$	−3.50105	−0.164132
$456$	−9.10121	−0.426203
$457$	0.679393	0.0317806	0.0158903	−	0.999874i	$-0.494942\pi$
$457$	0.679393	0.0317806	0.0158903	+	0.999874i	$0.494942\pi$
$458$	3.89113	0.181820
$459$	−15.8727	−0.740875
$460$	2.00000	0.0932505
$461$	33.9310	1.58033	0.790163	−	0.612897i	$-0.209996\pi$
$461$	33.9310	1.58033	0.790163	+	0.612897i	$0.209996\pi$
$462$	0	0
$463$	−8.67985	−0.403387	−0.201693	−	0.979449i	$-0.564644\pi$
$463$	−8.67985	−0.403387	−0.201693	+	0.979449i	$0.564644\pi$
$464$	1.30628	0.0606426
$465$	−11.6834	−0.541805
$466$	−6.92431	−0.320763
$467$	4.58156	0.212009	0.106005	−	0.994366i	$-0.466194\pi$
$467$	4.58156	0.212009	0.106005	+	0.994366i	$0.466194\pi$
$468$	4.16376	0.192470
$469$	−8.44661	−0.390028
$470$	−12.2258	−0.563932
$471$	−37.8561	−1.74432
$472$	10.0613	0.463107
$473$	0	0
$474$	−1.02125	−0.0469075
$475$	4.44661	0.204024
$476$	−4.28284	−0.196304
$477$	1.29807	0.0594344
$478$	26.6477	1.21884
$479$	−38.8930	−1.77707	−0.888534	−	0.458811i	$-0.848275\pi$
$479$	−38.8930	−1.77707	−0.888534	+	0.458811i	$0.848275\pi$
$480$	2.04678	0.0934221
$481$	−1.14670	−0.0522849
$482$	0.0928140	0.00422756
$483$	4.09355	0.186263
$484$	0	0
$485$	9.79837	0.444921
$486$	11.7103	0.531190
$487$	19.7593	0.895377	0.447689	−	0.894189i	$-0.352247\pi$
$487$	19.7593	0.895377	0.447689	+	0.894189i	$0.352247\pi$
$488$	13.0021	0.588577
$489$	−13.3638	−0.604334
$490$	−1.00000	−0.0451754
$491$	30.7082	1.38584	0.692920	−	0.721014i	$-0.256324\pi$
$491$	30.7082	1.38584	0.692920	+	0.721014i	$0.256324\pi$
$492$	7.93198	0.357601
$493$	−5.59460	−0.251968
$494$	15.5678	0.700427
$495$	0	0
$496$	−5.70820	−0.256306
$497$	−5.87963	−0.263737
$498$	25.8149	1.15679
$499$	−18.5980	−0.832563	−0.416281	−	0.909236i	$-0.636667\pi$
$499$	−18.5980	−0.832563	−0.416281	+	0.909236i	$0.636667\pi$
$500$	−1.00000	−0.0447214
$501$	−24.5613	−1.09732
$502$	21.0063	0.937556
$503$	1.53678	0.0685216	0.0342608	−	0.999413i	$-0.489092\pi$
$503$	1.53678	0.0685216	0.0342608	+	0.999413i	$0.489092\pi$
$504$	1.18929	0.0529753
$505$	−17.3874	−0.773731
$506$	0	0
$507$	1.52011	0.0675103
$508$	−9.94218	−0.441113
$509$	−19.0488	−0.844322	−0.422161	−	0.906521i	$-0.638728\pi$
$509$	−19.0488	−0.844322	−0.422161	+	0.906521i	$0.638728\pi$
$510$	−8.76602	−0.388166
$511$	−7.44187	−0.329209
$512$	1.00000	0.0441942
$513$	16.4796	0.727594
$514$	−4.23060	−0.186604
$515$	−6.86988	−0.302723
$516$	5.40202	0.237811
$517$	0	0
$518$	−0.327530	−0.0143908
$519$	−15.4399	−0.677738
$520$	−3.50105	−0.153531
$521$	18.1769	0.796344	0.398172	−	0.917311i	$-0.369645\pi$
$521$	18.1769	0.796344	0.398172	+	0.917311i	$0.369645\pi$
$522$	1.55355	0.0679970
$523$	−17.2935	−0.756192	−0.378096	−	0.925766i	$-0.623421\pi$
$523$	−17.2935	−0.756192	−0.378096	+	0.925766i	$0.623421\pi$
$524$	−7.65112	−0.334241
$525$	−2.04678	−0.0893286
$526$	16.4789	0.718514
$527$	24.4473	1.06494
$528$	0	0
$529$	−19.0000	−0.826087
$530$	−1.09146	−0.0474101
$531$	11.9658	0.519270
$532$	4.44661	0.192785
$533$	−13.5678	−0.587686
$534$	−2.85822	−0.123687
$535$	−1.75025	−0.0756697
$536$	−8.44661	−0.364838
$537$	−13.6483	−0.588967
$538$	20.2848	0.874541
$539$	0	0
$540$	−3.70611	−0.159486
$541$	24.4783	1.05241	0.526203	−	0.850359i	$-0.323615\pi$
$541$	24.4783	1.05241	0.526203	+	0.850359i	$0.323615\pi$
$542$	13.8443	0.594662
$543$	6.67726	0.286549
$544$	−4.28284	−0.183625
$545$	5.88436	0.252058
$546$	−7.16586	−0.306670
$547$	39.8203	1.70259	0.851297	−	0.524684i	$-0.175817\pi$
$547$	39.8203	1.70259	0.851297	+	0.524684i	$0.175817\pi$
$548$	2.16038	0.0922870
$549$	15.4633	0.659957
$550$	0	0
$551$	5.80852	0.247451
$552$	4.09355	0.174233
$553$	0.498955	0.0212177
$554$	−8.50369	−0.361287
$555$	−0.670380	−0.0284560
$556$	−21.1314	−0.896170
$557$	22.9552	0.972644	0.486322	−	0.873780i	$-0.338338\pi$
$557$	22.9552	0.972644	0.486322	+	0.873780i	$0.338338\pi$
$558$	−6.78872	−0.287389
$559$	−9.24025	−0.390821
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	2.06245	0.0869994
$563$	−15.9140	−0.670696	−0.335348	−	0.942094i	$-0.608854\pi$
$563$	−15.9140	−0.670696	−0.335348	+	0.942094i	$0.608854\pi$
$564$	−25.0234	−1.05368
$565$	−17.1279	−0.720578
$566$	27.3214	1.14840
$567$	−11.1535	−0.468401
$568$	−5.87963	−0.246704
$569$	17.7349	0.743484	0.371742	−	0.928336i	$-0.378761\pi$
$569$	17.7349	0.743484	0.371742	+	0.928336i	$0.378761\pi$
$570$	9.10121	0.381208
$571$	1.05280	0.0440584	0.0220292	−	0.999757i	$-0.492987\pi$
$571$	1.05280	0.0440584	0.0220292	+	0.999757i	$0.492987\pi$
$572$	0	0
$573$	28.0479	1.17172
$574$	−3.87535	−0.161754
$575$	−2.00000	−0.0834058
$576$	1.18929	0.0495538
$577$	−12.8223	−0.533798	−0.266899	−	0.963725i	$-0.585999\pi$
$577$	−12.8223	−0.533798	−0.266899	+	0.963725i	$0.585999\pi$
$578$	1.34275	0.0558511
$579$	22.8148	0.948150
$580$	−1.30628	−0.0542404
$581$	−12.6125	−0.523253
$582$	20.0551	0.831310
$583$	0	0
$584$	−7.44187	−0.307947
$585$	−4.16376	−0.172150
$586$	19.0599	0.787358
$587$	−9.76035	−0.402853	−0.201426	−	0.979504i	$-0.564558\pi$
$587$	−9.76035	−0.402853	−0.201426	+	0.979504i	$0.564558\pi$
$588$	−2.04678	−0.0844076
$589$	−25.3822	−1.04585
$590$	−10.0613	−0.414216
$591$	−7.18939	−0.295732
$592$	−0.327530	−0.0134614
$593$	25.4142	1.04364	0.521818	−	0.853057i	$-0.325254\pi$
$593$	25.4142	1.04364	0.521818	+	0.853057i	$0.325254\pi$
$594$	0	0
$595$	4.28284	0.175579
$596$	16.0089	0.655751
$597$	−39.9464	−1.63490
$598$	−7.00209	−0.286337
$599$	−41.3324	−1.68879	−0.844397	−	0.535718i	$-0.820041\pi$
$599$	−41.3324	−1.68879	−0.844397	+	0.535718i	$0.820041\pi$
$600$	−2.04678	−0.0835593
$601$	39.2422	1.60072	0.800361	−	0.599518i	$-0.204641\pi$
$601$	39.2422	1.60072	0.800361	+	0.599518i	$0.204641\pi$
$602$	−2.63928	−0.107569
$603$	−10.0455	−0.409084
$604$	−12.9386	−0.526466
$605$	0	0
$606$	−35.5882	−1.44567
$607$	7.82091	0.317441	0.158721	−	0.987324i	$-0.449263\pi$
$607$	7.82091	0.317441	0.158721	+	0.987324i	$0.449263\pi$
$608$	4.44661	0.180334
$609$	−2.67366	−0.108342
$610$	−13.0021	−0.526439
$611$	42.8029	1.73162
$612$	−5.09355	−0.205895
$613$	1.22930	0.0496511	0.0248256	−	0.999692i	$-0.492097\pi$
$613$	1.22930	0.0496511	0.0248256	+	0.999692i	$0.492097\pi$
$614$	−27.8834	−1.12528
$615$	−7.93198	−0.319848
$616$	0	0
$617$	−8.43129	−0.339431	−0.169715	−	0.985493i	$-0.554285\pi$
$617$	−8.43129	−0.339431	−0.169715	+	0.985493i	$0.554285\pi$
$618$	−14.0611	−0.565620
$619$	7.19522	0.289200	0.144600	−	0.989490i	$-0.453810\pi$
$619$	7.19522	0.289200	0.144600	+	0.989490i	$0.453810\pi$
$620$	5.70820	0.229247
$621$	−7.41223	−0.297442
$622$	21.0336	0.843372
$623$	1.39645	0.0559476
$624$	−7.16586	−0.286864
$625$	1.00000	0.0400000
$626$	3.94821	0.157802
$627$	0	0
$628$	18.4955	0.738050
$629$	1.40276	0.0559317
$630$	−1.18929	−0.0473825
$631$	9.16167	0.364721	0.182360	−	0.983232i	$-0.441626\pi$
$631$	9.16167	0.364721	0.182360	+	0.983232i	$0.441626\pi$
$632$	0.498955	0.0198474
$633$	−29.3702	−1.16736
$634$	−28.1225	−1.11689
$635$	9.94218	0.394543
$636$	−2.23398	−0.0885830
$637$	3.50105	0.138716
$638$	0	0
$639$	−6.99260	−0.276623
$640$	−1.00000	−0.0395285
$641$	−7.82633	−0.309121	−0.154561	−	0.987983i	$-0.549396\pi$
$641$	−7.82633	−0.309121	−0.154561	+	0.987983i	$0.549396\pi$
$642$	−3.58236	−0.141385
$643$	−42.1180	−1.66097	−0.830487	−	0.557038i	$-0.811938\pi$
$643$	−42.1180	−1.66097	−0.830487	+	0.557038i	$0.811938\pi$
$644$	−2.00000	−0.0788110
$645$	−5.40202	−0.212704
$646$	−19.0441	−0.749281
$647$	−20.5480	−0.807825	−0.403913	−	0.914798i	$-0.632350\pi$
$647$	−20.5480	−0.807825	−0.403913	+	0.914798i	$0.632350\pi$
$648$	−11.1535	−0.438149
$649$	0	0
$650$	3.50105	0.137322
$651$	11.6834	0.457909
$652$	6.52922	0.255704
$653$	−29.0694	−1.13757	−0.568786	−	0.822485i	$-0.692587\pi$
$653$	−29.0694	−1.13757	−0.568786	+	0.822485i	$0.692587\pi$
$654$	12.0440	0.470957
$655$	7.65112	0.298954
$656$	−3.87535	−0.151307
$657$	−8.85056	−0.345293
$658$	12.2258	0.476610
$659$	16.3882	0.638395	0.319198	−	0.947688i	$-0.396587\pi$
$659$	16.3882	0.638395	0.319198	+	0.947688i	$0.396587\pi$
$660$	0	0
$661$	37.7386	1.46786	0.733932	−	0.679223i	$-0.237683\pi$
$661$	37.7386	1.46786	0.733932	+	0.679223i	$0.237683\pi$
$662$	30.2738	1.17662
$663$	30.6902	1.19191
$664$	−12.6125	−0.489459
$665$	−4.44661	−0.172432
$666$	−0.389529	−0.0150939
$667$	−2.61256	−0.101159
$668$	12.0000	0.464294
$669$	−27.2182	−1.05232
$670$	8.44661	0.326321
$671$	0	0
$672$	−2.04678	−0.0789561
$673$	21.0258	0.810486	0.405243	−	0.914209i	$-0.367187\pi$
$673$	21.0258	0.810486	0.405243	+	0.914209i	$0.367187\pi$
$674$	−32.0764	−1.23554
$675$	3.70611	0.142648
$676$	−0.742683	−0.0285647
$677$	−6.89196	−0.264880	−0.132440	−	0.991191i	$-0.542281\pi$
$677$	−6.89196	−0.264880	−0.132440	+	0.991191i	$0.542281\pi$
$678$	−35.0570	−1.34636
$679$	−9.79837	−0.376027
$680$	4.28284	0.164240
$681$	−38.7127	−1.48348
$682$	0	0
$683$	−1.37212	−0.0525026	−0.0262513	−	0.999655i	$-0.508357\pi$
$683$	−1.37212	−0.0525026	−0.0262513	+	0.999655i	$0.508357\pi$
$684$	5.28832	0.202204
$685$	−2.16038	−0.0825440
$686$	1.00000	0.0381802
$687$	−7.96427	−0.303856
$688$	−2.63928	−0.100622
$689$	3.82126	0.145578
$690$	−4.09355	−0.155839
$691$	−9.39673	−0.357469	−0.178734	−	0.983897i	$-0.557200\pi$
$691$	−9.39673	−0.357469	−0.178734	+	0.983897i	$0.557200\pi$
$692$	7.54354	0.286762
$693$	0	0
$694$	16.6585	0.632348
$695$	21.1314	0.801559
$696$	−2.67366	−0.101345
$697$	16.5975	0.628676
$698$	−5.44944	−0.206264
$699$	14.1725	0.536054
$700$	1.00000	0.0377964
$701$	13.5330	0.511133	0.255566	−	0.966792i	$-0.417738\pi$
$701$	13.5330	0.511133	0.255566	+	0.966792i	$0.417738\pi$
$702$	12.9753	0.489720
$703$	−1.45640	−0.0549290
$704$	0	0
$705$	25.0234	0.942436
$706$	12.9302	0.486636
$707$	17.3874	0.653922
$708$	−20.5931	−0.773938
$709$	−40.1756	−1.50883	−0.754413	−	0.656400i	$-0.772078\pi$
$709$	−40.1756	−1.50883	−0.754413	+	0.656400i	$0.772078\pi$
$710$	5.87963	0.220659
$711$	0.593403	0.0222544
$712$	1.39645	0.0523342
$713$	11.4164	0.427548
$714$	8.76602	0.328060
$715$	0	0
$716$	6.66819	0.249202
$717$	−54.5420	−2.03691
$718$	30.5347	1.13954
$719$	13.1839	0.491677	0.245838	−	0.969311i	$-0.420937\pi$
$719$	13.1839	0.491677	0.245838	+	0.969311i	$0.420937\pi$
$720$	−1.18929	−0.0443223
$721$	6.86988	0.255848
$722$	0.772330	0.0287432
$723$	−0.189970	−0.00706504
$724$	−3.26233	−0.121244
$725$	1.30628	0.0485141
$726$	0	0
$727$	−40.5403	−1.50356	−0.751779	−	0.659415i	$-0.770804\pi$
$727$	−40.5403	−1.50356	−0.751779	+	0.659415i	$0.770804\pi$
$728$	3.50105	0.129757
$729$	9.49203	0.351557
$730$	7.44187	0.275436
$731$	11.3036	0.418080
$732$	−26.6124	−0.983621
$733$	22.8799	0.845089	0.422545	−	0.906342i	$-0.361137\pi$
$733$	22.8799	0.845089	0.422545	+	0.906342i	$0.361137\pi$
$734$	27.8437	1.02773
$735$	2.04678	0.0754965
$736$	−2.00000	−0.0737210
$737$	0	0
$738$	−4.60892	−0.169657
$739$	30.1038	1.10738	0.553692	−	0.832721i	$-0.313218\pi$
$739$	30.1038	1.10738	0.553692	+	0.832721i	$0.313218\pi$
$740$	0.327530	0.0120402
$741$	−31.8638	−1.17054
$742$	1.09146	0.0400688
$743$	−29.6655	−1.08832	−0.544161	−	0.838981i	$-0.683152\pi$
$743$	−29.6655	−1.08832	−0.544161	+	0.838981i	$0.683152\pi$
$744$	11.6834	0.428335
$745$	−16.0089	−0.586521
$746$	−14.5146	−0.531418
$747$	−14.9999	−0.548818
$748$	0	0
$749$	1.75025	0.0639526
$750$	2.04678	0.0747377
$751$	47.4328	1.73085	0.865423	−	0.501041i	$-0.167049\pi$
$751$	47.4328	1.73085	0.865423	+	0.501041i	$0.167049\pi$
$752$	12.2258	0.445828
$753$	−42.9951	−1.56683
$754$	4.57335	0.166552
$755$	12.9386	0.470885
$756$	3.70611	0.134790
$757$	−26.5518	−0.965043	−0.482522	−	0.875884i	$-0.660279\pi$
$757$	−26.5518	−0.965043	−0.482522	+	0.875884i	$0.660279\pi$
$758$	−11.9208	−0.432982
$759$	0	0
$760$	−4.44661	−0.161295
$761$	48.7468	1.76707	0.883534	−	0.468366i	$-0.155157\pi$
$761$	48.7468	1.76707	0.883534	+	0.468366i	$0.155157\pi$
$762$	20.3494	0.737182
$763$	−5.88436	−0.213028
$764$	−13.7035	−0.495774
$765$	5.09355	0.184158
$766$	31.7926	1.14871
$767$	35.2249	1.27190
$768$	−2.04678	−0.0738567
$769$	25.8289	0.931415	0.465708	−	0.884939i	$-0.345800\pi$
$769$	25.8289	0.931415	0.465708	+	0.884939i	$0.345800\pi$
$770$	0	0
$771$	8.65908	0.311849
$772$	−11.1467	−0.401178
$773$	38.6257	1.38927	0.694634	−	0.719363i	$-0.255566\pi$
$773$	38.6257	1.38927	0.694634	+	0.719363i	$0.255566\pi$
$774$	−3.13888	−0.112825
$775$	−5.70820	−0.205045
$776$	−9.79837	−0.351741
$777$	0.670380	0.0240497
$778$	3.03309	0.108742
$779$	−17.2322	−0.617407
$780$	7.16586	0.256579
$781$	0	0
$782$	8.56569	0.306308
$783$	4.84123	0.173011
$784$	1.00000	0.0357143
$785$	−18.4955	−0.660132
$786$	15.6601	0.558579
$787$	−10.5712	−0.376821	−0.188411	−	0.982090i	$-0.560334\pi$
$787$	−10.5712	−0.376821	−0.188411	+	0.982090i	$0.560334\pi$
$788$	3.51254	0.125129
$789$	−33.7286	−1.20077
$790$	−0.498955	−0.0177520
$791$	17.1279	0.608999
$792$	0	0
$793$	45.5209	1.61649
$794$	−28.8927	−1.02536
$795$	2.23398	0.0792310
$796$	19.5167	0.691752
$797$	−40.3031	−1.42761	−0.713805	−	0.700345i	$-0.753029\pi$
$797$	−40.3031	−1.42761	−0.713805	+	0.700345i	$0.753029\pi$
$798$	−9.10121	−0.322179
$799$	−52.3610	−1.85240
$800$	1.00000	0.0353553
$801$	1.66079	0.0586810
$802$	36.2856	1.28129
$803$	0	0
$804$	17.2883	0.609712
$805$	2.00000	0.0704907
$806$	−19.9847	−0.703930
$807$	−41.5185	−1.46152
$808$	17.3874	0.611688
$809$	−14.7984	−0.520283	−0.260142	−	0.965570i	$-0.583769\pi$
$809$	−14.7984	−0.520283	−0.260142	+	0.965570i	$0.583769\pi$
$810$	11.1535	0.391893
$811$	30.8043	1.08169	0.540843	−	0.841124i	$-0.318105\pi$
$811$	30.8043	1.08169	0.540843	+	0.841124i	$0.318105\pi$
$812$	1.30628	0.0458415
$813$	−28.3361	−0.993790
$814$	0	0
$815$	−6.52922	−0.228708
$816$	8.76602	0.306872
$817$	−11.7359	−0.410586
$818$	−5.94636	−0.207910
$819$	4.16376	0.145494
$820$	3.87535	0.135333
$821$	−26.5412	−0.926296	−0.463148	−	0.886281i	$-0.653280\pi$
$821$	−26.5412	−0.926296	−0.463148	+	0.886281i	$0.653280\pi$
$822$	−4.42182	−0.154229
$823$	39.8431	1.38884	0.694422	−	0.719568i	$-0.255660\pi$
$823$	39.8431	1.38884	0.694422	+	0.719568i	$0.255660\pi$
$824$	6.86988	0.239324
$825$	0	0
$826$	10.0613	0.350076
$827$	−36.6888	−1.27579	−0.637896	−	0.770122i	$-0.720195\pi$
$827$	−36.6888	−1.27579	−0.637896	+	0.770122i	$0.720195\pi$
$828$	−2.37858	−0.0826615
$829$	16.8821	0.586339	0.293169	−	0.956061i	$-0.405290\pi$
$829$	16.8821	0.586339	0.293169	+	0.956061i	$0.405290\pi$
$830$	12.6125	0.437785
$831$	17.4051	0.603778
$832$	3.50105	0.121377
$833$	−4.28284	−0.148392
$834$	43.2512	1.49767
$835$	−12.0000	−0.415277
$836$	0	0
$837$	−21.1553	−0.731233
$838$	7.10047	0.245282
$839$	−40.8091	−1.40889	−0.704444	−	0.709760i	$-0.748804\pi$
$839$	−40.8091	−1.40889	−0.704444	+	0.709760i	$0.748804\pi$
$840$	2.04678	0.0706205
$841$	−27.2936	−0.941160
$842$	−21.7460	−0.749416
$843$	−4.22138	−0.145392
$844$	14.3495	0.493931
$845$	0.742683	0.0255491
$846$	14.5400	0.499896
$847$	0	0
$848$	1.09146	0.0374810
$849$	−55.9208	−1.91920
$850$	−4.28284	−0.146900
$851$	0.655059	0.0224552
$852$	12.0343	0.412288
$853$	0.0955818	0.00327266	0.00163633	−	0.999999i	$-0.499479\pi$
$853$	0.0955818	0.00327266	0.00163633	+	0.999999i	$0.499479\pi$
$854$	13.0021	0.444922
$855$	−5.28832	−0.180857
$856$	1.75025	0.0598222
$857$	−51.9594	−1.77490	−0.887449	−	0.460906i	$-0.847525\pi$
$857$	−51.9594	−1.77490	−0.887449	+	0.460906i	$0.847525\pi$
$858$	0	0
$859$	35.1186	1.19823	0.599115	−	0.800663i	$-0.295519\pi$
$859$	35.1186	1.19823	0.599115	+	0.800663i	$0.295519\pi$
$860$	2.63928	0.0899988
$861$	7.93198	0.270321
$862$	18.7940	0.640127
$863$	−23.0325	−0.784037	−0.392018	−	0.919957i	$-0.628223\pi$
$863$	−23.0325	−0.784037	−0.392018	+	0.919957i	$0.628223\pi$
$864$	3.70611	0.126085
$865$	−7.54354	−0.256488
$866$	−32.4488	−1.10265
$867$	−2.74831	−0.0933376
$868$	−5.70820	−0.193749
$869$	0	0
$870$	2.67366	0.0906457
$871$	−29.5720	−1.00201
$872$	−5.88436	−0.199270
$873$	−11.6531	−0.394399
$874$	−8.89322	−0.300818
$875$	−1.00000	−0.0338062
$876$	15.2319	0.514637
$877$	30.7254	1.03752	0.518762	−	0.854919i	$-0.326393\pi$
$877$	30.7254	1.03752	0.518762	+	0.854919i	$0.326393\pi$
$878$	17.1535	0.578901
$879$	−39.0114	−1.31582
$880$	0	0
$881$	−42.3788	−1.42778	−0.713890	−	0.700258i	$-0.753068\pi$
$881$	−42.3788	−1.42778	−0.713890	+	0.700258i	$0.753068\pi$
$882$	1.18929	0.0400455
$883$	38.3269	1.28980	0.644901	−	0.764266i	$-0.276899\pi$
$883$	38.3269	1.28980	0.644901	+	0.764266i	$0.276899\pi$
$884$	−14.9944	−0.504317
$885$	20.5931	0.692231
$886$	4.23861	0.142399
$887$	25.5816	0.858948	0.429474	−	0.903079i	$-0.358699\pi$
$887$	25.5816	0.858948	0.429474	+	0.903079i	$0.358699\pi$
$888$	0.670380	0.0224965
$889$	−9.94218	−0.333450
$890$	−1.39645	−0.0468091
$891$	0	0
$892$	13.2981	0.445252
$893$	54.3632	1.81919
$894$	−32.7667	−1.09588
$895$	−6.66819	−0.222893
$896$	1.00000	0.0334077
$897$	14.3317	0.478522
$898$	13.1589	0.439119
$899$	−7.45652	−0.248689
$900$	1.18929	0.0396431
$901$	−4.67456	−0.155732
$902$	0	0
$903$	5.40202	0.179768
$904$	17.1279	0.569667
$905$	3.26233	0.108444
$906$	26.4825	0.879822
$907$	−49.7760	−1.65279	−0.826393	−	0.563094i	$-0.809611\pi$
$907$	−49.7760	−1.65279	−0.826393	+	0.563094i	$0.809611\pi$
$908$	18.9140	0.627683
$909$	20.6787	0.685871
$910$	−3.50105	−0.116059
$911$	22.3261	0.739696	0.369848	−	0.929092i	$-0.379410\pi$
$911$	22.3261	0.739696	0.369848	+	0.929092i	$0.379410\pi$
$912$	−9.10121	−0.301371
$913$	0	0
$914$	0.679393	0.0224723
$915$	26.6124	0.879778
$916$	3.89113	0.128566
$917$	−7.65112	−0.252662
$918$	−15.8727	−0.523877
$919$	17.9670	0.592677	0.296338	−	0.955083i	$-0.404234\pi$
$919$	17.9670	0.592677	0.296338	+	0.955083i	$0.404234\pi$
$920$	2.00000	0.0659380
$921$	57.0710	1.88055
$922$	33.9310	1.11746
$923$	−20.5848	−0.677558
$924$	0	0
$925$	−0.327530	−0.0107691
$926$	−8.67985	−0.285238
$927$	8.17029	0.268348
$928$	1.30628	0.0428808
$929$	13.7491	0.451092	0.225546	−	0.974233i	$-0.427583\pi$
$929$	13.7491	0.451092	0.225546	+	0.974233i	$0.427583\pi$
$930$	−11.6834	−0.383114
$931$	4.44661	0.145732
$932$	−6.92431	−0.226813
$933$	−43.0512	−1.40943
$934$	4.58156	0.149913
$935$	0	0
$936$	4.16376	0.136097
$937$	8.38824	0.274032	0.137016	−	0.990569i	$-0.456249\pi$
$937$	8.38824	0.274032	0.137016	+	0.990569i	$0.456249\pi$
$938$	−8.44661	−0.275792
$939$	−8.08110	−0.263717
$940$	−12.2258	−0.398760
$941$	28.1650	0.918153	0.459077	−	0.888397i	$-0.348180\pi$
$941$	28.1650	0.918153	0.459077	+	0.888397i	$0.348180\pi$
$942$	−37.8561	−1.23342
$943$	7.75070	0.252398
$944$	10.0613	0.327466
$945$	−3.70611	−0.120560
$946$	0	0
$947$	−0.901036	−0.0292797	−0.0146399	−	0.999893i	$-0.504660\pi$
$947$	−0.901036	−0.0292797	−0.0146399	+	0.999893i	$0.504660\pi$
$948$	−1.02125	−0.0331686
$949$	−26.0543	−0.845760
$950$	4.44661	0.144267
$951$	57.5605	1.86653
$952$	−4.28284	−0.138808
$953$	5.45845	0.176817	0.0884083	−	0.996084i	$-0.471822\pi$
$953$	5.45845	0.176817	0.0884083	+	0.996084i	$0.471822\pi$
$954$	1.29807	0.0420265
$955$	13.7035	0.443434
$956$	26.6477	0.861850
$957$	0	0
$958$	−38.8930	−1.25658
$959$	2.16038	0.0697624
$960$	2.04678	0.0660594
$961$	1.58359	0.0510836
$962$	−1.14670	−0.0369710
$963$	2.08155	0.0670771
$964$	0.0928140	0.00298934
$965$	11.1467	0.358825
$966$	4.09355	0.131708
$967$	36.7612	1.18216	0.591079	−	0.806613i	$-0.298702\pi$
$967$	36.7612	1.18216	0.591079	+	0.806613i	$0.298702\pi$
$968$	0	0
$969$	38.9791	1.25219
$970$	9.79837	0.314607
$971$	−47.2855	−1.51746	−0.758731	−	0.651404i	$-0.774180\pi$
$971$	−47.2855	−1.51746	−0.758731	+	0.651404i	$0.774180\pi$
$972$	11.7103	0.375608
$973$	−21.1314	−0.677441
$974$	19.7593	0.633127
$975$	−7.16586	−0.229491
$976$	13.0021	0.416187
$977$	−44.3011	−1.41732	−0.708658	−	0.705552i	$-0.750699\pi$
$977$	−44.3011	−1.41732	−0.708658	+	0.705552i	$0.750699\pi$
$978$	−13.3638	−0.427329
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−6.99823	−0.223436
$982$	30.7082	0.979937
$983$	−2.98404	−0.0951761	−0.0475880	−	0.998867i	$-0.515153\pi$
$983$	−2.98404	−0.0951761	−0.0475880	+	0.998867i	$0.515153\pi$
$984$	7.93198	0.252862
$985$	−3.51254	−0.111919
$986$	−5.59460	−0.178168
$987$	−25.0234	−0.796504
$988$	15.5678	0.495277
$989$	5.27857	0.167849
$990$	0	0
$991$	−47.4047	−1.50586	−0.752930	−	0.658100i	$-0.771360\pi$
$991$	−47.4047	−1.50586	−0.752930	+	0.658100i	$0.771360\pi$
$992$	−5.70820	−0.181236
$993$	−61.9637	−1.96636
$994$	−5.87963	−0.186491
$995$	−19.5167	−0.618722
$996$	25.8149	0.817976
$997$	−16.7474	−0.530394	−0.265197	−	0.964194i	$-0.585437\pi$
$997$	−16.7474	−0.530394	−0.265197	+	0.964194i	$0.585437\pi$
$998$	−18.5980	−0.588711
$999$	−1.21386	−0.0384049

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.ct.1.1		4
11.2	odd	10		770.2.n.e.631.1	yes	8
11.6	odd	10		770.2.n.e.421.1	✓	8
11.10	odd	2		8470.2.a.cq.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.e.421.1	✓	8	11.6	odd	10
770.2.n.e.631.1	yes	8	11.2	odd	10
8470.2.a.cq.1.1		4	11.10	odd	2
8470.2.a.ct.1.1		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:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.04678 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.04678 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.18929 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.04678 q^{3} +1.00000 q^{4} -1.00000 q^{5} -2.04678 q^{6} +1.00000 q^{7} +1.00000 q^{8} +1.18929 q^{9} -1.00000 q^{10} -2.04678 q^{12} +3.50105 q^{13} +1.00000 q^{14} +2.04678 q^{15} +1.00000 q^{16} -4.28284 q^{17} +1.18929 q^{18} +4.44661 q^{19} -1.00000 q^{20} -2.04678 q^{21} -2.00000 q^{23} -2.04678 q^{24} +1.00000 q^{25} +3.50105 q^{26} +3.70611 q^{27} +1.00000 q^{28} +1.30628 q^{29} +2.04678 q^{30} -5.70820 q^{31} +1.00000 q^{32} -4.28284 q^{34} -1.00000 q^{35} +1.18929 q^{36} -0.327530 q^{37} +4.44661 q^{38} -7.16586 q^{39} -1.00000 q^{40} -3.87535 q^{41} -2.04678 q^{42} -2.63928 q^{43} -1.18929 q^{45} -2.00000 q^{46} +12.2258 q^{47} -2.04678 q^{48} +1.00000 q^{49} +1.00000 q^{50} +8.76602 q^{51} +3.50105 q^{52} +1.09146 q^{53} +3.70611 q^{54} +1.00000 q^{56} -9.10121 q^{57} +1.30628 q^{58} +10.0613 q^{59} +2.04678 q^{60} +13.0021 q^{61} -5.70820 q^{62} +1.18929 q^{63} +1.00000 q^{64} -3.50105 q^{65} -8.44661 q^{67} -4.28284 q^{68} +4.09355 q^{69} -1.00000 q^{70} -5.87963 q^{71} +1.18929 q^{72} -7.44187 q^{73} -0.327530 q^{74} -2.04678 q^{75} +4.44661 q^{76} -7.16586 q^{78} +0.498955 q^{79} -1.00000 q^{80} -11.1535 q^{81} -3.87535 q^{82} -12.6125 q^{83} -2.04678 q^{84} +4.28284 q^{85} -2.63928 q^{86} -2.67366 q^{87} +1.39645 q^{89} -1.18929 q^{90} +3.50105 q^{91} -2.00000 q^{92} +11.6834 q^{93} +12.2258 q^{94} -4.44661 q^{95} -2.04678 q^{96} -9.79837 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} + 4 q^{7} + 4 q^{8} + 6 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 + 4 * q^7 + 4 * q^8 + 6 * q^9	\( 4 q + 4 q^{2} + 2 q^{3} + 4 q^{4} - 4 q^{5} + 2 q^{6} + 4 q^{7} + 4 q^{8} + 6 q^{9} - 4 q^{10} + 2 q^{12} + q^{13} + 4 q^{14} - 2 q^{15} + 4 q^{16} + 2 q^{17} + 6 q^{18} - 3 q^{19} - 4 q^{20} + 2 q^{21} - 8 q^{23} + 2 q^{24} + 4 q^{25} + q^{26} + 14 q^{27} + 4 q^{28} + 15 q^{29} - 2 q^{30} + 4 q^{31} + 4 q^{32} + 2 q^{34} - 4 q^{35} + 6 q^{36} + 2 q^{37} - 3 q^{38} - q^{39} - 4 q^{40} + 11 q^{41} + 2 q^{42} + 7 q^{43} - 6 q^{45} - 8 q^{46} + 5 q^{47} + 2 q^{48} + 4 q^{49} + 4 q^{50} + 18 q^{51} + q^{52} + 10 q^{53} + 14 q^{54} + 4 q^{56} - 34 q^{57} + 15 q^{58} + 13 q^{59} - 2 q^{60} + 26 q^{61} + 4 q^{62} + 6 q^{63} + 4 q^{64} - q^{65} - 13 q^{67} + 2 q^{68} - 4 q^{69} - 4 q^{70} - 13 q^{71} + 6 q^{72} - 18 q^{73} + 2 q^{74} + 2 q^{75} - 3 q^{76} - q^{78} + 15 q^{79} - 4 q^{80} - 8 q^{81} + 11 q^{82} - 2 q^{83} + 2 q^{84} - 2 q^{85} + 7 q^{86} + 26 q^{87} - 7 q^{89} - 6 q^{90} + q^{91} - 8 q^{92} + 2 q^{93} + 5 q^{94} + 3 q^{95} + 2 q^{96} + 10 q^{97} + 4 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 2 * q^3 + 4 * q^4 - 4 * q^5 + 2 * q^6 + 4 * q^7 + 4 * q^8 + 6 * q^9 - 4 * q^10 + 2 * q^12 + q^13 + 4 * q^14 - 2 * q^15 + 4 * q^16 + 2 * q^17 + 6 * q^18 - 3 * q^19 - 4 * q^20 + 2 * q^21 - 8 * q^23 + 2 * q^24 + 4 * q^25 + q^26 + 14 * q^27 + 4 * q^28 + 15 * q^29 - 2 * q^30 + 4 * q^31 + 4 * q^32 + 2 * q^34 - 4 * q^35 + 6 * q^36 + 2 * q^37 - 3 * q^38 - q^39 - 4 * q^40 + 11 * q^41 + 2 * q^42 + 7 * q^43 - 6 * q^45 - 8 * q^46 + 5 * q^47 + 2 * q^48 + 4 * q^49 + 4 * q^50 + 18 * q^51 + q^52 + 10 * q^53 + 14 * q^54 + 4 * q^56 - 34 * q^57 + 15 * q^58 + 13 * q^59 - 2 * q^60 + 26 * q^61 + 4 * q^62 + 6 * q^63 + 4 * q^64 - q^65 - 13 * q^67 + 2 * q^68 - 4 * q^69 - 4 * q^70 - 13 * q^71 + 6 * q^72 - 18 * q^73 + 2 * q^74 + 2 * q^75 - 3 * q^76 - q^78 + 15 * q^79 - 4 * q^80 - 8 * q^81 + 11 * q^82 - 2 * q^83 + 2 * q^84 - 2 * q^85 + 7 * q^86 + 26 * q^87 - 7 * q^89 - 6 * q^90 + q^91 - 8 * q^92 + 2 * q^93 + 5 * q^94 + 3 * q^95 + 2 * q^96 + 10 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

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Database

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