LMFDB - Embedded newform 8470.2.a.cf.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +2.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.61803 q^{6} +1.00000 q^{7} +1.00000 q^{8} +3.85410 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +2.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.61803 q^{6} +1.00000 q^{7} +1.00000 q^{8} +3.85410 q^{9} -1.00000 q^{10} +2.61803 q^{12} +2.00000 q^{13} +1.00000 q^{14} -2.61803 q^{15} +1.00000 q^{16} -1.61803 q^{17} +3.85410 q^{18} +6.85410 q^{19} -1.00000 q^{20} +2.61803 q^{21} +6.00000 q^{23} +2.61803 q^{24} +1.00000 q^{25} +2.00000 q^{26} +2.23607 q^{27} +1.00000 q^{28} +3.23607 q^{29} -2.61803 q^{30} -1.23607 q^{31} +1.00000 q^{32} -1.61803 q^{34} -1.00000 q^{35} +3.85410 q^{36} -6.47214 q^{37} +6.85410 q^{38} +5.23607 q^{39} -1.00000 q^{40} -10.3262 q^{41} +2.61803 q^{42} +1.85410 q^{43} -3.85410 q^{45} +6.00000 q^{46} +9.23607 q^{47} +2.61803 q^{48} +1.00000 q^{49} +1.00000 q^{50} -4.23607 q^{51} +2.00000 q^{52} +1.23607 q^{53} +2.23607 q^{54} +1.00000 q^{56} +17.9443 q^{57} +3.23607 q^{58} -7.61803 q^{59} -2.61803 q^{60} -3.52786 q^{61} -1.23607 q^{62} +3.85410 q^{63} +1.00000 q^{64} -2.00000 q^{65} +6.09017 q^{67} -1.61803 q^{68} +15.7082 q^{69} -1.00000 q^{70} -9.70820 q^{71} +3.85410 q^{72} +13.8541 q^{73} -6.47214 q^{74} +2.61803 q^{75} +6.85410 q^{76} +5.23607 q^{78} -8.47214 q^{79} -1.00000 q^{80} -5.70820 q^{81} -10.3262 q^{82} +10.3262 q^{83} +2.61803 q^{84} +1.61803 q^{85} +1.85410 q^{86} +8.47214 q^{87} +13.0902 q^{89} -3.85410 q^{90} +2.00000 q^{91} +6.00000 q^{92} -3.23607 q^{93} +9.23607 q^{94} -6.85410 q^{95} +2.61803 q^{96} -9.56231 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 2 q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - 2 * q^5 + 3 * q^6 + 2 * q^7 + 2 * q^8 + q^9	$ 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 2 q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + q^{9} - 2 q^{10} + 3 q^{12} + 4 q^{13} + 2 q^{14} - 3 q^{15} + 2 q^{16} - q^{17} + q^{18} + 7 q^{19} - 2 q^{20} + 3 q^{21} + 12 q^{23} + 3 q^{24} + 2 q^{25} + 4 q^{26} + 2 q^{28} + 2 q^{29} - 3 q^{30} + 2 q^{31} + 2 q^{32} - q^{34} - 2 q^{35} + q^{36} - 4 q^{37} + 7 q^{38} + 6 q^{39} - 2 q^{40} - 5 q^{41} + 3 q^{42} - 3 q^{43} - q^{45} + 12 q^{46} + 14 q^{47} + 3 q^{48} + 2 q^{49} + 2 q^{50} - 4 q^{51} + 4 q^{52} - 2 q^{53} + 2 q^{56} + 18 q^{57} + 2 q^{58} - 13 q^{59} - 3 q^{60} - 16 q^{61} + 2 q^{62} + q^{63} + 2 q^{64} - 4 q^{65} + q^{67} - q^{68} + 18 q^{69} - 2 q^{70} - 6 q^{71} + q^{72} + 21 q^{73} - 4 q^{74} + 3 q^{75} + 7 q^{76} + 6 q^{78} - 8 q^{79} - 2 q^{80} + 2 q^{81} - 5 q^{82} + 5 q^{83} + 3 q^{84} + q^{85} - 3 q^{86} + 8 q^{87} + 15 q^{89} - q^{90} + 4 q^{91} + 12 q^{92} - 2 q^{93} + 14 q^{94} - 7 q^{95} + 3 q^{96} + q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - 2 * q^5 + 3 * q^6 + 2 * q^7 + 2 * q^8 + q^9 - 2 * q^10 + 3 * q^12 + 4 * q^13 + 2 * q^14 - 3 * q^15 + 2 * q^16 - q^17 + q^18 + 7 * q^19 - 2 * q^20 + 3 * q^21 + 12 * q^23 + 3 * q^24 + 2 * q^25 + 4 * q^26 + 2 * q^28 + 2 * q^29 - 3 * q^30 + 2 * q^31 + 2 * q^32 - q^34 - 2 * q^35 + q^36 - 4 * q^37 + 7 * q^38 + 6 * q^39 - 2 * q^40 - 5 * q^41 + 3 * q^42 - 3 * q^43 - q^45 + 12 * q^46 + 14 * q^47 + 3 * q^48 + 2 * q^49 + 2 * q^50 - 4 * q^51 + 4 * q^52 - 2 * q^53 + 2 * q^56 + 18 * q^57 + 2 * q^58 - 13 * q^59 - 3 * q^60 - 16 * q^61 + 2 * q^62 + q^63 + 2 * q^64 - 4 * q^65 + q^67 - q^68 + 18 * q^69 - 2 * q^70 - 6 * q^71 + q^72 + 21 * q^73 - 4 * q^74 + 3 * q^75 + 7 * q^76 + 6 * q^78 - 8 * q^79 - 2 * q^80 + 2 * q^81 - 5 * q^82 + 5 * q^83 + 3 * q^84 + q^85 - 3 * q^86 + 8 * q^87 + 15 * q^89 - q^90 + 4 * q^91 + 12 * q^92 - 2 * q^93 + 14 * q^94 - 7 * q^95 + 3 * q^96 + q^97 + 2 * 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
$3$	2.61803	1.51152	0.755761	−	0.654847i	$-0.227267\pi$
$3$	2.61803	1.51152	0.755761	+	0.654847i	$0.227267\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	2.61803	1.06881
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	3.85410	1.28470
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	2.61803	0.755761
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	1.00000	0.267261
$15$	−2.61803	−0.675973
$16$	1.00000	0.250000
$17$	−1.61803	−0.392431	−0.196215	−	0.980561i	$-0.562865\pi$
$17$	−1.61803	−0.392431	−0.196215	+	0.980561i	$0.562865\pi$
$18$	3.85410	0.908421
$19$	6.85410	1.57244	0.786219	−	0.617947i	$-0.212036\pi$
$19$	6.85410	1.57244	0.786219	+	0.617947i	$0.212036\pi$
$20$	−1.00000	−0.223607
$21$	2.61803	0.571302
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	2.61803	0.534404
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	2.23607	0.430331
$28$	1.00000	0.188982
$29$	3.23607	0.600923	0.300461	−	0.953794i	$-0.402859\pi$
$29$	3.23607	0.600923	0.300461	+	0.953794i	$0.402859\pi$
$30$	−2.61803	−0.477985
$31$	−1.23607	−0.222004	−0.111002	−	0.993820i	$-0.535406\pi$
$31$	−1.23607	−0.222004	−0.111002	+	0.993820i	$0.535406\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.61803	−0.277491
$35$	−1.00000	−0.169031
$36$	3.85410	0.642350
$37$	−6.47214	−1.06401	−0.532006	−	0.846740i	$-0.678562\pi$
$37$	−6.47214	−1.06401	−0.532006	+	0.846740i	$0.678562\pi$
$38$	6.85410	1.11188
$39$	5.23607	0.838442
$40$	−1.00000	−0.158114
$41$	−10.3262	−1.61269	−0.806344	−	0.591447i	$-0.798557\pi$
$41$	−10.3262	−1.61269	−0.806344	+	0.591447i	$0.798557\pi$
$42$	2.61803	0.403971
$43$	1.85410	0.282748	0.141374	−	0.989956i	$-0.454848\pi$
$43$	1.85410	0.282748	0.141374	+	0.989956i	$0.454848\pi$
$44$	0	0
$45$	−3.85410	−0.574536
$46$	6.00000	0.884652
$47$	9.23607	1.34722	0.673609	−	0.739087i	$-0.264743\pi$
$47$	9.23607	1.34722	0.673609	+	0.739087i	$0.264743\pi$
$48$	2.61803	0.377881
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−4.23607	−0.593168
$52$	2.00000	0.277350
$53$	1.23607	0.169787	0.0848935	−	0.996390i	$-0.472945\pi$
$53$	1.23607	0.169787	0.0848935	+	0.996390i	$0.472945\pi$
$54$	2.23607	0.304290
$55$	0	0
$56$	1.00000	0.133631
$57$	17.9443	2.37678
$58$	3.23607	0.424917
$59$	−7.61803	−0.991784	−0.495892	−	0.868384i	$-0.665159\pi$
$59$	−7.61803	−0.991784	−0.495892	+	0.868384i	$0.665159\pi$
$60$	−2.61803	−0.337987
$61$	−3.52786	−0.451697	−0.225848	−	0.974162i	$-0.572515\pi$
$61$	−3.52786	−0.451697	−0.225848	+	0.974162i	$0.572515\pi$
$62$	−1.23607	−0.156981
$63$	3.85410	0.485571
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	0	0
$67$	6.09017	0.744033	0.372016	−	0.928226i	$-0.378667\pi$
$67$	6.09017	0.744033	0.372016	+	0.928226i	$0.378667\pi$
$68$	−1.61803	−0.196215
$69$	15.7082	1.89105
$70$	−1.00000	−0.119523
$71$	−9.70820	−1.15215	−0.576076	−	0.817396i	$-0.695417\pi$
$71$	−9.70820	−1.15215	−0.576076	+	0.817396i	$0.695417\pi$
$72$	3.85410	0.454210
$73$	13.8541	1.62150	0.810750	−	0.585393i	$-0.199060\pi$
$73$	13.8541	1.62150	0.810750	+	0.585393i	$0.199060\pi$
$74$	−6.47214	−0.752371
$75$	2.61803	0.302305
$76$	6.85410	0.786219
$77$	0	0
$78$	5.23607	0.592868
$79$	−8.47214	−0.953190	−0.476595	−	0.879123i	$-0.658129\pi$
$79$	−8.47214	−0.953190	−0.476595	+	0.879123i	$0.658129\pi$
$80$	−1.00000	−0.111803
$81$	−5.70820	−0.634245
$82$	−10.3262	−1.14034
$83$	10.3262	1.13345	0.566726	−	0.823906i	$-0.308210\pi$
$83$	10.3262	1.13345	0.566726	+	0.823906i	$0.308210\pi$
$84$	2.61803	0.285651
$85$	1.61803	0.175500
$86$	1.85410	0.199933
$87$	8.47214	0.908308
$88$	0	0
$89$	13.0902	1.38756	0.693778	−	0.720189i	$-0.255945\pi$
$89$	13.0902	1.38756	0.693778	+	0.720189i	$0.255945\pi$
$90$	−3.85410	−0.406258
$91$	2.00000	0.209657
$92$	6.00000	0.625543
$93$	−3.23607	−0.335565
$94$	9.23607	0.952628
$95$	−6.85410	−0.703216
$96$	2.61803	0.267202
$97$	−9.56231	−0.970905	−0.485453	−	0.874263i	$-0.661345\pi$
$97$	−9.56231	−0.970905	−0.485453	+	0.874263i	$0.661345\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	8.94427	0.889988	0.444994	−	0.895533i	$-0.353206\pi$
$101$	8.94427	0.889988	0.444994	+	0.895533i	$0.353206\pi$
$102$	−4.23607	−0.419433
$103$	−16.9443	−1.66957	−0.834784	−	0.550577i	$-0.814408\pi$
$103$	−16.9443	−1.66957	−0.834784	+	0.550577i	$0.814408\pi$
$104$	2.00000	0.196116
$105$	−2.61803	−0.255494
$106$	1.23607	0.120058
$107$	15.3820	1.48703	0.743515	−	0.668719i	$-0.233157\pi$
$107$	15.3820	1.48703	0.743515	+	0.668719i	$0.233157\pi$
$108$	2.23607	0.215166
$109$	−0.472136	−0.0452224	−0.0226112	−	0.999744i	$-0.507198\pi$
$109$	−0.472136	−0.0452224	−0.0226112	+	0.999744i	$0.507198\pi$
$110$	0	0
$111$	−16.9443	−1.60828
$112$	1.00000	0.0944911
$113$	−9.32624	−0.877339	−0.438669	−	0.898649i	$-0.644550\pi$
$113$	−9.32624	−0.877339	−0.438669	+	0.898649i	$0.644550\pi$
$114$	17.9443	1.68064
$115$	−6.00000	−0.559503
$116$	3.23607	0.300461
$117$	7.70820	0.712624
$118$	−7.61803	−0.701297
$119$	−1.61803	−0.148325
$120$	−2.61803	−0.238993
$121$	0	0
$122$	−3.52786	−0.319398
$123$	−27.0344	−2.43761
$124$	−1.23607	−0.111002
$125$	−1.00000	−0.0894427
$126$	3.85410	0.343351
$127$	−11.4164	−1.01304	−0.506521	−	0.862228i	$-0.669069\pi$
$127$	−11.4164	−1.01304	−0.506521	+	0.862228i	$0.669069\pi$
$128$	1.00000	0.0883883
$129$	4.85410	0.427380
$130$	−2.00000	−0.175412
$131$	−7.79837	−0.681347	−0.340674	−	0.940182i	$-0.610655\pi$
$131$	−7.79837	−0.681347	−0.340674	+	0.940182i	$0.610655\pi$
$132$	0	0
$133$	6.85410	0.594326
$134$	6.09017	0.526111
$135$	−2.23607	−0.192450
$136$	−1.61803	−0.138745
$137$	−1.09017	−0.0931395	−0.0465698	−	0.998915i	$-0.514829\pi$
$137$	−1.09017	−0.0931395	−0.0465698	+	0.998915i	$0.514829\pi$
$138$	15.7082	1.33717
$139$	18.4721	1.56679	0.783393	−	0.621527i	$-0.213487\pi$
$139$	18.4721	1.56679	0.783393	+	0.621527i	$0.213487\pi$
$140$	−1.00000	−0.0845154
$141$	24.1803	2.03635
$142$	−9.70820	−0.814694
$143$	0	0
$144$	3.85410	0.321175
$145$	−3.23607	−0.268741
$146$	13.8541	1.14657
$147$	2.61803	0.215932
$148$	−6.47214	−0.532006
$149$	24.1803	1.98093	0.990465	−	0.137762i	$-0.0439910\pi$
$149$	24.1803	1.98093	0.990465	+	0.137762i	$0.0439910\pi$
$150$	2.61803	0.213762
$151$	2.29180	0.186504	0.0932519	−	0.995643i	$-0.470274\pi$
$151$	2.29180	0.186504	0.0932519	+	0.995643i	$0.470274\pi$
$152$	6.85410	0.555941
$153$	−6.23607	−0.504156
$154$	0	0
$155$	1.23607	0.0992834
$156$	5.23607	0.419221
$157$	0.291796	0.0232879	0.0116439	−	0.999932i	$-0.496294\pi$
$157$	0.291796	0.0232879	0.0116439	+	0.999932i	$0.496294\pi$
$158$	−8.47214	−0.674007
$159$	3.23607	0.256637
$160$	−1.00000	−0.0790569
$161$	6.00000	0.472866
$162$	−5.70820	−0.448479
$163$	2.85410	0.223551	0.111775	−	0.993734i	$-0.464346\pi$
$163$	2.85410	0.223551	0.111775	+	0.993734i	$0.464346\pi$
$164$	−10.3262	−0.806344
$165$	0	0
$166$	10.3262	0.801471
$167$	6.47214	0.500829	0.250414	−	0.968139i	$-0.419433\pi$
$167$	6.47214	0.500829	0.250414	+	0.968139i	$0.419433\pi$
$168$	2.61803	0.201986
$169$	−9.00000	−0.692308
$170$	1.61803	0.124098
$171$	26.4164	2.02011
$172$	1.85410	0.141374
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	8.47214	0.642271
$175$	1.00000	0.0755929
$176$	0	0
$177$	−19.9443	−1.49910
$178$	13.0902	0.981150
$179$	−6.38197	−0.477011	−0.238505	−	0.971141i	$-0.576657\pi$
$179$	−6.38197	−0.477011	−0.238505	+	0.971141i	$0.576657\pi$
$180$	−3.85410	−0.287268
$181$	3.52786	0.262224	0.131112	−	0.991368i	$-0.458145\pi$
$181$	3.52786	0.262224	0.131112	+	0.991368i	$0.458145\pi$
$182$	2.00000	0.148250
$183$	−9.23607	−0.682750
$184$	6.00000	0.442326
$185$	6.47214	0.475841
$186$	−3.23607	−0.237280
$187$	0	0
$188$	9.23607	0.673609
$189$	2.23607	0.162650
$190$	−6.85410	−0.497249
$191$	−1.52786	−0.110552	−0.0552762	−	0.998471i	$-0.517604\pi$
$191$	−1.52786	−0.110552	−0.0552762	+	0.998471i	$0.517604\pi$
$192$	2.61803	0.188940
$193$	−2.00000	−0.143963	−0.0719816	−	0.997406i	$-0.522932\pi$
$193$	−2.00000	−0.143963	−0.0719816	+	0.997406i	$0.522932\pi$
$194$	−9.56231	−0.686534
$195$	−5.23607	−0.374963
$196$	1.00000	0.0714286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	−19.4164	−1.37639	−0.688196	−	0.725525i	$-0.741597\pi$
$199$	−19.4164	−1.37639	−0.688196	+	0.725525i	$0.741597\pi$
$200$	1.00000	0.0707107
$201$	15.9443	1.12462
$202$	8.94427	0.629317
$203$	3.23607	0.227127
$204$	−4.23607	−0.296584
$205$	10.3262	0.721216
$206$	−16.9443	−1.18056
$207$	23.1246	1.60727
$208$	2.00000	0.138675
$209$	0	0
$210$	−2.61803	−0.180662
$211$	11.2705	0.775894	0.387947	−	0.921682i	$-0.373184\pi$
$211$	11.2705	0.775894	0.387947	+	0.921682i	$0.373184\pi$
$212$	1.23607	0.0848935
$213$	−25.4164	−1.74150
$214$	15.3820	1.05149
$215$	−1.85410	−0.126449
$216$	2.23607	0.152145
$217$	−1.23607	−0.0839098
$218$	−0.472136	−0.0319771
$219$	36.2705	2.45093
$220$	0	0
$221$	−3.23607	−0.217681
$222$	−16.9443	−1.13723
$223$	6.29180	0.421330	0.210665	−	0.977558i	$-0.432437\pi$
$223$	6.29180	0.421330	0.210665	+	0.977558i	$0.432437\pi$
$224$	1.00000	0.0668153
$225$	3.85410	0.256940
$226$	−9.32624	−0.620372
$227$	−24.2705	−1.61089	−0.805445	−	0.592670i	$-0.798074\pi$
$227$	−24.2705	−1.61089	−0.805445	+	0.592670i	$0.798074\pi$
$228$	17.9443	1.18839
$229$	11.1246	0.735135	0.367568	−	0.929997i	$-0.380191\pi$
$229$	11.1246	0.735135	0.367568	+	0.929997i	$0.380191\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	3.23607	0.212458
$233$	10.9098	0.714727	0.357363	−	0.933965i	$-0.383676\pi$
$233$	10.9098	0.714727	0.357363	+	0.933965i	$0.383676\pi$
$234$	7.70820	0.503901
$235$	−9.23607	−0.602495
$236$	−7.61803	−0.495892
$237$	−22.1803	−1.44077
$238$	−1.61803	−0.104882
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	−2.61803	−0.168993
$241$	−6.27051	−0.403919	−0.201960	−	0.979394i	$-0.564731\pi$
$241$	−6.27051	−0.403919	−0.201960	+	0.979394i	$0.564731\pi$
$242$	0	0
$243$	−21.6525	−1.38901
$244$	−3.52786	−0.225848
$245$	−1.00000	−0.0638877
$246$	−27.0344	−1.72365
$247$	13.7082	0.872232
$248$	−1.23607	−0.0784904
$249$	27.0344	1.71324
$250$	−1.00000	−0.0632456
$251$	26.8328	1.69367	0.846836	−	0.531854i	$-0.178504\pi$
$251$	26.8328	1.69367	0.846836	+	0.531854i	$0.178504\pi$
$252$	3.85410	0.242786
$253$	0	0
$254$	−11.4164	−0.716329
$255$	4.23607	0.265273
$256$	1.00000	0.0625000
$257$	29.5623	1.84405	0.922023	−	0.387135i	$-0.126535\pi$
$257$	29.5623	1.84405	0.922023	+	0.387135i	$0.126535\pi$
$258$	4.85410	0.302203
$259$	−6.47214	−0.402159
$260$	−2.00000	−0.124035
$261$	12.4721	0.772006
$262$	−7.79837	−0.481785
$263$	20.1803	1.24437	0.622187	−	0.782869i	$-0.286245\pi$
$263$	20.1803	1.24437	0.622187	+	0.782869i	$0.286245\pi$
$264$	0	0
$265$	−1.23607	−0.0759311
$266$	6.85410	0.420252
$267$	34.2705	2.09732
$268$	6.09017	0.372016
$269$	−12.1803	−0.742648	−0.371324	−	0.928503i	$-0.621096\pi$
$269$	−12.1803	−0.742648	−0.371324	+	0.928503i	$0.621096\pi$
$270$	−2.23607	−0.136083
$271$	−9.41641	−0.572006	−0.286003	−	0.958229i	$-0.592327\pi$
$271$	−9.41641	−0.572006	−0.286003	+	0.958229i	$0.592327\pi$
$272$	−1.61803	−0.0981077
$273$	5.23607	0.316901
$274$	−1.09017	−0.0658596
$275$	0	0
$276$	15.7082	0.945523
$277$	6.29180	0.378037	0.189019	−	0.981973i	$-0.439469\pi$
$277$	6.29180	0.378037	0.189019	+	0.981973i	$0.439469\pi$
$278$	18.4721	1.10789
$279$	−4.76393	−0.285209
$280$	−1.00000	−0.0597614
$281$	−24.0344	−1.43377	−0.716887	−	0.697189i	$-0.754434\pi$
$281$	−24.0344	−1.43377	−0.716887	+	0.697189i	$0.754434\pi$
$282$	24.1803	1.43992
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	−9.70820	−0.576076
$285$	−17.9443	−1.06293
$286$	0	0
$287$	−10.3262	−0.609539
$288$	3.85410	0.227105
$289$	−14.3820	−0.845998
$290$	−3.23607	−0.190028
$291$	−25.0344	−1.46754
$292$	13.8541	0.810750
$293$	16.4721	0.962312	0.481156	−	0.876635i	$-0.340217\pi$
$293$	16.4721	0.962312	0.481156	+	0.876635i	$0.340217\pi$
$294$	2.61803	0.152687
$295$	7.61803	0.443539
$296$	−6.47214	−0.376185
$297$	0	0
$298$	24.1803	1.40073
$299$	12.0000	0.693978
$300$	2.61803	0.151152
$301$	1.85410	0.106869
$302$	2.29180	0.131878
$303$	23.4164	1.34524
$304$	6.85410	0.393110
$305$	3.52786	0.202005
$306$	−6.23607	−0.356492
$307$	−27.5623	−1.57306	−0.786532	−	0.617550i	$-0.788125\pi$
$307$	−27.5623	−1.57306	−0.786532	+	0.617550i	$0.788125\pi$
$308$	0	0
$309$	−44.3607	−2.52359
$310$	1.23607	0.0702039
$311$	−29.1246	−1.65151	−0.825753	−	0.564032i	$-0.809249\pi$
$311$	−29.1246	−1.65151	−0.825753	+	0.564032i	$0.809249\pi$
$312$	5.23607	0.296434
$313$	−0.854102	−0.0482767	−0.0241383	−	0.999709i	$-0.507684\pi$
$313$	−0.854102	−0.0482767	−0.0241383	+	0.999709i	$0.507684\pi$
$314$	0.291796	0.0164670
$315$	−3.85410	−0.217154
$316$	−8.47214	−0.476595
$317$	−20.7639	−1.16622	−0.583109	−	0.812394i	$-0.698164\pi$
$317$	−20.7639	−1.16622	−0.583109	+	0.812394i	$0.698164\pi$
$318$	3.23607	0.181470
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	40.2705	2.24768
$322$	6.00000	0.334367
$323$	−11.0902	−0.617074
$324$	−5.70820	−0.317122
$325$	2.00000	0.110940
$326$	2.85410	0.158074
$327$	−1.23607	−0.0683547
$328$	−10.3262	−0.570171
$329$	9.23607	0.509201
$330$	0	0
$331$	6.03444	0.331683	0.165841	−	0.986152i	$-0.446966\pi$
$331$	6.03444	0.331683	0.165841	+	0.986152i	$0.446966\pi$
$332$	10.3262	0.566726
$333$	−24.9443	−1.36694
$334$	6.47214	0.354140
$335$	−6.09017	−0.332742
$336$	2.61803	0.142825
$337$	−15.6180	−0.850769	−0.425384	−	0.905013i	$-0.639861\pi$
$337$	−15.6180	−0.850769	−0.425384	+	0.905013i	$0.639861\pi$
$338$	−9.00000	−0.489535
$339$	−24.4164	−1.32612
$340$	1.61803	0.0877502
$341$	0	0
$342$	26.4164	1.42844
$343$	1.00000	0.0539949
$344$	1.85410	0.0999665
$345$	−15.7082	−0.845701
$346$	12.0000	0.645124
$347$	5.32624	0.285927	0.142964	−	0.989728i	$-0.454337\pi$
$347$	5.32624	0.285927	0.142964	+	0.989728i	$0.454337\pi$
$348$	8.47214	0.454154
$349$	5.52786	0.295900	0.147950	−	0.988995i	$-0.452733\pi$
$349$	5.52786	0.295900	0.147950	+	0.988995i	$0.452733\pi$
$350$	1.00000	0.0534522
$351$	4.47214	0.238705
$352$	0	0
$353$	−28.4508	−1.51429	−0.757143	−	0.653249i	$-0.773405\pi$
$353$	−28.4508	−1.51429	−0.757143	+	0.653249i	$0.773405\pi$
$354$	−19.9443	−1.06003
$355$	9.70820	0.515258
$356$	13.0902	0.693778
$357$	−4.23607	−0.224196
$358$	−6.38197	−0.337297
$359$	18.6525	0.984440	0.492220	−	0.870471i	$-0.336186\pi$
$359$	18.6525	0.984440	0.492220	+	0.870471i	$0.336186\pi$
$360$	−3.85410	−0.203129
$361$	27.9787	1.47256
$362$	3.52786	0.185420
$363$	0	0
$364$	2.00000	0.104828
$365$	−13.8541	−0.725157
$366$	−9.23607	−0.482777
$367$	28.1803	1.47100	0.735501	−	0.677524i	$-0.236947\pi$
$367$	28.1803	1.47100	0.735501	+	0.677524i	$0.236947\pi$
$368$	6.00000	0.312772
$369$	−39.7984	−2.07182
$370$	6.47214	0.336470
$371$	1.23607	0.0641735
$372$	−3.23607	−0.167782
$373$	−24.4721	−1.26712	−0.633560	−	0.773694i	$-0.718407\pi$
$373$	−24.4721	−1.26712	−0.633560	+	0.773694i	$0.718407\pi$
$374$	0	0
$375$	−2.61803	−0.135195
$376$	9.23607	0.476314
$377$	6.47214	0.333332
$378$	2.23607	0.115011
$379$	−27.8541	−1.43077	−0.715385	−	0.698731i	$-0.753748\pi$
$379$	−27.8541	−1.43077	−0.715385	+	0.698731i	$0.753748\pi$
$380$	−6.85410	−0.351608
$381$	−29.8885	−1.53124
$382$	−1.52786	−0.0781723
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	2.61803	0.133601
$385$	0	0
$386$	−2.00000	−0.101797
$387$	7.14590	0.363246
$388$	−9.56231	−0.485453
$389$	−32.8328	−1.66469	−0.832345	−	0.554258i	$-0.813002\pi$
$389$	−32.8328	−1.66469	−0.832345	+	0.554258i	$0.813002\pi$
$390$	−5.23607	−0.265139
$391$	−9.70820	−0.490965
$392$	1.00000	0.0505076
$393$	−20.4164	−1.02987
$394$	6.00000	0.302276
$395$	8.47214	0.426279
$396$	0	0
$397$	−16.9443	−0.850409	−0.425204	−	0.905097i	$-0.639798\pi$
$397$	−16.9443	−0.850409	−0.425204	+	0.905097i	$0.639798\pi$
$398$	−19.4164	−0.973257
$399$	17.9443	0.898337
$400$	1.00000	0.0500000
$401$	−26.5066	−1.32368	−0.661838	−	0.749647i	$-0.730223\pi$
$401$	−26.5066	−1.32368	−0.661838	+	0.749647i	$0.730223\pi$
$402$	15.9443	0.795228
$403$	−2.47214	−0.123146
$404$	8.94427	0.444994
$405$	5.70820	0.283643
$406$	3.23607	0.160603
$407$	0	0
$408$	−4.23607	−0.209717
$409$	27.8885	1.37900	0.689500	−	0.724286i	$-0.257830\pi$
$409$	27.8885	1.37900	0.689500	+	0.724286i	$0.257830\pi$
$410$	10.3262	0.509977
$411$	−2.85410	−0.140782
$412$	−16.9443	−0.834784
$413$	−7.61803	−0.374859
$414$	23.1246	1.13651
$415$	−10.3262	−0.506895
$416$	2.00000	0.0980581
$417$	48.3607	2.36823
$418$	0	0
$419$	12.3262	0.602176	0.301088	−	0.953596i	$-0.402650\pi$
$419$	12.3262	0.602176	0.301088	+	0.953596i	$0.402650\pi$
$420$	−2.61803	−0.127747
$421$	−6.36068	−0.310001	−0.155000	−	0.987914i	$-0.549538\pi$
$421$	−6.36068	−0.310001	−0.155000	+	0.987914i	$0.549538\pi$
$422$	11.2705	0.548640
$423$	35.5967	1.73077
$424$	1.23607	0.0600288
$425$	−1.61803	−0.0784862
$426$	−25.4164	−1.23143
$427$	−3.52786	−0.170725
$428$	15.3820	0.743515
$429$	0	0
$430$	−1.85410	−0.0894127
$431$	31.2361	1.50459	0.752294	−	0.658827i	$-0.228947\pi$
$431$	31.2361	1.50459	0.752294	+	0.658827i	$0.228947\pi$
$432$	2.23607	0.107583
$433$	16.4508	0.790577	0.395289	−	0.918557i	$-0.370645\pi$
$433$	16.4508	0.790577	0.395289	+	0.918557i	$0.370645\pi$
$434$	−1.23607	−0.0593332
$435$	−8.47214	−0.406208
$436$	−0.472136	−0.0226112
$437$	41.1246	1.96726
$438$	36.2705	1.73307
$439$	−28.6525	−1.36751	−0.683754	−	0.729713i	$-0.739654\pi$
$439$	−28.6525	−1.36751	−0.683754	+	0.729713i	$0.739654\pi$
$440$	0	0
$441$	3.85410	0.183529
$442$	−3.23607	−0.153924
$443$	28.0344	1.33196	0.665978	−	0.745971i	$-0.268014\pi$
$443$	28.0344	1.33196	0.665978	+	0.745971i	$0.268014\pi$
$444$	−16.9443	−0.804140
$445$	−13.0902	−0.620534
$446$	6.29180	0.297925
$447$	63.3050	2.99422
$448$	1.00000	0.0472456
$449$	24.4508	1.15391	0.576953	−	0.816777i	$-0.304241\pi$
$449$	24.4508	1.15391	0.576953	+	0.816777i	$0.304241\pi$
$450$	3.85410	0.181684
$451$	0	0
$452$	−9.32624	−0.438669
$453$	6.00000	0.281905
$454$	−24.2705	−1.13907
$455$	−2.00000	−0.0937614
$456$	17.9443	0.840318
$457$	−31.2705	−1.46277	−0.731386	−	0.681963i	$-0.761126\pi$
$457$	−31.2705	−1.46277	−0.731386	+	0.681963i	$0.761126\pi$
$458$	11.1246	0.519819
$459$	−3.61803	−0.168875
$460$	−6.00000	−0.279751
$461$	6.94427	0.323427	0.161713	−	0.986838i	$-0.448298\pi$
$461$	6.94427	0.323427	0.161713	+	0.986838i	$0.448298\pi$
$462$	0	0
$463$	−7.41641	−0.344670	−0.172335	−	0.985038i	$-0.555131\pi$
$463$	−7.41641	−0.344670	−0.172335	+	0.985038i	$0.555131\pi$
$464$	3.23607	0.150231
$465$	3.23607	0.150069
$466$	10.9098	0.505388
$467$	−17.8885	−0.827783	−0.413892	−	0.910326i	$-0.635831\pi$
$467$	−17.8885	−0.827783	−0.413892	+	0.910326i	$0.635831\pi$
$468$	7.70820	0.356312
$469$	6.09017	0.281218
$470$	−9.23607	−0.426028
$471$	0.763932	0.0352001
$472$	−7.61803	−0.350648
$473$	0	0
$474$	−22.1803	−1.01878
$475$	6.85410	0.314488
$476$	−1.61803	−0.0741625
$477$	4.76393	0.218125
$478$	0	0
$479$	−37.8885	−1.73117	−0.865586	−	0.500761i	$-0.833054\pi$
$479$	−37.8885	−1.73117	−0.865586	+	0.500761i	$0.833054\pi$
$480$	−2.61803	−0.119496
$481$	−12.9443	−0.590208
$482$	−6.27051	−0.285614
$483$	15.7082	0.714748
$484$	0	0
$485$	9.56231	0.434202
$486$	−21.6525	−0.982176
$487$	−26.8328	−1.21591	−0.607955	−	0.793971i	$-0.708010\pi$
$487$	−26.8328	−1.21591	−0.607955	+	0.793971i	$0.708010\pi$
$488$	−3.52786	−0.159699
$489$	7.47214	0.337902
$490$	−1.00000	−0.0451754
$491$	−32.5066	−1.46700	−0.733501	−	0.679689i	$-0.762115\pi$
$491$	−32.5066	−1.46700	−0.733501	+	0.679689i	$0.762115\pi$
$492$	−27.0344	−1.21881
$493$	−5.23607	−0.235821
$494$	13.7082	0.616761
$495$	0	0
$496$	−1.23607	−0.0555011
$497$	−9.70820	−0.435472
$498$	27.0344	1.21144
$499$	−15.2016	−0.680518	−0.340259	−	0.940332i	$-0.610515\pi$
$499$	−15.2016	−0.680518	−0.340259	+	0.940332i	$0.610515\pi$
$500$	−1.00000	−0.0447214
$501$	16.9443	0.757014
$502$	26.8328	1.19761
$503$	−17.1246	−0.763549	−0.381774	−	0.924256i	$-0.624687\pi$
$503$	−17.1246	−0.763549	−0.381774	+	0.924256i	$0.624687\pi$
$504$	3.85410	0.171675
$505$	−8.94427	−0.398015
$506$	0	0
$507$	−23.5623	−1.04644
$508$	−11.4164	−0.506521
$509$	−25.5279	−1.13150	−0.565751	−	0.824576i	$-0.691414\pi$
$509$	−25.5279	−1.13150	−0.565751	+	0.824576i	$0.691414\pi$
$510$	4.23607	0.187576
$511$	13.8541	0.612869
$512$	1.00000	0.0441942
$513$	15.3262	0.676670
$514$	29.5623	1.30394
$515$	16.9443	0.746654
$516$	4.85410	0.213690
$517$	0	0
$518$	−6.47214	−0.284369
$519$	31.4164	1.37903
$520$	−2.00000	−0.0877058
$521$	35.7984	1.56836	0.784178	−	0.620536i	$-0.213085\pi$
$521$	35.7984	1.56836	0.784178	+	0.620536i	$0.213085\pi$
$522$	12.4721	0.545891
$523$	5.97871	0.261431	0.130715	−	0.991420i	$-0.458273\pi$
$523$	5.97871	0.261431	0.130715	+	0.991420i	$0.458273\pi$
$524$	−7.79837	−0.340674
$525$	2.61803	0.114260
$526$	20.1803	0.879905
$527$	2.00000	0.0871214
$528$	0	0
$529$	13.0000	0.565217
$530$	−1.23607	−0.0536914
$531$	−29.3607	−1.27414
$532$	6.85410	0.297163
$533$	−20.6525	−0.894558
$534$	34.2705	1.48303
$535$	−15.3820	−0.665020
$536$	6.09017	0.263055
$537$	−16.7082	−0.721012
$538$	−12.1803	−0.525132
$539$	0	0
$540$	−2.23607	−0.0962250
$541$	14.8328	0.637713	0.318856	−	0.947803i	$-0.396701\pi$
$541$	14.8328	0.637713	0.318856	+	0.947803i	$0.396701\pi$
$542$	−9.41641	−0.404469
$543$	9.23607	0.396358
$544$	−1.61803	−0.0693726
$545$	0.472136	0.0202241
$546$	5.23607	0.224083
$547$	−17.9098	−0.765769	−0.382885	−	0.923796i	$-0.625069\pi$
$547$	−17.9098	−0.765769	−0.382885	+	0.923796i	$0.625069\pi$
$548$	−1.09017	−0.0465698
$549$	−13.5967	−0.580295
$550$	0	0
$551$	22.1803	0.944914
$552$	15.7082	0.668586
$553$	−8.47214	−0.360272
$554$	6.29180	0.267313
$555$	16.9443	0.719244
$556$	18.4721	0.783393
$557$	38.1803	1.61775	0.808876	−	0.587979i	$-0.200076\pi$
$557$	38.1803	1.61775	0.808876	+	0.587979i	$0.200076\pi$
$558$	−4.76393	−0.201673
$559$	3.70820	0.156840
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−24.0344	−1.01383
$563$	3.49342	0.147230	0.0736151	−	0.997287i	$-0.476546\pi$
$563$	3.49342	0.147230	0.0736151	+	0.997287i	$0.476546\pi$
$564$	24.1803	1.01818
$565$	9.32624	0.392358
$566$	−12.0000	−0.504398
$567$	−5.70820	−0.239722
$568$	−9.70820	−0.407347
$569$	−8.09017	−0.339158	−0.169579	−	0.985517i	$-0.554241\pi$
$569$	−8.09017	−0.339158	−0.169579	+	0.985517i	$0.554241\pi$
$570$	−17.9443	−0.751603
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	0	0
$573$	−4.00000	−0.167102
$574$	−10.3262	−0.431009
$575$	6.00000	0.250217
$576$	3.85410	0.160588
$577$	25.2148	1.04971	0.524853	−	0.851193i	$-0.324120\pi$
$577$	25.2148	1.04971	0.524853	+	0.851193i	$0.324120\pi$
$578$	−14.3820	−0.598211
$579$	−5.23607	−0.217604
$580$	−3.23607	−0.134370
$581$	10.3262	0.428405
$582$	−25.0344	−1.03771
$583$	0	0
$584$	13.8541	0.573287
$585$	−7.70820	−0.318695
$586$	16.4721	0.680458
$587$	2.74265	0.113201	0.0566006	−	0.998397i	$-0.481974\pi$
$587$	2.74265	0.113201	0.0566006	+	0.998397i	$0.481974\pi$
$588$	2.61803	0.107966
$589$	−8.47214	−0.349088
$590$	7.61803	0.313629
$591$	15.7082	0.646149
$592$	−6.47214	−0.266003
$593$	12.0902	0.496484	0.248242	−	0.968698i	$-0.420147\pi$
$593$	12.0902	0.496484	0.248242	+	0.968698i	$0.420147\pi$
$594$	0	0
$595$	1.61803	0.0663329
$596$	24.1803	0.990465
$597$	−50.8328	−2.08045
$598$	12.0000	0.490716
$599$	−25.2361	−1.03112	−0.515559	−	0.856854i	$-0.672416\pi$
$599$	−25.2361	−1.03112	−0.515559	+	0.856854i	$0.672416\pi$
$600$	2.61803	0.106881
$601$	−29.0902	−1.18661	−0.593306	−	0.804977i	$-0.702178\pi$
$601$	−29.0902	−1.18661	−0.593306	+	0.804977i	$0.702178\pi$
$602$	1.85410	0.0755676
$603$	23.4721	0.955859
$604$	2.29180	0.0932519
$605$	0	0
$606$	23.4164	0.951227
$607$	−29.5967	−1.20129	−0.600647	−	0.799514i	$-0.705090\pi$
$607$	−29.5967	−1.20129	−0.600647	+	0.799514i	$0.705090\pi$
$608$	6.85410	0.277971
$609$	8.47214	0.343308
$610$	3.52786	0.142839
$611$	18.4721	0.747303
$612$	−6.23607	−0.252078
$613$	−37.4164	−1.51123	−0.755617	−	0.655013i	$-0.772663\pi$
$613$	−37.4164	−1.51123	−0.755617	+	0.655013i	$0.772663\pi$
$614$	−27.5623	−1.11232
$615$	27.0344	1.09013
$616$	0	0
$617$	23.0344	0.927332	0.463666	−	0.886010i	$-0.346534\pi$
$617$	23.0344	0.927332	0.463666	+	0.886010i	$0.346534\pi$
$618$	−44.3607	−1.78445
$619$	18.6738	0.750562	0.375281	−	0.926911i	$-0.377546\pi$
$619$	18.6738	0.750562	0.375281	+	0.926911i	$0.377546\pi$
$620$	1.23607	0.0496417
$621$	13.4164	0.538382
$622$	−29.1246	−1.16779
$623$	13.0902	0.524447
$624$	5.23607	0.209610
$625$	1.00000	0.0400000
$626$	−0.854102	−0.0341368
$627$	0	0
$628$	0.291796	0.0116439
$629$	10.4721	0.417551
$630$	−3.85410	−0.153551
$631$	−23.7082	−0.943809	−0.471904	−	0.881650i	$-0.656433\pi$
$631$	−23.7082	−0.943809	−0.471904	+	0.881650i	$0.656433\pi$
$632$	−8.47214	−0.337003
$633$	29.5066	1.17278
$634$	−20.7639	−0.824641
$635$	11.4164	0.453046
$636$	3.23607	0.128318
$637$	2.00000	0.0792429
$638$	0	0
$639$	−37.4164	−1.48017
$640$	−1.00000	−0.0395285
$641$	40.3262	1.59279	0.796395	−	0.604776i	$-0.206738\pi$
$641$	40.3262	1.59279	0.796395	+	0.604776i	$0.206738\pi$
$642$	40.2705	1.58935
$643$	−14.9787	−0.590703	−0.295351	−	0.955389i	$-0.595437\pi$
$643$	−14.9787	−0.590703	−0.295351	+	0.955389i	$0.595437\pi$
$644$	6.00000	0.236433
$645$	−4.85410	−0.191130
$646$	−11.0902	−0.436337
$647$	0.180340	0.00708989	0.00354495	−	0.999994i	$-0.498872\pi$
$647$	0.180340	0.00708989	0.00354495	+	0.999994i	$0.498872\pi$
$648$	−5.70820	−0.224239
$649$	0	0
$650$	2.00000	0.0784465
$651$	−3.23607	−0.126832
$652$	2.85410	0.111775
$653$	26.8328	1.05005	0.525025	−	0.851087i	$-0.324056\pi$
$653$	26.8328	1.05005	0.525025	+	0.851087i	$0.324056\pi$
$654$	−1.23607	−0.0483341
$655$	7.79837	0.304708
$656$	−10.3262	−0.403172
$657$	53.3951	2.08314
$658$	9.23607	0.360059
$659$	4.09017	0.159330	0.0796652	−	0.996822i	$-0.474615\pi$
$659$	4.09017	0.159330	0.0796652	+	0.996822i	$0.474615\pi$
$660$	0	0
$661$	−45.8885	−1.78486	−0.892429	−	0.451188i	$-0.851000\pi$
$661$	−45.8885	−1.78486	−0.892429	+	0.451188i	$0.851000\pi$
$662$	6.03444	0.234535
$663$	−8.47214	−0.329030
$664$	10.3262	0.400736
$665$	−6.85410	−0.265791
$666$	−24.9443	−0.966571
$667$	19.4164	0.751806
$668$	6.47214	0.250414
$669$	16.4721	0.636850
$670$	−6.09017	−0.235284
$671$	0	0
$672$	2.61803	0.100993
$673$	−38.8541	−1.49772	−0.748858	−	0.662731i	$-0.769397\pi$
$673$	−38.8541	−1.49772	−0.748858	+	0.662731i	$0.769397\pi$
$674$	−15.6180	−0.601584
$675$	2.23607	0.0860663
$676$	−9.00000	−0.346154
$677$	−13.5967	−0.522565	−0.261283	−	0.965262i	$-0.584145\pi$
$677$	−13.5967	−0.522565	−0.261283	+	0.965262i	$0.584145\pi$
$678$	−24.4164	−0.937706
$679$	−9.56231	−0.366968
$680$	1.61803	0.0620488
$681$	−63.5410	−2.43490
$682$	0	0
$683$	−31.0557	−1.18831	−0.594157	−	0.804349i	$-0.702514\pi$
$683$	−31.0557	−1.18831	−0.594157	+	0.804349i	$0.702514\pi$
$684$	26.4164	1.01006
$685$	1.09017	0.0416533
$686$	1.00000	0.0381802
$687$	29.1246	1.11117
$688$	1.85410	0.0706870
$689$	2.47214	0.0941809
$690$	−15.7082	−0.598001
$691$	21.1459	0.804428	0.402214	−	0.915546i	$-0.368241\pi$
$691$	21.1459	0.804428	0.402214	+	0.915546i	$0.368241\pi$
$692$	12.0000	0.456172
$693$	0	0
$694$	5.32624	0.202181
$695$	−18.4721	−0.700688
$696$	8.47214	0.321135
$697$	16.7082	0.632868
$698$	5.52786	0.209233
$699$	28.5623	1.08033
$700$	1.00000	0.0377964
$701$	−12.1803	−0.460045	−0.230023	−	0.973185i	$-0.573880\pi$
$701$	−12.1803	−0.460045	−0.230023	+	0.973185i	$0.573880\pi$
$702$	4.47214	0.168790
$703$	−44.3607	−1.67309
$704$	0	0
$705$	−24.1803	−0.910684
$706$	−28.4508	−1.07076
$707$	8.94427	0.336384
$708$	−19.9443	−0.749552
$709$	−24.9443	−0.936802	−0.468401	−	0.883516i	$-0.655170\pi$
$709$	−24.9443	−0.936802	−0.468401	+	0.883516i	$0.655170\pi$
$710$	9.70820	0.364342
$711$	−32.6525	−1.22456
$712$	13.0902	0.490575
$713$	−7.41641	−0.277747
$714$	−4.23607	−0.158531
$715$	0	0
$716$	−6.38197	−0.238505
$717$	0	0
$718$	18.6525	0.696104
$719$	32.3607	1.20685	0.603425	−	0.797420i	$-0.293802\pi$
$719$	32.3607	1.20685	0.603425	+	0.797420i	$0.293802\pi$
$720$	−3.85410	−0.143634
$721$	−16.9443	−0.631038
$722$	27.9787	1.04126
$723$	−16.4164	−0.610533
$724$	3.52786	0.131112
$725$	3.23607	0.120185
$726$	0	0
$727$	38.8328	1.44023	0.720115	−	0.693855i	$-0.244089\pi$
$727$	38.8328	1.44023	0.720115	+	0.693855i	$0.244089\pi$
$728$	2.00000	0.0741249
$729$	−39.5623	−1.46527
$730$	−13.8541	−0.512763
$731$	−3.00000	−0.110959
$732$	−9.23607	−0.341375
$733$	−13.2361	−0.488885	−0.244443	−	0.969664i	$-0.578605\pi$
$733$	−13.2361	−0.488885	−0.244443	+	0.969664i	$0.578605\pi$
$734$	28.1803	1.04016
$735$	−2.61803	−0.0965676
$736$	6.00000	0.221163
$737$	0	0
$738$	−39.7984	−1.46500
$739$	−24.6738	−0.907639	−0.453820	−	0.891094i	$-0.649939\pi$
$739$	−24.6738	−0.907639	−0.453820	+	0.891094i	$0.649939\pi$
$740$	6.47214	0.237920
$741$	35.8885	1.31840
$742$	1.23607	0.0453775
$743$	−12.1115	−0.444326	−0.222163	−	0.975010i	$-0.571312\pi$
$743$	−12.1115	−0.444326	−0.222163	+	0.975010i	$0.571312\pi$
$744$	−3.23607	−0.118640
$745$	−24.1803	−0.885899
$746$	−24.4721	−0.895989
$747$	39.7984	1.45615
$748$	0	0
$749$	15.3820	0.562045
$750$	−2.61803	−0.0955971
$751$	−32.6525	−1.19151	−0.595753	−	0.803168i	$-0.703146\pi$
$751$	−32.6525	−1.19151	−0.595753	+	0.803168i	$0.703146\pi$
$752$	9.23607	0.336805
$753$	70.2492	2.56002
$754$	6.47214	0.235701
$755$	−2.29180	−0.0834070
$756$	2.23607	0.0813250
$757$	19.8885	0.722861	0.361431	−	0.932399i	$-0.382288\pi$
$757$	19.8885	0.722861	0.361431	+	0.932399i	$0.382288\pi$
$758$	−27.8541	−1.01171
$759$	0	0
$760$	−6.85410	−0.248624
$761$	−31.7984	−1.15269	−0.576345	−	0.817206i	$-0.695522\pi$
$761$	−31.7984	−1.15269	−0.576345	+	0.817206i	$0.695522\pi$
$762$	−29.8885	−1.08275
$763$	−0.472136	−0.0170925
$764$	−1.52786	−0.0552762
$765$	6.23607	0.225466
$766$	−24.0000	−0.867155
$767$	−15.2361	−0.550143
$768$	2.61803	0.0944702
$769$	−7.88854	−0.284468	−0.142234	−	0.989833i	$-0.545429\pi$
$769$	−7.88854	−0.284468	−0.142234	+	0.989833i	$0.545429\pi$
$770$	0	0
$771$	77.3951	2.78732
$772$	−2.00000	−0.0719816
$773$	−24.6525	−0.886688	−0.443344	−	0.896352i	$-0.646208\pi$
$773$	−24.6525	−0.886688	−0.443344	+	0.896352i	$0.646208\pi$
$774$	7.14590	0.256854
$775$	−1.23607	−0.0444009
$776$	−9.56231	−0.343267
$777$	−16.9443	−0.607872
$778$	−32.8328	−1.17711
$779$	−70.7771	−2.53585
$780$	−5.23607	−0.187481
$781$	0	0
$782$	−9.70820	−0.347165
$783$	7.23607	0.258596
$784$	1.00000	0.0357143
$785$	−0.291796	−0.0104146
$786$	−20.4164	−0.728229
$787$	51.1033	1.82164	0.910818	−	0.412807i	$-0.135452\pi$
$787$	51.1033	1.82164	0.910818	+	0.412807i	$0.135452\pi$
$788$	6.00000	0.213741
$789$	52.8328	1.88090
$790$	8.47214	0.301425
$791$	−9.32624	−0.331603
$792$	0	0
$793$	−7.05573	−0.250556
$794$	−16.9443	−0.601330
$795$	−3.23607	−0.114772
$796$	−19.4164	−0.688196
$797$	−6.36068	−0.225307	−0.112653	−	0.993634i	$-0.535935\pi$
$797$	−6.36068	−0.225307	−0.112653	+	0.993634i	$0.535935\pi$
$798$	17.9443	0.635220
$799$	−14.9443	−0.528690
$800$	1.00000	0.0353553
$801$	50.4508	1.78259
$802$	−26.5066	−0.935980
$803$	0	0
$804$	15.9443	0.562311
$805$	−6.00000	−0.211472
$806$	−2.47214	−0.0870773
$807$	−31.8885	−1.12253
$808$	8.94427	0.314658
$809$	−48.8115	−1.71612	−0.858061	−	0.513548i	$-0.828331\pi$
$809$	−48.8115	−1.71612	−0.858061	+	0.513548i	$0.828331\pi$
$810$	5.70820	0.200566
$811$	−29.9230	−1.05074	−0.525369	−	0.850874i	$-0.676073\pi$
$811$	−29.9230	−1.05074	−0.525369	+	0.850874i	$0.676073\pi$
$812$	3.23607	0.113564
$813$	−24.6525	−0.864600
$814$	0	0
$815$	−2.85410	−0.0999748
$816$	−4.23607	−0.148292
$817$	12.7082	0.444604
$818$	27.8885	0.975100
$819$	7.70820	0.269346
$820$	10.3262	0.360608
$821$	31.0557	1.08385	0.541926	−	0.840426i	$-0.317695\pi$
$821$	31.0557	1.08385	0.541926	+	0.840426i	$0.317695\pi$
$822$	−2.85410	−0.0995482
$823$	−2.83282	−0.0987457	−0.0493729	−	0.998780i	$-0.515722\pi$
$823$	−2.83282	−0.0987457	−0.0493729	+	0.998780i	$0.515722\pi$
$824$	−16.9443	−0.590282
$825$	0	0
$826$	−7.61803	−0.265065
$827$	−40.7426	−1.41676	−0.708380	−	0.705831i	$-0.750574\pi$
$827$	−40.7426	−1.41676	−0.708380	+	0.705831i	$0.750574\pi$
$828$	23.1246	0.803636
$829$	34.1803	1.18713	0.593566	−	0.804785i	$-0.297720\pi$
$829$	34.1803	1.18713	0.593566	+	0.804785i	$0.297720\pi$
$830$	−10.3262	−0.358429
$831$	16.4721	0.571412
$832$	2.00000	0.0693375
$833$	−1.61803	−0.0560616
$834$	48.3607	1.67459
$835$	−6.47214	−0.223978
$836$	0	0
$837$	−2.76393	−0.0955355
$838$	12.3262	0.425803
$839$	4.47214	0.154395	0.0771976	−	0.997016i	$-0.475403\pi$
$839$	4.47214	0.154395	0.0771976	+	0.997016i	$0.475403\pi$
$840$	−2.61803	−0.0903308
$841$	−18.5279	−0.638892
$842$	−6.36068	−0.219204
$843$	−62.9230	−2.16718
$844$	11.2705	0.387947
$845$	9.00000	0.309609
$846$	35.5967	1.22384
$847$	0	0
$848$	1.23607	0.0424467
$849$	−31.4164	−1.07821
$850$	−1.61803	−0.0554981
$851$	−38.8328	−1.33117
$852$	−25.4164	−0.870752
$853$	−4.29180	−0.146948	−0.0734741	−	0.997297i	$-0.523409\pi$
$853$	−4.29180	−0.146948	−0.0734741	+	0.997297i	$0.523409\pi$
$854$	−3.52786	−0.120721
$855$	−26.4164	−0.903422
$856$	15.3820	0.525745
$857$	−9.27051	−0.316675	−0.158337	−	0.987385i	$-0.550613\pi$
$857$	−9.27051	−0.316675	−0.158337	+	0.987385i	$0.550613\pi$
$858$	0	0
$859$	−7.43769	−0.253771	−0.126885	−	0.991917i	$-0.540498\pi$
$859$	−7.43769	−0.253771	−0.126885	+	0.991917i	$0.540498\pi$
$860$	−1.85410	−0.0632244
$861$	−27.0344	−0.921331
$862$	31.2361	1.06390
$863$	6.76393	0.230247	0.115123	−	0.993351i	$-0.463274\pi$
$863$	6.76393	0.230247	0.115123	+	0.993351i	$0.463274\pi$
$864$	2.23607	0.0760726
$865$	−12.0000	−0.408012
$866$	16.4508	0.559023
$867$	−37.6525	−1.27875
$868$	−1.23607	−0.0419549
$869$	0	0
$870$	−8.47214	−0.287232
$871$	12.1803	0.412715
$872$	−0.472136	−0.0159885
$873$	−36.8541	−1.24732
$874$	41.1246	1.39106
$875$	−1.00000	−0.0338062
$876$	36.2705	1.22547
$877$	18.0000	0.607817	0.303908	−	0.952701i	$-0.401708\pi$
$877$	18.0000	0.607817	0.303908	+	0.952701i	$0.401708\pi$
$878$	−28.6525	−0.966974
$879$	43.1246	1.45456
$880$	0	0
$881$	4.14590	0.139679	0.0698394	−	0.997558i	$-0.477751\pi$
$881$	4.14590	0.139679	0.0698394	+	0.997558i	$0.477751\pi$
$882$	3.85410	0.129774
$883$	2.43769	0.0820349	0.0410175	−	0.999158i	$-0.486940\pi$
$883$	2.43769	0.0820349	0.0410175	+	0.999158i	$0.486940\pi$
$884$	−3.23607	−0.108841
$885$	19.9443	0.670419
$886$	28.0344	0.941835
$887$	−4.36068	−0.146417	−0.0732086	−	0.997317i	$-0.523324\pi$
$887$	−4.36068	−0.146417	−0.0732086	+	0.997317i	$0.523324\pi$
$888$	−16.9443	−0.568613
$889$	−11.4164	−0.382894
$890$	−13.0902	−0.438783
$891$	0	0
$892$	6.29180	0.210665
$893$	63.3050	2.11842
$894$	63.3050	2.11723
$895$	6.38197	0.213326
$896$	1.00000	0.0334077
$897$	31.4164	1.04896
$898$	24.4508	0.815935
$899$	−4.00000	−0.133407
$900$	3.85410	0.128470
$901$	−2.00000	−0.0666297
$902$	0	0
$903$	4.85410	0.161534
$904$	−9.32624	−0.310186
$905$	−3.52786	−0.117270
$906$	6.00000	0.199337
$907$	22.2705	0.739480	0.369740	−	0.929135i	$-0.379447\pi$
$907$	22.2705	0.739480	0.369740	+	0.929135i	$0.379447\pi$
$908$	−24.2705	−0.805445
$909$	34.4721	1.14337
$910$	−2.00000	−0.0662994
$911$	−51.5967	−1.70948	−0.854738	−	0.519059i	$-0.826282\pi$
$911$	−51.5967	−1.70948	−0.854738	+	0.519059i	$0.826282\pi$
$912$	17.9443	0.594194
$913$	0	0
$914$	−31.2705	−1.03434
$915$	9.23607	0.305335
$916$	11.1246	0.367568
$917$	−7.79837	−0.257525
$918$	−3.61803	−0.119413
$919$	−8.83282	−0.291368	−0.145684	−	0.989331i	$-0.546538\pi$
$919$	−8.83282	−0.291368	−0.145684	+	0.989331i	$0.546538\pi$
$920$	−6.00000	−0.197814
$921$	−72.1591	−2.37772
$922$	6.94427	0.228697
$923$	−19.4164	−0.639099
$924$	0	0
$925$	−6.47214	−0.212803
$926$	−7.41641	−0.243718
$927$	−65.3050	−2.14490
$928$	3.23607	0.106229
$929$	−3.25735	−0.106870	−0.0534352	−	0.998571i	$-0.517017\pi$
$929$	−3.25735	−0.106870	−0.0534352	+	0.998571i	$0.517017\pi$
$930$	3.23607	0.106115
$931$	6.85410	0.224634
$932$	10.9098	0.357363
$933$	−76.2492	−2.49629
$934$	−17.8885	−0.585331
$935$	0	0
$936$	7.70820	0.251951
$937$	33.3951	1.09097	0.545486	−	0.838120i	$-0.316345\pi$
$937$	33.3951	1.09097	0.545486	+	0.838120i	$0.316345\pi$
$938$	6.09017	0.198851
$939$	−2.23607	−0.0729713
$940$	−9.23607	−0.301247
$941$	34.1803	1.11425	0.557124	−	0.830430i	$-0.311905\pi$
$941$	34.1803	1.11425	0.557124	+	0.830430i	$0.311905\pi$
$942$	0.763932	0.0248903
$943$	−61.9574	−2.01761
$944$	−7.61803	−0.247946
$945$	−2.23607	−0.0727393
$946$	0	0
$947$	−41.9098	−1.36189	−0.680943	−	0.732336i	$-0.738430\pi$
$947$	−41.9098	−1.36189	−0.680943	+	0.732336i	$0.738430\pi$
$948$	−22.1803	−0.720384
$949$	27.7082	0.899446
$950$	6.85410	0.222376
$951$	−54.3607	−1.76277
$952$	−1.61803	−0.0524408
$953$	29.4508	0.954007	0.477003	−	0.878902i	$-0.341723\pi$
$953$	29.4508	0.954007	0.477003	+	0.878902i	$0.341723\pi$
$954$	4.76393	0.154238
$955$	1.52786	0.0494405
$956$	0	0
$957$	0	0
$958$	−37.8885	−1.22412
$959$	−1.09017	−0.0352034
$960$	−2.61803	−0.0844967
$961$	−29.4721	−0.950714
$962$	−12.9443	−0.417340
$963$	59.2837	1.91039
$964$	−6.27051	−0.201960
$965$	2.00000	0.0643823
$966$	15.7082	0.505403
$967$	27.0132	0.868685	0.434342	−	0.900748i	$-0.356981\pi$
$967$	27.0132	0.868685	0.434342	+	0.900748i	$0.356981\pi$
$968$	0	0
$969$	−29.0344	−0.932721
$970$	9.56231	0.307027
$971$	−26.8328	−0.861106	−0.430553	−	0.902565i	$-0.641681\pi$
$971$	−26.8328	−0.861106	−0.430553	+	0.902565i	$0.641681\pi$
$972$	−21.6525	−0.694503
$973$	18.4721	0.592189
$974$	−26.8328	−0.859779
$975$	5.23607	0.167688
$976$	−3.52786	−0.112924
$977$	43.3050	1.38545	0.692724	−	0.721203i	$-0.256410\pi$
$977$	43.3050	1.38545	0.692724	+	0.721203i	$0.256410\pi$
$978$	7.47214	0.238933
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−1.81966	−0.0580973
$982$	−32.5066	−1.03733
$983$	−14.6525	−0.467341	−0.233671	−	0.972316i	$-0.575074\pi$
$983$	−14.6525	−0.467341	−0.233671	+	0.972316i	$0.575074\pi$
$984$	−27.0344	−0.861827
$985$	−6.00000	−0.191176
$986$	−5.23607	−0.166750
$987$	24.1803	0.769669
$988$	13.7082	0.436116
$989$	11.1246	0.353742
$990$	0	0
$991$	20.6525	0.656048	0.328024	−	0.944669i	$-0.393617\pi$
$991$	20.6525	0.656048	0.328024	+	0.944669i	$0.393617\pi$
$992$	−1.23607	−0.0392452
$993$	15.7984	0.501346
$994$	−9.70820	−0.307926
$995$	19.4164	0.615542
$996$	27.0344	0.856619
$997$	−23.1246	−0.732364	−0.366182	−	0.930543i	$-0.619335\pi$
$997$	−23.1246	−0.732364	−0.366182	+	0.930543i	$0.619335\pi$
$998$	−15.2016	−0.481199
$999$	−14.4721	−0.457878

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.cf.1.2		2
11.7	odd	10		770.2.n.b.71.1	✓	4
11.8	odd	10		770.2.n.b.141.1	yes	4
11.10	odd	2		8470.2.a.bt.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.b.71.1	✓	4	11.7	odd	10
770.2.n.b.141.1	yes	4	11.8	odd	10
8470.2.a.bt.1.2		2	11.10	odd	2
8470.2.a.cf.1.2		2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +2.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.61803 q^{6} +1.00000 q^{7} +1.00000 q^{8} +3.85410 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +2.61803 q^{3} +1.00000 q^{4} -1.00000 q^{5} +2.61803 q^{6} +1.00000 q^{7} +1.00000 q^{8} +3.85410 q^{9} -1.00000 q^{10} +2.61803 q^{12} +2.00000 q^{13} +1.00000 q^{14} -2.61803 q^{15} +1.00000 q^{16} -1.61803 q^{17} +3.85410 q^{18} +6.85410 q^{19} -1.00000 q^{20} +2.61803 q^{21} +6.00000 q^{23} +2.61803 q^{24} +1.00000 q^{25} +2.00000 q^{26} +2.23607 q^{27} +1.00000 q^{28} +3.23607 q^{29} -2.61803 q^{30} -1.23607 q^{31} +1.00000 q^{32} -1.61803 q^{34} -1.00000 q^{35} +3.85410 q^{36} -6.47214 q^{37} +6.85410 q^{38} +5.23607 q^{39} -1.00000 q^{40} -10.3262 q^{41} +2.61803 q^{42} +1.85410 q^{43} -3.85410 q^{45} +6.00000 q^{46} +9.23607 q^{47} +2.61803 q^{48} +1.00000 q^{49} +1.00000 q^{50} -4.23607 q^{51} +2.00000 q^{52} +1.23607 q^{53} +2.23607 q^{54} +1.00000 q^{56} +17.9443 q^{57} +3.23607 q^{58} -7.61803 q^{59} -2.61803 q^{60} -3.52786 q^{61} -1.23607 q^{62} +3.85410 q^{63} +1.00000 q^{64} -2.00000 q^{65} +6.09017 q^{67} -1.61803 q^{68} +15.7082 q^{69} -1.00000 q^{70} -9.70820 q^{71} +3.85410 q^{72} +13.8541 q^{73} -6.47214 q^{74} +2.61803 q^{75} +6.85410 q^{76} +5.23607 q^{78} -8.47214 q^{79} -1.00000 q^{80} -5.70820 q^{81} -10.3262 q^{82} +10.3262 q^{83} +2.61803 q^{84} +1.61803 q^{85} +1.85410 q^{86} +8.47214 q^{87} +13.0902 q^{89} -3.85410 q^{90} +2.00000 q^{91} +6.00000 q^{92} -3.23607 q^{93} +9.23607 q^{94} -6.85410 q^{95} +2.61803 q^{96} -9.56231 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 2 q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - 2 * q^5 + 3 * q^6 + 2 * q^7 + 2 * q^8 + q^9	\( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} - 2 q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + q^{9} - 2 q^{10} + 3 q^{12} + 4 q^{13} + 2 q^{14} - 3 q^{15} + 2 q^{16} - q^{17} + q^{18} + 7 q^{19} - 2 q^{20} + 3 q^{21} + 12 q^{23} + 3 q^{24} + 2 q^{25} + 4 q^{26} + 2 q^{28} + 2 q^{29} - 3 q^{30} + 2 q^{31} + 2 q^{32} - q^{34} - 2 q^{35} + q^{36} - 4 q^{37} + 7 q^{38} + 6 q^{39} - 2 q^{40} - 5 q^{41} + 3 q^{42} - 3 q^{43} - q^{45} + 12 q^{46} + 14 q^{47} + 3 q^{48} + 2 q^{49} + 2 q^{50} - 4 q^{51} + 4 q^{52} - 2 q^{53} + 2 q^{56} + 18 q^{57} + 2 q^{58} - 13 q^{59} - 3 q^{60} - 16 q^{61} + 2 q^{62} + q^{63} + 2 q^{64} - 4 q^{65} + q^{67} - q^{68} + 18 q^{69} - 2 q^{70} - 6 q^{71} + q^{72} + 21 q^{73} - 4 q^{74} + 3 q^{75} + 7 q^{76} + 6 q^{78} - 8 q^{79} - 2 q^{80} + 2 q^{81} - 5 q^{82} + 5 q^{83} + 3 q^{84} + q^{85} - 3 q^{86} + 8 q^{87} + 15 q^{89} - q^{90} + 4 q^{91} + 12 q^{92} - 2 q^{93} + 14 q^{94} - 7 q^{95} + 3 q^{96} + q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 3 * q^3 + 2 * q^4 - 2 * q^5 + 3 * q^6 + 2 * q^7 + 2 * q^8 + q^9 - 2 * q^10 + 3 * q^12 + 4 * q^13 + 2 * q^14 - 3 * q^15 + 2 * q^16 - q^17 + q^18 + 7 * q^19 - 2 * q^20 + 3 * q^21 + 12 * q^23 + 3 * q^24 + 2 * q^25 + 4 * q^26 + 2 * q^28 + 2 * q^29 - 3 * q^30 + 2 * q^31 + 2 * q^32 - q^34 - 2 * q^35 + q^36 - 4 * q^37 + 7 * q^38 + 6 * q^39 - 2 * q^40 - 5 * q^41 + 3 * q^42 - 3 * q^43 - q^45 + 12 * q^46 + 14 * q^47 + 3 * q^48 + 2 * q^49 + 2 * q^50 - 4 * q^51 + 4 * q^52 - 2 * q^53 + 2 * q^56 + 18 * q^57 + 2 * q^58 - 13 * q^59 - 3 * q^60 - 16 * q^61 + 2 * q^62 + q^63 + 2 * q^64 - 4 * q^65 + q^67 - q^68 + 18 * q^69 - 2 * q^70 - 6 * q^71 + q^72 + 21 * q^73 - 4 * q^74 + 3 * q^75 + 7 * q^76 + 6 * q^78 - 8 * q^79 - 2 * q^80 + 2 * q^81 - 5 * q^82 + 5 * q^83 + 3 * q^84 + q^85 - 3 * q^86 + 8 * q^87 + 15 * q^89 - q^90 + 4 * q^91 + 12 * q^92 - 2 * q^93 + 14 * q^94 - 7 * q^95 + 3 * q^96 + q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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