LMFDB - Embedded newform 8470.2.a.dg.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$67.6332905120$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 16x^{6} + 69x^{4} - 10x^{3} - 70x^{2} + 10x + 5 $ x^8 - 16x^6 + 69x^4 - 10x^3 - 70x^2 + 10*x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.365778$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.365778 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.365778 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.86621 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.365778 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.365778 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.86621 q^{9} +1.00000 q^{10} +0.365778 q^{12} +2.20049 q^{13} -1.00000 q^{14} -0.365778 q^{15} +1.00000 q^{16} -6.26330 q^{17} +2.86621 q^{18} +6.00967 q^{19} -1.00000 q^{20} +0.365778 q^{21} +0.452126 q^{23} -0.365778 q^{24} +1.00000 q^{25} -2.20049 q^{26} -2.14573 q^{27} +1.00000 q^{28} +1.90770 q^{29} +0.365778 q^{30} -6.05116 q^{31} -1.00000 q^{32} +6.26330 q^{34} -1.00000 q^{35} -2.86621 q^{36} +3.54127 q^{37} -6.00967 q^{38} +0.804890 q^{39} +1.00000 q^{40} +0.0206760 q^{41} -0.365778 q^{42} -3.71582 q^{43} +2.86621 q^{45} -0.452126 q^{46} +8.63329 q^{47} +0.365778 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.29098 q^{51} +2.20049 q^{52} +8.03517 q^{53} +2.14573 q^{54} -1.00000 q^{56} +2.19820 q^{57} -1.90770 q^{58} -12.7279 q^{59} -0.365778 q^{60} -6.02013 q^{61} +6.05116 q^{62} -2.86621 q^{63} +1.00000 q^{64} -2.20049 q^{65} +8.19469 q^{67} -6.26330 q^{68} +0.165378 q^{69} +1.00000 q^{70} +15.7605 q^{71} +2.86621 q^{72} -12.7842 q^{73} -3.54127 q^{74} +0.365778 q^{75} +6.00967 q^{76} -0.804890 q^{78} -10.7445 q^{79} -1.00000 q^{80} +7.81376 q^{81} -0.0206760 q^{82} -10.9679 q^{83} +0.365778 q^{84} +6.26330 q^{85} +3.71582 q^{86} +0.697795 q^{87} -9.56469 q^{89} -2.86621 q^{90} +2.20049 q^{91} +0.452126 q^{92} -2.21338 q^{93} -8.63329 q^{94} -6.00967 q^{95} -0.365778 q^{96} -4.22569 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} + 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^5 + 8 * q^7 - 8 * q^8 + 8 * q^9	$ 8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} + 8 q^{7} - 8 q^{8} + 8 q^{9} + 8 q^{10} + q^{13} - 8 q^{14} + 8 q^{16} - 6 q^{17} - 8 q^{18} - 5 q^{19} - 8 q^{20} + 10 q^{23} + 8 q^{25} - q^{26} + 8 q^{28} - 3 q^{29} - 8 q^{31} - 8 q^{32} + 6 q^{34} - 8 q^{35} + 8 q^{36} - 6 q^{37} + 5 q^{38} - 35 q^{39} + 8 q^{40} - 11 q^{41} + 5 q^{43} - 8 q^{45} - 10 q^{46} - 15 q^{47} + 8 q^{49} - 8 q^{50} + 6 q^{51} + q^{52} - 16 q^{53} - 8 q^{56} - 38 q^{57} + 3 q^{58} - 9 q^{59} - 32 q^{61} + 8 q^{62} + 8 q^{63} + 8 q^{64} - q^{65} + 33 q^{67} - 6 q^{68} - 22 q^{69} + 8 q^{70} + 11 q^{71} - 8 q^{72} + 34 q^{73} + 6 q^{74} - 5 q^{76} + 35 q^{78} - 31 q^{79} - 8 q^{80} + 20 q^{81} + 11 q^{82} - 50 q^{83} + 6 q^{85} - 5 q^{86} + 12 q^{87} + q^{89} + 8 q^{90} + q^{91} + 10 q^{92} + 26 q^{93} + 15 q^{94} + 5 q^{95} - 4 q^{97} - 8 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^5 + 8 * q^7 - 8 * q^8 + 8 * q^9 + 8 * q^10 + q^13 - 8 * q^14 + 8 * q^16 - 6 * q^17 - 8 * q^18 - 5 * q^19 - 8 * q^20 + 10 * q^23 + 8 * q^25 - q^26 + 8 * q^28 - 3 * q^29 - 8 * q^31 - 8 * q^32 + 6 * q^34 - 8 * q^35 + 8 * q^36 - 6 * q^37 + 5 * q^38 - 35 * q^39 + 8 * q^40 - 11 * q^41 + 5 * q^43 - 8 * q^45 - 10 * q^46 - 15 * q^47 + 8 * q^49 - 8 * q^50 + 6 * q^51 + q^52 - 16 * q^53 - 8 * q^56 - 38 * q^57 + 3 * q^58 - 9 * q^59 - 32 * q^61 + 8 * q^62 + 8 * q^63 + 8 * q^64 - q^65 + 33 * q^67 - 6 * q^68 - 22 * q^69 + 8 * q^70 + 11 * q^71 - 8 * q^72 + 34 * q^73 + 6 * q^74 - 5 * q^76 + 35 * q^78 - 31 * q^79 - 8 * q^80 + 20 * q^81 + 11 * q^82 - 50 * q^83 + 6 * q^85 - 5 * q^86 + 12 * q^87 + q^89 + 8 * q^90 + q^91 + 10 * q^92 + 26 * q^93 + 15 * q^94 + 5 * q^95 - 4 * q^97 - 8 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0.365778	0.211182	0.105591	−	0.994410i	$-0.466327\pi$
$3$	0.365778	0.211182	0.105591	+	0.994410i	$0.466327\pi$
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.365778	−0.149328
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	−2.86621	−0.955402
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	0.365778	0.105591
$13$	2.20049	0.610306	0.305153	−	0.952303i	$-0.401292\pi$
$13$	2.20049	0.610306	0.305153	+	0.952303i	$0.401292\pi$
$14$	−1.00000	−0.267261
$15$	−0.365778	−0.0944434
$16$	1.00000	0.250000
$17$	−6.26330	−1.51907	−0.759537	−	0.650464i	$-0.774575\pi$
$17$	−6.26330	−1.51907	−0.759537	+	0.650464i	$0.774575\pi$
$18$	2.86621	0.675571
$19$	6.00967	1.37871	0.689356	−	0.724422i	$-0.257893\pi$
$19$	6.00967	1.37871	0.689356	+	0.724422i	$0.257893\pi$
$20$	−1.00000	−0.223607
$21$	0.365778	0.0798193
$22$	0	0
$23$	0.452126	0.0942748	0.0471374	−	0.998888i	$-0.484990\pi$
$23$	0.452126	0.0942748	0.0471374	+	0.998888i	$0.484990\pi$
$24$	−0.365778	−0.0746641
$25$	1.00000	0.200000
$26$	−2.20049	−0.431551
$27$	−2.14573	−0.412946
$28$	1.00000	0.188982
$29$	1.90770	0.354251	0.177126	−	0.984188i	$-0.443320\pi$
$29$	1.90770	0.354251	0.177126	+	0.984188i	$0.443320\pi$
$30$	0.365778	0.0667816
$31$	−6.05116	−1.08682	−0.543411	−	0.839467i	$-0.682867\pi$
$31$	−6.05116	−1.08682	−0.543411	+	0.839467i	$0.682867\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	6.26330	1.07415
$35$	−1.00000	−0.169031
$36$	−2.86621	−0.477701
$37$	3.54127	0.582181	0.291090	−	0.956696i	$-0.405982\pi$
$37$	3.54127	0.582181	0.291090	+	0.956696i	$0.405982\pi$
$38$	−6.00967	−0.974897
$39$	0.804890	0.128886
$40$	1.00000	0.158114
$41$	0.0206760	0.00322905	0.00161452	−	0.999999i	$-0.499486\pi$
$41$	0.0206760	0.00322905	0.00161452	+	0.999999i	$0.499486\pi$
$42$	−0.365778	−0.0564407
$43$	−3.71582	−0.566657	−0.283329	−	0.959023i	$-0.591439\pi$
$43$	−3.71582	−0.566657	−0.283329	+	0.959023i	$0.591439\pi$
$44$	0	0
$45$	2.86621	0.427269
$46$	−0.452126	−0.0666624
$47$	8.63329	1.25929	0.629647	−	0.776881i	$-0.283199\pi$
$47$	8.63329	1.25929	0.629647	+	0.776881i	$0.283199\pi$
$48$	0.365778	0.0527955
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−2.29098	−0.320801
$52$	2.20049	0.305153
$53$	8.03517	1.10372	0.551858	−	0.833938i	$-0.313919\pi$
$53$	8.03517	1.10372	0.551858	+	0.833938i	$0.313919\pi$
$54$	2.14573	0.291997
$55$	0	0
$56$	−1.00000	−0.133631
$57$	2.19820	0.291159
$58$	−1.90770	−0.250493
$59$	−12.7279	−1.65703	−0.828516	−	0.559965i	$-0.810814\pi$
$59$	−12.7279	−1.65703	−0.828516	+	0.559965i	$0.810814\pi$
$60$	−0.365778	−0.0472217
$61$	−6.02013	−0.770798	−0.385399	−	0.922750i	$-0.625936\pi$
$61$	−6.02013	−0.770798	−0.385399	+	0.922750i	$0.625936\pi$
$62$	6.05116	0.768499
$63$	−2.86621	−0.361108
$64$	1.00000	0.125000
$65$	−2.20049	−0.272937
$66$	0	0
$67$	8.19469	1.00114	0.500570	−	0.865696i	$-0.333124\pi$
$67$	8.19469	1.00114	0.500570	+	0.865696i	$0.333124\pi$
$68$	−6.26330	−0.759537
$69$	0.165378	0.0199091
$70$	1.00000	0.119523
$71$	15.7605	1.87043	0.935216	−	0.354077i	$-0.115205\pi$
$71$	15.7605	1.87043	0.935216	+	0.354077i	$0.115205\pi$
$72$	2.86621	0.337786
$73$	−12.7842	−1.49628	−0.748139	−	0.663542i	$-0.769053\pi$
$73$	−12.7842	−1.49628	−0.748139	+	0.663542i	$0.769053\pi$
$74$	−3.54127	−0.411664
$75$	0.365778	0.0422364
$76$	6.00967	0.689356
$77$	0	0
$78$	−0.804890	−0.0911358
$79$	−10.7445	−1.20885	−0.604425	−	0.796662i	$-0.706597\pi$
$79$	−10.7445	−1.20885	−0.604425	+	0.796662i	$0.706597\pi$
$80$	−1.00000	−0.111803
$81$	7.81376	0.868196
$82$	−0.0206760	−0.00228328
$83$	−10.9679	−1.20388	−0.601942	−	0.798540i	$-0.705606\pi$
$83$	−10.9679	−1.20388	−0.601942	+	0.798540i	$0.705606\pi$
$84$	0.365778	0.0399096
$85$	6.26330	0.679351
$86$	3.71582	0.400687
$87$	0.697795	0.0748115
$88$	0	0
$89$	−9.56469	−1.01385	−0.506927	−	0.861989i	$-0.669219\pi$
$89$	−9.56469	−1.01385	−0.506927	+	0.861989i	$0.669219\pi$
$90$	−2.86621	−0.302125
$91$	2.20049	0.230674
$92$	0.452126	0.0471374
$93$	−2.21338	−0.229517
$94$	−8.63329	−0.890455
$95$	−6.00967	−0.616579
$96$	−0.365778	−0.0373320
$97$	−4.22569	−0.429053	−0.214527	−	0.976718i	$-0.568821\pi$
$97$	−4.22569	−0.429053	−0.214527	+	0.976718i	$0.568821\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	10.0515	1.00016	0.500080	−	0.865979i	$-0.333304\pi$
$101$	10.0515	1.00016	0.500080	+	0.865979i	$0.333304\pi$
$102$	2.29098	0.226841
$103$	14.6726	1.44573	0.722867	−	0.690987i	$-0.242824\pi$
$103$	14.6726	1.44573	0.722867	+	0.690987i	$0.242824\pi$
$104$	−2.20049	−0.215776
$105$	−0.365778	−0.0356963
$106$	−8.03517	−0.780445
$107$	1.85065	0.178909	0.0894545	−	0.995991i	$-0.471488\pi$
$107$	1.85065	0.178909	0.0894545	+	0.995991i	$0.471488\pi$
$108$	−2.14573	−0.206473
$109$	16.3800	1.56892	0.784459	−	0.620181i	$-0.212941\pi$
$109$	16.3800	1.56892	0.784459	+	0.620181i	$0.212941\pi$
$110$	0	0
$111$	1.29532	0.122946
$112$	1.00000	0.0944911
$113$	−10.7686	−1.01303	−0.506514	−	0.862232i	$-0.669066\pi$
$113$	−10.7686	−1.01303	−0.506514	+	0.862232i	$0.669066\pi$
$114$	−2.19820	−0.205881
$115$	−0.452126	−0.0421610
$116$	1.90770	0.177126
$117$	−6.30706	−0.583087
$118$	12.7279	1.17170
$119$	−6.26330	−0.574156
$120$	0.365778	0.0333908
$121$	0	0
$122$	6.02013	0.545037
$123$	0.00756282	0.000681917 0
$124$	−6.05116	−0.543411
$125$	−1.00000	−0.0894427
$126$	2.86621	0.255342
$127$	−4.44315	−0.394266	−0.197133	−	0.980377i	$-0.563163\pi$
$127$	−4.44315	−0.394266	−0.197133	+	0.980377i	$0.563163\pi$
$128$	−1.00000	−0.0883883
$129$	−1.35916	−0.119668
$130$	2.20049	0.192996
$131$	2.54949	0.222750	0.111375	−	0.993778i	$-0.464474\pi$
$131$	2.54949	0.222750	0.111375	+	0.993778i	$0.464474\pi$
$132$	0	0
$133$	6.00967	0.521104
$134$	−8.19469	−0.707913
$135$	2.14573	0.184675
$136$	6.26330	0.537074
$137$	−15.1630	−1.29546	−0.647730	−	0.761870i	$-0.724281\pi$
$137$	−15.1630	−1.29546	−0.647730	+	0.761870i	$0.724281\pi$
$138$	−0.165378	−0.0140779
$139$	−4.56534	−0.387227	−0.193614	−	0.981078i	$-0.562021\pi$
$139$	−4.56534	−0.387227	−0.193614	+	0.981078i	$0.562021\pi$
$140$	−1.00000	−0.0845154
$141$	3.15786	0.265940
$142$	−15.7605	−1.32260
$143$	0	0
$144$	−2.86621	−0.238851
$145$	−1.90770	−0.158426
$146$	12.7842	1.05803
$147$	0.365778	0.0301688
$148$	3.54127	0.291090
$149$	−13.9113	−1.13966	−0.569830	−	0.821762i	$-0.692991\pi$
$149$	−13.9113	−1.13966	−0.569830	+	0.821762i	$0.692991\pi$
$150$	−0.365778	−0.0298656
$151$	−3.78025	−0.307632	−0.153816	−	0.988099i	$-0.549156\pi$
$151$	−3.78025	−0.307632	−0.153816	+	0.988099i	$0.549156\pi$
$152$	−6.00967	−0.487449
$153$	17.9519	1.45133
$154$	0	0
$155$	6.05116	0.486041
$156$	0.804890	0.0644428
$157$	12.6782	1.01183	0.505916	−	0.862583i	$-0.331154\pi$
$157$	12.6782	1.01183	0.505916	+	0.862583i	$0.331154\pi$
$158$	10.7445	0.854787
$159$	2.93909	0.233085
$160$	1.00000	0.0790569
$161$	0.452126	0.0356325
$162$	−7.81376	−0.613907
$163$	18.8534	1.47671	0.738356	−	0.674412i	$-0.235603\pi$
$163$	18.8534	1.47671	0.738356	+	0.674412i	$0.235603\pi$
$164$	0.0206760	0.00161452
$165$	0	0
$166$	10.9679	0.851274
$167$	−15.5834	−1.20588	−0.602939	−	0.797787i	$-0.706004\pi$
$167$	−15.5834	−1.20588	−0.602939	+	0.797787i	$0.706004\pi$
$168$	−0.365778	−0.0282204
$169$	−8.15785	−0.627527
$170$	−6.26330	−0.480374
$171$	−17.2250	−1.31723
$172$	−3.71582	−0.283329
$173$	−6.79145	−0.516344	−0.258172	−	0.966099i	$-0.583120\pi$
$173$	−6.79145	−0.516344	−0.258172	+	0.966099i	$0.583120\pi$
$174$	−0.697795	−0.0528997
$175$	1.00000	0.0755929
$176$	0	0
$177$	−4.65559	−0.349935
$178$	9.56469	0.716904
$179$	−23.7293	−1.77361	−0.886807	−	0.462139i	$-0.847082\pi$
$179$	−23.7293	−1.77361	−0.886807	+	0.462139i	$0.847082\pi$
$180$	2.86621	0.213634
$181$	14.5332	1.08024	0.540122	−	0.841587i	$-0.318378\pi$
$181$	14.5332	1.08024	0.540122	+	0.841587i	$0.318378\pi$
$182$	−2.20049	−0.163111
$183$	−2.20203	−0.162779
$184$	−0.452126	−0.0333312
$185$	−3.54127	−0.260359
$186$	2.21338	0.162293
$187$	0	0
$188$	8.63329	0.629647
$189$	−2.14573	−0.156079
$190$	6.00967	0.435987
$191$	−6.74284	−0.487895	−0.243947	−	0.969789i	$-0.578442\pi$
$191$	−6.74284	−0.487895	−0.243947	+	0.969789i	$0.578442\pi$
$192$	0.365778	0.0263977
$193$	10.4918	0.755220	0.377610	−	0.925965i	$-0.376746\pi$
$193$	10.4918	0.755220	0.377610	+	0.925965i	$0.376746\pi$
$194$	4.22569	0.303387
$195$	−0.804890	−0.0576394
$196$	1.00000	0.0714286
$197$	8.40324	0.598706	0.299353	−	0.954142i	$-0.403229\pi$
$197$	8.40324	0.598706	0.299353	+	0.954142i	$0.403229\pi$
$198$	0	0
$199$	5.41088	0.383567	0.191783	−	0.981437i	$-0.438573\pi$
$199$	5.41088	0.383567	0.191783	+	0.981437i	$0.438573\pi$
$200$	−1.00000	−0.0707107
$201$	2.99744	0.211423
$202$	−10.0515	−0.707220
$203$	1.90770	0.133894
$204$	−2.29098	−0.160401
$205$	−0.0206760	−0.00144407
$206$	−14.6726	−1.02229
$207$	−1.29589	−0.0900704
$208$	2.20049	0.152576
$209$	0	0
$210$	0.365778	0.0252411
$211$	−8.53391	−0.587499	−0.293749	−	0.955882i	$-0.594903\pi$
$211$	−8.53391	−0.587499	−0.293749	+	0.955882i	$0.594903\pi$
$212$	8.03517	0.551858
$213$	5.76486	0.395001
$214$	−1.85065	−0.126508
$215$	3.71582	0.253417
$216$	2.14573	0.145998
$217$	−6.05116	−0.410780
$218$	−16.3800	−1.10939
$219$	−4.67618	−0.315987
$220$	0	0
$221$	−13.7823	−0.927100
$222$	−1.29532	−0.0869360
$223$	−19.6449	−1.31552	−0.657760	−	0.753228i	$-0.728496\pi$
$223$	−19.6449	−1.31552	−0.657760	+	0.753228i	$0.728496\pi$
$224$	−1.00000	−0.0668153
$225$	−2.86621	−0.191080
$226$	10.7686	0.716318
$227$	−2.37742	−0.157795	−0.0788975	−	0.996883i	$-0.525140\pi$
$227$	−2.37742	−0.157795	−0.0788975	+	0.996883i	$0.525140\pi$
$228$	2.19820	0.145580
$229$	9.73039	0.643002	0.321501	−	0.946909i	$-0.395813\pi$
$229$	9.73039	0.643002	0.321501	+	0.946909i	$0.395813\pi$
$230$	0.452126	0.0298123
$231$	0	0
$232$	−1.90770	−0.125247
$233$	−18.4540	−1.20896	−0.604482	−	0.796619i	$-0.706620\pi$
$233$	−18.4540	−1.20896	−0.604482	+	0.796619i	$0.706620\pi$
$234$	6.30706	0.412305
$235$	−8.63329	−0.563173
$236$	−12.7279	−0.828516
$237$	−3.93010	−0.255287
$238$	6.26330	0.405990
$239$	−17.6068	−1.13889	−0.569445	−	0.822030i	$-0.692842\pi$
$239$	−17.6068	−1.13889	−0.569445	+	0.822030i	$0.692842\pi$
$240$	−0.365778	−0.0236109
$241$	−17.4835	−1.12621	−0.563105	−	0.826385i	$-0.690393\pi$
$241$	−17.4835	−1.12621	−0.563105	+	0.826385i	$0.690393\pi$
$242$	0	0
$243$	9.29528	0.596293
$244$	−6.02013	−0.385399
$245$	−1.00000	−0.0638877
$246$	−0.00756282	−0.000482188 0
$247$	13.2242	0.841436
$248$	6.05116	0.384249
$249$	−4.01182	−0.254239
$250$	1.00000	0.0632456
$251$	−22.9585	−1.44913	−0.724563	−	0.689209i	$-0.757958\pi$
$251$	−22.9585	−1.44913	−0.724563	+	0.689209i	$0.757958\pi$
$252$	−2.86621	−0.180554
$253$	0	0
$254$	4.44315	0.278788
$255$	2.29098	0.143467
$256$	1.00000	0.0625000
$257$	14.2609	0.889569	0.444784	−	0.895638i	$-0.353280\pi$
$257$	14.2609	0.889569	0.444784	+	0.895638i	$0.353280\pi$
$258$	1.35916	0.0846179
$259$	3.54127	0.220044
$260$	−2.20049	−0.136469
$261$	−5.46787	−0.338452
$262$	−2.54949	−0.157508
$263$	−9.29262	−0.573007	−0.286504	−	0.958079i	$-0.592493\pi$
$263$	−9.29262	−0.573007	−0.286504	+	0.958079i	$0.592493\pi$
$264$	0	0
$265$	−8.03517	−0.493597
$266$	−6.00967	−0.368476
$267$	−3.49855	−0.214108
$268$	8.19469	0.500570
$269$	1.17894	0.0718811	0.0359405	−	0.999354i	$-0.488557\pi$
$269$	1.17894	0.0718811	0.0359405	+	0.999354i	$0.488557\pi$
$270$	−2.14573	−0.130585
$271$	8.12467	0.493539	0.246769	−	0.969074i	$-0.420631\pi$
$271$	8.12467	0.493539	0.246769	+	0.969074i	$0.420631\pi$
$272$	−6.26330	−0.379769
$273$	0.804890	0.0487142
$274$	15.1630	0.916028
$275$	0	0
$276$	0.165378	0.00995457
$277$	−8.23042	−0.494518	−0.247259	−	0.968949i	$-0.579530\pi$
$277$	−8.23042	−0.494518	−0.247259	+	0.968949i	$0.579530\pi$
$278$	4.56534	0.273811
$279$	17.3439	1.03835
$280$	1.00000	0.0597614
$281$	−21.0726	−1.25709	−0.628543	−	0.777775i	$-0.716348\pi$
$281$	−21.0726	−1.25709	−0.628543	+	0.777775i	$0.716348\pi$
$282$	−3.15786	−0.188048
$283$	−0.468381	−0.0278423	−0.0139212	−	0.999903i	$-0.504431\pi$
$283$	−0.468381	−0.0278423	−0.0139212	+	0.999903i	$0.504431\pi$
$284$	15.7605	0.935216
$285$	−2.19820	−0.130210
$286$	0	0
$287$	0.0206760	0.00122047
$288$	2.86621	0.168893
$289$	22.2290	1.30759
$290$	1.90770	0.112024
$291$	−1.54566	−0.0906083
$292$	−12.7842	−0.748139
$293$	6.71911	0.392535	0.196267	−	0.980550i	$-0.437118\pi$
$293$	6.71911	0.392535	0.196267	+	0.980550i	$0.437118\pi$
$294$	−0.365778	−0.0213326
$295$	12.7279	0.741047
$296$	−3.54127	−0.205832
$297$	0	0
$298$	13.9113	0.805862
$299$	0.994899	0.0575365
$300$	0.365778	0.0211182
$301$	−3.71582	−0.214176
$302$	3.78025	0.217529
$303$	3.67661	0.211216
$304$	6.00967	0.344678
$305$	6.02013	0.344712
$306$	−17.9519	−1.02624
$307$	−22.7190	−1.29664	−0.648321	−	0.761367i	$-0.724529\pi$
$307$	−22.7190	−1.29664	−0.648321	+	0.761367i	$0.724529\pi$
$308$	0	0
$309$	5.36691	0.305313
$310$	−6.05116	−0.343683
$311$	−21.2105	−1.20274	−0.601368	−	0.798972i	$-0.705377\pi$
$311$	−21.2105	−1.20274	−0.601368	+	0.798972i	$0.705377\pi$
$312$	−0.804890	−0.0455679
$313$	−3.70075	−0.209179	−0.104589	−	0.994515i	$-0.533353\pi$
$313$	−3.70075	−0.209179	−0.104589	+	0.994515i	$0.533353\pi$
$314$	−12.6782	−0.715474
$315$	2.86621	0.161492
$316$	−10.7445	−0.604425
$317$	−7.71067	−0.433074	−0.216537	−	0.976274i	$-0.569476\pi$
$317$	−7.71067	−0.433074	−0.216537	+	0.976274i	$0.569476\pi$
$318$	−2.93909	−0.164816
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	0.676926	0.0377824
$322$	−0.452126	−0.0251960
$323$	−37.6404	−2.09437
$324$	7.81376	0.434098
$325$	2.20049	0.122061
$326$	−18.8534	−1.04419
$327$	5.99143	0.331327
$328$	−0.0206760	−0.00114164
$329$	8.63329	0.475968
$330$	0	0
$331$	32.2493	1.77258	0.886291	−	0.463128i	$-0.153273\pi$
$331$	32.2493	1.77258	0.886291	+	0.463128i	$0.153273\pi$
$332$	−10.9679	−0.601942
$333$	−10.1500	−0.556217
$334$	15.5834	0.852684
$335$	−8.19469	−0.447724
$336$	0.365778	0.0199548
$337$	−33.2152	−1.80935	−0.904673	−	0.426106i	$-0.859885\pi$
$337$	−33.2152	−1.80935	−0.904673	+	0.426106i	$0.859885\pi$
$338$	8.15785	0.443728
$339$	−3.93892	−0.213933
$340$	6.26330	0.339675
$341$	0	0
$342$	17.2250	0.931419
$343$	1.00000	0.0539949
$344$	3.71582	0.200344
$345$	−0.165378	−0.00890364
$346$	6.79145	0.365111
$347$	3.37780	0.181330	0.0906649	−	0.995881i	$-0.471101\pi$
$347$	3.37780	0.181330	0.0906649	+	0.995881i	$0.471101\pi$
$348$	0.697795	0.0374057
$349$	31.4117	1.68143	0.840716	−	0.541477i	$-0.182135\pi$
$349$	31.4117	1.68143	0.840716	+	0.541477i	$0.182135\pi$
$350$	−1.00000	−0.0534522
$351$	−4.72165	−0.252023
$352$	0	0
$353$	34.9041	1.85776	0.928879	−	0.370383i	$-0.120774\pi$
$353$	34.9041	1.85776	0.928879	+	0.370383i	$0.120774\pi$
$354$	4.65559	0.247442
$355$	−15.7605	−0.836483
$356$	−9.56469	−0.506927
$357$	−2.29098	−0.121251
$358$	23.7293	1.25414
$359$	−32.4765	−1.71404	−0.857022	−	0.515280i	$-0.827688\pi$
$359$	−32.4765	−1.71404	−0.857022	+	0.515280i	$0.827688\pi$
$360$	−2.86621	−0.151062
$361$	17.1161	0.900849
$362$	−14.5332	−0.763848
$363$	0	0
$364$	2.20049	0.115337
$365$	12.7842	0.669156
$366$	2.20203	0.115102
$367$	−28.7289	−1.49963	−0.749817	−	0.661645i	$-0.769859\pi$
$367$	−28.7289	−1.49963	−0.749817	+	0.661645i	$0.769859\pi$
$368$	0.452126	0.0235687
$369$	−0.0592617	−0.00308504
$370$	3.54127	0.184102
$371$	8.03517	0.417165
$372$	−2.21338	−0.114758
$373$	36.4802	1.88887	0.944437	−	0.328691i	$-0.106608\pi$
$373$	36.4802	1.88887	0.944437	+	0.328691i	$0.106608\pi$
$374$	0	0
$375$	−0.365778	−0.0188887
$376$	−8.63329	−0.445228
$377$	4.19788	0.216202
$378$	2.14573	0.110364
$379$	0.299074	0.0153624	0.00768120	−	0.999970i	$-0.497555\pi$
$379$	0.299074	0.0153624	0.00768120	+	0.999970i	$0.497555\pi$
$380$	−6.00967	−0.308290
$381$	−1.62521	−0.0832619
$382$	6.74284	0.344994
$383$	−15.7020	−0.802337	−0.401168	−	0.916004i	$-0.631396\pi$
$383$	−15.7020	−0.802337	−0.401168	+	0.916004i	$0.631396\pi$
$384$	−0.365778	−0.0186660
$385$	0	0
$386$	−10.4918	−0.534021
$387$	10.6503	0.541386
$388$	−4.22569	−0.214527
$389$	−9.66386	−0.489977	−0.244989	−	0.969526i	$-0.578784\pi$
$389$	−9.66386	−0.489977	−0.244989	+	0.969526i	$0.578784\pi$
$390$	0.804890	0.0407572
$391$	−2.83180	−0.143211
$392$	−1.00000	−0.0505076
$393$	0.932548	0.0470408
$394$	−8.40324	−0.423349
$395$	10.7445	0.540615
$396$	0	0
$397$	17.7947	0.893090	0.446545	−	0.894761i	$-0.352654\pi$
$397$	17.7947	0.893090	0.446545	+	0.894761i	$0.352654\pi$
$398$	−5.41088	−0.271223
$399$	2.19820	0.110048
$400$	1.00000	0.0500000
$401$	37.5083	1.87308	0.936538	−	0.350566i	$-0.114011\pi$
$401$	37.5083	1.87308	0.936538	+	0.350566i	$0.114011\pi$
$402$	−2.99744	−0.149498
$403$	−13.3155	−0.663293
$404$	10.0515	0.500080
$405$	−7.81376	−0.388269
$406$	−1.90770	−0.0946776
$407$	0	0
$408$	2.29098	0.113420
$409$	−21.9950	−1.08758	−0.543791	−	0.839221i	$-0.683012\pi$
$409$	−21.9950	−1.08758	−0.543791	+	0.839221i	$0.683012\pi$
$410$	0.0206760	0.00102112
$411$	−5.54628	−0.273578
$412$	14.6726	0.722867
$413$	−12.7279	−0.626299
$414$	1.29589	0.0636894
$415$	10.9679	0.538393
$416$	−2.20049	−0.107888
$417$	−1.66990	−0.0817754
$418$	0	0
$419$	−14.6014	−0.713327	−0.356664	−	0.934233i	$-0.616086\pi$
$419$	−14.6014	−0.713327	−0.356664	+	0.934233i	$0.616086\pi$
$420$	−0.365778	−0.0178481
$421$	−36.1140	−1.76009	−0.880044	−	0.474893i	$-0.842487\pi$
$421$	−36.1140	−1.76009	−0.880044	+	0.474893i	$0.842487\pi$
$422$	8.53391	0.415424
$423$	−24.7448	−1.20313
$424$	−8.03517	−0.390223
$425$	−6.26330	−0.303815
$426$	−5.76486	−0.279308
$427$	−6.02013	−0.291334
$428$	1.85065	0.0894545
$429$	0	0
$430$	−3.71582	−0.179193
$431$	−1.92226	−0.0925922	−0.0462961	−	0.998928i	$-0.514742\pi$
$431$	−1.92226	−0.0925922	−0.0462961	+	0.998928i	$0.514742\pi$
$432$	−2.14573	−0.103236
$433$	−8.11232	−0.389853	−0.194927	−	0.980818i	$-0.562447\pi$
$433$	−8.11232	−0.389853	−0.194927	+	0.980818i	$0.562447\pi$
$434$	6.05116	0.290465
$435$	−0.697795	−0.0334567
$436$	16.3800	0.784459
$437$	2.71713	0.129978
$438$	4.67618	0.223437
$439$	−3.97670	−0.189797	−0.0948987	−	0.995487i	$-0.530253\pi$
$439$	−3.97670	−0.189797	−0.0948987	+	0.995487i	$0.530253\pi$
$440$	0	0
$441$	−2.86621	−0.136486
$442$	13.7823	0.655559
$443$	−7.61310	−0.361709	−0.180855	−	0.983510i	$-0.557886\pi$
$443$	−7.61310	−0.361709	−0.180855	+	0.983510i	$0.557886\pi$
$444$	1.29532	0.0614730
$445$	9.56469	0.453410
$446$	19.6449	0.930213
$447$	−5.08846	−0.240676
$448$	1.00000	0.0472456
$449$	3.27630	0.154618	0.0773090	−	0.997007i	$-0.475367\pi$
$449$	3.27630	0.154618	0.0773090	+	0.997007i	$0.475367\pi$
$450$	2.86621	0.135114
$451$	0	0
$452$	−10.7686	−0.506514
$453$	−1.38273	−0.0649664
$454$	2.37742	0.111578
$455$	−2.20049	−0.103161
$456$	−2.19820	−0.102940
$457$	−8.41968	−0.393856	−0.196928	−	0.980418i	$-0.563097\pi$
$457$	−8.41968	−0.393856	−0.196928	+	0.980418i	$0.563097\pi$
$458$	−9.73039	−0.454671
$459$	13.4393	0.627295
$460$	−0.452126	−0.0210805
$461$	22.7631	1.06018	0.530091	−	0.847941i	$-0.322158\pi$
$461$	22.7631	1.06018	0.530091	+	0.847941i	$0.322158\pi$
$462$	0	0
$463$	24.6401	1.14512	0.572561	−	0.819862i	$-0.305950\pi$
$463$	24.6401	1.14512	0.572561	+	0.819862i	$0.305950\pi$
$464$	1.90770	0.0885628
$465$	2.21338	0.102643
$466$	18.4540	0.854867
$467$	4.38924	0.203110	0.101555	−	0.994830i	$-0.467618\pi$
$467$	4.38924	0.203110	0.101555	+	0.994830i	$0.467618\pi$
$468$	−6.30706	−0.291544
$469$	8.19469	0.378396
$470$	8.63329	0.398224
$471$	4.63741	0.213681
$472$	12.7279	0.585849
$473$	0	0
$474$	3.93010	0.180516
$475$	6.00967	0.275743
$476$	−6.26330	−0.287078
$477$	−23.0305	−1.05449
$478$	17.6068	0.805316
$479$	10.1427	0.463433	0.231716	−	0.972783i	$-0.425566\pi$
$479$	10.1427	0.463433	0.231716	+	0.972783i	$0.425566\pi$
$480$	0.365778	0.0166954
$481$	7.79252	0.355308
$482$	17.4835	0.796351
$483$	0.165378	0.00752495
$484$	0	0
$485$	4.22569	0.191879
$486$	−9.29528	−0.421643
$487$	−22.6195	−1.02499	−0.512494	−	0.858691i	$-0.671278\pi$
$487$	−22.6195	−1.02499	−0.512494	+	0.858691i	$0.671278\pi$
$488$	6.02013	0.272518
$489$	6.89615	0.311855
$490$	1.00000	0.0451754
$491$	−2.55144	−0.115145	−0.0575724	−	0.998341i	$-0.518336\pi$
$491$	−2.55144	−0.115145	−0.0575724	+	0.998341i	$0.518336\pi$
$492$	0.00756282	0.000340958 0
$493$	−11.9485	−0.538134
$494$	−13.2242	−0.594985
$495$	0	0
$496$	−6.05116	−0.271705
$497$	15.7605	0.706957
$498$	4.01182	0.179774
$499$	4.35625	0.195012	0.0975062	−	0.995235i	$-0.468913\pi$
$499$	4.35625	0.195012	0.0975062	+	0.995235i	$0.468913\pi$
$500$	−1.00000	−0.0447214
$501$	−5.70005	−0.254660
$502$	22.9585	1.02469
$503$	−23.2545	−1.03687	−0.518433	−	0.855118i	$-0.673485\pi$
$503$	−23.2545	−1.03687	−0.518433	+	0.855118i	$0.673485\pi$
$504$	2.86621	0.127671
$505$	−10.0515	−0.447285
$506$	0	0
$507$	−2.98396	−0.132522
$508$	−4.44315	−0.197133
$509$	−2.94487	−0.130529	−0.0652646	−	0.997868i	$-0.520789\pi$
$509$	−2.94487	−0.130529	−0.0652646	+	0.997868i	$0.520789\pi$
$510$	−2.29098	−0.101446
$511$	−12.7842	−0.565540
$512$	−1.00000	−0.0441942
$513$	−12.8951	−0.569333
$514$	−14.2609	−0.629020
$515$	−14.6726	−0.646552
$516$	−1.35916	−0.0598339
$517$	0	0
$518$	−3.54127	−0.155594
$519$	−2.48416	−0.109043
$520$	2.20049	0.0964978
$521$	−35.1983	−1.54207	−0.771034	−	0.636794i	$-0.780260\pi$
$521$	−35.1983	−1.54207	−0.771034	+	0.636794i	$0.780260\pi$
$522$	5.46787	0.239322
$523$	−4.34730	−0.190094	−0.0950470	−	0.995473i	$-0.530300\pi$
$523$	−4.34730	−0.190094	−0.0950470	+	0.995473i	$0.530300\pi$
$524$	2.54949	0.111375
$525$	0.365778	0.0159639
$526$	9.29262	0.405177
$527$	37.9003	1.65096
$528$	0	0
$529$	−22.7956	−0.991112
$530$	8.03517	0.349026
$531$	36.4808	1.58313
$532$	6.00967	0.260552
$533$	0.0454973	0.00197071
$534$	3.49855	0.151397
$535$	−1.85065	−0.0800105
$536$	−8.19469	−0.353957
$537$	−8.67967	−0.374555
$538$	−1.17894	−0.0508276
$539$	0	0
$540$	2.14573	0.0923374
$541$	−19.7931	−0.850973	−0.425486	−	0.904965i	$-0.639897\pi$
$541$	−19.7931	−0.850973	−0.425486	+	0.904965i	$0.639897\pi$
$542$	−8.12467	−0.348985
$543$	5.31592	0.228128
$544$	6.26330	0.268537
$545$	−16.3800	−0.701641
$546$	−0.804890	−0.0344461
$547$	−12.4719	−0.533260	−0.266630	−	0.963799i	$-0.585910\pi$
$547$	−12.4719	−0.533260	−0.266630	+	0.963799i	$0.585910\pi$
$548$	−15.1630	−0.647730
$549$	17.2549	0.736423
$550$	0	0
$551$	11.4647	0.488411
$552$	−0.165378	−0.00703894
$553$	−10.7445	−0.456903
$554$	8.23042	0.349677
$555$	−1.29532	−0.0549831
$556$	−4.56534	−0.193614
$557$	−30.3225	−1.28480	−0.642402	−	0.766368i	$-0.722062\pi$
$557$	−30.3225	−1.28480	−0.642402	+	0.766368i	$0.722062\pi$
$558$	−17.3439	−0.734225
$559$	−8.17662	−0.345834
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	21.0726	0.888894
$563$	19.1404	0.806671	0.403336	−	0.915052i	$-0.367851\pi$
$563$	19.1404	0.806671	0.403336	+	0.915052i	$0.367851\pi$
$564$	3.15786	0.132970
$565$	10.7686	0.453040
$566$	0.468381	0.0196875
$567$	7.81376	0.328147
$568$	−15.7605	−0.661298
$569$	−9.22473	−0.386721	−0.193360	−	0.981128i	$-0.561939\pi$
$569$	−9.22473	−0.386721	−0.193360	+	0.981128i	$0.561939\pi$
$570$	2.19820	0.0920726
$571$	−26.6182	−1.11394	−0.556968	−	0.830534i	$-0.688035\pi$
$571$	−26.6182	−1.11394	−0.556968	+	0.830534i	$0.688035\pi$
$572$	0	0
$573$	−2.46638	−0.103035
$574$	−0.0206760	−0.000863000 0
$575$	0.452126	0.0188550
$576$	−2.86621	−0.119425
$577$	−8.14831	−0.339219	−0.169609	−	0.985511i	$-0.554251\pi$
$577$	−8.14831	−0.339219	−0.169609	+	0.985511i	$0.554251\pi$
$578$	−22.2290	−0.924604
$579$	3.83768	0.159489
$580$	−1.90770	−0.0792130
$581$	−10.9679	−0.455025
$582$	1.54566	0.0640698
$583$	0	0
$584$	12.7842	0.529014
$585$	6.30706	0.260765
$586$	−6.71911	−0.277564
$587$	−18.1817	−0.750440	−0.375220	−	0.926936i	$-0.622433\pi$
$587$	−18.1817	−0.750440	−0.375220	+	0.926936i	$0.622433\pi$
$588$	0.365778	0.0150844
$589$	−36.3655	−1.49841
$590$	−12.7279	−0.524000
$591$	3.07372	0.126436
$592$	3.54127	0.145545
$593$	14.6040	0.599715	0.299857	−	0.953984i	$-0.403061\pi$
$593$	14.6040	0.599715	0.299857	+	0.953984i	$0.403061\pi$
$594$	0	0
$595$	6.26330	0.256770
$596$	−13.9113	−0.569830
$597$	1.97918	0.0810024
$598$	−0.994899	−0.0406844
$599$	−15.9571	−0.651990	−0.325995	−	0.945371i	$-0.605699\pi$
$599$	−15.9571	−0.651990	−0.325995	+	0.945371i	$0.605699\pi$
$600$	−0.365778	−0.0149328
$601$	−22.0047	−0.897591	−0.448796	−	0.893634i	$-0.648147\pi$
$601$	−22.0047	−0.897591	−0.448796	+	0.893634i	$0.648147\pi$
$602$	3.71582	0.151446
$603$	−23.4877	−0.956492
$604$	−3.78025	−0.153816
$605$	0	0
$606$	−3.67661	−0.149352
$607$	17.4148	0.706846	0.353423	−	0.935464i	$-0.385018\pi$
$607$	17.4148	0.706846	0.353423	+	0.935464i	$0.385018\pi$
$608$	−6.00967	−0.243724
$609$	0.697795	0.0282761
$610$	−6.02013	−0.243748
$611$	18.9975	0.768555
$612$	17.9519	0.725664
$613$	18.4109	0.743611	0.371805	−	0.928311i	$-0.378739\pi$
$613$	18.4109	0.743611	0.371805	+	0.928311i	$0.378739\pi$
$614$	22.7190	0.916864
$615$	−0.00756282	−0.000304963 0
$616$	0	0
$617$	−5.54431	−0.223206	−0.111603	−	0.993753i	$-0.535598\pi$
$617$	−5.54431	−0.223206	−0.111603	+	0.993753i	$0.535598\pi$
$618$	−5.36691	−0.215889
$619$	−27.3331	−1.09861	−0.549305	−	0.835622i	$-0.685108\pi$
$619$	−27.3331	−1.09861	−0.549305	+	0.835622i	$0.685108\pi$
$620$	6.05116	0.243021
$621$	−0.970140	−0.0389304
$622$	21.2105	0.850463
$623$	−9.56469	−0.383201
$624$	0.804890	0.0322214
$625$	1.00000	0.0400000
$626$	3.70075	0.147912
$627$	0	0
$628$	12.6782	0.505916
$629$	−22.1800	−0.884376
$630$	−2.86621	−0.114192
$631$	33.1472	1.31957	0.659785	−	0.751454i	$-0.270647\pi$
$631$	33.1472	1.31957	0.659785	+	0.751454i	$0.270647\pi$
$632$	10.7445	0.427393
$633$	−3.12152	−0.124069
$634$	7.71067	0.306230
$635$	4.44315	0.176321
$636$	2.93909	0.116542
$637$	2.20049	0.0871865
$638$	0	0
$639$	−45.1730	−1.78702
$640$	1.00000	0.0395285
$641$	1.55259	0.0613236	0.0306618	−	0.999530i	$-0.490239\pi$
$641$	1.55259	0.0613236	0.0306618	+	0.999530i	$0.490239\pi$
$642$	−0.676926	−0.0267162
$643$	7.60419	0.299880	0.149940	−	0.988695i	$-0.452092\pi$
$643$	7.60419	0.299880	0.149940	+	0.988695i	$0.452092\pi$
$644$	0.452126	0.0178163
$645$	1.35916	0.0535171
$646$	37.6404	1.48094
$647$	8.45292	0.332319	0.166159	−	0.986099i	$-0.446863\pi$
$647$	8.45292	0.332319	0.166159	+	0.986099i	$0.446863\pi$
$648$	−7.81376	−0.306953
$649$	0	0
$650$	−2.20049	−0.0863103
$651$	−2.21338	−0.0867493
$652$	18.8534	0.738356
$653$	32.5646	1.27435	0.637175	−	0.770719i	$-0.280103\pi$
$653$	32.5646	1.27435	0.637175	+	0.770719i	$0.280103\pi$
$654$	−5.99143	−0.234284
$655$	−2.54949	−0.0996169
$656$	0.0206760	0.000807262 0
$657$	36.6422	1.42955
$658$	−8.63329	−0.336561
$659$	−37.9922	−1.47997	−0.739983	−	0.672626i	$-0.765166\pi$
$659$	−37.9922	−1.47997	−0.739983	+	0.672626i	$0.765166\pi$
$660$	0	0
$661$	−15.0403	−0.584999	−0.292499	−	0.956266i	$-0.594487\pi$
$661$	−15.0403	−0.584999	−0.292499	+	0.956266i	$0.594487\pi$
$662$	−32.2493	−1.25341
$663$	−5.04127	−0.195787
$664$	10.9679	0.425637
$665$	−6.00967	−0.233045
$666$	10.1500	0.393305
$667$	0.862522	0.0333970
$668$	−15.5834	−0.602939
$669$	−7.18567	−0.277814
$670$	8.19469	0.316588
$671$	0	0
$672$	−0.365778	−0.0141102
$673$	38.3805	1.47946	0.739729	−	0.672905i	$-0.234954\pi$
$673$	38.3805	1.47946	0.739729	+	0.672905i	$0.234954\pi$
$674$	33.2152	1.27940
$675$	−2.14573	−0.0825891
$676$	−8.15785	−0.313763
$677$	30.8004	1.18375	0.591877	−	0.806028i	$-0.298387\pi$
$677$	30.8004	1.18375	0.591877	+	0.806028i	$0.298387\pi$
$678$	3.93892	0.151274
$679$	−4.22569	−0.162167
$680$	−6.26330	−0.240187
$681$	−0.869608	−0.0333234
$682$	0	0
$683$	46.0974	1.76387	0.881935	−	0.471371i	$-0.156241\pi$
$683$	46.0974	1.76387	0.881935	+	0.471371i	$0.156241\pi$
$684$	−17.2250	−0.658613
$685$	15.1630	0.579347
$686$	−1.00000	−0.0381802
$687$	3.55916	0.135790
$688$	−3.71582	−0.141664
$689$	17.6813	0.673604
$690$	0.165378	0.00629582
$691$	−26.6683	−1.01451	−0.507255	−	0.861796i	$-0.669340\pi$
$691$	−26.6683	−1.01451	−0.507255	+	0.861796i	$0.669340\pi$
$692$	−6.79145	−0.258172
$693$	0	0
$694$	−3.37780	−0.128220
$695$	4.56534	0.173173
$696$	−0.697795	−0.0264498
$697$	−0.129500	−0.00490517
$698$	−31.4117	−1.18895
$699$	−6.75008	−0.255311
$700$	1.00000	0.0377964
$701$	−15.2735	−0.576872	−0.288436	−	0.957499i	$-0.593135\pi$
$701$	−15.2735	−0.576872	−0.288436	+	0.957499i	$0.593135\pi$
$702$	4.72165	0.178207
$703$	21.2818	0.802660
$704$	0	0
$705$	−3.15786	−0.118932
$706$	−34.9041	−1.31363
$707$	10.0515	0.378025
$708$	−4.65559	−0.174968
$709$	−23.2892	−0.874645	−0.437323	−	0.899305i	$-0.644073\pi$
$709$	−23.2892	−0.874645	−0.437323	+	0.899305i	$0.644073\pi$
$710$	15.7605	0.591483
$711$	30.7960	1.15494
$712$	9.56469	0.358452
$713$	−2.73589	−0.102460
$714$	2.29098	0.0857377
$715$	0	0
$716$	−23.7293	−0.886807
$717$	−6.44018	−0.240513
$718$	32.4765	1.21201
$719$	6.64924	0.247975	0.123987	−	0.992284i	$-0.460432\pi$
$719$	6.64924	0.247975	0.123987	+	0.992284i	$0.460432\pi$
$720$	2.86621	0.106817
$721$	14.6726	0.546436
$722$	−17.1161	−0.636996
$723$	−6.39507	−0.237835
$724$	14.5332	0.540122
$725$	1.90770	0.0708502
$726$	0	0
$727$	−36.8003	−1.36485	−0.682424	−	0.730956i	$-0.739074\pi$
$727$	−36.8003	−1.36485	−0.682424	+	0.730956i	$0.739074\pi$
$728$	−2.20049	−0.0815555
$729$	−20.0413	−0.742269
$730$	−12.7842	−0.473165
$731$	23.2733	0.860795
$732$	−2.20203	−0.0813893
$733$	6.60763	0.244058	0.122029	−	0.992527i	$-0.461060\pi$
$733$	6.60763	0.244058	0.122029	+	0.992527i	$0.461060\pi$
$734$	28.7289	1.06040
$735$	−0.365778	−0.0134919
$736$	−0.452126	−0.0166656
$737$	0	0
$738$	0.0592617	0.00218145
$739$	14.1761	0.521476	0.260738	−	0.965410i	$-0.416034\pi$
$739$	14.1761	0.521476	0.260738	+	0.965410i	$0.416034\pi$
$740$	−3.54127	−0.130180
$741$	4.83712	0.177696
$742$	−8.03517	−0.294980
$743$	21.8533	0.801719	0.400859	−	0.916140i	$-0.368712\pi$
$743$	21.8533	0.801719	0.400859	+	0.916140i	$0.368712\pi$
$744$	2.21338	0.0811465
$745$	13.9113	0.509672
$746$	−36.4802	−1.33564
$747$	31.4363	1.15019
$748$	0	0
$749$	1.85065	0.0676213
$750$	0.365778	0.0133563
$751$	−18.3253	−0.668700	−0.334350	−	0.942449i	$-0.608517\pi$
$751$	−18.3253	−0.668700	−0.334350	+	0.942449i	$0.608517\pi$
$752$	8.63329	0.314824
$753$	−8.39770	−0.306029
$754$	−4.19788	−0.152878
$755$	3.78025	0.137577
$756$	−2.14573	−0.0780394
$757$	−11.6388	−0.423020	−0.211510	−	0.977376i	$-0.567838\pi$
$757$	−11.6388	−0.423020	−0.211510	+	0.977376i	$0.567838\pi$
$758$	−0.299074	−0.0108629
$759$	0	0
$760$	6.00967	0.217994
$761$	−25.4547	−0.922731	−0.461366	−	0.887210i	$-0.652640\pi$
$761$	−25.4547	−0.922731	−0.461366	+	0.887210i	$0.652640\pi$
$762$	1.62521	0.0588750
$763$	16.3800	0.592995
$764$	−6.74284	−0.243947
$765$	−17.9519	−0.649053
$766$	15.7020	0.567338
$767$	−28.0076	−1.01130
$768$	0.365778	0.0131989
$769$	−41.1644	−1.48442	−0.742212	−	0.670165i	$-0.766223\pi$
$769$	−41.1644	−1.48442	−0.742212	+	0.670165i	$0.766223\pi$
$770$	0	0
$771$	5.21631	0.187861
$772$	10.4918	0.377610
$773$	3.93221	0.141432	0.0707159	−	0.997496i	$-0.477472\pi$
$773$	3.93221	0.141432	0.0707159	+	0.997496i	$0.477472\pi$
$774$	−10.6503	−0.382817
$775$	−6.05116	−0.217364
$776$	4.22569	0.151693
$777$	1.29532	0.0464692
$778$	9.66386	0.346466
$779$	0.124256	0.00445193
$780$	−0.804890	−0.0288197
$781$	0	0
$782$	2.83180	0.101265
$783$	−4.09341	−0.146286
$784$	1.00000	0.0357143
$785$	−12.6782	−0.452505
$786$	−0.932548	−0.0332629
$787$	34.2138	1.21959	0.609796	−	0.792558i	$-0.291251\pi$
$787$	34.2138	1.21959	0.609796	+	0.792558i	$0.291251\pi$
$788$	8.40324	0.299353
$789$	−3.39903	−0.121009
$790$	−10.7445	−0.382272
$791$	−10.7686	−0.382888
$792$	0	0
$793$	−13.2472	−0.470423
$794$	−17.7947	−0.631510
$795$	−2.93909	−0.104239
$796$	5.41088	0.191783
$797$	0.946359	0.0335217	0.0167609	−	0.999860i	$-0.494665\pi$
$797$	0.946359	0.0335217	0.0167609	+	0.999860i	$0.494665\pi$
$798$	−2.19820	−0.0778156
$799$	−54.0729	−1.91296
$800$	−1.00000	−0.0353553
$801$	27.4144	0.968639
$802$	−37.5083	−1.32446
$803$	0	0
$804$	2.99744	0.105711
$805$	−0.452126	−0.0159354
$806$	13.3155	0.469019
$807$	0.431229	0.0151800
$808$	−10.0515	−0.353610
$809$	12.0510	0.423692	0.211846	−	0.977303i	$-0.432053\pi$
$809$	12.0510	0.423692	0.211846	+	0.977303i	$0.432053\pi$
$810$	7.81376	0.274548
$811$	−31.6751	−1.11226	−0.556132	−	0.831094i	$-0.687715\pi$
$811$	−31.6751	−1.11226	−0.556132	+	0.831094i	$0.687715\pi$
$812$	1.90770	0.0669472
$813$	2.97182	0.104226
$814$	0	0
$815$	−18.8534	−0.660405
$816$	−2.29098	−0.0802003
$817$	−22.3309	−0.781258
$818$	21.9950	0.769037
$819$	−6.30706	−0.220386
$820$	−0.0206760	−0.000722037 0
$821$	−24.4294	−0.852591	−0.426296	−	0.904584i	$-0.640182\pi$
$821$	−24.4294	−0.852591	−0.426296	+	0.904584i	$0.640182\pi$
$822$	5.54628	0.193449
$823$	−28.4472	−0.991607	−0.495803	−	0.868435i	$-0.665126\pi$
$823$	−28.4472	−0.991607	−0.495803	+	0.868435i	$0.665126\pi$
$824$	−14.6726	−0.511144
$825$	0	0
$826$	12.7279	0.442860
$827$	−43.6508	−1.51789	−0.758943	−	0.651157i	$-0.774284\pi$
$827$	−43.6508	−1.51789	−0.758943	+	0.651157i	$0.774284\pi$
$828$	−1.29589	−0.0450352
$829$	0.554570	0.0192610	0.00963050	−	0.999954i	$-0.496934\pi$
$829$	0.554570	0.0192610	0.00963050	+	0.999954i	$0.496934\pi$
$830$	−10.9679	−0.380701
$831$	−3.01051	−0.104433
$832$	2.20049	0.0762882
$833$	−6.26330	−0.217011
$834$	1.66990	0.0578239
$835$	15.5834	0.539285
$836$	0	0
$837$	12.9842	0.448798
$838$	14.6014	0.504399
$839$	−2.69760	−0.0931315	−0.0465658	−	0.998915i	$-0.514828\pi$
$839$	−2.69760	−0.0931315	−0.0465658	+	0.998915i	$0.514828\pi$
$840$	0.365778	0.0126205
$841$	−25.3607	−0.874506
$842$	36.1140	1.24457
$843$	−7.70789	−0.265474
$844$	−8.53391	−0.293749
$845$	8.15785	0.280639
$846$	24.7448	0.850743
$847$	0	0
$848$	8.03517	0.275929
$849$	−0.171323	−0.00587980
$850$	6.26330	0.214830
$851$	1.60110	0.0548850
$852$	5.76486	0.197501
$853$	−9.81741	−0.336142	−0.168071	−	0.985775i	$-0.553754\pi$
$853$	−9.81741	−0.336142	−0.168071	+	0.985775i	$0.553754\pi$
$854$	6.02013	0.206005
$855$	17.2250	0.589081
$856$	−1.85065	−0.0632539
$857$	−38.1531	−1.30329	−0.651643	−	0.758526i	$-0.725920\pi$
$857$	−38.1531	−1.30329	−0.651643	+	0.758526i	$0.725920\pi$
$858$	0	0
$859$	7.51538	0.256421	0.128211	−	0.991747i	$-0.459077\pi$
$859$	7.51538	0.256421	0.128211	+	0.991747i	$0.459077\pi$
$860$	3.71582	0.126708
$861$	0.00756282	0.000257740 0
$862$	1.92226	0.0654726
$863$	57.5800	1.96005	0.980024	−	0.198880i	$-0.0637305\pi$
$863$	57.5800	1.96005	0.980024	+	0.198880i	$0.0637305\pi$
$864$	2.14573	0.0729992
$865$	6.79145	0.230916
$866$	8.11232	0.275668
$867$	8.13087	0.276139
$868$	−6.05116	−0.205390
$869$	0	0
$870$	0.697795	0.0236575
$871$	18.0323	0.611002
$872$	−16.3800	−0.554696
$873$	12.1117	0.409919
$874$	−2.71713	−0.0919083
$875$	−1.00000	−0.0338062
$876$	−4.67618	−0.157994
$877$	29.0321	0.980343	0.490172	−	0.871626i	$-0.336934\pi$
$877$	29.0321	0.980343	0.490172	+	0.871626i	$0.336934\pi$
$878$	3.97670	0.134207
$879$	2.45770	0.0828963
$880$	0	0
$881$	−48.6656	−1.63958	−0.819792	−	0.572661i	$-0.805911\pi$
$881$	−48.6656	−1.63958	−0.819792	+	0.572661i	$0.805911\pi$
$882$	2.86621	0.0965102
$883$	47.0038	1.58180	0.790902	−	0.611943i	$-0.209612\pi$
$883$	47.0038	1.58180	0.790902	+	0.611943i	$0.209612\pi$
$884$	−13.7823	−0.463550
$885$	4.65559	0.156496
$886$	7.61310	0.255767
$887$	−30.2777	−1.01662	−0.508312	−	0.861173i	$-0.669730\pi$
$887$	−30.2777	−1.01662	−0.508312	+	0.861173i	$0.669730\pi$
$888$	−1.29532	−0.0434680
$889$	−4.44315	−0.149019
$890$	−9.56469	−0.320609
$891$	0	0
$892$	−19.6449	−0.657760
$893$	51.8832	1.73621
$894$	5.08846	0.170183
$895$	23.7293	0.793185
$896$	−1.00000	−0.0334077
$897$	0.363912	0.0121507
$898$	−3.27630	−0.109331
$899$	−11.5438	−0.385008
$900$	−2.86621	−0.0955402
$901$	−50.3267	−1.67663
$902$	0	0
$903$	−1.35916	−0.0452302
$904$	10.7686	0.358159
$905$	−14.5332	−0.483100
$906$	1.38273	0.0459382
$907$	48.9263	1.62457	0.812285	−	0.583261i	$-0.198223\pi$
$907$	48.9263	1.62457	0.812285	+	0.583261i	$0.198223\pi$
$908$	−2.37742	−0.0788975
$909$	−28.8096	−0.955556
$910$	2.20049	0.0729455
$911$	42.4585	1.40671	0.703357	−	0.710837i	$-0.251684\pi$
$911$	42.4585	1.40671	0.703357	+	0.710837i	$0.251684\pi$
$912$	2.19820	0.0727898
$913$	0	0
$914$	8.41968	0.278498
$915$	2.20203	0.0727968
$916$	9.73039	0.321501
$917$	2.54949	0.0841917
$918$	−13.4393	−0.443565
$919$	−40.5258	−1.33682	−0.668412	−	0.743791i	$-0.733026\pi$
$919$	−40.5258	−1.33682	−0.668412	+	0.743791i	$0.733026\pi$
$920$	0.452126	0.0149062
$921$	−8.31011	−0.273827
$922$	−22.7631	−0.749662
$923$	34.6809	1.14154
$924$	0	0
$925$	3.54127	0.116436
$926$	−24.6401	−0.809724
$927$	−42.0547	−1.38126
$928$	−1.90770	−0.0626234
$929$	40.3461	1.32371	0.661856	−	0.749631i	$-0.269769\pi$
$929$	40.3461	1.32371	0.661856	+	0.749631i	$0.269769\pi$
$930$	−2.21338	−0.0725796
$931$	6.00967	0.196959
$932$	−18.4540	−0.604482
$933$	−7.75833	−0.253996
$934$	−4.38924	−0.143620
$935$	0	0
$936$	6.30706	0.206153
$937$	−33.7379	−1.10217	−0.551084	−	0.834450i	$-0.685786\pi$
$937$	−33.7379	−1.10217	−0.551084	+	0.834450i	$0.685786\pi$
$938$	−8.19469	−0.267566
$939$	−1.35365	−0.0441747
$940$	−8.63329	−0.281587
$941$	−12.3247	−0.401773	−0.200887	−	0.979614i	$-0.564382\pi$
$941$	−12.3247	−0.401773	−0.200887	+	0.979614i	$0.564382\pi$
$942$	−4.63741	−0.151095
$943$	0.00934817	0.000304418 0
$944$	−12.7279	−0.414258
$945$	2.14573	0.0698005
$946$	0	0
$947$	−0.817385	−0.0265615	−0.0132807	−	0.999912i	$-0.504228\pi$
$947$	−0.817385	−0.0265615	−0.0132807	+	0.999912i	$0.504228\pi$
$948$	−3.93010	−0.127644
$949$	−28.1315	−0.913188
$950$	−6.00967	−0.194979
$951$	−2.82039	−0.0914575
$952$	6.26330	0.202995
$953$	−6.96194	−0.225519	−0.112760	−	0.993622i	$-0.535969\pi$
$953$	−6.96194	−0.225519	−0.112760	+	0.993622i	$0.535969\pi$
$954$	23.0305	0.745639
$955$	6.74284	0.218193
$956$	−17.6068	−0.569445
$957$	0	0
$958$	−10.1427	−0.327696
$959$	−15.1630	−0.489638
$960$	−0.365778	−0.0118054
$961$	5.61659	0.181180
$962$	−7.79252	−0.251241
$963$	−5.30434	−0.170930
$964$	−17.4835	−0.563105
$965$	−10.4918	−0.337744
$966$	−0.165378	−0.00532094
$967$	5.35673	0.172261	0.0861304	−	0.996284i	$-0.472550\pi$
$967$	5.35673	0.172261	0.0861304	+	0.996284i	$0.472550\pi$
$968$	0	0
$969$	−13.7680	−0.442293
$970$	−4.22569	−0.135679
$971$	−17.4088	−0.558676	−0.279338	−	0.960193i	$-0.590115\pi$
$971$	−17.4088	−0.558676	−0.279338	+	0.960193i	$0.590115\pi$
$972$	9.29528	0.298146
$973$	−4.56534	−0.146358
$974$	22.6195	0.724776
$975$	0.804890	0.0257771
$976$	−6.02013	−0.192700
$977$	5.42269	0.173487	0.0867437	−	0.996231i	$-0.472354\pi$
$977$	5.42269	0.173487	0.0867437	+	0.996231i	$0.472354\pi$
$978$	−6.89615	−0.220515
$979$	0	0
$980$	−1.00000	−0.0319438
$981$	−46.9484	−1.49895
$982$	2.55144	0.0814197
$983$	12.9202	0.412092	0.206046	−	0.978542i	$-0.433940\pi$
$983$	12.9202	0.412092	0.206046	+	0.978542i	$0.433940\pi$
$984$	−0.00756282	−0.000241094 0
$985$	−8.40324	−0.267750
$986$	11.9485	0.380518
$987$	3.15786	0.100516
$988$	13.2242	0.420718
$989$	−1.68002	−0.0534215
$990$	0	0
$991$	−26.6040	−0.845104	−0.422552	−	0.906339i	$-0.638866\pi$
$991$	−26.6040	−0.845104	−0.422552	+	0.906339i	$0.638866\pi$
$992$	6.05116	0.192125
$993$	11.7961	0.374337
$994$	−15.7605	−0.499894
$995$	−5.41088	−0.171536
$996$	−4.01182	−0.127119
$997$	57.0633	1.80721	0.903606	−	0.428365i	$-0.140910\pi$
$997$	57.0633	1.80721	0.903606	+	0.428365i	$0.140910\pi$
$998$	−4.35625	−0.137895
$999$	−7.59860	−0.240409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.dg.1.5		8
11.7	odd	10		770.2.n.k.71.3	✓	16
11.8	odd	10		770.2.n.k.141.3	yes	16
11.10	odd	2		8470.2.a.dh.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.k.71.3	✓	16	11.7	odd	10
770.2.n.k.141.3	yes	16	11.8	odd	10
8470.2.a.dg.1.5		8	1.1	even	1	trivial
8470.2.a.dh.1.5		8	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.365778 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.365778 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.86621 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.365778 q^{3} +1.00000 q^{4} -1.00000 q^{5} -0.365778 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.86621 q^{9} +1.00000 q^{10} +0.365778 q^{12} +2.20049 q^{13} -1.00000 q^{14} -0.365778 q^{15} +1.00000 q^{16} -6.26330 q^{17} +2.86621 q^{18} +6.00967 q^{19} -1.00000 q^{20} +0.365778 q^{21} +0.452126 q^{23} -0.365778 q^{24} +1.00000 q^{25} -2.20049 q^{26} -2.14573 q^{27} +1.00000 q^{28} +1.90770 q^{29} +0.365778 q^{30} -6.05116 q^{31} -1.00000 q^{32} +6.26330 q^{34} -1.00000 q^{35} -2.86621 q^{36} +3.54127 q^{37} -6.00967 q^{38} +0.804890 q^{39} +1.00000 q^{40} +0.0206760 q^{41} -0.365778 q^{42} -3.71582 q^{43} +2.86621 q^{45} -0.452126 q^{46} +8.63329 q^{47} +0.365778 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.29098 q^{51} +2.20049 q^{52} +8.03517 q^{53} +2.14573 q^{54} -1.00000 q^{56} +2.19820 q^{57} -1.90770 q^{58} -12.7279 q^{59} -0.365778 q^{60} -6.02013 q^{61} +6.05116 q^{62} -2.86621 q^{63} +1.00000 q^{64} -2.20049 q^{65} +8.19469 q^{67} -6.26330 q^{68} +0.165378 q^{69} +1.00000 q^{70} +15.7605 q^{71} +2.86621 q^{72} -12.7842 q^{73} -3.54127 q^{74} +0.365778 q^{75} +6.00967 q^{76} -0.804890 q^{78} -10.7445 q^{79} -1.00000 q^{80} +7.81376 q^{81} -0.0206760 q^{82} -10.9679 q^{83} +0.365778 q^{84} +6.26330 q^{85} +3.71582 q^{86} +0.697795 q^{87} -9.56469 q^{89} -2.86621 q^{90} +2.20049 q^{91} +0.452126 q^{92} -2.21338 q^{93} -8.63329 q^{94} -6.00967 q^{95} -0.365778 q^{96} -4.22569 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} + 8 q^{7} - 8 q^{8} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^5 + 8 * q^7 - 8 * q^8 + 8 * q^9	\( 8 q - 8 q^{2} + 8 q^{4} - 8 q^{5} + 8 q^{7} - 8 q^{8} + 8 q^{9} + 8 q^{10} + q^{13} - 8 q^{14} + 8 q^{16} - 6 q^{17} - 8 q^{18} - 5 q^{19} - 8 q^{20} + 10 q^{23} + 8 q^{25} - q^{26} + 8 q^{28} - 3 q^{29} - 8 q^{31} - 8 q^{32} + 6 q^{34} - 8 q^{35} + 8 q^{36} - 6 q^{37} + 5 q^{38} - 35 q^{39} + 8 q^{40} - 11 q^{41} + 5 q^{43} - 8 q^{45} - 10 q^{46} - 15 q^{47} + 8 q^{49} - 8 q^{50} + 6 q^{51} + q^{52} - 16 q^{53} - 8 q^{56} - 38 q^{57} + 3 q^{58} - 9 q^{59} - 32 q^{61} + 8 q^{62} + 8 q^{63} + 8 q^{64} - q^{65} + 33 q^{67} - 6 q^{68} - 22 q^{69} + 8 q^{70} + 11 q^{71} - 8 q^{72} + 34 q^{73} + 6 q^{74} - 5 q^{76} + 35 q^{78} - 31 q^{79} - 8 q^{80} + 20 q^{81} + 11 q^{82} - 50 q^{83} + 6 q^{85} - 5 q^{86} + 12 q^{87} + q^{89} + 8 q^{90} + q^{91} + 10 q^{92} + 26 q^{93} + 15 q^{94} + 5 q^{95} - 4 q^{97} - 8 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 - 8 * q^5 + 8 * q^7 - 8 * q^8 + 8 * q^9 + 8 * q^10 + q^13 - 8 * q^14 + 8 * q^16 - 6 * q^17 - 8 * q^18 - 5 * q^19 - 8 * q^20 + 10 * q^23 + 8 * q^25 - q^26 + 8 * q^28 - 3 * q^29 - 8 * q^31 - 8 * q^32 + 6 * q^34 - 8 * q^35 + 8 * q^36 - 6 * q^37 + 5 * q^38 - 35 * q^39 + 8 * q^40 - 11 * q^41 + 5 * q^43 - 8 * q^45 - 10 * q^46 - 15 * q^47 + 8 * q^49 - 8 * q^50 + 6 * q^51 + q^52 - 16 * q^53 - 8 * q^56 - 38 * q^57 + 3 * q^58 - 9 * q^59 - 32 * q^61 + 8 * q^62 + 8 * q^63 + 8 * q^64 - q^65 + 33 * q^67 - 6 * q^68 - 22 * q^69 + 8 * q^70 + 11 * q^71 - 8 * q^72 + 34 * q^73 + 6 * q^74 - 5 * q^76 + 35 * q^78 - 31 * q^79 - 8 * q^80 + 20 * q^81 + 11 * q^82 - 50 * q^83 + 6 * q^85 - 5 * q^86 + 12 * q^87 + q^89 + 8 * q^90 + q^91 + 10 * q^92 + 26 * q^93 + 15 * q^94 + 5 * q^95 - 4 * q^97 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more