LMFDB - Embedded newform 8470.2.a.dc.1.4

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

Embedding invariants

Embedding label			1.4
Root			$-0.245893$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +0.245893 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.245893 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.93954 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.245893 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.245893 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.93954 q^{9} +1.00000 q^{10} +0.245893 q^{12} +1.15197 q^{13} -1.00000 q^{14} +0.245893 q^{15} +1.00000 q^{16} -6.55125 q^{17} -2.93954 q^{18} +3.43477 q^{19} +1.00000 q^{20} -0.245893 q^{21} +8.30827 q^{23} +0.245893 q^{24} +1.00000 q^{25} +1.15197 q^{26} -1.46049 q^{27} -1.00000 q^{28} +2.93346 q^{29} +0.245893 q^{30} -6.33956 q^{31} +1.00000 q^{32} -6.55125 q^{34} -1.00000 q^{35} -2.93954 q^{36} -3.86743 q^{37} +3.43477 q^{38} +0.283261 q^{39} +1.00000 q^{40} +10.5558 q^{41} -0.245893 q^{42} +2.95752 q^{43} -2.93954 q^{45} +8.30827 q^{46} -6.17003 q^{47} +0.245893 q^{48} +1.00000 q^{49} +1.00000 q^{50} -1.61090 q^{51} +1.15197 q^{52} +5.35680 q^{53} -1.46049 q^{54} -1.00000 q^{56} +0.844585 q^{57} +2.93346 q^{58} -2.24164 q^{59} +0.245893 q^{60} +9.98548 q^{61} -6.33956 q^{62} +2.93954 q^{63} +1.00000 q^{64} +1.15197 q^{65} +14.2147 q^{67} -6.55125 q^{68} +2.04294 q^{69} -1.00000 q^{70} -0.380220 q^{71} -2.93954 q^{72} +7.23216 q^{73} -3.86743 q^{74} +0.245893 q^{75} +3.43477 q^{76} +0.283261 q^{78} +2.27381 q^{79} +1.00000 q^{80} +8.45949 q^{81} +10.5558 q^{82} +11.3292 q^{83} -0.245893 q^{84} -6.55125 q^{85} +2.95752 q^{86} +0.721318 q^{87} -4.48891 q^{89} -2.93954 q^{90} -1.15197 q^{91} +8.30827 q^{92} -1.55885 q^{93} -6.17003 q^{94} +3.43477 q^{95} +0.245893 q^{96} +3.45447 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 + 6 * q^8 + 3 * q^9	$ 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} + 3 q^{9} + 6 q^{10} - q^{12} + 9 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} + 9 q^{17} + 3 q^{18} + 12 q^{19} + 6 q^{20} + q^{21} + 4 q^{23} - q^{24} + 6 q^{25} + 9 q^{26} - 4 q^{27} - 6 q^{28} + 15 q^{29} - q^{30} + 8 q^{31} + 6 q^{32} + 9 q^{34} - 6 q^{35} + 3 q^{36} - 4 q^{37} + 12 q^{38} - 19 q^{39} + 6 q^{40} + 4 q^{41} + q^{42} + 30 q^{43} + 3 q^{45} + 4 q^{46} - 7 q^{47} - q^{48} + 6 q^{49} + 6 q^{50} + 16 q^{51} + 9 q^{52} - 6 q^{53} - 4 q^{54} - 6 q^{56} - 14 q^{57} + 15 q^{58} + 4 q^{59} - q^{60} - 14 q^{61} + 8 q^{62} - 3 q^{63} + 6 q^{64} + 9 q^{65} + 18 q^{67} + 9 q^{68} - 10 q^{69} - 6 q^{70} + 23 q^{71} + 3 q^{72} + 23 q^{73} - 4 q^{74} - q^{75} + 12 q^{76} - 19 q^{78} + 21 q^{79} + 6 q^{80} - 18 q^{81} + 4 q^{82} + 25 q^{83} + q^{84} + 9 q^{85} + 30 q^{86} + 14 q^{87} - 18 q^{89} + 3 q^{90} - 9 q^{91} + 4 q^{92} - 24 q^{93} - 7 q^{94} + 12 q^{95} - q^{96} + 7 q^{97} + 6 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 + 6 * q^8 + 3 * q^9 + 6 * q^10 - q^12 + 9 * q^13 - 6 * q^14 - q^15 + 6 * q^16 + 9 * q^17 + 3 * q^18 + 12 * q^19 + 6 * q^20 + q^21 + 4 * q^23 - q^24 + 6 * q^25 + 9 * q^26 - 4 * q^27 - 6 * q^28 + 15 * q^29 - q^30 + 8 * q^31 + 6 * q^32 + 9 * q^34 - 6 * q^35 + 3 * q^36 - 4 * q^37 + 12 * q^38 - 19 * q^39 + 6 * q^40 + 4 * q^41 + q^42 + 30 * q^43 + 3 * q^45 + 4 * q^46 - 7 * q^47 - q^48 + 6 * q^49 + 6 * q^50 + 16 * q^51 + 9 * q^52 - 6 * q^53 - 4 * q^54 - 6 * q^56 - 14 * q^57 + 15 * q^58 + 4 * q^59 - q^60 - 14 * q^61 + 8 * q^62 - 3 * q^63 + 6 * q^64 + 9 * q^65 + 18 * q^67 + 9 * q^68 - 10 * q^69 - 6 * q^70 + 23 * q^71 + 3 * q^72 + 23 * q^73 - 4 * q^74 - q^75 + 12 * q^76 - 19 * q^78 + 21 * q^79 + 6 * q^80 - 18 * q^81 + 4 * q^82 + 25 * q^83 + q^84 + 9 * q^85 + 30 * q^86 + 14 * q^87 - 18 * q^89 + 3 * q^90 - 9 * q^91 + 4 * q^92 - 24 * q^93 - 7 * q^94 + 12 * q^95 - q^96 + 7 * q^97 + 6 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.245893	0.141966	0.0709831	−	0.997478i	$-0.477386\pi$
$3$	0.245893	0.141966	0.0709831	+	0.997478i	$0.477386\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0.245893	0.100385
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	−2.93954	−0.979846
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	0.245893	0.0709831
$13$	1.15197	0.319499	0.159750	−	0.987158i	$-0.448931\pi$
$13$	1.15197	0.319499	0.159750	+	0.987158i	$0.448931\pi$
$14$	−1.00000	−0.267261
$15$	0.245893	0.0634892
$16$	1.00000	0.250000
$17$	−6.55125	−1.58891	−0.794455	−	0.607323i	$-0.792244\pi$
$17$	−6.55125	−1.58891	−0.794455	+	0.607323i	$0.792244\pi$
$18$	−2.93954	−0.692855
$19$	3.43477	0.787990	0.393995	−	0.919113i	$-0.371093\pi$
$19$	3.43477	0.787990	0.393995	+	0.919113i	$0.371093\pi$
$20$	1.00000	0.223607
$21$	−0.245893	−0.0536582
$22$	0	0
$23$	8.30827	1.73239	0.866197	−	0.499703i	$-0.166558\pi$
$23$	8.30827	1.73239	0.866197	+	0.499703i	$0.166558\pi$
$24$	0.245893	0.0501927
$25$	1.00000	0.200000
$26$	1.15197	0.225920
$27$	−1.46049	−0.281071
$28$	−1.00000	−0.188982
$29$	2.93346	0.544731	0.272365	−	0.962194i	$-0.412194\pi$
$29$	2.93346	0.544731	0.272365	+	0.962194i	$0.412194\pi$
$30$	0.245893	0.0448937
$31$	−6.33956	−1.13862	−0.569310	−	0.822123i	$-0.692789\pi$
$31$	−6.33956	−1.13862	−0.569310	+	0.822123i	$0.692789\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.55125	−1.12353
$35$	−1.00000	−0.169031
$36$	−2.93954	−0.489923
$37$	−3.86743	−0.635801	−0.317901	−	0.948124i	$-0.602978\pi$
$37$	−3.86743	−0.635801	−0.317901	+	0.948124i	$0.602978\pi$
$38$	3.43477	0.557193
$39$	0.283261	0.0453581
$40$	1.00000	0.158114
$41$	10.5558	1.64854	0.824271	−	0.566195i	$-0.191585\pi$
$41$	10.5558	1.64854	0.824271	+	0.566195i	$0.191585\pi$
$42$	−0.245893	−0.0379421
$43$	2.95752	0.451018	0.225509	−	0.974241i	$-0.427596\pi$
$43$	2.95752	0.451018	0.225509	+	0.974241i	$0.427596\pi$
$44$	0	0
$45$	−2.93954	−0.438200
$46$	8.30827	1.22499
$47$	−6.17003	−0.899991	−0.449996	−	0.893031i	$-0.648574\pi$
$47$	−6.17003	−0.899991	−0.449996	+	0.893031i	$0.648574\pi$
$48$	0.245893	0.0354916
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−1.61090	−0.225572
$52$	1.15197	0.159750
$53$	5.35680	0.735813	0.367906	−	0.929863i	$-0.380075\pi$
$53$	5.35680	0.735813	0.367906	+	0.929863i	$0.380075\pi$
$54$	−1.46049	−0.198747
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0.844585	0.111868
$58$	2.93346	0.385183
$59$	−2.24164	−0.291837	−0.145919	−	0.989297i	$-0.546614\pi$
$59$	−2.24164	−0.291837	−0.145919	+	0.989297i	$0.546614\pi$
$60$	0.245893	0.0317446
$61$	9.98548	1.27851	0.639255	−	0.768995i	$-0.279243\pi$
$61$	9.98548	1.27851	0.639255	+	0.768995i	$0.279243\pi$
$62$	−6.33956	−0.805125
$63$	2.93954	0.370347
$64$	1.00000	0.125000
$65$	1.15197	0.142884
$66$	0	0
$67$	14.2147	1.73660	0.868299	−	0.496041i	$-0.165213\pi$
$67$	14.2147	1.73660	0.868299	+	0.496041i	$0.165213\pi$
$68$	−6.55125	−0.794455
$69$	2.04294	0.245941
$70$	−1.00000	−0.119523
$71$	−0.380220	−0.0451238	−0.0225619	−	0.999745i	$-0.507182\pi$
$71$	−0.380220	−0.0451238	−0.0225619	+	0.999745i	$0.507182\pi$
$72$	−2.93954	−0.346428
$73$	7.23216	0.846460	0.423230	−	0.906022i	$-0.360896\pi$
$73$	7.23216	0.846460	0.423230	+	0.906022i	$0.360896\pi$
$74$	−3.86743	−0.449579
$75$	0.245893	0.0283933
$76$	3.43477	0.393995
$77$	0	0
$78$	0.283261	0.0320730
$79$	2.27381	0.255824	0.127912	−	0.991786i	$-0.459173\pi$
$79$	2.27381	0.255824	0.127912	+	0.991786i	$0.459173\pi$
$80$	1.00000	0.111803
$81$	8.45949	0.939943
$82$	10.5558	1.16570
$83$	11.3292	1.24355	0.621773	−	0.783198i	$-0.286413\pi$
$83$	11.3292	1.24355	0.621773	+	0.783198i	$0.286413\pi$
$84$	−0.245893	−0.0268291
$85$	−6.55125	−0.710582
$86$	2.95752	0.318918
$87$	0.721318	0.0773334
$88$	0	0
$89$	−4.48891	−0.475823	−0.237912	−	0.971287i	$-0.576463\pi$
$89$	−4.48891	−0.475823	−0.237912	+	0.971287i	$0.576463\pi$
$90$	−2.93954	−0.309854
$91$	−1.15197	−0.120759
$92$	8.30827	0.866197
$93$	−1.55885	−0.161646
$94$	−6.17003	−0.636390
$95$	3.43477	0.352400
$96$	0.245893	0.0250963
$97$	3.45447	0.350748	0.175374	−	0.984502i	$-0.443886\pi$
$97$	3.45447	0.350748	0.175374	+	0.984502i	$0.443886\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−2.71712	−0.270364	−0.135182	−	0.990821i	$-0.543162\pi$
$101$	−2.71712	−0.270364	−0.135182	+	0.990821i	$0.543162\pi$
$102$	−1.61090	−0.159503
$103$	−9.90941	−0.976403	−0.488201	−	0.872731i	$-0.662347\pi$
$103$	−9.90941	−0.976403	−0.488201	+	0.872731i	$0.662347\pi$
$104$	1.15197	0.112960
$105$	−0.245893	−0.0239967
$106$	5.35680	0.520298
$107$	−5.18688	−0.501434	−0.250717	−	0.968060i	$-0.580666\pi$
$107$	−5.18688	−0.501434	−0.250717	+	0.968060i	$0.580666\pi$
$108$	−1.46049	−0.140536
$109$	−12.1283	−1.16168	−0.580839	−	0.814018i	$-0.697275\pi$
$109$	−12.1283	−1.16168	−0.580839	+	0.814018i	$0.697275\pi$
$110$	0	0
$111$	−0.950972	−0.0902623
$112$	−1.00000	−0.0944911
$113$	14.5100	1.36499	0.682493	−	0.730893i	$-0.260896\pi$
$113$	14.5100	1.36499	0.682493	+	0.730893i	$0.260896\pi$
$114$	0.844585	0.0791026
$115$	8.30827	0.774750
$116$	2.93346	0.272365
$117$	−3.38626	−0.313060
$118$	−2.24164	−0.206360
$119$	6.55125	0.600552
$120$	0.245893	0.0224468
$121$	0	0
$122$	9.98548	0.904043
$123$	2.59560	0.234037
$124$	−6.33956	−0.569310
$125$	1.00000	0.0894427
$126$	2.93954	0.261875
$127$	0.818399	0.0726212	0.0363106	−	0.999341i	$-0.488439\pi$
$127$	0.818399	0.0726212	0.0363106	+	0.999341i	$0.488439\pi$
$128$	1.00000	0.0883883
$129$	0.727233	0.0640293
$130$	1.15197	0.101034
$131$	−9.76900	−0.853522	−0.426761	−	0.904365i	$-0.640345\pi$
$131$	−9.76900	−0.853522	−0.426761	+	0.904365i	$0.640345\pi$
$132$	0	0
$133$	−3.43477	−0.297832
$134$	14.2147	1.22796
$135$	−1.46049	−0.125699
$136$	−6.55125	−0.561765
$137$	−6.68920	−0.571497	−0.285749	−	0.958305i	$-0.592242\pi$
$137$	−6.68920	−0.571497	−0.285749	+	0.958305i	$0.592242\pi$
$138$	2.04294	0.173907
$139$	8.85349	0.750943	0.375471	−	0.926834i	$-0.377481\pi$
$139$	8.85349	0.750943	0.375471	+	0.926834i	$0.377481\pi$
$140$	−1.00000	−0.0845154
$141$	−1.51717	−0.127768
$142$	−0.380220	−0.0319074
$143$	0	0
$144$	−2.93954	−0.244961
$145$	2.93346	0.243611
$146$	7.23216	0.598538
$147$	0.245893	0.0202809
$148$	−3.86743	−0.317901
$149$	4.32495	0.354314	0.177157	−	0.984183i	$-0.443310\pi$
$149$	4.32495	0.354314	0.177157	+	0.984183i	$0.443310\pi$
$150$	0.245893	0.0200771
$151$	17.5142	1.42529	0.712644	−	0.701526i	$-0.247498\pi$
$151$	17.5142	1.42529	0.712644	+	0.701526i	$0.247498\pi$
$152$	3.43477	0.278596
$153$	19.2576	1.55689
$154$	0	0
$155$	−6.33956	−0.509206
$156$	0.283261	0.0226790
$157$	8.20435	0.654778	0.327389	−	0.944890i	$-0.393831\pi$
$157$	8.20435	0.654778	0.327389	+	0.944890i	$0.393831\pi$
$158$	2.27381	0.180895
$159$	1.31720	0.104461
$160$	1.00000	0.0790569
$161$	−8.30827	−0.654783
$162$	8.45949	0.664640
$163$	−11.4586	−0.897504	−0.448752	−	0.893656i	$-0.648131\pi$
$163$	−11.4586	−0.897504	−0.448752	+	0.893656i	$0.648131\pi$
$164$	10.5558	0.824271
$165$	0	0
$166$	11.3292	0.879320
$167$	0.763127	0.0590525	0.0295263	−	0.999564i	$-0.490600\pi$
$167$	0.763127	0.0590525	0.0295263	+	0.999564i	$0.490600\pi$
$168$	−0.245893	−0.0189710
$169$	−11.6730	−0.897920
$170$	−6.55125	−0.502458
$171$	−10.0966	−0.772108
$172$	2.95752	0.225509
$173$	8.76752	0.666582	0.333291	−	0.942824i	$-0.391841\pi$
$173$	8.76752	0.666582	0.333291	+	0.942824i	$0.391841\pi$
$174$	0.721318	0.0546830
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−0.551204	−0.0414310
$178$	−4.48891	−0.336458
$179$	15.0279	1.12324	0.561619	−	0.827396i	$-0.310179\pi$
$179$	15.0279	1.12324	0.561619	+	0.827396i	$0.310179\pi$
$180$	−2.93954	−0.219100
$181$	21.1888	1.57495	0.787474	−	0.616348i	$-0.211389\pi$
$181$	21.1888	1.57495	0.787474	+	0.616348i	$0.211389\pi$
$182$	−1.15197	−0.0853897
$183$	2.45536	0.181505
$184$	8.30827	0.612494
$185$	−3.86743	−0.284339
$186$	−1.55885	−0.114301
$187$	0	0
$188$	−6.17003	−0.449996
$189$	1.46049	0.106235
$190$	3.43477	0.249184
$191$	4.05200	0.293192	0.146596	−	0.989196i	$-0.453168\pi$
$191$	4.05200	0.293192	0.146596	+	0.989196i	$0.453168\pi$
$192$	0.245893	0.0177458
$193$	20.5136	1.47660	0.738302	−	0.674470i	$-0.235628\pi$
$193$	20.5136	1.47660	0.738302	+	0.674470i	$0.235628\pi$
$194$	3.45447	0.248017
$195$	0.283261	0.0202848
$196$	1.00000	0.0714286
$197$	20.4398	1.45628	0.728139	−	0.685429i	$-0.240386\pi$
$197$	20.4398	1.45628	0.728139	+	0.685429i	$0.240386\pi$
$198$	0	0
$199$	−3.23576	−0.229377	−0.114688	−	0.993402i	$-0.536587\pi$
$199$	−3.23576	−0.229377	−0.114688	+	0.993402i	$0.536587\pi$
$200$	1.00000	0.0707107
$201$	3.49529	0.246538
$202$	−2.71712	−0.191176
$203$	−2.93346	−0.205889
$204$	−1.61090	−0.112786
$205$	10.5558	0.737251
$206$	−9.90941	−0.690421
$207$	−24.4225	−1.69748
$208$	1.15197	0.0798748
$209$	0	0
$210$	−0.245893	−0.0169682
$211$	12.3479	0.850064	0.425032	−	0.905178i	$-0.360263\pi$
$211$	12.3479	0.850064	0.425032	+	0.905178i	$0.360263\pi$
$212$	5.35680	0.367906
$213$	−0.0934934	−0.00640606
$214$	−5.18688	−0.354568
$215$	2.95752	0.201701
$216$	−1.46049	−0.0993737
$217$	6.33956	0.430358
$218$	−12.1283	−0.821431
$219$	1.77834	0.120169
$220$	0	0
$221$	−7.54684	−0.507655
$222$	−0.950972	−0.0638251
$223$	8.30846	0.556376	0.278188	−	0.960527i	$-0.410266\pi$
$223$	8.30846	0.556376	0.278188	+	0.960527i	$0.410266\pi$
$224$	−1.00000	−0.0668153
$225$	−2.93954	−0.195969
$226$	14.5100	0.965190
$227$	15.7302	1.04405	0.522023	−	0.852931i	$-0.325177\pi$
$227$	15.7302	1.04405	0.522023	+	0.852931i	$0.325177\pi$
$228$	0.844585	0.0559340
$229$	−23.7309	−1.56818	−0.784090	−	0.620647i	$-0.786870\pi$
$229$	−23.7309	−1.56818	−0.784090	+	0.620647i	$0.786870\pi$
$230$	8.30827	0.547831
$231$	0	0
$232$	2.93346	0.192591
$233$	−11.9023	−0.779745	−0.389872	−	0.920869i	$-0.627481\pi$
$233$	−11.9023	−0.779745	−0.389872	+	0.920869i	$0.627481\pi$
$234$	−3.38626	−0.221367
$235$	−6.17003	−0.402488
$236$	−2.24164	−0.145919
$237$	0.559114	0.0363184
$238$	6.55125	0.424654
$239$	21.8184	1.41132	0.705659	−	0.708552i	$-0.250651\pi$
$239$	21.8184	1.41132	0.705659	+	0.708552i	$0.250651\pi$
$240$	0.245893	0.0158723
$241$	14.7413	0.949569	0.474785	−	0.880102i	$-0.342526\pi$
$241$	14.7413	0.949569	0.474785	+	0.880102i	$0.342526\pi$
$242$	0	0
$243$	6.46159	0.414511
$244$	9.98548	0.639255
$245$	1.00000	0.0638877
$246$	2.59560	0.165489
$247$	3.95675	0.251762
$248$	−6.33956	−0.402563
$249$	2.78578	0.176542
$250$	1.00000	0.0632456
$251$	−12.8410	−0.810517	−0.405258	−	0.914202i	$-0.632818\pi$
$251$	−12.8410	−0.810517	−0.405258	+	0.914202i	$0.632818\pi$
$252$	2.93954	0.185173
$253$	0	0
$254$	0.818399	0.0513509
$255$	−1.61090	−0.100879
$256$	1.00000	0.0625000
$257$	5.70646	0.355959	0.177980	−	0.984034i	$-0.443044\pi$
$257$	5.70646	0.355959	0.177980	+	0.984034i	$0.443044\pi$
$258$	0.727233	0.0452756
$259$	3.86743	0.240310
$260$	1.15197	0.0714422
$261$	−8.62303	−0.533752
$262$	−9.76900	−0.603531
$263$	−12.9095	−0.796033	−0.398016	−	0.917378i	$-0.630301\pi$
$263$	−12.9095	−0.796033	−0.398016	+	0.917378i	$0.630301\pi$
$264$	0	0
$265$	5.35680	0.329065
$266$	−3.43477	−0.210599
$267$	−1.10379	−0.0675508
$268$	14.2147	0.868299
$269$	19.2650	1.17461	0.587304	−	0.809367i	$-0.300189\pi$
$269$	19.2650	1.17461	0.587304	+	0.809367i	$0.300189\pi$
$270$	−1.46049	−0.0888825
$271$	−27.8182	−1.68983	−0.844917	−	0.534897i	$-0.820351\pi$
$271$	−27.8182	−1.68983	−0.844917	+	0.534897i	$0.820351\pi$
$272$	−6.55125	−0.397228
$273$	−0.283261	−0.0171437
$274$	−6.68920	−0.404110
$275$	0	0
$276$	2.04294	0.122971
$277$	−12.3700	−0.743240	−0.371620	−	0.928385i	$-0.621198\pi$
$277$	−12.3700	−0.743240	−0.371620	+	0.928385i	$0.621198\pi$
$278$	8.85349	0.530997
$279$	18.6354	1.11567
$280$	−1.00000	−0.0597614
$281$	−21.9883	−1.31171	−0.655856	−	0.754886i	$-0.727692\pi$
$281$	−21.9883	−1.31171	−0.655856	+	0.754886i	$0.727692\pi$
$282$	−1.51717	−0.0903459
$283$	13.8205	0.821542	0.410771	−	0.911739i	$-0.365260\pi$
$283$	13.8205	0.821542	0.410771	+	0.911739i	$0.365260\pi$
$284$	−0.380220	−0.0225619
$285$	0.844585	0.0500289
$286$	0	0
$287$	−10.5558	−0.623091
$288$	−2.93954	−0.173214
$289$	25.9188	1.52464
$290$	2.93346	0.172259
$291$	0.849430	0.0497945
$292$	7.23216	0.423230
$293$	18.2052	1.06356	0.531779	−	0.846883i	$-0.321524\pi$
$293$	18.2052	1.06356	0.531779	+	0.846883i	$0.321524\pi$
$294$	0.245893	0.0143408
$295$	−2.24164	−0.130513
$296$	−3.86743	−0.224790
$297$	0	0
$298$	4.32495	0.250538
$299$	9.57087	0.553498
$300$	0.245893	0.0141966
$301$	−2.95752	−0.170469
$302$	17.5142	1.00783
$303$	−0.668121	−0.0383825
$304$	3.43477	0.196997
$305$	9.98548	0.571767
$306$	19.2576	1.10089
$307$	−23.7729	−1.35679	−0.678395	−	0.734697i	$-0.737324\pi$
$307$	−23.7729	−1.35679	−0.678395	+	0.734697i	$0.737324\pi$
$308$	0	0
$309$	−2.43665	−0.138616
$310$	−6.33956	−0.360063
$311$	−18.2509	−1.03492	−0.517458	−	0.855709i	$-0.673121\pi$
$311$	−18.2509	−1.03492	−0.517458	+	0.855709i	$0.673121\pi$
$312$	0.283261	0.0160365
$313$	−16.4004	−0.927003	−0.463502	−	0.886096i	$-0.653407\pi$
$313$	−16.4004	−0.927003	−0.463502	+	0.886096i	$0.653407\pi$
$314$	8.20435	0.462998
$315$	2.93954	0.165624
$316$	2.27381	0.127912
$317$	−26.2832	−1.47621	−0.738106	−	0.674684i	$-0.764280\pi$
$317$	−26.2832	−1.47621	−0.738106	+	0.674684i	$0.764280\pi$
$318$	1.31720	0.0738648
$319$	0	0
$320$	1.00000	0.0559017
$321$	−1.27542	−0.0711868
$322$	−8.30827	−0.463002
$323$	−22.5020	−1.25205
$324$	8.45949	0.469971
$325$	1.15197	0.0638998
$326$	−11.4586	−0.634631
$327$	−2.98226	−0.164919
$328$	10.5558	0.582848
$329$	6.17003	0.340165
$330$	0	0
$331$	24.6604	1.35546	0.677729	−	0.735312i	$-0.262964\pi$
$331$	24.6604	1.35546	0.677729	+	0.735312i	$0.262964\pi$
$332$	11.3292	0.621773
$333$	11.3684	0.622987
$334$	0.763127	0.0417565
$335$	14.2147	0.776631
$336$	−0.245893	−0.0134146
$337$	7.67814	0.418255	0.209128	−	0.977888i	$-0.432938\pi$
$337$	7.67814	0.418255	0.209128	+	0.977888i	$0.432938\pi$
$338$	−11.6730	−0.634926
$339$	3.56790	0.193782
$340$	−6.55125	−0.355291
$341$	0	0
$342$	−10.0966	−0.545963
$343$	−1.00000	−0.0539949
$344$	2.95752	0.159459
$345$	2.04294	0.109988
$346$	8.76752	0.471345
$347$	−18.3330	−0.984169	−0.492084	−	0.870547i	$-0.663765\pi$
$347$	−18.3330	−0.984169	−0.492084	+	0.870547i	$0.663765\pi$
$348$	0.721318	0.0386667
$349$	6.32456	0.338546	0.169273	−	0.985569i	$-0.445858\pi$
$349$	6.32456	0.338546	0.169273	+	0.985569i	$0.445858\pi$
$350$	−1.00000	−0.0534522
$351$	−1.68244	−0.0898020
$352$	0	0
$353$	−18.7476	−0.997834	−0.498917	−	0.866650i	$-0.666269\pi$
$353$	−18.7476	−0.997834	−0.498917	+	0.866650i	$0.666269\pi$
$354$	−0.551204	−0.0292961
$355$	−0.380220	−0.0201800
$356$	−4.48891	−0.237912
$357$	1.61090	0.0852581
$358$	15.0279	0.794250
$359$	−20.8408	−1.09994	−0.549968	−	0.835186i	$-0.685360\pi$
$359$	−20.8408	−1.09994	−0.549968	+	0.835186i	$0.685360\pi$
$360$	−2.93954	−0.154927
$361$	−7.20237	−0.379072
$362$	21.1888	1.11366
$363$	0	0
$364$	−1.15197	−0.0603796
$365$	7.23216	0.378549
$366$	2.45536	0.128344
$367$	−25.6396	−1.33838	−0.669188	−	0.743093i	$-0.733358\pi$
$367$	−25.6396	−1.33838	−0.669188	+	0.743093i	$0.733358\pi$
$368$	8.30827	0.433098
$369$	−31.0292	−1.61532
$370$	−3.86743	−0.201058
$371$	−5.35680	−0.278111
$372$	−1.55885	−0.0808228
$373$	34.9239	1.80829	0.904146	−	0.427223i	$-0.140508\pi$
$373$	34.9239	1.80829	0.904146	+	0.427223i	$0.140508\pi$
$374$	0	0
$375$	0.245893	0.0126978
$376$	−6.17003	−0.318195
$377$	3.37926	0.174041
$378$	1.46049	0.0751195
$379$	1.32409	0.0680138	0.0340069	−	0.999422i	$-0.489173\pi$
$379$	1.32409	0.0680138	0.0340069	+	0.999422i	$0.489173\pi$
$380$	3.43477	0.176200
$381$	0.201238	0.0103098
$382$	4.05200	0.207318
$383$	2.42235	0.123776	0.0618880	−	0.998083i	$-0.480288\pi$
$383$	2.42235	0.123776	0.0618880	+	0.998083i	$0.480288\pi$
$384$	0.245893	0.0125482
$385$	0	0
$386$	20.5136	1.04412
$387$	−8.69374	−0.441928
$388$	3.45447	0.175374
$389$	−21.2547	−1.07766	−0.538828	−	0.842416i	$-0.681133\pi$
$389$	−21.2547	−1.07766	−0.538828	+	0.842416i	$0.681133\pi$
$390$	0.283261	0.0143435
$391$	−54.4295	−2.75262
$392$	1.00000	0.0505076
$393$	−2.40213	−0.121171
$394$	20.4398	1.02974
$395$	2.27381	0.114408
$396$	0	0
$397$	12.8715	0.646003	0.323001	−	0.946398i	$-0.395308\pi$
$397$	12.8715	0.646003	0.323001	+	0.946398i	$0.395308\pi$
$398$	−3.23576	−0.162194
$399$	−0.844585	−0.0422821
$400$	1.00000	0.0500000
$401$	−11.1726	−0.557933	−0.278966	−	0.960301i	$-0.589992\pi$
$401$	−11.1726	−0.557933	−0.278966	+	0.960301i	$0.589992\pi$
$402$	3.49529	0.174329
$403$	−7.30299	−0.363788
$404$	−2.71712	−0.135182
$405$	8.45949	0.420355
$406$	−2.93346	−0.145585
$407$	0	0
$408$	−1.61090	−0.0797516
$409$	−14.7289	−0.728297	−0.364148	−	0.931341i	$-0.618640\pi$
$409$	−14.7289	−0.728297	−0.364148	+	0.931341i	$0.618640\pi$
$410$	10.5558	0.521315
$411$	−1.64483	−0.0811333
$412$	−9.90941	−0.488201
$413$	2.24164	0.110304
$414$	−24.4225	−1.20030
$415$	11.3292	0.556131
$416$	1.15197	0.0564800
$417$	2.17701	0.106609
$418$	0	0
$419$	−14.1913	−0.693290	−0.346645	−	0.937996i	$-0.612679\pi$
$419$	−14.1913	−0.693290	−0.346645	+	0.937996i	$0.612679\pi$
$420$	−0.245893	−0.0119983
$421$	−26.0595	−1.27006	−0.635030	−	0.772487i	$-0.719012\pi$
$421$	−26.0595	−1.27006	−0.635030	+	0.772487i	$0.719012\pi$
$422$	12.3479	0.601086
$423$	18.1370	0.881853
$424$	5.35680	0.260149
$425$	−6.55125	−0.317782
$426$	−0.0934934	−0.00452977
$427$	−9.98548	−0.483231
$428$	−5.18688	−0.250717
$429$	0	0
$430$	2.95752	0.142624
$431$	18.8146	0.906270	0.453135	−	0.891442i	$-0.350306\pi$
$431$	18.8146	0.906270	0.453135	+	0.891442i	$0.350306\pi$
$432$	−1.46049	−0.0702678
$433$	20.4028	0.980497	0.490248	−	0.871583i	$-0.336906\pi$
$433$	20.4028	0.980497	0.490248	+	0.871583i	$0.336906\pi$
$434$	6.33956	0.304309
$435$	0.721318	0.0345845
$436$	−12.1283	−0.580839
$437$	28.5370	1.36511
$438$	1.77834	0.0849722
$439$	−17.5292	−0.836621	−0.418311	−	0.908304i	$-0.637378\pi$
$439$	−17.5292	−0.836621	−0.418311	+	0.908304i	$0.637378\pi$
$440$	0	0
$441$	−2.93954	−0.139978
$442$	−7.54684	−0.358967
$443$	−26.1950	−1.24456	−0.622281	−	0.782794i	$-0.713794\pi$
$443$	−26.1950	−1.24456	−0.622281	+	0.782794i	$0.713794\pi$
$444$	−0.950972	−0.0451312
$445$	−4.48891	−0.212795
$446$	8.30846	0.393417
$447$	1.06347	0.0503006
$448$	−1.00000	−0.0472456
$449$	−9.49700	−0.448191	−0.224096	−	0.974567i	$-0.571943\pi$
$449$	−9.49700	−0.448191	−0.224096	+	0.974567i	$0.571943\pi$
$450$	−2.93954	−0.138571
$451$	0	0
$452$	14.5100	0.682493
$453$	4.30662	0.202343
$454$	15.7302	0.738253
$455$	−1.15197	−0.0540052
$456$	0.844585	0.0395513
$457$	−17.5412	−0.820545	−0.410272	−	0.911963i	$-0.634566\pi$
$457$	−17.5412	−0.820545	−0.410272	+	0.911963i	$0.634566\pi$
$458$	−23.7309	−1.10887
$459$	9.56802	0.446597
$460$	8.30827	0.387375
$461$	22.6720	1.05594	0.527970	−	0.849263i	$-0.322953\pi$
$461$	22.6720	1.05594	0.527970	+	0.849263i	$0.322953\pi$
$462$	0	0
$463$	−26.9554	−1.25272	−0.626362	−	0.779532i	$-0.715457\pi$
$463$	−26.9554	−1.25272	−0.626362	+	0.779532i	$0.715457\pi$
$464$	2.93346	0.136183
$465$	−1.55885	−0.0722901
$466$	−11.9023	−0.551363
$467$	14.7718	0.683557	0.341778	−	0.939781i	$-0.388971\pi$
$467$	14.7718	0.683557	0.341778	+	0.939781i	$0.388971\pi$
$468$	−3.38626	−0.156530
$469$	−14.2147	−0.656373
$470$	−6.17003	−0.284602
$471$	2.01739	0.0929565
$472$	−2.24164	−0.103180
$473$	0	0
$474$	0.559114	0.0256810
$475$	3.43477	0.157598
$476$	6.55125	0.300276
$477$	−15.7465	−0.720983
$478$	21.8184	0.997952
$479$	31.8220	1.45398	0.726992	−	0.686646i	$-0.240918\pi$
$479$	31.8220	1.45398	0.726992	+	0.686646i	$0.240918\pi$
$480$	0.245893	0.0112234
$481$	−4.45516	−0.203138
$482$	14.7413	0.671447
$483$	−2.04294	−0.0929571
$484$	0	0
$485$	3.45447	0.156859
$486$	6.46159	0.293104
$487$	24.8664	1.12681	0.563403	−	0.826182i	$-0.309492\pi$
$487$	24.8664	1.12681	0.563403	+	0.826182i	$0.309492\pi$
$488$	9.98548	0.452022
$489$	−2.81758	−0.127415
$490$	1.00000	0.0451754
$491$	41.8496	1.88864	0.944322	−	0.329022i	$-0.106719\pi$
$491$	41.8496	1.88864	0.944322	+	0.329022i	$0.106719\pi$
$492$	2.59560	0.117019
$493$	−19.2178	−0.865528
$494$	3.95675	0.178023
$495$	0	0
$496$	−6.33956	−0.284655
$497$	0.380220	0.0170552
$498$	2.78578	0.124834
$499$	24.8351	1.11177	0.555885	−	0.831259i	$-0.312379\pi$
$499$	24.8351	1.11177	0.555885	+	0.831259i	$0.312379\pi$
$500$	1.00000	0.0447214
$501$	0.187647	0.00838347
$502$	−12.8410	−0.573122
$503$	−8.92909	−0.398128	−0.199064	−	0.979986i	$-0.563790\pi$
$503$	−8.92909	−0.398128	−0.199064	+	0.979986i	$0.563790\pi$
$504$	2.93954	0.130937
$505$	−2.71712	−0.120910
$506$	0	0
$507$	−2.87030	−0.127474
$508$	0.818399	0.0363106
$509$	22.8685	1.01363	0.506813	−	0.862056i	$-0.330823\pi$
$509$	22.8685	1.01363	0.506813	+	0.862056i	$0.330823\pi$
$510$	−1.61090	−0.0713320
$511$	−7.23216	−0.319932
$512$	1.00000	0.0441942
$513$	−5.01644	−0.221481
$514$	5.70646	0.251701
$515$	−9.90941	−0.436661
$516$	0.727233	0.0320147
$517$	0	0
$518$	3.86743	0.169925
$519$	2.15587	0.0946322
$520$	1.15197	0.0505172
$521$	−34.4131	−1.50767	−0.753833	−	0.657066i	$-0.771797\pi$
$521$	−34.4131	−1.50767	−0.753833	+	0.657066i	$0.771797\pi$
$522$	−8.62303	−0.377420
$523$	6.70316	0.293109	0.146554	−	0.989203i	$-0.453182\pi$
$523$	6.70316	0.293109	0.146554	+	0.989203i	$0.453182\pi$
$524$	−9.76900	−0.426761
$525$	−0.245893	−0.0107316
$526$	−12.9095	−0.562880
$527$	41.5320	1.80916
$528$	0	0
$529$	46.0273	2.00119
$530$	5.35680	0.232684
$531$	6.58939	0.285955
$532$	−3.43477	−0.148916
$533$	12.1600	0.526708
$534$	−1.10379	−0.0477657
$535$	−5.18688	−0.224248
$536$	14.2147	0.613980
$537$	3.69525	0.159462
$538$	19.2650	0.830573
$539$	0	0
$540$	−1.46049	−0.0628495
$541$	16.4021	0.705181	0.352590	−	0.935778i	$-0.385301\pi$
$541$	16.4021	0.705181	0.352590	+	0.935778i	$0.385301\pi$
$542$	−27.8182	−1.19489
$543$	5.21016	0.223589
$544$	−6.55125	−0.280882
$545$	−12.1283	−0.519518
$546$	−0.283261	−0.0121225
$547$	6.99852	0.299235	0.149618	−	0.988744i	$-0.452196\pi$
$547$	6.99852	0.299235	0.149618	+	0.988744i	$0.452196\pi$
$548$	−6.68920	−0.285749
$549$	−29.3527	−1.25274
$550$	0	0
$551$	10.0758	0.429242
$552$	2.04294	0.0869534
$553$	−2.27381	−0.0966923
$554$	−12.3700	−0.525550
$555$	−0.950972	−0.0403665
$556$	8.85349	0.375471
$557$	−14.1845	−0.601016	−0.300508	−	0.953779i	$-0.597156\pi$
$557$	−14.1845	−0.601016	−0.300508	+	0.953779i	$0.597156\pi$
$558$	18.6354	0.788898
$559$	3.40698	0.144100
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−21.9883	−0.927520
$563$	26.6253	1.12212	0.561062	−	0.827774i	$-0.310393\pi$
$563$	26.6253	1.12212	0.561062	+	0.827774i	$0.310393\pi$
$564$	−1.51717	−0.0638842
$565$	14.5100	0.610440
$566$	13.8205	0.580918
$567$	−8.45949	−0.355265
$568$	−0.380220	−0.0159537
$569$	1.29976	0.0544889	0.0272445	−	0.999629i	$-0.491327\pi$
$569$	1.29976	0.0544889	0.0272445	+	0.999629i	$0.491327\pi$
$570$	0.844585	0.0353758
$571$	24.2338	1.01415	0.507076	−	0.861901i	$-0.330726\pi$
$571$	24.2338	1.01415	0.507076	+	0.861901i	$0.330726\pi$
$572$	0	0
$573$	0.996357	0.0416234
$574$	−10.5558	−0.440592
$575$	8.30827	0.346479
$576$	−2.93954	−0.122481
$577$	−37.9967	−1.58183	−0.790913	−	0.611929i	$-0.790394\pi$
$577$	−37.9967	−1.58183	−0.790913	+	0.611929i	$0.790394\pi$
$578$	25.9188	1.07808
$579$	5.04416	0.209628
$580$	2.93346	0.121805
$581$	−11.3292	−0.470016
$582$	0.849430	0.0352100
$583$	0	0
$584$	7.23216	0.299269
$585$	−3.38626	−0.140005
$586$	18.2052	0.752049
$587$	5.54502	0.228868	0.114434	−	0.993431i	$-0.463495\pi$
$587$	5.54502	0.228868	0.114434	+	0.993431i	$0.463495\pi$
$588$	0.245893	0.0101404
$589$	−21.7749	−0.897220
$590$	−2.24164	−0.0922870
$591$	5.02601	0.206742
$592$	−3.86743	−0.158950
$593$	45.5378	1.87001	0.935006	−	0.354633i	$-0.115394\pi$
$593$	45.5378	1.87001	0.935006	+	0.354633i	$0.115394\pi$
$594$	0	0
$595$	6.55125	0.268575
$596$	4.32495	0.177157
$597$	−0.795650	−0.0325638
$598$	9.57087	0.391382
$599$	38.2388	1.56240	0.781198	−	0.624284i	$-0.214609\pi$
$599$	38.2388	1.56240	0.781198	+	0.624284i	$0.214609\pi$
$600$	0.245893	0.0100385
$601$	−41.8737	−1.70807	−0.854033	−	0.520219i	$-0.825850\pi$
$601$	−41.8737	−1.70807	−0.854033	+	0.520219i	$0.825850\pi$
$602$	−2.95752	−0.120540
$603$	−41.7846	−1.70160
$604$	17.5142	0.712644
$605$	0	0
$606$	−0.668121	−0.0271406
$607$	−6.87012	−0.278850	−0.139425	−	0.990233i	$-0.544525\pi$
$607$	−6.87012	−0.278850	−0.139425	+	0.990233i	$0.544525\pi$
$608$	3.43477	0.139298
$609$	−0.721318	−0.0292293
$610$	9.98548	0.404300
$611$	−7.10769	−0.287546
$612$	19.2576	0.778443
$613$	−37.8887	−1.53031	−0.765155	−	0.643847i	$-0.777337\pi$
$613$	−37.8887	−1.53031	−0.765155	+	0.643847i	$0.777337\pi$
$614$	−23.7729	−0.959396
$615$	2.59560	0.104665
$616$	0	0
$617$	19.7135	0.793636	0.396818	−	0.917897i	$-0.370114\pi$
$617$	19.7135	0.793636	0.396818	+	0.917897i	$0.370114\pi$
$618$	−2.43665	−0.0980165
$619$	45.3200	1.82156	0.910782	−	0.412888i	$-0.135480\pi$
$619$	45.3200	1.82156	0.910782	+	0.412888i	$0.135480\pi$
$620$	−6.33956	−0.254603
$621$	−12.1341	−0.486926
$622$	−18.2509	−0.731796
$623$	4.48891	0.179844
$624$	0.283261	0.0113395
$625$	1.00000	0.0400000
$626$	−16.4004	−0.655490
$627$	0	0
$628$	8.20435	0.327389
$629$	25.3365	1.01023
$630$	2.93954	0.117114
$631$	−29.3153	−1.16702	−0.583512	−	0.812104i	$-0.698322\pi$
$631$	−29.3153	−1.16702	−0.583512	+	0.812104i	$0.698322\pi$
$632$	2.27381	0.0904474
$633$	3.03626	0.120680
$634$	−26.2832	−1.04384
$635$	0.818399	0.0324772
$636$	1.31720	0.0522303
$637$	1.15197	0.0456427
$638$	0	0
$639$	1.11767	0.0442144
$640$	1.00000	0.0395285
$641$	−21.9740	−0.867923	−0.433961	−	0.900932i	$-0.642885\pi$
$641$	−21.9740	−0.867923	−0.433961	+	0.900932i	$0.642885\pi$
$642$	−1.27542	−0.0503366
$643$	−46.4736	−1.83274	−0.916370	−	0.400332i	$-0.868895\pi$
$643$	−46.4736	−1.83274	−0.916370	+	0.400332i	$0.868895\pi$
$644$	−8.30827	−0.327392
$645$	0.727233	0.0286348
$646$	−22.5020	−0.885330
$647$	19.3483	0.760660	0.380330	−	0.924851i	$-0.375810\pi$
$647$	19.3483	0.760660	0.380330	+	0.924851i	$0.375810\pi$
$648$	8.45949	0.332320
$649$	0	0
$650$	1.15197	0.0451840
$651$	1.55885	0.0610963
$652$	−11.4586	−0.448752
$653$	36.9129	1.44452	0.722258	−	0.691624i	$-0.243105\pi$
$653$	36.9129	1.44452	0.722258	+	0.691624i	$0.243105\pi$
$654$	−2.98226	−0.116615
$655$	−9.76900	−0.381706
$656$	10.5558	0.412136
$657$	−21.2592	−0.829400
$658$	6.17003	0.240533
$659$	24.3913	0.950151	0.475075	−	0.879945i	$-0.342421\pi$
$659$	24.3913	0.950151	0.475075	+	0.879945i	$0.342421\pi$
$660$	0	0
$661$	−5.60864	−0.218151	−0.109075	−	0.994033i	$-0.534789\pi$
$661$	−5.60864	−0.218151	−0.109075	+	0.994033i	$0.534789\pi$
$662$	24.6604	0.958453
$663$	−1.85571	−0.0720699
$664$	11.3292	0.439660
$665$	−3.43477	−0.133195
$666$	11.3684	0.440518
$667$	24.3720	0.943688
$668$	0.763127	0.0295263
$669$	2.04299	0.0789866
$670$	14.2147	0.549161
$671$	0	0
$672$	−0.245893	−0.00948552
$673$	−27.6564	−1.06608	−0.533038	−	0.846091i	$-0.678950\pi$
$673$	−27.6564	−1.06608	−0.533038	+	0.846091i	$0.678950\pi$
$674$	7.67814	0.295751
$675$	−1.46049	−0.0562143
$676$	−11.6730	−0.448960
$677$	28.6810	1.10230	0.551149	−	0.834407i	$-0.314189\pi$
$677$	28.6810	1.10230	0.551149	+	0.834407i	$0.314189\pi$
$678$	3.56790	0.137024
$679$	−3.45447	−0.132570
$680$	−6.55125	−0.251229
$681$	3.86793	0.148219
$682$	0	0
$683$	−38.9631	−1.49088	−0.745440	−	0.666572i	$-0.767761\pi$
$683$	−38.9631	−1.49088	−0.745440	+	0.666572i	$0.767761\pi$
$684$	−10.0966	−0.386054
$685$	−6.68920	−0.255581
$686$	−1.00000	−0.0381802
$687$	−5.83525	−0.222629
$688$	2.95752	0.112754
$689$	6.17087	0.235091
$690$	2.04294	0.0777735
$691$	−15.0471	−0.572418	−0.286209	−	0.958167i	$-0.592395\pi$
$691$	−15.0471	−0.572418	−0.286209	+	0.958167i	$0.592395\pi$
$692$	8.76752	0.333291
$693$	0	0
$694$	−18.3330	−0.695912
$695$	8.85349	0.335832
$696$	0.721318	0.0273415
$697$	−69.1538	−2.61939
$698$	6.32456	0.239388
$699$	−2.92669	−0.110697
$700$	−1.00000	−0.0377964
$701$	−26.9497	−1.01788	−0.508938	−	0.860803i	$-0.669962\pi$
$701$	−26.9497	−1.01788	−0.508938	+	0.860803i	$0.669962\pi$
$702$	−1.68244	−0.0634996
$703$	−13.2837	−0.501005
$704$	0	0
$705$	−1.51717	−0.0571398
$706$	−18.7476	−0.705575
$707$	2.71712	0.102188
$708$	−0.551204	−0.0207155
$709$	−28.0992	−1.05529	−0.527644	−	0.849466i	$-0.676924\pi$
$709$	−28.0992	−1.05529	−0.527644	+	0.849466i	$0.676924\pi$
$710$	−0.380220	−0.0142694
$711$	−6.68395	−0.250668
$712$	−4.48891	−0.168229
$713$	−52.6708	−1.97254
$714$	1.61090	0.0602866
$715$	0	0
$716$	15.0279	0.561619
$717$	5.36500	0.200360
$718$	−20.8408	−0.777772
$719$	−34.1887	−1.27502	−0.637511	−	0.770441i	$-0.720036\pi$
$719$	−34.1887	−1.27502	−0.637511	+	0.770441i	$0.720036\pi$
$720$	−2.93954	−0.109550
$721$	9.90941	0.369046
$722$	−7.20237	−0.268045
$723$	3.62477	0.134807
$724$	21.1888	0.787474
$725$	2.93346	0.108946
$726$	0	0
$727$	46.8368	1.73708	0.868541	−	0.495618i	$-0.165058\pi$
$727$	46.8368	1.73708	0.868541	+	0.495618i	$0.165058\pi$
$728$	−1.15197	−0.0426949
$729$	−23.7896	−0.881096
$730$	7.23216	0.267674
$731$	−19.3754	−0.716627
$732$	2.45536	0.0907526
$733$	4.80903	0.177625	0.0888127	−	0.996048i	$-0.471693\pi$
$733$	4.80903	0.177625	0.0888127	+	0.996048i	$0.471693\pi$
$734$	−25.6396	−0.946374
$735$	0.245893	0.00906989
$736$	8.30827	0.306247
$737$	0	0
$738$	−31.0292	−1.14220
$739$	−7.60017	−0.279577	−0.139788	−	0.990181i	$-0.544642\pi$
$739$	−7.60017	−0.279577	−0.139788	+	0.990181i	$0.544642\pi$
$740$	−3.86743	−0.142169
$741$	0.972936	0.0357417
$742$	−5.35680	−0.196654
$743$	51.5981	1.89295	0.946475	−	0.322777i	$-0.104616\pi$
$743$	51.5981	1.89295	0.946475	+	0.322777i	$0.104616\pi$
$744$	−1.55885	−0.0571503
$745$	4.32495	0.158454
$746$	34.9239	1.27866
$747$	−33.3027	−1.21848
$748$	0	0
$749$	5.18688	0.189524
$750$	0.245893	0.00897874
$751$	45.3313	1.65416	0.827081	−	0.562082i	$-0.190000\pi$
$751$	45.3313	1.65416	0.827081	+	0.562082i	$0.190000\pi$
$752$	−6.17003	−0.224998
$753$	−3.15751	−0.115066
$754$	3.37926	0.123066
$755$	17.5142	0.637408
$756$	1.46049	0.0531175
$757$	39.5221	1.43646	0.718228	−	0.695808i	$-0.244954\pi$
$757$	39.5221	1.43646	0.718228	+	0.695808i	$0.244954\pi$
$758$	1.32409	0.0480930
$759$	0	0
$760$	3.43477	0.124592
$761$	−27.1122	−0.982818	−0.491409	−	0.870929i	$-0.663518\pi$
$761$	−27.1122	−0.982818	−0.491409	+	0.870929i	$0.663518\pi$
$762$	0.201238	0.00729010
$763$	12.1283	0.439073
$764$	4.05200	0.146596
$765$	19.2576	0.696261
$766$	2.42235	0.0875229
$767$	−2.58231	−0.0932416
$768$	0.245893	0.00887289
$769$	−18.0791	−0.651948	−0.325974	−	0.945379i	$-0.605692\pi$
$769$	−18.0791	−0.651948	−0.325974	+	0.945379i	$0.605692\pi$
$770$	0	0
$771$	1.40318	0.0505342
$772$	20.5136	0.738302
$773$	−37.7084	−1.35628	−0.678138	−	0.734935i	$-0.737213\pi$
$773$	−37.7084	−1.35628	−0.678138	+	0.734935i	$0.737213\pi$
$774$	−8.69374	−0.312490
$775$	−6.33956	−0.227724
$776$	3.45447	0.124008
$777$	0.950972	0.0341159
$778$	−21.2547	−0.762018
$779$	36.2568	1.29903
$780$	0.283261	0.0101424
$781$	0	0
$782$	−54.4295	−1.94639
$783$	−4.28429	−0.153108
$784$	1.00000	0.0357143
$785$	8.20435	0.292826
$786$	−2.40213	−0.0856810
$787$	−4.74651	−0.169195	−0.0845974	−	0.996415i	$-0.526960\pi$
$787$	−4.74651	−0.169195	−0.0845974	+	0.996415i	$0.526960\pi$
$788$	20.4398	0.728139
$789$	−3.17435	−0.113010
$790$	2.27381	0.0808986
$791$	−14.5100	−0.515916
$792$	0	0
$793$	11.5030	0.408483
$794$	12.8715	0.456793
$795$	1.31720	0.0467162
$796$	−3.23576	−0.114688
$797$	−25.4843	−0.902701	−0.451350	−	0.892347i	$-0.649058\pi$
$797$	−25.4843	−0.902701	−0.451350	+	0.892347i	$0.649058\pi$
$798$	−0.844585	−0.0298980
$799$	40.4214	1.43001
$800$	1.00000	0.0353553
$801$	13.1953	0.466233
$802$	−11.1726	−0.394518
$803$	0	0
$804$	3.49529	0.123269
$805$	−8.30827	−0.292828
$806$	−7.30299	−0.257237
$807$	4.73712	0.166755
$808$	−2.71712	−0.0955880
$809$	7.24097	0.254579	0.127289	−	0.991866i	$-0.459372\pi$
$809$	7.24097	0.254579	0.127289	+	0.991866i	$0.459372\pi$
$810$	8.45949	0.297236
$811$	−12.4529	−0.437281	−0.218640	−	0.975806i	$-0.570162\pi$
$811$	−12.4529	−0.437281	−0.218640	+	0.975806i	$0.570162\pi$
$812$	−2.93346	−0.102944
$813$	−6.84029	−0.239900
$814$	0	0
$815$	−11.4586	−0.401376
$816$	−1.61090	−0.0563929
$817$	10.1584	0.355397
$818$	−14.7289	−0.514984
$819$	3.38626	0.118325
$820$	10.5558	0.368625
$821$	16.5682	0.578233	0.289116	−	0.957294i	$-0.406639\pi$
$821$	16.5682	0.578233	0.289116	+	0.957294i	$0.406639\pi$
$822$	−1.64483	−0.0573699
$823$	−28.8974	−1.00730	−0.503651	−	0.863907i	$-0.668010\pi$
$823$	−28.8974	−1.00730	−0.503651	+	0.863907i	$0.668010\pi$
$824$	−9.90941	−0.345211
$825$	0	0
$826$	2.24164	0.0779967
$827$	22.5139	0.782884	0.391442	−	0.920203i	$-0.371976\pi$
$827$	22.5139	0.782884	0.391442	+	0.920203i	$0.371976\pi$
$828$	−24.4225	−0.848739
$829$	−27.8760	−0.968172	−0.484086	−	0.875020i	$-0.660848\pi$
$829$	−27.8760	−0.968172	−0.484086	+	0.875020i	$0.660848\pi$
$830$	11.3292	0.393244
$831$	−3.04169	−0.105515
$832$	1.15197	0.0399374
$833$	−6.55125	−0.226987
$834$	2.17701	0.0753836
$835$	0.763127	0.0264091
$836$	0	0
$837$	9.25886	0.320033
$838$	−14.1913	−0.490230
$839$	32.0359	1.10600	0.553002	−	0.833180i	$-0.313482\pi$
$839$	32.0359	1.10600	0.553002	+	0.833180i	$0.313482\pi$
$840$	−0.245893	−0.00848411
$841$	−20.3948	−0.703268
$842$	−26.0595	−0.898068
$843$	−5.40676	−0.186219
$844$	12.3479	0.425032
$845$	−11.6730	−0.401562
$846$	18.1370	0.623564
$847$	0	0
$848$	5.35680	0.183953
$849$	3.39835	0.116631
$850$	−6.55125	−0.224706
$851$	−32.1316	−1.10146
$852$	−0.0934934	−0.00320303
$853$	−20.3366	−0.696312	−0.348156	−	0.937437i	$-0.613192\pi$
$853$	−20.3366	−0.696312	−0.348156	+	0.937437i	$0.613192\pi$
$854$	−9.98548	−0.341696
$855$	−10.0966	−0.345297
$856$	−5.18688	−0.177284
$857$	−52.8958	−1.80689	−0.903444	−	0.428706i	$-0.858969\pi$
$857$	−52.8958	−1.80689	−0.903444	+	0.428706i	$0.858969\pi$
$858$	0	0
$859$	−56.9360	−1.94263	−0.971315	−	0.237797i	$-0.923575\pi$
$859$	−56.9360	−1.94263	−0.971315	+	0.237797i	$0.923575\pi$
$860$	2.95752	0.100851
$861$	−2.59560	−0.0884579
$862$	18.8146	0.640829
$863$	−16.5500	−0.563367	−0.281683	−	0.959507i	$-0.590893\pi$
$863$	−16.5500	−0.563367	−0.281683	+	0.959507i	$0.590893\pi$
$864$	−1.46049	−0.0496869
$865$	8.76752	0.298105
$866$	20.4028	0.693316
$867$	6.37325	0.216447
$868$	6.33956	0.215179
$869$	0	0
$870$	0.721318	0.0244550
$871$	16.3749	0.554842
$872$	−12.1283	−0.410715
$873$	−10.1545	−0.343679
$874$	28.5370	0.965277
$875$	−1.00000	−0.0338062
$876$	1.77834	0.0600844
$877$	48.1835	1.62704	0.813521	−	0.581535i	$-0.197548\pi$
$877$	48.1835	1.62704	0.813521	+	0.581535i	$0.197548\pi$
$878$	−17.5292	−0.591581
$879$	4.47652	0.150989
$880$	0	0
$881$	−42.1700	−1.42074	−0.710372	−	0.703827i	$-0.751473\pi$
$881$	−42.1700	−1.42074	−0.710372	+	0.703827i	$0.751473\pi$
$882$	−2.93954	−0.0989794
$883$	39.2105	1.31954	0.659769	−	0.751468i	$-0.270654\pi$
$883$	39.2105	1.31954	0.659769	+	0.751468i	$0.270654\pi$
$884$	−7.54684	−0.253828
$885$	−0.551204	−0.0185285
$886$	−26.1950	−0.880038
$887$	−45.0473	−1.51254	−0.756270	−	0.654260i	$-0.772980\pi$
$887$	−45.0473	−1.51254	−0.756270	+	0.654260i	$0.772980\pi$
$888$	−0.950972	−0.0319125
$889$	−0.818399	−0.0274482
$890$	−4.48891	−0.150468
$891$	0	0
$892$	8.30846	0.278188
$893$	−21.1926	−0.709184
$894$	1.06347	0.0355679
$895$	15.0279	0.502328
$896$	−1.00000	−0.0334077
$897$	2.35341	0.0785780
$898$	−9.49700	−0.316919
$899$	−18.5969	−0.620241
$900$	−2.93954	−0.0979846
$901$	−35.0937	−1.16914
$902$	0	0
$903$	−0.727233	−0.0242008
$904$	14.5100	0.482595
$905$	21.1888	0.704338
$906$	4.30662	0.143078
$907$	36.8904	1.22493	0.612463	−	0.790499i	$-0.290179\pi$
$907$	36.8904	1.22493	0.612463	+	0.790499i	$0.290179\pi$
$908$	15.7302	0.522023
$909$	7.98708	0.264915
$910$	−1.15197	−0.0381874
$911$	−20.6591	−0.684468	−0.342234	−	0.939615i	$-0.611183\pi$
$911$	−20.6591	−0.684468	−0.342234	+	0.939615i	$0.611183\pi$
$912$	0.844585	0.0279670
$913$	0	0
$914$	−17.5412	−0.580213
$915$	2.45536	0.0811716
$916$	−23.7309	−0.784090
$917$	9.76900	0.322601
$918$	9.56802	0.315792
$919$	−40.3012	−1.32941	−0.664706	−	0.747105i	$-0.731443\pi$
$919$	−40.3012	−1.32941	−0.664706	+	0.747105i	$0.731443\pi$
$920$	8.30827	0.273915
$921$	−5.84558	−0.192618
$922$	22.6720	0.746662
$923$	−0.438002	−0.0144170
$924$	0	0
$925$	−3.86743	−0.127160
$926$	−26.9554	−0.885809
$927$	29.1291	0.956724
$928$	2.93346	0.0962957
$929$	−7.46775	−0.245009	−0.122504	−	0.992468i	$-0.539093\pi$
$929$	−7.46775	−0.245009	−0.122504	+	0.992468i	$0.539093\pi$
$930$	−1.55885	−0.0511168
$931$	3.43477	0.112570
$932$	−11.9023	−0.389872
$933$	−4.48777	−0.146923
$934$	14.7718	0.483348
$935$	0	0
$936$	−3.38626	−0.110683
$937$	23.6513	0.772656	0.386328	−	0.922362i	$-0.373743\pi$
$937$	23.6513	0.772656	0.386328	+	0.922362i	$0.373743\pi$
$938$	−14.2147	−0.464126
$939$	−4.03273	−0.131603
$940$	−6.17003	−0.201244
$941$	34.0226	1.10911	0.554553	−	0.832148i	$-0.312889\pi$
$941$	34.0226	1.10911	0.554553	+	0.832148i	$0.312889\pi$
$942$	2.01739	0.0657301
$943$	87.7006	2.85592
$944$	−2.24164	−0.0729593
$945$	1.46049	0.0475097
$946$	0	0
$947$	16.1828	0.525871	0.262935	−	0.964813i	$-0.415309\pi$
$947$	16.1828	0.525871	0.262935	+	0.964813i	$0.415309\pi$
$948$	0.559114	0.0181592
$949$	8.33123	0.270443
$950$	3.43477	0.111439
$951$	−6.46285	−0.209572
$952$	6.55125	0.212327
$953$	−55.1879	−1.78771	−0.893855	−	0.448356i	$-0.852010\pi$
$953$	−55.1879	−1.78771	−0.893855	+	0.448356i	$0.852010\pi$
$954$	−15.7465	−0.509812
$955$	4.05200	0.131120
$956$	21.8184	0.705659
$957$	0	0
$958$	31.8220	1.02812
$959$	6.68920	0.216006
$960$	0.245893	0.00793616
$961$	9.19006	0.296453
$962$	−4.45516	−0.143640
$963$	15.2470	0.491328
$964$	14.7413	0.474785
$965$	20.5136	0.660358
$966$	−2.04294	−0.0657306
$967$	48.0059	1.54376	0.771882	−	0.635765i	$-0.219315\pi$
$967$	48.0059	1.54376	0.771882	+	0.635765i	$0.219315\pi$
$968$	0	0
$969$	−5.53308	−0.177748
$970$	3.45447	0.110916
$971$	8.41693	0.270112	0.135056	−	0.990838i	$-0.456879\pi$
$971$	8.41693	0.270112	0.135056	+	0.990838i	$0.456879\pi$
$972$	6.46159	0.207256
$973$	−8.85349	−0.283830
$974$	24.8664	0.796772
$975$	0.283261	0.00907162
$976$	9.98548	0.319627
$977$	33.7222	1.07887	0.539435	−	0.842027i	$-0.318638\pi$
$977$	33.7222	1.07887	0.539435	+	0.842027i	$0.318638\pi$
$978$	−2.81758	−0.0900962
$979$	0	0
$980$	1.00000	0.0319438
$981$	35.6515	1.13827
$982$	41.8496	1.33547
$983$	16.3099	0.520205	0.260102	−	0.965581i	$-0.416244\pi$
$983$	16.3099	0.520205	0.260102	+	0.965581i	$0.416244\pi$
$984$	2.59560	0.0827447
$985$	20.4398	0.651267
$986$	−19.2178	−0.612021
$987$	1.51717	0.0482919
$988$	3.95675	0.125881
$989$	24.5719	0.781340
$990$	0	0
$991$	1.23518	0.0392367	0.0196183	−	0.999808i	$-0.493755\pi$
$991$	1.23518	0.0392367	0.0196183	+	0.999808i	$0.493755\pi$
$992$	−6.33956	−0.201281
$993$	6.06381	0.192429
$994$	0.380220	0.0120599
$995$	−3.23576	−0.102581
$996$	2.78578	0.0882708
$997$	8.45647	0.267819	0.133909	−	0.990994i	$-0.457247\pi$
$997$	8.45647	0.267819	0.133909	+	0.990994i	$0.457247\pi$
$998$	24.8351	0.786140
$999$	5.64834	0.178705

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.dc.1.4		6
11.2	odd	10		770.2.n.j.631.2	yes	12
11.6	odd	10		770.2.n.j.421.2	✓	12
11.10	odd	2		8470.2.a.cw.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.j.421.2	✓	12	11.6	odd	10
770.2.n.j.631.2	yes	12	11.2	odd	10
8470.2.a.cw.1.4		6	11.10	odd	2
8470.2.a.dc.1.4		6	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.245893 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.245893 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.93954 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.245893 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.245893 q^{6} -1.00000 q^{7} +1.00000 q^{8} -2.93954 q^{9} +1.00000 q^{10} +0.245893 q^{12} +1.15197 q^{13} -1.00000 q^{14} +0.245893 q^{15} +1.00000 q^{16} -6.55125 q^{17} -2.93954 q^{18} +3.43477 q^{19} +1.00000 q^{20} -0.245893 q^{21} +8.30827 q^{23} +0.245893 q^{24} +1.00000 q^{25} +1.15197 q^{26} -1.46049 q^{27} -1.00000 q^{28} +2.93346 q^{29} +0.245893 q^{30} -6.33956 q^{31} +1.00000 q^{32} -6.55125 q^{34} -1.00000 q^{35} -2.93954 q^{36} -3.86743 q^{37} +3.43477 q^{38} +0.283261 q^{39} +1.00000 q^{40} +10.5558 q^{41} -0.245893 q^{42} +2.95752 q^{43} -2.93954 q^{45} +8.30827 q^{46} -6.17003 q^{47} +0.245893 q^{48} +1.00000 q^{49} +1.00000 q^{50} -1.61090 q^{51} +1.15197 q^{52} +5.35680 q^{53} -1.46049 q^{54} -1.00000 q^{56} +0.844585 q^{57} +2.93346 q^{58} -2.24164 q^{59} +0.245893 q^{60} +9.98548 q^{61} -6.33956 q^{62} +2.93954 q^{63} +1.00000 q^{64} +1.15197 q^{65} +14.2147 q^{67} -6.55125 q^{68} +2.04294 q^{69} -1.00000 q^{70} -0.380220 q^{71} -2.93954 q^{72} +7.23216 q^{73} -3.86743 q^{74} +0.245893 q^{75} +3.43477 q^{76} +0.283261 q^{78} +2.27381 q^{79} +1.00000 q^{80} +8.45949 q^{81} +10.5558 q^{82} +11.3292 q^{83} -0.245893 q^{84} -6.55125 q^{85} +2.95752 q^{86} +0.721318 q^{87} -4.48891 q^{89} -2.93954 q^{90} -1.15197 q^{91} +8.30827 q^{92} -1.55885 q^{93} -6.17003 q^{94} +3.43477 q^{95} +0.245893 q^{96} +3.45447 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} + 3 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 + 6 * q^8 + 3 * q^9	\( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} + 3 q^{9} + 6 q^{10} - q^{12} + 9 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} + 9 q^{17} + 3 q^{18} + 12 q^{19} + 6 q^{20} + q^{21} + 4 q^{23} - q^{24} + 6 q^{25} + 9 q^{26} - 4 q^{27} - 6 q^{28} + 15 q^{29} - q^{30} + 8 q^{31} + 6 q^{32} + 9 q^{34} - 6 q^{35} + 3 q^{36} - 4 q^{37} + 12 q^{38} - 19 q^{39} + 6 q^{40} + 4 q^{41} + q^{42} + 30 q^{43} + 3 q^{45} + 4 q^{46} - 7 q^{47} - q^{48} + 6 q^{49} + 6 q^{50} + 16 q^{51} + 9 q^{52} - 6 q^{53} - 4 q^{54} - 6 q^{56} - 14 q^{57} + 15 q^{58} + 4 q^{59} - q^{60} - 14 q^{61} + 8 q^{62} - 3 q^{63} + 6 q^{64} + 9 q^{65} + 18 q^{67} + 9 q^{68} - 10 q^{69} - 6 q^{70} + 23 q^{71} + 3 q^{72} + 23 q^{73} - 4 q^{74} - q^{75} + 12 q^{76} - 19 q^{78} + 21 q^{79} + 6 q^{80} - 18 q^{81} + 4 q^{82} + 25 q^{83} + q^{84} + 9 q^{85} + 30 q^{86} + 14 q^{87} - 18 q^{89} + 3 q^{90} - 9 q^{91} + 4 q^{92} - 24 q^{93} - 7 q^{94} + 12 q^{95} - q^{96} + 7 q^{97} + 6 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 + 6 * q^8 + 3 * q^9 + 6 * q^10 - q^12 + 9 * q^13 - 6 * q^14 - q^15 + 6 * q^16 + 9 * q^17 + 3 * q^18 + 12 * q^19 + 6 * q^20 + q^21 + 4 * q^23 - q^24 + 6 * q^25 + 9 * q^26 - 4 * q^27 - 6 * q^28 + 15 * q^29 - q^30 + 8 * q^31 + 6 * q^32 + 9 * q^34 - 6 * q^35 + 3 * q^36 - 4 * q^37 + 12 * q^38 - 19 * q^39 + 6 * q^40 + 4 * q^41 + q^42 + 30 * q^43 + 3 * q^45 + 4 * q^46 - 7 * q^47 - q^48 + 6 * q^49 + 6 * q^50 + 16 * q^51 + 9 * q^52 - 6 * q^53 - 4 * q^54 - 6 * q^56 - 14 * q^57 + 15 * q^58 + 4 * q^59 - q^60 - 14 * q^61 + 8 * q^62 - 3 * q^63 + 6 * q^64 + 9 * q^65 + 18 * q^67 + 9 * q^68 - 10 * q^69 - 6 * q^70 + 23 * q^71 + 3 * q^72 + 23 * q^73 - 4 * q^74 - q^75 + 12 * q^76 - 19 * q^78 + 21 * q^79 + 6 * q^80 - 18 * q^81 + 4 * q^82 + 25 * q^83 + q^84 + 9 * q^85 + 30 * q^86 + 14 * q^87 - 18 * q^89 + 3 * q^90 - 9 * q^91 + 4 * q^92 - 24 * q^93 - 7 * q^94 + 12 * q^95 - q^96 + 7 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more