LMFDB - Embedded newform 8470.2.a.cj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$67.6332905120$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.733.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 8 $ x^3 - x^2 - 7*x + 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.69639$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.42586 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.42586 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.966927 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.42586 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.42586 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.966927 q^{9} -1.00000 q^{10} +1.42586 q^{12} -3.96693 q^{13} -1.00000 q^{14} +1.42586 q^{15} +1.00000 q^{16} -0.425859 q^{17} +0.966927 q^{18} +3.54107 q^{19} +1.00000 q^{20} +1.42586 q^{21} +2.57414 q^{23} -1.42586 q^{24} +1.00000 q^{25} +3.96693 q^{26} -5.65628 q^{27} +1.00000 q^{28} -2.00000 q^{29} -1.42586 q^{30} +2.54107 q^{31} -1.00000 q^{32} +0.425859 q^{34} +1.00000 q^{35} -0.966927 q^{36} +4.85172 q^{37} -3.54107 q^{38} -5.65628 q^{39} -1.00000 q^{40} -1.39279 q^{41} -1.42586 q^{42} +1.57414 q^{43} -0.966927 q^{45} -2.57414 q^{46} +6.54107 q^{47} +1.42586 q^{48} +1.00000 q^{49} -1.00000 q^{50} -0.607214 q^{51} -3.96693 q^{52} +0.722424 q^{53} +5.65628 q^{54} -1.00000 q^{56} +5.04906 q^{57} +2.00000 q^{58} +11.2445 q^{59} +1.42586 q^{60} -5.81864 q^{61} -2.54107 q^{62} -0.966927 q^{63} +1.00000 q^{64} -3.96693 q^{65} +5.57414 q^{67} -0.425859 q^{68} +3.67036 q^{69} -1.00000 q^{70} +1.88479 q^{71} +0.966927 q^{72} -1.57414 q^{73} -4.85172 q^{74} +1.42586 q^{75} +3.54107 q^{76} +5.65628 q^{78} -1.54107 q^{79} +1.00000 q^{80} -5.16427 q^{81} +1.39279 q^{82} -0.574141 q^{83} +1.42586 q^{84} -0.425859 q^{85} -1.57414 q^{86} -2.85172 q^{87} -2.54107 q^{89} +0.966927 q^{90} -3.96693 q^{91} +2.57414 q^{92} +3.62320 q^{93} -6.54107 q^{94} +3.54107 q^{95} -1.42586 q^{96} -6.67036 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 13 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^5 - 2 * q^6 + 3 * q^7 - 3 * q^8 + 13 * q^9	$ 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 13 q^{9} - 3 q^{10} + 2 q^{12} + 4 q^{13} - 3 q^{14} + 2 q^{15} + 3 q^{16} + q^{17} - 13 q^{18} - 3 q^{19} + 3 q^{20} + 2 q^{21} + 10 q^{23} - 2 q^{24} + 3 q^{25} - 4 q^{26} + 8 q^{27} + 3 q^{28} - 6 q^{29} - 2 q^{30} - 6 q^{31} - 3 q^{32} - q^{34} + 3 q^{35} + 13 q^{36} + 10 q^{37} + 3 q^{38} + 8 q^{39} - 3 q^{40} + 14 q^{41} - 2 q^{42} + 7 q^{43} + 13 q^{45} - 10 q^{46} + 6 q^{47} + 2 q^{48} + 3 q^{49} - 3 q^{50} - 20 q^{51} + 4 q^{52} + 9 q^{53} - 8 q^{54} - 3 q^{56} - 28 q^{57} + 6 q^{58} + 11 q^{59} + 2 q^{60} + 3 q^{61} + 6 q^{62} + 13 q^{63} + 3 q^{64} + 4 q^{65} + 19 q^{67} + q^{68} - 14 q^{69} - 3 q^{70} + 17 q^{71} - 13 q^{72} - 7 q^{73} - 10 q^{74} + 2 q^{75} - 3 q^{76} - 8 q^{78} + 9 q^{79} + 3 q^{80} + 39 q^{81} - 14 q^{82} - 4 q^{83} + 2 q^{84} + q^{85} - 7 q^{86} - 4 q^{87} + 6 q^{89} - 13 q^{90} + 4 q^{91} + 10 q^{92} - 30 q^{93} - 6 q^{94} - 3 q^{95} - 2 q^{96} + 5 q^{97} - 3 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^5 - 2 * q^6 + 3 * q^7 - 3 * q^8 + 13 * q^9 - 3 * q^10 + 2 * q^12 + 4 * q^13 - 3 * q^14 + 2 * q^15 + 3 * q^16 + q^17 - 13 * q^18 - 3 * q^19 + 3 * q^20 + 2 * q^21 + 10 * q^23 - 2 * q^24 + 3 * q^25 - 4 * q^26 + 8 * q^27 + 3 * q^28 - 6 * q^29 - 2 * q^30 - 6 * q^31 - 3 * q^32 - q^34 + 3 * q^35 + 13 * q^36 + 10 * q^37 + 3 * q^38 + 8 * q^39 - 3 * q^40 + 14 * q^41 - 2 * q^42 + 7 * q^43 + 13 * q^45 - 10 * q^46 + 6 * q^47 + 2 * q^48 + 3 * q^49 - 3 * q^50 - 20 * q^51 + 4 * q^52 + 9 * q^53 - 8 * q^54 - 3 * q^56 - 28 * q^57 + 6 * q^58 + 11 * q^59 + 2 * q^60 + 3 * q^61 + 6 * q^62 + 13 * q^63 + 3 * q^64 + 4 * q^65 + 19 * q^67 + q^68 - 14 * q^69 - 3 * q^70 + 17 * q^71 - 13 * q^72 - 7 * q^73 - 10 * q^74 + 2 * q^75 - 3 * q^76 - 8 * q^78 + 9 * q^79 + 3 * q^80 + 39 * q^81 - 14 * q^82 - 4 * q^83 + 2 * q^84 + q^85 - 7 * q^86 - 4 * q^87 + 6 * q^89 - 13 * q^90 + 4 * q^91 + 10 * q^92 - 30 * q^93 - 6 * q^94 - 3 * q^95 - 2 * q^96 + 5 * q^97 - 3 * 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
$3$	1.42586	0.823220	0.411610	−	0.911360i	$-0.364967\pi$
$3$	1.42586	0.823220	0.411610	+	0.911360i	$0.364967\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.42586	−0.582104
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	−0.966927	−0.322309
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	1.42586	0.411610
$13$	−3.96693	−1.10023	−0.550114	−	0.835090i	$-0.685416\pi$
$13$	−3.96693	−1.10023	−0.550114	+	0.835090i	$0.685416\pi$
$14$	−1.00000	−0.267261
$15$	1.42586	0.368155
$16$	1.00000	0.250000
$17$	−0.425859	−0.103286	−0.0516430	−	0.998666i	$-0.516446\pi$
$17$	−0.425859	−0.103286	−0.0516430	+	0.998666i	$0.516446\pi$
$18$	0.966927	0.227907
$19$	3.54107	0.812377	0.406188	−	0.913789i	$-0.366858\pi$
$19$	3.54107	0.812377	0.406188	+	0.913789i	$0.366858\pi$
$20$	1.00000	0.223607
$21$	1.42586	0.311148
$22$	0	0
$23$	2.57414	0.536746	0.268373	−	0.963315i	$-0.413514\pi$
$23$	2.57414	0.536746	0.268373	+	0.963315i	$0.413514\pi$
$24$	−1.42586	−0.291052
$25$	1.00000	0.200000
$26$	3.96693	0.777978
$27$	−5.65628	−1.08855
$28$	1.00000	0.188982
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	−1.42586	−0.260325
$31$	2.54107	0.456389	0.228195	−	0.973616i	$-0.426718\pi$
$31$	2.54107	0.456389	0.228195	+	0.973616i	$0.426718\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.425859	0.0730342
$35$	1.00000	0.169031
$36$	−0.966927	−0.161154
$37$	4.85172	0.797617	0.398809	−	0.917034i	$-0.369424\pi$
$37$	4.85172	0.797617	0.398809	+	0.917034i	$0.369424\pi$
$38$	−3.54107	−0.574437
$39$	−5.65628	−0.905729
$40$	−1.00000	−0.158114
$41$	−1.39279	−0.217517	−0.108758	−	0.994068i	$-0.534687\pi$
$41$	−1.39279	−0.217517	−0.108758	+	0.994068i	$0.534687\pi$
$42$	−1.42586	−0.220015
$43$	1.57414	0.240054	0.120027	−	0.992771i	$-0.461702\pi$
$43$	1.57414	0.240054	0.120027	+	0.992771i	$0.461702\pi$
$44$	0	0
$45$	−0.966927	−0.144141
$46$	−2.57414	−0.379536
$47$	6.54107	0.954113	0.477056	−	0.878873i	$-0.341704\pi$
$47$	6.54107	0.954113	0.477056	+	0.878873i	$0.341704\pi$
$48$	1.42586	0.205805
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−0.607214	−0.0850270
$52$	−3.96693	−0.550114
$53$	0.722424	0.0992325	0.0496163	−	0.998768i	$-0.484200\pi$
$53$	0.722424	0.0992325	0.0496163	+	0.998768i	$0.484200\pi$
$54$	5.65628	0.769722
$55$	0	0
$56$	−1.00000	−0.133631
$57$	5.04906	0.668765
$58$	2.00000	0.262613
$59$	11.2445	1.46391	0.731955	−	0.681353i	$-0.238608\pi$
$59$	11.2445	1.46391	0.731955	+	0.681353i	$0.238608\pi$
$60$	1.42586	0.184078
$61$	−5.81864	−0.745001	−0.372501	−	0.928032i	$-0.621500\pi$
$61$	−5.81864	−0.745001	−0.372501	+	0.928032i	$0.621500\pi$
$62$	−2.54107	−0.322716
$63$	−0.966927	−0.121821
$64$	1.00000	0.125000
$65$	−3.96693	−0.492037
$66$	0	0
$67$	5.57414	0.680990	0.340495	−	0.940246i	$-0.389405\pi$
$67$	5.57414	0.680990	0.340495	+	0.940246i	$0.389405\pi$
$68$	−0.425859	−0.0516430
$69$	3.67036	0.441860
$70$	−1.00000	−0.119523
$71$	1.88479	0.223684	0.111842	−	0.993726i	$-0.464325\pi$
$71$	1.88479	0.223684	0.111842	+	0.993726i	$0.464325\pi$
$72$	0.966927	0.113953
$73$	−1.57414	−0.184239	−0.0921196	−	0.995748i	$-0.529364\pi$
$73$	−1.57414	−0.184239	−0.0921196	+	0.995748i	$0.529364\pi$
$74$	−4.85172	−0.564001
$75$	1.42586	0.164644
$76$	3.54107	0.406188
$77$	0	0
$78$	5.65628	0.640447
$79$	−1.54107	−0.173384	−0.0866919	−	0.996235i	$-0.527630\pi$
$79$	−1.54107	−0.173384	−0.0866919	+	0.996235i	$0.527630\pi$
$80$	1.00000	0.111803
$81$	−5.16427	−0.573808
$82$	1.39279	0.153807
$83$	−0.574141	−0.0630202	−0.0315101	−	0.999503i	$-0.510032\pi$
$83$	−0.574141	−0.0630202	−0.0315101	+	0.999503i	$0.510032\pi$
$84$	1.42586	0.155574
$85$	−0.425859	−0.0461909
$86$	−1.57414	−0.169744
$87$	−2.85172	−0.305736
$88$	0	0
$89$	−2.54107	−0.269353	−0.134676	−	0.990890i	$-0.542999\pi$
$89$	−2.54107	−0.269353	−0.134676	+	0.990890i	$0.542999\pi$
$90$	0.966927	0.101923
$91$	−3.96693	−0.415847
$92$	2.57414	0.268373
$93$	3.62320	0.375709
$94$	−6.54107	−0.674660
$95$	3.54107	0.363306
$96$	−1.42586	−0.145526
$97$	−6.67036	−0.677273	−0.338636	−	0.940917i	$-0.609966\pi$
$97$	−6.67036	−0.677273	−0.338636	+	0.940917i	$0.609966\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−1.11521	−0.110967	−0.0554837	−	0.998460i	$-0.517670\pi$
$101$	−1.11521	−0.110967	−0.0554837	+	0.998460i	$0.517670\pi$
$102$	0.607214	0.0601232
$103$	17.5080	1.72511	0.862557	−	0.505960i	$-0.168862\pi$
$103$	17.5080	1.72511	0.862557	+	0.505960i	$0.168862\pi$
$104$	3.96693	0.388989
$105$	1.42586	0.139150
$106$	−0.722424	−0.0701680
$107$	1.03307	0.0998710	0.0499355	−	0.998752i	$-0.484098\pi$
$107$	1.03307	0.0998710	0.0499355	+	0.998752i	$0.484098\pi$
$108$	−5.65628	−0.544276
$109$	−9.32664	−0.893330	−0.446665	−	0.894701i	$-0.647388\pi$
$109$	−9.32664	−0.893330	−0.446665	+	0.894701i	$0.647388\pi$
$110$	0	0
$111$	6.91786	0.656615
$112$	1.00000	0.0944911
$113$	16.7525	1.57594	0.787971	−	0.615712i	$-0.211132\pi$
$113$	16.7525	1.57594	0.787971	+	0.615712i	$0.211132\pi$
$114$	−5.04906	−0.472888
$115$	2.57414	0.240040
$116$	−2.00000	−0.185695
$117$	3.83573	0.354613
$118$	−11.2445	−1.03514
$119$	−0.425859	−0.0390384
$120$	−1.42586	−0.130163
$121$	0	0
$122$	5.81864	0.526795
$123$	−1.98592	−0.179064
$124$	2.54107	0.228195
$125$	1.00000	0.0894427
$126$	0.966927	0.0861407
$127$	4.21143	0.373704	0.186852	−	0.982388i	$-0.440172\pi$
$127$	4.21143	0.373704	0.186852	+	0.982388i	$0.440172\pi$
$128$	−1.00000	−0.0883883
$129$	2.24450	0.197617
$130$	3.96693	0.347922
$131$	6.27758	0.548474	0.274237	−	0.961662i	$-0.411575\pi$
$131$	6.27758	0.548474	0.274237	+	0.961662i	$0.411575\pi$
$132$	0	0
$133$	3.54107	0.307050
$134$	−5.57414	−0.481532
$135$	−5.65628	−0.486815
$136$	0.425859	0.0365171
$137$	−13.9008	−1.18762	−0.593812	−	0.804604i	$-0.702378\pi$
$137$	−13.9008	−1.18762	−0.593812	+	0.804604i	$0.702378\pi$
$138$	−3.67036	−0.312442
$139$	6.68935	0.567383	0.283692	−	0.958916i	$-0.408441\pi$
$139$	6.68935	0.567383	0.283692	+	0.958916i	$0.408441\pi$
$140$	1.00000	0.0845154
$141$	9.32664	0.785445
$142$	−1.88479	−0.158168
$143$	0	0
$144$	−0.966927	−0.0805772
$145$	−2.00000	−0.166091
$146$	1.57414	0.130277
$147$	1.42586	0.117603
$148$	4.85172	0.398809
$149$	17.0821	1.39942	0.699712	−	0.714425i	$-0.253312\pi$
$149$	17.0821	1.39942	0.699712	+	0.714425i	$0.253312\pi$
$150$	−1.42586	−0.116421
$151$	−12.5741	−1.02327	−0.511635	−	0.859203i	$-0.670960\pi$
$151$	−12.5741	−1.02327	−0.511635	+	0.859203i	$0.670960\pi$
$152$	−3.54107	−0.287219
$153$	0.411774	0.0332900
$154$	0	0
$155$	2.54107	0.204103
$156$	−5.65628	−0.452865
$157$	−12.7525	−1.01776	−0.508880	−	0.860837i	$-0.669940\pi$
$157$	−12.7525	−1.01776	−0.508880	+	0.860837i	$0.669940\pi$
$158$	1.54107	0.122601
$159$	1.03007	0.0816902
$160$	−1.00000	−0.0790569
$161$	2.57414	0.202871
$162$	5.16427	0.405744
$163$	3.26349	0.255616	0.127808	−	0.991799i	$-0.459206\pi$
$163$	3.26349	0.255616	0.127808	+	0.991799i	$0.459206\pi$
$164$	−1.39279	−0.108758
$165$	0	0
$166$	0.574141	0.0445620
$167$	3.27758	0.253626	0.126813	−	0.991927i	$-0.459525\pi$
$167$	3.27758	0.253626	0.126813	+	0.991927i	$0.459525\pi$
$168$	−1.42586	−0.110007
$169$	2.73651	0.210501
$170$	0.425859	0.0326619
$171$	−3.42395	−0.261836
$172$	1.57414	0.120027
$173$	11.9339	0.907314	0.453657	−	0.891176i	$-0.350119\pi$
$173$	11.9339	0.907314	0.453657	+	0.891176i	$0.350119\pi$
$174$	2.85172	0.216188
$175$	1.00000	0.0755929
$176$	0	0
$177$	16.0331	1.20512
$178$	2.54107	0.190461
$179$	−4.31065	−0.322193	−0.161097	−	0.986939i	$-0.551503\pi$
$179$	−4.31065	−0.322193	−0.161097	+	0.986939i	$0.551503\pi$
$180$	−0.966927	−0.0720705
$181$	−0.214429	−0.0159384	−0.00796919	−	0.999968i	$-0.502537\pi$
$181$	−0.214429	−0.0159384	−0.00796919	+	0.999968i	$0.502537\pi$
$182$	3.96693	0.294048
$183$	−8.29656	−0.613300
$184$	−2.57414	−0.189768
$185$	4.85172	0.356705
$186$	−3.62320	−0.265666
$187$	0	0
$188$	6.54107	0.477056
$189$	−5.65628	−0.411434
$190$	−3.54107	−0.256896
$191$	24.9970	1.80872	0.904360	−	0.426771i	$-0.140349\pi$
$191$	24.9970	1.80872	0.904360	+	0.426771i	$0.140349\pi$
$192$	1.42586	0.102902
$193$	20.7525	1.49380	0.746899	−	0.664938i	$-0.231542\pi$
$193$	20.7525	1.49380	0.746899	+	0.664938i	$0.231542\pi$
$194$	6.67036	0.478904
$195$	−5.65628	−0.405054
$196$	1.00000	0.0714286
$197$	7.26349	0.517502	0.258751	−	0.965944i	$-0.416689\pi$
$197$	7.26349	0.517502	0.258751	+	0.965944i	$0.416689\pi$
$198$	0	0
$199$	19.5711	1.38736	0.693681	−	0.720283i	$-0.255988\pi$
$199$	19.5711	1.38736	0.693681	+	0.720283i	$0.255988\pi$
$200$	−1.00000	−0.0707107
$201$	7.94794	0.560604
$202$	1.11521	0.0784659
$203$	−2.00000	−0.140372
$204$	−0.607214	−0.0425135
$205$	−1.39279	−0.0972764
$206$	−17.5080	−1.21984
$207$	−2.48901	−0.172998
$208$	−3.96693	−0.275057
$209$	0	0
$210$	−1.42586	−0.0983936
$211$	9.63729	0.663458	0.331729	−	0.943375i	$-0.392368\pi$
$211$	9.63729	0.663458	0.331729	+	0.943375i	$0.392368\pi$
$212$	0.722424	0.0496163
$213$	2.68745	0.184141
$214$	−1.03307	−0.0706194
$215$	1.57414	0.107356
$216$	5.65628	0.384861
$217$	2.54107	0.172499
$218$	9.32664	0.631680
$219$	−2.24450	−0.151669
$220$	0	0
$221$	1.68935	0.113638
$222$	−6.91786	−0.464297
$223$	4.35971	0.291948	0.145974	−	0.989288i	$-0.453368\pi$
$223$	4.35971	0.291948	0.145974	+	0.989288i	$0.453368\pi$
$224$	−1.00000	−0.0668153
$225$	−0.966927	−0.0644618
$226$	−16.7525	−1.11436
$227$	22.7856	1.51233	0.756166	−	0.654380i	$-0.227070\pi$
$227$	22.7856	1.51233	0.756166	+	0.654380i	$0.227070\pi$
$228$	5.04906	0.334382
$229$	15.2635	1.00864	0.504320	−	0.863517i	$-0.331743\pi$
$229$	15.2635	1.00864	0.504320	+	0.863517i	$0.331743\pi$
$230$	−2.57414	−0.169734
$231$	0	0
$232$	2.00000	0.131306
$233$	−3.73651	−0.244787	−0.122393	−	0.992482i	$-0.539057\pi$
$233$	−3.73651	−0.244787	−0.122393	+	0.992482i	$0.539057\pi$
$234$	−3.83573	−0.250749
$235$	6.54107	0.426692
$236$	11.2445	0.731955
$237$	−2.19735	−0.142733
$238$	0.425859	0.0276043
$239$	−1.78857	−0.115693	−0.0578465	−	0.998325i	$-0.518423\pi$
$239$	−1.78857	−0.115693	−0.0578465	+	0.998325i	$0.518423\pi$
$240$	1.42586	0.0920388
$241$	17.6373	1.13612	0.568059	−	0.822988i	$-0.307695\pi$
$241$	17.6373	1.13612	0.568059	+	0.822988i	$0.307695\pi$
$242$	0	0
$243$	9.60531	0.616181
$244$	−5.81864	−0.372501
$245$	1.00000	0.0638877
$246$	1.98592	0.126617
$247$	−14.0472	−0.893799
$248$	−2.54107	−0.161358
$249$	−0.818644	−0.0518795
$250$	−1.00000	−0.0632456
$251$	6.96693	0.439749	0.219874	−	0.975528i	$-0.429435\pi$
$251$	6.96693	0.439749	0.219874	+	0.975528i	$0.429435\pi$
$252$	−0.966927	−0.0609107
$253$	0	0
$254$	−4.21143	−0.264249
$255$	−0.607214	−0.0380252
$256$	1.00000	0.0625000
$257$	17.2114	1.07362	0.536810	−	0.843703i	$-0.319629\pi$
$257$	17.2114	1.07362	0.536810	+	0.843703i	$0.319629\pi$
$258$	−2.24450	−0.139737
$259$	4.85172	0.301471
$260$	−3.96693	−0.246018
$261$	1.93385	0.119703
$262$	−6.27758	−0.387830
$263$	3.65628	0.225456	0.112728	−	0.993626i	$-0.464041\pi$
$263$	3.65628	0.225456	0.112728	+	0.993626i	$0.464041\pi$
$264$	0	0
$265$	0.722424	0.0443781
$266$	−3.54107	−0.217117
$267$	−3.62320	−0.221736
$268$	5.57414	0.340495
$269$	−11.4890	−0.700497	−0.350249	−	0.936657i	$-0.613903\pi$
$269$	−11.4890	−0.700497	−0.350249	+	0.936657i	$0.613903\pi$
$270$	5.65628	0.344230
$271$	6.55515	0.398197	0.199099	−	0.979979i	$-0.436199\pi$
$271$	6.55515	0.398197	0.199099	+	0.979979i	$0.436199\pi$
$272$	−0.425859	−0.0258215
$273$	−5.65628	−0.342333
$274$	13.9008	0.839777
$275$	0	0
$276$	3.67036	0.220930
$277$	5.62320	0.337866	0.168933	−	0.985628i	$-0.445968\pi$
$277$	5.62320	0.337866	0.168933	+	0.985628i	$0.445968\pi$
$278$	−6.68935	−0.401201
$279$	−2.45703	−0.147098
$280$	−1.00000	−0.0597614
$281$	−9.23042	−0.550641	−0.275320	−	0.961353i	$-0.588784\pi$
$281$	−9.23042	−0.550641	−0.275320	+	0.961353i	$0.588784\pi$
$282$	−9.32664	−0.555393
$283$	4.50799	0.267972	0.133986	−	0.990983i	$-0.457222\pi$
$283$	4.50799	0.267972	0.133986	+	0.990983i	$0.457222\pi$
$284$	1.88479	0.111842
$285$	5.04906	0.299081
$286$	0	0
$287$	−1.39279	−0.0822135
$288$	0.966927	0.0569767
$289$	−16.8186	−0.989332
$290$	2.00000	0.117444
$291$	−9.51099	−0.557544
$292$	−1.57414	−0.0921196
$293$	−25.8346	−1.50928	−0.754638	−	0.656142i	$-0.772187\pi$
$293$	−25.8346	−1.50928	−0.754638	+	0.656142i	$0.772187\pi$
$294$	−1.42586	−0.0831578
$295$	11.2445	0.654680
$296$	−4.85172	−0.282000
$297$	0	0
$298$	−17.0821	−0.989542
$299$	−10.2114	−0.590542
$300$	1.42586	0.0823220
$301$	1.57414	0.0907320
$302$	12.5741	0.723561
$303$	−1.59013	−0.0913506
$304$	3.54107	0.203094
$305$	−5.81864	−0.333175
$306$	−0.411774	−0.0235396
$307$	−11.1483	−0.636266	−0.318133	−	0.948046i	$-0.603056\pi$
$307$	−11.1483	−0.636266	−0.318133	+	0.948046i	$0.603056\pi$
$308$	0	0
$309$	24.9639	1.42015
$310$	−2.54107	−0.144323
$311$	−13.9479	−0.790915	−0.395458	−	0.918484i	$-0.629414\pi$
$311$	−13.9479	−0.790915	−0.395458	+	0.918484i	$0.629414\pi$
$312$	5.65628	0.320224
$313$	−34.4890	−1.94943	−0.974717	−	0.223443i	$-0.928270\pi$
$313$	−34.4890	−1.94943	−0.974717	+	0.223443i	$0.928270\pi$
$314$	12.7525	0.719665
$315$	−0.966927	−0.0544802
$316$	−1.54107	−0.0866919
$317$	−13.5080	−0.758685	−0.379342	−	0.925256i	$-0.623850\pi$
$317$	−13.5080	−0.758685	−0.379342	+	0.925256i	$0.623850\pi$
$318$	−1.03007	−0.0577637
$319$	0	0
$320$	1.00000	0.0559017
$321$	1.47302	0.0822158
$322$	−2.57414	−0.143451
$323$	−1.50799	−0.0839071
$324$	−5.16427	−0.286904
$325$	−3.96693	−0.220046
$326$	−3.26349	−0.180748
$327$	−13.2985	−0.735407
$328$	1.39279	0.0769037
$329$	6.54107	0.360621
$330$	0	0
$331$	30.1122	1.65512	0.827558	−	0.561380i	$-0.189729\pi$
$331$	30.1122	1.65512	0.827558	+	0.561380i	$0.189729\pi$
$332$	−0.574141	−0.0315101
$333$	−4.69126	−0.257079
$334$	−3.27758	−0.179341
$335$	5.57414	0.304548
$336$	1.42586	0.0777870
$337$	12.5221	0.682121	0.341061	−	0.940041i	$-0.389214\pi$
$337$	12.5221	0.682121	0.341061	+	0.940041i	$0.389214\pi$
$338$	−2.73651	−0.148846
$339$	23.8867	1.29735
$340$	−0.425859	−0.0230954
$341$	0	0
$342$	3.42395	0.185146
$343$	1.00000	0.0539949
$344$	−1.57414	−0.0848720
$345$	3.67036	0.197606
$346$	−11.9339	−0.641568
$347$	20.3738	1.09372	0.546861	−	0.837223i	$-0.315822\pi$
$347$	20.3738	1.09372	0.546861	+	0.837223i	$0.315822\pi$
$348$	−2.85172	−0.152868
$349$	−18.2635	−0.977622	−0.488811	−	0.872390i	$-0.662569\pi$
$349$	−18.2635	−0.977622	−0.488811	+	0.872390i	$0.662569\pi$
$350$	−1.00000	−0.0534522
$351$	22.4380	1.19765
$352$	0	0
$353$	11.5080	0.612509	0.306254	−	0.951950i	$-0.400924\pi$
$353$	11.5080	0.612509	0.306254	+	0.951950i	$0.400924\pi$
$354$	−16.0331	−0.852148
$355$	1.88479	0.100034
$356$	−2.54107	−0.134676
$357$	−0.607214	−0.0321372
$358$	4.31065	0.227825
$359$	−25.2635	−1.33336	−0.666678	−	0.745346i	$-0.732284\pi$
$359$	−25.2635	−1.33336	−0.666678	+	0.745346i	$0.732284\pi$
$360$	0.966927	0.0509615
$361$	−6.46084	−0.340044
$362$	0.214429	0.0112701
$363$	0	0
$364$	−3.96693	−0.207923
$365$	−1.57414	−0.0823943
$366$	8.29656	0.433668
$367$	20.1293	1.05074	0.525370	−	0.850874i	$-0.323927\pi$
$367$	20.1293	1.05074	0.525370	+	0.850874i	$0.323927\pi$
$368$	2.57414	0.134186
$369$	1.34672	0.0701075
$370$	−4.85172	−0.252229
$371$	0.722424	0.0375064
$372$	3.62320	0.187854
$373$	−22.9669	−1.18918	−0.594591	−	0.804028i	$-0.702686\pi$
$373$	−22.9669	−1.18918	−0.594591	+	0.804028i	$0.702686\pi$
$374$	0	0
$375$	1.42586	0.0736310
$376$	−6.54107	−0.337330
$377$	7.93385	0.408614
$378$	5.65628	0.290928
$379$	−19.8677	−1.02054	−0.510268	−	0.860016i	$-0.670454\pi$
$379$	−19.8677	−1.02054	−0.510268	+	0.860016i	$0.670454\pi$
$380$	3.54107	0.181653
$381$	6.00490	0.307641
$382$	−24.9970	−1.27896
$383$	16.0631	0.820788	0.410394	−	0.911908i	$-0.365391\pi$
$383$	16.0631	0.820788	0.410394	+	0.911908i	$0.365391\pi$
$384$	−1.42586	−0.0727631
$385$	0	0
$386$	−20.7525	−1.05627
$387$	−1.52208	−0.0773716
$388$	−6.67036	−0.338636
$389$	−16.7997	−0.851776	−0.425888	−	0.904776i	$-0.640038\pi$
$389$	−16.7997	−0.851776	−0.425888	+	0.904776i	$0.640038\pi$
$390$	5.65628	0.286417
$391$	−1.09622	−0.0554383
$392$	−1.00000	−0.0505076
$393$	8.95094	0.451515
$394$	−7.26349	−0.365929
$395$	−1.54107	−0.0775395
$396$	0	0
$397$	7.93385	0.398189	0.199094	−	0.979980i	$-0.436200\pi$
$397$	7.93385	0.398189	0.199094	+	0.979980i	$0.436200\pi$
$398$	−19.5711	−0.981013
$399$	5.04906	0.252769
$400$	1.00000	0.0500000
$401$	−23.7034	−1.18369	−0.591847	−	0.806051i	$-0.701601\pi$
$401$	−23.7034	−1.18369	−0.591847	+	0.806051i	$0.701601\pi$
$402$	−7.94794	−0.396407
$403$	−10.0802	−0.502132
$404$	−1.11521	−0.0554837
$405$	−5.16427	−0.256615
$406$	2.00000	0.0992583
$407$	0	0
$408$	0.607214	0.0300616
$409$	−13.6373	−0.674321	−0.337160	−	0.941447i	$-0.609466\pi$
$409$	−13.6373	−0.674321	−0.337160	+	0.941447i	$0.609466\pi$
$410$	1.39279	0.0687848
$411$	−19.8205	−0.977676
$412$	17.5080	0.862557
$413$	11.2445	0.553306
$414$	2.48901	0.122328
$415$	−0.574141	−0.0281835
$416$	3.96693	0.194495
$417$	9.53807	0.467081
$418$	0	0
$419$	25.7715	1.25902	0.629510	−	0.776993i	$-0.283256\pi$
$419$	25.7715	1.25902	0.629510	+	0.776993i	$0.283256\pi$
$420$	1.42586	0.0695748
$421$	29.5711	1.44121	0.720605	−	0.693346i	$-0.243864\pi$
$421$	29.5711	1.44121	0.720605	+	0.693346i	$0.243864\pi$
$422$	−9.63729	−0.469136
$423$	−6.32473	−0.307519
$424$	−0.722424	−0.0350840
$425$	−0.425859	−0.0206572
$426$	−2.68745	−0.130207
$427$	−5.81864	−0.281584
$428$	1.03307	0.0499355
$429$	0	0
$430$	−1.57414	−0.0759118
$431$	21.7997	1.05005	0.525026	−	0.851086i	$-0.324056\pi$
$431$	21.7997	1.05005	0.525026	+	0.851086i	$0.324056\pi$
$432$	−5.65628	−0.272138
$433$	−25.5571	−1.22819	−0.614097	−	0.789231i	$-0.710480\pi$
$433$	−25.5571	−1.22819	−0.614097	+	0.789231i	$0.710480\pi$
$434$	−2.54107	−0.121975
$435$	−2.85172	−0.136729
$436$	−9.32664	−0.446665
$437$	9.11521	0.436040
$438$	2.24450	0.107247
$439$	−9.82164	−0.468761	−0.234381	−	0.972145i	$-0.575306\pi$
$439$	−9.82164	−0.468761	−0.234381	+	0.972145i	$0.575306\pi$
$440$	0	0
$441$	−0.966927	−0.0460441
$442$	−1.68935	−0.0803542
$443$	3.37870	0.160527	0.0802635	−	0.996774i	$-0.474424\pi$
$443$	3.37870	0.160527	0.0802635	+	0.996774i	$0.474424\pi$
$444$	6.91786	0.328307
$445$	−2.54107	−0.120458
$446$	−4.35971	−0.206438
$447$	24.3567	1.15203
$448$	1.00000	0.0472456
$449$	7.29656	0.344346	0.172173	−	0.985067i	$-0.444921\pi$
$449$	7.29656	0.344346	0.172173	+	0.985067i	$0.444921\pi$
$450$	0.966927	0.0455814
$451$	0	0
$452$	16.7525	0.787971
$453$	−17.9289	−0.842376
$454$	−22.7856	−1.06938
$455$	−3.96693	−0.185972
$456$	−5.04906	−0.236444
$457$	35.5381	1.66240	0.831200	−	0.555973i	$-0.187654\pi$
$457$	35.5381	1.66240	0.831200	+	0.555973i	$0.187654\pi$
$458$	−15.2635	−0.713216
$459$	2.40878	0.112432
$460$	2.57414	0.120020
$461$	26.6704	1.24216	0.621081	−	0.783746i	$-0.286694\pi$
$461$	26.6704	1.24216	0.621081	+	0.783746i	$0.286694\pi$
$462$	0	0
$463$	17.9528	0.834339	0.417170	−	0.908829i	$-0.363022\pi$
$463$	17.9528	0.834339	0.417170	+	0.908829i	$0.363022\pi$
$464$	−2.00000	−0.0928477
$465$	3.62320	0.168022
$466$	3.73651	0.173090
$467$	−18.8327	−0.871475	−0.435737	−	0.900074i	$-0.643512\pi$
$467$	−18.8327	−0.871475	−0.435737	+	0.900074i	$0.643512\pi$
$468$	3.83573	0.177307
$469$	5.57414	0.257390
$470$	−6.54107	−0.301717
$471$	−18.1833	−0.837840
$472$	−11.2445	−0.517570
$473$	0	0
$474$	2.19735	0.100927
$475$	3.54107	0.162475
$476$	−0.425859	−0.0195192
$477$	−0.698531	−0.0319835
$478$	1.78857	0.0818073
$479$	−0.310649	−0.0141939	−0.00709697	−	0.999975i	$-0.502259\pi$
$479$	−0.310649	−0.0141939	−0.00709697	+	0.999975i	$0.502259\pi$
$480$	−1.42586	−0.0650813
$481$	−19.2464	−0.877561
$482$	−17.6373	−0.803356
$483$	3.67036	0.167007
$484$	0	0
$485$	−6.67036	−0.302886
$486$	−9.60531	−0.435706
$487$	−12.5362	−0.568068	−0.284034	−	0.958814i	$-0.591673\pi$
$487$	−12.5362	−0.568068	−0.284034	+	0.958814i	$0.591673\pi$
$488$	5.81864	0.263398
$489$	4.65328	0.210429
$490$	−1.00000	−0.0451754
$491$	−7.71752	−0.348287	−0.174143	−	0.984720i	$-0.555716\pi$
$491$	−7.71752	−0.348287	−0.174143	+	0.984720i	$0.555716\pi$
$492$	−1.98592	−0.0895320
$493$	0.851718	0.0383594
$494$	14.0472	0.632012
$495$	0	0
$496$	2.54107	0.114097
$497$	1.88479	0.0845444
$498$	0.818644	0.0366843
$499$	−18.9179	−0.846880	−0.423440	−	0.905924i	$-0.639178\pi$
$499$	−18.9179	−0.846880	−0.423440	+	0.905924i	$0.639178\pi$
$500$	1.00000	0.0447214
$501$	4.67336	0.208790
$502$	−6.96693	−0.310949
$503$	16.4259	0.732393	0.366196	−	0.930538i	$-0.380660\pi$
$503$	16.4259	0.732393	0.366196	+	0.930538i	$0.380660\pi$
$504$	0.966927	0.0430703
$505$	−1.11521	−0.0496262
$506$	0	0
$507$	3.90187	0.173288
$508$	4.21143	0.186852
$509$	−7.71942	−0.342157	−0.171079	−	0.985257i	$-0.554725\pi$
$509$	−7.71942	−0.342157	−0.171079	+	0.985257i	$0.554725\pi$
$510$	0.607214	0.0268879
$511$	−1.57414	−0.0696359
$512$	−1.00000	−0.0441942
$513$	−20.0293	−0.884314
$514$	−17.2114	−0.759164
$515$	17.5080	0.771494
$516$	2.24450	0.0988087
$517$	0	0
$518$	−4.85172	−0.213172
$519$	17.0160	0.746919
$520$	3.96693	0.173961
$521$	31.0821	1.36173	0.680867	−	0.732407i	$-0.261603\pi$
$521$	31.0821	1.36173	0.680867	+	0.732407i	$0.261603\pi$
$522$	−1.93385	−0.0846425
$523$	34.6723	1.51611	0.758056	−	0.652189i	$-0.226149\pi$
$523$	34.6723	1.51611	0.758056	+	0.652189i	$0.226149\pi$
$524$	6.27758	0.274237
$525$	1.42586	0.0622296
$526$	−3.65628	−0.159421
$527$	−1.08214	−0.0471386
$528$	0	0
$529$	−16.3738	−0.711904
$530$	−0.722424	−0.0313801
$531$	−10.8726	−0.471831
$532$	3.54107	0.153525
$533$	5.52508	0.239318
$534$	3.62320	0.156791
$535$	1.03307	0.0446636
$536$	−5.57414	−0.240766
$537$	−6.14638	−0.265236
$538$	11.4890	0.495326
$539$	0	0
$540$	−5.65628	−0.243407
$541$	18.4890	0.794904	0.397452	−	0.917623i	$-0.369894\pi$
$541$	18.4890	0.794904	0.397452	+	0.917623i	$0.369894\pi$
$542$	−6.55515	−0.281568
$543$	−0.305745	−0.0131208
$544$	0.425859	0.0182585
$545$	−9.32664	−0.399509
$546$	5.65628	0.242066
$547$	−29.5741	−1.26450	−0.632249	−	0.774765i	$-0.717868\pi$
$547$	−29.5741	−1.26450	−0.632249	+	0.774765i	$0.717868\pi$
$548$	−13.9008	−0.593812
$549$	5.62620	0.240120
$550$	0	0
$551$	−7.08214	−0.301709
$552$	−3.67036	−0.156221
$553$	−1.54107	−0.0655329
$554$	−5.62320	−0.238907
$555$	6.91786	0.293647
$556$	6.68935	0.283692
$557$	−22.7715	−0.964859	−0.482429	−	0.875935i	$-0.660246\pi$
$557$	−22.7715	−0.964859	−0.482429	+	0.875935i	$0.660246\pi$
$558$	2.45703	0.104014
$559$	−6.24450	−0.264114
$560$	1.00000	0.0422577
$561$	0	0
$562$	9.23042	0.389362
$563$	18.4418	0.777231	0.388616	−	0.921400i	$-0.372953\pi$
$563$	18.4418	0.777231	0.388616	+	0.921400i	$0.372953\pi$
$564$	9.32664	0.392722
$565$	16.7525	0.704783
$566$	−4.50799	−0.189485
$567$	−5.16427	−0.216879
$568$	−1.88479	−0.0790841
$569$	−31.3567	−1.31454	−0.657271	−	0.753654i	$-0.728289\pi$
$569$	−31.3567	−1.31454	−0.657271	+	0.753654i	$0.728289\pi$
$570$	−5.04906	−0.211482
$571$	24.7194	1.03448	0.517238	−	0.855842i	$-0.326960\pi$
$571$	24.7194	1.03448	0.517238	+	0.855842i	$0.326960\pi$
$572$	0	0
$573$	35.6422	1.48897
$574$	1.39279	0.0581337
$575$	2.57414	0.107349
$576$	−0.966927	−0.0402886
$577$	−35.9970	−1.49857	−0.749287	−	0.662245i	$-0.769604\pi$
$577$	−35.9970	−1.49857	−0.749287	+	0.662245i	$0.769604\pi$
$578$	16.8186	0.699563
$579$	29.5901	1.22972
$580$	−2.00000	−0.0830455
$581$	−0.574141	−0.0238194
$582$	9.51099	0.394243
$583$	0	0
$584$	1.57414	0.0651384
$585$	3.83573	0.158588
$586$	25.8346	1.06722
$587$	−34.3757	−1.41884	−0.709419	−	0.704787i	$-0.751042\pi$
$587$	−34.3757	−1.41884	−0.709419	+	0.704787i	$0.751042\pi$
$588$	1.42586	0.0588014
$589$	8.99809	0.370760
$590$	−11.2445	−0.462929
$591$	10.3567	0.426018
$592$	4.85172	0.199404
$593$	−8.53807	−0.350616	−0.175308	−	0.984514i	$-0.556092\pi$
$593$	−8.53807	−0.350616	−0.175308	+	0.984514i	$0.556092\pi$
$594$	0	0
$595$	−0.425859	−0.0174585
$596$	17.0821	0.699712
$597$	27.9057	1.14210
$598$	10.2114	0.417576
$599$	6.75550	0.276022	0.138011	−	0.990431i	$-0.455929\pi$
$599$	6.75550	0.276022	0.138011	+	0.990431i	$0.455929\pi$
$600$	−1.42586	−0.0582104
$601$	−4.17836	−0.170439	−0.0852194	−	0.996362i	$-0.527159\pi$
$601$	−4.17836	−0.170439	−0.0852194	+	0.996362i	$0.527159\pi$
$602$	−1.57414	−0.0641572
$603$	−5.38979	−0.219489
$604$	−12.5741	−0.511635
$605$	0	0
$606$	1.59013	0.0645947
$607$	−42.4058	−1.72120	−0.860599	−	0.509284i	$-0.829910\pi$
$607$	−42.4058	−1.72120	−0.860599	+	0.509284i	$0.829910\pi$
$608$	−3.54107	−0.143609
$609$	−2.85172	−0.115557
$610$	5.81864	0.235590
$611$	−25.9479	−1.04974
$612$	0.411774	0.0166450
$613$	25.7384	1.03956	0.519782	−	0.854299i	$-0.326013\pi$
$613$	25.7384	1.03956	0.519782	+	0.854299i	$0.326013\pi$
$614$	11.1483	0.449908
$615$	−1.98592	−0.0800798
$616$	0	0
$617$	−28.1643	−1.13385	−0.566925	−	0.823769i	$-0.691867\pi$
$617$	−28.1643	−1.13385	−0.566925	+	0.823769i	$0.691867\pi$
$618$	−24.9639	−1.00420
$619$	−29.6183	−1.19046	−0.595230	−	0.803555i	$-0.702939\pi$
$619$	−29.6183	−1.19046	−0.595230	+	0.803555i	$0.702939\pi$
$620$	2.54107	0.102052
$621$	−14.5601	−0.584275
$622$	13.9479	0.559261
$623$	−2.54107	−0.101806
$624$	−5.65628	−0.226432
$625$	1.00000	0.0400000
$626$	34.4890	1.37846
$627$	0	0
$628$	−12.7525	−0.508880
$629$	−2.06615	−0.0823827
$630$	0.966927	0.0385233
$631$	−9.52208	−0.379068	−0.189534	−	0.981874i	$-0.560698\pi$
$631$	−9.52208	−0.379068	−0.189534	+	0.981874i	$0.560698\pi$
$632$	1.54107	0.0613004
$633$	13.7414	0.546172
$634$	13.5080	0.536471
$635$	4.21143	0.167125
$636$	1.03007	0.0408451
$637$	−3.96693	−0.157175
$638$	0	0
$639$	−1.82245	−0.0720952
$640$	−1.00000	−0.0395285
$641$	32.8016	1.29558	0.647792	−	0.761817i	$-0.275693\pi$
$641$	32.8016	1.29558	0.647792	+	0.761817i	$0.275693\pi$
$642$	−1.47302	−0.0581353
$643$	40.9780	1.61602	0.808008	−	0.589172i	$-0.200546\pi$
$643$	40.9780	1.61602	0.808008	+	0.589172i	$0.200546\pi$
$644$	2.57414	0.101435
$645$	2.24450	0.0883772
$646$	1.50799	0.0593313
$647$	28.6563	1.12659	0.563297	−	0.826254i	$-0.309533\pi$
$647$	28.6563	1.12659	0.563297	+	0.826254i	$0.309533\pi$
$648$	5.16427	0.202872
$649$	0	0
$650$	3.96693	0.155596
$651$	3.62320	0.142005
$652$	3.26349	0.127808
$653$	5.21143	0.203939	0.101970	−	0.994788i	$-0.467486\pi$
$653$	5.21143	0.203939	0.101970	+	0.994788i	$0.467486\pi$
$654$	13.2985	0.520011
$655$	6.27758	0.245285
$656$	−1.39279	−0.0543791
$657$	1.52208	0.0593820
$658$	−6.54107	−0.254997
$659$	44.6674	1.73999	0.869997	−	0.493058i	$-0.164121\pi$
$659$	44.6674	1.73999	0.869997	+	0.493058i	$0.164121\pi$
$660$	0	0
$661$	−39.2415	−1.52632	−0.763159	−	0.646211i	$-0.776353\pi$
$661$	−39.2415	−1.52632	−0.763159	+	0.646211i	$0.776353\pi$
$662$	−30.1122	−1.17034
$663$	2.40878	0.0935491
$664$	0.574141	0.0222810
$665$	3.54107	0.137317
$666$	4.69126	0.181782
$667$	−5.14828	−0.199342
$668$	3.27758	0.126813
$669$	6.21633	0.240337
$670$	−5.57414	−0.215348
$671$	0	0
$672$	−1.42586	−0.0550037
$673$	3.11521	0.120082	0.0600412	−	0.998196i	$-0.480877\pi$
$673$	3.11521	0.120082	0.0600412	+	0.998196i	$0.480877\pi$
$674$	−12.5221	−0.482332
$675$	−5.65628	−0.217710
$676$	2.73651	0.105250
$677$	23.0772	0.886930	0.443465	−	0.896292i	$-0.353749\pi$
$677$	23.0772	0.886930	0.443465	+	0.896292i	$0.353749\pi$
$678$	−23.8867	−0.917363
$679$	−6.67036	−0.255985
$680$	0.425859	0.0163309
$681$	32.4890	1.24498
$682$	0	0
$683$	−28.9669	−1.10839	−0.554194	−	0.832387i	$-0.686974\pi$
$683$	−28.9669	−1.10839	−0.554194	+	0.832387i	$0.686974\pi$
$684$	−3.42395	−0.130918
$685$	−13.9008	−0.531122
$686$	−1.00000	−0.0381802
$687$	21.7636	0.830332
$688$	1.57414	0.0600136
$689$	−2.86580	−0.109178
$690$	−3.67036	−0.139728
$691$	−23.1973	−0.882469	−0.441234	−	0.897392i	$-0.645459\pi$
$691$	−23.1973	−0.882469	−0.441234	+	0.897392i	$0.645459\pi$
$692$	11.9339	0.453657
$693$	0	0
$694$	−20.3738	−0.773379
$695$	6.68935	0.253742
$696$	2.85172	0.108094
$697$	0.593130	0.0224664
$698$	18.2635	0.691283
$699$	−5.32773	−0.201513
$700$	1.00000	0.0377964
$701$	14.5411	0.549209	0.274604	−	0.961557i	$-0.411453\pi$
$701$	14.5411	0.549209	0.274604	+	0.961557i	$0.411453\pi$
$702$	−22.4380	−0.846869
$703$	17.1803	0.647966
$704$	0	0
$705$	9.32664	0.351262
$706$	−11.5080	−0.433109
$707$	−1.11521	−0.0419418
$708$	16.0331	0.602560
$709$	−20.1263	−0.755859	−0.377929	−	0.925834i	$-0.623364\pi$
$709$	−20.1263	−0.755859	−0.377929	+	0.925834i	$0.623364\pi$
$710$	−1.88479	−0.0707349
$711$	1.49010	0.0558831
$712$	2.54107	0.0952306
$713$	6.54107	0.244965
$714$	0.607214	0.0227244
$715$	0	0
$716$	−4.31065	−0.161097
$717$	−2.55025	−0.0952408
$718$	25.2635	0.942825
$719$	−49.8016	−1.85728	−0.928642	−	0.370976i	$-0.879023\pi$
$719$	−49.8016	−1.85728	−0.928642	+	0.370976i	$0.879023\pi$
$720$	−0.966927	−0.0360352
$721$	17.5080	0.652032
$722$	6.46084	0.240447
$723$	25.1483	0.935275
$724$	−0.214429	−0.00796919
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−21.5050	−0.797576	−0.398788	−	0.917043i	$-0.630569\pi$
$727$	−21.5050	−0.797576	−0.398788	+	0.917043i	$0.630569\pi$
$728$	3.96693	0.147024
$729$	29.1886	1.08106
$730$	1.57414	0.0582616
$731$	−0.670362	−0.0247942
$732$	−8.29656	−0.306650
$733$	−2.45593	−0.0907120	−0.0453560	−	0.998971i	$-0.514442\pi$
$733$	−2.45593	−0.0907120	−0.0453560	+	0.998971i	$0.514442\pi$
$734$	−20.1293	−0.742986
$735$	1.42586	0.0525936
$736$	−2.57414	−0.0948841
$737$	0	0
$738$	−1.34672	−0.0495735
$739$	−0.310649	−0.0114274	−0.00571371	−	0.999984i	$-0.501819\pi$
$739$	−0.310649	−0.0114274	−0.00571371	+	0.999984i	$0.501819\pi$
$740$	4.85172	0.178353
$741$	−20.0293	−0.735793
$742$	−0.722424	−0.0265210
$743$	13.3407	0.489424	0.244712	−	0.969596i	$-0.421307\pi$
$743$	13.3407	0.489424	0.244712	+	0.969596i	$0.421307\pi$
$744$	−3.62320	−0.132833
$745$	17.0821	0.625841
$746$	22.9669	0.840879
$747$	0.555153	0.0203120
$748$	0	0
$749$	1.03307	0.0377477
$750$	−1.42586	−0.0520650
$751$	5.10112	0.186143	0.0930713	−	0.995659i	$-0.470332\pi$
$751$	5.10112	0.186143	0.0930713	+	0.995659i	$0.470332\pi$
$752$	6.54107	0.238528
$753$	9.93385	0.362010
$754$	−7.93385	−0.288934
$755$	−12.5741	−0.457620
$756$	−5.65628	−0.205717
$757$	10.6844	0.388333	0.194166	−	0.980969i	$-0.437800\pi$
$757$	10.6844	0.388333	0.194166	+	0.980969i	$0.437800\pi$
$758$	19.8677	0.721628
$759$	0	0
$760$	−3.54107	−0.128448
$761$	33.9198	1.22959	0.614795	−	0.788687i	$-0.289239\pi$
$761$	33.9198	1.22959	0.614795	+	0.788687i	$0.289239\pi$
$762$	−6.00490	−0.217535
$763$	−9.32664	−0.337647
$764$	24.9970	0.904360
$765$	0.411774	0.0148877
$766$	−16.0631	−0.580385
$767$	−44.6061	−1.61063
$768$	1.42586	0.0514512
$769$	−36.7194	−1.32414	−0.662068	−	0.749444i	$-0.730321\pi$
$769$	−36.7194	−1.32414	−0.662068	+	0.749444i	$0.730321\pi$
$770$	0	0
$771$	24.5411	0.883825
$772$	20.7525	0.746899
$773$	−13.3077	−0.478643	−0.239321	−	0.970940i	$-0.576925\pi$
$773$	−13.3077	−0.478643	−0.239321	+	0.970940i	$0.576925\pi$
$774$	1.52208	0.0547100
$775$	2.54107	0.0912779
$776$	6.67036	0.239452
$777$	6.91786	0.248177
$778$	16.7997	0.602297
$779$	−4.93195	−0.176705
$780$	−5.65628	−0.202527
$781$	0	0
$782$	1.09622	0.0392008
$783$	11.3126	0.404278
$784$	1.00000	0.0357143
$785$	−12.7525	−0.455156
$786$	−8.95094	−0.319269
$787$	−31.5050	−1.12303	−0.561516	−	0.827466i	$-0.689782\pi$
$787$	−31.5050	−1.12303	−0.561516	+	0.827466i	$0.689782\pi$
$788$	7.26349	0.258751
$789$	5.21334	0.185600
$790$	1.54107	0.0548287
$791$	16.7525	0.595650
$792$	0	0
$793$	23.0821	0.819671
$794$	−7.93385	−0.281562
$795$	1.03007	0.0365330
$796$	19.5711	0.693681
$797$	14.2635	0.505239	0.252619	−	0.967566i	$-0.418708\pi$
$797$	14.2635	0.505239	0.252619	+	0.967566i	$0.418708\pi$
$798$	−5.04906	−0.178735
$799$	−2.78557	−0.0985464
$800$	−1.00000	−0.0353553
$801$	2.45703	0.0868148
$802$	23.7034	0.836997
$803$	0	0
$804$	7.94794	0.280302
$805$	2.57414	0.0907266
$806$	10.0802	0.355061
$807$	−16.3817	−0.576663
$808$	1.11521	0.0392329
$809$	11.3898	0.400444	0.200222	−	0.979751i	$-0.435834\pi$
$809$	11.3898	0.400444	0.200222	+	0.979751i	$0.435834\pi$
$810$	5.16427	0.181454
$811$	10.1924	0.357905	0.178953	−	0.983858i	$-0.442729\pi$
$811$	10.1924	0.357905	0.178953	+	0.983858i	$0.442729\pi$
$812$	−2.00000	−0.0701862
$813$	9.34672	0.327804
$814$	0	0
$815$	3.26349	0.114315
$816$	−0.607214	−0.0212568
$817$	5.57414	0.195015
$818$	13.6373	0.476817
$819$	3.83573	0.134031
$820$	−1.39279	−0.0486382
$821$	−15.4730	−0.540012	−0.270006	−	0.962859i	$-0.587026\pi$
$821$	−15.4730	−0.540012	−0.270006	+	0.962859i	$0.587026\pi$
$822$	19.8205	0.691321
$823$	37.3126	1.30063	0.650317	−	0.759663i	$-0.274636\pi$
$823$	37.3126	1.30063	0.650317	+	0.759663i	$0.274636\pi$
$824$	−17.5080	−0.609920
$825$	0	0
$826$	−11.2445	−0.391246
$827$	13.0851	0.455015	0.227507	−	0.973776i	$-0.426942\pi$
$827$	13.0851	0.455015	0.227507	+	0.973776i	$0.426942\pi$
$828$	−2.48901	−0.0864989
$829$	−1.77449	−0.0616304	−0.0308152	−	0.999525i	$-0.509810\pi$
$829$	−1.77449	−0.0616304	−0.0308152	+	0.999525i	$0.509810\pi$
$830$	0.574141	0.0199287
$831$	8.01789	0.278138
$832$	−3.96693	−0.137528
$833$	−0.425859	−0.0147551
$834$	−9.53807	−0.330276
$835$	3.27758	0.113425
$836$	0	0
$837$	−14.3730	−0.496803
$838$	−25.7715	−0.890261
$839$	−26.6674	−0.920660	−0.460330	−	0.887748i	$-0.652269\pi$
$839$	−26.6674	−0.920660	−0.460330	+	0.887748i	$0.652269\pi$
$840$	−1.42586	−0.0491968
$841$	−25.0000	−0.862069
$842$	−29.5711	−1.01909
$843$	−13.1613	−0.453298
$844$	9.63729	0.331729
$845$	2.73651	0.0941387
$846$	6.32473	0.217449
$847$	0	0
$848$	0.722424	0.0248081
$849$	6.42776	0.220600
$850$	0.425859	0.0146068
$851$	12.4890	0.428118
$852$	2.68745	0.0920704
$853$	48.8506	1.67261	0.836307	−	0.548262i	$-0.184710\pi$
$853$	48.8506	1.67261	0.836307	+	0.548262i	$0.184710\pi$
$854$	5.81864	0.199110
$855$	−3.42395	−0.117097
$856$	−1.03307	−0.0353097
$857$	−11.0442	−0.377261	−0.188631	−	0.982048i	$-0.560405\pi$
$857$	−11.0442	−0.377261	−0.188631	+	0.982048i	$0.560405\pi$
$858$	0	0
$859$	−1.39578	−0.0476236	−0.0238118	−	0.999716i	$-0.507580\pi$
$859$	−1.39578	−0.0476236	−0.0238118	+	0.999716i	$0.507580\pi$
$860$	1.57414	0.0536778
$861$	−1.98592	−0.0676798
$862$	−21.7997	−0.742499
$863$	27.0442	0.920594	0.460297	−	0.887765i	$-0.347743\pi$
$863$	27.0442	0.920594	0.460297	+	0.887765i	$0.347743\pi$
$864$	5.65628	0.192430
$865$	11.9339	0.405763
$866$	25.5571	0.868464
$867$	−23.9810	−0.814438
$868$	2.54107	0.0862495
$869$	0	0
$870$	2.85172	0.0966823
$871$	−22.1122	−0.749244
$872$	9.32664	0.315840
$873$	6.44975	0.218291
$874$	−9.11521	−0.308327
$875$	1.00000	0.0338062
$876$	−2.24450	−0.0758347
$877$	−17.3437	−0.585656	−0.292828	−	0.956165i	$-0.594596\pi$
$877$	−17.3437	−0.585656	−0.292828	+	0.956165i	$0.594596\pi$
$878$	9.82164	0.331464
$879$	−36.8365	−1.24247
$880$	0	0
$881$	9.67527	0.325968	0.162984	−	0.986629i	$-0.447888\pi$
$881$	9.67527	0.325968	0.162984	+	0.986629i	$0.447888\pi$
$882$	0.966927	0.0325581
$883$	32.3077	1.08724	0.543620	−	0.839332i	$-0.317053\pi$
$883$	32.3077	1.08724	0.543620	+	0.839332i	$0.317053\pi$
$884$	1.68935	0.0568190
$885$	16.0331	0.538946
$886$	−3.37870	−0.113510
$887$	38.4088	1.28964	0.644820	−	0.764334i	$-0.276932\pi$
$887$	38.4088	1.28964	0.644820	+	0.764334i	$0.276932\pi$
$888$	−6.91786	−0.232148
$889$	4.21143	0.141247
$890$	2.54107	0.0851768
$891$	0	0
$892$	4.35971	0.145974
$893$	23.1624	0.775099
$894$	−24.3567	−0.814610
$895$	−4.31065	−0.144089
$896$	−1.00000	−0.0334077
$897$	−14.5601	−0.486146
$898$	−7.29656	−0.243490
$899$	−5.08214	−0.169499
$900$	−0.966927	−0.0322309
$901$	−0.307650	−0.0102493
$902$	0	0
$903$	2.24450	0.0746924
$904$	−16.7525	−0.557180
$905$	−0.214429	−0.00712786
$906$	17.9289	0.595649
$907$	−11.9509	−0.396824	−0.198412	−	0.980119i	$-0.563578\pi$
$907$	−11.9509	−0.396824	−0.198412	+	0.980119i	$0.563578\pi$
$908$	22.7856	0.756166
$909$	1.07833	0.0357658
$910$	3.96693	0.131502
$911$	29.9639	0.992749	0.496375	−	0.868108i	$-0.334664\pi$
$911$	29.9639	0.992749	0.496375	+	0.868108i	$0.334664\pi$
$912$	5.04906	0.167191
$913$	0	0
$914$	−35.5381	−1.17549
$915$	−8.29656	−0.274276
$916$	15.2635	0.504320
$917$	6.27758	0.207304
$918$	−2.40878	−0.0795014
$919$	−49.4388	−1.63084	−0.815419	−	0.578872i	$-0.803493\pi$
$919$	−49.4388	−1.63084	−0.815419	+	0.578872i	$0.803493\pi$
$920$	−2.57414	−0.0848669
$921$	−15.8959	−0.523787
$922$	−26.6704	−0.878342
$923$	−7.47683	−0.246103
$924$	0	0
$925$	4.85172	0.159523
$926$	−17.9528	−0.589967
$927$	−16.9289	−0.556020
$928$	2.00000	0.0656532
$929$	−9.83573	−0.322700	−0.161350	−	0.986897i	$-0.551585\pi$
$929$	−9.83573	−0.322700	−0.161350	+	0.986897i	$0.551585\pi$
$930$	−3.62320	−0.118810
$931$	3.54107	0.116054
$932$	−3.73651	−0.122393
$933$	−19.8878	−0.651097
$934$	18.8327	0.616226
$935$	0	0
$936$	−3.83573	−0.125375
$937$	13.2746	0.433662	0.216831	−	0.976209i	$-0.430428\pi$
$937$	13.2746	0.433662	0.216831	+	0.976209i	$0.430428\pi$
$938$	−5.57414	−0.182002
$939$	−49.1765	−1.60481
$940$	6.54107	0.213346
$941$	−45.9449	−1.49776	−0.748881	−	0.662704i	$-0.769409\pi$
$941$	−45.9449	−1.49776	−0.748881	+	0.662704i	$0.769409\pi$
$942$	18.1833	0.592443
$943$	−3.58523	−0.116751
$944$	11.2445	0.365977
$945$	−5.65628	−0.183999
$946$	0	0
$947$	60.5351	1.96713	0.983563	−	0.180567i	$-0.0577932\pi$
$947$	60.5351	1.96713	0.983563	+	0.180567i	$0.0577932\pi$
$948$	−2.19735	−0.0713665
$949$	6.24450	0.202705
$950$	−3.54107	−0.114887
$951$	−19.2605	−0.624564
$952$	0.425859	0.0138022
$953$	−49.0491	−1.58885	−0.794427	−	0.607359i	$-0.792229\pi$
$953$	−49.0491	−1.58885	−0.794427	+	0.607359i	$0.792229\pi$
$954$	0.698531	0.0226158
$955$	24.9970	0.808884
$956$	−1.78857	−0.0578465
$957$	0	0
$958$	0.310649	0.0100366
$959$	−13.9008	−0.448880
$960$	1.42586	0.0460194
$961$	−24.5430	−0.791709
$962$	19.2464	0.620529
$963$	−0.998906	−0.0321893
$964$	17.6373	0.568059
$965$	20.7525	0.668047
$966$	−3.67036	−0.118092
$967$	−26.4229	−0.849702	−0.424851	−	0.905263i	$-0.639673\pi$
$967$	−26.4229	−0.849702	−0.424851	+	0.905263i	$0.639673\pi$
$968$	0	0
$969$	−2.15019	−0.0690740
$970$	6.67036	0.214172
$971$	−26.6232	−0.854379	−0.427190	−	0.904162i	$-0.640496\pi$
$971$	−26.6232	−0.854379	−0.427190	+	0.904162i	$0.640496\pi$
$972$	9.60531	0.308090
$973$	6.68935	0.214451
$974$	12.5362	0.401685
$975$	−5.65628	−0.181146
$976$	−5.81864	−0.186250
$977$	35.7967	1.14524	0.572618	−	0.819822i	$-0.305928\pi$
$977$	35.7967	1.14524	0.572618	+	0.819822i	$0.305928\pi$
$978$	−4.65328	−0.148795
$979$	0	0
$980$	1.00000	0.0319438
$981$	9.01818	0.287928
$982$	7.71752	0.246276
$983$	47.1423	1.50361	0.751803	−	0.659388i	$-0.229185\pi$
$983$	47.1423	1.50361	0.751803	+	0.659388i	$0.229185\pi$
$984$	1.98592	0.0633087
$985$	7.26349	0.231434
$986$	−0.851718	−0.0271242
$987$	9.32664	0.296870
$988$	−14.0472	−0.446900
$989$	4.05206	0.128848
$990$	0	0
$991$	−17.0141	−0.540470	−0.270235	−	0.962794i	$-0.587101\pi$
$991$	−17.0141	−0.540470	−0.270235	+	0.962794i	$0.587101\pi$
$992$	−2.54107	−0.0806790
$993$	42.9358	1.36253
$994$	−1.88479	−0.0597819
$995$	19.5711	0.620447
$996$	−0.818644	−0.0259397
$997$	29.9008	0.946967	0.473484	−	0.880803i	$-0.342996\pi$
$997$	29.9008	0.946967	0.473484	+	0.880803i	$0.342996\pi$
$998$	18.9179	0.598835
$999$	−27.4427	−0.868247

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.cj.1.2	✓	3
11.10	odd	2		8470.2.a.cm.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8470.2.a.cj.1.2	✓	3	1.1	even	1	trivial
8470.2.a.cm.1.2	yes	3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.42586 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.42586 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.966927 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.42586 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.42586 q^{6} +1.00000 q^{7} -1.00000 q^{8} -0.966927 q^{9} -1.00000 q^{10} +1.42586 q^{12} -3.96693 q^{13} -1.00000 q^{14} +1.42586 q^{15} +1.00000 q^{16} -0.425859 q^{17} +0.966927 q^{18} +3.54107 q^{19} +1.00000 q^{20} +1.42586 q^{21} +2.57414 q^{23} -1.42586 q^{24} +1.00000 q^{25} +3.96693 q^{26} -5.65628 q^{27} +1.00000 q^{28} -2.00000 q^{29} -1.42586 q^{30} +2.54107 q^{31} -1.00000 q^{32} +0.425859 q^{34} +1.00000 q^{35} -0.966927 q^{36} +4.85172 q^{37} -3.54107 q^{38} -5.65628 q^{39} -1.00000 q^{40} -1.39279 q^{41} -1.42586 q^{42} +1.57414 q^{43} -0.966927 q^{45} -2.57414 q^{46} +6.54107 q^{47} +1.42586 q^{48} +1.00000 q^{49} -1.00000 q^{50} -0.607214 q^{51} -3.96693 q^{52} +0.722424 q^{53} +5.65628 q^{54} -1.00000 q^{56} +5.04906 q^{57} +2.00000 q^{58} +11.2445 q^{59} +1.42586 q^{60} -5.81864 q^{61} -2.54107 q^{62} -0.966927 q^{63} +1.00000 q^{64} -3.96693 q^{65} +5.57414 q^{67} -0.425859 q^{68} +3.67036 q^{69} -1.00000 q^{70} +1.88479 q^{71} +0.966927 q^{72} -1.57414 q^{73} -4.85172 q^{74} +1.42586 q^{75} +3.54107 q^{76} +5.65628 q^{78} -1.54107 q^{79} +1.00000 q^{80} -5.16427 q^{81} +1.39279 q^{82} -0.574141 q^{83} +1.42586 q^{84} -0.425859 q^{85} -1.57414 q^{86} -2.85172 q^{87} -2.54107 q^{89} +0.966927 q^{90} -3.96693 q^{91} +2.57414 q^{92} +3.62320 q^{93} -6.54107 q^{94} +3.54107 q^{95} -1.42586 q^{96} -6.67036 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 13 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^5 - 2 * q^6 + 3 * q^7 - 3 * q^8 + 13 * q^9	\( 3 q - 3 q^{2} + 2 q^{3} + 3 q^{4} + 3 q^{5} - 2 q^{6} + 3 q^{7} - 3 q^{8} + 13 q^{9} - 3 q^{10} + 2 q^{12} + 4 q^{13} - 3 q^{14} + 2 q^{15} + 3 q^{16} + q^{17} - 13 q^{18} - 3 q^{19} + 3 q^{20} + 2 q^{21} + 10 q^{23} - 2 q^{24} + 3 q^{25} - 4 q^{26} + 8 q^{27} + 3 q^{28} - 6 q^{29} - 2 q^{30} - 6 q^{31} - 3 q^{32} - q^{34} + 3 q^{35} + 13 q^{36} + 10 q^{37} + 3 q^{38} + 8 q^{39} - 3 q^{40} + 14 q^{41} - 2 q^{42} + 7 q^{43} + 13 q^{45} - 10 q^{46} + 6 q^{47} + 2 q^{48} + 3 q^{49} - 3 q^{50} - 20 q^{51} + 4 q^{52} + 9 q^{53} - 8 q^{54} - 3 q^{56} - 28 q^{57} + 6 q^{58} + 11 q^{59} + 2 q^{60} + 3 q^{61} + 6 q^{62} + 13 q^{63} + 3 q^{64} + 4 q^{65} + 19 q^{67} + q^{68} - 14 q^{69} - 3 q^{70} + 17 q^{71} - 13 q^{72} - 7 q^{73} - 10 q^{74} + 2 q^{75} - 3 q^{76} - 8 q^{78} + 9 q^{79} + 3 q^{80} + 39 q^{81} - 14 q^{82} - 4 q^{83} + 2 q^{84} + q^{85} - 7 q^{86} - 4 q^{87} + 6 q^{89} - 13 q^{90} + 4 q^{91} + 10 q^{92} - 30 q^{93} - 6 q^{94} - 3 q^{95} - 2 q^{96} + 5 q^{97} - 3 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 2 * q^3 + 3 * q^4 + 3 * q^5 - 2 * q^6 + 3 * q^7 - 3 * q^8 + 13 * q^9 - 3 * q^10 + 2 * q^12 + 4 * q^13 - 3 * q^14 + 2 * q^15 + 3 * q^16 + q^17 - 13 * q^18 - 3 * q^19 + 3 * q^20 + 2 * q^21 + 10 * q^23 - 2 * q^24 + 3 * q^25 - 4 * q^26 + 8 * q^27 + 3 * q^28 - 6 * q^29 - 2 * q^30 - 6 * q^31 - 3 * q^32 - q^34 + 3 * q^35 + 13 * q^36 + 10 * q^37 + 3 * q^38 + 8 * q^39 - 3 * q^40 + 14 * q^41 - 2 * q^42 + 7 * q^43 + 13 * q^45 - 10 * q^46 + 6 * q^47 + 2 * q^48 + 3 * q^49 - 3 * q^50 - 20 * q^51 + 4 * q^52 + 9 * q^53 - 8 * q^54 - 3 * q^56 - 28 * q^57 + 6 * q^58 + 11 * q^59 + 2 * q^60 + 3 * q^61 + 6 * q^62 + 13 * q^63 + 3 * q^64 + 4 * q^65 + 19 * q^67 + q^68 - 14 * q^69 - 3 * q^70 + 17 * q^71 - 13 * q^72 - 7 * q^73 - 10 * q^74 + 2 * q^75 - 3 * q^76 - 8 * q^78 + 9 * q^79 + 3 * q^80 + 39 * q^81 - 14 * q^82 - 4 * q^83 + 2 * q^84 + q^85 - 7 * q^86 - 4 * q^87 + 6 * q^89 - 13 * q^90 + 4 * q^91 + 10 * q^92 - 30 * q^93 - 6 * q^94 - 3 * q^95 - 2 * q^96 + 5 * q^97 - 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more