LMFDB - Embedded newform 867.2.d.c.577.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [867,2,Mod(577,867)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(867, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("867.577");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 867 = 3 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	867.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$6.92302985525$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			577.1
Root			$-1.56155i$ of defining polynomial
Character	$\chi$	$=$	867.577
Dual form			867.2.d.c.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56155 q^{2} -1.00000i q^{3} +0.438447 q^{4} -0.561553i q^{5} +1.56155i q^{6} +2.43845 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.56155 q^{2} -1.00000i q^{3} +0.438447 q^{4} -0.561553i q^{5} +1.56155i q^{6} +2.43845 q^{8} -1.00000 q^{9} +0.876894i q^{10} +2.56155i q^{11} -0.438447i q^{12} +4.56155 q^{13} -0.561553 q^{15} -4.68466 q^{16} +1.56155 q^{18} -7.68466 q^{19} -0.246211i q^{20} -4.00000i q^{22} +6.56155i q^{23} -2.43845i q^{24} +4.68466 q^{25} -7.12311 q^{26} +1.00000i q^{27} +8.24621i q^{29} +0.876894 q^{30} -5.12311i q^{31} +2.43845 q^{32} +2.56155 q^{33} -0.438447 q^{36} +3.12311i q^{37} +12.0000 q^{38} -4.56155i q^{39} -1.36932i q^{40} -0.561553i q^{41} +7.68466 q^{43} +1.12311i q^{44} +0.561553i q^{45} -10.2462i q^{46} -2.87689 q^{47} +4.68466i q^{48} +7.00000 q^{49} -7.31534 q^{50} +2.00000 q^{52} +4.24621 q^{53} -1.56155i q^{54} +1.43845 q^{55} +7.68466i q^{57} -12.8769i q^{58} +1.12311 q^{59} -0.246211 q^{60} -0.876894i q^{61} +8.00000i q^{62} +5.56155 q^{64} -2.56155i q^{65} -4.00000 q^{66} +4.00000 q^{67} +6.56155 q^{69} +10.2462i q^{71} -2.43845 q^{72} +4.24621i q^{73} -4.87689i q^{74} -4.68466i q^{75} -3.36932 q^{76} +7.12311i q^{78} -15.3693i q^{79} +2.63068i q^{80} +1.00000 q^{81} +0.876894i q^{82} +9.12311 q^{83} -12.0000 q^{86} +8.24621 q^{87} +6.24621i q^{88} +7.12311 q^{89} -0.876894i q^{90} +2.87689i q^{92} -5.12311 q^{93} +4.49242 q^{94} +4.31534i q^{95} -2.43845i q^{96} -11.1231i q^{97} -10.9309 q^{98} -2.56155i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 10 q^{4} + 18 q^{8} - 4 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 + 10 * q^4 + 18 * q^8 - 4 * q^9	$ 4 q + 2 q^{2} + 10 q^{4} + 18 q^{8} - 4 q^{9} + 10 q^{13} + 6 q^{15} + 6 q^{16} - 2 q^{18} - 6 q^{19} - 6 q^{25} - 12 q^{26} + 20 q^{30} + 18 q^{32} + 2 q^{33} - 10 q^{36} + 48 q^{38} + 6 q^{43} - 28 q^{47} + 28 q^{49} - 54 q^{50} + 8 q^{52} - 16 q^{53} + 14 q^{55} - 12 q^{59} + 32 q^{60} + 14 q^{64} - 16 q^{66} + 16 q^{67} + 18 q^{69} - 18 q^{72} + 36 q^{76} + 4 q^{81} + 20 q^{83} - 48 q^{86} + 12 q^{89} - 4 q^{93} - 48 q^{94} + 14 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 10 * q^4 + 18 * q^8 - 4 * q^9 + 10 * q^13 + 6 * q^15 + 6 * q^16 - 2 * q^18 - 6 * q^19 - 6 * q^25 - 12 * q^26 + 20 * q^30 + 18 * q^32 + 2 * q^33 - 10 * q^36 + 48 * q^38 + 6 * q^43 - 28 * q^47 + 28 * q^49 - 54 * q^50 + 8 * q^52 - 16 * q^53 + 14 * q^55 - 12 * q^59 + 32 * q^60 + 14 * q^64 - 16 * q^66 + 16 * q^67 + 18 * q^69 - 18 * q^72 + 36 * q^76 + 4 * q^81 + 20 * q^83 - 48 * q^86 + 12 * q^89 - 4 * q^93 - 48 * q^94 + 14 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/867\mathbb{Z}\right)^\times$.

$n$	$290$	$292$
$\chi(n)$	$1$	$-1$

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 $

$2$	−1.56155	−1.10418	−0.552092	−	0.833783i	$-0.686170\pi$
$2$	−1.56155	−1.10418	−0.552092	+	0.833783i	$0.686170\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0.438447	0.219224
$5$	− 0.561553i	− 0.251134i	−0.992085	−	0.125567i	$-0.959925\pi$
$5$	− 0.561553i	− 0.251134i	0.992085	−	0.125567i	$-0.0400750\pi$
$6$	1.56155i	0.637501i
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	2.43845	0.862121
$9$	−1.00000	−0.333333
$10$	0.876894i	0.277298i
$11$	2.56155i	0.772337i	0.922428	+	0.386169i	$0.126202\pi$
$11$	2.56155i	0.772337i	−0.922428	+	0.386169i	$0.873798\pi$
$12$	− 0.438447i	− 0.126569i
$13$	4.56155	1.26515	0.632574	−	0.774500i	$-0.281999\pi$
$13$	4.56155	1.26515	0.632574	+	0.774500i	$0.281999\pi$
$14$	0	0
$15$	−0.561553	−0.144992
$16$	−4.68466	−1.17116
$17$	0	0
$17$	0	0
$18$	1.56155	0.368062
$19$	−7.68466	−1.76298	−0.881491	−	0.472201i	$-0.843460\pi$
$19$	−7.68466	−1.76298	−0.881491	+	0.472201i	$0.843460\pi$
$20$	− 0.246211i	− 0.0550545i
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	6.56155i	1.36818i	0.729398	+	0.684089i	$0.239800\pi$
$23$	6.56155i	1.36818i	−0.729398	+	0.684089i	$0.760200\pi$
$24$	− 2.43845i	− 0.497746i
$25$	4.68466	0.936932
$26$	−7.12311	−1.39696
$27$	1.00000i	0.192450i
$28$	0	0
$29$	8.24621i	1.53128i	0.643268	+	0.765641i	$0.277578\pi$
$29$	8.24621i	1.53128i	−0.643268	+	0.765641i	$0.722422\pi$
$30$	0.876894	0.160098
$31$	− 5.12311i	− 0.920137i	−0.887883	−	0.460068i	$-0.847825\pi$
$31$	− 5.12311i	− 0.920137i	0.887883	−	0.460068i	$-0.152175\pi$
$32$	2.43845	0.431061
$33$	2.56155	0.445909
$34$	0	0
$35$	0	0
$36$	−0.438447	−0.0730745
$37$	3.12311i	0.513435i	0.966486	+	0.256718i	$0.0826411\pi$
$37$	3.12311i	0.513435i	−0.966486	+	0.256718i	$0.917359\pi$
$38$	12.0000	1.94666
$39$	− 4.56155i	− 0.730433i
$40$	− 1.36932i	− 0.216508i
$41$	− 0.561553i	− 0.0876998i	−0.999038	−	0.0438499i	$-0.986038\pi$
$41$	− 0.561553i	− 0.0876998i	0.999038	−	0.0438499i	$-0.0139623\pi$
$42$	0	0
$43$	7.68466	1.17190	0.585950	−	0.810347i	$-0.300722\pi$
$43$	7.68466	1.17190	0.585950	+	0.810347i	$0.300722\pi$
$44$	1.12311i	0.169315i
$45$	0.561553i	0.0837114i
$46$	− 10.2462i	− 1.51072i
$47$	−2.87689	−0.419638	−0.209819	−	0.977740i	$-0.567288\pi$
$47$	−2.87689	−0.419638	−0.209819	+	0.977740i	$0.567288\pi$
$48$	4.68466i	0.676172i
$49$	7.00000	1.00000
$50$	−7.31534	−1.03455
$51$	0	0
$52$	2.00000	0.277350
$53$	4.24621	0.583262	0.291631	−	0.956531i	$-0.405802\pi$
$53$	4.24621	0.583262	0.291631	+	0.956531i	$0.405802\pi$
$54$	− 1.56155i	− 0.212500i
$55$	1.43845	0.193960
$56$	0	0
$57$	7.68466i	1.01786i
$58$	− 12.8769i	− 1.69082i
$59$	1.12311	0.146216	0.0731079	−	0.997324i	$-0.476708\pi$
$59$	1.12311	0.146216	0.0731079	+	0.997324i	$0.476708\pi$
$60$	−0.246211	−0.0317857
$61$	− 0.876894i	− 0.112275i	−0.998423	−	0.0561374i	$-0.982122\pi$
$61$	− 0.876894i	− 0.112275i	0.998423	−	0.0561374i	$-0.0178785\pi$
$62$	8.00000i	1.01600i
$63$	0	0
$64$	5.56155	0.695194
$65$	− 2.56155i	− 0.317722i
$66$	−4.00000	−0.492366
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	6.56155	0.789918
$70$	0	0
$71$	10.2462i	1.21600i	0.793936	+	0.608001i	$0.208028\pi$
$71$	10.2462i	1.21600i	−0.793936	+	0.608001i	$0.791972\pi$
$72$	−2.43845	−0.287374
$73$	4.24621i	0.496981i	0.968634	+	0.248491i	$0.0799345\pi$
$73$	4.24621i	0.496981i	−0.968634	+	0.248491i	$0.920065\pi$
$74$	− 4.87689i	− 0.566927i
$75$	− 4.68466i	− 0.540938i
$76$	−3.36932	−0.386487
$77$	0	0
$78$	7.12311i	0.806533i
$79$	− 15.3693i	− 1.72918i	−0.502475	−	0.864592i	$-0.667577\pi$
$79$	− 15.3693i	− 1.72918i	0.502475	−	0.864592i	$-0.332423\pi$
$80$	2.63068i	0.294119i
$81$	1.00000	0.111111
$82$	0.876894i	0.0968368i
$83$	9.12311	1.00139	0.500695	−	0.865624i	$-0.333078\pi$
$83$	9.12311	1.00139	0.500695	+	0.865624i	$0.333078\pi$
$84$	0	0
$85$	0	0
$86$	−12.0000	−1.29399
$87$	8.24621	0.884087
$88$	6.24621i	0.665848i
$89$	7.12311	0.755048	0.377524	−	0.926000i	$-0.376776\pi$
$89$	7.12311	0.755048	0.377524	+	0.926000i	$0.376776\pi$
$90$	− 0.876894i	− 0.0924328i
$91$	0	0
$92$	2.87689i	0.299937i
$93$	−5.12311	−0.531241
$94$	4.49242	0.463358
$95$	4.31534i	0.442745i
$96$	− 2.43845i	− 0.248873i
$97$	− 11.1231i	− 1.12938i	−0.825303	−	0.564690i	$-0.808996\pi$
$97$	− 11.1231i	− 1.12938i	0.825303	−	0.564690i	$-0.191004\pi$
$98$	−10.9309	−1.10418
$99$	− 2.56155i	− 0.257446i
$100$	2.05398	0.205398
$101$	19.1231	1.90282	0.951410	−	0.307927i	$-0.0996352\pi$
$101$	19.1231	1.90282	0.951410	+	0.307927i	$0.0996352\pi$
$102$	0	0
$103$	4.31534	0.425203	0.212602	−	0.977139i	$-0.431806\pi$
$103$	4.31534	0.425203	0.212602	+	0.977139i	$0.431806\pi$
$104$	11.1231	1.09071
$105$	0	0
$106$	−6.63068	−0.644029
$107$	7.68466i	0.742904i	0.928452	+	0.371452i	$0.121140\pi$
$107$	7.68466i	0.742904i	−0.928452	+	0.371452i	$0.878860\pi$
$108$	0.438447i	0.0421896i
$109$	15.1231i	1.44853i	0.689521	+	0.724265i	$0.257821\pi$
$109$	15.1231i	1.44853i	−0.689521	+	0.724265i	$0.742179\pi$
$110$	−2.24621	−0.214168
$111$	3.12311	0.296432
$112$	0	0
$113$	4.56155i	0.429115i	0.976711	+	0.214557i	$0.0688309\pi$
$113$	4.56155i	0.429115i	−0.976711	+	0.214557i	$0.931169\pi$
$114$	− 12.0000i	− 1.12390i
$115$	3.68466	0.343596
$116$	3.61553i	0.335693i
$117$	−4.56155	−0.421716
$118$	−1.75379	−0.161449
$119$	0	0
$120$	−1.36932	−0.125001
$121$	4.43845	0.403495
$122$	1.36932i	0.123972i
$123$	−0.561553	−0.0506335
$124$	− 2.24621i	− 0.201716i
$125$	− 5.43845i	− 0.486430i
$126$	0	0
$127$	0.807764	0.0716775	0.0358387	−	0.999358i	$-0.488590\pi$
$127$	0.807764	0.0716775	0.0358387	+	0.999358i	$0.488590\pi$
$128$	−13.5616	−1.19868
$129$	− 7.68466i	− 0.676596i
$130$	4.00000i	0.350823i
$131$	18.5616i	1.62173i	0.585233	+	0.810865i	$0.301003\pi$
$131$	18.5616i	1.62173i	−0.585233	+	0.810865i	$0.698997\pi$
$132$	1.12311	0.0977538
$133$	0	0
$134$	−6.24621	−0.539590
$135$	0.561553	0.0483308
$136$	0	0
$137$	−16.2462	−1.38801	−0.694004	−	0.719971i	$-0.744155\pi$
$137$	−16.2462	−1.38801	−0.694004	+	0.719971i	$0.744155\pi$
$138$	−10.2462	−0.872215
$139$	− 9.12311i	− 0.773812i	−0.922119	−	0.386906i	$-0.873544\pi$
$139$	− 9.12311i	− 0.773812i	0.922119	−	0.386906i	$-0.126456\pi$
$140$	0	0
$141$	2.87689i	0.242278i
$142$	− 16.0000i	− 1.34269i
$143$	11.6847i	0.977120i
$144$	4.68466	0.390388
$145$	4.63068	0.384557
$146$	− 6.63068i	− 0.548759i
$147$	− 7.00000i	− 0.577350i
$148$	1.36932i	0.112557i
$149$	−4.24621	−0.347863	−0.173932	−	0.984758i	$-0.555647\pi$
$149$	−4.24621	−0.347863	−0.173932	+	0.984758i	$0.555647\pi$
$150$	7.31534i	0.597295i
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	−18.7386	−1.51990
$153$	0	0
$154$	0	0
$155$	−2.87689	−0.231078
$156$	− 2.00000i	− 0.160128i
$157$	−5.68466	−0.453685	−0.226843	−	0.973931i	$-0.572840\pi$
$157$	−5.68466	−0.453685	−0.226843	+	0.973931i	$0.572840\pi$
$158$	24.0000i	1.90934i
$159$	− 4.24621i	− 0.336746i
$160$	− 1.36932i	− 0.108254i
$161$	0	0
$162$	−1.56155	−0.122687
$163$	6.87689i	0.538640i	0.963051	+	0.269320i	$0.0867989\pi$
$163$	6.87689i	0.538640i	−0.963051	+	0.269320i	$0.913201\pi$
$164$	− 0.246211i	− 0.0192259i
$165$	− 1.43845i	− 0.111983i
$166$	−14.2462	−1.10572
$167$	− 0.807764i	− 0.0625067i	−0.999511	−	0.0312533i	$-0.990050\pi$
$167$	− 0.807764i	− 0.0625067i	0.999511	−	0.0312533i	$-0.00994986\pi$
$168$	0	0
$169$	7.80776	0.600597
$170$	0	0
$171$	7.68466	0.587661
$172$	3.36932	0.256908
$173$	− 18.8078i	− 1.42993i	−0.699161	−	0.714964i	$-0.746443\pi$
$173$	− 18.8078i	− 1.42993i	0.699161	−	0.714964i	$-0.253557\pi$
$174$	−12.8769	−0.976195
$175$	0	0
$176$	− 12.0000i	− 0.904534i
$177$	− 1.12311i	− 0.0844178i
$178$	−11.1231	−0.833712
$179$	9.12311	0.681893	0.340946	−	0.940083i	$-0.389253\pi$
$179$	9.12311	0.681893	0.340946	+	0.940083i	$0.389253\pi$
$180$	0.246211i	0.0183515i
$181$	− 6.00000i	− 0.445976i	−0.974821	−	0.222988i	$-0.928419\pi$
$181$	− 6.00000i	− 0.445976i	0.974821	−	0.222988i	$-0.0715812\pi$
$182$	0	0
$183$	−0.876894	−0.0648219
$184$	16.0000i	1.17954i
$185$	1.75379	0.128941
$186$	8.00000	0.586588
$187$	0	0
$188$	−1.26137	−0.0919946
$189$	0	0
$190$	− 6.73863i	− 0.488872i
$191$	−13.1231	−0.949555	−0.474777	−	0.880106i	$-0.657471\pi$
$191$	−13.1231	−0.949555	−0.474777	+	0.880106i	$0.657471\pi$
$192$	− 5.56155i	− 0.401371i
$193$	24.2462i	1.74528i	0.488364	+	0.872640i	$0.337594\pi$
$193$	24.2462i	1.74528i	−0.488364	+	0.872640i	$0.662406\pi$
$194$	17.3693i	1.24704i
$195$	−2.56155	−0.183437
$196$	3.06913	0.219224
$197$	− 19.9309i	− 1.42002i	−0.704194	−	0.710008i	$-0.748691\pi$
$197$	− 19.9309i	− 1.42002i	0.704194	−	0.710008i	$-0.251309\pi$
$198$	4.00000i	0.284268i
$199$	16.0000i	1.13421i	0.823646	+	0.567105i	$0.191937\pi$
$199$	16.0000i	1.13421i	−0.823646	+	0.567105i	$0.808063\pi$
$200$	11.4233	0.807749
$201$	− 4.00000i	− 0.282138i
$202$	−29.8617	−2.10106
$203$	0	0
$204$	0	0
$205$	−0.315342	−0.0220244
$206$	−6.73863	−0.469503
$207$	− 6.56155i	− 0.456059i
$208$	−21.3693	−1.48170
$209$	− 19.6847i	− 1.36162i
$210$	0	0
$211$	11.3693i	0.782696i	0.920243	+	0.391348i	$0.127991\pi$
$211$	11.3693i	0.782696i	−0.920243	+	0.391348i	$0.872009\pi$
$212$	1.86174	0.127865
$213$	10.2462	0.702059
$214$	− 12.0000i	− 0.820303i
$215$	− 4.31534i	− 0.294304i
$216$	2.43845i	0.165915i
$217$	0	0
$218$	− 23.6155i	− 1.59945i
$219$	4.24621	0.286932
$220$	0.630683	0.0425206
$221$	0	0
$222$	−4.87689	−0.327316
$223$	−13.9309	−0.932880	−0.466440	−	0.884553i	$-0.654464\pi$
$223$	−13.9309	−0.932880	−0.466440	+	0.884553i	$0.654464\pi$
$224$	0	0
$225$	−4.68466	−0.312311
$226$	− 7.12311i	− 0.473822i
$227$	− 23.0540i	− 1.53015i	−0.643944	−	0.765073i	$-0.722703\pi$
$227$	− 23.0540i	− 1.53015i	0.643944	−	0.765073i	$-0.277297\pi$
$228$	3.36932i	0.223138i
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−5.75379	−0.379394
$231$	0	0
$232$	20.1080i	1.32015i
$233$	0.561553i	0.0367885i	0.999831	+	0.0183943i	$0.00585541\pi$
$233$	0.561553i	0.0367885i	−0.999831	+	0.0183943i	$0.994145\pi$
$234$	7.12311	0.465652
$235$	1.61553i	0.105385i
$236$	0.492423	0.0320540
$237$	−15.3693	−0.998344
$238$	0	0
$239$	10.2462	0.662772	0.331386	−	0.943495i	$-0.392484\pi$
$239$	10.2462	0.662772	0.331386	+	0.943495i	$0.392484\pi$
$240$	2.63068	0.169810
$241$	− 21.3693i	− 1.37652i	−0.725465	−	0.688259i	$-0.758375\pi$
$241$	− 21.3693i	− 1.37652i	0.725465	−	0.688259i	$-0.241625\pi$
$242$	−6.93087	−0.445533
$243$	− 1.00000i	− 0.0641500i
$244$	− 0.384472i	− 0.0246133i
$245$	− 3.93087i	− 0.251134i
$246$	0.876894	0.0559087
$247$	−35.0540	−2.23043
$248$	− 12.4924i	− 0.793270i
$249$	− 9.12311i	− 0.578153i
$250$	8.49242i	0.537108i
$251$	−24.4924	−1.54595	−0.772974	−	0.634438i	$-0.781232\pi$
$251$	−24.4924	−1.54595	−0.772974	+	0.634438i	$0.781232\pi$
$252$	0	0
$253$	−16.8078	−1.05670
$254$	−1.26137	−0.0791452
$255$	0	0
$256$	10.0540	0.628373
$257$	−9.36932	−0.584442	−0.292221	−	0.956351i	$-0.594394\pi$
$257$	−9.36932	−0.584442	−0.292221	+	0.956351i	$0.594394\pi$
$258$	12.0000i	0.747087i
$259$	0	0
$260$	− 1.12311i	− 0.0696521i
$261$	− 8.24621i	− 0.510428i
$262$	− 28.9848i	− 1.79069i
$263$	−12.4924	−0.770316	−0.385158	−	0.922851i	$-0.625853\pi$
$263$	−12.4924	−0.770316	−0.385158	+	0.922851i	$0.625853\pi$
$264$	6.24621	0.384428
$265$	− 2.38447i	− 0.146477i
$266$	0	0
$267$	− 7.12311i	− 0.435927i
$268$	1.75379	0.107130
$269$	20.5616i	1.25366i	0.779156	+	0.626830i	$0.215648\pi$
$269$	20.5616i	1.25366i	−0.779156	+	0.626830i	$0.784352\pi$
$270$	−0.876894	−0.0533661
$271$	−0.807764	−0.0490682	−0.0245341	−	0.999699i	$-0.507810\pi$
$271$	−0.807764	−0.0490682	−0.0245341	+	0.999699i	$0.507810\pi$
$272$	0	0
$273$	0	0
$274$	25.3693	1.53262
$275$	12.0000i	0.723627i
$276$	2.87689	0.173169
$277$	6.00000i	0.360505i	0.983620	+	0.180253i	$0.0576915\pi$
$277$	6.00000i	0.360505i	−0.983620	+	0.180253i	$0.942309\pi$
$278$	14.2462i	0.854431i
$279$	5.12311i	0.306712i
$280$	0	0
$281$	19.1231	1.14079	0.570394	−	0.821371i	$-0.306790\pi$
$281$	19.1231	1.14079	0.570394	+	0.821371i	$0.306790\pi$
$282$	− 4.49242i	− 0.267520i
$283$	3.36932i	0.200285i	0.994973	+	0.100143i	$0.0319299\pi$
$283$	3.36932i	0.200285i	−0.994973	+	0.100143i	$0.968070\pi$
$284$	4.49242i	0.266576i
$285$	4.31534	0.255619
$286$	− 18.2462i	− 1.07892i
$287$	0	0
$288$	−2.43845	−0.143687
$289$	0	0
$290$	−7.23106	−0.424622
$291$	−11.1231	−0.652048
$292$	1.86174i	0.108950i
$293$	−7.12311	−0.416136	−0.208068	−	0.978114i	$-0.566718\pi$
$293$	−7.12311	−0.416136	−0.208068	+	0.978114i	$0.566718\pi$
$294$	10.9309i	0.637501i
$295$	− 0.630683i	− 0.0367198i
$296$	7.61553i	0.442644i
$297$	−2.56155	−0.148636
$298$	6.63068	0.384105
$299$	29.9309i	1.73095i
$300$	− 2.05398i	− 0.118586i
$301$	0	0
$302$	12.4924	0.718858
$303$	− 19.1231i	− 1.09859i
$304$	36.0000	2.06474
$305$	−0.492423	−0.0281960
$306$	0	0
$307$	−0.492423	−0.0281040	−0.0140520	−	0.999901i	$-0.504473\pi$
$307$	−0.492423	−0.0281040	−0.0140520	+	0.999901i	$0.504473\pi$
$308$	0	0
$309$	− 4.31534i	− 0.245491i
$310$	4.49242	0.255152
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	− 11.1231i	− 0.629722i
$313$	7.61553i	0.430455i	0.976564	+	0.215228i	$0.0690493\pi$
$313$	7.61553i	0.430455i	−0.976564	+	0.215228i	$0.930951\pi$
$314$	8.87689	0.500952
$315$	0	0
$316$	− 6.73863i	− 0.379078i
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	6.63068i	0.371830i
$319$	−21.1231	−1.18267
$320$	− 3.12311i	− 0.174587i
$321$	7.68466	0.428916
$322$	0	0
$323$	0	0
$324$	0.438447	0.0243582
$325$	21.3693	1.18536
$326$	− 10.7386i	− 0.594758i
$327$	15.1231	0.836310
$328$	− 1.36932i	− 0.0756079i
$329$	0	0
$330$	2.24621i	0.123650i
$331$	6.06913	0.333590	0.166795	−	0.985992i	$-0.446658\pi$
$331$	6.06913	0.333590	0.166795	+	0.985992i	$0.446658\pi$
$332$	4.00000	0.219529
$333$	− 3.12311i	− 0.171145i
$334$	1.26137i	0.0690189i
$335$	− 2.24621i	− 0.122724i
$336$	0	0
$337$	32.7386i	1.78339i	0.452640	+	0.891694i	$0.350482\pi$
$337$	32.7386i	1.78339i	−0.452640	+	0.891694i	$0.649518\pi$
$338$	−12.1922	−0.663170
$339$	4.56155	0.247750
$340$	0	0
$341$	13.1231	0.710656
$342$	−12.0000	−0.648886
$343$	0	0
$344$	18.7386	1.01032
$345$	− 3.68466i	− 0.198375i
$346$	29.3693i	1.57890i
$347$	− 24.4924i	− 1.31482i	−0.753532	−	0.657411i	$-0.771652\pi$
$347$	− 24.4924i	− 1.31482i	0.753532	−	0.657411i	$-0.228348\pi$
$348$	3.61553	0.193813
$349$	−7.43845	−0.398171	−0.199085	−	0.979982i	$-0.563797\pi$
$349$	−7.43845	−0.398171	−0.199085	+	0.979982i	$0.563797\pi$
$350$	0	0
$351$	4.56155i	0.243478i
$352$	6.24621i	0.332924i
$353$	22.4924	1.19715	0.598575	−	0.801066i	$-0.295734\pi$
$353$	22.4924	1.19715	0.598575	+	0.801066i	$0.295734\pi$
$354$	1.75379i	0.0932128i
$355$	5.75379	0.305379
$356$	3.12311	0.165524
$357$	0	0
$358$	−14.2462	−0.752936
$359$	2.24621	0.118550	0.0592752	−	0.998242i	$-0.481121\pi$
$359$	2.24621	0.118550	0.0592752	+	0.998242i	$0.481121\pi$
$360$	1.36932i	0.0721693i
$361$	40.0540	2.10810
$362$	9.36932i	0.492440i
$363$	− 4.43845i	− 0.232958i
$364$	0	0
$365$	2.38447	0.124809
$366$	1.36932	0.0715753
$367$	18.2462i	0.952444i	0.879325	+	0.476222i	$0.157994\pi$
$367$	18.2462i	0.952444i	−0.879325	+	0.476222i	$0.842006\pi$
$368$	− 30.7386i	− 1.60236i
$369$	0.561553i	0.0292333i
$370$	−2.73863	−0.142375
$371$	0	0
$372$	−2.24621	−0.116461
$373$	16.2462	0.841197	0.420598	−	0.907247i	$-0.361820\pi$
$373$	16.2462	0.841197	0.420598	+	0.907247i	$0.361820\pi$
$374$	0	0
$375$	−5.43845	−0.280840
$376$	−7.01515	−0.361779
$377$	37.6155i	1.93730i
$378$	0	0
$379$	− 12.0000i	− 0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$	− 12.0000i	− 0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$	1.89205i	0.0970601i
$381$	− 0.807764i	− 0.0413830i
$382$	20.4924	1.04848
$383$	10.2462	0.523557	0.261778	−	0.965128i	$-0.415691\pi$
$383$	10.2462	0.523557	0.261778	+	0.965128i	$0.415691\pi$
$384$	13.5616i	0.692060i
$385$	0	0
$386$	− 37.8617i	− 1.92711i
$387$	−7.68466	−0.390633
$388$	− 4.87689i	− 0.247587i
$389$	21.8617	1.10843	0.554217	−	0.832372i	$-0.313018\pi$
$389$	21.8617	1.10843	0.554217	+	0.832372i	$0.313018\pi$
$390$	4.00000	0.202548
$391$	0	0
$392$	17.0691	0.862121
$393$	18.5616	0.936306
$394$	31.1231i	1.56796i
$395$	−8.63068	−0.434257
$396$	− 1.12311i	− 0.0564382i
$397$	− 5.36932i	− 0.269478i	−0.990881	−	0.134739i	$-0.956980\pi$
$397$	− 5.36932i	− 0.269478i	0.990881	−	0.134739i	$-0.0430196\pi$
$398$	− 24.9848i	− 1.25238i
$399$	0	0
$400$	−21.9460	−1.09730
$401$	6.17708i	0.308469i	0.988034	+	0.154234i	$0.0492911\pi$
$401$	6.17708i	0.308469i	−0.988034	+	0.154234i	$0.950709\pi$
$402$	6.24621i	0.311533i
$403$	− 23.3693i	− 1.16411i
$404$	8.38447	0.417143
$405$	− 0.561553i	− 0.0279038i
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	−2.31534	−0.114486	−0.0572431	−	0.998360i	$-0.518231\pi$
$409$	−2.31534	−0.114486	−0.0572431	+	0.998360i	$0.518231\pi$
$410$	0.492423	0.0243190
$411$	16.2462i	0.801367i
$412$	1.89205	0.0932146
$413$	0	0
$414$	10.2462i	0.503574i
$415$	− 5.12311i	− 0.251483i
$416$	11.1231	0.545355
$417$	−9.12311	−0.446760
$418$	30.7386i	1.50348i
$419$	− 32.4924i	− 1.58736i	−0.608336	−	0.793679i	$-0.708163\pi$
$419$	− 32.4924i	− 1.58736i	0.608336	−	0.793679i	$-0.291837\pi$
$420$	0	0
$421$	28.5616	1.39200	0.696002	−	0.718039i	$-0.254960\pi$
$421$	28.5616	1.39200	0.696002	+	0.718039i	$0.254960\pi$
$422$	− 17.7538i	− 0.864241i
$423$	2.87689	0.139879
$424$	10.3542	0.502843
$425$	0	0
$426$	−16.0000	−0.775203
$427$	0	0
$428$	3.36932i	0.162862i
$429$	11.6847	0.564141
$430$	6.73863i	0.324966i
$431$	24.0000i	1.15604i	0.816023	+	0.578020i	$0.196174\pi$
$431$	24.0000i	1.15604i	−0.816023	+	0.578020i	$0.803826\pi$
$432$	− 4.68466i	− 0.225391i
$433$	−14.3153	−0.687951	−0.343976	−	0.938979i	$-0.611774\pi$
$433$	−14.3153	−0.687951	−0.343976	+	0.938979i	$0.611774\pi$
$434$	0	0
$435$	− 4.63068i	− 0.222024i
$436$	6.63068i	0.317552i
$437$	− 50.4233i	− 2.41207i
$438$	−6.63068	−0.316826
$439$	− 5.75379i	− 0.274613i	−0.990529	−	0.137307i	$-0.956155\pi$
$439$	− 5.75379i	− 0.274613i	0.990529	−	0.137307i	$-0.0438446\pi$
$440$	3.50758	0.167217
$441$	−7.00000	−0.333333
$442$	0	0
$443$	−22.8769	−1.08691	−0.543457	−	0.839437i	$-0.682885\pi$
$443$	−22.8769	−1.08691	−0.543457	+	0.839437i	$0.682885\pi$
$444$	1.36932	0.0649849
$445$	− 4.00000i	− 0.189618i
$446$	21.7538	1.03007
$447$	4.24621i	0.200839i
$448$	0	0
$449$	12.7386i	0.601173i	0.953755	+	0.300587i	$0.0971825\pi$
$449$	12.7386i	0.601173i	−0.953755	+	0.300587i	$0.902818\pi$
$450$	7.31534	0.344849
$451$	1.43845	0.0677338
$452$	2.00000i	0.0940721i
$453$	8.00000i	0.375873i
$454$	36.0000i	1.68956i
$455$	0	0
$456$	18.7386i	0.877517i
$457$	6.80776	0.318454	0.159227	−	0.987242i	$-0.449100\pi$
$457$	6.80776	0.318454	0.159227	+	0.987242i	$0.449100\pi$
$458$	9.36932	0.437799
$459$	0	0
$460$	1.61553	0.0753244
$461$	−8.24621	−0.384064	−0.192032	−	0.981389i	$-0.561508\pi$
$461$	−8.24621	−0.384064	−0.192032	+	0.981389i	$0.561508\pi$
$462$	0	0
$463$	24.9848	1.16114	0.580572	−	0.814209i	$-0.302829\pi$
$463$	24.9848	1.16114	0.580572	+	0.814209i	$0.302829\pi$
$464$	− 38.6307i	− 1.79338i
$465$	2.87689i	0.133413i
$466$	− 0.876894i	− 0.0406213i
$467$	3.36932	0.155913	0.0779567	−	0.996957i	$-0.475160\pi$
$467$	3.36932	0.155913	0.0779567	+	0.996957i	$0.475160\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	− 2.52273i	− 0.116365i
$471$	5.68466i	0.261935i
$472$	2.73863	0.126056
$473$	19.6847i	0.905102i
$474$	24.0000	1.10236
$475$	−36.0000	−1.65179
$476$	0	0
$477$	−4.24621	−0.194421
$478$	−16.0000	−0.731823
$479$	− 29.3002i	− 1.33876i	−0.742920	−	0.669380i	$-0.766560\pi$
$479$	− 29.3002i	− 1.33876i	0.742920	−	0.669380i	$-0.233440\pi$
$480$	−1.36932	−0.0625005
$481$	14.2462i	0.649571i
$482$	33.3693i	1.51993i
$483$	0	0
$484$	1.94602	0.0884557
$485$	−6.24621	−0.283626
$486$	1.56155i	0.0708335i
$487$	7.36932i	0.333936i	0.985962	+	0.166968i	$0.0533976\pi$
$487$	7.36932i	0.333936i	−0.985962	+	0.166968i	$0.946602\pi$
$488$	− 2.13826i	− 0.0967945i
$489$	6.87689	0.310984
$490$	6.13826i	0.277298i
$491$	−3.36932	−0.152055	−0.0760276	−	0.997106i	$-0.524224\pi$
$491$	−3.36932	−0.152055	−0.0760276	+	0.997106i	$0.524224\pi$
$492$	−0.246211	−0.0111001
$493$	0	0
$494$	54.7386	2.46281
$495$	−1.43845	−0.0646534
$496$	24.0000i	1.07763i
$497$	0	0
$498$	14.2462i	0.638388i
$499$	11.3693i	0.508961i	0.967078	+	0.254480i	$0.0819044\pi$
$499$	11.3693i	0.508961i	−0.967078	+	0.254480i	$0.918096\pi$
$500$	− 2.38447i	− 0.106637i
$501$	−0.807764	−0.0360882
$502$	38.2462	1.70701
$503$	25.4384i	1.13424i	0.823634	+	0.567122i	$0.191943\pi$
$503$	25.4384i	1.13424i	−0.823634	+	0.567122i	$0.808057\pi$
$504$	0	0
$505$	− 10.7386i	− 0.477863i
$506$	26.2462	1.16679
$507$	− 7.80776i	− 0.346755i
$508$	0.354162	0.0157134
$509$	16.8769	0.748055	0.374028	−	0.927418i	$-0.377977\pi$
$509$	16.8769	0.748055	0.374028	+	0.927418i	$0.377977\pi$
$510$	0	0
$511$	0	0
$512$	11.4233	0.504843
$513$	− 7.68466i	− 0.339286i
$514$	14.6307	0.645332
$515$	− 2.42329i	− 0.106783i
$516$	− 3.36932i	− 0.148326i
$517$	− 7.36932i	− 0.324102i
$518$	0	0
$519$	−18.8078	−0.825569
$520$	− 6.24621i	− 0.273914i
$521$	31.4384i	1.37734i	0.725073	+	0.688672i	$0.241806\pi$
$521$	31.4384i	1.37734i	−0.725073	+	0.688672i	$0.758194\pi$
$522$	12.8769i	0.563606i
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	8.13826i	0.355522i
$525$	0	0
$526$	19.5076	0.850571
$527$	0	0
$528$	−12.0000	−0.522233
$529$	−20.0540	−0.871912
$530$	3.72348i	0.161738i
$531$	−1.12311	−0.0487386
$532$	0	0
$533$	− 2.56155i	− 0.110953i
$534$	11.1231i	0.481344i
$535$	4.31534	0.186568
$536$	9.75379	0.421300
$537$	− 9.12311i	− 0.393691i
$538$	− 32.1080i	− 1.38427i
$539$	17.9309i	0.772337i
$540$	0.246211	0.0105952
$541$	− 40.1080i	− 1.72438i	−0.506589	−	0.862188i	$-0.669094\pi$
$541$	− 40.1080i	− 1.72438i	0.506589	−	0.862188i	$-0.330906\pi$
$542$	1.26137	0.0541803
$543$	−6.00000	−0.257485
$544$	0	0
$545$	8.49242	0.363775
$546$	0	0
$547$	28.0000i	1.19719i	0.801050	+	0.598597i	$0.204275\pi$
$547$	28.0000i	1.19719i	−0.801050	+	0.598597i	$0.795725\pi$
$548$	−7.12311	−0.304284
$549$	0.876894i	0.0374249i
$550$	− 18.7386i	− 0.799018i
$551$	− 63.3693i	− 2.69962i
$552$	16.0000	0.681005
$553$	0	0
$554$	− 9.36932i	− 0.398064i
$555$	− 1.75379i	− 0.0744442i
$556$	− 4.00000i	− 0.169638i
$557$	−6.49242	−0.275093	−0.137546	−	0.990495i	$-0.543922\pi$
$557$	−6.49242	−0.275093	−0.137546	+	0.990495i	$0.543922\pi$
$558$	− 8.00000i	− 0.338667i
$559$	35.0540	1.48263
$560$	0	0
$561$	0	0
$562$	−29.8617	−1.25964
$563$	−22.8769	−0.964146	−0.482073	−	0.876131i	$-0.660116\pi$
$563$	−22.8769	−0.964146	−0.482073	+	0.876131i	$0.660116\pi$
$564$	1.26137i	0.0531131i
$565$	2.56155	0.107765
$566$	− 5.26137i	− 0.221152i
$567$	0	0
$568$	24.9848i	1.04834i
$569$	−12.8769	−0.539827	−0.269914	−	0.962885i	$-0.586995\pi$
$569$	−12.8769	−0.539827	−0.269914	+	0.962885i	$0.586995\pi$
$570$	−6.73863	−0.282250
$571$	− 18.7386i	− 0.784187i	−0.919925	−	0.392094i	$-0.871751\pi$
$571$	− 18.7386i	− 0.784187i	0.919925	−	0.392094i	$-0.128249\pi$
$572$	5.12311i	0.214208i
$573$	13.1231i	0.548226i
$574$	0	0
$575$	30.7386i	1.28189i
$576$	−5.56155	−0.231731
$577$	−41.0540	−1.70910	−0.854550	−	0.519370i	$-0.826167\pi$
$577$	−41.0540	−1.70910	−0.854550	+	0.519370i	$0.826167\pi$
$578$	0	0
$579$	24.2462	1.00764
$580$	2.03031	0.0843040
$581$	0	0
$582$	17.3693	0.719981
$583$	10.8769i	0.450475i
$584$	10.3542i	0.428458i
$585$	2.56155i	0.105907i
$586$	11.1231	0.459491
$587$	−36.9848	−1.52653	−0.763264	−	0.646087i	$-0.776404\pi$
$587$	−36.9848	−1.52653	−0.763264	+	0.646087i	$0.776404\pi$
$588$	− 3.06913i	− 0.126569i
$589$	39.3693i	1.62218i
$590$	0.984845i	0.0405454i
$591$	−19.9309	−0.819846
$592$	− 14.6307i	− 0.601317i
$593$	−44.2462	−1.81697	−0.908487	−	0.417913i	$-0.862762\pi$
$593$	−44.2462	−1.81697	−0.908487	+	0.417913i	$0.862762\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−1.86174	−0.0762598
$597$	16.0000	0.654836
$598$	− 46.7386i	− 1.91128i
$599$	−41.6155	−1.70036	−0.850182	−	0.526489i	$-0.823508\pi$
$599$	−41.6155	−1.70036	−0.850182	+	0.526489i	$0.823508\pi$
$600$	− 11.4233i	− 0.466354i
$601$	− 34.9848i	− 1.42706i	−0.700624	−	0.713531i	$-0.747095\pi$
$601$	− 34.9848i	− 1.42706i	0.700624	−	0.713531i	$-0.252905\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	−3.50758	−0.142721
$605$	− 2.49242i	− 0.101331i
$606$	29.8617i	1.21305i
$607$	15.3693i	0.623821i	0.950111	+	0.311911i	$0.100969\pi$
$607$	15.3693i	0.623821i	−0.950111	+	0.311911i	$0.899031\pi$
$608$	−18.7386	−0.759952
$609$	0	0
$610$	0.768944	0.0311336
$611$	−13.1231	−0.530904
$612$	0	0
$613$	2.31534	0.0935158	0.0467579	−	0.998906i	$-0.485111\pi$
$613$	2.31534	0.0935158	0.0467579	+	0.998906i	$0.485111\pi$
$614$	0.768944	0.0310320
$615$	0.315342i	0.0127158i
$616$	0	0
$617$	− 27.7538i	− 1.11733i	−0.829395	−	0.558663i	$-0.811315\pi$
$617$	− 27.7538i	− 1.11733i	0.829395	−	0.558663i	$-0.188685\pi$
$618$	6.73863i	0.271068i
$619$	19.3693i	0.778519i	0.921128	+	0.389259i	$0.127269\pi$
$619$	19.3693i	0.778519i	−0.921128	+	0.389259i	$0.872731\pi$
$620$	−1.26137	−0.0506577
$621$	−6.56155	−0.263306
$622$	0	0
$623$	0	0
$624$	21.3693i	0.855457i
$625$	20.3693	0.814773
$626$	− 11.8920i	− 0.475302i
$627$	−19.6847	−0.786130
$628$	−2.49242	−0.0994585
$629$	0	0
$630$	0	0
$631$	−11.6847	−0.465159	−0.232579	−	0.972577i	$-0.574717\pi$
$631$	−11.6847	−0.465159	−0.232579	+	0.972577i	$0.574717\pi$
$632$	− 37.4773i	− 1.49077i
$633$	11.3693	0.451890
$634$	− 28.1080i	− 1.11631i
$635$	− 0.453602i	− 0.0180007i
$636$	− 1.86174i	− 0.0738228i
$637$	31.9309	1.26515
$638$	32.9848	1.30588
$639$	− 10.2462i	− 0.405334i
$640$	7.61553i	0.301030i
$641$	− 0.0691303i	− 0.00273048i	−0.999999	−	0.00136524i	$-0.999565\pi$
$641$	− 0.0691303i	− 0.00273048i	0.999999	−	0.00136524i	$-0.000434570\pi$
$642$	−12.0000	−0.473602
$643$	− 30.2462i	− 1.19279i	−0.802690	−	0.596397i	$-0.796598\pi$
$643$	− 30.2462i	− 1.19279i	0.802690	−	0.596397i	$-0.203402\pi$
$644$	0	0
$645$	−4.31534	−0.169916
$646$	0	0
$647$	15.3693	0.604230	0.302115	−	0.953271i	$-0.402307\pi$
$647$	15.3693	0.604230	0.302115	+	0.953271i	$0.402307\pi$
$648$	2.43845	0.0957913
$649$	2.87689i	0.112928i
$650$	−33.3693	−1.30885
$651$	0	0
$652$	3.01515i	0.118083i
$653$	4.06913i	0.159237i	0.996825	+	0.0796187i	$0.0253703\pi$
$653$	4.06913i	0.159237i	−0.996825	+	0.0796187i	$0.974630\pi$
$654$	−23.6155	−0.923440
$655$	10.4233	0.407272
$656$	2.63068i	0.102711i
$657$	− 4.24621i	− 0.165660i
$658$	0	0
$659$	47.8617	1.86443	0.932214	−	0.361907i	$-0.117874\pi$
$659$	47.8617	1.86443	0.932214	+	0.361907i	$0.117874\pi$
$660$	− 0.630683i	− 0.0245493i
$661$	−25.6847	−0.999017	−0.499509	−	0.866309i	$-0.666486\pi$
$661$	−25.6847	−0.999017	−0.499509	+	0.866309i	$0.666486\pi$
$662$	−9.47727	−0.368344
$663$	0	0
$664$	22.2462	0.863320
$665$	0	0
$666$	4.87689i	0.188976i
$667$	−54.1080	−2.09507
$668$	− 0.354162i	− 0.0137029i
$669$	13.9309i	0.538599i
$670$	3.50758i	0.135510i
$671$	2.24621	0.0867140
$672$	0	0
$673$	− 48.7386i	− 1.87874i	−0.342910	−	0.939368i	$-0.611413\pi$
$673$	− 48.7386i	− 1.87874i	0.342910	−	0.939368i	$-0.388587\pi$
$674$	− 51.1231i	− 1.96919i
$675$	4.68466i	0.180313i
$676$	3.42329	0.131665
$677$	− 13.6847i	− 0.525944i	−0.964803	−	0.262972i	$-0.915297\pi$
$677$	− 13.6847i	− 0.525944i	0.964803	−	0.262972i	$-0.0847027\pi$
$678$	−7.12311	−0.273561
$679$	0	0
$680$	0	0
$681$	−23.0540	−0.883430
$682$	−20.4924	−0.784695
$683$	− 5.43845i	− 0.208096i	−0.994572	−	0.104048i	$-0.966820\pi$
$683$	− 5.43845i	− 0.208096i	0.994572	−	0.104048i	$-0.0331796\pi$
$684$	3.36932	0.128829
$685$	9.12311i	0.348576i
$686$	0	0
$687$	6.00000i	0.228914i
$688$	−36.0000	−1.37249
$689$	19.3693	0.737912
$690$	5.75379i	0.219043i
$691$	− 36.9848i	− 1.40697i	−0.710710	−	0.703485i	$-0.751626\pi$
$691$	− 36.9848i	− 1.40697i	0.710710	−	0.703485i	$-0.248374\pi$
$692$	− 8.24621i	− 0.313474i
$693$	0	0
$694$	38.2462i	1.45181i
$695$	−5.12311	−0.194330
$696$	20.1080	0.762190
$697$	0	0
$698$	11.6155	0.439654
$699$	0.561553	0.0212399
$700$	0	0
$701$	−9.36932	−0.353874	−0.176937	−	0.984222i	$-0.556619\pi$
$701$	−9.36932	−0.353874	−0.176937	+	0.984222i	$0.556619\pi$
$702$	− 7.12311i	− 0.268844i
$703$	− 24.0000i	− 0.905177i
$704$	14.2462i	0.536924i
$705$	1.61553	0.0608443
$706$	−35.1231	−1.32188
$707$	0	0
$708$	− 0.492423i	− 0.0185064i
$709$	4.73863i	0.177963i	0.996033	+	0.0889816i	$0.0283612\pi$
$709$	4.73863i	0.177963i	−0.996033	+	0.0889816i	$0.971639\pi$
$710$	−8.98485	−0.337195
$711$	15.3693i	0.576394i
$712$	17.3693	0.650943
$713$	33.6155	1.25891
$714$	0	0
$715$	6.56155	0.245388
$716$	4.00000	0.149487
$717$	− 10.2462i	− 0.382652i
$718$	−3.50758	−0.130902
$719$	8.80776i	0.328474i	0.986421	+	0.164237i	$0.0525162\pi$
$719$	8.80776i	0.328474i	−0.986421	+	0.164237i	$0.947484\pi$
$720$	− 2.63068i	− 0.0980398i
$721$	0	0
$722$	−62.5464	−2.32774
$723$	−21.3693	−0.794733
$724$	− 2.63068i	− 0.0977686i
$725$	38.6307i	1.43471i
$726$	6.93087i	0.257229i
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	−3.72348	−0.137812
$731$	0	0
$732$	−0.384472	−0.0142105
$733$	28.2462	1.04330	0.521649	−	0.853160i	$-0.325317\pi$
$733$	28.2462	1.04330	0.521649	+	0.853160i	$0.325317\pi$
$734$	− 28.4924i	− 1.05167i
$735$	−3.93087	−0.144992
$736$	16.0000i	0.589768i
$737$	10.2462i	0.377424i
$738$	− 0.876894i	− 0.0322789i
$739$	8.31534	0.305885	0.152942	−	0.988235i	$-0.451125\pi$
$739$	8.31534	0.305885	0.152942	+	0.988235i	$0.451125\pi$
$740$	0.768944	0.0282669
$741$	35.0540i	1.28774i
$742$	0	0
$743$	− 4.49242i	− 0.164811i	−0.996599	−	0.0824055i	$-0.973740\pi$
$743$	− 4.49242i	− 0.164811i	0.996599	−	0.0824055i	$-0.0262603\pi$
$744$	−12.4924	−0.457994
$745$	2.38447i	0.0873603i
$746$	−25.3693	−0.928837
$747$	−9.12311	−0.333797
$748$	0	0
$749$	0	0
$750$	8.49242	0.310099
$751$	− 0.630683i	− 0.0230140i	−0.999934	−	0.0115070i	$-0.996337\pi$
$751$	− 0.630683i	− 0.0230140i	0.999934	−	0.0115070i	$-0.00366287\pi$
$752$	13.4773	0.491465
$753$	24.4924i	0.892553i
$754$	− 58.7386i	− 2.13913i
$755$	4.49242i	0.163496i
$756$	0	0
$757$	21.0540	0.765220	0.382610	−	0.923910i	$-0.375025\pi$
$757$	21.0540	0.765220	0.382610	+	0.923910i	$0.375025\pi$
$758$	18.7386i	0.680618i
$759$	16.8078i	0.610083i
$760$	10.5227i	0.381700i
$761$	−32.2462	−1.16892	−0.584462	−	0.811421i	$-0.698694\pi$
$761$	−32.2462	−1.16892	−0.584462	+	0.811421i	$0.698694\pi$
$762$	1.26137i	0.0456945i
$763$	0	0
$764$	−5.75379	−0.208165
$765$	0	0
$766$	−16.0000	−0.578103
$767$	5.12311	0.184985
$768$	− 10.0540i	− 0.362792i
$769$	−29.5464	−1.06547	−0.532735	−	0.846282i	$-0.678836\pi$
$769$	−29.5464	−1.06547	−0.532735	+	0.846282i	$0.678836\pi$
$770$	0	0
$771$	9.36932i	0.337428i
$772$	10.6307i	0.382607i
$773$	33.3693	1.20021	0.600105	−	0.799921i	$-0.295125\pi$
$773$	33.3693	1.20021	0.600105	+	0.799921i	$0.295125\pi$
$774$	12.0000	0.431331
$775$	− 24.0000i	− 0.862105i
$776$	− 27.1231i	− 0.973663i
$777$	0	0
$778$	−34.1383	−1.22392
$779$	4.31534i	0.154613i
$780$	−1.12311	−0.0402136
$781$	−26.2462	−0.939163
$782$	0	0
$783$	−8.24621	−0.294696
$784$	−32.7926	−1.17116
$785$	3.19224i	0.113936i
$786$	−28.9848	−1.03386
$787$	6.24621i	0.222653i	0.993784	+	0.111327i	$0.0355100\pi$
$787$	6.24621i	0.222653i	−0.993784	+	0.111327i	$0.964490\pi$
$788$	− 8.73863i	− 0.311301i
$789$	12.4924i	0.444742i
$790$	13.4773	0.479500
$791$	0	0
$792$	− 6.24621i	− 0.221949i
$793$	− 4.00000i	− 0.142044i
$794$	8.38447i	0.297554i
$795$	−2.38447	−0.0845685
$796$	7.01515i	0.248646i
$797$	−31.6155	−1.11988	−0.559940	−	0.828533i	$-0.689176\pi$
$797$	−31.6155	−1.11988	−0.559940	+	0.828533i	$0.689176\pi$
$798$	0	0
$799$	0	0
$800$	11.4233	0.403874
$801$	−7.12311	−0.251683
$802$	− 9.64584i	− 0.340606i
$803$	−10.8769	−0.383837
$804$	− 1.75379i	− 0.0618514i
$805$	0	0
$806$	36.4924i	1.28539i
$807$	20.5616	0.723801
$808$	46.6307	1.64046
$809$	− 53.0540i	− 1.86528i	−0.360810	−	0.932639i	$-0.617500\pi$
$809$	− 53.0540i	− 1.86528i	0.360810	−	0.932639i	$-0.382500\pi$
$810$	0.876894i	0.0308109i
$811$	− 20.6307i	− 0.724441i	−0.932092	−	0.362221i	$-0.882019\pi$
$811$	− 20.6307i	− 0.724441i	0.932092	−	0.362221i	$-0.117981\pi$
$812$	0	0
$813$	0.807764i	0.0283295i
$814$	12.4924	0.437859
$815$	3.86174	0.135271
$816$	0	0
$817$	−59.0540	−2.06604
$818$	3.61553	0.126414
$819$	0	0
$820$	−0.138261	−0.00482827
$821$	− 16.5616i	− 0.578002i	−0.957329	−	0.289001i	$-0.906677\pi$
$821$	− 16.5616i	− 0.578002i	0.957329	−	0.289001i	$-0.0933231\pi$
$822$	− 25.3693i	− 0.884857i
$823$	− 36.4924i	− 1.27205i	−0.771670	−	0.636023i	$-0.780578\pi$
$823$	− 36.4924i	− 1.27205i	0.771670	−	0.636023i	$-0.219422\pi$
$824$	10.5227	0.366577
$825$	12.0000	0.417786
$826$	0	0
$827$	14.4233i	0.501547i	0.968046	+	0.250774i	$0.0806849\pi$
$827$	14.4233i	0.501547i	−0.968046	+	0.250774i	$0.919315\pi$
$828$	− 2.87689i	− 0.0999790i
$829$	50.4924	1.75367	0.876837	−	0.480787i	$-0.159649\pi$
$829$	50.4924	1.75367	0.876837	+	0.480787i	$0.159649\pi$
$830$	8.00000i	0.277684i
$831$	6.00000	0.208138
$832$	25.3693	0.879523
$833$	0	0
$834$	14.2462	0.493306
$835$	−0.453602	−0.0156976
$836$	− 8.63068i	− 0.298498i
$837$	5.12311	0.177080
$838$	50.7386i	1.75274i
$839$	− 11.0540i	− 0.381626i	−0.981626	−	0.190813i	$-0.938888\pi$
$839$	− 11.0540i	− 0.381626i	0.981626	−	0.190813i	$-0.0611123\pi$
$840$	0	0
$841$	−39.0000	−1.34483
$842$	−44.6004	−1.53703
$843$	− 19.1231i	− 0.658635i
$844$	4.98485i	0.171585i
$845$	− 4.38447i	− 0.150830i
$846$	−4.49242	−0.154453
$847$	0	0
$848$	−19.8920	−0.683096
$849$	3.36932	0.115635
$850$	0	0
$851$	−20.4924	−0.702471
$852$	4.49242	0.153908
$853$	20.7386i	0.710077i	0.934852	+	0.355039i	$0.115532\pi$
$853$	20.7386i	0.710077i	−0.934852	+	0.355039i	$0.884468\pi$
$854$	0	0
$855$	− 4.31534i	− 0.147582i
$856$	18.7386i	0.640473i
$857$	6.00000i	0.204956i	0.994735	+	0.102478i	$0.0326771\pi$
$857$	6.00000i	0.204956i	−0.994735	+	0.102478i	$0.967323\pi$
$858$	−18.2462	−0.622915
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	− 1.89205i	− 0.0645183i
$861$	0	0
$862$	− 37.4773i	− 1.27648i
$863$	−26.2462	−0.893431	−0.446716	−	0.894676i	$-0.647406\pi$
$863$	−26.2462	−0.893431	−0.446716	+	0.894676i	$0.647406\pi$
$864$	2.43845i	0.0829577i
$865$	−10.5616	−0.359104
$866$	22.3542	0.759625
$867$	0	0
$868$	0	0
$869$	39.3693	1.33551
$870$	7.23106i	0.245156i
$871$	18.2462	0.618249
$872$	36.8769i	1.24881i
$873$	11.1231i	0.376460i
$874$	78.7386i	2.66337i
$875$	0	0
$876$	1.86174	0.0629023
$877$	34.0000i	1.14810i	0.818821	+	0.574049i	$0.194628\pi$
$877$	34.0000i	1.14810i	−0.818821	+	0.574049i	$0.805372\pi$
$878$	8.98485i	0.303224i
$879$	7.12311i	0.240256i
$880$	−6.73863	−0.227159
$881$	23.7538i	0.800285i	0.916453	+	0.400143i	$0.131039\pi$
$881$	23.7538i	0.800285i	−0.916453	+	0.400143i	$0.868961\pi$
$882$	10.9309	0.368062
$883$	38.4233	1.29305	0.646523	−	0.762894i	$-0.276222\pi$
$883$	38.4233	1.29305	0.646523	+	0.762894i	$0.276222\pi$
$884$	0	0
$885$	−0.630683	−0.0212002
$886$	35.7235	1.20015
$887$	22.5616i	0.757543i	0.925490	+	0.378771i	$0.123653\pi$
$887$	22.5616i	0.757543i	−0.925490	+	0.378771i	$0.876347\pi$
$888$	7.61553	0.255560
$889$	0	0
$890$	6.24621i	0.209373i
$891$	2.56155i	0.0858152i
$892$	−6.10795	−0.204509
$893$	22.1080	0.739814
$894$	− 6.63068i	− 0.221763i
$895$	− 5.12311i	− 0.171247i
$896$	0	0
$897$	29.9309	0.999363
$898$	− 19.8920i	− 0.663806i
$899$	42.2462	1.40899
$900$	−2.05398	−0.0684658
$901$	0	0
$902$	−2.24621	−0.0747907
$903$	0	0
$904$	11.1231i	0.369949i
$905$	−3.36932	−0.112000
$906$	− 12.4924i	− 0.415033i
$907$	− 47.8617i	− 1.58922i	−0.607118	−	0.794611i	$-0.707675\pi$
$907$	− 47.8617i	− 1.58922i	0.607118	−	0.794611i	$-0.292325\pi$
$908$	− 10.1080i	− 0.335444i
$909$	−19.1231	−0.634273
$910$	0	0
$911$	29.3002i	0.970758i	0.874304	+	0.485379i	$0.161318\pi$
$911$	29.3002i	0.970758i	−0.874304	+	0.485379i	$0.838682\pi$
$912$	− 36.0000i	− 1.19208i
$913$	23.3693i	0.773412i
$914$	−10.6307	−0.351632
$915$	0.492423i	0.0162790i
$916$	−2.63068	−0.0869202
$917$	0	0
$918$	0	0
$919$	−4.31534	−0.142350	−0.0711750	−	0.997464i	$-0.522675\pi$
$919$	−4.31534	−0.142350	−0.0711750	+	0.997464i	$0.522675\pi$
$920$	8.98485	0.296222
$921$	0.492423i	0.0162259i
$922$	12.8769	0.424078
$923$	46.7386i	1.53842i
$924$	0	0
$925$	14.6307i	0.481054i
$926$	−39.0152	−1.28212
$927$	−4.31534	−0.141734
$928$	20.1080i	0.660076i
$929$	− 31.9309i	− 1.04762i	−0.851836	−	0.523809i	$-0.824511\pi$
$929$	− 31.9309i	− 1.04762i	0.851836	−	0.523809i	$-0.175489\pi$
$930$	− 4.49242i	− 0.147312i
$931$	−53.7926	−1.76298
$932$	0.246211i	0.00806492i
$933$	0	0
$934$	−5.26137	−0.172157
$935$	0	0
$936$	−11.1231	−0.363570
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	7.61553	0.248523
$940$	0.708324i	0.0231030i
$941$	− 30.0000i	− 0.977972i	−0.872292	−	0.488986i	$-0.837367\pi$
$941$	− 30.0000i	− 0.977972i	0.872292	−	0.488986i	$-0.162633\pi$
$942$	− 8.87689i	− 0.289225i
$943$	3.68466	0.119989
$944$	−5.26137	−0.171243
$945$	0	0
$946$	− 30.7386i	− 0.999399i
$947$	12.0000i	0.389948i	0.980808	+	0.194974i	$0.0624622\pi$
$947$	12.0000i	0.389948i	−0.980808	+	0.194974i	$0.937538\pi$
$948$	−6.73863	−0.218861
$949$	19.3693i	0.628755i
$950$	56.2159	1.82388
$951$	18.0000	0.583690
$952$	0	0
$953$	−54.3542	−1.76070	−0.880352	−	0.474321i	$-0.842694\pi$
$953$	−54.3542	−1.76070	−0.880352	+	0.474321i	$0.842694\pi$
$954$	6.63068	0.214676
$955$	7.36932i	0.238465i
$956$	4.49242	0.145295
$957$	21.1231i	0.682813i
$958$	45.7538i	1.47824i
$959$	0	0
$960$	−3.12311	−0.100798
$961$	4.75379	0.153348
$962$	− 22.2462i	− 0.717247i
$963$	− 7.68466i	− 0.247635i
$964$	− 9.36932i	− 0.301765i
$965$	13.6155	0.438299
$966$	0	0
$967$	−46.5616	−1.49732	−0.748659	−	0.662955i	$-0.769302\pi$
$967$	−46.5616	−1.49732	−0.748659	+	0.662955i	$0.769302\pi$
$968$	10.8229	0.347862
$969$	0	0
$970$	9.75379	0.313175
$971$	2.38447	0.0765213	0.0382607	−	0.999268i	$-0.487818\pi$
$971$	2.38447	0.0765213	0.0382607	+	0.999268i	$0.487818\pi$
$972$	− 0.438447i	− 0.0140632i
$973$	0	0
$974$	− 11.5076i	− 0.368727i
$975$	− 21.3693i	− 0.684366i
$976$	4.10795i	0.131492i
$977$	8.24621	0.263820	0.131910	−	0.991262i	$-0.457889\pi$
$977$	8.24621	0.263820	0.131910	+	0.991262i	$0.457889\pi$
$978$	−10.7386	−0.343384
$979$	18.2462i	0.583151i
$980$	− 1.72348i	− 0.0550545i
$981$	− 15.1231i	− 0.482844i
$982$	5.26137	0.167897
$983$	2.06913i	0.0659950i	0.999455	+	0.0329975i	$0.0105053\pi$
$983$	2.06913i	0.0659950i	−0.999455	+	0.0329975i	$0.989495\pi$
$984$	−1.36932	−0.0436522
$985$	−11.1922	−0.356614
$986$	0	0
$987$	0	0
$988$	−15.3693	−0.488963
$989$	50.4233i	1.60337i
$990$	2.24621	0.0713893
$991$	− 6.73863i	− 0.214060i	−0.994256	−	0.107030i	$-0.965866\pi$
$991$	− 6.73863i	− 0.214060i	0.994256	−	0.107030i	$-0.0341341\pi$
$992$	− 12.4924i	− 0.396635i
$993$	− 6.06913i	− 0.192598i
$994$	0	0
$995$	8.98485	0.284839
$996$	− 4.00000i	− 0.126745i
$997$	10.0000i	0.316703i	0.987383	+	0.158352i	$0.0506179\pi$
$997$	10.0000i	0.316703i	−0.987383	+	0.158352i	$0.949382\pi$
$998$	− 17.7538i	− 0.561986i
$999$	−3.12311	−0.0988107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	867.2.d.c.577.1		4
17.2	even	8		867.2.e.f.829.1		8
17.3	odd	16		867.2.h.j.688.2		16
17.4	even	4		51.2.a.b.1.2	✓	2
17.5	odd	16		867.2.h.j.757.4		16
17.6	odd	16		867.2.h.j.712.1		16
17.7	odd	16		867.2.h.j.733.3		16
17.8	even	8		867.2.e.f.616.3		8
17.9	even	8		867.2.e.f.616.4		8
17.10	odd	16		867.2.h.j.733.4		16
17.11	odd	16		867.2.h.j.712.2		16
17.12	odd	16		867.2.h.j.757.3		16
17.13	even	4		867.2.a.f.1.2		2
17.14	odd	16		867.2.h.j.688.1		16
17.15	even	8		867.2.e.f.829.2		8
17.16	even	2	inner	867.2.d.c.577.2		4
51.38	odd	4		153.2.a.e.1.1		2
51.47	odd	4		2601.2.a.t.1.1		2
68.55	odd	4		816.2.a.m.1.1		2
85.4	even	4		1275.2.a.n.1.1		2
85.38	odd	4		1275.2.b.d.1174.2		4
85.72	odd	4		1275.2.b.d.1174.3		4
119.55	odd	4		2499.2.a.o.1.2		2
136.21	even	4		3264.2.a.bl.1.2		2
136.123	odd	4		3264.2.a.bg.1.2		2
187.21	odd	4		6171.2.a.p.1.1		2
204.191	even	4		2448.2.a.v.1.2		2
221.38	even	4		8619.2.a.q.1.1		2
255.89	odd	4		3825.2.a.s.1.2		2
357.293	even	4		7497.2.a.v.1.1		2
408.293	odd	4		9792.2.a.cy.1.1		2
408.395	even	4		9792.2.a.cz.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.a.b.1.2	✓	2	17.4	even	4
153.2.a.e.1.1		2	51.38	odd	4
816.2.a.m.1.1		2	68.55	odd	4
867.2.a.f.1.2		2	17.13	even	4
867.2.d.c.577.1		4	1.1	even	1	trivial
867.2.d.c.577.2		4	17.16	even	2	inner
867.2.e.f.616.3		8	17.8	even	8
867.2.e.f.616.4		8	17.9	even	8
867.2.e.f.829.1		8	17.2	even	8
867.2.e.f.829.2		8	17.15	even	8
867.2.h.j.688.1		16	17.14	odd	16
867.2.h.j.688.2		16	17.3	odd	16
867.2.h.j.712.1		16	17.6	odd	16
867.2.h.j.712.2		16	17.11	odd	16
867.2.h.j.733.3		16	17.7	odd	16
867.2.h.j.733.4		16	17.10	odd	16
867.2.h.j.757.3		16	17.12	odd	16
867.2.h.j.757.4		16	17.5	odd	16
1275.2.a.n.1.1		2	85.4	even	4
1275.2.b.d.1174.2		4	85.38	odd	4
1275.2.b.d.1174.3		4	85.72	odd	4
2448.2.a.v.1.2		2	204.191	even	4
2499.2.a.o.1.2		2	119.55	odd	4
2601.2.a.t.1.1		2	51.47	odd	4
3264.2.a.bg.1.2		2	136.123	odd	4
3264.2.a.bl.1.2		2	136.21	even	4
3825.2.a.s.1.2		2	255.89	odd	4
6171.2.a.p.1.1		2	187.21	odd	4
7497.2.a.v.1.1		2	357.293	even	4
8619.2.a.q.1.1		2	221.38	even	4
9792.2.a.cy.1.1		2	408.293	odd	4
9792.2.a.cz.1.1		2	408.395	even	4

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:	\( N \)	\(=\)	\( 867 = 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	867.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.56155 q^{2} -1.00000i q^{3} +0.438447 q^{4} -0.561553i q^{5} +1.56155i q^{6} +2.43845 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.56155 q^{2} -1.00000i q^{3} +0.438447 q^{4} -0.561553i q^{5} +1.56155i q^{6} +2.43845 q^{8} -1.00000 q^{9} +0.876894i q^{10} +2.56155i q^{11} -0.438447i q^{12} +4.56155 q^{13} -0.561553 q^{15} -4.68466 q^{16} +1.56155 q^{18} -7.68466 q^{19} -0.246211i q^{20} -4.00000i q^{22} +6.56155i q^{23} -2.43845i q^{24} +4.68466 q^{25} -7.12311 q^{26} +1.00000i q^{27} +8.24621i q^{29} +0.876894 q^{30} -5.12311i q^{31} +2.43845 q^{32} +2.56155 q^{33} -0.438447 q^{36} +3.12311i q^{37} +12.0000 q^{38} -4.56155i q^{39} -1.36932i q^{40} -0.561553i q^{41} +7.68466 q^{43} +1.12311i q^{44} +0.561553i q^{45} -10.2462i q^{46} -2.87689 q^{47} +4.68466i q^{48} +7.00000 q^{49} -7.31534 q^{50} +2.00000 q^{52} +4.24621 q^{53} -1.56155i q^{54} +1.43845 q^{55} +7.68466i q^{57} -12.8769i q^{58} +1.12311 q^{59} -0.246211 q^{60} -0.876894i q^{61} +8.00000i q^{62} +5.56155 q^{64} -2.56155i q^{65} -4.00000 q^{66} +4.00000 q^{67} +6.56155 q^{69} +10.2462i q^{71} -2.43845 q^{72} +4.24621i q^{73} -4.87689i q^{74} -4.68466i q^{75} -3.36932 q^{76} +7.12311i q^{78} -15.3693i q^{79} +2.63068i q^{80} +1.00000 q^{81} +0.876894i q^{82} +9.12311 q^{83} -12.0000 q^{86} +8.24621 q^{87} +6.24621i q^{88} +7.12311 q^{89} -0.876894i q^{90} +2.87689i q^{92} -5.12311 q^{93} +4.49242 q^{94} +4.31534i q^{95} -2.43845i q^{96} -11.1231i q^{97} -10.9309 q^{98} -2.56155i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 10 q^{4} + 18 q^{8} - 4 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 + 10 * q^4 + 18 * q^8 - 4 * q^9	\( 4 q + 2 q^{2} + 10 q^{4} + 18 q^{8} - 4 q^{9} + 10 q^{13} + 6 q^{15} + 6 q^{16} - 2 q^{18} - 6 q^{19} - 6 q^{25} - 12 q^{26} + 20 q^{30} + 18 q^{32} + 2 q^{33} - 10 q^{36} + 48 q^{38} + 6 q^{43} - 28 q^{47} + 28 q^{49} - 54 q^{50} + 8 q^{52} - 16 q^{53} + 14 q^{55} - 12 q^{59} + 32 q^{60} + 14 q^{64} - 16 q^{66} + 16 q^{67} + 18 q^{69} - 18 q^{72} + 36 q^{76} + 4 q^{81} + 20 q^{83} - 48 q^{86} + 12 q^{89} - 4 q^{93} - 48 q^{94} + 14 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 10 * q^4 + 18 * q^8 - 4 * q^9 + 10 * q^13 + 6 * q^15 + 6 * q^16 - 2 * q^18 - 6 * q^19 - 6 * q^25 - 12 * q^26 + 20 * q^30 + 18 * q^32 + 2 * q^33 - 10 * q^36 + 48 * q^38 + 6 * q^43 - 28 * q^47 + 28 * q^49 - 54 * q^50 + 8 * q^52 - 16 * q^53 + 14 * q^55 - 12 * q^59 + 32 * q^60 + 14 * q^64 - 16 * q^66 + 16 * q^67 + 18 * q^69 - 18 * q^72 + 36 * q^76 + 4 * q^81 + 20 * q^83 - 48 * q^86 + 12 * q^89 - 4 * q^93 - 48 * q^94 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more