LMFDB - Embedded newform 3483.2.a.t.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3483,2,Mod(1,3483)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3483, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3483.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3483 = 3^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3483.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:	$27.8118950240$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 6 x^{19} - 13 x^{18} + 137 x^{17} - 24 x^{16} - 1247 x^{15} + 1257 x^{14} + 5756 x^{13} + \cdots - 81 $ x^20 - 6x^19 - 13x^18 + 137x^17 - 24x^16 - 1247x^15 + 1257x^14 + 5756x^13 - 8778x^12 - 13888x^11 + 28633x^10 + 15048x^9 - 48400x^8 - 548x^7 + 40321x^6 - 10543x^5 - 13966x^4 + 5450x^3 + 1344x^2 - 441*x - 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 387)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Root			$-1.82281$ of defining polynomial
Character	$\chi$	$=$	3483.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.82281 q^{2} +1.32263 q^{4} -0.432883 q^{5} -0.691058 q^{7} -1.23472 q^{8} +O(q^{10})$	\(q+1.82281 q^{2} +1.32263 q^{4} -0.432883 q^{5} -0.691058 q^{7} -1.23472 q^{8} -0.789063 q^{10} +2.53559 q^{11} -7.20272 q^{13} -1.25967 q^{14} -4.89591 q^{16} +4.46895 q^{17} +6.05843 q^{19} -0.572545 q^{20} +4.62190 q^{22} -4.11156 q^{23} -4.81261 q^{25} -13.1292 q^{26} -0.914014 q^{28} +6.46148 q^{29} -2.53897 q^{31} -6.45487 q^{32} +8.14604 q^{34} +0.299147 q^{35} -9.14340 q^{37} +11.0434 q^{38} +0.534488 q^{40} +0.910760 q^{41} -1.00000 q^{43} +3.35365 q^{44} -7.49459 q^{46} -9.59426 q^{47} -6.52244 q^{49} -8.77247 q^{50} -9.52653 q^{52} -8.85806 q^{53} -1.09761 q^{55} +0.853259 q^{56} +11.7780 q^{58} -13.0226 q^{59} +0.793492 q^{61} -4.62805 q^{62} -1.97418 q^{64} +3.11794 q^{65} +11.3997 q^{67} +5.91077 q^{68} +0.545288 q^{70} -13.2707 q^{71} -4.01445 q^{73} -16.6667 q^{74} +8.01307 q^{76} -1.75224 q^{77} -11.8858 q^{79} +2.11936 q^{80} +1.66014 q^{82} -9.76179 q^{83} -1.93454 q^{85} -1.82281 q^{86} -3.13073 q^{88} +8.04947 q^{89} +4.97749 q^{91} -5.43807 q^{92} -17.4885 q^{94} -2.62259 q^{95} +14.9889 q^{97} -11.8892 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 6 q^{2} + 22 q^{4} - 17 q^{5} + 3 q^{7} - 15 q^{8}+O(q^{10}) $ 20 * q - 6 * q^2 + 22 * q^4 - 17 * q^5 + 3 * q^7 - 15 * q^8	$ 20 q - 6 q^{2} + 22 q^{4} - 17 q^{5} + 3 q^{7} - 15 q^{8} - 2 q^{10} - 10 q^{11} - q^{13} - 10 q^{14} + 22 q^{16} - 20 q^{17} + 8 q^{19} - 30 q^{20} + 15 q^{22} - 19 q^{23} + 19 q^{25} - 25 q^{26} - 3 q^{28} - 25 q^{29} - 11 q^{31} - 36 q^{32} + 9 q^{34} + 9 q^{37} - 28 q^{38} + 12 q^{40} - 12 q^{41} - 20 q^{43} - 5 q^{44} + 4 q^{46} - 38 q^{47} + 37 q^{49} - 36 q^{50} - 8 q^{52} - 69 q^{53} - 9 q^{55} - 30 q^{56} - 27 q^{58} - 31 q^{59} + 19 q^{61} - 32 q^{62} + 11 q^{64} - 47 q^{65} + 9 q^{67} - 68 q^{68} - 6 q^{70} - 21 q^{71} - 2 q^{73} + 16 q^{74} + 37 q^{76} - 85 q^{77} - 4 q^{79} - 61 q^{80} + q^{82} - 19 q^{83} - 6 q^{85} + 6 q^{86} + 60 q^{88} - 54 q^{89} - 3 q^{91} - 85 q^{92} - 19 q^{94} + 11 q^{95} + 2 q^{97} - 5 q^{98}+O(q^{100}) $ 20 * q - 6 * q^2 + 22 * q^4 - 17 * q^5 + 3 * q^7 - 15 * q^8 - 2 * q^10 - 10 * q^11 - q^13 - 10 * q^14 + 22 * q^16 - 20 * q^17 + 8 * q^19 - 30 * q^20 + 15 * q^22 - 19 * q^23 + 19 * q^25 - 25 * q^26 - 3 * q^28 - 25 * q^29 - 11 * q^31 - 36 * q^32 + 9 * q^34 + 9 * q^37 - 28 * q^38 + 12 * q^40 - 12 * q^41 - 20 * q^43 - 5 * q^44 + 4 * q^46 - 38 * q^47 + 37 * q^49 - 36 * q^50 - 8 * q^52 - 69 * q^53 - 9 * q^55 - 30 * q^56 - 27 * q^58 - 31 * q^59 + 19 * q^61 - 32 * q^62 + 11 * q^64 - 47 * q^65 + 9 * q^67 - 68 * q^68 - 6 * q^70 - 21 * q^71 - 2 * q^73 + 16 * q^74 + 37 * q^76 - 85 * q^77 - 4 * q^79 - 61 * q^80 + q^82 - 19 * q^83 - 6 * q^85 + 6 * q^86 + 60 * q^88 - 54 * q^89 - 3 * q^91 - 85 * q^92 - 19 * q^94 + 11 * q^95 + 2 * q^97 - 5 * 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 $

$2$	1.82281	1.28892	0.644460	−	0.764638i	$-0.277082\pi$
$2$	1.82281	1.28892	0.644460	+	0.764638i	$0.277082\pi$
$3$	0	0
$3$	0	0
$4$	1.32263	0.661315
$5$	−0.432883	−0.193591	−0.0967957	−	0.995304i	$-0.530859\pi$
$5$	−0.432883	−0.193591	−0.0967957	+	0.995304i	$0.530859\pi$
$6$	0	0
$7$	−0.691058	−0.261195	−0.130598	−	0.991435i	$-0.541690\pi$
$7$	−0.691058	−0.261195	−0.130598	+	0.991435i	$0.541690\pi$
$8$	−1.23472	−0.436538
$9$	0	0
$10$	−0.789063	−0.249524
$11$	2.53559	0.764509	0.382255	−	0.924057i	$-0.375148\pi$
$11$	2.53559	0.764509	0.382255	+	0.924057i	$0.375148\pi$
$12$	0	0
$13$	−7.20272	−1.99767	−0.998837	−	0.0482173i	$-0.984646\pi$
$13$	−7.20272	−1.99767	−0.998837	+	0.0482173i	$0.984646\pi$
$14$	−1.25967	−0.336660
$15$	0	0
$16$	−4.89591	−1.22398
$17$	4.46895	1.08388	0.541940	−	0.840417i	$-0.317690\pi$
$17$	4.46895	1.08388	0.541940	+	0.840417i	$0.317690\pi$
$18$	0	0
$19$	6.05843	1.38990	0.694950	−	0.719058i	$-0.255427\pi$
$19$	6.05843	1.38990	0.694950	+	0.719058i	$0.255427\pi$
$20$	−0.572545	−0.128025
$21$	0	0
$22$	4.62190	0.985391
$23$	−4.11156	−0.857319	−0.428660	−	0.903466i	$-0.641014\pi$
$23$	−4.11156	−0.857319	−0.428660	+	0.903466i	$0.641014\pi$
$24$	0	0
$25$	−4.81261	−0.962522
$26$	−13.1292	−2.57484
$27$	0	0
$28$	−0.914014	−0.172732
$29$	6.46148	1.19987	0.599933	−	0.800050i	$-0.295194\pi$
$29$	6.46148	1.19987	0.599933	+	0.800050i	$0.295194\pi$
$30$	0	0
$31$	−2.53897	−0.456012	−0.228006	−	0.973660i	$-0.573221\pi$
$31$	−2.53897	−0.456012	−0.228006	+	0.973660i	$0.573221\pi$
$32$	−6.45487	−1.14107
$33$	0	0
$34$	8.14604	1.39704
$35$	0.299147	0.0505651
$36$	0	0
$37$	−9.14340	−1.50317	−0.751583	−	0.659638i	$-0.770709\pi$
$37$	−9.14340	−1.50317	−0.751583	+	0.659638i	$0.770709\pi$
$38$	11.0434	1.79147
$39$	0	0
$40$	0.534488	0.0845099
$41$	0.910760	0.142237	0.0711184	−	0.997468i	$-0.477343\pi$
$41$	0.910760	0.142237	0.0711184	+	0.997468i	$0.477343\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	3.35365	0.505582
$45$	0	0
$46$	−7.49459	−1.10502
$47$	−9.59426	−1.39947	−0.699733	−	0.714404i	$-0.746698\pi$
$47$	−9.59426	−1.39947	−0.699733	+	0.714404i	$0.746698\pi$
$48$	0	0
$49$	−6.52244	−0.931777
$50$	−8.77247	−1.24061
$51$	0	0
$52$	−9.52653	−1.32109
$53$	−8.85806	−1.21675	−0.608374	−	0.793650i	$-0.708178\pi$
$53$	−8.85806	−1.21675	−0.608374	+	0.793650i	$0.708178\pi$
$54$	0	0
$55$	−1.09761	−0.148002
$56$	0.853259	0.114022
$57$	0	0
$58$	11.7780	1.54653
$59$	−13.0226	−1.69540	−0.847699	−	0.530478i	$-0.822013\pi$
$59$	−13.0226	−1.69540	−0.847699	+	0.530478i	$0.822013\pi$
$60$	0	0
$61$	0.793492	0.101596	0.0507981	−	0.998709i	$-0.483824\pi$
$61$	0.793492	0.101596	0.0507981	+	0.998709i	$0.483824\pi$
$62$	−4.62805	−0.587763
$63$	0	0
$64$	−1.97418	−0.246772
$65$	3.11794	0.386732
$66$	0	0
$67$	11.3997	1.39270	0.696349	−	0.717704i	$-0.254807\pi$
$67$	11.3997	1.39270	0.696349	+	0.717704i	$0.254807\pi$
$68$	5.91077	0.716786
$69$	0	0
$70$	0.545288	0.0651744
$71$	−13.2707	−1.57495	−0.787473	−	0.616349i	$-0.788611\pi$
$71$	−13.2707	−1.57495	−0.787473	+	0.616349i	$0.788611\pi$
$72$	0	0
$73$	−4.01445	−0.469856	−0.234928	−	0.972013i	$-0.575485\pi$
$73$	−4.01445	−0.469856	−0.234928	+	0.972013i	$0.575485\pi$
$74$	−16.6667	−1.93746
$75$	0	0
$76$	8.01307	0.919162
$77$	−1.75224	−0.199686
$78$	0	0
$79$	−11.8858	−1.33726	−0.668628	−	0.743597i	$-0.733118\pi$
$79$	−11.8858	−1.33726	−0.668628	+	0.743597i	$0.733118\pi$
$80$	2.11936	0.236951
$81$	0	0
$82$	1.66014	0.183332
$83$	−9.76179	−1.07150	−0.535748	−	0.844378i	$-0.679970\pi$
$83$	−9.76179	−1.07150	−0.535748	+	0.844378i	$0.679970\pi$
$84$	0	0
$85$	−1.93454	−0.209830
$86$	−1.82281	−0.196558
$87$	0	0
$88$	−3.13073	−0.333737
$89$	8.04947	0.853242	0.426621	−	0.904431i	$-0.359704\pi$
$89$	8.04947	0.853242	0.426621	+	0.904431i	$0.359704\pi$
$90$	0	0
$91$	4.97749	0.521783
$92$	−5.43807	−0.566958
$93$	0	0
$94$	−17.4885	−1.80380
$95$	−2.62259	−0.269073
$96$	0	0
$97$	14.9889	1.52189	0.760947	−	0.648815i	$-0.224735\pi$
$97$	14.9889	1.52189	0.760947	+	0.648815i	$0.224735\pi$
$98$	−11.8892	−1.20099
$99$	0	0
$100$	−6.36531	−0.636531
$101$	−3.10620	−0.309078	−0.154539	−	0.987987i	$-0.549389\pi$
$101$	−3.10620	−0.309078	−0.154539	+	0.987987i	$0.549389\pi$
$102$	0	0
$103$	6.96258	0.686044	0.343022	−	0.939327i	$-0.388550\pi$
$103$	6.96258	0.686044	0.343022	+	0.939327i	$0.388550\pi$
$104$	8.89330	0.872060
$105$	0	0
$106$	−16.1466	−1.56829
$107$	8.10561	0.783599	0.391800	−	0.920051i	$-0.371853\pi$
$107$	8.10561	0.783599	0.391800	+	0.920051i	$0.371853\pi$
$108$	0	0
$109$	−0.0751888	−0.00720178	−0.00360089	−	0.999994i	$-0.501146\pi$
$109$	−0.0751888	−0.00720178	−0.00360089	+	0.999994i	$0.501146\pi$
$110$	−2.00074	−0.190763
$111$	0	0
$112$	3.38336	0.319697
$113$	7.68859	0.723282	0.361641	−	0.932317i	$-0.382217\pi$
$113$	7.68859	0.723282	0.361641	+	0.932317i	$0.382217\pi$
$114$	0	0
$115$	1.77983	0.165970
$116$	8.54615	0.793490
$117$	0	0
$118$	−23.7377	−2.18523
$119$	−3.08830	−0.283104
$120$	0	0
$121$	−4.57078	−0.415526
$122$	1.44638	0.130949
$123$	0	0
$124$	−3.35811	−0.301567
$125$	4.24772	0.379927
$126$	0	0
$127$	0.411999	0.0365590	0.0182795	−	0.999833i	$-0.494181\pi$
$127$	0.411999	0.0365590	0.0182795	+	0.999833i	$0.494181\pi$
$128$	9.31120	0.823001
$129$	0	0
$130$	5.68340	0.498467
$131$	3.03110	0.264828	0.132414	−	0.991194i	$-0.457727\pi$
$131$	3.03110	0.264828	0.132414	+	0.991194i	$0.457727\pi$
$132$	0	0
$133$	−4.18673	−0.363035
$134$	20.7795	1.79508
$135$	0	0
$136$	−5.51788	−0.473155
$137$	−10.6888	−0.913209	−0.456604	−	0.889670i	$-0.650935\pi$
$137$	−10.6888	−0.913209	−0.456604	+	0.889670i	$0.650935\pi$
$138$	0	0
$139$	−22.7296	−1.92790	−0.963951	−	0.266080i	$-0.914271\pi$
$139$	−22.7296	−1.92790	−0.963951	+	0.266080i	$0.914271\pi$
$140$	0.395661	0.0334395
$141$	0	0
$142$	−24.1900	−2.02998
$143$	−18.2631	−1.52724
$144$	0	0
$145$	−2.79707	−0.232284
$146$	−7.31757	−0.605607
$147$	0	0
$148$	−12.0933	−0.994066
$149$	14.9854	1.22765	0.613824	−	0.789443i	$-0.289631\pi$
$149$	14.9854	1.22765	0.613824	+	0.789443i	$0.289631\pi$
$150$	0	0
$151$	8.03500	0.653879	0.326939	−	0.945045i	$-0.393983\pi$
$151$	8.03500	0.653879	0.326939	+	0.945045i	$0.393983\pi$
$152$	−7.48044	−0.606744
$153$	0	0
$154$	−3.19400	−0.257380
$155$	1.09908	0.0882799
$156$	0	0
$157$	6.89260	0.550089	0.275045	−	0.961431i	$-0.411307\pi$
$157$	6.89260	0.550089	0.275045	+	0.961431i	$0.411307\pi$
$158$	−21.6655	−1.72362
$159$	0	0
$160$	2.79421	0.220902
$161$	2.84132	0.223928
$162$	0	0
$163$	6.87193	0.538251	0.269125	−	0.963105i	$-0.413265\pi$
$163$	6.87193	0.538251	0.269125	+	0.963105i	$0.413265\pi$
$164$	1.20460	0.0940634
$165$	0	0
$166$	−17.7939	−1.38107
$167$	15.5912	1.20649	0.603243	−	0.797557i	$-0.293875\pi$
$167$	15.5912	1.20649	0.603243	+	0.797557i	$0.293875\pi$
$168$	0	0
$169$	38.8791	2.99070
$170$	−3.52629	−0.270454
$171$	0	0
$172$	−1.32263	−0.100850
$173$	−3.07147	−0.233519	−0.116760	−	0.993160i	$-0.537251\pi$
$173$	−3.07147	−0.233519	−0.116760	+	0.993160i	$0.537251\pi$
$174$	0	0
$175$	3.32579	0.251406
$176$	−12.4140	−0.935742
$177$	0	0
$178$	14.6726	1.09976
$179$	6.81694	0.509522	0.254761	−	0.967004i	$-0.418003\pi$
$179$	6.81694	0.509522	0.254761	+	0.967004i	$0.418003\pi$
$180$	0	0
$181$	−6.26156	−0.465418	−0.232709	−	0.972546i	$-0.574759\pi$
$181$	−6.26156	−0.465418	−0.232709	+	0.972546i	$0.574759\pi$
$182$	9.07301	0.672536
$183$	0	0
$184$	5.07661	0.374252
$185$	3.95803	0.291000
$186$	0	0
$187$	11.3314	0.828636
$188$	−12.6897	−0.925488
$189$	0	0
$190$	−4.78049	−0.346813
$191$	−9.78970	−0.708358	−0.354179	−	0.935178i	$-0.615240\pi$
$191$	−9.78970	−0.708358	−0.354179	+	0.935178i	$0.615240\pi$
$192$	0	0
$193$	6.45556	0.464681	0.232341	−	0.972635i	$-0.425362\pi$
$193$	6.45556	0.464681	0.232341	+	0.972635i	$0.425362\pi$
$194$	27.3219	1.96160
$195$	0	0
$196$	−8.62678	−0.616198
$197$	−9.52049	−0.678307	−0.339154	−	0.940731i	$-0.610141\pi$
$197$	−9.52049	−0.678307	−0.339154	+	0.940731i	$0.610141\pi$
$198$	0	0
$199$	16.8531	1.19469	0.597344	−	0.801985i	$-0.296223\pi$
$199$	16.8531	1.19469	0.597344	+	0.801985i	$0.296223\pi$
$200$	5.94221	0.420177
$201$	0	0
$202$	−5.66200	−0.398377
$203$	−4.46526	−0.313400
$204$	0	0
$205$	−0.394253	−0.0275358
$206$	12.6915	0.884256
$207$	0	0
$208$	35.2638	2.44511
$209$	15.3617	1.06259
$210$	0	0
$211$	1.31253	0.0903583	0.0451792	−	0.998979i	$-0.485614\pi$
$211$	1.31253	0.0903583	0.0451792	+	0.998979i	$0.485614\pi$
$212$	−11.7159	−0.804654
$213$	0	0
$214$	14.7750	1.01000
$215$	0.432883	0.0295224
$216$	0	0
$217$	1.75457	0.119108
$218$	−0.137055	−0.00928252
$219$	0	0
$220$	−1.45174	−0.0978762
$221$	−32.1886	−2.16524
$222$	0	0
$223$	−1.89997	−0.127231	−0.0636157	−	0.997974i	$-0.520263\pi$
$223$	−1.89997	−0.127231	−0.0636157	+	0.997974i	$0.520263\pi$
$224$	4.46069	0.298042
$225$	0	0
$226$	14.0148	0.932252
$227$	−12.7393	−0.845535	−0.422767	−	0.906238i	$-0.638941\pi$
$227$	−12.7393	−0.845535	−0.422767	+	0.906238i	$0.638941\pi$
$228$	0	0
$229$	23.7441	1.56905	0.784526	−	0.620096i	$-0.212906\pi$
$229$	23.7441	1.56905	0.784526	+	0.620096i	$0.212906\pi$
$230$	3.24428	0.213922
$231$	0	0
$232$	−7.97809	−0.523787
$233$	−7.78814	−0.510218	−0.255109	−	0.966912i	$-0.582111\pi$
$233$	−7.78814	−0.510218	−0.255109	+	0.966912i	$0.582111\pi$
$234$	0	0
$235$	4.15319	0.270925
$236$	−17.2241	−1.12119
$237$	0	0
$238$	−5.62939	−0.364899
$239$	−3.41301	−0.220769	−0.110385	−	0.993889i	$-0.535208\pi$
$239$	−3.41301	−0.220769	−0.110385	+	0.993889i	$0.535208\pi$
$240$	0	0
$241$	−12.3354	−0.794593	−0.397296	−	0.917690i	$-0.630051\pi$
$241$	−12.3354	−0.794593	−0.397296	+	0.917690i	$0.630051\pi$
$242$	−8.33166	−0.535579
$243$	0	0
$244$	1.04950	0.0671871
$245$	2.82346	0.180384
$246$	0	0
$247$	−43.6372	−2.77657
$248$	3.13490	0.199066
$249$	0	0
$250$	7.74277	0.489696
$251$	7.32241	0.462186	0.231093	−	0.972932i	$-0.425770\pi$
$251$	7.32241	0.462186	0.231093	+	0.972932i	$0.425770\pi$
$252$	0	0
$253$	−10.4252	−0.655429
$254$	0.750996	0.0471217
$255$	0	0
$256$	20.9209	1.30756
$257$	4.99494	0.311576	0.155788	−	0.987791i	$-0.450208\pi$
$257$	4.99494	0.311576	0.155788	+	0.987791i	$0.450208\pi$
$258$	0	0
$259$	6.31862	0.392620
$260$	4.12388	0.255752
$261$	0	0
$262$	5.52511	0.341342
$263$	−18.1100	−1.11671	−0.558355	−	0.829602i	$-0.688567\pi$
$263$	−18.1100	−1.11671	−0.558355	+	0.829602i	$0.688567\pi$
$264$	0	0
$265$	3.83451	0.235552
$266$	−7.63160	−0.467923
$267$	0	0
$268$	15.0776	0.921012
$269$	12.3502	0.753004	0.376502	−	0.926416i	$-0.377127\pi$
$269$	12.3502	0.753004	0.376502	+	0.926416i	$0.377127\pi$
$270$	0	0
$271$	−11.1352	−0.676418	−0.338209	−	0.941071i	$-0.609821\pi$
$271$	−11.1352	−0.676418	−0.338209	+	0.941071i	$0.609821\pi$
$272$	−21.8796	−1.32664
$273$	0	0
$274$	−19.4837	−1.17705
$275$	−12.2028	−0.735857
$276$	0	0
$277$	15.8511	0.952399	0.476200	−	0.879337i	$-0.342014\pi$
$277$	15.8511	0.952399	0.476200	+	0.879337i	$0.342014\pi$
$278$	−41.4318	−2.48491
$279$	0	0
$280$	−0.369362	−0.0220736
$281$	−8.89152	−0.530424	−0.265212	−	0.964190i	$-0.585442\pi$
$281$	−8.89152	−0.530424	−0.265212	+	0.964190i	$0.585442\pi$
$282$	0	0
$283$	−7.86993	−0.467819	−0.233909	−	0.972258i	$-0.575152\pi$
$283$	−7.86993	−0.467819	−0.233909	+	0.972258i	$0.575152\pi$
$284$	−17.5523	−1.04154
$285$	0	0
$286$	−33.2902	−1.96849
$287$	−0.629388	−0.0371516
$288$	0	0
$289$	2.97154	0.174796
$290$	−5.09852	−0.299395
$291$	0	0
$292$	−5.30963	−0.310723
$293$	−33.5108	−1.95772	−0.978859	−	0.204534i	$-0.934432\pi$
$293$	−33.5108	−1.95772	−0.978859	+	0.204534i	$0.934432\pi$
$294$	0	0
$295$	5.63727	0.328214
$296$	11.2895	0.656189
$297$	0	0
$298$	27.3154	1.58234
$299$	29.6144	1.71264
$300$	0	0
$301$	0.691058	0.0398319
$302$	14.6463	0.842798
$303$	0	0
$304$	−29.6615	−1.70121
$305$	−0.343489	−0.0196681
$306$	0	0
$307$	12.9225	0.737524	0.368762	−	0.929524i	$-0.379782\pi$
$307$	12.9225	0.737524	0.368762	+	0.929524i	$0.379782\pi$
$308$	−2.31756	−0.132055
$309$	0	0
$310$	2.00340	0.113786
$311$	−13.5621	−0.769036	−0.384518	−	0.923117i	$-0.625632\pi$
$311$	−13.5621	−0.769036	−0.384518	+	0.923117i	$0.625632\pi$
$312$	0	0
$313$	−23.3819	−1.32162	−0.660812	−	0.750552i	$-0.729788\pi$
$313$	−23.3819	−1.32162	−0.660812	+	0.750552i	$0.729788\pi$
$314$	12.5639	0.709021
$315$	0	0
$316$	−15.7205	−0.884348
$317$	1.61143	0.0905067	0.0452533	−	0.998976i	$-0.485590\pi$
$317$	1.61143	0.0905067	0.0452533	+	0.998976i	$0.485590\pi$
$318$	0	0
$319$	16.3837	0.917309
$320$	0.854590	0.0477730
$321$	0	0
$322$	5.17919	0.288625
$323$	27.0748	1.50648
$324$	0	0
$325$	34.6639	1.92281
$326$	12.5262	0.693762
$327$	0	0
$328$	−1.12453	−0.0620917
$329$	6.63019	0.365534
$330$	0	0
$331$	34.2553	1.88284	0.941422	−	0.337232i	$-0.109491\pi$
$331$	34.2553	1.88284	0.941422	+	0.337232i	$0.109491\pi$
$332$	−12.9112	−0.708597
$333$	0	0
$334$	28.4198	1.55506
$335$	−4.93475	−0.269614
$336$	0	0
$337$	15.3568	0.836539	0.418269	−	0.908323i	$-0.362637\pi$
$337$	15.3568	0.836539	0.418269	+	0.908323i	$0.362637\pi$
$338$	70.8692	3.85477
$339$	0	0
$340$	−2.55867	−0.138764
$341$	−6.43778	−0.348625
$342$	0	0
$343$	9.34478	0.504571
$344$	1.23472	0.0665714
$345$	0	0
$346$	−5.59869	−0.300988
$347$	−1.93444	−0.103846	−0.0519232	−	0.998651i	$-0.516535\pi$
$347$	−1.93444	−0.103846	−0.0519232	+	0.998651i	$0.516535\pi$
$348$	0	0
$349$	5.58879	0.299161	0.149580	−	0.988750i	$-0.452208\pi$
$349$	5.58879	0.299161	0.149580	+	0.988750i	$0.452208\pi$
$350$	6.06228	0.324043
$351$	0	0
$352$	−16.3669	−0.872360
$353$	−4.53839	−0.241554	−0.120777	−	0.992680i	$-0.538539\pi$
$353$	−4.53839	−0.241554	−0.120777	+	0.992680i	$0.538539\pi$
$354$	0	0
$355$	5.74468	0.304896
$356$	10.6465	0.564262
$357$	0	0
$358$	12.4260	0.656733
$359$	−11.5921	−0.611809	−0.305904	−	0.952062i	$-0.598959\pi$
$359$	−11.5921	−0.611809	−0.305904	+	0.952062i	$0.598959\pi$
$360$	0	0
$361$	17.7046	0.931821
$362$	−11.4136	−0.599887
$363$	0	0
$364$	6.58338	0.345063
$365$	1.73779	0.0909600
$366$	0	0
$367$	−7.23499	−0.377663	−0.188832	−	0.982009i	$-0.560470\pi$
$367$	−7.23499	−0.377663	−0.188832	+	0.982009i	$0.560470\pi$
$368$	20.1298	1.04934
$369$	0	0
$370$	7.21472	0.375076
$371$	6.12143	0.317809
$372$	0	0
$373$	11.7599	0.608906	0.304453	−	0.952527i	$-0.401526\pi$
$373$	11.7599	0.608906	0.304453	+	0.952527i	$0.401526\pi$
$374$	20.6550	1.06805
$375$	0	0
$376$	11.8462	0.610920
$377$	−46.5402	−2.39694
$378$	0	0
$379$	−5.11108	−0.262539	−0.131269	−	0.991347i	$-0.541905\pi$
$379$	−5.11108	−0.262539	−0.131269	+	0.991347i	$0.541905\pi$
$380$	−3.46872	−0.177942
$381$	0	0
$382$	−17.8447	−0.913017
$383$	16.1519	0.825321	0.412661	−	0.910885i	$-0.364600\pi$
$383$	16.1519	0.825321	0.412661	+	0.910885i	$0.364600\pi$
$384$	0	0
$385$	0.758515	0.0386575
$386$	11.7672	0.598937
$387$	0	0
$388$	19.8248	1.00645
$389$	17.2323	0.873714	0.436857	−	0.899531i	$-0.356092\pi$
$389$	17.2323	0.873714	0.436857	+	0.899531i	$0.356092\pi$
$390$	0	0
$391$	−18.3744	−0.929232
$392$	8.05336	0.406756
$393$	0	0
$394$	−17.3540	−0.874284
$395$	5.14516	0.258881
$396$	0	0
$397$	−29.6151	−1.48634	−0.743170	−	0.669103i	$-0.766679\pi$
$397$	−29.6151	−1.48634	−0.743170	+	0.669103i	$0.766679\pi$
$398$	30.7201	1.53986
$399$	0	0
$400$	23.5621	1.17811
$401$	−14.3050	−0.714358	−0.357179	−	0.934036i	$-0.616261\pi$
$401$	−14.3050	−0.714358	−0.357179	+	0.934036i	$0.616261\pi$
$402$	0	0
$403$	18.2874	0.910962
$404$	−4.10835	−0.204398
$405$	0	0
$406$	−8.13931	−0.403947
$407$	−23.1839	−1.14918
$408$	0	0
$409$	16.8663	0.833983	0.416992	−	0.908910i	$-0.363084\pi$
$409$	16.8663	0.833983	0.416992	+	0.908910i	$0.363084\pi$
$410$	−0.718647	−0.0354915
$411$	0	0
$412$	9.20892	0.453691
$413$	8.99937	0.442830
$414$	0	0
$415$	4.22572	0.207432
$416$	46.4926	2.27949
$417$	0	0
$418$	28.0014	1.36960
$419$	−1.44141	−0.0704176	−0.0352088	−	0.999380i	$-0.511210\pi$
$419$	−1.44141	−0.0704176	−0.0352088	+	0.999380i	$0.511210\pi$
$420$	0	0
$421$	−1.38698	−0.0675972	−0.0337986	−	0.999429i	$-0.510760\pi$
$421$	−1.38698	−0.0675972	−0.0337986	+	0.999429i	$0.510760\pi$
$422$	2.39249	0.116465
$423$	0	0
$424$	10.9372	0.531157
$425$	−21.5073	−1.04326
$426$	0	0
$427$	−0.548348	−0.0265364
$428$	10.7207	0.518206
$429$	0	0
$430$	0.789063	0.0380520
$431$	−19.8758	−0.957383	−0.478692	−	0.877983i	$-0.658889\pi$
$431$	−19.8758	−0.957383	−0.478692	+	0.877983i	$0.658889\pi$
$432$	0	0
$433$	−3.09155	−0.148570	−0.0742851	−	0.997237i	$-0.523667\pi$
$433$	−3.09155	−0.148570	−0.0742851	+	0.997237i	$0.523667\pi$
$434$	3.19825	0.153521
$435$	0	0
$436$	−0.0994470	−0.00476265
$437$	−24.9096	−1.19159
$438$	0	0
$439$	8.90197	0.424868	0.212434	−	0.977175i	$-0.431861\pi$
$439$	8.90197	0.424868	0.212434	+	0.977175i	$0.431861\pi$
$440$	1.35524	0.0646086
$441$	0	0
$442$	−58.6736	−2.79082
$443$	−10.9765	−0.521510	−0.260755	−	0.965405i	$-0.583971\pi$
$443$	−10.9765	−0.521510	−0.260755	+	0.965405i	$0.583971\pi$
$444$	0	0
$445$	−3.48448	−0.165180
$446$	−3.46328	−0.163991
$447$	0	0
$448$	1.36427	0.0644558
$449$	18.7937	0.886931	0.443465	−	0.896292i	$-0.353749\pi$
$449$	18.7937	0.886931	0.443465	+	0.896292i	$0.353749\pi$
$450$	0	0
$451$	2.30931	0.108741
$452$	10.1692	0.478317
$453$	0	0
$454$	−23.2212	−1.08983
$455$	−2.15467	−0.101013
$456$	0	0
$457$	19.7161	0.922279	0.461140	−	0.887328i	$-0.347441\pi$
$457$	19.7161	0.922279	0.461140	+	0.887328i	$0.347441\pi$
$458$	43.2809	2.02238
$459$	0	0
$460$	2.35405	0.109758
$461$	−40.1745	−1.87111	−0.935556	−	0.353177i	$-0.885101\pi$
$461$	−40.1745	−1.87111	−0.935556	+	0.353177i	$0.885101\pi$
$462$	0	0
$463$	22.2931	1.03605	0.518024	−	0.855366i	$-0.326668\pi$
$463$	22.2931	1.03605	0.518024	+	0.855366i	$0.326668\pi$
$464$	−31.6348	−1.46861
$465$	0	0
$466$	−14.1963	−0.657630
$467$	2.81433	0.130232	0.0651158	−	0.997878i	$-0.479258\pi$
$467$	2.81433	0.130232	0.0651158	+	0.997878i	$0.479258\pi$
$468$	0	0
$469$	−7.87786	−0.363766
$470$	7.57048	0.349200
$471$	0	0
$472$	16.0792	0.740105
$473$	−2.53559	−0.116587
$474$	0	0
$475$	−29.1569	−1.33781
$476$	−4.08468	−0.187221
$477$	0	0
$478$	−6.22126	−0.284554
$479$	15.6891	0.716855	0.358428	−	0.933558i	$-0.383313\pi$
$479$	15.6891	0.716855	0.358428	+	0.933558i	$0.383313\pi$
$480$	0	0
$481$	65.8573	3.00284
$482$	−22.4851	−1.02417
$483$	0	0
$484$	−6.04545	−0.274793
$485$	−6.48845	−0.294625
$486$	0	0
$487$	−8.31905	−0.376972	−0.188486	−	0.982076i	$-0.560358\pi$
$487$	−8.31905	−0.376972	−0.188486	+	0.982076i	$0.560358\pi$
$488$	−0.979736	−0.0443506
$489$	0	0
$490$	5.14662	0.232501
$491$	21.4599	0.968474	0.484237	−	0.874937i	$-0.339097\pi$
$491$	21.4599	0.968474	0.484237	+	0.874937i	$0.339097\pi$
$492$	0	0
$493$	28.8761	1.30051
$494$	−79.5422	−3.57877
$495$	0	0
$496$	12.4305	0.558148
$497$	9.17085	0.411369
$498$	0	0
$499$	−8.86371	−0.396794	−0.198397	−	0.980122i	$-0.563574\pi$
$499$	−8.86371	−0.396794	−0.198397	+	0.980122i	$0.563574\pi$
$500$	5.61816	0.251252
$501$	0	0
$502$	13.3473	0.595721
$503$	−18.4384	−0.822126	−0.411063	−	0.911607i	$-0.634842\pi$
$503$	−18.4384	−0.822126	−0.411063	+	0.911607i	$0.634842\pi$
$504$	0	0
$505$	1.34462	0.0598349
$506$	−19.0032	−0.844795
$507$	0	0
$508$	0.544923	0.0241770
$509$	−5.39526	−0.239141	−0.119570	−	0.992826i	$-0.538152\pi$
$509$	−5.39526	−0.239141	−0.119570	+	0.992826i	$0.538152\pi$
$510$	0	0
$511$	2.77422	0.122724
$512$	19.5124	0.862333
$513$	0	0
$514$	9.10483	0.401597
$515$	−3.01399	−0.132812
$516$	0	0
$517$	−24.3271	−1.06991
$518$	11.5176	0.506056
$519$	0	0
$520$	−3.84976	−0.168823
$521$	34.0231	1.49058	0.745290	−	0.666741i	$-0.232311\pi$
$521$	34.0231	1.49058	0.745290	+	0.666741i	$0.232311\pi$
$522$	0	0
$523$	17.5996	0.769576	0.384788	−	0.923005i	$-0.374275\pi$
$523$	17.5996	0.769576	0.384788	+	0.923005i	$0.374275\pi$
$524$	4.00902	0.175135
$525$	0	0
$526$	−33.0110	−1.43935
$527$	−11.3465	−0.494262
$528$	0	0
$529$	−6.09508	−0.265003
$530$	6.98957	0.303608
$531$	0	0
$532$	−5.53749	−0.240081
$533$	−6.55994	−0.284143
$534$	0	0
$535$	−3.50879	−0.151698
$536$	−14.0754	−0.607965
$537$	0	0
$538$	22.5120	0.970562
$539$	−16.5382	−0.712352
$540$	0	0
$541$	19.6744	0.845868	0.422934	−	0.906160i	$-0.361000\pi$
$541$	19.6744	0.845868	0.422934	+	0.906160i	$0.361000\pi$
$542$	−20.2974	−0.871848
$543$	0	0
$544$	−28.8465	−1.23678
$545$	0.0325480	0.00139420
$546$	0	0
$547$	−14.3233	−0.612421	−0.306211	−	0.951964i	$-0.599061\pi$
$547$	−14.3233	−0.612421	−0.306211	+	0.951964i	$0.599061\pi$
$548$	−14.1374	−0.603919
$549$	0	0
$550$	−22.2434	−0.948461
$551$	39.1464	1.66769
$552$	0	0
$553$	8.21377	0.349285
$554$	28.8935	1.22757
$555$	0	0
$556$	−30.0629	−1.27495
$557$	1.21426	0.0514499	0.0257250	−	0.999669i	$-0.491811\pi$
$557$	1.21426	0.0514499	0.0257250	+	0.999669i	$0.491811\pi$
$558$	0	0
$559$	7.20272	0.304642
$560$	−1.46460	−0.0618906
$561$	0	0
$562$	−16.2075	−0.683674
$563$	31.6272	1.33293	0.666464	−	0.745537i	$-0.267807\pi$
$563$	31.6272	1.33293	0.666464	+	0.745537i	$0.267807\pi$
$564$	0	0
$565$	−3.32826	−0.140021
$566$	−14.3454	−0.602981
$567$	0	0
$568$	16.3856	0.687524
$569$	−47.0142	−1.97094	−0.985468	−	0.169859i	$-0.945669\pi$
$569$	−47.0142	−1.97094	−0.985468	+	0.169859i	$0.945669\pi$
$570$	0	0
$571$	31.6050	1.32263	0.661314	−	0.750109i	$-0.269999\pi$
$571$	31.6050	1.32263	0.661314	+	0.750109i	$0.269999\pi$
$572$	−24.1554	−1.00999
$573$	0	0
$574$	−1.14725	−0.0478854
$575$	19.7873	0.825189
$576$	0	0
$577$	−35.2559	−1.46772	−0.733860	−	0.679300i	$-0.762283\pi$
$577$	−35.2559	−1.46772	−0.733860	+	0.679300i	$0.762283\pi$
$578$	5.41655	0.225299
$579$	0	0
$580$	−3.69949	−0.153613
$581$	6.74596	0.279870
$582$	0	0
$583$	−22.4604	−0.930216
$584$	4.95670	0.205110
$585$	0	0
$586$	−61.0837	−2.52334
$587$	−28.4801	−1.17550	−0.587749	−	0.809043i	$-0.699986\pi$
$587$	−28.4801	−1.17550	−0.587749	+	0.809043i	$0.699986\pi$
$588$	0	0
$589$	−15.3821	−0.633810
$590$	10.2757	0.423042
$591$	0	0
$592$	44.7653	1.83984
$593$	−4.79696	−0.196987	−0.0984937	−	0.995138i	$-0.531402\pi$
$593$	−4.79696	−0.196987	−0.0984937	+	0.995138i	$0.531402\pi$
$594$	0	0
$595$	1.33688	0.0548065
$596$	19.8201	0.811862
$597$	0	0
$598$	53.9814	2.20746
$599$	5.06413	0.206915	0.103457	−	0.994634i	$-0.467009\pi$
$599$	5.06413	0.206915	0.103457	+	0.994634i	$0.467009\pi$
$600$	0	0
$601$	−35.9289	−1.46557	−0.732785	−	0.680460i	$-0.761780\pi$
$601$	−35.9289	−1.46557	−0.732785	+	0.680460i	$0.761780\pi$
$602$	1.25967	0.0513401
$603$	0	0
$604$	10.6273	0.432420
$605$	1.97862	0.0804421
$606$	0	0
$607$	−5.25566	−0.213321	−0.106660	−	0.994296i	$-0.534016\pi$
$607$	−5.25566	−0.213321	−0.106660	+	0.994296i	$0.534016\pi$
$608$	−39.1064	−1.58597
$609$	0	0
$610$	−0.626115	−0.0253507
$611$	69.1047	2.79568
$612$	0	0
$613$	−21.9713	−0.887412	−0.443706	−	0.896172i	$-0.646337\pi$
$613$	−21.9713	−0.887412	−0.443706	+	0.896172i	$0.646337\pi$
$614$	23.5552	0.950610
$615$	0	0
$616$	2.16352	0.0871706
$617$	−31.7381	−1.27773	−0.638864	−	0.769320i	$-0.720595\pi$
$617$	−31.7381	−1.27773	−0.638864	+	0.769320i	$0.720595\pi$
$618$	0	0
$619$	−36.4514	−1.46511	−0.732554	−	0.680709i	$-0.761672\pi$
$619$	−36.4514	−1.46511	−0.732554	+	0.680709i	$0.761672\pi$
$620$	1.45367	0.0583808
$621$	0	0
$622$	−24.7211	−0.991226
$623$	−5.56264	−0.222863
$624$	0	0
$625$	22.2243	0.888972
$626$	−42.6207	−1.70347
$627$	0	0
$628$	9.11636	0.363782
$629$	−40.8614	−1.62925
$630$	0	0
$631$	−6.25533	−0.249021	−0.124510	−	0.992218i	$-0.539736\pi$
$631$	−6.25533	−0.249021	−0.124510	+	0.992218i	$0.539736\pi$
$632$	14.6756	0.583763
$633$	0	0
$634$	2.93732	0.116656
$635$	−0.178348	−0.00707751
$636$	0	0
$637$	46.9793	1.86139
$638$	29.8643	1.18234
$639$	0	0
$640$	−4.03066	−0.159326
$641$	39.3213	1.55310	0.776550	−	0.630056i	$-0.216968\pi$
$641$	39.3213	1.55310	0.776550	+	0.630056i	$0.216968\pi$
$642$	0	0
$643$	9.00103	0.354966	0.177483	−	0.984124i	$-0.443205\pi$
$643$	9.00103	0.354966	0.177483	+	0.984124i	$0.443205\pi$
$644$	3.75802	0.148087
$645$	0	0
$646$	49.3523	1.94174
$647$	49.7161	1.95454	0.977271	−	0.211996i	$-0.0679964\pi$
$647$	49.7161	1.95454	0.977271	+	0.211996i	$0.0679964\pi$
$648$	0	0
$649$	−33.0200	−1.29615
$650$	63.1856	2.47834
$651$	0	0
$652$	9.08902	0.355953
$653$	−27.8105	−1.08831	−0.544155	−	0.838985i	$-0.683150\pi$
$653$	−27.8105	−1.08831	−0.544155	+	0.838985i	$0.683150\pi$
$654$	0	0
$655$	−1.31211	−0.0512684
$656$	−4.45900	−0.174095
$657$	0	0
$658$	12.0856	0.471144
$659$	8.76051	0.341261	0.170631	−	0.985335i	$-0.445420\pi$
$659$	8.76051	0.341261	0.170631	+	0.985335i	$0.445420\pi$
$660$	0	0
$661$	−4.63090	−0.180121	−0.0900606	−	0.995936i	$-0.528706\pi$
$661$	−4.63090	−0.180121	−0.0900606	+	0.995936i	$0.528706\pi$
$662$	62.4409	2.42683
$663$	0	0
$664$	12.0530	0.467748
$665$	1.81236	0.0702805
$666$	0	0
$667$	−26.5668	−1.02867
$668$	20.6214	0.797868
$669$	0	0
$670$	−8.99510	−0.347511
$671$	2.01197	0.0776712
$672$	0	0
$673$	10.5345	0.406076	0.203038	−	0.979171i	$-0.434918\pi$
$673$	10.5345	0.406076	0.203038	+	0.979171i	$0.434918\pi$
$674$	27.9925	1.07823
$675$	0	0
$676$	51.4227	1.97780
$677$	−2.02418	−0.0777957	−0.0388978	−	0.999243i	$-0.512385\pi$
$677$	−2.02418	−0.0777957	−0.0388978	+	0.999243i	$0.512385\pi$
$678$	0	0
$679$	−10.3582	−0.397511
$680$	2.38860	0.0915986
$681$	0	0
$682$	−11.7348	−0.449350
$683$	−30.3719	−1.16215	−0.581075	−	0.813850i	$-0.697368\pi$
$683$	−30.3719	−1.16215	−0.581075	+	0.813850i	$0.697368\pi$
$684$	0	0
$685$	4.62702	0.176789
$686$	17.0338	0.650352
$687$	0	0
$688$	4.89591	0.186655
$689$	63.8021	2.43067
$690$	0	0
$691$	−18.3536	−0.698205	−0.349102	−	0.937085i	$-0.613513\pi$
$691$	−18.3536	−0.698205	−0.349102	+	0.937085i	$0.613513\pi$
$692$	−4.06241	−0.154430
$693$	0	0
$694$	−3.52612	−0.133850
$695$	9.83928	0.373225
$696$	0	0
$697$	4.07014	0.154168
$698$	10.1873	0.385595
$699$	0	0
$700$	4.39879	0.166259
$701$	−4.98491	−0.188277	−0.0941387	−	0.995559i	$-0.530010\pi$
$701$	−4.98491	−0.188277	−0.0941387	+	0.995559i	$0.530010\pi$
$702$	0	0
$703$	−55.3947	−2.08925
$704$	−5.00571	−0.188660
$705$	0	0
$706$	−8.27261	−0.311344
$707$	2.14656	0.0807298
$708$	0	0
$709$	27.8837	1.04719	0.523597	−	0.851966i	$-0.324590\pi$
$709$	27.8837	1.04719	0.523597	+	0.851966i	$0.324590\pi$
$710$	10.4715	0.392987
$711$	0	0
$712$	−9.93880	−0.372472
$713$	10.4391	0.390948
$714$	0	0
$715$	7.90581	0.295660
$716$	9.01629	0.336954
$717$	0	0
$718$	−21.1302	−0.788573
$719$	−10.7704	−0.401668	−0.200834	−	0.979625i	$-0.564365\pi$
$719$	−10.7704	−0.401668	−0.200834	+	0.979625i	$0.564365\pi$
$720$	0	0
$721$	−4.81155	−0.179191
$722$	32.2721	1.20104
$723$	0	0
$724$	−8.28173	−0.307788
$725$	−31.0966	−1.15490
$726$	0	0
$727$	2.46886	0.0915651	0.0457826	−	0.998951i	$-0.485422\pi$
$727$	2.46886	0.0915651	0.0457826	+	0.998951i	$0.485422\pi$
$728$	−6.14578	−0.227778
$729$	0	0
$730$	3.16766	0.117240
$731$	−4.46895	−0.165290
$732$	0	0
$733$	−12.4103	−0.458383	−0.229192	−	0.973381i	$-0.573608\pi$
$733$	−12.4103	−0.458383	−0.229192	+	0.973381i	$0.573608\pi$
$734$	−13.1880	−0.486778
$735$	0	0
$736$	26.5396	0.978263
$737$	28.9050	1.06473
$738$	0	0
$739$	38.5237	1.41712	0.708559	−	0.705651i	$-0.249346\pi$
$739$	38.5237	1.41712	0.708559	+	0.705651i	$0.249346\pi$
$740$	5.23501	0.192443
$741$	0	0
$742$	11.1582	0.409630
$743$	36.8075	1.35034	0.675168	−	0.737664i	$-0.264071\pi$
$743$	36.8075	1.35034	0.675168	+	0.737664i	$0.264071\pi$
$744$	0	0
$745$	−6.48691	−0.237662
$746$	21.4361	0.784831
$747$	0	0
$748$	14.9873	0.547990
$749$	−5.60145	−0.204672
$750$	0	0
$751$	14.1985	0.518111	0.259055	−	0.965862i	$-0.416589\pi$
$751$	14.1985	0.518111	0.259055	+	0.965862i	$0.416589\pi$
$752$	46.9726	1.71292
$753$	0	0
$754$	−84.8339	−3.08947
$755$	−3.47822	−0.126585
$756$	0	0
$757$	10.5676	0.384085	0.192042	−	0.981387i	$-0.438489\pi$
$757$	10.5676	0.384085	0.192042	+	0.981387i	$0.438489\pi$
$758$	−9.31652	−0.338391
$759$	0	0
$760$	3.23816	0.117460
$761$	27.4842	0.996302	0.498151	−	0.867090i	$-0.334012\pi$
$761$	27.4842	0.996302	0.498151	+	0.867090i	$0.334012\pi$
$762$	0	0
$763$	0.0519598	0.00188107
$764$	−12.9482	−0.468448
$765$	0	0
$766$	29.4417	1.06377
$767$	93.7981	3.38685
$768$	0	0
$769$	−11.0083	−0.396969	−0.198485	−	0.980104i	$-0.563602\pi$
$769$	−11.0083	−0.396969	−0.198485	+	0.980104i	$0.563602\pi$
$770$	1.38263	0.0498264
$771$	0	0
$772$	8.53831	0.307301
$773$	−40.1139	−1.44280	−0.721398	−	0.692521i	$-0.756500\pi$
$773$	−40.1139	−1.44280	−0.721398	+	0.692521i	$0.756500\pi$
$774$	0	0
$775$	12.2191	0.438921
$776$	−18.5070	−0.664364
$777$	0	0
$778$	31.4112	1.12615
$779$	5.51778	0.197695
$780$	0	0
$781$	−33.6492	−1.20406
$782$	−33.4929	−1.19771
$783$	0	0
$784$	31.9333	1.14047
$785$	−2.98369	−0.106492
$786$	0	0
$787$	−15.6905	−0.559306	−0.279653	−	0.960101i	$-0.590219\pi$
$787$	−15.6905	−0.559306	−0.279653	+	0.960101i	$0.590219\pi$
$788$	−12.5921	−0.448575
$789$	0	0
$790$	9.37865	0.333677
$791$	−5.31326	−0.188918
$792$	0	0
$793$	−5.71529	−0.202956
$794$	−53.9827	−1.91577
$795$	0	0
$796$	22.2905	0.790065
$797$	−42.7721	−1.51507	−0.757533	−	0.652797i	$-0.773596\pi$
$797$	−42.7721	−1.51507	−0.757533	+	0.652797i	$0.773596\pi$
$798$	0	0
$799$	−42.8763	−1.51685
$800$	31.0648	1.09831
$801$	0	0
$802$	−26.0753	−0.920751
$803$	−10.1790	−0.359209
$804$	0	0
$805$	−1.22996	−0.0433505
$806$	33.3345	1.17416
$807$	0	0
$808$	3.83527	0.134924
$809$	45.5750	1.60233	0.801166	−	0.598442i	$-0.204213\pi$
$809$	45.5750	1.60233	0.801166	+	0.598442i	$0.204213\pi$
$810$	0	0
$811$	19.9527	0.700634	0.350317	−	0.936631i	$-0.386074\pi$
$811$	19.9527	0.700634	0.350317	+	0.936631i	$0.386074\pi$
$812$	−5.90588	−0.207256
$813$	0	0
$814$	−42.2598	−1.48121
$815$	−2.97474	−0.104201
$816$	0	0
$817$	−6.05843	−0.211958
$818$	30.7440	1.07494
$819$	0	0
$820$	−0.521451	−0.0182098
$821$	47.1393	1.64517	0.822587	−	0.568640i	$-0.192530\pi$
$821$	47.1393	1.64517	0.822587	+	0.568640i	$0.192530\pi$
$822$	0	0
$823$	22.2522	0.775663	0.387831	−	0.921730i	$-0.373224\pi$
$823$	22.2522	0.775663	0.387831	+	0.921730i	$0.373224\pi$
$824$	−8.59681	−0.299484
$825$	0	0
$826$	16.4041	0.570772
$827$	−46.9285	−1.63186	−0.815931	−	0.578149i	$-0.803775\pi$
$827$	−46.9285	−1.63186	−0.815931	+	0.578149i	$0.803775\pi$
$828$	0	0
$829$	48.4646	1.68324	0.841622	−	0.540067i	$-0.181601\pi$
$829$	48.4646	1.68324	0.841622	+	0.540067i	$0.181601\pi$
$830$	7.70267	0.267364
$831$	0	0
$832$	14.2195	0.492971
$833$	−29.1485	−1.00993
$834$	0	0
$835$	−6.74919	−0.233565
$836$	20.3179	0.702708
$837$	0	0
$838$	−2.62742	−0.0907627
$839$	9.26196	0.319758	0.159879	−	0.987137i	$-0.448890\pi$
$839$	9.26196	0.319758	0.159879	+	0.987137i	$0.448890\pi$
$840$	0	0
$841$	12.7507	0.439681
$842$	−2.52820	−0.0871275
$843$	0	0
$844$	1.73599	0.0597553
$845$	−16.8301	−0.578974
$846$	0	0
$847$	3.15867	0.108533
$848$	43.3683	1.48927
$849$	0	0
$850$	−39.2038	−1.34468
$851$	37.5936	1.28869
$852$	0	0
$853$	−30.7761	−1.05375	−0.526877	−	0.849941i	$-0.676637\pi$
$853$	−30.7761	−1.05375	−0.526877	+	0.849941i	$0.676637\pi$
$854$	−0.999534	−0.0342034
$855$	0	0
$856$	−10.0081	−0.342071
$857$	−20.4236	−0.697656	−0.348828	−	0.937187i	$-0.613420\pi$
$857$	−20.4236	−0.697656	−0.348828	+	0.937187i	$0.613420\pi$
$858$	0	0
$859$	−35.6527	−1.21645	−0.608226	−	0.793764i	$-0.708119\pi$
$859$	−35.6527	−1.21645	−0.608226	+	0.793764i	$0.708119\pi$
$860$	0.572545	0.0195236
$861$	0	0
$862$	−36.2298	−1.23399
$863$	9.21342	0.313628	0.156814	−	0.987628i	$-0.449878\pi$
$863$	9.21342	0.313628	0.156814	+	0.987628i	$0.449878\pi$
$864$	0	0
$865$	1.32959	0.0452073
$866$	−5.63530	−0.191495
$867$	0	0
$868$	2.32065	0.0787680
$869$	−30.1375	−1.02235
$870$	0	0
$871$	−82.1089	−2.78216
$872$	0.0928368	0.00314385
$873$	0	0
$874$	−45.4054	−1.53586
$875$	−2.93542	−0.0992352
$876$	0	0
$877$	−57.8367	−1.95301	−0.976504	−	0.215501i	$-0.930861\pi$
$877$	−57.8367	−1.95301	−0.976504	+	0.215501i	$0.930861\pi$
$878$	16.2266	0.547621
$879$	0	0
$880$	5.37382	0.181152
$881$	−28.9309	−0.974706	−0.487353	−	0.873205i	$-0.662038\pi$
$881$	−28.9309	−0.974706	−0.487353	+	0.873205i	$0.662038\pi$
$882$	0	0
$883$	43.7957	1.47384	0.736922	−	0.675978i	$-0.236278\pi$
$883$	43.7957	1.47384	0.736922	+	0.675978i	$0.236278\pi$
$884$	−42.5736	−1.43191
$885$	0	0
$886$	−20.0081	−0.672185
$887$	−30.1524	−1.01242	−0.506210	−	0.862410i	$-0.668954\pi$
$887$	−30.1524	−1.01242	−0.506210	+	0.862410i	$0.668954\pi$
$888$	0	0
$889$	−0.284715	−0.00954904
$890$	−6.35154	−0.212904
$891$	0	0
$892$	−2.51296	−0.0841401
$893$	−58.1262	−1.94512
$894$	0	0
$895$	−2.95094	−0.0986390
$896$	−6.43457	−0.214964
$897$	0	0
$898$	34.2574	1.14318
$899$	−16.4055	−0.547153
$900$	0	0
$901$	−39.5863	−1.31881
$902$	4.20944	0.140159
$903$	0	0
$904$	−9.49322	−0.315740
$905$	2.71053	0.0901009
$906$	0	0
$907$	−41.6983	−1.38457	−0.692284	−	0.721625i	$-0.743396\pi$
$907$	−41.6983	−1.38457	−0.692284	+	0.721625i	$0.743396\pi$
$908$	−16.8493	−0.559165
$909$	0	0
$910$	−3.92756	−0.130197
$911$	8.80971	0.291879	0.145939	−	0.989294i	$-0.453380\pi$
$911$	8.80971	0.291879	0.145939	+	0.989294i	$0.453380\pi$
$912$	0	0
$913$	−24.7519	−0.819169
$914$	35.9386	1.18874
$915$	0	0
$916$	31.4046	1.03764
$917$	−2.09466	−0.0691718
$918$	0	0
$919$	−10.9520	−0.361273	−0.180636	−	0.983550i	$-0.557816\pi$
$919$	−10.9520	−0.361273	−0.180636	+	0.983550i	$0.557816\pi$
$920$	−2.19758	−0.0724520
$921$	0	0
$922$	−73.2304	−2.41171
$923$	95.5853	3.14623
$924$	0	0
$925$	44.0036	1.44683
$926$	40.6360	1.33538
$927$	0	0
$928$	−41.7080	−1.36913
$929$	28.2079	0.925470	0.462735	−	0.886497i	$-0.346868\pi$
$929$	28.2079	0.925470	0.462735	+	0.886497i	$0.346868\pi$
$930$	0	0
$931$	−39.5158	−1.29508
$932$	−10.3008	−0.337415
$933$	0	0
$934$	5.12998	0.167858
$935$	−4.90519	−0.160417
$936$	0	0
$937$	−32.3092	−1.05550	−0.527748	−	0.849401i	$-0.676963\pi$
$937$	−32.3092	−1.05550	−0.527748	+	0.849401i	$0.676963\pi$
$938$	−14.3598	−0.468865
$939$	0	0
$940$	5.49314	0.179167
$941$	23.1402	0.754349	0.377175	−	0.926142i	$-0.376896\pi$
$941$	23.1402	0.754349	0.377175	+	0.926142i	$0.376896\pi$
$942$	0	0
$943$	−3.74464	−0.121942
$944$	63.7575	2.07513
$945$	0	0
$946$	−4.62190	−0.150271
$947$	26.2737	0.853780	0.426890	−	0.904303i	$-0.359609\pi$
$947$	26.2737	0.853780	0.426890	+	0.904303i	$0.359609\pi$
$948$	0	0
$949$	28.9149	0.938619
$950$	−53.1474	−1.72433
$951$	0	0
$952$	3.81318	0.123586
$953$	−8.01441	−0.259612	−0.129806	−	0.991539i	$-0.541435\pi$
$953$	−8.01441	−0.259612	−0.129806	+	0.991539i	$0.541435\pi$
$954$	0	0
$955$	4.23780	0.137132
$956$	−4.51415	−0.145998
$957$	0	0
$958$	28.5983	0.923969
$959$	7.38660	0.238526
$960$	0	0
$961$	−24.5537	−0.792053
$962$	120.045	3.87042
$963$	0	0
$964$	−16.3152	−0.525476
$965$	−2.79450	−0.0899582
$966$	0	0
$967$	24.4688	0.786865	0.393432	−	0.919354i	$-0.371288\pi$
$967$	24.4688	0.786865	0.393432	+	0.919354i	$0.371288\pi$
$968$	5.64361	0.181393
$969$	0	0
$970$	−11.8272	−0.379749
$971$	−20.7426	−0.665662	−0.332831	−	0.942987i	$-0.608004\pi$
$971$	−20.7426	−0.665662	−0.332831	+	0.942987i	$0.608004\pi$
$972$	0	0
$973$	15.7075	0.503559
$974$	−15.1640	−0.485887
$975$	0	0
$976$	−3.88486	−0.124351
$977$	−4.62330	−0.147912	−0.0739562	−	0.997261i	$-0.523562\pi$
$977$	−4.62330	−0.147912	−0.0739562	+	0.997261i	$0.523562\pi$
$978$	0	0
$979$	20.4101	0.652311
$980$	3.73439	0.119291
$981$	0	0
$982$	39.1174	1.24829
$983$	−43.9207	−1.40085	−0.700427	−	0.713725i	$-0.747007\pi$
$983$	−43.9207	−1.40085	−0.700427	+	0.713725i	$0.747007\pi$
$984$	0	0
$985$	4.12126	0.131314
$986$	52.6355	1.67626
$987$	0	0
$988$	−57.7158	−1.83619
$989$	4.11156	0.130740
$990$	0	0
$991$	30.0775	0.955442	0.477721	−	0.878511i	$-0.341463\pi$
$991$	30.0775	0.955442	0.477721	+	0.878511i	$0.341463\pi$
$992$	16.3887	0.520342
$993$	0	0
$994$	16.7167	0.530221
$995$	−7.29545	−0.231281
$996$	0	0
$997$	−25.4260	−0.805250	−0.402625	−	0.915365i	$-0.631902\pi$
$997$	−25.4260	−0.805250	−0.402625	+	0.915365i	$0.631902\pi$
$998$	−16.1568	−0.511436
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3483.2.a.t.1.17		20
3.2	odd	2		3483.2.a.u.1.4		20
9.2	odd	6		1161.2.f.d.388.17		40
9.4	even	3		387.2.f.d.259.4	yes	40
9.5	odd	6		1161.2.f.d.775.17		40
9.7	even	3		387.2.f.d.130.4	✓	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
387.2.f.d.130.4	✓	40	9.7	even	3
387.2.f.d.259.4	yes	40	9.4	even	3
1161.2.f.d.388.17		40	9.2	odd	6
1161.2.f.d.775.17		40	9.5	odd	6
3483.2.a.t.1.17		20	1.1	even	1	trivial
3483.2.a.u.1.4		20	3.2	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:	\( N \)	\(=\)	\( 3483 = 3^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3483.a (trivial)

\(f(q)\)	\(=\)	\(q+1.82281 q^{2} +1.32263 q^{4} -0.432883 q^{5} -0.691058 q^{7} -1.23472 q^{8} +O(q^{10})\)	\(q+1.82281 q^{2} +1.32263 q^{4} -0.432883 q^{5} -0.691058 q^{7} -1.23472 q^{8} -0.789063 q^{10} +2.53559 q^{11} -7.20272 q^{13} -1.25967 q^{14} -4.89591 q^{16} +4.46895 q^{17} +6.05843 q^{19} -0.572545 q^{20} +4.62190 q^{22} -4.11156 q^{23} -4.81261 q^{25} -13.1292 q^{26} -0.914014 q^{28} +6.46148 q^{29} -2.53897 q^{31} -6.45487 q^{32} +8.14604 q^{34} +0.299147 q^{35} -9.14340 q^{37} +11.0434 q^{38} +0.534488 q^{40} +0.910760 q^{41} -1.00000 q^{43} +3.35365 q^{44} -7.49459 q^{46} -9.59426 q^{47} -6.52244 q^{49} -8.77247 q^{50} -9.52653 q^{52} -8.85806 q^{53} -1.09761 q^{55} +0.853259 q^{56} +11.7780 q^{58} -13.0226 q^{59} +0.793492 q^{61} -4.62805 q^{62} -1.97418 q^{64} +3.11794 q^{65} +11.3997 q^{67} +5.91077 q^{68} +0.545288 q^{70} -13.2707 q^{71} -4.01445 q^{73} -16.6667 q^{74} +8.01307 q^{76} -1.75224 q^{77} -11.8858 q^{79} +2.11936 q^{80} +1.66014 q^{82} -9.76179 q^{83} -1.93454 q^{85} -1.82281 q^{86} -3.13073 q^{88} +8.04947 q^{89} +4.97749 q^{91} -5.43807 q^{92} -17.4885 q^{94} -2.62259 q^{95} +14.9889 q^{97} -11.8892 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 6 q^{2} + 22 q^{4} - 17 q^{5} + 3 q^{7} - 15 q^{8}+O(q^{10}) \) 20 * q - 6 * q^2 + 22 * q^4 - 17 * q^5 + 3 * q^7 - 15 * q^8	\( 20 q - 6 q^{2} + 22 q^{4} - 17 q^{5} + 3 q^{7} - 15 q^{8} - 2 q^{10} - 10 q^{11} - q^{13} - 10 q^{14} + 22 q^{16} - 20 q^{17} + 8 q^{19} - 30 q^{20} + 15 q^{22} - 19 q^{23} + 19 q^{25} - 25 q^{26} - 3 q^{28} - 25 q^{29} - 11 q^{31} - 36 q^{32} + 9 q^{34} + 9 q^{37} - 28 q^{38} + 12 q^{40} - 12 q^{41} - 20 q^{43} - 5 q^{44} + 4 q^{46} - 38 q^{47} + 37 q^{49} - 36 q^{50} - 8 q^{52} - 69 q^{53} - 9 q^{55} - 30 q^{56} - 27 q^{58} - 31 q^{59} + 19 q^{61} - 32 q^{62} + 11 q^{64} - 47 q^{65} + 9 q^{67} - 68 q^{68} - 6 q^{70} - 21 q^{71} - 2 q^{73} + 16 q^{74} + 37 q^{76} - 85 q^{77} - 4 q^{79} - 61 q^{80} + q^{82} - 19 q^{83} - 6 q^{85} + 6 q^{86} + 60 q^{88} - 54 q^{89} - 3 q^{91} - 85 q^{92} - 19 q^{94} + 11 q^{95} + 2 q^{97} - 5 q^{98}+O(q^{100}) \) 20 * q - 6 * q^2 + 22 * q^4 - 17 * q^5 + 3 * q^7 - 15 * q^8 - 2 * q^10 - 10 * q^11 - q^13 - 10 * q^14 + 22 * q^16 - 20 * q^17 + 8 * q^19 - 30 * q^20 + 15 * q^22 - 19 * q^23 + 19 * q^25 - 25 * q^26 - 3 * q^28 - 25 * q^29 - 11 * q^31 - 36 * q^32 + 9 * q^34 + 9 * q^37 - 28 * q^38 + 12 * q^40 - 12 * q^41 - 20 * q^43 - 5 * q^44 + 4 * q^46 - 38 * q^47 + 37 * q^49 - 36 * q^50 - 8 * q^52 - 69 * q^53 - 9 * q^55 - 30 * q^56 - 27 * q^58 - 31 * q^59 + 19 * q^61 - 32 * q^62 + 11 * q^64 - 47 * q^65 + 9 * q^67 - 68 * q^68 - 6 * q^70 - 21 * q^71 - 2 * q^73 + 16 * q^74 + 37 * q^76 - 85 * q^77 - 4 * q^79 - 61 * q^80 + q^82 - 19 * q^83 - 6 * q^85 + 6 * q^86 + 60 * q^88 - 54 * q^89 - 3 * q^91 - 85 * q^92 - 19 * q^94 + 11 * q^95 + 2 * q^97 - 5 * q^98

Introduction

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