LMFDB - Embedded newform 8925.2.a.cx.1.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8925,2,Mod(1,8925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8925 = 3 \cdot 5^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [14,3,14,17,0,3,14,15,14,0,4,17,-1,3,0,19,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$71.2664838040$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 18 x^{12} + 54 x^{11} + 124 x^{10} - 366 x^{9} - 416 x^{8} + 1164 x^{7} + 727 x^{6} + \cdots - 40 $ x^14 - 3x^13 - 18x^12 + 54x^11 + 124x^10 - 366x^9 - 416x^8 + 1164x^7 + 727x^6 - 1783x^5 - 662x^4 + 1242x^3 + 284x^2 - 300*x - 40
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 1785)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$2.13433$ of defining polynomial
Character	$\chi$	$=$	8925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.13433 q^{2} +1.00000 q^{3} +2.55536 q^{4} +2.13433 q^{6} +1.00000 q^{7} +1.18533 q^{8} +1.00000 q^{9} -5.50983 q^{11} +2.55536 q^{12} +0.643407 q^{13} +2.13433 q^{14} -2.58084 q^{16} -1.00000 q^{17} +2.13433 q^{18} +5.16312 q^{19} +1.00000 q^{21} -11.7598 q^{22} +4.28012 q^{23} +1.18533 q^{24} +1.37324 q^{26} +1.00000 q^{27} +2.55536 q^{28} +8.85121 q^{29} -4.74477 q^{31} -7.87903 q^{32} -5.50983 q^{33} -2.13433 q^{34} +2.55536 q^{36} +7.79937 q^{37} +11.0198 q^{38} +0.643407 q^{39} +7.36348 q^{41} +2.13433 q^{42} +12.0566 q^{43} -14.0796 q^{44} +9.13519 q^{46} +10.0239 q^{47} -2.58084 q^{48} +1.00000 q^{49} -1.00000 q^{51} +1.64414 q^{52} -10.3163 q^{53} +2.13433 q^{54} +1.18533 q^{56} +5.16312 q^{57} +18.8914 q^{58} +14.0781 q^{59} +1.22749 q^{61} -10.1269 q^{62} +1.00000 q^{63} -11.6548 q^{64} -11.7598 q^{66} -9.34997 q^{67} -2.55536 q^{68} +4.28012 q^{69} -3.39468 q^{71} +1.18533 q^{72} -4.21709 q^{73} +16.6464 q^{74} +13.1936 q^{76} -5.50983 q^{77} +1.37324 q^{78} -7.23677 q^{79} +1.00000 q^{81} +15.7161 q^{82} -8.49261 q^{83} +2.55536 q^{84} +25.7327 q^{86} +8.85121 q^{87} -6.53097 q^{88} +10.0122 q^{89} +0.643407 q^{91} +10.9373 q^{92} -4.74477 q^{93} +21.3944 q^{94} -7.87903 q^{96} -1.82896 q^{97} +2.13433 q^{98} -5.50983 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 3 q^{2} + 14 q^{3} + 17 q^{4} + 3 q^{6} + 14 q^{7} + 15 q^{8} + 14 q^{9} + 4 q^{11} + 17 q^{12} - q^{13} + 3 q^{14} + 19 q^{16} - 14 q^{17} + 3 q^{18} - 6 q^{19} + 14 q^{21} + 12 q^{22} + 7 q^{23}+ \cdots + 4 q^{99}+O(q^{100}) $ 14 * q + 3 * q^2 + 14 * q^3 + 17 * q^4 + 3 * q^6 + 14 * q^7 + 15 * q^8 + 14 * q^9 + 4 * q^11 + 17 * q^12 - q^13 + 3 * q^14 + 19 * q^16 - 14 * q^17 + 3 * q^18 - 6 * q^19 + 14 * q^21 + 12 * q^22 + 7 * q^23 + 15 * q^24 - 2 * q^26 + 14 * q^27 + 17 * q^28 + 20 * q^29 - 7 * q^31 + 33 * q^32 + 4 * q^33 - 3 * q^34 + 17 * q^36 + 9 * q^37 + 18 * q^38 - q^39 + 25 * q^41 + 3 * q^42 + 28 * q^43 + 24 * q^44 - 10 * q^46 + 29 * q^47 + 19 * q^48 + 14 * q^49 - 14 * q^51 + 4 * q^52 + 26 * q^53 + 3 * q^54 + 15 * q^56 - 6 * q^57 + 2 * q^58 + 14 * q^59 - 9 * q^61 + 28 * q^62 + 14 * q^63 + 27 * q^64 + 12 * q^66 + 32 * q^67 - 17 * q^68 + 7 * q^69 + 2 * q^71 + 15 * q^72 + 18 * q^73 + 64 * q^74 - 42 * q^76 + 4 * q^77 - 2 * q^78 - 4 * q^79 + 14 * q^81 + 42 * q^82 + 51 * q^83 + 17 * q^84 + 44 * q^86 + 20 * q^87 + 32 * q^88 + 38 * q^89 - q^91 - 16 * q^92 - 7 * q^93 + 8 * q^94 + 33 * q^96 + 14 * q^97 + 3 * q^98 + 4 * q^99

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 $

$2$	2.13433	1.50920	0.754600	−	0.656186i	$-0.227831\pi$
$2$	2.13433	1.50920	0.754600	+	0.656186i	$0.227831\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.55536	1.27768
$5$	0	0
$5$	0	0
$6$	2.13433	0.871336
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.18533	0.419077
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.50983	−1.66128	−0.830639	−	0.556812i	$-0.812024\pi$
$11$	−5.50983	−1.66128	−0.830639	+	0.556812i	$0.812024\pi$
$12$	2.55536	0.737670
$13$	0.643407	0.178449	0.0892244	−	0.996012i	$-0.471561\pi$
$13$	0.643407	0.178449	0.0892244	+	0.996012i	$0.471561\pi$
$14$	2.13433	0.570424
$15$	0	0
$16$	−2.58084	−0.645211
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	2.13433	0.503066
$19$	5.16312	1.18450	0.592250	−	0.805754i	$-0.298240\pi$
$19$	5.16312	1.18450	0.592250	+	0.805754i	$0.298240\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	−11.7598	−2.50720
$23$	4.28012	0.892467	0.446234	−	0.894916i	$-0.352765\pi$
$23$	4.28012	0.892467	0.446234	+	0.894916i	$0.352765\pi$
$24$	1.18533	0.241954
$25$	0	0
$26$	1.37324	0.269315
$27$	1.00000	0.192450
$28$	2.55536	0.482918
$29$	8.85121	1.64363	0.821815	−	0.569755i	$-0.192962\pi$
$29$	8.85121	1.64363	0.821815	+	0.569755i	$0.192962\pi$
$30$	0	0
$31$	−4.74477	−0.852186	−0.426093	−	0.904679i	$-0.640110\pi$
$31$	−4.74477	−0.852186	−0.426093	+	0.904679i	$0.640110\pi$
$32$	−7.87903	−1.39283
$33$	−5.50983	−0.959139
$34$	−2.13433	−0.366035
$35$	0	0
$36$	2.55536	0.425894
$37$	7.79937	1.28221	0.641104	−	0.767454i	$-0.278477\pi$
$37$	7.79937	1.28221	0.641104	+	0.767454i	$0.278477\pi$
$38$	11.0198	1.78765
$39$	0.643407	0.103027
$40$	0	0
$41$	7.36348	1.14998	0.574992	−	0.818159i	$-0.305005\pi$
$41$	7.36348	1.14998	0.574992	+	0.818159i	$0.305005\pi$
$42$	2.13433	0.329334
$43$	12.0566	1.83861	0.919303	−	0.393549i	$-0.128753\pi$
$43$	12.0566	1.83861	0.919303	+	0.393549i	$0.128753\pi$
$44$	−14.0796	−2.12258
$45$	0	0
$46$	9.13519	1.34691
$47$	10.0239	1.46214	0.731072	−	0.682301i	$-0.239021\pi$
$47$	10.0239	1.46214	0.731072	+	0.682301i	$0.239021\pi$
$48$	−2.58084	−0.372513
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.00000	−0.140028
$52$	1.64414	0.228001
$53$	−10.3163	−1.41705	−0.708523	−	0.705687i	$-0.750638\pi$
$53$	−10.3163	−1.41705	−0.708523	+	0.705687i	$0.750638\pi$
$54$	2.13433	0.290445
$55$	0	0
$56$	1.18533	0.158396
$57$	5.16312	0.683872
$58$	18.8914	2.48056
$59$	14.0781	1.83281	0.916404	−	0.400254i	$-0.131078\pi$
$59$	14.0781	1.83281	0.916404	+	0.400254i	$0.131078\pi$
$60$	0	0
$61$	1.22749	0.157165	0.0785823	−	0.996908i	$-0.474961\pi$
$61$	1.22749	0.157165	0.0785823	+	0.996908i	$0.474961\pi$
$62$	−10.1269	−1.28612
$63$	1.00000	0.125988
$64$	−11.6548	−1.45685
$65$	0	0
$66$	−11.7598	−1.44753
$67$	−9.34997	−1.14228	−0.571141	−	0.820852i	$-0.693499\pi$
$67$	−9.34997	−1.14228	−0.571141	+	0.820852i	$0.693499\pi$
$68$	−2.55536	−0.309883
$69$	4.28012	0.515266
$70$	0	0
$71$	−3.39468	−0.402874	−0.201437	−	0.979501i	$-0.564561\pi$
$71$	−3.39468	−0.402874	−0.201437	+	0.979501i	$0.564561\pi$
$72$	1.18533	0.139692
$73$	−4.21709	−0.493573	−0.246787	−	0.969070i	$-0.579375\pi$
$73$	−4.21709	−0.493573	−0.246787	+	0.969070i	$0.579375\pi$
$74$	16.6464	1.93511
$75$	0	0
$76$	13.1936	1.51341
$77$	−5.50983	−0.627904
$78$	1.37324	0.155489
$79$	−7.23677	−0.814200	−0.407100	−	0.913384i	$-0.633460\pi$
$79$	−7.23677	−0.814200	−0.407100	+	0.913384i	$0.633460\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	15.7161	1.73555
$83$	−8.49261	−0.932185	−0.466092	−	0.884736i	$-0.654339\pi$
$83$	−8.49261	−0.932185	−0.466092	+	0.884736i	$0.654339\pi$
$84$	2.55536	0.278813
$85$	0	0
$86$	25.7327	2.77482
$87$	8.85121	0.948950
$88$	−6.53097	−0.696203
$89$	10.0122	1.06129	0.530643	−	0.847595i	$-0.321951\pi$
$89$	10.0122	1.06129	0.530643	+	0.847595i	$0.321951\pi$
$90$	0	0
$91$	0.643407	0.0674473
$92$	10.9373	1.14029
$93$	−4.74477	−0.492010
$94$	21.3944	2.20666
$95$	0	0
$96$	−7.87903	−0.804150
$97$	−1.82896	−0.185702	−0.0928511	−	0.995680i	$-0.529598\pi$
$97$	−1.82896	−0.185702	−0.0928511	+	0.995680i	$0.529598\pi$
$98$	2.13433	0.215600
$99$	−5.50983	−0.553759
$100$	0	0
$101$	−1.10831	−0.110281	−0.0551405	−	0.998479i	$-0.517561\pi$
$101$	−1.10831	−0.110281	−0.0551405	+	0.998479i	$0.517561\pi$
$102$	−2.13433	−0.211330
$103$	7.56063	0.744971	0.372485	−	0.928038i	$-0.378506\pi$
$103$	7.56063	0.744971	0.372485	+	0.928038i	$0.378506\pi$
$104$	0.762648	0.0747838
$105$	0	0
$106$	−22.0183	−2.13861
$107$	4.28328	0.414080	0.207040	−	0.978332i	$-0.433617\pi$
$107$	4.28328	0.414080	0.207040	+	0.978332i	$0.433617\pi$
$108$	2.55536	0.245890
$109$	12.1978	1.16834	0.584169	−	0.811632i	$-0.301421\pi$
$109$	12.1978	1.16834	0.584169	+	0.811632i	$0.301421\pi$
$110$	0	0
$111$	7.79937	0.740283
$112$	−2.58084	−0.243867
$113$	4.91040	0.461932	0.230966	−	0.972962i	$-0.425811\pi$
$113$	4.91040	0.461932	0.230966	+	0.972962i	$0.425811\pi$
$114$	11.0198	1.03210
$115$	0	0
$116$	22.6181	2.10003
$117$	0.643407	0.0594830
$118$	30.0472	2.76607
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	19.3583	1.75984
$122$	2.61988	0.237193
$123$	7.36348	0.663943
$124$	−12.1246	−1.08882
$125$	0	0
$126$	2.13433	0.190141
$127$	2.34771	0.208326	0.104163	−	0.994560i	$-0.466784\pi$
$127$	2.34771	0.208326	0.104163	+	0.994560i	$0.466784\pi$
$128$	−9.11704	−0.805840
$129$	12.0566	1.06152
$130$	0	0
$131$	13.9915	1.22244	0.611220	−	0.791461i	$-0.290679\pi$
$131$	13.9915	1.22244	0.611220	+	0.791461i	$0.290679\pi$
$132$	−14.0796	−1.22547
$133$	5.16312	0.447699
$134$	−19.9559	−1.72393
$135$	0	0
$136$	−1.18533	−0.101641
$137$	12.7146	1.08628	0.543141	−	0.839642i	$-0.317235\pi$
$137$	12.7146	1.08628	0.543141	+	0.839642i	$0.317235\pi$
$138$	9.13519	0.777639
$139$	−0.625840	−0.0530831	−0.0265415	−	0.999648i	$-0.508449\pi$
$139$	−0.625840	−0.0530831	−0.0265415	+	0.999648i	$0.508449\pi$
$140$	0	0
$141$	10.0239	0.844169
$142$	−7.24536	−0.608017
$143$	−3.54506	−0.296453
$144$	−2.58084	−0.215070
$145$	0	0
$146$	−9.00067	−0.744901
$147$	1.00000	0.0824786
$148$	19.9302	1.63825
$149$	−1.48444	−0.121610	−0.0608049	−	0.998150i	$-0.519367\pi$
$149$	−1.48444	−0.121610	−0.0608049	+	0.998150i	$0.519367\pi$
$150$	0	0
$151$	2.05728	0.167419	0.0837096	−	0.996490i	$-0.473323\pi$
$151$	2.05728	0.167419	0.0837096	+	0.996490i	$0.473323\pi$
$152$	6.11999	0.496397
$153$	−1.00000	−0.0808452
$154$	−11.7598	−0.947632
$155$	0	0
$156$	1.64414	0.131636
$157$	−14.9185	−1.19063	−0.595314	−	0.803493i	$-0.702972\pi$
$157$	−14.9185	−1.19063	−0.595314	+	0.803493i	$0.702972\pi$
$158$	−15.4456	−1.22879
$159$	−10.3163	−0.818132
$160$	0	0
$161$	4.28012	0.337321
$162$	2.13433	0.167689
$163$	15.0326	1.17744	0.588720	−	0.808337i	$-0.299632\pi$
$163$	15.0326	1.17744	0.588720	+	0.808337i	$0.299632\pi$
$164$	18.8164	1.46931
$165$	0	0
$166$	−18.1260	−1.40685
$167$	6.96721	0.539139	0.269569	−	0.962981i	$-0.413119\pi$
$167$	6.96721	0.539139	0.269569	+	0.962981i	$0.413119\pi$
$168$	1.18533	0.0914501
$169$	−12.5860	−0.968156
$170$	0	0
$171$	5.16312	0.394834
$172$	30.8089	2.34915
$173$	9.32109	0.708669	0.354335	−	0.935119i	$-0.384707\pi$
$173$	9.32109	0.708669	0.354335	+	0.935119i	$0.384707\pi$
$174$	18.8914	1.43215
$175$	0	0
$176$	14.2200	1.07187
$177$	14.0781	1.05817
$178$	21.3692	1.60169
$179$	−20.5559	−1.53642	−0.768211	−	0.640196i	$-0.778853\pi$
$179$	−20.5559	−1.53642	−0.768211	+	0.640196i	$0.778853\pi$
$180$	0	0
$181$	−16.7234	−1.24304	−0.621522	−	0.783397i	$-0.713485\pi$
$181$	−16.7234	−1.24304	−0.621522	+	0.783397i	$0.713485\pi$
$182$	1.37324	0.101791
$183$	1.22749	0.0907390
$184$	5.07335	0.374013
$185$	0	0
$186$	−10.1269	−0.742541
$187$	5.50983	0.402919
$188$	25.6148	1.86815
$189$	1.00000	0.0727393
$190$	0	0
$191$	−8.24929	−0.596898	−0.298449	−	0.954426i	$-0.596469\pi$
$191$	−8.24929	−0.596898	−0.298449	+	0.954426i	$0.596469\pi$
$192$	−11.6548	−0.841110
$193$	−25.0098	−1.80024	−0.900121	−	0.435640i	$-0.856522\pi$
$193$	−25.0098	−1.80024	−0.900121	+	0.435640i	$0.856522\pi$
$194$	−3.90359	−0.280262
$195$	0	0
$196$	2.55536	0.182526
$197$	−7.51729	−0.535585	−0.267793	−	0.963477i	$-0.586294\pi$
$197$	−7.51729	−0.535585	−0.267793	+	0.963477i	$0.586294\pi$
$198$	−11.7598	−0.835733
$199$	−2.41445	−0.171156	−0.0855779	−	0.996331i	$-0.527274\pi$
$199$	−2.41445	−0.171156	−0.0855779	+	0.996331i	$0.527274\pi$
$200$	0	0
$201$	−9.34997	−0.659496
$202$	−2.36550	−0.166436
$203$	8.85121	0.621233
$204$	−2.55536	−0.178911
$205$	0	0
$206$	16.1369	1.12431
$207$	4.28012	0.297489
$208$	−1.66053	−0.115137
$209$	−28.4479	−1.96778
$210$	0	0
$211$	−26.6246	−1.83291	−0.916456	−	0.400136i	$-0.868963\pi$
$211$	−26.6246	−1.83291	−0.916456	+	0.400136i	$0.868963\pi$
$212$	−26.3618	−1.81053
$213$	−3.39468	−0.232599
$214$	9.14192	0.624929
$215$	0	0
$216$	1.18533	0.0806514
$217$	−4.74477	−0.322096
$218$	26.0341	1.76325
$219$	−4.21709	−0.284965
$220$	0	0
$221$	−0.643407	−0.0432802
$222$	16.6464	1.11723
$223$	−21.6013	−1.44653	−0.723265	−	0.690571i	$-0.757359\pi$
$223$	−21.6013	−1.44653	−0.723265	+	0.690571i	$0.757359\pi$
$224$	−7.87903	−0.526440
$225$	0	0
$226$	10.4804	0.697147
$227$	−5.66803	−0.376201	−0.188100	−	0.982150i	$-0.560233\pi$
$227$	−5.66803	−0.376201	−0.188100	+	0.982150i	$0.560233\pi$
$228$	13.1936	0.873770
$229$	3.34447	0.221009	0.110504	−	0.993876i	$-0.464753\pi$
$229$	3.34447	0.221009	0.110504	+	0.993876i	$0.464753\pi$
$230$	0	0
$231$	−5.50983	−0.362520
$232$	10.4916	0.688807
$233$	−22.3728	−1.46569	−0.732846	−	0.680394i	$-0.761808\pi$
$233$	−22.3728	−1.46569	−0.732846	+	0.680394i	$0.761808\pi$
$234$	1.37324	0.0897716
$235$	0	0
$236$	35.9746	2.34175
$237$	−7.23677	−0.470079
$238$	−2.13433	−0.138348
$239$	16.3289	1.05623	0.528115	−	0.849173i	$-0.322899\pi$
$239$	16.3289	1.05623	0.528115	+	0.849173i	$0.322899\pi$
$240$	0	0
$241$	−30.3048	−1.95211	−0.976053	−	0.217533i	$-0.930199\pi$
$241$	−30.3048	−1.95211	−0.976053	+	0.217533i	$0.930199\pi$
$242$	41.3169	2.65595
$243$	1.00000	0.0641500
$244$	3.13670	0.200806
$245$	0	0
$246$	15.7161	1.00202
$247$	3.32198	0.211373
$248$	−5.62411	−0.357132
$249$	−8.49261	−0.538197
$250$	0	0
$251$	20.4528	1.29097	0.645485	−	0.763773i	$-0.276655\pi$
$251$	20.4528	1.29097	0.645485	+	0.763773i	$0.276655\pi$
$252$	2.55536	0.160973
$253$	−23.5828	−1.48264
$254$	5.01079	0.314405
$255$	0	0
$256$	3.85075	0.240672
$257$	12.8868	0.803854	0.401927	−	0.915672i	$-0.368341\pi$
$257$	12.8868	0.803854	0.401927	+	0.915672i	$0.368341\pi$
$258$	25.7327	1.60205
$259$	7.79937	0.484629
$260$	0	0
$261$	8.85121	0.547876
$262$	29.8624	1.84490
$263$	−8.19887	−0.505564	−0.252782	−	0.967523i	$-0.581346\pi$
$263$	−8.19887	−0.505564	−0.252782	+	0.967523i	$0.581346\pi$
$264$	−6.53097	−0.401953
$265$	0	0
$266$	11.0198	0.675667
$267$	10.0122	0.612734
$268$	−23.8926	−1.45947
$269$	7.05528	0.430168	0.215084	−	0.976596i	$-0.430997\pi$
$269$	7.05528	0.430168	0.215084	+	0.976596i	$0.430997\pi$
$270$	0	0
$271$	16.0618	0.975686	0.487843	−	0.872931i	$-0.337784\pi$
$271$	16.0618	0.975686	0.487843	+	0.872931i	$0.337784\pi$
$272$	2.58084	0.156487
$273$	0.643407	0.0389407
$274$	27.1371	1.63941
$275$	0	0
$276$	10.9373	0.658346
$277$	−16.5942	−0.997050	−0.498525	−	0.866875i	$-0.666125\pi$
$277$	−16.5942	−0.997050	−0.498525	+	0.866875i	$0.666125\pi$
$278$	−1.33575	−0.0801129
$279$	−4.74477	−0.284062
$280$	0	0
$281$	30.6476	1.82828	0.914142	−	0.405394i	$-0.132866\pi$
$281$	30.6476	1.82828	0.914142	+	0.405394i	$0.132866\pi$
$282$	21.3944	1.27402
$283$	−15.2805	−0.908333	−0.454166	−	0.890917i	$-0.650063\pi$
$283$	−15.2805	−0.908333	−0.454166	+	0.890917i	$0.650063\pi$
$284$	−8.67463	−0.514745
$285$	0	0
$286$	−7.56633	−0.447407
$287$	7.36348	0.434653
$288$	−7.87903	−0.464276
$289$	1.00000	0.0588235
$290$	0	0
$291$	−1.82896	−0.107215
$292$	−10.7762	−0.630630
$293$	−22.3430	−1.30529	−0.652646	−	0.757663i	$-0.726341\pi$
$293$	−22.3430	−1.30529	−0.652646	+	0.757663i	$0.726341\pi$
$294$	2.13433	0.124477
$295$	0	0
$296$	9.24481	0.537344
$297$	−5.50983	−0.319713
$298$	−3.16828	−0.183533
$299$	2.75386	0.159260
$300$	0	0
$301$	12.0566	0.694928
$302$	4.39092	0.252669
$303$	−1.10831	−0.0636708
$304$	−13.3252	−0.764253
$305$	0	0
$306$	−2.13433	−0.122012
$307$	−17.1138	−0.976735	−0.488368	−	0.872638i	$-0.662408\pi$
$307$	−17.1138	−0.976735	−0.488368	+	0.872638i	$0.662408\pi$
$308$	−14.0796	−0.802261
$309$	7.56063	0.430109
$310$	0	0
$311$	−12.2190	−0.692876	−0.346438	−	0.938073i	$-0.612609\pi$
$311$	−12.2190	−0.692876	−0.346438	+	0.938073i	$0.612609\pi$
$312$	0.762648	0.0431765
$313$	9.24803	0.522729	0.261365	−	0.965240i	$-0.415827\pi$
$313$	9.24803	0.522729	0.261365	+	0.965240i	$0.415827\pi$
$314$	−31.8410	−1.79689
$315$	0	0
$316$	−18.4926	−1.04029
$317$	4.99443	0.280515	0.140258	−	0.990115i	$-0.455207\pi$
$317$	4.99443	0.280515	0.140258	+	0.990115i	$0.455207\pi$
$318$	−22.0183	−1.23472
$319$	−48.7687	−2.73052
$320$	0	0
$321$	4.28328	0.239069
$322$	9.13519	0.509084
$323$	−5.16312	−0.287284
$324$	2.55536	0.141965
$325$	0	0
$326$	32.0844	1.77699
$327$	12.1978	0.674540
$328$	8.72815	0.481932
$329$	10.0239	0.552638
$330$	0	0
$331$	−22.1145	−1.21552	−0.607761	−	0.794120i	$-0.707932\pi$
$331$	−22.1145	−1.21552	−0.607761	+	0.794120i	$0.707932\pi$
$332$	−21.7017	−1.19104
$333$	7.79937	0.427403
$334$	14.8703	0.813668
$335$	0	0
$336$	−2.58084	−0.140797
$337$	22.4262	1.22163	0.610815	−	0.791773i	$-0.290842\pi$
$337$	22.4262	1.22163	0.610815	+	0.791773i	$0.290842\pi$
$338$	−26.8627	−1.46114
$339$	4.91040	0.266696
$340$	0	0
$341$	26.1429	1.41572
$342$	11.0198	0.595882
$343$	1.00000	0.0539949
$344$	14.2910	0.770518
$345$	0	0
$346$	19.8943	1.06952
$347$	3.56283	0.191263	0.0956313	−	0.995417i	$-0.469513\pi$
$347$	3.56283	0.191263	0.0956313	+	0.995417i	$0.469513\pi$
$348$	22.6181	1.21246
$349$	−6.30005	−0.337234	−0.168617	−	0.985682i	$-0.553930\pi$
$349$	−6.30005	−0.337234	−0.168617	+	0.985682i	$0.553930\pi$
$350$	0	0
$351$	0.643407	0.0343425
$352$	43.4122	2.31388
$353$	14.5193	0.772786	0.386393	−	0.922334i	$-0.373721\pi$
$353$	14.5193	0.772786	0.386393	+	0.922334i	$0.373721\pi$
$354$	30.0472	1.59699
$355$	0	0
$356$	25.5847	1.35599
$357$	−1.00000	−0.0529256
$358$	−43.8732	−2.31877
$359$	10.9678	0.578858	0.289429	−	0.957200i	$-0.406535\pi$
$359$	10.9678	0.578858	0.289429	+	0.957200i	$0.406535\pi$
$360$	0	0
$361$	7.65779	0.403042
$362$	−35.6933	−1.87600
$363$	19.3583	1.01605
$364$	1.64414	0.0861762
$365$	0	0
$366$	2.61988	0.136943
$367$	−20.1871	−1.05376	−0.526879	−	0.849940i	$-0.676638\pi$
$367$	−20.1871	−1.05376	−0.526879	+	0.849940i	$0.676638\pi$
$368$	−11.0463	−0.575830
$369$	7.36348	0.383328
$370$	0	0
$371$	−10.3163	−0.535593
$372$	−12.1246	−0.628632
$373$	27.6334	1.43080	0.715402	−	0.698713i	$-0.246244\pi$
$373$	27.6334	1.43080	0.715402	+	0.698713i	$0.246244\pi$
$374$	11.7598	0.608085
$375$	0	0
$376$	11.8817	0.612751
$377$	5.69493	0.293304
$378$	2.13433	0.109778
$379$	8.77721	0.450855	0.225428	−	0.974260i	$-0.427622\pi$
$379$	8.77721	0.450855	0.225428	+	0.974260i	$0.427622\pi$
$380$	0	0
$381$	2.34771	0.120277
$382$	−17.6067	−0.900837
$383$	−7.40695	−0.378477	−0.189239	−	0.981931i	$-0.560602\pi$
$383$	−7.40695	−0.378477	−0.189239	+	0.981931i	$0.560602\pi$
$384$	−9.11704	−0.465252
$385$	0	0
$386$	−53.3791	−2.71692
$387$	12.0566	0.612869
$388$	−4.67365	−0.237268
$389$	26.2306	1.32995	0.664973	−	0.746867i	$-0.268443\pi$
$389$	26.2306	1.32995	0.664973	+	0.746867i	$0.268443\pi$
$390$	0	0
$391$	−4.28012	−0.216455
$392$	1.18533	0.0598681
$393$	13.9915	0.705776
$394$	−16.0444	−0.808304
$395$	0	0
$396$	−14.0796	−0.707528
$397$	−29.6923	−1.49022	−0.745108	−	0.666944i	$-0.767602\pi$
$397$	−29.6923	−1.49022	−0.745108	+	0.666944i	$0.767602\pi$
$398$	−5.15323	−0.258308
$399$	5.16312	0.258479
$400$	0	0
$401$	−15.6946	−0.783753	−0.391876	−	0.920018i	$-0.628174\pi$
$401$	−15.6946	−0.783753	−0.391876	+	0.920018i	$0.628174\pi$
$402$	−19.9559	−0.995311
$403$	−3.05282	−0.152072
$404$	−2.83214	−0.140904
$405$	0	0
$406$	18.8914	0.937565
$407$	−42.9732	−2.13010
$408$	−1.18533	−0.0586825
$409$	−32.5578	−1.60988	−0.804939	−	0.593357i	$-0.797802\pi$
$409$	−32.5578	−1.60988	−0.804939	+	0.593357i	$0.797802\pi$
$410$	0	0
$411$	12.7146	0.627165
$412$	19.3201	0.951835
$413$	14.0781	0.692737
$414$	9.13519	0.448970
$415$	0	0
$416$	−5.06942	−0.248549
$417$	−0.625840	−0.0306475
$418$	−60.7173	−2.96978
$419$	−14.7975	−0.722904	−0.361452	−	0.932391i	$-0.617719\pi$
$419$	−14.7975	−0.722904	−0.361452	+	0.932391i	$0.617719\pi$
$420$	0	0
$421$	8.93468	0.435450	0.217725	−	0.976010i	$-0.430136\pi$
$421$	8.93468	0.435450	0.217725	+	0.976010i	$0.430136\pi$
$422$	−56.8256	−2.76623
$423$	10.0239	0.487381
$424$	−12.2282	−0.593852
$425$	0	0
$426$	−7.24536	−0.351039
$427$	1.22749	0.0594026
$428$	10.9453	0.529062
$429$	−3.54506	−0.171157
$430$	0	0
$431$	2.30891	0.111216	0.0556082	−	0.998453i	$-0.482290\pi$
$431$	2.30891	0.111216	0.0556082	+	0.998453i	$0.482290\pi$
$432$	−2.58084	−0.124171
$433$	7.10020	0.341214	0.170607	−	0.985339i	$-0.445427\pi$
$433$	7.10020	0.341214	0.170607	+	0.985339i	$0.445427\pi$
$434$	−10.1269	−0.486107
$435$	0	0
$436$	31.1698	1.49276
$437$	22.0988	1.05713
$438$	−9.00067	−0.430069
$439$	10.6936	0.510375	0.255188	−	0.966892i	$-0.417863\pi$
$439$	10.6936	0.510375	0.255188	+	0.966892i	$0.417863\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−1.37324	−0.0653184
$443$	−16.3278	−0.775759	−0.387880	−	0.921710i	$-0.626792\pi$
$443$	−16.3278	−0.775759	−0.387880	+	0.921710i	$0.626792\pi$
$444$	19.9302	0.945846
$445$	0	0
$446$	−46.1043	−2.18310
$447$	−1.48444	−0.0702115
$448$	−11.6548	−0.550636
$449$	9.73297	0.459327	0.229664	−	0.973270i	$-0.426237\pi$
$449$	9.73297	0.459327	0.229664	+	0.973270i	$0.426237\pi$
$450$	0	0
$451$	−40.5716	−1.91044
$452$	12.5479	0.590201
$453$	2.05728	0.0966595
$454$	−12.0975	−0.567762
$455$	0	0
$456$	6.11999	0.286595
$457$	27.5677	1.28956	0.644781	−	0.764367i	$-0.276949\pi$
$457$	27.5677	1.28956	0.644781	+	0.764367i	$0.276949\pi$
$458$	7.13820	0.333546
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	−8.80131	−0.409918	−0.204959	−	0.978771i	$-0.565706\pi$
$461$	−8.80131	−0.409918	−0.204959	+	0.978771i	$0.565706\pi$
$462$	−11.7598	−0.547116
$463$	18.3765	0.854030	0.427015	−	0.904245i	$-0.359565\pi$
$463$	18.3765	0.854030	0.427015	+	0.904245i	$0.359565\pi$
$464$	−22.8436	−1.06049
$465$	0	0
$466$	−47.7510	−2.21202
$467$	33.9479	1.57092	0.785460	−	0.618912i	$-0.212426\pi$
$467$	33.9479	1.57092	0.785460	+	0.618912i	$0.212426\pi$
$468$	1.64414	0.0760003
$469$	−9.34997	−0.431742
$470$	0	0
$471$	−14.9185	−0.687409
$472$	16.6871	0.768088
$473$	−66.4296	−3.05444
$474$	−15.4456	−0.709442
$475$	0	0
$476$	−2.55536	−0.117125
$477$	−10.3163	−0.472349
$478$	34.8513	1.59406
$479$	32.8894	1.50275	0.751376	−	0.659874i	$-0.229390\pi$
$479$	32.8894	1.50275	0.751376	+	0.659874i	$0.229390\pi$
$480$	0	0
$481$	5.01816	0.228809
$482$	−64.6805	−2.94612
$483$	4.28012	0.194752
$484$	49.4674	2.24852
$485$	0	0
$486$	2.13433	0.0968152
$487$	40.9669	1.85639	0.928195	−	0.372095i	$-0.121361\pi$
$487$	40.9669	1.85639	0.928195	+	0.372095i	$0.121361\pi$
$488$	1.45498	0.0658641
$489$	15.0326	0.679796
$490$	0	0
$491$	12.4761	0.563038	0.281519	−	0.959556i	$-0.409162\pi$
$491$	12.4761	0.563038	0.281519	+	0.959556i	$0.409162\pi$
$492$	18.8164	0.848308
$493$	−8.85121	−0.398639
$494$	7.09021	0.319004
$495$	0	0
$496$	12.2455	0.549840
$497$	−3.39468	−0.152272
$498$	−18.1260	−0.812246
$499$	−26.6977	−1.19515	−0.597577	−	0.801811i	$-0.703870\pi$
$499$	−26.6977	−1.19515	−0.597577	+	0.801811i	$0.703870\pi$
$500$	0	0
$501$	6.96721	0.311272
$502$	43.6531	1.94833
$503$	−25.1092	−1.11956	−0.559782	−	0.828640i	$-0.689115\pi$
$503$	−25.1092	−1.11956	−0.559782	+	0.828640i	$0.689115\pi$
$504$	1.18533	0.0527987
$505$	0	0
$506$	−50.3334	−2.23759
$507$	−12.5860	−0.558965
$508$	5.99925	0.266174
$509$	−20.8537	−0.924325	−0.462163	−	0.886795i	$-0.652926\pi$
$509$	−20.8537	−0.924325	−0.462163	+	0.886795i	$0.652926\pi$
$510$	0	0
$511$	−4.21709	−0.186553
$512$	26.4529	1.16906
$513$	5.16312	0.227957
$514$	27.5046	1.21318
$515$	0	0
$516$	30.8089	1.35629
$517$	−55.2303	−2.42903
$518$	16.6464	0.731402
$519$	9.32109	0.409150
$520$	0	0
$521$	−30.2822	−1.32669	−0.663343	−	0.748315i	$-0.730863\pi$
$521$	−30.2822	−1.32669	−0.663343	+	0.748315i	$0.730863\pi$
$522$	18.8914	0.826854
$523$	13.0009	0.568490	0.284245	−	0.958752i	$-0.408257\pi$
$523$	13.0009	0.568490	0.284245	+	0.958752i	$0.408257\pi$
$524$	35.7533	1.56189
$525$	0	0
$526$	−17.4991	−0.762996
$527$	4.74477	0.206685
$528$	14.2200	0.618847
$529$	−4.68055	−0.203502
$530$	0	0
$531$	14.0781	0.610936
$532$	13.1936	0.572017
$533$	4.73771	0.205213
$534$	21.3692	0.924737
$535$	0	0
$536$	−11.0828	−0.478704
$537$	−20.5559	−0.887054
$538$	15.0583	0.649210
$539$	−5.50983	−0.237325
$540$	0	0
$541$	−13.5508	−0.582593	−0.291296	−	0.956633i	$-0.594087\pi$
$541$	−13.5508	−0.582593	−0.291296	+	0.956633i	$0.594087\pi$
$542$	34.2812	1.47250
$543$	−16.7234	−0.717671
$544$	7.87903	0.337811
$545$	0	0
$546$	1.37324	0.0587693
$547$	28.3384	1.21166	0.605832	−	0.795593i	$-0.292841\pi$
$547$	28.3384	1.21166	0.605832	+	0.795593i	$0.292841\pi$
$548$	32.4904	1.38792
$549$	1.22749	0.0523882
$550$	0	0
$551$	45.6999	1.94688
$552$	5.07335	0.215936
$553$	−7.23677	−0.307739
$554$	−35.4175	−1.50475
$555$	0	0
$556$	−1.59925	−0.0678232
$557$	21.6314	0.916552	0.458276	−	0.888810i	$-0.348467\pi$
$557$	21.6314	0.916552	0.458276	+	0.888810i	$0.348467\pi$
$558$	−10.1269	−0.428706
$559$	7.75726	0.328097
$560$	0	0
$561$	5.50983	0.232625
$562$	65.4122	2.75925
$563$	−7.56964	−0.319022	−0.159511	−	0.987196i	$-0.550992\pi$
$563$	−7.56964	−0.319022	−0.159511	+	0.987196i	$0.550992\pi$
$564$	25.6148	1.07858
$565$	0	0
$566$	−32.6137	−1.37085
$567$	1.00000	0.0419961
$568$	−4.02381	−0.168835
$569$	−34.8187	−1.45968	−0.729838	−	0.683621i	$-0.760404\pi$
$569$	−34.8187	−1.45968	−0.729838	+	0.683621i	$0.760404\pi$
$570$	0	0
$571$	41.8796	1.75260	0.876302	−	0.481761i	$-0.160003\pi$
$571$	41.8796	1.75260	0.876302	+	0.481761i	$0.160003\pi$
$572$	−9.05893	−0.378773
$573$	−8.24929	−0.344619
$574$	15.7161	0.655978
$575$	0	0
$576$	−11.6548	−0.485615
$577$	−27.0110	−1.12448	−0.562242	−	0.826973i	$-0.690061\pi$
$577$	−27.0110	−1.12448	−0.562242	+	0.826973i	$0.690061\pi$
$578$	2.13433	0.0887764
$579$	−25.0098	−1.03937
$580$	0	0
$581$	−8.49261	−0.352333
$582$	−3.90359	−0.161809
$583$	56.8409	2.35411
$584$	−4.99864	−0.206845
$585$	0	0
$586$	−47.6873	−1.96995
$587$	33.2666	1.37306	0.686529	−	0.727102i	$-0.259133\pi$
$587$	33.2666	1.37306	0.686529	+	0.727102i	$0.259133\pi$
$588$	2.55536	0.105381
$589$	−24.4978	−1.00941
$590$	0	0
$591$	−7.51729	−0.309220
$592$	−20.1290	−0.827295
$593$	−6.67275	−0.274017	−0.137009	−	0.990570i	$-0.543749\pi$
$593$	−6.67275	−0.274017	−0.137009	+	0.990570i	$0.543749\pi$
$594$	−11.7598	−0.482511
$595$	0	0
$596$	−3.79328	−0.155379
$597$	−2.41445	−0.0988169
$598$	5.87764	0.240355
$599$	29.8095	1.21798	0.608991	−	0.793177i	$-0.291575\pi$
$599$	29.8095	1.21798	0.608991	+	0.793177i	$0.291575\pi$
$600$	0	0
$601$	39.2586	1.60139	0.800695	−	0.599072i	$-0.204464\pi$
$601$	39.2586	1.60139	0.800695	+	0.599072i	$0.204464\pi$
$602$	25.7327	1.04878
$603$	−9.34997	−0.380760
$604$	5.25710	0.213908
$605$	0	0
$606$	−2.36550	−0.0960919
$607$	−10.1194	−0.410735	−0.205368	−	0.978685i	$-0.565839\pi$
$607$	−10.1194	−0.410735	−0.205368	+	0.978685i	$0.565839\pi$
$608$	−40.6804	−1.64981
$609$	8.85121	0.358669
$610$	0	0
$611$	6.44947	0.260918
$612$	−2.55536	−0.103294
$613$	−14.2612	−0.576006	−0.288003	−	0.957629i	$-0.592991\pi$
$613$	−14.2612	−0.576006	−0.288003	+	0.957629i	$0.592991\pi$
$614$	−36.5265	−1.47409
$615$	0	0
$616$	−6.53097	−0.263140
$617$	28.7376	1.15693	0.578466	−	0.815706i	$-0.303651\pi$
$617$	28.7376	1.15693	0.578466	+	0.815706i	$0.303651\pi$
$618$	16.1369	0.649120
$619$	−11.0280	−0.443252	−0.221626	−	0.975132i	$-0.571136\pi$
$619$	−11.0280	−0.443252	−0.221626	+	0.975132i	$0.571136\pi$
$620$	0	0
$621$	4.28012	0.171755
$622$	−26.0794	−1.04569
$623$	10.0122	0.401128
$624$	−1.66053	−0.0664745
$625$	0	0
$626$	19.7383	0.788903
$627$	−28.4479	−1.13610
$628$	−38.1222	−1.52124
$629$	−7.79937	−0.310981
$630$	0	0
$631$	−14.4693	−0.576012	−0.288006	−	0.957629i	$-0.592992\pi$
$631$	−14.4693	−0.576012	−0.288006	+	0.957629i	$0.592992\pi$
$632$	−8.57795	−0.341213
$633$	−26.6246	−1.05823
$634$	10.6598	0.423353
$635$	0	0
$636$	−26.3618	−1.04531
$637$	0.643407	0.0254927
$638$	−104.089	−4.12090
$639$	−3.39468	−0.134291
$640$	0	0
$641$	−2.58221	−0.101991	−0.0509956	−	0.998699i	$-0.516239\pi$
$641$	−2.58221	−0.101991	−0.0509956	+	0.998699i	$0.516239\pi$
$642$	9.14192	0.360803
$643$	1.51754	0.0598460	0.0299230	−	0.999552i	$-0.490474\pi$
$643$	1.51754	0.0598460	0.0299230	+	0.999552i	$0.490474\pi$
$644$	10.9373	0.430989
$645$	0	0
$646$	−11.0198	−0.433568
$647$	−17.4697	−0.686806	−0.343403	−	0.939188i	$-0.611580\pi$
$647$	−17.4697	−0.686806	−0.343403	+	0.939188i	$0.611580\pi$
$648$	1.18533	0.0465641
$649$	−77.5678	−3.04480
$650$	0	0
$651$	−4.74477	−0.185962
$652$	38.4137	1.50439
$653$	−5.53281	−0.216516	−0.108258	−	0.994123i	$-0.534527\pi$
$653$	−5.53281	−0.216516	−0.108258	+	0.994123i	$0.534527\pi$
$654$	26.0341	1.01802
$655$	0	0
$656$	−19.0040	−0.741982
$657$	−4.21709	−0.164524
$658$	21.3944	0.834041
$659$	15.7015	0.611644	0.305822	−	0.952089i	$-0.401069\pi$
$659$	15.7015	0.611644	0.305822	+	0.952089i	$0.401069\pi$
$660$	0	0
$661$	7.53083	0.292915	0.146458	−	0.989217i	$-0.453213\pi$
$661$	7.53083	0.292915	0.146458	+	0.989217i	$0.453213\pi$
$662$	−47.1996	−1.83447
$663$	−0.643407	−0.0249878
$664$	−10.0665	−0.390657
$665$	0	0
$666$	16.6464	0.645036
$667$	37.8843	1.46689
$668$	17.8037	0.688848
$669$	−21.6013	−0.835154
$670$	0	0
$671$	−6.76329	−0.261094
$672$	−7.87903	−0.303940
$673$	17.1440	0.660852	0.330426	−	0.943832i	$-0.392808\pi$
$673$	17.1440	0.660852	0.330426	+	0.943832i	$0.392808\pi$
$674$	47.8648	1.84368
$675$	0	0
$676$	−32.1619	−1.23700
$677$	−19.5065	−0.749697	−0.374849	−	0.927086i	$-0.622305\pi$
$677$	−19.5065	−0.749697	−0.374849	+	0.927086i	$0.622305\pi$
$678$	10.4804	0.402498
$679$	−1.82896	−0.0701889
$680$	0	0
$681$	−5.66803	−0.217200
$682$	55.7976	2.13660
$683$	45.8792	1.75552	0.877760	−	0.479101i	$-0.159037\pi$
$683$	45.8792	1.75552	0.877760	+	0.479101i	$0.159037\pi$
$684$	13.1936	0.504472
$685$	0	0
$686$	2.13433	0.0814891
$687$	3.34447	0.127599
$688$	−31.1161	−1.18629
$689$	−6.63755	−0.252870
$690$	0	0
$691$	−33.2317	−1.26419	−0.632097	−	0.774890i	$-0.717805\pi$
$691$	−33.2317	−1.26419	−0.632097	+	0.774890i	$0.717805\pi$
$692$	23.8188	0.905454
$693$	−5.50983	−0.209301
$694$	7.60425	0.288653
$695$	0	0
$696$	10.4916	0.397683
$697$	−7.36348	−0.278912
$698$	−13.4464	−0.508953
$699$	−22.3728	−0.846218
$700$	0	0
$701$	29.0462	1.09706	0.548531	−	0.836130i	$-0.315187\pi$
$701$	29.0462	1.09706	0.548531	+	0.836130i	$0.315187\pi$
$702$	1.37324	0.0518297
$703$	40.2691	1.51878
$704$	64.2158	2.42022
$705$	0	0
$706$	30.9890	1.16629
$707$	−1.10831	−0.0416823
$708$	35.9746	1.35201
$709$	−1.29333	−0.0485720	−0.0242860	−	0.999705i	$-0.507731\pi$
$709$	−1.29333	−0.0485720	−0.0242860	+	0.999705i	$0.507731\pi$
$710$	0	0
$711$	−7.23677	−0.271400
$712$	11.8677	0.444761
$713$	−20.3082	−0.760548
$714$	−2.13433	−0.0798753
$715$	0	0
$716$	−52.5279	−1.96306
$717$	16.3289	0.609814
$718$	23.4089	0.873612
$719$	−14.5480	−0.542549	−0.271274	−	0.962502i	$-0.587445\pi$
$719$	−14.5480	−0.542549	−0.271274	+	0.962502i	$0.587445\pi$
$720$	0	0
$721$	7.56063	0.281572
$722$	16.3443	0.608270
$723$	−30.3048	−1.12705
$724$	−42.7345	−1.58821
$725$	0	0
$726$	41.3169	1.53342
$727$	3.10087	0.115005	0.0575024	−	0.998345i	$-0.481686\pi$
$727$	3.10087	0.115005	0.0575024	+	0.998345i	$0.481686\pi$
$728$	0.762648	0.0282656
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.0566	−0.445928
$732$	3.13670	0.115936
$733$	−38.5676	−1.42453	−0.712264	−	0.701912i	$-0.752330\pi$
$733$	−38.5676	−1.42453	−0.712264	+	0.701912i	$0.752330\pi$
$734$	−43.0859	−1.59033
$735$	0	0
$736$	−33.7232	−1.24305
$737$	51.5168	1.89765
$738$	15.7161	0.578518
$739$	9.89925	0.364150	0.182075	−	0.983285i	$-0.441719\pi$
$739$	9.89925	0.364150	0.182075	+	0.983285i	$0.441719\pi$
$740$	0	0
$741$	3.32198	0.122036
$742$	−22.0183	−0.808317
$743$	−11.2455	−0.412556	−0.206278	−	0.978493i	$-0.566135\pi$
$743$	−11.2455	−0.412556	−0.206278	+	0.978493i	$0.566135\pi$
$744$	−5.62411	−0.206190
$745$	0	0
$746$	58.9788	2.15937
$747$	−8.49261	−0.310728
$748$	14.0796	0.514802
$749$	4.28328	0.156508
$750$	0	0
$751$	−20.2184	−0.737779	−0.368890	−	0.929473i	$-0.620262\pi$
$751$	−20.2184	−0.737779	−0.368890	+	0.929473i	$0.620262\pi$
$752$	−25.8702	−0.943391
$753$	20.4528	0.745342
$754$	12.1549	0.442654
$755$	0	0
$756$	2.55536	0.0929377
$757$	25.9911	0.944664	0.472332	−	0.881421i	$-0.343412\pi$
$757$	25.9911	0.944664	0.472332	+	0.881421i	$0.343412\pi$
$758$	18.7335	0.680430
$759$	−23.5828	−0.856000
$760$	0	0
$761$	−38.8080	−1.40679	−0.703394	−	0.710800i	$-0.748333\pi$
$761$	−38.8080	−1.40679	−0.703394	+	0.710800i	$0.748333\pi$
$762$	5.01079	0.181522
$763$	12.1978	0.441590
$764$	−21.0799	−0.762645
$765$	0	0
$766$	−15.8089	−0.571198
$767$	9.05792	0.327063
$768$	3.85075	0.138952
$769$	17.2403	0.621701	0.310850	−	0.950459i	$-0.399386\pi$
$769$	17.2403	0.621701	0.310850	+	0.950459i	$0.399386\pi$
$770$	0	0
$771$	12.8868	0.464105
$772$	−63.9090	−2.30014
$773$	39.3307	1.41463	0.707314	−	0.706899i	$-0.249907\pi$
$773$	39.3307	1.41463	0.707314	+	0.706899i	$0.249907\pi$
$774$	25.7327	0.924941
$775$	0	0
$776$	−2.16791	−0.0778235
$777$	7.79937	0.279801
$778$	55.9848	2.00715
$779$	38.0185	1.36216
$780$	0	0
$781$	18.7041	0.669285
$782$	−9.13519	−0.326674
$783$	8.85121	0.316317
$784$	−2.58084	−0.0921730
$785$	0	0
$786$	29.8624	1.06516
$787$	28.6413	1.02095	0.510476	−	0.859892i	$-0.329469\pi$
$787$	28.6413	1.02095	0.510476	+	0.859892i	$0.329469\pi$
$788$	−19.2094	−0.684307
$789$	−8.19887	−0.291887
$790$	0	0
$791$	4.91040	0.174594
$792$	−6.53097	−0.232068
$793$	0.789778	0.0280458
$794$	−63.3732	−2.24903
$795$	0	0
$796$	−6.16980	−0.218683
$797$	23.4749	0.831523	0.415762	−	0.909474i	$-0.363515\pi$
$797$	23.4749	0.831523	0.415762	+	0.909474i	$0.363515\pi$
$798$	11.0198	0.390097
$799$	−10.0239	−0.354622
$800$	0	0
$801$	10.0122	0.353762
$802$	−33.4975	−1.18284
$803$	23.2355	0.819962
$804$	−23.8926	−0.842626
$805$	0	0
$806$	−6.51572	−0.229506
$807$	7.05528	0.248358
$808$	−1.31371	−0.0462162
$809$	−8.05154	−0.283077	−0.141539	−	0.989933i	$-0.545205\pi$
$809$	−8.05154	−0.283077	−0.141539	+	0.989933i	$0.545205\pi$
$810$	0	0
$811$	−46.0746	−1.61790	−0.808949	−	0.587879i	$-0.799963\pi$
$811$	−46.0746	−1.61790	−0.808949	+	0.587879i	$0.799963\pi$
$812$	22.6181	0.793739
$813$	16.0618	0.563313
$814$	−91.7190	−3.21475
$815$	0	0
$816$	2.58084	0.0903476
$817$	62.2494	2.17783
$818$	−69.4890	−2.42963
$819$	0.643407	0.0224824
$820$	0	0
$821$	−22.2778	−0.777500	−0.388750	−	0.921343i	$-0.627093\pi$
$821$	−22.2778	−0.777500	−0.388750	+	0.921343i	$0.627093\pi$
$822$	27.1371	0.946516
$823$	40.2435	1.40280	0.701401	−	0.712767i	$-0.252558\pi$
$823$	40.2435	1.40280	0.701401	+	0.712767i	$0.252558\pi$
$824$	8.96183	0.312200
$825$	0	0
$826$	30.0472	1.04548
$827$	−12.3160	−0.428270	−0.214135	−	0.976804i	$-0.568693\pi$
$827$	−12.3160	−0.428270	−0.214135	+	0.976804i	$0.568693\pi$
$828$	10.9373	0.380096
$829$	−39.7303	−1.37989	−0.689945	−	0.723862i	$-0.742365\pi$
$829$	−39.7303	−1.37989	−0.689945	+	0.723862i	$0.742365\pi$
$830$	0	0
$831$	−16.5942	−0.575647
$832$	−7.49875	−0.259972
$833$	−1.00000	−0.0346479
$834$	−1.33575	−0.0462532
$835$	0	0
$836$	−72.6948	−2.51420
$837$	−4.74477	−0.164003
$838$	−31.5827	−1.09101
$839$	30.7153	1.06041	0.530205	−	0.847870i	$-0.322115\pi$
$839$	30.7153	1.06041	0.530205	+	0.847870i	$0.322115\pi$
$840$	0	0
$841$	49.3440	1.70152
$842$	19.0696	0.657180
$843$	30.6476	1.05556
$844$	−68.0355	−2.34188
$845$	0	0
$846$	21.3944	0.735555
$847$	19.3583	0.665158
$848$	26.6246	0.914294
$849$	−15.2805	−0.524426
$850$	0	0
$851$	33.3822	1.14433
$852$	−8.67463	−0.297188
$853$	32.8606	1.12513	0.562564	−	0.826754i	$-0.309815\pi$
$853$	32.8606	1.12513	0.562564	+	0.826754i	$0.309815\pi$
$854$	2.61988	0.0896504
$855$	0	0
$856$	5.07709	0.173531
$857$	−38.1857	−1.30440	−0.652199	−	0.758048i	$-0.726153\pi$
$857$	−38.1857	−1.30440	−0.652199	+	0.758048i	$0.726153\pi$
$858$	−7.56633	−0.258310
$859$	−21.2418	−0.724759	−0.362380	−	0.932031i	$-0.618036\pi$
$859$	−21.2418	−0.724759	−0.362380	+	0.932031i	$0.618036\pi$
$860$	0	0
$861$	7.36348	0.250947
$862$	4.92798	0.167848
$863$	−9.43641	−0.321219	−0.160610	−	0.987018i	$-0.551346\pi$
$863$	−9.43641	−0.321219	−0.160610	+	0.987018i	$0.551346\pi$
$864$	−7.87903	−0.268050
$865$	0	0
$866$	15.1542	0.514959
$867$	1.00000	0.0339618
$868$	−12.1246	−0.411536
$869$	39.8734	1.35261
$870$	0	0
$871$	−6.01583	−0.203839
$872$	14.4584	0.489624
$873$	−1.82896	−0.0619007
$874$	47.1661	1.59542
$875$	0	0
$876$	−10.7762	−0.364094
$877$	−8.76375	−0.295931	−0.147965	−	0.988993i	$-0.547272\pi$
$877$	−8.76375	−0.295931	−0.147965	+	0.988993i	$0.547272\pi$
$878$	22.8236	0.770258
$879$	−22.3430	−0.753611
$880$	0	0
$881$	11.9829	0.403715	0.201858	−	0.979415i	$-0.435302\pi$
$881$	11.9829	0.403715	0.201858	+	0.979415i	$0.435302\pi$
$882$	2.13433	0.0718666
$883$	−2.93723	−0.0988458	−0.0494229	−	0.998778i	$-0.515738\pi$
$883$	−2.93723	−0.0988458	−0.0494229	+	0.998778i	$0.515738\pi$
$884$	−1.64414	−0.0552983
$885$	0	0
$886$	−34.8490	−1.17078
$887$	−24.6824	−0.828755	−0.414377	−	0.910105i	$-0.636001\pi$
$887$	−24.6824	−0.828755	−0.414377	+	0.910105i	$0.636001\pi$
$888$	9.24481	0.310236
$889$	2.34771	0.0787397
$890$	0	0
$891$	−5.50983	−0.184586
$892$	−55.1992	−1.84820
$893$	51.7548	1.73191
$894$	−3.16828	−0.105963
$895$	0	0
$896$	−9.11704	−0.304579
$897$	2.75386	0.0919487
$898$	20.7734	0.693216
$899$	−41.9970	−1.40068
$900$	0	0
$901$	10.3163	0.343684
$902$	−86.5931	−2.88324
$903$	12.0566	0.401217
$904$	5.82044	0.193585
$905$	0	0
$906$	4.39092	0.145878
$907$	−52.1775	−1.73253	−0.866263	−	0.499587i	$-0.833485\pi$
$907$	−52.1775	−1.73253	−0.866263	+	0.499587i	$0.833485\pi$
$908$	−14.4839	−0.480665
$909$	−1.10831	−0.0367603
$910$	0	0
$911$	−2.72441	−0.0902639	−0.0451319	−	0.998981i	$-0.514371\pi$
$911$	−2.72441	−0.0902639	−0.0451319	+	0.998981i	$0.514371\pi$
$912$	−13.3252	−0.441242
$913$	46.7929	1.54862
$914$	58.8385	1.94621
$915$	0	0
$916$	8.54633	0.282379
$917$	13.9915	0.462039
$918$	−2.13433	−0.0704434
$919$	−47.2218	−1.55770	−0.778851	−	0.627209i	$-0.784197\pi$
$919$	−47.2218	−1.55770	−0.778851	+	0.627209i	$0.784197\pi$
$920$	0	0
$921$	−17.1138	−0.563918
$922$	−18.7849	−0.618647
$923$	−2.18416	−0.0718924
$924$	−14.0796	−0.463186
$925$	0	0
$926$	39.2216	1.28890
$927$	7.56063	0.248324
$928$	−69.7390	−2.28929
$929$	−39.4868	−1.29552	−0.647759	−	0.761845i	$-0.724294\pi$
$929$	−39.4868	−1.29552	−0.647759	+	0.761845i	$0.724294\pi$
$930$	0	0
$931$	5.16312	0.169214
$932$	−57.1707	−1.87269
$933$	−12.2190	−0.400032
$934$	72.4560	2.37083
$935$	0	0
$936$	0.762648	0.0249279
$937$	25.4530	0.831512	0.415756	−	0.909476i	$-0.363517\pi$
$937$	25.4530	0.831512	0.415756	+	0.909476i	$0.363517\pi$
$938$	−19.9559	−0.651584
$939$	9.24803	0.301798
$940$	0	0
$941$	−28.8529	−0.940577	−0.470288	−	0.882513i	$-0.655850\pi$
$941$	−28.8529	−0.940577	−0.470288	+	0.882513i	$0.655850\pi$
$942$	−31.8410	−1.03744
$943$	31.5166	1.02632
$944$	−36.3333	−1.18255
$945$	0	0
$946$	−141.783	−4.60975
$947$	−2.25238	−0.0731924	−0.0365962	−	0.999330i	$-0.511652\pi$
$947$	−2.25238	−0.0731924	−0.0365962	+	0.999330i	$0.511652\pi$
$948$	−18.4926	−0.600611
$949$	−2.71331	−0.0880776
$950$	0	0
$951$	4.99443	0.161956
$952$	−1.18533	−0.0384167
$953$	−11.6851	−0.378518	−0.189259	−	0.981927i	$-0.560609\pi$
$953$	−11.6851	−0.378518	−0.189259	+	0.981927i	$0.560609\pi$
$954$	−22.0183	−0.712868
$955$	0	0
$956$	41.7263	1.34952
$957$	−48.7687	−1.57647
$958$	70.1967	2.26795
$959$	12.7146	0.410576
$960$	0	0
$961$	−8.48716	−0.273779
$962$	10.7104	0.345318
$963$	4.28328	0.138027
$964$	−77.4399	−2.49417
$965$	0	0
$966$	9.13519	0.293920
$967$	41.7296	1.34193	0.670967	−	0.741487i	$-0.265879\pi$
$967$	41.7296	1.34193	0.670967	+	0.741487i	$0.265879\pi$
$968$	22.9459	0.737510
$969$	−5.16312	−0.165863
$970$	0	0
$971$	−23.2889	−0.747377	−0.373688	−	0.927554i	$-0.621907\pi$
$971$	−23.2889	−0.747377	−0.373688	+	0.927554i	$0.621907\pi$
$972$	2.55536	0.0819633
$973$	−0.625840	−0.0200635
$974$	87.4369	2.80166
$975$	0	0
$976$	−3.16797	−0.101404
$977$	4.86330	0.155591	0.0777954	−	0.996969i	$-0.475212\pi$
$977$	4.86330	0.155591	0.0777954	+	0.996969i	$0.475212\pi$
$978$	32.0844	1.02595
$979$	−55.1653	−1.76309
$980$	0	0
$981$	12.1978	0.389446
$982$	26.6281	0.849737
$983$	37.5117	1.19644	0.598218	−	0.801333i	$-0.295876\pi$
$983$	37.5117	1.19644	0.598218	+	0.801333i	$0.295876\pi$
$984$	8.72815	0.278243
$985$	0	0
$986$	−18.8914	−0.601625
$987$	10.0239	0.319066
$988$	8.48888	0.270067
$989$	51.6035	1.64090
$990$	0	0
$991$	14.4990	0.460577	0.230288	−	0.973122i	$-0.426033\pi$
$991$	14.4990	0.460577	0.230288	+	0.973122i	$0.426033\pi$
$992$	37.3842	1.18695
$993$	−22.1145	−0.701782
$994$	−7.24536	−0.229809
$995$	0	0
$996$	−21.7017	−0.687645
$997$	2.33799	0.0740450	0.0370225	−	0.999314i	$-0.488213\pi$
$997$	2.33799	0.0740450	0.0370225	+	0.999314i	$0.488213\pi$
$998$	−56.9818	−1.80373
$999$	7.79937	0.246761

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8925.2.a.cx.1.11		14
5.2	odd	4		1785.2.g.g.1429.23	yes	28
5.3	odd	4		1785.2.g.g.1429.6	✓	28
5.4	even	2		8925.2.a.cu.1.4		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1785.2.g.g.1429.6	✓	28	5.3	odd	4
1785.2.g.g.1429.23	yes	28	5.2	odd	4
8925.2.a.cu.1.4		14	5.4	even	2
8925.2.a.cx.1.11		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8925 = 3 \cdot 5^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8925.a (trivial)

\(f(q)\)	\(=\)	\(q+2.13433 q^{2} +1.00000 q^{3} +2.55536 q^{4} +2.13433 q^{6} +1.00000 q^{7} +1.18533 q^{8} +1.00000 q^{9} -5.50983 q^{11} +2.55536 q^{12} +0.643407 q^{13} +2.13433 q^{14} -2.58084 q^{16} -1.00000 q^{17} +2.13433 q^{18} +5.16312 q^{19} +1.00000 q^{21} -11.7598 q^{22} +4.28012 q^{23} +1.18533 q^{24} +1.37324 q^{26} +1.00000 q^{27} +2.55536 q^{28} +8.85121 q^{29} -4.74477 q^{31} -7.87903 q^{32} -5.50983 q^{33} -2.13433 q^{34} +2.55536 q^{36} +7.79937 q^{37} +11.0198 q^{38} +0.643407 q^{39} +7.36348 q^{41} +2.13433 q^{42} +12.0566 q^{43} -14.0796 q^{44} +9.13519 q^{46} +10.0239 q^{47} -2.58084 q^{48} +1.00000 q^{49} -1.00000 q^{51} +1.64414 q^{52} -10.3163 q^{53} +2.13433 q^{54} +1.18533 q^{56} +5.16312 q^{57} +18.8914 q^{58} +14.0781 q^{59} +1.22749 q^{61} -10.1269 q^{62} +1.00000 q^{63} -11.6548 q^{64} -11.7598 q^{66} -9.34997 q^{67} -2.55536 q^{68} +4.28012 q^{69} -3.39468 q^{71} +1.18533 q^{72} -4.21709 q^{73} +16.6464 q^{74} +13.1936 q^{76} -5.50983 q^{77} +1.37324 q^{78} -7.23677 q^{79} +1.00000 q^{81} +15.7161 q^{82} -8.49261 q^{83} +2.55536 q^{84} +25.7327 q^{86} +8.85121 q^{87} -6.53097 q^{88} +10.0122 q^{89} +0.643407 q^{91} +10.9373 q^{92} -4.74477 q^{93} +21.3944 q^{94} -7.87903 q^{96} -1.82896 q^{97} +2.13433 q^{98} -5.50983 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 3 q^{2} + 14 q^{3} + 17 q^{4} + 3 q^{6} + 14 q^{7} + 15 q^{8} + 14 q^{9} + 4 q^{11} + 17 q^{12} - q^{13} + 3 q^{14} + 19 q^{16} - 14 q^{17} + 3 q^{18} - 6 q^{19} + 14 q^{21} + 12 q^{22} + 7 q^{23}+ \cdots + 4 q^{99}+O(q^{100}) \) 14 * q + 3 * q^2 + 14 * q^3 + 17 * q^4 + 3 * q^6 + 14 * q^7 + 15 * q^8 + 14 * q^9 + 4 * q^11 + 17 * q^12 - q^13 + 3 * q^14 + 19 * q^16 - 14 * q^17 + 3 * q^18 - 6 * q^19 + 14 * q^21 + 12 * q^22 + 7 * q^23 + 15 * q^24 - 2 * q^26 + 14 * q^27 + 17 * q^28 + 20 * q^29 - 7 * q^31 + 33 * q^32 + 4 * q^33 - 3 * q^34 + 17 * q^36 + 9 * q^37 + 18 * q^38 - q^39 + 25 * q^41 + 3 * q^42 + 28 * q^43 + 24 * q^44 - 10 * q^46 + 29 * q^47 + 19 * q^48 + 14 * q^49 - 14 * q^51 + 4 * q^52 + 26 * q^53 + 3 * q^54 + 15 * q^56 - 6 * q^57 + 2 * q^58 + 14 * q^59 - 9 * q^61 + 28 * q^62 + 14 * q^63 + 27 * q^64 + 12 * q^66 + 32 * q^67 - 17 * q^68 + 7 * q^69 + 2 * q^71 + 15 * q^72 + 18 * q^73 + 64 * q^74 - 42 * q^76 + 4 * q^77 - 2 * q^78 - 4 * q^79 + 14 * q^81 + 42 * q^82 + 51 * q^83 + 17 * q^84 + 44 * q^86 + 20 * q^87 + 32 * q^88 + 38 * q^89 - q^91 - 16 * q^92 - 7 * q^93 + 8 * q^94 + 33 * q^96 + 14 * q^97 + 3 * q^98 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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