LMFDB - Embedded newform 9196.2.a.x.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9196,2,Mod(1,9196)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9196, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9196.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9196 = 2^{2} \cdot 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9196.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:	$73.4304296988$
Analytic rank:	$0$
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} - 26 x^{18} + 208 x^{17} + 185 x^{16} - 2910 x^{15} + 687 x^{14} + 21067 x^{13} + \cdots - 3520 $ x^20 - 6x^19 - 26x^18 + 208x^17 + 185x^16 - 2910x^15 + 687x^14 + 21067x^13 - 16530x^12 - 83962x^11 + 93139x^10 + 180932x^9 - 242362x^8 - 194923x^7 + 303963x^6 + 96332x^5 - 176108x^4 - 24984x^3 + 44976x^2 + 3840*x - 3520
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 836)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.33451$ of defining polynomial
Character	$\chi$	$=$	9196.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.33451 q^{3} -3.18058 q^{5} -2.36116 q^{7} +2.44994 q^{9} +1.55121 q^{13} +7.42511 q^{15} -6.41700 q^{17} -1.00000 q^{19} +5.51215 q^{21} +6.02916 q^{23} +5.11612 q^{25} +1.28411 q^{27} +2.89575 q^{29} -7.92142 q^{31} +7.50986 q^{35} +10.8063 q^{37} -3.62131 q^{39} -7.48426 q^{41} -11.4166 q^{43} -7.79225 q^{45} +5.57862 q^{47} -1.42493 q^{49} +14.9806 q^{51} -11.2927 q^{53} +2.33451 q^{57} -15.1137 q^{59} -6.31845 q^{61} -5.78470 q^{63} -4.93375 q^{65} -4.35479 q^{67} -14.0751 q^{69} +4.78575 q^{71} -14.3122 q^{73} -11.9436 q^{75} +1.49793 q^{79} -10.3476 q^{81} -4.13113 q^{83} +20.4098 q^{85} -6.76016 q^{87} +3.05209 q^{89} -3.66265 q^{91} +18.4926 q^{93} +3.18058 q^{95} -9.16951 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 6 q^{3} + 2 q^{5} + 4 q^{7} + 28 q^{9} - 5 q^{13} + 14 q^{15} - 6 q^{17} - 20 q^{19} - q^{21} + 18 q^{23} + 44 q^{25} + 24 q^{27} + q^{29} + 23 q^{31} - 36 q^{35} + q^{37} + 21 q^{39} + 6 q^{41}+ \cdots + 9 q^{97}+O(q^{100}) $ 20 * q + 6 * q^3 + 2 * q^5 + 4 * q^7 + 28 * q^9 - 5 * q^13 + 14 * q^15 - 6 * q^17 - 20 * q^19 - q^21 + 18 * q^23 + 44 * q^25 + 24 * q^27 + q^29 + 23 * q^31 - 36 * q^35 + q^37 + 21 * q^39 + 6 * q^41 - 8 * q^43 - 15 * q^45 + 34 * q^47 + 42 * q^49 + 14 * q^51 - 10 * q^53 - 6 * q^57 + 52 * q^59 - 2 * q^61 + 59 * q^63 + 25 * q^65 + 14 * q^67 + 15 * q^69 + 88 * q^71 - 13 * q^73 + 15 * q^75 - 2 * q^79 + 48 * q^81 - 2 * q^83 + 11 * q^85 + 37 * q^87 + 46 * q^89 - 4 * q^91 + 2 * q^93 - 2 * q^95 + 9 * q^97

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 $

$2$	0	0
$2$	0	0
$3$	−2.33451	−1.34783	−0.673915	−	0.738809i	$-0.735389\pi$
$3$	−2.33451	−1.34783	−0.673915	+	0.738809i	$0.735389\pi$
$4$	0	0
$5$	−3.18058	−1.42240	−0.711200	−	0.702989i	$-0.751848\pi$
$5$	−3.18058	−1.42240	−0.711200	+	0.702989i	$0.751848\pi$
$6$	0	0
$7$	−2.36116	−0.892434	−0.446217	−	0.894925i	$-0.647229\pi$
$7$	−2.36116	−0.892434	−0.446217	+	0.894925i	$0.647229\pi$
$8$	0	0
$9$	2.44994	0.816648
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.55121	0.430228	0.215114	−	0.976589i	$-0.430988\pi$
$13$	1.55121	0.430228	0.215114	+	0.976589i	$0.430988\pi$
$14$	0	0
$15$	7.42511	1.91716
$16$	0	0
$17$	−6.41700	−1.55635	−0.778176	−	0.628047i	$-0.783855\pi$
$17$	−6.41700	−1.55635	−0.778176	+	0.628047i	$0.783855\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	5.51215	1.20285
$22$	0	0
$23$	6.02916	1.25717	0.628583	−	0.777742i	$-0.283635\pi$
$23$	6.02916	1.25717	0.628583	+	0.777742i	$0.283635\pi$
$24$	0	0
$25$	5.11612	1.02322
$26$	0	0
$27$	1.28411	0.247128
$28$	0	0
$29$	2.89575	0.537727	0.268863	−	0.963178i	$-0.413352\pi$
$29$	2.89575	0.537727	0.268863	+	0.963178i	$0.413352\pi$
$30$	0	0
$31$	−7.92142	−1.42273	−0.711364	−	0.702823i	$-0.751922\pi$
$31$	−7.92142	−1.42273	−0.711364	+	0.702823i	$0.751922\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	7.50986	1.26940
$36$	0	0
$37$	10.8063	1.77654	0.888271	−	0.459320i	$-0.151907\pi$
$37$	10.8063	1.77654	0.888271	+	0.459320i	$0.151907\pi$
$38$	0	0
$39$	−3.62131	−0.579874
$40$	0	0
$41$	−7.48426	−1.16885	−0.584423	−	0.811449i	$-0.698679\pi$
$41$	−7.48426	−1.16885	−0.584423	+	0.811449i	$0.698679\pi$
$42$	0	0
$43$	−11.4166	−1.74102	−0.870508	−	0.492154i	$-0.836210\pi$
$43$	−11.4166	−1.74102	−0.870508	+	0.492154i	$0.836210\pi$
$44$	0	0
$45$	−7.79225	−1.16160
$46$	0	0
$47$	5.57862	0.813726	0.406863	−	0.913489i	$-0.366623\pi$
$47$	5.57862	0.813726	0.406863	+	0.913489i	$0.366623\pi$
$48$	0	0
$49$	−1.42493	−0.203562
$50$	0	0
$51$	14.9806	2.09770
$52$	0	0
$53$	−11.2927	−1.55118	−0.775589	−	0.631238i	$-0.782547\pi$
$53$	−11.2927	−1.55118	−0.775589	+	0.631238i	$0.782547\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.33451	0.309214
$58$	0	0
$59$	−15.1137	−1.96764	−0.983819	−	0.179165i	$-0.942660\pi$
$59$	−15.1137	−1.96764	−0.983819	+	0.179165i	$0.942660\pi$
$60$	0	0
$61$	−6.31845	−0.808995	−0.404497	−	0.914539i	$-0.632554\pi$
$61$	−6.31845	−0.808995	−0.404497	+	0.914539i	$0.632554\pi$
$62$	0	0
$63$	−5.78470	−0.728804
$64$	0	0
$65$	−4.93375	−0.611956
$66$	0	0
$67$	−4.35479	−0.532022	−0.266011	−	0.963970i	$-0.585706\pi$
$67$	−4.35479	−0.532022	−0.266011	+	0.963970i	$0.585706\pi$
$68$	0	0
$69$	−14.0751	−1.69445
$70$	0	0
$71$	4.78575	0.567965	0.283982	−	0.958830i	$-0.408344\pi$
$71$	4.78575	0.567965	0.283982	+	0.958830i	$0.408344\pi$
$72$	0	0
$73$	−14.3122	−1.67511	−0.837555	−	0.546352i	$-0.816016\pi$
$73$	−14.3122	−1.67511	−0.837555	+	0.546352i	$0.816016\pi$
$74$	0	0
$75$	−11.9436	−1.37913
$76$	0	0
$77$	0	0
$78$	0	0
$79$	1.49793	0.168530	0.0842652	−	0.996443i	$-0.473146\pi$
$79$	1.49793	0.168530	0.0842652	+	0.996443i	$0.473146\pi$
$80$	0	0
$81$	−10.3476	−1.14973
$82$	0	0
$83$	−4.13113	−0.453450	−0.226725	−	0.973959i	$-0.572802\pi$
$83$	−4.13113	−0.453450	−0.226725	+	0.973959i	$0.572802\pi$
$84$	0	0
$85$	20.4098	2.21376
$86$	0	0
$87$	−6.76016	−0.724765
$88$	0	0
$89$	3.05209	0.323521	0.161760	−	0.986830i	$-0.448283\pi$
$89$	3.05209	0.323521	0.161760	+	0.986830i	$0.448283\pi$
$90$	0	0
$91$	−3.66265	−0.383950
$92$	0	0
$93$	18.4926	1.91760
$94$	0	0
$95$	3.18058	0.326321
$96$	0	0
$97$	−9.16951	−0.931023	−0.465511	−	0.885042i	$-0.654130\pi$
$97$	−9.16951	−0.931023	−0.465511	+	0.885042i	$0.654130\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.80029	−0.179136	−0.0895679	−	0.995981i	$-0.528549\pi$
$101$	−1.80029	−0.179136	−0.0895679	+	0.995981i	$0.528549\pi$
$102$	0	0
$103$	−10.9187	−1.07585	−0.537927	−	0.842991i	$-0.680793\pi$
$103$	−10.9187	−1.07585	−0.537927	+	0.842991i	$0.680793\pi$
$104$	0	0
$105$	−17.5319	−1.71093
$106$	0	0
$107$	−19.8174	−1.91582	−0.957911	−	0.287064i	$-0.907321\pi$
$107$	−19.8174	−1.91582	−0.957911	+	0.287064i	$0.907321\pi$
$108$	0	0
$109$	10.2603	0.982759	0.491380	−	0.870945i	$-0.336493\pi$
$109$	10.2603	0.982759	0.491380	+	0.870945i	$0.336493\pi$
$110$	0	0
$111$	−25.2274	−2.39448
$112$	0	0
$113$	−2.27935	−0.214423	−0.107211	−	0.994236i	$-0.534192\pi$
$113$	−2.27935	−0.214423	−0.107211	+	0.994236i	$0.534192\pi$
$114$	0	0
$115$	−19.1763	−1.78819
$116$	0	0
$117$	3.80037	0.351345
$118$	0	0
$119$	15.1516	1.38894
$120$	0	0
$121$	0	0
$122$	0	0
$123$	17.4721	1.57541
$124$	0	0
$125$	−0.369323	−0.0330333
$126$	0	0
$127$	10.0852	0.894913	0.447457	−	0.894306i	$-0.352330\pi$
$127$	10.0852	0.894913	0.447457	+	0.894306i	$0.352330\pi$
$128$	0	0
$129$	26.6522	2.34660
$130$	0	0
$131$	14.8159	1.29447	0.647234	−	0.762291i	$-0.275925\pi$
$131$	14.8159	1.29447	0.647234	+	0.762291i	$0.275925\pi$
$132$	0	0
$133$	2.36116	0.204738
$134$	0	0
$135$	−4.08423	−0.351515
$136$	0	0
$137$	3.19282	0.272781	0.136390	−	0.990655i	$-0.456450\pi$
$137$	3.19282	0.272781	0.136390	+	0.990655i	$0.456450\pi$
$138$	0	0
$139$	−1.95005	−0.165401	−0.0827007	−	0.996574i	$-0.526355\pi$
$139$	−1.95005	−0.165401	−0.0827007	+	0.996574i	$0.526355\pi$
$140$	0	0
$141$	−13.0234	−1.09677
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−9.21017	−0.764863
$146$	0	0
$147$	3.32652	0.274367
$148$	0	0
$149$	−15.1828	−1.24383	−0.621913	−	0.783087i	$-0.713644\pi$
$149$	−15.1828	−1.24383	−0.621913	+	0.783087i	$0.713644\pi$
$150$	0	0
$151$	6.34819	0.516608	0.258304	−	0.966064i	$-0.416836\pi$
$151$	6.34819	0.516608	0.258304	+	0.966064i	$0.416836\pi$
$152$	0	0
$153$	−15.7213	−1.27099
$154$	0	0
$155$	25.1947	2.02369
$156$	0	0
$157$	−5.90156	−0.470996	−0.235498	−	0.971875i	$-0.575672\pi$
$157$	−5.90156	−0.470996	−0.235498	+	0.971875i	$0.575672\pi$
$158$	0	0
$159$	26.3630	2.09073
$160$	0	0
$161$	−14.2358	−1.12194
$162$	0	0
$163$	−13.7879	−1.07995	−0.539974	−	0.841681i	$-0.681566\pi$
$163$	−13.7879	−1.07995	−0.539974	+	0.841681i	$0.681566\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−11.5017	−0.890028	−0.445014	−	0.895524i	$-0.646801\pi$
$167$	−11.5017	−0.890028	−0.445014	+	0.895524i	$0.646801\pi$
$168$	0	0
$169$	−10.5938	−0.814904
$170$	0	0
$171$	−2.44994	−0.187352
$172$	0	0
$173$	15.5256	1.18039	0.590195	−	0.807261i	$-0.299051\pi$
$173$	15.5256	1.18039	0.590195	+	0.807261i	$0.299051\pi$
$174$	0	0
$175$	−12.0800	−0.913160
$176$	0	0
$177$	35.2831	2.65204
$178$	0	0
$179$	−1.95564	−0.146171	−0.0730857	−	0.997326i	$-0.523285\pi$
$179$	−1.95564	−0.146171	−0.0730857	+	0.997326i	$0.523285\pi$
$180$	0	0
$181$	−13.2011	−0.981233	−0.490616	−	0.871376i	$-0.663228\pi$
$181$	−13.2011	−0.981233	−0.490616	+	0.871376i	$0.663228\pi$
$182$	0	0
$183$	14.7505	1.09039
$184$	0	0
$185$	−34.3703	−2.52695
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−3.03200	−0.220545
$190$	0	0
$191$	20.1308	1.45661	0.728305	−	0.685253i	$-0.240308\pi$
$191$	20.1308	1.45661	0.728305	+	0.685253i	$0.240308\pi$
$192$	0	0
$193$	−15.9501	−1.14812	−0.574058	−	0.818815i	$-0.694631\pi$
$193$	−15.9501	−1.14812	−0.574058	+	0.818815i	$0.694631\pi$
$194$	0	0
$195$	11.5179	0.824814
$196$	0	0
$197$	−2.74250	−0.195395	−0.0976976	−	0.995216i	$-0.531148\pi$
$197$	−2.74250	−0.195395	−0.0976976	+	0.995216i	$0.531148\pi$
$198$	0	0
$199$	−14.6253	−1.03676	−0.518379	−	0.855151i	$-0.673464\pi$
$199$	−14.6253	−1.03676	−0.518379	+	0.855151i	$0.673464\pi$
$200$	0	0
$201$	10.1663	0.717075
$202$	0	0
$203$	−6.83732	−0.479886
$204$	0	0
$205$	23.8043	1.66257
$206$	0	0
$207$	14.7711	1.02666
$208$	0	0
$209$	0	0
$210$	0	0
$211$	5.51901	0.379945	0.189972	−	0.981789i	$-0.439160\pi$
$211$	5.51901	0.379945	0.189972	+	0.981789i	$0.439160\pi$
$212$	0	0
$213$	−11.1724	−0.765520
$214$	0	0
$215$	36.3115	2.47642
$216$	0	0
$217$	18.7037	1.26969
$218$	0	0
$219$	33.4119	2.25777
$220$	0	0
$221$	−9.95411	−0.669586
$222$	0	0
$223$	9.59470	0.642508	0.321254	−	0.946993i	$-0.395896\pi$
$223$	9.59470	0.642508	0.321254	+	0.946993i	$0.395896\pi$
$224$	0	0
$225$	12.5342	0.835613
$226$	0	0
$227$	−4.91969	−0.326531	−0.163266	−	0.986582i	$-0.552203\pi$
$227$	−4.91969	−0.326531	−0.163266	+	0.986582i	$0.552203\pi$
$228$	0	0
$229$	−6.02114	−0.397888	−0.198944	−	0.980011i	$-0.563751\pi$
$229$	−6.02114	−0.397888	−0.198944	+	0.980011i	$0.563751\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.8685	−1.23612	−0.618059	−	0.786131i	$-0.712081\pi$
$233$	−18.8685	−1.23612	−0.618059	+	0.786131i	$0.712081\pi$
$234$	0	0
$235$	−17.7433	−1.15744
$236$	0	0
$237$	−3.49694	−0.227150
$238$	0	0
$239$	−0.222128	−0.0143683	−0.00718413	−	0.999974i	$-0.502287\pi$
$239$	−0.222128	−0.0143683	−0.00718413	+	0.999974i	$0.502287\pi$
$240$	0	0
$241$	−8.24782	−0.531289	−0.265644	−	0.964071i	$-0.585585\pi$
$241$	−8.24782	−0.531289	−0.265644	+	0.964071i	$0.585585\pi$
$242$	0	0
$243$	20.3043	1.30252
$244$	0	0
$245$	4.53211	0.289546
$246$	0	0
$247$	−1.55121	−0.0987010
$248$	0	0
$249$	9.64417	0.611174
$250$	0	0
$251$	−8.22262	−0.519007	−0.259504	−	0.965742i	$-0.583559\pi$
$251$	−8.22262	−0.519007	−0.259504	+	0.965742i	$0.583559\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−47.6469	−2.98377
$256$	0	0
$257$	2.52235	0.157340	0.0786700	−	0.996901i	$-0.474933\pi$
$257$	2.52235	0.157340	0.0786700	+	0.996901i	$0.474933\pi$
$258$	0	0
$259$	−25.5153	−1.58545
$260$	0	0
$261$	7.09442	0.439133
$262$	0	0
$263$	3.80875	0.234857	0.117429	−	0.993081i	$-0.462535\pi$
$263$	3.80875	0.234857	0.117429	+	0.993081i	$0.462535\pi$
$264$	0	0
$265$	35.9175	2.20640
$266$	0	0
$267$	−7.12514	−0.436051
$268$	0	0
$269$	−20.0510	−1.22253	−0.611267	−	0.791425i	$-0.709340\pi$
$269$	−20.0510	−1.22253	−0.611267	+	0.791425i	$0.709340\pi$
$270$	0	0
$271$	−18.3780	−1.11638	−0.558191	−	0.829712i	$-0.688505\pi$
$271$	−18.3780	−1.11638	−0.558191	+	0.829712i	$0.688505\pi$
$272$	0	0
$273$	8.55050	0.517500
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−13.6835	−0.822162	−0.411081	−	0.911599i	$-0.634849\pi$
$277$	−13.6835	−0.822162	−0.411081	+	0.911599i	$0.634849\pi$
$278$	0	0
$279$	−19.4070	−1.16187
$280$	0	0
$281$	1.77827	0.106083	0.0530414	−	0.998592i	$-0.483108\pi$
$281$	1.77827	0.106083	0.0530414	+	0.998592i	$0.483108\pi$
$282$	0	0
$283$	22.2154	1.32057	0.660284	−	0.751016i	$-0.270436\pi$
$283$	22.2154	1.32057	0.660284	+	0.751016i	$0.270436\pi$
$284$	0	0
$285$	−7.42511	−0.439826
$286$	0	0
$287$	17.6715	1.04312
$288$	0	0
$289$	24.1779	1.42223
$290$	0	0
$291$	21.4063	1.25486
$292$	0	0
$293$	−21.0125	−1.22756	−0.613781	−	0.789476i	$-0.710352\pi$
$293$	−21.0125	−1.22756	−0.613781	+	0.789476i	$0.710352\pi$
$294$	0	0
$295$	48.0704	2.79877
$296$	0	0
$297$	0	0
$298$	0	0
$299$	9.35248	0.540868
$300$	0	0
$301$	26.9564	1.55374
$302$	0	0
$303$	4.20280	0.241445
$304$	0	0
$305$	20.0964	1.15071
$306$	0	0
$307$	−21.2809	−1.21457	−0.607283	−	0.794485i	$-0.707741\pi$
$307$	−21.2809	−1.21457	−0.607283	+	0.794485i	$0.707741\pi$
$308$	0	0
$309$	25.4899	1.45007
$310$	0	0
$311$	−3.04208	−0.172501	−0.0862503	−	0.996273i	$-0.527488\pi$
$311$	−3.04208	−0.172501	−0.0862503	+	0.996273i	$0.527488\pi$
$312$	0	0
$313$	0.635065	0.0358960	0.0179480	−	0.999839i	$-0.494287\pi$
$313$	0.635065	0.0358960	0.0179480	+	0.999839i	$0.494287\pi$
$314$	0	0
$315$	18.3987	1.03665
$316$	0	0
$317$	−11.7310	−0.658881	−0.329440	−	0.944176i	$-0.606860\pi$
$317$	−11.7310	−0.658881	−0.329440	+	0.944176i	$0.606860\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	46.2640	2.58220
$322$	0	0
$323$	6.41700	0.357051
$324$	0	0
$325$	7.93617	0.440219
$326$	0	0
$327$	−23.9528	−1.32459
$328$	0	0
$329$	−13.1720	−0.726197
$330$	0	0
$331$	−0.468473	−0.0257496	−0.0128748	−	0.999917i	$-0.504098\pi$
$331$	−0.468473	−0.0257496	−0.0128748	+	0.999917i	$0.504098\pi$
$332$	0	0
$333$	26.4748	1.45081
$334$	0	0
$335$	13.8508	0.756748
$336$	0	0
$337$	−5.66986	−0.308857	−0.154428	−	0.988004i	$-0.549354\pi$
$337$	−5.66986	−0.308857	−0.154428	+	0.988004i	$0.549354\pi$
$338$	0	0
$339$	5.32116	0.289006
$340$	0	0
$341$	0	0
$342$	0	0
$343$	19.8926	1.07410
$344$	0	0
$345$	44.7672	2.41018
$346$	0	0
$347$	−23.4994	−1.26151	−0.630757	−	0.775981i	$-0.717255\pi$
$347$	−23.4994	−1.26151	−0.630757	+	0.775981i	$0.717255\pi$
$348$	0	0
$349$	7.03599	0.376628	0.188314	−	0.982109i	$-0.439698\pi$
$349$	7.03599	0.376628	0.188314	+	0.982109i	$0.439698\pi$
$350$	0	0
$351$	1.99193	0.106321
$352$	0	0
$353$	7.30547	0.388831	0.194415	−	0.980919i	$-0.437719\pi$
$353$	7.30547	0.388831	0.194415	+	0.980919i	$0.437719\pi$
$354$	0	0
$355$	−15.2215	−0.807873
$356$	0	0
$357$	−35.3715	−1.87206
$358$	0	0
$359$	14.9887	0.791071	0.395535	−	0.918451i	$-0.370559\pi$
$359$	14.9887	0.791071	0.395535	+	0.918451i	$0.370559\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	45.5210	2.38268
$366$	0	0
$367$	29.5862	1.54439	0.772194	−	0.635387i	$-0.219160\pi$
$367$	29.5862	1.54439	0.772194	+	0.635387i	$0.219160\pi$
$368$	0	0
$369$	−18.3360	−0.954535
$370$	0	0
$371$	26.6640	1.38432
$372$	0	0
$373$	−18.9139	−0.979325	−0.489663	−	0.871912i	$-0.662880\pi$
$373$	−18.9139	−0.979325	−0.489663	+	0.871912i	$0.662880\pi$
$374$	0	0
$375$	0.862189	0.0445233
$376$	0	0
$377$	4.49191	0.231345
$378$	0	0
$379$	−2.25390	−0.115775	−0.0578876	−	0.998323i	$-0.518436\pi$
$379$	−2.25390	−0.115775	−0.0578876	+	0.998323i	$0.518436\pi$
$380$	0	0
$381$	−23.5439	−1.20619
$382$	0	0
$383$	5.37133	0.274462	0.137231	−	0.990539i	$-0.456180\pi$
$383$	5.37133	0.274462	0.137231	+	0.990539i	$0.456180\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−27.9700	−1.42180
$388$	0	0
$389$	2.30019	0.116624	0.0583121	−	0.998298i	$-0.481428\pi$
$389$	2.30019	0.116624	0.0583121	+	0.998298i	$0.481428\pi$
$390$	0	0
$391$	−38.6891	−1.95659
$392$	0	0
$393$	−34.5878	−1.74472
$394$	0	0
$395$	−4.76430	−0.239718
$396$	0	0
$397$	−26.1429	−1.31207	−0.656037	−	0.754729i	$-0.727768\pi$
$397$	−26.1429	−1.31207	−0.656037	+	0.754729i	$0.727768\pi$
$398$	0	0
$399$	−5.51215	−0.275953
$400$	0	0
$401$	10.7720	0.537927	0.268963	−	0.963150i	$-0.413319\pi$
$401$	10.7720	0.537927	0.268963	+	0.963150i	$0.413319\pi$
$402$	0	0
$403$	−12.2878	−0.612098
$404$	0	0
$405$	32.9114	1.63538
$406$	0	0
$407$	0	0
$408$	0	0
$409$	26.4549	1.30811	0.654055	−	0.756447i	$-0.273066\pi$
$409$	26.4549	1.30811	0.654055	+	0.756447i	$0.273066\pi$
$410$	0	0
$411$	−7.45367	−0.367662
$412$	0	0
$413$	35.6859	1.75599
$414$	0	0
$415$	13.1394	0.644988
$416$	0	0
$417$	4.55242	0.222933
$418$	0	0
$419$	31.5810	1.54283	0.771416	−	0.636332i	$-0.219549\pi$
$419$	31.5810	1.54283	0.771416	+	0.636332i	$0.219549\pi$
$420$	0	0
$421$	−21.5369	−1.04964	−0.524822	−	0.851212i	$-0.675868\pi$
$421$	−21.5369	−1.04964	−0.524822	+	0.851212i	$0.675868\pi$
$422$	0	0
$423$	13.6673	0.664528
$424$	0	0
$425$	−32.8301	−1.59250
$426$	0	0
$427$	14.9189	0.721974
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−19.4419	−0.936481	−0.468241	−	0.883601i	$-0.655112\pi$
$431$	−19.4419	−0.936481	−0.468241	+	0.883601i	$0.655112\pi$
$432$	0	0
$433$	−8.23518	−0.395758	−0.197879	−	0.980226i	$-0.563405\pi$
$433$	−8.23518	−0.395758	−0.197879	+	0.980226i	$0.563405\pi$
$434$	0	0
$435$	21.5012	1.03091
$436$	0	0
$437$	−6.02916	−0.288414
$438$	0	0
$439$	−16.4188	−0.783628	−0.391814	−	0.920044i	$-0.628152\pi$
$439$	−16.4188	−0.783628	−0.391814	+	0.920044i	$0.628152\pi$
$440$	0	0
$441$	−3.49100	−0.166238
$442$	0	0
$443$	36.1348	1.71682	0.858408	−	0.512968i	$-0.171454\pi$
$443$	36.1348	1.71682	0.858408	+	0.512968i	$0.171454\pi$
$444$	0	0
$445$	−9.70743	−0.460176
$446$	0	0
$447$	35.4445	1.67647
$448$	0	0
$449$	40.5156	1.91205	0.956024	−	0.293288i	$-0.0947494\pi$
$449$	40.5156	1.91205	0.956024	+	0.293288i	$0.0947494\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−14.8199	−0.696301
$454$	0	0
$455$	11.6494	0.546131
$456$	0	0
$457$	−15.0631	−0.704623	−0.352312	−	0.935883i	$-0.614604\pi$
$457$	−15.0631	−0.704623	−0.352312	+	0.935883i	$0.614604\pi$
$458$	0	0
$459$	−8.24016	−0.384618
$460$	0	0
$461$	28.7591	1.33945	0.669723	−	0.742611i	$-0.266413\pi$
$461$	28.7591	1.33945	0.669723	+	0.742611i	$0.266413\pi$
$462$	0	0
$463$	25.1929	1.17081	0.585407	−	0.810740i	$-0.300935\pi$
$463$	25.1929	1.17081	0.585407	+	0.810740i	$0.300935\pi$
$464$	0	0
$465$	−58.8174	−2.72759
$466$	0	0
$467$	7.70903	0.356731	0.178366	−	0.983964i	$-0.442919\pi$
$467$	7.70903	0.356731	0.178366	+	0.983964i	$0.442919\pi$
$468$	0	0
$469$	10.2823	0.474794
$470$	0	0
$471$	13.7773	0.634822
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−5.11612	−0.234744
$476$	0	0
$477$	−27.6666	−1.26677
$478$	0	0
$479$	−8.97151	−0.409919	−0.204959	−	0.978770i	$-0.565706\pi$
$479$	−8.97151	−0.409919	−0.204959	+	0.978770i	$0.565706\pi$
$480$	0	0
$481$	16.7628	0.764318
$482$	0	0
$483$	33.2336	1.51218
$484$	0	0
$485$	29.1644	1.32429
$486$	0	0
$487$	8.27098	0.374794	0.187397	−	0.982284i	$-0.439995\pi$
$487$	8.27098	0.374794	0.187397	+	0.982284i	$0.439995\pi$
$488$	0	0
$489$	32.1879	1.45559
$490$	0	0
$491$	18.2579	0.823966	0.411983	−	0.911192i	$-0.364836\pi$
$491$	18.2579	0.823966	0.411983	+	0.911192i	$0.364836\pi$
$492$	0	0
$493$	−18.5820	−0.836892
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−11.2999	−0.506871
$498$	0	0
$499$	18.9410	0.847914	0.423957	−	0.905682i	$-0.360641\pi$
$499$	18.9410	0.847914	0.423957	+	0.905682i	$0.360641\pi$
$500$	0	0
$501$	26.8508	1.19961
$502$	0	0
$503$	31.0787	1.38573	0.692865	−	0.721067i	$-0.256348\pi$
$503$	31.0787	1.38573	0.692865	+	0.721067i	$0.256348\pi$
$504$	0	0
$505$	5.72598	0.254803
$506$	0	0
$507$	24.7312	1.09835
$508$	0	0
$509$	−12.4184	−0.550434	−0.275217	−	0.961382i	$-0.588750\pi$
$509$	−12.4184	−0.550434	−0.275217	+	0.961382i	$0.588750\pi$
$510$	0	0
$511$	33.7933	1.49493
$512$	0	0
$513$	−1.28411	−0.0566950
$514$	0	0
$515$	34.7280	1.53030
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−36.2447	−1.59096
$520$	0	0
$521$	−21.6227	−0.947306	−0.473653	−	0.880712i	$-0.657065\pi$
$521$	−21.6227	−0.947306	−0.473653	+	0.880712i	$0.657065\pi$
$522$	0	0
$523$	12.8448	0.561666	0.280833	−	0.959757i	$-0.409389\pi$
$523$	12.8448	0.561666	0.280833	+	0.959757i	$0.409389\pi$
$524$	0	0
$525$	28.2008	1.23078
$526$	0	0
$527$	50.8318	2.21427
$528$	0	0
$529$	13.3508	0.580468
$530$	0	0
$531$	−37.0277	−1.60687
$532$	0	0
$533$	−11.6097	−0.502870
$534$	0	0
$535$	63.0310	2.72507
$536$	0	0
$537$	4.56546	0.197014
$538$	0	0
$539$	0	0
$540$	0	0
$541$	27.8385	1.19687	0.598435	−	0.801171i	$-0.295790\pi$
$541$	27.8385	1.19687	0.598435	+	0.801171i	$0.295790\pi$
$542$	0	0
$543$	30.8182	1.32254
$544$	0	0
$545$	−32.6338	−1.39788
$546$	0	0
$547$	−34.8419	−1.48973	−0.744867	−	0.667213i	$-0.767487\pi$
$547$	−34.8419	−1.48973	−0.744867	+	0.667213i	$0.767487\pi$
$548$	0	0
$549$	−15.4798	−0.660664
$550$	0	0
$551$	−2.89575	−0.123363
$552$	0	0
$553$	−3.53685	−0.150402
$554$	0	0
$555$	80.2378	3.40591
$556$	0	0
$557$	26.1540	1.10818	0.554090	−	0.832457i	$-0.313067\pi$
$557$	26.1540	1.10818	0.554090	+	0.832457i	$0.313067\pi$
$558$	0	0
$559$	−17.7095	−0.749034
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.3166	−0.603372	−0.301686	−	0.953407i	$-0.597549\pi$
$563$	−14.3166	−0.603372	−0.301686	+	0.953407i	$0.597549\pi$
$564$	0	0
$565$	7.24966	0.304995
$566$	0	0
$567$	24.4323	1.02606
$568$	0	0
$569$	−7.50946	−0.314813	−0.157406	−	0.987534i	$-0.550313\pi$
$569$	−7.50946	−0.314813	−0.157406	+	0.987534i	$0.550313\pi$
$570$	0	0
$571$	−10.8984	−0.456085	−0.228042	−	0.973651i	$-0.573232\pi$
$571$	−10.8984	−0.456085	−0.228042	+	0.973651i	$0.573232\pi$
$572$	0	0
$573$	−46.9955	−1.96326
$574$	0	0
$575$	30.8459	1.28636
$576$	0	0
$577$	24.9640	1.03927	0.519633	−	0.854390i	$-0.326069\pi$
$577$	24.9640	1.03927	0.519633	+	0.854390i	$0.326069\pi$
$578$	0	0
$579$	37.2357	1.54746
$580$	0	0
$581$	9.75425	0.404675
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−12.0874	−0.499753
$586$	0	0
$587$	28.5644	1.17898	0.589489	−	0.807777i	$-0.299329\pi$
$587$	28.5644	1.17898	0.589489	+	0.807777i	$0.299329\pi$
$588$	0	0
$589$	7.92142	0.326396
$590$	0	0
$591$	6.40240	0.263360
$592$	0	0
$593$	33.9740	1.39514	0.697572	−	0.716515i	$-0.254264\pi$
$593$	33.9740	1.39514	0.697572	+	0.716515i	$0.254264\pi$
$594$	0	0
$595$	−48.1908	−1.97563
$596$	0	0
$597$	34.1429	1.39737
$598$	0	0
$599$	25.2316	1.03093	0.515467	−	0.856910i	$-0.327619\pi$
$599$	25.2316	1.03093	0.515467	+	0.856910i	$0.327619\pi$
$600$	0	0
$601$	−18.1024	−0.738410	−0.369205	−	0.929348i	$-0.620370\pi$
$601$	−18.1024	−0.738410	−0.369205	+	0.929348i	$0.620370\pi$
$602$	0	0
$603$	−10.6690	−0.434474
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−24.1342	−0.979578	−0.489789	−	0.871841i	$-0.662926\pi$
$607$	−24.1342	−0.979578	−0.489789	+	0.871841i	$0.662926\pi$
$608$	0	0
$609$	15.9618	0.646805
$610$	0	0
$611$	8.65361	0.350088
$612$	0	0
$613$	−19.5388	−0.789165	−0.394582	−	0.918861i	$-0.629111\pi$
$613$	−19.5388	−0.789165	−0.394582	+	0.918861i	$0.629111\pi$
$614$	0	0
$615$	−55.5715	−2.24086
$616$	0	0
$617$	−29.8390	−1.20127	−0.600637	−	0.799522i	$-0.705086\pi$
$617$	−29.8390	−1.20127	−0.600637	+	0.799522i	$0.705086\pi$
$618$	0	0
$619$	31.3616	1.26053	0.630264	−	0.776381i	$-0.282947\pi$
$619$	31.3616	1.26053	0.630264	+	0.776381i	$0.282947\pi$
$620$	0	0
$621$	7.74213	0.310681
$622$	0	0
$623$	−7.20647	−0.288721
$624$	0	0
$625$	−24.4059	−0.976237
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−69.3439	−2.76492
$630$	0	0
$631$	−8.32475	−0.331403	−0.165702	−	0.986176i	$-0.552989\pi$
$631$	−8.32475	−0.331403	−0.165702	+	0.986176i	$0.552989\pi$
$632$	0	0
$633$	−12.8842	−0.512101
$634$	0	0
$635$	−32.0767	−1.27293
$636$	0	0
$637$	−2.21037	−0.0875779
$638$	0	0
$639$	11.7248	0.463827
$640$	0	0
$641$	−15.8607	−0.626459	−0.313230	−	0.949677i	$-0.601411\pi$
$641$	−15.8607	−0.626459	−0.313230	+	0.949677i	$0.601411\pi$
$642$	0	0
$643$	−38.4857	−1.51773	−0.758864	−	0.651249i	$-0.774245\pi$
$643$	−38.4857	−1.51773	−0.758864	+	0.651249i	$0.774245\pi$
$644$	0	0
$645$	−84.7696	−3.33780
$646$	0	0
$647$	49.6332	1.95128	0.975640	−	0.219376i	$-0.0704021\pi$
$647$	49.6332	1.95128	0.975640	+	0.219376i	$0.0704021\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−43.6641	−1.71133
$652$	0	0
$653$	13.6918	0.535801	0.267900	−	0.963447i	$-0.413670\pi$
$653$	13.6918	0.535801	0.267900	+	0.963447i	$0.413670\pi$
$654$	0	0
$655$	−47.1231	−1.84125
$656$	0	0
$657$	−35.0640	−1.36798
$658$	0	0
$659$	−12.0345	−0.468797	−0.234398	−	0.972141i	$-0.575312\pi$
$659$	−12.0345	−0.468797	−0.234398	+	0.972141i	$0.575312\pi$
$660$	0	0
$661$	7.95293	0.309333	0.154667	−	0.987967i	$-0.450570\pi$
$661$	7.95293	0.309333	0.154667	+	0.987967i	$0.450570\pi$
$662$	0	0
$663$	23.2380	0.902488
$664$	0	0
$665$	−7.50986	−0.291220
$666$	0	0
$667$	17.4589	0.676012
$668$	0	0
$669$	−22.3989	−0.865993
$670$	0	0
$671$	0	0
$672$	0	0
$673$	16.4781	0.635184	0.317592	−	0.948228i	$-0.397126\pi$
$673$	16.4781	0.635184	0.317592	+	0.948228i	$0.397126\pi$
$674$	0	0
$675$	6.56968	0.252867
$676$	0	0
$677$	−49.3868	−1.89809	−0.949045	−	0.315141i	$-0.897948\pi$
$677$	−49.3868	−1.89809	−0.949045	+	0.315141i	$0.897948\pi$
$678$	0	0
$679$	21.6507	0.830876
$680$	0	0
$681$	11.4851	0.440109
$682$	0	0
$683$	14.1002	0.539528	0.269764	−	0.962927i	$-0.413054\pi$
$683$	14.1002	0.539528	0.269764	+	0.962927i	$0.413054\pi$
$684$	0	0
$685$	−10.1550	−0.388004
$686$	0	0
$687$	14.0564	0.536286
$688$	0	0
$689$	−17.5174	−0.667360
$690$	0	0
$691$	5.78097	0.219919	0.109959	−	0.993936i	$-0.464928\pi$
$691$	5.78097	0.219919	0.109959	+	0.993936i	$0.464928\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6.20231	0.235267
$696$	0	0
$697$	48.0265	1.81913
$698$	0	0
$699$	44.0488	1.66608
$700$	0	0
$701$	−41.0485	−1.55038	−0.775191	−	0.631727i	$-0.782346\pi$
$701$	−41.0485	−1.55038	−0.775191	+	0.631727i	$0.782346\pi$
$702$	0	0
$703$	−10.8063	−0.407567
$704$	0	0
$705$	41.4219	1.56004
$706$	0	0
$707$	4.25077	0.159867
$708$	0	0
$709$	3.34601	0.125662	0.0628310	−	0.998024i	$-0.479987\pi$
$709$	3.34601	0.125662	0.0628310	+	0.998024i	$0.479987\pi$
$710$	0	0
$711$	3.66985	0.137630
$712$	0	0
$713$	−47.7595	−1.78861
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0.518560	0.0193660
$718$	0	0
$719$	11.0179	0.410898	0.205449	−	0.978668i	$-0.434135\pi$
$719$	11.0179	0.410898	0.205449	+	0.978668i	$0.434135\pi$
$720$	0	0
$721$	25.7809	0.960129
$722$	0	0
$723$	19.2546	0.716087
$724$	0	0
$725$	14.8150	0.550215
$726$	0	0
$727$	−42.0402	−1.55918	−0.779592	−	0.626287i	$-0.784574\pi$
$727$	−42.0402	−1.55918	−0.779592	+	0.626287i	$0.784574\pi$
$728$	0	0
$729$	−16.3577	−0.605841
$730$	0	0
$731$	73.2604	2.70963
$732$	0	0
$733$	−0.188990	−0.00698051	−0.00349026	−	0.999994i	$-0.501111\pi$
$733$	−0.188990	−0.00698051	−0.00349026	+	0.999994i	$0.501111\pi$
$734$	0	0
$735$	−10.5803	−0.390259
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−27.1249	−0.997804	−0.498902	−	0.866658i	$-0.666263\pi$
$739$	−27.1249	−0.997804	−0.498902	+	0.866658i	$0.666263\pi$
$740$	0	0
$741$	3.62131	0.133032
$742$	0	0
$743$	21.6111	0.792833	0.396416	−	0.918071i	$-0.370254\pi$
$743$	21.6111	0.792833	0.396416	+	0.918071i	$0.370254\pi$
$744$	0	0
$745$	48.2902	1.76922
$746$	0	0
$747$	−10.1210	−0.370309
$748$	0	0
$749$	46.7921	1.70975
$750$	0	0
$751$	50.5538	1.84473	0.922367	−	0.386314i	$-0.126252\pi$
$751$	50.5538	1.84473	0.922367	+	0.386314i	$0.126252\pi$
$752$	0	0
$753$	19.1958	0.699534
$754$	0	0
$755$	−20.1910	−0.734824
$756$	0	0
$757$	−32.0972	−1.16659	−0.583297	−	0.812259i	$-0.698238\pi$
$757$	−32.0972	−1.16659	−0.583297	+	0.812259i	$0.698238\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.59422	−0.0940402	−0.0470201	−	0.998894i	$-0.514972\pi$
$761$	−2.59422	−0.0940402	−0.0470201	+	0.998894i	$0.514972\pi$
$762$	0	0
$763$	−24.2262	−0.877048
$764$	0	0
$765$	50.0029	1.80786
$766$	0	0
$767$	−23.4445	−0.846533
$768$	0	0
$769$	23.8468	0.859937	0.429968	−	0.902844i	$-0.358525\pi$
$769$	23.8468	0.859937	0.429968	+	0.902844i	$0.358525\pi$
$770$	0	0
$771$	−5.88846	−0.212068
$772$	0	0
$773$	−22.3893	−0.805285	−0.402643	−	0.915357i	$-0.631908\pi$
$773$	−22.3893	−0.805285	−0.402643	+	0.915357i	$0.631908\pi$
$774$	0	0
$775$	−40.5269	−1.45577
$776$	0	0
$777$	59.5658	2.13691
$778$	0	0
$779$	7.48426	0.268152
$780$	0	0
$781$	0	0
$782$	0	0
$783$	3.71847	0.132887
$784$	0	0
$785$	18.7704	0.669944
$786$	0	0
$787$	−27.7732	−0.990007	−0.495003	−	0.868891i	$-0.664833\pi$
$787$	−27.7732	−0.990007	−0.495003	+	0.868891i	$0.664833\pi$
$788$	0	0
$789$	−8.89156	−0.316548
$790$	0	0
$791$	5.38190	0.191358
$792$	0	0
$793$	−9.80124	−0.348052
$794$	0	0
$795$	−83.8499	−2.97385
$796$	0	0
$797$	10.2456	0.362918	0.181459	−	0.983398i	$-0.441918\pi$
$797$	10.2456	0.362918	0.181459	+	0.983398i	$0.441918\pi$
$798$	0	0
$799$	−35.7980	−1.26644
$800$	0	0
$801$	7.47745	0.264203
$802$	0	0
$803$	0	0
$804$	0	0
$805$	45.2782	1.59585
$806$	0	0
$807$	46.8094	1.64777
$808$	0	0
$809$	−49.4163	−1.73739	−0.868693	−	0.495351i	$-0.835039\pi$
$809$	−49.4163	−1.73739	−0.868693	+	0.495351i	$0.835039\pi$
$810$	0	0
$811$	−18.7685	−0.659053	−0.329526	−	0.944146i	$-0.606889\pi$
$811$	−18.7685	−0.659053	−0.329526	+	0.944146i	$0.606889\pi$
$812$	0	0
$813$	42.9036	1.50470
$814$	0	0
$815$	43.8535	1.53612
$816$	0	0
$817$	11.4166	0.399417
$818$	0	0
$819$	−8.97328	−0.313552
$820$	0	0
$821$	36.0259	1.25731	0.628657	−	0.777683i	$-0.283605\pi$
$821$	36.0259	1.25731	0.628657	+	0.777683i	$0.283605\pi$
$822$	0	0
$823$	−31.7653	−1.10727	−0.553634	−	0.832760i	$-0.686759\pi$
$823$	−31.7653	−1.10727	−0.553634	+	0.832760i	$0.686759\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	8.65678	0.301026	0.150513	−	0.988608i	$-0.451908\pi$
$827$	8.65678	0.301026	0.150513	+	0.988608i	$0.451908\pi$
$828$	0	0
$829$	−36.2894	−1.26038	−0.630192	−	0.776440i	$-0.717024\pi$
$829$	−36.2894	−1.26038	−0.630192	+	0.776440i	$0.717024\pi$
$830$	0	0
$831$	31.9443	1.10813
$832$	0	0
$833$	9.14378	0.316813
$834$	0	0
$835$	36.5821	1.26598
$836$	0	0
$837$	−10.1720	−0.351596
$838$	0	0
$839$	12.4457	0.429673	0.214836	−	0.976650i	$-0.431078\pi$
$839$	12.4457	0.429673	0.214836	+	0.976650i	$0.431078\pi$
$840$	0	0
$841$	−20.6146	−0.710850
$842$	0	0
$843$	−4.15139	−0.142982
$844$	0	0
$845$	33.6943	1.15912
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−51.8621	−1.77990
$850$	0	0
$851$	65.1528	2.23341
$852$	0	0
$853$	−21.4135	−0.733185	−0.366593	−	0.930382i	$-0.619476\pi$
$853$	−21.4135	−0.733185	−0.366593	+	0.930382i	$0.619476\pi$
$854$	0	0
$855$	7.79225	0.266489
$856$	0	0
$857$	−54.7988	−1.87189	−0.935945	−	0.352146i	$-0.885452\pi$
$857$	−54.7988	−1.87189	−0.935945	+	0.352146i	$0.885452\pi$
$858$	0	0
$859$	−2.23130	−0.0761310	−0.0380655	−	0.999275i	$-0.512120\pi$
$859$	−2.23130	−0.0761310	−0.0380655	+	0.999275i	$0.512120\pi$
$860$	0	0
$861$	−41.2544	−1.40595
$862$	0	0
$863$	−11.6857	−0.397786	−0.198893	−	0.980021i	$-0.563735\pi$
$863$	−11.6857	−0.397786	−0.198893	+	0.980021i	$0.563735\pi$
$864$	0	0
$865$	−49.3805	−1.67899
$866$	0	0
$867$	−56.4436	−1.91692
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−6.75518	−0.228891
$872$	0	0
$873$	−22.4648	−0.760318
$874$	0	0
$875$	0.872031	0.0294800
$876$	0	0
$877$	−5.24146	−0.176992	−0.0884958	−	0.996077i	$-0.528206\pi$
$877$	−5.24146	−0.176992	−0.0884958	+	0.996077i	$0.528206\pi$
$878$	0	0
$879$	49.0539	1.65455
$880$	0	0
$881$	4.47861	0.150888	0.0754441	−	0.997150i	$-0.475963\pi$
$881$	4.47861	0.150888	0.0754441	+	0.997150i	$0.475963\pi$
$882$	0	0
$883$	−47.6539	−1.60368	−0.801841	−	0.597538i	$-0.796146\pi$
$883$	−47.6539	−1.60368	−0.801841	+	0.597538i	$0.796146\pi$
$884$	0	0
$885$	−112.221	−3.77227
$886$	0	0
$887$	−41.0165	−1.37720	−0.688599	−	0.725142i	$-0.741774\pi$
$887$	−41.0165	−1.37720	−0.688599	+	0.725142i	$0.741774\pi$
$888$	0	0
$889$	−23.8127	−0.798651
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.57862	−0.186682
$894$	0	0
$895$	6.22008	0.207914
$896$	0	0
$897$	−21.8335	−0.728999
$898$	0	0
$899$	−22.9384	−0.765040
$900$	0	0
$901$	72.4656	2.41418
$902$	0	0
$903$	−62.9301	−2.09418
$904$	0	0
$905$	41.9873	1.39571
$906$	0	0
$907$	26.6430	0.884666	0.442333	−	0.896851i	$-0.354151\pi$
$907$	26.6430	0.884666	0.442333	+	0.896851i	$0.354151\pi$
$908$	0	0
$909$	−4.41061	−0.146291
$910$	0	0
$911$	11.8236	0.391733	0.195866	−	0.980631i	$-0.437248\pi$
$911$	11.8236	0.391733	0.195866	+	0.980631i	$0.437248\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−46.9152	−1.55097
$916$	0	0
$917$	−34.9826	−1.15523
$918$	0	0
$919$	−11.9843	−0.395326	−0.197663	−	0.980270i	$-0.563335\pi$
$919$	−11.9843	−0.395326	−0.197663	+	0.980270i	$0.563335\pi$
$920$	0	0
$921$	49.6806	1.63703
$922$	0	0
$923$	7.42370	0.244354
$924$	0	0
$925$	55.2862	1.81780
$926$	0	0
$927$	−26.7503	−0.878594
$928$	0	0
$929$	−43.8181	−1.43763	−0.718813	−	0.695203i	$-0.755314\pi$
$929$	−43.8181	−1.43763	−0.718813	+	0.695203i	$0.755314\pi$
$930$	0	0
$931$	1.42493	0.0467002
$932$	0	0
$933$	7.10177	0.232502
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42.9413	1.40283	0.701415	−	0.712753i	$-0.252552\pi$
$937$	42.9413	1.40283	0.701415	+	0.712753i	$0.252552\pi$
$938$	0	0
$939$	−1.48257	−0.0483817
$940$	0	0
$941$	29.1174	0.949200	0.474600	−	0.880202i	$-0.342593\pi$
$941$	29.1174	0.949200	0.474600	+	0.880202i	$0.342593\pi$
$942$	0	0
$943$	−45.1238	−1.46943
$944$	0	0
$945$	9.64352	0.313704
$946$	0	0
$947$	−52.0067	−1.68999	−0.844996	−	0.534773i	$-0.820397\pi$
$947$	−52.0067	−1.68999	−0.844996	+	0.534773i	$0.820397\pi$
$948$	0	0
$949$	−22.2011	−0.720679
$950$	0	0
$951$	27.3862	0.888060
$952$	0	0
$953$	18.9109	0.612583	0.306292	−	0.951938i	$-0.400912\pi$
$953$	18.9109	0.612583	0.306292	+	0.951938i	$0.400912\pi$
$954$	0	0
$955$	−64.0276	−2.07188
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−7.53875	−0.243439
$960$	0	0
$961$	31.7489	1.02416
$962$	0	0
$963$	−48.5516	−1.56455
$964$	0	0
$965$	50.7307	1.63308
$966$	0	0
$967$	34.5614	1.11142	0.555710	−	0.831376i	$-0.312446\pi$
$967$	34.5614	1.11142	0.555710	+	0.831376i	$0.312446\pi$
$968$	0	0
$969$	−14.9806	−0.481245
$970$	0	0
$971$	34.3839	1.10343	0.551716	−	0.834032i	$-0.313973\pi$
$971$	34.3839	1.10343	0.551716	+	0.834032i	$0.313973\pi$
$972$	0	0
$973$	4.60439	0.147610
$974$	0	0
$975$	−18.5271	−0.593341
$976$	0	0
$977$	32.9207	1.05323	0.526614	−	0.850105i	$-0.323461\pi$
$977$	32.9207	1.05323	0.526614	+	0.850105i	$0.323461\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	25.1372	0.802568
$982$	0	0
$983$	−18.6801	−0.595804	−0.297902	−	0.954596i	$-0.596287\pi$
$983$	−18.6801	−0.595804	−0.297902	+	0.954596i	$0.596287\pi$
$984$	0	0
$985$	8.72276	0.277930
$986$	0	0
$987$	30.7502	0.978790
$988$	0	0
$989$	−68.8325	−2.18875
$990$	0	0
$991$	40.6048	1.28985	0.644927	−	0.764244i	$-0.276888\pi$
$991$	40.6048	1.28985	0.644927	+	0.764244i	$0.276888\pi$
$992$	0	0
$993$	1.09366	0.0347061
$994$	0	0
$995$	46.5169	1.47469
$996$	0	0
$997$	−22.4881	−0.712204	−0.356102	−	0.934447i	$-0.615894\pi$
$997$	−22.4881	−0.712204	−0.356102	+	0.934447i	$0.615894\pi$
$998$	0	0
$999$	13.8765	0.439033

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.x.1.4		20
11.3	even	5		836.2.j.c.229.9	✓	40
11.4	even	5		836.2.j.c.533.9	yes	40
11.10	odd	2		9196.2.a.w.1.4		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.j.c.229.9	✓	40	11.3	even	5
836.2.j.c.533.9	yes	40	11.4	even	5
9196.2.a.w.1.4		20	11.10	odd	2
9196.2.a.x.1.4		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9196 = 2^{2} \cdot 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9196.a (trivial)

\(f(q)\)	\(=\)	\(q-2.33451 q^{3} -3.18058 q^{5} -2.36116 q^{7} +2.44994 q^{9} +1.55121 q^{13} +7.42511 q^{15} -6.41700 q^{17} -1.00000 q^{19} +5.51215 q^{21} +6.02916 q^{23} +5.11612 q^{25} +1.28411 q^{27} +2.89575 q^{29} -7.92142 q^{31} +7.50986 q^{35} +10.8063 q^{37} -3.62131 q^{39} -7.48426 q^{41} -11.4166 q^{43} -7.79225 q^{45} +5.57862 q^{47} -1.42493 q^{49} +14.9806 q^{51} -11.2927 q^{53} +2.33451 q^{57} -15.1137 q^{59} -6.31845 q^{61} -5.78470 q^{63} -4.93375 q^{65} -4.35479 q^{67} -14.0751 q^{69} +4.78575 q^{71} -14.3122 q^{73} -11.9436 q^{75} +1.49793 q^{79} -10.3476 q^{81} -4.13113 q^{83} +20.4098 q^{85} -6.76016 q^{87} +3.05209 q^{89} -3.66265 q^{91} +18.4926 q^{93} +3.18058 q^{95} -9.16951 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 6 q^{3} + 2 q^{5} + 4 q^{7} + 28 q^{9} - 5 q^{13} + 14 q^{15} - 6 q^{17} - 20 q^{19} - q^{21} + 18 q^{23} + 44 q^{25} + 24 q^{27} + q^{29} + 23 q^{31} - 36 q^{35} + q^{37} + 21 q^{39} + 6 q^{41}+ \cdots + 9 q^{97}+O(q^{100}) \) 20 * q + 6 * q^3 + 2 * q^5 + 4 * q^7 + 28 * q^9 - 5 * q^13 + 14 * q^15 - 6 * q^17 - 20 * q^19 - q^21 + 18 * q^23 + 44 * q^25 + 24 * q^27 + q^29 + 23 * q^31 - 36 * q^35 + q^37 + 21 * q^39 + 6 * q^41 - 8 * q^43 - 15 * q^45 + 34 * q^47 + 42 * q^49 + 14 * q^51 - 10 * q^53 - 6 * q^57 + 52 * q^59 - 2 * q^61 + 59 * q^63 + 25 * q^65 + 14 * q^67 + 15 * q^69 + 88 * q^71 - 13 * q^73 + 15 * q^75 - 2 * q^79 + 48 * q^81 - 2 * q^83 + 11 * q^85 + 37 * q^87 + 46 * q^89 - 4 * q^91 + 2 * q^93 - 2 * q^95 + 9 * q^97

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