LMFDB - Embedded newform 9196.2.a.u.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9196,2,Mod(1,9196)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [14,0,-2,0,-6,0,-4,0,12,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$73.4304296988$
Analytic rank:	$1$
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} - 2 x^{13} - 25 x^{12} + 52 x^{11} + 222 x^{10} - 492 x^{9} - 800 x^{8} + 1984 x^{7} + 854 x^{6} + \cdots + 13 $ x^14 - 2x^13 - 25x^12 + 52x^11 + 222x^10 - 492x^9 - 800x^8 + 1984x^7 + 854x^6 - 2946x^5 + 300x^4 + 878x^3 - 194x^2 - 52*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.267324$ 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)

$f(q)$	$=$	$q-0.267324 q^{3} +3.78137 q^{5} -2.20748 q^{7} -2.92854 q^{9} -1.08019 q^{13} -1.01085 q^{15} +2.61660 q^{17} +1.00000 q^{19} +0.590112 q^{21} +1.70282 q^{23} +9.29875 q^{25} +1.58484 q^{27} -7.08406 q^{29} -2.86036 q^{31} -8.34728 q^{35} +3.73304 q^{37} +0.288760 q^{39} -2.75213 q^{41} -6.77495 q^{43} -11.0739 q^{45} -2.53752 q^{47} -2.12705 q^{49} -0.699481 q^{51} -10.7575 q^{53} -0.267324 q^{57} +8.28389 q^{59} +7.41064 q^{61} +6.46468 q^{63} -4.08458 q^{65} +5.90019 q^{67} -0.455205 q^{69} -10.2320 q^{71} -10.2358 q^{73} -2.48578 q^{75} -4.39977 q^{79} +8.36195 q^{81} -1.18562 q^{83} +9.89433 q^{85} +1.89374 q^{87} +13.6214 q^{89} +2.38449 q^{91} +0.764644 q^{93} +3.78137 q^{95} -2.27485 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 2 q^{3} - 6 q^{5} - 4 q^{7} + 12 q^{9} - 4 q^{13} + 2 q^{15} - 8 q^{17} + 14 q^{19} - 18 q^{21} - 2 q^{23} + 8 q^{25} + 10 q^{27} + 4 q^{29} - 2 q^{31} - 24 q^{35} - 10 q^{37} - 24 q^{39} - 4 q^{41}+ \cdots - 20 q^{97}+O(q^{100}) $ 14 * q - 2 * q^3 - 6 * q^5 - 4 * q^7 + 12 * q^9 - 4 * q^13 + 2 * q^15 - 8 * q^17 + 14 * q^19 - 18 * q^21 - 2 * q^23 + 8 * q^25 + 10 * q^27 + 4 * q^29 - 2 * q^31 - 24 * q^35 - 10 * q^37 - 24 * q^39 - 4 * q^41 - 32 * q^43 - 20 * q^45 - 4 * q^47 - 14 * q^49 - 22 * q^51 + 6 * q^53 - 2 * q^57 - 6 * q^59 + 4 * q^61 + 16 * q^63 - 54 * q^65 + 6 * q^67 + 4 * q^69 - 6 * q^71 + 24 * q^73 - 10 * q^75 - 48 * q^79 + 2 * q^81 + 40 * q^85 - 28 * q^87 + 2 * q^89 + 28 * q^91 + 34 * q^93 - 6 * q^95 - 20 * 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
$3$	−0.267324	−0.154340	−0.0771699	−	0.997018i	$-0.524588\pi$
$3$	−0.267324	−0.154340	−0.0771699	+	0.997018i	$0.524588\pi$
$4$	0	0
$5$	3.78137	1.69108	0.845540	−	0.533913i	$-0.179279\pi$
$5$	3.78137	1.69108	0.845540	+	0.533913i	$0.179279\pi$
$6$	0	0
$7$	−2.20748	−0.834348	−0.417174	−	0.908827i	$-0.636979\pi$
$7$	−2.20748	−0.834348	−0.417174	+	0.908827i	$0.636979\pi$
$8$	0	0
$9$	−2.92854	−0.976179
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.08019	−0.299590	−0.149795	−	0.988717i	$-0.547861\pi$
$13$	−1.08019	−0.299590	−0.149795	+	0.988717i	$0.547861\pi$
$14$	0	0
$15$	−1.01085	−0.261001
$16$	0	0
$17$	2.61660	0.634619	0.317310	−	0.948322i	$-0.397221\pi$
$17$	2.61660	0.634619	0.317310	+	0.948322i	$0.397221\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.590112	0.128773
$22$	0	0
$23$	1.70282	0.355063	0.177531	−	0.984115i	$-0.443189\pi$
$23$	1.70282	0.355063	0.177531	+	0.984115i	$0.443189\pi$
$24$	0	0
$25$	9.29875	1.85975
$26$	0	0
$27$	1.58484	0.305003
$28$	0	0
$29$	−7.08406	−1.31548	−0.657739	−	0.753246i	$-0.728487\pi$
$29$	−7.08406	−1.31548	−0.657739	+	0.753246i	$0.728487\pi$
$30$	0	0
$31$	−2.86036	−0.513736	−0.256868	−	0.966446i	$-0.582691\pi$
$31$	−2.86036	−0.513736	−0.256868	+	0.966446i	$0.582691\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−8.34728	−1.41095
$36$	0	0
$37$	3.73304	0.613707	0.306854	−	0.951757i	$-0.400724\pi$
$37$	3.73304	0.613707	0.306854	+	0.951757i	$0.400724\pi$
$38$	0	0
$39$	0.288760	0.0462386
$40$	0	0
$41$	−2.75213	−0.429810	−0.214905	−	0.976635i	$-0.568944\pi$
$41$	−2.75213	−0.429810	−0.214905	+	0.976635i	$0.568944\pi$
$42$	0	0
$43$	−6.77495	−1.03317	−0.516585	−	0.856236i	$-0.672797\pi$
$43$	−6.77495	−1.03317	−0.516585	+	0.856236i	$0.672797\pi$
$44$	0	0
$45$	−11.0739	−1.65080
$46$	0	0
$47$	−2.53752	−0.370135	−0.185067	−	0.982726i	$-0.559250\pi$
$47$	−2.53752	−0.370135	−0.185067	+	0.982726i	$0.559250\pi$
$48$	0	0
$49$	−2.12705	−0.303864
$50$	0	0
$51$	−0.699481	−0.0979469
$52$	0	0
$53$	−10.7575	−1.47766	−0.738830	−	0.673891i	$-0.764622\pi$
$53$	−10.7575	−1.47766	−0.738830	+	0.673891i	$0.764622\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.267324	−0.0354080
$58$	0	0
$59$	8.28389	1.07847	0.539235	−	0.842155i	$-0.318713\pi$
$59$	8.28389	1.07847	0.539235	+	0.842155i	$0.318713\pi$
$60$	0	0
$61$	7.41064	0.948835	0.474418	−	0.880300i	$-0.342659\pi$
$61$	7.41064	0.948835	0.474418	+	0.880300i	$0.342659\pi$
$62$	0	0
$63$	6.46468	0.814473
$64$	0	0
$65$	−4.08458	−0.506630
$66$	0	0
$67$	5.90019	0.720823	0.360411	−	0.932794i	$-0.382636\pi$
$67$	5.90019	0.720823	0.360411	+	0.932794i	$0.382636\pi$
$68$	0	0
$69$	−0.455205	−0.0548003
$70$	0	0
$71$	−10.2320	−1.21431	−0.607157	−	0.794582i	$-0.707690\pi$
$71$	−10.2320	−1.21431	−0.607157	+	0.794582i	$0.707690\pi$
$72$	0	0
$73$	−10.2358	−1.19801	−0.599005	−	0.800746i	$-0.704437\pi$
$73$	−10.2358	−1.19801	−0.599005	+	0.800746i	$0.704437\pi$
$74$	0	0
$75$	−2.48578	−0.287033
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.39977	−0.495013	−0.247506	−	0.968886i	$-0.579611\pi$
$79$	−4.39977	−0.495013	−0.247506	+	0.968886i	$0.579611\pi$
$80$	0	0
$81$	8.36195	0.929105
$82$	0	0
$83$	−1.18562	−0.130138	−0.0650692	−	0.997881i	$-0.520727\pi$
$83$	−1.18562	−0.130138	−0.0650692	+	0.997881i	$0.520727\pi$
$84$	0	0
$85$	9.89433	1.07319
$86$	0	0
$87$	1.89374	0.203030
$88$	0	0
$89$	13.6214	1.44386	0.721930	−	0.691966i	$-0.243255\pi$
$89$	13.6214	1.44386	0.721930	+	0.691966i	$0.243255\pi$
$90$	0	0
$91$	2.38449	0.249962
$92$	0	0
$93$	0.764644	0.0792899
$94$	0	0
$95$	3.78137	0.387960
$96$	0	0
$97$	−2.27485	−0.230976	−0.115488	−	0.993309i	$-0.536843\pi$
$97$	−2.27485	−0.230976	−0.115488	+	0.993309i	$0.536843\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.73819	−0.172956	−0.0864780	−	0.996254i	$-0.527561\pi$
$101$	−1.73819	−0.172956	−0.0864780	+	0.996254i	$0.527561\pi$
$102$	0	0
$103$	2.55388	0.251641	0.125821	−	0.992053i	$-0.459844\pi$
$103$	2.55388	0.251641	0.125821	+	0.992053i	$0.459844\pi$
$104$	0	0
$105$	2.23143	0.217765
$106$	0	0
$107$	−15.6970	−1.51749	−0.758743	−	0.651390i	$-0.774186\pi$
$107$	−15.6970	−1.51749	−0.758743	+	0.651390i	$0.774186\pi$
$108$	0	0
$109$	12.6627	1.21287	0.606433	−	0.795135i	$-0.292600\pi$
$109$	12.6627	1.21287	0.606433	+	0.795135i	$0.292600\pi$
$110$	0	0
$111$	−0.997931	−0.0947194
$112$	0	0
$113$	9.95727	0.936701	0.468351	−	0.883543i	$-0.344848\pi$
$113$	9.95727	0.936701	0.468351	+	0.883543i	$0.344848\pi$
$114$	0	0
$115$	6.43899	0.600439
$116$	0	0
$117$	3.16337	0.292453
$118$	0	0
$119$	−5.77609	−0.529493
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0.735710	0.0663368
$124$	0	0
$125$	16.2551	1.45390
$126$	0	0
$127$	−2.84519	−0.252470	−0.126235	−	0.992000i	$-0.540289\pi$
$127$	−2.84519	−0.252470	−0.126235	+	0.992000i	$0.540289\pi$
$128$	0	0
$129$	1.81111	0.159459
$130$	0	0
$131$	−20.5293	−1.79365	−0.896827	−	0.442381i	$-0.854134\pi$
$131$	−20.5293	−1.79365	−0.896827	+	0.442381i	$0.854134\pi$
$132$	0	0
$133$	−2.20748	−0.191413
$134$	0	0
$135$	5.99287	0.515784
$136$	0	0
$137$	−13.3636	−1.14173	−0.570863	−	0.821046i	$-0.693391\pi$
$137$	−13.3636	−1.14173	−0.570863	+	0.821046i	$0.693391\pi$
$138$	0	0
$139$	−18.3087	−1.55292	−0.776462	−	0.630165i	$-0.782987\pi$
$139$	−18.3087	−1.55292	−0.776462	+	0.630165i	$0.782987\pi$
$140$	0	0
$141$	0.678339	0.0571265
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−26.7874	−2.22458
$146$	0	0
$147$	0.568611	0.0468983
$148$	0	0
$149$	−0.829122	−0.0679243	−0.0339621	−	0.999423i	$-0.510813\pi$
$149$	−0.829122	−0.0679243	−0.0339621	+	0.999423i	$0.510813\pi$
$150$	0	0
$151$	−14.5812	−1.18660	−0.593299	−	0.804982i	$-0.702175\pi$
$151$	−14.5812	−1.18660	−0.593299	+	0.804982i	$0.702175\pi$
$152$	0	0
$153$	−7.66282	−0.619502
$154$	0	0
$155$	−10.8161	−0.868768
$156$	0	0
$157$	0.923144	0.0736749	0.0368375	−	0.999321i	$-0.488272\pi$
$157$	0.923144	0.0736749	0.0368375	+	0.999321i	$0.488272\pi$
$158$	0	0
$159$	2.87575	0.228062
$160$	0	0
$161$	−3.75894	−0.296246
$162$	0	0
$163$	11.0056	0.862025	0.431013	−	0.902346i	$-0.358156\pi$
$163$	11.0056	0.862025	0.431013	+	0.902346i	$0.358156\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.28230	−0.176610	−0.0883048	−	0.996093i	$-0.528145\pi$
$167$	−2.28230	−0.176610	−0.0883048	+	0.996093i	$0.528145\pi$
$168$	0	0
$169$	−11.8332	−0.910246
$170$	0	0
$171$	−2.92854	−0.223951
$172$	0	0
$173$	13.7974	1.04900	0.524499	−	0.851411i	$-0.324253\pi$
$173$	13.7974	1.04900	0.524499	+	0.851411i	$0.324253\pi$
$174$	0	0
$175$	−20.5268	−1.55168
$176$	0	0
$177$	−2.21449	−0.166451
$178$	0	0
$179$	−13.6449	−1.01987	−0.509935	−	0.860213i	$-0.670330\pi$
$179$	−13.6449	−1.01987	−0.509935	+	0.860213i	$0.670330\pi$
$180$	0	0
$181$	18.2920	1.35963	0.679816	−	0.733383i	$-0.262060\pi$
$181$	18.2920	1.35963	0.679816	+	0.733383i	$0.262060\pi$
$182$	0	0
$183$	−1.98104	−0.146443
$184$	0	0
$185$	14.1160	1.03783
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−3.49850	−0.254479
$190$	0	0
$191$	12.8005	0.926214	0.463107	−	0.886302i	$-0.346735\pi$
$191$	12.8005	0.926214	0.463107	+	0.886302i	$0.346735\pi$
$192$	0	0
$193$	4.73360	0.340732	0.170366	−	0.985381i	$-0.445505\pi$
$193$	4.73360	0.340732	0.170366	+	0.985381i	$0.445505\pi$
$194$	0	0
$195$	1.09191	0.0781931
$196$	0	0
$197$	−18.3771	−1.30931	−0.654656	−	0.755927i	$-0.727186\pi$
$197$	−18.3771	−1.30931	−0.654656	+	0.755927i	$0.727186\pi$
$198$	0	0
$199$	−24.0810	−1.70706	−0.853529	−	0.521046i	$-0.825542\pi$
$199$	−24.0810	−1.70706	−0.853529	+	0.521046i	$0.825542\pi$
$200$	0	0
$201$	−1.57726	−0.111252
$202$	0	0
$203$	15.6379	1.09757
$204$	0	0
$205$	−10.4068	−0.726843
$206$	0	0
$207$	−4.98677	−0.346605
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0.206543	0.0142190	0.00710949	−	0.999975i	$-0.497737\pi$
$211$	0.206543	0.0142190	0.00710949	+	0.999975i	$0.497737\pi$
$212$	0	0
$213$	2.73526	0.187417
$214$	0	0
$215$	−25.6186	−1.74717
$216$	0	0
$217$	6.31418	0.428634
$218$	0	0
$219$	2.73628	0.184900
$220$	0	0
$221$	−2.82642	−0.190125
$222$	0	0
$223$	−25.9944	−1.74071	−0.870357	−	0.492422i	$-0.836112\pi$
$223$	−25.9944	−1.74071	−0.870357	+	0.492422i	$0.836112\pi$
$224$	0	0
$225$	−27.2317	−1.81545
$226$	0	0
$227$	−3.72912	−0.247511	−0.123755	−	0.992313i	$-0.539494\pi$
$227$	−3.72912	−0.247511	−0.123755	+	0.992313i	$0.539494\pi$
$228$	0	0
$229$	10.8118	0.714466	0.357233	−	0.934015i	$-0.383720\pi$
$229$	10.8118	0.714466	0.357233	+	0.934015i	$0.383720\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−12.1778	−0.797797	−0.398898	−	0.916995i	$-0.630607\pi$
$233$	−12.1778	−0.797797	−0.398898	+	0.916995i	$0.630607\pi$
$234$	0	0
$235$	−9.59528	−0.625927
$236$	0	0
$237$	1.17617	0.0764001
$238$	0	0
$239$	−18.9208	−1.22389	−0.611944	−	0.790901i	$-0.709612\pi$
$239$	−18.9208	−1.22389	−0.611944	+	0.790901i	$0.709612\pi$
$240$	0	0
$241$	4.28508	0.276026	0.138013	−	0.990430i	$-0.455928\pi$
$241$	4.28508	0.276026	0.138013	+	0.990430i	$0.455928\pi$
$242$	0	0
$243$	−6.98988	−0.448401
$244$	0	0
$245$	−8.04315	−0.513858
$246$	0	0
$247$	−1.08019	−0.0687306
$248$	0	0
$249$	0.316944	0.0200855
$250$	0	0
$251$	−14.0418	−0.886311	−0.443156	−	0.896445i	$-0.646141\pi$
$251$	−14.0418	−0.886311	−0.443156	+	0.896445i	$0.646141\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−2.64500	−0.165636
$256$	0	0
$257$	−0.966932	−0.0603156	−0.0301578	−	0.999545i	$-0.509601\pi$
$257$	−0.966932	−0.0603156	−0.0301578	+	0.999545i	$0.509601\pi$
$258$	0	0
$259$	−8.24059	−0.512045
$260$	0	0
$261$	20.7459	1.28414
$262$	0	0
$263$	17.4928	1.07865	0.539326	−	0.842097i	$-0.318679\pi$
$263$	17.4928	1.07865	0.539326	+	0.842097i	$0.318679\pi$
$264$	0	0
$265$	−40.6782	−2.49884
$266$	0	0
$267$	−3.64132	−0.222845
$268$	0	0
$269$	−16.3634	−0.997692	−0.498846	−	0.866691i	$-0.666243\pi$
$269$	−16.3634	−0.997692	−0.498846	+	0.866691i	$0.666243\pi$
$270$	0	0
$271$	20.0585	1.21847	0.609234	−	0.792990i	$-0.291477\pi$
$271$	20.0585	1.21847	0.609234	+	0.792990i	$0.291477\pi$
$272$	0	0
$273$	−0.637431	−0.0385791
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−15.1655	−0.911209	−0.455605	−	0.890182i	$-0.650577\pi$
$277$	−15.1655	−0.911209	−0.455605	+	0.890182i	$0.650577\pi$
$278$	0	0
$279$	8.37667	0.501498
$280$	0	0
$281$	17.7065	1.05628	0.528141	−	0.849157i	$-0.322889\pi$
$281$	17.7065	1.05628	0.528141	+	0.849157i	$0.322889\pi$
$282$	0	0
$283$	−13.2854	−0.789735	−0.394867	−	0.918738i	$-0.629209\pi$
$283$	−13.2854	−0.789735	−0.394867	+	0.918738i	$0.629209\pi$
$284$	0	0
$285$	−1.01085	−0.0598777
$286$	0	0
$287$	6.07526	0.358611
$288$	0	0
$289$	−10.1534	−0.597259
$290$	0	0
$291$	0.608122	0.0356487
$292$	0	0
$293$	18.3318	1.07095	0.535477	−	0.844550i	$-0.320132\pi$
$293$	18.3318	1.07095	0.535477	+	0.844550i	$0.320132\pi$
$294$	0	0
$295$	31.3244	1.82378
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.83936	−0.106373
$300$	0	0
$301$	14.9555	0.862023
$302$	0	0
$303$	0.464659	0.0266940
$304$	0	0
$305$	28.0224	1.60456
$306$	0	0
$307$	−12.9669	−0.740063	−0.370031	−	0.929019i	$-0.620653\pi$
$307$	−12.9669	−0.740063	−0.370031	+	0.929019i	$0.620653\pi$
$308$	0	0
$309$	−0.682715	−0.0388383
$310$	0	0
$311$	−21.4404	−1.21577	−0.607886	−	0.794024i	$-0.707982\pi$
$311$	−21.4404	−1.21577	−0.607886	+	0.794024i	$0.707982\pi$
$312$	0	0
$313$	−8.08851	−0.457190	−0.228595	−	0.973522i	$-0.573413\pi$
$313$	−8.08851	−0.457190	−0.228595	+	0.973522i	$0.573413\pi$
$314$	0	0
$315$	24.4453	1.37734
$316$	0	0
$317$	32.7880	1.84156	0.920780	−	0.390083i	$-0.127554\pi$
$317$	32.7880	1.84156	0.920780	+	0.390083i	$0.127554\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	4.19619	0.234208
$322$	0	0
$323$	2.61660	0.145592
$324$	0	0
$325$	−10.0444	−0.557162
$326$	0	0
$327$	−3.38504	−0.187193
$328$	0	0
$329$	5.60151	0.308821
$330$	0	0
$331$	11.0146	0.605420	0.302710	−	0.953083i	$-0.402109\pi$
$331$	11.0146	0.605420	0.302710	+	0.953083i	$0.402109\pi$
$332$	0	0
$333$	−10.9323	−0.599088
$334$	0	0
$335$	22.3108	1.21897
$336$	0	0
$337$	−15.6294	−0.851385	−0.425693	−	0.904868i	$-0.639970\pi$
$337$	−15.6294	−0.851385	−0.425693	+	0.904868i	$0.639970\pi$
$338$	0	0
$339$	−2.66182	−0.144570
$340$	0	0
$341$	0	0
$342$	0	0
$343$	20.1477	1.08788
$344$	0	0
$345$	−1.72130	−0.0926716
$346$	0	0
$347$	3.98533	0.213944	0.106972	−	0.994262i	$-0.465885\pi$
$347$	3.98533	0.213944	0.106972	+	0.994262i	$0.465885\pi$
$348$	0	0
$349$	−4.30004	−0.230176	−0.115088	−	0.993355i	$-0.536715\pi$
$349$	−4.30004	−0.230176	−0.115088	+	0.993355i	$0.536715\pi$
$350$	0	0
$351$	−1.71192	−0.0913757
$352$	0	0
$353$	17.1224	0.911332	0.455666	−	0.890151i	$-0.349401\pi$
$353$	17.1224	0.911332	0.455666	+	0.890151i	$0.349401\pi$
$354$	0	0
$355$	−38.6909	−2.05350
$356$	0	0
$357$	1.54409	0.0817218
$358$	0	0
$359$	16.9806	0.896203	0.448101	−	0.893983i	$-0.352100\pi$
$359$	16.9806	0.896203	0.448101	+	0.893983i	$0.352100\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−38.7053	−2.02593
$366$	0	0
$367$	8.61076	0.449478	0.224739	−	0.974419i	$-0.427847\pi$
$367$	8.61076	0.449478	0.224739	+	0.974419i	$0.427847\pi$
$368$	0	0
$369$	8.05971	0.419572
$370$	0	0
$371$	23.7470	1.23288
$372$	0	0
$373$	−8.41225	−0.435570	−0.217785	−	0.975997i	$-0.569883\pi$
$373$	−8.41225	−0.435570	−0.217785	+	0.975997i	$0.569883\pi$
$374$	0	0
$375$	−4.34539	−0.224395
$376$	0	0
$377$	7.65210	0.394103
$378$	0	0
$379$	9.95201	0.511200	0.255600	−	0.966783i	$-0.417727\pi$
$379$	9.95201	0.511200	0.255600	+	0.966783i	$0.417727\pi$
$380$	0	0
$381$	0.760588	0.0389661
$382$	0	0
$383$	2.02439	0.103441	0.0517207	−	0.998662i	$-0.483529\pi$
$383$	2.02439	0.103441	0.0517207	+	0.998662i	$0.483529\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	19.8407	1.00856
$388$	0	0
$389$	23.4981	1.19140	0.595700	−	0.803207i	$-0.296875\pi$
$389$	23.4981	1.19140	0.595700	+	0.803207i	$0.296875\pi$
$390$	0	0
$391$	4.45560	0.225330
$392$	0	0
$393$	5.48798	0.276832
$394$	0	0
$395$	−16.6372	−0.837106
$396$	0	0
$397$	−30.5686	−1.53419	−0.767096	−	0.641532i	$-0.778299\pi$
$397$	−30.5686	−1.53419	−0.767096	+	0.641532i	$0.778299\pi$
$398$	0	0
$399$	0.590112	0.0295426
$400$	0	0
$401$	3.42757	0.171164	0.0855822	−	0.996331i	$-0.472725\pi$
$401$	3.42757	0.171164	0.0855822	+	0.996331i	$0.472725\pi$
$402$	0	0
$403$	3.08972	0.153910
$404$	0	0
$405$	31.6196	1.57119
$406$	0	0
$407$	0	0
$408$	0	0
$409$	9.24223	0.456999	0.228499	−	0.973544i	$-0.426618\pi$
$409$	9.24223	0.456999	0.228499	+	0.973544i	$0.426618\pi$
$410$	0	0
$411$	3.57240	0.176214
$412$	0	0
$413$	−18.2865	−0.899820
$414$	0	0
$415$	−4.48326	−0.220074
$416$	0	0
$417$	4.89436	0.239678
$418$	0	0
$419$	−21.8617	−1.06802	−0.534008	−	0.845479i	$-0.679315\pi$
$419$	−21.8617	−1.06802	−0.534008	+	0.845479i	$0.679315\pi$
$420$	0	0
$421$	−15.1631	−0.739006	−0.369503	−	0.929230i	$-0.620472\pi$
$421$	−15.1631	−0.739006	−0.369503	+	0.929230i	$0.620472\pi$
$422$	0	0
$423$	7.43121	0.361318
$424$	0	0
$425$	24.3311	1.18023
$426$	0	0
$427$	−16.3588	−0.791658
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−34.1392	−1.64443	−0.822213	−	0.569180i	$-0.807261\pi$
$431$	−34.1392	−1.64443	−0.822213	+	0.569180i	$0.807261\pi$
$432$	0	0
$433$	−4.12689	−0.198326	−0.0991628	−	0.995071i	$-0.531616\pi$
$433$	−4.12689	−0.198326	−0.0991628	+	0.995071i	$0.531616\pi$
$434$	0	0
$435$	7.16093	0.343341
$436$	0	0
$437$	1.70282	0.0814570
$438$	0	0
$439$	−33.4222	−1.59515	−0.797577	−	0.603217i	$-0.793885\pi$
$439$	−33.4222	−1.59515	−0.797577	+	0.603217i	$0.793885\pi$
$440$	0	0
$441$	6.22914	0.296626
$442$	0	0
$443$	−20.5059	−0.974266	−0.487133	−	0.873328i	$-0.661957\pi$
$443$	−20.5059	−0.974266	−0.487133	+	0.873328i	$0.661957\pi$
$444$	0	0
$445$	51.5074	2.44168
$446$	0	0
$447$	0.221644	0.0104834
$448$	0	0
$449$	−3.32297	−0.156821	−0.0784104	−	0.996921i	$-0.524984\pi$
$449$	−3.32297	−0.156821	−0.0784104	+	0.996921i	$0.524984\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	3.89790	0.183139
$454$	0	0
$455$	9.01662	0.422706
$456$	0	0
$457$	6.68250	0.312594	0.156297	−	0.987710i	$-0.450044\pi$
$457$	6.68250	0.312594	0.156297	+	0.987710i	$0.450044\pi$
$458$	0	0
$459$	4.14690	0.193561
$460$	0	0
$461$	12.1313	0.565012	0.282506	−	0.959266i	$-0.408834\pi$
$461$	12.1313	0.565012	0.282506	+	0.959266i	$0.408834\pi$
$462$	0	0
$463$	−24.0312	−1.11683	−0.558413	−	0.829563i	$-0.688589\pi$
$463$	−24.0312	−1.11683	−0.558413	+	0.829563i	$0.688589\pi$
$464$	0	0
$465$	2.89140	0.134085
$466$	0	0
$467$	28.8225	1.33375	0.666873	−	0.745171i	$-0.267632\pi$
$467$	28.8225	1.33375	0.666873	+	0.745171i	$0.267632\pi$
$468$	0	0
$469$	−13.0245	−0.601417
$470$	0	0
$471$	−0.246779	−0.0113710
$472$	0	0
$473$	0	0
$474$	0	0
$475$	9.29875	0.426656
$476$	0	0
$477$	31.5038	1.44246
$478$	0	0
$479$	21.9551	1.00315	0.501576	−	0.865113i	$-0.332754\pi$
$479$	21.9551	1.00315	0.501576	+	0.865113i	$0.332754\pi$
$480$	0	0
$481$	−4.03237	−0.183860
$482$	0	0
$483$	1.00486	0.0457225
$484$	0	0
$485$	−8.60203	−0.390598
$486$	0	0
$487$	19.4521	0.881458	0.440729	−	0.897640i	$-0.354720\pi$
$487$	19.4521	0.881458	0.440729	+	0.897640i	$0.354720\pi$
$488$	0	0
$489$	−2.94206	−0.133045
$490$	0	0
$491$	−31.0732	−1.40231	−0.701157	−	0.713007i	$-0.747333\pi$
$491$	−31.0732	−1.40231	−0.701157	+	0.713007i	$0.747333\pi$
$492$	0	0
$493$	−18.5362	−0.834827
$494$	0	0
$495$	0	0
$496$	0	0
$497$	22.5869	1.01316
$498$	0	0
$499$	−10.7364	−0.480629	−0.240314	−	0.970695i	$-0.577251\pi$
$499$	−10.7364	−0.480629	−0.240314	+	0.970695i	$0.577251\pi$
$500$	0	0
$501$	0.610114	0.0272579
$502$	0	0
$503$	−3.37082	−0.150297	−0.0751487	−	0.997172i	$-0.523943\pi$
$503$	−3.37082	−0.150297	−0.0751487	+	0.997172i	$0.523943\pi$
$504$	0	0
$505$	−6.57272	−0.292482
$506$	0	0
$507$	3.16330	0.140487
$508$	0	0
$509$	−35.6084	−1.57831	−0.789157	−	0.614192i	$-0.789482\pi$
$509$	−35.6084	−1.57831	−0.789157	+	0.614192i	$0.789482\pi$
$510$	0	0
$511$	22.5953	0.999556
$512$	0	0
$513$	1.58484	0.0699725
$514$	0	0
$515$	9.65717	0.425546
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−3.68838	−0.161902
$520$	0	0
$521$	2.37619	0.104103	0.0520513	−	0.998644i	$-0.483424\pi$
$521$	2.37619	0.104103	0.0520513	+	0.998644i	$0.483424\pi$
$522$	0	0
$523$	−44.5098	−1.94628	−0.973140	−	0.230215i	$-0.926057\pi$
$523$	−44.5098	−1.94628	−0.973140	+	0.230215i	$0.926057\pi$
$524$	0	0
$525$	5.48730	0.239486
$526$	0	0
$527$	−7.48442	−0.326027
$528$	0	0
$529$	−20.1004	−0.873931
$530$	0	0
$531$	−24.2597	−1.05278
$532$	0	0
$533$	2.97281	0.128767
$534$	0	0
$535$	−59.3562	−2.56619
$536$	0	0
$537$	3.64762	0.157406
$538$	0	0
$539$	0	0
$540$	0	0
$541$	24.8687	1.06919	0.534595	−	0.845109i	$-0.320464\pi$
$541$	24.8687	1.06919	0.534595	+	0.845109i	$0.320464\pi$
$542$	0	0
$543$	−4.88989	−0.209845
$544$	0	0
$545$	47.8823	2.05105
$546$	0	0
$547$	−3.61284	−0.154474	−0.0772370	−	0.997013i	$-0.524610\pi$
$547$	−3.61284	−0.154474	−0.0772370	+	0.997013i	$0.524610\pi$
$548$	0	0
$549$	−21.7023	−0.926233
$550$	0	0
$551$	−7.08406	−0.301791
$552$	0	0
$553$	9.71239	0.413013
$554$	0	0
$555$	−3.77355	−0.160178
$556$	0	0
$557$	0.884220	0.0374656	0.0187328	−	0.999825i	$-0.494037\pi$
$557$	0.884220	0.0374656	0.0187328	+	0.999825i	$0.494037\pi$
$558$	0	0
$559$	7.31820	0.309527
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−15.7311	−0.662988	−0.331494	−	0.943457i	$-0.607553\pi$
$563$	−15.7311	−0.662988	−0.331494	+	0.943457i	$0.607553\pi$
$564$	0	0
$565$	37.6521	1.58404
$566$	0	0
$567$	−18.4588	−0.775197
$568$	0	0
$569$	9.72101	0.407526	0.203763	−	0.979020i	$-0.434683\pi$
$569$	9.72101	0.407526	0.203763	+	0.979020i	$0.434683\pi$
$570$	0	0
$571$	15.2407	0.637804	0.318902	−	0.947788i	$-0.396686\pi$
$571$	15.2407	0.637804	0.318902	+	0.947788i	$0.396686\pi$
$572$	0	0
$573$	−3.42189	−0.142952
$574$	0	0
$575$	15.8341	0.660328
$576$	0	0
$577$	0.867254	0.0361042	0.0180521	−	0.999837i	$-0.494254\pi$
$577$	0.867254	0.0361042	0.0180521	+	0.999837i	$0.494254\pi$
$578$	0	0
$579$	−1.26541	−0.0525885
$580$	0	0
$581$	2.61722	0.108581
$582$	0	0
$583$	0	0
$584$	0	0
$585$	11.9618	0.494562
$586$	0	0
$587$	−3.13461	−0.129379	−0.0646895	−	0.997905i	$-0.520606\pi$
$587$	−3.13461	−0.129379	−0.0646895	+	0.997905i	$0.520606\pi$
$588$	0	0
$589$	−2.86036	−0.117859
$590$	0	0
$591$	4.91263	0.202079
$592$	0	0
$593$	9.49179	0.389781	0.194891	−	0.980825i	$-0.437565\pi$
$593$	9.49179	0.389781	0.194891	+	0.980825i	$0.437565\pi$
$594$	0	0
$595$	−21.8415	−0.895415
$596$	0	0
$597$	6.43744	0.263467
$598$	0	0
$599$	9.32986	0.381208	0.190604	−	0.981667i	$-0.438955\pi$
$599$	9.32986	0.381208	0.190604	+	0.981667i	$0.438955\pi$
$600$	0	0
$601$	41.2338	1.68196	0.840981	−	0.541065i	$-0.181979\pi$
$601$	41.2338	1.68196	0.840981	+	0.541065i	$0.181979\pi$
$602$	0	0
$603$	−17.2789	−0.703652
$604$	0	0
$605$	0	0
$606$	0	0
$607$	27.0481	1.09785	0.548925	−	0.835872i	$-0.315037\pi$
$607$	27.0481	1.09785	0.548925	+	0.835872i	$0.315037\pi$
$608$	0	0
$609$	−4.18039	−0.169398
$610$	0	0
$611$	2.74099	0.110889
$612$	0	0
$613$	−31.8532	−1.28654	−0.643269	−	0.765640i	$-0.722422\pi$
$613$	−31.8532	−1.28654	−0.643269	+	0.765640i	$0.722422\pi$
$614$	0	0
$615$	2.78199	0.112181
$616$	0	0
$617$	−25.3344	−1.01993	−0.509963	−	0.860196i	$-0.670341\pi$
$617$	−25.3344	−1.01993	−0.509963	+	0.860196i	$0.670341\pi$
$618$	0	0
$619$	19.8583	0.798173	0.399087	−	0.916913i	$-0.369327\pi$
$619$	19.8583	0.798173	0.399087	+	0.916913i	$0.369327\pi$
$620$	0	0
$621$	2.69870	0.108295
$622$	0	0
$623$	−30.0688	−1.20468
$624$	0	0
$625$	14.9730	0.598918
$626$	0	0
$627$	0	0
$628$	0	0
$629$	9.76787	0.389470
$630$	0	0
$631$	−23.5275	−0.936615	−0.468308	−	0.883565i	$-0.655136\pi$
$631$	−23.5275	−0.936615	−0.468308	+	0.883565i	$0.655136\pi$
$632$	0	0
$633$	−0.0552139	−0.00219455
$634$	0	0
$635$	−10.7587	−0.426946
$636$	0	0
$637$	2.29761	0.0910345
$638$	0	0
$639$	29.9648	1.18539
$640$	0	0
$641$	41.5323	1.64043	0.820214	−	0.572056i	$-0.193854\pi$
$641$	41.5323	1.64043	0.820214	+	0.572056i	$0.193854\pi$
$642$	0	0
$643$	19.7315	0.778135	0.389067	−	0.921209i	$-0.372797\pi$
$643$	19.7315	0.778135	0.389067	+	0.921209i	$0.372797\pi$
$644$	0	0
$645$	6.84847	0.269658
$646$	0	0
$647$	−18.9367	−0.744479	−0.372239	−	0.928137i	$-0.621410\pi$
$647$	−18.9367	−0.744479	−0.372239	+	0.928137i	$0.621410\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−1.68793	−0.0661553
$652$	0	0
$653$	1.12045	0.0438466	0.0219233	−	0.999760i	$-0.493021\pi$
$653$	1.12045	0.0438466	0.0219233	+	0.999760i	$0.493021\pi$
$654$	0	0
$655$	−77.6289	−3.03321
$656$	0	0
$657$	29.9759	1.16947
$658$	0	0
$659$	−10.9076	−0.424899	−0.212450	−	0.977172i	$-0.568144\pi$
$659$	−10.9076	−0.424899	−0.212450	+	0.977172i	$0.568144\pi$
$660$	0	0
$661$	−3.75082	−0.145890	−0.0729450	−	0.997336i	$-0.523240\pi$
$661$	−3.75082	−0.145890	−0.0729450	+	0.997336i	$0.523240\pi$
$662$	0	0
$663$	0.755570	0.0293439
$664$	0	0
$665$	−8.34728	−0.323694
$666$	0	0
$667$	−12.0629	−0.467077
$668$	0	0
$669$	6.94893	0.268661
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−39.7035	−1.53046	−0.765228	−	0.643759i	$-0.777374\pi$
$673$	−39.7035	−1.53046	−0.765228	+	0.643759i	$0.777374\pi$
$674$	0	0
$675$	14.7370	0.567229
$676$	0	0
$677$	45.6752	1.75544	0.877720	−	0.479174i	$-0.159064\pi$
$677$	45.6752	1.75544	0.877720	+	0.479174i	$0.159064\pi$
$678$	0	0
$679$	5.02167	0.192714
$680$	0	0
$681$	0.996885	0.0382007
$682$	0	0
$683$	28.9548	1.10793	0.553963	−	0.832541i	$-0.313115\pi$
$683$	28.9548	1.10793	0.553963	+	0.832541i	$0.313115\pi$
$684$	0	0
$685$	−50.5325	−1.93075
$686$	0	0
$687$	−2.89027	−0.110271
$688$	0	0
$689$	11.6201	0.442692
$690$	0	0
$691$	33.1628	1.26157	0.630787	−	0.775956i	$-0.282732\pi$
$691$	33.1628	1.26157	0.630787	+	0.775956i	$0.282732\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−69.2319	−2.62612
$696$	0	0
$697$	−7.20122	−0.272766
$698$	0	0
$699$	3.25543	0.123132
$700$	0	0
$701$	37.6787	1.42311	0.711553	−	0.702632i	$-0.247992\pi$
$701$	37.6787	1.42311	0.711553	+	0.702632i	$0.247992\pi$
$702$	0	0
$703$	3.73304	0.140794
$704$	0	0
$705$	2.56505	0.0966054
$706$	0	0
$707$	3.83701	0.144305
$708$	0	0
$709$	−17.9471	−0.674016	−0.337008	−	0.941502i	$-0.609415\pi$
$709$	−17.9471	−0.674016	−0.337008	+	0.941502i	$0.609415\pi$
$710$	0	0
$711$	12.8849	0.483221
$712$	0	0
$713$	−4.87068	−0.182408
$714$	0	0
$715$	0	0
$716$	0	0
$717$	5.05800	0.188895
$718$	0	0
$719$	47.2012	1.76031	0.880153	−	0.474690i	$-0.157440\pi$
$719$	47.2012	1.76031	0.880153	+	0.474690i	$0.157440\pi$
$720$	0	0
$721$	−5.63763	−0.209956
$722$	0	0
$723$	−1.14550	−0.0426018
$724$	0	0
$725$	−65.8729	−2.44646
$726$	0	0
$727$	−37.5639	−1.39317	−0.696584	−	0.717475i	$-0.745298\pi$
$727$	−37.5639	−1.39317	−0.696584	+	0.717475i	$0.745298\pi$
$728$	0	0
$729$	−23.2173	−0.859899
$730$	0	0
$731$	−17.7273	−0.655669
$732$	0	0
$733$	−23.5701	−0.870583	−0.435291	−	0.900290i	$-0.643355\pi$
$733$	−23.5701	−0.870583	−0.435291	+	0.900290i	$0.643355\pi$
$734$	0	0
$735$	2.15013	0.0793087
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−4.36640	−0.160621	−0.0803103	−	0.996770i	$-0.525591\pi$
$739$	−4.36640	−0.160621	−0.0803103	+	0.996770i	$0.525591\pi$
$740$	0	0
$741$	0.288760	0.0106079
$742$	0	0
$743$	2.51725	0.0923488	0.0461744	−	0.998933i	$-0.485297\pi$
$743$	2.51725	0.0923488	0.0461744	+	0.998933i	$0.485297\pi$
$744$	0	0
$745$	−3.13521	−0.114865
$746$	0	0
$747$	3.47213	0.127038
$748$	0	0
$749$	34.6508	1.26611
$750$	0	0
$751$	−10.7548	−0.392449	−0.196224	−	0.980559i	$-0.562868\pi$
$751$	−10.7548	−0.392449	−0.196224	+	0.980559i	$0.562868\pi$
$752$	0	0
$753$	3.75372	0.136793
$754$	0	0
$755$	−55.1367	−2.00663
$756$	0	0
$757$	−42.6256	−1.54925	−0.774627	−	0.632418i	$-0.782062\pi$
$757$	−42.6256	−1.54925	−0.774627	+	0.632418i	$0.782062\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.67338	0.169410	0.0847049	−	0.996406i	$-0.473005\pi$
$761$	4.67338	0.169410	0.0847049	+	0.996406i	$0.473005\pi$
$762$	0	0
$763$	−27.9526	−1.01195
$764$	0	0
$765$	−28.9759	−1.04763
$766$	0	0
$767$	−8.94814	−0.323099
$768$	0	0
$769$	9.54946	0.344362	0.172181	−	0.985065i	$-0.444919\pi$
$769$	9.54946	0.344362	0.172181	+	0.985065i	$0.444919\pi$
$770$	0	0
$771$	0.258484	0.00930909
$772$	0	0
$773$	−5.29914	−0.190597	−0.0952984	−	0.995449i	$-0.530381\pi$
$773$	−5.29914	−0.190597	−0.0952984	+	0.995449i	$0.530381\pi$
$774$	0	0
$775$	−26.5978	−0.955420
$776$	0	0
$777$	2.20291	0.0790290
$778$	0	0
$779$	−2.75213	−0.0986052
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−11.2271	−0.401224
$784$	0	0
$785$	3.49075	0.124590
$786$	0	0
$787$	51.4527	1.83409	0.917046	−	0.398782i	$-0.130567\pi$
$787$	51.4527	1.83409	0.917046	+	0.398782i	$0.130567\pi$
$788$	0	0
$789$	−4.67625	−0.166479
$790$	0	0
$791$	−21.9804	−0.781535
$792$	0	0
$793$	−8.00487	−0.284261
$794$	0	0
$795$	10.8743	0.385671
$796$	0	0
$797$	−26.0638	−0.923226	−0.461613	−	0.887081i	$-0.652729\pi$
$797$	−26.0638	−0.923226	−0.461613	+	0.887081i	$0.652729\pi$
$798$	0	0
$799$	−6.63967	−0.234894
$800$	0	0
$801$	−39.8907	−1.40947
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−14.2139	−0.500975
$806$	0	0
$807$	4.37432	0.153984
$808$	0	0
$809$	34.3187	1.20658	0.603291	−	0.797521i	$-0.293856\pi$
$809$	34.3187	1.20658	0.603291	+	0.797521i	$0.293856\pi$
$810$	0	0
$811$	−32.4926	−1.14097	−0.570484	−	0.821309i	$-0.693244\pi$
$811$	−32.4926	−1.14097	−0.570484	+	0.821309i	$0.693244\pi$
$812$	0	0
$813$	−5.36213	−0.188058
$814$	0	0
$815$	41.6162	1.45775
$816$	0	0
$817$	−6.77495	−0.237025
$818$	0	0
$819$	−6.98306	−0.244008
$820$	0	0
$821$	−2.04163	−0.0712532	−0.0356266	−	0.999365i	$-0.511343\pi$
$821$	−2.04163	−0.0712532	−0.0356266	+	0.999365i	$0.511343\pi$
$822$	0	0
$823$	−13.5640	−0.472810	−0.236405	−	0.971655i	$-0.575969\pi$
$823$	−13.5640	−0.472810	−0.236405	+	0.971655i	$0.575969\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−55.6132	−1.93386	−0.966929	−	0.255044i	$-0.917910\pi$
$827$	−55.6132	−1.93386	−0.966929	+	0.255044i	$0.917910\pi$
$828$	0	0
$829$	11.7891	0.409453	0.204727	−	0.978819i	$-0.434369\pi$
$829$	11.7891	0.409453	0.204727	+	0.978819i	$0.434369\pi$
$830$	0	0
$831$	4.05412	0.140636
$832$	0	0
$833$	−5.56563	−0.192838
$834$	0	0
$835$	−8.63022	−0.298661
$836$	0	0
$837$	−4.53322	−0.156691
$838$	0	0
$839$	55.4798	1.91538	0.957688	−	0.287807i	$-0.0929263\pi$
$839$	55.4798	1.91538	0.957688	+	0.287807i	$0.0929263\pi$
$840$	0	0
$841$	21.1839	0.730480
$842$	0	0
$843$	−4.73338	−0.163026
$844$	0	0
$845$	−44.7457	−1.53930
$846$	0	0
$847$	0	0
$848$	0	0
$849$	3.55151	0.121887
$850$	0	0
$851$	6.35669	0.217905
$852$	0	0
$853$	−12.5734	−0.430507	−0.215253	−	0.976558i	$-0.569058\pi$
$853$	−12.5734	−0.430507	−0.215253	+	0.976558i	$0.569058\pi$
$854$	0	0
$855$	−11.0739	−0.378719
$856$	0	0
$857$	10.5267	0.359587	0.179793	−	0.983704i	$-0.442457\pi$
$857$	10.5267	0.359587	0.179793	+	0.983704i	$0.442457\pi$
$858$	0	0
$859$	−3.49588	−0.119278	−0.0596389	−	0.998220i	$-0.518995\pi$
$859$	−3.49588	−0.119278	−0.0596389	+	0.998220i	$0.518995\pi$
$860$	0	0
$861$	−1.62406	−0.0553479
$862$	0	0
$863$	51.7773	1.76252	0.881259	−	0.472634i	$-0.156697\pi$
$863$	51.7773	1.76252	0.881259	+	0.472634i	$0.156697\pi$
$864$	0	0
$865$	52.1731	1.77394
$866$	0	0
$867$	2.71425	0.0921807
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−6.37330	−0.215951
$872$	0	0
$873$	6.66197	0.225474
$874$	0	0
$875$	−35.8829	−1.21306
$876$	0	0
$877$	35.0022	1.18194	0.590969	−	0.806694i	$-0.298745\pi$
$877$	35.0022	1.18194	0.590969	+	0.806694i	$0.298745\pi$
$878$	0	0
$879$	−4.90053	−0.165291
$880$	0	0
$881$	3.50850	0.118204	0.0591022	−	0.998252i	$-0.481176\pi$
$881$	3.50850	0.118204	0.0591022	+	0.998252i	$0.481176\pi$
$882$	0	0
$883$	47.9969	1.61523	0.807613	−	0.589713i	$-0.200759\pi$
$883$	47.9969	1.61523	0.807613	+	0.589713i	$0.200759\pi$
$884$	0	0
$885$	−8.37378	−0.281482
$886$	0	0
$887$	8.55985	0.287412	0.143706	−	0.989620i	$-0.454098\pi$
$887$	8.55985	0.287412	0.143706	+	0.989620i	$0.454098\pi$
$888$	0	0
$889$	6.28069	0.210648
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−2.53752	−0.0849147
$894$	0	0
$895$	−51.5965	−1.72468
$896$	0	0
$897$	0.491706	0.0164176
$898$	0	0
$899$	20.2630	0.675808
$900$	0	0
$901$	−28.1482	−0.937752
$902$	0	0
$903$	−3.99798	−0.133044
$904$	0	0
$905$	69.1687	2.29925
$906$	0	0
$907$	−28.4156	−0.943523	−0.471762	−	0.881726i	$-0.656382\pi$
$907$	−28.4156	−0.943523	−0.471762	+	0.881726i	$0.656382\pi$
$908$	0	0
$909$	5.09035	0.168836
$910$	0	0
$911$	−16.6776	−0.552552	−0.276276	−	0.961078i	$-0.589100\pi$
$911$	−16.6776	−0.552552	−0.276276	+	0.961078i	$0.589100\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−7.49106	−0.247647
$916$	0	0
$917$	45.3180	1.49653
$918$	0	0
$919$	8.44101	0.278443	0.139222	−	0.990261i	$-0.455540\pi$
$919$	8.44101	0.278443	0.139222	+	0.990261i	$0.455540\pi$
$920$	0	0
$921$	3.46638	0.114221
$922$	0	0
$923$	11.0524	0.363796
$924$	0	0
$925$	34.7126	1.14134
$926$	0	0
$927$	−7.47914	−0.245647
$928$	0	0
$929$	−11.7442	−0.385315	−0.192658	−	0.981266i	$-0.561711\pi$
$929$	−11.7442	−0.385315	−0.192658	+	0.981266i	$0.561711\pi$
$930$	0	0
$931$	−2.12705	−0.0697111
$932$	0	0
$933$	5.73153	0.187642
$934$	0	0
$935$	0	0
$936$	0	0
$937$	51.7386	1.69023	0.845113	−	0.534587i	$-0.179533\pi$
$937$	51.7386	1.69023	0.845113	+	0.534587i	$0.179533\pi$
$938$	0	0
$939$	2.16226	0.0705625
$940$	0	0
$941$	39.6474	1.29247	0.646233	−	0.763140i	$-0.276343\pi$
$941$	39.6474	1.29247	0.646233	+	0.763140i	$0.276343\pi$
$942$	0	0
$943$	−4.68638	−0.152609
$944$	0	0
$945$	−13.2291	−0.430343
$946$	0	0
$947$	−23.6506	−0.768541	−0.384270	−	0.923221i	$-0.625547\pi$
$947$	−23.6506	−0.768541	−0.384270	+	0.923221i	$0.625547\pi$
$948$	0	0
$949$	11.0566	0.358911
$950$	0	0
$951$	−8.76504	−0.284226
$952$	0	0
$953$	−55.4629	−1.79662	−0.898310	−	0.439362i	$-0.855204\pi$
$953$	−55.4629	−1.79662	−0.898310	+	0.439362i	$0.855204\pi$
$954$	0	0
$955$	48.4035	1.56630
$956$	0	0
$957$	0	0
$958$	0	0
$959$	29.4997	0.952596
$960$	0	0
$961$	−22.8183	−0.736075
$962$	0	0
$963$	45.9693	1.48134
$964$	0	0
$965$	17.8995	0.576204
$966$	0	0
$967$	46.7483	1.50333	0.751663	−	0.659548i	$-0.229252\pi$
$967$	46.7483	1.50333	0.751663	+	0.659548i	$0.229252\pi$
$968$	0	0
$969$	−0.699481	−0.0224706
$970$	0	0
$971$	−10.5727	−0.339294	−0.169647	−	0.985505i	$-0.554263\pi$
$971$	−10.5727	−0.339294	−0.169647	+	0.985505i	$0.554263\pi$
$972$	0	0
$973$	40.4160	1.29568
$974$	0	0
$975$	2.68511	0.0859922
$976$	0	0
$977$	30.0515	0.961431	0.480716	−	0.876876i	$-0.340377\pi$
$977$	30.0515	0.961431	0.480716	+	0.876876i	$0.340377\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−37.0832	−1.18397
$982$	0	0
$983$	−23.1493	−0.738349	−0.369175	−	0.929360i	$-0.620360\pi$
$983$	−23.1493	−0.738349	−0.369175	+	0.929360i	$0.620360\pi$
$984$	0	0
$985$	−69.4904	−2.21415
$986$	0	0
$987$	−1.49742	−0.0476633
$988$	0	0
$989$	−11.5365	−0.366840
$990$	0	0
$991$	44.7833	1.42259	0.711294	−	0.702894i	$-0.248109\pi$
$991$	44.7833	1.42259	0.711294	+	0.702894i	$0.248109\pi$
$992$	0	0
$993$	−2.94448	−0.0934403
$994$	0	0
$995$	−91.0592	−2.88677
$996$	0	0
$997$	−53.1983	−1.68481	−0.842403	−	0.538848i	$-0.818860\pi$
$997$	−53.1983	−1.68481	−0.842403	+	0.538848i	$0.818860\pi$
$998$	0	0
$999$	5.91627	0.187183

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.u.1.8	✓	14
11.10	odd	2		9196.2.a.v.1.8	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9196.2.a.u.1.8	✓	14	1.1	even	1	trivial
9196.2.a.v.1.8	yes	14	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.267324 q^{3} +3.78137 q^{5} -2.20748 q^{7} -2.92854 q^{9} -1.08019 q^{13} -1.01085 q^{15} +2.61660 q^{17} +1.00000 q^{19} +0.590112 q^{21} +1.70282 q^{23} +9.29875 q^{25} +1.58484 q^{27} -7.08406 q^{29} -2.86036 q^{31} -8.34728 q^{35} +3.73304 q^{37} +0.288760 q^{39} -2.75213 q^{41} -6.77495 q^{43} -11.0739 q^{45} -2.53752 q^{47} -2.12705 q^{49} -0.699481 q^{51} -10.7575 q^{53} -0.267324 q^{57} +8.28389 q^{59} +7.41064 q^{61} +6.46468 q^{63} -4.08458 q^{65} +5.90019 q^{67} -0.455205 q^{69} -10.2320 q^{71} -10.2358 q^{73} -2.48578 q^{75} -4.39977 q^{79} +8.36195 q^{81} -1.18562 q^{83} +9.89433 q^{85} +1.89374 q^{87} +13.6214 q^{89} +2.38449 q^{91} +0.764644 q^{93} +3.78137 q^{95} -2.27485 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{3} - 6 q^{5} - 4 q^{7} + 12 q^{9} - 4 q^{13} + 2 q^{15} - 8 q^{17} + 14 q^{19} - 18 q^{21} - 2 q^{23} + 8 q^{25} + 10 q^{27} + 4 q^{29} - 2 q^{31} - 24 q^{35} - 10 q^{37} - 24 q^{39} - 4 q^{41}+ \cdots - 20 q^{97}+O(q^{100}) \) 14 * q - 2 * q^3 - 6 * q^5 - 4 * q^7 + 12 * q^9 - 4 * q^13 + 2 * q^15 - 8 * q^17 + 14 * q^19 - 18 * q^21 - 2 * q^23 + 8 * q^25 + 10 * q^27 + 4 * q^29 - 2 * q^31 - 24 * q^35 - 10 * q^37 - 24 * q^39 - 4 * q^41 - 32 * q^43 - 20 * q^45 - 4 * q^47 - 14 * q^49 - 22 * q^51 + 6 * q^53 - 2 * q^57 - 6 * q^59 + 4 * q^61 + 16 * q^63 - 54 * q^65 + 6 * q^67 + 4 * q^69 - 6 * q^71 + 24 * q^73 - 10 * q^75 - 48 * q^79 + 2 * q^81 + 40 * q^85 - 28 * q^87 + 2 * q^89 + 28 * q^91 + 34 * q^93 - 6 * q^95 - 20 * q^97

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