LMFDB - Embedded newform 9196.2.a.u.1.7

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			1.7
Root			$0.317193$ 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)

$f(q)$	$=$	$q-0.317193 q^{3} -0.877997 q^{5} +2.82891 q^{7} -2.89939 q^{9} -2.34917 q^{13} +0.278495 q^{15} +2.83540 q^{17} +1.00000 q^{19} -0.897311 q^{21} -2.35099 q^{23} -4.22912 q^{25} +1.87125 q^{27} -2.13112 q^{29} +6.84533 q^{31} -2.48378 q^{35} +0.900818 q^{37} +0.745139 q^{39} +0.541889 q^{41} +11.2750 q^{43} +2.54566 q^{45} -10.2520 q^{47} +1.00273 q^{49} -0.899369 q^{51} -5.45175 q^{53} -0.317193 q^{57} +4.66778 q^{59} +5.60226 q^{61} -8.20211 q^{63} +2.06256 q^{65} -8.36602 q^{67} +0.745719 q^{69} +5.63271 q^{71} +3.87935 q^{73} +1.34145 q^{75} +2.29580 q^{79} +8.10462 q^{81} -4.51212 q^{83} -2.48947 q^{85} +0.675978 q^{87} -17.2031 q^{89} -6.64558 q^{91} -2.17129 q^{93} -0.877997 q^{95} -3.48740 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.317193	−0.183132	−0.0915658	−	0.995799i	$-0.529187\pi$
$3$	−0.317193	−0.183132	−0.0915658	+	0.995799i	$0.529187\pi$
$4$	0	0
$5$	−0.877997	−0.392652	−0.196326	−	0.980539i	$-0.562901\pi$
$5$	−0.877997	−0.392652	−0.196326	+	0.980539i	$0.562901\pi$
$6$	0	0
$7$	2.82891	1.06923	0.534614	−	0.845097i	$-0.320457\pi$
$7$	2.82891	1.06923	0.534614	+	0.845097i	$0.320457\pi$
$8$	0	0
$9$	−2.89939	−0.966463
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.34917	−0.651541	−0.325771	−	0.945449i	$-0.605624\pi$
$13$	−2.34917	−0.651541	−0.325771	+	0.945449i	$0.605624\pi$
$14$	0	0
$15$	0.278495	0.0719071
$16$	0	0
$17$	2.83540	0.687685	0.343843	−	0.939027i	$-0.388271\pi$
$17$	2.83540	0.687685	0.343843	+	0.939027i	$0.388271\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.897311	−0.195809
$22$	0	0
$23$	−2.35099	−0.490216	−0.245108	−	0.969496i	$-0.578823\pi$
$23$	−2.35099	−0.490216	−0.245108	+	0.969496i	$0.578823\pi$
$24$	0	0
$25$	−4.22912	−0.845824
$26$	0	0
$27$	1.87125	0.360122
$28$	0	0
$29$	−2.13112	−0.395739	−0.197870	−	0.980228i	$-0.563402\pi$
$29$	−2.13112	−0.395739	−0.197870	+	0.980228i	$0.563402\pi$
$30$	0	0
$31$	6.84533	1.22946	0.614729	−	0.788738i	$-0.289265\pi$
$31$	6.84533	1.22946	0.614729	+	0.788738i	$0.289265\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.48378	−0.419835
$36$	0	0
$37$	0.900818	0.148093	0.0740467	−	0.997255i	$-0.476409\pi$
$37$	0.900818	0.148093	0.0740467	+	0.997255i	$0.476409\pi$
$38$	0	0
$39$	0.745139	0.119318
$40$	0	0
$41$	0.541889	0.0846288	0.0423144	−	0.999104i	$-0.486527\pi$
$41$	0.541889	0.0846288	0.0423144	+	0.999104i	$0.486527\pi$
$42$	0	0
$43$	11.2750	1.71941	0.859707	−	0.510787i	$-0.170646\pi$
$43$	11.2750	1.71941	0.859707	+	0.510787i	$0.170646\pi$
$44$	0	0
$45$	2.54566	0.379484
$46$	0	0
$47$	−10.2520	−1.49541	−0.747704	−	0.664032i	$-0.768844\pi$
$47$	−10.2520	−1.49541	−0.747704	+	0.664032i	$0.768844\pi$
$48$	0	0
$49$	1.00273	0.143248
$50$	0	0
$51$	−0.899369	−0.125937
$52$	0	0
$53$	−5.45175	−0.748855	−0.374428	−	0.927256i	$-0.622161\pi$
$53$	−5.45175	−0.748855	−0.374428	+	0.927256i	$0.622161\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.317193	−0.0420133
$58$	0	0
$59$	4.66778	0.607693	0.303846	−	0.952721i	$-0.401729\pi$
$59$	4.66778	0.607693	0.303846	+	0.952721i	$0.401729\pi$
$60$	0	0
$61$	5.60226	0.717296	0.358648	−	0.933473i	$-0.383238\pi$
$61$	5.60226	0.717296	0.358648	+	0.933473i	$0.383238\pi$
$62$	0	0
$63$	−8.20211	−1.03337
$64$	0	0
$65$	2.06256	0.255829
$66$	0	0
$67$	−8.36602	−1.02207	−0.511036	−	0.859559i	$-0.670738\pi$
$67$	−8.36602	−1.02207	−0.511036	+	0.859559i	$0.670738\pi$
$68$	0	0
$69$	0.745719	0.0897740
$70$	0	0
$71$	5.63271	0.668479	0.334240	−	0.942488i	$-0.391520\pi$
$71$	5.63271	0.668479	0.334240	+	0.942488i	$0.391520\pi$
$72$	0	0
$73$	3.87935	0.454044	0.227022	−	0.973890i	$-0.427101\pi$
$73$	3.87935	0.454044	0.227022	+	0.973890i	$0.427101\pi$
$74$	0	0
$75$	1.34145	0.154897
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.29580	0.258298	0.129149	−	0.991625i	$-0.458775\pi$
$79$	2.29580	0.258298	0.129149	+	0.991625i	$0.458775\pi$
$80$	0	0
$81$	8.10462	0.900513
$82$	0	0
$83$	−4.51212	−0.495269	−0.247635	−	0.968853i	$-0.579653\pi$
$83$	−4.51212	−0.495269	−0.247635	+	0.968853i	$0.579653\pi$
$84$	0	0
$85$	−2.48947	−0.270021
$86$	0	0
$87$	0.675978	0.0724724
$88$	0	0
$89$	−17.2031	−1.82353	−0.911765	−	0.410713i	$-0.865280\pi$
$89$	−17.2031	−1.82353	−0.911765	+	0.410713i	$0.865280\pi$
$90$	0	0
$91$	−6.64558	−0.696646
$92$	0	0
$93$	−2.17129	−0.225153
$94$	0	0
$95$	−0.877997	−0.0900806
$96$	0	0
$97$	−3.48740	−0.354092	−0.177046	−	0.984203i	$-0.556654\pi$
$97$	−3.48740	−0.354092	−0.177046	+	0.984203i	$0.556654\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−11.1658	−1.11104	−0.555521	−	0.831502i	$-0.687481\pi$
$101$	−11.1658	−1.11104	−0.555521	+	0.831502i	$0.687481\pi$
$102$	0	0
$103$	5.59731	0.551519	0.275760	−	0.961227i	$-0.411071\pi$
$103$	5.59731	0.551519	0.275760	+	0.961227i	$0.411071\pi$
$104$	0	0
$105$	0.787837	0.0768850
$106$	0	0
$107$	12.3542	1.19432	0.597162	−	0.802120i	$-0.296295\pi$
$107$	12.3542	1.19432	0.597162	+	0.802120i	$0.296295\pi$
$108$	0	0
$109$	−18.2381	−1.74689	−0.873446	−	0.486921i	$-0.838120\pi$
$109$	−18.2381	−1.74689	−0.873446	+	0.486921i	$0.838120\pi$
$110$	0	0
$111$	−0.285733	−0.0271206
$112$	0	0
$113$	−8.76910	−0.824927	−0.412464	−	0.910974i	$-0.635332\pi$
$113$	−8.76910	−0.824927	−0.412464	+	0.910974i	$0.635332\pi$
$114$	0	0
$115$	2.06416	0.192484
$116$	0	0
$117$	6.81114	0.629690
$118$	0	0
$119$	8.02109	0.735292
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−0.171884	−0.0154982
$124$	0	0
$125$	8.10314	0.724767
$126$	0	0
$127$	12.2389	1.08602	0.543012	−	0.839725i	$-0.317284\pi$
$127$	12.2389	1.08602	0.543012	+	0.839725i	$0.317284\pi$
$128$	0	0
$129$	−3.57634	−0.314879
$130$	0	0
$131$	−7.89245	−0.689567	−0.344783	−	0.938682i	$-0.612048\pi$
$131$	−7.89245	−0.689567	−0.344783	+	0.938682i	$0.612048\pi$
$132$	0	0
$133$	2.82891	0.245298
$134$	0	0
$135$	−1.64295	−0.141403
$136$	0	0
$137$	−14.1177	−1.20615	−0.603077	−	0.797683i	$-0.706059\pi$
$137$	−14.1177	−1.20615	−0.603077	+	0.797683i	$0.706059\pi$
$138$	0	0
$139$	−5.80757	−0.492592	−0.246296	−	0.969195i	$-0.579214\pi$
$139$	−5.80757	−0.492592	−0.246296	+	0.969195i	$0.579214\pi$
$140$	0	0
$141$	3.25187	0.273857
$142$	0	0
$143$	0	0
$144$	0	0
$145$	1.87112	0.155388
$146$	0	0
$147$	−0.318060	−0.0262332
$148$	0	0
$149$	−16.4524	−1.34783	−0.673917	−	0.738807i	$-0.735389\pi$
$149$	−16.4524	−1.34783	−0.673917	+	0.738807i	$0.735389\pi$
$150$	0	0
$151$	−0.505516	−0.0411383	−0.0205692	−	0.999788i	$-0.506548\pi$
$151$	−0.505516	−0.0411383	−0.0205692	+	0.999788i	$0.506548\pi$
$152$	0	0
$153$	−8.22092	−0.664622
$154$	0	0
$155$	−6.01018	−0.482750
$156$	0	0
$157$	7.58210	0.605117	0.302559	−	0.953131i	$-0.402159\pi$
$157$	7.58210	0.605117	0.302559	+	0.953131i	$0.402159\pi$
$158$	0	0
$159$	1.72926	0.137139
$160$	0	0
$161$	−6.65075	−0.524152
$162$	0	0
$163$	13.9486	1.09254	0.546269	−	0.837610i	$-0.316048\pi$
$163$	13.9486	1.09254	0.546269	+	0.837610i	$0.316048\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.42273	−0.419623	−0.209812	−	0.977742i	$-0.567285\pi$
$167$	−5.42273	−0.419623	−0.209812	+	0.977742i	$0.567285\pi$
$168$	0	0
$169$	−7.48142	−0.575494
$170$	0	0
$171$	−2.89939	−0.221722
$172$	0	0
$173$	6.35089	0.482849	0.241425	−	0.970420i	$-0.422385\pi$
$173$	6.35089	0.482849	0.241425	+	0.970420i	$0.422385\pi$
$174$	0	0
$175$	−11.9638	−0.904378
$176$	0	0
$177$	−1.48059	−0.111288
$178$	0	0
$179$	12.5833	0.940519	0.470260	−	0.882528i	$-0.344160\pi$
$179$	12.5833	0.940519	0.470260	+	0.882528i	$0.344160\pi$
$180$	0	0
$181$	−1.44772	−0.107608	−0.0538039	−	0.998552i	$-0.517135\pi$
$181$	−1.44772	−0.107608	−0.0538039	+	0.998552i	$0.517135\pi$
$182$	0	0
$183$	−1.77700	−0.131360
$184$	0	0
$185$	−0.790915	−0.0581493
$186$	0	0
$187$	0	0
$188$	0	0
$189$	5.29359	0.385052
$190$	0	0
$191$	14.7977	1.07072	0.535360	−	0.844624i	$-0.320176\pi$
$191$	14.7977	1.07072	0.535360	+	0.844624i	$0.320176\pi$
$192$	0	0
$193$	−13.1943	−0.949745	−0.474873	−	0.880054i	$-0.657506\pi$
$193$	−13.1943	−0.949745	−0.474873	+	0.880054i	$0.657506\pi$
$194$	0	0
$195$	−0.654231	−0.0468504
$196$	0	0
$197$	2.08986	0.148896	0.0744480	−	0.997225i	$-0.476281\pi$
$197$	2.08986	0.148896	0.0744480	+	0.997225i	$0.476281\pi$
$198$	0	0
$199$	8.60211	0.609787	0.304894	−	0.952386i	$-0.401379\pi$
$199$	8.60211	0.609787	0.304894	+	0.952386i	$0.401379\pi$
$200$	0	0
$201$	2.65365	0.187174
$202$	0	0
$203$	−6.02875	−0.423135
$204$	0	0
$205$	−0.475777	−0.0332297
$206$	0	0
$207$	6.81644	0.473775
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−10.4197	−0.717322	−0.358661	−	0.933468i	$-0.616767\pi$
$211$	−10.4197	−0.717322	−0.358661	+	0.933468i	$0.616767\pi$
$212$	0	0
$213$	−1.78666	−0.122420
$214$	0	0
$215$	−9.89938	−0.675132
$216$	0	0
$217$	19.3648	1.31457
$218$	0	0
$219$	−1.23051	−0.0831498
$220$	0	0
$221$	−6.66082	−0.448055
$222$	0	0
$223$	−20.0287	−1.34122	−0.670610	−	0.741811i	$-0.733967\pi$
$223$	−20.0287	−1.34122	−0.670610	+	0.741811i	$0.733967\pi$
$224$	0	0
$225$	12.2619	0.817458
$226$	0	0
$227$	−17.0996	−1.13494	−0.567470	−	0.823394i	$-0.692078\pi$
$227$	−17.0996	−1.13494	−0.567470	+	0.823394i	$0.692078\pi$
$228$	0	0
$229$	−16.0695	−1.06190	−0.530952	−	0.847402i	$-0.678166\pi$
$229$	−16.0695	−1.06190	−0.530952	+	0.847402i	$0.678166\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.4881	−1.47324	−0.736622	−	0.676305i	$-0.763580\pi$
$233$	−22.4881	−1.47324	−0.736622	+	0.676305i	$0.763580\pi$
$234$	0	0
$235$	9.00123	0.587175
$236$	0	0
$237$	−0.728213	−0.0473025
$238$	0	0
$239$	9.96162	0.644364	0.322182	−	0.946678i	$-0.395584\pi$
$239$	9.96162	0.644364	0.322182	+	0.946678i	$0.395584\pi$
$240$	0	0
$241$	−21.1466	−1.36217	−0.681085	−	0.732205i	$-0.738491\pi$
$241$	−21.1466	−1.36217	−0.681085	+	0.732205i	$0.738491\pi$
$242$	0	0
$243$	−8.18447	−0.525034
$244$	0	0
$245$	−0.880397	−0.0562465
$246$	0	0
$247$	−2.34917	−0.149474
$248$	0	0
$249$	1.43121	0.0906995
$250$	0	0
$251$	1.30438	0.0823315	0.0411658	−	0.999152i	$-0.486893\pi$
$251$	1.30438	0.0823315	0.0411658	+	0.999152i	$0.486893\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0.789644	0.0494494
$256$	0	0
$257$	14.5417	0.907088	0.453544	−	0.891234i	$-0.350159\pi$
$257$	14.5417	0.907088	0.453544	+	0.891234i	$0.350159\pi$
$258$	0	0
$259$	2.54833	0.158346
$260$	0	0
$261$	6.17895	0.382467
$262$	0	0
$263$	−3.67772	−0.226778	−0.113389	−	0.993551i	$-0.536171\pi$
$263$	−3.67772	−0.226778	−0.113389	+	0.993551i	$0.536171\pi$
$264$	0	0
$265$	4.78662	0.294040
$266$	0	0
$267$	5.45672	0.333946
$268$	0	0
$269$	16.4799	1.00480	0.502398	−	0.864636i	$-0.332451\pi$
$269$	16.4799	1.00480	0.502398	+	0.864636i	$0.332451\pi$
$270$	0	0
$271$	−3.23598	−0.196572	−0.0982860	−	0.995158i	$-0.531336\pi$
$271$	−3.23598	−0.196572	−0.0982860	+	0.995158i	$0.531336\pi$
$272$	0	0
$273$	2.10793	0.127578
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4.47856	−0.269091	−0.134545	−	0.990907i	$-0.542957\pi$
$277$	−4.47856	−0.269091	−0.134545	+	0.990907i	$0.542957\pi$
$278$	0	0
$279$	−19.8473	−1.18823
$280$	0	0
$281$	22.4665	1.34024	0.670120	−	0.742252i	$-0.266242\pi$
$281$	22.4665	1.34024	0.670120	+	0.742252i	$0.266242\pi$
$282$	0	0
$283$	8.74896	0.520071	0.260036	−	0.965599i	$-0.416266\pi$
$283$	8.74896	0.520071	0.260036	+	0.965599i	$0.416266\pi$
$284$	0	0
$285$	0.278495	0.0164966
$286$	0	0
$287$	1.53295	0.0904875
$288$	0	0
$289$	−8.96052	−0.527089
$290$	0	0
$291$	1.10618	0.0648454
$292$	0	0
$293$	−9.01052	−0.526400	−0.263200	−	0.964741i	$-0.584778\pi$
$293$	−9.01052	−0.526400	−0.263200	+	0.964741i	$0.584778\pi$
$294$	0	0
$295$	−4.09830	−0.238612
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.52287	0.319396
$300$	0	0
$301$	31.8958	1.83845
$302$	0	0
$303$	3.54173	0.203467
$304$	0	0
$305$	−4.91877	−0.281648
$306$	0	0
$307$	−25.1681	−1.43642	−0.718209	−	0.695828i	$-0.755038\pi$
$307$	−25.1681	−1.43642	−0.718209	+	0.695828i	$0.755038\pi$
$308$	0	0
$309$	−1.77543	−0.101001
$310$	0	0
$311$	2.66921	0.151357	0.0756784	−	0.997132i	$-0.475888\pi$
$311$	2.66921	0.151357	0.0756784	+	0.997132i	$0.475888\pi$
$312$	0	0
$313$	13.5639	0.766675	0.383338	−	0.923608i	$-0.374775\pi$
$313$	13.5639	0.766675	0.383338	+	0.923608i	$0.374775\pi$
$314$	0	0
$315$	7.20143	0.405755
$316$	0	0
$317$	−13.3402	−0.749262	−0.374631	−	0.927174i	$-0.622231\pi$
$317$	−13.3402	−0.749262	−0.374631	+	0.927174i	$0.622231\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−3.91867	−0.218719
$322$	0	0
$323$	2.83540	0.157766
$324$	0	0
$325$	9.93490	0.551089
$326$	0	0
$327$	5.78500	0.319911
$328$	0	0
$329$	−29.0020	−1.59893
$330$	0	0
$331$	−27.3697	−1.50438	−0.752189	−	0.658948i	$-0.771002\pi$
$331$	−27.3697	−1.50438	−0.752189	+	0.658948i	$0.771002\pi$
$332$	0	0
$333$	−2.61182	−0.143127
$334$	0	0
$335$	7.34535	0.401319
$336$	0	0
$337$	25.8029	1.40557	0.702787	−	0.711401i	$-0.251939\pi$
$337$	25.8029	1.40557	0.702787	+	0.711401i	$0.251939\pi$
$338$	0	0
$339$	2.78150	0.151070
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−16.9657	−0.916063
$344$	0	0
$345$	−0.654739	−0.0352500
$346$	0	0
$347$	−27.6620	−1.48498	−0.742488	−	0.669859i	$-0.766355\pi$
$347$	−27.6620	−1.48498	−0.742488	+	0.669859i	$0.766355\pi$
$348$	0	0
$349$	−8.88316	−0.475505	−0.237752	−	0.971326i	$-0.576411\pi$
$349$	−8.88316	−0.475505	−0.237752	+	0.971326i	$0.576411\pi$
$350$	0	0
$351$	−4.39587	−0.234634
$352$	0	0
$353$	−20.4136	−1.08650	−0.543252	−	0.839570i	$-0.682807\pi$
$353$	−20.4136	−1.08650	−0.543252	+	0.839570i	$0.682807\pi$
$354$	0	0
$355$	−4.94550	−0.262480
$356$	0	0
$357$	−2.54424	−0.134655
$358$	0	0
$359$	−37.8649	−1.99843	−0.999215	−	0.0396106i	$-0.987388\pi$
$359$	−37.8649	−1.99843	−0.999215	+	0.0396106i	$0.987388\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.40606	−0.178281
$366$	0	0
$367$	−6.55003	−0.341909	−0.170955	−	0.985279i	$-0.554685\pi$
$367$	−6.55003	−0.341909	−0.170955	+	0.985279i	$0.554685\pi$
$368$	0	0
$369$	−1.57115	−0.0817906
$370$	0	0
$371$	−15.4225	−0.800697
$372$	0	0
$373$	0.901613	0.0466838	0.0233419	−	0.999728i	$-0.492569\pi$
$373$	0.901613	0.0466838	0.0233419	+	0.999728i	$0.492569\pi$
$374$	0	0
$375$	−2.57026	−0.132728
$376$	0	0
$377$	5.00636	0.257840
$378$	0	0
$379$	21.4714	1.10291	0.551455	−	0.834204i	$-0.314073\pi$
$379$	21.4714	1.10291	0.551455	+	0.834204i	$0.314073\pi$
$380$	0	0
$381$	−3.88209	−0.198885
$382$	0	0
$383$	14.7379	0.753072	0.376536	−	0.926402i	$-0.377115\pi$
$383$	14.7379	0.753072	0.376536	+	0.926402i	$0.377115\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−32.6905	−1.66175
$388$	0	0
$389$	8.43484	0.427664	0.213832	−	0.976871i	$-0.431406\pi$
$389$	8.43484	0.427664	0.213832	+	0.976871i	$0.431406\pi$
$390$	0	0
$391$	−6.66600	−0.337114
$392$	0	0
$393$	2.50343	0.126281
$394$	0	0
$395$	−2.01571	−0.101421
$396$	0	0
$397$	14.2657	0.715975	0.357988	−	0.933726i	$-0.383463\pi$
$397$	14.2657	0.715975	0.357988	+	0.933726i	$0.383463\pi$
$398$	0	0
$399$	−0.897311	−0.0449218
$400$	0	0
$401$	−21.7936	−1.08832	−0.544159	−	0.838982i	$-0.683151\pi$
$401$	−21.7936	−1.08832	−0.544159	+	0.838982i	$0.683151\pi$
$402$	0	0
$403$	−16.0808	−0.801042
$404$	0	0
$405$	−7.11583	−0.353589
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−25.0553	−1.23890	−0.619451	−	0.785035i	$-0.712645\pi$
$409$	−25.0553	−1.23890	−0.619451	+	0.785035i	$0.712645\pi$
$410$	0	0
$411$	4.47803	0.220885
$412$	0	0
$413$	13.2047	0.649762
$414$	0	0
$415$	3.96163	0.194469
$416$	0	0
$417$	1.84212	0.0902092
$418$	0	0
$419$	9.17365	0.448162	0.224081	−	0.974571i	$-0.428062\pi$
$419$	9.17365	0.448162	0.224081	+	0.974571i	$0.428062\pi$
$420$	0	0
$421$	−23.7192	−1.15600	−0.578001	−	0.816036i	$-0.696167\pi$
$421$	−23.7192	−1.15600	−0.578001	+	0.816036i	$0.696167\pi$
$422$	0	0
$423$	29.7245	1.44526
$424$	0	0
$425$	−11.9912	−0.581661
$426$	0	0
$427$	15.8483	0.766953
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.74001	0.228318	0.114159	−	0.993462i	$-0.463583\pi$
$431$	4.74001	0.228318	0.114159	+	0.993462i	$0.463583\pi$
$432$	0	0
$433$	4.28966	0.206148	0.103074	−	0.994674i	$-0.467132\pi$
$433$	4.28966	0.206148	0.103074	+	0.994674i	$0.467132\pi$
$434$	0	0
$435$	−0.593506	−0.0284565
$436$	0	0
$437$	−2.35099	−0.112463
$438$	0	0
$439$	−14.5044	−0.692257	−0.346129	−	0.938187i	$-0.612504\pi$
$439$	−14.5044	−0.692257	−0.346129	+	0.938187i	$0.612504\pi$
$440$	0	0
$441$	−2.90731	−0.138443
$442$	0	0
$443$	40.3396	1.91659	0.958297	−	0.285773i	$-0.0922504\pi$
$443$	40.3396	1.91659	0.958297	+	0.285773i	$0.0922504\pi$
$444$	0	0
$445$	15.1043	0.716013
$446$	0	0
$447$	5.21859	0.246831
$448$	0	0
$449$	7.55299	0.356448	0.178224	−	0.983990i	$-0.442965\pi$
$449$	7.55299	0.356448	0.178224	+	0.983990i	$0.442965\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0.160346	0.00753373
$454$	0	0
$455$	5.83480	0.273540
$456$	0	0
$457$	14.9425	0.698979	0.349489	−	0.936940i	$-0.386355\pi$
$457$	14.9425	0.698979	0.349489	+	0.936940i	$0.386355\pi$
$458$	0	0
$459$	5.30573	0.247650
$460$	0	0
$461$	2.32084	0.108092	0.0540461	−	0.998538i	$-0.482788\pi$
$461$	2.32084	0.108092	0.0540461	+	0.998538i	$0.482788\pi$
$462$	0	0
$463$	−21.5839	−1.00309	−0.501544	−	0.865132i	$-0.667235\pi$
$463$	−21.5839	−1.00309	−0.501544	+	0.865132i	$0.667235\pi$
$464$	0	0
$465$	1.90639	0.0884067
$466$	0	0
$467$	−4.03378	−0.186661	−0.0933306	−	0.995635i	$-0.529751\pi$
$467$	−4.03378	−0.186661	−0.0933306	+	0.995635i	$0.529751\pi$
$468$	0	0
$469$	−23.6667	−1.09283
$470$	0	0
$471$	−2.40499	−0.110816
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.22912	−0.194045
$476$	0	0
$477$	15.8067	0.723741
$478$	0	0
$479$	−5.18610	−0.236959	−0.118479	−	0.992957i	$-0.537802\pi$
$479$	−5.18610	−0.236959	−0.118479	+	0.992957i	$0.537802\pi$
$480$	0	0
$481$	−2.11617	−0.0964890
$482$	0	0
$483$	2.10957	0.0959889
$484$	0	0
$485$	3.06193	0.139035
$486$	0	0
$487$	−17.0907	−0.774455	−0.387228	−	0.921984i	$-0.626567\pi$
$487$	−17.0907	−0.774455	−0.387228	+	0.921984i	$0.626567\pi$
$488$	0	0
$489$	−4.42440	−0.200078
$490$	0	0
$491$	25.9792	1.17243	0.586213	−	0.810157i	$-0.300618\pi$
$491$	25.9792	1.17243	0.586213	+	0.810157i	$0.300618\pi$
$492$	0	0
$493$	−6.04258	−0.272144
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.9344	0.714757
$498$	0	0
$499$	24.7696	1.10884	0.554420	−	0.832237i	$-0.312940\pi$
$499$	24.7696	1.10884	0.554420	+	0.832237i	$0.312940\pi$
$500$	0	0
$501$	1.72005	0.0768463
$502$	0	0
$503$	−11.7214	−0.522630	−0.261315	−	0.965254i	$-0.584156\pi$
$503$	−11.7214	−0.522630	−0.261315	+	0.965254i	$0.584156\pi$
$504$	0	0
$505$	9.80357	0.436253
$506$	0	0
$507$	2.37306	0.105391
$508$	0	0
$509$	−24.1144	−1.06885	−0.534426	−	0.845215i	$-0.679472\pi$
$509$	−24.1144	−1.06885	−0.534426	+	0.845215i	$0.679472\pi$
$510$	0	0
$511$	10.9743	0.485476
$512$	0	0
$513$	1.87125	0.0826176
$514$	0	0
$515$	−4.91442	−0.216555
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−2.01446	−0.0884250
$520$	0	0
$521$	27.3411	1.19783	0.598917	−	0.800811i	$-0.295598\pi$
$521$	27.3411	1.19783	0.598917	+	0.800811i	$0.295598\pi$
$522$	0	0
$523$	−16.0181	−0.700421	−0.350211	−	0.936671i	$-0.613890\pi$
$523$	−16.0181	−0.700421	−0.350211	+	0.936671i	$0.613890\pi$
$524$	0	0
$525$	3.79484	0.165620
$526$	0	0
$527$	19.4092	0.845480
$528$	0	0
$529$	−17.4728	−0.759689
$530$	0	0
$531$	−13.5337	−0.587312
$532$	0	0
$533$	−1.27299	−0.0551391
$534$	0	0
$535$	−10.8469	−0.468954
$536$	0	0
$537$	−3.99134	−0.172239
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−0.120066	−0.00516206	−0.00258103	−	0.999997i	$-0.500822\pi$
$541$	−0.120066	−0.00516206	−0.00258103	+	0.999997i	$0.500822\pi$
$542$	0	0
$543$	0.459206	0.0197064
$544$	0	0
$545$	16.0130	0.685921
$546$	0	0
$547$	13.3381	0.570295	0.285148	−	0.958484i	$-0.407957\pi$
$547$	13.3381	0.570295	0.285148	+	0.958484i	$0.407957\pi$
$548$	0	0
$549$	−16.2431	−0.693240
$550$	0	0
$551$	−2.13112	−0.0907888
$552$	0	0
$553$	6.49462	0.276179
$554$	0	0
$555$	0.250873	0.0106490
$556$	0	0
$557$	−20.2841	−0.859464	−0.429732	−	0.902956i	$-0.641392\pi$
$557$	−20.2841	−0.859464	−0.429732	+	0.902956i	$0.641392\pi$
$558$	0	0
$559$	−26.4867	−1.12027
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−3.67872	−0.155039	−0.0775197	−	0.996991i	$-0.524700\pi$
$563$	−3.67872	−0.155039	−0.0775197	+	0.996991i	$0.524700\pi$
$564$	0	0
$565$	7.69924	0.323910
$566$	0	0
$567$	22.9272	0.962853
$568$	0	0
$569$	4.37233	0.183298	0.0916488	−	0.995791i	$-0.470786\pi$
$569$	4.37233	0.183298	0.0916488	+	0.995791i	$0.470786\pi$
$570$	0	0
$571$	−3.98335	−0.166698	−0.0833490	−	0.996520i	$-0.526562\pi$
$571$	−3.98335	−0.166698	−0.0833490	+	0.996520i	$0.526562\pi$
$572$	0	0
$573$	−4.69372	−0.196083
$574$	0	0
$575$	9.94263	0.414636
$576$	0	0
$577$	−40.1538	−1.67162	−0.835812	−	0.549016i	$-0.815003\pi$
$577$	−40.1538	−1.67162	−0.835812	+	0.549016i	$0.815003\pi$
$578$	0	0
$579$	4.18514	0.173928
$580$	0	0
$581$	−12.7644	−0.529556
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−5.98016	−0.247249
$586$	0	0
$587$	17.4130	0.718710	0.359355	−	0.933201i	$-0.382997\pi$
$587$	17.4130	0.718710	0.359355	+	0.933201i	$0.382997\pi$
$588$	0	0
$589$	6.84533	0.282057
$590$	0	0
$591$	−0.662888	−0.0272676
$592$	0	0
$593$	3.18264	0.130696	0.0653478	−	0.997863i	$-0.479184\pi$
$593$	3.18264	0.130696	0.0653478	+	0.997863i	$0.479184\pi$
$594$	0	0
$595$	−7.04249	−0.288714
$596$	0	0
$597$	−2.72853	−0.111671
$598$	0	0
$599$	−33.0196	−1.34914	−0.674571	−	0.738210i	$-0.735671\pi$
$599$	−33.0196	−1.34914	−0.674571	+	0.738210i	$0.735671\pi$
$600$	0	0
$601$	32.1352	1.31082	0.655411	−	0.755272i	$-0.272495\pi$
$601$	32.1352	1.31082	0.655411	+	0.755272i	$0.272495\pi$
$602$	0	0
$603$	24.2563	0.987795
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−12.8889	−0.523142	−0.261571	−	0.965184i	$-0.584241\pi$
$607$	−12.8889	−0.523142	−0.261571	+	0.965184i	$0.584241\pi$
$608$	0	0
$609$	1.91228	0.0774895
$610$	0	0
$611$	24.0836	0.974320
$612$	0	0
$613$	−19.2889	−0.779071	−0.389535	−	0.921011i	$-0.627364\pi$
$613$	−19.2889	−0.779071	−0.389535	+	0.921011i	$0.627364\pi$
$614$	0	0
$615$	0.150913	0.00608541
$616$	0	0
$617$	−43.9564	−1.76962	−0.884809	−	0.465954i	$-0.845711\pi$
$617$	−43.9564	−1.76962	−0.884809	+	0.465954i	$0.845711\pi$
$618$	0	0
$619$	−41.4171	−1.66469	−0.832347	−	0.554255i	$-0.813003\pi$
$619$	−41.4171	−1.66469	−0.832347	+	0.554255i	$0.813003\pi$
$620$	0	0
$621$	−4.39929	−0.176537
$622$	0	0
$623$	−48.6661	−1.94977
$624$	0	0
$625$	14.0311	0.561243
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.55418	0.101842
$630$	0	0
$631$	5.15798	0.205336	0.102668	−	0.994716i	$-0.467262\pi$
$631$	5.15798	0.205336	0.102668	+	0.994716i	$0.467262\pi$
$632$	0	0
$633$	3.30506	0.131364
$634$	0	0
$635$	−10.7457	−0.426430
$636$	0	0
$637$	−2.35559	−0.0933317
$638$	0	0
$639$	−16.3314	−0.646061
$640$	0	0
$641$	−19.5232	−0.771119	−0.385560	−	0.922683i	$-0.625992\pi$
$641$	−19.5232	−0.771119	−0.385560	+	0.922683i	$0.625992\pi$
$642$	0	0
$643$	29.2197	1.15231	0.576156	−	0.817340i	$-0.304552\pi$
$643$	29.2197	1.15231	0.576156	+	0.817340i	$0.304552\pi$
$644$	0	0
$645$	3.14002	0.123638
$646$	0	0
$647$	−31.3809	−1.23371	−0.616855	−	0.787076i	$-0.711594\pi$
$647$	−31.3809	−1.23371	−0.616855	+	0.787076i	$0.711594\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−6.14240	−0.240739
$652$	0	0
$653$	5.41554	0.211926	0.105963	−	0.994370i	$-0.466207\pi$
$653$	5.41554	0.211926	0.105963	+	0.994370i	$0.466207\pi$
$654$	0	0
$655$	6.92955	0.270760
$656$	0	0
$657$	−11.2478	−0.438817
$658$	0	0
$659$	22.0425	0.858654	0.429327	−	0.903149i	$-0.358751\pi$
$659$	22.0425	0.858654	0.429327	+	0.903149i	$0.358751\pi$
$660$	0	0
$661$	24.2761	0.944230	0.472115	−	0.881537i	$-0.343491\pi$
$661$	24.2761	0.944230	0.472115	+	0.881537i	$0.343491\pi$
$662$	0	0
$663$	2.11277	0.0820531
$664$	0	0
$665$	−2.48378	−0.0963167
$666$	0	0
$667$	5.01025	0.193998
$668$	0	0
$669$	6.35296	0.245620
$670$	0	0
$671$	0	0
$672$	0	0
$673$	8.64827	0.333366	0.166683	−	0.986010i	$-0.446694\pi$
$673$	8.64827	0.333366	0.166683	+	0.986010i	$0.446694\pi$
$674$	0	0
$675$	−7.91373	−0.304600
$676$	0	0
$677$	−44.8883	−1.72520	−0.862599	−	0.505888i	$-0.831165\pi$
$677$	−44.8883	−1.72520	−0.862599	+	0.505888i	$0.831165\pi$
$678$	0	0
$679$	−9.86554	−0.378605
$680$	0	0
$681$	5.42388	0.207844
$682$	0	0
$683$	−26.1230	−0.999569	−0.499785	−	0.866150i	$-0.666588\pi$
$683$	−26.1230	−0.999569	−0.499785	+	0.866150i	$0.666588\pi$
$684$	0	0
$685$	12.3953	0.473599
$686$	0	0
$687$	5.09715	0.194468
$688$	0	0
$689$	12.8071	0.487910
$690$	0	0
$691$	9.26699	0.352533	0.176266	−	0.984343i	$-0.443598\pi$
$691$	9.26699	0.352533	0.176266	+	0.984343i	$0.443598\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.09903	0.193417
$696$	0	0
$697$	1.53647	0.0581980
$698$	0	0
$699$	7.13307	0.269798
$700$	0	0
$701$	7.11234	0.268630	0.134315	−	0.990939i	$-0.457117\pi$
$701$	7.11234	0.268630	0.134315	+	0.990939i	$0.457117\pi$
$702$	0	0
$703$	0.900818	0.0339750
$704$	0	0
$705$	−2.85513	−0.107530
$706$	0	0
$707$	−31.5871	−1.18796
$708$	0	0
$709$	13.0058	0.488443	0.244222	−	0.969719i	$-0.421468\pi$
$709$	13.0058	0.488443	0.244222	+	0.969719i	$0.421468\pi$
$710$	0	0
$711$	−6.65642	−0.249635
$712$	0	0
$713$	−16.0933	−0.602700
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−3.15976	−0.118003
$718$	0	0
$719$	−27.1251	−1.01160	−0.505798	−	0.862652i	$-0.668802\pi$
$719$	−27.1251	−1.01160	−0.505798	+	0.862652i	$0.668802\pi$
$720$	0	0
$721$	15.8343	0.589700
$722$	0	0
$723$	6.70755	0.249456
$724$	0	0
$725$	9.01277	0.334726
$726$	0	0
$727$	−39.9376	−1.48120	−0.740602	−	0.671944i	$-0.765460\pi$
$727$	−39.9376	−1.48120	−0.740602	+	0.671944i	$0.765460\pi$
$728$	0	0
$729$	−21.7178	−0.804363
$730$	0	0
$731$	31.9690	1.18242
$732$	0	0
$733$	29.8601	1.10291	0.551454	−	0.834205i	$-0.314073\pi$
$733$	29.8601	1.10291	0.551454	+	0.834205i	$0.314073\pi$
$734$	0	0
$735$	0.279256	0.0103005
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−23.1977	−0.853340	−0.426670	−	0.904407i	$-0.640313\pi$
$739$	−23.1977	−0.853340	−0.426670	+	0.904407i	$0.640313\pi$
$740$	0	0
$741$	0.745139	0.0273734
$742$	0	0
$743$	−22.3718	−0.820740	−0.410370	−	0.911919i	$-0.634601\pi$
$743$	−22.3718	−0.820740	−0.410370	+	0.911919i	$0.634601\pi$
$744$	0	0
$745$	14.4452	0.529230
$746$	0	0
$747$	13.0824	0.478659
$748$	0	0
$749$	34.9489	1.27701
$750$	0	0
$751$	43.4017	1.58375	0.791875	−	0.610684i	$-0.209105\pi$
$751$	43.4017	1.58375	0.791875	+	0.610684i	$0.209105\pi$
$752$	0	0
$753$	−0.413740	−0.0150775
$754$	0	0
$755$	0.443842	0.0161531
$756$	0	0
$757$	10.3956	0.377833	0.188916	−	0.981993i	$-0.439502\pi$
$757$	10.3956	0.377833	0.188916	+	0.981993i	$0.439502\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.6502	−0.676069	−0.338034	−	0.941134i	$-0.609762\pi$
$761$	−18.6502	−0.676069	−0.338034	+	0.941134i	$0.609762\pi$
$762$	0	0
$763$	−51.5939	−1.86782
$764$	0	0
$765$	7.21795	0.260965
$766$	0	0
$767$	−10.9654	−0.395937
$768$	0	0
$769$	18.4248	0.664414	0.332207	−	0.943207i	$-0.392207\pi$
$769$	18.4248	0.664414	0.332207	+	0.943207i	$0.392207\pi$
$770$	0	0
$771$	−4.61254	−0.166117
$772$	0	0
$773$	−32.0028	−1.15106	−0.575530	−	0.817781i	$-0.695204\pi$
$773$	−32.0028	−1.15106	−0.575530	+	0.817781i	$0.695204\pi$
$774$	0	0
$775$	−28.9497	−1.03991
$776$	0	0
$777$	−0.808314	−0.0289981
$778$	0	0
$779$	0.541889	0.0194152
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−3.98785	−0.142514
$784$	0	0
$785$	−6.65706	−0.237601
$786$	0	0
$787$	−41.4204	−1.47648	−0.738239	−	0.674539i	$-0.764342\pi$
$787$	−41.4204	−1.47648	−0.738239	+	0.674539i	$0.764342\pi$
$788$	0	0
$789$	1.16655	0.0415302
$790$	0	0
$791$	−24.8070	−0.882035
$792$	0	0
$793$	−13.1606	−0.467348
$794$	0	0
$795$	−1.51828	−0.0538480
$796$	0	0
$797$	−4.03532	−0.142938	−0.0714691	−	0.997443i	$-0.522769\pi$
$797$	−4.03532	−0.142938	−0.0714691	+	0.997443i	$0.522769\pi$
$798$	0	0
$799$	−29.0685	−1.02837
$800$	0	0
$801$	49.8786	1.76237
$802$	0	0
$803$	0	0
$804$	0	0
$805$	5.83934	0.205810
$806$	0	0
$807$	−5.22731	−0.184010
$808$	0	0
$809$	−51.6820	−1.81704	−0.908521	−	0.417838i	$-0.862788\pi$
$809$	−51.6820	−1.81704	−0.908521	+	0.417838i	$0.862788\pi$
$810$	0	0
$811$	−45.8234	−1.60908	−0.804539	−	0.593900i	$-0.797587\pi$
$811$	−45.8234	−1.60908	−0.804539	+	0.593900i	$0.797587\pi$
$812$	0	0
$813$	1.02643	0.0359985
$814$	0	0
$815$	−12.2468	−0.428987
$816$	0	0
$817$	11.2750	0.394461
$818$	0	0
$819$	19.2681	0.673282
$820$	0	0
$821$	−0.804568	−0.0280796	−0.0140398	−	0.999901i	$-0.504469\pi$
$821$	−0.804568	−0.0280796	−0.0140398	+	0.999901i	$0.504469\pi$
$822$	0	0
$823$	3.97536	0.138572	0.0692861	−	0.997597i	$-0.477928\pi$
$823$	3.97536	0.138572	0.0692861	+	0.997597i	$0.477928\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.02261	−0.278974	−0.139487	−	0.990224i	$-0.544545\pi$
$827$	−8.02261	−0.278974	−0.139487	+	0.990224i	$0.544545\pi$
$828$	0	0
$829$	49.4246	1.71659	0.858293	−	0.513160i	$-0.171525\pi$
$829$	49.4246	1.71659	0.858293	+	0.513160i	$0.171525\pi$
$830$	0	0
$831$	1.42057	0.0492790
$832$	0	0
$833$	2.84315	0.0985093
$834$	0	0
$835$	4.76114	0.164766
$836$	0	0
$837$	12.8093	0.442754
$838$	0	0
$839$	−41.9061	−1.44676	−0.723379	−	0.690451i	$-0.757412\pi$
$839$	−41.9061	−1.44676	−0.723379	+	0.690451i	$0.757412\pi$
$840$	0	0
$841$	−24.4583	−0.843390
$842$	0	0
$843$	−7.12623	−0.245440
$844$	0	0
$845$	6.56867	0.225969
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2.77511	−0.0952415
$850$	0	0
$851$	−2.11781	−0.0725978
$852$	0	0
$853$	−3.57735	−0.122486	−0.0612431	−	0.998123i	$-0.519506\pi$
$853$	−3.57735	−0.122486	−0.0612431	+	0.998123i	$0.519506\pi$
$854$	0	0
$855$	2.54566	0.0870596
$856$	0	0
$857$	−36.2460	−1.23814	−0.619070	−	0.785336i	$-0.712490\pi$
$857$	−36.2460	−1.23814	−0.619070	+	0.785336i	$0.712490\pi$
$858$	0	0
$859$	9.66571	0.329790	0.164895	−	0.986311i	$-0.447272\pi$
$859$	9.66571	0.329790	0.164895	+	0.986311i	$0.447272\pi$
$860$	0	0
$861$	−0.486243	−0.0165711
$862$	0	0
$863$	5.72843	0.194998	0.0974990	−	0.995236i	$-0.468916\pi$
$863$	5.72843	0.194998	0.0974990	+	0.995236i	$0.468916\pi$
$864$	0	0
$865$	−5.57606	−0.189592
$866$	0	0
$867$	2.84222	0.0965267
$868$	0	0
$869$	0	0
$870$	0	0
$871$	19.6532	0.665922
$872$	0	0
$873$	10.1113	0.342216
$874$	0	0
$875$	22.9231	0.774941
$876$	0	0
$877$	52.2622	1.76477	0.882384	−	0.470530i	$-0.155937\pi$
$877$	52.2622	1.76477	0.882384	+	0.470530i	$0.155937\pi$
$878$	0	0
$879$	2.85808	0.0964005
$880$	0	0
$881$	27.5956	0.929719	0.464859	−	0.885385i	$-0.346105\pi$
$881$	27.5956	0.929719	0.464859	+	0.885385i	$0.346105\pi$
$882$	0	0
$883$	10.7870	0.363012	0.181506	−	0.983390i	$-0.441903\pi$
$883$	10.7870	0.363012	0.181506	+	0.983390i	$0.441903\pi$
$884$	0	0
$885$	1.29995	0.0436974
$886$	0	0
$887$	−36.0127	−1.20919	−0.604593	−	0.796534i	$-0.706664\pi$
$887$	−36.0127	−1.20919	−0.604593	+	0.796534i	$0.706664\pi$
$888$	0	0
$889$	34.6227	1.16121
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−10.2520	−0.343070
$894$	0	0
$895$	−11.0481	−0.369297
$896$	0	0
$897$	−1.75182	−0.0584915
$898$	0	0
$899$	−14.5882	−0.486545
$900$	0	0
$901$	−15.4579	−0.514977
$902$	0	0
$903$	−10.1171	−0.336678
$904$	0	0
$905$	1.27109	0.0422525
$906$	0	0
$907$	3.65075	0.121221	0.0606106	−	0.998161i	$-0.480695\pi$
$907$	3.65075	0.121221	0.0606106	+	0.998161i	$0.480695\pi$
$908$	0	0
$909$	32.3741	1.07378
$910$	0	0
$911$	18.0769	0.598914	0.299457	−	0.954110i	$-0.403194\pi$
$911$	18.0769	0.598914	0.299457	+	0.954110i	$0.403194\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	1.56020	0.0515787
$916$	0	0
$917$	−22.3270	−0.737304
$918$	0	0
$919$	35.3625	1.16650	0.583250	−	0.812293i	$-0.301781\pi$
$919$	35.3625	1.16650	0.583250	+	0.812293i	$0.301781\pi$
$920$	0	0
$921$	7.98314	0.263054
$922$	0	0
$923$	−13.2322	−0.435542
$924$	0	0
$925$	−3.80967	−0.125261
$926$	0	0
$927$	−16.2288	−0.533023
$928$	0	0
$929$	−11.7918	−0.386877	−0.193439	−	0.981112i	$-0.561964\pi$
$929$	−11.7918	−0.386877	−0.193439	+	0.981112i	$0.561964\pi$
$930$	0	0
$931$	1.00273	0.0328633
$932$	0	0
$933$	−0.846655	−0.0277182
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−41.6450	−1.36048	−0.680242	−	0.732987i	$-0.738125\pi$
$937$	−41.6450	−1.36048	−0.680242	+	0.732987i	$0.738125\pi$
$938$	0	0
$939$	−4.30237	−0.140402
$940$	0	0
$941$	10.3339	0.336875	0.168437	−	0.985712i	$-0.446128\pi$
$941$	10.3339	0.336875	0.168437	+	0.985712i	$0.446128\pi$
$942$	0	0
$943$	−1.27398	−0.0414864
$944$	0	0
$945$	−4.64776	−0.151192
$946$	0	0
$947$	−9.49384	−0.308508	−0.154254	−	0.988031i	$-0.549297\pi$
$947$	−9.49384	−0.308508	−0.154254	+	0.988031i	$0.549297\pi$
$948$	0	0
$949$	−9.11324	−0.295828
$950$	0	0
$951$	4.23143	0.137214
$952$	0	0
$953$	−15.6512	−0.506991	−0.253495	−	0.967337i	$-0.581580\pi$
$953$	−15.6512	−0.506991	−0.253495	+	0.967337i	$0.581580\pi$
$954$	0	0
$955$	−12.9923	−0.420421
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−39.9376	−1.28965
$960$	0	0
$961$	15.8586	0.511567
$962$	0	0
$963$	−35.8196	−1.15427
$964$	0	0
$965$	11.5845	0.372920
$966$	0	0
$967$	−15.1075	−0.485826	−0.242913	−	0.970048i	$-0.578103\pi$
$967$	−15.1075	−0.485826	−0.242913	+	0.970048i	$0.578103\pi$
$968$	0	0
$969$	−0.899369	−0.0288919
$970$	0	0
$971$	59.8099	1.91939	0.959696	−	0.281039i	$-0.0906791\pi$
$971$	59.8099	1.91939	0.959696	+	0.281039i	$0.0906791\pi$
$972$	0	0
$973$	−16.4291	−0.526693
$974$	0	0
$975$	−3.15128	−0.100922
$976$	0	0
$977$	−21.1265	−0.675898	−0.337949	−	0.941164i	$-0.609733\pi$
$977$	−21.1265	−0.675898	−0.337949	+	0.941164i	$0.609733\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	52.8793	1.68831
$982$	0	0
$983$	−19.4586	−0.620634	−0.310317	−	0.950633i	$-0.600435\pi$
$983$	−19.4586	−0.620634	−0.310317	+	0.950633i	$0.600435\pi$
$984$	0	0
$985$	−1.83489	−0.0584644
$986$	0	0
$987$	9.19924	0.292815
$988$	0	0
$989$	−26.5073	−0.842884
$990$	0	0
$991$	17.5327	0.556945	0.278473	−	0.960444i	$-0.410172\pi$
$991$	17.5327	0.556945	0.278473	+	0.960444i	$0.410172\pi$
$992$	0	0
$993$	8.68150	0.275499
$994$	0	0
$995$	−7.55263	−0.239434
$996$	0	0
$997$	62.3254	1.97387	0.986933	−	0.161130i	$-0.0515140\pi$
$997$	62.3254	1.97387	0.986933	+	0.161130i	$0.0515140\pi$
$998$	0	0
$999$	1.68565	0.0533317

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9196.2.a.u.1.7	✓	14
11.10	odd	2		9196.2.a.v.1.7	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9196.2.a.u.1.7	✓	14	1.1	even	1	trivial
9196.2.a.v.1.7	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)

\(f(q)\)	\(=\)	\(q-0.317193 q^{3} -0.877997 q^{5} +2.82891 q^{7} -2.89939 q^{9} -2.34917 q^{13} +0.278495 q^{15} +2.83540 q^{17} +1.00000 q^{19} -0.897311 q^{21} -2.35099 q^{23} -4.22912 q^{25} +1.87125 q^{27} -2.13112 q^{29} +6.84533 q^{31} -2.48378 q^{35} +0.900818 q^{37} +0.745139 q^{39} +0.541889 q^{41} +11.2750 q^{43} +2.54566 q^{45} -10.2520 q^{47} +1.00273 q^{49} -0.899369 q^{51} -5.45175 q^{53} -0.317193 q^{57} +4.66778 q^{59} +5.60226 q^{61} -8.20211 q^{63} +2.06256 q^{65} -8.36602 q^{67} +0.745719 q^{69} +5.63271 q^{71} +3.87935 q^{73} +1.34145 q^{75} +2.29580 q^{79} +8.10462 q^{81} -4.51212 q^{83} -2.48947 q^{85} +0.675978 q^{87} -17.2031 q^{89} -6.64558 q^{91} -2.17129 q^{93} -0.877997 q^{95} -3.48740 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 2 q^{3} - 6 q^{5} - 4 q^{7} + 12 q^{9} - 4 q^{13} + 2 q^{15} - 8 q^{17} + 14 q^{19} - 18 q^{21} - 2 q^{23} + 8 q^{25} + 10 q^{27} + 4 q^{29} - 2 q^{31} - 24 q^{35} - 10 q^{37} - 24 q^{39} - 4 q^{41}+ \cdots - 20 q^{97}+O(q^{100}) \) 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

Introduction

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Database

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