LMFDB - Embedded newform 4275.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, 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("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.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:	$34.1360468641$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{2} +3.82843 q^{4} -3.41421 q^{7} +4.41421 q^{8} +1.41421 q^{11} -2.58579 q^{13} -8.24264 q^{14} +3.00000 q^{16} -6.82843 q^{17} +1.00000 q^{19} +3.41421 q^{22} -3.65685 q^{23} -6.24264 q^{26} -13.0711 q^{28} -5.07107 q^{29} -10.4853 q^{31} -1.58579 q^{32} -16.4853 q^{34} +3.07107 q^{37} +2.41421 q^{38} +4.58579 q^{41} -3.41421 q^{43} +5.41421 q^{44} -8.82843 q^{46} +11.6569 q^{47} +4.65685 q^{49} -9.89949 q^{52} +4.00000 q^{53} -15.0711 q^{56} -12.2426 q^{58} +8.48528 q^{59} -5.65685 q^{61} -25.3137 q^{62} -9.82843 q^{64} -12.0000 q^{67} -26.1421 q^{68} -12.4853 q^{71} +2.00000 q^{73} +7.41421 q^{74} +3.82843 q^{76} -4.82843 q^{77} +11.3137 q^{79} +11.0711 q^{82} +6.48528 q^{83} -8.24264 q^{86} +6.24264 q^{88} +14.7279 q^{89} +8.82843 q^{91} -14.0000 q^{92} +28.1421 q^{94} -4.24264 q^{97} +11.2426 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 6 q^{8} - 8 q^{13} - 8 q^{14} + 6 q^{16} - 8 q^{17} + 2 q^{19} + 4 q^{22} + 4 q^{23} - 4 q^{26} - 12 q^{28} + 4 q^{29} - 4 q^{31} - 6 q^{32} - 16 q^{34} - 8 q^{37} + 2 q^{38}+ \cdots + 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^7 + 6 * q^8 - 8 * q^13 - 8 * q^14 + 6 * q^16 - 8 * q^17 + 2 * q^19 + 4 * q^22 + 4 * q^23 - 4 * q^26 - 12 * q^28 + 4 * q^29 - 4 * q^31 - 6 * q^32 - 16 * q^34 - 8 * q^37 + 2 * q^38 + 12 * q^41 - 4 * q^43 + 8 * q^44 - 12 * q^46 + 12 * q^47 - 2 * q^49 + 8 * q^53 - 16 * q^56 - 16 * q^58 - 28 * q^62 - 14 * q^64 - 24 * q^67 - 24 * q^68 - 8 * q^71 + 4 * q^73 + 12 * q^74 + 2 * q^76 - 4 * q^77 + 8 * q^82 - 4 * q^83 - 8 * q^86 + 4 * q^88 + 4 * q^89 + 12 * q^91 - 28 * q^92 + 28 * q^94 + 14 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.41421	−1.29045	−0.645226	−	0.763992i	$-0.723237\pi$
$7$	−3.41421	−1.29045	−0.645226	+	0.763992i	$0.723237\pi$
$8$	4.41421	1.56066
$9$	0	0
$10$	0	0
$11$	1.41421	0.426401	0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421	0.426401	0.213201	+	0.977008i	$0.431611\pi$
$12$	0	0
$13$	−2.58579	−0.717168	−0.358584	−	0.933497i	$-0.616740\pi$
$13$	−2.58579	−0.717168	−0.358584	+	0.933497i	$0.616740\pi$
$14$	−8.24264	−2.20294
$15$	0	0
$16$	3.00000	0.750000
$17$	−6.82843	−1.65614	−0.828068	−	0.560627i	$-0.810560\pi$
$17$	−6.82843	−1.65614	−0.828068	+	0.560627i	$0.810560\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	3.41421	0.727913
$23$	−3.65685	−0.762507	−0.381253	−	0.924471i	$-0.624507\pi$
$23$	−3.65685	−0.762507	−0.381253	+	0.924471i	$0.624507\pi$
$24$	0	0
$25$	0	0
$26$	−6.24264	−1.22428
$27$	0	0
$28$	−13.0711	−2.47020
$29$	−5.07107	−0.941674	−0.470837	−	0.882220i	$-0.656048\pi$
$29$	−5.07107	−0.941674	−0.470837	+	0.882220i	$0.656048\pi$
$30$	0	0
$31$	−10.4853	−1.88321	−0.941606	−	0.336717i	$-0.890684\pi$
$31$	−10.4853	−1.88321	−0.941606	+	0.336717i	$0.890684\pi$
$32$	−1.58579	−0.280330
$33$	0	0
$34$	−16.4853	−2.82720
$35$	0	0
$36$	0	0
$37$	3.07107	0.504880	0.252440	−	0.967612i	$-0.418767\pi$
$37$	3.07107	0.504880	0.252440	+	0.967612i	$0.418767\pi$
$38$	2.41421	0.391637
$39$	0	0
$40$	0	0
$41$	4.58579	0.716180	0.358090	−	0.933687i	$-0.383428\pi$
$41$	4.58579	0.716180	0.358090	+	0.933687i	$0.383428\pi$
$42$	0	0
$43$	−3.41421	−0.520663	−0.260331	−	0.965519i	$-0.583832\pi$
$43$	−3.41421	−0.520663	−0.260331	+	0.965519i	$0.583832\pi$
$44$	5.41421	0.816223
$45$	0	0
$46$	−8.82843	−1.30168
$47$	11.6569	1.70033	0.850163	−	0.526519i	$-0.176503\pi$
$47$	11.6569	1.70033	0.850163	+	0.526519i	$0.176503\pi$
$48$	0	0
$49$	4.65685	0.665265
$50$	0	0
$51$	0	0
$52$	−9.89949	−1.37281
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	0	0
$56$	−15.0711	−2.01396
$57$	0	0
$58$	−12.2426	−1.60754
$59$	8.48528	1.10469	0.552345	−	0.833616i	$-0.313733\pi$
$59$	8.48528	1.10469	0.552345	+	0.833616i	$0.313733\pi$
$60$	0	0
$61$	−5.65685	−0.724286	−0.362143	−	0.932123i	$-0.617955\pi$
$61$	−5.65685	−0.724286	−0.362143	+	0.932123i	$0.617955\pi$
$62$	−25.3137	−3.21484
$63$	0	0
$64$	−9.82843	−1.22855
$65$	0	0
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	−26.1421	−3.17020
$69$	0	0
$70$	0	0
$71$	−12.4853	−1.48173	−0.740865	−	0.671654i	$-0.765584\pi$
$71$	−12.4853	−1.48173	−0.740865	+	0.671654i	$0.765584\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	7.41421	0.861885
$75$	0	0
$76$	3.82843	0.439151
$77$	−4.82843	−0.550250
$78$	0	0
$79$	11.3137	1.27289	0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137	1.27289	0.636446	+	0.771321i	$0.280404\pi$
$80$	0	0
$81$	0	0
$82$	11.0711	1.22259
$83$	6.48528	0.711852	0.355926	−	0.934514i	$-0.384165\pi$
$83$	6.48528	0.711852	0.355926	+	0.934514i	$0.384165\pi$
$84$	0	0
$85$	0	0
$86$	−8.24264	−0.888827
$87$	0	0
$88$	6.24264	0.665468
$89$	14.7279	1.56116	0.780578	−	0.625058i	$-0.214925\pi$
$89$	14.7279	1.56116	0.780578	+	0.625058i	$0.214925\pi$
$90$	0	0
$91$	8.82843	0.925471
$92$	−14.0000	−1.45960
$93$	0	0
$94$	28.1421	2.90264
$95$	0	0
$96$	0	0
$97$	−4.24264	−0.430775	−0.215387	−	0.976529i	$-0.569101\pi$
$97$	−4.24264	−0.430775	−0.215387	+	0.976529i	$0.569101\pi$
$98$	11.2426	1.13568
$99$	0	0
$100$	0	0
$101$	−0.828427	−0.0824316	−0.0412158	−	0.999150i	$-0.513123\pi$
$101$	−0.828427	−0.0824316	−0.0412158	+	0.999150i	$0.513123\pi$
$102$	0	0
$103$	−9.65685	−0.951518	−0.475759	−	0.879576i	$-0.657827\pi$
$103$	−9.65685	−0.951518	−0.475759	+	0.879576i	$0.657827\pi$
$104$	−11.4142	−1.11926
$105$	0	0
$106$	9.65685	0.937957
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	8.82843	0.845610	0.422805	−	0.906221i	$-0.361046\pi$
$109$	8.82843	0.845610	0.422805	+	0.906221i	$0.361046\pi$
$110$	0	0
$111$	0	0
$112$	−10.2426	−0.967839
$113$	−4.48528	−0.421940	−0.210970	−	0.977493i	$-0.567662\pi$
$113$	−4.48528	−0.421940	−0.210970	+	0.977493i	$0.567662\pi$
$114$	0	0
$115$	0	0
$116$	−19.4142	−1.80256
$117$	0	0
$118$	20.4853	1.88582
$119$	23.3137	2.13716
$120$	0	0
$121$	−9.00000	−0.818182
$122$	−13.6569	−1.23643
$123$	0	0
$124$	−40.1421	−3.60487
$125$	0	0
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−20.5563	−1.81694
$129$	0	0
$130$	0	0
$131$	8.72792	0.762562	0.381281	−	0.924459i	$-0.375483\pi$
$131$	8.72792	0.762562	0.381281	+	0.924459i	$0.375483\pi$
$132$	0	0
$133$	−3.41421	−0.296050
$134$	−28.9706	−2.50268
$135$	0	0
$136$	−30.1421	−2.58467
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	0	0
$139$	6.82843	0.579180	0.289590	−	0.957151i	$-0.406481\pi$
$139$	6.82843	0.579180	0.289590	+	0.957151i	$0.406481\pi$
$140$	0	0
$141$	0	0
$142$	−30.1421	−2.52947
$143$	−3.65685	−0.305802
$144$	0	0
$145$	0	0
$146$	4.82843	0.399603
$147$	0	0
$148$	11.7574	0.966449
$149$	−7.65685	−0.627274	−0.313637	−	0.949543i	$-0.601547\pi$
$149$	−7.65685	−0.627274	−0.313637	+	0.949543i	$0.601547\pi$
$150$	0	0
$151$	−21.7990	−1.77398	−0.886988	−	0.461792i	$-0.847207\pi$
$151$	−21.7990	−1.77398	−0.886988	+	0.461792i	$0.847207\pi$
$152$	4.41421	0.358040
$153$	0	0
$154$	−11.6569	−0.939336
$155$	0	0
$156$	0	0
$157$	−17.7990	−1.42051	−0.710257	−	0.703942i	$-0.751421\pi$
$157$	−17.7990	−1.42051	−0.710257	+	0.703942i	$0.751421\pi$
$158$	27.3137	2.17296
$159$	0	0
$160$	0	0
$161$	12.4853	0.983978
$162$	0	0
$163$	11.8995	0.932040	0.466020	−	0.884774i	$-0.345687\pi$
$163$	11.8995	0.932040	0.466020	+	0.884774i	$0.345687\pi$
$164$	17.5563	1.37092
$165$	0	0
$166$	15.6569	1.21521
$167$	10.0000	0.773823	0.386912	−	0.922117i	$-0.373542\pi$
$167$	10.0000	0.773823	0.386912	+	0.922117i	$0.373542\pi$
$168$	0	0
$169$	−6.31371	−0.485670
$170$	0	0
$171$	0	0
$172$	−13.0711	−0.996660
$173$	22.1421	1.68344	0.841718	−	0.539918i	$-0.181545\pi$
$173$	22.1421	1.68344	0.841718	+	0.539918i	$0.181545\pi$
$174$	0	0
$175$	0	0
$176$	4.24264	0.319801
$177$	0	0
$178$	35.5563	2.66506
$179$	−22.8284	−1.70628	−0.853138	−	0.521685i	$-0.825304\pi$
$179$	−22.8284	−1.70628	−0.853138	+	0.521685i	$0.825304\pi$
$180$	0	0
$181$	−24.8284	−1.84548	−0.922741	−	0.385420i	$-0.874057\pi$
$181$	−24.8284	−1.84548	−0.922741	+	0.385420i	$0.874057\pi$
$182$	21.3137	1.57988
$183$	0	0
$184$	−16.1421	−1.19001
$185$	0	0
$186$	0	0
$187$	−9.65685	−0.706179
$188$	44.6274	3.25479
$189$	0	0
$190$	0	0
$191$	17.8995	1.29516	0.647581	−	0.761997i	$-0.275781\pi$
$191$	17.8995	1.29516	0.647581	+	0.761997i	$0.275781\pi$
$192$	0	0
$193$	0.928932	0.0668660	0.0334330	−	0.999441i	$-0.489356\pi$
$193$	0.928932	0.0668660	0.0334330	+	0.999441i	$0.489356\pi$
$194$	−10.2426	−0.735379
$195$	0	0
$196$	17.8284	1.27346
$197$	−9.17157	−0.653448	−0.326724	−	0.945120i	$-0.605945\pi$
$197$	−9.17157	−0.653448	−0.326724	+	0.945120i	$0.605945\pi$
$198$	0	0
$199$	0.485281	0.0344007	0.0172003	−	0.999852i	$-0.494525\pi$
$199$	0.485281	0.0344007	0.0172003	+	0.999852i	$0.494525\pi$
$200$	0	0
$201$	0	0
$202$	−2.00000	−0.140720
$203$	17.3137	1.21518
$204$	0	0
$205$	0	0
$206$	−23.3137	−1.62434
$207$	0	0
$208$	−7.75736	−0.537876
$209$	1.41421	0.0978232
$210$	0	0
$211$	7.31371	0.503496	0.251748	−	0.967793i	$-0.418995\pi$
$211$	7.31371	0.503496	0.251748	+	0.967793i	$0.418995\pi$
$212$	15.3137	1.05175
$213$	0	0
$214$	19.3137	1.32026
$215$	0	0
$216$	0	0
$217$	35.7990	2.43019
$218$	21.3137	1.44355
$219$	0	0
$220$	0	0
$221$	17.6569	1.18773
$222$	0	0
$223$	−17.6569	−1.18239	−0.591195	−	0.806529i	$-0.701344\pi$
$223$	−17.6569	−1.18239	−0.591195	+	0.806529i	$0.701344\pi$
$224$	5.41421	0.361752
$225$	0	0
$226$	−10.8284	−0.720296
$227$	14.9706	0.993631	0.496816	−	0.867856i	$-0.334503\pi$
$227$	14.9706	0.993631	0.496816	+	0.867856i	$0.334503\pi$
$228$	0	0
$229$	9.65685	0.638143	0.319071	−	0.947731i	$-0.396629\pi$
$229$	9.65685	0.638143	0.319071	+	0.947731i	$0.396629\pi$
$230$	0	0
$231$	0	0
$232$	−22.3848	−1.46963
$233$	−19.6569	−1.28776	−0.643882	−	0.765125i	$-0.722677\pi$
$233$	−19.6569	−1.28776	−0.643882	+	0.765125i	$0.722677\pi$
$234$	0	0
$235$	0	0
$236$	32.4853	2.11461
$237$	0	0
$238$	56.2843	3.64837
$239$	−5.41421	−0.350216	−0.175108	−	0.984549i	$-0.556028\pi$
$239$	−5.41421	−0.350216	−0.175108	+	0.984549i	$0.556028\pi$
$240$	0	0
$241$	18.9706	1.22200	0.611001	−	0.791630i	$-0.290767\pi$
$241$	18.9706	1.22200	0.611001	+	0.791630i	$0.290767\pi$
$242$	−21.7279	−1.39672
$243$	0	0
$244$	−21.6569	−1.38644
$245$	0	0
$246$	0	0
$247$	−2.58579	−0.164530
$248$	−46.2843	−2.93905
$249$	0	0
$250$	0	0
$251$	−27.0711	−1.70871	−0.854355	−	0.519689i	$-0.826048\pi$
$251$	−27.0711	−1.70871	−0.854355	+	0.519689i	$0.826048\pi$
$252$	0	0
$253$	−5.17157	−0.325134
$254$	−19.3137	−1.21185
$255$	0	0
$256$	−29.9706	−1.87316
$257$	6.82843	0.425946	0.212973	−	0.977058i	$-0.431685\pi$
$257$	6.82843	0.425946	0.212973	+	0.977058i	$0.431685\pi$
$258$	0	0
$259$	−10.4853	−0.651524
$260$	0	0
$261$	0	0
$262$	21.0711	1.30177
$263$	−3.85786	−0.237886	−0.118943	−	0.992901i	$-0.537951\pi$
$263$	−3.85786	−0.237886	−0.118943	+	0.992901i	$0.537951\pi$
$264$	0	0
$265$	0	0
$266$	−8.24264	−0.505389
$267$	0	0
$268$	−45.9411	−2.80630
$269$	28.3848	1.73065	0.865325	−	0.501211i	$-0.167112\pi$
$269$	28.3848	1.73065	0.865325	+	0.501211i	$0.167112\pi$
$270$	0	0
$271$	−25.1716	−1.52906	−0.764532	−	0.644586i	$-0.777030\pi$
$271$	−25.1716	−1.52906	−0.764532	+	0.644586i	$0.777030\pi$
$272$	−20.4853	−1.24210
$273$	0	0
$274$	−33.7990	−2.04187
$275$	0	0
$276$	0	0
$277$	22.9706	1.38017	0.690084	−	0.723730i	$-0.257574\pi$
$277$	22.9706	1.38017	0.690084	+	0.723730i	$0.257574\pi$
$278$	16.4853	0.988721
$279$	0	0
$280$	0	0
$281$	−10.7279	−0.639974	−0.319987	−	0.947422i	$-0.603679\pi$
$281$	−10.7279	−0.639974	−0.319987	+	0.947422i	$0.603679\pi$
$282$	0	0
$283$	2.24264	0.133311	0.0666556	−	0.997776i	$-0.478767\pi$
$283$	2.24264	0.133311	0.0666556	+	0.997776i	$0.478767\pi$
$284$	−47.7990	−2.83635
$285$	0	0
$286$	−8.82843	−0.522036
$287$	−15.6569	−0.924195
$288$	0	0
$289$	29.6274	1.74279
$290$	0	0
$291$	0	0
$292$	7.65685	0.448084
$293$	7.79899	0.455622	0.227811	−	0.973705i	$-0.426843\pi$
$293$	7.79899	0.455622	0.227811	+	0.973705i	$0.426843\pi$
$294$	0	0
$295$	0	0
$296$	13.5563	0.787947
$297$	0	0
$298$	−18.4853	−1.07082
$299$	9.45584	0.546846
$300$	0	0
$301$	11.6569	0.671890
$302$	−52.6274	−3.02837
$303$	0	0
$304$	3.00000	0.172062
$305$	0	0
$306$	0	0
$307$	31.7990	1.81486	0.907432	−	0.420199i	$-0.138040\pi$
$307$	31.7990	1.81486	0.907432	+	0.420199i	$0.138040\pi$
$308$	−18.4853	−1.05330
$309$	0	0
$310$	0	0
$311$	23.7574	1.34716	0.673578	−	0.739116i	$-0.264756\pi$
$311$	23.7574	1.34716	0.673578	+	0.739116i	$0.264756\pi$
$312$	0	0
$313$	26.4853	1.49704	0.748518	−	0.663114i	$-0.230766\pi$
$313$	26.4853	1.49704	0.748518	+	0.663114i	$0.230766\pi$
$314$	−42.9706	−2.42497
$315$	0	0
$316$	43.3137	2.43659
$317$	−11.3137	−0.635441	−0.317721	−	0.948184i	$-0.602917\pi$
$317$	−11.3137	−0.635441	−0.317721	+	0.948184i	$0.602917\pi$
$318$	0	0
$319$	−7.17157	−0.401531
$320$	0	0
$321$	0	0
$322$	30.1421	1.67976
$323$	−6.82843	−0.379944
$324$	0	0
$325$	0	0
$326$	28.7279	1.59109
$327$	0	0
$328$	20.2426	1.11771
$329$	−39.7990	−2.19419
$330$	0	0
$331$	12.8284	0.705114	0.352557	−	0.935790i	$-0.385312\pi$
$331$	12.8284	0.705114	0.352557	+	0.935790i	$0.385312\pi$
$332$	24.8284	1.36264
$333$	0	0
$334$	24.1421	1.32100
$335$	0	0
$336$	0	0
$337$	−19.7574	−1.07625	−0.538126	−	0.842864i	$-0.680868\pi$
$337$	−19.7574	−1.07625	−0.538126	+	0.842864i	$0.680868\pi$
$338$	−15.2426	−0.829090
$339$	0	0
$340$	0	0
$341$	−14.8284	−0.803004
$342$	0	0
$343$	8.00000	0.431959
$344$	−15.0711	−0.812578
$345$	0	0
$346$	53.4558	2.87380
$347$	−6.48528	−0.348148	−0.174074	−	0.984733i	$-0.555693\pi$
$347$	−6.48528	−0.348148	−0.174074	+	0.984733i	$0.555693\pi$
$348$	0	0
$349$	6.68629	0.357909	0.178954	−	0.983857i	$-0.442729\pi$
$349$	6.68629	0.357909	0.178954	+	0.983857i	$0.442729\pi$
$350$	0	0
$351$	0	0
$352$	−2.24264	−0.119533
$353$	−7.65685	−0.407533	−0.203767	−	0.979019i	$-0.565318\pi$
$353$	−7.65685	−0.407533	−0.203767	+	0.979019i	$0.565318\pi$
$354$	0	0
$355$	0	0
$356$	56.3848	2.98839
$357$	0	0
$358$	−55.1127	−2.91280
$359$	9.89949	0.522475	0.261238	−	0.965275i	$-0.415869\pi$
$359$	9.89949	0.522475	0.261238	+	0.965275i	$0.415869\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−59.9411	−3.15044
$363$	0	0
$364$	33.7990	1.77155
$365$	0	0
$366$	0	0
$367$	16.5858	0.865771	0.432886	−	0.901449i	$-0.357495\pi$
$367$	16.5858	0.865771	0.432886	+	0.901449i	$0.357495\pi$
$368$	−10.9706	−0.571880
$369$	0	0
$370$	0	0
$371$	−13.6569	−0.709029
$372$	0	0
$373$	−9.89949	−0.512576	−0.256288	−	0.966600i	$-0.582500\pi$
$373$	−9.89949	−0.512576	−0.256288	+	0.966600i	$0.582500\pi$
$374$	−23.3137	−1.20552
$375$	0	0
$376$	51.4558	2.65363
$377$	13.1127	0.675338
$378$	0	0
$379$	−4.14214	−0.212767	−0.106384	−	0.994325i	$-0.533927\pi$
$379$	−4.14214	−0.212767	−0.106384	+	0.994325i	$0.533927\pi$
$380$	0	0
$381$	0	0
$382$	43.2132	2.21098
$383$	12.0000	0.613171	0.306586	−	0.951843i	$-0.400813\pi$
$383$	12.0000	0.613171	0.306586	+	0.951843i	$0.400813\pi$
$384$	0	0
$385$	0	0
$386$	2.24264	0.114147
$387$	0	0
$388$	−16.2426	−0.824595
$389$	18.9706	0.961846	0.480923	−	0.876763i	$-0.340302\pi$
$389$	18.9706	0.961846	0.480923	+	0.876763i	$0.340302\pi$
$390$	0	0
$391$	24.9706	1.26282
$392$	20.5563	1.03825
$393$	0	0
$394$	−22.1421	−1.11550
$395$	0	0
$396$	0	0
$397$	−24.3431	−1.22175	−0.610874	−	0.791728i	$-0.709182\pi$
$397$	−24.3431	−1.22175	−0.610874	+	0.791728i	$0.709182\pi$
$398$	1.17157	0.0587256
$399$	0	0
$400$	0	0
$401$	18.0416	0.900956	0.450478	−	0.892788i	$-0.351254\pi$
$401$	18.0416	0.900956	0.450478	+	0.892788i	$0.351254\pi$
$402$	0	0
$403$	27.1127	1.35058
$404$	−3.17157	−0.157792
$405$	0	0
$406$	41.7990	2.07445
$407$	4.34315	0.215282
$408$	0	0
$409$	26.4853	1.30961	0.654806	−	0.755797i	$-0.272750\pi$
$409$	26.4853	1.30961	0.654806	+	0.755797i	$0.272750\pi$
$410$	0	0
$411$	0	0
$412$	−36.9706	−1.82141
$413$	−28.9706	−1.42555
$414$	0	0
$415$	0	0
$416$	4.10051	0.201044
$417$	0	0
$418$	3.41421	0.166995
$419$	−18.8701	−0.921863	−0.460931	−	0.887436i	$-0.652485\pi$
$419$	−18.8701	−0.921863	−0.460931	+	0.887436i	$0.652485\pi$
$420$	0	0
$421$	−37.3137	−1.81856	−0.909279	−	0.416186i	$-0.863366\pi$
$421$	−37.3137	−1.81856	−0.909279	+	0.416186i	$0.863366\pi$
$422$	17.6569	0.859522
$423$	0	0
$424$	17.6569	0.857493
$425$	0	0
$426$	0	0
$427$	19.3137	0.934656
$428$	30.6274	1.48043
$429$	0	0
$430$	0	0
$431$	−20.4853	−0.986741	−0.493371	−	0.869819i	$-0.664235\pi$
$431$	−20.4853	−0.986741	−0.493371	+	0.869819i	$0.664235\pi$
$432$	0	0
$433$	−15.0711	−0.724269	−0.362135	−	0.932126i	$-0.617952\pi$
$433$	−15.0711	−0.724269	−0.362135	+	0.932126i	$0.617952\pi$
$434$	86.4264	4.14860
$435$	0	0
$436$	33.7990	1.61868
$437$	−3.65685	−0.174931
$438$	0	0
$439$	−32.9706	−1.57360	−0.786800	−	0.617209i	$-0.788263\pi$
$439$	−32.9706	−1.57360	−0.786800	+	0.617209i	$0.788263\pi$
$440$	0	0
$441$	0	0
$442$	42.6274	2.02758
$443$	21.3137	1.01264	0.506322	−	0.862344i	$-0.331005\pi$
$443$	21.3137	1.01264	0.506322	+	0.862344i	$0.331005\pi$
$444$	0	0
$445$	0	0
$446$	−42.6274	−2.01847
$447$	0	0
$448$	33.5563	1.58539
$449$	11.8995	0.561572	0.280786	−	0.959770i	$-0.409405\pi$
$449$	11.8995	0.561572	0.280786	+	0.959770i	$0.409405\pi$
$450$	0	0
$451$	6.48528	0.305380
$452$	−17.1716	−0.807683
$453$	0	0
$454$	36.1421	1.69623
$455$	0	0
$456$	0	0
$457$	−23.1716	−1.08392	−0.541960	−	0.840404i	$-0.682318\pi$
$457$	−23.1716	−1.08392	−0.541960	+	0.840404i	$0.682318\pi$
$458$	23.3137	1.08938
$459$	0	0
$460$	0	0
$461$	−33.3137	−1.55157	−0.775787	−	0.630995i	$-0.782647\pi$
$461$	−33.3137	−1.55157	−0.775787	+	0.630995i	$0.782647\pi$
$462$	0	0
$463$	27.8995	1.29660	0.648300	−	0.761385i	$-0.275480\pi$
$463$	27.8995	1.29660	0.648300	+	0.761385i	$0.275480\pi$
$464$	−15.2132	−0.706255
$465$	0	0
$466$	−47.4558	−2.19835
$467$	27.6569	1.27981	0.639903	−	0.768455i	$-0.278974\pi$
$467$	27.6569	1.27981	0.639903	+	0.768455i	$0.278974\pi$
$468$	0	0
$469$	40.9706	1.89184
$470$	0	0
$471$	0	0
$472$	37.4558	1.72404
$473$	−4.82843	−0.222011
$474$	0	0
$475$	0	0
$476$	89.2548	4.09099
$477$	0	0
$478$	−13.0711	−0.597857
$479$	−38.1838	−1.74466	−0.872330	−	0.488917i	$-0.837392\pi$
$479$	−38.1838	−1.74466	−0.872330	+	0.488917i	$0.837392\pi$
$480$	0	0
$481$	−7.94113	−0.362084
$482$	45.7990	2.08609
$483$	0	0
$484$	−34.4558	−1.56617
$485$	0	0
$486$	0	0
$487$	−2.82843	−0.128168	−0.0640841	−	0.997944i	$-0.520413\pi$
$487$	−2.82843	−0.128168	−0.0640841	+	0.997944i	$0.520413\pi$
$488$	−24.9706	−1.13036
$489$	0	0
$490$	0	0
$491$	−2.10051	−0.0947945	−0.0473972	−	0.998876i	$-0.515093\pi$
$491$	−2.10051	−0.0947945	−0.0473972	+	0.998876i	$0.515093\pi$
$492$	0	0
$493$	34.6274	1.55954
$494$	−6.24264	−0.280870
$495$	0	0
$496$	−31.4558	−1.41241
$497$	42.6274	1.91210
$498$	0	0
$499$	15.7990	0.707260	0.353630	−	0.935385i	$-0.384947\pi$
$499$	15.7990	0.707260	0.353630	+	0.935385i	$0.384947\pi$
$500$	0	0
$501$	0	0
$502$	−65.3553	−2.91695
$503$	−4.82843	−0.215289	−0.107644	−	0.994189i	$-0.534331\pi$
$503$	−4.82843	−0.215289	−0.107644	+	0.994189i	$0.534331\pi$
$504$	0	0
$505$	0	0
$506$	−12.4853	−0.555038
$507$	0	0
$508$	−30.6274	−1.35887
$509$	4.38478	0.194352	0.0971759	−	0.995267i	$-0.469019\pi$
$509$	4.38478	0.194352	0.0971759	+	0.995267i	$0.469019\pi$
$510$	0	0
$511$	−6.82843	−0.302072
$512$	−31.2426	−1.38074
$513$	0	0
$514$	16.4853	0.727135
$515$	0	0
$516$	0	0
$517$	16.4853	0.725022
$518$	−25.3137	−1.11222
$519$	0	0
$520$	0	0
$521$	−12.3848	−0.542587	−0.271293	−	0.962497i	$-0.587451\pi$
$521$	−12.3848	−0.542587	−0.271293	+	0.962497i	$0.587451\pi$
$522$	0	0
$523$	−23.7990	−1.04066	−0.520329	−	0.853966i	$-0.674191\pi$
$523$	−23.7990	−1.04066	−0.520329	+	0.853966i	$0.674191\pi$
$524$	33.4142	1.45971
$525$	0	0
$526$	−9.31371	−0.406097
$527$	71.5980	3.11886
$528$	0	0
$529$	−9.62742	−0.418583
$530$	0	0
$531$	0	0
$532$	−13.0711	−0.566703
$533$	−11.8579	−0.513621
$534$	0	0
$535$	0	0
$536$	−52.9706	−2.28798
$537$	0	0
$538$	68.5269	2.95440
$539$	6.58579	0.283670
$540$	0	0
$541$	12.6274	0.542895	0.271448	−	0.962453i	$-0.412498\pi$
$541$	12.6274	0.542895	0.271448	+	0.962453i	$0.412498\pi$
$542$	−60.7696	−2.61028
$543$	0	0
$544$	10.8284	0.464265
$545$	0	0
$546$	0	0
$547$	5.85786	0.250464	0.125232	−	0.992127i	$-0.460032\pi$
$547$	5.85786	0.250464	0.125232	+	0.992127i	$0.460032\pi$
$548$	−53.5980	−2.28959
$549$	0	0
$550$	0	0
$551$	−5.07107	−0.216035
$552$	0	0
$553$	−38.6274	−1.64260
$554$	55.4558	2.35609
$555$	0	0
$556$	26.1421	1.10867
$557$	10.0000	0.423714	0.211857	−	0.977301i	$-0.432049\pi$
$557$	10.0000	0.423714	0.211857	+	0.977301i	$0.432049\pi$
$558$	0	0
$559$	8.82843	0.373403
$560$	0	0
$561$	0	0
$562$	−25.8995	−1.09250
$563$	−14.2843	−0.602010	−0.301005	−	0.953623i	$-0.597322\pi$
$563$	−14.2843	−0.602010	−0.301005	+	0.953623i	$0.597322\pi$
$564$	0	0
$565$	0	0
$566$	5.41421	0.227576
$567$	0	0
$568$	−55.1127	−2.31248
$569$	−18.7279	−0.785115	−0.392558	−	0.919727i	$-0.628410\pi$
$569$	−18.7279	−0.785115	−0.392558	+	0.919727i	$0.628410\pi$
$570$	0	0
$571$	−19.7990	−0.828562	−0.414281	−	0.910149i	$-0.635967\pi$
$571$	−19.7990	−0.828562	−0.414281	+	0.910149i	$0.635967\pi$
$572$	−14.0000	−0.585369
$573$	0	0
$574$	−37.7990	−1.57770
$575$	0	0
$576$	0	0
$577$	−1.79899	−0.0748929	−0.0374465	−	0.999299i	$-0.511922\pi$
$577$	−1.79899	−0.0748929	−0.0374465	+	0.999299i	$0.511922\pi$
$578$	71.5269	2.97513
$579$	0	0
$580$	0	0
$581$	−22.1421	−0.918611
$582$	0	0
$583$	5.65685	0.234283
$584$	8.82843	0.365323
$585$	0	0
$586$	18.8284	0.777795
$587$	0.343146	0.0141631	0.00708157	−	0.999975i	$-0.497746\pi$
$587$	0.343146	0.0141631	0.00708157	+	0.999975i	$0.497746\pi$
$588$	0	0
$589$	−10.4853	−0.432038
$590$	0	0
$591$	0	0
$592$	9.21320	0.378660
$593$	−6.68629	−0.274573	−0.137287	−	0.990531i	$-0.543838\pi$
$593$	−6.68629	−0.274573	−0.137287	+	0.990531i	$0.543838\pi$
$594$	0	0
$595$	0	0
$596$	−29.3137	−1.20074
$597$	0	0
$598$	22.8284	0.933524
$599$	−45.9411	−1.87710	−0.938552	−	0.345139i	$-0.887832\pi$
$599$	−45.9411	−1.87710	−0.938552	+	0.345139i	$0.887832\pi$
$600$	0	0
$601$	−23.1716	−0.945188	−0.472594	−	0.881280i	$-0.656682\pi$
$601$	−23.1716	−0.945188	−0.472594	+	0.881280i	$0.656682\pi$
$602$	28.1421	1.14699
$603$	0	0
$604$	−83.4558	−3.39577
$605$	0	0
$606$	0	0
$607$	−26.1421	−1.06108	−0.530538	−	0.847661i	$-0.678010\pi$
$607$	−26.1421	−1.06108	−0.530538	+	0.847661i	$0.678010\pi$
$608$	−1.58579	−0.0643121
$609$	0	0
$610$	0	0
$611$	−30.1421	−1.21942
$612$	0	0
$613$	37.1127	1.49897	0.749484	−	0.662023i	$-0.230302\pi$
$613$	37.1127	1.49897	0.749484	+	0.662023i	$0.230302\pi$
$614$	76.7696	3.09817
$615$	0	0
$616$	−21.3137	−0.858754
$617$	−18.1421	−0.730375	−0.365187	−	0.930934i	$-0.618995\pi$
$617$	−18.1421	−0.730375	−0.365187	+	0.930934i	$0.618995\pi$
$618$	0	0
$619$	−4.20101	−0.168853	−0.0844264	−	0.996430i	$-0.526906\pi$
$619$	−4.20101	−0.168853	−0.0844264	+	0.996430i	$0.526906\pi$
$620$	0	0
$621$	0	0
$622$	57.3553	2.29974
$623$	−50.2843	−2.01460
$624$	0	0
$625$	0	0
$626$	63.9411	2.55560
$627$	0	0
$628$	−68.1421	−2.71917
$629$	−20.9706	−0.836151
$630$	0	0
$631$	−22.6274	−0.900783	−0.450392	−	0.892831i	$-0.648716\pi$
$631$	−22.6274	−0.900783	−0.450392	+	0.892831i	$0.648716\pi$
$632$	49.9411	1.98655
$633$	0	0
$634$	−27.3137	−1.08477
$635$	0	0
$636$	0	0
$637$	−12.0416	−0.477107
$638$	−17.3137	−0.685456
$639$	0	0
$640$	0	0
$641$	11.4142	0.450834	0.225417	−	0.974262i	$-0.427625\pi$
$641$	11.4142	0.450834	0.225417	+	0.974262i	$0.427625\pi$
$642$	0	0
$643$	42.0416	1.65796	0.828980	−	0.559278i	$-0.188922\pi$
$643$	42.0416	1.65796	0.828980	+	0.559278i	$0.188922\pi$
$644$	47.7990	1.88354
$645$	0	0
$646$	−16.4853	−0.648605
$647$	−17.1127	−0.672770	−0.336385	−	0.941725i	$-0.609204\pi$
$647$	−17.1127	−0.672770	−0.336385	+	0.941725i	$0.609204\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	45.5563	1.78412
$653$	−42.4264	−1.66027	−0.830137	−	0.557560i	$-0.811738\pi$
$653$	−42.4264	−1.66027	−0.830137	+	0.557560i	$0.811738\pi$
$654$	0	0
$655$	0	0
$656$	13.7574	0.537135
$657$	0	0
$658$	−96.0833	−3.74572
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	1.51472	0.0589157	0.0294579	−	0.999566i	$-0.490622\pi$
$661$	1.51472	0.0589157	0.0294579	+	0.999566i	$0.490622\pi$
$662$	30.9706	1.20371
$663$	0	0
$664$	28.6274	1.11096
$665$	0	0
$666$	0	0
$667$	18.5442	0.718033
$668$	38.2843	1.48126
$669$	0	0
$670$	0	0
$671$	−8.00000	−0.308837
$672$	0	0
$673$	−2.10051	−0.0809685	−0.0404843	−	0.999180i	$-0.512890\pi$
$673$	−2.10051	−0.0809685	−0.0404843	+	0.999180i	$0.512890\pi$
$674$	−47.6985	−1.83728
$675$	0	0
$676$	−24.1716	−0.929676
$677$	−11.0294	−0.423896	−0.211948	−	0.977281i	$-0.567981\pi$
$677$	−11.0294	−0.423896	−0.211948	+	0.977281i	$0.567981\pi$
$678$	0	0
$679$	14.4853	0.555894
$680$	0	0
$681$	0	0
$682$	−35.7990	−1.37081
$683$	−5.65685	−0.216454	−0.108227	−	0.994126i	$-0.534517\pi$
$683$	−5.65685	−0.216454	−0.108227	+	0.994126i	$0.534517\pi$
$684$	0	0
$685$	0	0
$686$	19.3137	0.737401
$687$	0	0
$688$	−10.2426	−0.390497
$689$	−10.3431	−0.394042
$690$	0	0
$691$	39.1127	1.48792	0.743959	−	0.668226i	$-0.232946\pi$
$691$	39.1127	1.48792	0.743959	+	0.668226i	$0.232946\pi$
$692$	84.7696	3.22245
$693$	0	0
$694$	−15.6569	−0.594326
$695$	0	0
$696$	0	0
$697$	−31.3137	−1.18609
$698$	16.1421	0.610989
$699$	0	0
$700$	0	0
$701$	11.6569	0.440273	0.220137	−	0.975469i	$-0.429350\pi$
$701$	11.6569	0.440273	0.220137	+	0.975469i	$0.429350\pi$
$702$	0	0
$703$	3.07107	0.115828
$704$	−13.8995	−0.523857
$705$	0	0
$706$	−18.4853	−0.695703
$707$	2.82843	0.106374
$708$	0	0
$709$	−12.6863	−0.476444	−0.238222	−	0.971211i	$-0.576565\pi$
$709$	−12.6863	−0.476444	−0.238222	+	0.971211i	$0.576565\pi$
$710$	0	0
$711$	0	0
$712$	65.0122	2.43643
$713$	38.3431	1.43596
$714$	0	0
$715$	0	0
$716$	−87.3970	−3.26618
$717$	0	0
$718$	23.8995	0.891921
$719$	47.5563	1.77355	0.886776	−	0.462199i	$-0.152939\pi$
$719$	47.5563	1.77355	0.886776	+	0.462199i	$0.152939\pi$
$720$	0	0
$721$	32.9706	1.22789
$722$	2.41421	0.0898477
$723$	0	0
$724$	−95.0538	−3.53265
$725$	0	0
$726$	0	0
$727$	7.41421	0.274978	0.137489	−	0.990503i	$-0.456097\pi$
$727$	7.41421	0.274978	0.137489	+	0.990503i	$0.456097\pi$
$728$	38.9706	1.44435
$729$	0	0
$730$	0	0
$731$	23.3137	0.862289
$732$	0	0
$733$	−21.3137	−0.787240	−0.393620	−	0.919273i	$-0.628777\pi$
$733$	−21.3137	−0.787240	−0.393620	+	0.919273i	$0.628777\pi$
$734$	40.0416	1.47796
$735$	0	0
$736$	5.79899	0.213754
$737$	−16.9706	−0.625119
$738$	0	0
$739$	1.65685	0.0609484	0.0304742	−	0.999536i	$-0.490298\pi$
$739$	1.65685	0.0609484	0.0304742	+	0.999536i	$0.490298\pi$
$740$	0	0
$741$	0	0
$742$	−32.9706	−1.21039
$743$	7.31371	0.268314	0.134157	−	0.990960i	$-0.457167\pi$
$743$	7.31371	0.268314	0.134157	+	0.990960i	$0.457167\pi$
$744$	0	0
$745$	0	0
$746$	−23.8995	−0.875023
$747$	0	0
$748$	−36.9706	−1.35178
$749$	−27.3137	−0.998021
$750$	0	0
$751$	25.1127	0.916375	0.458188	−	0.888855i	$-0.348499\pi$
$751$	25.1127	0.916375	0.458188	+	0.888855i	$0.348499\pi$
$752$	34.9706	1.27525
$753$	0	0
$754$	31.6569	1.15287
$755$	0	0
$756$	0	0
$757$	−11.1716	−0.406038	−0.203019	−	0.979175i	$-0.565075\pi$
$757$	−11.1716	−0.406038	−0.203019	+	0.979175i	$0.565075\pi$
$758$	−10.0000	−0.363216
$759$	0	0
$760$	0	0
$761$	−46.2843	−1.67780	−0.838902	−	0.544283i	$-0.816802\pi$
$761$	−46.2843	−1.67780	−0.838902	+	0.544283i	$0.816802\pi$
$762$	0	0
$763$	−30.1421	−1.09122
$764$	68.5269	2.47922
$765$	0	0
$766$	28.9706	1.04675
$767$	−21.9411	−0.792248
$768$	0	0
$769$	−24.3431	−0.877836	−0.438918	−	0.898527i	$-0.644638\pi$
$769$	−24.3431	−0.877836	−0.438918	+	0.898527i	$0.644638\pi$
$770$	0	0
$771$	0	0
$772$	3.55635	0.127996
$773$	0.970563	0.0349087	0.0174544	−	0.999848i	$-0.494444\pi$
$773$	0.970563	0.0349087	0.0174544	+	0.999848i	$0.494444\pi$
$774$	0	0
$775$	0	0
$776$	−18.7279	−0.672293
$777$	0	0
$778$	45.7990	1.64197
$779$	4.58579	0.164303
$780$	0	0
$781$	−17.6569	−0.631812
$782$	60.2843	2.15576
$783$	0	0
$784$	13.9706	0.498949
$785$	0	0
$786$	0	0
$787$	37.4558	1.33516	0.667578	−	0.744540i	$-0.267331\pi$
$787$	37.4558	1.33516	0.667578	+	0.744540i	$0.267331\pi$
$788$	−35.1127	−1.25084
$789$	0	0
$790$	0	0
$791$	15.3137	0.544493
$792$	0	0
$793$	14.6274	0.519435
$794$	−58.7696	−2.08565
$795$	0	0
$796$	1.85786	0.0658503
$797$	5.17157	0.183187	0.0915933	−	0.995797i	$-0.470804\pi$
$797$	5.17157	0.183187	0.0915933	+	0.995797i	$0.470804\pi$
$798$	0	0
$799$	−79.5980	−2.81597
$800$	0	0
$801$	0	0
$802$	43.5563	1.53803
$803$	2.82843	0.0998130
$804$	0	0
$805$	0	0
$806$	65.4558	2.30558
$807$	0	0
$808$	−3.65685	−0.128648
$809$	−26.6863	−0.938240	−0.469120	−	0.883134i	$-0.655429\pi$
$809$	−26.6863	−0.938240	−0.469120	+	0.883134i	$0.655429\pi$
$810$	0	0
$811$	7.31371	0.256819	0.128410	−	0.991721i	$-0.459013\pi$
$811$	7.31371	0.256819	0.128410	+	0.991721i	$0.459013\pi$
$812$	66.2843	2.32612
$813$	0	0
$814$	10.4853	0.367509
$815$	0	0
$816$	0	0
$817$	−3.41421	−0.119448
$818$	63.9411	2.23565
$819$	0	0
$820$	0	0
$821$	−0.544156	−0.0189912	−0.00949559	−	0.999955i	$-0.503023\pi$
$821$	−0.544156	−0.0189912	−0.00949559	+	0.999955i	$0.503023\pi$
$822$	0	0
$823$	−22.7279	−0.792246	−0.396123	−	0.918198i	$-0.629645\pi$
$823$	−22.7279	−0.792246	−0.396123	+	0.918198i	$0.629645\pi$
$824$	−42.6274	−1.48500
$825$	0	0
$826$	−69.9411	−2.43356
$827$	3.37258	0.117276	0.0586381	−	0.998279i	$-0.481324\pi$
$827$	3.37258	0.117276	0.0586381	+	0.998279i	$0.481324\pi$
$828$	0	0
$829$	−21.5147	−0.747237	−0.373619	−	0.927582i	$-0.621883\pi$
$829$	−21.5147	−0.747237	−0.373619	+	0.927582i	$0.621883\pi$
$830$	0	0
$831$	0	0
$832$	25.4142	0.881079
$833$	−31.7990	−1.10177
$834$	0	0
$835$	0	0
$836$	5.41421	0.187254
$837$	0	0
$838$	−45.5563	−1.57372
$839$	35.1127	1.21222	0.606112	−	0.795379i	$-0.292728\pi$
$839$	35.1127	1.21222	0.606112	+	0.795379i	$0.292728\pi$
$840$	0	0
$841$	−3.28427	−0.113251
$842$	−90.0833	−3.10447
$843$	0	0
$844$	28.0000	0.963800
$845$	0	0
$846$	0	0
$847$	30.7279	1.05582
$848$	12.0000	0.412082
$849$	0	0
$850$	0	0
$851$	−11.2304	−0.384975
$852$	0	0
$853$	26.4853	0.906839	0.453419	−	0.891297i	$-0.350204\pi$
$853$	26.4853	0.906839	0.453419	+	0.891297i	$0.350204\pi$
$854$	46.6274	1.59556
$855$	0	0
$856$	35.3137	1.20700
$857$	−29.9411	−1.02277	−0.511385	−	0.859352i	$-0.670867\pi$
$857$	−29.9411	−1.02277	−0.511385	+	0.859352i	$0.670867\pi$
$858$	0	0
$859$	−41.9411	−1.43101	−0.715506	−	0.698606i	$-0.753804\pi$
$859$	−41.9411	−1.43101	−0.715506	+	0.698606i	$0.753804\pi$
$860$	0	0
$861$	0	0
$862$	−49.4558	−1.68447
$863$	−8.68629	−0.295685	−0.147842	−	0.989011i	$-0.547233\pi$
$863$	−8.68629	−0.295685	−0.147842	+	0.989011i	$0.547233\pi$
$864$	0	0
$865$	0	0
$866$	−36.3848	−1.23641
$867$	0	0
$868$	137.054	4.65191
$869$	16.0000	0.542763
$870$	0	0
$871$	31.0294	1.05139
$872$	38.9706	1.31971
$873$	0	0
$874$	−8.82843	−0.298626
$875$	0	0
$876$	0	0
$877$	1.89949	0.0641414	0.0320707	−	0.999486i	$-0.489790\pi$
$877$	1.89949	0.0641414	0.0320707	+	0.999486i	$0.489790\pi$
$878$	−79.5980	−2.68630
$879$	0	0
$880$	0	0
$881$	−3.45584	−0.116430	−0.0582152	−	0.998304i	$-0.518541\pi$
$881$	−3.45584	−0.116430	−0.0582152	+	0.998304i	$0.518541\pi$
$882$	0	0
$883$	18.5269	0.623480	0.311740	−	0.950167i	$-0.399088\pi$
$883$	18.5269	0.623480	0.311740	+	0.950167i	$0.399088\pi$
$884$	67.5980	2.27357
$885$	0	0
$886$	51.4558	1.72869
$887$	−8.00000	−0.268614	−0.134307	−	0.990940i	$-0.542881\pi$
$887$	−8.00000	−0.268614	−0.134307	+	0.990940i	$0.542881\pi$
$888$	0	0
$889$	27.3137	0.916072
$890$	0	0
$891$	0	0
$892$	−67.5980	−2.26335
$893$	11.6569	0.390082
$894$	0	0
$895$	0	0
$896$	70.1838	2.34468
$897$	0	0
$898$	28.7279	0.958663
$899$	53.1716	1.77337
$900$	0	0
$901$	−27.3137	−0.909952
$902$	15.6569	0.521316
$903$	0	0
$904$	−19.7990	−0.658505
$905$	0	0
$906$	0	0
$907$	−9.85786	−0.327325	−0.163663	−	0.986516i	$-0.552331\pi$
$907$	−9.85786	−0.327325	−0.163663	+	0.986516i	$0.552331\pi$
$908$	57.3137	1.90202
$909$	0	0
$910$	0	0
$911$	0.686292	0.0227379	0.0113689	−	0.999935i	$-0.496381\pi$
$911$	0.686292	0.0227379	0.0113689	+	0.999935i	$0.496381\pi$
$912$	0	0
$913$	9.17157	0.303535
$914$	−55.9411	−1.85037
$915$	0	0
$916$	36.9706	1.22154
$917$	−29.7990	−0.984049
$918$	0	0
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	0	0
$921$	0	0
$922$	−80.4264	−2.64870
$923$	32.2843	1.06265
$924$	0	0
$925$	0	0
$926$	67.3553	2.21343
$927$	0	0
$928$	8.04163	0.263979
$929$	0.544156	0.0178532	0.00892659	−	0.999960i	$-0.497159\pi$
$929$	0.544156	0.0178532	0.00892659	+	0.999960i	$0.497159\pi$
$930$	0	0
$931$	4.65685	0.152622
$932$	−75.2548	−2.46505
$933$	0	0
$934$	66.7696	2.18477
$935$	0	0
$936$	0	0
$937$	−54.7696	−1.78924	−0.894622	−	0.446824i	$-0.852555\pi$
$937$	−54.7696	−1.78924	−0.894622	+	0.446824i	$0.852555\pi$
$938$	98.9117	3.22958
$939$	0	0
$940$	0	0
$941$	−13.5563	−0.441924	−0.220962	−	0.975282i	$-0.570920\pi$
$941$	−13.5563	−0.441924	−0.220962	+	0.975282i	$0.570920\pi$
$942$	0	0
$943$	−16.7696	−0.546092
$944$	25.4558	0.828517
$945$	0	0
$946$	−11.6569	−0.378997
$947$	7.17157	0.233045	0.116522	−	0.993188i	$-0.462825\pi$
$947$	7.17157	0.233045	0.116522	+	0.993188i	$0.462825\pi$
$948$	0	0
$949$	−5.17157	−0.167876
$950$	0	0
$951$	0	0
$952$	102.912	3.33539
$953$	−34.1421	−1.10597	−0.552986	−	0.833190i	$-0.686512\pi$
$953$	−34.1421	−1.10597	−0.552986	+	0.833190i	$0.686512\pi$
$954$	0	0
$955$	0	0
$956$	−20.7279	−0.670389
$957$	0	0
$958$	−92.1838	−2.97832
$959$	47.7990	1.54351
$960$	0	0
$961$	78.9411	2.54649
$962$	−19.1716	−0.618116
$963$	0	0
$964$	72.6274	2.33917
$965$	0	0
$966$	0	0
$967$	8.10051	0.260495	0.130247	−	0.991482i	$-0.458423\pi$
$967$	8.10051	0.260495	0.130247	+	0.991482i	$0.458423\pi$
$968$	−39.7279	−1.27690
$969$	0	0
$970$	0	0
$971$	−6.34315	−0.203561	−0.101781	−	0.994807i	$-0.532454\pi$
$971$	−6.34315	−0.203561	−0.101781	+	0.994807i	$0.532454\pi$
$972$	0	0
$973$	−23.3137	−0.747403
$974$	−6.82843	−0.218797
$975$	0	0
$976$	−16.9706	−0.543214
$977$	−39.5980	−1.26685	−0.633426	−	0.773803i	$-0.718352\pi$
$977$	−39.5980	−1.26685	−0.633426	+	0.773803i	$0.718352\pi$
$978$	0	0
$979$	20.8284	0.665679
$980$	0	0
$981$	0	0
$982$	−5.07107	−0.161824
$983$	−35.9411	−1.14634	−0.573172	−	0.819435i	$-0.694287\pi$
$983$	−35.9411	−1.14634	−0.573172	+	0.819435i	$0.694287\pi$
$984$	0	0
$985$	0	0
$986$	83.5980	2.66230
$987$	0	0
$988$	−9.89949	−0.314945
$989$	12.4853	0.397009
$990$	0	0
$991$	−50.3431	−1.59920	−0.799601	−	0.600531i	$-0.794956\pi$
$991$	−50.3431	−1.59920	−0.799601	+	0.600531i	$0.794956\pi$
$992$	16.6274	0.527921
$993$	0	0
$994$	102.912	3.26416
$995$	0	0
$996$	0	0
$997$	33.5147	1.06142	0.530711	−	0.847553i	$-0.321925\pi$
$997$	33.5147	1.06142	0.530711	+	0.847553i	$0.321925\pi$
$998$	38.1421	1.20737
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.x.1.2		2
3.2	odd	2		1425.2.a.l.1.1		2
5.4	even	2		855.2.a.e.1.1		2
15.2	even	4		1425.2.c.j.799.1		4
15.8	even	4		1425.2.c.j.799.4		4
15.14	odd	2		285.2.a.f.1.2	✓	2
60.59	even	2		4560.2.a.bj.1.1		2
285.284	even	2		5415.2.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.f.1.2	✓	2	15.14	odd	2
855.2.a.e.1.1		2	5.4	even	2
1425.2.a.l.1.1		2	3.2	odd	2
1425.2.c.j.799.1		4	15.2	even	4
1425.2.c.j.799.4		4	15.8	even	4
4275.2.a.x.1.2		2	1.1	even	1	trivial
4560.2.a.bj.1.1		2	60.59	even	2
5415.2.a.p.1.1		2	285.284	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} +3.82843 q^{4} -3.41421 q^{7} +4.41421 q^{8} +1.41421 q^{11} -2.58579 q^{13} -8.24264 q^{14} +3.00000 q^{16} -6.82843 q^{17} +1.00000 q^{19} +3.41421 q^{22} -3.65685 q^{23} -6.24264 q^{26} -13.0711 q^{28} -5.07107 q^{29} -10.4853 q^{31} -1.58579 q^{32} -16.4853 q^{34} +3.07107 q^{37} +2.41421 q^{38} +4.58579 q^{41} -3.41421 q^{43} +5.41421 q^{44} -8.82843 q^{46} +11.6569 q^{47} +4.65685 q^{49} -9.89949 q^{52} +4.00000 q^{53} -15.0711 q^{56} -12.2426 q^{58} +8.48528 q^{59} -5.65685 q^{61} -25.3137 q^{62} -9.82843 q^{64} -12.0000 q^{67} -26.1421 q^{68} -12.4853 q^{71} +2.00000 q^{73} +7.41421 q^{74} +3.82843 q^{76} -4.82843 q^{77} +11.3137 q^{79} +11.0711 q^{82} +6.48528 q^{83} -8.24264 q^{86} +6.24264 q^{88} +14.7279 q^{89} +8.82843 q^{91} -14.0000 q^{92} +28.1421 q^{94} -4.24264 q^{97} +11.2426 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 4 q^{7} + 6 q^{8} - 8 q^{13} - 8 q^{14} + 6 q^{16} - 8 q^{17} + 2 q^{19} + 4 q^{22} + 4 q^{23} - 4 q^{26} - 12 q^{28} + 4 q^{29} - 4 q^{31} - 6 q^{32} - 16 q^{34} - 8 q^{37} + 2 q^{38}+ \cdots + 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 4 * q^7 + 6 * q^8 - 8 * q^13 - 8 * q^14 + 6 * q^16 - 8 * q^17 + 2 * q^19 + 4 * q^22 + 4 * q^23 - 4 * q^26 - 12 * q^28 + 4 * q^29 - 4 * q^31 - 6 * q^32 - 16 * q^34 - 8 * q^37 + 2 * q^38 + 12 * q^41 - 4 * q^43 + 8 * q^44 - 12 * q^46 + 12 * q^47 - 2 * q^49 + 8 * q^53 - 16 * q^56 - 16 * q^58 - 28 * q^62 - 14 * q^64 - 24 * q^67 - 24 * q^68 - 8 * q^71 + 4 * q^73 + 12 * q^74 + 2 * q^76 - 4 * q^77 + 8 * q^82 - 4 * q^83 - 8 * q^86 + 4 * q^88 + 4 * q^89 + 12 * q^91 - 28 * q^92 + 28 * q^94 + 14 * q^98

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