LMFDB - Embedded newform 2601.2.a.bg.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2601,2,Mod(1,2601)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2601, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2601.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2601 = 3^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2601.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:	$20.7690895657$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 153)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.84776$ of defining polynomial
Character	$\chi$	$=$	2601.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{4} -3.37849 q^{5} -3.69552 q^{7} -3.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} -1.00000 q^{4} -3.37849 q^{5} -3.69552 q^{7} -3.00000 q^{8} -3.37849 q^{10} -5.22625 q^{11} -4.24264 q^{13} -3.69552 q^{14} -1.00000 q^{16} +2.82843 q^{19} +3.37849 q^{20} -5.22625 q^{22} +0.634051 q^{23} +6.41421 q^{25} -4.24264 q^{26} +3.69552 q^{28} -4.46088 q^{29} +1.53073 q^{31} +5.00000 q^{32} +12.4853 q^{35} -2.48181 q^{37} +2.82843 q^{38} +10.1355 q^{40} -4.46088 q^{41} -5.65685 q^{43} +5.22625 q^{44} +0.634051 q^{46} -2.82843 q^{47} +6.65685 q^{49} +6.41421 q^{50} +4.24264 q^{52} -0.242641 q^{53} +17.6569 q^{55} +11.0866 q^{56} -4.46088 q^{58} -4.48528 q^{59} +12.4860 q^{61} +1.53073 q^{62} +7.00000 q^{64} +14.3337 q^{65} -4.48528 q^{67} +12.4853 q^{70} -0.634051 q^{71} -9.23880 q^{73} -2.48181 q^{74} -2.82843 q^{76} +19.3137 q^{77} +8.02509 q^{79} +3.37849 q^{80} -4.46088 q^{82} -12.4853 q^{83} -5.65685 q^{86} +15.6788 q^{88} +10.5858 q^{89} +15.6788 q^{91} -0.634051 q^{92} -2.82843 q^{94} -9.55582 q^{95} -9.68714 q^{97} +6.65685 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 - 4 * q^4 - 12 * q^8	$ 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8} - 4 q^{16} + 20 q^{25} + 20 q^{32} + 16 q^{35} + 4 q^{49} + 20 q^{50} + 16 q^{53} + 48 q^{55} + 16 q^{59} + 28 q^{64} + 16 q^{67} + 16 q^{70} + 32 q^{77} - 16 q^{83} + 48 q^{89} + 4 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 - 4 * q^4 - 12 * q^8 - 4 * q^16 + 20 * q^25 + 20 * q^32 + 16 * q^35 + 4 * q^49 + 20 * q^50 + 16 * q^53 + 48 * q^55 + 16 * q^59 + 28 * q^64 + 16 * q^67 + 16 * q^70 + 32 * q^77 - 16 * q^83 + 48 * q^89 + 4 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	−3.37849	−1.51091	−0.755454	−	0.655202i	$-0.772584\pi$
$5$	−3.37849	−1.51091	−0.755454	+	0.655202i	$0.772584\pi$
$6$	0	0
$7$	−3.69552	−1.39677	−0.698387	−	0.715720i	$-0.746099\pi$
$7$	−3.69552	−1.39677	−0.698387	+	0.715720i	$0.746099\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	−3.37849	−1.06837
$11$	−5.22625	−1.57577	−0.787887	−	0.615820i	$-0.788825\pi$
$11$	−5.22625	−1.57577	−0.787887	+	0.615820i	$0.788825\pi$
$12$	0	0
$13$	−4.24264	−1.17670	−0.588348	−	0.808608i	$-0.700222\pi$
$13$	−4.24264	−1.17670	−0.588348	+	0.808608i	$0.700222\pi$
$14$	−3.69552	−0.987669
$15$	0	0
$16$	−1.00000	−0.250000
$17$	0	0
$17$	0	0
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	3.37849	0.755454
$21$	0	0
$22$	−5.22625	−1.11424
$23$	0.634051	0.132209	0.0661044	−	0.997813i	$-0.478943\pi$
$23$	0.634051	0.132209	0.0661044	+	0.997813i	$0.478943\pi$
$24$	0	0
$25$	6.41421	1.28284
$26$	−4.24264	−0.832050
$27$	0	0
$28$	3.69552	0.698387
$29$	−4.46088	−0.828366	−0.414183	−	0.910194i	$-0.635933\pi$
$29$	−4.46088	−0.828366	−0.414183	+	0.910194i	$0.635933\pi$
$30$	0	0
$31$	1.53073	0.274928	0.137464	−	0.990507i	$-0.456105\pi$
$31$	1.53073	0.274928	0.137464	+	0.990507i	$0.456105\pi$
$32$	5.00000	0.883883
$33$	0	0
$34$	0	0
$35$	12.4853	2.11040
$36$	0	0
$37$	−2.48181	−0.408007	−0.204004	−	0.978970i	$-0.565395\pi$
$37$	−2.48181	−0.408007	−0.204004	+	0.978970i	$0.565395\pi$
$38$	2.82843	0.458831
$39$	0	0
$40$	10.1355	1.60256
$41$	−4.46088	−0.696673	−0.348337	−	0.937370i	$-0.613253\pi$
$41$	−4.46088	−0.696673	−0.348337	+	0.937370i	$0.613253\pi$
$42$	0	0
$43$	−5.65685	−0.862662	−0.431331	−	0.902194i	$-0.641956\pi$
$43$	−5.65685	−0.862662	−0.431331	+	0.902194i	$0.641956\pi$
$44$	5.22625	0.787887
$45$	0	0
$46$	0.634051	0.0934857
$47$	−2.82843	−0.412568	−0.206284	−	0.978492i	$-0.566137\pi$
$47$	−2.82843	−0.412568	−0.206284	+	0.978492i	$0.566137\pi$
$48$	0	0
$49$	6.65685	0.950979
$50$	6.41421	0.907107
$51$	0	0
$52$	4.24264	0.588348
$53$	−0.242641	−0.0333293	−0.0166646	−	0.999861i	$-0.505305\pi$
$53$	−0.242641	−0.0333293	−0.0166646	+	0.999861i	$0.505305\pi$
$54$	0	0
$55$	17.6569	2.38085
$56$	11.0866	1.48150
$57$	0	0
$58$	−4.46088	−0.585743
$59$	−4.48528	−0.583934	−0.291967	−	0.956428i	$-0.594310\pi$
$59$	−4.48528	−0.583934	−0.291967	+	0.956428i	$0.594310\pi$
$60$	0	0
$61$	12.4860	1.59866	0.799332	−	0.600889i	$-0.205187\pi$
$61$	12.4860	1.59866	0.799332	+	0.600889i	$0.205187\pi$
$62$	1.53073	0.194403
$63$	0	0
$64$	7.00000	0.875000
$65$	14.3337	1.77788
$66$	0	0
$67$	−4.48528	−0.547964	−0.273982	−	0.961735i	$-0.588341\pi$
$67$	−4.48528	−0.547964	−0.273982	+	0.961735i	$0.588341\pi$
$68$	0	0
$69$	0	0
$70$	12.4853	1.49228
$71$	−0.634051	−0.0752480	−0.0376240	−	0.999292i	$-0.511979\pi$
$71$	−0.634051	−0.0752480	−0.0376240	+	0.999292i	$0.511979\pi$
$72$	0	0
$73$	−9.23880	−1.08132	−0.540660	−	0.841241i	$-0.681825\pi$
$73$	−9.23880	−1.08132	−0.540660	+	0.841241i	$0.681825\pi$
$74$	−2.48181	−0.288505
$75$	0	0
$76$	−2.82843	−0.324443
$77$	19.3137	2.20100
$78$	0	0
$79$	8.02509	0.902893	0.451446	−	0.892298i	$-0.350908\pi$
$79$	8.02509	0.902893	0.451446	+	0.892298i	$0.350908\pi$
$80$	3.37849	0.377727
$81$	0	0
$82$	−4.46088	−0.492622
$83$	−12.4853	−1.37044	−0.685219	−	0.728337i	$-0.740293\pi$
$83$	−12.4853	−1.37044	−0.685219	+	0.728337i	$0.740293\pi$
$84$	0	0
$85$	0	0
$86$	−5.65685	−0.609994
$87$	0	0
$88$	15.6788	1.67136
$89$	10.5858	1.12209	0.561046	−	0.827785i	$-0.310399\pi$
$89$	10.5858	1.12209	0.561046	+	0.827785i	$0.310399\pi$
$90$	0	0
$91$	15.6788	1.64358
$92$	−0.634051	−0.0661044
$93$	0	0
$94$	−2.82843	−0.291730
$95$	−9.55582	−0.980407
$96$	0	0
$97$	−9.68714	−0.983580	−0.491790	−	0.870714i	$-0.663657\pi$
$97$	−9.68714	−0.983580	−0.491790	+	0.870714i	$0.663657\pi$
$98$	6.65685	0.672444
$99$	0	0
$100$	−6.41421	−0.641421
$101$	−7.75736	−0.771886	−0.385943	−	0.922523i	$-0.626124\pi$
$101$	−7.75736	−0.771886	−0.385943	+	0.922523i	$0.626124\pi$
$102$	0	0
$103$	−0.485281	−0.0478162	−0.0239081	−	0.999714i	$-0.507611\pi$
$103$	−0.485281	−0.0478162	−0.0239081	+	0.999714i	$0.507611\pi$
$104$	12.7279	1.24808
$105$	0	0
$106$	−0.242641	−0.0235673
$107$	−4.32957	−0.418555	−0.209278	−	0.977856i	$-0.567111\pi$
$107$	−4.32957	−0.418555	−0.209278	+	0.977856i	$0.567111\pi$
$108$	0	0
$109$	−12.4860	−1.19594	−0.597970	−	0.801519i	$-0.704026\pi$
$109$	−12.4860	−1.19594	−0.597970	+	0.801519i	$0.704026\pi$
$110$	17.6569	1.68351
$111$	0	0
$112$	3.69552	0.349194
$113$	−2.93015	−0.275645	−0.137823	−	0.990457i	$-0.544010\pi$
$113$	−2.93015	−0.275645	−0.137823	+	0.990457i	$0.544010\pi$
$114$	0	0
$115$	−2.14214	−0.199755
$116$	4.46088	0.414183
$117$	0	0
$118$	−4.48528	−0.412904
$119$	0	0
$120$	0	0
$121$	16.3137	1.48306
$122$	12.4860	1.13043
$123$	0	0
$124$	−1.53073	−0.137464
$125$	−4.77791	−0.427349
$126$	0	0
$127$	17.6569	1.56679	0.783396	−	0.621523i	$-0.213486\pi$
$127$	17.6569	1.56679	0.783396	+	0.621523i	$0.213486\pi$
$128$	−3.00000	−0.265165
$129$	0	0
$130$	14.3337	1.25715
$131$	−17.8435	−1.55900	−0.779499	−	0.626404i	$-0.784526\pi$
$131$	−17.8435	−1.55900	−0.779499	+	0.626404i	$0.784526\pi$
$132$	0	0
$133$	−10.4525	−0.906347
$134$	−4.48528	−0.387469
$135$	0	0
$136$	0	0
$137$	−16.2426	−1.38770	−0.693851	−	0.720118i	$-0.744087\pi$
$137$	−16.2426	−1.38770	−0.693851	+	0.720118i	$0.744087\pi$
$138$	0	0
$139$	−12.6173	−1.07018	−0.535092	−	0.844794i	$-0.679723\pi$
$139$	−12.6173	−1.07018	−0.535092	+	0.844794i	$0.679723\pi$
$140$	−12.4853	−1.05520
$141$	0	0
$142$	−0.634051	−0.0532084
$143$	22.1731	1.85421
$144$	0	0
$145$	15.0711	1.25158
$146$	−9.23880	−0.764608
$147$	0	0
$148$	2.48181	0.204004
$149$	−5.31371	−0.435316	−0.217658	−	0.976025i	$-0.569842\pi$
$149$	−5.31371	−0.435316	−0.217658	+	0.976025i	$0.569842\pi$
$150$	0	0
$151$	17.6569	1.43689	0.718447	−	0.695581i	$-0.244853\pi$
$151$	17.6569	1.43689	0.718447	+	0.695581i	$0.244853\pi$
$152$	−8.48528	−0.688247
$153$	0	0
$154$	19.3137	1.55634
$155$	−5.17157	−0.415391
$156$	0	0
$157$	0	0		−	1.00000i	$-0.5\pi$
$157$	0	0			1.00000i	$0.5\pi$
$158$	8.02509	0.638442
$159$	0	0
$160$	−16.8925	−1.33547
$161$	−2.34315	−0.184666
$162$	0	0
$163$	−0.896683	−0.0702336	−0.0351168	−	0.999383i	$-0.511180\pi$
$163$	−0.896683	−0.0702336	−0.0351168	+	0.999383i	$0.511180\pi$
$164$	4.46088	0.348337
$165$	0	0
$166$	−12.4853	−0.969046
$167$	−5.86030	−0.453484	−0.226742	−	0.973955i	$-0.572807\pi$
$167$	−5.86030	−0.453484	−0.226742	+	0.973955i	$0.572807\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	0	0
$172$	5.65685	0.431331
$173$	0.317025	0.0241030	0.0120515	−	0.999927i	$-0.496164\pi$
$173$	0.317025	0.0241030	0.0120515	+	0.999927i	$0.496164\pi$
$174$	0	0
$175$	−23.7038	−1.79184
$176$	5.22625	0.393944
$177$	0	0
$178$	10.5858	0.793438
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	−8.60474	−0.639586	−0.319793	−	0.947487i	$-0.603613\pi$
$181$	−8.60474	−0.639586	−0.319793	+	0.947487i	$0.603613\pi$
$182$	15.6788	1.16219
$183$	0	0
$184$	−1.90215	−0.140229
$185$	8.38478	0.616461
$186$	0	0
$187$	0	0
$188$	2.82843	0.206284
$189$	0	0
$190$	−9.55582	−0.693252
$191$	−9.17157	−0.663632	−0.331816	−	0.943344i	$-0.607661\pi$
$191$	−9.17157	−0.663632	−0.331816	+	0.943344i	$0.607661\pi$
$192$	0	0
$193$	5.35757	0.385646	0.192823	−	0.981234i	$-0.438236\pi$
$193$	5.35757	0.385646	0.192823	+	0.981234i	$0.438236\pi$
$194$	−9.68714	−0.695496
$195$	0	0
$196$	−6.65685	−0.475490
$197$	26.6340	1.89759	0.948797	−	0.315887i	$-0.102302\pi$
$197$	26.6340	1.89759	0.948797	+	0.315887i	$0.102302\pi$
$198$	0	0
$199$	3.69552	0.261968	0.130984	−	0.991384i	$-0.458186\pi$
$199$	3.69552	0.261968	0.130984	+	0.991384i	$0.458186\pi$
$200$	−19.2426	−1.36066
$201$	0	0
$202$	−7.75736	−0.545806
$203$	16.4853	1.15704
$204$	0	0
$205$	15.0711	1.05261
$206$	−0.485281	−0.0338112
$207$	0	0
$208$	4.24264	0.294174
$209$	−14.7821	−1.02250
$210$	0	0
$211$	23.0698	1.58819	0.794095	−	0.607794i	$-0.207945\pi$
$211$	23.0698	1.58819	0.794095	+	0.607794i	$0.207945\pi$
$212$	0.242641	0.0166646
$213$	0	0
$214$	−4.32957	−0.295963
$215$	19.1116	1.30340
$216$	0	0
$217$	−5.65685	−0.384012
$218$	−12.4860	−0.845657
$219$	0	0
$220$	−17.6569	−1.19042
$221$	0	0
$222$	0	0
$223$	14.8284	0.992985	0.496492	−	0.868041i	$-0.334621\pi$
$223$	14.8284	0.992985	0.496492	+	0.868041i	$0.334621\pi$
$224$	−18.4776	−1.23459
$225$	0	0
$226$	−2.93015	−0.194911
$227$	11.7206	0.777924	0.388962	−	0.921254i	$-0.372834\pi$
$227$	11.7206	0.777924	0.388962	+	0.921254i	$0.372834\pi$
$228$	0	0
$229$	22.6274	1.49526	0.747631	−	0.664114i	$-0.231191\pi$
$229$	22.6274	1.49526	0.747631	+	0.664114i	$0.231191\pi$
$230$	−2.14214	−0.141248
$231$	0	0
$232$	13.3827	0.878614
$233$	14.4650	0.947637	0.473818	−	0.880623i	$-0.342875\pi$
$233$	14.4650	0.947637	0.473818	+	0.880623i	$0.342875\pi$
$234$	0	0
$235$	9.55582	0.623353
$236$	4.48528	0.291967
$237$	0	0
$238$	0	0
$239$	3.51472	0.227348	0.113674	−	0.993518i	$-0.463738\pi$
$239$	3.51472	0.227348	0.113674	+	0.993518i	$0.463738\pi$
$240$	0	0
$241$	5.35757	0.345111	0.172556	−	0.985000i	$-0.444798\pi$
$241$	5.35757	0.345111	0.172556	+	0.985000i	$0.444798\pi$
$242$	16.3137	1.04868
$243$	0	0
$244$	−12.4860	−0.799332
$245$	−22.4901	−1.43684
$246$	0	0
$247$	−12.0000	−0.763542
$248$	−4.59220	−0.291605
$249$	0	0
$250$	−4.77791	−0.302182
$251$	22.6274	1.42823	0.714115	−	0.700028i	$-0.246829\pi$
$251$	22.6274	1.42823	0.714115	+	0.700028i	$0.246829\pi$
$252$	0	0
$253$	−3.31371	−0.208331
$254$	17.6569	1.10789
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−30.9706	−1.93189	−0.965945	−	0.258746i	$-0.916691\pi$
$257$	−30.9706	−1.93189	−0.965945	+	0.258746i	$0.916691\pi$
$258$	0	0
$259$	9.17157	0.569894
$260$	−14.3337	−0.888940
$261$	0	0
$262$	−17.8435	−1.10238
$263$	−28.4853	−1.75648	−0.878239	−	0.478222i	$-0.841281\pi$
$263$	−28.4853	−1.75648	−0.878239	+	0.478222i	$0.841281\pi$
$264$	0	0
$265$	0.819760	0.0503574
$266$	−10.4525	−0.640884
$267$	0	0
$268$	4.48528	0.273982
$269$	−14.6508	−0.893272	−0.446636	−	0.894716i	$-0.647378\pi$
$269$	−14.6508	−0.893272	−0.446636	+	0.894716i	$0.647378\pi$
$270$	0	0
$271$	4.00000	0.242983	0.121491	−	0.992592i	$-0.461232\pi$
$271$	4.00000	0.242983	0.121491	+	0.992592i	$0.461232\pi$
$272$	0	0
$273$	0	0
$274$	−16.2426	−0.981254
$275$	−33.5223	−2.02147
$276$	0	0
$277$	12.9343	0.777148	0.388574	−	0.921418i	$-0.372968\pi$
$277$	12.9343	0.777148	0.388574	+	0.921418i	$0.372968\pi$
$278$	−12.6173	−0.756735
$279$	0	0
$280$	−37.4558	−2.23841
$281$	−16.7279	−0.997904	−0.498952	−	0.866630i	$-0.666282\pi$
$281$	−16.7279	−0.997904	−0.498952	+	0.866630i	$0.666282\pi$
$282$	0	0
$283$	6.12293	0.363971	0.181985	−	0.983301i	$-0.441748\pi$
$283$	6.12293	0.363971	0.181985	+	0.983301i	$0.441748\pi$
$284$	0.634051	0.0376240
$285$	0	0
$286$	22.1731	1.31112
$287$	16.4853	0.973095
$288$	0	0
$289$	0	0
$290$	15.0711	0.885004
$291$	0	0
$292$	9.23880	0.540660
$293$	−29.6569	−1.73257	−0.866286	−	0.499548i	$-0.833499\pi$
$293$	−29.6569	−1.73257	−0.866286	+	0.499548i	$0.833499\pi$
$294$	0	0
$295$	15.1535	0.882270
$296$	7.44543	0.432757
$297$	0	0
$298$	−5.31371	−0.307815
$299$	−2.69005	−0.155570
$300$	0	0
$301$	20.9050	1.20494
$302$	17.6569	1.01604
$303$	0	0
$304$	−2.82843	−0.162221
$305$	−42.1838	−2.41544
$306$	0	0
$307$	−8.97056	−0.511977	−0.255989	−	0.966680i	$-0.582401\pi$
$307$	−8.97056	−0.511977	−0.255989	+	0.966680i	$0.582401\pi$
$308$	−19.3137	−1.10050
$309$	0	0
$310$	−5.17157	−0.293726
$311$	6.75699	0.383154	0.191577	−	0.981478i	$-0.438640\pi$
$311$	6.75699	0.383154	0.191577	+	0.981478i	$0.438640\pi$
$312$	0	0
$313$	11.0322	0.623575	0.311787	−	0.950152i	$-0.399072\pi$
$313$	11.0322	0.623575	0.311787	+	0.950152i	$0.399072\pi$
$314$	0	0
$315$	0	0
$316$	−8.02509	−0.451446
$317$	−12.6717	−0.711713	−0.355856	−	0.934541i	$-0.615811\pi$
$317$	−12.6717	−0.711713	−0.355856	+	0.934541i	$0.615811\pi$
$318$	0	0
$319$	23.3137	1.30532
$320$	−23.6494	−1.32204
$321$	0	0
$322$	−2.34315	−0.130578
$323$	0	0
$324$	0	0
$325$	−27.2132	−1.50952
$326$	−0.896683	−0.0496627
$327$	0	0
$328$	13.3827	0.738934
$329$	10.4525	0.576265
$330$	0	0
$331$	−23.3137	−1.28144	−0.640719	−	0.767776i	$-0.721363\pi$
$331$	−23.3137	−1.28144	−0.640719	+	0.767776i	$0.721363\pi$
$332$	12.4853	0.685219
$333$	0	0
$334$	−5.86030	−0.320661
$335$	15.1535	0.827924
$336$	0	0
$337$	−32.1229	−1.74984	−0.874922	−	0.484263i	$-0.839088\pi$
$337$	−32.1229	−1.74984	−0.874922	+	0.484263i	$0.839088\pi$
$338$	5.00000	0.271964
$339$	0	0
$340$	0	0
$341$	−8.00000	−0.433224
$342$	0	0
$343$	1.26810	0.0684710
$344$	16.9706	0.914991
$345$	0	0
$346$	0.317025	0.0170434
$347$	−2.16478	−0.116212	−0.0581059	−	0.998310i	$-0.518506\pi$
$347$	−2.16478	−0.116212	−0.0581059	+	0.998310i	$0.518506\pi$
$348$	0	0
$349$	7.07107	0.378506	0.189253	−	0.981928i	$-0.439393\pi$
$349$	7.07107	0.378506	0.189253	+	0.981928i	$0.439393\pi$
$350$	−23.7038	−1.26702
$351$	0	0
$352$	−26.1313	−1.39280
$353$	5.65685	0.301084	0.150542	−	0.988604i	$-0.451898\pi$
$353$	5.65685	0.301084	0.150542	+	0.988604i	$0.451898\pi$
$354$	0	0
$355$	2.14214	0.113693
$356$	−10.5858	−0.561046
$357$	0	0
$358$	−16.0000	−0.845626
$359$	32.9706	1.74012	0.870060	−	0.492946i	$-0.164080\pi$
$359$	32.9706	1.74012	0.870060	+	0.492946i	$0.164080\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	−8.60474	−0.452255
$363$	0	0
$364$	−15.6788	−0.821790
$365$	31.2132	1.63377
$366$	0	0
$367$	12.8799	0.672326	0.336163	−	0.941804i	$-0.390871\pi$
$367$	12.8799	0.672326	0.336163	+	0.941804i	$0.390871\pi$
$368$	−0.634051	−0.0330522
$369$	0	0
$370$	8.38478	0.435904
$371$	0.896683	0.0465535
$372$	0	0
$373$	37.2132	1.92683	0.963413	−	0.268020i	$-0.0863694\pi$
$373$	37.2132	1.92683	0.963413	+	0.268020i	$0.0863694\pi$
$374$	0	0
$375$	0	0
$376$	8.48528	0.437595
$377$	18.9259	0.974735
$378$	0	0
$379$	13.8854	0.713245	0.356622	−	0.934249i	$-0.383928\pi$
$379$	13.8854	0.713245	0.356622	+	0.934249i	$0.383928\pi$
$380$	9.55582	0.490203
$381$	0	0
$382$	−9.17157	−0.469258
$383$	−7.51472	−0.383984	−0.191992	−	0.981396i	$-0.561495\pi$
$383$	−7.51472	−0.383984	−0.191992	+	0.981396i	$0.561495\pi$
$384$	0	0
$385$	−65.2512	−3.32551
$386$	5.35757	0.272693
$387$	0	0
$388$	9.68714	0.491790
$389$	24.9706	1.26606	0.633029	−	0.774128i	$-0.281811\pi$
$389$	24.9706	1.26606	0.633029	+	0.774128i	$0.281811\pi$
$390$	0	0
$391$	0	0
$392$	−19.9706	−1.00867
$393$	0	0
$394$	26.6340	1.34180
$395$	−27.1127	−1.36419
$396$	0	0
$397$	−0.579658	−0.0290922	−0.0145461	−	0.999894i	$-0.504630\pi$
$397$	−0.579658	−0.0290922	−0.0145461	+	0.999894i	$0.504630\pi$
$398$	3.69552	0.185240
$399$	0	0
$400$	−6.41421	−0.320711
$401$	13.6453	0.681413	0.340707	−	0.940170i	$-0.389334\pi$
$401$	13.6453	0.681413	0.340707	+	0.940170i	$0.389334\pi$
$402$	0	0
$403$	−6.49435	−0.323507
$404$	7.75736	0.385943
$405$	0	0
$406$	16.4853	0.818151
$407$	12.9706	0.642927
$408$	0	0
$409$	−32.9706	−1.63029	−0.815145	−	0.579257i	$-0.803343\pi$
$409$	−32.9706	−1.63029	−0.815145	+	0.579257i	$0.803343\pi$
$410$	15.0711	0.744307
$411$	0	0
$412$	0.485281	0.0239081
$413$	16.5754	0.815624
$414$	0	0
$415$	42.1814	2.07061
$416$	−21.2132	−1.04006
$417$	0	0
$418$	−14.7821	−0.723015
$419$	23.4412	1.14518	0.572589	−	0.819843i	$-0.305939\pi$
$419$	23.4412	1.14518	0.572589	+	0.819843i	$0.305939\pi$
$420$	0	0
$421$	−36.7279	−1.79001	−0.895005	−	0.446057i	$-0.852828\pi$
$421$	−36.7279	−1.79001	−0.895005	+	0.446057i	$0.852828\pi$
$422$	23.0698	1.12302
$423$	0	0
$424$	0.727922	0.0353510
$425$	0	0
$426$	0	0
$427$	−46.1421	−2.23297
$428$	4.32957	0.209278
$429$	0	0
$430$	19.1116	0.921645
$431$	−34.5278	−1.66314	−0.831572	−	0.555417i	$-0.812559\pi$
$431$	−34.5278	−1.66314	−0.831572	+	0.555417i	$0.812559\pi$
$432$	0	0
$433$	11.6569	0.560193	0.280096	−	0.959972i	$-0.409634\pi$
$433$	11.6569	0.560193	0.280096	+	0.959972i	$0.409634\pi$
$434$	−5.65685	−0.271538
$435$	0	0
$436$	12.4860	0.597970
$437$	1.79337	0.0857883
$438$	0	0
$439$	23.7038	1.13132	0.565661	−	0.824638i	$-0.308621\pi$
$439$	23.7038	1.13132	0.565661	+	0.824638i	$0.308621\pi$
$440$	−52.9706	−2.52527
$441$	0	0
$442$	0	0
$443$	−28.9706	−1.37643	−0.688216	−	0.725505i	$-0.741606\pi$
$443$	−28.9706	−1.37643	−0.688216	+	0.725505i	$0.741606\pi$
$444$	0	0
$445$	−35.7640	−1.69538
$446$	14.8284	0.702146
$447$	0	0
$448$	−25.8686	−1.22218
$449$	23.6494	1.11609	0.558043	−	0.829812i	$-0.311552\pi$
$449$	23.6494	1.11609	0.558043	+	0.829812i	$0.311552\pi$
$450$	0	0
$451$	23.3137	1.09780
$452$	2.93015	0.137823
$453$	0	0
$454$	11.7206	0.550075
$455$	−52.9706	−2.48330
$456$	0	0
$457$	−28.2843	−1.32308	−0.661541	−	0.749909i	$-0.730097\pi$
$457$	−28.2843	−1.32308	−0.661541	+	0.749909i	$0.730097\pi$
$458$	22.6274	1.05731
$459$	0	0
$460$	2.14214	0.0998776
$461$	−7.27208	−0.338694	−0.169347	−	0.985556i	$-0.554166\pi$
$461$	−7.27208	−0.338694	−0.169347	+	0.985556i	$0.554166\pi$
$462$	0	0
$463$	−20.4853	−0.952032	−0.476016	−	0.879437i	$-0.657920\pi$
$463$	−20.4853	−0.952032	−0.476016	+	0.879437i	$0.657920\pi$
$464$	4.46088	0.207091
$465$	0	0
$466$	14.4650	0.670080
$467$	32.9706	1.52570	0.762848	−	0.646578i	$-0.223801\pi$
$467$	32.9706	1.52570	0.762848	+	0.646578i	$0.223801\pi$
$468$	0	0
$469$	16.5754	0.765383
$470$	9.55582	0.440777
$471$	0	0
$472$	13.4558	0.619355
$473$	29.5641	1.35936
$474$	0	0
$475$	18.1421	0.832418
$476$	0	0
$477$	0	0
$478$	3.51472	0.160759
$479$	8.55035	0.390676	0.195338	−	0.980736i	$-0.437420\pi$
$479$	8.55035	0.390676	0.195338	+	0.980736i	$0.437420\pi$
$480$	0	0
$481$	10.5294	0.480101
$482$	5.35757	0.244031
$483$	0	0
$484$	−16.3137	−0.741532
$485$	32.7279	1.48610
$486$	0	0
$487$	4.22078	0.191262	0.0956310	−	0.995417i	$-0.469513\pi$
$487$	4.22078	0.191262	0.0956310	+	0.995417i	$0.469513\pi$
$488$	−37.4579	−1.69564
$489$	0	0
$490$	−22.4901	−1.01600
$491$	−37.9411	−1.71226	−0.856130	−	0.516761i	$-0.827137\pi$
$491$	−37.9411	−1.71226	−0.856130	+	0.516761i	$0.827137\pi$
$492$	0	0
$493$	0	0
$494$	−12.0000	−0.539906
$495$	0	0
$496$	−1.53073	−0.0687320
$497$	2.34315	0.105104
$498$	0	0
$499$	−37.4804	−1.67786	−0.838928	−	0.544243i	$-0.816817\pi$
$499$	−37.4804	−1.67786	−0.838928	+	0.544243i	$0.816817\pi$
$500$	4.77791	0.213675
$501$	0	0
$502$	22.6274	1.00991
$503$	9.44703	0.421222	0.210611	−	0.977570i	$-0.432455\pi$
$503$	9.44703	0.421222	0.210611	+	0.977570i	$0.432455\pi$
$504$	0	0
$505$	26.2082	1.16625
$506$	−3.31371	−0.147312
$507$	0	0
$508$	−17.6569	−0.783396
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	34.1421	1.51036
$512$	−11.0000	−0.486136
$513$	0	0
$514$	−30.9706	−1.36605
$515$	1.63952	0.0722459
$516$	0	0
$517$	14.7821	0.650115
$518$	9.17157	0.402976
$519$	0	0
$520$	−43.0012	−1.88573
$521$	31.6745	1.38769	0.693843	−	0.720126i	$-0.255916\pi$
$521$	31.6745	1.38769	0.693843	+	0.720126i	$0.255916\pi$
$522$	0	0
$523$	8.48528	0.371035	0.185518	−	0.982641i	$-0.440604\pi$
$523$	8.48528	0.371035	0.185518	+	0.982641i	$0.440604\pi$
$524$	17.8435	0.779499
$525$	0	0
$526$	−28.4853	−1.24202
$527$	0	0
$528$	0	0
$529$	−22.5980	−0.982521
$530$	0.819760	0.0356081
$531$	0	0
$532$	10.4525	0.453174
$533$	18.9259	0.819773
$534$	0	0
$535$	14.6274	0.632398
$536$	13.4558	0.581204
$537$	0	0
$538$	−14.6508	−0.631639
$539$	−34.7904	−1.49853
$540$	0	0
$541$	24.6549	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$541$	24.6549	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$542$	4.00000	0.171815
$543$	0	0
$544$	0	0
$545$	42.1838	1.80695
$546$	0	0
$547$	−25.6060	−1.09483	−0.547417	−	0.836860i	$-0.684389\pi$
$547$	−25.6060	−1.09483	−0.547417	+	0.836860i	$0.684389\pi$
$548$	16.2426	0.693851
$549$	0	0
$550$	−33.5223	−1.42940
$551$	−12.6173	−0.537515
$552$	0	0
$553$	−29.6569	−1.26114
$554$	12.9343	0.549526
$555$	0	0
$556$	12.6173	0.535092
$557$	−12.0416	−0.510220	−0.255110	−	0.966912i	$-0.582112\pi$
$557$	−12.0416	−0.510220	−0.255110	+	0.966912i	$0.582112\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	−12.4853	−0.527599
$561$	0	0
$562$	−16.7279	−0.705625
$563$	12.4853	0.526192	0.263096	−	0.964770i	$-0.415256\pi$
$563$	12.4853	0.526192	0.263096	+	0.964770i	$0.415256\pi$
$564$	0	0
$565$	9.89949	0.416475
$566$	6.12293	0.257366
$567$	0	0
$568$	1.90215	0.0798125
$569$	8.24264	0.345549	0.172775	−	0.984961i	$-0.444727\pi$
$569$	8.24264	0.345549	0.172775	+	0.984961i	$0.444727\pi$
$570$	0	0
$571$	−16.5754	−0.693661	−0.346830	−	0.937928i	$-0.612742\pi$
$571$	−16.5754	−0.693661	−0.346830	+	0.937928i	$0.612742\pi$
$572$	−22.1731	−0.927104
$573$	0	0
$574$	16.4853	0.688082
$575$	4.06694	0.169603
$576$	0	0
$577$	−20.7279	−0.862915	−0.431457	−	0.902133i	$-0.642000\pi$
$577$	−20.7279	−0.862915	−0.431457	+	0.902133i	$0.642000\pi$
$578$	0	0
$579$	0	0
$580$	−15.0711	−0.625792
$581$	46.1396	1.91419
$582$	0	0
$583$	1.26810	0.0525194
$584$	27.7164	1.14691
$585$	0	0
$586$	−29.6569	−1.22511
$587$	−20.9706	−0.865548	−0.432774	−	0.901503i	$-0.642465\pi$
$587$	−20.9706	−0.865548	−0.432774	+	0.901503i	$0.642465\pi$
$588$	0	0
$589$	4.32957	0.178397
$590$	15.1535	0.623859
$591$	0	0
$592$	2.48181	0.102002
$593$	7.27208	0.298628	0.149314	−	0.988790i	$-0.452293\pi$
$593$	7.27208	0.298628	0.149314	+	0.988790i	$0.452293\pi$
$594$	0	0
$595$	0	0
$596$	5.31371	0.217658
$597$	0	0
$598$	−2.69005	−0.110004
$599$	40.2843	1.64597	0.822985	−	0.568063i	$-0.192307\pi$
$599$	40.2843	1.64597	0.822985	+	0.568063i	$0.192307\pi$
$600$	0	0
$601$	−39.7765	−1.62252	−0.811260	−	0.584686i	$-0.801218\pi$
$601$	−39.7765	−1.62252	−0.811260	+	0.584686i	$0.801218\pi$
$602$	20.9050	0.852024
$603$	0	0
$604$	−17.6569	−0.718447
$605$	−55.1157	−2.24077
$606$	0	0
$607$	−15.7875	−0.640797	−0.320398	−	0.947283i	$-0.603817\pi$
$607$	−15.7875	−0.640797	−0.320398	+	0.947283i	$0.603817\pi$
$608$	14.1421	0.573539
$609$	0	0
$610$	−42.1838	−1.70797
$611$	12.0000	0.485468
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	−8.97056	−0.362022
$615$	0	0
$616$	−57.9411	−2.33451
$617$	16.8925	0.680065	0.340032	−	0.940414i	$-0.389562\pi$
$617$	16.8925	0.680065	0.340032	+	0.940414i	$0.389562\pi$
$618$	0	0
$619$	32.2542	1.29641	0.648203	−	0.761468i	$-0.275521\pi$
$619$	32.2542	1.29641	0.648203	+	0.761468i	$0.275521\pi$
$620$	5.17157	0.207695
$621$	0	0
$622$	6.75699	0.270930
$623$	−39.1200	−1.56731
$624$	0	0
$625$	−15.9289	−0.637157
$626$	11.0322	0.440934
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	14.8284	0.590310	0.295155	−	0.955449i	$-0.404629\pi$
$631$	14.8284	0.590310	0.295155	+	0.955449i	$0.404629\pi$
$632$	−24.0753	−0.957662
$633$	0	0
$634$	−12.6717	−0.503257
$635$	−59.6536	−2.36728
$636$	0	0
$637$	−28.2426	−1.11901
$638$	23.3137	0.922999
$639$	0	0
$640$	10.1355	0.400640
$641$	5.28064	0.208573	0.104286	−	0.994547i	$-0.466744\pi$
$641$	5.28064	0.208573	0.104286	+	0.994547i	$0.466744\pi$
$642$	0	0
$643$	−16.0502	−0.632957	−0.316479	−	0.948600i	$-0.602501\pi$
$643$	−16.0502	−0.632957	−0.316479	+	0.948600i	$0.602501\pi$
$644$	2.34315	0.0923329
$645$	0	0
$646$	0	0
$647$	27.5147	1.08172	0.540858	−	0.841114i	$-0.318100\pi$
$647$	27.5147	1.08172	0.540858	+	0.841114i	$0.318100\pi$
$648$	0	0
$649$	23.4412	0.920148
$650$	−27.2132	−1.06739
$651$	0	0
$652$	0.896683	0.0351168
$653$	0.131316	0.00513880	0.00256940	−	0.999997i	$-0.499182\pi$
$653$	0.131316	0.00513880	0.00256940	+	0.999997i	$0.499182\pi$
$654$	0	0
$655$	60.2843	2.35550
$656$	4.46088	0.174168
$657$	0	0
$658$	10.4525	0.407481
$659$	26.1421	1.01835	0.509177	−	0.860662i	$-0.329950\pi$
$659$	26.1421	1.01835	0.509177	+	0.860662i	$0.329950\pi$
$660$	0	0
$661$	9.89949	0.385046	0.192523	−	0.981292i	$-0.438333\pi$
$661$	9.89949	0.385046	0.192523	+	0.981292i	$0.438333\pi$
$662$	−23.3137	−0.906113
$663$	0	0
$664$	37.4558	1.45357
$665$	35.3137	1.36941
$666$	0	0
$667$	−2.82843	−0.109517
$668$	5.86030	0.226742
$669$	0	0
$670$	15.1535	0.585430
$671$	−65.2548	−2.51913
$672$	0	0
$673$	8.68167	0.334654	0.167327	−	0.985901i	$-0.446486\pi$
$673$	8.68167	0.334654	0.167327	+	0.985901i	$0.446486\pi$
$674$	−32.1229	−1.23733
$675$	0	0
$676$	−5.00000	−0.192308
$677$	−9.50143	−0.365170	−0.182585	−	0.983190i	$-0.558446\pi$
$677$	−9.50143	−0.365170	−0.182585	+	0.983190i	$0.558446\pi$
$678$	0	0
$679$	35.7990	1.37384
$680$	0	0
$681$	0	0
$682$	−8.00000	−0.306336
$683$	−5.22625	−0.199977	−0.0999885	−	0.994989i	$-0.531881\pi$
$683$	−5.22625	−0.199977	−0.0999885	+	0.994989i	$0.531881\pi$
$684$	0	0
$685$	54.8756	2.09669
$686$	1.26810	0.0484163
$687$	0	0
$688$	5.65685	0.215666
$689$	1.02944	0.0392184
$690$	0	0
$691$	−31.7289	−1.20703	−0.603513	−	0.797353i	$-0.706233\pi$
$691$	−31.7289	−1.20703	−0.603513	+	0.797353i	$0.706233\pi$
$692$	−0.317025	−0.0120515
$693$	0	0
$694$	−2.16478	−0.0821741
$695$	42.6274	1.61695
$696$	0	0
$697$	0	0
$698$	7.07107	0.267644
$699$	0	0
$700$	23.7038	0.895921
$701$	−13.4142	−0.506648	−0.253324	−	0.967382i	$-0.581524\pi$
$701$	−13.4142	−0.506648	−0.253324	+	0.967382i	$0.581524\pi$
$702$	0	0
$703$	−7.01962	−0.264750
$704$	−36.5838	−1.37880
$705$	0	0
$706$	5.65685	0.212899
$707$	28.6675	1.07815
$708$	0	0
$709$	−33.2053	−1.24705	−0.623525	−	0.781803i	$-0.714300\pi$
$709$	−33.2053	−1.24705	−0.623525	+	0.781803i	$0.714300\pi$
$710$	2.14214	0.0803929
$711$	0	0
$712$	−31.7574	−1.19016
$713$	0.970563	0.0363479
$714$	0	0
$715$	−74.9117	−2.80154
$716$	16.0000	0.597948
$717$	0	0
$718$	32.9706	1.23045
$719$	4.06694	0.151671	0.0758356	−	0.997120i	$-0.475838\pi$
$719$	4.06694	0.151671	0.0758356	+	0.997120i	$0.475838\pi$
$720$	0	0
$721$	1.79337	0.0667884
$722$	−11.0000	−0.409378
$723$	0	0
$724$	8.60474	0.319793
$725$	−28.6131	−1.06266
$726$	0	0
$727$	16.9706	0.629403	0.314702	−	0.949191i	$-0.398096\pi$
$727$	16.9706	0.629403	0.314702	+	0.949191i	$0.398096\pi$
$728$	−47.0363	−1.74328
$729$	0	0
$730$	31.2132	1.15525
$731$	0	0
$732$	0	0
$733$	−42.3848	−1.56552	−0.782759	−	0.622325i	$-0.786188\pi$
$733$	−42.3848	−1.56552	−0.782759	+	0.622325i	$0.786188\pi$
$734$	12.8799	0.475407
$735$	0	0
$736$	3.17025	0.116857
$737$	23.4412	0.863468
$738$	0	0
$739$	21.1716	0.778809	0.389404	−	0.921067i	$-0.372681\pi$
$739$	21.1716	0.778809	0.389404	+	0.921067i	$0.372681\pi$
$740$	−8.38478	−0.308231
$741$	0	0
$742$	0.896683	0.0329183
$743$	18.1062	0.664251	0.332126	−	0.943235i	$-0.392234\pi$
$743$	18.1062	0.664251	0.332126	+	0.943235i	$0.392234\pi$
$744$	0	0
$745$	17.9523	0.657722
$746$	37.2132	1.36247
$747$	0	0
$748$	0	0
$749$	16.0000	0.584627
$750$	0	0
$751$	−42.0726	−1.53525	−0.767626	−	0.640898i	$-0.778562\pi$
$751$	−42.0726	−1.53525	−0.767626	+	0.640898i	$0.778562\pi$
$752$	2.82843	0.103142
$753$	0	0
$754$	18.9259	0.689242
$755$	−59.6536	−2.17102
$756$	0	0
$757$	−12.6863	−0.461091	−0.230546	−	0.973062i	$-0.574051\pi$
$757$	−12.6863	−0.461091	−0.230546	+	0.973062i	$0.574051\pi$
$758$	13.8854	0.504340
$759$	0	0
$760$	28.6675	1.03988
$761$	38.8701	1.40904	0.704519	−	0.709685i	$-0.251163\pi$
$761$	38.8701	1.40904	0.704519	+	0.709685i	$0.251163\pi$
$762$	0	0
$763$	46.1421	1.67046
$764$	9.17157	0.331816
$765$	0	0
$766$	−7.51472	−0.271518
$767$	19.0294	0.687113
$768$	0	0
$769$	−2.78680	−0.100494	−0.0502472	−	0.998737i	$-0.516001\pi$
$769$	−2.78680	−0.100494	−0.0502472	+	0.998737i	$0.516001\pi$
$770$	−65.2512	−2.35149
$771$	0	0
$772$	−5.35757	−0.192823
$773$	−24.9706	−0.898129	−0.449064	−	0.893499i	$-0.648243\pi$
$773$	−24.9706	−0.898129	−0.449064	+	0.893499i	$0.648243\pi$
$774$	0	0
$775$	9.81845	0.352689
$776$	29.0614	1.04324
$777$	0	0
$778$	24.9706	0.895238
$779$	−12.6173	−0.452061
$780$	0	0
$781$	3.31371	0.118574
$782$	0	0
$783$	0	0
$784$	−6.65685	−0.237745
$785$	0	0
$786$	0	0
$787$	23.0698	0.822349	0.411175	−	0.911557i	$-0.365119\pi$
$787$	23.0698	0.822349	0.411175	+	0.911557i	$0.365119\pi$
$788$	−26.6340	−0.948797
$789$	0	0
$790$	−27.1127	−0.964627
$791$	10.8284	0.385015
$792$	0	0
$793$	−52.9735	−1.88114
$794$	−0.579658	−0.0205713
$795$	0	0
$796$	−3.69552	−0.130984
$797$	−25.2132	−0.893097	−0.446549	−	0.894759i	$-0.647347\pi$
$797$	−25.2132	−0.893097	−0.446549	+	0.894759i	$0.647347\pi$
$798$	0	0
$799$	0	0
$800$	32.0711	1.13388
$801$	0	0
$802$	13.6453	0.481832
$803$	48.2843	1.70391
$804$	0	0
$805$	7.91630	0.279013
$806$	−6.49435	−0.228754
$807$	0	0
$808$	23.2721	0.818709
$809$	29.8812	1.05057	0.525283	−	0.850928i	$-0.323959\pi$
$809$	29.8812	1.05057	0.525283	+	0.850928i	$0.323959\pi$
$810$	0	0
$811$	−16.5754	−0.582042	−0.291021	−	0.956717i	$-0.593995\pi$
$811$	−16.5754	−0.582042	−0.291021	+	0.956717i	$0.593995\pi$
$812$	−16.4853	−0.578520
$813$	0	0
$814$	12.9706	0.454618
$815$	3.02944	0.106117
$816$	0	0
$817$	−16.0000	−0.559769
$818$	−32.9706	−1.15279
$819$	0	0
$820$	−15.0711	−0.526305
$821$	−9.31572	−0.325121	−0.162560	−	0.986699i	$-0.551975\pi$
$821$	−9.31572	−0.325121	−0.162560	+	0.986699i	$0.551975\pi$
$822$	0	0
$823$	2.79884	0.0975613	0.0487806	−	0.998810i	$-0.484466\pi$
$823$	2.79884	0.0975613	0.0487806	+	0.998810i	$0.484466\pi$
$824$	1.45584	0.0507167
$825$	0	0
$826$	16.5754	0.576733
$827$	−45.7682	−1.59152	−0.795758	−	0.605615i	$-0.792927\pi$
$827$	−45.7682	−1.59152	−0.795758	+	0.605615i	$0.792927\pi$
$828$	0	0
$829$	0	0		−	1.00000i	$-0.5\pi$
$829$	0	0			1.00000i	$0.5\pi$
$830$	42.1814	1.46414
$831$	0	0
$832$	−29.6985	−1.02961
$833$	0	0
$834$	0	0
$835$	19.7990	0.685172
$836$	14.7821	0.511249
$837$	0	0
$838$	23.4412	0.809763
$839$	38.4859	1.32868	0.664341	−	0.747430i	$-0.268712\pi$
$839$	38.4859	1.32868	0.664341	+	0.747430i	$0.268712\pi$
$840$	0	0
$841$	−9.10051	−0.313811
$842$	−36.7279	−1.26573
$843$	0	0
$844$	−23.0698	−0.794095
$845$	−16.8925	−0.581118
$846$	0	0
$847$	−60.2876	−2.07151
$848$	0.242641	0.00833232
$849$	0	0
$850$	0	0
$851$	−1.57359	−0.0539421
$852$	0	0
$853$	8.60474	0.294621	0.147310	−	0.989090i	$-0.452938\pi$
$853$	8.60474	0.294621	0.147310	+	0.989090i	$0.452938\pi$
$854$	−46.1421	−1.57895
$855$	0	0
$856$	12.9887	0.443945
$857$	44.4775	1.51932	0.759662	−	0.650318i	$-0.225364\pi$
$857$	44.4775	1.51932	0.759662	+	0.650318i	$0.225364\pi$
$858$	0	0
$859$	−10.6274	−0.362603	−0.181301	−	0.983428i	$-0.558031\pi$
$859$	−10.6274	−0.362603	−0.181301	+	0.983428i	$0.558031\pi$
$860$	−19.1116	−0.651702
$861$	0	0
$862$	−34.5278	−1.17602
$863$	−16.2843	−0.554323	−0.277162	−	0.960823i	$-0.589394\pi$
$863$	−16.2843	−0.554323	−0.277162	+	0.960823i	$0.589394\pi$
$864$	0	0
$865$	−1.07107	−0.0364174
$866$	11.6569	0.396116
$867$	0	0
$868$	5.65685	0.192006
$869$	−41.9411	−1.42276
$870$	0	0
$871$	19.0294	0.644788
$872$	37.4579	1.26849
$873$	0	0
$874$	1.79337	0.0606615
$875$	17.6569	0.596911
$876$	0	0
$877$	25.4747	0.860219	0.430109	−	0.902777i	$-0.358475\pi$
$877$	25.4747	0.860219	0.430109	+	0.902777i	$0.358475\pi$
$878$	23.7038	0.799966
$879$	0	0
$880$	−17.6569	−0.595212
$881$	−35.4788	−1.19531	−0.597656	−	0.801752i	$-0.703901\pi$
$881$	−35.4788	−1.19531	−0.597656	+	0.801752i	$0.703901\pi$
$882$	0	0
$883$	29.4558	0.991268	0.495634	−	0.868531i	$-0.334936\pi$
$883$	29.4558	0.991268	0.495634	+	0.868531i	$0.334936\pi$
$884$	0	0
$885$	0	0
$886$	−28.9706	−0.973285
$887$	−38.4859	−1.29223	−0.646115	−	0.763240i	$-0.723607\pi$
$887$	−38.4859	−1.29223	−0.646115	+	0.763240i	$0.723607\pi$
$888$	0	0
$889$	−65.2512	−2.18846
$890$	−35.7640	−1.19881
$891$	0	0
$892$	−14.8284	−0.496492
$893$	−8.00000	−0.267710
$894$	0	0
$895$	54.0559	1.80689
$896$	11.0866	0.370376
$897$	0	0
$898$	23.6494	0.789192
$899$	−6.82843	−0.227741
$900$	0	0
$901$	0	0
$902$	23.3137	0.776262
$903$	0	0
$904$	8.79045	0.292366
$905$	29.0711	0.966355
$906$	0	0
$907$	38.2233	1.26918	0.634592	−	0.772848i	$-0.281168\pi$
$907$	38.2233	1.26918	0.634592	+	0.772848i	$0.281168\pi$
$908$	−11.7206	−0.388962
$909$	0	0
$910$	−52.9706	−1.75596
$911$	−31.0949	−1.03022	−0.515110	−	0.857124i	$-0.672249\pi$
$911$	−31.0949	−1.03022	−0.515110	+	0.857124i	$0.672249\pi$
$912$	0	0
$913$	65.2512	2.15950
$914$	−28.2843	−0.935561
$915$	0	0
$916$	−22.6274	−0.747631
$917$	65.9411	2.17757
$918$	0	0
$919$	−20.9706	−0.691755	−0.345878	−	0.938280i	$-0.612419\pi$
$919$	−20.9706	−0.691755	−0.345878	+	0.938280i	$0.612419\pi$
$920$	6.42641	0.211872
$921$	0	0
$922$	−7.27208	−0.239493
$923$	2.69005	0.0885440
$924$	0	0
$925$	−15.9189	−0.523409
$926$	−20.4853	−0.673188
$927$	0	0
$928$	−22.3044	−0.732179
$929$	−1.58513	−0.0520063	−0.0260032	−	0.999662i	$-0.508278\pi$
$929$	−1.58513	−0.0520063	−0.0260032	+	0.999662i	$0.508278\pi$
$930$	0	0
$931$	18.8284	0.617077
$932$	−14.4650	−0.473818
$933$	0	0
$934$	32.9706	1.07883
$935$	0	0
$936$	0	0
$937$	24.0416	0.785406	0.392703	−	0.919665i	$-0.371540\pi$
$937$	24.0416	0.785406	0.392703	+	0.919665i	$0.371540\pi$
$938$	16.5754	0.541207
$939$	0	0
$940$	−9.55582	−0.311677
$941$	42.1270	1.37330	0.686651	−	0.726987i	$-0.259080\pi$
$941$	42.1270	1.37330	0.686651	+	0.726987i	$0.259080\pi$
$942$	0	0
$943$	−2.82843	−0.0921063
$944$	4.48528	0.145983
$945$	0	0
$946$	29.5641	0.961213
$947$	−37.3266	−1.21295	−0.606476	−	0.795102i	$-0.707417\pi$
$947$	−37.3266	−1.21295	−0.606476	+	0.795102i	$0.707417\pi$
$948$	0	0
$949$	39.1969	1.27238
$950$	18.1421	0.588609
$951$	0	0
$952$	0	0
$953$	−9.21320	−0.298445	−0.149222	−	0.988804i	$-0.547677\pi$
$953$	−9.21320	−0.298445	−0.149222	+	0.988804i	$0.547677\pi$
$954$	0	0
$955$	30.9861	1.00269
$956$	−3.51472	−0.113674
$957$	0	0
$958$	8.55035	0.276249
$959$	60.0250	1.93831
$960$	0	0
$961$	−28.6569	−0.924415
$962$	10.5294	0.339482
$963$	0	0
$964$	−5.35757	−0.172556
$965$	−18.1005	−0.582676
$966$	0	0
$967$	7.11270	0.228729	0.114364	−	0.993439i	$-0.463517\pi$
$967$	7.11270	0.228729	0.114364	+	0.993439i	$0.463517\pi$
$968$	−48.9411	−1.57303
$969$	0	0
$970$	32.7279	1.05083
$971$	−23.5147	−0.754623	−0.377312	−	0.926086i	$-0.623151\pi$
$971$	−23.5147	−0.754623	−0.377312	+	0.926086i	$0.623151\pi$
$972$	0	0
$973$	46.6274	1.49481
$974$	4.22078	0.135243
$975$	0	0
$976$	−12.4860	−0.399666
$977$	16.7279	0.535174	0.267587	−	0.963534i	$-0.413774\pi$
$977$	16.7279	0.535174	0.267587	+	0.963534i	$0.413774\pi$
$978$	0	0
$979$	−55.3240	−1.76816
$980$	22.4901	0.718421
$981$	0	0
$982$	−37.9411	−1.21075
$983$	34.6816	1.10617	0.553086	−	0.833124i	$-0.313450\pi$
$983$	34.6816	1.10617	0.553086	+	0.833124i	$0.313450\pi$
$984$	0	0
$985$	−89.9828	−2.86709
$986$	0	0
$987$	0	0
$988$	12.0000	0.381771
$989$	−3.58673	−0.114051
$990$	0	0
$991$	−19.3743	−0.615444	−0.307722	−	0.951476i	$-0.599567\pi$
$991$	−19.3743	−0.615444	−0.307722	+	0.951476i	$0.599567\pi$
$992$	7.65367	0.243004
$993$	0	0
$994$	2.34315	0.0743201
$995$	−12.4853	−0.395810
$996$	0	0
$997$	−0.765367	−0.0242394	−0.0121197	−	0.999927i	$-0.503858\pi$
$997$	−0.765367	−0.0242394	−0.0121197	+	0.999927i	$0.503858\pi$
$998$	−37.4804	−1.18642
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2601.2.a.bg.1.1		4
3.2	odd	2		2601.2.a.ba.1.4		4
17.10	odd	16		153.2.l.a.100.1	✓	4
17.12	odd	16		153.2.l.a.127.1	yes	4
17.16	even	2	inner	2601.2.a.bg.1.4		4
51.29	even	16		153.2.l.b.127.1	yes	4
51.44	even	16		153.2.l.b.100.1	yes	4
51.50	odd	2		2601.2.a.ba.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.2.l.a.100.1	✓	4	17.10	odd	16
153.2.l.a.127.1	yes	4	17.12	odd	16
153.2.l.b.100.1	yes	4	51.44	even	16
153.2.l.b.127.1	yes	4	51.29	even	16
2601.2.a.ba.1.1		4	51.50	odd	2
2601.2.a.ba.1.4		4	3.2	odd	2
2601.2.a.bg.1.1		4	1.1	even	1	trivial
2601.2.a.bg.1.4		4	17.16	even	2	inner

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:	\( N \)	\(=\)	\( 2601 = 3^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2601.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{4} -3.37849 q^{5} -3.69552 q^{7} -3.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{4} -3.37849 q^{5} -3.69552 q^{7} -3.00000 q^{8} -3.37849 q^{10} -5.22625 q^{11} -4.24264 q^{13} -3.69552 q^{14} -1.00000 q^{16} +2.82843 q^{19} +3.37849 q^{20} -5.22625 q^{22} +0.634051 q^{23} +6.41421 q^{25} -4.24264 q^{26} +3.69552 q^{28} -4.46088 q^{29} +1.53073 q^{31} +5.00000 q^{32} +12.4853 q^{35} -2.48181 q^{37} +2.82843 q^{38} +10.1355 q^{40} -4.46088 q^{41} -5.65685 q^{43} +5.22625 q^{44} +0.634051 q^{46} -2.82843 q^{47} +6.65685 q^{49} +6.41421 q^{50} +4.24264 q^{52} -0.242641 q^{53} +17.6569 q^{55} +11.0866 q^{56} -4.46088 q^{58} -4.48528 q^{59} +12.4860 q^{61} +1.53073 q^{62} +7.00000 q^{64} +14.3337 q^{65} -4.48528 q^{67} +12.4853 q^{70} -0.634051 q^{71} -9.23880 q^{73} -2.48181 q^{74} -2.82843 q^{76} +19.3137 q^{77} +8.02509 q^{79} +3.37849 q^{80} -4.46088 q^{82} -12.4853 q^{83} -5.65685 q^{86} +15.6788 q^{88} +10.5858 q^{89} +15.6788 q^{91} -0.634051 q^{92} -2.82843 q^{94} -9.55582 q^{95} -9.68714 q^{97} +6.65685 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 - 4 * q^4 - 12 * q^8	\( 4 q + 4 q^{2} - 4 q^{4} - 12 q^{8} - 4 q^{16} + 20 q^{25} + 20 q^{32} + 16 q^{35} + 4 q^{49} + 20 q^{50} + 16 q^{53} + 48 q^{55} + 16 q^{59} + 28 q^{64} + 16 q^{67} + 16 q^{70} + 32 q^{77} - 16 q^{83} + 48 q^{89} + 4 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 4 * q^4 - 12 * q^8 - 4 * q^16 + 20 * q^25 + 20 * q^32 + 16 * q^35 + 4 * q^49 + 20 * q^50 + 16 * q^53 + 48 * q^55 + 16 * q^59 + 28 * q^64 + 16 * q^67 + 16 * q^70 + 32 * q^77 - 16 * q^83 + 48 * q^89 + 4 * q^98

Introduction

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