LMFDB - Embedded newform 6292.2.a.r.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6292,2,Mod(1,6292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6292, 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("6292.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6292 = 2^{2} \cdot 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6292.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:	$50.2418729518$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.725.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 3x^{2} + x + 1 $ x^4 - x^3 - 3*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.737640$ of defining polynomial
Character	$\chi$	$=$	6292.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.61803 q^{3} -1.47528 q^{5} -3.00509 q^{7} -0.381966 q^{9} +O(q^{10})$	$q+1.61803 q^{3} -1.47528 q^{5} -3.00509 q^{7} -0.381966 q^{9} -1.00000 q^{13} -2.38705 q^{15} +5.03564 q^{17} -1.85725 q^{19} -4.86233 q^{21} -7.39214 q^{23} -2.82355 q^{25} -5.47214 q^{27} +8.14784 q^{29} +8.33447 q^{31} +4.43335 q^{35} +4.47214 q^{37} -1.61803 q^{39} -6.71958 q^{41} -1.32115 q^{43} +0.563507 q^{45} -3.03564 q^{47} +2.03055 q^{49} +8.14784 q^{51} +0.683938 q^{53} -3.00509 q^{57} +1.06590 q^{59} +12.8369 q^{61} +1.14784 q^{63} +1.47528 q^{65} -9.22227 q^{67} -11.9607 q^{69} -0.911774 q^{71} +14.4384 q^{73} -4.56860 q^{75} +2.13767 q^{79} -7.70820 q^{81} +13.0764 q^{83} -7.42899 q^{85} +13.1835 q^{87} +9.80975 q^{89} +3.00509 q^{91} +13.4855 q^{93} +2.73996 q^{95} +7.97139 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 2 q^{5} + 6 q^{7} - 6 q^{9}+O(q^{10}) $ 4 * q + 2 * q^3 - 2 * q^5 + 6 * q^7 - 6 * q^9	$ 4 q + 2 q^{3} - 2 q^{5} + 6 q^{7} - 6 q^{9} - 4 q^{13} + 4 q^{15} + 8 q^{17} - 8 q^{19} - 2 q^{21} + 2 q^{23} + 8 q^{25} - 4 q^{27} + 14 q^{29} - 2 q^{31} - 2 q^{35} - 2 q^{39} - 10 q^{41} + 8 q^{43} + 8 q^{45} + 14 q^{49} + 14 q^{51} - 2 q^{53} + 6 q^{57} + 4 q^{59} + 10 q^{61} - 14 q^{63} + 2 q^{65} - 8 q^{67} - 4 q^{69} + 6 q^{71} + 20 q^{73} - 6 q^{75} + 26 q^{79} - 4 q^{81} + 10 q^{83} - 32 q^{85} + 22 q^{87} - 6 q^{91} + 14 q^{93} + 36 q^{95} - 6 q^{97}+O(q^{100}) $ 4 * q + 2 * q^3 - 2 * q^5 + 6 * q^7 - 6 * q^9 - 4 * q^13 + 4 * q^15 + 8 * q^17 - 8 * q^19 - 2 * q^21 + 2 * q^23 + 8 * q^25 - 4 * q^27 + 14 * q^29 - 2 * q^31 - 2 * q^35 - 2 * q^39 - 10 * q^41 + 8 * q^43 + 8 * q^45 + 14 * q^49 + 14 * q^51 - 2 * q^53 + 6 * q^57 + 4 * q^59 + 10 * q^61 - 14 * q^63 + 2 * q^65 - 8 * q^67 - 4 * q^69 + 6 * q^71 + 20 * q^73 - 6 * q^75 + 26 * q^79 - 4 * q^81 + 10 * q^83 - 32 * q^85 + 22 * q^87 - 6 * q^91 + 14 * q^93 + 36 * q^95 - 6 * q^97

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 $

$2$	0	0
$2$	0	0
$3$	1.61803	0.934172	0.467086	−	0.884212i	$-0.345304\pi$
$3$	1.61803	0.934172	0.467086	+	0.884212i	$0.345304\pi$
$4$	0	0
$5$	−1.47528	−0.659766	−0.329883	−	0.944022i	$-0.607009\pi$
$5$	−1.47528	−0.659766	−0.329883	+	0.944022i	$0.607009\pi$
$6$	0	0
$7$	−3.00509	−1.13582	−0.567908	−	0.823092i	$-0.692247\pi$
$7$	−3.00509	−1.13582	−0.567908	+	0.823092i	$0.692247\pi$
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−2.38705	−0.616335
$16$	0	0
$17$	5.03564	1.22132	0.610661	−	0.791892i	$-0.290904\pi$
$17$	5.03564	1.22132	0.610661	+	0.791892i	$0.290904\pi$
$18$	0	0
$19$	−1.85725	−0.426082	−0.213041	−	0.977043i	$-0.568337\pi$
$19$	−1.85725	−0.426082	−0.213041	+	0.977043i	$0.568337\pi$
$20$	0	0
$21$	−4.86233	−1.06105
$22$	0	0
$23$	−7.39214	−1.54137	−0.770684	−	0.637217i	$-0.780085\pi$
$23$	−7.39214	−1.54137	−0.770684	+	0.637217i	$0.780085\pi$
$24$	0	0
$25$	−2.82355	−0.564709
$26$	0	0
$27$	−5.47214	−1.05311
$28$	0	0
$29$	8.14784	1.51302	0.756508	−	0.653984i	$-0.226904\pi$
$29$	8.14784	1.51302	0.756508	+	0.653984i	$0.226904\pi$
$30$	0	0
$31$	8.33447	1.49692	0.748458	−	0.663183i	$-0.230795\pi$
$31$	8.33447	1.49692	0.748458	+	0.663183i	$0.230795\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.43335	0.749373
$36$	0	0
$37$	4.47214	0.735215	0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214	0.735215	0.367607	+	0.929981i	$0.380177\pi$
$38$	0	0
$39$	−1.61803	−0.259093
$40$	0	0
$41$	−6.71958	−1.04942	−0.524711	−	0.851280i	$-0.675827\pi$
$41$	−6.71958	−1.04942	−0.524711	+	0.851280i	$0.675827\pi$
$42$	0	0
$43$	−1.32115	−0.201473	−0.100737	−	0.994913i	$-0.532120\pi$
$43$	−1.32115	−0.201473	−0.100737	+	0.994913i	$0.532120\pi$
$44$	0	0
$45$	0.563507	0.0840027
$46$	0	0
$47$	−3.03564	−0.442794	−0.221397	−	0.975184i	$-0.571062\pi$
$47$	−3.03564	−0.442794	−0.221397	+	0.975184i	$0.571062\pi$
$48$	0	0
$49$	2.03055	0.290079
$50$	0	0
$51$	8.14784	1.14093
$52$	0	0
$53$	0.683938	0.0939462	0.0469731	−	0.998896i	$-0.485042\pi$
$53$	0.683938	0.0939462	0.0469731	+	0.998896i	$0.485042\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.00509	−0.398034
$58$	0	0
$59$	1.06590	0.138769	0.0693845	−	0.997590i	$-0.477896\pi$
$59$	1.06590	0.138769	0.0693845	+	0.997590i	$0.477896\pi$
$60$	0	0
$61$	12.8369	1.64359	0.821796	−	0.569781i	$-0.192972\pi$
$61$	12.8369	1.64359	0.821796	+	0.569781i	$0.192972\pi$
$62$	0	0
$63$	1.14784	0.144614
$64$	0	0
$65$	1.47528	0.182986
$66$	0	0
$67$	−9.22227	−1.12668	−0.563340	−	0.826225i	$-0.690484\pi$
$67$	−9.22227	−1.12668	−0.563340	+	0.826225i	$0.690484\pi$
$68$	0	0
$69$	−11.9607	−1.43990
$70$	0	0
$71$	−0.911774	−0.108208	−0.0541038	−	0.998535i	$-0.517230\pi$
$71$	−0.911774	−0.108208	−0.0541038	+	0.998535i	$0.517230\pi$
$72$	0	0
$73$	14.4384	1.68989	0.844946	−	0.534852i	$-0.179633\pi$
$73$	14.4384	1.68989	0.844946	+	0.534852i	$0.179633\pi$
$74$	0	0
$75$	−4.56860	−0.527536
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.13767	0.240506	0.120253	−	0.992743i	$-0.461629\pi$
$79$	2.13767	0.240506	0.120253	+	0.992743i	$0.461629\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	13.0764	1.43532	0.717659	−	0.696394i	$-0.245214\pi$
$83$	13.0764	1.43532	0.717659	+	0.696394i	$0.245214\pi$
$84$	0	0
$85$	−7.42899	−0.805787
$86$	0	0
$87$	13.1835	1.41342
$88$	0	0
$89$	9.80975	1.03983	0.519916	−	0.854218i	$-0.325963\pi$
$89$	9.80975	1.03983	0.519916	+	0.854218i	$0.325963\pi$
$90$	0	0
$91$	3.00509	0.315019
$92$	0	0
$93$	13.4855	1.39838
$94$	0	0
$95$	2.73996	0.281114
$96$	0	0
$97$	7.97139	0.809372	0.404686	−	0.914456i	$-0.367381\pi$
$97$	7.97139	0.809372	0.404686	+	0.914456i	$0.367381\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.8725	1.77838	0.889191	−	0.457537i	$-0.151268\pi$
$101$	17.8725	1.77838	0.889191	+	0.457537i	$0.151268\pi$
$102$	0	0
$103$	−7.97962	−0.786255	−0.393128	−	0.919484i	$-0.628607\pi$
$103$	−7.97962	−0.786255	−0.393128	+	0.919484i	$0.628607\pi$
$104$	0	0
$105$	7.17331	0.700043
$106$	0	0
$107$	1.57683	0.152438	0.0762189	−	0.997091i	$-0.475715\pi$
$107$	1.57683	0.152438	0.0762189	+	0.997091i	$0.475715\pi$
$108$	0	0
$109$	−8.21404	−0.786762	−0.393381	−	0.919375i	$-0.628695\pi$
$109$	−8.21404	−0.786762	−0.393381	+	0.919375i	$0.628695\pi$
$110$	0	0
$111$	7.23607	0.686817
$112$	0	0
$113$	2.38197	0.224077	0.112038	−	0.993704i	$-0.464262\pi$
$113$	2.38197	0.224077	0.112038	+	0.993704i	$0.464262\pi$
$114$	0	0
$115$	10.9055	1.01694
$116$	0	0
$117$	0.381966	0.0353128
$118$	0	0
$119$	−15.1326	−1.38720
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−10.8725	−0.980341
$124$	0	0
$125$	11.5419	1.03234
$126$	0	0
$127$	14.5046	1.28708	0.643539	−	0.765413i	$-0.277465\pi$
$127$	14.5046	1.28708	0.643539	+	0.765413i	$0.277465\pi$
$128$	0	0
$129$	−2.13767	−0.188211
$130$	0	0
$131$	0.816515	0.0713393	0.0356696	−	0.999364i	$-0.488644\pi$
$131$	0.816515	0.0713393	0.0356696	+	0.999364i	$0.488644\pi$
$132$	0	0
$133$	5.58119	0.483951
$134$	0	0
$135$	8.07294	0.694808
$136$	0	0
$137$	−21.1937	−1.81070	−0.905348	−	0.424670i	$-0.860390\pi$
$137$	−21.1937	−1.81070	−0.905348	+	0.424670i	$0.860390\pi$
$138$	0	0
$139$	−12.6226	−1.07064	−0.535319	−	0.844650i	$-0.679809\pi$
$139$	−12.6226	−1.07064	−0.535319	+	0.844650i	$0.679809\pi$
$140$	0	0
$141$	−4.91177	−0.413646
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−12.0204	−0.998236
$146$	0	0
$147$	3.28551	0.270984
$148$	0	0
$149$	−15.3721	−1.25933	−0.629664	−	0.776868i	$-0.716807\pi$
$149$	−15.3721	−1.25933	−0.629664	+	0.776868i	$0.716807\pi$
$150$	0	0
$151$	−16.2957	−1.32612	−0.663062	−	0.748565i	$-0.730743\pi$
$151$	−16.2957	−1.32612	−0.663062	+	0.748565i	$0.730743\pi$
$152$	0	0
$153$	−1.92344	−0.155501
$154$	0	0
$155$	−12.2957	−0.987613
$156$	0	0
$157$	23.5889	1.88260	0.941301	−	0.337567i	$-0.109604\pi$
$157$	23.5889	1.88260	0.941301	+	0.337567i	$0.109604\pi$
$158$	0	0
$159$	1.10664	0.0877619
$160$	0	0
$161$	22.2140	1.75071
$162$	0	0
$163$	3.85216	0.301724	0.150862	−	0.988555i	$-0.451795\pi$
$163$	3.85216	0.301724	0.150862	+	0.988555i	$0.451795\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.8674	1.69215	0.846076	−	0.533062i	$-0.178959\pi$
$167$	21.8674	1.69215	0.846076	+	0.533062i	$0.178959\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.709405	0.0542496
$172$	0	0
$173$	11.0458	0.839798	0.419899	−	0.907571i	$-0.362065\pi$
$173$	11.0458	0.839798	0.419899	+	0.907571i	$0.362065\pi$
$174$	0	0
$175$	8.48501	0.641406
$176$	0	0
$177$	1.72467	0.129634
$178$	0	0
$179$	9.02185	0.674325	0.337162	−	0.941447i	$-0.390533\pi$
$179$	9.02185	0.674325	0.337162	+	0.941447i	$0.390533\pi$
$180$	0	0
$181$	10.8807	0.808759	0.404380	−	0.914591i	$-0.367487\pi$
$181$	10.8807	0.808759	0.404380	+	0.914591i	$0.367487\pi$
$182$	0	0
$183$	20.7705	1.53540
$184$	0	0
$185$	−6.59766	−0.485069
$186$	0	0
$187$	0	0
$188$	0	0
$189$	16.4443	1.19614
$190$	0	0
$191$	−4.68394	−0.338918	−0.169459	−	0.985537i	$-0.554202\pi$
$191$	−4.68394	−0.338918	−0.169459	+	0.985537i	$0.554202\pi$
$192$	0	0
$193$	−6.50536	−0.468266	−0.234133	−	0.972205i	$-0.575225\pi$
$193$	−6.50536	−0.468266	−0.234133	+	0.972205i	$0.575225\pi$
$194$	0	0
$195$	2.38705	0.170940
$196$	0	0
$197$	−10.1631	−0.724091	−0.362046	−	0.932160i	$-0.617922\pi$
$197$	−10.1631	−0.724091	−0.362046	+	0.932160i	$0.617922\pi$
$198$	0	0
$199$	22.4799	1.59356	0.796779	−	0.604271i	$-0.206536\pi$
$199$	22.4799	1.59356	0.796779	+	0.604271i	$0.206536\pi$
$200$	0	0
$201$	−14.9219	−1.05251
$202$	0	0
$203$	−24.4850	−1.71851
$204$	0	0
$205$	9.91327	0.692373
$206$	0	0
$207$	2.82355	0.196250
$208$	0	0
$209$	0	0
$210$	0	0
$211$	14.1580	0.974678	0.487339	−	0.873213i	$-0.337968\pi$
$211$	14.1580	0.974678	0.487339	+	0.873213i	$0.337968\pi$
$212$	0	0
$213$	−1.47528	−0.101085
$214$	0	0
$215$	1.94907	0.132925
$216$	0	0
$217$	−25.0458	−1.70022
$218$	0	0
$219$	23.3619	1.57865
$220$	0	0
$221$	−5.03564	−0.338734
$222$	0	0
$223$	−7.59062	−0.508306	−0.254153	−	0.967164i	$-0.581797\pi$
$223$	−7.59062	−0.508306	−0.254153	+	0.967164i	$0.581797\pi$
$224$	0	0
$225$	1.07850	0.0718999
$226$	0	0
$227$	23.8165	1.58076	0.790378	−	0.612620i	$-0.209884\pi$
$227$	23.8165	1.58076	0.790378	+	0.612620i	$0.209884\pi$
$228$	0	0
$229$	1.58268	0.104587	0.0522934	−	0.998632i	$-0.483347\pi$
$229$	1.58268	0.104587	0.0522934	+	0.998632i	$0.483347\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.42899	0.486689	0.243345	−	0.969940i	$-0.421755\pi$
$233$	7.42899	0.486689	0.243345	+	0.969940i	$0.421755\pi$
$234$	0	0
$235$	4.47843	0.292140
$236$	0	0
$237$	3.45881	0.224674
$238$	0	0
$239$	−11.7961	−0.763029	−0.381514	−	0.924363i	$-0.624597\pi$
$239$	−11.7961	−0.763029	−0.381514	+	0.924363i	$0.624597\pi$
$240$	0	0
$241$	9.36188	0.603052	0.301526	−	0.953458i	$-0.402504\pi$
$241$	9.36188	0.603052	0.301526	+	0.953458i	$0.402504\pi$
$242$	0	0
$243$	3.94427	0.253025
$244$	0	0
$245$	−2.99564	−0.191384
$246$	0	0
$247$	1.85725	0.118174
$248$	0	0
$249$	21.1580	1.34083
$250$	0	0
$251$	−8.81484	−0.556388	−0.278194	−	0.960525i	$-0.589736\pi$
$251$	−8.81484	−0.556388	−0.278194	+	0.960525i	$0.589736\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−12.0204	−0.752744
$256$	0	0
$257$	11.4799	0.716096	0.358048	−	0.933703i	$-0.383442\pi$
$257$	11.4799	0.716096	0.358048	+	0.933703i	$0.383442\pi$
$258$	0	0
$259$	−13.4392	−0.835069
$260$	0	0
$261$	−3.11220	−0.192640
$262$	0	0
$263$	18.6890	1.15241	0.576207	−	0.817304i	$-0.304532\pi$
$263$	18.6890	1.15241	0.576207	+	0.817304i	$0.304532\pi$
$264$	0	0
$265$	−1.00900	−0.0619824
$266$	0	0
$267$	15.8725	0.971382
$268$	0	0
$269$	16.0509	0.978641	0.489321	−	0.872104i	$-0.337245\pi$
$269$	16.0509	0.978641	0.489321	+	0.872104i	$0.337245\pi$
$270$	0	0
$271$	−4.49955	−0.273328	−0.136664	−	0.990617i	$-0.543638\pi$
$271$	−4.49955	−0.273328	−0.136664	+	0.990617i	$0.543638\pi$
$272$	0	0
$273$	4.86233	0.294282
$274$	0	0
$275$	0	0
$276$	0	0
$277$	25.2038	1.51435	0.757176	−	0.653211i	$-0.226579\pi$
$277$	25.2038	1.51435	0.757176	+	0.653211i	$0.226579\pi$
$278$	0	0
$279$	−3.18348	−0.190590
$280$	0	0
$281$	−13.8776	−0.827868	−0.413934	−	0.910307i	$-0.635846\pi$
$281$	−13.8776	−0.827868	−0.413934	+	0.910307i	$0.635846\pi$
$282$	0	0
$283$	11.8061	0.701802	0.350901	−	0.936413i	$-0.385875\pi$
$283$	11.8061	0.701802	0.350901	+	0.936413i	$0.385875\pi$
$284$	0	0
$285$	4.43335	0.262609
$286$	0	0
$287$	20.1929	1.19195
$288$	0	0
$289$	8.35770	0.491629
$290$	0	0
$291$	12.8980	0.756093
$292$	0	0
$293$	−8.09164	−0.472719	−0.236359	−	0.971666i	$-0.575954\pi$
$293$	−8.09164	−0.472719	−0.236359	+	0.971666i	$0.575954\pi$
$294$	0	0
$295$	−1.57251	−0.0915550
$296$	0	0
$297$	0	0
$298$	0	0
$299$	7.39214	0.427499
$300$	0	0
$301$	3.97017	0.228837
$302$	0	0
$303$	28.9183	1.66131
$304$	0	0
$305$	−18.9380	−1.08439
$306$	0	0
$307$	23.7247	1.35404	0.677019	−	0.735965i	$-0.263271\pi$
$307$	23.7247	1.35404	0.677019	+	0.735965i	$0.263271\pi$
$308$	0	0
$309$	−12.9113	−0.734498
$310$	0	0
$311$	−25.4077	−1.44074	−0.720369	−	0.693591i	$-0.756028\pi$
$311$	−25.4077	−1.44074	−0.720369	+	0.693591i	$0.756028\pi$
$312$	0	0
$313$	5.05240	0.285579	0.142789	−	0.989753i	$-0.454393\pi$
$313$	5.05240	0.285579	0.142789	+	0.989753i	$0.454393\pi$
$314$	0	0
$315$	−1.69339	−0.0954116
$316$	0	0
$317$	0.0867318	0.00487135	0.00243567	−	0.999997i	$-0.499225\pi$
$317$	0.0867318	0.00487135	0.00243567	+	0.999997i	$0.499225\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	2.55136	0.142403
$322$	0	0
$323$	−9.35243	−0.520383
$324$	0	0
$325$	2.82355	0.156622
$326$	0	0
$327$	−13.2906	−0.734972
$328$	0	0
$329$	9.12237	0.502933
$330$	0	0
$331$	−15.4404	−0.848680	−0.424340	−	0.905503i	$-0.639494\pi$
$331$	−15.4404	−0.848680	−0.424340	+	0.905503i	$0.639494\pi$
$332$	0	0
$333$	−1.70820	−0.0936090
$334$	0	0
$335$	13.6054	0.743345
$336$	0	0
$337$	5.85216	0.318787	0.159394	−	0.987215i	$-0.449046\pi$
$337$	5.85216	0.318787	0.159394	+	0.987215i	$0.449046\pi$
$338$	0	0
$339$	3.85410	0.209326
$340$	0	0
$341$	0	0
$342$	0	0
$343$	14.9336	0.806340
$344$	0	0
$345$	17.6454	0.949999
$346$	0	0
$347$	−13.0255	−0.699244	−0.349622	−	0.936891i	$-0.613690\pi$
$347$	−13.0255	−0.699244	−0.349622	+	0.936891i	$0.613690\pi$
$348$	0	0
$349$	−13.6372	−0.729983	−0.364992	−	0.931011i	$-0.618928\pi$
$349$	−13.6372	−0.729983	−0.364992	+	0.931011i	$0.618928\pi$
$350$	0	0
$351$	5.47214	0.292081
$352$	0	0
$353$	−3.55451	−0.189187	−0.0945936	−	0.995516i	$-0.530155\pi$
$353$	−3.55451	−0.189187	−0.0945936	+	0.995516i	$0.530155\pi$
$354$	0	0
$355$	1.34512	0.0713917
$356$	0	0
$357$	−24.4850	−1.29588
$358$	0	0
$359$	−5.86161	−0.309364	−0.154682	−	0.987964i	$-0.549435\pi$
$359$	−5.86161	−0.309364	−0.154682	+	0.987964i	$0.549435\pi$
$360$	0	0
$361$	−15.5506	−0.818454
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−21.3007	−1.11493
$366$	0	0
$367$	19.8697	1.03719	0.518594	−	0.855021i	$-0.326456\pi$
$367$	19.8697	1.03719	0.518594	+	0.855021i	$0.326456\pi$
$368$	0	0
$369$	2.56665	0.133615
$370$	0	0
$371$	−2.05530	−0.106706
$372$	0	0
$373$	24.4850	1.26778	0.633892	−	0.773421i	$-0.281456\pi$
$373$	24.4850	1.26778	0.633892	+	0.773421i	$0.281456\pi$
$374$	0	0
$375$	18.6752	0.964385
$376$	0	0
$377$	−8.14784	−0.419635
$378$	0	0
$379$	14.3897	0.739150	0.369575	−	0.929201i	$-0.379503\pi$
$379$	14.3897	0.739150	0.369575	+	0.929201i	$0.379503\pi$
$380$	0	0
$381$	23.4690	1.20235
$382$	0	0
$383$	3.67406	0.187735	0.0938677	−	0.995585i	$-0.470077\pi$
$383$	3.67406	0.187735	0.0938677	+	0.995585i	$0.470077\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.504634	0.0256520
$388$	0	0
$389$	−10.6297	−0.538947	−0.269474	−	0.963008i	$-0.586850\pi$
$389$	−10.6297	−0.538947	−0.269474	+	0.963008i	$0.586850\pi$
$390$	0	0
$391$	−37.2242	−1.88251
$392$	0	0
$393$	1.32115	0.0666432
$394$	0	0
$395$	−3.15366	−0.158678
$396$	0	0
$397$	−15.0560	−0.755639	−0.377819	−	0.925879i	$-0.623326\pi$
$397$	−15.0560	−0.755639	−0.377819	+	0.925879i	$0.623326\pi$
$398$	0	0
$399$	9.03055	0.452093
$400$	0	0
$401$	−5.37278	−0.268304	−0.134152	−	0.990961i	$-0.542831\pi$
$401$	−5.37278	−0.268304	−0.134152	+	0.990961i	$0.542831\pi$
$402$	0	0
$403$	−8.33447	−0.415170
$404$	0	0
$405$	11.3718	0.565067
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−24.3873	−1.20588	−0.602938	−	0.797788i	$-0.706003\pi$
$409$	−24.3873	−1.20588	−0.602938	+	0.797788i	$0.706003\pi$
$410$	0	0
$411$	−34.2921	−1.69150
$412$	0	0
$413$	−3.20314	−0.157616
$414$	0	0
$415$	−19.2913	−0.946974
$416$	0	0
$417$	−20.4239	−1.00016
$418$	0	0
$419$	−24.0611	−1.17546	−0.587730	−	0.809057i	$-0.699978\pi$
$419$	−24.0611	−1.17546	−0.587730	+	0.809057i	$0.699978\pi$
$420$	0	0
$421$	−24.9847	−1.21768	−0.608840	−	0.793293i	$-0.708365\pi$
$421$	−24.9847	−1.21768	−0.608840	+	0.793293i	$0.708365\pi$
$422$	0	0
$423$	1.15951	0.0563774
$424$	0	0
$425$	−14.2184	−0.689693
$426$	0	0
$427$	−38.5759	−1.86682
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.27024	0.0611854	0.0305927	−	0.999532i	$-0.490261\pi$
$431$	1.27024	0.0611854	0.0305927	+	0.999532i	$0.490261\pi$
$432$	0	0
$433$	−35.8076	−1.72080	−0.860402	−	0.509616i	$-0.829787\pi$
$433$	−35.8076	−1.72080	−0.860402	+	0.509616i	$0.829787\pi$
$434$	0	0
$435$	−19.4493	−0.932524
$436$	0	0
$437$	13.7290	0.656749
$438$	0	0
$439$	33.2351	1.58623	0.793113	−	0.609075i	$-0.208459\pi$
$439$	33.2351	1.58623	0.793113	+	0.609075i	$0.208459\pi$
$440$	0	0
$441$	−0.775603	−0.0369335
$442$	0	0
$443$	8.48234	0.403008	0.201504	−	0.979488i	$-0.435417\pi$
$443$	8.48234	0.403008	0.201504	+	0.979488i	$0.435417\pi$
$444$	0	0
$445$	−14.4721	−0.686045
$446$	0	0
$447$	−24.8725	−1.17643
$448$	0	0
$449$	21.1973	1.00036	0.500181	−	0.865921i	$-0.333267\pi$
$449$	21.1973	1.00036	0.500181	+	0.865921i	$0.333267\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−26.3670	−1.23883
$454$	0	0
$455$	−4.43335	−0.207839
$456$	0	0
$457$	−7.65320	−0.358002	−0.179001	−	0.983849i	$-0.557286\pi$
$457$	−7.65320	−0.358002	−0.179001	+	0.983849i	$0.557286\pi$
$458$	0	0
$459$	−27.5557	−1.28619
$460$	0	0
$461$	−5.01090	−0.233381	−0.116691	−	0.993168i	$-0.537229\pi$
$461$	−5.01090	−0.233381	−0.116691	+	0.993168i	$0.537229\pi$
$462$	0	0
$463$	−30.8020	−1.43149	−0.715744	−	0.698362i	$-0.753912\pi$
$463$	−30.8020	−1.43149	−0.715744	+	0.698362i	$0.753912\pi$
$464$	0	0
$465$	−19.8948	−0.922601
$466$	0	0
$467$	−21.7184	−1.00501	−0.502503	−	0.864575i	$-0.667588\pi$
$467$	−21.7184	−1.00501	−0.502503	+	0.864575i	$0.667588\pi$
$468$	0	0
$469$	27.7137	1.27970
$470$	0	0
$471$	38.1677	1.75868
$472$	0	0
$473$	0	0
$474$	0	0
$475$	5.24402	0.240612
$476$	0	0
$477$	−0.261241	−0.0119614
$478$	0	0
$479$	−10.4997	−0.479745	−0.239872	−	0.970804i	$-0.577106\pi$
$479$	−10.4997	−0.479745	−0.239872	+	0.970804i	$0.577106\pi$
$480$	0	0
$481$	−4.47214	−0.203912
$482$	0	0
$483$	35.9431	1.63547
$484$	0	0
$485$	−11.7600	−0.533996
$486$	0	0
$487$	9.59451	0.434769	0.217384	−	0.976086i	$-0.430247\pi$
$487$	9.59451	0.434769	0.217384	+	0.976086i	$0.430247\pi$
$488$	0	0
$489$	6.23292	0.281863
$490$	0	0
$491$	−7.93362	−0.358039	−0.179020	−	0.983845i	$-0.557293\pi$
$491$	−7.93362	−0.358039	−0.179020	+	0.983845i	$0.557293\pi$
$492$	0	0
$493$	41.0296	1.84788
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.73996	0.122904
$498$	0	0
$499$	−14.8369	−0.664189	−0.332095	−	0.943246i	$-0.607755\pi$
$499$	−14.8369	−0.664189	−0.332095	+	0.943246i	$0.607755\pi$
$500$	0	0
$501$	35.3822	1.58076
$502$	0	0
$503$	23.2804	1.03802	0.519011	−	0.854768i	$-0.326300\pi$
$503$	23.2804	1.03802	0.519011	+	0.854768i	$0.326300\pi$
$504$	0	0
$505$	−26.3670	−1.17331
$506$	0	0
$507$	1.61803	0.0718594
$508$	0	0
$509$	−5.38124	−0.238519	−0.119260	−	0.992863i	$-0.538052\pi$
$509$	−5.38124	−0.238519	−0.119260	+	0.992863i	$0.538052\pi$
$510$	0	0
$511$	−43.3888	−1.91941
$512$	0	0
$513$	10.1631	0.448712
$514$	0	0
$515$	11.7722	0.518744
$516$	0	0
$517$	0	0
$518$	0	0
$519$	17.8725	0.784516
$520$	0	0
$521$	28.7606	1.26003	0.630013	−	0.776585i	$-0.283050\pi$
$521$	28.7606	1.26003	0.630013	+	0.776585i	$0.283050\pi$
$522$	0	0
$523$	−14.2490	−0.623063	−0.311532	−	0.950236i	$-0.600842\pi$
$523$	−14.2490	−0.623063	−0.311532	+	0.950236i	$0.600842\pi$
$524$	0	0
$525$	13.7290	0.599184
$526$	0	0
$527$	41.9694	1.82822
$528$	0	0
$529$	31.6438	1.37582
$530$	0	0
$531$	−0.407139	−0.0176683
$532$	0	0
$533$	6.71958	0.291057
$534$	0	0
$535$	−2.32626	−0.100573
$536$	0	0
$537$	14.5977	0.629935
$538$	0	0
$539$	0	0
$540$	0	0
$541$	17.0051	0.731106	0.365553	−	0.930791i	$-0.380880\pi$
$541$	17.0051	0.731106	0.365553	+	0.930791i	$0.380880\pi$
$542$	0	0
$543$	17.6054	0.755521
$544$	0	0
$545$	12.1180	0.519079
$546$	0	0
$547$	−38.4850	−1.64550	−0.822750	−	0.568404i	$-0.807561\pi$
$547$	−38.4850	−1.64550	−0.822750	+	0.568404i	$0.807561\pi$
$548$	0	0
$549$	−4.90325	−0.209265
$550$	0	0
$551$	−15.1326	−0.644668
$552$	0	0
$553$	−6.42387	−0.273171
$554$	0	0
$555$	−10.6752	−0.453138
$556$	0	0
$557$	41.9532	1.77762	0.888808	−	0.458280i	$-0.151534\pi$
$557$	41.9532	1.77762	0.888808	+	0.458280i	$0.151534\pi$
$558$	0	0
$559$	1.32115	0.0558787
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.42899	−0.313094	−0.156547	−	0.987670i	$-0.550036\pi$
$563$	−7.42899	−0.313094	−0.156547	+	0.987670i	$0.550036\pi$
$564$	0	0
$565$	−3.51407	−0.147838
$566$	0	0
$567$	23.1638	0.972790
$568$	0	0
$569$	−19.9540	−0.836514	−0.418257	−	0.908329i	$-0.637359\pi$
$569$	−19.9540	−0.836514	−0.418257	+	0.908329i	$0.637359\pi$
$570$	0	0
$571$	4.87763	0.204122	0.102061	−	0.994778i	$-0.467456\pi$
$571$	4.87763	0.204122	0.102061	+	0.994778i	$0.467456\pi$
$572$	0	0
$573$	−7.57877	−0.316608
$574$	0	0
$575$	20.8721	0.870425
$576$	0	0
$577$	−35.0684	−1.45992	−0.729958	−	0.683491i	$-0.760461\pi$
$577$	−35.0684	−1.45992	−0.729958	+	0.683491i	$0.760461\pi$
$578$	0	0
$579$	−10.5259	−0.437441
$580$	0	0
$581$	−39.2957	−1.63026
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−0.563507	−0.0232981
$586$	0	0
$587$	25.8163	1.06555	0.532777	−	0.846256i	$-0.321149\pi$
$587$	25.8163	1.06555	0.532777	+	0.846256i	$0.321149\pi$
$588$	0	0
$589$	−15.4792	−0.637808
$590$	0	0
$591$	−16.4443	−0.676426
$592$	0	0
$593$	−21.5965	−0.886860	−0.443430	−	0.896309i	$-0.646239\pi$
$593$	−21.5965	−0.886860	−0.443430	+	0.896309i	$0.646239\pi$
$594$	0	0
$595$	22.3248	0.915226
$596$	0	0
$597$	36.3732	1.48866
$598$	0	0
$599$	47.3771	1.93578	0.967888	−	0.251381i	$-0.0808846\pi$
$599$	47.3771	1.93578	0.967888	+	0.251381i	$0.0808846\pi$
$600$	0	0
$601$	−14.5461	−0.593347	−0.296674	−	0.954979i	$-0.595877\pi$
$601$	−14.5461	−0.593347	−0.296674	+	0.954979i	$0.595877\pi$
$602$	0	0
$603$	3.52259	0.143451
$604$	0	0
$605$	0	0
$606$	0	0
$607$	37.4799	1.52126	0.760630	−	0.649185i	$-0.224890\pi$
$607$	37.4799	1.52126	0.760630	+	0.649185i	$0.224890\pi$
$608$	0	0
$609$	−39.6175	−1.60538
$610$	0	0
$611$	3.03564	0.122809
$612$	0	0
$613$	−16.5199	−0.667232	−0.333616	−	0.942709i	$-0.608269\pi$
$613$	−16.5199	−0.667232	−0.333616	+	0.942709i	$0.608269\pi$
$614$	0	0
$615$	16.0400	0.646795
$616$	0	0
$617$	7.45461	0.300111	0.150056	−	0.988678i	$-0.452055\pi$
$617$	7.45461	0.300111	0.150056	+	0.988678i	$0.452055\pi$
$618$	0	0
$619$	−47.3018	−1.90122	−0.950609	−	0.310390i	$-0.899540\pi$
$619$	−47.3018	−1.90122	−0.950609	+	0.310390i	$0.899540\pi$
$620$	0	0
$621$	40.4508	1.62324
$622$	0	0
$623$	−29.4792	−1.18106
$624$	0	0
$625$	−2.90985	−0.116394
$626$	0	0
$627$	0	0
$628$	0	0
$629$	22.5201	0.897934
$630$	0	0
$631$	17.8926	0.712293	0.356147	−	0.934430i	$-0.384090\pi$
$631$	17.8926	0.712293	0.356147	+	0.934430i	$0.384090\pi$
$632$	0	0
$633$	22.9082	0.910517
$634$	0	0
$635$	−21.3984	−0.849170
$636$	0	0
$637$	−2.03055	−0.0804535
$638$	0	0
$639$	0.348267	0.0137772
$640$	0	0
$641$	41.2744	1.63024	0.815120	−	0.579292i	$-0.196671\pi$
$641$	41.2744	1.63024	0.815120	+	0.579292i	$0.196671\pi$
$642$	0	0
$643$	20.0911	0.792314	0.396157	−	0.918183i	$-0.370344\pi$
$643$	20.0911	0.792314	0.396157	+	0.918183i	$0.370344\pi$
$644$	0	0
$645$	3.15366	0.124175
$646$	0	0
$647$	−6.94280	−0.272950	−0.136475	−	0.990644i	$-0.543577\pi$
$647$	−6.94280	−0.272950	−0.136475	+	0.990644i	$0.543577\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−40.5250	−1.58830
$652$	0	0
$653$	41.8226	1.63664	0.818322	−	0.574760i	$-0.194904\pi$
$653$	41.8226	1.63664	0.818322	+	0.574760i	$0.194904\pi$
$654$	0	0
$655$	−1.20459	−0.0470672
$656$	0	0
$657$	−5.51499	−0.215160
$658$	0	0
$659$	−26.6226	−1.03707	−0.518536	−	0.855056i	$-0.673523\pi$
$659$	−26.6226	−1.03707	−0.518536	+	0.855056i	$0.673523\pi$
$660$	0	0
$661$	29.0001	1.12797	0.563987	−	0.825784i	$-0.309267\pi$
$661$	29.0001	1.12797	0.563987	+	0.825784i	$0.309267\pi$
$662$	0	0
$663$	−8.14784	−0.316436
$664$	0	0
$665$	−8.23382	−0.319294
$666$	0	0
$667$	−60.2300	−2.33212
$668$	0	0
$669$	−12.2819	−0.474845
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−22.2802	−0.858840	−0.429420	−	0.903105i	$-0.641282\pi$
$673$	−22.2802	−0.858840	−0.429420	+	0.903105i	$0.641282\pi$
$674$	0	0
$675$	15.4508	0.594703
$676$	0	0
$677$	−18.5914	−0.714524	−0.357262	−	0.934004i	$-0.616290\pi$
$677$	−18.5914	−0.714524	−0.357262	+	0.934004i	$0.616290\pi$
$678$	0	0
$679$	−23.9547	−0.919298
$680$	0	0
$681$	38.5359	1.47670
$682$	0	0
$683$	9.90906	0.379160	0.189580	−	0.981865i	$-0.439287\pi$
$683$	9.90906	0.379160	0.189580	+	0.981865i	$0.439287\pi$
$684$	0	0
$685$	31.2666	1.19464
$686$	0	0
$687$	2.56084	0.0977020
$688$	0	0
$689$	−0.683938	−0.0260560
$690$	0	0
$691$	13.1490	0.500212	0.250106	−	0.968218i	$-0.419535\pi$
$691$	13.1490	0.500212	0.250106	+	0.968218i	$0.419535\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	18.6219	0.706371
$696$	0	0
$697$	−33.8374	−1.28168
$698$	0	0
$699$	12.0204	0.454651
$700$	0	0
$701$	18.5310	0.699907	0.349953	−	0.936767i	$-0.386197\pi$
$701$	18.5310	0.699907	0.349953	+	0.936767i	$0.386197\pi$
$702$	0	0
$703$	−8.30586	−0.313261
$704$	0	0
$705$	7.24624	0.272909
$706$	0	0
$707$	−53.7085	−2.01991
$708$	0	0
$709$	9.60107	0.360576	0.180288	−	0.983614i	$-0.442297\pi$
$709$	9.60107	0.360576	0.180288	+	0.983614i	$0.442297\pi$
$710$	0	0
$711$	−0.816515	−0.0306217
$712$	0	0
$713$	−61.6096	−2.30730
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−19.0866	−0.712800
$718$	0	0
$719$	15.5695	0.580645	0.290323	−	0.956929i	$-0.406237\pi$
$719$	15.5695	0.580645	0.290323	+	0.956929i	$0.406237\pi$
$720$	0	0
$721$	23.9795	0.893042
$722$	0	0
$723$	15.1478	0.563354
$724$	0	0
$725$	−23.0058	−0.854415
$726$	0	0
$727$	−26.2007	−0.971731	−0.485865	−	0.874034i	$-0.661495\pi$
$727$	−26.2007	−0.971731	−0.485865	+	0.874034i	$0.661495\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	−6.65284	−0.246064
$732$	0	0
$733$	10.0875	0.372589	0.186294	−	0.982494i	$-0.440352\pi$
$733$	10.0875	0.372589	0.186294	+	0.982494i	$0.440352\pi$
$734$	0	0
$735$	−4.84704	−0.178786
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−40.9134	−1.50502	−0.752512	−	0.658579i	$-0.771158\pi$
$739$	−40.9134	−1.50502	−0.752512	+	0.658579i	$0.771158\pi$
$740$	0	0
$741$	3.00509	0.110395
$742$	0	0
$743$	−43.9185	−1.61121	−0.805606	−	0.592451i	$-0.798160\pi$
$743$	−43.9185	−1.61121	−0.805606	+	0.592451i	$0.798160\pi$
$744$	0	0
$745$	22.6781	0.830861
$746$	0	0
$747$	−4.99473	−0.182748
$748$	0	0
$749$	−4.73851	−0.173141
$750$	0	0
$751$	−35.0956	−1.28065	−0.640327	−	0.768102i	$-0.721201\pi$
$751$	−35.0956	−1.28065	−0.640327	+	0.768102i	$0.721201\pi$
$752$	0	0
$753$	−14.2627	−0.519762
$754$	0	0
$755$	24.0407	0.874931
$756$	0	0
$757$	8.75468	0.318194	0.159097	−	0.987263i	$-0.449142\pi$
$757$	8.75468	0.318194	0.159097	+	0.987263i	$0.449142\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	38.8825	1.40949	0.704745	−	0.709461i	$-0.251061\pi$
$761$	38.8825	1.40949	0.704745	+	0.709461i	$0.251061\pi$
$762$	0	0
$763$	24.6839	0.893618
$764$	0	0
$765$	2.83762	0.102594
$766$	0	0
$767$	−1.06590	−0.0384876
$768$	0	0
$769$	−46.0144	−1.65932	−0.829660	−	0.558269i	$-0.811466\pi$
$769$	−46.0144	−1.65932	−0.829660	+	0.558269i	$0.811466\pi$
$770$	0	0
$771$	18.5749	0.668957
$772$	0	0
$773$	−31.4535	−1.13131	−0.565653	−	0.824644i	$-0.691376\pi$
$773$	−31.4535	−1.13131	−0.565653	+	0.824644i	$0.691376\pi$
$774$	0	0
$775$	−23.5328	−0.845322
$776$	0	0
$777$	−21.7450	−0.780098
$778$	0	0
$779$	12.4799	0.447140
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−44.5861	−1.59338
$784$	0	0
$785$	−34.8003	−1.24208
$786$	0	0
$787$	−29.5863	−1.05464	−0.527318	−	0.849668i	$-0.676803\pi$
$787$	−29.5863	−1.05464	−0.527318	+	0.849668i	$0.676803\pi$
$788$	0	0
$789$	30.2395	1.07655
$790$	0	0
$791$	−7.15802	−0.254510
$792$	0	0
$793$	−12.8369	−0.455851
$794$	0	0
$795$	−1.63260	−0.0579023
$796$	0	0
$797$	−16.6306	−0.589086	−0.294543	−	0.955638i	$-0.595167\pi$
$797$	−16.6306	−0.589086	−0.294543	+	0.955638i	$0.595167\pi$
$798$	0	0
$799$	−15.2864	−0.540794
$800$	0	0
$801$	−3.74699	−0.132393
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−32.7719	−1.15506
$806$	0	0
$807$	25.9709	0.914219
$808$	0	0
$809$	53.3057	1.87413	0.937063	−	0.349159i	$-0.113533\pi$
$809$	53.3057	1.87413	0.937063	+	0.349159i	$0.113533\pi$
$810$	0	0
$811$	−18.5752	−0.652263	−0.326132	−	0.945324i	$-0.605745\pi$
$811$	−18.5752	−0.652263	−0.326132	+	0.945324i	$0.605745\pi$
$812$	0	0
$813$	−7.28042	−0.255335
$814$	0	0
$815$	−5.68301	−0.199067
$816$	0	0
$817$	2.45370	0.0858441
$818$	0	0
$819$	−1.14784	−0.0401088
$820$	0	0
$821$	26.7136	0.932311	0.466155	−	0.884703i	$-0.345639\pi$
$821$	26.7136	0.932311	0.466155	+	0.884703i	$0.345639\pi$
$822$	0	0
$823$	4.05186	0.141239	0.0706194	−	0.997503i	$-0.477502\pi$
$823$	4.05186	0.141239	0.0706194	+	0.997503i	$0.477502\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−17.5921	−0.611737	−0.305868	−	0.952074i	$-0.598947\pi$
$827$	−17.5921	−0.611737	−0.305868	+	0.952074i	$0.598947\pi$
$828$	0	0
$829$	−41.3514	−1.43619	−0.718096	−	0.695944i	$-0.754986\pi$
$829$	−41.3514	−1.43619	−0.718096	+	0.695944i	$0.754986\pi$
$830$	0	0
$831$	40.7807	1.41467
$832$	0	0
$833$	10.2251	0.354280
$834$	0	0
$835$	−32.2606	−1.11642
$836$	0	0
$837$	−45.6074	−1.57642
$838$	0	0
$839$	50.1905	1.73277	0.866385	−	0.499377i	$-0.166438\pi$
$839$	50.1905	1.73277	0.866385	+	0.499377i	$0.166438\pi$
$840$	0	0
$841$	37.3873	1.28922
$842$	0	0
$843$	−22.4544	−0.773372
$844$	0	0
$845$	−1.47528	−0.0507512
$846$	0	0
$847$	0	0
$848$	0	0
$849$	19.1027	0.655604
$850$	0	0
$851$	−33.0587	−1.13324
$852$	0	0
$853$	−3.37787	−0.115656	−0.0578281	−	0.998327i	$-0.518418\pi$
$853$	−3.37787	−0.115656	−0.0578281	+	0.998327i	$0.518418\pi$
$854$	0	0
$855$	−1.04657	−0.0357920
$856$	0	0
$857$	4.12819	0.141016	0.0705081	−	0.997511i	$-0.477538\pi$
$857$	4.12819	0.141016	0.0705081	+	0.997511i	$0.477538\pi$
$858$	0	0
$859$	44.8517	1.53032	0.765160	−	0.643840i	$-0.222660\pi$
$859$	44.8517	1.53032	0.765160	+	0.643840i	$0.222660\pi$
$860$	0	0
$861$	32.6729	1.11349
$862$	0	0
$863$	58.1987	1.98111	0.990554	−	0.137125i	$-0.0437861\pi$
$863$	58.1987	1.98111	0.990554	+	0.137125i	$0.0437861\pi$
$864$	0	0
$865$	−16.2957	−0.554070
$866$	0	0
$867$	13.5230	0.459267
$868$	0	0
$869$	0	0
$870$	0	0
$871$	9.22227	0.312485
$872$	0	0
$873$	−3.04480	−0.103051
$874$	0	0
$875$	−34.6845	−1.17255
$876$	0	0
$877$	40.9378	1.38237	0.691186	−	0.722677i	$-0.257089\pi$
$877$	40.9378	1.38237	0.691186	+	0.722677i	$0.257089\pi$
$878$	0	0
$879$	−13.0925	−0.441601
$880$	0	0
$881$	15.1435	0.510197	0.255099	−	0.966915i	$-0.417892\pi$
$881$	15.1435	0.510197	0.255099	+	0.966915i	$0.417892\pi$
$882$	0	0
$883$	8.12310	0.273364	0.136682	−	0.990615i	$-0.456356\pi$
$883$	8.12310	0.273364	0.136682	+	0.990615i	$0.456356\pi$
$884$	0	0
$885$	−2.54437	−0.0855281
$886$	0	0
$887$	19.1375	0.642573	0.321286	−	0.946982i	$-0.395885\pi$
$887$	19.1375	0.642573	0.321286	+	0.946982i	$0.395885\pi$
$888$	0	0
$889$	−43.5877	−1.46188
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.63794	0.188666
$894$	0	0
$895$	−13.3098	−0.444896
$896$	0	0
$897$	11.9607	0.399357
$898$	0	0
$899$	67.9079	2.26486
$900$	0	0
$901$	3.44407	0.114739
$902$	0	0
$903$	6.42387	0.213773
$904$	0	0
$905$	−16.0522	−0.533592
$906$	0	0
$907$	33.7552	1.12082	0.560412	−	0.828214i	$-0.310643\pi$
$907$	33.7552	1.12082	0.560412	+	0.828214i	$0.310643\pi$
$908$	0	0
$909$	−6.82669	−0.226427
$910$	0	0
$911$	−39.9283	−1.32288	−0.661442	−	0.749996i	$-0.730055\pi$
$911$	−39.9283	−1.32288	−0.661442	+	0.749996i	$0.730055\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−30.6423	−1.01300
$916$	0	0
$917$	−2.45370	−0.0810283
$918$	0	0
$919$	8.99909	0.296853	0.148426	−	0.988923i	$-0.452579\pi$
$919$	8.99909	0.296853	0.148426	+	0.988923i	$0.452579\pi$
$920$	0	0
$921$	38.3873	1.26491
$922$	0	0
$923$	0.911774	0.0300114
$924$	0	0
$925$	−12.6273	−0.415183
$926$	0	0
$927$	3.04794	0.100108
$928$	0	0
$929$	−27.0160	−0.886366	−0.443183	−	0.896431i	$-0.646151\pi$
$929$	−27.0160	−0.886366	−0.443183	+	0.896431i	$0.646151\pi$
$930$	0	0
$931$	−3.77124	−0.123597
$932$	0	0
$933$	−41.1105	−1.34590
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−19.7050	−0.643735	−0.321867	−	0.946785i	$-0.604311\pi$
$937$	−19.7050	−0.643735	−0.321867	+	0.946785i	$0.604311\pi$
$938$	0	0
$939$	8.17496	0.266780
$940$	0	0
$941$	−38.2955	−1.24840	−0.624199	−	0.781266i	$-0.714574\pi$
$941$	−38.2955	−1.24840	−0.624199	+	0.781266i	$0.714574\pi$
$942$	0	0
$943$	49.6721	1.61755
$944$	0	0
$945$	−24.2599	−0.789174
$946$	0	0
$947$	52.3922	1.70252	0.851259	−	0.524745i	$-0.175839\pi$
$947$	52.3922	1.70252	0.851259	+	0.524745i	$0.175839\pi$
$948$	0	0
$949$	−14.4384	−0.468691
$950$	0	0
$951$	0.140335	0.00455068
$952$	0	0
$953$	−4.31007	−0.139617	−0.0698084	−	0.997560i	$-0.522239\pi$
$953$	−4.31007	−0.139617	−0.0698084	+	0.997560i	$0.522239\pi$
$954$	0	0
$955$	6.91012	0.223606
$956$	0	0
$957$	0	0
$958$	0	0
$959$	63.6888	2.05662
$960$	0	0
$961$	38.4634	1.24075
$962$	0	0
$963$	−0.602295	−0.0194087
$964$	0	0
$965$	9.59723	0.308946
$966$	0	0
$967$	9.41863	0.302883	0.151441	−	0.988466i	$-0.451609\pi$
$967$	9.41863	0.302883	0.151441	+	0.988466i	$0.451609\pi$
$968$	0	0
$969$	−15.1326	−0.486128
$970$	0	0
$971$	−0.476007	−0.0152758	−0.00763790	−	0.999971i	$-0.502431\pi$
$971$	−0.476007	−0.0152758	−0.00763790	+	0.999971i	$0.502431\pi$
$972$	0	0
$973$	37.9322	1.21605
$974$	0	0
$975$	4.56860	0.146312
$976$	0	0
$977$	28.3597	0.907308	0.453654	−	0.891178i	$-0.350120\pi$
$977$	28.3597	0.907308	0.453654	+	0.891178i	$0.350120\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	3.13748	0.100172
$982$	0	0
$983$	−17.1868	−0.548173	−0.274087	−	0.961705i	$-0.588376\pi$
$983$	−17.1868	−0.548173	−0.274087	+	0.961705i	$0.588376\pi$
$984$	0	0
$985$	14.9934	0.477731
$986$	0	0
$987$	14.7603	0.469826
$988$	0	0
$989$	9.76613	0.310545
$990$	0	0
$991$	−15.3466	−0.487501	−0.243750	−	0.969838i	$-0.578378\pi$
$991$	−15.3466	−0.487501	−0.243750	+	0.969838i	$0.578378\pi$
$992$	0	0
$993$	−24.9831	−0.792813
$994$	0	0
$995$	−33.1642	−1.05137
$996$	0	0
$997$	54.2131	1.71695	0.858473	−	0.512859i	$-0.171413\pi$
$997$	54.2131	1.71695	0.858473	+	0.512859i	$0.171413\pi$
$998$	0	0
$999$	−24.4721	−0.774264

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6292.2.a.r.1.3	yes	4
11.10	odd	2		6292.2.a.q.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6292.2.a.q.1.3	✓	4	11.10	odd	2
6292.2.a.r.1.3	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6292 = 2^{2} \cdot 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6292.a (trivial)

\(f(q)\)	\(=\)	\(q+1.61803 q^{3} -1.47528 q^{5} -3.00509 q^{7} -0.381966 q^{9} +O(q^{10})\)	\(q+1.61803 q^{3} -1.47528 q^{5} -3.00509 q^{7} -0.381966 q^{9} -1.00000 q^{13} -2.38705 q^{15} +5.03564 q^{17} -1.85725 q^{19} -4.86233 q^{21} -7.39214 q^{23} -2.82355 q^{25} -5.47214 q^{27} +8.14784 q^{29} +8.33447 q^{31} +4.43335 q^{35} +4.47214 q^{37} -1.61803 q^{39} -6.71958 q^{41} -1.32115 q^{43} +0.563507 q^{45} -3.03564 q^{47} +2.03055 q^{49} +8.14784 q^{51} +0.683938 q^{53} -3.00509 q^{57} +1.06590 q^{59} +12.8369 q^{61} +1.14784 q^{63} +1.47528 q^{65} -9.22227 q^{67} -11.9607 q^{69} -0.911774 q^{71} +14.4384 q^{73} -4.56860 q^{75} +2.13767 q^{79} -7.70820 q^{81} +13.0764 q^{83} -7.42899 q^{85} +13.1835 q^{87} +9.80975 q^{89} +3.00509 q^{91} +13.4855 q^{93} +2.73996 q^{95} +7.97139 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 2 q^{5} + 6 q^{7} - 6 q^{9}+O(q^{10}) \) 4 * q + 2 * q^3 - 2 * q^5 + 6 * q^7 - 6 * q^9	\( 4 q + 2 q^{3} - 2 q^{5} + 6 q^{7} - 6 q^{9} - 4 q^{13} + 4 q^{15} + 8 q^{17} - 8 q^{19} - 2 q^{21} + 2 q^{23} + 8 q^{25} - 4 q^{27} + 14 q^{29} - 2 q^{31} - 2 q^{35} - 2 q^{39} - 10 q^{41} + 8 q^{43} + 8 q^{45} + 14 q^{49} + 14 q^{51} - 2 q^{53} + 6 q^{57} + 4 q^{59} + 10 q^{61} - 14 q^{63} + 2 q^{65} - 8 q^{67} - 4 q^{69} + 6 q^{71} + 20 q^{73} - 6 q^{75} + 26 q^{79} - 4 q^{81} + 10 q^{83} - 32 q^{85} + 22 q^{87} - 6 q^{91} + 14 q^{93} + 36 q^{95} - 6 q^{97}+O(q^{100}) \) 4 * q + 2 * q^3 - 2 * q^5 + 6 * q^7 - 6 * q^9 - 4 * q^13 + 4 * q^15 + 8 * q^17 - 8 * q^19 - 2 * q^21 + 2 * q^23 + 8 * q^25 - 4 * q^27 + 14 * q^29 - 2 * q^31 - 2 * q^35 - 2 * q^39 - 10 * q^41 + 8 * q^43 + 8 * q^45 + 14 * q^49 + 14 * q^51 - 2 * q^53 + 6 * q^57 + 4 * q^59 + 10 * q^61 - 14 * q^63 + 2 * q^65 - 8 * q^67 - 4 * q^69 + 6 * q^71 + 20 * q^73 - 6 * q^75 + 26 * q^79 - 4 * q^81 + 10 * q^83 - 32 * q^85 + 22 * q^87 - 6 * q^91 + 14 * q^93 + 36 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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