LMFDB - Embedded newform 6292.2.a.s.1.4

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:	$1$
Dimension:	$5$
Coefficient field:	5.5.223824.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 4x^{3} + 6x^{2} + 2x - 2 $ x^5 - 2x^4 - 4x^3 + 6x^2 + 2x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.72348$ 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)

$f(q)$	$=$	$q+1.35630 q^{3} +3.25690 q^{5} -0.970383 q^{7} -1.16044 q^{9} +O(q^{10})$	$q+1.35630 q^{3} +3.25690 q^{5} -0.970383 q^{7} -1.16044 q^{9} -1.00000 q^{13} +4.41734 q^{15} -6.38773 q^{17} -2.52255 q^{19} -1.31613 q^{21} -5.53762 q^{23} +5.60740 q^{25} -5.64282 q^{27} -4.41259 q^{29} +5.83556 q^{31} -3.16044 q^{35} +4.80906 q^{37} -1.35630 q^{39} -7.96951 q^{41} +3.69894 q^{43} -3.77945 q^{45} -9.82994 q^{47} -6.05836 q^{49} -8.66369 q^{51} +7.96164 q^{53} -3.42134 q^{57} -2.40187 q^{59} -5.61900 q^{61} +1.12607 q^{63} -3.25690 q^{65} +7.79271 q^{67} -7.51069 q^{69} -12.4430 q^{71} +7.32001 q^{73} +7.60534 q^{75} +6.60446 q^{79} -4.17204 q^{81} -6.60740 q^{83} -20.8042 q^{85} -5.98481 q^{87} -5.25127 q^{89} +0.970383 q^{91} +7.91479 q^{93} -8.21568 q^{95} -9.99513 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 5 * q - q^3 - 2 * q^7 + 2 * q^9	$ 5 q - q^{3} - 2 q^{7} + 2 q^{9} - 5 q^{13} - 2 q^{15} - 5 q^{17} - 2 q^{19} + 4 q^{21} + 7 q^{23} - q^{25} - 7 q^{27} - 3 q^{29} + 10 q^{31} - 8 q^{35} + q^{39} - 8 q^{41} + 5 q^{43} + 8 q^{45} - 6 q^{47} - 7 q^{49} - 3 q^{51} + 15 q^{53} - 20 q^{57} - 12 q^{59} - 9 q^{61} + 10 q^{67} - 7 q^{69} - 14 q^{71} - 8 q^{73} + 21 q^{75} - 13 q^{79} - 23 q^{81} - 4 q^{83} - 24 q^{85} + 7 q^{87} + 14 q^{89} + 2 q^{91} - 22 q^{93} - 12 q^{95} + 6 q^{97}+O(q^{100}) $ 5 * q - q^3 - 2 * q^7 + 2 * q^9 - 5 * q^13 - 2 * q^15 - 5 * q^17 - 2 * q^19 + 4 * q^21 + 7 * q^23 - q^25 - 7 * q^27 - 3 * q^29 + 10 * q^31 - 8 * q^35 + q^39 - 8 * q^41 + 5 * q^43 + 8 * q^45 - 6 * q^47 - 7 * q^49 - 3 * q^51 + 15 * q^53 - 20 * q^57 - 12 * q^59 - 9 * q^61 + 10 * q^67 - 7 * q^69 - 14 * q^71 - 8 * q^73 + 21 * q^75 - 13 * q^79 - 23 * q^81 - 4 * q^83 - 24 * q^85 + 7 * q^87 + 14 * q^89 + 2 * q^91 - 22 * q^93 - 12 * 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
$3$	1.35630	0.783062	0.391531	−	0.920165i	$-0.371946\pi$
$3$	1.35630	0.783062	0.391531	+	0.920165i	$0.371946\pi$
$4$	0	0
$5$	3.25690	1.45653	0.728265	−	0.685295i	$-0.240327\pi$
$5$	3.25690	1.45653	0.728265	+	0.685295i	$0.240327\pi$
$6$	0	0
$7$	−0.970383	−0.366770	−0.183385	−	0.983041i	$-0.558706\pi$
$7$	−0.970383	−0.366770	−0.183385	+	0.983041i	$0.558706\pi$
$8$	0	0
$9$	−1.16044	−0.386814
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	4.41734	1.14055
$16$	0	0
$17$	−6.38773	−1.54925	−0.774626	−	0.632420i	$-0.782062\pi$
$17$	−6.38773	−1.54925	−0.774626	+	0.632420i	$0.782062\pi$
$18$	0	0
$19$	−2.52255	−0.578712	−0.289356	−	0.957222i	$-0.593441\pi$
$19$	−2.52255	−0.578712	−0.289356	+	0.957222i	$0.593441\pi$
$20$	0	0
$21$	−1.31613	−0.287204
$22$	0	0
$23$	−5.53762	−1.15467	−0.577337	−	0.816506i	$-0.695908\pi$
$23$	−5.53762	−1.15467	−0.577337	+	0.816506i	$0.695908\pi$
$24$	0	0
$25$	5.60740	1.12148
$26$	0	0
$27$	−5.64282	−1.08596
$28$	0	0
$29$	−4.41259	−0.819398	−0.409699	−	0.912221i	$-0.634366\pi$
$29$	−4.41259	−0.819398	−0.409699	+	0.912221i	$0.634366\pi$
$30$	0	0
$31$	5.83556	1.04810	0.524049	−	0.851688i	$-0.324421\pi$
$31$	5.83556	1.04810	0.524049	+	0.851688i	$0.324421\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.16044	−0.534212
$36$	0	0
$37$	4.80906	0.790605	0.395303	−	0.918551i	$-0.370640\pi$
$37$	4.80906	0.790605	0.395303	+	0.918551i	$0.370640\pi$
$38$	0	0
$39$	−1.35630	−0.217182
$40$	0	0
$41$	−7.96951	−1.24463	−0.622314	−	0.782768i	$-0.713807\pi$
$41$	−7.96951	−1.24463	−0.622314	+	0.782768i	$0.713807\pi$
$42$	0	0
$43$	3.69894	0.564083	0.282041	−	0.959402i	$-0.408988\pi$
$43$	3.69894	0.564083	0.282041	+	0.959402i	$0.408988\pi$
$44$	0	0
$45$	−3.77945	−0.563407
$46$	0	0
$47$	−9.82994	−1.43384	−0.716922	−	0.697154i	$-0.754450\pi$
$47$	−9.82994	−1.43384	−0.716922	+	0.697154i	$0.754450\pi$
$48$	0	0
$49$	−6.05836	−0.865479
$50$	0	0
$51$	−8.66369	−1.21316
$52$	0	0
$53$	7.96164	1.09362	0.546808	−	0.837258i	$-0.315843\pi$
$53$	7.96164	1.09362	0.546808	+	0.837258i	$0.315843\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.42134	−0.453167
$58$	0	0
$59$	−2.40187	−0.312697	−0.156348	−	0.987702i	$-0.549972\pi$
$59$	−2.40187	−0.312697	−0.156348	+	0.987702i	$0.549972\pi$
$60$	0	0
$61$	−5.61900	−0.719440	−0.359720	−	0.933060i	$-0.617128\pi$
$61$	−5.61900	−0.719440	−0.359720	+	0.933060i	$0.617128\pi$
$62$	0	0
$63$	1.12607	0.141872
$64$	0	0
$65$	−3.25690	−0.403969
$66$	0	0
$67$	7.79271	0.952031	0.476016	−	0.879437i	$-0.342081\pi$
$67$	7.79271	0.952031	0.476016	+	0.879437i	$0.342081\pi$
$68$	0	0
$69$	−7.51069	−0.904180
$70$	0	0
$71$	−12.4430	−1.47671	−0.738354	−	0.674413i	$-0.764397\pi$
$71$	−12.4430	−1.47671	−0.738354	+	0.674413i	$0.764397\pi$
$72$	0	0
$73$	7.32001	0.856742	0.428371	−	0.903603i	$-0.359088\pi$
$73$	7.32001	0.856742	0.428371	+	0.903603i	$0.359088\pi$
$74$	0	0
$75$	7.60534	0.878189
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.60446	0.743060	0.371530	−	0.928421i	$-0.378833\pi$
$79$	6.60446	0.743060	0.371530	+	0.928421i	$0.378833\pi$
$80$	0	0
$81$	−4.17204	−0.463560
$82$	0	0
$83$	−6.60740	−0.725257	−0.362628	−	0.931934i	$-0.618121\pi$
$83$	−6.60740	−0.725257	−0.362628	+	0.931934i	$0.618121\pi$
$84$	0	0
$85$	−20.8042	−2.25653
$86$	0	0
$87$	−5.98481	−0.641639
$88$	0	0
$89$	−5.25127	−0.556634	−0.278317	−	0.960489i	$-0.589777\pi$
$89$	−5.25127	−0.556634	−0.278317	+	0.960489i	$0.589777\pi$
$90$	0	0
$91$	0.970383	0.101724
$92$	0	0
$93$	7.91479	0.820726
$94$	0	0
$95$	−8.21568	−0.842911
$96$	0	0
$97$	−9.99513	−1.01485	−0.507426	−	0.861695i	$-0.669403\pi$
$97$	−9.99513	−1.01485	−0.507426	+	0.861695i	$0.669403\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.87393	−0.584477	−0.292239	−	0.956345i	$-0.594400\pi$
$101$	−5.87393	−0.584477	−0.292239	+	0.956345i	$0.594400\pi$
$102$	0	0
$103$	10.0297	0.988253	0.494126	−	0.869390i	$-0.335488\pi$
$103$	10.0297	0.988253	0.494126	+	0.869390i	$0.335488\pi$
$104$	0	0
$105$	−4.28652	−0.418321
$106$	0	0
$107$	−12.6419	−1.22214	−0.611071	−	0.791575i	$-0.709261\pi$
$107$	−12.6419	−1.22214	−0.611071	+	0.791575i	$0.709261\pi$
$108$	0	0
$109$	−1.21093	−0.115986	−0.0579931	−	0.998317i	$-0.518470\pi$
$109$	−1.21093	−0.115986	−0.0579931	+	0.998317i	$0.518470\pi$
$110$	0	0
$111$	6.52255	0.619093
$112$	0	0
$113$	13.8930	1.30695	0.653474	−	0.756949i	$-0.273311\pi$
$113$	13.8930	1.30695	0.653474	+	0.756949i	$0.273311\pi$
$114$	0	0
$115$	−18.0355	−1.68182
$116$	0	0
$117$	1.16044	0.107283
$118$	0	0
$119$	6.19854	0.568220
$120$	0	0
$121$	0	0
$122$	0	0
$123$	−10.8091	−0.974620
$124$	0	0
$125$	1.97825	0.176940
$126$	0	0
$127$	21.4143	1.90021	0.950105	−	0.311930i	$-0.100975\pi$
$127$	21.4143	1.90021	0.950105	+	0.311930i	$0.100975\pi$
$128$	0	0
$129$	5.01688	0.441712
$130$	0	0
$131$	−6.34382	−0.554262	−0.277131	−	0.960832i	$-0.589384\pi$
$131$	−6.34382	−0.554262	−0.277131	+	0.960832i	$0.589384\pi$
$132$	0	0
$133$	2.44784	0.212254
$134$	0	0
$135$	−18.3781	−1.58174
$136$	0	0
$137$	−11.8730	−1.01438	−0.507191	−	0.861833i	$-0.669316\pi$
$137$	−11.8730	−1.01438	−0.507191	+	0.861833i	$0.669316\pi$
$138$	0	0
$139$	5.45728	0.462880	0.231440	−	0.972849i	$-0.425656\pi$
$139$	5.45728	0.462880	0.231440	+	0.972849i	$0.425656\pi$
$140$	0	0
$141$	−13.3324	−1.12279
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−14.3714	−1.19348
$146$	0	0
$147$	−8.21696	−0.677724
$148$	0	0
$149$	−10.1463	−0.831217	−0.415609	−	0.909544i	$-0.636431\pi$
$149$	−10.1463	−0.831217	−0.415609	+	0.909544i	$0.636431\pi$
$150$	0	0
$151$	−13.9274	−1.13340	−0.566698	−	0.823925i	$-0.691779\pi$
$151$	−13.9274	−1.13340	−0.566698	+	0.823925i	$0.691779\pi$
$152$	0	0
$153$	7.41259	0.599273
$154$	0	0
$155$	19.0059	1.52659
$156$	0	0
$157$	19.0727	1.52217	0.761083	−	0.648654i	$-0.224668\pi$
$157$	19.0727	1.52217	0.761083	+	0.648654i	$0.224668\pi$
$158$	0	0
$159$	10.7984	0.856368
$160$	0	0
$161$	5.37361	0.423500
$162$	0	0
$163$	−7.09634	−0.555828	−0.277914	−	0.960606i	$-0.589643\pi$
$163$	−7.09634	−0.555828	−0.277914	+	0.960606i	$0.589643\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.4833	−1.73981	−0.869905	−	0.493219i	$-0.835820\pi$
$167$	−22.4833	−1.73981	−0.869905	+	0.493219i	$0.835820\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.92727	0.223854
$172$	0	0
$173$	17.6426	1.34134	0.670672	−	0.741754i	$-0.266006\pi$
$173$	17.6426	1.34134	0.670672	+	0.741754i	$0.266006\pi$
$174$	0	0
$175$	−5.44133	−0.411326
$176$	0	0
$177$	−3.25766	−0.244861
$178$	0	0
$179$	0.0562891	0.00420724	0.00210362	−	0.999998i	$-0.499330\pi$
$179$	0.0562891	0.00420724	0.00210362	+	0.999998i	$0.499330\pi$
$180$	0	0
$181$	−6.65425	−0.494606	−0.247303	−	0.968938i	$-0.579544\pi$
$181$	−6.65425	−0.494606	−0.247303	+	0.968938i	$0.579544\pi$
$182$	0	0
$183$	−7.62107	−0.563366
$184$	0	0
$185$	15.6626	1.15154
$186$	0	0
$187$	0	0
$188$	0	0
$189$	5.47570	0.398299
$190$	0	0
$191$	7.25766	0.525146	0.262573	−	0.964912i	$-0.415429\pi$
$191$	7.25766	0.525146	0.262573	+	0.964912i	$0.415429\pi$
$192$	0	0
$193$	−17.7721	−1.27926	−0.639632	−	0.768681i	$-0.720913\pi$
$193$	−17.7721	−1.27926	−0.639632	+	0.768681i	$0.720913\pi$
$194$	0	0
$195$	−4.41734	−0.316333
$196$	0	0
$197$	24.3360	1.73387	0.866934	−	0.498422i	$-0.166087\pi$
$197$	24.3360	1.73387	0.866934	+	0.498422i	$0.166087\pi$
$198$	0	0
$199$	8.99895	0.637919	0.318959	−	0.947768i	$-0.396667\pi$
$199$	8.99895	0.637919	0.318959	+	0.947768i	$0.396667\pi$
$200$	0	0
$201$	10.5693	0.745499
$202$	0	0
$203$	4.28191	0.300531
$204$	0	0
$205$	−25.9559	−1.81284
$206$	0	0
$207$	6.42609	0.446644
$208$	0	0
$209$	0	0
$210$	0	0
$211$	2.53867	0.174769	0.0873845	−	0.996175i	$-0.472149\pi$
$211$	2.53867	0.174769	0.0873845	+	0.996175i	$0.472149\pi$
$212$	0	0
$213$	−16.8764	−1.15635
$214$	0	0
$215$	12.0471	0.821603
$216$	0	0
$217$	−5.66274	−0.384412
$218$	0	0
$219$	9.92815	0.670882
$220$	0	0
$221$	6.38773	0.429685
$222$	0	0
$223$	22.2165	1.48773	0.743864	−	0.668331i	$-0.232991\pi$
$223$	22.2165	1.48773	0.743864	+	0.668331i	$0.232991\pi$
$224$	0	0
$225$	−6.50707	−0.433805
$226$	0	0
$227$	2.50054	0.165967	0.0829833	−	0.996551i	$-0.473555\pi$
$227$	2.50054	0.165967	0.0829833	+	0.996551i	$0.473555\pi$
$228$	0	0
$229$	0.00976249	0.000645124 0	0.000322562	−	1.00000i	$-0.499897\pi$
$229$	0.00976249	0.000645124 0	0.000322562		1.00000i	$0.499897\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.2967	−0.674557	−0.337278	−	0.941405i	$-0.609506\pi$
$233$	−10.2967	−0.674557	−0.337278	+	0.941405i	$0.609506\pi$
$234$	0	0
$235$	−32.0151	−2.08844
$236$	0	0
$237$	8.95765	0.581862
$238$	0	0
$239$	23.5134	1.52096	0.760478	−	0.649363i	$-0.224964\pi$
$239$	23.5134	1.52096	0.760478	+	0.649363i	$0.224964\pi$
$240$	0	0
$241$	0.834946	0.0537836	0.0268918	−	0.999638i	$-0.491439\pi$
$241$	0.834946	0.0537836	0.0268918	+	0.999638i	$0.491439\pi$
$242$	0	0
$243$	11.2699	0.722965
$244$	0	0
$245$	−19.7315	−1.26060
$246$	0	0
$247$	2.52255	0.160506
$248$	0	0
$249$	−8.96164	−0.567921
$250$	0	0
$251$	3.32680	0.209986	0.104993	−	0.994473i	$-0.466518\pi$
$251$	3.32680	0.209986	0.104993	+	0.994473i	$0.466518\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−28.2168	−1.76700
$256$	0	0
$257$	−21.7782	−1.35849	−0.679243	−	0.733913i	$-0.737692\pi$
$257$	−21.7782	−1.35849	−0.679243	+	0.733913i	$0.737692\pi$
$258$	0	0
$259$	−4.66664	−0.289971
$260$	0	0
$261$	5.12056	0.316955
$262$	0	0
$263$	−6.17100	−0.380520	−0.190260	−	0.981734i	$-0.560933\pi$
$263$	−6.17100	−0.380520	−0.190260	+	0.981734i	$0.560933\pi$
$264$	0	0
$265$	25.9303	1.59288
$266$	0	0
$267$	−7.12231	−0.435879
$268$	0	0
$269$	−29.6640	−1.80865	−0.904323	−	0.426849i	$-0.859624\pi$
$269$	−29.6640	−1.80865	−0.904323	+	0.426849i	$0.859624\pi$
$270$	0	0
$271$	15.0186	0.912315	0.456157	−	0.889899i	$-0.349225\pi$
$271$	15.0186	0.912315	0.456157	+	0.889899i	$0.349225\pi$
$272$	0	0
$273$	1.31613	0.0796560
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−28.2905	−1.69981	−0.849905	−	0.526936i	$-0.823341\pi$
$277$	−28.2905	−1.69981	−0.849905	+	0.526936i	$0.823341\pi$
$278$	0	0
$279$	−6.77184	−0.405419
$280$	0	0
$281$	−12.1828	−0.726763	−0.363382	−	0.931640i	$-0.618378\pi$
$281$	−12.1828	−0.726763	−0.363382	+	0.931640i	$0.618378\pi$
$282$	0	0
$283$	−1.99411	−0.118538	−0.0592688	−	0.998242i	$-0.518877\pi$
$283$	−1.99411	−0.118538	−0.0592688	+	0.998242i	$0.518877\pi$
$284$	0	0
$285$	−11.1430	−0.660052
$286$	0	0
$287$	7.73348	0.456493
$288$	0	0
$289$	23.8031	1.40018
$290$	0	0
$291$	−13.5564	−0.794692
$292$	0	0
$293$	11.9608	0.698755	0.349378	−	0.936982i	$-0.386393\pi$
$293$	11.9608	0.698755	0.349378	+	0.936982i	$0.386393\pi$
$294$	0	0
$295$	−7.82264	−0.455452
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.53762	0.320249
$300$	0	0
$301$	−3.58939	−0.206889
$302$	0	0
$303$	−7.96682	−0.457682
$304$	0	0
$305$	−18.3005	−1.04789
$306$	0	0
$307$	20.4833	1.16904	0.584522	−	0.811378i	$-0.301282\pi$
$307$	20.4833	1.16904	0.584522	+	0.811378i	$0.301282\pi$
$308$	0	0
$309$	13.6033	0.773863
$310$	0	0
$311$	−16.2316	−0.920409	−0.460205	−	0.887813i	$-0.652224\pi$
$311$	−16.2316	−0.920409	−0.460205	+	0.887813i	$0.652224\pi$
$312$	0	0
$313$	−18.7672	−1.06078	−0.530392	−	0.847752i	$-0.677955\pi$
$313$	−18.7672	−1.06078	−0.530392	+	0.847752i	$0.677955\pi$
$314$	0	0
$315$	3.66751	0.206641
$316$	0	0
$317$	21.6346	1.21512	0.607561	−	0.794273i	$-0.292148\pi$
$317$	21.6346	1.21512	0.607561	+	0.794273i	$0.292148\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−17.1463	−0.957013
$322$	0	0
$323$	16.1133	0.896570
$324$	0	0
$325$	−5.60740	−0.311043
$326$	0	0
$327$	−1.64239	−0.0908243
$328$	0	0
$329$	9.53881	0.525891
$330$	0	0
$331$	8.87141	0.487617	0.243808	−	0.969823i	$-0.421603\pi$
$331$	8.87141	0.487617	0.243808	+	0.969823i	$0.421603\pi$
$332$	0	0
$333$	−5.58064	−0.305817
$334$	0	0
$335$	25.3801	1.38666
$336$	0	0
$337$	22.8572	1.24511	0.622554	−	0.782577i	$-0.286095\pi$
$337$	22.8572	1.24511	0.622554	+	0.782577i	$0.286095\pi$
$338$	0	0
$339$	18.8432	1.02342
$340$	0	0
$341$	0	0
$342$	0	0
$343$	12.6716	0.684203
$344$	0	0
$345$	−24.4616	−1.31697
$346$	0	0
$347$	16.1361	0.866229	0.433115	−	0.901339i	$-0.357415\pi$
$347$	16.1361	0.866229	0.433115	+	0.901339i	$0.357415\pi$
$348$	0	0
$349$	10.6299	0.569004	0.284502	−	0.958675i	$-0.408172\pi$
$349$	10.6299	0.569004	0.284502	+	0.958675i	$0.408172\pi$
$350$	0	0
$351$	5.64282	0.301191
$352$	0	0
$353$	0.876782	0.0466664	0.0233332	−	0.999728i	$-0.492572\pi$
$353$	0.876782	0.0466664	0.0233332	+	0.999728i	$0.492572\pi$
$354$	0	0
$355$	−40.5255	−2.15087
$356$	0	0
$357$	8.40710	0.444951
$358$	0	0
$359$	21.3591	1.12729	0.563645	−	0.826017i	$-0.309399\pi$
$359$	21.3591	1.12729	0.563645	+	0.826017i	$0.309399\pi$
$360$	0	0
$361$	−12.6368	−0.665093
$362$	0	0
$363$	0	0
$364$	0	0
$365$	23.8405	1.24787
$366$	0	0
$367$	−10.3009	−0.537702	−0.268851	−	0.963182i	$-0.586644\pi$
$367$	−10.3009	−0.537702	−0.268851	+	0.963182i	$0.586644\pi$
$368$	0	0
$369$	9.24816	0.481440
$370$	0	0
$371$	−7.72584	−0.401106
$372$	0	0
$373$	−26.2393	−1.35862	−0.679310	−	0.733852i	$-0.737721\pi$
$373$	−26.2393	−1.35862	−0.679310	+	0.733852i	$0.737721\pi$
$374$	0	0
$375$	2.68311	0.138555
$376$	0	0
$377$	4.41259	0.227260
$378$	0	0
$379$	−13.2738	−0.681828	−0.340914	−	0.940094i	$-0.610737\pi$
$379$	−13.2738	−0.681828	−0.340914	+	0.940094i	$0.610737\pi$
$380$	0	0
$381$	29.0442	1.48798
$382$	0	0
$383$	5.60688	0.286498	0.143249	−	0.989687i	$-0.454245\pi$
$383$	5.60688	0.286498	0.143249	+	0.989687i	$0.454245\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.29241	−0.218195
$388$	0	0
$389$	4.19481	0.212685	0.106343	−	0.994330i	$-0.466086\pi$
$389$	4.19481	0.212685	0.106343	+	0.994330i	$0.466086\pi$
$390$	0	0
$391$	35.3728	1.78888
$392$	0	0
$393$	−8.60415	−0.434022
$394$	0	0
$395$	21.5101	1.08229
$396$	0	0
$397$	−10.1508	−0.509455	−0.254728	−	0.967013i	$-0.581986\pi$
$397$	−10.1508	−0.509455	−0.254728	+	0.967013i	$0.581986\pi$
$398$	0	0
$399$	3.32001	0.166208
$400$	0	0
$401$	−19.7686	−0.987195	−0.493597	−	0.869690i	$-0.664318\pi$
$401$	−19.7686	−0.987195	−0.493597	+	0.869690i	$0.664318\pi$
$402$	0	0
$403$	−5.83556	−0.290690
$404$	0	0
$405$	−13.5879	−0.675190
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−13.1794	−0.651680	−0.325840	−	0.945425i	$-0.605647\pi$
$409$	−13.1794	−0.651680	−0.325840	+	0.945425i	$0.605647\pi$
$410$	0	0
$411$	−16.1034	−0.794325
$412$	0	0
$413$	2.33073	0.114688
$414$	0	0
$415$	−21.5197	−1.05636
$416$	0	0
$417$	7.40172	0.362464
$418$	0	0
$419$	27.2399	1.33075	0.665377	−	0.746507i	$-0.268271\pi$
$419$	27.2399	1.33075	0.665377	+	0.746507i	$0.268271\pi$
$420$	0	0
$421$	15.3711	0.749144	0.374572	−	0.927198i	$-0.377790\pi$
$421$	15.3711	0.749144	0.374572	+	0.927198i	$0.377790\pi$
$422$	0	0
$423$	11.4071	0.554631
$424$	0	0
$425$	−35.8186	−1.73746
$426$	0	0
$427$	5.45259	0.263869
$428$	0	0
$429$	0	0
$430$	0	0
$431$	34.4141	1.65767	0.828834	−	0.559495i	$-0.189005\pi$
$431$	34.4141	1.65767	0.828834	+	0.559495i	$0.189005\pi$
$432$	0	0
$433$	−6.65903	−0.320012	−0.160006	−	0.987116i	$-0.551151\pi$
$433$	−6.65903	−0.320012	−0.160006	+	0.987116i	$0.551151\pi$
$434$	0	0
$435$	−19.4919	−0.934567
$436$	0	0
$437$	13.9689	0.668223
$438$	0	0
$439$	26.5088	1.26519	0.632597	−	0.774481i	$-0.281989\pi$
$439$	26.5088	1.26519	0.632597	+	0.774481i	$0.281989\pi$
$440$	0	0
$441$	7.03038	0.334780
$442$	0	0
$443$	−7.47788	−0.355285	−0.177643	−	0.984095i	$-0.556847\pi$
$443$	−7.47788	−0.355285	−0.177643	+	0.984095i	$0.556847\pi$
$444$	0	0
$445$	−17.1029	−0.810754
$446$	0	0
$447$	−13.7615	−0.650895
$448$	0	0
$449$	19.4786	0.919250	0.459625	−	0.888113i	$-0.347984\pi$
$449$	19.4786	0.919250	0.459625	+	0.888113i	$0.347984\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−18.8898	−0.887520
$454$	0	0
$455$	3.16044	0.148164
$456$	0	0
$457$	3.22565	0.150889	0.0754447	−	0.997150i	$-0.475962\pi$
$457$	3.22565	0.150889	0.0754447	+	0.997150i	$0.475962\pi$
$458$	0	0
$459$	36.0448	1.68243
$460$	0	0
$461$	25.7378	1.19873	0.599365	−	0.800476i	$-0.295420\pi$
$461$	25.7378	1.19873	0.599365	+	0.800476i	$0.295420\pi$
$462$	0	0
$463$	−30.4543	−1.41533	−0.707666	−	0.706547i	$-0.750252\pi$
$463$	−30.4543	−1.41533	−0.707666	+	0.706547i	$0.750252\pi$
$464$	0	0
$465$	25.7777	1.19541
$466$	0	0
$467$	17.6288	0.815765	0.407882	−	0.913035i	$-0.366267\pi$
$467$	17.6288	0.815765	0.407882	+	0.913035i	$0.366267\pi$
$468$	0	0
$469$	−7.56192	−0.349177
$470$	0	0
$471$	25.8684	1.19195
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−14.1449	−0.649014
$476$	0	0
$477$	−9.23903	−0.423026
$478$	0	0
$479$	18.5694	0.848459	0.424230	−	0.905555i	$-0.360545\pi$
$479$	18.5694	0.848459	0.424230	+	0.905555i	$0.360545\pi$
$480$	0	0
$481$	−4.80906	−0.219274
$482$	0	0
$483$	7.28825	0.331627
$484$	0	0
$485$	−32.5531	−1.47816
$486$	0	0
$487$	−37.0409	−1.67848	−0.839241	−	0.543759i	$-0.817001\pi$
$487$	−37.0409	−1.67848	−0.839241	+	0.543759i	$0.817001\pi$
$488$	0	0
$489$	−9.62479	−0.435248
$490$	0	0
$491$	−35.3252	−1.59421	−0.797103	−	0.603843i	$-0.793635\pi$
$491$	−35.3252	−1.59421	−0.797103	+	0.603843i	$0.793635\pi$
$492$	0	0
$493$	28.1864	1.26945
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.0745	0.541613
$498$	0	0
$499$	27.4898	1.23061	0.615306	−	0.788288i	$-0.289032\pi$
$499$	27.4898	1.23061	0.615306	+	0.788288i	$0.289032\pi$
$500$	0	0
$501$	−30.4942	−1.36238
$502$	0	0
$503$	17.7456	0.791238	0.395619	−	0.918415i	$-0.370530\pi$
$503$	17.7456	0.791238	0.395619	+	0.918415i	$0.370530\pi$
$504$	0	0
$505$	−19.1308	−0.851309
$506$	0	0
$507$	1.35630	0.0602355
$508$	0	0
$509$	9.46990	0.419746	0.209873	−	0.977729i	$-0.432695\pi$
$509$	9.46990	0.419746	0.209873	+	0.977729i	$0.432695\pi$
$510$	0	0
$511$	−7.10322	−0.314228
$512$	0	0
$513$	14.2343	0.628459
$514$	0	0
$515$	32.6656	1.43942
$516$	0	0
$517$	0	0
$518$	0	0
$519$	23.9287	1.05035
$520$	0	0
$521$	18.5859	0.814264	0.407132	−	0.913369i	$-0.366529\pi$
$521$	18.5859	0.814264	0.407132	+	0.913369i	$0.366529\pi$
$522$	0	0
$523$	−25.7590	−1.12636	−0.563182	−	0.826333i	$-0.690423\pi$
$523$	−25.7590	−1.12636	−0.563182	+	0.826333i	$0.690423\pi$
$524$	0	0
$525$	−7.38009	−0.322094
$526$	0	0
$527$	−37.2760	−1.62377
$528$	0	0
$529$	7.66521	0.333270
$530$	0	0
$531$	2.78723	0.120955
$532$	0	0
$533$	7.96951	0.345198
$534$	0	0
$535$	−41.1736	−1.78009
$536$	0	0
$537$	0.0763450	0.00329453
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−40.2152	−1.72899	−0.864493	−	0.502645i	$-0.832360\pi$
$541$	−40.2152	−1.72899	−0.864493	+	0.502645i	$0.832360\pi$
$542$	0	0
$543$	−9.02518	−0.387307
$544$	0	0
$545$	−3.94388	−0.168937
$546$	0	0
$547$	−27.3022	−1.16736	−0.583678	−	0.811985i	$-0.698387\pi$
$547$	−27.3022	−1.16736	−0.583678	+	0.811985i	$0.698387\pi$
$548$	0	0
$549$	6.52053	0.278290
$550$	0	0
$551$	11.1310	0.474195
$552$	0	0
$553$	−6.40886	−0.272532
$554$	0	0
$555$	21.2433	0.901727
$556$	0	0
$557$	−41.9383	−1.77698	−0.888491	−	0.458894i	$-0.848246\pi$
$557$	−41.9383	−1.77698	−0.888491	+	0.458894i	$0.848246\pi$
$558$	0	0
$559$	−3.69894	−0.156448
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20.9908	−0.884656	−0.442328	−	0.896853i	$-0.645847\pi$
$563$	−20.9908	−0.884656	−0.442328	+	0.896853i	$0.645847\pi$
$564$	0	0
$565$	45.2483	1.90361
$566$	0	0
$567$	4.04848	0.170020
$568$	0	0
$569$	−39.0614	−1.63754	−0.818769	−	0.574123i	$-0.805343\pi$
$569$	−39.0614	−1.63754	−0.818769	+	0.574123i	$0.805343\pi$
$570$	0	0
$571$	41.4061	1.73279	0.866397	−	0.499357i	$-0.166430\pi$
$571$	41.4061	1.73279	0.866397	+	0.499357i	$0.166430\pi$
$572$	0	0
$573$	9.84358	0.411222
$574$	0	0
$575$	−31.0517	−1.29494
$576$	0	0
$577$	31.9138	1.32859	0.664294	−	0.747471i	$-0.268732\pi$
$577$	31.9138	1.32859	0.664294	+	0.747471i	$0.268732\pi$
$578$	0	0
$579$	−24.1044	−1.00174
$580$	0	0
$581$	6.41171	0.266003
$582$	0	0
$583$	0	0
$584$	0	0
$585$	3.77945	0.156261
$586$	0	0
$587$	−26.1456	−1.07914	−0.539572	−	0.841939i	$-0.681414\pi$
$587$	−26.1456	−1.07914	−0.539572	+	0.841939i	$0.681414\pi$
$588$	0	0
$589$	−14.7205	−0.606547
$590$	0	0
$591$	33.0070	1.35773
$592$	0	0
$593$	−5.35442	−0.219880	−0.109940	−	0.993938i	$-0.535066\pi$
$593$	−5.35442	−0.219880	−0.109940	+	0.993938i	$0.535066\pi$
$594$	0	0
$595$	20.1880	0.827629
$596$	0	0
$597$	12.2053	0.499530
$598$	0	0
$599$	−15.1755	−0.620055	−0.310028	−	0.950728i	$-0.600338\pi$
$599$	−15.1755	−0.620055	−0.310028	+	0.950728i	$0.600338\pi$
$600$	0	0
$601$	5.00900	0.204322	0.102161	−	0.994768i	$-0.467424\pi$
$601$	5.00900	0.204322	0.102161	+	0.994768i	$0.467424\pi$
$602$	0	0
$603$	−9.04299	−0.368259
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−6.04259	−0.245261	−0.122631	−	0.992452i	$-0.539133\pi$
$607$	−6.04259	−0.245261	−0.122631	+	0.992452i	$0.539133\pi$
$608$	0	0
$609$	5.80756	0.235334
$610$	0	0
$611$	9.82994	0.397677
$612$	0	0
$613$	−20.1081	−0.812157	−0.406078	−	0.913838i	$-0.633104\pi$
$613$	−20.1081	−0.812157	−0.406078	+	0.913838i	$0.633104\pi$
$614$	0	0
$615$	−35.2040	−1.41956
$616$	0	0
$617$	−17.0351	−0.685806	−0.342903	−	0.939371i	$-0.611410\pi$
$617$	−17.0351	−0.685806	−0.342903	+	0.939371i	$0.611410\pi$
$618$	0	0
$619$	−13.9825	−0.562003	−0.281001	−	0.959707i	$-0.590667\pi$
$619$	−13.9825	−0.562003	−0.281001	+	0.959707i	$0.590667\pi$
$620$	0	0
$621$	31.2478	1.25393
$622$	0	0
$623$	5.09575	0.204157
$624$	0	0
$625$	−21.5940	−0.863762
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−30.7190	−1.22485
$630$	0	0
$631$	−19.0096	−0.756761	−0.378381	−	0.925650i	$-0.623519\pi$
$631$	−19.0096	−0.756761	−0.378381	+	0.925650i	$0.623519\pi$
$632$	0	0
$633$	3.44320	0.136855
$634$	0	0
$635$	69.7442	2.76771
$636$	0	0
$637$	6.05836	0.240041
$638$	0	0
$639$	14.4394	0.571212
$640$	0	0
$641$	−41.9068	−1.65522	−0.827609	−	0.561305i	$-0.810299\pi$
$641$	−41.9068	−1.65522	−0.827609	+	0.561305i	$0.810299\pi$
$642$	0	0
$643$	−7.08244	−0.279304	−0.139652	−	0.990201i	$-0.544598\pi$
$643$	−7.08244	−0.279304	−0.139652	+	0.990201i	$0.544598\pi$
$644$	0	0
$645$	16.3395	0.643366
$646$	0	0
$647$	22.6361	0.889915	0.444958	−	0.895552i	$-0.353219\pi$
$647$	22.6361	0.889915	0.444958	+	0.895552i	$0.353219\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−7.68038	−0.301018
$652$	0	0
$653$	−31.6317	−1.23784	−0.618921	−	0.785453i	$-0.712430\pi$
$653$	−31.6317	−1.23784	−0.618921	+	0.785453i	$0.712430\pi$
$654$	0	0
$655$	−20.6612	−0.807300
$656$	0	0
$657$	−8.49445	−0.331400
$658$	0	0
$659$	4.24951	0.165537	0.0827686	−	0.996569i	$-0.473624\pi$
$659$	4.24951	0.165537	0.0827686	+	0.996569i	$0.473624\pi$
$660$	0	0
$661$	−25.4854	−0.991267	−0.495634	−	0.868532i	$-0.665064\pi$
$661$	−25.4854	−0.991267	−0.495634	+	0.868532i	$0.665064\pi$
$662$	0	0
$663$	8.66369	0.336470
$664$	0	0
$665$	7.97236	0.309155
$666$	0	0
$667$	24.4352	0.946136
$668$	0	0
$669$	30.1323	1.16498
$670$	0	0
$671$	0	0
$672$	0	0
$673$	37.1631	1.43253	0.716265	−	0.697828i	$-0.245850\pi$
$673$	37.1631	1.43253	0.716265	+	0.697828i	$0.245850\pi$
$674$	0	0
$675$	−31.6416	−1.21788
$676$	0	0
$677$	−20.5426	−0.789516	−0.394758	−	0.918785i	$-0.629171\pi$
$677$	−20.5426	−0.789516	−0.394758	+	0.918785i	$0.629171\pi$
$678$	0	0
$679$	9.69911	0.372218
$680$	0	0
$681$	3.39149	0.129962
$682$	0	0
$683$	−9.33738	−0.357285	−0.178643	−	0.983914i	$-0.557171\pi$
$683$	−9.33738	−0.357285	−0.178643	+	0.983914i	$0.557171\pi$
$684$	0	0
$685$	−38.6693	−1.47748
$686$	0	0
$687$	0.0132409	0.000505172 0
$688$	0	0
$689$	−7.96164	−0.303314
$690$	0	0
$691$	22.2073	0.844805	0.422402	−	0.906408i	$-0.361187\pi$
$691$	22.2073	0.844805	0.422402	+	0.906408i	$0.361187\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.7738	0.674199
$696$	0	0
$697$	50.9070	1.92824
$698$	0	0
$699$	−13.9654	−0.528220
$700$	0	0
$701$	−31.6963	−1.19715	−0.598576	−	0.801066i	$-0.704267\pi$
$701$	−31.6963	−1.19715	−0.598576	+	0.801066i	$0.704267\pi$
$702$	0	0
$703$	−12.1311	−0.457533
$704$	0	0
$705$	−43.4222	−1.63537
$706$	0	0
$707$	5.69996	0.214369
$708$	0	0
$709$	−28.9255	−1.08632	−0.543160	−	0.839629i	$-0.682772\pi$
$709$	−28.9255	−1.08632	−0.543160	+	0.839629i	$0.682772\pi$
$710$	0	0
$711$	−7.66410	−0.287426
$712$	0	0
$713$	−32.3151	−1.21021
$714$	0	0
$715$	0	0
$716$	0	0
$717$	31.8913	1.19100
$718$	0	0
$719$	18.7280	0.698436	0.349218	−	0.937041i	$-0.386447\pi$
$719$	18.7280	0.698436	0.349218	+	0.937041i	$0.386447\pi$
$720$	0	0
$721$	−9.73263	−0.362462
$722$	0	0
$723$	1.13244	0.0421159
$724$	0	0
$725$	−24.7432	−0.918939
$726$	0	0
$727$	0.675640	0.0250581	0.0125290	−	0.999922i	$-0.496012\pi$
$727$	0.675640	0.0250581	0.0125290	+	0.999922i	$0.496012\pi$
$728$	0	0
$729$	27.8015	1.02969
$730$	0	0
$731$	−23.6278	−0.873906
$732$	0	0
$733$	6.16976	0.227885	0.113943	−	0.993487i	$-0.463652\pi$
$733$	6.16976	0.227885	0.113943	+	0.993487i	$0.463652\pi$
$734$	0	0
$735$	−26.7618	−0.987125
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−42.8033	−1.57454	−0.787272	−	0.616606i	$-0.788507\pi$
$739$	−42.8033	−1.57454	−0.787272	+	0.616606i	$0.788507\pi$
$740$	0	0
$741$	3.42134	0.125686
$742$	0	0
$743$	13.7412	0.504117	0.252058	−	0.967712i	$-0.418893\pi$
$743$	13.7412	0.504117	0.252058	+	0.967712i	$0.418893\pi$
$744$	0	0
$745$	−33.0455	−1.21069
$746$	0	0
$747$	7.66751	0.280540
$748$	0	0
$749$	12.2675	0.448246
$750$	0	0
$751$	48.6339	1.77467	0.887337	−	0.461121i	$-0.152552\pi$
$751$	48.6339	1.77467	0.887337	+	0.461121i	$0.152552\pi$
$752$	0	0
$753$	4.51215	0.164432
$754$	0	0
$755$	−45.3602	−1.65083
$756$	0	0
$757$	−35.4703	−1.28919	−0.644596	−	0.764524i	$-0.722974\pi$
$757$	−35.4703	−1.28919	−0.644596	+	0.764524i	$0.722974\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	45.8693	1.66276	0.831381	−	0.555703i	$-0.187551\pi$
$761$	45.8693	1.66276	0.831381	+	0.555703i	$0.187551\pi$
$762$	0	0
$763$	1.17507	0.0425403
$764$	0	0
$765$	24.1421	0.872859
$766$	0	0
$767$	2.40187	0.0867264
$768$	0	0
$769$	−16.3468	−0.589481	−0.294740	−	0.955577i	$-0.595233\pi$
$769$	−16.3468	−0.589481	−0.294740	+	0.955577i	$0.595233\pi$
$770$	0	0
$771$	−29.5378	−1.06378
$772$	0	0
$773$	8.65819	0.311414	0.155707	−	0.987803i	$-0.450235\pi$
$773$	8.65819	0.311414	0.155707	+	0.987803i	$0.450235\pi$
$774$	0	0
$775$	32.7224	1.17542
$776$	0	0
$777$	−6.32937	−0.227065
$778$	0	0
$779$	20.1034	0.720281
$780$	0	0
$781$	0	0
$782$	0	0
$783$	24.8995	0.889834
$784$	0	0
$785$	62.1179	2.21708
$786$	0	0
$787$	3.54307	0.126297	0.0631485	−	0.998004i	$-0.479886\pi$
$787$	3.54307	0.126297	0.0631485	+	0.998004i	$0.479886\pi$
$788$	0	0
$789$	−8.36974	−0.297971
$790$	0	0
$791$	−13.4816	−0.479350
$792$	0	0
$793$	5.61900	0.199537
$794$	0	0
$795$	35.1693	1.24733
$796$	0	0
$797$	−47.1969	−1.67180	−0.835899	−	0.548883i	$-0.815053\pi$
$797$	−47.1969	−1.67180	−0.835899	+	0.548883i	$0.815053\pi$
$798$	0	0
$799$	62.7909	2.22138
$800$	0	0
$801$	6.09380	0.215314
$802$	0	0
$803$	0	0
$804$	0	0
$805$	17.5013	0.616841
$806$	0	0
$807$	−40.2334	−1.41628
$808$	0	0
$809$	44.2339	1.55518	0.777590	−	0.628771i	$-0.216442\pi$
$809$	44.2339	1.55518	0.777590	+	0.628771i	$0.216442\pi$
$810$	0	0
$811$	−21.0133	−0.737875	−0.368938	−	0.929454i	$-0.620278\pi$
$811$	−21.0133	−0.737875	−0.368938	+	0.929454i	$0.620278\pi$
$812$	0	0
$813$	20.3698	0.714399
$814$	0	0
$815$	−23.1121	−0.809581
$816$	0	0
$817$	−9.33074	−0.326441
$818$	0	0
$819$	−1.12607	−0.0393482
$820$	0	0
$821$	−34.8858	−1.21752	−0.608761	−	0.793354i	$-0.708333\pi$
$821$	−34.8858	−1.21752	−0.608761	+	0.793354i	$0.708333\pi$
$822$	0	0
$823$	−45.0167	−1.56919	−0.784593	−	0.620012i	$-0.787128\pi$
$823$	−45.0167	−1.56919	−0.784593	+	0.620012i	$0.787128\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26.8873	0.934963	0.467482	−	0.884003i	$-0.345161\pi$
$827$	26.8873	0.934963	0.467482	+	0.884003i	$0.345161\pi$
$828$	0	0
$829$	53.0488	1.84246	0.921231	−	0.389015i	$-0.127185\pi$
$829$	53.0488	1.84246	0.921231	+	0.389015i	$0.127185\pi$
$830$	0	0
$831$	−38.3705	−1.33106
$832$	0	0
$833$	38.6991	1.34085
$834$	0	0
$835$	−73.2259	−2.53409
$836$	0	0
$837$	−32.9290	−1.13819
$838$	0	0
$839$	−44.1937	−1.52574	−0.762868	−	0.646555i	$-0.776209\pi$
$839$	−44.1937	−1.52574	−0.762868	+	0.646555i	$0.776209\pi$
$840$	0	0
$841$	−9.52903	−0.328587
$842$	0	0
$843$	−16.5235	−0.569100
$844$	0	0
$845$	3.25690	0.112041
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−2.70462	−0.0928223
$850$	0	0
$851$	−26.6308	−0.912890
$852$	0	0
$853$	−23.7607	−0.813550	−0.406775	−	0.913528i	$-0.633347\pi$
$853$	−23.7607	−0.813550	−0.406775	+	0.913528i	$0.633347\pi$
$854$	0	0
$855$	9.53383	0.326050
$856$	0	0
$857$	−13.3542	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$857$	−13.3542	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$858$	0	0
$859$	−47.2379	−1.61174	−0.805869	−	0.592094i	$-0.798301\pi$
$859$	−47.2379	−1.61174	−0.805869	+	0.592094i	$0.798301\pi$
$860$	0	0
$861$	10.4889	0.357462
$862$	0	0
$863$	25.2165	0.858380	0.429190	−	0.903214i	$-0.358799\pi$
$863$	25.2165	0.858380	0.429190	+	0.903214i	$0.358799\pi$
$864$	0	0
$865$	57.4602	1.95371
$866$	0	0
$867$	32.2842	1.09643
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−7.79271	−0.264046
$872$	0	0
$873$	11.5988	0.392559
$874$	0	0
$875$	−1.91966	−0.0648964
$876$	0	0
$877$	−11.2579	−0.380151	−0.190076	−	0.981769i	$-0.560873\pi$
$877$	−11.2579	−0.380151	−0.190076	+	0.981769i	$0.560873\pi$
$878$	0	0
$879$	16.2224	0.547168
$880$	0	0
$881$	41.9676	1.41392	0.706962	−	0.707251i	$-0.250065\pi$
$881$	41.9676	1.41392	0.706962	+	0.707251i	$0.250065\pi$
$882$	0	0
$883$	3.09628	0.104198	0.0520990	−	0.998642i	$-0.483409\pi$
$883$	3.09628	0.104198	0.0520990	+	0.998642i	$0.483409\pi$
$884$	0	0
$885$	−10.6099	−0.356647
$886$	0	0
$887$	−21.1459	−0.710011	−0.355005	−	0.934864i	$-0.615521\pi$
$887$	−21.1459	−0.710011	−0.355005	+	0.934864i	$0.615521\pi$
$888$	0	0
$889$	−20.7801	−0.696941
$890$	0	0
$891$	0	0
$892$	0	0
$893$	24.7965	0.829782
$894$	0	0
$895$	0.183328	0.00612798
$896$	0	0
$897$	7.51069	0.250775
$898$	0	0
$899$	−25.7500	−0.858809
$900$	0	0
$901$	−50.8568	−1.69428
$902$	0	0
$903$	−4.86830	−0.162007
$904$	0	0
$905$	−21.6722	−0.720409
$906$	0	0
$907$	−42.5545	−1.41300	−0.706500	−	0.707713i	$-0.749727\pi$
$907$	−42.5545	−1.41300	−0.706500	+	0.707713i	$0.749727\pi$
$908$	0	0
$909$	6.81635	0.226084
$910$	0	0
$911$	−32.4046	−1.07361	−0.536806	−	0.843705i	$-0.680369\pi$
$911$	−32.4046	−1.07361	−0.536806	+	0.843705i	$0.680369\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−24.8211	−0.820559
$916$	0	0
$917$	6.15594	0.203287
$918$	0	0
$919$	36.4551	1.20254	0.601272	−	0.799045i	$-0.294661\pi$
$919$	36.4551	1.20254	0.601272	+	0.799045i	$0.294661\pi$
$920$	0	0
$921$	27.7816	0.915434
$922$	0	0
$923$	12.4430	0.409565
$924$	0	0
$925$	26.9664	0.886648
$926$	0	0
$927$	−11.6389	−0.382270
$928$	0	0
$929$	36.2316	1.18872	0.594360	−	0.804199i	$-0.297406\pi$
$929$	36.2316	1.18872	0.594360	+	0.804199i	$0.297406\pi$
$930$	0	0
$931$	15.2825	0.500863
$932$	0	0
$933$	−22.0150	−0.720737
$934$	0	0
$935$	0	0
$936$	0	0
$937$	18.5449	0.605836	0.302918	−	0.953017i	$-0.402039\pi$
$937$	18.5449	0.605836	0.302918	+	0.953017i	$0.402039\pi$
$938$	0	0
$939$	−25.4540	−0.830660
$940$	0	0
$941$	−13.6537	−0.445097	−0.222548	−	0.974922i	$-0.571437\pi$
$941$	−13.6537	−0.445097	−0.222548	+	0.974922i	$0.571437\pi$
$942$	0	0
$943$	44.1321	1.43714
$944$	0	0
$945$	17.8338	0.580134
$946$	0	0
$947$	−25.8355	−0.839542	−0.419771	−	0.907630i	$-0.637890\pi$
$947$	−25.8355	−0.839542	−0.419771	+	0.907630i	$0.637890\pi$
$948$	0	0
$949$	−7.32001	−0.237618
$950$	0	0
$951$	29.3431	0.951515
$952$	0	0
$953$	54.5460	1.76692	0.883459	−	0.468509i	$-0.155208\pi$
$953$	54.5460	1.76692	0.883459	+	0.468509i	$0.155208\pi$
$954$	0	0
$955$	23.6375	0.764891
$956$	0	0
$957$	0	0
$958$	0	0
$959$	11.5214	0.372046
$960$	0	0
$961$	3.05381	0.0985100
$962$	0	0
$963$	14.6703	0.472742
$964$	0	0
$965$	−57.8820	−1.86329
$966$	0	0
$967$	26.4913	0.851903	0.425952	−	0.904746i	$-0.359939\pi$
$967$	26.4913	0.851903	0.425952	+	0.904746i	$0.359939\pi$
$968$	0	0
$969$	21.8546	0.702070
$970$	0	0
$971$	8.89511	0.285458	0.142729	−	0.989762i	$-0.454412\pi$
$971$	8.89511	0.285458	0.142729	+	0.989762i	$0.454412\pi$
$972$	0	0
$973$	−5.29565	−0.169771
$974$	0	0
$975$	−7.60534	−0.243566
$976$	0	0
$977$	2.58181	0.0825995	0.0412998	−	0.999147i	$-0.486850\pi$
$977$	2.58181	0.0825995	0.0412998	+	0.999147i	$0.486850\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	1.40522	0.0448651
$982$	0	0
$983$	27.1597	0.866259	0.433129	−	0.901332i	$-0.357409\pi$
$983$	27.1597	0.866259	0.433129	+	0.901332i	$0.357409\pi$
$984$	0	0
$985$	79.2600	2.52543
$986$	0	0
$987$	12.9375	0.411805
$988$	0	0
$989$	−20.4833	−0.651331
$990$	0	0
$991$	44.1900	1.40374	0.701871	−	0.712304i	$-0.252348\pi$
$991$	44.1900	1.40374	0.701871	+	0.712304i	$0.252348\pi$
$992$	0	0
$993$	12.0323	0.381834
$994$	0	0
$995$	29.3087	0.929148
$996$	0	0
$997$	22.2376	0.704271	0.352135	−	0.935949i	$-0.385456\pi$
$997$	22.2376	0.704271	0.352135	+	0.935949i	$0.385456\pi$
$998$	0	0
$999$	−27.1367	−0.858567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6292.2.a.s.1.4	✓	5
11.10	odd	2		6292.2.a.t.1.4	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6292.2.a.s.1.4	✓	5	1.1	even	1	trivial
6292.2.a.t.1.4	yes	5	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.35630 q^{3} +3.25690 q^{5} -0.970383 q^{7} -1.16044 q^{9} +O(q^{10})\)	\(q+1.35630 q^{3} +3.25690 q^{5} -0.970383 q^{7} -1.16044 q^{9} -1.00000 q^{13} +4.41734 q^{15} -6.38773 q^{17} -2.52255 q^{19} -1.31613 q^{21} -5.53762 q^{23} +5.60740 q^{25} -5.64282 q^{27} -4.41259 q^{29} +5.83556 q^{31} -3.16044 q^{35} +4.80906 q^{37} -1.35630 q^{39} -7.96951 q^{41} +3.69894 q^{43} -3.77945 q^{45} -9.82994 q^{47} -6.05836 q^{49} -8.66369 q^{51} +7.96164 q^{53} -3.42134 q^{57} -2.40187 q^{59} -5.61900 q^{61} +1.12607 q^{63} -3.25690 q^{65} +7.79271 q^{67} -7.51069 q^{69} -12.4430 q^{71} +7.32001 q^{73} +7.60534 q^{75} +6.60446 q^{79} -4.17204 q^{81} -6.60740 q^{83} -20.8042 q^{85} -5.98481 q^{87} -5.25127 q^{89} +0.970383 q^{91} +7.91479 q^{93} -8.21568 q^{95} -9.99513 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 5 * q - q^3 - 2 * q^7 + 2 * q^9	\( 5 q - q^{3} - 2 q^{7} + 2 q^{9} - 5 q^{13} - 2 q^{15} - 5 q^{17} - 2 q^{19} + 4 q^{21} + 7 q^{23} - q^{25} - 7 q^{27} - 3 q^{29} + 10 q^{31} - 8 q^{35} + q^{39} - 8 q^{41} + 5 q^{43} + 8 q^{45} - 6 q^{47} - 7 q^{49} - 3 q^{51} + 15 q^{53} - 20 q^{57} - 12 q^{59} - 9 q^{61} + 10 q^{67} - 7 q^{69} - 14 q^{71} - 8 q^{73} + 21 q^{75} - 13 q^{79} - 23 q^{81} - 4 q^{83} - 24 q^{85} + 7 q^{87} + 14 q^{89} + 2 q^{91} - 22 q^{93} - 12 q^{95} + 6 q^{97}+O(q^{100}) \) 5 * q - q^3 - 2 * q^7 + 2 * q^9 - 5 * q^13 - 2 * q^15 - 5 * q^17 - 2 * q^19 + 4 * q^21 + 7 * q^23 - q^25 - 7 * q^27 - 3 * q^29 + 10 * q^31 - 8 * q^35 + q^39 - 8 * q^41 + 5 * q^43 + 8 * q^45 - 6 * q^47 - 7 * q^49 - 3 * q^51 + 15 * q^53 - 20 * q^57 - 12 * q^59 - 9 * q^61 + 10 * q^67 - 7 * q^69 - 14 * q^71 - 8 * q^73 + 21 * q^75 - 13 * q^79 - 23 * q^81 - 4 * q^83 - 24 * q^85 + 7 * q^87 + 14 * q^89 + 2 * q^91 - 22 * q^93 - 12 * q^95 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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