LMFDB - Embedded newform 6292.2.a.t.1.1

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:	$5$
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			1.1
Root			$0.521948$ 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)

$f(q)$	$=$	$q-2.61377 q^{3} +1.06033 q^{5} -1.72757 q^{7} +3.83180 q^{9} +O(q^{10})$	$q-2.61377 q^{3} +1.06033 q^{5} -1.72757 q^{7} +3.83180 q^{9} +1.00000 q^{13} -2.77147 q^{15} -3.49904 q^{17} -3.12331 q^{19} +4.51547 q^{21} -0.525979 q^{23} -3.87569 q^{25} -2.17413 q^{27} +10.4076 q^{29} -1.10329 q^{31} -1.83180 q^{35} -0.335411 q^{37} -2.61377 q^{39} -2.16721 q^{41} -1.13382 q^{43} +4.06298 q^{45} -8.63614 q^{47} -4.01550 q^{49} +9.14568 q^{51} +6.51641 q^{53} +8.16363 q^{57} -9.20273 q^{59} +2.76882 q^{61} -6.61970 q^{63} +1.06033 q^{65} +9.19915 q^{67} +1.37479 q^{69} +3.97899 q^{71} +10.1032 q^{73} +10.1302 q^{75} -1.69054 q^{79} -5.81271 q^{81} -2.87569 q^{83} -3.71014 q^{85} -27.2031 q^{87} -8.79976 q^{89} -1.72757 q^{91} +2.88376 q^{93} -3.31175 q^{95} +9.37473 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.61377	−1.50906	−0.754531	−	0.656265i	$-0.772135\pi$
$3$	−2.61377	−1.50906	−0.754531	+	0.656265i	$0.772135\pi$
$4$	0	0
$5$	1.06033	0.474195	0.237098	−	0.971486i	$-0.423804\pi$
$5$	1.06033	0.474195	0.237098	+	0.971486i	$0.423804\pi$
$6$	0	0
$7$	−1.72757	−0.652960	−0.326480	−	0.945204i	$-0.605863\pi$
$7$	−1.72757	−0.652960	−0.326480	+	0.945204i	$0.605863\pi$
$8$	0	0
$9$	3.83180	1.27727
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.77147	−0.715590
$16$	0	0
$17$	−3.49904	−0.848641	−0.424321	−	0.905512i	$-0.639487\pi$
$17$	−3.49904	−0.848641	−0.424321	+	0.905512i	$0.639487\pi$
$18$	0	0
$19$	−3.12331	−0.716537	−0.358269	−	0.933619i	$-0.616633\pi$
$19$	−3.12331	−0.716537	−0.358269	+	0.933619i	$0.616633\pi$
$20$	0	0
$21$	4.51547	0.985357
$22$	0	0
$23$	−0.525979	−0.109674	−0.0548371	−	0.998495i	$-0.517464\pi$
$23$	−0.525979	−0.109674	−0.0548371	+	0.998495i	$0.517464\pi$
$24$	0	0
$25$	−3.87569	−0.775139
$26$	0	0
$27$	−2.17413	−0.418412
$28$	0	0
$29$	10.4076	1.93264	0.966322	−	0.257337i	$-0.0828449\pi$
$29$	10.4076	1.93264	0.966322	+	0.257337i	$0.0828449\pi$
$30$	0	0
$31$	−1.10329	−0.198157	−0.0990787	−	0.995080i	$-0.531590\pi$
$31$	−1.10329	−0.198157	−0.0990787	+	0.995080i	$0.531590\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.83180	−0.309631
$36$	0	0
$37$	−0.335411	−0.0551412	−0.0275706	−	0.999620i	$-0.508777\pi$
$37$	−0.335411	−0.0551412	−0.0275706	+	0.999620i	$0.508777\pi$
$38$	0	0
$39$	−2.61377	−0.418538
$40$	0	0
$41$	−2.16721	−0.338461	−0.169231	−	0.985576i	$-0.554128\pi$
$41$	−2.16721	−0.338461	−0.169231	+	0.985576i	$0.554128\pi$
$42$	0	0
$43$	−1.13382	−0.172906	−0.0864529	−	0.996256i	$-0.527553\pi$
$43$	−1.13382	−0.172906	−0.0864529	+	0.996256i	$0.527553\pi$
$44$	0	0
$45$	4.06298	0.605673
$46$	0	0
$47$	−8.63614	−1.25971	−0.629855	−	0.776713i	$-0.716886\pi$
$47$	−8.63614	−1.25971	−0.629855	+	0.776713i	$0.716886\pi$
$48$	0	0
$49$	−4.01550	−0.573643
$50$	0	0
$51$	9.14568	1.28065
$52$	0	0
$53$	6.51641	0.895097	0.447549	−	0.894260i	$-0.352297\pi$
$53$	6.51641	0.895097	0.447549	+	0.894260i	$0.352297\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	8.16363	1.08130
$58$	0	0
$59$	−9.20273	−1.19809	−0.599047	−	0.800714i	$-0.704454\pi$
$59$	−9.20273	−1.19809	−0.599047	+	0.800714i	$0.704454\pi$
$60$	0	0
$61$	2.76882	0.354511	0.177255	−	0.984165i	$-0.443278\pi$
$61$	2.76882	0.354511	0.177255	+	0.984165i	$0.443278\pi$
$62$	0	0
$63$	−6.61970	−0.834004
$64$	0	0
$65$	1.06033	0.131518
$66$	0	0
$67$	9.19915	1.12385	0.561927	−	0.827187i	$-0.310060\pi$
$67$	9.19915	1.12385	0.561927	+	0.827187i	$0.310060\pi$
$68$	0	0
$69$	1.37479	0.165505
$70$	0	0
$71$	3.97899	0.472219	0.236110	−	0.971726i	$-0.424128\pi$
$71$	3.97899	0.472219	0.236110	+	0.971726i	$0.424128\pi$
$72$	0	0
$73$	10.1032	1.18249	0.591247	−	0.806490i	$-0.298636\pi$
$73$	10.1032	1.18249	0.591247	+	0.806490i	$0.298636\pi$
$74$	0	0
$75$	10.1302	1.16973
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−1.69054	−0.190201	−0.0951003	−	0.995468i	$-0.530317\pi$
$79$	−1.69054	−0.190201	−0.0951003	+	0.995468i	$0.530317\pi$
$80$	0	0
$81$	−5.81271	−0.645857
$82$	0	0
$83$	−2.87569	−0.315649	−0.157824	−	0.987467i	$-0.550448\pi$
$83$	−2.87569	−0.315649	−0.157824	+	0.987467i	$0.550448\pi$
$84$	0	0
$85$	−3.71014	−0.402421
$86$	0	0
$87$	−27.2031	−2.91648
$88$	0	0
$89$	−8.79976	−0.932773	−0.466387	−	0.884581i	$-0.654444\pi$
$89$	−8.79976	−0.932773	−0.466387	+	0.884581i	$0.654444\pi$
$90$	0	0
$91$	−1.72757	−0.181099
$92$	0	0
$93$	2.88376	0.299032
$94$	0	0
$95$	−3.31175	−0.339778
$96$	0	0
$97$	9.37473	0.951860	0.475930	−	0.879483i	$-0.342112\pi$
$97$	9.37473	0.951860	0.475930	+	0.879483i	$0.342112\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.380298	0.0378411	0.0189206	−	0.999821i	$-0.493977\pi$
$101$	0.380298	0.0378411	0.0189206	+	0.999821i	$0.493977\pi$
$102$	0	0
$103$	−10.7645	−1.06066	−0.530331	−	0.847791i	$-0.677932\pi$
$103$	−10.7645	−1.06066	−0.530331	+	0.847791i	$0.677932\pi$
$104$	0	0
$105$	4.78790	0.467251
$106$	0	0
$107$	1.73449	0.167680	0.0838399	−	0.996479i	$-0.473282\pi$
$107$	1.73449	0.167680	0.0838399	+	0.996479i	$0.473282\pi$
$108$	0	0
$109$	2.86732	0.274639	0.137320	−	0.990527i	$-0.456151\pi$
$109$	2.86732	0.274639	0.137320	+	0.990527i	$0.456151\pi$
$110$	0	0
$111$	0.876687	0.0832114
$112$	0	0
$113$	−2.52743	−0.237761	−0.118880	−	0.992909i	$-0.537931\pi$
$113$	−2.52743	−0.237761	−0.118880	+	0.992909i	$0.537931\pi$
$114$	0	0
$115$	−0.557712	−0.0520069
$116$	0	0
$117$	3.83180	0.354250
$118$	0	0
$119$	6.04483	0.554129
$120$	0	0
$121$	0	0
$122$	0	0
$123$	5.66459	0.510759
$124$	0	0
$125$	−9.41119	−0.841762
$126$	0	0
$127$	13.7590	1.22092	0.610458	−	0.792048i	$-0.290985\pi$
$127$	13.7590	1.22092	0.610458	+	0.792048i	$0.290985\pi$
$128$	0	0
$129$	2.96354	0.260925
$130$	0	0
$131$	−11.6979	−1.02205	−0.511026	−	0.859565i	$-0.670734\pi$
$131$	−11.6979	−1.02205	−0.511026	+	0.859565i	$0.670734\pi$
$132$	0	0
$133$	5.39574	0.467870
$134$	0	0
$135$	−2.30530	−0.198409
$136$	0	0
$137$	1.05934	0.0905056	0.0452528	−	0.998976i	$-0.485591\pi$
$137$	1.05934	0.0905056	0.0452528	+	0.998976i	$0.485591\pi$
$138$	0	0
$139$	−14.7845	−1.25400	−0.627001	−	0.779018i	$-0.715718\pi$
$139$	−14.7845	−1.25400	−0.627001	+	0.779018i	$0.715718\pi$
$140$	0	0
$141$	22.5729	1.90098
$142$	0	0
$143$	0	0
$144$	0	0
$145$	11.0355	0.916450
$146$	0	0
$147$	10.4956	0.865663
$148$	0	0
$149$	−11.5336	−0.944867	−0.472433	−	0.881366i	$-0.656624\pi$
$149$	−11.5336	−0.944867	−0.472433	+	0.881366i	$0.656624\pi$
$150$	0	0
$151$	−12.9789	−1.05621	−0.528105	−	0.849179i	$-0.677097\pi$
$151$	−12.9789	−1.05621	−0.528105	+	0.849179i	$0.677097\pi$
$152$	0	0
$153$	−13.4076	−1.08394
$154$	0	0
$155$	−1.16986	−0.0939652
$156$	0	0
$157$	−1.00530	−0.0802314	−0.0401157	−	0.999195i	$-0.512773\pi$
$157$	−1.00530	−0.0802314	−0.0401157	+	0.999195i	$0.512773\pi$
$158$	0	0
$159$	−17.0324	−1.35076
$160$	0	0
$161$	0.908665	0.0716129
$162$	0	0
$163$	22.6617	1.77500	0.887500	−	0.460809i	$-0.152441\pi$
$163$	22.6617	1.77500	0.887500	+	0.460809i	$0.152441\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.95346	0.615457	0.307728	−	0.951474i	$-0.400431\pi$
$167$	7.95346	0.615457	0.307728	+	0.951474i	$0.400431\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−11.9679	−0.915209
$172$	0	0
$173$	5.92494	0.450465	0.225233	−	0.974305i	$-0.427686\pi$
$173$	5.92494	0.450465	0.225233	+	0.974305i	$0.427686\pi$
$174$	0	0
$175$	6.69554	0.506135
$176$	0	0
$177$	24.0538	1.80800
$178$	0	0
$179$	10.0214	0.749033	0.374516	−	0.927220i	$-0.377809\pi$
$179$	10.0214	0.749033	0.374516	+	0.927220i	$0.377809\pi$
$180$	0	0
$181$	−0.756957	−0.0562642	−0.0281321	−	0.999604i	$-0.508956\pi$
$181$	−0.756957	−0.0562642	−0.0281321	+	0.999604i	$0.508956\pi$
$182$	0	0
$183$	−7.23706	−0.534979
$184$	0	0
$185$	−0.355647	−0.0261477
$186$	0	0
$187$	0	0
$188$	0	0
$189$	3.75597	0.273206
$190$	0	0
$191$	−20.0538	−1.45104	−0.725522	−	0.688199i	$-0.758402\pi$
$191$	−20.0538	−1.45104	−0.725522	+	0.688199i	$0.758402\pi$
$192$	0	0
$193$	17.1917	1.23748	0.618741	−	0.785595i	$-0.287643\pi$
$193$	17.1917	1.23748	0.618741	+	0.785595i	$0.287643\pi$
$194$	0	0
$195$	−2.77147	−0.198469
$196$	0	0
$197$	8.50642	0.606057	0.303029	−	0.952981i	$-0.402002\pi$
$197$	8.50642	0.606057	0.303029	+	0.952981i	$0.402002\pi$
$198$	0	0
$199$	−7.50133	−0.531755	−0.265877	−	0.964007i	$-0.585662\pi$
$199$	−7.50133	−0.531755	−0.265877	+	0.964007i	$0.585662\pi$
$200$	0	0
$201$	−24.0445	−1.69597
$202$	0	0
$203$	−17.9799	−1.26194
$204$	0	0
$205$	−2.29796	−0.160497
$206$	0	0
$207$	−2.01544	−0.140083
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−14.0273	−0.965680	−0.482840	−	0.875709i	$-0.660395\pi$
$211$	−14.0273	−0.965680	−0.482840	+	0.875709i	$0.660395\pi$
$212$	0	0
$213$	−10.4002	−0.712608
$214$	0	0
$215$	−1.20223	−0.0819911
$216$	0	0
$217$	1.90602	0.129389
$218$	0	0
$219$	−26.4075	−1.78446
$220$	0	0
$221$	−3.49904	−0.235371
$222$	0	0
$223$	7.09050	0.474815	0.237408	−	0.971410i	$-0.423702\pi$
$223$	7.09050	0.474815	0.237408	+	0.971410i	$0.423702\pi$
$224$	0	0
$225$	−14.8509	−0.990059
$226$	0	0
$227$	11.1415	0.739485	0.369742	−	0.929134i	$-0.379446\pi$
$227$	11.1415	0.739485	0.369742	+	0.929134i	$0.379446\pi$
$228$	0	0
$229$	−29.7512	−1.96601	−0.983007	−	0.183570i	$-0.941235\pi$
$229$	−29.7512	−1.96601	−0.983007	+	0.183570i	$0.941235\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	15.5546	1.01901	0.509507	−	0.860466i	$-0.329828\pi$
$233$	15.5546	1.01901	0.509507	+	0.860466i	$0.329828\pi$
$234$	0	0
$235$	−9.15718	−0.597348
$236$	0	0
$237$	4.41868	0.287024
$238$	0	0
$239$	10.0401	0.649440	0.324720	−	0.945810i	$-0.394730\pi$
$239$	10.0401	0.649440	0.324720	+	0.945810i	$0.394730\pi$
$240$	0	0
$241$	16.9360	1.09094	0.545471	−	0.838130i	$-0.316351\pi$
$241$	16.9360	1.09094	0.545471	+	0.838130i	$0.316351\pi$
$242$	0	0
$243$	21.7155	1.39305
$244$	0	0
$245$	−4.25777	−0.272019
$246$	0	0
$247$	−3.12331	−0.198732
$248$	0	0
$249$	7.51641	0.476333
$250$	0	0
$251$	29.2125	1.84387	0.921937	−	0.387340i	$-0.126606\pi$
$251$	29.2125	1.84387	0.921937	+	0.387340i	$0.126606\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	9.69746	0.607279
$256$	0	0
$257$	22.6634	1.41370	0.706852	−	0.707362i	$-0.250115\pi$
$257$	22.6634	1.41370	0.706852	+	0.707362i	$0.250115\pi$
$258$	0	0
$259$	0.579445	0.0360050
$260$	0	0
$261$	39.8798	2.46850
$262$	0	0
$263$	−8.68861	−0.535763	−0.267881	−	0.963452i	$-0.586324\pi$
$263$	−8.68861	−0.535763	−0.267881	+	0.963452i	$0.586324\pi$
$264$	0	0
$265$	6.90956	0.424451
$266$	0	0
$267$	23.0006	1.40761
$268$	0	0
$269$	−3.26053	−0.198798	−0.0993989	−	0.995048i	$-0.531692\pi$
$269$	−3.26053	−0.198798	−0.0993989	+	0.995048i	$0.531692\pi$
$270$	0	0
$271$	−10.8783	−0.660813	−0.330406	−	0.943839i	$-0.607186\pi$
$271$	−10.8783	−0.660813	−0.330406	+	0.943839i	$0.607186\pi$
$272$	0	0
$273$	4.51547	0.273289
$274$	0	0
$275$	0	0
$276$	0	0
$277$	29.6856	1.78364	0.891818	−	0.452395i	$-0.149430\pi$
$277$	29.6856	1.78364	0.891818	+	0.452395i	$0.149430\pi$
$278$	0	0
$279$	−4.22760	−0.253100
$280$	0	0
$281$	13.0176	0.776564	0.388282	−	0.921541i	$-0.373069\pi$
$281$	13.0176	0.776564	0.388282	+	0.921541i	$0.373069\pi$
$282$	0	0
$283$	11.1325	0.661757	0.330878	−	0.943673i	$-0.392655\pi$
$283$	11.1325	0.661757	0.330878	+	0.943673i	$0.392655\pi$
$284$	0	0
$285$	8.65616	0.512747
$286$	0	0
$287$	3.74401	0.221002
$288$	0	0
$289$	−4.75674	−0.279808
$290$	0	0
$291$	−24.5034	−1.43641
$292$	0	0
$293$	−3.07677	−0.179747	−0.0898734	−	0.995953i	$-0.528646\pi$
$293$	−3.07677	−0.179747	−0.0898734	+	0.995953i	$0.528646\pi$
$294$	0	0
$295$	−9.75795	−0.568130
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.525979	−0.0304181
$300$	0	0
$301$	1.95875	0.112901
$302$	0	0
$303$	−0.994013	−0.0571046
$304$	0	0
$305$	2.93587	0.168107
$306$	0	0
$307$	−5.95346	−0.339782	−0.169891	−	0.985463i	$-0.554342\pi$
$307$	−5.95346	−0.339782	−0.169891	+	0.985463i	$0.554342\pi$
$308$	0	0
$309$	28.1361	1.60060
$310$	0	0
$311$	3.61730	0.205118	0.102559	−	0.994727i	$-0.467297\pi$
$311$	3.61730	0.205118	0.102559	+	0.994727i	$0.467297\pi$
$312$	0	0
$313$	26.8323	1.51665	0.758326	−	0.651876i	$-0.226018\pi$
$313$	26.8323	1.51665	0.758326	+	0.651876i	$0.226018\pi$
$314$	0	0
$315$	−7.01908	−0.395481
$316$	0	0
$317$	−31.1522	−1.74968	−0.874841	−	0.484409i	$-0.839035\pi$
$317$	−31.1522	−1.74968	−0.874841	+	0.484409i	$0.839035\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−4.53357	−0.253039
$322$	0	0
$323$	10.9286	0.608083
$324$	0	0
$325$	−3.87569	−0.214985
$326$	0	0
$327$	−7.49452	−0.414448
$328$	0	0
$329$	14.9195	0.822541
$330$	0	0
$331$	13.6152	0.748358	0.374179	−	0.927356i	$-0.377925\pi$
$331$	13.6152	0.748358	0.374179	+	0.927356i	$0.377925\pi$
$332$	0	0
$333$	−1.28523	−0.0704300
$334$	0	0
$335$	9.75415	0.532926
$336$	0	0
$337$	21.6404	1.17883	0.589413	−	0.807832i	$-0.299359\pi$
$337$	21.6404	1.17883	0.589413	+	0.807832i	$0.299359\pi$
$338$	0	0
$339$	6.60613	0.358796
$340$	0	0
$341$	0	0
$342$	0	0
$343$	19.0301	1.02753
$344$	0	0
$345$	1.45773	0.0784817
$346$	0	0
$347$	22.5619	1.21119	0.605593	−	0.795774i	$-0.292936\pi$
$347$	22.5619	1.21119	0.605593	+	0.795774i	$0.292936\pi$
$348$	0	0
$349$	28.9731	1.55090	0.775449	−	0.631411i	$-0.217524\pi$
$349$	28.9731	1.55090	0.775449	+	0.631411i	$0.217524\pi$
$350$	0	0
$351$	−2.17413	−0.116047
$352$	0	0
$353$	3.26879	0.173980	0.0869901	−	0.996209i	$-0.472275\pi$
$353$	3.26879	0.173980	0.0869901	+	0.996209i	$0.472275\pi$
$354$	0	0
$355$	4.21905	0.223924
$356$	0	0
$357$	−15.7998	−0.836214
$358$	0	0
$359$	30.7712	1.62404	0.812021	−	0.583628i	$-0.198368\pi$
$359$	30.7712	1.62404	0.812021	+	0.583628i	$0.198368\pi$
$360$	0	0
$361$	−9.24491	−0.486574
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.7128	0.560733
$366$	0	0
$367$	−3.80449	−0.198593	−0.0992965	−	0.995058i	$-0.531659\pi$
$367$	−3.80449	−0.198593	−0.0992965	+	0.995058i	$0.531659\pi$
$368$	0	0
$369$	−8.30431	−0.432305
$370$	0	0
$371$	−11.2576	−0.584463
$372$	0	0
$373$	−6.00551	−0.310954	−0.155477	−	0.987840i	$-0.549691\pi$
$373$	−6.00551	−0.310954	−0.155477	+	0.987840i	$0.549691\pi$
$374$	0	0
$375$	24.5987	1.27027
$376$	0	0
$377$	10.4076	0.536019
$378$	0	0
$379$	−3.09679	−0.159071	−0.0795357	−	0.996832i	$-0.525344\pi$
$379$	−3.09679	−0.159071	−0.0795357	+	0.996832i	$0.525344\pi$
$380$	0	0
$381$	−35.9630	−1.84244
$382$	0	0
$383$	−2.44671	−0.125021	−0.0625105	−	0.998044i	$-0.519911\pi$
$383$	−2.44671	−0.125021	−0.0625105	+	0.998044i	$0.519911\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.34457	−0.220847
$388$	0	0
$389$	−11.2833	−0.572086	−0.286043	−	0.958217i	$-0.592340\pi$
$389$	−11.2833	−0.572086	−0.286043	+	0.958217i	$0.592340\pi$
$390$	0	0
$391$	1.84042	0.0930740
$392$	0	0
$393$	30.5757	1.54234
$394$	0	0
$395$	−1.79253	−0.0901922
$396$	0	0
$397$	1.02746	0.0515667	0.0257834	−	0.999668i	$-0.491792\pi$
$397$	1.02746	0.0515667	0.0257834	+	0.999668i	$0.491792\pi$
$398$	0	0
$399$	−14.1032	−0.706045
$400$	0	0
$401$	36.2161	1.80855	0.904274	−	0.426953i	$-0.140413\pi$
$401$	36.2161	1.80855	0.904274	+	0.426953i	$0.140413\pi$
$402$	0	0
$403$	−1.10329	−0.0549590
$404$	0	0
$405$	−6.16341	−0.306262
$406$	0	0
$407$	0	0
$408$	0	0
$409$	33.2073	1.64200	0.820998	−	0.570931i	$-0.193418\pi$
$409$	33.2073	1.64200	0.820998	+	0.570931i	$0.193418\pi$
$410$	0	0
$411$	−2.76887	−0.136579
$412$	0	0
$413$	15.8984	0.782307
$414$	0	0
$415$	−3.04919	−0.149679
$416$	0	0
$417$	38.6432	1.89237
$418$	0	0
$419$	−10.5673	−0.516247	−0.258124	−	0.966112i	$-0.583104\pi$
$419$	−10.5673	−0.516247	−0.258124	+	0.966112i	$0.583104\pi$
$420$	0	0
$421$	35.7207	1.74092	0.870460	−	0.492239i	$-0.163821\pi$
$421$	35.7207	1.74092	0.870460	+	0.492239i	$0.163821\pi$
$422$	0	0
$423$	−33.0919	−1.60899
$424$	0	0
$425$	13.5612	0.657815
$426$	0	0
$427$	−4.78333	−0.231481
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.8581	1.00470	0.502350	−	0.864664i	$-0.332469\pi$
$431$	20.8581	1.00470	0.502350	+	0.864664i	$0.332469\pi$
$432$	0	0
$433$	−7.87002	−0.378209	−0.189105	−	0.981957i	$-0.560559\pi$
$433$	−7.87002	−0.378209	−0.189105	+	0.981957i	$0.560559\pi$
$434$	0	0
$435$	−28.8443	−1.38298
$436$	0	0
$437$	1.64280	0.0785856
$438$	0	0
$439$	6.31575	0.301435	0.150717	−	0.988577i	$-0.451842\pi$
$439$	6.31575	0.301435	0.150717	+	0.988577i	$0.451842\pi$
$440$	0	0
$441$	−15.3866	−0.732695
$442$	0	0
$443$	−18.7920	−0.892833	−0.446417	−	0.894825i	$-0.647300\pi$
$443$	−18.7920	−0.892833	−0.446417	+	0.894825i	$0.647300\pi$
$444$	0	0
$445$	−9.33068	−0.442316
$446$	0	0
$447$	30.1461	1.42586
$448$	0	0
$449$	18.1325	0.855727	0.427863	−	0.903843i	$-0.359266\pi$
$449$	18.1325	0.855727	0.427863	+	0.903843i	$0.359266\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	33.9240	1.59389
$454$	0	0
$455$	−1.83180	−0.0858761
$456$	0	0
$457$	−16.0073	−0.748789	−0.374395	−	0.927269i	$-0.622149\pi$
$457$	−16.0073	−0.748789	−0.374395	+	0.927269i	$0.622149\pi$
$458$	0	0
$459$	7.60737	0.355081
$460$	0	0
$461$	36.9903	1.72281	0.861405	−	0.507918i	$-0.169585\pi$
$461$	36.9903	1.72281	0.861405	+	0.507918i	$0.169585\pi$
$462$	0	0
$463$	−33.3133	−1.54820	−0.774100	−	0.633063i	$-0.781798\pi$
$463$	−33.3133	−1.54820	−0.774100	+	0.633063i	$0.781798\pi$
$464$	0	0
$465$	3.05774	0.141799
$466$	0	0
$467$	−38.4745	−1.78039	−0.890193	−	0.455583i	$-0.849431\pi$
$467$	−38.4745	−1.78039	−0.890193	+	0.455583i	$0.849431\pi$
$468$	0	0
$469$	−15.8922	−0.733832
$470$	0	0
$471$	2.62762	0.121074
$472$	0	0
$473$	0	0
$474$	0	0
$475$	12.1050	0.555416
$476$	0	0
$477$	24.9696	1.14328
$478$	0	0
$479$	31.6992	1.44837	0.724186	−	0.689605i	$-0.242216\pi$
$479$	31.6992	1.44837	0.724186	+	0.689605i	$0.242216\pi$
$480$	0	0
$481$	−0.335411	−0.0152934
$482$	0	0
$483$	−2.37504	−0.108068
$484$	0	0
$485$	9.94033	0.451367
$486$	0	0
$487$	30.9972	1.40462	0.702308	−	0.711873i	$-0.252153\pi$
$487$	30.9972	1.40462	0.702308	+	0.711873i	$0.252153\pi$
$488$	0	0
$489$	−59.2324	−2.67858
$490$	0	0
$491$	11.6106	0.523980	0.261990	−	0.965071i	$-0.415621\pi$
$491$	11.6106	0.523980	0.261990	+	0.965071i	$0.415621\pi$
$492$	0	0
$493$	−36.4166	−1.64012
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.87398	−0.308340
$498$	0	0
$499$	23.9084	1.07029	0.535144	−	0.844761i	$-0.320257\pi$
$499$	23.9084	1.07029	0.535144	+	0.844761i	$0.320257\pi$
$500$	0	0
$501$	−20.7885	−0.928762
$502$	0	0
$503$	30.1818	1.34574	0.672870	−	0.739761i	$-0.265061\pi$
$503$	30.1818	1.34574	0.672870	+	0.739761i	$0.265061\pi$
$504$	0	0
$505$	0.403243	0.0179441
$506$	0	0
$507$	−2.61377	−0.116082
$508$	0	0
$509$	21.6382	0.959095	0.479547	−	0.877516i	$-0.340801\pi$
$509$	21.6382	0.959095	0.479547	+	0.877516i	$0.340801\pi$
$510$	0	0
$511$	−17.4541	−0.772122
$512$	0	0
$513$	6.79049	0.299808
$514$	0	0
$515$	−11.4140	−0.502961
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−15.4864	−0.679780
$520$	0	0
$521$	−0.322830	−0.0141435	−0.00707173	−	0.999975i	$-0.502251\pi$
$521$	−0.322830	−0.0141435	−0.00707173	+	0.999975i	$0.502251\pi$
$522$	0	0
$523$	24.9912	1.09279	0.546395	−	0.837528i	$-0.316000\pi$
$523$	24.9912	1.09279	0.546395	+	0.837528i	$0.316000\pi$
$524$	0	0
$525$	−17.5006	−0.763789
$526$	0	0
$527$	3.86046	0.168164
$528$	0	0
$529$	−22.7233	−0.987972
$530$	0	0
$531$	−35.2630	−1.53028
$532$	0	0
$533$	−2.16721	−0.0938723
$534$	0	0
$535$	1.83914	0.0795129
$536$	0	0
$537$	−26.1936	−1.13034
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−33.0263	−1.41991	−0.709956	−	0.704246i	$-0.751285\pi$
$541$	−33.0263	−1.41991	−0.709956	+	0.704246i	$0.751285\pi$
$542$	0	0
$543$	1.97851	0.0849061
$544$	0	0
$545$	3.04031	0.130233
$546$	0	0
$547$	28.2325	1.20713	0.603567	−	0.797312i	$-0.293746\pi$
$547$	28.2325	1.20713	0.603567	+	0.797312i	$0.293746\pi$
$548$	0	0
$549$	10.6096	0.452805
$550$	0	0
$551$	−32.5062	−1.38481
$552$	0	0
$553$	2.92053	0.124193
$554$	0	0
$555$	0.929579	0.0394584
$556$	0	0
$557$	−21.7209	−0.920344	−0.460172	−	0.887830i	$-0.652212\pi$
$557$	−21.7209	−0.920344	−0.460172	+	0.887830i	$0.652212\pi$
$558$	0	0
$559$	−1.13382	−0.0479554
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.53794	0.233396	0.116698	−	0.993167i	$-0.462769\pi$
$563$	5.53794	0.233396	0.116698	+	0.993167i	$0.462769\pi$
$564$	0	0
$565$	−2.67992	−0.112745
$566$	0	0
$567$	10.0419	0.421719
$568$	0	0
$569$	14.3142	0.600084	0.300042	−	0.953926i	$-0.402999\pi$
$569$	14.3142	0.600084	0.300042	+	0.953926i	$0.402999\pi$
$570$	0	0
$571$	−22.2153	−0.929681	−0.464840	−	0.885395i	$-0.653888\pi$
$571$	−22.2153	−0.929681	−0.464840	+	0.885395i	$0.653888\pi$
$572$	0	0
$573$	52.4161	2.18971
$574$	0	0
$575$	2.03853	0.0850127
$576$	0	0
$577$	−13.1097	−0.545763	−0.272882	−	0.962048i	$-0.587977\pi$
$577$	−13.1097	−0.545763	−0.272882	+	0.962048i	$0.587977\pi$
$578$	0	0
$579$	−44.9350	−1.86744
$580$	0	0
$581$	4.96797	0.206106
$582$	0	0
$583$	0	0
$584$	0	0
$585$	4.06298	0.167984
$586$	0	0
$587$	13.2708	0.547745	0.273873	−	0.961766i	$-0.411695\pi$
$587$	13.2708	0.547745	0.273873	+	0.961766i	$0.411695\pi$
$588$	0	0
$589$	3.44593	0.141987
$590$	0	0
$591$	−22.2338	−0.914578
$592$	0	0
$593$	−6.16026	−0.252971	−0.126486	−	0.991968i	$-0.540370\pi$
$593$	−6.16026	−0.252971	−0.126486	+	0.991968i	$0.540370\pi$
$594$	0	0
$595$	6.40953	0.262765
$596$	0	0
$597$	19.6068	0.802451
$598$	0	0
$599$	38.3239	1.56587	0.782935	−	0.622103i	$-0.213722\pi$
$599$	38.3239	1.56587	0.782935	+	0.622103i	$0.213722\pi$
$600$	0	0
$601$	−0.362983	−0.0148064	−0.00740320	−	0.999973i	$-0.502357\pi$
$601$	−0.362983	−0.0148064	−0.00740320	+	0.999973i	$0.502357\pi$
$602$	0	0
$603$	35.2493	1.43546
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−23.6258	−0.958941	−0.479471	−	0.877558i	$-0.659171\pi$
$607$	−23.6258	−0.958941	−0.479471	+	0.877558i	$0.659171\pi$
$608$	0	0
$609$	46.9953	1.90434
$610$	0	0
$611$	−8.63614	−0.349381
$612$	0	0
$613$	38.7914	1.56677	0.783385	−	0.621537i	$-0.213491\pi$
$613$	38.7914	1.56677	0.783385	+	0.621537i	$0.213491\pi$
$614$	0	0
$615$	6.00635	0.242199
$616$	0	0
$617$	−38.8976	−1.56596	−0.782979	−	0.622048i	$-0.786301\pi$
$617$	−38.8976	−1.56596	−0.782979	+	0.622048i	$0.786301\pi$
$618$	0	0
$619$	−0.0934779	−0.00375719	−0.00187860	−	0.999998i	$-0.500598\pi$
$619$	−0.0934779	−0.00375719	−0.00187860	+	0.999998i	$0.500598\pi$
$620$	0	0
$621$	1.14355	0.0458890
$622$	0	0
$623$	15.2022	0.609064
$624$	0	0
$625$	9.39949	0.375979
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.17361	0.0467951
$630$	0	0
$631$	−25.1957	−1.00302	−0.501512	−	0.865151i	$-0.667223\pi$
$631$	−25.1957	−1.00302	−0.501512	+	0.865151i	$0.667223\pi$
$632$	0	0
$633$	36.6642	1.45727
$634$	0	0
$635$	14.5892	0.578953
$636$	0	0
$637$	−4.01550	−0.159100
$638$	0	0
$639$	15.2467	0.603149
$640$	0	0
$641$	−1.97618	−0.0780546	−0.0390273	−	0.999238i	$-0.512426\pi$
$641$	−1.97618	−0.0780546	−0.0390273	+	0.999238i	$0.512426\pi$
$642$	0	0
$643$	−31.1013	−1.22651	−0.613257	−	0.789884i	$-0.710141\pi$
$643$	−31.1013	−1.22651	−0.613257	+	0.789884i	$0.710141\pi$
$644$	0	0
$645$	3.14234	0.123730
$646$	0	0
$647$	20.8670	0.820365	0.410182	−	0.912003i	$-0.365465\pi$
$647$	20.8670	0.820365	0.410182	+	0.912003i	$0.365465\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−4.98189	−0.195256
$652$	0	0
$653$	−32.5554	−1.27399	−0.636995	−	0.770868i	$-0.719823\pi$
$653$	−32.5554	−1.27399	−0.636995	+	0.770868i	$0.719823\pi$
$654$	0	0
$655$	−12.4037	−0.484652
$656$	0	0
$657$	38.7136	1.51036
$658$	0	0
$659$	−32.8071	−1.27798	−0.638991	−	0.769214i	$-0.720648\pi$
$659$	−32.8071	−1.27798	−0.638991	+	0.769214i	$0.720648\pi$
$660$	0	0
$661$	44.8367	1.74394	0.871972	−	0.489555i	$-0.162841\pi$
$661$	44.8367	1.74394	0.871972	+	0.489555i	$0.162841\pi$
$662$	0	0
$663$	9.14568	0.355189
$664$	0	0
$665$	5.72128	0.221862
$666$	0	0
$667$	−5.47418	−0.211961
$668$	0	0
$669$	−18.5329	−0.716525
$670$	0	0
$671$	0	0
$672$	0	0
$673$	15.7962	0.608897	0.304449	−	0.952529i	$-0.401528\pi$
$673$	15.7962	0.608897	0.304449	+	0.952529i	$0.401528\pi$
$674$	0	0
$675$	8.42627	0.324327
$676$	0	0
$677$	27.8031	1.06856	0.534279	−	0.845308i	$-0.320583\pi$
$677$	27.8031	1.06856	0.534279	+	0.845308i	$0.320583\pi$
$678$	0	0
$679$	−16.1955	−0.621527
$680$	0	0
$681$	−29.1212	−1.11593
$682$	0	0
$683$	−45.6478	−1.74666	−0.873332	−	0.487125i	$-0.838045\pi$
$683$	−45.6478	−1.74666	−0.873332	+	0.487125i	$0.838045\pi$
$684$	0	0
$685$	1.12325	0.0429173
$686$	0	0
$687$	77.7628	2.96683
$688$	0	0
$689$	6.51641	0.248255
$690$	0	0
$691$	30.0556	1.14337	0.571685	−	0.820474i	$-0.306290\pi$
$691$	30.0556	1.14337	0.571685	+	0.820474i	$0.306290\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.6765	−0.594642
$696$	0	0
$697$	7.58315	0.287232
$698$	0	0
$699$	−40.6561	−1.53776
$700$	0	0
$701$	−38.1936	−1.44255	−0.721277	−	0.692647i	$-0.756444\pi$
$701$	−38.1936	−1.44255	−0.721277	+	0.692647i	$0.756444\pi$
$702$	0	0
$703$	1.04759	0.0395107
$704$	0	0
$705$	23.9348	0.901435
$706$	0	0
$707$	−0.656992	−0.0247087
$708$	0	0
$709$	−23.0886	−0.867112	−0.433556	−	0.901127i	$-0.642741\pi$
$709$	−23.0886	−0.867112	−0.433556	+	0.901127i	$0.642741\pi$
$710$	0	0
$711$	−6.47781	−0.242937
$712$	0	0
$713$	0.580309	0.0217327
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−26.2425	−0.980045
$718$	0	0
$719$	9.25299	0.345078	0.172539	−	0.985003i	$-0.444803\pi$
$719$	9.25299	0.345078	0.172539	+	0.985003i	$0.444803\pi$
$720$	0	0
$721$	18.5965	0.692570
$722$	0	0
$723$	−44.2668	−1.64630
$724$	0	0
$725$	−40.3367	−1.49807
$726$	0	0
$727$	−8.05757	−0.298839	−0.149419	−	0.988774i	$-0.547740\pi$
$727$	−8.05757	−0.298839	−0.149419	+	0.988774i	$0.547740\pi$
$728$	0	0
$729$	−39.3212	−1.45634
$730$	0	0
$731$	3.96727	0.146735
$732$	0	0
$733$	5.63758	0.208229	0.104114	−	0.994565i	$-0.466799\pi$
$733$	5.63758	0.208229	0.104114	+	0.994565i	$0.466799\pi$
$734$	0	0
$735$	11.1288	0.410493
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−0.187170	−0.00688515	−0.00344257	−	0.999994i	$-0.501096\pi$
$739$	−0.187170	−0.00688515	−0.00344257	+	0.999994i	$0.501096\pi$
$740$	0	0
$741$	8.16363	0.299898
$742$	0	0
$743$	31.4934	1.15538	0.577691	−	0.816256i	$-0.303954\pi$
$743$	31.4934	1.15538	0.577691	+	0.816256i	$0.303954\pi$
$744$	0	0
$745$	−12.2294	−0.448051
$746$	0	0
$747$	−11.0191	−0.403167
$748$	0	0
$749$	−2.99646	−0.109488
$750$	0	0
$751$	35.5738	1.29811	0.649053	−	0.760743i	$-0.275165\pi$
$751$	35.5738	1.29811	0.649053	+	0.760743i	$0.275165\pi$
$752$	0	0
$753$	−76.3547	−2.78252
$754$	0	0
$755$	−13.7620	−0.500850
$756$	0	0
$757$	−33.9475	−1.23384	−0.616922	−	0.787024i	$-0.711621\pi$
$757$	−33.9475	−1.23384	−0.616922	+	0.787024i	$0.711621\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−22.9696	−0.832648	−0.416324	−	0.909216i	$-0.636682\pi$
$761$	−22.9696	−0.832648	−0.416324	+	0.909216i	$0.636682\pi$
$762$	0	0
$763$	−4.95350	−0.179329
$764$	0	0
$765$	−14.2165	−0.513999
$766$	0	0
$767$	−9.20273	−0.332291
$768$	0	0
$769$	19.3851	0.699044	0.349522	−	0.936928i	$-0.386344\pi$
$769$	19.3851	0.699044	0.349522	+	0.936928i	$0.386344\pi$
$770$	0	0
$771$	−59.2369	−2.13336
$772$	0	0
$773$	18.8249	0.677083	0.338542	−	0.940951i	$-0.390066\pi$
$773$	18.8249	0.677083	0.338542	+	0.940951i	$0.390066\pi$
$774$	0	0
$775$	4.27603	0.153599
$776$	0	0
$777$	−1.51454	−0.0543337
$778$	0	0
$779$	6.76887	0.242520
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−22.6275	−0.808641
$784$	0	0
$785$	−1.06595	−0.0380453
$786$	0	0
$787$	16.2751	0.580146	0.290073	−	0.957005i	$-0.406320\pi$
$787$	16.2751	0.580146	0.290073	+	0.957005i	$0.406320\pi$
$788$	0	0
$789$	22.7100	0.808499
$790$	0	0
$791$	4.36632	0.155248
$792$	0	0
$793$	2.76882	0.0983236
$794$	0	0
$795$	−18.0600	−0.640522
$796$	0	0
$797$	35.1108	1.24369	0.621844	−	0.783141i	$-0.286384\pi$
$797$	35.1108	1.24369	0.621844	+	0.783141i	$0.286384\pi$
$798$	0	0
$799$	30.2182	1.06904
$800$	0	0
$801$	−33.7189	−1.19140
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0.963487	0.0339585
$806$	0	0
$807$	8.52227	0.299998
$808$	0	0
$809$	−3.30349	−0.116145	−0.0580723	−	0.998312i	$-0.518495\pi$
$809$	−3.30349	−0.116145	−0.0580723	+	0.998312i	$0.518495\pi$
$810$	0	0
$811$	−49.2759	−1.73031	−0.865155	−	0.501504i	$-0.832780\pi$
$811$	−49.2759	−1.73031	−0.865155	+	0.501504i	$0.832780\pi$
$812$	0	0
$813$	28.4335	0.997207
$814$	0	0
$815$	24.0289	0.841696
$816$	0	0
$817$	3.54127	0.123893
$818$	0	0
$819$	−6.61970	−0.231311
$820$	0	0
$821$	41.8865	1.46185	0.730925	−	0.682458i	$-0.239089\pi$
$821$	41.8865	1.46185	0.730925	+	0.682458i	$0.239089\pi$
$822$	0	0
$823$	55.6961	1.94144	0.970722	−	0.240207i	$-0.0772154\pi$
$823$	55.6961	1.94144	0.970722	+	0.240207i	$0.0772154\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	25.5966	0.890079	0.445040	−	0.895511i	$-0.353190\pi$
$827$	25.5966	0.890079	0.445040	+	0.895511i	$0.353190\pi$
$828$	0	0
$829$	−11.4191	−0.396603	−0.198302	−	0.980141i	$-0.563543\pi$
$829$	−11.4191	−0.396603	−0.198302	+	0.980141i	$0.563543\pi$
$830$	0	0
$831$	−77.5914	−2.69162
$832$	0	0
$833$	14.0504	0.486817
$834$	0	0
$835$	8.43331	0.291847
$836$	0	0
$837$	2.39871	0.0829114
$838$	0	0
$839$	−24.6701	−0.851706	−0.425853	−	0.904792i	$-0.640026\pi$
$839$	−24.6701	−0.851706	−0.425853	+	0.904792i	$0.640026\pi$
$840$	0	0
$841$	79.3182	2.73511
$842$	0	0
$843$	−34.0250	−1.17188
$844$	0	0
$845$	1.06033	0.0364765
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−29.0977	−0.998631
$850$	0	0
$851$	0.176419	0.00604756
$852$	0	0
$853$	−43.8247	−1.50053	−0.750264	−	0.661139i	$-0.770074\pi$
$853$	−43.8247	−1.50053	−0.750264	+	0.661139i	$0.770074\pi$
$854$	0	0
$855$	−12.6900	−0.433988
$856$	0	0
$857$	21.3921	0.730740	0.365370	−	0.930862i	$-0.380942\pi$
$857$	21.3921	0.730740	0.365370	+	0.930862i	$0.380942\pi$
$858$	0	0
$859$	14.5083	0.495016	0.247508	−	0.968886i	$-0.420388\pi$
$859$	14.5083	0.495016	0.247508	+	0.968886i	$0.420388\pi$
$860$	0	0
$861$	−9.78598	−0.333505
$862$	0	0
$863$	4.73341	0.161127	0.0805636	−	0.996749i	$-0.474328\pi$
$863$	4.73341	0.161127	0.0805636	+	0.996749i	$0.474328\pi$
$864$	0	0
$865$	6.28241	0.213608
$866$	0	0
$867$	12.4330	0.422248
$868$	0	0
$869$	0	0
$870$	0	0
$871$	9.19915	0.311701
$872$	0	0
$873$	35.9221	1.21578
$874$	0	0
$875$	16.2585	0.549637
$876$	0	0
$877$	−35.8413	−1.21028	−0.605138	−	0.796120i	$-0.706882\pi$
$877$	−35.8413	−1.21028	−0.605138	+	0.796120i	$0.706882\pi$
$878$	0	0
$879$	8.04197	0.271249
$880$	0	0
$881$	−26.6377	−0.897446	−0.448723	−	0.893671i	$-0.648121\pi$
$881$	−26.6377	−0.897446	−0.448723	+	0.893671i	$0.648121\pi$
$882$	0	0
$883$	25.4445	0.856275	0.428138	−	0.903714i	$-0.359170\pi$
$883$	25.4445	0.856275	0.428138	+	0.903714i	$0.359170\pi$
$884$	0	0
$885$	25.5051	0.857343
$886$	0	0
$887$	−29.6943	−0.997038	−0.498519	−	0.866879i	$-0.666123\pi$
$887$	−29.6943	−0.997038	−0.498519	+	0.866879i	$0.666123\pi$
$888$	0	0
$889$	−23.7697	−0.797210
$890$	0	0
$891$	0	0
$892$	0	0
$893$	26.9734	0.902629
$894$	0	0
$895$	10.6260	0.355188
$896$	0	0
$897$	1.37479	0.0459028
$898$	0	0
$899$	−11.4826	−0.382968
$900$	0	0
$901$	−22.8012	−0.759616
$902$	0	0
$903$	−5.11973	−0.170374
$904$	0	0
$905$	−0.802626	−0.0266802
$906$	0	0
$907$	54.1084	1.79664	0.898320	−	0.439341i	$-0.144788\pi$
$907$	54.1084	1.79664	0.898320	+	0.439341i	$0.144788\pi$
$908$	0	0
$909$	1.45723	0.0483332
$910$	0	0
$911$	25.3492	0.839856	0.419928	−	0.907558i	$-0.362055\pi$
$911$	25.3492	0.839856	0.419928	+	0.907558i	$0.362055\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−7.67369	−0.253684
$916$	0	0
$917$	20.2090	0.667359
$918$	0	0
$919$	23.9048	0.788547	0.394273	−	0.918993i	$-0.370996\pi$
$919$	23.9048	0.788547	0.394273	+	0.918993i	$0.370996\pi$
$920$	0	0
$921$	15.5610	0.512751
$922$	0	0
$923$	3.97899	0.130970
$924$	0	0
$925$	1.29995	0.0427421
$926$	0	0
$927$	−41.2476	−1.35475
$928$	0	0
$929$	−23.2687	−0.763422	−0.381711	−	0.924282i	$-0.624665\pi$
$929$	−23.2687	−0.763422	−0.381711	+	0.924282i	$0.624665\pi$
$930$	0	0
$931$	12.5417	0.411037
$932$	0	0
$933$	−9.45479	−0.309536
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−35.1206	−1.14734	−0.573669	−	0.819087i	$-0.694480\pi$
$937$	−35.1206	−1.14734	−0.573669	+	0.819087i	$0.694480\pi$
$938$	0	0
$939$	−70.1335	−2.28872
$940$	0	0
$941$	17.2164	0.561239	0.280620	−	0.959819i	$-0.409460\pi$
$941$	17.2164	0.561239	0.280620	+	0.959819i	$0.409460\pi$
$942$	0	0
$943$	1.13991	0.0371205
$944$	0	0
$945$	3.98257	0.129553
$946$	0	0
$947$	−2.50223	−0.0813115	−0.0406557	−	0.999173i	$-0.512945\pi$
$947$	−2.50223	−0.0813115	−0.0406557	+	0.999173i	$0.512945\pi$
$948$	0	0
$949$	10.1032	0.327965
$950$	0	0
$951$	81.4248	2.64038
$952$	0	0
$953$	11.5106	0.372865	0.186433	−	0.982468i	$-0.440307\pi$
$953$	11.5106	0.372865	0.186433	+	0.982468i	$0.440307\pi$
$954$	0	0
$955$	−21.2637	−0.688078
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.83009	−0.0590965
$960$	0	0
$961$	−29.7827	−0.960734
$962$	0	0
$963$	6.64623	0.214172
$964$	0	0
$965$	18.2289	0.586808
$966$	0	0
$967$	32.6770	1.05082	0.525411	−	0.850849i	$-0.323912\pi$
$967$	32.6770	1.05082	0.525411	+	0.850849i	$0.323912\pi$
$968$	0	0
$969$	−28.5648	−0.917634
$970$	0	0
$971$	−19.5333	−0.626854	−0.313427	−	0.949612i	$-0.601477\pi$
$971$	−19.5333	−0.626854	−0.313427	+	0.949612i	$0.601477\pi$
$972$	0	0
$973$	25.5412	0.818813
$974$	0	0
$975$	10.1302	0.324425
$976$	0	0
$977$	27.9811	0.895193	0.447597	−	0.894236i	$-0.352280\pi$
$977$	27.9811	0.895193	0.447597	+	0.894236i	$0.352280\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	10.9870	0.350788
$982$	0	0
$983$	−37.6128	−1.19966	−0.599830	−	0.800127i	$-0.704765\pi$
$983$	−37.6128	−1.19966	−0.599830	+	0.800127i	$0.704765\pi$
$984$	0	0
$985$	9.01963	0.287389
$986$	0	0
$987$	−38.9962	−1.24126
$988$	0	0
$989$	0.596365	0.0189633
$990$	0	0
$991$	−4.89539	−0.155507	−0.0777537	−	0.996973i	$-0.524775\pi$
$991$	−4.89539	−0.155507	−0.0777537	+	0.996973i	$0.524775\pi$
$992$	0	0
$993$	−35.5870	−1.12932
$994$	0	0
$995$	−7.95390	−0.252156
$996$	0	0
$997$	−60.0103	−1.90055	−0.950273	−	0.311418i	$-0.899196\pi$
$997$	−60.0103	−1.90055	−0.950273	+	0.311418i	$0.899196\pi$
$998$	0	0
$999$	0.729227	0.0230717

Twists

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

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

\(f(q)\)	\(=\)	\(q-2.61377 q^{3} +1.06033 q^{5} -1.72757 q^{7} +3.83180 q^{9} +O(q^{10})\)	\(q-2.61377 q^{3} +1.06033 q^{5} -1.72757 q^{7} +3.83180 q^{9} +1.00000 q^{13} -2.77147 q^{15} -3.49904 q^{17} -3.12331 q^{19} +4.51547 q^{21} -0.525979 q^{23} -3.87569 q^{25} -2.17413 q^{27} +10.4076 q^{29} -1.10329 q^{31} -1.83180 q^{35} -0.335411 q^{37} -2.61377 q^{39} -2.16721 q^{41} -1.13382 q^{43} +4.06298 q^{45} -8.63614 q^{47} -4.01550 q^{49} +9.14568 q^{51} +6.51641 q^{53} +8.16363 q^{57} -9.20273 q^{59} +2.76882 q^{61} -6.61970 q^{63} +1.06033 q^{65} +9.19915 q^{67} +1.37479 q^{69} +3.97899 q^{71} +10.1032 q^{73} +10.1302 q^{75} -1.69054 q^{79} -5.81271 q^{81} -2.87569 q^{83} -3.71014 q^{85} -27.2031 q^{87} -8.79976 q^{89} -1.72757 q^{91} +2.88376 q^{93} -3.31175 q^{95} +9.37473 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 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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