LMFDB - Embedded newform 5292.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(1,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, 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("5292.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.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:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	5292.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +O(q^{10})$	$q-1.00000 q^{5} +5.29150 q^{11} +5.29150 q^{13} +3.00000 q^{17} +5.29150 q^{19} -4.00000 q^{25} +5.29150 q^{29} -5.29150 q^{31} -3.00000 q^{37} +9.00000 q^{41} +1.00000 q^{43} -1.00000 q^{47} -10.5830 q^{53} -5.29150 q^{55} +11.0000 q^{59} -5.29150 q^{61} -5.29150 q^{65} +4.00000 q^{67} -10.5830 q^{71} +10.5830 q^{73} -11.0000 q^{79} +9.00000 q^{83} -3.00000 q^{85} -14.0000 q^{89} -5.29150 q^{95} -5.29150 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5}+O(q^{10}) $ 2 * q - 2 * q^5	$ 2 q - 2 q^{5} + 6 q^{17} - 8 q^{25} - 6 q^{37} + 18 q^{41} + 2 q^{43} - 2 q^{47} + 22 q^{59} + 8 q^{67} - 22 q^{79} + 18 q^{83} - 6 q^{85} - 28 q^{89}+O(q^{100}) $ 2 * q - 2 * q^5 + 6 * q^17 - 8 * q^25 - 6 * q^37 + 18 * q^41 + 2 * q^43 - 2 * q^47 + 22 * q^59 + 8 * q^67 - 22 * q^79 + 18 * q^83 - 6 * q^85 - 28 * q^89

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.29150	1.59545	0.797724	−	0.603023i	$-0.206037\pi$
$11$	5.29150	1.59545	0.797724	+	0.603023i	$0.206037\pi$
$12$	0	0
$13$	5.29150	1.46760	0.733799	−	0.679366i	$-0.237745\pi$
$13$	5.29150	1.46760	0.733799	+	0.679366i	$0.237745\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	5.29150	1.21395	0.606977	−	0.794719i	$-0.292382\pi$
$19$	5.29150	1.21395	0.606977	+	0.794719i	$0.292382\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.29150	0.982607	0.491304	−	0.870988i	$-0.336521\pi$
$29$	5.29150	0.982607	0.491304	+	0.870988i	$0.336521\pi$
$30$	0	0
$31$	−5.29150	−0.950382	−0.475191	−	0.879883i	$-0.657621\pi$
$31$	−5.29150	−0.950382	−0.475191	+	0.879883i	$0.657621\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	9.00000	1.40556	0.702782	−	0.711405i	$-0.251941\pi$
$41$	9.00000	1.40556	0.702782	+	0.711405i	$0.251941\pi$
$42$	0	0
$43$	1.00000	0.152499	0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000	0.152499	0.0762493	+	0.997089i	$0.475706\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.00000	−0.145865	−0.0729325	−	0.997337i	$-0.523236\pi$
$47$	−1.00000	−0.145865	−0.0729325	+	0.997337i	$0.523236\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.5830	−1.45369	−0.726844	−	0.686803i	$-0.759014\pi$
$53$	−10.5830	−1.45369	−0.726844	+	0.686803i	$0.759014\pi$
$54$	0	0
$55$	−5.29150	−0.713506
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.0000	1.43208	0.716039	−	0.698060i	$-0.245953\pi$
$59$	11.0000	1.43208	0.716039	+	0.698060i	$0.245953\pi$
$60$	0	0
$61$	−5.29150	−0.677507	−0.338754	−	0.940875i	$-0.610005\pi$
$61$	−5.29150	−0.677507	−0.338754	+	0.940875i	$0.610005\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.29150	−0.656330
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.5830	−1.25597	−0.627986	−	0.778225i	$-0.716120\pi$
$71$	−10.5830	−1.25597	−0.627986	+	0.778225i	$0.716120\pi$
$72$	0	0
$73$	10.5830	1.23865	0.619324	−	0.785136i	$-0.287407\pi$
$73$	10.5830	1.23865	0.619324	+	0.785136i	$0.287407\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.29150	−0.542897
$96$	0	0
$97$	−5.29150	−0.537271	−0.268635	−	0.963242i	$-0.586573\pi$
$97$	−5.29150	−0.537271	−0.268635	+	0.963242i	$0.586573\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	14.0000	1.39305	0.696526	−	0.717532i	$-0.254728\pi$
$101$	14.0000	1.39305	0.696526	+	0.717532i	$0.254728\pi$
$102$	0	0
$103$	−10.5830	−1.04277	−0.521387	−	0.853320i	$-0.674585\pi$
$103$	−10.5830	−1.04277	−0.521387	+	0.853320i	$0.674585\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.5830	1.02310	0.511549	−	0.859254i	$-0.329072\pi$
$107$	10.5830	1.02310	0.511549	+	0.859254i	$0.329072\pi$
$108$	0	0
$109$	−5.00000	−0.478913	−0.239457	−	0.970907i	$-0.576969\pi$
$109$	−5.00000	−0.478913	−0.239457	+	0.970907i	$0.576969\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−5.29150	−0.497783	−0.248891	−	0.968531i	$-0.580066\pi$
$113$	−5.29150	−0.497783	−0.248891	+	0.968531i	$0.580066\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	17.0000	1.54545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	9.00000	0.804984
$126$	0	0
$127$	−1.00000	−0.0887357	−0.0443678	−	0.999015i	$-0.514127\pi$
$127$	−1.00000	−0.0887357	−0.0443678	+	0.999015i	$0.514127\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.29150	0.452084	0.226042	−	0.974118i	$-0.427421\pi$
$137$	5.29150	0.452084	0.226042	+	0.974118i	$0.427421\pi$
$138$	0	0
$139$	−15.8745	−1.34646	−0.673229	−	0.739434i	$-0.735093\pi$
$139$	−15.8745	−1.34646	−0.673229	+	0.739434i	$0.735093\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	28.0000	2.34148
$144$	0	0
$145$	−5.29150	−0.439435
$146$	0	0
$147$	0	0
$148$	0	0
$149$	5.29150	0.433497	0.216748	−	0.976228i	$-0.430455\pi$
$149$	5.29150	0.433497	0.216748	+	0.976228i	$0.430455\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.29150	0.425024
$156$	0	0
$157$	5.29150	0.422308	0.211154	−	0.977453i	$-0.432278\pi$
$157$	5.29150	0.422308	0.211154	+	0.977453i	$0.432278\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3.00000	0.234978	0.117489	−	0.993074i	$-0.462515\pi$
$163$	3.00000	0.234978	0.117489	+	0.993074i	$0.462515\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.0000	1.47026	0.735132	−	0.677924i	$-0.237120\pi$
$167$	19.0000	1.47026	0.735132	+	0.677924i	$0.237120\pi$
$168$	0	0
$169$	15.0000	1.15385
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.8745	1.18652	0.593258	−	0.805012i	$-0.297841\pi$
$179$	15.8745	1.18652	0.593258	+	0.805012i	$0.297841\pi$
$180$	0	0
$181$	10.5830	0.786629	0.393314	−	0.919404i	$-0.371328\pi$
$181$	10.5830	0.786629	0.393314	+	0.919404i	$0.371328\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.00000	0.220564
$186$	0	0
$187$	15.8745	1.16086
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	23.0000	1.65558	0.827788	−	0.561041i	$-0.189599\pi$
$193$	23.0000	1.65558	0.827788	+	0.561041i	$0.189599\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.8745	1.13101	0.565506	−	0.824744i	$-0.308681\pi$
$197$	15.8745	1.13101	0.565506	+	0.824744i	$0.308681\pi$
$198$	0	0
$199$	−21.1660	−1.50042	−0.750209	−	0.661200i	$-0.770047\pi$
$199$	−21.1660	−1.50042	−0.750209	+	0.661200i	$0.770047\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−9.00000	−0.628587
$206$	0	0
$207$	0	0
$208$	0	0
$209$	28.0000	1.93680
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	15.8745	1.06783
$222$	0	0
$223$	26.4575	1.77173	0.885863	−	0.463947i	$-0.153567\pi$
$223$	26.4575	1.77173	0.885863	+	0.463947i	$0.153567\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	28.0000	1.85843	0.929213	−	0.369546i	$-0.120487\pi$
$227$	28.0000	1.85843	0.929213	+	0.369546i	$0.120487\pi$
$228$	0	0
$229$	21.1660	1.39869	0.699345	−	0.714785i	$-0.253475\pi$
$229$	21.1660	1.39869	0.699345	+	0.714785i	$0.253475\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−26.4575	−1.73329	−0.866645	−	0.498926i	$-0.833728\pi$
$233$	−26.4575	−1.73329	−0.866645	+	0.498926i	$0.833728\pi$
$234$	0	0
$235$	1.00000	0.0652328
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.8745	1.02684	0.513418	−	0.858138i	$-0.328379\pi$
$239$	15.8745	1.02684	0.513418	+	0.858138i	$0.328379\pi$
$240$	0	0
$241$	−5.29150	−0.340856	−0.170428	−	0.985370i	$-0.554515\pi$
$241$	−5.29150	−0.340856	−0.170428	+	0.985370i	$0.554515\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	28.0000	1.78160
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5.00000	0.315597	0.157799	−	0.987471i	$-0.449560\pi$
$251$	5.00000	0.315597	0.157799	+	0.987471i	$0.449560\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−26.4575	−1.63144	−0.815720	−	0.578447i	$-0.803659\pi$
$263$	−26.4575	−1.63144	−0.815720	+	0.578447i	$0.803659\pi$
$264$	0	0
$265$	10.5830	0.650109
$266$	0	0
$267$	0	0
$268$	0	0
$269$	11.0000	0.670682	0.335341	−	0.942097i	$-0.391148\pi$
$269$	11.0000	0.670682	0.335341	+	0.942097i	$0.391148\pi$
$270$	0	0
$271$	−10.5830	−0.642872	−0.321436	−	0.946931i	$-0.604165\pi$
$271$	−10.5830	−0.642872	−0.321436	+	0.946931i	$0.604165\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−21.1660	−1.27636
$276$	0	0
$277$	−23.0000	−1.38194	−0.690968	−	0.722885i	$-0.742815\pi$
$277$	−23.0000	−1.38194	−0.690968	+	0.722885i	$0.742815\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−31.7490	−1.89399	−0.946994	−	0.321252i	$-0.895896\pi$
$281$	−31.7490	−1.89399	−0.946994	+	0.321252i	$0.895896\pi$
$282$	0	0
$283$	26.4575	1.57274	0.786368	−	0.617758i	$-0.211959\pi$
$283$	26.4575	1.57274	0.786368	+	0.617758i	$0.211959\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.00000	0.525786	0.262893	−	0.964825i	$-0.415323\pi$
$293$	9.00000	0.525786	0.262893	+	0.964825i	$0.415323\pi$
$294$	0	0
$295$	−11.0000	−0.640445
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.29150	0.302991
$306$	0	0
$307$	10.5830	0.604004	0.302002	−	0.953307i	$-0.402345\pi$
$307$	10.5830	0.604004	0.302002	+	0.953307i	$0.402345\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	31.0000	1.75785	0.878924	−	0.476961i	$-0.158262\pi$
$311$	31.0000	1.75785	0.878924	+	0.476961i	$0.158262\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.7490	1.78320	0.891601	−	0.452822i	$-0.149583\pi$
$317$	31.7490	1.78320	0.891601	+	0.452822i	$0.149583\pi$
$318$	0	0
$319$	28.0000	1.56770
$320$	0	0
$321$	0	0
$322$	0	0
$323$	15.8745	0.883281
$324$	0	0
$325$	−21.1660	−1.17408
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−31.0000	−1.70391	−0.851957	−	0.523612i	$-0.824584\pi$
$331$	−31.0000	−1.70391	−0.851957	+	0.523612i	$0.824584\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−13.0000	−0.708155	−0.354078	−	0.935216i	$-0.615205\pi$
$337$	−13.0000	−0.708155	−0.354078	+	0.935216i	$0.615205\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−28.0000	−1.51629
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.8745	−0.852188	−0.426094	−	0.904679i	$-0.640111\pi$
$347$	−15.8745	−0.852188	−0.426094	+	0.904679i	$0.640111\pi$
$348$	0	0
$349$	−31.7490	−1.69949	−0.849743	−	0.527197i	$-0.823243\pi$
$349$	−31.7490	−1.69949	−0.849743	+	0.527197i	$0.823243\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.00000	0.159674	0.0798369	−	0.996808i	$-0.474560\pi$
$353$	3.00000	0.159674	0.0798369	+	0.996808i	$0.474560\pi$
$354$	0	0
$355$	10.5830	0.561688
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.5830	0.558550	0.279275	−	0.960211i	$-0.409906\pi$
$359$	10.5830	0.558550	0.279275	+	0.960211i	$0.409906\pi$
$360$	0	0
$361$	9.00000	0.473684
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10.5830	−0.553940
$366$	0	0
$367$	10.5830	0.552428	0.276214	−	0.961096i	$-0.410920\pi$
$367$	10.5830	0.552428	0.276214	+	0.961096i	$0.410920\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	31.0000	1.60512	0.802560	−	0.596572i	$-0.203471\pi$
$373$	31.0000	1.60512	0.802560	+	0.596572i	$0.203471\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	28.0000	1.44207
$378$	0	0
$379$	15.0000	0.770498	0.385249	−	0.922813i	$-0.374116\pi$
$379$	15.0000	0.770498	0.385249	+	0.922813i	$0.374116\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	15.0000	0.766464	0.383232	−	0.923652i	$-0.374811\pi$
$383$	15.0000	0.766464	0.383232	+	0.923652i	$0.374811\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−15.8745	−0.804869	−0.402435	−	0.915449i	$-0.631836\pi$
$389$	−15.8745	−0.804869	−0.402435	+	0.915449i	$0.631836\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	11.0000	0.553470
$396$	0	0
$397$	−10.5830	−0.531146	−0.265573	−	0.964091i	$-0.585561\pi$
$397$	−10.5830	−0.531146	−0.265573	+	0.964091i	$0.585561\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	31.7490	1.58547	0.792735	−	0.609566i	$-0.208656\pi$
$401$	31.7490	1.58547	0.792735	+	0.609566i	$0.208656\pi$
$402$	0	0
$403$	−28.0000	−1.39478
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−15.8745	−0.786870
$408$	0	0
$409$	−31.7490	−1.56989	−0.784944	−	0.619567i	$-0.787308\pi$
$409$	−31.7490	−1.56989	−0.784944	+	0.619567i	$0.787308\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−9.00000	−0.441793
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5.00000	0.244266	0.122133	−	0.992514i	$-0.461027\pi$
$419$	5.00000	0.244266	0.122133	+	0.992514i	$0.461027\pi$
$420$	0	0
$421$	30.0000	1.46211	0.731055	−	0.682318i	$-0.239028\pi$
$421$	30.0000	1.46211	0.731055	+	0.682318i	$0.239028\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−12.0000	−0.582086
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.5830	0.509765	0.254883	−	0.966972i	$-0.417963\pi$
$431$	10.5830	0.509765	0.254883	+	0.966972i	$0.417963\pi$
$432$	0	0
$433$	−26.4575	−1.27147	−0.635733	−	0.771909i	$-0.719302\pi$
$433$	−26.4575	−1.27147	−0.635733	+	0.771909i	$0.719302\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−31.7490	−1.51530	−0.757649	−	0.652662i	$-0.773652\pi$
$439$	−31.7490	−1.51530	−0.757649	+	0.652662i	$0.773652\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15.8745	−0.754221	−0.377110	−	0.926168i	$-0.623082\pi$
$443$	−15.8745	−0.754221	−0.377110	+	0.926168i	$0.623082\pi$
$444$	0	0
$445$	14.0000	0.663664
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−15.8745	−0.749164	−0.374582	−	0.927194i	$-0.622214\pi$
$449$	−15.8745	−0.749164	−0.374582	+	0.927194i	$0.622214\pi$
$450$	0	0
$451$	47.6235	2.24250
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.00000	0.419172	0.209586	−	0.977790i	$-0.432788\pi$
$461$	9.00000	0.419172	0.209586	+	0.977790i	$0.432788\pi$
$462$	0	0
$463$	−1.00000	−0.0464739	−0.0232370	−	0.999730i	$-0.507397\pi$
$463$	−1.00000	−0.0464739	−0.0232370	+	0.999730i	$0.507397\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.0000	1.29569	0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000	1.29569	0.647843	+	0.761774i	$0.275671\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.29150	0.243304
$474$	0	0
$475$	−21.1660	−0.971163
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.00000	−0.137073	−0.0685367	−	0.997649i	$-0.521833\pi$
$479$	−3.00000	−0.137073	−0.0685367	+	0.997649i	$0.521833\pi$
$480$	0	0
$481$	−15.8745	−0.723815
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.29150	0.240275
$486$	0	0
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−15.8745	−0.716407	−0.358203	−	0.933644i	$-0.616611\pi$
$491$	−15.8745	−0.716407	−0.358203	+	0.933644i	$0.616611\pi$
$492$	0	0
$493$	15.8745	0.714952
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	39.0000	1.74588	0.872940	−	0.487828i	$-0.162211\pi$
$499$	39.0000	1.74588	0.872940	+	0.487828i	$0.162211\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5.00000	−0.222939	−0.111469	−	0.993768i	$-0.535556\pi$
$503$	−5.00000	−0.222939	−0.111469	+	0.993768i	$0.535556\pi$
$504$	0	0
$505$	−14.0000	−0.622992
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−41.0000	−1.81729	−0.908647	−	0.417566i	$-0.862883\pi$
$509$	−41.0000	−1.81729	−0.908647	+	0.417566i	$0.862883\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.5830	0.466343
$516$	0	0
$517$	−5.29150	−0.232720
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−17.0000	−0.744784	−0.372392	−	0.928076i	$-0.621462\pi$
$521$	−17.0000	−0.744784	−0.372392	+	0.928076i	$0.621462\pi$
$522$	0	0
$523$	5.29150	0.231381	0.115691	−	0.993285i	$-0.463092\pi$
$523$	5.29150	0.231381	0.115691	+	0.993285i	$0.463092\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−15.8745	−0.691504
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	47.6235	2.06280
$534$	0	0
$535$	−10.5830	−0.457543
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−11.0000	−0.472927	−0.236463	−	0.971640i	$-0.575988\pi$
$541$	−11.0000	−0.472927	−0.236463	+	0.971640i	$0.575988\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.00000	0.214176
$546$	0	0
$547$	13.0000	0.555840	0.277920	−	0.960604i	$-0.410355\pi$
$547$	13.0000	0.555840	0.277920	+	0.960604i	$0.410355\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	28.0000	1.19284
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	0	0		−	1.00000i	$-0.5\pi$
$557$	0	0			1.00000i	$0.5\pi$
$558$	0	0
$559$	5.29150	0.223807
$560$	0	0
$561$	0	0
$562$	0	0
$563$	28.0000	1.18006	0.590030	−	0.807382i	$-0.299116\pi$
$563$	28.0000	1.18006	0.590030	+	0.807382i	$0.299116\pi$
$564$	0	0
$565$	5.29150	0.222615
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−31.7490	−1.33099	−0.665494	−	0.746403i	$-0.731779\pi$
$569$	−31.7490	−1.33099	−0.665494	+	0.746403i	$0.731779\pi$
$570$	0	0
$571$	33.0000	1.38101	0.690504	−	0.723329i	$-0.257389\pi$
$571$	33.0000	1.38101	0.690504	+	0.723329i	$0.257389\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	10.5830	0.440576	0.220288	−	0.975435i	$-0.429300\pi$
$577$	10.5830	0.440576	0.220288	+	0.975435i	$0.429300\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−56.0000	−2.31928
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	−28.0000	−1.15372
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15.0000	0.615976	0.307988	−	0.951390i	$-0.400344\pi$
$593$	15.0000	0.615976	0.307988	+	0.951390i	$0.400344\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	15.8745	0.648615	0.324307	−	0.945952i	$-0.394869\pi$
$599$	15.8745	0.648615	0.324307	+	0.945952i	$0.394869\pi$
$600$	0	0
$601$	37.0405	1.51091	0.755457	−	0.655198i	$-0.227415\pi$
$601$	37.0405	1.51091	0.755457	+	0.655198i	$0.227415\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−17.0000	−0.691148
$606$	0	0
$607$	−15.8745	−0.644326	−0.322163	−	0.946684i	$-0.604410\pi$
$607$	−15.8745	−0.644326	−0.322163	+	0.946684i	$0.604410\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.29150	−0.214071
$612$	0	0
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.3320	1.70422	0.852111	−	0.523360i	$-0.175322\pi$
$617$	42.3320	1.70422	0.852111	+	0.523360i	$0.175322\pi$
$618$	0	0
$619$	−37.0405	−1.48878	−0.744392	−	0.667743i	$-0.767261\pi$
$619$	−37.0405	−1.48878	−0.744392	+	0.667743i	$0.767261\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−9.00000	−0.358854
$630$	0	0
$631$	27.0000	1.07485	0.537427	−	0.843311i	$-0.319397\pi$
$631$	27.0000	1.07485	0.537427	+	0.843311i	$0.319397\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.00000	0.0396838
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	26.4575	1.04501	0.522504	−	0.852637i	$-0.324998\pi$
$641$	26.4575	1.04501	0.522504	+	0.852637i	$0.324998\pi$
$642$	0	0
$643$	31.7490	1.25206	0.626029	−	0.779799i	$-0.284679\pi$
$643$	31.7490	1.25206	0.626029	+	0.779799i	$0.284679\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	58.2065	2.28481
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−26.4575	−1.03536	−0.517681	−	0.855574i	$-0.673205\pi$
$653$	−26.4575	−1.03536	−0.517681	+	0.855574i	$0.673205\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	10.5830	0.412255	0.206128	−	0.978525i	$-0.433914\pi$
$659$	10.5830	0.412255	0.206128	+	0.978525i	$0.433914\pi$
$660$	0	0
$661$	5.29150	0.205816	0.102908	−	0.994691i	$-0.467185\pi$
$661$	5.29150	0.205816	0.102908	+	0.994691i	$0.467185\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−28.0000	−1.08093
$672$	0	0
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−42.0000	−1.61419	−0.807096	−	0.590421i	$-0.798962\pi$
$677$	−42.0000	−1.61419	−0.807096	+	0.590421i	$0.798962\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−37.0405	−1.41732	−0.708658	−	0.705552i	$-0.750699\pi$
$683$	−37.0405	−1.41732	−0.708658	+	0.705552i	$0.750699\pi$
$684$	0	0
$685$	−5.29150	−0.202178
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−56.0000	−2.13343
$690$	0	0
$691$	26.4575	1.00649	0.503246	−	0.864143i	$-0.332139\pi$
$691$	26.4575	1.00649	0.503246	+	0.864143i	$0.332139\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	15.8745	0.602154
$696$	0	0
$697$	27.0000	1.02270
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−5.29150	−0.199857	−0.0999286	−	0.994995i	$-0.531861\pi$
$701$	−5.29150	−0.199857	−0.0999286	+	0.994995i	$0.531861\pi$
$702$	0	0
$703$	−15.8745	−0.598718
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−17.0000	−0.638448	−0.319224	−	0.947679i	$-0.603422\pi$
$709$	−17.0000	−0.638448	−0.319224	+	0.947679i	$0.603422\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−28.0000	−1.04714
$716$	0	0
$717$	0	0
$718$	0	0
$719$	15.0000	0.559406	0.279703	−	0.960087i	$-0.409764\pi$
$719$	15.0000	0.559406	0.279703	+	0.960087i	$0.409764\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−21.1660	−0.786086
$726$	0	0
$727$	5.29150	0.196251	0.0981255	−	0.995174i	$-0.468715\pi$
$727$	5.29150	0.196251	0.0981255	+	0.995174i	$0.468715\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.00000	0.110959
$732$	0	0
$733$	−21.1660	−0.781784	−0.390892	−	0.920436i	$-0.627833\pi$
$733$	−21.1660	−0.781784	−0.390892	+	0.920436i	$0.627833\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.1660	0.779660
$738$	0	0
$739$	4.00000	0.147142	0.0735712	−	0.997290i	$-0.476560\pi$
$739$	4.00000	0.147142	0.0735712	+	0.997290i	$0.476560\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−15.8745	−0.582379	−0.291190	−	0.956665i	$-0.594051\pi$
$743$	−15.8745	−0.582379	−0.291190	+	0.956665i	$0.594051\pi$
$744$	0	0
$745$	−5.29150	−0.193866
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.00000	0.181969
$756$	0	0
$757$	−41.0000	−1.49017	−0.745085	−	0.666969i	$-0.767591\pi$
$757$	−41.0000	−1.49017	−0.745085	+	0.666969i	$0.767591\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	29.0000	1.05125	0.525625	−	0.850717i	$-0.323832\pi$
$761$	29.0000	1.05125	0.525625	+	0.850717i	$0.323832\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	58.2065	2.10172
$768$	0	0
$769$	−10.5830	−0.381633	−0.190816	−	0.981626i	$-0.561114\pi$
$769$	−10.5830	−0.381633	−0.190816	+	0.981626i	$0.561114\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−39.0000	−1.40273	−0.701366	−	0.712801i	$-0.747426\pi$
$773$	−39.0000	−1.40273	−0.701366	+	0.712801i	$0.747426\pi$
$774$	0	0
$775$	21.1660	0.760306
$776$	0	0
$777$	0	0
$778$	0	0
$779$	47.6235	1.70629
$780$	0	0
$781$	−56.0000	−2.00384
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.29150	−0.188862
$786$	0	0
$787$	−21.1660	−0.754487	−0.377243	−	0.926114i	$-0.623128\pi$
$787$	−21.1660	−0.754487	−0.377243	+	0.926114i	$0.623128\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−28.0000	−0.994309
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−42.0000	−1.48772	−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000	−1.48772	−0.743858	+	0.668338i	$0.767006\pi$
$798$	0	0
$799$	−3.00000	−0.106132
$800$	0	0
$801$	0	0
$802$	0	0
$803$	56.0000	1.97620
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−31.7490	−1.11624	−0.558118	−	0.829762i	$-0.688476\pi$
$809$	−31.7490	−1.11624	−0.558118	+	0.829762i	$0.688476\pi$
$810$	0	0
$811$	5.29150	0.185810	0.0929049	−	0.995675i	$-0.470385\pi$
$811$	5.29150	0.185810	0.0929049	+	0.995675i	$0.470385\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−3.00000	−0.105085
$816$	0	0
$817$	5.29150	0.185126
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.5830	−0.369349	−0.184675	−	0.982800i	$-0.559123\pi$
$821$	−10.5830	−0.369349	−0.184675	+	0.982800i	$0.559123\pi$
$822$	0	0
$823$	19.0000	0.662298	0.331149	−	0.943578i	$-0.392564\pi$
$823$	19.0000	0.662298	0.331149	+	0.943578i	$0.392564\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−47.6235	−1.65603	−0.828016	−	0.560704i	$-0.810530\pi$
$827$	−47.6235	−1.65603	−0.828016	+	0.560704i	$0.810530\pi$
$828$	0	0
$829$	−5.29150	−0.183781	−0.0918907	−	0.995769i	$-0.529291\pi$
$829$	−5.29150	−0.183781	−0.0918907	+	0.995769i	$0.529291\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−19.0000	−0.657522
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9.00000	−0.310715	−0.155357	−	0.987858i	$-0.549653\pi$
$839$	−9.00000	−0.310715	−0.155357	+	0.987858i	$0.549653\pi$
$840$	0	0
$841$	−1.00000	−0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−15.0000	−0.516016
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	52.9150	1.81178	0.905888	−	0.423517i	$-0.139205\pi$
$853$	52.9150	1.81178	0.905888	+	0.423517i	$0.139205\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−39.0000	−1.33221	−0.666107	−	0.745856i	$-0.732041\pi$
$857$	−39.0000	−1.33221	−0.666107	+	0.745856i	$0.732041\pi$
$858$	0	0
$859$	0	0		−	1.00000i	$-0.5\pi$
$859$	0	0			1.00000i	$0.5\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−58.2065	−1.97452
$870$	0	0
$871$	21.1660	0.717183
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	45.0000	1.51954	0.759771	−	0.650191i	$-0.225311\pi$
$877$	45.0000	1.51954	0.759771	+	0.650191i	$0.225311\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	0	0
$883$	43.0000	1.44707	0.723533	−	0.690290i	$-0.242517\pi$
$883$	43.0000	1.44707	0.723533	+	0.690290i	$0.242517\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	27.0000	0.906571	0.453286	−	0.891365i	$-0.350252\pi$
$887$	27.0000	0.906571	0.453286	+	0.891365i	$0.350252\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.29150	−0.177073
$894$	0	0
$895$	−15.8745	−0.530626
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−28.0000	−0.933852
$900$	0	0
$901$	−31.7490	−1.05771
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.5830	−0.351791
$906$	0	0
$907$	−5.00000	−0.166022	−0.0830111	−	0.996549i	$-0.526454\pi$
$907$	−5.00000	−0.166022	−0.0830111	+	0.996549i	$0.526454\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	26.4575	0.876577	0.438288	−	0.898834i	$-0.355585\pi$
$911$	26.4575	0.876577	0.438288	+	0.898834i	$0.355585\pi$
$912$	0	0
$913$	47.6235	1.57611
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	39.0000	1.28649	0.643246	−	0.765660i	$-0.277587\pi$
$919$	39.0000	1.28649	0.643246	+	0.765660i	$0.277587\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−56.0000	−1.84326
$924$	0	0
$925$	12.0000	0.394558
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−15.0000	−0.492134	−0.246067	−	0.969253i	$-0.579138\pi$
$929$	−15.0000	−0.492134	−0.246067	+	0.969253i	$0.579138\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−15.8745	−0.519152
$936$	0	0
$937$	15.8745	0.518598	0.259299	−	0.965797i	$-0.416509\pi$
$937$	15.8745	0.518598	0.259299	+	0.965797i	$0.416509\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−45.0000	−1.46696	−0.733479	−	0.679712i	$-0.762105\pi$
$941$	−45.0000	−1.46696	−0.733479	+	0.679712i	$0.762105\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21.1660	0.687803	0.343901	−	0.939006i	$-0.388251\pi$
$947$	21.1660	0.687803	0.343901	+	0.939006i	$0.388251\pi$
$948$	0	0
$949$	56.0000	1.81784
$950$	0	0
$951$	0	0
$952$	0	0
$953$	0	0		−	1.00000i	$-0.5\pi$
$953$	0	0			1.00000i	$0.5\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−3.00000	−0.0967742
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−23.0000	−0.740396
$966$	0	0
$967$	16.0000	0.514525	0.257263	−	0.966342i	$-0.417179\pi$
$967$	16.0000	0.514525	0.257263	+	0.966342i	$0.417179\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	29.0000	0.930654	0.465327	−	0.885139i	$-0.345937\pi$
$971$	29.0000	0.930654	0.465327	+	0.885139i	$0.345937\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−42.3320	−1.35432	−0.677161	−	0.735835i	$-0.736790\pi$
$977$	−42.3320	−1.35432	−0.677161	+	0.735835i	$0.736790\pi$
$978$	0	0
$979$	−74.0810	−2.36764
$980$	0	0
$981$	0	0
$982$	0	0
$983$	11.0000	0.350846	0.175423	−	0.984493i	$-0.443871\pi$
$983$	11.0000	0.350846	0.175423	+	0.984493i	$0.443871\pi$
$984$	0	0
$985$	−15.8745	−0.505804
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−19.0000	−0.603555	−0.301777	−	0.953378i	$-0.597580\pi$
$991$	−19.0000	−0.603555	−0.301777	+	0.953378i	$0.597580\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	21.1660	0.671008
$996$	0	0
$997$	5.29150	0.167584	0.0837918	−	0.996483i	$-0.473297\pi$
$997$	5.29150	0.167584	0.0837918	+	0.996483i	$0.473297\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.a.n.1.2	yes	2
3.2	odd	2		5292.2.a.t.1.1	yes	2
7.6	odd	2		5292.2.a.t.1.2	yes	2
21.20	even	2	inner	5292.2.a.n.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5292.2.a.n.1.1	✓	2	21.20	even	2	inner
5292.2.a.n.1.2	yes	2	1.1	even	1	trivial
5292.2.a.t.1.1	yes	2	3.2	odd	2
5292.2.a.t.1.2	yes	2	7.6	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:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +O(q^{10})\)	\(q-1.00000 q^{5} +5.29150 q^{11} +5.29150 q^{13} +3.00000 q^{17} +5.29150 q^{19} -4.00000 q^{25} +5.29150 q^{29} -5.29150 q^{31} -3.00000 q^{37} +9.00000 q^{41} +1.00000 q^{43} -1.00000 q^{47} -10.5830 q^{53} -5.29150 q^{55} +11.0000 q^{59} -5.29150 q^{61} -5.29150 q^{65} +4.00000 q^{67} -10.5830 q^{71} +10.5830 q^{73} -11.0000 q^{79} +9.00000 q^{83} -3.00000 q^{85} -14.0000 q^{89} -5.29150 q^{95} -5.29150 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5}+O(q^{10}) \) 2 * q - 2 * q^5	\( 2 q - 2 q^{5} + 6 q^{17} - 8 q^{25} - 6 q^{37} + 18 q^{41} + 2 q^{43} - 2 q^{47} + 22 q^{59} + 8 q^{67} - 22 q^{79} + 18 q^{83} - 6 q^{85} - 28 q^{89}+O(q^{100}) \) 2 * q - 2 * q^5 + 6 * q^17 - 8 * q^25 - 6 * q^37 + 18 * q^41 + 2 * q^43 - 2 * q^47 + 22 * q^59 + 8 * q^67 - 22 * q^79 + 18 * q^83 - 6 * q^85 - 28 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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