LMFDB - Embedded newform 4536.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4536,2,Mod(1,4536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4536, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4536.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4536 = 2^{3} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4536.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:	$36.2201423569$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.25488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} - 6x + 6 $ x^4 - 9x^2 - 6x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.28657$ of defining polynomial
Character	$\chi$	$=$	4536.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.445480 q^{5} +1.00000 q^{7} +O(q^{10})$	$q+0.445480 q^{5} +1.00000 q^{7} -0.960457 q^{11} -3.80155 q^{13} +1.22841 q^{17} +4.24703 q^{19} -2.24703 q^{23} -4.80155 q^{25} +7.80155 q^{29} +0.782926 q^{31} +0.445480 q^{35} +0.217074 q^{37} +5.46410 q^{41} -1.33745 q^{43} +2.32611 q^{47} +1.00000 q^{49} -5.85971 q^{53} -0.427864 q^{55} +8.46511 q^{59} +9.72246 q^{61} -1.69351 q^{65} -6.37368 q^{67} +12.8897 q^{71} +2.80054 q^{73} -0.960457 q^{77} +6.59176 q^{79} -1.79022 q^{83} +0.547230 q^{85} +4.87334 q^{89} -3.80155 q^{91} +1.89197 q^{95} +7.02995 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^5 + 4 * q^7	$ 4 q + 4 q^{5} + 4 q^{7} + 6 q^{11} + 6 q^{13} + 2 q^{17} - 2 q^{19} + 10 q^{23} + 2 q^{25} + 10 q^{29} - 2 q^{31} + 4 q^{35} + 6 q^{37} + 8 q^{41} + 2 q^{43} + 10 q^{47} + 4 q^{49} + 4 q^{53} - 6 q^{55} + 6 q^{59} - 2 q^{61} + 24 q^{65} + 4 q^{73} + 6 q^{77} - 8 q^{79} + 6 q^{83} + 8 q^{85} + 26 q^{89} + 6 q^{91} - 2 q^{95} + 4 q^{97}+O(q^{100}) $ 4 * q + 4 * q^5 + 4 * q^7 + 6 * q^11 + 6 * q^13 + 2 * q^17 - 2 * q^19 + 10 * q^23 + 2 * q^25 + 10 * q^29 - 2 * q^31 + 4 * q^35 + 6 * q^37 + 8 * q^41 + 2 * q^43 + 10 * q^47 + 4 * q^49 + 4 * q^53 - 6 * q^55 + 6 * q^59 - 2 * q^61 + 24 * q^65 + 4 * q^73 + 6 * q^77 - 8 * q^79 + 6 * q^83 + 8 * q^85 + 26 * q^89 + 6 * q^91 - 2 * q^95 + 4 * 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0.445480	0.199225	0.0996124	−	0.995026i	$-0.468240\pi$
$5$	0.445480	0.199225	0.0996124	+	0.995026i	$0.468240\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−0.960457	−0.289589	−0.144794	−	0.989462i	$-0.546252\pi$
$11$	−0.960457	−0.289589	−0.144794	+	0.989462i	$0.546252\pi$
$12$	0	0
$13$	−3.80155	−1.05436	−0.527180	−	0.849754i	$-0.676751\pi$
$13$	−3.80155	−1.05436	−0.527180	+	0.849754i	$0.676751\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.22841	0.297932	0.148966	−	0.988842i	$-0.452405\pi$
$17$	1.22841	0.297932	0.148966	+	0.988842i	$0.452405\pi$
$18$	0	0
$19$	4.24703	0.974335	0.487167	−	0.873309i	$-0.338030\pi$
$19$	4.24703	0.974335	0.487167	+	0.873309i	$0.338030\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.24703	−0.468538	−0.234269	−	0.972172i	$-0.575270\pi$
$23$	−2.24703	−0.468538	−0.234269	+	0.972172i	$0.575270\pi$
$24$	0	0
$25$	−4.80155	−0.960310
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.80155	1.44871	0.724356	−	0.689427i	$-0.242137\pi$
$29$	7.80155	1.44871	0.724356	+	0.689427i	$0.242137\pi$
$30$	0	0
$31$	0.782926	0.140618	0.0703088	−	0.997525i	$-0.477602\pi$
$31$	0.782926	0.140618	0.0703088	+	0.997525i	$0.477602\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.445480	0.0752999
$36$	0	0
$37$	0.217074	0.0356868	0.0178434	−	0.999841i	$-0.494320\pi$
$37$	0.217074	0.0356868	0.0178434	+	0.999841i	$0.494320\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.46410	0.853349	0.426675	−	0.904405i	$-0.359685\pi$
$41$	5.46410	0.853349	0.426675	+	0.904405i	$0.359685\pi$
$42$	0	0
$43$	−1.33745	−0.203959	−0.101979	−	0.994787i	$-0.532518\pi$
$43$	−1.33745	−0.203959	−0.101979	+	0.994787i	$0.532518\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.32611	0.339299	0.169649	−	0.985504i	$-0.445737\pi$
$47$	2.32611	0.339299	0.169649	+	0.985504i	$0.445737\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.85971	−0.804893	−0.402447	−	0.915443i	$-0.631840\pi$
$53$	−5.85971	−0.804893	−0.402447	+	0.915443i	$0.631840\pi$
$54$	0	0
$55$	−0.427864	−0.0576932
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.46511	1.10206	0.551032	−	0.834484i	$-0.314234\pi$
$59$	8.46511	1.10206	0.551032	+	0.834484i	$0.314234\pi$
$60$	0	0
$61$	9.72246	1.24483	0.622417	−	0.782686i	$-0.286151\pi$
$61$	9.72246	1.24483	0.622417	+	0.782686i	$0.286151\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.69351	−0.210054
$66$	0	0
$67$	−6.37368	−0.778669	−0.389335	−	0.921096i	$-0.627295\pi$
$67$	−6.37368	−0.778669	−0.389335	+	0.921096i	$0.627295\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.8897	1.52972	0.764861	−	0.644195i	$-0.222808\pi$
$71$	12.8897	1.52972	0.764861	+	0.644195i	$0.222808\pi$
$72$	0	0
$73$	2.80054	0.327779	0.163889	−	0.986479i	$-0.447596\pi$
$73$	2.80054	0.327779	0.163889	+	0.986479i	$0.447596\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.960457	−0.109454
$78$	0	0
$79$	6.59176	0.741631	0.370816	−	0.928706i	$-0.379078\pi$
$79$	6.59176	0.741631	0.370816	+	0.928706i	$0.379078\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.79022	−0.196502	−0.0982508	−	0.995162i	$-0.531325\pi$
$83$	−1.79022	−0.196502	−0.0982508	+	0.995162i	$0.531325\pi$
$84$	0	0
$85$	0.547230	0.0593555
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.87334	0.516573	0.258287	−	0.966068i	$-0.416842\pi$
$89$	4.87334	0.516573	0.258287	+	0.966068i	$0.416842\pi$
$90$	0	0
$91$	−3.80155	−0.398510
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.89197	0.194112
$96$	0	0
$97$	7.02995	0.713784	0.356892	−	0.934146i	$-0.383836\pi$
$97$	7.02995	0.713784	0.356892	+	0.934146i	$0.383836\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	5.38502	0.535829	0.267915	−	0.963443i	$-0.413666\pi$
$101$	5.38502	0.535829	0.267915	+	0.963443i	$0.413666\pi$
$102$	0	0
$103$	−18.6104	−1.83374	−0.916868	−	0.399191i	$-0.869291\pi$
$103$	−18.6104	−1.83374	−0.916868	+	0.399191i	$0.869291\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.75067	0.845959	0.422980	−	0.906139i	$-0.360984\pi$
$107$	8.75067	0.845959	0.422980	+	0.906139i	$0.360984\pi$
$108$	0	0
$109$	15.2739	1.46298	0.731489	−	0.681853i	$-0.238826\pi$
$109$	15.2739	1.46298	0.731489	+	0.681853i	$0.238826\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	10.0871	0.948916	0.474458	−	0.880278i	$-0.342644\pi$
$113$	10.0871	0.948916	0.474458	+	0.880278i	$0.342644\pi$
$114$	0	0
$115$	−1.00101	−0.0933443
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.22841	0.112608
$120$	0	0
$121$	−10.0775	−0.916138
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−4.36639	−0.390542
$126$	0	0
$127$	3.61342	0.320639	0.160320	−	0.987065i	$-0.448748\pi$
$127$	3.61342	0.320639	0.160320	+	0.987065i	$0.448748\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−6.25532	−0.546530	−0.273265	−	0.961939i	$-0.588104\pi$
$131$	−6.25532	−0.546530	−0.273265	+	0.961939i	$0.588104\pi$
$132$	0	0
$133$	4.24703	0.368264
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.7393	0.917524	0.458762	−	0.888559i	$-0.348293\pi$
$137$	10.7393	0.917524	0.458762	+	0.888559i	$0.348293\pi$
$138$	0	0
$139$	−13.6094	−1.15433	−0.577166	−	0.816627i	$-0.695842\pi$
$139$	−13.6094	−1.15433	−0.577166	+	0.816627i	$0.695842\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.65122	0.305331
$144$	0	0
$145$	3.47543	0.288619
$146$	0	0
$147$	0	0
$148$	0	0
$149$	13.1899	1.08056	0.540278	−	0.841487i	$-0.318319\pi$
$149$	13.1899	1.08056	0.540278	+	0.841487i	$0.318319\pi$
$150$	0	0
$151$	−3.68421	−0.299817	−0.149908	−	0.988700i	$-0.547898\pi$
$151$	−3.68421	−0.299817	−0.149908	+	0.988700i	$0.547898\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.348778	0.0280145
$156$	0	0
$157$	−5.16390	−0.412124	−0.206062	−	0.978539i	$-0.566065\pi$
$157$	−5.16390	−0.412124	−0.206062	+	0.978539i	$0.566065\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2.24703	−0.177091
$162$	0	0
$163$	−0.901286	−0.0705942	−0.0352971	−	0.999377i	$-0.511238\pi$
$163$	−0.901286	−0.0705942	−0.0352971	+	0.999377i	$0.511238\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−14.9592	−1.15757	−0.578787	−	0.815478i	$-0.696474\pi$
$167$	−14.9592	−1.15757	−0.578787	+	0.815478i	$0.696474\pi$
$168$	0	0
$169$	1.45176	0.111674
$170$	0	0
$171$	0	0
$172$	0	0
$173$	5.63361	0.428315	0.214158	−	0.976799i	$-0.431299\pi$
$173$	5.63361	0.428315	0.214158	+	0.976799i	$0.431299\pi$
$174$	0	0
$175$	−4.80155	−0.362963
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.0662	0.752382	0.376191	−	0.926542i	$-0.377234\pi$
$179$	10.0662	0.752382	0.376191	+	0.926542i	$0.377234\pi$
$180$	0	0
$181$	20.7474	1.54214	0.771070	−	0.636751i	$-0.219722\pi$
$181$	20.7474	1.54214	0.771070	+	0.636751i	$0.219722\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.0967021	0.00710968
$186$	0	0
$187$	−1.17983	−0.0862778
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.14129	0.661441	0.330720	−	0.943729i	$-0.392708\pi$
$191$	9.14129	0.661441	0.330720	+	0.943729i	$0.392708\pi$
$192$	0	0
$193$	−12.2936	−0.884912	−0.442456	−	0.896790i	$-0.645893\pi$
$193$	−12.2936	−0.884912	−0.442456	+	0.896790i	$0.645893\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.10001	0.149619	0.0748097	−	0.997198i	$-0.476165\pi$
$197$	2.10001	0.149619	0.0748097	+	0.997198i	$0.476165\pi$
$198$	0	0
$199$	11.3933	0.807650	0.403825	−	0.914836i	$-0.367680\pi$
$199$	11.3933	0.807650	0.403825	+	0.914836i	$0.367680\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.80155	0.547561
$204$	0	0
$205$	2.43415	0.170008
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−4.07909	−0.282156
$210$	0	0
$211$	−2.33644	−0.160847	−0.0804236	−	0.996761i	$-0.525627\pi$
$211$	−2.33644	−0.160847	−0.0804236	+	0.996761i	$0.525627\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.595805	−0.0406336
$216$	0	0
$217$	0.782926	0.0531485
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.66984	−0.314128
$222$	0	0
$223$	−8.03724	−0.538214	−0.269107	−	0.963110i	$-0.586728\pi$
$223$	−8.03724	−0.538214	−0.269107	+	0.963110i	$0.586728\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−6.82645	−0.453088	−0.226544	−	0.974001i	$-0.572743\pi$
$227$	−6.82645	−0.453088	−0.226544	+	0.974001i	$0.572743\pi$
$228$	0	0
$229$	11.6646	0.770816	0.385408	−	0.922746i	$-0.374061\pi$
$229$	11.6646	0.770816	0.385408	+	0.922746i	$0.374061\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.5499	−0.756660	−0.378330	−	0.925671i	$-0.623502\pi$
$233$	−11.5499	−0.756660	−0.378330	+	0.925671i	$0.623502\pi$
$234$	0	0
$235$	1.03624	0.0675967
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.2085	1.04844	0.524220	−	0.851583i	$-0.324357\pi$
$239$	16.2085	1.04844	0.524220	+	0.851583i	$0.324357\pi$
$240$	0	0
$241$	5.31579	0.342420	0.171210	−	0.985235i	$-0.445232\pi$
$241$	5.31579	0.342420	0.171210	+	0.985235i	$0.445232\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.445480	0.0284607
$246$	0	0
$247$	−16.1453	−1.02730
$248$	0	0
$249$	0	0
$250$	0	0
$251$	18.6393	1.17650	0.588252	−	0.808678i	$-0.299816\pi$
$251$	18.6393	1.17650	0.588252	+	0.808678i	$0.299816\pi$
$252$	0	0
$253$	2.15817	0.135683
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.4026	0.836033	0.418017	−	0.908439i	$-0.362725\pi$
$257$	13.4026	0.836033	0.418017	+	0.908439i	$0.362725\pi$
$258$	0	0
$259$	0.217074	0.0134883
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−11.9113	−0.734484	−0.367242	−	0.930125i	$-0.619698\pi$
$263$	−11.9113	−0.734484	−0.367242	+	0.930125i	$0.619698\pi$
$264$	0	0
$265$	−2.61038	−0.160355
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.58447	0.401462	0.200731	−	0.979646i	$-0.435668\pi$
$269$	6.58447	0.401462	0.200731	+	0.979646i	$0.435668\pi$
$270$	0	0
$271$	−26.0889	−1.58479	−0.792393	−	0.610012i	$-0.791165\pi$
$271$	−26.0889	−1.58479	−0.792393	+	0.610012i	$0.791165\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.61168	0.278095
$276$	0	0
$277$	11.4764	0.689552	0.344776	−	0.938685i	$-0.387955\pi$
$277$	11.4764	0.689552	0.344776	+	0.938685i	$0.387955\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.04987	0.480215	0.240107	−	0.970746i	$-0.422817\pi$
$281$	8.04987	0.480215	0.240107	+	0.970746i	$0.422817\pi$
$282$	0	0
$283$	−21.1271	−1.25588	−0.627938	−	0.778263i	$-0.716101\pi$
$283$	−21.1271	−1.25588	−0.627938	+	0.778263i	$0.716101\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.46410	0.322536
$288$	0	0
$289$	−15.4910	−0.911236
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8.14831	−0.476030	−0.238015	−	0.971262i	$-0.576497\pi$
$293$	−8.14831	−0.476030	−0.238015	+	0.971262i	$0.576497\pi$
$294$	0	0
$295$	3.77104	0.219558
$296$	0	0
$297$	0	0
$298$	0	0
$299$	8.54218	0.494007
$300$	0	0
$301$	−1.33745	−0.0770891
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.33116	0.248002
$306$	0	0
$307$	33.3159	1.90144	0.950720	−	0.310052i	$-0.100346\pi$
$307$	33.3159	1.90144	0.950720	+	0.310052i	$0.100346\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.16794	−0.122933	−0.0614663	−	0.998109i	$-0.519578\pi$
$311$	−2.16794	−0.122933	−0.0614663	+	0.998109i	$0.519578\pi$
$312$	0	0
$313$	11.6708	0.659675	0.329838	−	0.944038i	$-0.393006\pi$
$313$	11.6708	0.659675	0.329838	+	0.944038i	$0.393006\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.4791	1.54338	0.771691	−	0.635998i	$-0.219411\pi$
$317$	27.4791	1.54338	0.771691	+	0.635998i	$0.219411\pi$
$318$	0	0
$319$	−7.49305	−0.419530
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.21707	0.290286
$324$	0	0
$325$	18.2533	1.01251
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.32611	0.128243
$330$	0	0
$331$	6.27598	0.344959	0.172479	−	0.985013i	$-0.444822\pi$
$331$	6.27598	0.344959	0.172479	+	0.985013i	$0.444822\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.83935	−0.155130
$336$	0	0
$337$	−5.21504	−0.284082	−0.142041	−	0.989861i	$-0.545366\pi$
$337$	−5.21504	−0.284082	−0.142041	+	0.989861i	$0.545366\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−0.751967	−0.0407213
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.58521	0.407195	0.203598	−	0.979055i	$-0.434737\pi$
$347$	7.58521	0.407195	0.203598	+	0.979055i	$0.434737\pi$
$348$	0	0
$349$	−12.2135	−0.653773	−0.326886	−	0.945064i	$-0.605999\pi$
$349$	−12.2135	−0.653773	−0.326886	+	0.945064i	$0.605999\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.4150	1.08658	0.543290	−	0.839545i	$-0.317179\pi$
$353$	20.4150	1.08658	0.543290	+	0.839545i	$0.317179\pi$
$354$	0	0
$355$	5.74209	0.304758
$356$	0	0
$357$	0	0
$358$	0	0
$359$	32.9476	1.73891	0.869453	−	0.494016i	$-0.164472\pi$
$359$	32.9476	1.73891	0.869453	+	0.494016i	$0.164472\pi$
$360$	0	0
$361$	−0.962757	−0.0506714
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.24759	0.0653016
$366$	0	0
$367$	−14.1246	−0.737299	−0.368650	−	0.929568i	$-0.620180\pi$
$367$	−14.1246	−0.737299	−0.368650	+	0.929568i	$0.620180\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−5.85971	−0.304221
$372$	0	0
$373$	12.5835	0.651547	0.325774	−	0.945448i	$-0.394375\pi$
$373$	12.5835	0.651547	0.325774	+	0.945448i	$0.394375\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−29.6580	−1.52746
$378$	0	0
$379$	−25.2812	−1.29861	−0.649305	−	0.760528i	$-0.724940\pi$
$379$	−25.2812	−1.29861	−0.649305	+	0.760528i	$0.724940\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13.3498	0.682142	0.341071	−	0.940038i	$-0.389210\pi$
$383$	13.3498	0.682142	0.341071	+	0.940038i	$0.389210\pi$
$384$	0	0
$385$	−0.427864	−0.0218060
$386$	0	0
$387$	0	0
$388$	0	0
$389$	25.4368	1.28970	0.644849	−	0.764310i	$-0.276920\pi$
$389$	25.4368	1.28970	0.644849	+	0.764310i	$0.276920\pi$
$390$	0	0
$391$	−2.76026	−0.139592
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.93650	0.147751
$396$	0	0
$397$	14.1194	0.708631	0.354315	−	0.935126i	$-0.384714\pi$
$397$	14.1194	0.708631	0.354315	+	0.935126i	$0.384714\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.8292	1.09010	0.545049	−	0.838404i	$-0.316511\pi$
$401$	21.8292	1.09010	0.545049	+	0.838404i	$0.316511\pi$
$402$	0	0
$403$	−2.97633	−0.148262
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.208490	−0.0103345
$408$	0	0
$409$	−5.09771	−0.252065	−0.126033	−	0.992026i	$-0.540224\pi$
$409$	−5.09771	−0.252065	−0.126033	+	0.992026i	$0.540224\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.46511	0.416541
$414$	0	0
$415$	−0.797505	−0.0391480
$416$	0	0
$417$	0	0
$418$	0	0
$419$	3.68218	0.179886	0.0899432	−	0.995947i	$-0.471331\pi$
$419$	3.68218	0.179886	0.0899432	+	0.995947i	$0.471331\pi$
$420$	0	0
$421$	28.7111	1.39929	0.699647	−	0.714488i	$-0.253340\pi$
$421$	28.7111	1.39929	0.699647	+	0.714488i	$0.253340\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.89825	−0.286107
$426$	0	0
$427$	9.72246	0.470503
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−0.932803	−0.0449316	−0.0224658	−	0.999748i	$-0.507152\pi$
$431$	−0.932803	−0.0449316	−0.0224658	+	0.999748i	$0.507152\pi$
$432$	0	0
$433$	32.8558	1.57895	0.789476	−	0.613782i	$-0.210353\pi$
$433$	32.8558	1.57895	0.789476	+	0.613782i	$0.210353\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9.54319	−0.456513
$438$	0	0
$439$	−15.2707	−0.728831	−0.364415	−	0.931237i	$-0.618731\pi$
$439$	−15.2707	−0.728831	−0.364415	+	0.931237i	$0.618731\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−13.6919	−0.650524	−0.325262	−	0.945624i	$-0.605453\pi$
$443$	−13.6919	−0.650524	−0.325262	+	0.945624i	$0.605453\pi$
$444$	0	0
$445$	2.17098	0.102914
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16.6882	−0.787564	−0.393782	−	0.919204i	$-0.628833\pi$
$449$	−16.6882	−0.787564	−0.393782	+	0.919204i	$0.628833\pi$
$450$	0	0
$451$	−5.24803	−0.247120
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.69351	−0.0793931
$456$	0	0
$457$	15.5313	0.726523	0.363262	−	0.931687i	$-0.381663\pi$
$457$	15.5313	0.726523	0.363262	+	0.931687i	$0.381663\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−21.8418	−1.01728	−0.508638	−	0.860981i	$-0.669851\pi$
$461$	−21.8418	−1.01728	−0.508638	+	0.860981i	$0.669851\pi$
$462$	0	0
$463$	−4.38177	−0.203638	−0.101819	−	0.994803i	$-0.532466\pi$
$463$	−4.38177	−0.203638	−0.101819	+	0.994803i	$0.532466\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	33.4739	1.54899	0.774493	−	0.632582i	$-0.218005\pi$
$467$	33.4739	1.54899	0.774493	+	0.632582i	$0.218005\pi$
$468$	0	0
$469$	−6.37368	−0.294309
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.28456	0.0590641
$474$	0	0
$475$	−20.3923	−0.935663
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−37.2724	−1.70302	−0.851509	−	0.524340i	$-0.824312\pi$
$479$	−37.2724	−1.70302	−0.851509	+	0.524340i	$0.824312\pi$
$480$	0	0
$481$	−0.825217	−0.0376267
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.13170	0.142203
$486$	0	0
$487$	12.9050	0.584781	0.292390	−	0.956299i	$-0.405549\pi$
$487$	12.9050	0.584781	0.292390	+	0.956299i	$0.405549\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−39.8971	−1.80053	−0.900266	−	0.435340i	$-0.856628\pi$
$491$	−39.8971	−1.80053	−0.900266	+	0.435340i	$0.856628\pi$
$492$	0	0
$493$	9.58347	0.431618
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.8897	0.578181
$498$	0	0
$499$	24.6620	1.10402	0.552011	−	0.833837i	$-0.313861\pi$
$499$	24.6620	1.10402	0.552011	+	0.833837i	$0.313861\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−30.1473	−1.34420	−0.672101	−	0.740460i	$-0.734608\pi$
$503$	−30.1473	−1.34420	−0.672101	+	0.740460i	$0.734608\pi$
$504$	0	0
$505$	2.39892	0.106750
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.37571	−0.105302	−0.0526508	−	0.998613i	$-0.516767\pi$
$509$	−2.37571	−0.105302	−0.0526508	+	0.998613i	$0.516767\pi$
$510$	0	0
$511$	2.80054	0.123889
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.29055	−0.365325
$516$	0	0
$517$	−2.23413	−0.0982570
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−45.0208	−1.97240	−0.986198	−	0.165573i	$-0.947053\pi$
$521$	−45.0208	−1.97240	−0.986198	+	0.165573i	$0.947053\pi$
$522$	0	0
$523$	−19.7773	−0.864802	−0.432401	−	0.901681i	$-0.642333\pi$
$523$	−19.7773	−0.864802	−0.432401	+	0.901681i	$0.642333\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.961751	0.0418945
$528$	0	0
$529$	−17.9509	−0.780472
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−20.7720	−0.899737
$534$	0	0
$535$	3.89825	0.168536
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−0.960457	−0.0413698
$540$	0	0
$541$	−8.78360	−0.377637	−0.188818	−	0.982012i	$-0.560466\pi$
$541$	−8.78360	−0.377637	−0.188818	+	0.982012i	$0.560466\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.80424	0.291461
$546$	0	0
$547$	15.9395	0.681525	0.340763	−	0.940149i	$-0.389315\pi$
$547$	15.9395	0.681525	0.340763	+	0.940149i	$0.389315\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	33.1334	1.41153
$552$	0	0
$553$	6.59176	0.280310
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−27.4432	−1.16280	−0.581402	−	0.813616i	$-0.697496\pi$
$557$	−27.4432	−1.16280	−0.581402	+	0.813616i	$0.697496\pi$
$558$	0	0
$559$	5.08436	0.215046
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−22.8511	−0.963060	−0.481530	−	0.876430i	$-0.659919\pi$
$563$	−22.8511	−0.963060	−0.481530	+	0.876430i	$0.659919\pi$
$564$	0	0
$565$	4.49361	0.189047
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24.4215	1.02380	0.511902	−	0.859044i	$-0.328941\pi$
$569$	24.4215	1.02380	0.511902	+	0.859044i	$0.328941\pi$
$570$	0	0
$571$	−14.6474	−0.612975	−0.306488	−	0.951875i	$-0.599154\pi$
$571$	−14.6474	−0.612975	−0.306488	+	0.951875i	$0.599154\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	10.7892	0.449941
$576$	0	0
$577$	−4.35034	−0.181107	−0.0905535	−	0.995892i	$-0.528864\pi$
$577$	−4.35034	−0.181107	−0.0905535	+	0.995892i	$0.528864\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.79022	−0.0742707
$582$	0	0
$583$	5.62800	0.233088
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−25.1007	−1.03602	−0.518009	−	0.855375i	$-0.673327\pi$
$587$	−25.1007	−1.03602	−0.518009	+	0.855375i	$0.673327\pi$
$588$	0	0
$589$	3.32511	0.137009
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−3.42382	−0.140600	−0.0702998	−	0.997526i	$-0.522396\pi$
$593$	−3.42382	−0.140600	−0.0702998	+	0.997526i	$0.522396\pi$
$594$	0	0
$595$	0.547230	0.0224343
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−27.7144	−1.13238	−0.566191	−	0.824274i	$-0.691583\pi$
$599$	−27.7144	−1.13238	−0.566191	+	0.824274i	$0.691583\pi$
$600$	0	0
$601$	−46.0021	−1.87647	−0.938233	−	0.346004i	$-0.887538\pi$
$601$	−46.0021	−1.87647	−0.938233	+	0.346004i	$0.887538\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−4.48934	−0.182517
$606$	0	0
$607$	−14.2470	−0.578269	−0.289135	−	0.957288i	$-0.593367\pi$
$607$	−14.2470	−0.578269	−0.289135	+	0.957288i	$0.593367\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.84283	−0.357743
$612$	0	0
$613$	−3.47644	−0.140412	−0.0702060	−	0.997533i	$-0.522366\pi$
$613$	−3.47644	−0.140412	−0.0702060	+	0.997533i	$0.522366\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.7541	1.31863	0.659315	−	0.751867i	$-0.270846\pi$
$617$	32.7541	1.31863	0.659315	+	0.751867i	$0.270846\pi$
$618$	0	0
$619$	−9.97881	−0.401082	−0.200541	−	0.979685i	$-0.564270\pi$
$619$	−9.97881	−0.401082	−0.200541	+	0.979685i	$0.564270\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	4.87334	0.195246
$624$	0	0
$625$	22.0626	0.882504
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.266655	0.0106322
$630$	0	0
$631$	−37.8435	−1.50653	−0.753263	−	0.657719i	$-0.771521\pi$
$631$	−37.8435	−1.50653	−0.753263	+	0.657719i	$0.771521\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.60971	0.0638793
$636$	0	0
$637$	−3.80155	−0.150623
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−36.4668	−1.44035	−0.720177	−	0.693790i	$-0.755939\pi$
$641$	−36.4668	−1.44035	−0.720177	+	0.693790i	$0.755939\pi$
$642$	0	0
$643$	−25.2337	−0.995119	−0.497559	−	0.867430i	$-0.665770\pi$
$643$	−25.2337	−0.995119	−0.497559	+	0.867430i	$0.665770\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.6703	0.498121	0.249060	−	0.968488i	$-0.419878\pi$
$647$	12.6703	0.498121	0.249060	+	0.968488i	$0.419878\pi$
$648$	0	0
$649$	−8.13037	−0.319145
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−39.9941	−1.56509	−0.782545	−	0.622594i	$-0.786079\pi$
$653$	−39.9941	−1.56509	−0.782545	+	0.622594i	$0.786079\pi$
$654$	0	0
$655$	−2.78662	−0.108882
$656$	0	0
$657$	0	0
$658$	0	0
$659$	37.9589	1.47867	0.739334	−	0.673339i	$-0.235141\pi$
$659$	37.9589	1.47867	0.739334	+	0.673339i	$0.235141\pi$
$660$	0	0
$661$	19.2238	0.747719	0.373860	−	0.927485i	$-0.378034\pi$
$661$	19.2238	0.747719	0.373860	+	0.927485i	$0.378034\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.89197	0.0733673
$666$	0	0
$667$	−17.5303	−0.678776
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9.33800	−0.360490
$672$	0	0
$673$	26.4615	1.02002	0.510008	−	0.860170i	$-0.329642\pi$
$673$	26.4615	1.02002	0.510008	+	0.860170i	$0.329642\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.13698	−0.235863	−0.117932	−	0.993022i	$-0.537626\pi$
$677$	−6.13698	−0.235863	−0.117932	+	0.993022i	$0.537626\pi$
$678$	0	0
$679$	7.02995	0.269785
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10.1005	−0.386483	−0.193242	−	0.981151i	$-0.561900\pi$
$683$	−10.1005	−0.386483	−0.193242	+	0.981151i	$0.561900\pi$
$684$	0	0
$685$	4.78416	0.182793
$686$	0	0
$687$	0	0
$688$	0	0
$689$	22.2760	0.848647
$690$	0	0
$691$	6.74938	0.256758	0.128379	−	0.991725i	$-0.459023\pi$
$691$	6.74938	0.256758	0.128379	+	0.991725i	$0.459023\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.06271	−0.229972
$696$	0	0
$697$	6.71214	0.254240
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.2923	1.06859	0.534293	−	0.845300i	$-0.320578\pi$
$701$	28.2923	1.06859	0.534293	+	0.845300i	$0.320578\pi$
$702$	0	0
$703$	0.921919	0.0347709
$704$	0	0
$705$	0	0
$706$	0	0
$707$	5.38502	0.202524
$708$	0	0
$709$	17.6992	0.664709	0.332354	−	0.943155i	$-0.392157\pi$
$709$	17.6992	0.664709	0.332354	+	0.943155i	$0.392157\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.75926	−0.0658847
$714$	0	0
$715$	1.62655	0.0608294
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3.15088	−0.117508	−0.0587541	−	0.998272i	$-0.518713\pi$
$719$	−3.15088	−0.117508	−0.0587541	+	0.998272i	$0.518713\pi$
$720$	0	0
$721$	−18.6104	−0.693087
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−37.4595	−1.39121
$726$	0	0
$727$	−33.6337	−1.24741	−0.623703	−	0.781661i	$-0.714372\pi$
$727$	−33.6337	−1.24741	−0.623703	+	0.781661i	$0.714372\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.64293	−0.0607658
$732$	0	0
$733$	39.9148	1.47429	0.737144	−	0.675736i	$-0.236174\pi$
$733$	39.9148	1.47429	0.737144	+	0.675736i	$0.236174\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.12165	0.225494
$738$	0	0
$739$	17.6636	0.649765	0.324882	−	0.945754i	$-0.394675\pi$
$739$	17.6636	0.649765	0.324882	+	0.945754i	$0.394675\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	14.1650	0.519662	0.259831	−	0.965654i	$-0.416333\pi$
$743$	14.1650	0.519662	0.259831	+	0.965654i	$0.416333\pi$
$744$	0	0
$745$	5.87582	0.215273
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8.75067	0.319743
$750$	0	0
$751$	25.0130	0.912738	0.456369	−	0.889791i	$-0.349150\pi$
$751$	25.0130	0.912738	0.456369	+	0.889791i	$0.349150\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.64124	−0.0597309
$756$	0	0
$757$	−3.89566	−0.141590	−0.0707951	−	0.997491i	$-0.522554\pi$
$757$	−3.89566	−0.141590	−0.0707951	+	0.997491i	$0.522554\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	42.9386	1.55652	0.778262	−	0.627940i	$-0.216102\pi$
$761$	42.9386	1.55652	0.778262	+	0.627940i	$0.216102\pi$
$762$	0	0
$763$	15.2739	0.552954
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−32.1805	−1.16197
$768$	0	0
$769$	4.06517	0.146594	0.0732968	−	0.997310i	$-0.476648\pi$
$769$	4.06517	0.146594	0.0732968	+	0.997310i	$0.476648\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.54263	0.235322	0.117661	−	0.993054i	$-0.462460\pi$
$773$	6.54263	0.235322	0.117661	+	0.993054i	$0.462460\pi$
$774$	0	0
$775$	−3.75926	−0.135036
$776$	0	0
$777$	0	0
$778$	0	0
$779$	23.2062	0.831448
$780$	0	0
$781$	−12.3800	−0.442990
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−2.30041	−0.0821053
$786$	0	0
$787$	11.3658	0.405148	0.202574	−	0.979267i	$-0.435069\pi$
$787$	11.3658	0.405148	0.202574	+	0.979267i	$0.435069\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10.0871	0.358656
$792$	0	0
$793$	−36.9604	−1.31250
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−36.2021	−1.28235	−0.641173	−	0.767396i	$-0.721552\pi$
$797$	−36.2021	−1.28235	−0.641173	+	0.767396i	$0.721552\pi$
$798$	0	0
$799$	2.85741	0.101088
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2.68980	−0.0949210
$804$	0	0
$805$	−1.00101	−0.0352808
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−34.0261	−1.19629	−0.598147	−	0.801387i	$-0.704096\pi$
$809$	−34.0261	−1.19629	−0.598147	+	0.801387i	$0.704096\pi$
$810$	0	0
$811$	16.9918	0.596663	0.298332	−	0.954462i	$-0.403570\pi$
$811$	16.9918	0.596663	0.298332	+	0.954462i	$0.403570\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.401505	−0.0140641
$816$	0	0
$817$	−5.68017	−0.198724
$818$	0	0
$819$	0	0
$820$	0	0
$821$	15.3621	0.536140	0.268070	−	0.963399i	$-0.413614\pi$
$821$	15.3621	0.536140	0.268070	+	0.963399i	$0.413614\pi$
$822$	0	0
$823$	46.7317	1.62896	0.814482	−	0.580189i	$-0.197021\pi$
$823$	46.7317	1.62896	0.814482	+	0.580189i	$0.197021\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−36.2953	−1.26211	−0.631056	−	0.775737i	$-0.717378\pi$
$827$	−36.2953	−1.26211	−0.631056	+	0.775737i	$0.717378\pi$
$828$	0	0
$829$	−2.73078	−0.0948437	−0.0474219	−	0.998875i	$-0.515101\pi$
$829$	−2.73078	−0.0948437	−0.0474219	+	0.998875i	$0.515101\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.22841	0.0425617
$834$	0	0
$835$	−6.66401	−0.230618
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.26193	−0.0780907	−0.0390453	−	0.999237i	$-0.512432\pi$
$839$	−2.26193	−0.0780907	−0.0390453	+	0.999237i	$0.512432\pi$
$840$	0	0
$841$	31.8641	1.09876
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.646732	0.0222482
$846$	0	0
$847$	−10.0775	−0.346268
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−0.487771	−0.0167206
$852$	0	0
$853$	5.59513	0.191574	0.0957868	−	0.995402i	$-0.469463\pi$
$853$	5.59513	0.191574	0.0957868	+	0.995402i	$0.469463\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−41.3482	−1.41243	−0.706214	−	0.707998i	$-0.749598\pi$
$857$	−41.3482	−1.41243	−0.706214	+	0.707998i	$0.749598\pi$
$858$	0	0
$859$	13.2787	0.453062	0.226531	−	0.974004i	$-0.427262\pi$
$859$	13.2787	0.453062	0.226531	+	0.974004i	$0.427262\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23.3694	0.795502	0.397751	−	0.917493i	$-0.369791\pi$
$863$	23.3694	0.795502	0.397751	+	0.917493i	$0.369791\pi$
$864$	0	0
$865$	2.50966	0.0853310
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−6.33110	−0.214768
$870$	0	0
$871$	24.2299	0.820997
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.36639	−0.147611
$876$	0	0
$877$	−7.55228	−0.255022	−0.127511	−	0.991837i	$-0.540699\pi$
$877$	−7.55228	−0.255022	−0.127511	+	0.991837i	$0.540699\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−39.6822	−1.33693	−0.668463	−	0.743745i	$-0.733048\pi$
$881$	−39.6822	−1.33693	−0.668463	+	0.743745i	$0.733048\pi$
$882$	0	0
$883$	−45.9283	−1.54561	−0.772806	−	0.634643i	$-0.781147\pi$
$883$	−45.9283	−1.54561	−0.772806	+	0.634643i	$0.781147\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−49.9054	−1.67566	−0.837830	−	0.545931i	$-0.816176\pi$
$887$	−49.9054	−1.67566	−0.837830	+	0.545931i	$0.816176\pi$
$888$	0	0
$889$	3.61342	0.121190
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9.87907	0.330590
$894$	0	0
$895$	4.48429	0.149893
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.10803	0.203714
$900$	0	0
$901$	−7.19811	−0.239804
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.24254	0.307232
$906$	0	0
$907$	−42.1386	−1.39919	−0.699595	−	0.714539i	$-0.746636\pi$
$907$	−42.1386	−1.39919	−0.699595	+	0.714539i	$0.746636\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−23.9808	−0.794520	−0.397260	−	0.917706i	$-0.630039\pi$
$911$	−23.9808	−0.794520	−0.397260	+	0.917706i	$0.630039\pi$
$912$	0	0
$913$	1.71942	0.0569047
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.25532	−0.206569
$918$	0	0
$919$	−57.9676	−1.91217	−0.956087	−	0.293083i	$-0.905319\pi$
$919$	−57.9676	−1.91217	−0.956087	+	0.293083i	$0.905319\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−49.0007	−1.61288
$924$	0	0
$925$	−1.04229	−0.0342703
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−19.0575	−0.625258	−0.312629	−	0.949875i	$-0.601210\pi$
$929$	−19.0575	−0.625258	−0.312629	+	0.949875i	$0.601210\pi$
$930$	0	0
$931$	4.24703	0.139191
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.525591	−0.0171887
$936$	0	0
$937$	30.7687	1.00517	0.502584	−	0.864528i	$-0.332383\pi$
$937$	30.7687	1.00517	0.502584	+	0.864528i	$0.332383\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−33.1283	−1.07995	−0.539977	−	0.841680i	$-0.681567\pi$
$941$	−33.1283	−1.07995	−0.539977	+	0.841680i	$0.681567\pi$
$942$	0	0
$943$	−12.2780	−0.399826
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29.6378	−0.963097	−0.481549	−	0.876419i	$-0.659926\pi$
$947$	−29.6378	−0.963097	−0.481549	+	0.876419i	$0.659926\pi$
$948$	0	0
$949$	−10.6464	−0.345597
$950$	0	0
$951$	0	0
$952$	0	0
$953$	27.2594	0.883017	0.441509	−	0.897257i	$-0.354443\pi$
$953$	27.2594	0.883017	0.441509	+	0.897257i	$0.354443\pi$
$954$	0	0
$955$	4.07226	0.131775
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10.7393	0.346791
$960$	0	0
$961$	−30.3870	−0.980227
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5.47655	−0.176296
$966$	0	0
$967$	12.2817	0.394952	0.197476	−	0.980308i	$-0.436726\pi$
$967$	12.2817	0.394952	0.197476	+	0.980308i	$0.436726\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−32.0197	−1.02756	−0.513781	−	0.857921i	$-0.671756\pi$
$971$	−32.0197	−1.02756	−0.513781	+	0.857921i	$0.671756\pi$
$972$	0	0
$973$	−13.6094	−0.436297
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−25.3105	−0.809756	−0.404878	−	0.914371i	$-0.632686\pi$
$977$	−25.3105	−0.809756	−0.404878	+	0.914371i	$0.632686\pi$
$978$	0	0
$979$	−4.68064	−0.149594
$980$	0	0
$981$	0	0
$982$	0	0
$983$	17.6776	0.563827	0.281914	−	0.959440i	$-0.409031\pi$
$983$	17.6776	0.563827	0.281914	+	0.959440i	$0.409031\pi$
$984$	0	0
$985$	0.935512	0.0298079
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.00528	0.0955623
$990$	0	0
$991$	46.4955	1.47698	0.738489	−	0.674265i	$-0.235540\pi$
$991$	46.4955	1.47698	0.738489	+	0.674265i	$0.235540\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.07549	0.160904
$996$	0	0
$997$	−56.0910	−1.77642	−0.888210	−	0.459438i	$-0.848051\pi$
$997$	−56.0910	−1.77642	−0.888210	+	0.459438i	$0.848051\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4536.2.a.bb.1.2	yes	4
3.2	odd	2		4536.2.a.w.1.3	✓	4
4.3	odd	2		9072.2.a.ck.1.2		4
12.11	even	2		9072.2.a.cf.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4536.2.a.w.1.3	✓	4	3.2	odd	2
4536.2.a.bb.1.2	yes	4	1.1	even	1	trivial
9072.2.a.cf.1.3		4	12.11	even	2
9072.2.a.ck.1.2		4	4.3	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:	\( N \)	\(=\)	\( 4536 = 2^{3} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4536.a (trivial)

\(f(q)\)	\(=\)	\(q+0.445480 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q+0.445480 q^{5} +1.00000 q^{7} -0.960457 q^{11} -3.80155 q^{13} +1.22841 q^{17} +4.24703 q^{19} -2.24703 q^{23} -4.80155 q^{25} +7.80155 q^{29} +0.782926 q^{31} +0.445480 q^{35} +0.217074 q^{37} +5.46410 q^{41} -1.33745 q^{43} +2.32611 q^{47} +1.00000 q^{49} -5.85971 q^{53} -0.427864 q^{55} +8.46511 q^{59} +9.72246 q^{61} -1.69351 q^{65} -6.37368 q^{67} +12.8897 q^{71} +2.80054 q^{73} -0.960457 q^{77} +6.59176 q^{79} -1.79022 q^{83} +0.547230 q^{85} +4.87334 q^{89} -3.80155 q^{91} +1.89197 q^{95} +7.02995 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^5 + 4 * q^7	\( 4 q + 4 q^{5} + 4 q^{7} + 6 q^{11} + 6 q^{13} + 2 q^{17} - 2 q^{19} + 10 q^{23} + 2 q^{25} + 10 q^{29} - 2 q^{31} + 4 q^{35} + 6 q^{37} + 8 q^{41} + 2 q^{43} + 10 q^{47} + 4 q^{49} + 4 q^{53} - 6 q^{55} + 6 q^{59} - 2 q^{61} + 24 q^{65} + 4 q^{73} + 6 q^{77} - 8 q^{79} + 6 q^{83} + 8 q^{85} + 26 q^{89} + 6 q^{91} - 2 q^{95} + 4 q^{97}+O(q^{100}) \) 4 * q + 4 * q^5 + 4 * q^7 + 6 * q^11 + 6 * q^13 + 2 * q^17 - 2 * q^19 + 10 * q^23 + 2 * q^25 + 10 * q^29 - 2 * q^31 + 4 * q^35 + 6 * q^37 + 8 * q^41 + 2 * q^43 + 10 * q^47 + 4 * q^49 + 4 * q^53 - 6 * q^55 + 6 * q^59 - 2 * q^61 + 24 * q^65 + 4 * q^73 + 6 * q^77 - 8 * q^79 + 6 * q^83 + 8 * q^85 + 26 * q^89 + 6 * q^91 - 2 * q^95 + 4 * q^97

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