LMFDB - Embedded newform 4400.2.a.cc.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(1,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, 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("4400.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.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:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 4 $ x^4 - 7*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.792287$ of defining polynomial
Character	$\chi$	$=$	4400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.792287 q^{3} -3.46410 q^{7} -2.37228 q^{9} +O(q^{10})$	$q+0.792287 q^{3} -3.46410 q^{7} -2.37228 q^{9} +1.00000 q^{11} +1.58457 q^{17} -4.00000 q^{19} -2.74456 q^{21} -0.792287 q^{23} -4.25639 q^{27} +8.74456 q^{29} -3.37228 q^{31} +0.792287 q^{33} -1.08724 q^{37} +8.74456 q^{41} -3.46410 q^{43} +6.63325 q^{47} +5.00000 q^{49} +1.25544 q^{51} +10.0974 q^{53} -3.16915 q^{57} +7.37228 q^{59} -0.744563 q^{61} +8.21782 q^{63} -9.30506 q^{67} -0.627719 q^{69} +10.1168 q^{71} -6.92820 q^{73} -3.46410 q^{77} -1.25544 q^{79} +3.74456 q^{81} -6.63325 q^{83} +6.92820 q^{87} +1.37228 q^{89} -2.67181 q^{93} +5.84096 q^{97} -2.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{9}+O(q^{10}) $ 4 * q + 2 * q^9	$ 4 q + 2 q^{9} + 4 q^{11} - 16 q^{19} + 12 q^{21} + 12 q^{29} - 2 q^{31} + 12 q^{41} + 20 q^{49} + 28 q^{51} + 18 q^{59} + 20 q^{61} - 14 q^{69} + 6 q^{71} - 28 q^{79} - 8 q^{81} - 6 q^{89} + 2 q^{99}+O(q^{100}) $ 4 * q + 2 * q^9 + 4 * q^11 - 16 * q^19 + 12 * q^21 + 12 * q^29 - 2 * q^31 + 12 * q^41 + 20 * q^49 + 28 * q^51 + 18 * q^59 + 20 * q^61 - 14 * q^69 + 6 * q^71 - 28 * q^79 - 8 * q^81 - 6 * q^89 + 2 * q^99

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
$3$	0.792287	0.457427	0.228714	−	0.973494i	$-0.426548\pi$
$3$	0.792287	0.457427	0.228714	+	0.973494i	$0.426548\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.46410	−1.30931	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	−3.46410	−1.30931	−0.654654	+	0.755929i	$0.727186\pi$
$8$	0	0
$9$	−2.37228	−0.790760
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.58457	0.384316	0.192158	−	0.981364i	$-0.438451\pi$
$17$	1.58457	0.384316	0.192158	+	0.981364i	$0.438451\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	−2.74456	−0.598913
$22$	0	0
$23$	−0.792287	−0.165203	−0.0826016	−	0.996583i	$-0.526323\pi$
$23$	−0.792287	−0.165203	−0.0826016	+	0.996583i	$0.526323\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−4.25639	−0.819142
$28$	0	0
$29$	8.74456	1.62382	0.811912	−	0.583779i	$-0.198427\pi$
$29$	8.74456	1.62382	0.811912	+	0.583779i	$0.198427\pi$
$30$	0	0
$31$	−3.37228	−0.605680	−0.302840	−	0.953041i	$-0.597935\pi$
$31$	−3.37228	−0.605680	−0.302840	+	0.953041i	$0.597935\pi$
$32$	0	0
$33$	0.792287	0.137919
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.08724	−0.178741	−0.0893706	−	0.995998i	$-0.528486\pi$
$37$	−1.08724	−0.178741	−0.0893706	+	0.995998i	$0.528486\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.74456	1.36567	0.682836	−	0.730572i	$-0.260747\pi$
$41$	8.74456	1.36567	0.682836	+	0.730572i	$0.260747\pi$
$42$	0	0
$43$	−3.46410	−0.528271	−0.264135	−	0.964486i	$-0.585087\pi$
$43$	−3.46410	−0.528271	−0.264135	+	0.964486i	$0.585087\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.63325	0.967559	0.483779	−	0.875190i	$-0.339264\pi$
$47$	6.63325	0.967559	0.483779	+	0.875190i	$0.339264\pi$
$48$	0	0
$49$	5.00000	0.714286
$50$	0	0
$51$	1.25544	0.175796
$52$	0	0
$53$	10.0974	1.38698	0.693489	−	0.720467i	$-0.256073\pi$
$53$	10.0974	1.38698	0.693489	+	0.720467i	$0.256073\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.16915	−0.419764
$58$	0	0
$59$	7.37228	0.959789	0.479895	−	0.877326i	$-0.340675\pi$
$59$	7.37228	0.959789	0.479895	+	0.877326i	$0.340675\pi$
$60$	0	0
$61$	−0.744563	−0.0953315	−0.0476657	−	0.998863i	$-0.515178\pi$
$61$	−0.744563	−0.0953315	−0.0476657	+	0.998863i	$0.515178\pi$
$62$	0	0
$63$	8.21782	1.03535
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.30506	−1.13679	−0.568397	−	0.822754i	$-0.692436\pi$
$67$	−9.30506	−1.13679	−0.568397	+	0.822754i	$0.692436\pi$
$68$	0	0
$69$	−0.627719	−0.0755684
$70$	0	0
$71$	10.1168	1.20065	0.600324	−	0.799757i	$-0.295038\pi$
$71$	10.1168	1.20065	0.600324	+	0.799757i	$0.295038\pi$
$72$	0	0
$73$	−6.92820	−0.810885	−0.405442	−	0.914121i	$-0.632883\pi$
$73$	−6.92820	−0.810885	−0.405442	+	0.914121i	$0.632883\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.46410	−0.394771
$78$	0	0
$79$	−1.25544	−0.141248	−0.0706239	−	0.997503i	$-0.522499\pi$
$79$	−1.25544	−0.141248	−0.0706239	+	0.997503i	$0.522499\pi$
$80$	0	0
$81$	3.74456	0.416063
$82$	0	0
$83$	−6.63325	−0.728094	−0.364047	−	0.931381i	$-0.618605\pi$
$83$	−6.63325	−0.728094	−0.364047	+	0.931381i	$0.618605\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	6.92820	0.742781
$88$	0	0
$89$	1.37228	0.145462	0.0727308	−	0.997352i	$-0.476829\pi$
$89$	1.37228	0.145462	0.0727308	+	0.997352i	$0.476829\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.67181	−0.277054
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.84096	0.593060	0.296530	−	0.955024i	$-0.404171\pi$
$97$	5.84096	0.593060	0.296530	+	0.955024i	$0.404171\pi$
$98$	0	0
$99$	−2.37228	−0.238423
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	10.3923	1.02398	0.511992	−	0.858990i	$-0.328908\pi$
$103$	10.3923	1.02398	0.511992	+	0.858990i	$0.328908\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.63325	0.641260	0.320630	−	0.947204i	$-0.396105\pi$
$107$	6.63325	0.641260	0.320630	+	0.947204i	$0.396105\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−0.861407	−0.0817611
$112$	0	0
$113$	−0.497333	−0.0467852	−0.0233926	−	0.999726i	$-0.507447\pi$
$113$	−0.497333	−0.0467852	−0.0233926	+	0.999726i	$0.507447\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.48913	−0.503187
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.92820	0.624695
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.21782	0.729214	0.364607	−	0.931162i	$-0.381203\pi$
$127$	8.21782	0.729214	0.364607	+	0.931162i	$0.381203\pi$
$128$	0	0
$129$	−2.74456	−0.241645
$130$	0	0
$131$	2.74456	0.239794	0.119897	−	0.992786i	$-0.461744\pi$
$131$	2.74456	0.239794	0.119897	+	0.992786i	$0.461744\pi$
$132$	0	0
$133$	13.8564	1.20150
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.3537	1.22632	0.613161	−	0.789958i	$-0.289898\pi$
$137$	14.3537	1.22632	0.613161	+	0.789958i	$0.289898\pi$
$138$	0	0
$139$	16.2337	1.37692	0.688462	−	0.725273i	$-0.258286\pi$
$139$	16.2337	1.37692	0.688462	+	0.725273i	$0.258286\pi$
$140$	0	0
$141$	5.25544	0.442588
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.96143	0.326734
$148$	0	0
$149$	−11.4891	−0.941226	−0.470613	−	0.882340i	$-0.655967\pi$
$149$	−11.4891	−0.941226	−0.470613	+	0.882340i	$0.655967\pi$
$150$	0	0
$151$	12.2337	0.995563	0.497782	−	0.867302i	$-0.334148\pi$
$151$	12.2337	0.995563	0.497782	+	0.867302i	$0.334148\pi$
$152$	0	0
$153$	−3.75906	−0.303902
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−24.4511	−1.95141	−0.975705	−	0.219090i	$-0.929691\pi$
$157$	−24.4511	−1.95141	−0.975705	+	0.219090i	$0.929691\pi$
$158$	0	0
$159$	8.00000	0.634441
$160$	0	0
$161$	2.74456	0.216302
$162$	0	0
$163$	3.46410	0.271329	0.135665	−	0.990755i	$-0.456683\pi$
$163$	3.46410	0.271329	0.135665	+	0.990755i	$0.456683\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.7359	1.21768	0.608842	−	0.793292i	$-0.291635\pi$
$167$	15.7359	1.21768	0.608842	+	0.793292i	$0.291635\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	9.48913	0.725652
$172$	0	0
$173$	−8.51278	−0.647214	−0.323607	−	0.946192i	$-0.604896\pi$
$173$	−8.51278	−0.647214	−0.323607	+	0.946192i	$0.604896\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.84096	0.439034
$178$	0	0
$179$	−12.8614	−0.961307	−0.480653	−	0.876911i	$-0.659600\pi$
$179$	−12.8614	−0.961307	−0.480653	+	0.876911i	$0.659600\pi$
$180$	0	0
$181$	24.1168	1.79259	0.896295	−	0.443457i	$-0.146248\pi$
$181$	24.1168	1.79259	0.896295	+	0.443457i	$0.146248\pi$
$182$	0	0
$183$	−0.589907	−0.0436072
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.58457	0.115876
$188$	0	0
$189$	14.7446	1.07251
$190$	0	0
$191$	19.3723	1.40173	0.700865	−	0.713294i	$-0.252798\pi$
$191$	19.3723	1.40173	0.700865	+	0.713294i	$0.252798\pi$
$192$	0	0
$193$	−16.4356	−1.18306	−0.591532	−	0.806282i	$-0.701477\pi$
$193$	−16.4356	−1.18306	−0.591532	+	0.806282i	$0.701477\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.51278	0.606510	0.303255	−	0.952909i	$-0.401927\pi$
$197$	8.51278	0.606510	0.303255	+	0.952909i	$0.401927\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−7.37228	−0.520001
$202$	0	0
$203$	−30.2921	−2.12609
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.87953	0.130636
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−1.48913	−0.102516	−0.0512578	−	0.998685i	$-0.516323\pi$
$211$	−1.48913	−0.102516	−0.0512578	+	0.998685i	$0.516323\pi$
$212$	0	0
$213$	8.01544	0.549209
$214$	0	0
$215$	0	0
$216$	0	0
$217$	11.6819	0.793021
$218$	0	0
$219$	−5.48913	−0.370921
$220$	0	0
$221$	0	0
$222$	0	0
$223$	2.37686	0.159166	0.0795832	−	0.996828i	$-0.474641\pi$
$223$	2.37686	0.159166	0.0795832	+	0.996828i	$0.474641\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16.7306	−1.11045	−0.555224	−	0.831701i	$-0.687368\pi$
$227$	−16.7306	−1.11045	−0.555224	+	0.831701i	$0.687368\pi$
$228$	0	0
$229$	14.6277	0.966627	0.483313	−	0.875447i	$-0.339433\pi$
$229$	14.6277	0.966627	0.483313	+	0.875447i	$0.339433\pi$
$230$	0	0
$231$	−2.74456	−0.180579
$232$	0	0
$233$	−3.75906	−0.246264	−0.123132	−	0.992390i	$-0.539294\pi$
$233$	−3.75906	−0.246264	−0.123132	+	0.992390i	$0.539294\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−0.994667	−0.0646105
$238$	0	0
$239$	14.7446	0.953746	0.476873	−	0.878972i	$-0.341770\pi$
$239$	14.7446	0.953746	0.476873	+	0.878972i	$0.341770\pi$
$240$	0	0
$241$	16.7446	1.07861	0.539306	−	0.842110i	$-0.318687\pi$
$241$	16.7446	1.07861	0.539306	+	0.842110i	$0.318687\pi$
$242$	0	0
$243$	15.7359	1.00946
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−5.25544	−0.333050
$250$	0	0
$251$	22.1168	1.39600	0.698001	−	0.716096i	$-0.254073\pi$
$251$	22.1168	1.39600	0.698001	+	0.716096i	$0.254073\pi$
$252$	0	0
$253$	−0.792287	−0.0498107
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.6873	0.666653	0.333326	−	0.942811i	$-0.391829\pi$
$257$	10.6873	0.666653	0.333326	+	0.942811i	$0.391829\pi$
$258$	0	0
$259$	3.76631	0.234027
$260$	0	0
$261$	−20.7446	−1.28406
$262$	0	0
$263$	−27.4179	−1.69066	−0.845329	−	0.534246i	$-0.820595\pi$
$263$	−27.4179	−1.69066	−0.845329	+	0.534246i	$0.820595\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.08724	0.0665380
$268$	0	0
$269$	−11.4891	−0.700504	−0.350252	−	0.936655i	$-0.613904\pi$
$269$	−11.4891	−0.700504	−0.350252	+	0.936655i	$0.613904\pi$
$270$	0	0
$271$	−13.4891	−0.819406	−0.409703	−	0.912219i	$-0.634368\pi$
$271$	−13.4891	−0.819406	−0.409703	+	0.912219i	$0.634368\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−11.6819	−0.701899	−0.350949	−	0.936394i	$-0.614141\pi$
$277$	−11.6819	−0.701899	−0.350949	+	0.936394i	$0.614141\pi$
$278$	0	0
$279$	8.00000	0.478947
$280$	0	0
$281$	−0.510875	−0.0304762	−0.0152381	−	0.999884i	$-0.504851\pi$
$281$	−0.510875	−0.0304762	−0.0152381	+	0.999884i	$0.504851\pi$
$282$	0	0
$283$	−15.1460	−0.900338	−0.450169	−	0.892943i	$-0.648636\pi$
$283$	−15.1460	−0.900338	−0.450169	+	0.892943i	$0.648636\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−30.2921	−1.78808
$288$	0	0
$289$	−14.4891	−0.852301
$290$	0	0
$291$	4.62772	0.271282
$292$	0	0
$293$	3.16915	0.185144	0.0925718	−	0.995706i	$-0.470491\pi$
$293$	3.16915	0.185144	0.0925718	+	0.995706i	$0.470491\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−4.25639	−0.246981
$298$	0	0
$299$	0	0
$300$	0	0
$301$	12.0000	0.691669
$302$	0	0
$303$	4.75372	0.273094
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−31.5817	−1.80246	−0.901231	−	0.433340i	$-0.857335\pi$
$307$	−31.5817	−1.80246	−0.901231	+	0.433340i	$0.857335\pi$
$308$	0	0
$309$	8.23369	0.468398
$310$	0	0
$311$	−5.48913	−0.311260	−0.155630	−	0.987815i	$-0.549741\pi$
$311$	−5.48913	−0.311260	−0.155630	+	0.987815i	$0.549741\pi$
$312$	0	0
$313$	−21.8719	−1.23627	−0.618135	−	0.786072i	$-0.712112\pi$
$313$	−21.8719	−1.23627	−0.618135	+	0.786072i	$0.712112\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32.9639	1.85144	0.925718	−	0.378215i	$-0.123462\pi$
$317$	32.9639	1.85144	0.925718	+	0.378215i	$0.123462\pi$
$318$	0	0
$319$	8.74456	0.489602
$320$	0	0
$321$	5.25544	0.293330
$322$	0	0
$323$	−6.33830	−0.352672
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.92287	0.438136
$328$	0	0
$329$	−22.9783	−1.26683
$330$	0	0
$331$	14.1168	0.775932	0.387966	−	0.921674i	$-0.373178\pi$
$331$	14.1168	0.775932	0.387966	+	0.921674i	$0.373178\pi$
$332$	0	0
$333$	2.57924	0.141342
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−32.4665	−1.76856	−0.884282	−	0.466952i	$-0.845352\pi$
$337$	−32.4665	−1.76856	−0.884282	+	0.466952i	$0.845352\pi$
$338$	0	0
$339$	−0.394031	−0.0214008
$340$	0	0
$341$	−3.37228	−0.182619
$342$	0	0
$343$	6.92820	0.374088
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.6641	1.21667	0.608337	−	0.793679i	$-0.291837\pi$
$347$	22.6641	1.21667	0.608337	+	0.793679i	$0.291837\pi$
$348$	0	0
$349$	15.4891	0.829114	0.414557	−	0.910023i	$-0.363937\pi$
$349$	15.4891	0.829114	0.414557	+	0.910023i	$0.363937\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.0410	1.33280	0.666399	−	0.745595i	$-0.267835\pi$
$353$	25.0410	1.33280	0.666399	+	0.745595i	$0.267835\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.34896	−0.230172
$358$	0	0
$359$	−29.4891	−1.55638	−0.778188	−	0.628031i	$-0.783861\pi$
$359$	−29.4891	−1.55638	−0.778188	+	0.628031i	$0.783861\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0.792287	0.0415843
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−25.7407	−1.34365	−0.671827	−	0.740708i	$-0.734490\pi$
$367$	−25.7407	−1.34365	−0.671827	+	0.740708i	$0.734490\pi$
$368$	0	0
$369$	−20.7446	−1.07992
$370$	0	0
$371$	−34.9783	−1.81598
$372$	0	0
$373$	−11.6819	−0.604867	−0.302434	−	0.953170i	$-0.597799\pi$
$373$	−11.6819	−0.604867	−0.302434	+	0.953170i	$0.597799\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0.627719	0.0322437	0.0161219	−	0.999870i	$-0.494868\pi$
$379$	0.627719	0.0322437	0.0161219	+	0.999870i	$0.494868\pi$
$380$	0	0
$381$	6.51087	0.333562
$382$	0	0
$383$	10.8896	0.556435	0.278217	−	0.960518i	$-0.410256\pi$
$383$	10.8896	0.556435	0.278217	+	0.960518i	$0.410256\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.21782	0.417735
$388$	0	0
$389$	18.8614	0.956311	0.478156	−	0.878275i	$-0.341305\pi$
$389$	18.8614	0.956311	0.478156	+	0.878275i	$0.341305\pi$
$390$	0	0
$391$	−1.25544	−0.0634902
$392$	0	0
$393$	2.17448	0.109688
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−16.4356	−0.824881	−0.412441	−	0.910984i	$-0.635324\pi$
$397$	−16.4356	−0.824881	−0.412441	+	0.910984i	$0.635324\pi$
$398$	0	0
$399$	10.9783	0.549600
$400$	0	0
$401$	−11.4891	−0.573740	−0.286870	−	0.957970i	$-0.592615\pi$
$401$	−11.4891	−0.573740	−0.286870	+	0.957970i	$0.592615\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.08724	−0.0538925
$408$	0	0
$409$	27.4891	1.35925	0.679625	−	0.733560i	$-0.262143\pi$
$409$	27.4891	1.35925	0.679625	+	0.733560i	$0.262143\pi$
$410$	0	0
$411$	11.3723	0.560953
$412$	0	0
$413$	−25.5383	−1.25666
$414$	0	0
$415$	0	0
$416$	0	0
$417$	12.8617	0.629842
$418$	0	0
$419$	22.9783	1.12256	0.561280	−	0.827626i	$-0.310309\pi$
$419$	22.9783	1.12256	0.561280	+	0.827626i	$0.310309\pi$
$420$	0	0
$421$	8.51087	0.414795	0.207397	−	0.978257i	$-0.433501\pi$
$421$	8.51087	0.414795	0.207397	+	0.978257i	$0.433501\pi$
$422$	0	0
$423$	−15.7359	−0.765107
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.57924	0.124818
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.7228	1.23902	0.619512	−	0.784987i	$-0.287330\pi$
$431$	25.7228	1.23902	0.619512	+	0.784987i	$0.287330\pi$
$432$	0	0
$433$	29.2048	1.40349	0.701747	−	0.712426i	$-0.252404\pi$
$433$	29.2048	1.40349	0.701747	+	0.712426i	$0.252404\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.16915	0.151601
$438$	0	0
$439$	−21.4891	−1.02562	−0.512810	−	0.858502i	$-0.671395\pi$
$439$	−21.4891	−1.02562	−0.512810	+	0.858502i	$0.671395\pi$
$440$	0	0
$441$	−11.8614	−0.564829
$442$	0	0
$443$	31.6742	1.50489	0.752444	−	0.658656i	$-0.228875\pi$
$443$	31.6742	1.50489	0.752444	+	0.658656i	$0.228875\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−9.10268	−0.430542
$448$	0	0
$449$	6.86141	0.323810	0.161905	−	0.986806i	$-0.448236\pi$
$449$	6.86141	0.323810	0.161905	+	0.986806i	$0.448236\pi$
$450$	0	0
$451$	8.74456	0.411765
$452$	0	0
$453$	9.69259	0.455398
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−20.7846	−0.972263	−0.486132	−	0.873886i	$-0.661592\pi$
$457$	−20.7846	−0.972263	−0.486132	+	0.873886i	$0.661592\pi$
$458$	0	0
$459$	−6.74456	−0.314809
$460$	0	0
$461$	2.23369	0.104033	0.0520166	−	0.998646i	$-0.483435\pi$
$461$	2.23369	0.104033	0.0520166	+	0.998646i	$0.483435\pi$
$462$	0	0
$463$	30.0897	1.39839	0.699193	−	0.714933i	$-0.253543\pi$
$463$	30.0897	1.39839	0.699193	+	0.714933i	$0.253543\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.72049	0.357262	0.178631	−	0.983916i	$-0.442833\pi$
$467$	7.72049	0.357262	0.178631	+	0.983916i	$0.442833\pi$
$468$	0	0
$469$	32.2337	1.48841
$470$	0	0
$471$	−19.3723	−0.892628
$472$	0	0
$473$	−3.46410	−0.159280
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−23.9538	−1.09677
$478$	0	0
$479$	5.48913	0.250805	0.125402	−	0.992106i	$-0.459978\pi$
$479$	5.48913	0.250805	0.125402	+	0.992106i	$0.459978\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	2.17448	0.0989423
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.13058	0.323118	0.161559	−	0.986863i	$-0.448348\pi$
$487$	7.13058	0.323118	0.161559	+	0.986863i	$0.448348\pi$
$488$	0	0
$489$	2.74456	0.124113
$490$	0	0
$491$	−6.51087	−0.293832	−0.146916	−	0.989149i	$-0.546935\pi$
$491$	−6.51087	−0.293832	−0.146916	+	0.989149i	$0.546935\pi$
$492$	0	0
$493$	13.8564	0.624061
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−35.0458	−1.57202
$498$	0	0
$499$	20.0000	0.895323	0.447661	−	0.894203i	$-0.352257\pi$
$499$	20.0000	0.895323	0.447661	+	0.894203i	$0.352257\pi$
$500$	0	0
$501$	12.4674	0.557001
$502$	0	0
$503$	−13.5615	−0.604675	−0.302338	−	0.953201i	$-0.597767\pi$
$503$	−13.5615	−0.604675	−0.302338	+	0.953201i	$0.597767\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−10.2997	−0.457427
$508$	0	0
$509$	22.6277	1.00296	0.501478	−	0.865170i	$-0.332790\pi$
$509$	22.6277	1.00296	0.501478	+	0.865170i	$0.332790\pi$
$510$	0	0
$511$	24.0000	1.06170
$512$	0	0
$513$	17.0256	0.751697
$514$	0	0
$515$	0	0
$516$	0	0
$517$	6.63325	0.291730
$518$	0	0
$519$	−6.74456	−0.296053
$520$	0	0
$521$	−21.6060	−0.946575	−0.473287	−	0.880908i	$-0.656933\pi$
$521$	−21.6060	−0.946575	−0.473287	+	0.880908i	$0.656933\pi$
$522$	0	0
$523$	−29.0024	−1.26819	−0.634094	−	0.773256i	$-0.718627\pi$
$523$	−29.0024	−1.26819	−0.634094	+	0.773256i	$0.718627\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.34363	−0.232772
$528$	0	0
$529$	−22.3723	−0.972708
$530$	0	0
$531$	−17.4891	−0.758963
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−10.1899	−0.439728
$538$	0	0
$539$	5.00000	0.215365
$540$	0	0
$541$	34.2337	1.47182	0.735911	−	0.677079i	$-0.236754\pi$
$541$	34.2337	1.47182	0.735911	+	0.677079i	$0.236754\pi$
$542$	0	0
$543$	19.1075	0.819980
$544$	0	0
$545$	0	0
$546$	0	0
$547$	29.0024	1.24005	0.620027	−	0.784580i	$-0.287122\pi$
$547$	29.0024	1.24005	0.620027	+	0.784580i	$0.287122\pi$
$548$	0	0
$549$	1.76631	0.0753844
$550$	0	0
$551$	−34.9783	−1.49012
$552$	0	0
$553$	4.34896	0.184937
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−0.994667	−0.0421454	−0.0210727	−	0.999778i	$-0.506708\pi$
$557$	−0.994667	−0.0421454	−0.0210727	+	0.999778i	$0.506708\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	1.25544	0.0530046
$562$	0	0
$563$	18.9051	0.796754	0.398377	−	0.917222i	$-0.369574\pi$
$563$	18.9051	0.796754	0.398377	+	0.917222i	$0.369574\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−12.9715	−0.544754
$568$	0	0
$569$	−27.2554	−1.14261	−0.571304	−	0.820739i	$-0.693562\pi$
$569$	−27.2554	−1.14261	−0.571304	+	0.820739i	$0.693562\pi$
$570$	0	0
$571$	−1.48913	−0.0623180	−0.0311590	−	0.999514i	$-0.509920\pi$
$571$	−1.48913	−0.0623180	−0.0311590	+	0.999514i	$0.509920\pi$
$572$	0	0
$573$	15.3484	0.641189
$574$	0	0
$575$	0	0
$576$	0	0
$577$	21.8719	0.910537	0.455269	−	0.890354i	$-0.349543\pi$
$577$	21.8719	0.910537	0.455269	+	0.890354i	$0.349543\pi$
$578$	0	0
$579$	−13.0217	−0.541165
$580$	0	0
$581$	22.9783	0.953298
$582$	0	0
$583$	10.0974	0.418190
$584$	0	0
$585$	0	0
$586$	0	0
$587$	41.2743	1.70357	0.851786	−	0.523890i	$-0.175520\pi$
$587$	41.2743	1.70357	0.851786	+	0.523890i	$0.175520\pi$
$588$	0	0
$589$	13.4891	0.555810
$590$	0	0
$591$	6.74456	0.277434
$592$	0	0
$593$	−22.7739	−0.935214	−0.467607	−	0.883937i	$-0.654884\pi$
$593$	−22.7739	−0.935214	−0.467607	+	0.883937i	$0.654884\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.33830	0.259409
$598$	0	0
$599$	−10.9783	−0.448559	−0.224280	−	0.974525i	$-0.572003\pi$
$599$	−10.9783	−0.448559	−0.224280	+	0.974525i	$0.572003\pi$
$600$	0	0
$601$	−38.4674	−1.56912	−0.784558	−	0.620055i	$-0.787110\pi$
$601$	−38.4674	−1.56912	−0.784558	+	0.620055i	$0.787110\pi$
$602$	0	0
$603$	22.0742	0.898932
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−3.46410	−0.140604	−0.0703018	−	0.997526i	$-0.522396\pi$
$607$	−3.46410	−0.140604	−0.0703018	+	0.997526i	$0.522396\pi$
$608$	0	0
$609$	−24.0000	−0.972529
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−4.34896	−0.175653	−0.0878265	−	0.996136i	$-0.527992\pi$
$613$	−4.34896	−0.175653	−0.0878265	+	0.996136i	$0.527992\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−17.0256	−0.685423	−0.342712	−	0.939441i	$-0.611345\pi$
$617$	−17.0256	−0.685423	−0.342712	+	0.939441i	$0.611345\pi$
$618$	0	0
$619$	−14.1168	−0.567404	−0.283702	−	0.958913i	$-0.591563\pi$
$619$	−14.1168	−0.567404	−0.283702	+	0.958913i	$0.591563\pi$
$620$	0	0
$621$	3.37228	0.135325
$622$	0	0
$623$	−4.75372	−0.190454
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−3.16915	−0.126564
$628$	0	0
$629$	−1.72281	−0.0686931
$630$	0	0
$631$	−23.6060	−0.939739	−0.469869	−	0.882736i	$-0.655699\pi$
$631$	−23.6060	−0.939739	−0.469869	+	0.882736i	$0.655699\pi$
$632$	0	0
$633$	−1.17981	−0.0468934
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−24.0000	−0.949425
$640$	0	0
$641$	−25.3723	−1.00214	−0.501072	−	0.865405i	$-0.667061\pi$
$641$	−25.3723	−1.00214	−0.501072	+	0.865405i	$0.667061\pi$
$642$	0	0
$643$	30.4944	1.20258	0.601292	−	0.799030i	$-0.294653\pi$
$643$	30.4944	1.20258	0.601292	+	0.799030i	$0.294653\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.9817	0.864188	0.432094	−	0.901829i	$-0.357775\pi$
$647$	21.9817	0.864188	0.432094	+	0.901829i	$0.357775\pi$
$648$	0	0
$649$	7.37228	0.289387
$650$	0	0
$651$	9.25544	0.362749
$652$	0	0
$653$	−30.7894	−1.20488	−0.602441	−	0.798163i	$-0.705805\pi$
$653$	−30.7894	−1.20488	−0.602441	+	0.798163i	$0.705805\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	16.4356	0.641216
$658$	0	0
$659$	−21.2554	−0.827994	−0.413997	−	0.910278i	$-0.635868\pi$
$659$	−21.2554	−0.827994	−0.413997	+	0.910278i	$0.635868\pi$
$660$	0	0
$661$	−16.3505	−0.635962	−0.317981	−	0.948097i	$-0.603005\pi$
$661$	−16.3505	−0.635962	−0.317981	+	0.948097i	$0.603005\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.92820	−0.268261
$668$	0	0
$669$	1.88316	0.0728070
$670$	0	0
$671$	−0.744563	−0.0287435
$672$	0	0
$673$	18.6101	0.717368	0.358684	−	0.933459i	$-0.383226\pi$
$673$	18.6101	0.717368	0.358684	+	0.933459i	$0.383226\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	50.0820	1.92481	0.962404	−	0.271623i	$-0.0875604\pi$
$677$	50.0820	1.92481	0.962404	+	0.271623i	$0.0875604\pi$
$678$	0	0
$679$	−20.2337	−0.776498
$680$	0	0
$681$	−13.2554	−0.507949
$682$	0	0
$683$	−17.9104	−0.685323	−0.342661	−	0.939459i	$-0.611328\pi$
$683$	−17.9104	−0.685323	−0.342661	+	0.939459i	$0.611328\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	11.5894	0.442161
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−44.8614	−1.70661	−0.853304	−	0.521413i	$-0.825405\pi$
$691$	−44.8614	−1.70661	−0.853304	+	0.521413i	$0.825405\pi$
$692$	0	0
$693$	8.21782	0.312169
$694$	0	0
$695$	0	0
$696$	0	0
$697$	13.8564	0.524849
$698$	0	0
$699$	−2.97825	−0.112648
$700$	0	0
$701$	−12.5109	−0.472529	−0.236265	−	0.971689i	$-0.575923\pi$
$701$	−12.5109	−0.472529	−0.236265	+	0.971689i	$0.575923\pi$
$702$	0	0
$703$	4.34896	0.164024
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−20.7846	−0.781686
$708$	0	0
$709$	23.8832	0.896951	0.448475	−	0.893795i	$-0.351967\pi$
$709$	23.8832	0.896951	0.448475	+	0.893795i	$0.351967\pi$
$710$	0	0
$711$	2.97825	0.111693
$712$	0	0
$713$	2.67181	0.100060
$714$	0	0
$715$	0	0
$716$	0	0
$717$	11.6819	0.436269
$718$	0	0
$719$	30.3505	1.13188	0.565942	−	0.824445i	$-0.308513\pi$
$719$	30.3505	1.13188	0.565942	+	0.824445i	$0.308513\pi$
$720$	0	0
$721$	−36.0000	−1.34071
$722$	0	0
$723$	13.2665	0.493386
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−14.0588	−0.521412	−0.260706	−	0.965418i	$-0.583955\pi$
$727$	−14.0588	−0.521412	−0.260706	+	0.965418i	$0.583955\pi$
$728$	0	0
$729$	1.23369	0.0456921
$730$	0	0
$731$	−5.48913	−0.203023
$732$	0	0
$733$	−9.50744	−0.351165	−0.175583	−	0.984465i	$-0.556181\pi$
$733$	−9.50744	−0.351165	−0.175583	+	0.984465i	$0.556181\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.30506	−0.342756
$738$	0	0
$739$	10.7446	0.395245	0.197623	−	0.980278i	$-0.436678\pi$
$739$	10.7446	0.395245	0.197623	+	0.980278i	$0.436678\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−11.3870	−0.417747	−0.208874	−	0.977943i	$-0.566980\pi$
$743$	−11.3870	−0.417747	−0.208874	+	0.977943i	$0.566980\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	15.7359	0.575748
$748$	0	0
$749$	−22.9783	−0.839607
$750$	0	0
$751$	−27.3723	−0.998829	−0.499414	−	0.866363i	$-0.666451\pi$
$751$	−27.3723	−0.998829	−0.499414	+	0.866363i	$0.666451\pi$
$752$	0	0
$753$	17.5229	0.638570
$754$	0	0
$755$	0	0
$756$	0	0
$757$	39.7995	1.44654	0.723269	−	0.690567i	$-0.242639\pi$
$757$	39.7995	1.44654	0.723269	+	0.690567i	$0.242639\pi$
$758$	0	0
$759$	−0.627719	−0.0227847
$760$	0	0
$761$	32.7446	1.18699	0.593495	−	0.804838i	$-0.297748\pi$
$761$	32.7446	1.18699	0.593495	+	0.804838i	$0.297748\pi$
$762$	0	0
$763$	−34.6410	−1.25409
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	29.2119	1.05341	0.526705	−	0.850048i	$-0.323427\pi$
$769$	29.2119	1.05341	0.526705	+	0.850048i	$0.323427\pi$
$770$	0	0
$771$	8.46738	0.304945
$772$	0	0
$773$	−17.6155	−0.633584	−0.316792	−	0.948495i	$-0.602606\pi$
$773$	−17.6155	−0.633584	−0.316792	+	0.948495i	$0.602606\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	2.98400	0.107050
$778$	0	0
$779$	−34.9783	−1.25323
$780$	0	0
$781$	10.1168	0.362009
$782$	0	0
$783$	−37.2203	−1.33014
$784$	0	0
$785$	0	0
$786$	0	0
$787$	15.1460	0.539898	0.269949	−	0.962875i	$-0.412993\pi$
$787$	15.1460	0.539898	0.269949	+	0.962875i	$0.412993\pi$
$788$	0	0
$789$	−21.7228	−0.773353
$790$	0	0
$791$	1.72281	0.0612562
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.25106	0.186002	0.0930010	−	0.995666i	$-0.470354\pi$
$797$	5.25106	0.186002	0.0930010	+	0.995666i	$0.470354\pi$
$798$	0	0
$799$	10.5109	0.371848
$800$	0	0
$801$	−3.25544	−0.115025
$802$	0	0
$803$	−6.92820	−0.244491
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.10268	−0.320430
$808$	0	0
$809$	−32.7446	−1.15124	−0.575619	−	0.817718i	$-0.695239\pi$
$809$	−32.7446	−1.15124	−0.575619	+	0.817718i	$0.695239\pi$
$810$	0	0
$811$	0.233688	0.00820589	0.00410295	−	0.999992i	$-0.498694\pi$
$811$	0.233688	0.00820589	0.00410295	+	0.999992i	$0.498694\pi$
$812$	0	0
$813$	−10.6873	−0.374819
$814$	0	0
$815$	0	0
$816$	0	0
$817$	13.8564	0.484774
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−18.0000	−0.628204	−0.314102	−	0.949389i	$-0.601703\pi$
$821$	−18.0000	−0.628204	−0.314102	+	0.949389i	$0.601703\pi$
$822$	0	0
$823$	−56.0328	−1.95318	−0.976590	−	0.215111i	$-0.930989\pi$
$823$	−56.0328	−1.95318	−0.976590	+	0.215111i	$0.930989\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−28.4125	−0.988000	−0.494000	−	0.869462i	$-0.664466\pi$
$827$	−28.4125	−0.988000	−0.494000	+	0.869462i	$0.664466\pi$
$828$	0	0
$829$	−20.3505	−0.706803	−0.353402	−	0.935472i	$-0.614975\pi$
$829$	−20.3505	−0.706803	−0.353402	+	0.935472i	$0.614975\pi$
$830$	0	0
$831$	−9.25544	−0.321068
$832$	0	0
$833$	7.92287	0.274511
$834$	0	0
$835$	0	0
$836$	0	0
$837$	14.3537	0.496138
$838$	0	0
$839$	10.1168	0.349272	0.174636	−	0.984633i	$-0.444125\pi$
$839$	10.1168	0.349272	0.174636	+	0.984633i	$0.444125\pi$
$840$	0	0
$841$	47.4674	1.63681
$842$	0	0
$843$	−0.404759	−0.0139407
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−3.46410	−0.119028
$848$	0	0
$849$	−12.0000	−0.411839
$850$	0	0
$851$	0.861407	0.0295286
$852$	0	0
$853$	35.0458	1.19994	0.599972	−	0.800021i	$-0.295178\pi$
$853$	35.0458	1.19994	0.599972	+	0.800021i	$0.295178\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−23.9538	−0.818245	−0.409122	−	0.912480i	$-0.634165\pi$
$857$	−23.9538	−0.818245	−0.409122	+	0.912480i	$0.634165\pi$
$858$	0	0
$859$	6.11684	0.208704	0.104352	−	0.994540i	$-0.466723\pi$
$859$	6.11684	0.208704	0.104352	+	0.994540i	$0.466723\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	0	0
$863$	2.87419	0.0978387	0.0489194	−	0.998803i	$-0.484422\pi$
$863$	2.87419	0.0978387	0.0489194	+	0.998803i	$0.484422\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−11.4795	−0.389866
$868$	0	0
$869$	−1.25544	−0.0425878
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−13.8564	−0.468968
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	0	0
$879$	2.51087	0.0846897
$880$	0	0
$881$	−6.86141	−0.231167	−0.115583	−	0.993298i	$-0.536874\pi$
$881$	−6.86141	−0.231167	−0.115583	+	0.993298i	$0.536874\pi$
$882$	0	0
$883$	−24.2487	−0.816034	−0.408017	−	0.912974i	$-0.633780\pi$
$883$	−24.2487	−0.816034	−0.408017	+	0.912974i	$0.633780\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	27.4179	0.920602	0.460301	−	0.887763i	$-0.347742\pi$
$887$	27.4179	0.920602	0.460301	+	0.887763i	$0.347742\pi$
$888$	0	0
$889$	−28.4674	−0.954765
$890$	0	0
$891$	3.74456	0.125448
$892$	0	0
$893$	−26.5330	−0.887893
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−29.4891	−0.983517
$900$	0	0
$901$	16.0000	0.533037
$902$	0	0
$903$	9.50744	0.316388
$904$	0	0
$905$	0	0
$906$	0	0
$907$	19.8997	0.660760	0.330380	−	0.943848i	$-0.392823\pi$
$907$	19.8997	0.660760	0.330380	+	0.943848i	$0.392823\pi$
$908$	0	0
$909$	−14.2337	−0.472102
$910$	0	0
$911$	53.4891	1.77217	0.886087	−	0.463519i	$-0.153413\pi$
$911$	53.4891	1.77217	0.886087	+	0.463519i	$0.153413\pi$
$912$	0	0
$913$	−6.63325	−0.219529
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−9.50744	−0.313963
$918$	0	0
$919$	28.2337	0.931343	0.465672	−	0.884958i	$-0.345813\pi$
$919$	28.2337	0.931343	0.465672	+	0.884958i	$0.345813\pi$
$920$	0	0
$921$	−25.0217	−0.824495
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−24.6535	−0.809726
$928$	0	0
$929$	−7.02175	−0.230376	−0.115188	−	0.993344i	$-0.536747\pi$
$929$	−7.02175	−0.230376	−0.115188	+	0.993344i	$0.536747\pi$
$930$	0	0
$931$	−20.0000	−0.655474
$932$	0	0
$933$	−4.34896	−0.142379
$934$	0	0
$935$	0	0
$936$	0	0
$937$	53.6559	1.75286	0.876431	−	0.481527i	$-0.159918\pi$
$937$	53.6559	1.75286	0.876431	+	0.481527i	$0.159918\pi$
$938$	0	0
$939$	−17.3288	−0.565503
$940$	0	0
$941$	−58.4674	−1.90598	−0.952991	−	0.302999i	$-0.902012\pi$
$941$	−58.4674	−1.90598	−0.952991	+	0.302999i	$0.902012\pi$
$942$	0	0
$943$	−6.92820	−0.225613
$944$	0	0
$945$	0	0
$946$	0	0
$947$	26.7354	0.868783	0.434392	−	0.900724i	$-0.356963\pi$
$947$	26.7354	0.868783	0.434392	+	0.900724i	$0.356963\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	26.1168	0.846897
$952$	0	0
$953$	31.2867	1.01348	0.506738	−	0.862100i	$-0.330851\pi$
$953$	31.2867	1.01348	0.506738	+	0.862100i	$0.330851\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.92820	0.223957
$958$	0	0
$959$	−49.7228	−1.60563
$960$	0	0
$961$	−19.6277	−0.633152
$962$	0	0
$963$	−15.7359	−0.507083
$964$	0	0
$965$	0	0
$966$	0	0
$967$	26.4232	0.849713	0.424856	−	0.905261i	$-0.360325\pi$
$967$	26.4232	0.849713	0.424856	+	0.905261i	$0.360325\pi$
$968$	0	0
$969$	−5.02175	−0.161322
$970$	0	0
$971$	9.09509	0.291875	0.145938	−	0.989294i	$-0.453380\pi$
$971$	9.09509	0.291875	0.145938	+	0.989294i	$0.453380\pi$
$972$	0	0
$973$	−56.2351	−1.80282
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.5793	−1.61818	−0.809088	−	0.587687i	$-0.800038\pi$
$977$	−50.5793	−1.61818	−0.809088	+	0.587687i	$0.800038\pi$
$978$	0	0
$979$	1.37228	0.0438583
$980$	0	0
$981$	−23.7228	−0.757411
$982$	0	0
$983$	24.7460	0.789276	0.394638	−	0.918837i	$-0.370870\pi$
$983$	24.7460	0.789276	0.394638	+	0.918837i	$0.370870\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−18.2054	−0.579483
$988$	0	0
$989$	2.74456	0.0872720
$990$	0	0
$991$	−18.9783	−0.602864	−0.301432	−	0.953488i	$-0.597465\pi$
$991$	−18.9783	−0.602864	−0.301432	+	0.953488i	$0.597465\pi$
$992$	0	0
$993$	11.1846	0.354932
$994$	0	0
$995$	0	0
$996$	0	0
$997$	2.17448	0.0688665	0.0344333	−	0.999407i	$-0.489037\pi$
$997$	2.17448	0.0688665	0.0344333	+	0.999407i	$0.489037\pi$
$998$	0	0
$999$	4.62772	0.146415

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.cc.1.3		4
4.3	odd	2		275.2.a.h.1.1		4
5.2	odd	4		880.2.b.h.529.2		4
5.3	odd	4		880.2.b.h.529.3		4
5.4	even	2	inner	4400.2.a.cc.1.2		4
12.11	even	2		2475.2.a.bi.1.4		4
20.3	even	4		55.2.b.a.34.4	yes	4
20.7	even	4		55.2.b.a.34.1	✓	4
20.19	odd	2		275.2.a.h.1.4		4
44.43	even	2		3025.2.a.ba.1.4		4
60.23	odd	4		495.2.c.a.199.1		4
60.47	odd	4		495.2.c.a.199.4		4
60.59	even	2		2475.2.a.bi.1.1		4
220.3	even	20		605.2.j.i.9.4		16
220.7	odd	20		605.2.j.j.269.1		16
220.27	even	20		605.2.j.i.124.4		16
220.43	odd	4		605.2.b.c.364.1		4
220.47	even	20		605.2.j.i.9.1		16
220.63	odd	20		605.2.j.j.9.1		16
220.83	odd	20		605.2.j.j.124.4		16
220.87	odd	4		605.2.b.c.364.4		4
220.103	even	20		605.2.j.i.269.1		16
220.107	odd	20		605.2.j.j.9.4		16
220.123	odd	20		605.2.j.j.444.1		16
220.127	odd	20		605.2.j.j.124.1		16
220.147	even	20		605.2.j.i.269.4		16
220.163	even	20		605.2.j.i.444.4		16
220.167	odd	20		605.2.j.j.444.4		16
220.183	odd	20		605.2.j.j.269.4		16
220.203	even	20		605.2.j.i.124.1		16
220.207	even	20		605.2.j.i.444.1		16
220.219	even	2		3025.2.a.ba.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.2.b.a.34.1	✓	4	20.7	even	4
55.2.b.a.34.4	yes	4	20.3	even	4
275.2.a.h.1.1		4	4.3	odd	2
275.2.a.h.1.4		4	20.19	odd	2
495.2.c.a.199.1		4	60.23	odd	4
495.2.c.a.199.4		4	60.47	odd	4
605.2.b.c.364.1		4	220.43	odd	4
605.2.b.c.364.4		4	220.87	odd	4
605.2.j.i.9.1		16	220.47	even	20
605.2.j.i.9.4		16	220.3	even	20
605.2.j.i.124.1		16	220.203	even	20
605.2.j.i.124.4		16	220.27	even	20
605.2.j.i.269.1		16	220.103	even	20
605.2.j.i.269.4		16	220.147	even	20
605.2.j.i.444.1		16	220.207	even	20
605.2.j.i.444.4		16	220.163	even	20
605.2.j.j.9.1		16	220.63	odd	20
605.2.j.j.9.4		16	220.107	odd	20
605.2.j.j.124.1		16	220.127	odd	20
605.2.j.j.124.4		16	220.83	odd	20
605.2.j.j.269.1		16	220.7	odd	20
605.2.j.j.269.4		16	220.183	odd	20
605.2.j.j.444.1		16	220.123	odd	20
605.2.j.j.444.4		16	220.167	odd	20
880.2.b.h.529.2		4	5.2	odd	4
880.2.b.h.529.3		4	5.3	odd	4
2475.2.a.bi.1.1		4	60.59	even	2
2475.2.a.bi.1.4		4	12.11	even	2
3025.2.a.ba.1.1		4	220.219	even	2
3025.2.a.ba.1.4		4	44.43	even	2
4400.2.a.cc.1.2		4	5.4	even	2	inner
4400.2.a.cc.1.3		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q+0.792287 q^{3} -3.46410 q^{7} -2.37228 q^{9} +O(q^{10})\)	\(q+0.792287 q^{3} -3.46410 q^{7} -2.37228 q^{9} +1.00000 q^{11} +1.58457 q^{17} -4.00000 q^{19} -2.74456 q^{21} -0.792287 q^{23} -4.25639 q^{27} +8.74456 q^{29} -3.37228 q^{31} +0.792287 q^{33} -1.08724 q^{37} +8.74456 q^{41} -3.46410 q^{43} +6.63325 q^{47} +5.00000 q^{49} +1.25544 q^{51} +10.0974 q^{53} -3.16915 q^{57} +7.37228 q^{59} -0.744563 q^{61} +8.21782 q^{63} -9.30506 q^{67} -0.627719 q^{69} +10.1168 q^{71} -6.92820 q^{73} -3.46410 q^{77} -1.25544 q^{79} +3.74456 q^{81} -6.63325 q^{83} +6.92820 q^{87} +1.37228 q^{89} -2.67181 q^{93} +5.84096 q^{97} -2.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{9}+O(q^{10}) \) 4 * q + 2 * q^9	\( 4 q + 2 q^{9} + 4 q^{11} - 16 q^{19} + 12 q^{21} + 12 q^{29} - 2 q^{31} + 12 q^{41} + 20 q^{49} + 28 q^{51} + 18 q^{59} + 20 q^{61} - 14 q^{69} + 6 q^{71} - 28 q^{79} - 8 q^{81} - 6 q^{89} + 2 q^{99}+O(q^{100}) \) 4 * q + 2 * q^9 + 4 * q^11 - 16 * q^19 + 12 * q^21 + 12 * q^29 - 2 * q^31 + 12 * q^41 + 20 * q^49 + 28 * q^51 + 18 * q^59 + 20 * q^61 - 14 * q^69 + 6 * q^71 - 28 * q^79 - 8 * q^81 - 6 * q^89 + 2 * q^99

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