LMFDB - Embedded newform 2475.2.a.bi.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2475,2,Mod(1,2475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2475, 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("2475.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2475.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:	$19.7629745003$
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:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 55)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.52434$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.52434 q^{2} +4.37228 q^{4} +3.46410 q^{7} +5.98844 q^{8} +1.00000 q^{11} +8.74456 q^{14} +6.37228 q^{16} -1.58457 q^{17} +4.00000 q^{19} +2.52434 q^{22} -0.792287 q^{23} +15.1460 q^{28} -8.74456 q^{29} +3.37228 q^{31} +4.10891 q^{32} -4.00000 q^{34} -1.08724 q^{37} +10.0974 q^{38} -8.74456 q^{41} +3.46410 q^{43} +4.37228 q^{44} -2.00000 q^{46} +6.63325 q^{47} +5.00000 q^{49} -10.0974 q^{53} +20.7446 q^{56} -22.0742 q^{58} +7.37228 q^{59} -0.744563 q^{61} +8.51278 q^{62} -2.37228 q^{64} +9.30506 q^{67} -6.92820 q^{68} +10.1168 q^{71} -6.92820 q^{73} -2.74456 q^{74} +17.4891 q^{76} +3.46410 q^{77} +1.25544 q^{79} -22.0742 q^{82} -6.63325 q^{83} +8.74456 q^{86} +5.98844 q^{88} -1.37228 q^{89} -3.46410 q^{92} +16.7446 q^{94} +5.84096 q^{97} +12.6217 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 6 q^{4} + 4 q^{11} + 12 q^{14} + 14 q^{16} + 16 q^{19} - 12 q^{29} + 2 q^{31} - 16 q^{34} - 12 q^{41} + 6 q^{44} - 8 q^{46} + 20 q^{49} + 60 q^{56} + 18 q^{59} + 20 q^{61} + 2 q^{64} + 6 q^{71} + 12 q^{74}+ \cdots + 44 q^{94}+O(q^{100}) $ 4 * q + 6 * q^4 + 4 * q^11 + 12 * q^14 + 14 * q^16 + 16 * q^19 - 12 * q^29 + 2 * q^31 - 16 * q^34 - 12 * q^41 + 6 * q^44 - 8 * q^46 + 20 * q^49 + 60 * q^56 + 18 * q^59 + 20 * q^61 + 2 * q^64 + 6 * q^71 + 12 * q^74 + 24 * q^76 + 28 * q^79 + 12 * q^86 + 6 * q^89 + 44 * q^94

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 $

$2$	2.52434	1.78498	0.892488	−	0.451071i	$-0.148958\pi$
$2$	2.52434	1.78498	0.892488	+	0.451071i	$0.148958\pi$
$3$	0	0
$3$	0	0
$4$	4.37228	2.18614
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.46410	1.30931	0.654654	−	0.755929i	$-0.272814\pi$
$7$	3.46410	1.30931	0.654654	+	0.755929i	$0.272814\pi$
$8$	5.98844	2.11723
$9$	0	0
$10$	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	$0.5\pi$
$14$	8.74456	2.33708
$15$	0	0
$16$	6.37228	1.59307
$17$	−1.58457	−0.384316	−0.192158	−	0.981364i	$-0.561549\pi$
$17$	−1.58457	−0.384316	−0.192158	+	0.981364i	$0.561549\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	2.52434	0.538191
$23$	−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
$27$	0	0
$28$	15.1460	2.86233
$29$	−8.74456	−1.62382	−0.811912	−	0.583779i	$-0.801573\pi$
$29$	−8.74456	−1.62382	−0.811912	+	0.583779i	$0.801573\pi$
$30$	0	0
$31$	3.37228	0.605680	0.302840	−	0.953041i	$-0.402065\pi$
$31$	3.37228	0.605680	0.302840	+	0.953041i	$0.402065\pi$
$32$	4.10891	0.726360
$33$	0	0
$34$	−4.00000	−0.685994
$35$	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	$0.528486\pi$
$38$	10.0974	1.63801
$39$	0	0
$40$	0	0
$41$	−8.74456	−1.36567	−0.682836	−	0.730572i	$-0.739253\pi$
$41$	−8.74456	−1.36567	−0.682836	+	0.730572i	$0.739253\pi$
$42$	0	0
$43$	3.46410	0.528271	0.264135	−	0.964486i	$-0.414913\pi$
$43$	3.46410	0.528271	0.264135	+	0.964486i	$0.414913\pi$
$44$	4.37228	0.659146
$45$	0	0
$46$	−2.00000	−0.294884
$47$	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
$51$	0	0
$52$	0	0
$53$	−10.0974	−1.38698	−0.693489	−	0.720467i	$-0.743927\pi$
$53$	−10.0974	−1.38698	−0.693489	+	0.720467i	$0.743927\pi$
$54$	0	0
$55$	0	0
$56$	20.7446	2.77211
$57$	0	0
$58$	−22.0742	−2.89849
$59$	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	$0.515178\pi$
$62$	8.51278	1.08112
$63$	0	0
$64$	−2.37228	−0.296535
$65$	0	0
$66$	0	0
$67$	9.30506	1.13679	0.568397	−	0.822754i	$-0.307564\pi$
$67$	9.30506	1.13679	0.568397	+	0.822754i	$0.307564\pi$
$68$	−6.92820	−0.840168
$69$	0	0
$70$	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	$0.632883\pi$
$74$	−2.74456	−0.319049
$75$	0	0
$76$	17.4891	2.00614
$77$	3.46410	0.394771
$78$	0	0
$79$	1.25544	0.141248	0.0706239	−	0.997503i	$-0.477501\pi$
$79$	1.25544	0.141248	0.0706239	+	0.997503i	$0.477501\pi$
$80$	0	0
$81$	0	0
$82$	−22.0742	−2.43769
$83$	−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
$86$	8.74456	0.942950
$87$	0	0
$88$	5.98844	0.638370
$89$	−1.37228	−0.145462	−0.0727308	−	0.997352i	$-0.523171\pi$
$89$	−1.37228	−0.145462	−0.0727308	+	0.997352i	$0.523171\pi$
$90$	0	0
$91$	0	0
$92$	−3.46410	−0.361158
$93$	0	0
$94$	16.7446	1.72707
$95$	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	$0.404171\pi$
$98$	12.6217	1.27498
$99$	0	0
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−10.3923	−1.02398	−0.511992	−	0.858990i	$-0.671092\pi$
$103$	−10.3923	−1.02398	−0.511992	+	0.858990i	$0.671092\pi$
$104$	0	0
$105$	0	0
$106$	−25.4891	−2.47572
$107$	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	0
$112$	22.0742	2.08582
$113$	0.497333	0.0467852	0.0233926	−	0.999726i	$-0.492553\pi$
$113$	0.497333	0.0467852	0.0233926	+	0.999726i	$0.492553\pi$
$114$	0	0
$115$	0	0
$116$	−38.2337	−3.54991
$117$	0	0
$118$	18.6101	1.71320
$119$	−5.48913	−0.503187
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.87953	−0.170164
$123$	0	0
$124$	14.7446	1.32410
$125$	0	0
$126$	0	0
$127$	−8.21782	−0.729214	−0.364607	−	0.931162i	$-0.618797\pi$
$127$	−8.21782	−0.729214	−0.364607	+	0.931162i	$0.618797\pi$
$128$	−14.2063	−1.25567
$129$	0	0
$130$	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
$134$	23.4891	2.02915
$135$	0	0
$136$	−9.48913	−0.813686
$137$	−14.3537	−1.22632	−0.613161	−	0.789958i	$-0.710102\pi$
$137$	−14.3537	−1.22632	−0.613161	+	0.789958i	$0.710102\pi$
$138$	0	0
$139$	−16.2337	−1.37692	−0.688462	−	0.725273i	$-0.741714\pi$
$139$	−16.2337	−1.37692	−0.688462	+	0.725273i	$0.741714\pi$
$140$	0	0
$141$	0	0
$142$	25.5383	2.14313
$143$	0	0
$144$	0	0
$145$	0	0
$146$	−17.4891	−1.44741
$147$	0	0
$148$	−4.75372	−0.390754
$149$	11.4891	0.941226	0.470613	−	0.882340i	$-0.344033\pi$
$149$	11.4891	0.941226	0.470613	+	0.882340i	$0.344033\pi$
$150$	0	0
$151$	−12.2337	−0.995563	−0.497782	−	0.867302i	$-0.665852\pi$
$151$	−12.2337	−0.995563	−0.497782	+	0.867302i	$0.665852\pi$
$152$	23.9538	1.94291
$153$	0	0
$154$	8.74456	0.704657
$155$	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	$0.929691\pi$
$158$	3.16915	0.252124
$159$	0	0
$160$	0	0
$161$	−2.74456	−0.216302
$162$	0	0
$163$	−3.46410	−0.271329	−0.135665	−	0.990755i	$-0.543317\pi$
$163$	−3.46410	−0.271329	−0.135665	+	0.990755i	$0.543317\pi$
$164$	−38.2337	−2.98555
$165$	0	0
$166$	−16.7446	−1.29963
$167$	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
$171$	0	0
$172$	15.1460	1.15487
$173$	8.51278	0.647214	0.323607	−	0.946192i	$-0.395104\pi$
$173$	8.51278	0.647214	0.323607	+	0.946192i	$0.395104\pi$
$174$	0	0
$175$	0	0
$176$	6.37228	0.480329
$177$	0	0
$178$	−3.46410	−0.259645
$179$	−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	0
$184$	−4.74456	−0.349774
$185$	0	0
$186$	0	0
$187$	−1.58457	−0.115876
$188$	29.0024	2.11522
$189$	0	0
$190$	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	$0.701477\pi$
$194$	14.7446	1.05860
$195$	0	0
$196$	21.8614	1.56153
$197$	−8.51278	−0.606510	−0.303255	−	0.952909i	$-0.598073\pi$
$197$	−8.51278	−0.606510	−0.303255	+	0.952909i	$0.598073\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	0	0
$202$	−15.1460	−1.06567
$203$	−30.2921	−2.12609
$204$	0	0
$205$	0	0
$206$	−26.2337	−1.82779
$207$	0	0
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	1.48913	0.102516	0.0512578	−	0.998685i	$-0.483677\pi$
$211$	1.48913	0.102516	0.0512578	+	0.998685i	$0.483677\pi$
$212$	−44.1485	−3.03213
$213$	0	0
$214$	16.7446	1.14463
$215$	0	0
$216$	0	0
$217$	11.6819	0.793021
$218$	25.2434	1.70970
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−2.37686	−0.159166	−0.0795832	−	0.996828i	$-0.525359\pi$
$223$	−2.37686	−0.159166	−0.0795832	+	0.996828i	$0.525359\pi$
$224$	14.2337	0.951028
$225$	0	0
$226$	1.25544	0.0835105
$227$	−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
$231$	0	0
$232$	−52.3663	−3.43801
$233$	3.75906	0.246264	0.123132	−	0.992390i	$-0.460706\pi$
$233$	3.75906	0.246264	0.123132	+	0.992390i	$0.460706\pi$
$234$	0	0
$235$	0	0
$236$	32.2337	2.09823
$237$	0	0
$238$	−13.8564	−0.898177
$239$	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	$0.318687\pi$
$242$	2.52434	0.162271
$243$	0	0
$244$	−3.25544	−0.208408
$245$	0	0
$246$	0	0
$247$	0	0
$248$	20.1947	1.28236
$249$	0	0
$250$	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
$254$	−20.7446	−1.30163
$255$	0	0
$256$	−31.1168	−1.94480
$257$	−10.6873	−0.666653	−0.333326	−	0.942811i	$-0.608171\pi$
$257$	−10.6873	−0.666653	−0.333326	+	0.942811i	$0.608171\pi$
$258$	0	0
$259$	−3.76631	−0.234027
$260$	0	0
$261$	0	0
$262$	6.92820	0.428026
$263$	−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
$266$	34.9783	2.14465
$267$	0	0
$268$	40.6844	2.48519
$269$	11.4891	0.700504	0.350252	−	0.936655i	$-0.386096\pi$
$269$	11.4891	0.700504	0.350252	+	0.936655i	$0.386096\pi$
$270$	0	0
$271$	13.4891	0.819406	0.409703	−	0.912219i	$-0.365632\pi$
$271$	13.4891	0.819406	0.409703	+	0.912219i	$0.365632\pi$
$272$	−10.0974	−0.612242
$273$	0	0
$274$	−36.2337	−2.18896
$275$	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	$0.614141\pi$
$278$	−40.9793	−2.45778
$279$	0	0
$280$	0	0
$281$	0.510875	0.0304762	0.0152381	−	0.999884i	$-0.495149\pi$
$281$	0.510875	0.0304762	0.0152381	+	0.999884i	$0.495149\pi$
$282$	0	0
$283$	15.1460	0.900338	0.450169	−	0.892943i	$-0.351364\pi$
$283$	15.1460	0.900338	0.450169	+	0.892943i	$0.351364\pi$
$284$	44.2337	2.62479
$285$	0	0
$286$	0	0
$287$	−30.2921	−1.78808
$288$	0	0
$289$	−14.4891	−0.852301
$290$	0	0
$291$	0	0
$292$	−30.2921	−1.77271
$293$	−3.16915	−0.185144	−0.0925718	−	0.995706i	$-0.529509\pi$
$293$	−3.16915	−0.185144	−0.0925718	+	0.995706i	$0.529509\pi$
$294$	0	0
$295$	0	0
$296$	−6.51087	−0.378437
$297$	0	0
$298$	29.0024	1.68007
$299$	0	0
$300$	0	0
$301$	12.0000	0.691669
$302$	−30.8820	−1.77706
$303$	0	0
$304$	25.4891	1.46190
$305$	0	0
$306$	0	0
$307$	31.5817	1.80246	0.901231	−	0.433340i	$-0.142665\pi$
$307$	31.5817	1.80246	0.901231	+	0.433340i	$0.142665\pi$
$308$	15.1460	0.863025
$309$	0	0
$310$	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	$0.712112\pi$
$314$	−61.7228	−3.48322
$315$	0	0
$316$	5.48913	0.308787
$317$	−32.9639	−1.85144	−0.925718	−	0.378215i	$-0.876538\pi$
$317$	−32.9639	−1.85144	−0.925718	+	0.378215i	$0.876538\pi$
$318$	0	0
$319$	−8.74456	−0.489602
$320$	0	0
$321$	0	0
$322$	−6.92820	−0.386094
$323$	−6.33830	−0.352672
$324$	0	0
$325$	0	0
$326$	−8.74456	−0.484317
$327$	0	0
$328$	−52.3663	−2.89144
$329$	22.9783	1.26683
$330$	0	0
$331$	−14.1168	−0.775932	−0.387966	−	0.921674i	$-0.626822\pi$
$331$	−14.1168	−0.775932	−0.387966	+	0.921674i	$0.626822\pi$
$332$	−29.0024	−1.59172
$333$	0	0
$334$	39.7228	2.17354
$335$	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	$0.845352\pi$
$338$	−32.8164	−1.78498
$339$	0	0
$340$	0	0
$341$	3.37228	0.182619
$342$	0	0
$343$	−6.92820	−0.374088
$344$	20.7446	1.11847
$345$	0	0
$346$	21.4891	1.15526
$347$	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
$352$	4.10891	0.219006
$353$	−25.0410	−1.33280	−0.666399	−	0.745595i	$-0.732165\pi$
$353$	−25.0410	−1.33280	−0.666399	+	0.745595i	$0.732165\pi$
$354$	0	0
$355$	0	0
$356$	−6.00000	−0.317999
$357$	0	0
$358$	−32.4665	−1.71591
$359$	−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
$362$	60.8791	3.19973
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	25.7407	1.34365	0.671827	−	0.740708i	$-0.265510\pi$
$367$	25.7407	1.34365	0.671827	+	0.740708i	$0.265510\pi$
$368$	−5.04868	−0.263180
$369$	0	0
$370$	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	$0.597799\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	39.7228	2.04855
$377$	0	0
$378$	0	0
$379$	−0.627719	−0.0322437	−0.0161219	−	0.999870i	$-0.505132\pi$
$379$	−0.627719	−0.0322437	−0.0161219	+	0.999870i	$0.505132\pi$
$380$	0	0
$381$	0	0
$382$	48.9022	2.50205
$383$	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
$386$	−41.4891	−2.11174
$387$	0	0
$388$	25.5383	1.29651
$389$	−18.8614	−0.956311	−0.478156	−	0.878275i	$-0.658695\pi$
$389$	−18.8614	−0.956311	−0.478156	+	0.878275i	$0.658695\pi$
$390$	0	0
$391$	1.25544	0.0634902
$392$	29.9422	1.51231
$393$	0	0
$394$	−21.4891	−1.08261
$395$	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	$0.635324\pi$
$398$	−20.1947	−1.01227
$399$	0	0
$400$	0	0
$401$	11.4891	0.573740	0.286870	−	0.957970i	$-0.407385\pi$
$401$	11.4891	0.573740	0.286870	+	0.957970i	$0.407385\pi$
$402$	0	0
$403$	0	0
$404$	−26.2337	−1.30517
$405$	0	0
$406$	−76.4674	−3.79501
$407$	−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
$411$	0	0
$412$	−45.4381	−2.23857
$413$	25.5383	1.25666
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	10.0974	0.493878
$419$	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	$0.433501\pi$
$422$	3.75906	0.182988
$423$	0	0
$424$	−60.4674	−2.93656
$425$	0	0
$426$	0	0
$427$	−2.57924	−0.124818
$428$	29.0024	1.40189
$429$	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	$0.252404\pi$
$434$	29.4891	1.41552
$435$	0	0
$436$	43.7228	2.09394
$437$	−3.16915	−0.151601
$438$	0	0
$439$	21.4891	1.02562	0.512810	−	0.858502i	$-0.328605\pi$
$439$	21.4891	1.02562	0.512810	+	0.858502i	$0.328605\pi$
$440$	0	0
$441$	0	0
$442$	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
$446$	−6.00000	−0.284108
$447$	0	0
$448$	−8.21782	−0.388256
$449$	−6.86141	−0.323810	−0.161905	−	0.986806i	$-0.551764\pi$
$449$	−6.86141	−0.323810	−0.161905	+	0.986806i	$0.551764\pi$
$450$	0	0
$451$	−8.74456	−0.411765
$452$	2.17448	0.102279
$453$	0	0
$454$	−42.2337	−1.98213
$455$	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	$0.661592\pi$
$458$	36.9253	1.72541
$459$	0	0
$460$	0	0
$461$	−2.23369	−0.104033	−0.0520166	−	0.998646i	$-0.516565\pi$
$461$	−2.23369	−0.104033	−0.0520166	+	0.998646i	$0.516565\pi$
$462$	0	0
$463$	−30.0897	−1.39839	−0.699193	−	0.714933i	$-0.746457\pi$
$463$	−30.0897	−1.39839	−0.699193	+	0.714933i	$0.746457\pi$
$464$	−55.7228	−2.58687
$465$	0	0
$466$	9.48913	0.439575
$467$	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
$471$	0	0
$472$	44.1485	2.03210
$473$	3.46410	0.159280
$474$	0	0
$475$	0	0
$476$	−24.0000	−1.10004
$477$	0	0
$478$	37.2203	1.70241
$479$	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
$482$	42.2689	1.92530
$483$	0	0
$484$	4.37228	0.198740
$485$	0	0
$486$	0	0
$487$	−7.13058	−0.323118	−0.161559	−	0.986863i	$-0.551652\pi$
$487$	−7.13058	−0.323118	−0.161559	+	0.986863i	$0.551652\pi$
$488$	−4.45877	−0.201839
$489$	0	0
$490$	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
$496$	21.4891	0.964890
$497$	35.0458	1.57202
$498$	0	0
$499$	−20.0000	−0.895323	−0.447661	−	0.894203i	$-0.647743\pi$
$499$	−20.0000	−0.895323	−0.447661	+	0.894203i	$0.647743\pi$
$500$	0	0
$501$	0	0
$502$	55.8304	2.49183
$503$	−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
$506$	−2.00000	−0.0889108
$507$	0	0
$508$	−35.9306	−1.59416
$509$	−22.6277	−1.00296	−0.501478	−	0.865170i	$-0.667210\pi$
$509$	−22.6277	−1.00296	−0.501478	+	0.865170i	$0.667210\pi$
$510$	0	0
$511$	−24.0000	−1.06170
$512$	−50.1369	−2.21576
$513$	0	0
$514$	−26.9783	−1.18996
$515$	0	0
$516$	0	0
$517$	6.63325	0.291730
$518$	−9.50744	−0.417733
$519$	0	0
$520$	0	0
$521$	21.6060	0.946575	0.473287	−	0.880908i	$-0.343067\pi$
$521$	21.6060	0.946575	0.473287	+	0.880908i	$0.343067\pi$
$522$	0	0
$523$	29.0024	1.26819	0.634094	−	0.773256i	$-0.281373\pi$
$523$	29.0024	1.26819	0.634094	+	0.773256i	$0.281373\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−69.2119	−3.01778
$527$	−5.34363	−0.232772
$528$	0	0
$529$	−22.3723	−0.972708
$530$	0	0
$531$	0	0
$532$	60.5841	2.62665
$533$	0	0
$534$	0	0
$535$	0	0
$536$	55.7228	2.40686
$537$	0	0
$538$	29.0024	1.25038
$539$	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	$0.236754\pi$
$542$	34.0511	1.46262
$543$	0	0
$544$	−6.51087	−0.279151
$545$	0	0
$546$	0	0
$547$	−29.0024	−1.24005	−0.620027	−	0.784580i	$-0.712878\pi$
$547$	−29.0024	−1.24005	−0.620027	+	0.784580i	$0.712878\pi$
$548$	−62.7586	−2.68091
$549$	0	0
$550$	0	0
$551$	−34.9783	−1.49012
$552$	0	0
$553$	4.34896	0.184937
$554$	−29.4891	−1.25287
$555$	0	0
$556$	−70.9783	−3.01015
$557$	0.994667	0.0421454	0.0210727	−	0.999778i	$-0.493292\pi$
$557$	0.994667	0.0421454	0.0210727	+	0.999778i	$0.493292\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	1.28962	0.0543994
$563$	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
$566$	38.2337	1.60708
$567$	0	0
$568$	60.5841	2.54205
$569$	27.2554	1.14261	0.571304	−	0.820739i	$-0.306438\pi$
$569$	27.2554	1.14261	0.571304	+	0.820739i	$0.306438\pi$
$570$	0	0
$571$	1.48913	0.0623180	0.0311590	−	0.999514i	$-0.490080\pi$
$571$	1.48913	0.0623180	0.0311590	+	0.999514i	$0.490080\pi$
$572$	0	0
$573$	0	0
$574$	−76.4674	−3.19169
$575$	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	$0.349543\pi$
$578$	−36.5754	−1.52134
$579$	0	0
$580$	0	0
$581$	−22.9783	−0.953298
$582$	0	0
$583$	−10.0974	−0.418190
$584$	−41.4891	−1.71683
$585$	0	0
$586$	−8.00000	−0.330477
$587$	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
$591$	0	0
$592$	−6.92820	−0.284747
$593$	22.7739	0.935214	0.467607	−	0.883937i	$-0.345116\pi$
$593$	22.7739	0.935214	0.467607	+	0.883937i	$0.345116\pi$
$594$	0	0
$595$	0	0
$596$	50.2337	2.05765
$597$	0	0
$598$	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	$0.787110\pi$
$602$	30.2921	1.23461
$603$	0	0
$604$	−53.4891	−2.17644
$605$	0	0
$606$	0	0
$607$	3.46410	0.140604	0.0703018	−	0.997526i	$-0.477604\pi$
$607$	3.46410	0.140604	0.0703018	+	0.997526i	$0.477604\pi$
$608$	16.4356	0.666554
$609$	0	0
$610$	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	$0.527992\pi$
$614$	79.7228	3.21735
$615$	0	0
$616$	20.7446	0.835822
$617$	17.0256	0.685423	0.342712	−	0.939441i	$-0.388655\pi$
$617$	17.0256	0.685423	0.342712	+	0.939441i	$0.388655\pi$
$618$	0	0
$619$	14.1168	0.567404	0.283702	−	0.958913i	$-0.408437\pi$
$619$	14.1168	0.567404	0.283702	+	0.958913i	$0.408437\pi$
$620$	0	0
$621$	0	0
$622$	−13.8564	−0.555591
$623$	−4.75372	−0.190454
$624$	0	0
$625$	0	0
$626$	−55.2119	−2.20671
$627$	0	0
$628$	−106.907	−4.26606
$629$	1.72281	0.0686931
$630$	0	0
$631$	23.6060	0.939739	0.469869	−	0.882736i	$-0.344301\pi$
$631$	23.6060	0.939739	0.469869	+	0.882736i	$0.344301\pi$
$632$	7.51811	0.299054
$633$	0	0
$634$	−83.2119	−3.30477
$635$	0	0
$636$	0	0
$637$	0	0
$638$	−22.0742	−0.873927
$639$	0	0
$640$	0	0
$641$	25.3723	1.00214	0.501072	−	0.865405i	$-0.332939\pi$
$641$	25.3723	1.00214	0.501072	+	0.865405i	$0.332939\pi$
$642$	0	0
$643$	−30.4944	−1.20258	−0.601292	−	0.799030i	$-0.705347\pi$
$643$	−30.4944	−1.20258	−0.601292	+	0.799030i	$0.705347\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−16.0000	−0.629512
$647$	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
$651$	0	0
$652$	−15.1460	−0.593164
$653$	30.7894	1.20488	0.602441	−	0.798163i	$-0.294195\pi$
$653$	30.7894	1.20488	0.602441	+	0.798163i	$0.294195\pi$
$654$	0	0
$655$	0	0
$656$	−55.7228	−2.17561
$657$	0	0
$658$	58.0049	2.26127
$659$	−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	$0.603005\pi$
$662$	−35.6357	−1.38502
$663$	0	0
$664$	−39.7228	−1.54154
$665$	0	0
$666$	0	0
$667$	6.92820	0.268261
$668$	68.8019	2.66203
$669$	0	0
$670$	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	$0.383226\pi$
$674$	−81.9565	−3.15685
$675$	0	0
$676$	−56.8397	−2.18614
$677$	−50.0820	−1.92481	−0.962404	−	0.271623i	$-0.912440\pi$
$677$	−50.0820	−1.92481	−0.962404	+	0.271623i	$0.912440\pi$
$678$	0	0
$679$	20.2337	0.776498
$680$	0	0
$681$	0	0
$682$	8.51278	0.325971
$683$	−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
$686$	−17.4891	−0.667738
$687$	0	0
$688$	22.0742	0.841572
$689$	0	0
$690$	0	0
$691$	44.8614	1.70661	0.853304	−	0.521413i	$-0.174595\pi$
$691$	44.8614	1.70661	0.853304	+	0.521413i	$0.174595\pi$
$692$	37.2203	1.41490
$693$	0	0
$694$	57.2119	2.17174
$695$	0	0
$696$	0	0
$697$	13.8564	0.524849
$698$	39.0998	1.47995
$699$	0	0
$700$	0	0
$701$	12.5109	0.472529	0.236265	−	0.971689i	$-0.424077\pi$
$701$	12.5109	0.472529	0.236265	+	0.971689i	$0.424077\pi$
$702$	0	0
$703$	−4.34896	−0.164024
$704$	−2.37228	−0.0894087
$705$	0	0
$706$	−63.2119	−2.37901
$707$	−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
$711$	0	0
$712$	−8.21782	−0.307976
$713$	−2.67181	−0.100060
$714$	0	0
$715$	0	0
$716$	−56.2337	−2.10155
$717$	0	0
$718$	−74.4405	−2.77810
$719$	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
$722$	−7.57301	−0.281838
$723$	0	0
$724$	105.446	3.91886
$725$	0	0
$726$	0	0
$727$	14.0588	0.521412	0.260706	−	0.965418i	$-0.416045\pi$
$727$	14.0588	0.521412	0.260706	+	0.965418i	$0.416045\pi$
$728$	0	0
$729$	0	0
$730$	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	$0.556181\pi$
$734$	64.9783	2.39839
$735$	0	0
$736$	−3.25544	−0.119997
$737$	9.30506	0.342756
$738$	0	0
$739$	−10.7446	−0.395245	−0.197623	−	0.980278i	$-0.563322\pi$
$739$	−10.7446	−0.395245	−0.197623	+	0.980278i	$0.563322\pi$
$740$	0	0
$741$	0	0
$742$	−88.2969	−3.24148
$743$	−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
$746$	−29.4891	−1.07967
$747$	0	0
$748$	−6.92820	−0.253320
$749$	22.9783	0.839607
$750$	0	0
$751$	27.3723	0.998829	0.499414	−	0.866363i	$-0.333549\pi$
$751$	27.3723	0.998829	0.499414	+	0.866363i	$0.333549\pi$
$752$	42.2689	1.54139
$753$	0	0
$754$	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	$0.242639\pi$
$758$	−1.58457	−0.0575543
$759$	0	0
$760$	0	0
$761$	−32.7446	−1.18699	−0.593495	−	0.804838i	$-0.702252\pi$
$761$	−32.7446	−1.18699	−0.593495	+	0.804838i	$0.702252\pi$
$762$	0	0
$763$	34.6410	1.25409
$764$	84.7011	3.06438
$765$	0	0
$766$	27.4891	0.993222
$767$	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
$771$	0	0
$772$	−71.8613	−2.58634
$773$	17.6155	0.633584	0.316792	−	0.948495i	$-0.397394\pi$
$773$	17.6155	0.633584	0.316792	+	0.948495i	$0.397394\pi$
$774$	0	0
$775$	0	0
$776$	34.9783	1.25565
$777$	0	0
$778$	−47.6126	−1.70699
$779$	−34.9783	−1.25323
$780$	0	0
$781$	10.1168	0.362009
$782$	3.16915	0.113328
$783$	0	0
$784$	31.8614	1.13791
$785$	0	0
$786$	0	0
$787$	−15.1460	−0.539898	−0.269949	−	0.962875i	$-0.587007\pi$
$787$	−15.1460	−0.539898	−0.269949	+	0.962875i	$0.587007\pi$
$788$	−37.2203	−1.32592
$789$	0	0
$790$	0	0
$791$	1.72281	0.0612562
$792$	0	0
$793$	0	0
$794$	−41.4891	−1.47239
$795$	0	0
$796$	−34.9783	−1.23977
$797$	−5.25106	−0.186002	−0.0930010	−	0.995666i	$-0.529646\pi$
$797$	−5.25106	−0.186002	−0.0930010	+	0.995666i	$0.529646\pi$
$798$	0	0
$799$	−10.5109	−0.371848
$800$	0	0
$801$	0	0
$802$	29.0024	1.02411
$803$	−6.92820	−0.244491
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−35.9306	−1.26404
$809$	32.7446	1.15124	0.575619	−	0.817718i	$-0.304761\pi$
$809$	32.7446	1.15124	0.575619	+	0.817718i	$0.304761\pi$
$810$	0	0
$811$	−0.233688	−0.00820589	−0.00410295	−	0.999992i	$-0.501306\pi$
$811$	−0.233688	−0.00820589	−0.00410295	+	0.999992i	$0.501306\pi$
$812$	−132.445	−4.64792
$813$	0	0
$814$	−2.74456	−0.0961969
$815$	0	0
$816$	0	0
$817$	13.8564	0.484774
$818$	69.3918	2.42623
$819$	0	0
$820$	0	0
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	56.0328	1.95318	0.976590	−	0.215111i	$-0.0690112\pi$
$823$	56.0328	1.95318	0.976590	+	0.215111i	$0.0690112\pi$
$824$	−62.2337	−2.16801
$825$	0	0
$826$	64.4674	2.24311
$827$	−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
$831$	0	0
$832$	0	0
$833$	−7.92287	−0.274511
$834$	0	0
$835$	0	0
$836$	17.4891	0.604874
$837$	0	0
$838$	58.0049	2.00374
$839$	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
$842$	21.4843	0.740399
$843$	0	0
$844$	6.51087	0.224114
$845$	0	0
$846$	0	0
$847$	3.46410	0.119028
$848$	−64.3432	−2.20955
$849$	0	0
$850$	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	$0.295178\pi$
$854$	−6.51087	−0.222798
$855$	0	0
$856$	39.7228	1.35770
$857$	23.9538	0.818245	0.409122	−	0.912480i	$-0.365835\pi$
$857$	23.9538	0.818245	0.409122	+	0.912480i	$0.365835\pi$
$858$	0	0
$859$	−6.11684	−0.208704	−0.104352	−	0.994540i	$-0.533277\pi$
$859$	−6.11684	−0.208704	−0.104352	+	0.994540i	$0.533277\pi$
$860$	0	0
$861$	0	0
$862$	64.9331	2.21163
$863$	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
$866$	73.7228	2.50520
$867$	0	0
$868$	51.0767	1.73365
$869$	1.25544	0.0425878
$870$	0	0
$871$	0	0
$872$	59.8844	2.02794
$873$	0	0
$874$	−8.00000	−0.270604
$875$	0	0
$876$	0	0
$877$	0	0		−	1.00000i	$-0.5\pi$
$877$	0	0			1.00000i	$0.5\pi$
$878$	54.2458	1.83071
$879$	0	0
$880$	0	0
$881$	6.86141	0.231167	0.115583	−	0.993298i	$-0.463126\pi$
$881$	6.86141	0.231167	0.115583	+	0.993298i	$0.463126\pi$
$882$	0	0
$883$	24.2487	0.816034	0.408017	−	0.912974i	$-0.366220\pi$
$883$	24.2487	0.816034	0.408017	+	0.912974i	$0.366220\pi$
$884$	0	0
$885$	0	0
$886$	79.9565	2.68619
$887$	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
$891$	0	0
$892$	−10.3923	−0.347960
$893$	26.5330	0.887893
$894$	0	0
$895$	0	0
$896$	−49.2119	−1.64406
$897$	0	0
$898$	−17.3205	−0.577993
$899$	−29.4891	−0.983517
$900$	0	0
$901$	16.0000	0.533037
$902$	−22.0742	−0.734991
$903$	0	0
$904$	2.97825	0.0990551
$905$	0	0
$906$	0	0
$907$	−19.8997	−0.660760	−0.330380	−	0.943848i	$-0.607177\pi$
$907$	−19.8997	−0.660760	−0.330380	+	0.943848i	$0.607177\pi$
$908$	−73.1509	−2.42760
$909$	0	0
$910$	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
$914$	−52.4674	−1.73547
$915$	0	0
$916$	63.9565	2.11318
$917$	9.50744	0.313963
$918$	0	0
$919$	−28.2337	−0.931343	−0.465672	−	0.884958i	$-0.654187\pi$
$919$	−28.2337	−0.931343	−0.465672	+	0.884958i	$0.654187\pi$
$920$	0	0
$921$	0	0
$922$	−5.63858	−0.185697
$923$	0	0
$924$	0	0
$925$	0	0
$926$	−75.9565	−2.49609
$927$	0	0
$928$	−35.9306	−1.17948
$929$	7.02175	0.230376	0.115188	−	0.993344i	$-0.463253\pi$
$929$	7.02175	0.230376	0.115188	+	0.993344i	$0.463253\pi$
$930$	0	0
$931$	20.0000	0.655474
$932$	16.4356	0.538368
$933$	0	0
$934$	19.4891	0.637704
$935$	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	$0.159918\pi$
$938$	81.3687	2.65678
$939$	0	0
$940$	0	0
$941$	58.4674	1.90598	0.952991	−	0.302999i	$-0.0979878\pi$
$941$	58.4674	1.90598	0.952991	+	0.302999i	$0.0979878\pi$
$942$	0	0
$943$	6.92820	0.225613
$944$	46.9783	1.52901
$945$	0	0
$946$	8.74456	0.284310
$947$	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
$951$	0	0
$952$	−32.8713	−1.06536
$953$	−31.2867	−1.01348	−0.506738	−	0.862100i	$-0.669149\pi$
$953$	−31.2867	−1.01348	−0.506738	+	0.862100i	$0.669149\pi$
$954$	0	0
$955$	0	0
$956$	64.4674	2.08502
$957$	0	0
$958$	13.8564	0.447680
$959$	−49.7228	−1.60563
$960$	0	0
$961$	−19.6277	−0.633152
$962$	0	0
$963$	0	0
$964$	73.2119	2.35800
$965$	0	0
$966$	0	0
$967$	−26.4232	−0.849713	−0.424856	−	0.905261i	$-0.639675\pi$
$967$	−26.4232	−0.849713	−0.424856	+	0.905261i	$0.639675\pi$
$968$	5.98844	0.192476
$969$	0	0
$970$	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
$974$	−18.0000	−0.576757
$975$	0	0
$976$	−4.74456	−0.151870
$977$	50.5793	1.61818	0.809088	−	0.587687i	$-0.199962\pi$
$977$	50.5793	1.61818	0.809088	+	0.587687i	$0.199962\pi$
$978$	0	0
$979$	−1.37228	−0.0438583
$980$	0	0
$981$	0	0
$982$	−16.4356	−0.524483
$983$	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
$986$	34.9783	1.11393
$987$	0	0
$988$	0	0
$989$	−2.74456	−0.0872720
$990$	0	0
$991$	18.9783	0.602864	0.301432	−	0.953488i	$-0.402535\pi$
$991$	18.9783	0.602864	0.301432	+	0.953488i	$0.402535\pi$
$992$	13.8564	0.439941
$993$	0	0
$994$	88.4674	2.80601
$995$	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	$0.489037\pi$
$998$	−50.4868	−1.59813
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.2.a.bi.1.4		4
3.2	odd	2		275.2.a.h.1.1		4
5.2	odd	4		495.2.c.a.199.4		4
5.3	odd	4		495.2.c.a.199.1		4
5.4	even	2	inner	2475.2.a.bi.1.1		4
12.11	even	2		4400.2.a.cc.1.3		4
15.2	even	4		55.2.b.a.34.1	✓	4
15.8	even	4		55.2.b.a.34.4	yes	4
15.14	odd	2		275.2.a.h.1.4		4
33.32	even	2		3025.2.a.ba.1.4		4
60.23	odd	4		880.2.b.h.529.3		4
60.47	odd	4		880.2.b.h.529.2		4
60.59	even	2		4400.2.a.cc.1.2		4
165.2	odd	20		605.2.j.j.444.4		16
165.8	odd	20		605.2.j.j.9.1		16
165.17	odd	20		605.2.j.j.124.1		16
165.32	odd	4		605.2.b.c.364.4		4
165.38	even	20		605.2.j.i.124.1		16
165.47	even	20		605.2.j.i.9.1		16
165.53	even	20		605.2.j.i.444.4		16
165.62	odd	20		605.2.j.j.269.1		16
165.68	odd	20		605.2.j.j.444.1		16
165.83	odd	20		605.2.j.j.124.4		16
165.92	even	20		605.2.j.i.269.4		16
165.98	odd	4		605.2.b.c.364.1		4
165.107	odd	20		605.2.j.j.9.4		16
165.113	even	20		605.2.j.i.9.4		16
165.128	odd	20		605.2.j.j.269.4		16
165.137	even	20		605.2.j.i.124.4		16
165.152	even	20		605.2.j.i.444.1		16
165.158	even	20		605.2.j.i.269.1		16
165.164	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	15.2	even	4
55.2.b.a.34.4	yes	4	15.8	even	4
275.2.a.h.1.1		4	3.2	odd	2
275.2.a.h.1.4		4	15.14	odd	2
495.2.c.a.199.1		4	5.3	odd	4
495.2.c.a.199.4		4	5.2	odd	4
605.2.b.c.364.1		4	165.98	odd	4
605.2.b.c.364.4		4	165.32	odd	4
605.2.j.i.9.1		16	165.47	even	20
605.2.j.i.9.4		16	165.113	even	20
605.2.j.i.124.1		16	165.38	even	20
605.2.j.i.124.4		16	165.137	even	20
605.2.j.i.269.1		16	165.158	even	20
605.2.j.i.269.4		16	165.92	even	20
605.2.j.i.444.1		16	165.152	even	20
605.2.j.i.444.4		16	165.53	even	20
605.2.j.j.9.1		16	165.8	odd	20
605.2.j.j.9.4		16	165.107	odd	20
605.2.j.j.124.1		16	165.17	odd	20
605.2.j.j.124.4		16	165.83	odd	20
605.2.j.j.269.1		16	165.62	odd	20
605.2.j.j.269.4		16	165.128	odd	20
605.2.j.j.444.1		16	165.68	odd	20
605.2.j.j.444.4		16	165.2	odd	20
880.2.b.h.529.2		4	60.47	odd	4
880.2.b.h.529.3		4	60.23	odd	4
2475.2.a.bi.1.1		4	5.4	even	2	inner
2475.2.a.bi.1.4		4	1.1	even	1	trivial
3025.2.a.ba.1.1		4	165.164	even	2
3025.2.a.ba.1.4		4	33.32	even	2
4400.2.a.cc.1.2		4	60.59	even	2
4400.2.a.cc.1.3		4	12.11	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+2.52434 q^{2} +4.37228 q^{4} +3.46410 q^{7} +5.98844 q^{8} +1.00000 q^{11} +8.74456 q^{14} +6.37228 q^{16} -1.58457 q^{17} +4.00000 q^{19} +2.52434 q^{22} -0.792287 q^{23} +15.1460 q^{28} -8.74456 q^{29} +3.37228 q^{31} +4.10891 q^{32} -4.00000 q^{34} -1.08724 q^{37} +10.0974 q^{38} -8.74456 q^{41} +3.46410 q^{43} +4.37228 q^{44} -2.00000 q^{46} +6.63325 q^{47} +5.00000 q^{49} -10.0974 q^{53} +20.7446 q^{56} -22.0742 q^{58} +7.37228 q^{59} -0.744563 q^{61} +8.51278 q^{62} -2.37228 q^{64} +9.30506 q^{67} -6.92820 q^{68} +10.1168 q^{71} -6.92820 q^{73} -2.74456 q^{74} +17.4891 q^{76} +3.46410 q^{77} +1.25544 q^{79} -22.0742 q^{82} -6.63325 q^{83} +8.74456 q^{86} +5.98844 q^{88} -1.37228 q^{89} -3.46410 q^{92} +16.7446 q^{94} +5.84096 q^{97} +12.6217 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 6 q^{4} + 4 q^{11} + 12 q^{14} + 14 q^{16} + 16 q^{19} - 12 q^{29} + 2 q^{31} - 16 q^{34} - 12 q^{41} + 6 q^{44} - 8 q^{46} + 20 q^{49} + 60 q^{56} + 18 q^{59} + 20 q^{61} + 2 q^{64} + 6 q^{71} + 12 q^{74}+ \cdots + 44 q^{94}+O(q^{100}) \) 4 * q + 6 * q^4 + 4 * q^11 + 12 * q^14 + 14 * q^16 + 16 * q^19 - 12 * q^29 + 2 * q^31 - 16 * q^34 - 12 * q^41 + 6 * q^44 - 8 * q^46 + 20 * q^49 + 60 * q^56 + 18 * q^59 + 20 * q^61 + 2 * q^64 + 6 * q^71 + 12 * q^74 + 24 * q^76 + 28 * q^79 + 12 * q^86 + 6 * q^89 + 44 * q^94

Introduction

L-functions

Modular forms

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