LMFDB - Embedded newform 5796.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5796,2,Mod(1,5796)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5796, 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("5796.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5796 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5796.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:	$46.2812930115$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.140608.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 26x^{2} + 52 $ x^4 - 26*x^2 + 52
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			$4.88023$ of defining polynomial
Character	$\chi$	$=$	5796.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.30278 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-1.30278 q^{5} +1.00000 q^{7} +5.88023 q^{11} +3.78039 q^{13} +5.60555 q^{17} +4.40262 q^{19} +1.00000 q^{23} -3.30278 q^{25} +0.202935 q^{29} -0.725320 q^{31} -1.30278 q^{35} +1.52238 q^{37} +6.47762 q^{41} +1.44952 q^{43} +1.00000 q^{49} -1.66062 q^{53} -7.66062 q^{55} -2.53269 q^{59} -5.18301 q^{61} -4.92500 q^{65} -5.57746 q^{67} +10.5775 q^{71} -11.6969 q^{73} +5.88023 q^{77} +8.23808 q^{79} +1.34968 q^{83} -7.30278 q^{85} +9.10801 q^{89} +3.78039 q^{91} -5.73562 q^{95} -16.5128 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q + 2 * q^5 + 4 * q^7	$ 4 q + 2 q^{5} + 4 q^{7} + 4 q^{11} + 2 q^{13} + 8 q^{17} + 4 q^{19} + 4 q^{23} - 6 q^{25} - 8 q^{31} + 2 q^{35} + 12 q^{37} + 20 q^{41} + 2 q^{43} + 4 q^{49} + 26 q^{53} + 2 q^{55} + 14 q^{59} + 6 q^{61} - 12 q^{65} - 10 q^{67} + 30 q^{71} + 16 q^{73} + 4 q^{77} - 12 q^{79} + 8 q^{83} - 22 q^{85} + 2 q^{89} + 2 q^{91} + 2 q^{95} - 16 q^{97}+O(q^{100}) $ 4 * q + 2 * q^5 + 4 * q^7 + 4 * q^11 + 2 * q^13 + 8 * q^17 + 4 * q^19 + 4 * q^23 - 6 * q^25 - 8 * q^31 + 2 * q^35 + 12 * q^37 + 20 * q^41 + 2 * q^43 + 4 * q^49 + 26 * q^53 + 2 * q^55 + 14 * q^59 + 6 * q^61 - 12 * q^65 - 10 * q^67 + 30 * q^71 + 16 * q^73 + 4 * q^77 - 12 * q^79 + 8 * q^83 - 22 * q^85 + 2 * q^89 + 2 * q^91 + 2 * q^95 - 16 * 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
$5$	−1.30278	−0.582619	−0.291309	−	0.956629i	$-0.594091\pi$
$5$	−1.30278	−0.582619	−0.291309	+	0.956629i	$0.594091\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.88023	1.77296	0.886478	−	0.462770i	$-0.153145\pi$
$11$	5.88023	1.77296	0.886478	+	0.462770i	$0.153145\pi$
$12$	0	0
$13$	3.78039	1.04849	0.524246	−	0.851567i	$-0.324347\pi$
$13$	3.78039	1.04849	0.524246	+	0.851567i	$0.324347\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.60555	1.35955	0.679773	−	0.733423i	$-0.262078\pi$
$17$	5.60555	1.35955	0.679773	+	0.733423i	$0.262078\pi$
$18$	0	0
$19$	4.40262	1.01003	0.505015	−	0.863111i	$-0.331487\pi$
$19$	4.40262	1.01003	0.505015	+	0.863111i	$0.331487\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.30278	−0.660555
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.202935	0.0376842	0.0188421	−	0.999822i	$-0.494002\pi$
$29$	0.202935	0.0376842	0.0188421	+	0.999822i	$0.494002\pi$
$30$	0	0
$31$	−0.725320	−0.130271	−0.0651357	−	0.997876i	$-0.520748\pi$
$31$	−0.725320	−0.130271	−0.0651357	+	0.997876i	$0.520748\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.30278	−0.220209
$36$	0	0
$37$	1.52238	0.250279	0.125139	−	0.992139i	$-0.460062\pi$
$37$	1.52238	0.250279	0.125139	+	0.992139i	$0.460062\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	6.47762	1.01163	0.505817	−	0.862641i	$-0.331191\pi$
$41$	6.47762	1.01163	0.505817	+	0.862641i	$0.331191\pi$
$42$	0	0
$43$	1.44952	0.221050	0.110525	−	0.993873i	$-0.464747\pi$
$43$	1.44952	0.221050	0.110525	+	0.993873i	$0.464747\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.66062	−0.228104	−0.114052	−	0.993475i	$-0.536383\pi$
$53$	−1.66062	−0.228104	−0.114052	+	0.993475i	$0.536383\pi$
$54$	0	0
$55$	−7.66062	−1.03296
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.53269	−0.329728	−0.164864	−	0.986316i	$-0.552718\pi$
$59$	−2.53269	−0.329728	−0.164864	+	0.986316i	$0.552718\pi$
$60$	0	0
$61$	−5.18301	−0.663616	−0.331808	−	0.943347i	$-0.607659\pi$
$61$	−5.18301	−0.663616	−0.331808	+	0.943347i	$0.607659\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.92500	−0.610871
$66$	0	0
$67$	−5.57746	−0.681395	−0.340697	−	0.940173i	$-0.610663\pi$
$67$	−5.57746	−0.681395	−0.340697	+	0.940173i	$0.610663\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.5775	1.25531	0.627656	−	0.778490i	$-0.284014\pi$
$71$	10.5775	1.25531	0.627656	+	0.778490i	$0.284014\pi$
$72$	0	0
$73$	−11.6969	−1.36902	−0.684508	−	0.729005i	$-0.739983\pi$
$73$	−11.6969	−1.36902	−0.684508	+	0.729005i	$0.739983\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.88023	0.670115
$78$	0	0
$79$	8.23808	0.926856	0.463428	−	0.886135i	$-0.346619\pi$
$79$	8.23808	0.926856	0.463428	+	0.886135i	$0.346619\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.34968	0.148146	0.0740732	−	0.997253i	$-0.476400\pi$
$83$	1.34968	0.148146	0.0740732	+	0.997253i	$0.476400\pi$
$84$	0	0
$85$	−7.30278	−0.792097
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.10801	0.965447	0.482723	−	0.875773i	$-0.339648\pi$
$89$	9.10801	0.965447	0.482723	+	0.875773i	$0.339648\pi$
$90$	0	0
$91$	3.78039	0.396293
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−5.73562	−0.588462
$96$	0	0
$97$	−16.5128	−1.67662	−0.838308	−	0.545197i	$-0.816455\pi$
$97$	−16.5128	−1.67662	−0.838308	+	0.545197i	$0.816455\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.0728659	−0.00725043	−0.00362522	−	0.999993i	$-0.501154\pi$
$101$	−0.0728659	−0.00725043	−0.00362522	+	0.999993i	$0.501154\pi$
$102$	0	0
$103$	−0.605551	−0.0596667	−0.0298334	−	0.999555i	$-0.509498\pi$
$103$	−0.605551	−0.0596667	−0.0298334	+	0.999555i	$0.509498\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.1016	−1.65328	−0.826639	−	0.562733i	$-0.809750\pi$
$107$	−17.1016	−1.65328	−0.826639	+	0.562733i	$0.809750\pi$
$108$	0	0
$109$	−9.83333	−0.941862	−0.470931	−	0.882170i	$-0.656082\pi$
$109$	−9.83333	−0.941862	−0.470931	+	0.882170i	$0.656082\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.31094	−0.311467	−0.155734	−	0.987799i	$-0.549774\pi$
$113$	−3.31094	−0.311467	−0.155734	+	0.987799i	$0.549774\pi$
$114$	0	0
$115$	−1.30278	−0.121484
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.60555	0.513860
$120$	0	0
$121$	23.5771	2.14337
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.8167	0.967471
$126$	0	0
$127$	−8.85213	−0.785500	−0.392750	−	0.919645i	$-0.628476\pi$
$127$	−8.85213	−0.785500	−0.392750	+	0.919645i	$0.628476\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.67730	0.233916	0.116958	−	0.993137i	$-0.462686\pi$
$131$	2.67730	0.233916	0.116958	+	0.993137i	$0.462686\pi$
$132$	0	0
$133$	4.40262	0.381755
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−3.44101	−0.293986	−0.146993	−	0.989138i	$-0.546959\pi$
$137$	−3.44101	−0.293986	−0.146993	+	0.989138i	$0.546959\pi$
$138$	0	0
$139$	11.7168	0.993807	0.496904	−	0.867806i	$-0.334470\pi$
$139$	11.7168	0.993807	0.496904	+	0.867806i	$0.334470\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	22.2296	1.85893
$144$	0	0
$145$	−0.264379	−0.0219555
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.39445	0.278084	0.139042	−	0.990286i	$-0.455598\pi$
$149$	3.39445	0.278084	0.139042	+	0.990286i	$0.455598\pi$
$150$	0	0
$151$	−21.6873	−1.76488	−0.882442	−	0.470421i	$-0.844102\pi$
$151$	−21.6873	−1.76488	−0.882442	+	0.470421i	$0.844102\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.944930	0.0758986
$156$	0	0
$157$	−20.2765	−1.61824	−0.809119	−	0.587644i	$-0.800055\pi$
$157$	−20.2765	−1.61824	−0.809119	+	0.587644i	$0.800055\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	10.1642	0.796122	0.398061	−	0.917359i	$-0.369683\pi$
$163$	10.1642	0.796122	0.398061	+	0.917359i	$0.369683\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.92826	0.458742	0.229371	−	0.973339i	$-0.426333\pi$
$167$	5.92826	0.458742	0.229371	+	0.973339i	$0.426333\pi$
$168$	0	0
$169$	1.29135	0.0993349
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−11.4492	−0.870465	−0.435232	−	0.900318i	$-0.643334\pi$
$173$	−11.4492	−0.870465	−0.435232	+	0.900318i	$0.643334\pi$
$174$	0	0
$175$	−3.30278	−0.249666
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.98007	0.521715	0.260858	−	0.965377i	$-0.415995\pi$
$179$	6.98007	0.521715	0.260858	+	0.965377i	$0.415995\pi$
$180$	0	0
$181$	−15.5575	−1.15638	−0.578191	−	0.815902i	$-0.696241\pi$
$181$	−15.5575	−1.15638	−0.578191	+	0.815902i	$0.696241\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.98333	−0.145817
$186$	0	0
$187$	32.9619	2.41042
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.16633	0.446180	0.223090	−	0.974798i	$-0.428386\pi$
$191$	6.16633	0.446180	0.223090	+	0.974798i	$0.428386\pi$
$192$	0	0
$193$	27.1712	1.95583	0.977915	−	0.209005i	$-0.0670226\pi$
$193$	27.1712	1.95583	0.977915	+	0.209005i	$0.0670226\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	26.3046	1.87412	0.937061	−	0.349164i	$-0.113535\pi$
$197$	26.3046	1.87412	0.937061	+	0.349164i	$0.113535\pi$
$198$	0	0
$199$	26.3575	1.86843	0.934217	−	0.356705i	$-0.116100\pi$
$199$	26.3575	1.86843	0.934217	+	0.356705i	$0.116100\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.202935	0.0142433
$204$	0	0
$205$	−8.43888	−0.589397
$206$	0	0
$207$	0	0
$208$	0	0
$209$	25.8884	1.79074
$210$	0	0
$211$	21.1827	1.45827	0.729137	−	0.684367i	$-0.239921\pi$
$211$	21.1827	1.45827	0.729137	+	0.684367i	$0.239921\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.88840	−0.128788
$216$	0	0
$217$	−0.725320	−0.0492380
$218$	0	0
$219$	0	0
$220$	0	0
$221$	21.1912	1.42547
$222$	0	0
$223$	1.05507	0.0706527	0.0353264	−	0.999376i	$-0.488753\pi$
$223$	1.05507	0.0706527	0.0353264	+	0.999376i	$0.488753\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.9737	0.993839	0.496920	−	0.867797i	$-0.334464\pi$
$227$	14.9737	0.993839	0.496920	+	0.867797i	$0.334464\pi$
$228$	0	0
$229$	14.4307	0.953608	0.476804	−	0.879010i	$-0.341795\pi$
$229$	14.4307	0.953608	0.476804	+	0.879010i	$0.341795\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	27.6062	1.80854	0.904272	−	0.426957i	$-0.140414\pi$
$233$	27.6062	1.80854	0.904272	+	0.426957i	$0.140414\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.34117	−0.539546	−0.269773	−	0.962924i	$-0.586949\pi$
$239$	−8.34117	−0.539546	−0.269773	+	0.962924i	$0.586949\pi$
$240$	0	0
$241$	−16.6585	−1.07307	−0.536534	−	0.843879i	$-0.680267\pi$
$241$	−16.6585	−1.07307	−0.536534	+	0.843879i	$0.680267\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.30278	−0.0832313
$246$	0	0
$247$	16.6436	1.05901
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−19.5771	−1.23570	−0.617848	−	0.786297i	$-0.711995\pi$
$251$	−19.5771	−1.23570	−0.617848	+	0.786297i	$0.711995\pi$
$252$	0	0
$253$	5.88023	0.369687
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.4670	1.33907	0.669537	−	0.742779i	$-0.266493\pi$
$257$	21.4670	1.33907	0.669537	+	0.742779i	$0.266493\pi$
$258$	0	0
$259$	1.52238	0.0945964
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−15.4907	−0.955197	−0.477599	−	0.878578i	$-0.658493\pi$
$263$	−15.4907	−0.955197	−0.477599	+	0.878578i	$0.658493\pi$
$264$	0	0
$265$	2.16342	0.132898
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−24.2377	−1.47780	−0.738900	−	0.673815i	$-0.764655\pi$
$269$	−24.2377	−1.47780	−0.738900	+	0.673815i	$0.764655\pi$
$270$	0	0
$271$	−30.4126	−1.84743	−0.923716	−	0.383077i	$-0.874864\pi$
$271$	−30.4126	−1.84743	−0.923716	+	0.383077i	$0.874864\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−19.4211	−1.17114
$276$	0	0
$277$	−1.89690	−0.113974	−0.0569870	−	0.998375i	$-0.518149\pi$
$277$	−1.89690	−0.113974	−0.0569870	+	0.998375i	$0.518149\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	11.9062	0.710264	0.355132	−	0.934816i	$-0.384436\pi$
$281$	11.9062	0.710264	0.355132	+	0.934816i	$0.384436\pi$
$282$	0	0
$283$	−2.37452	−0.141151	−0.0705753	−	0.997506i	$-0.522483\pi$
$283$	−2.37452	−0.141151	−0.0705753	+	0.997506i	$0.522483\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.47762	0.382362
$288$	0	0
$289$	14.4222	0.848365
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.41078	−0.0824188	−0.0412094	−	0.999151i	$-0.513121\pi$
$293$	−1.41078	−0.0824188	−0.0412094	+	0.999151i	$0.513121\pi$
$294$	0	0
$295$	3.29952	0.192106
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.78039	0.218626
$300$	0	0
$301$	1.44952	0.0835489
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.75229	0.386635
$306$	0	0
$307$	−15.7345	−0.898015	−0.449008	−	0.893528i	$-0.648222\pi$
$307$	−15.7345	−0.898015	−0.449008	+	0.893528i	$0.648222\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−14.7502	−0.836405	−0.418202	−	0.908354i	$-0.637340\pi$
$311$	−14.7502	−0.836405	−0.418202	+	0.908354i	$0.637340\pi$
$312$	0	0
$313$	8.45555	0.477936	0.238968	−	0.971027i	$-0.423191\pi$
$313$	8.45555	0.477936	0.238968	+	0.971027i	$0.423191\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.6866	0.600218	0.300109	−	0.953905i	$-0.402977\pi$
$317$	10.6866	0.600218	0.300109	+	0.953905i	$0.402977\pi$
$318$	0	0
$319$	1.19331	0.0668124
$320$	0	0
$321$	0	0
$322$	0	0
$323$	24.6791	1.37318
$324$	0	0
$325$	−12.4858	−0.692587
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−11.9105	−0.654658	−0.327329	−	0.944910i	$-0.606149\pi$
$331$	−11.9105	−0.654658	−0.327329	+	0.944910i	$0.606149\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.26617	0.396993
$336$	0	0
$337$	32.3315	1.76121	0.880606	−	0.473850i	$-0.157136\pi$
$337$	32.3315	1.76121	0.880606	+	0.473850i	$0.157136\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.26505	−0.230965
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.7124	1.43400	0.716999	−	0.697074i	$-0.245515\pi$
$347$	26.7124	1.43400	0.716999	+	0.697074i	$0.245515\pi$
$348$	0	0
$349$	8.87172	0.474893	0.237446	−	0.971401i	$-0.423690\pi$
$349$	8.87172	0.474893	0.237446	+	0.971401i	$0.423690\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−13.2665	−0.706105	−0.353053	−	0.935603i	$-0.614856\pi$
$353$	−13.2665	−0.706105	−0.353053	+	0.935603i	$0.614856\pi$
$354$	0	0
$355$	−13.7801	−0.731369
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7.51713	−0.396739	−0.198370	−	0.980127i	$-0.563565\pi$
$359$	−7.51713	−0.396739	−0.198370	+	0.980127i	$0.563565\pi$
$360$	0	0
$361$	0.383027	0.0201593
$362$	0	0
$363$	0	0
$364$	0	0
$365$	15.2384	0.797615
$366$	0	0
$367$	−3.03301	−0.158322	−0.0791609	−	0.996862i	$-0.525224\pi$
$367$	−3.03301	−0.158322	−0.0791609	+	0.996862i	$0.525224\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.66062	−0.0862152
$372$	0	0
$373$	13.4492	0.696372	0.348186	−	0.937425i	$-0.386798\pi$
$373$	13.4492	0.696372	0.348186	+	0.937425i	$0.386798\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.767175	0.0395115
$378$	0	0
$379$	10.0562	0.516552	0.258276	−	0.966071i	$-0.416846\pi$
$379$	10.0562	0.516552	0.258276	+	0.966071i	$0.416846\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	9.02060	0.460931	0.230466	−	0.973080i	$-0.425975\pi$
$383$	9.02060	0.460931	0.230466	+	0.973080i	$0.425975\pi$
$384$	0	0
$385$	−7.66062	−0.390421
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−18.5021	−0.938095	−0.469047	−	0.883173i	$-0.655403\pi$
$389$	−18.5021	−0.938095	−0.469047	+	0.883173i	$0.655403\pi$
$390$	0	0
$391$	5.60555	0.283485
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10.7324	−0.540004
$396$	0	0
$397$	−18.1265	−0.909742	−0.454871	−	0.890557i	$-0.650315\pi$
$397$	−18.1265	−0.909742	−0.454871	+	0.890557i	$0.650315\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9.42041	0.470433	0.235216	−	0.971943i	$-0.424420\pi$
$401$	9.42041	0.470433	0.235216	+	0.971943i	$0.424420\pi$
$402$	0	0
$403$	−2.74199	−0.136588
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.95198	0.443733
$408$	0	0
$409$	−10.9812	−0.542985	−0.271493	−	0.962441i	$-0.587517\pi$
$409$	−10.9812	−0.542985	−0.271493	+	0.962441i	$0.587517\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.53269	−0.124625
$414$	0	0
$415$	−1.75833	−0.0863130
$416$	0	0
$417$	0	0
$418$	0	0
$419$	18.1080	0.884634	0.442317	−	0.896859i	$-0.354157\pi$
$419$	18.1080	0.884634	0.442317	+	0.896859i	$0.354157\pi$
$420$	0	0
$421$	−4.55048	−0.221777	−0.110888	−	0.993833i	$-0.535370\pi$
$421$	−4.55048	−0.221777	−0.110888	+	0.993833i	$0.535370\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−18.5139	−0.898055
$426$	0	0
$427$	−5.18301	−0.250823
$428$	0	0
$429$	0	0
$430$	0	0
$431$	25.1578	1.21181	0.605905	−	0.795537i	$-0.292811\pi$
$431$	25.1578	1.21181	0.605905	+	0.795537i	$0.292811\pi$
$432$	0	0
$433$	15.7206	0.755484	0.377742	−	0.925911i	$-0.376701\pi$
$433$	15.7206	0.755484	0.377742	+	0.925911i	$0.376701\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.40262	0.210606
$438$	0	0
$439$	−7.99261	−0.381467	−0.190733	−	0.981642i	$-0.561087\pi$
$439$	−7.99261	−0.381467	−0.190733	+	0.981642i	$0.561087\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	23.1173	1.09834	0.549168	−	0.835712i	$-0.314945\pi$
$443$	23.1173	1.09834	0.549168	+	0.835712i	$0.314945\pi$
$444$	0	0
$445$	−11.8657	−0.562488
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−12.1180	−0.571882	−0.285941	−	0.958247i	$-0.592306\pi$
$449$	−12.1180	−0.571882	−0.285941	+	0.958247i	$0.592306\pi$
$450$	0	0
$451$	38.0899	1.79358
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.92500	−0.230888
$456$	0	0
$457$	−12.8425	−0.600747	−0.300374	−	0.953822i	$-0.597111\pi$
$457$	−12.8425	−0.600747	−0.300374	+	0.953822i	$0.597111\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	22.0870	1.02869	0.514346	−	0.857583i	$-0.328035\pi$
$461$	22.0870	1.02869	0.514346	+	0.857583i	$0.328035\pi$
$462$	0	0
$463$	−33.9360	−1.57714	−0.788569	−	0.614946i	$-0.789178\pi$
$463$	−33.9360	−1.57714	−0.788569	+	0.614946i	$0.789178\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−41.6758	−1.92853	−0.964264	−	0.264944i	$-0.914647\pi$
$467$	−41.6758	−1.92853	−0.964264	+	0.264944i	$0.914647\pi$
$468$	0	0
$469$	−5.57746	−0.257543
$470$	0	0
$471$	0	0
$472$	0	0
$473$	8.52351	0.391911
$474$	0	0
$475$	−14.5409	−0.667180
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.26014	−0.423107	−0.211553	−	0.977366i	$-0.567852\pi$
$479$	−9.26014	−0.423107	−0.211553	+	0.977366i	$0.567852\pi$
$480$	0	0
$481$	5.75521	0.262415
$482$	0	0
$483$	0	0
$484$	0	0
$485$	21.5124	0.976829
$486$	0	0
$487$	−10.4670	−0.474304	−0.237152	−	0.971473i	$-0.576214\pi$
$487$	−10.4670	−0.474304	−0.237152	+	0.971473i	$0.576214\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9.52842	0.430012	0.215006	−	0.976613i	$-0.431023\pi$
$491$	9.52842	0.430012	0.215006	+	0.976613i	$0.431023\pi$
$492$	0	0
$493$	1.13756	0.0512333
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.5775	0.474464
$498$	0	0
$499$	5.19264	0.232454	0.116227	−	0.993223i	$-0.462920\pi$
$499$	5.19264	0.232454	0.116227	+	0.993223i	$0.462920\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	3.81452	0.170081	0.0850405	−	0.996377i	$-0.472898\pi$
$503$	3.81452	0.170081	0.0850405	+	0.996377i	$0.472898\pi$
$504$	0	0
$505$	0.0949280	0.00422424
$506$	0	0
$507$	0	0
$508$	0	0
$509$	29.3546	1.30112	0.650560	−	0.759455i	$-0.274534\pi$
$509$	29.3546	1.30112	0.650560	+	0.759455i	$0.274534\pi$
$510$	0	0
$511$	−11.6969	−0.517440
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.788897	0.0347630
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−31.5771	−1.38342	−0.691709	−	0.722176i	$-0.743142\pi$
$521$	−31.5771	−1.38342	−0.691709	+	0.722176i	$0.743142\pi$
$522$	0	0
$523$	−29.1712	−1.27557	−0.637785	−	0.770215i	$-0.720149\pi$
$523$	−29.1712	−1.27557	−0.637785	+	0.770215i	$0.720149\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.06582	−0.177110
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	24.4879	1.06069
$534$	0	0
$535$	22.2796	0.963231
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.88023	0.253279
$540$	0	0
$541$	−34.1663	−1.46893	−0.734463	−	0.678649i	$-0.762566\pi$
$541$	−34.1663	−1.46893	−0.734463	+	0.678649i	$0.762566\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.8106	0.548747
$546$	0	0
$547$	19.1756	0.819890	0.409945	−	0.912110i	$-0.365548\pi$
$547$	19.1756	0.819890	0.409945	+	0.912110i	$0.365548\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.893447	0.0380621
$552$	0	0
$553$	8.23808	0.350319
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19.3611	0.820356	0.410178	−	0.912005i	$-0.365467\pi$
$557$	19.3611	0.820356	0.410178	+	0.912005i	$0.365467\pi$
$558$	0	0
$559$	5.47975	0.231769
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.2296	−1.18973	−0.594867	−	0.803824i	$-0.702795\pi$
$563$	−28.2296	−1.18973	−0.594867	+	0.803824i	$0.702795\pi$
$564$	0	0
$565$	4.31342	0.181467
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23.3774	−0.980033	−0.490017	−	0.871713i	$-0.663009\pi$
$569$	−23.3774	−0.980033	−0.490017	+	0.871713i	$0.663009\pi$
$570$	0	0
$571$	−29.6489	−1.24077	−0.620383	−	0.784299i	$-0.713023\pi$
$571$	−29.6489	−1.24077	−0.620383	+	0.784299i	$0.713023\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.30278	−0.137735
$576$	0	0
$577$	−24.2928	−1.01132	−0.505661	−	0.862732i	$-0.668751\pi$
$577$	−24.2928	−1.01132	−0.505661	+	0.862732i	$0.668751\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.34968	0.0559941
$582$	0	0
$583$	−9.76484	−0.404418
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−43.7423	−1.80544	−0.902720	−	0.430230i	$-0.858433\pi$
$587$	−43.7423	−1.80544	−0.902720	+	0.430230i	$0.858433\pi$
$588$	0	0
$589$	−3.19331	−0.131578
$590$	0	0
$591$	0	0
$592$	0	0
$593$	44.1471	1.81290	0.906452	−	0.422310i	$-0.138781\pi$
$593$	44.1471	1.81290	0.906452	+	0.422310i	$0.138781\pi$
$594$	0	0
$595$	−7.30278	−0.299385
$596$	0	0
$597$	0	0
$598$	0	0
$599$	13.2768	0.542476	0.271238	−	0.962512i	$-0.412567\pi$
$599$	13.2768	0.542476	0.271238	+	0.962512i	$0.412567\pi$
$600$	0	0
$601$	30.2868	1.23542	0.617712	−	0.786405i	$-0.288060\pi$
$601$	30.2868	1.23542	0.617712	+	0.786405i	$0.288060\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−30.7157	−1.24877
$606$	0	0
$607$	−38.3127	−1.55507	−0.777533	−	0.628842i	$-0.783529\pi$
$607$	−38.3127	−1.55507	−0.777533	+	0.628842i	$0.783529\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	22.2943	0.900457	0.450229	−	0.892913i	$-0.351343\pi$
$613$	22.2943	0.900457	0.450229	+	0.892913i	$0.351343\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−43.7569	−1.76159	−0.880793	−	0.473502i	$-0.842990\pi$
$617$	−43.7569	−1.76159	−0.880793	+	0.473502i	$0.842990\pi$
$618$	0	0
$619$	8.92287	0.358640	0.179320	−	0.983791i	$-0.442610\pi$
$619$	8.92287	0.358640	0.179320	+	0.983791i	$0.442610\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.10801	0.364905
$624$	0	0
$625$	2.42221	0.0968882
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.53381	0.340265
$630$	0	0
$631$	12.2047	0.485863	0.242931	−	0.970043i	$-0.421891\pi$
$631$	12.2047	0.485863	0.242931	+	0.970043i	$0.421891\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.5323	0.457647
$636$	0	0
$637$	3.78039	0.149785
$638$	0	0
$639$	0	0
$640$	0	0
$641$	42.6787	1.68571	0.842855	−	0.538141i	$-0.180873\pi$
$641$	42.6787	1.68571	0.842855	+	0.538141i	$0.180873\pi$
$642$	0	0
$643$	−3.27255	−0.129057	−0.0645283	−	0.997916i	$-0.520554\pi$
$643$	−3.27255	−0.129057	−0.0645283	+	0.997916i	$0.520554\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−23.5213	−0.924716	−0.462358	−	0.886693i	$-0.652997\pi$
$647$	−23.5213	−0.924716	−0.462358	+	0.886693i	$0.652997\pi$
$648$	0	0
$649$	−14.8928	−0.584593
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−0.168330	−0.00658725	−0.00329362	−	0.999995i	$-0.501048\pi$
$653$	−0.168330	−0.00658725	−0.00329362	+	0.999995i	$0.501048\pi$
$654$	0	0
$655$	−3.48792	−0.136284
$656$	0	0
$657$	0	0
$658$	0	0
$659$	0.669128	0.0260655	0.0130328	−	0.999915i	$-0.495851\pi$
$659$	0.669128	0.0260655	0.0130328	+	0.999915i	$0.495851\pi$
$660$	0	0
$661$	7.81767	0.304072	0.152036	−	0.988375i	$-0.451417\pi$
$661$	7.81767	0.304072	0.152036	+	0.988375i	$0.451417\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.73562	−0.222418
$666$	0	0
$667$	0.202935	0.00785769
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−30.4773	−1.17656
$672$	0	0
$673$	29.0931	1.12146	0.560729	−	0.828000i	$-0.310521\pi$
$673$	29.0931	1.12146	0.560729	+	0.828000i	$0.310521\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.01847	0.192875	0.0964377	−	0.995339i	$-0.469255\pi$
$677$	5.01847	0.192875	0.0964377	+	0.995339i	$0.469255\pi$
$678$	0	0
$679$	−16.5128	−0.633701
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−3.97973	−0.152280	−0.0761401	−	0.997097i	$-0.524260\pi$
$683$	−3.97973	−0.152280	−0.0761401	+	0.997097i	$0.524260\pi$
$684$	0	0
$685$	4.48287	0.171282
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−6.27780	−0.239165
$690$	0	0
$691$	14.6062	0.555647	0.277823	−	0.960632i	$-0.410387\pi$
$691$	14.6062	0.555647	0.277823	+	0.960632i	$0.410387\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−15.2644	−0.579011
$696$	0	0
$697$	36.3106	1.37536
$698$	0	0
$699$	0	0
$700$	0	0
$701$	36.8717	1.39263	0.696313	−	0.717738i	$-0.254823\pi$
$701$	36.8717	1.39263	0.696313	+	0.717738i	$0.254823\pi$
$702$	0	0
$703$	6.70248	0.252789
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.0728659	−0.00274041
$708$	0	0
$709$	−18.9165	−0.710424	−0.355212	−	0.934786i	$-0.615591\pi$
$709$	−18.9165	−0.710424	−0.355212	+	0.934786i	$0.615591\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.725320	−0.0271635
$714$	0	0
$715$	−28.9601	−1.08305
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−29.4492	−1.09827	−0.549135	−	0.835734i	$-0.685043\pi$
$719$	−29.4492	−1.09827	−0.549135	+	0.835734i	$0.685043\pi$
$720$	0	0
$721$	−0.605551	−0.0225519
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.670250	−0.0248925
$726$	0	0
$727$	5.11899	0.189853	0.0949264	−	0.995484i	$-0.469738\pi$
$727$	5.11899	0.189853	0.0949264	+	0.995484i	$0.469738\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8.12535	0.300527
$732$	0	0
$733$	−7.56078	−0.279264	−0.139632	−	0.990203i	$-0.544592\pi$
$733$	−7.56078	−0.279264	−0.139632	+	0.990203i	$0.544592\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−32.7967	−1.20808
$738$	0	0
$739$	−5.20619	−0.191513	−0.0957564	−	0.995405i	$-0.530527\pi$
$739$	−5.20619	−0.191513	−0.0957564	+	0.995405i	$0.530527\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9.54824	−0.350291	−0.175145	−	0.984543i	$-0.556040\pi$
$743$	−9.54824	−0.350291	−0.175145	+	0.984543i	$0.556040\pi$
$744$	0	0
$745$	−4.42221	−0.162017
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−17.1016	−0.624880
$750$	0	0
$751$	−12.6052	−0.459971	−0.229985	−	0.973194i	$-0.573868\pi$
$751$	−12.6052	−0.459971	−0.229985	+	0.973194i	$0.573868\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	28.2536	1.02825
$756$	0	0
$757$	−42.3874	−1.54060	−0.770298	−	0.637684i	$-0.779893\pi$
$757$	−42.3874	−1.54060	−0.770298	+	0.637684i	$0.779893\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−39.2977	−1.42454	−0.712271	−	0.701905i	$-0.752333\pi$
$761$	−39.2977	−1.42454	−0.712271	+	0.701905i	$0.752333\pi$
$762$	0	0
$763$	−9.83333	−0.355990
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.57454	−0.345717
$768$	0	0
$769$	43.3333	1.56264	0.781320	−	0.624131i	$-0.214547\pi$
$769$	43.3333	1.56264	0.781320	+	0.624131i	$0.214547\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	13.1827	0.474148	0.237074	−	0.971492i	$-0.423812\pi$
$773$	13.1827	0.474148	0.237074	+	0.971492i	$0.423812\pi$
$774$	0	0
$775$	2.39557	0.0860514
$776$	0	0
$777$	0	0
$778$	0	0
$779$	28.5185	1.02178
$780$	0	0
$781$	62.1979	2.22562
$782$	0	0
$783$	0	0
$784$	0	0
$785$	26.4157	0.942817
$786$	0	0
$787$	−8.75979	−0.312253	−0.156126	−	0.987737i	$-0.549901\pi$
$787$	−8.75979	−0.312253	−0.156126	+	0.987737i	$0.549901\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.31094	−0.117724
$792$	0	0
$793$	−19.5938	−0.695796
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.2559	0.788343	0.394172	−	0.919037i	$-0.371032\pi$
$797$	22.2559	0.788343	0.394172	+	0.919037i	$0.371032\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−68.7804	−2.42721
$804$	0	0
$805$	−1.30278	−0.0459168
$806$	0	0
$807$	0	0
$808$	0	0
$809$	39.0722	1.37370	0.686852	−	0.726797i	$-0.258992\pi$
$809$	39.0722	1.37370	0.686852	+	0.726797i	$0.258992\pi$
$810$	0	0
$811$	6.99528	0.245638	0.122819	−	0.992429i	$-0.460807\pi$
$811$	6.99528	0.245638	0.122819	+	0.992429i	$0.460807\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−13.2417	−0.463836
$816$	0	0
$817$	6.38168	0.223267
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−51.5601	−1.79946	−0.899730	−	0.436447i	$-0.856237\pi$
$821$	−51.5601	−1.79946	−0.899730	+	0.436447i	$0.856237\pi$
$822$	0	0
$823$	21.8727	0.762436	0.381218	−	0.924485i	$-0.375505\pi$
$823$	21.8727	0.762436	0.381218	+	0.924485i	$0.375505\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−4.40966	−0.153339	−0.0766695	−	0.997057i	$-0.524429\pi$
$827$	−4.40966	−0.153339	−0.0766695	+	0.997057i	$0.524429\pi$
$828$	0	0
$829$	24.3727	0.846500	0.423250	−	0.906013i	$-0.360889\pi$
$829$	24.3727	0.846500	0.423250	+	0.906013i	$0.360889\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.60555	0.194221
$834$	0	0
$835$	−7.72319	−0.267272
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−29.6639	−1.02411	−0.512055	−	0.858952i	$-0.671116\pi$
$839$	−29.6639	−1.02411	−0.512055	+	0.858952i	$0.671116\pi$
$840$	0	0
$841$	−28.9588	−0.998580
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.68234	−0.0578744
$846$	0	0
$847$	23.5771	0.810119
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.52238	0.0521867
$852$	0	0
$853$	7.78331	0.266495	0.133248	−	0.991083i	$-0.457459\pi$
$853$	7.78331	0.266495	0.133248	+	0.991083i	$0.457459\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−36.5722	−1.24928	−0.624641	−	0.780912i	$-0.714755\pi$
$857$	−36.5722	−1.24928	−0.624641	+	0.780912i	$0.714755\pi$
$858$	0	0
$859$	9.41078	0.321092	0.160546	−	0.987028i	$-0.448675\pi$
$859$	9.41078	0.321092	0.160546	+	0.987028i	$0.448675\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−54.9546	−1.87067	−0.935337	−	0.353757i	$-0.884904\pi$
$863$	−54.9546	−1.87067	−0.935337	+	0.353757i	$0.884904\pi$
$864$	0	0
$865$	14.9157	0.507149
$866$	0	0
$867$	0	0
$868$	0	0
$869$	48.4418	1.64328
$870$	0	0
$871$	−21.0850	−0.714437
$872$	0	0
$873$	0	0
$874$	0	0
$875$	10.8167	0.365670
$876$	0	0
$877$	41.0107	1.38483	0.692417	−	0.721497i	$-0.256546\pi$
$877$	41.0107	1.38483	0.692417	+	0.721497i	$0.256546\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	23.6170	0.795676	0.397838	−	0.917456i	$-0.369761\pi$
$881$	23.6170	0.795676	0.397838	+	0.917456i	$0.369761\pi$
$882$	0	0
$883$	3.71783	0.125115	0.0625574	−	0.998041i	$-0.480074\pi$
$883$	3.71783	0.125115	0.0625574	+	0.998041i	$0.480074\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	8.19788	0.275258	0.137629	−	0.990484i	$-0.456052\pi$
$887$	8.19788	0.275258	0.137629	+	0.990484i	$0.456052\pi$
$888$	0	0
$889$	−8.85213	−0.296891
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−9.09347	−0.303961
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.147193	−0.00490917
$900$	0	0
$901$	−9.30870	−0.310118
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.2680	0.673730
$906$	0	0
$907$	−17.9370	−0.595587	−0.297793	−	0.954630i	$-0.596251\pi$
$907$	−17.9370	−0.595587	−0.297793	+	0.954630i	$0.596251\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−52.1330	−1.72724	−0.863621	−	0.504141i	$-0.831809\pi$
$911$	−52.1330	−1.72724	−0.863621	+	0.504141i	$0.831809\pi$
$912$	0	0
$913$	7.93642	0.262657
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.67730	0.0884121
$918$	0	0
$919$	55.1136	1.81803	0.909015	−	0.416764i	$-0.136836\pi$
$919$	55.1136	1.81803	0.909015	+	0.416764i	$0.136836\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	39.9869	1.31619
$924$	0	0
$925$	−5.02810	−0.165323
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−16.3287	−0.535729	−0.267864	−	0.963457i	$-0.586318\pi$
$929$	−16.3287	−0.535729	−0.267864	+	0.963457i	$0.586318\pi$
$930$	0	0
$931$	4.40262	0.144290
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−42.9420	−1.40435
$936$	0	0
$937$	16.5454	0.540515	0.270258	−	0.962788i	$-0.412891\pi$
$937$	16.5454	0.540515	0.270258	+	0.962788i	$0.412891\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	12.5990	0.410717	0.205359	−	0.978687i	$-0.434164\pi$
$941$	12.5990	0.410717	0.205359	+	0.978687i	$0.434164\pi$
$942$	0	0
$943$	6.47762	0.210940
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−40.9510	−1.33073	−0.665364	−	0.746519i	$-0.731724\pi$
$947$	−40.9510	−1.33073	−0.665364	+	0.746519i	$0.731724\pi$
$948$	0	0
$949$	−44.2188	−1.43540
$950$	0	0
$951$	0	0
$952$	0	0
$953$	30.3461	0.983006	0.491503	−	0.870876i	$-0.336448\pi$
$953$	30.3461	0.983006	0.491503	+	0.870876i	$0.336448\pi$
$954$	0	0
$955$	−8.03335	−0.259953
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.44101	−0.111116
$960$	0	0
$961$	−30.4739	−0.983029
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−35.3980	−1.13950
$966$	0	0
$967$	−34.2160	−1.10031	−0.550156	−	0.835062i	$-0.685432\pi$
$967$	−34.2160	−1.10031	−0.550156	+	0.835062i	$0.685432\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	35.5465	1.14074	0.570371	−	0.821387i	$-0.306799\pi$
$971$	35.5465	1.14074	0.570371	+	0.821387i	$0.306799\pi$
$972$	0	0
$973$	11.7168	0.375624
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.1072	0.579301	0.289651	−	0.957132i	$-0.406461\pi$
$977$	18.1072	0.579301	0.289651	+	0.957132i	$0.406461\pi$
$978$	0	0
$979$	53.5572	1.71170
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−49.5885	−1.58163	−0.790814	−	0.612056i	$-0.790343\pi$
$983$	−49.5885	−1.58163	−0.790814	+	0.612056i	$0.790343\pi$
$984$	0	0
$985$	−34.2690	−1.09190
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.44952	0.0460920
$990$	0	0
$991$	45.8897	1.45773	0.728867	−	0.684655i	$-0.240047\pi$
$991$	45.8897	1.45773	0.728867	+	0.684655i	$0.240047\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−34.3379	−1.08859
$996$	0	0
$997$	−19.6577	−0.622566	−0.311283	−	0.950317i	$-0.600759\pi$
$997$	−19.6577	−0.622566	−0.311283	+	0.950317i	$0.600759\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5796.2.a.r.1.2	yes	4
3.2	odd	2		5796.2.a.q.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5796.2.a.q.1.3	✓	4	3.2	odd	2
5796.2.a.r.1.2	yes	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:	\( N \)	\(=\)	\( 5796 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5796.a (trivial)

\(f(q)\)	\(=\)	\(q-1.30278 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-1.30278 q^{5} +1.00000 q^{7} +5.88023 q^{11} +3.78039 q^{13} +5.60555 q^{17} +4.40262 q^{19} +1.00000 q^{23} -3.30278 q^{25} +0.202935 q^{29} -0.725320 q^{31} -1.30278 q^{35} +1.52238 q^{37} +6.47762 q^{41} +1.44952 q^{43} +1.00000 q^{49} -1.66062 q^{53} -7.66062 q^{55} -2.53269 q^{59} -5.18301 q^{61} -4.92500 q^{65} -5.57746 q^{67} +10.5775 q^{71} -11.6969 q^{73} +5.88023 q^{77} +8.23808 q^{79} +1.34968 q^{83} -7.30278 q^{85} +9.10801 q^{89} +3.78039 q^{91} -5.73562 q^{95} -16.5128 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q + 2 * q^5 + 4 * q^7	\( 4 q + 2 q^{5} + 4 q^{7} + 4 q^{11} + 2 q^{13} + 8 q^{17} + 4 q^{19} + 4 q^{23} - 6 q^{25} - 8 q^{31} + 2 q^{35} + 12 q^{37} + 20 q^{41} + 2 q^{43} + 4 q^{49} + 26 q^{53} + 2 q^{55} + 14 q^{59} + 6 q^{61} - 12 q^{65} - 10 q^{67} + 30 q^{71} + 16 q^{73} + 4 q^{77} - 12 q^{79} + 8 q^{83} - 22 q^{85} + 2 q^{89} + 2 q^{91} + 2 q^{95} - 16 q^{97}+O(q^{100}) \) 4 * q + 2 * q^5 + 4 * q^7 + 4 * q^11 + 2 * q^13 + 8 * q^17 + 4 * q^19 + 4 * q^23 - 6 * q^25 - 8 * q^31 + 2 * q^35 + 12 * q^37 + 20 * q^41 + 2 * q^43 + 4 * q^49 + 26 * q^53 + 2 * q^55 + 14 * q^59 + 6 * q^61 - 12 * q^65 - 10 * q^67 + 30 * q^71 + 16 * q^73 + 4 * q^77 - 12 * q^79 + 8 * q^83 - 22 * q^85 + 2 * q^89 + 2 * q^91 + 2 * q^95 - 16 * q^97

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