LMFDB - Embedded newform 6044.2.a.a.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6044,2,Mod(1,6044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6044 = 2^{2} \cdot 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6044.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:	$48.2615829817$
Analytic rank:	$1$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	6044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.97134 q^{3} -2.23756 q^{5} +4.75272 q^{7} +5.82886 q^{9} +O(q^{10})$	$q-2.97134 q^{3} -2.23756 q^{5} +4.75272 q^{7} +5.82886 q^{9} +5.30019 q^{11} +2.25013 q^{13} +6.64855 q^{15} +3.41140 q^{17} -7.44027 q^{19} -14.1220 q^{21} -6.31900 q^{23} +0.00667165 q^{25} -8.40552 q^{27} -4.25129 q^{29} +7.90432 q^{31} -15.7487 q^{33} -10.6345 q^{35} -7.03989 q^{37} -6.68589 q^{39} -3.34927 q^{41} -9.17337 q^{43} -13.0424 q^{45} +11.0060 q^{47} +15.5884 q^{49} -10.1364 q^{51} +0.639981 q^{53} -11.8595 q^{55} +22.1076 q^{57} -11.0201 q^{59} -6.62403 q^{61} +27.7030 q^{63} -5.03479 q^{65} +5.06487 q^{67} +18.7759 q^{69} -13.1718 q^{71} -13.5588 q^{73} -0.0198238 q^{75} +25.1903 q^{77} +2.41814 q^{79} +7.48907 q^{81} -6.87723 q^{83} -7.63322 q^{85} +12.6320 q^{87} +13.3555 q^{89} +10.6942 q^{91} -23.4864 q^{93} +16.6481 q^{95} -16.3378 q^{97} +30.8941 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	$ 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) $ 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * 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
$3$	−2.97134	−1.71550	−0.857752	−	0.514064i	$-0.828140\pi$
$3$	−2.97134	−1.71550	−0.857752	+	0.514064i	$0.828140\pi$
$4$	0	0
$5$	−2.23756	−1.00067	−0.500333	−	0.865833i	$-0.666789\pi$
$5$	−2.23756	−1.00067	−0.500333	+	0.865833i	$0.666789\pi$
$6$	0	0
$7$	4.75272	1.79636	0.898180	−	0.439628i	$-0.144890\pi$
$7$	4.75272	1.79636	0.898180	+	0.439628i	$0.144890\pi$
$8$	0	0
$9$	5.82886	1.94295
$10$	0	0
$11$	5.30019	1.59807	0.799033	−	0.601287i	$-0.205345\pi$
$11$	5.30019	1.59807	0.799033	+	0.601287i	$0.205345\pi$
$12$	0	0
$13$	2.25013	0.624073	0.312037	−	0.950070i	$-0.398989\pi$
$13$	2.25013	0.624073	0.312037	+	0.950070i	$0.398989\pi$
$14$	0	0
$15$	6.64855	1.71665
$16$	0	0
$17$	3.41140	0.827387	0.413693	−	0.910416i	$-0.364239\pi$
$17$	3.41140	0.827387	0.413693	+	0.910416i	$0.364239\pi$
$18$	0	0
$19$	−7.44027	−1.70692	−0.853458	−	0.521162i	$-0.825499\pi$
$19$	−7.44027	−1.70692	−0.853458	+	0.521162i	$0.825499\pi$
$20$	0	0
$21$	−14.1220	−3.08166
$22$	0	0
$23$	−6.31900	−1.31760	−0.658801	−	0.752317i	$-0.728936\pi$
$23$	−6.31900	−1.31760	−0.658801	+	0.752317i	$0.728936\pi$
$24$	0	0
$25$	0.00667165	0.00133433
$26$	0	0
$27$	−8.40552	−1.61764
$28$	0	0
$29$	−4.25129	−0.789444	−0.394722	−	0.918801i	$-0.629159\pi$
$29$	−4.25129	−0.789444	−0.394722	+	0.918801i	$0.629159\pi$
$30$	0	0
$31$	7.90432	1.41966	0.709828	−	0.704375i	$-0.248772\pi$
$31$	7.90432	1.41966	0.709828	+	0.704375i	$0.248772\pi$
$32$	0	0
$33$	−15.7487	−2.74149
$34$	0	0
$35$	−10.6345	−1.79756
$36$	0	0
$37$	−7.03989	−1.15735	−0.578676	−	0.815558i	$-0.696430\pi$
$37$	−7.03989	−1.15735	−0.578676	+	0.815558i	$0.696430\pi$
$38$	0	0
$39$	−6.68589	−1.07060
$40$	0	0
$41$	−3.34927	−0.523068	−0.261534	−	0.965194i	$-0.584228\pi$
$41$	−3.34927	−0.523068	−0.261534	+	0.965194i	$0.584228\pi$
$42$	0	0
$43$	−9.17337	−1.39893	−0.699463	−	0.714669i	$-0.746577\pi$
$43$	−9.17337	−1.39893	−0.699463	+	0.714669i	$0.746577\pi$
$44$	0	0
$45$	−13.0424	−1.94425
$46$	0	0
$47$	11.0060	1.60539	0.802695	−	0.596389i	$-0.203398\pi$
$47$	11.0060	1.60539	0.802695	+	0.596389i	$0.203398\pi$
$48$	0	0
$49$	15.5884	2.22691
$50$	0	0
$51$	−10.1364	−1.41939
$52$	0	0
$53$	0.639981	0.0879082	0.0439541	−	0.999034i	$-0.486004\pi$
$53$	0.639981	0.0879082	0.0439541	+	0.999034i	$0.486004\pi$
$54$	0	0
$55$	−11.8595	−1.59913
$56$	0	0
$57$	22.1076	2.92822
$58$	0	0
$59$	−11.0201	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$59$	−11.0201	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$60$	0	0
$61$	−6.62403	−0.848120	−0.424060	−	0.905634i	$-0.639395\pi$
$61$	−6.62403	−0.848120	−0.424060	+	0.905634i	$0.639395\pi$
$62$	0	0
$63$	27.7030	3.49025
$64$	0	0
$65$	−5.03479	−0.624489
$66$	0	0
$67$	5.06487	0.618772	0.309386	−	0.950937i	$-0.399876\pi$
$67$	5.06487	0.618772	0.309386	+	0.950937i	$0.399876\pi$
$68$	0	0
$69$	18.7759	2.26035
$70$	0	0
$71$	−13.1718	−1.56320	−0.781600	−	0.623780i	$-0.785596\pi$
$71$	−13.1718	−1.56320	−0.781600	+	0.623780i	$0.785596\pi$
$72$	0	0
$73$	−13.5588	−1.58693	−0.793467	−	0.608614i	$-0.791726\pi$
$73$	−13.5588	−1.58693	−0.793467	+	0.608614i	$0.791726\pi$
$74$	0	0
$75$	−0.0198238	−0.00228905
$76$	0	0
$77$	25.1903	2.87070
$78$	0	0
$79$	2.41814	0.272062	0.136031	−	0.990705i	$-0.456565\pi$
$79$	2.41814	0.272062	0.136031	+	0.990705i	$0.456565\pi$
$80$	0	0
$81$	7.48907	0.832118
$82$	0	0
$83$	−6.87723	−0.754874	−0.377437	−	0.926035i	$-0.623195\pi$
$83$	−6.87723	−0.754874	−0.377437	+	0.926035i	$0.623195\pi$
$84$	0	0
$85$	−7.63322	−0.827939
$86$	0	0
$87$	12.6320	1.35429
$88$	0	0
$89$	13.3555	1.41568	0.707842	−	0.706371i	$-0.249669\pi$
$89$	13.3555	1.41568	0.707842	+	0.706371i	$0.249669\pi$
$90$	0	0
$91$	10.6942	1.12106
$92$	0	0
$93$	−23.4864	−2.43543
$94$	0	0
$95$	16.6481	1.70805
$96$	0	0
$97$	−16.3378	−1.65886	−0.829428	−	0.558614i	$-0.811333\pi$
$97$	−16.3378	−1.65886	−0.829428	+	0.558614i	$0.811333\pi$
$98$	0	0
$99$	30.8941	3.10497
$100$	0	0
$101$	−3.98375	−0.396398	−0.198199	−	0.980162i	$-0.563509\pi$
$101$	−3.98375	−0.396398	−0.198199	+	0.980162i	$0.563509\pi$
$102$	0	0
$103$	0.0887451	0.00874431	0.00437216	−	0.999990i	$-0.498608\pi$
$103$	0.0887451	0.00874431	0.00437216	+	0.999990i	$0.498608\pi$
$104$	0	0
$105$	31.5987	3.08372
$106$	0	0
$107$	9.89874	0.956948	0.478474	−	0.878102i	$-0.341190\pi$
$107$	9.89874	0.956948	0.478474	+	0.878102i	$0.341190\pi$
$108$	0	0
$109$	−6.34727	−0.607958	−0.303979	−	0.952679i	$-0.598315\pi$
$109$	−6.34727	−0.607958	−0.303979	+	0.952679i	$0.598315\pi$
$110$	0	0
$111$	20.9179	1.98544
$112$	0	0
$113$	−0.913049	−0.0858924	−0.0429462	−	0.999077i	$-0.513674\pi$
$113$	−0.913049	−0.0858924	−0.0429462	+	0.999077i	$0.513674\pi$
$114$	0	0
$115$	14.1391	1.31848
$116$	0	0
$117$	13.1157	1.21255
$118$	0	0
$119$	16.2134	1.48628
$120$	0	0
$121$	17.0920	1.55382
$122$	0	0
$123$	9.95183	0.897326
$124$	0	0
$125$	11.1729	0.999332
$126$	0	0
$127$	0.537189	0.0476678	0.0238339	−	0.999716i	$-0.492413\pi$
$127$	0.537189	0.0476678	0.0238339	+	0.999716i	$0.492413\pi$
$128$	0	0
$129$	27.2572	2.39986
$130$	0	0
$131$	19.3992	1.69492	0.847460	−	0.530860i	$-0.178131\pi$
$131$	19.3992	1.69492	0.847460	+	0.530860i	$0.178131\pi$
$132$	0	0
$133$	−35.3615	−3.06623
$134$	0	0
$135$	18.8078	1.61872
$136$	0	0
$137$	−16.7790	−1.43353	−0.716763	−	0.697317i	$-0.754377\pi$
$137$	−16.7790	−1.43353	−0.716763	+	0.697317i	$0.754377\pi$
$138$	0	0
$139$	−14.7119	−1.24785	−0.623924	−	0.781485i	$-0.714463\pi$
$139$	−14.7119	−1.24785	−0.623924	+	0.781485i	$0.714463\pi$
$140$	0	0
$141$	−32.7026	−2.75405
$142$	0	0
$143$	11.9261	0.997310
$144$	0	0
$145$	9.51250	0.789970
$146$	0	0
$147$	−46.3183	−3.82027
$148$	0	0
$149$	9.54020	0.781564	0.390782	−	0.920483i	$-0.372205\pi$
$149$	9.54020	0.781564	0.390782	+	0.920483i	$0.372205\pi$
$150$	0	0
$151$	−4.72505	−0.384519	−0.192259	−	0.981344i	$-0.561582\pi$
$151$	−4.72505	−0.384519	−0.192259	+	0.981344i	$0.561582\pi$
$152$	0	0
$153$	19.8846	1.60758
$154$	0	0
$155$	−17.6864	−1.42060
$156$	0	0
$157$	−4.28565	−0.342032	−0.171016	−	0.985268i	$-0.554705\pi$
$157$	−4.28565	−0.342032	−0.171016	+	0.985268i	$0.554705\pi$
$158$	0	0
$159$	−1.90160	−0.150807
$160$	0	0
$161$	−30.0325	−2.36689
$162$	0	0
$163$	−15.7001	−1.22973	−0.614864	−	0.788633i	$-0.710789\pi$
$163$	−15.7001	−1.22973	−0.614864	+	0.788633i	$0.710789\pi$
$164$	0	0
$165$	35.2386	2.74332
$166$	0	0
$167$	14.5812	1.12833	0.564163	−	0.825663i	$-0.309199\pi$
$167$	14.5812	1.12833	0.564163	+	0.825663i	$0.309199\pi$
$168$	0	0
$169$	−7.93693	−0.610533
$170$	0	0
$171$	−43.3683	−3.31646
$172$	0	0
$173$	25.3370	1.92634	0.963169	−	0.268897i	$-0.0866592\pi$
$173$	25.3370	1.92634	0.963169	+	0.268897i	$0.0866592\pi$
$174$	0	0
$175$	0.0317085	0.00239694
$176$	0	0
$177$	32.7444	2.46122
$178$	0	0
$179$	−19.9888	−1.49403	−0.747016	−	0.664807i	$-0.768514\pi$
$179$	−19.9888	−1.49403	−0.747016	+	0.664807i	$0.768514\pi$
$180$	0	0
$181$	1.93917	0.144137	0.0720687	−	0.997400i	$-0.477040\pi$
$181$	1.93917	0.144137	0.0720687	+	0.997400i	$0.477040\pi$
$182$	0	0
$183$	19.6822	1.45495
$184$	0	0
$185$	15.7522	1.15812
$186$	0	0
$187$	18.0811	1.32222
$188$	0	0
$189$	−39.9491	−2.90587
$190$	0	0
$191$	−19.5630	−1.41553	−0.707765	−	0.706448i	$-0.750296\pi$
$191$	−19.5630	−1.41553	−0.707765	+	0.706448i	$0.750296\pi$
$192$	0	0
$193$	5.22841	0.376349	0.188175	−	0.982136i	$-0.439743\pi$
$193$	5.22841	0.376349	0.188175	+	0.982136i	$0.439743\pi$
$194$	0	0
$195$	14.9601	1.07131
$196$	0	0
$197$	20.0126	1.42584	0.712918	−	0.701248i	$-0.247373\pi$
$197$	20.0126	1.42584	0.712918	+	0.701248i	$0.247373\pi$
$198$	0	0
$199$	−5.27503	−0.373937	−0.186969	−	0.982366i	$-0.559866\pi$
$199$	−5.27503	−0.373937	−0.186969	+	0.982366i	$0.559866\pi$
$200$	0	0
$201$	−15.0495	−1.06151
$202$	0	0
$203$	−20.2052	−1.41813
$204$	0	0
$205$	7.49419	0.523417
$206$	0	0
$207$	−36.8326	−2.56004
$208$	0	0
$209$	−39.4348	−2.72776
$210$	0	0
$211$	−5.73837	−0.395046	−0.197523	−	0.980298i	$-0.563290\pi$
$211$	−5.73837	−0.395046	−0.197523	+	0.980298i	$0.563290\pi$
$212$	0	0
$213$	39.1378	2.68168
$214$	0	0
$215$	20.5260	1.39986
$216$	0	0
$217$	37.5670	2.55021
$218$	0	0
$219$	40.2877	2.72239
$220$	0	0
$221$	7.67609	0.516350
$222$	0	0
$223$	−10.0786	−0.674911	−0.337455	−	0.941341i	$-0.609566\pi$
$223$	−10.0786	−0.674911	−0.337455	+	0.941341i	$0.609566\pi$
$224$	0	0
$225$	0.0388882	0.00259254
$226$	0	0
$227$	−14.5279	−0.964251	−0.482126	−	0.876102i	$-0.660135\pi$
$227$	−14.5279	−0.964251	−0.482126	+	0.876102i	$0.660135\pi$
$228$	0	0
$229$	4.75361	0.314128	0.157064	−	0.987588i	$-0.449797\pi$
$229$	4.75361	0.314128	0.157064	+	0.987588i	$0.449797\pi$
$230$	0	0
$231$	−74.8490	−4.92470
$232$	0	0
$233$	−13.3923	−0.877356	−0.438678	−	0.898644i	$-0.644553\pi$
$233$	−13.3923	−0.877356	−0.438678	+	0.898644i	$0.644553\pi$
$234$	0	0
$235$	−24.6266	−1.60646
$236$	0	0
$237$	−7.18512	−0.466724
$238$	0	0
$239$	−19.4699	−1.25941	−0.629703	−	0.776836i	$-0.716823\pi$
$239$	−19.4699	−1.25941	−0.629703	+	0.776836i	$0.716823\pi$
$240$	0	0
$241$	17.8686	1.15102	0.575510	−	0.817795i	$-0.304804\pi$
$241$	17.8686	1.15102	0.575510	+	0.817795i	$0.304804\pi$
$242$	0	0
$243$	2.96399	0.190140
$244$	0	0
$245$	−34.8799	−2.22839
$246$	0	0
$247$	−16.7416	−1.06524
$248$	0	0
$249$	20.4346	1.29499
$250$	0	0
$251$	9.69639	0.612031	0.306015	−	0.952027i	$-0.401004\pi$
$251$	9.69639	0.612031	0.306015	+	0.952027i	$0.401004\pi$
$252$	0	0
$253$	−33.4919	−2.10562
$254$	0	0
$255$	22.6809	1.42033
$256$	0	0
$257$	−14.0648	−0.877340	−0.438670	−	0.898648i	$-0.644550\pi$
$257$	−14.0648	−0.877340	−0.438670	+	0.898648i	$0.644550\pi$
$258$	0	0
$259$	−33.4586	−2.07902
$260$	0	0
$261$	−24.7802	−1.53385
$262$	0	0
$263$	−25.5798	−1.57732	−0.788658	−	0.614832i	$-0.789224\pi$
$263$	−25.5798	−1.57732	−0.788658	+	0.614832i	$0.789224\pi$
$264$	0	0
$265$	−1.43200	−0.0879668
$266$	0	0
$267$	−39.6838	−2.42861
$268$	0	0
$269$	0.310081	0.0189060	0.00945298	−	0.999955i	$-0.496991\pi$
$269$	0.310081	0.0189060	0.00945298	+	0.999955i	$0.496991\pi$
$270$	0	0
$271$	−14.4218	−0.876061	−0.438031	−	0.898960i	$-0.644324\pi$
$271$	−14.4218	−0.876061	−0.438031	+	0.898960i	$0.644324\pi$
$272$	0	0
$273$	−31.7762	−1.92318
$274$	0	0
$275$	0.0353610	0.00213235
$276$	0	0
$277$	23.7727	1.42836	0.714182	−	0.699960i	$-0.246799\pi$
$277$	23.7727	1.42836	0.714182	+	0.699960i	$0.246799\pi$
$278$	0	0
$279$	46.0732	2.75833
$280$	0	0
$281$	5.96671	0.355944	0.177972	−	0.984036i	$-0.443046\pi$
$281$	5.96671	0.355944	0.177972	+	0.984036i	$0.443046\pi$
$282$	0	0
$283$	−20.4119	−1.21336	−0.606682	−	0.794945i	$-0.707500\pi$
$283$	−20.4119	−1.21336	−0.606682	+	0.794945i	$0.707500\pi$
$284$	0	0
$285$	−49.4670	−2.93017
$286$	0	0
$287$	−15.9182	−0.939619
$288$	0	0
$289$	−5.36233	−0.315431
$290$	0	0
$291$	48.5453	2.84577
$292$	0	0
$293$	1.62410	0.0948810	0.0474405	−	0.998874i	$-0.484894\pi$
$293$	1.62410	0.0948810	0.0474405	+	0.998874i	$0.484894\pi$
$294$	0	0
$295$	24.6581	1.43565
$296$	0	0
$297$	−44.5508	−2.58510
$298$	0	0
$299$	−14.2186	−0.822280
$300$	0	0
$301$	−43.5985	−2.51297
$302$	0	0
$303$	11.8371	0.680022
$304$	0	0
$305$	14.8217	0.848685
$306$	0	0
$307$	28.9713	1.65348	0.826740	−	0.562584i	$-0.190193\pi$
$307$	28.9713	1.65348	0.826740	+	0.562584i	$0.190193\pi$
$308$	0	0
$309$	−0.263692	−0.0150009
$310$	0	0
$311$	−13.7770	−0.781221	−0.390611	−	0.920556i	$-0.627736\pi$
$311$	−13.7770	−0.781221	−0.390611	+	0.920556i	$0.627736\pi$
$312$	0	0
$313$	11.1145	0.628227	0.314114	−	0.949385i	$-0.398293\pi$
$313$	11.1145	0.628227	0.314114	+	0.949385i	$0.398293\pi$
$314$	0	0
$315$	−61.9870	−3.49257
$316$	0	0
$317$	−3.90744	−0.219463	−0.109732	−	0.993961i	$-0.534999\pi$
$317$	−3.90744	−0.219463	−0.109732	+	0.993961i	$0.534999\pi$
$318$	0	0
$319$	−22.5326	−1.26158
$320$	0	0
$321$	−29.4125	−1.64165
$322$	0	0
$323$	−25.3818	−1.41228
$324$	0	0
$325$	0.0150121	0.000832720 0
$326$	0	0
$327$	18.8599	1.04295
$328$	0	0
$329$	52.3085	2.88386
$330$	0	0
$331$	−22.7863	−1.25245	−0.626223	−	0.779644i	$-0.715400\pi$
$331$	−22.7863	−1.25245	−0.626223	+	0.779644i	$0.715400\pi$
$332$	0	0
$333$	−41.0346	−2.24868
$334$	0	0
$335$	−11.3329	−0.619185
$336$	0	0
$337$	15.9828	0.870639	0.435319	−	0.900276i	$-0.356635\pi$
$337$	15.9828	0.870639	0.435319	+	0.900276i	$0.356635\pi$
$338$	0	0
$339$	2.71298	0.147349
$340$	0	0
$341$	41.8943	2.26871
$342$	0	0
$343$	40.8181	2.20397
$344$	0	0
$345$	−42.0122	−2.26186
$346$	0	0
$347$	−1.59211	−0.0854690	−0.0427345	−	0.999086i	$-0.513607\pi$
$347$	−1.59211	−0.0854690	−0.0427345	+	0.999086i	$0.513607\pi$
$348$	0	0
$349$	−20.4387	−1.09406	−0.547028	−	0.837114i	$-0.684241\pi$
$349$	−20.4387	−1.09406	−0.547028	+	0.837114i	$0.684241\pi$
$350$	0	0
$351$	−18.9135	−1.00953
$352$	0	0
$353$	13.8440	0.736842	0.368421	−	0.929659i	$-0.379899\pi$
$353$	13.8440	0.736842	0.368421	+	0.929659i	$0.379899\pi$
$354$	0	0
$355$	29.4726	1.56424
$356$	0	0
$357$	−48.1757	−2.54973
$358$	0	0
$359$	1.20155	0.0634156	0.0317078	−	0.999497i	$-0.489905\pi$
$359$	1.20155	0.0634156	0.0317078	+	0.999497i	$0.489905\pi$
$360$	0	0
$361$	36.3577	1.91356
$362$	0	0
$363$	−50.7861	−2.66558
$364$	0	0
$365$	30.3385	1.58799
$366$	0	0
$367$	−11.2536	−0.587432	−0.293716	−	0.955893i	$-0.594892\pi$
$367$	−11.2536	−0.587432	−0.293716	+	0.955893i	$0.594892\pi$
$368$	0	0
$369$	−19.5224	−1.01630
$370$	0	0
$371$	3.04165	0.157915
$372$	0	0
$373$	−14.1674	−0.733562	−0.366781	−	0.930307i	$-0.619540\pi$
$373$	−14.1674	−0.733562	−0.366781	+	0.930307i	$0.619540\pi$
$374$	0	0
$375$	−33.1984	−1.71436
$376$	0	0
$377$	−9.56593	−0.492671
$378$	0	0
$379$	−36.8333	−1.89200	−0.946000	−	0.324166i	$-0.894916\pi$
$379$	−36.8333	−1.89200	−0.946000	+	0.324166i	$0.894916\pi$
$380$	0	0
$381$	−1.59617	−0.0817743
$382$	0	0
$383$	17.6924	0.904040	0.452020	−	0.892008i	$-0.350704\pi$
$383$	17.6924	0.904040	0.452020	+	0.892008i	$0.350704\pi$
$384$	0	0
$385$	−56.3648	−2.87262
$386$	0	0
$387$	−53.4703	−2.71805
$388$	0	0
$389$	10.5813	0.536491	0.268245	−	0.963351i	$-0.413556\pi$
$389$	10.5813	0.536491	0.268245	+	0.963351i	$0.413556\pi$
$390$	0	0
$391$	−21.5567	−1.09017
$392$	0	0
$393$	−57.6417	−2.90764
$394$	0	0
$395$	−5.41073	−0.272244
$396$	0	0
$397$	−15.2708	−0.766418	−0.383209	−	0.923662i	$-0.625181\pi$
$397$	−15.2708	−0.766418	−0.383209	+	0.923662i	$0.625181\pi$
$398$	0	0
$399$	105.071	5.26014
$400$	0	0
$401$	36.4092	1.81819	0.909094	−	0.416592i	$-0.136776\pi$
$401$	36.4092	1.81819	0.909094	+	0.416592i	$0.136776\pi$
$402$	0	0
$403$	17.7857	0.885970
$404$	0	0
$405$	−16.7572	−0.832673
$406$	0	0
$407$	−37.3127	−1.84952
$408$	0	0
$409$	31.0367	1.53466	0.767332	−	0.641250i	$-0.221584\pi$
$409$	31.0367	1.53466	0.767332	+	0.641250i	$0.221584\pi$
$410$	0	0
$411$	49.8561	2.45922
$412$	0	0
$413$	−52.3753	−2.57722
$414$	0	0
$415$	15.3882	0.755378
$416$	0	0
$417$	43.7141	2.14069
$418$	0	0
$419$	21.0972	1.03067	0.515333	−	0.856990i	$-0.327668\pi$
$419$	21.0972	1.03067	0.515333	+	0.856990i	$0.327668\pi$
$420$	0	0
$421$	30.9947	1.51059	0.755294	−	0.655386i	$-0.227494\pi$
$421$	30.9947	1.51059	0.755294	+	0.655386i	$0.227494\pi$
$422$	0	0
$423$	64.1525	3.11920
$424$	0	0
$425$	0.0227597	0.00110401
$426$	0	0
$427$	−31.4822	−1.52353
$428$	0	0
$429$	−35.4365	−1.71089
$430$	0	0
$431$	27.3184	1.31588	0.657940	−	0.753070i	$-0.271428\pi$
$431$	27.3184	1.31588	0.657940	+	0.753070i	$0.271428\pi$
$432$	0	0
$433$	9.92601	0.477014	0.238507	−	0.971141i	$-0.423342\pi$
$433$	9.92601	0.477014	0.238507	+	0.971141i	$0.423342\pi$
$434$	0	0
$435$	−28.2649	−1.35520
$436$	0	0
$437$	47.0151	2.24904
$438$	0	0
$439$	3.77575	0.180207	0.0901035	−	0.995932i	$-0.471280\pi$
$439$	3.77575	0.180207	0.0901035	+	0.995932i	$0.471280\pi$
$440$	0	0
$441$	90.8624	4.32678
$442$	0	0
$443$	27.7445	1.31818	0.659091	−	0.752063i	$-0.270941\pi$
$443$	27.7445	1.31818	0.659091	+	0.752063i	$0.270941\pi$
$444$	0	0
$445$	−29.8838	−1.41663
$446$	0	0
$447$	−28.3472	−1.34078
$448$	0	0
$449$	−24.2622	−1.14501	−0.572503	−	0.819903i	$-0.694027\pi$
$449$	−24.2622	−1.14501	−0.572503	+	0.819903i	$0.694027\pi$
$450$	0	0
$451$	−17.7518	−0.835898
$452$	0	0
$453$	14.0397	0.659644
$454$	0	0
$455$	−23.9290	−1.12181
$456$	0	0
$457$	2.19277	0.102573	0.0512866	−	0.998684i	$-0.483668\pi$
$457$	2.19277	0.102573	0.0512866	+	0.998684i	$0.483668\pi$
$458$	0	0
$459$	−28.6746	−1.33842
$460$	0	0
$461$	−10.4814	−0.488170	−0.244085	−	0.969754i	$-0.578488\pi$
$461$	−10.4814	−0.488170	−0.244085	+	0.969754i	$0.578488\pi$
$462$	0	0
$463$	−19.6361	−0.912568	−0.456284	−	0.889834i	$-0.650820\pi$
$463$	−19.6361	−0.912568	−0.456284	+	0.889834i	$0.650820\pi$
$464$	0	0
$465$	52.5522	2.43705
$466$	0	0
$467$	−22.0177	−1.01886	−0.509429	−	0.860513i	$-0.670143\pi$
$467$	−22.0177	−1.01886	−0.509429	+	0.860513i	$0.670143\pi$
$468$	0	0
$469$	24.0719	1.11154
$470$	0	0
$471$	12.7341	0.586758
$472$	0	0
$473$	−48.6206	−2.23558
$474$	0	0
$475$	−0.0496389	−0.00227759
$476$	0	0
$477$	3.73037	0.170802
$478$	0	0
$479$	−8.59554	−0.392740	−0.196370	−	0.980530i	$-0.562915\pi$
$479$	−8.59554	−0.392740	−0.196370	+	0.980530i	$0.562915\pi$
$480$	0	0
$481$	−15.8407	−0.722272
$482$	0	0
$483$	89.2366	4.06041
$484$	0	0
$485$	36.5569	1.65996
$486$	0	0
$487$	−29.7448	−1.34786	−0.673932	−	0.738794i	$-0.735396\pi$
$487$	−29.7448	−1.34786	−0.673932	+	0.738794i	$0.735396\pi$
$488$	0	0
$489$	46.6504	2.10960
$490$	0	0
$491$	24.2771	1.09561	0.547805	−	0.836606i	$-0.315464\pi$
$491$	24.2771	1.09561	0.547805	+	0.836606i	$0.315464\pi$
$492$	0	0
$493$	−14.5028	−0.653175
$494$	0	0
$495$	−69.1273	−3.10704
$496$	0	0
$497$	−62.6017	−2.80807
$498$	0	0
$499$	−43.1098	−1.92986	−0.964930	−	0.262508i	$-0.915450\pi$
$499$	−43.1098	−1.92986	−0.964930	+	0.262508i	$0.915450\pi$
$500$	0	0
$501$	−43.3257	−1.93565
$502$	0	0
$503$	−7.92336	−0.353285	−0.176643	−	0.984275i	$-0.556524\pi$
$503$	−7.92336	−0.353285	−0.176643	+	0.984275i	$0.556524\pi$
$504$	0	0
$505$	8.91387	0.396662
$506$	0	0
$507$	23.5833	1.04737
$508$	0	0
$509$	8.78132	0.389225	0.194613	−	0.980880i	$-0.437655\pi$
$509$	8.78132	0.389225	0.194613	+	0.980880i	$0.437655\pi$
$510$	0	0
$511$	−64.4410	−2.85070
$512$	0	0
$513$	62.5394	2.76118
$514$	0	0
$515$	−0.198572	−0.00875014
$516$	0	0
$517$	58.3339	2.56552
$518$	0	0
$519$	−75.2849	−3.30464
$520$	0	0
$521$	4.80563	0.210539	0.105269	−	0.994444i	$-0.466430\pi$
$521$	4.80563	0.210539	0.105269	+	0.994444i	$0.466430\pi$
$522$	0	0
$523$	31.4510	1.37526	0.687629	−	0.726063i	$-0.258652\pi$
$523$	31.4510	1.37526	0.687629	+	0.726063i	$0.258652\pi$
$524$	0	0
$525$	−0.0942168	−0.00411196
$526$	0	0
$527$	26.9648	1.17461
$528$	0	0
$529$	16.9298	0.736077
$530$	0	0
$531$	−64.2345	−2.78754
$532$	0	0
$533$	−7.53629	−0.326433
$534$	0	0
$535$	−22.1490	−0.957586
$536$	0	0
$537$	59.3935	2.56302
$538$	0	0
$539$	82.6212	3.55875
$540$	0	0
$541$	31.0122	1.33332	0.666659	−	0.745363i	$-0.267724\pi$
$541$	31.0122	1.33332	0.666659	+	0.745363i	$0.267724\pi$
$542$	0	0
$543$	−5.76194	−0.247268
$544$	0	0
$545$	14.2024	0.608363
$546$	0	0
$547$	6.64375	0.284066	0.142033	−	0.989862i	$-0.454636\pi$
$547$	6.64375	0.284066	0.142033	+	0.989862i	$0.454636\pi$
$548$	0	0
$549$	−38.6106	−1.64786
$550$	0	0
$551$	31.6307	1.34751
$552$	0	0
$553$	11.4928	0.488722
$554$	0	0
$555$	−46.8051	−1.98677
$556$	0	0
$557$	−27.7003	−1.17370	−0.586851	−	0.809695i	$-0.699632\pi$
$557$	−27.7003	−1.17370	−0.586851	+	0.809695i	$0.699632\pi$
$558$	0	0
$559$	−20.6412	−0.873032
$560$	0	0
$561$	−53.7250	−2.26827
$562$	0	0
$563$	−11.9687	−0.504420	−0.252210	−	0.967673i	$-0.581157\pi$
$563$	−11.9687	−0.504420	−0.252210	+	0.967673i	$0.581157\pi$
$564$	0	0
$565$	2.04300	0.0859497
$566$	0	0
$567$	35.5934	1.49478
$568$	0	0
$569$	11.6301	0.487560	0.243780	−	0.969831i	$-0.421612\pi$
$569$	11.6301	0.487560	0.243780	+	0.969831i	$0.421612\pi$
$570$	0	0
$571$	−24.3831	−1.02040	−0.510200	−	0.860056i	$-0.670428\pi$
$571$	−24.3831	−1.02040	−0.510200	+	0.860056i	$0.670428\pi$
$572$	0	0
$573$	58.1284	2.42835
$574$	0	0
$575$	−0.0421582	−0.00175812
$576$	0	0
$577$	2.35877	0.0981967	0.0490984	−	0.998794i	$-0.484365\pi$
$577$	2.35877	0.0981967	0.0490984	+	0.998794i	$0.484365\pi$
$578$	0	0
$579$	−15.5354	−0.645628
$580$	0	0
$581$	−32.6856	−1.35603
$582$	0	0
$583$	3.39202	0.140483
$584$	0	0
$585$	−29.3471	−1.21335
$586$	0	0
$587$	−25.1638	−1.03862	−0.519310	−	0.854586i	$-0.673811\pi$
$587$	−25.1638	−1.03862	−0.519310	+	0.854586i	$0.673811\pi$
$588$	0	0
$589$	−58.8103	−2.42323
$590$	0	0
$591$	−59.4641	−2.44603
$592$	0	0
$593$	−19.9929	−0.821011	−0.410506	−	0.911858i	$-0.634648\pi$
$593$	−19.9929	−0.821011	−0.410506	+	0.911858i	$0.634648\pi$
$594$	0	0
$595$	−36.2786	−1.48728
$596$	0	0
$597$	15.6739	0.641491
$598$	0	0
$599$	5.14337	0.210153	0.105076	−	0.994464i	$-0.466491\pi$
$599$	5.14337	0.210153	0.105076	+	0.994464i	$0.466491\pi$
$600$	0	0
$601$	−16.9013	−0.689417	−0.344708	−	0.938710i	$-0.612022\pi$
$601$	−16.9013	−0.689417	−0.344708	+	0.938710i	$0.612022\pi$
$602$	0	0
$603$	29.5224	1.20225
$604$	0	0
$605$	−38.2443	−1.55485
$606$	0	0
$607$	−7.59083	−0.308102	−0.154051	−	0.988063i	$-0.549232\pi$
$607$	−7.59083	−0.308102	−0.154051	+	0.988063i	$0.549232\pi$
$608$	0	0
$609$	60.0365	2.43280
$610$	0	0
$611$	24.7649	1.00188
$612$	0	0
$613$	29.5597	1.19390	0.596952	−	0.802277i	$-0.296378\pi$
$613$	29.5597	1.19390	0.596952	+	0.802277i	$0.296378\pi$
$614$	0	0
$615$	−22.2678	−0.897924
$616$	0	0
$617$	40.3067	1.62269	0.811343	−	0.584571i	$-0.198737\pi$
$617$	40.3067	1.62269	0.811343	+	0.584571i	$0.198737\pi$
$618$	0	0
$619$	9.79001	0.393494	0.196747	−	0.980454i	$-0.436962\pi$
$619$	9.79001	0.393494	0.196747	+	0.980454i	$0.436962\pi$
$620$	0	0
$621$	53.1145	2.13141
$622$	0	0
$623$	63.4751	2.54308
$624$	0	0
$625$	−25.0333	−1.00133
$626$	0	0
$627$	117.174	4.67949
$628$	0	0
$629$	−24.0159	−0.957577
$630$	0	0
$631$	0.612815	0.0243958	0.0121979	−	0.999926i	$-0.496117\pi$
$631$	0.612815	0.0243958	0.0121979	+	0.999926i	$0.496117\pi$
$632$	0	0
$633$	17.0506	0.677702
$634$	0	0
$635$	−1.20199	−0.0476996
$636$	0	0
$637$	35.0758	1.38975
$638$	0	0
$639$	−76.7764	−3.03723
$640$	0	0
$641$	−19.5220	−0.771072	−0.385536	−	0.922693i	$-0.625983\pi$
$641$	−19.5220	−0.771072	−0.385536	+	0.922693i	$0.625983\pi$
$642$	0	0
$643$	46.8670	1.84826	0.924128	−	0.382084i	$-0.124793\pi$
$643$	46.8670	1.84826	0.924128	+	0.382084i	$0.124793\pi$
$644$	0	0
$645$	−60.9896	−2.40146
$646$	0	0
$647$	20.6498	0.811826	0.405913	−	0.913912i	$-0.366954\pi$
$647$	20.6498	0.811826	0.405913	+	0.913912i	$0.366954\pi$
$648$	0	0
$649$	−58.4084	−2.29273
$650$	0	0
$651$	−111.624	−4.37490
$652$	0	0
$653$	−39.7852	−1.55692	−0.778458	−	0.627697i	$-0.783998\pi$
$653$	−39.7852	−1.55692	−0.778458	+	0.627697i	$0.783998\pi$
$654$	0	0
$655$	−43.4069	−1.69605
$656$	0	0
$657$	−79.0322	−3.08334
$658$	0	0
$659$	−15.1726	−0.591041	−0.295521	−	0.955336i	$-0.595493\pi$
$659$	−15.1726	−0.591041	−0.295521	+	0.955336i	$0.595493\pi$
$660$	0	0
$661$	−6.17218	−0.240070	−0.120035	−	0.992770i	$-0.538301\pi$
$661$	−6.17218	−0.240070	−0.120035	+	0.992770i	$0.538301\pi$
$662$	0	0
$663$	−22.8083	−0.885800
$664$	0	0
$665$	79.1236	3.06828
$666$	0	0
$667$	26.8639	1.04017
$668$	0	0
$669$	29.9469	1.15781
$670$	0	0
$671$	−35.1086	−1.35535
$672$	0	0
$673$	−29.0013	−1.11792	−0.558959	−	0.829195i	$-0.688799\pi$
$673$	−29.0013	−1.11792	−0.558959	+	0.829195i	$0.688799\pi$
$674$	0	0
$675$	−0.0560787	−0.00215847
$676$	0	0
$677$	−8.97850	−0.345072	−0.172536	−	0.985003i	$-0.555196\pi$
$677$	−8.97850	−0.345072	−0.172536	+	0.985003i	$0.555196\pi$
$678$	0	0
$679$	−77.6492	−2.97990
$680$	0	0
$681$	43.1674	1.65418
$682$	0	0
$683$	−1.05659	−0.0404294	−0.0202147	−	0.999796i	$-0.506435\pi$
$683$	−1.05659	−0.0404294	−0.0202147	+	0.999796i	$0.506435\pi$
$684$	0	0
$685$	37.5440	1.43448
$686$	0	0
$687$	−14.1246	−0.538887
$688$	0	0
$689$	1.44004	0.0548611
$690$	0	0
$691$	36.2160	1.37772	0.688861	−	0.724893i	$-0.258111\pi$
$691$	36.2160	1.37772	0.688861	+	0.724893i	$0.258111\pi$
$692$	0	0
$693$	146.831	5.57764
$694$	0	0
$695$	32.9188	1.24868
$696$	0	0
$697$	−11.4257	−0.432780
$698$	0	0
$699$	39.7930	1.50511
$700$	0	0
$701$	8.08068	0.305203	0.152602	−	0.988288i	$-0.451235\pi$
$701$	8.08068	0.305203	0.152602	+	0.988288i	$0.451235\pi$
$702$	0	0
$703$	52.3787	1.97550
$704$	0	0
$705$	73.1740	2.75589
$706$	0	0
$707$	−18.9336	−0.712073
$708$	0	0
$709$	7.03033	0.264030	0.132015	−	0.991248i	$-0.457855\pi$
$709$	7.03033	0.264030	0.132015	+	0.991248i	$0.457855\pi$
$710$	0	0
$711$	14.0950	0.528604
$712$	0	0
$713$	−49.9474	−1.87054
$714$	0	0
$715$	−26.6853	−0.997975
$716$	0	0
$717$	57.8518	2.16051
$718$	0	0
$719$	−3.20900	−0.119675	−0.0598377	−	0.998208i	$-0.519058\pi$
$719$	−3.20900	−0.119675	−0.0598377	+	0.998208i	$0.519058\pi$
$720$	0	0
$721$	0.421781	0.0157079
$722$	0	0
$723$	−53.0938	−1.97458
$724$	0	0
$725$	−0.0283631	−0.00105338
$726$	0	0
$727$	−40.1606	−1.48948	−0.744738	−	0.667357i	$-0.767426\pi$
$727$	−40.1606	−1.48948	−0.744738	+	0.667357i	$0.767426\pi$
$728$	0	0
$729$	−31.2742	−1.15830
$730$	0	0
$731$	−31.2941	−1.15745
$732$	0	0
$733$	−16.1707	−0.597277	−0.298639	−	0.954366i	$-0.596533\pi$
$733$	−16.1707	−0.597277	−0.298639	+	0.954366i	$0.596533\pi$
$734$	0	0
$735$	103.640	3.82282
$736$	0	0
$737$	26.8447	0.988839
$738$	0	0
$739$	−44.3862	−1.63277	−0.816386	−	0.577507i	$-0.804026\pi$
$739$	−44.3862	−1.63277	−0.816386	+	0.577507i	$0.804026\pi$
$740$	0	0
$741$	49.7449	1.82742
$742$	0	0
$743$	37.0009	1.35743	0.678715	−	0.734402i	$-0.262537\pi$
$743$	37.0009	1.35743	0.678715	+	0.734402i	$0.262537\pi$
$744$	0	0
$745$	−21.3468	−0.782085
$746$	0	0
$747$	−40.0865	−1.46669
$748$	0	0
$749$	47.0460	1.71902
$750$	0	0
$751$	−49.3330	−1.80019	−0.900094	−	0.435697i	$-0.856502\pi$
$751$	−49.3330	−1.80019	−0.900094	+	0.435697i	$0.856502\pi$
$752$	0	0
$753$	−28.8113	−1.04994
$754$	0	0
$755$	10.5726	0.384775
$756$	0	0
$757$	7.19293	0.261432	0.130716	−	0.991420i	$-0.458272\pi$
$757$	7.19293	0.261432	0.130716	+	0.991420i	$0.458272\pi$
$758$	0	0
$759$	99.5158	3.61219
$760$	0	0
$761$	−20.9531	−0.759549	−0.379774	−	0.925079i	$-0.623998\pi$
$761$	−20.9531	−0.759549	−0.379774	+	0.925079i	$0.623998\pi$
$762$	0	0
$763$	−30.1668	−1.09211
$764$	0	0
$765$	−44.4930	−1.60865
$766$	0	0
$767$	−24.7966	−0.895352
$768$	0	0
$769$	3.89319	0.140392	0.0701960	−	0.997533i	$-0.477638\pi$
$769$	3.89319	0.140392	0.0701960	+	0.997533i	$0.477638\pi$
$770$	0	0
$771$	41.7914	1.50508
$772$	0	0
$773$	3.46372	0.124581	0.0622907	−	0.998058i	$-0.480159\pi$
$773$	3.46372	0.124581	0.0622907	+	0.998058i	$0.480159\pi$
$774$	0	0
$775$	0.0527349	0.00189429
$776$	0	0
$777$	99.4170	3.56657
$778$	0	0
$779$	24.9195	0.892833
$780$	0	0
$781$	−69.8128	−2.49810
$782$	0	0
$783$	35.7343	1.27704
$784$	0	0
$785$	9.58940	0.342261
$786$	0	0
$787$	23.9621	0.854158	0.427079	−	0.904214i	$-0.359543\pi$
$787$	23.9621	0.854158	0.427079	+	0.904214i	$0.359543\pi$
$788$	0	0
$789$	76.0062	2.70589
$790$	0	0
$791$	−4.33947	−0.154294
$792$	0	0
$793$	−14.9049	−0.529289
$794$	0	0
$795$	4.25495	0.150907
$796$	0	0
$797$	33.3948	1.18290	0.591452	−	0.806340i	$-0.298555\pi$
$797$	33.3948	1.18290	0.591452	+	0.806340i	$0.298555\pi$
$798$	0	0
$799$	37.5459	1.32828
$800$	0	0
$801$	77.8476	2.75061
$802$	0	0
$803$	−71.8640	−2.53602
$804$	0	0
$805$	67.1994	2.36847
$806$	0	0
$807$	−0.921356	−0.0324333
$808$	0	0
$809$	−13.6118	−0.478564	−0.239282	−	0.970950i	$-0.576912\pi$
$809$	−13.6118	−0.478564	−0.239282	+	0.970950i	$0.576912\pi$
$810$	0	0
$811$	−28.4580	−0.999297	−0.499648	−	0.866228i	$-0.666537\pi$
$811$	−28.4580	−0.999297	−0.499648	+	0.866228i	$0.666537\pi$
$812$	0	0
$813$	42.8520	1.50289
$814$	0	0
$815$	35.1299	1.23055
$816$	0	0
$817$	68.2524	2.38785
$818$	0	0
$819$	62.3352	2.17817
$820$	0	0
$821$	−26.4829	−0.924261	−0.462131	−	0.886812i	$-0.652915\pi$
$821$	−26.4829	−0.924261	−0.462131	+	0.886812i	$0.652915\pi$
$822$	0	0
$823$	−37.3579	−1.30221	−0.651107	−	0.758986i	$-0.725695\pi$
$823$	−37.3579	−1.30221	−0.651107	+	0.758986i	$0.725695\pi$
$824$	0	0
$825$	−0.105070	−0.00365805
$826$	0	0
$827$	3.12884	0.108800	0.0544002	−	0.998519i	$-0.482675\pi$
$827$	3.12884	0.108800	0.0544002	+	0.998519i	$0.482675\pi$
$828$	0	0
$829$	3.64201	0.126492	0.0632461	−	0.997998i	$-0.479855\pi$
$829$	3.64201	0.126492	0.0632461	+	0.997998i	$0.479855\pi$
$830$	0	0
$831$	−70.6369	−2.45037
$832$	0	0
$833$	53.1782	1.84251
$834$	0	0
$835$	−32.6263	−1.12908
$836$	0	0
$837$	−66.4399	−2.29650
$838$	0	0
$839$	29.1357	1.00587	0.502937	−	0.864323i	$-0.332253\pi$
$839$	29.1357	1.00587	0.502937	+	0.864323i	$0.332253\pi$
$840$	0	0
$841$	−10.9266	−0.376778
$842$	0	0
$843$	−17.7291	−0.610624
$844$	0	0
$845$	17.7593	0.610940
$846$	0	0
$847$	81.2334	2.79121
$848$	0	0
$849$	60.6508	2.08153
$850$	0	0
$851$	44.4851	1.52493
$852$	0	0
$853$	−27.1373	−0.929164	−0.464582	−	0.885530i	$-0.653795\pi$
$853$	−27.1373	−0.929164	−0.464582	+	0.885530i	$0.653795\pi$
$854$	0	0
$855$	97.0392	3.31867
$856$	0	0
$857$	40.6208	1.38758	0.693790	−	0.720178i	$-0.255940\pi$
$857$	40.6208	1.38758	0.693790	+	0.720178i	$0.255940\pi$
$858$	0	0
$859$	−27.5070	−0.938527	−0.469264	−	0.883058i	$-0.655481\pi$
$859$	−27.5070	−0.938527	−0.469264	+	0.883058i	$0.655481\pi$
$860$	0	0
$861$	47.2983	1.61192
$862$	0	0
$863$	49.9215	1.69935	0.849673	−	0.527310i	$-0.176799\pi$
$863$	49.9215	1.69935	0.849673	+	0.527310i	$0.176799\pi$
$864$	0	0
$865$	−56.6931	−1.92762
$866$	0	0
$867$	15.9333	0.541123
$868$	0	0
$869$	12.8166	0.434773
$870$	0	0
$871$	11.3966	0.386159
$872$	0	0
$873$	−95.2310	−3.22308
$874$	0	0
$875$	53.1015	1.79516
$876$	0	0
$877$	4.23979	0.143168	0.0715838	−	0.997435i	$-0.477195\pi$
$877$	4.23979	0.143168	0.0715838	+	0.997435i	$0.477195\pi$
$878$	0	0
$879$	−4.82576	−0.162769
$880$	0	0
$881$	−17.4964	−0.589470	−0.294735	−	0.955579i	$-0.595231\pi$
$881$	−17.4964	−0.589470	−0.294735	+	0.955579i	$0.595231\pi$
$882$	0	0
$883$	6.67417	0.224604	0.112302	−	0.993674i	$-0.464178\pi$
$883$	6.67417	0.224604	0.112302	+	0.993674i	$0.464178\pi$
$884$	0	0
$885$	−73.2675	−2.46286
$886$	0	0
$887$	−7.87790	−0.264514	−0.132257	−	0.991215i	$-0.542222\pi$
$887$	−7.87790	−0.264514	−0.132257	+	0.991215i	$0.542222\pi$
$888$	0	0
$889$	2.55311	0.0856285
$890$	0	0
$891$	39.6934	1.32978
$892$	0	0
$893$	−81.8877	−2.74027
$894$	0	0
$895$	44.7261	1.49503
$896$	0	0
$897$	42.2482	1.41063
$898$	0	0
$899$	−33.6035	−1.12074
$900$	0	0
$901$	2.18323	0.0727341
$902$	0	0
$903$	129.546	4.31102
$904$	0	0
$905$	−4.33901	−0.144234
$906$	0	0
$907$	11.0079	0.365510	0.182755	−	0.983158i	$-0.441498\pi$
$907$	11.0079	0.365510	0.182755	+	0.983158i	$0.441498\pi$
$908$	0	0
$909$	−23.2207	−0.770183
$910$	0	0
$911$	19.6527	0.651122	0.325561	−	0.945521i	$-0.394447\pi$
$911$	19.6527	0.651122	0.325561	+	0.945521i	$0.394447\pi$
$912$	0	0
$913$	−36.4506	−1.20634
$914$	0	0
$915$	−44.0402	−1.45592
$916$	0	0
$917$	92.1992	3.04468
$918$	0	0
$919$	−44.8406	−1.47916	−0.739578	−	0.673071i	$-0.764975\pi$
$919$	−44.8406	−1.47916	−0.739578	+	0.673071i	$0.764975\pi$
$920$	0	0
$921$	−86.0836	−2.83655
$922$	0	0
$923$	−29.6381	−0.975551
$924$	0	0
$925$	−0.0469677	−0.00154429
$926$	0	0
$927$	0.517283	0.0169898
$928$	0	0
$929$	−30.8366	−1.01172	−0.505858	−	0.862617i	$-0.668824\pi$
$929$	−30.8366	−1.01172	−0.505858	+	0.862617i	$0.668824\pi$
$930$	0	0
$931$	−115.982	−3.80115
$932$	0	0
$933$	40.9361	1.34019
$934$	0	0
$935$	−40.4575	−1.32310
$936$	0	0
$937$	−1.37484	−0.0449140	−0.0224570	−	0.999748i	$-0.507149\pi$
$937$	−1.37484	−0.0449140	−0.0224570	+	0.999748i	$0.507149\pi$
$938$	0	0
$939$	−33.0249	−1.07773
$940$	0	0
$941$	19.7262	0.643055	0.321528	−	0.946900i	$-0.395804\pi$
$941$	19.7262	0.643055	0.321528	+	0.946900i	$0.395804\pi$
$942$	0	0
$943$	21.1640	0.689196
$944$	0	0
$945$	89.3885	2.90781
$946$	0	0
$947$	−0.497544	−0.0161680	−0.00808400	−	0.999967i	$-0.502573\pi$
$947$	−0.497544	−0.0161680	−0.00808400	+	0.999967i	$0.502573\pi$
$948$	0	0
$949$	−30.5089	−0.990362
$950$	0	0
$951$	11.6103	0.376490
$952$	0	0
$953$	49.9337	1.61751	0.808756	−	0.588144i	$-0.200141\pi$
$953$	49.9337	1.61751	0.808756	+	0.588144i	$0.200141\pi$
$954$	0	0
$955$	43.7734	1.41647
$956$	0	0
$957$	66.9520	2.16425
$958$	0	0
$959$	−79.7459	−2.57513
$960$	0	0
$961$	31.4782	1.01543
$962$	0	0
$963$	57.6984	1.85931
$964$	0	0
$965$	−11.6989	−0.376600
$966$	0	0
$967$	26.4280	0.849867	0.424934	−	0.905224i	$-0.360297\pi$
$967$	26.4280	0.849867	0.424934	+	0.905224i	$0.360297\pi$
$968$	0	0
$969$	75.4179	2.42277
$970$	0	0
$971$	−37.1681	−1.19278	−0.596391	−	0.802694i	$-0.703399\pi$
$971$	−37.1681	−1.19278	−0.596391	+	0.802694i	$0.703399\pi$
$972$	0	0
$973$	−69.9216	−2.24158
$974$	0	0
$975$	−0.0446060	−0.00142853
$976$	0	0
$977$	11.6497	0.372706	0.186353	−	0.982483i	$-0.440333\pi$
$977$	11.6497	0.372706	0.186353	+	0.982483i	$0.440333\pi$
$978$	0	0
$979$	70.7868	2.26236
$980$	0	0
$981$	−36.9974	−1.18123
$982$	0	0
$983$	7.17208	0.228754	0.114377	−	0.993437i	$-0.463513\pi$
$983$	7.17208	0.228754	0.114377	+	0.993437i	$0.463513\pi$
$984$	0	0
$985$	−44.7793	−1.42679
$986$	0	0
$987$	−155.426	−4.94727
$988$	0	0
$989$	57.9665	1.84323
$990$	0	0
$991$	5.09162	0.161741	0.0808704	−	0.996725i	$-0.474230\pi$
$991$	5.09162	0.161741	0.0808704	+	0.996725i	$0.474230\pi$
$992$	0	0
$993$	67.7058	2.14858
$994$	0	0
$995$	11.8032	0.374186
$996$	0	0
$997$	−10.0876	−0.319477	−0.159739	−	0.987159i	$-0.551065\pi$
$997$	−10.0876	−0.319477	−0.159739	+	0.987159i	$0.551065\pi$
$998$	0	0
$999$	59.1740	1.87218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.a.1.6	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.a.1.6	✓	63	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:	\( N \)	\(=\)	\( 6044 = 2^{2} \cdot 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6044.a (trivial)

\(f(q)\)	\(=\)	\(q-2.97134 q^{3} -2.23756 q^{5} +4.75272 q^{7} +5.82886 q^{9} +O(q^{10})\)	\(q-2.97134 q^{3} -2.23756 q^{5} +4.75272 q^{7} +5.82886 q^{9} +5.30019 q^{11} +2.25013 q^{13} +6.64855 q^{15} +3.41140 q^{17} -7.44027 q^{19} -14.1220 q^{21} -6.31900 q^{23} +0.00667165 q^{25} -8.40552 q^{27} -4.25129 q^{29} +7.90432 q^{31} -15.7487 q^{33} -10.6345 q^{35} -7.03989 q^{37} -6.68589 q^{39} -3.34927 q^{41} -9.17337 q^{43} -13.0424 q^{45} +11.0060 q^{47} +15.5884 q^{49} -10.1364 q^{51} +0.639981 q^{53} -11.8595 q^{55} +22.1076 q^{57} -11.0201 q^{59} -6.62403 q^{61} +27.7030 q^{63} -5.03479 q^{65} +5.06487 q^{67} +18.7759 q^{69} -13.1718 q^{71} -13.5588 q^{73} -0.0198238 q^{75} +25.1903 q^{77} +2.41814 q^{79} +7.48907 q^{81} -6.87723 q^{83} -7.63322 q^{85} +12.6320 q^{87} +13.3555 q^{89} +10.6942 q^{91} -23.4864 q^{93} +16.6481 q^{95} -16.3378 q^{97} +30.8941 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9}+O(q^{10}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9	\( 63 q - 7 q^{3} - 7 q^{5} - 22 q^{7} + 62 q^{9} - 21 q^{11} - 19 q^{13} - 30 q^{15} - 5 q^{17} - 59 q^{19} - 30 q^{21} - 24 q^{23} + 60 q^{25} - 34 q^{27} - 28 q^{29} - 48 q^{31} - q^{33} - 44 q^{35} - 29 q^{37} - 75 q^{39} - 3 q^{41} - 88 q^{43} - 21 q^{45} - 21 q^{47} + 63 q^{49} - 85 q^{51} - 24 q^{53} - 85 q^{55} - 35 q^{59} - 78 q^{61} - 74 q^{63} - 13 q^{65} - 68 q^{67} - 43 q^{69} - 59 q^{71} - q^{73} - 45 q^{75} - 33 q^{77} - 140 q^{79} + 51 q^{81} - 27 q^{83} - 84 q^{85} - 61 q^{87} - 2 q^{89} - 92 q^{91} - 51 q^{93} - 51 q^{95} - 10 q^{97} - 115 q^{99}+O(q^{100}) \) 63 * q - 7 * q^3 - 7 * q^5 - 22 * q^7 + 62 * q^9 - 21 * q^11 - 19 * q^13 - 30 * q^15 - 5 * q^17 - 59 * q^19 - 30 * q^21 - 24 * q^23 + 60 * q^25 - 34 * q^27 - 28 * q^29 - 48 * q^31 - q^33 - 44 * q^35 - 29 * q^37 - 75 * q^39 - 3 * q^41 - 88 * q^43 - 21 * q^45 - 21 * q^47 + 63 * q^49 - 85 * q^51 - 24 * q^53 - 85 * q^55 - 35 * q^59 - 78 * q^61 - 74 * q^63 - 13 * q^65 - 68 * q^67 - 43 * q^69 - 59 * q^71 - q^73 - 45 * q^75 - 33 * q^77 - 140 * q^79 + 51 * q^81 - 27 * q^83 - 84 * q^85 - 61 * q^87 - 2 * q^89 - 92 * q^91 - 51 * q^93 - 51 * q^95 - 10 * q^97 - 115 * q^99

Introduction

L-functions

Modular forms

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Database

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