LMFDB - Embedded newform 6021.2.a.t.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6021,2,Mod(1,6021)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6021, 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("6021.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6021 = 3^{3} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6021.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:	$48.0779270570$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	6021.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.16137 q^{2} +2.67154 q^{4} -3.69668 q^{5} +0.733541 q^{7} -1.45144 q^{8} +O(q^{10})$	\(q-2.16137 q^{2} +2.67154 q^{4} -3.69668 q^{5} +0.733541 q^{7} -1.45144 q^{8} +7.98990 q^{10} -3.53604 q^{11} +1.36940 q^{13} -1.58546 q^{14} -2.20597 q^{16} -5.69708 q^{17} +1.50153 q^{19} -9.87580 q^{20} +7.64271 q^{22} -0.915753 q^{23} +8.66541 q^{25} -2.95978 q^{26} +1.95968 q^{28} +7.92130 q^{29} -0.00306304 q^{31} +7.67080 q^{32} +12.3135 q^{34} -2.71166 q^{35} -1.45037 q^{37} -3.24536 q^{38} +5.36550 q^{40} -10.4542 q^{41} -11.1885 q^{43} -9.44666 q^{44} +1.97928 q^{46} +1.60345 q^{47} -6.46192 q^{49} -18.7292 q^{50} +3.65839 q^{52} +9.46155 q^{53} +13.0716 q^{55} -1.06469 q^{56} -17.1209 q^{58} -2.36330 q^{59} -12.8040 q^{61} +0.00662037 q^{62} -12.1675 q^{64} -5.06222 q^{65} -4.65542 q^{67} -15.2200 q^{68} +5.86092 q^{70} -9.19586 q^{71} -1.20173 q^{73} +3.13478 q^{74} +4.01138 q^{76} -2.59383 q^{77} +11.7563 q^{79} +8.15474 q^{80} +22.5953 q^{82} +1.22658 q^{83} +21.0603 q^{85} +24.1826 q^{86} +5.13235 q^{88} +13.1777 q^{89} +1.00451 q^{91} -2.44647 q^{92} -3.46565 q^{94} -5.55066 q^{95} -7.58863 q^{97} +13.9666 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 46 q^{4} + 16 q^{7}+O(q^{10}) $ 40 * q + 46 * q^4 + 16 * q^7	$ 40 q + 46 q^{4} + 16 q^{7} + 22 q^{10} + 14 q^{13} + 50 q^{16} + 64 q^{19} + 12 q^{22} + 40 q^{25} + 48 q^{28} + 54 q^{31} + 32 q^{34} + 24 q^{37} + 40 q^{40} + 24 q^{43} + 52 q^{46} + 64 q^{49} + 18 q^{52} + 36 q^{55} + 8 q^{58} + 58 q^{61} + 120 q^{64} + 52 q^{67} - 30 q^{70} + 50 q^{73} + 112 q^{76} + 60 q^{79} + 50 q^{82} + 38 q^{85} + 16 q^{88} + 118 q^{91} + 44 q^{94} + 38 q^{97}+O(q^{100}) $ 40 * q + 46 * q^4 + 16 * q^7 + 22 * q^10 + 14 * q^13 + 50 * q^16 + 64 * q^19 + 12 * q^22 + 40 * q^25 + 48 * q^28 + 54 * q^31 + 32 * q^34 + 24 * q^37 + 40 * q^40 + 24 * q^43 + 52 * q^46 + 64 * q^49 + 18 * q^52 + 36 * q^55 + 8 * q^58 + 58 * q^61 + 120 * q^64 + 52 * q^67 - 30 * q^70 + 50 * q^73 + 112 * q^76 + 60 * q^79 + 50 * q^82 + 38 * q^85 + 16 * q^88 + 118 * q^91 + 44 * q^94 + 38 * 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 $

$2$	−2.16137	−1.52832	−0.764161	−	0.645026i	$-0.776847\pi$
$2$	−2.16137	−1.52832	−0.764161	+	0.645026i	$0.776847\pi$
$3$	0	0
$3$	0	0
$4$	2.67154	1.33577
$5$	−3.69668	−1.65320	−0.826602	−	0.562787i	$-0.809729\pi$
$5$	−3.69668	−1.65320	−0.826602	+	0.562787i	$0.809729\pi$
$6$	0	0
$7$	0.733541	0.277253	0.138626	−	0.990345i	$-0.455731\pi$
$7$	0.733541	0.277253	0.138626	+	0.990345i	$0.455731\pi$
$8$	−1.45144	−0.513162
$9$	0	0
$10$	7.98990	2.52663
$11$	−3.53604	−1.06616	−0.533078	−	0.846066i	$-0.678965\pi$
$11$	−3.53604	−1.06616	−0.533078	+	0.846066i	$0.678965\pi$
$12$	0	0
$13$	1.36940	0.379802	0.189901	−	0.981803i	$-0.439183\pi$
$13$	1.36940	0.379802	0.189901	+	0.981803i	$0.439183\pi$
$14$	−1.58546	−0.423731
$15$	0	0
$16$	−2.20597	−0.551492
$17$	−5.69708	−1.38174	−0.690872	−	0.722977i	$-0.742773\pi$
$17$	−5.69708	−1.38174	−0.690872	+	0.722977i	$0.742773\pi$
$18$	0	0
$19$	1.50153	0.344474	0.172237	−	0.985056i	$-0.444901\pi$
$19$	1.50153	0.344474	0.172237	+	0.985056i	$0.444901\pi$
$20$	−9.87580	−2.20830
$21$	0	0
$22$	7.64271	1.62943
$23$	−0.915753	−0.190948	−0.0954739	−	0.995432i	$-0.530437\pi$
$23$	−0.915753	−0.190948	−0.0954739	+	0.995432i	$0.530437\pi$
$24$	0	0
$25$	8.66541	1.73308
$26$	−2.95978	−0.580460
$27$	0	0
$28$	1.95968	0.370345
$29$	7.92130	1.47095	0.735474	−	0.677553i	$-0.236960\pi$
$29$	7.92130	1.47095	0.735474	+	0.677553i	$0.236960\pi$
$30$	0	0
$31$	−0.00306304	−0.000550138 0	−0.000275069	−	1.00000i	$-0.500088\pi$
$31$	−0.00306304	−0.000550138 0	−0.000275069		1.00000i	$0.500088\pi$
$32$	7.67080	1.35602
$33$	0	0
$34$	12.3135	2.11175
$35$	−2.71166	−0.458355
$36$	0	0
$37$	−1.45037	−0.238439	−0.119219	−	0.992868i	$-0.538039\pi$
$37$	−1.45037	−0.238439	−0.119219	+	0.992868i	$0.538039\pi$
$38$	−3.24536	−0.526467
$39$	0	0
$40$	5.36550	0.848361
$41$	−10.4542	−1.63266	−0.816332	−	0.577582i	$-0.803996\pi$
$41$	−10.4542	−1.63266	−0.816332	+	0.577582i	$0.803996\pi$
$42$	0	0
$43$	−11.1885	−1.70624	−0.853118	−	0.521718i	$-0.825291\pi$
$43$	−11.1885	−1.70624	−0.853118	+	0.521718i	$0.825291\pi$
$44$	−9.44666	−1.42414
$45$	0	0
$46$	1.97928	0.291830
$47$	1.60345	0.233887	0.116943	−	0.993139i	$-0.462690\pi$
$47$	1.60345	0.233887	0.116943	+	0.993139i	$0.462690\pi$
$48$	0	0
$49$	−6.46192	−0.923131
$50$	−18.7292	−2.64871
$51$	0	0
$52$	3.65839	0.507328
$53$	9.46155	1.29964	0.649822	−	0.760087i	$-0.274843\pi$
$53$	9.46155	1.29964	0.649822	+	0.760087i	$0.274843\pi$
$54$	0	0
$55$	13.0716	1.76257
$56$	−1.06469	−0.142275
$57$	0	0
$58$	−17.1209	−2.24808
$59$	−2.36330	−0.307676	−0.153838	−	0.988096i	$-0.549163\pi$
$59$	−2.36330	−0.307676	−0.153838	+	0.988096i	$0.549163\pi$
$60$	0	0
$61$	−12.8040	−1.63939	−0.819693	−	0.572804i	$-0.805856\pi$
$61$	−12.8040	−1.63939	−0.819693	+	0.572804i	$0.805856\pi$
$62$	0.00662037	0.000840787 0
$63$	0	0
$64$	−12.1675	−1.52094
$65$	−5.06222	−0.627891
$66$	0	0
$67$	−4.65542	−0.568750	−0.284375	−	0.958713i	$-0.591786\pi$
$67$	−4.65542	−0.568750	−0.284375	+	0.958713i	$0.591786\pi$
$68$	−15.2200	−1.84569
$69$	0	0
$70$	5.86092	0.700514
$71$	−9.19586	−1.09135	−0.545674	−	0.837998i	$-0.683726\pi$
$71$	−9.19586	−1.09135	−0.545674	+	0.837998i	$0.683726\pi$
$72$	0	0
$73$	−1.20173	−0.140652	−0.0703262	−	0.997524i	$-0.522404\pi$
$73$	−1.20173	−0.140652	−0.0703262	+	0.997524i	$0.522404\pi$
$74$	3.13478	0.364411
$75$	0	0
$76$	4.01138	0.460137
$77$	−2.59383	−0.295595
$78$	0	0
$79$	11.7563	1.32268	0.661341	−	0.750085i	$-0.269987\pi$
$79$	11.7563	1.32268	0.661341	+	0.750085i	$0.269987\pi$
$80$	8.15474	0.911728
$81$	0	0
$82$	22.5953	2.49524
$83$	1.22658	0.134634	0.0673171	−	0.997732i	$-0.478556\pi$
$83$	1.22658	0.134634	0.0673171	+	0.997732i	$0.478556\pi$
$84$	0	0
$85$	21.0603	2.28431
$86$	24.1826	2.60768
$87$	0	0
$88$	5.13235	0.547111
$89$	13.1777	1.39683	0.698417	−	0.715691i	$-0.253888\pi$
$89$	13.1777	1.39683	0.698417	+	0.715691i	$0.253888\pi$
$90$	0	0
$91$	1.00451	0.105301
$92$	−2.44647	−0.255062
$93$	0	0
$94$	−3.46565	−0.357454
$95$	−5.55066	−0.569486
$96$	0	0
$97$	−7.58863	−0.770509	−0.385254	−	0.922810i	$-0.625886\pi$
$97$	−7.58863	−0.770509	−0.385254	+	0.922810i	$0.625886\pi$
$98$	13.9666	1.41084
$99$	0	0
$100$	23.1500	2.31500
$101$	−5.42810	−0.540117	−0.270058	−	0.962844i	$-0.587043\pi$
$101$	−5.42810	−0.540117	−0.270058	+	0.962844i	$0.587043\pi$
$102$	0	0
$103$	−6.61242	−0.651541	−0.325770	−	0.945449i	$-0.605624\pi$
$103$	−6.61242	−0.651541	−0.325770	+	0.945449i	$0.605624\pi$
$104$	−1.98760	−0.194900
$105$	0	0
$106$	−20.4499	−1.98627
$107$	−2.34678	−0.226872	−0.113436	−	0.993545i	$-0.536186\pi$
$107$	−2.34678	−0.226872	−0.113436	+	0.993545i	$0.536186\pi$
$108$	0	0
$109$	15.9160	1.52447	0.762237	−	0.647298i	$-0.224101\pi$
$109$	15.9160	1.52447	0.762237	+	0.647298i	$0.224101\pi$
$110$	−28.2526	−2.69378
$111$	0	0
$112$	−1.61817	−0.152902
$113$	−16.2398	−1.52771	−0.763857	−	0.645386i	$-0.776697\pi$
$113$	−16.2398	−1.52771	−0.763857	+	0.645386i	$0.776697\pi$
$114$	0	0
$115$	3.38524	0.315675
$116$	21.1620	1.96485
$117$	0	0
$118$	5.10798	0.470228
$119$	−4.17904	−0.383092
$120$	0	0
$121$	1.50359	0.136690
$122$	27.6742	2.50551
$123$	0	0
$124$	−0.00818301	−0.000734856 0
$125$	−13.5498	−1.21193
$126$	0	0
$127$	6.50514	0.577238	0.288619	−	0.957444i	$-0.406804\pi$
$127$	6.50514	0.577238	0.288619	+	0.957444i	$0.406804\pi$
$128$	10.9570	0.968469
$129$	0	0
$130$	10.9413	0.959619
$131$	1.68819	0.147498	0.0737488	−	0.997277i	$-0.476504\pi$
$131$	1.68819	0.147498	0.0737488	+	0.997277i	$0.476504\pi$
$132$	0	0
$133$	1.10143	0.0955063
$134$	10.0621	0.869233
$135$	0	0
$136$	8.26897	0.709058
$137$	−9.62748	−0.822531	−0.411266	−	0.911516i	$-0.634913\pi$
$137$	−9.62748	−0.822531	−0.411266	+	0.911516i	$0.634913\pi$
$138$	0	0
$139$	−10.0047	−0.848584	−0.424292	−	0.905525i	$-0.639477\pi$
$139$	−10.0047	−0.848584	−0.424292	+	0.905525i	$0.639477\pi$
$140$	−7.24431	−0.612256
$141$	0	0
$142$	19.8757	1.66793
$143$	−4.84224	−0.404929
$144$	0	0
$145$	−29.2825	−2.43178
$146$	2.59740	0.214962
$147$	0	0
$148$	−3.87471	−0.318499
$149$	−4.13420	−0.338687	−0.169344	−	0.985557i	$-0.554165\pi$
$149$	−4.13420	−0.338687	−0.169344	+	0.985557i	$0.554165\pi$
$150$	0	0
$151$	3.96140	0.322374	0.161187	−	0.986924i	$-0.448468\pi$
$151$	3.96140	0.322374	0.161187	+	0.986924i	$0.448468\pi$
$152$	−2.17938	−0.176771
$153$	0	0
$154$	5.60624	0.451764
$155$	0.0113231	0.000909489 0
$156$	0	0
$157$	−4.92130	−0.392763	−0.196381	−	0.980528i	$-0.562919\pi$
$157$	−4.92130	−0.392763	−0.196381	+	0.980528i	$0.562919\pi$
$158$	−25.4097	−2.02149
$159$	0	0
$160$	−28.3565	−2.24177
$161$	−0.671743	−0.0529407
$162$	0	0
$163$	3.22168	0.252342	0.126171	−	0.992009i	$-0.459731\pi$
$163$	3.22168	0.252342	0.126171	+	0.992009i	$0.459731\pi$
$164$	−27.9287	−2.18086
$165$	0	0
$166$	−2.65109	−0.205764
$167$	17.3833	1.34516	0.672580	−	0.740025i	$-0.265186\pi$
$167$	17.3833	1.34516	0.672580	+	0.740025i	$0.265186\pi$
$168$	0	0
$169$	−11.1248	−0.855750
$170$	−45.5191	−3.49115
$171$	0	0
$172$	−29.8906	−2.27914
$173$	−2.81617	−0.214109	−0.107055	−	0.994253i	$-0.534142\pi$
$173$	−2.81617	−0.214109	−0.107055	+	0.994253i	$0.534142\pi$
$174$	0	0
$175$	6.35644	0.480501
$176$	7.80039	0.587977
$177$	0	0
$178$	−28.4820	−2.13481
$179$	−17.4249	−1.30240	−0.651201	−	0.758906i	$-0.725734\pi$
$179$	−17.4249	−1.30240	−0.651201	+	0.758906i	$0.725734\pi$
$180$	0	0
$181$	2.70337	0.200940	0.100470	−	0.994940i	$-0.467965\pi$
$181$	2.70337	0.200940	0.100470	+	0.994940i	$0.467965\pi$
$182$	−2.17112	−0.160934
$183$	0	0
$184$	1.32916	0.0979871
$185$	5.36153	0.394188
$186$	0	0
$187$	20.1451	1.47316
$188$	4.28367	0.312419
$189$	0	0
$190$	11.9970	0.870357
$191$	−6.62519	−0.479382	−0.239691	−	0.970849i	$-0.577046\pi$
$191$	−6.62519	−0.479382	−0.239691	+	0.970849i	$0.577046\pi$
$192$	0	0
$193$	−12.9367	−0.931203	−0.465601	−	0.884994i	$-0.654162\pi$
$193$	−12.9367	−0.931203	−0.465601	+	0.884994i	$0.654162\pi$
$194$	16.4019	1.17759
$195$	0	0
$196$	−17.2632	−1.23309
$197$	−20.3551	−1.45024	−0.725120	−	0.688622i	$-0.758216\pi$
$197$	−20.3551	−1.45024	−0.725120	+	0.688622i	$0.758216\pi$
$198$	0	0
$199$	23.5481	1.66928	0.834639	−	0.550797i	$-0.185676\pi$
$199$	23.5481	1.66928	0.834639	+	0.550797i	$0.185676\pi$
$200$	−12.5773	−0.889351
$201$	0	0
$202$	11.7322	0.825472
$203$	5.81060	0.407824
$204$	0	0
$205$	38.6456	2.69913
$206$	14.2919	0.995764
$207$	0	0
$208$	−3.02084	−0.209458
$209$	−5.30946	−0.367263
$210$	0	0
$211$	−24.3112	−1.67365	−0.836827	−	0.547467i	$-0.815592\pi$
$211$	−24.3112	−1.67365	−0.836827	+	0.547467i	$0.815592\pi$
$212$	25.2769	1.73602
$213$	0	0
$214$	5.07227	0.346733
$215$	41.3604	2.82076
$216$	0	0
$217$	−0.00224686	−0.000152527 0
$218$	−34.4004	−2.32989
$219$	0	0
$220$	34.9212	2.35439
$221$	−7.80156	−0.524790
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	5.62685	0.375960
$225$	0	0
$226$	35.1003	2.33484
$227$	8.79491	0.583739	0.291869	−	0.956458i	$-0.405723\pi$
$227$	8.79491	0.583739	0.291869	+	0.956458i	$0.405723\pi$
$228$	0	0
$229$	−0.794654	−0.0525122	−0.0262561	−	0.999655i	$-0.508359\pi$
$229$	−0.794654	−0.0525122	−0.0262561	+	0.999655i	$0.508359\pi$
$230$	−7.31677	−0.482454
$231$	0	0
$232$	−11.4973	−0.754834
$233$	28.4492	1.86377	0.931883	−	0.362758i	$-0.118165\pi$
$233$	28.4492	1.86377	0.931883	+	0.362758i	$0.118165\pi$
$234$	0	0
$235$	−5.92743	−0.386663
$236$	−6.31365	−0.410983
$237$	0	0
$238$	9.03247	0.585488
$239$	−17.6215	−1.13984	−0.569921	−	0.821699i	$-0.693026\pi$
$239$	−17.6215	−1.13984	−0.569921	+	0.821699i	$0.693026\pi$
$240$	0	0
$241$	−3.06635	−0.197521	−0.0987604	−	0.995111i	$-0.531488\pi$
$241$	−3.06635	−0.197521	−0.0987604	+	0.995111i	$0.531488\pi$
$242$	−3.24981	−0.208906
$243$	0	0
$244$	−34.2064	−2.18984
$245$	23.8876	1.52612
$246$	0	0
$247$	2.05619	0.130832
$248$	0.00444581	0.000282309 0
$249$	0	0
$250$	29.2863	1.85223
$251$	−20.3645	−1.28539	−0.642697	−	0.766120i	$-0.722185\pi$
$251$	−20.3645	−1.28539	−0.642697	+	0.766120i	$0.722185\pi$
$252$	0	0
$253$	3.23814	0.203580
$254$	−14.0600	−0.882206
$255$	0	0
$256$	0.652933	0.0408083
$257$	15.4616	0.964467	0.482233	−	0.876043i	$-0.339826\pi$
$257$	15.4616	0.964467	0.482233	+	0.876043i	$0.339826\pi$
$258$	0	0
$259$	−1.06390	−0.0661077
$260$	−13.5239	−0.838716
$261$	0	0
$262$	−3.64880	−0.225424
$263$	−1.64279	−0.101299	−0.0506494	−	0.998716i	$-0.516129\pi$
$263$	−1.64279	−0.101299	−0.0506494	+	0.998716i	$0.516129\pi$
$264$	0	0
$265$	−34.9763	−2.14858
$266$	−2.38061	−0.145964
$267$	0	0
$268$	−12.4371	−0.759718
$269$	19.3325	1.17872	0.589362	−	0.807869i	$-0.299379\pi$
$269$	19.3325	1.17872	0.589362	+	0.807869i	$0.299379\pi$
$270$	0	0
$271$	−19.2096	−1.16690	−0.583450	−	0.812149i	$-0.698298\pi$
$271$	−19.2096	−1.16690	−0.583450	+	0.812149i	$0.698298\pi$
$272$	12.5676	0.762021
$273$	0	0
$274$	20.8086	1.25709
$275$	−30.6412	−1.84774
$276$	0	0
$277$	−18.2857	−1.09868	−0.549342	−	0.835598i	$-0.685122\pi$
$277$	−18.2857	−1.09868	−0.549342	+	0.835598i	$0.685122\pi$
$278$	21.6238	1.29691
$279$	0	0
$280$	3.93582	0.235210
$281$	0.487387	0.0290751	0.0145375	−	0.999894i	$-0.495372\pi$
$281$	0.487387	0.0290751	0.0145375	+	0.999894i	$0.495372\pi$
$282$	0	0
$283$	−8.03245	−0.477480	−0.238740	−	0.971084i	$-0.576734\pi$
$283$	−8.03245	−0.477480	−0.238740	+	0.971084i	$0.576734\pi$
$284$	−24.5671	−1.45779
$285$	0	0
$286$	10.4659	0.618861
$287$	−7.66855	−0.452660
$288$	0	0
$289$	15.4567	0.909218
$290$	63.2904	3.71654
$291$	0	0
$292$	−3.21048	−0.187879
$293$	−10.0295	−0.585927	−0.292964	−	0.956124i	$-0.594641\pi$
$293$	−10.0295	−0.585927	−0.292964	+	0.956124i	$0.594641\pi$
$294$	0	0
$295$	8.73636	0.508651
$296$	2.10512	0.122358
$297$	0	0
$298$	8.93555	0.517623
$299$	−1.25403	−0.0725224
$300$	0	0
$301$	−8.20726	−0.473058
$302$	−8.56207	−0.492692
$303$	0	0
$304$	−3.31232	−0.189975
$305$	47.3323	2.71024
$306$	0	0
$307$	11.0530	0.630829	0.315414	−	0.948954i	$-0.397856\pi$
$307$	11.0530	0.630829	0.315414	+	0.948954i	$0.397856\pi$
$308$	−6.92951	−0.394846
$309$	0	0
$310$	−0.0244733	−0.00138999
$311$	25.0215	1.41884	0.709419	−	0.704787i	$-0.248957\pi$
$311$	25.0215	1.41884	0.709419	+	0.704787i	$0.248957\pi$
$312$	0	0
$313$	−4.42267	−0.249984	−0.124992	−	0.992158i	$-0.539891\pi$
$313$	−4.42267	−0.249984	−0.124992	+	0.992158i	$0.539891\pi$
$314$	10.6368	0.600268
$315$	0	0
$316$	31.4073	1.76680
$317$	15.2452	0.856258	0.428129	−	0.903718i	$-0.359173\pi$
$317$	15.2452	0.856258	0.428129	+	0.903718i	$0.359173\pi$
$318$	0	0
$319$	−28.0100	−1.56826
$320$	44.9794	2.51443
$321$	0	0
$322$	1.45189	0.0809105
$323$	−8.55432	−0.475975
$324$	0	0
$325$	11.8664	0.658229
$326$	−6.96326	−0.385659
$327$	0	0
$328$	15.1736	0.837821
$329$	1.17619	0.0648457
$330$	0	0
$331$	34.3187	1.88632	0.943162	−	0.332333i	$-0.107836\pi$
$331$	34.3187	1.88632	0.943162	+	0.332333i	$0.107836\pi$
$332$	3.27684	0.179840
$333$	0	0
$334$	−37.5718	−2.05584
$335$	17.2096	0.940260
$336$	0	0
$337$	30.6168	1.66781	0.833903	−	0.551911i	$-0.186101\pi$
$337$	30.6168	1.66781	0.833903	+	0.551911i	$0.186101\pi$
$338$	24.0447	1.30786
$339$	0	0
$340$	56.2632	3.05130
$341$	0.0108310	0.000586533 0
$342$	0	0
$343$	−9.87487	−0.533193
$344$	16.2395	0.875575
$345$	0	0
$346$	6.08679	0.327228
$347$	−12.5403	−0.673198	−0.336599	−	0.941648i	$-0.609277\pi$
$347$	−12.5403	−0.673198	−0.336599	+	0.941648i	$0.609277\pi$
$348$	0	0
$349$	29.4992	1.57906	0.789528	−	0.613715i	$-0.210326\pi$
$349$	29.4992	1.57906	0.789528	+	0.613715i	$0.210326\pi$
$350$	−13.7386	−0.734361
$351$	0	0
$352$	−27.1243	−1.44573
$353$	27.0375	1.43906	0.719531	−	0.694460i	$-0.244357\pi$
$353$	27.0375	1.43906	0.719531	+	0.694460i	$0.244357\pi$
$354$	0	0
$355$	33.9941	1.80422
$356$	35.2047	1.86585
$357$	0	0
$358$	37.6618	1.99049
$359$	−5.09119	−0.268703	−0.134351	−	0.990934i	$-0.542895\pi$
$359$	−5.09119	−0.268703	−0.134351	+	0.990934i	$0.542895\pi$
$360$	0	0
$361$	−16.7454	−0.881338
$362$	−5.84300	−0.307101
$363$	0	0
$364$	2.68358	0.140658
$365$	4.44242	0.232527
$366$	0	0
$367$	8.23945	0.430096	0.215048	−	0.976603i	$-0.431009\pi$
$367$	8.23945	0.430096	0.215048	+	0.976603i	$0.431009\pi$
$368$	2.02012	0.105306
$369$	0	0
$370$	−11.5883	−0.602446
$371$	6.94044	0.360329
$372$	0	0
$373$	−14.2958	−0.740211	−0.370106	−	0.928990i	$-0.620678\pi$
$373$	−14.2958	−0.740211	−0.370106	+	0.928990i	$0.620678\pi$
$374$	−43.5411	−2.25146
$375$	0	0
$376$	−2.32731	−0.120022
$377$	10.8474	0.558669
$378$	0	0
$379$	7.71313	0.396197	0.198098	−	0.980182i	$-0.436523\pi$
$379$	7.71313	0.396197	0.198098	+	0.980182i	$0.436523\pi$
$380$	−14.8288	−0.760701
$381$	0	0
$382$	14.3195	0.732649
$383$	−8.56957	−0.437885	−0.218942	−	0.975738i	$-0.570261\pi$
$383$	−8.56957	−0.437885	−0.218942	+	0.975738i	$0.570261\pi$
$384$	0	0
$385$	9.58855	0.488678
$386$	27.9610	1.42318
$387$	0	0
$388$	−20.2733	−1.02922
$389$	18.1244	0.918942	0.459471	−	0.888193i	$-0.348039\pi$
$389$	18.1244	0.918942	0.459471	+	0.888193i	$0.348039\pi$
$390$	0	0
$391$	5.21712	0.263841
$392$	9.37909	0.473715
$393$	0	0
$394$	43.9950	2.21643
$395$	−43.4591	−2.18666
$396$	0	0
$397$	20.4711	1.02742	0.513708	−	0.857965i	$-0.328272\pi$
$397$	20.4711	1.02742	0.513708	+	0.857965i	$0.328272\pi$
$398$	−50.8962	−2.55120
$399$	0	0
$400$	−19.1156	−0.955781
$401$	−39.0300	−1.94906	−0.974531	−	0.224251i	$-0.928007\pi$
$401$	−39.0300	−1.94906	−0.974531	+	0.224251i	$0.928007\pi$
$402$	0	0
$403$	−0.00419451	−0.000208944 0
$404$	−14.5014	−0.721470
$405$	0	0
$406$	−12.5589	−0.623286
$407$	5.12855	0.254213
$408$	0	0
$409$	23.9590	1.18470	0.592348	−	0.805682i	$-0.298201\pi$
$409$	23.9590	1.18470	0.592348	+	0.805682i	$0.298201\pi$
$410$	−83.5276	−4.12514
$411$	0	0
$412$	−17.6653	−0.870308
$413$	−1.73358	−0.0853039
$414$	0	0
$415$	−4.53425	−0.222578
$416$	10.5044	0.515019
$417$	0	0
$418$	11.4757	0.561296
$419$	−2.11776	−0.103459	−0.0517296	−	0.998661i	$-0.516473\pi$
$419$	−2.11776	−0.103459	−0.0517296	+	0.998661i	$0.516473\pi$
$420$	0	0
$421$	−16.1636	−0.787766	−0.393883	−	0.919161i	$-0.628868\pi$
$421$	−16.1636	−0.787766	−0.393883	+	0.919161i	$0.628868\pi$
$422$	52.5457	2.55788
$423$	0	0
$424$	−13.7329	−0.666927
$425$	−49.3675	−2.39468
$426$	0	0
$427$	−9.39227	−0.454524
$428$	−6.26951	−0.303048
$429$	0	0
$430$	−89.3953	−4.31102
$431$	26.3852	1.27093	0.635465	−	0.772130i	$-0.280809\pi$
$431$	26.3852	1.27093	0.635465	+	0.772130i	$0.280809\pi$
$432$	0	0
$433$	−7.29321	−0.350489	−0.175245	−	0.984525i	$-0.556072\pi$
$433$	−7.29321	−0.350489	−0.175245	+	0.984525i	$0.556072\pi$
$434$	0.00485631	0.000233110 0
$435$	0	0
$436$	42.5201	2.03634
$437$	−1.37503	−0.0657765
$438$	0	0
$439$	9.75014	0.465349	0.232674	−	0.972555i	$-0.425252\pi$
$439$	9.75014	0.465349	0.232674	+	0.972555i	$0.425252\pi$
$440$	−18.9726	−0.904485
$441$	0	0
$442$	16.8621	0.802048
$443$	−25.0794	−1.19156	−0.595779	−	0.803148i	$-0.703157\pi$
$443$	−25.0794	−1.19156	−0.595779	+	0.803148i	$0.703157\pi$
$444$	0	0
$445$	−48.7137	−2.30925
$446$	−2.16137	−0.102344
$447$	0	0
$448$	−8.92539	−0.421685
$449$	−4.66218	−0.220022	−0.110011	−	0.993930i	$-0.535089\pi$
$449$	−4.66218	−0.220022	−0.110011	+	0.993930i	$0.535089\pi$
$450$	0	0
$451$	36.9663	1.74068
$452$	−43.3852	−2.04067
$453$	0	0
$454$	−19.0091	−0.892141
$455$	−3.71334	−0.174084
$456$	0	0
$457$	−11.9063	−0.556953	−0.278476	−	0.960443i	$-0.589829\pi$
$457$	−11.9063	−0.556953	−0.278476	+	0.960443i	$0.589829\pi$
$458$	1.71754	0.0802556
$459$	0	0
$460$	9.04380	0.421669
$461$	−8.69503	−0.404968	−0.202484	−	0.979286i	$-0.564901\pi$
$461$	−8.69503	−0.404968	−0.202484	+	0.979286i	$0.564901\pi$
$462$	0	0
$463$	−17.1114	−0.795233	−0.397617	−	0.917552i	$-0.630163\pi$
$463$	−17.1114	−0.795233	−0.397617	+	0.917552i	$0.630163\pi$
$464$	−17.4741	−0.811216
$465$	0	0
$466$	−61.4893	−2.84844
$467$	7.60297	0.351824	0.175912	−	0.984406i	$-0.443713\pi$
$467$	7.60297	0.351824	0.175912	+	0.984406i	$0.443713\pi$
$468$	0	0
$469$	−3.41494	−0.157687
$470$	12.8114	0.590945
$471$	0	0
$472$	3.43019	0.157887
$473$	39.5631	1.81912
$474$	0	0
$475$	13.0114	0.597002
$476$	−11.1645	−0.511722
$477$	0	0
$478$	38.0867	1.74205
$479$	37.2521	1.70209	0.851047	−	0.525090i	$-0.175968\pi$
$479$	37.2521	1.70209	0.851047	+	0.525090i	$0.175968\pi$
$480$	0	0
$481$	−1.98613	−0.0905596
$482$	6.62752	0.301875
$483$	0	0
$484$	4.01688	0.182586
$485$	28.0527	1.27381
$486$	0	0
$487$	−12.9228	−0.585587	−0.292794	−	0.956176i	$-0.594585\pi$
$487$	−12.9228	−0.585587	−0.292794	+	0.956176i	$0.594585\pi$
$488$	18.5843	0.841270
$489$	0	0
$490$	−51.6301	−2.33241
$491$	3.00718	0.135712	0.0678560	−	0.997695i	$-0.478384\pi$
$491$	3.00718	0.135712	0.0678560	+	0.997695i	$0.478384\pi$
$492$	0	0
$493$	−45.1283	−2.03247
$494$	−4.44419	−0.199953
$495$	0	0
$496$	0.00675696	0.000303396 0
$497$	−6.74554	−0.302579
$498$	0	0
$499$	16.7872	0.751497	0.375749	−	0.926722i	$-0.377386\pi$
$499$	16.7872	0.751497	0.375749	+	0.926722i	$0.377386\pi$
$500$	−36.1989	−1.61886
$501$	0	0
$502$	44.0152	1.96450
$503$	28.9180	1.28939	0.644694	−	0.764441i	$-0.276985\pi$
$503$	28.9180	1.28939	0.644694	+	0.764441i	$0.276985\pi$
$504$	0	0
$505$	20.0659	0.892923
$506$	−6.99883	−0.311136
$507$	0	0
$508$	17.3787	0.771056
$509$	38.0324	1.68576	0.842878	−	0.538105i	$-0.180860\pi$
$509$	38.0324	1.68576	0.842878	+	0.538105i	$0.180860\pi$
$510$	0	0
$511$	−0.881522	−0.0389962
$512$	−23.3252	−1.03084
$513$	0	0
$514$	−33.4182	−1.47402
$515$	24.4440	1.07713
$516$	0	0
$517$	−5.66986	−0.249360
$518$	2.29949	0.101034
$519$	0	0
$520$	7.34750	0.322209
$521$	11.8416	0.518788	0.259394	−	0.965772i	$-0.416477\pi$
$521$	11.8416	0.518788	0.259394	+	0.965772i	$0.416477\pi$
$522$	0	0
$523$	8.33854	0.364619	0.182310	−	0.983241i	$-0.441643\pi$
$523$	8.33854	0.364619	0.182310	+	0.983241i	$0.441643\pi$
$524$	4.51005	0.197023
$525$	0	0
$526$	3.55068	0.154817
$527$	0.0174504	0.000760150 0
$528$	0	0
$529$	−22.1614	−0.963539
$530$	75.5968	3.28372
$531$	0	0
$532$	2.94252	0.127574
$533$	−14.3159	−0.620090
$534$	0	0
$535$	8.67529	0.375065
$536$	6.75706	0.291861
$537$	0	0
$538$	−41.7847	−1.80147
$539$	22.8496	0.984202
$540$	0	0
$541$	36.8073	1.58247	0.791235	−	0.611512i	$-0.209438\pi$
$541$	36.8073	1.58247	0.791235	+	0.611512i	$0.209438\pi$
$542$	41.5191	1.78340
$543$	0	0
$544$	−43.7012	−1.87367
$545$	−58.8362	−2.52027
$546$	0	0
$547$	8.60725	0.368020	0.184010	−	0.982924i	$-0.441092\pi$
$547$	8.60725	0.368020	0.184010	+	0.982924i	$0.441092\pi$
$548$	−25.7202	−1.09871
$549$	0	0
$550$	66.2272	2.82394
$551$	11.8940	0.506703
$552$	0	0
$553$	8.62370	0.366717
$554$	39.5223	1.67914
$555$	0	0
$556$	−26.7278	−1.13351
$557$	22.9911	0.974165	0.487082	−	0.873356i	$-0.338061\pi$
$557$	22.9911	0.974165	0.487082	+	0.873356i	$0.338061\pi$
$558$	0	0
$559$	−15.3216	−0.648033
$560$	5.98184	0.252779
$561$	0	0
$562$	−1.05343	−0.0444361
$563$	−3.21057	−0.135309	−0.0676547	−	0.997709i	$-0.521552\pi$
$563$	−3.21057	−0.135309	−0.0676547	+	0.997709i	$0.521552\pi$
$564$	0	0
$565$	60.0333	2.52562
$566$	17.3611	0.729743
$567$	0	0
$568$	13.3472	0.560038
$569$	37.4699	1.57082	0.785409	−	0.618977i	$-0.212452\pi$
$569$	37.4699	1.57082	0.785409	+	0.618977i	$0.212452\pi$
$570$	0	0
$571$	31.9771	1.33820	0.669100	−	0.743173i	$-0.266680\pi$
$571$	31.9771	1.33820	0.669100	+	0.743173i	$0.266680\pi$
$572$	−12.9362	−0.540891
$573$	0	0
$574$	16.5746	0.691811
$575$	−7.93538	−0.330928
$576$	0	0
$577$	9.08869	0.378367	0.189183	−	0.981942i	$-0.439416\pi$
$577$	9.08869	0.378367	0.189183	+	0.981942i	$0.439416\pi$
$578$	−33.4077	−1.38958
$579$	0	0
$580$	−78.2292	−3.24829
$581$	0.899744	0.0373277
$582$	0	0
$583$	−33.4564	−1.38562
$584$	1.74425	0.0721774
$585$	0	0
$586$	21.6774	0.895485
$587$	−44.4262	−1.83367	−0.916834	−	0.399269i	$-0.869264\pi$
$587$	−44.4262	−1.83367	−0.916834	+	0.399269i	$0.869264\pi$
$588$	0	0
$589$	−0.00459923	−0.000189508 0
$590$	−18.8825	−0.777382
$591$	0	0
$592$	3.19946	0.131497
$593$	16.4375	0.675006	0.337503	−	0.941325i	$-0.390418\pi$
$593$	16.4375	0.675006	0.337503	+	0.941325i	$0.390418\pi$
$594$	0	0
$595$	15.4486	0.633329
$596$	−11.0447	−0.452407
$597$	0	0
$598$	2.71043	0.110838
$599$	24.8430	1.01506	0.507529	−	0.861635i	$-0.330559\pi$
$599$	24.8430	1.01506	0.507529	+	0.861635i	$0.330559\pi$
$600$	0	0
$601$	−4.44051	−0.181132	−0.0905661	−	0.995890i	$-0.528868\pi$
$601$	−4.44051	−0.181132	−0.0905661	+	0.995890i	$0.528868\pi$
$602$	17.7389	0.722985
$603$	0	0
$604$	10.5830	0.430617
$605$	−5.55827	−0.225976
$606$	0	0
$607$	8.44082	0.342602	0.171301	−	0.985219i	$-0.445203\pi$
$607$	8.44082	0.342602	0.171301	+	0.985219i	$0.445203\pi$
$608$	11.5179	0.467113
$609$	0	0
$610$	−102.303	−4.14212
$611$	2.19576	0.0888308
$612$	0	0
$613$	21.7294	0.877643	0.438821	−	0.898574i	$-0.355396\pi$
$613$	21.7294	0.877643	0.438821	+	0.898574i	$0.355396\pi$
$614$	−23.8897	−0.964110
$615$	0	0
$616$	3.76479	0.151688
$617$	−4.94404	−0.199039	−0.0995197	−	0.995036i	$-0.531731\pi$
$617$	−4.94404	−0.199039	−0.0995197	+	0.995036i	$0.531731\pi$
$618$	0	0
$619$	18.1634	0.730049	0.365025	−	0.930998i	$-0.381061\pi$
$619$	18.1634	0.730049	0.365025	+	0.930998i	$0.381061\pi$
$620$	0.0302499	0.00121487
$621$	0	0
$622$	−54.0808	−2.16844
$623$	9.66640	0.387276
$624$	0	0
$625$	6.76229	0.270492
$626$	9.55905	0.382056
$627$	0	0
$628$	−13.1474	−0.524640
$629$	8.26285	0.329461
$630$	0	0
$631$	−2.43582	−0.0969683	−0.0484841	−	0.998824i	$-0.515439\pi$
$631$	−2.43582	−0.0969683	−0.0484841	+	0.998824i	$0.515439\pi$
$632$	−17.0635	−0.678750
$633$	0	0
$634$	−32.9506	−1.30864
$635$	−24.0474	−0.954292
$636$	0	0
$637$	−8.84893	−0.350607
$638$	60.5401	2.39681
$639$	0	0
$640$	−40.5044	−1.60108
$641$	3.31626	0.130984	0.0654921	−	0.997853i	$-0.479138\pi$
$641$	3.31626	0.130984	0.0654921	+	0.997853i	$0.479138\pi$
$642$	0	0
$643$	36.3974	1.43537	0.717687	−	0.696366i	$-0.245201\pi$
$643$	36.3974	1.43537	0.717687	+	0.696366i	$0.245201\pi$
$644$	−1.79458	−0.0707165
$645$	0	0
$646$	18.4891	0.727443
$647$	−31.1921	−1.22629	−0.613144	−	0.789971i	$-0.710095\pi$
$647$	−31.1921	−1.22629	−0.613144	+	0.789971i	$0.710095\pi$
$648$	0	0
$649$	8.35673	0.328030
$650$	−25.6477	−1.00599
$651$	0	0
$652$	8.60684	0.337070
$653$	47.4322	1.85617	0.928083	−	0.372372i	$-0.121456\pi$
$653$	47.4322	1.85617	0.928083	+	0.372372i	$0.121456\pi$
$654$	0	0
$655$	−6.24068	−0.243844
$656$	23.0615	0.900401
$657$	0	0
$658$	−2.54220	−0.0991051
$659$	37.6748	1.46760	0.733801	−	0.679365i	$-0.237744\pi$
$659$	37.6748	1.46760	0.733801	+	0.679365i	$0.237744\pi$
$660$	0	0
$661$	−5.56333	−0.216388	−0.108194	−	0.994130i	$-0.534507\pi$
$661$	−5.56333	−0.216388	−0.108194	+	0.994130i	$0.534507\pi$
$662$	−74.1754	−2.88291
$663$	0	0
$664$	−1.78030	−0.0690891
$665$	−4.07164	−0.157891
$666$	0	0
$667$	−7.25395	−0.280874
$668$	46.4401	1.79682
$669$	0	0
$670$	−37.1963	−1.43702
$671$	45.2755	1.74784
$672$	0	0
$673$	−41.2849	−1.59142	−0.795708	−	0.605681i	$-0.792901\pi$
$673$	−41.2849	−1.59142	−0.795708	+	0.605681i	$0.792901\pi$
$674$	−66.1744	−2.54894
$675$	0	0
$676$	−29.7202	−1.14308
$677$	−10.1524	−0.390190	−0.195095	−	0.980784i	$-0.562502\pi$
$677$	−10.1524	−0.390190	−0.195095	+	0.980784i	$0.562502\pi$
$678$	0	0
$679$	−5.56657	−0.213625
$680$	−30.5677	−1.17222
$681$	0	0
$682$	−0.0234099	−0.000896411 0
$683$	16.4232	0.628416	0.314208	−	0.949354i	$-0.398261\pi$
$683$	16.4232	0.628416	0.314208	+	0.949354i	$0.398261\pi$
$684$	0	0
$685$	35.5897	1.35981
$686$	21.3433	0.814890
$687$	0	0
$688$	24.6816	0.940975
$689$	12.9566	0.493608
$690$	0	0
$691$	42.8724	1.63094	0.815472	−	0.578796i	$-0.196477\pi$
$691$	42.8724	1.63094	0.815472	+	0.578796i	$0.196477\pi$
$692$	−7.52349	−0.286000
$693$	0	0
$694$	27.1043	1.02886
$695$	36.9840	1.40288
$696$	0	0
$697$	59.5581	2.25593
$698$	−63.7588	−2.41331
$699$	0	0
$700$	16.9814	0.641838
$701$	35.2208	1.33027	0.665136	−	0.746723i	$-0.268374\pi$
$701$	35.2208	1.33027	0.665136	+	0.746723i	$0.268374\pi$
$702$	0	0
$703$	−2.17776	−0.0821359
$704$	43.0249	1.62156
$705$	0	0
$706$	−58.4382	−2.19935
$707$	−3.98174	−0.149749
$708$	0	0
$709$	−32.3220	−1.21388	−0.606939	−	0.794748i	$-0.707603\pi$
$709$	−32.3220	−1.21388	−0.606939	+	0.794748i	$0.707603\pi$
$710$	−73.4740	−2.75743
$711$	0	0
$712$	−19.1267	−0.716802
$713$	0.00280498	0.000105048 0
$714$	0	0
$715$	17.9002	0.669430
$716$	−46.5514	−1.73971
$717$	0	0
$718$	11.0040	0.410664
$719$	−13.4939	−0.503238	−0.251619	−	0.967826i	$-0.580963\pi$
$719$	−13.4939	−0.503238	−0.251619	+	0.967826i	$0.580963\pi$
$720$	0	0
$721$	−4.85048	−0.180641
$722$	36.1931	1.34697
$723$	0	0
$724$	7.22216	0.268409
$725$	68.6413	2.54927
$726$	0	0
$727$	−16.1047	−0.597290	−0.298645	−	0.954364i	$-0.596535\pi$
$727$	−16.1047	−0.597290	−0.298645	+	0.954364i	$0.596535\pi$
$728$	−1.45798	−0.0540365
$729$	0	0
$730$	−9.60174	−0.355376
$731$	63.7420	2.35758
$732$	0	0
$733$	34.3357	1.26822	0.634109	−	0.773244i	$-0.281367\pi$
$733$	34.3357	1.26822	0.634109	+	0.773244i	$0.281367\pi$
$734$	−17.8085	−0.657325
$735$	0	0
$736$	−7.02456	−0.258929
$737$	16.4618	0.606377
$738$	0	0
$739$	−27.9999	−1.02999	−0.514997	−	0.857192i	$-0.672207\pi$
$739$	−27.9999	−1.02999	−0.514997	+	0.857192i	$0.672207\pi$
$740$	14.3235	0.526543
$741$	0	0
$742$	−15.0009	−0.550699
$743$	−37.9147	−1.39095	−0.695477	−	0.718548i	$-0.744807\pi$
$743$	−37.9147	−1.39095	−0.695477	+	0.718548i	$0.744807\pi$
$744$	0	0
$745$	15.2828	0.559919
$746$	30.8987	1.13128
$747$	0	0
$748$	53.8184	1.96779
$749$	−1.72146	−0.0629008
$750$	0	0
$751$	13.0960	0.477881	0.238940	−	0.971034i	$-0.423200\pi$
$751$	13.0960	0.477881	0.238940	+	0.971034i	$0.423200\pi$
$752$	−3.53715	−0.128987
$753$	0	0
$754$	−23.4453	−0.853827
$755$	−14.6440	−0.532950
$756$	0	0
$757$	21.1477	0.768626	0.384313	−	0.923203i	$-0.374438\pi$
$757$	21.1477	0.768626	0.384313	+	0.923203i	$0.374438\pi$
$758$	−16.6710	−0.605516
$759$	0	0
$760$	8.05645	0.292238
$761$	40.7154	1.47593	0.737966	−	0.674837i	$-0.235786\pi$
$761$	40.7154	1.47593	0.737966	+	0.674837i	$0.235786\pi$
$762$	0	0
$763$	11.6750	0.422664
$764$	−17.6994	−0.640343
$765$	0	0
$766$	18.5220	0.669229
$767$	−3.23630	−0.116856
$768$	0	0
$769$	31.4018	1.13238	0.566190	−	0.824275i	$-0.308417\pi$
$769$	31.4018	1.13238	0.566190	+	0.824275i	$0.308417\pi$
$770$	−20.7244	−0.746857
$771$	0	0
$772$	−34.5608	−1.24387
$773$	29.6112	1.06504	0.532520	−	0.846417i	$-0.321245\pi$
$773$	29.6112	1.06504	0.532520	+	0.846417i	$0.321245\pi$
$774$	0	0
$775$	−0.0265425	−0.000953433 0
$776$	11.0144	0.395396
$777$	0	0
$778$	−39.1735	−1.40444
$779$	−15.6972	−0.562410
$780$	0	0
$781$	32.5169	1.16355
$782$	−11.2761	−0.403234
$783$	0	0
$784$	14.2548	0.509099
$785$	18.1925	0.649317
$786$	0	0
$787$	3.43142	0.122317	0.0611585	−	0.998128i	$-0.480521\pi$
$787$	3.43142	0.122317	0.0611585	+	0.998128i	$0.480521\pi$
$788$	−54.3794	−1.93718
$789$	0	0
$790$	93.9313	3.34193
$791$	−11.9126	−0.423562
$792$	0	0
$793$	−17.5338	−0.622642
$794$	−44.2457	−1.57022
$795$	0	0
$796$	62.9095	2.22977
$797$	−43.1802	−1.52952	−0.764760	−	0.644315i	$-0.777142\pi$
$797$	−43.1802	−1.52952	−0.764760	+	0.644315i	$0.777142\pi$
$798$	0	0
$799$	−9.13497	−0.323172
$800$	66.4706	2.35009
$801$	0	0
$802$	84.3583	2.97880
$803$	4.24938	0.149958
$804$	0	0
$805$	2.48321	0.0875218
$806$	0.00906591	0.000319333 0
$807$	0	0
$808$	7.87857	0.277167
$809$	−31.9945	−1.12487	−0.562433	−	0.826843i	$-0.690134\pi$
$809$	−31.9945	−1.12487	−0.562433	+	0.826843i	$0.690134\pi$
$810$	0	0
$811$	−46.8537	−1.64526	−0.822628	−	0.568581i	$-0.807493\pi$
$811$	−46.8537	−1.64526	−0.822628	+	0.568581i	$0.807493\pi$
$812$	15.5232	0.544758
$813$	0	0
$814$	−11.0847	−0.388519
$815$	−11.9095	−0.417172
$816$	0	0
$817$	−16.7999	−0.587754
$818$	−51.7844	−1.81060
$819$	0	0
$820$	103.243	3.60541
$821$	−41.1796	−1.43718	−0.718588	−	0.695436i	$-0.755211\pi$
$821$	−41.1796	−1.43718	−0.718588	+	0.695436i	$0.755211\pi$
$822$	0	0
$823$	22.7846	0.794220	0.397110	−	0.917771i	$-0.370013\pi$
$823$	22.7846	0.794220	0.397110	+	0.917771i	$0.370013\pi$
$824$	9.59753	0.334346
$825$	0	0
$826$	3.74691	0.130372
$827$	48.5912	1.68968	0.844841	−	0.535017i	$-0.179695\pi$
$827$	48.5912	1.68968	0.844841	+	0.535017i	$0.179695\pi$
$828$	0	0
$829$	52.7082	1.83063	0.915316	−	0.402737i	$-0.131941\pi$
$829$	52.7082	1.83063	0.915316	+	0.402737i	$0.131941\pi$
$830$	9.80022	0.340170
$831$	0	0
$832$	−16.6622	−0.577657
$833$	36.8141	1.27553
$834$	0	0
$835$	−64.2604	−2.22382
$836$	−14.1844	−0.490578
$837$	0	0
$838$	4.57726	0.158119
$839$	18.4325	0.636361	0.318180	−	0.948030i	$-0.396928\pi$
$839$	18.4325	0.636361	0.318180	+	0.948030i	$0.396928\pi$
$840$	0	0
$841$	33.7470	1.16369
$842$	34.9356	1.20396
$843$	0	0
$844$	−64.9484	−2.23561
$845$	41.1246	1.41473
$846$	0	0
$847$	1.10294	0.0378975
$848$	−20.8719	−0.716743
$849$	0	0
$850$	106.702	3.65984
$851$	1.32818	0.0455293
$852$	0	0
$853$	−29.7874	−1.01990	−0.509951	−	0.860203i	$-0.670337\pi$
$853$	−29.7874	−1.01990	−0.509951	+	0.860203i	$0.670337\pi$
$854$	20.3002	0.694658
$855$	0	0
$856$	3.40621	0.116422
$857$	−16.8357	−0.575098	−0.287549	−	0.957766i	$-0.592840\pi$
$857$	−16.8357	−0.575098	−0.287549	+	0.957766i	$0.592840\pi$
$858$	0	0
$859$	0.871249	0.0297266	0.0148633	−	0.999890i	$-0.495269\pi$
$859$	0.871249	0.0297266	0.0148633	+	0.999890i	$0.495269\pi$
$860$	110.496	3.76788
$861$	0	0
$862$	−57.0282	−1.94239
$863$	−45.9092	−1.56277	−0.781384	−	0.624050i	$-0.785486\pi$
$863$	−45.9092	−1.56277	−0.781384	+	0.624050i	$0.785486\pi$
$864$	0	0
$865$	10.4105	0.353966
$866$	15.7633	0.535660
$867$	0	0
$868$	−0.00600258	−0.000203741 0
$869$	−41.5706	−1.41019
$870$	0	0
$871$	−6.37512	−0.216013
$872$	−23.1011	−0.782302
$873$	0	0
$874$	2.97195	0.100528
$875$	−9.93936	−0.336012
$876$	0	0
$877$	20.9226	0.706505	0.353252	−	0.935528i	$-0.385076\pi$
$877$	20.9226	0.706505	0.353252	+	0.935528i	$0.385076\pi$
$878$	−21.0737	−0.711202
$879$	0	0
$880$	−28.8355	−0.972045
$881$	−11.5804	−0.390152	−0.195076	−	0.980788i	$-0.562495\pi$
$881$	−11.5804	−0.390152	−0.195076	+	0.980788i	$0.562495\pi$
$882$	0	0
$883$	5.17478	0.174145	0.0870726	−	0.996202i	$-0.472249\pi$
$883$	5.17478	0.174145	0.0870726	+	0.996202i	$0.472249\pi$
$884$	−20.8422	−0.700997
$885$	0	0
$886$	54.2060	1.82109
$887$	−57.7208	−1.93808	−0.969038	−	0.246912i	$-0.920584\pi$
$887$	−57.7208	−1.93808	−0.969038	+	0.246912i	$0.920584\pi$
$888$	0	0
$889$	4.77179	0.160041
$890$	105.289	3.52928
$891$	0	0
$892$	2.67154	0.0894496
$893$	2.40762	0.0805679
$894$	0	0
$895$	64.4144	2.15313
$896$	8.03740	0.268511
$897$	0	0
$898$	10.0767	0.336264
$899$	−0.0242632	−0.000809224 0
$900$	0	0
$901$	−53.9032	−1.79578
$902$	−79.8980	−2.66031
$903$	0	0
$904$	23.5711	0.783964
$905$	−9.99350	−0.332195
$906$	0	0
$907$	17.2706	0.573460	0.286730	−	0.958011i	$-0.407432\pi$
$907$	17.2706	0.573460	0.286730	+	0.958011i	$0.407432\pi$
$908$	23.4959	0.779740
$909$	0	0
$910$	8.02592	0.266057
$911$	51.4627	1.70504	0.852518	−	0.522698i	$-0.175074\pi$
$911$	51.4627	1.70504	0.852518	+	0.522698i	$0.175074\pi$
$912$	0	0
$913$	−4.33722	−0.143541
$914$	25.7339	0.851203
$915$	0	0
$916$	−2.12295	−0.0701442
$917$	1.23836	0.0408941
$918$	0	0
$919$	−42.6692	−1.40753	−0.703763	−	0.710435i	$-0.748498\pi$
$919$	−42.6692	−1.40753	−0.703763	+	0.710435i	$0.748498\pi$
$920$	−4.91348	−0.161993
$921$	0	0
$922$	18.7932	0.618921
$923$	−12.5928	−0.414496
$924$	0	0
$925$	−12.5680	−0.413234
$926$	36.9841	1.21537
$927$	0	0
$928$	60.7627	1.99463
$929$	27.1513	0.890806	0.445403	−	0.895330i	$-0.353060\pi$
$929$	27.1513	0.890806	0.445403	+	0.895330i	$0.353060\pi$
$930$	0	0
$931$	−9.70274	−0.317995
$932$	76.0030	2.48956
$933$	0	0
$934$	−16.4329	−0.537700
$935$	−74.4699	−2.43543
$936$	0	0
$937$	6.20634	0.202752	0.101376	−	0.994848i	$-0.467675\pi$
$937$	6.20634	0.202752	0.101376	+	0.994848i	$0.467675\pi$
$938$	7.38097	0.240997
$939$	0	0
$940$	−15.8353	−0.516492
$941$	−28.9440	−0.943547	−0.471774	−	0.881720i	$-0.656386\pi$
$941$	−28.9440	−0.943547	−0.471774	+	0.881720i	$0.656386\pi$
$942$	0	0
$943$	9.57343	0.311754
$944$	5.21337	0.169681
$945$	0	0
$946$	−85.5107	−2.78019
$947$	−12.7704	−0.414982	−0.207491	−	0.978237i	$-0.566530\pi$
$947$	−12.7704	−0.414982	−0.207491	+	0.978237i	$0.566530\pi$
$948$	0	0
$949$	−1.64565	−0.0534201
$950$	−28.1224	−0.912411
$951$	0	0
$952$	6.06563	0.196588
$953$	−22.6216	−0.732785	−0.366392	−	0.930460i	$-0.619407\pi$
$953$	−22.6216	−0.732785	−0.366392	+	0.930460i	$0.619407\pi$
$954$	0	0
$955$	24.4912	0.792515
$956$	−47.0766	−1.52256
$957$	0	0
$958$	−80.5158	−2.60135
$959$	−7.06216	−0.228049
$960$	0	0
$961$	−31.0000	−1.00000
$962$	4.29276	0.138404
$963$	0	0
$964$	−8.19186	−0.263842
$965$	47.8227	1.53947
$966$	0	0
$967$	−26.4031	−0.849066	−0.424533	−	0.905413i	$-0.639562\pi$
$967$	−26.4031	−0.849066	−0.424533	+	0.905413i	$0.639562\pi$
$968$	−2.18236	−0.0701439
$969$	0	0
$970$	−60.6324	−1.94679
$971$	−29.4152	−0.943977	−0.471989	−	0.881605i	$-0.656464\pi$
$971$	−29.4152	−0.943977	−0.471989	+	0.881605i	$0.656464\pi$
$972$	0	0
$973$	−7.33883	−0.235272
$974$	27.9310	0.894966
$975$	0	0
$976$	28.2452	0.904107
$977$	−30.8432	−0.986761	−0.493381	−	0.869813i	$-0.664239\pi$
$977$	−30.8432	−0.986761	−0.493381	+	0.869813i	$0.664239\pi$
$978$	0	0
$979$	−46.5969	−1.48924
$980$	63.8166	2.03855
$981$	0	0
$982$	−6.49963	−0.207412
$983$	52.1331	1.66279	0.831393	−	0.555684i	$-0.187544\pi$
$983$	52.1331	1.66279	0.831393	+	0.555684i	$0.187544\pi$
$984$	0	0
$985$	75.2462	2.39754
$986$	97.5390	3.10628
$987$	0	0
$988$	5.49318	0.174761
$989$	10.2459	0.325802
$990$	0	0
$991$	−26.2264	−0.833108	−0.416554	−	0.909111i	$-0.636762\pi$
$991$	−26.2264	−0.833108	−0.416554	+	0.909111i	$0.636762\pi$
$992$	−0.0234959	−0.000745997 0
$993$	0	0
$994$	14.5796	0.462438
$995$	−87.0496	−2.75966
$996$	0	0
$997$	−53.5617	−1.69632	−0.848159	−	0.529742i	$-0.822289\pi$
$997$	−53.5617	−1.69632	−0.848159	+	0.529742i	$0.822289\pi$
$998$	−36.2834	−1.14853
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6021.2.a.t.1.6	✓	40
3.2	odd	2	inner	6021.2.a.t.1.35	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6021.2.a.t.1.6	✓	40	1.1	even	1	trivial
6021.2.a.t.1.35	yes	40	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 6021 = 3^{3} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6021.a (trivial)

\(f(q)\)	\(=\)	\(q-2.16137 q^{2} +2.67154 q^{4} -3.69668 q^{5} +0.733541 q^{7} -1.45144 q^{8} +O(q^{10})\)	\(q-2.16137 q^{2} +2.67154 q^{4} -3.69668 q^{5} +0.733541 q^{7} -1.45144 q^{8} +7.98990 q^{10} -3.53604 q^{11} +1.36940 q^{13} -1.58546 q^{14} -2.20597 q^{16} -5.69708 q^{17} +1.50153 q^{19} -9.87580 q^{20} +7.64271 q^{22} -0.915753 q^{23} +8.66541 q^{25} -2.95978 q^{26} +1.95968 q^{28} +7.92130 q^{29} -0.00306304 q^{31} +7.67080 q^{32} +12.3135 q^{34} -2.71166 q^{35} -1.45037 q^{37} -3.24536 q^{38} +5.36550 q^{40} -10.4542 q^{41} -11.1885 q^{43} -9.44666 q^{44} +1.97928 q^{46} +1.60345 q^{47} -6.46192 q^{49} -18.7292 q^{50} +3.65839 q^{52} +9.46155 q^{53} +13.0716 q^{55} -1.06469 q^{56} -17.1209 q^{58} -2.36330 q^{59} -12.8040 q^{61} +0.00662037 q^{62} -12.1675 q^{64} -5.06222 q^{65} -4.65542 q^{67} -15.2200 q^{68} +5.86092 q^{70} -9.19586 q^{71} -1.20173 q^{73} +3.13478 q^{74} +4.01138 q^{76} -2.59383 q^{77} +11.7563 q^{79} +8.15474 q^{80} +22.5953 q^{82} +1.22658 q^{83} +21.0603 q^{85} +24.1826 q^{86} +5.13235 q^{88} +13.1777 q^{89} +1.00451 q^{91} -2.44647 q^{92} -3.46565 q^{94} -5.55066 q^{95} -7.58863 q^{97} +13.9666 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 46 q^{4} + 16 q^{7}+O(q^{10}) \) 40 * q + 46 * q^4 + 16 * q^7	\( 40 q + 46 q^{4} + 16 q^{7} + 22 q^{10} + 14 q^{13} + 50 q^{16} + 64 q^{19} + 12 q^{22} + 40 q^{25} + 48 q^{28} + 54 q^{31} + 32 q^{34} + 24 q^{37} + 40 q^{40} + 24 q^{43} + 52 q^{46} + 64 q^{49} + 18 q^{52} + 36 q^{55} + 8 q^{58} + 58 q^{61} + 120 q^{64} + 52 q^{67} - 30 q^{70} + 50 q^{73} + 112 q^{76} + 60 q^{79} + 50 q^{82} + 38 q^{85} + 16 q^{88} + 118 q^{91} + 44 q^{94} + 38 q^{97}+O(q^{100}) \) 40 * q + 46 * q^4 + 16 * q^7 + 22 * q^10 + 14 * q^13 + 50 * q^16 + 64 * q^19 + 12 * q^22 + 40 * q^25 + 48 * q^28 + 54 * q^31 + 32 * q^34 + 24 * q^37 + 40 * q^40 + 24 * q^43 + 52 * q^46 + 64 * q^49 + 18 * q^52 + 36 * q^55 + 8 * q^58 + 58 * q^61 + 120 * q^64 + 52 * q^67 - 30 * q^70 + 50 * q^73 + 112 * q^76 + 60 * q^79 + 50 * q^82 + 38 * q^85 + 16 * q^88 + 118 * q^91 + 44 * q^94 + 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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