LMFDB - Embedded newform 6021.2.a.t.1.5

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			1.5
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)

$f(q)$	$=$	$q-2.32144 q^{2} +3.38909 q^{4} -1.08423 q^{5} +4.03069 q^{7} -3.22470 q^{8} +O(q^{10})$	\(q-2.32144 q^{2} +3.38909 q^{4} -1.08423 q^{5} +4.03069 q^{7} -3.22470 q^{8} +2.51697 q^{10} -3.49476 q^{11} -3.72073 q^{13} -9.35701 q^{14} +0.707766 q^{16} +4.48254 q^{17} +6.33001 q^{19} -3.67455 q^{20} +8.11289 q^{22} -7.10533 q^{23} -3.82445 q^{25} +8.63747 q^{26} +13.6604 q^{28} -0.519983 q^{29} -4.96809 q^{31} +4.80636 q^{32} -10.4059 q^{34} -4.37019 q^{35} +4.93356 q^{37} -14.6947 q^{38} +3.49631 q^{40} -3.38864 q^{41} +4.04743 q^{43} -11.8441 q^{44} +16.4946 q^{46} -3.12585 q^{47} +9.24645 q^{49} +8.87823 q^{50} -12.6099 q^{52} -8.30126 q^{53} +3.78912 q^{55} -12.9978 q^{56} +1.20711 q^{58} +3.99159 q^{59} +5.30912 q^{61} +11.5331 q^{62} -12.5732 q^{64} +4.03413 q^{65} -12.8573 q^{67} +15.1917 q^{68} +10.1451 q^{70} +4.05577 q^{71} -7.06713 q^{73} -11.4530 q^{74} +21.4530 q^{76} -14.0863 q^{77} +10.0440 q^{79} -0.767380 q^{80} +7.86652 q^{82} +13.4518 q^{83} -4.86009 q^{85} -9.39587 q^{86} +11.2696 q^{88} -2.13957 q^{89} -14.9971 q^{91} -24.0806 q^{92} +7.25648 q^{94} -6.86318 q^{95} +15.7277 q^{97} -21.4651 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.32144	−1.64151	−0.820754	−	0.571282i	$-0.806446\pi$
$2$	−2.32144	−1.64151	−0.820754	+	0.571282i	$0.806446\pi$
$3$	0	0
$3$	0	0
$4$	3.38909	1.69455
$5$	−1.08423	−0.484882	−0.242441	−	0.970166i	$-0.577948\pi$
$5$	−1.08423	−0.484882	−0.242441	+	0.970166i	$0.577948\pi$
$6$	0	0
$7$	4.03069	1.52346	0.761729	−	0.647896i	$-0.224351\pi$
$7$	4.03069	1.52346	0.761729	+	0.647896i	$0.224351\pi$
$8$	−3.22470	−1.14010
$9$	0	0
$10$	2.51697	0.795937
$11$	−3.49476	−1.05371	−0.526855	−	0.849955i	$-0.676629\pi$
$11$	−3.49476	−1.05371	−0.526855	+	0.849955i	$0.676629\pi$
$12$	0	0
$13$	−3.72073	−1.03195	−0.515973	−	0.856605i	$-0.672570\pi$
$13$	−3.72073	−1.03195	−0.515973	+	0.856605i	$0.672570\pi$
$14$	−9.35701	−2.50077
$15$	0	0
$16$	0.707766	0.176942
$17$	4.48254	1.08717	0.543587	−	0.839353i	$-0.317066\pi$
$17$	4.48254	1.08717	0.543587	+	0.839353i	$0.317066\pi$
$18$	0	0
$19$	6.33001	1.45220	0.726102	−	0.687587i	$-0.241330\pi$
$19$	6.33001	1.45220	0.726102	+	0.687587i	$0.241330\pi$
$20$	−3.67455	−0.821655
$21$	0	0
$22$	8.11289	1.72967
$23$	−7.10533	−1.48156	−0.740781	−	0.671746i	$-0.765545\pi$
$23$	−7.10533	−1.48156	−0.740781	+	0.671746i	$0.765545\pi$
$24$	0	0
$25$	−3.82445	−0.764890
$26$	8.63747	1.69395
$27$	0	0
$28$	13.6604	2.58157
$29$	−0.519983	−0.0965585	−0.0482792	−	0.998834i	$-0.515374\pi$
$29$	−0.519983	−0.0965585	−0.0482792	+	0.998834i	$0.515374\pi$
$30$	0	0
$31$	−4.96809	−0.892295	−0.446147	−	0.894960i	$-0.647204\pi$
$31$	−4.96809	−0.892295	−0.446147	+	0.894960i	$0.647204\pi$
$32$	4.80636	0.849653
$33$	0	0
$34$	−10.4059	−1.78461
$35$	−4.37019	−0.738697
$36$	0	0
$37$	4.93356	0.811072	0.405536	−	0.914079i	$-0.367085\pi$
$37$	4.93356	0.811072	0.405536	+	0.914079i	$0.367085\pi$
$38$	−14.6947	−2.38380
$39$	0	0
$40$	3.49631	0.552815
$41$	−3.38864	−0.529216	−0.264608	−	0.964356i	$-0.585243\pi$
$41$	−3.38864	−0.529216	−0.264608	+	0.964356i	$0.585243\pi$
$42$	0	0
$43$	4.04743	0.617227	0.308614	−	0.951187i	$-0.400135\pi$
$43$	4.04743	0.617227	0.308614	+	0.951187i	$0.400135\pi$
$44$	−11.8441	−1.78556
$45$	0	0
$46$	16.4946	2.43200
$47$	−3.12585	−0.455952	−0.227976	−	0.973667i	$-0.573211\pi$
$47$	−3.12585	−0.455952	−0.227976	+	0.973667i	$0.573211\pi$
$48$	0	0
$49$	9.24645	1.32092
$50$	8.87823	1.25557
$51$	0	0
$52$	−12.6099	−1.74868
$53$	−8.30126	−1.14027	−0.570133	−	0.821552i	$-0.693108\pi$
$53$	−8.30126	−1.14027	−0.570133	+	0.821552i	$0.693108\pi$
$54$	0	0
$55$	3.78912	0.510925
$56$	−12.9978	−1.73690
$57$	0	0
$58$	1.20711	0.158501
$59$	3.99159	0.519661	0.259830	−	0.965654i	$-0.416333\pi$
$59$	3.99159	0.519661	0.259830	+	0.965654i	$0.416333\pi$
$60$	0	0
$61$	5.30912	0.679763	0.339882	−	0.940468i	$-0.389613\pi$
$61$	5.30912	0.679763	0.339882	+	0.940468i	$0.389613\pi$
$62$	11.5331	1.46471
$63$	0	0
$64$	−12.5732	−1.57165
$65$	4.03413	0.500372
$66$	0	0
$67$	−12.8573	−1.57077	−0.785387	−	0.619006i	$-0.787536\pi$
$67$	−12.8573	−1.57077	−0.785387	+	0.619006i	$0.787536\pi$
$68$	15.1917	1.84227
$69$	0	0
$70$	10.1451	1.21258
$71$	4.05577	0.481331	0.240665	−	0.970608i	$-0.422634\pi$
$71$	4.05577	0.481331	0.240665	+	0.970608i	$0.422634\pi$
$72$	0	0
$73$	−7.06713	−0.827145	−0.413572	−	0.910471i	$-0.635719\pi$
$73$	−7.06713	−0.827145	−0.413572	+	0.910471i	$0.635719\pi$
$74$	−11.4530	−1.33138
$75$	0	0
$76$	21.4530	2.46083
$77$	−14.0863	−1.60528
$78$	0	0
$79$	10.0440	1.13004	0.565021	−	0.825076i	$-0.308868\pi$
$79$	10.0440	1.13004	0.565021	+	0.825076i	$0.308868\pi$
$80$	−0.767380	−0.0857957
$81$	0	0
$82$	7.86652	0.868712
$83$	13.4518	1.47652	0.738262	−	0.674514i	$-0.235647\pi$
$83$	13.4518	1.47652	0.738262	+	0.674514i	$0.235647\pi$
$84$	0	0
$85$	−4.86009	−0.527151
$86$	−9.39587	−1.01318
$87$	0	0
$88$	11.2696	1.20134
$89$	−2.13957	−0.226793	−0.113397	−	0.993550i	$-0.536173\pi$
$89$	−2.13957	−0.226793	−0.113397	+	0.993550i	$0.536173\pi$
$90$	0	0
$91$	−14.9971	−1.57213
$92$	−24.0806	−2.51058
$93$	0	0
$94$	7.25648	0.748449
$95$	−6.86318	−0.704147
$96$	0	0
$97$	15.7277	1.59691	0.798454	−	0.602056i	$-0.205652\pi$
$97$	15.7277	1.59691	0.798454	+	0.602056i	$0.205652\pi$
$98$	−21.4651	−2.16830
$99$	0	0
$100$	−12.9614	−1.29614
$101$	−5.30496	−0.527863	−0.263932	−	0.964541i	$-0.585019\pi$
$101$	−5.30496	−0.527863	−0.263932	+	0.964541i	$0.585019\pi$
$102$	0	0
$103$	19.6899	1.94010	0.970051	−	0.242900i	$-0.0780987\pi$
$103$	19.6899	1.94010	0.970051	+	0.242900i	$0.0780987\pi$
$104$	11.9982	1.17652
$105$	0	0
$106$	19.2709	1.87175
$107$	14.4573	1.39764	0.698819	−	0.715298i	$-0.253709\pi$
$107$	14.4573	1.39764	0.698819	+	0.715298i	$0.253709\pi$
$108$	0	0
$109$	−1.68293	−0.161196	−0.0805978	−	0.996747i	$-0.525683\pi$
$109$	−1.68293	−0.161196	−0.0805978	+	0.996747i	$0.525683\pi$
$110$	−8.79623	−0.838687
$111$	0	0
$112$	2.85278	0.269563
$113$	18.3226	1.72364	0.861822	−	0.507212i	$-0.169324\pi$
$113$	18.3226	1.72364	0.861822	+	0.507212i	$0.169324\pi$
$114$	0	0
$115$	7.70380	0.718383
$116$	−1.76227	−0.163623
$117$	0	0
$118$	−9.26624	−0.853027
$119$	18.0677	1.65626
$120$	0	0
$121$	1.21337	0.110306
$122$	−12.3248	−1.11584
$123$	0	0
$124$	−16.8373	−1.51204
$125$	9.56772	0.855763
$126$	0	0
$127$	−13.2862	−1.17896	−0.589480	−	0.807783i	$-0.700667\pi$
$127$	−13.2862	−1.17896	−0.589480	+	0.807783i	$0.700667\pi$
$128$	19.5753	1.73023
$129$	0	0
$130$	−9.36499	−0.821364
$131$	−3.60250	−0.314752	−0.157376	−	0.987539i	$-0.550303\pi$
$131$	−3.60250	−0.314752	−0.157376	+	0.987539i	$0.550303\pi$
$132$	0	0
$133$	25.5143	2.21237
$134$	29.8475	2.57844
$135$	0	0
$136$	−14.4548	−1.23949
$137$	14.4067	1.23085	0.615424	−	0.788196i	$-0.288985\pi$
$137$	14.4067	1.23085	0.615424	+	0.788196i	$0.288985\pi$
$138$	0	0
$139$	−10.7017	−0.907706	−0.453853	−	0.891077i	$-0.649951\pi$
$139$	−10.7017	−0.907706	−0.453853	+	0.891077i	$0.649951\pi$
$140$	−14.8110	−1.25176
$141$	0	0
$142$	−9.41523	−0.790108
$143$	13.0031	1.08737
$144$	0	0
$145$	0.563781	0.0468195
$146$	16.4059	1.35776
$147$	0	0
$148$	16.7203	1.37440
$149$	21.3639	1.75020	0.875100	−	0.483942i	$-0.160795\pi$
$149$	21.3639	1.75020	0.875100	+	0.483942i	$0.160795\pi$
$150$	0	0
$151$	−7.76483	−0.631893	−0.315947	−	0.948777i	$-0.602322\pi$
$151$	−7.76483	−0.631893	−0.315947	+	0.948777i	$0.602322\pi$
$152$	−20.4124	−1.65566
$153$	0	0
$154$	32.7005	2.63508
$155$	5.38654	0.432658
$156$	0	0
$157$	18.4784	1.47474	0.737369	−	0.675491i	$-0.236068\pi$
$157$	18.4784	1.47474	0.737369	+	0.675491i	$0.236068\pi$
$158$	−23.3166	−1.85497
$159$	0	0
$160$	−5.21119	−0.411981
$161$	−28.6394	−2.25710
$162$	0	0
$163$	2.20321	0.172569	0.0862843	−	0.996271i	$-0.472501\pi$
$163$	2.20321	0.172569	0.0862843	+	0.996271i	$0.472501\pi$
$164$	−11.4844	−0.896782
$165$	0	0
$166$	−31.2275	−2.42373
$167$	−14.0799	−1.08954	−0.544769	−	0.838586i	$-0.683383\pi$
$167$	−14.0799	−1.08954	−0.544769	+	0.838586i	$0.683383\pi$
$168$	0	0
$169$	0.843860	0.0649123
$170$	11.2824	0.865323
$171$	0	0
$172$	13.7171	1.04592
$173$	−15.8233	−1.20302	−0.601512	−	0.798863i	$-0.705435\pi$
$173$	−15.8233	−1.20302	−0.601512	+	0.798863i	$0.705435\pi$
$174$	0	0
$175$	−15.4152	−1.16528
$176$	−2.47347	−0.186445
$177$	0	0
$178$	4.96688	0.372283
$179$	16.0477	1.19946	0.599731	−	0.800202i	$-0.295274\pi$
$179$	16.0477	1.19946	0.599731	+	0.800202i	$0.295274\pi$
$180$	0	0
$181$	−0.515468	−0.0383145	−0.0191572	−	0.999816i	$-0.506098\pi$
$181$	−0.515468	−0.0383145	−0.0191572	+	0.999816i	$0.506098\pi$
$182$	34.8149	2.58066
$183$	0	0
$184$	22.9125	1.68913
$185$	−5.34911	−0.393274
$186$	0	0
$187$	−15.6654	−1.14557
$188$	−10.5938	−0.772632
$189$	0	0
$190$	15.9325	1.15586
$191$	−23.0849	−1.67037	−0.835183	−	0.549971i	$-0.814639\pi$
$191$	−23.0849	−1.67037	−0.835183	+	0.549971i	$0.814639\pi$
$192$	0	0
$193$	−20.4961	−1.47534	−0.737670	−	0.675161i	$-0.764074\pi$
$193$	−20.4961	−1.47534	−0.737670	+	0.675161i	$0.764074\pi$
$194$	−36.5110	−2.62134
$195$	0	0
$196$	31.3371	2.23836
$197$	−18.5195	−1.31946	−0.659730	−	0.751503i	$-0.729329\pi$
$197$	−18.5195	−1.31946	−0.659730	+	0.751503i	$0.729329\pi$
$198$	0	0
$199$	23.6537	1.67676	0.838382	−	0.545083i	$-0.183502\pi$
$199$	23.6537	1.67676	0.838382	+	0.545083i	$0.183502\pi$
$200$	12.3327	0.872053
$201$	0	0
$202$	12.3152	0.866492
$203$	−2.09589	−0.147103
$204$	0	0
$205$	3.67406	0.256607
$206$	−45.7089	−3.18469
$207$	0	0
$208$	−2.63341	−0.182594
$209$	−22.1219	−1.53020
$210$	0	0
$211$	−1.43537	−0.0988146	−0.0494073	−	0.998779i	$-0.515733\pi$
$211$	−1.43537	−0.0988146	−0.0494073	+	0.998779i	$0.515733\pi$
$212$	−28.1337	−1.93223
$213$	0	0
$214$	−33.5618	−2.29423
$215$	−4.38834	−0.299282
$216$	0	0
$217$	−20.0248	−1.35937
$218$	3.90683	0.264604
$219$	0	0
$220$	12.8417	0.865786
$221$	−16.6783	−1.12191
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	19.3729	1.29441
$225$	0	0
$226$	−42.5348	−2.82937
$227$	6.42761	0.426616	0.213308	−	0.976985i	$-0.431576\pi$
$227$	6.42761	0.426616	0.213308	+	0.976985i	$0.431576\pi$
$228$	0	0
$229$	−0.891358	−0.0589026	−0.0294513	−	0.999566i	$-0.509376\pi$
$229$	−0.891358	−0.0589026	−0.0294513	+	0.999566i	$0.509376\pi$
$230$	−17.8839	−1.17923
$231$	0	0
$232$	1.67679	0.110087
$233$	29.5192	1.93387	0.966934	−	0.255027i	$-0.0820845\pi$
$233$	29.5192	1.93387	0.966934	+	0.255027i	$0.0820845\pi$
$234$	0	0
$235$	3.38914	0.221083
$236$	13.5279	0.880589
$237$	0	0
$238$	−41.9431	−2.71877
$239$	20.2612	1.31059	0.655294	−	0.755374i	$-0.272545\pi$
$239$	20.2612	1.31059	0.655294	+	0.755374i	$0.272545\pi$
$240$	0	0
$241$	26.6655	1.71768	0.858839	−	0.512245i	$-0.171186\pi$
$241$	26.6655	1.71768	0.858839	+	0.512245i	$0.171186\pi$
$242$	−2.81676	−0.181068
$243$	0	0
$244$	17.9931	1.15189
$245$	−10.0253	−0.640491
$246$	0	0
$247$	−23.5523	−1.49860
$248$	16.0206	1.01731
$249$	0	0
$250$	−22.2109	−1.40474
$251$	5.75397	0.363187	0.181594	−	0.983374i	$-0.441874\pi$
$251$	5.75397	0.363187	0.181594	+	0.983374i	$0.441874\pi$
$252$	0	0
$253$	24.8314	1.56114
$254$	30.8431	1.93527
$255$	0	0
$256$	−20.2964	−1.26853
$257$	−17.0675	−1.06464	−0.532320	−	0.846543i	$-0.678680\pi$
$257$	−17.0675	−1.06464	−0.532320	+	0.846543i	$0.678680\pi$
$258$	0	0
$259$	19.8856	1.23563
$260$	13.6720	0.847903
$261$	0	0
$262$	8.36299	0.516667
$263$	−3.76638	−0.232245	−0.116122	−	0.993235i	$-0.537046\pi$
$263$	−3.76638	−0.232245	−0.116122	+	0.993235i	$0.537046\pi$
$264$	0	0
$265$	9.00047	0.552894
$266$	−59.2299	−3.63162
$267$	0	0
$268$	−43.5747	−2.66175
$269$	8.76591	0.534467	0.267233	−	0.963632i	$-0.413891\pi$
$269$	8.76591	0.534467	0.267233	+	0.963632i	$0.413891\pi$
$270$	0	0
$271$	−17.3746	−1.05543	−0.527714	−	0.849422i	$-0.676951\pi$
$271$	−17.3746	−1.05543	−0.527714	+	0.849422i	$0.676951\pi$
$272$	3.17259	0.192366
$273$	0	0
$274$	−33.4443	−2.02045
$275$	13.3655	0.805972
$276$	0	0
$277$	14.2617	0.856901	0.428450	−	0.903565i	$-0.359060\pi$
$277$	14.2617	0.856901	0.428450	+	0.903565i	$0.359060\pi$
$278$	24.8434	1.49001
$279$	0	0
$280$	14.0925	0.842190
$281$	0.227724	0.0135849	0.00679243	−	0.999977i	$-0.497838\pi$
$281$	0.227724	0.0135849	0.00679243	+	0.999977i	$0.497838\pi$
$282$	0	0
$283$	11.7378	0.697741	0.348871	−	0.937171i	$-0.386565\pi$
$283$	11.7378	0.697741	0.348871	+	0.937171i	$0.386565\pi$
$284$	13.7454	0.815638
$285$	0	0
$286$	−30.1859	−1.78493
$287$	−13.6585	−0.806238
$288$	0	0
$289$	3.09313	0.181949
$290$	−1.30878	−0.0768545
$291$	0	0
$292$	−23.9512	−1.40164
$293$	−16.4490	−0.960958	−0.480479	−	0.877006i	$-0.659537\pi$
$293$	−16.4490	−0.960958	−0.480479	+	0.877006i	$0.659537\pi$
$294$	0	0
$295$	−4.32779	−0.251974
$296$	−15.9092	−0.924706
$297$	0	0
$298$	−49.5951	−2.87297
$299$	26.4370	1.52889
$300$	0	0
$301$	16.3139	0.940319
$302$	18.0256	1.03726
$303$	0	0
$304$	4.48016	0.256955
$305$	−5.75630	−0.329605
$306$	0	0
$307$	−9.40896	−0.536998	−0.268499	−	0.963280i	$-0.586528\pi$
$307$	−9.40896	−0.536998	−0.268499	+	0.963280i	$0.586528\pi$
$308$	−47.7398	−2.72023
$309$	0	0
$310$	−12.5045	−0.710210
$311$	2.21548	0.125628	0.0628141	−	0.998025i	$-0.479992\pi$
$311$	2.21548	0.125628	0.0628141	+	0.998025i	$0.479992\pi$
$312$	0	0
$313$	−7.87381	−0.445054	−0.222527	−	0.974927i	$-0.571431\pi$
$313$	−7.87381	−0.445054	−0.222527	+	0.974927i	$0.571431\pi$
$314$	−42.8965	−2.42079
$315$	0	0
$316$	34.0402	1.91491
$317$	17.2995	0.971636	0.485818	−	0.874060i	$-0.338522\pi$
$317$	17.2995	0.971636	0.485818	+	0.874060i	$0.338522\pi$
$318$	0	0
$319$	1.81722	0.101745
$320$	13.6322	0.762066
$321$	0	0
$322$	66.4846	3.70504
$323$	28.3745	1.57880
$324$	0	0
$325$	14.2298	0.789325
$326$	−5.11462	−0.283273
$327$	0	0
$328$	10.9273	0.603361
$329$	−12.5993	−0.694624
$330$	0	0
$331$	−17.3691	−0.954692	−0.477346	−	0.878715i	$-0.658401\pi$
$331$	−17.3691	−0.954692	−0.477346	+	0.878715i	$0.658401\pi$
$332$	45.5893	2.50204
$333$	0	0
$334$	32.6857	1.78848
$335$	13.9403	0.761639
$336$	0	0
$337$	−5.73761	−0.312548	−0.156274	−	0.987714i	$-0.549948\pi$
$337$	−5.73761	−0.312548	−0.156274	+	0.987714i	$0.549948\pi$
$338$	−1.95897	−0.106554
$339$	0	0
$340$	−16.4713	−0.893282
$341$	17.3623	0.940220
$342$	0	0
$343$	9.05474	0.488910
$344$	−13.0517	−0.703703
$345$	0	0
$346$	36.7329	1.97477
$347$	−1.73910	−0.0933600	−0.0466800	−	0.998910i	$-0.514864\pi$
$347$	−1.73910	−0.0933600	−0.0466800	+	0.998910i	$0.514864\pi$
$348$	0	0
$349$	−17.5805	−0.941061	−0.470530	−	0.882384i	$-0.655937\pi$
$349$	−17.5805	−0.941061	−0.470530	+	0.882384i	$0.655937\pi$
$350$	35.7854	1.91281
$351$	0	0
$352$	−16.7971	−0.895288
$353$	−15.8221	−0.842124	−0.421062	−	0.907032i	$-0.638342\pi$
$353$	−15.8221	−0.842124	−0.421062	+	0.907032i	$0.638342\pi$
$354$	0	0
$355$	−4.39738	−0.233389
$356$	−7.25119	−0.384312
$357$	0	0
$358$	−37.2538	−1.96893
$359$	−4.96315	−0.261945	−0.130973	−	0.991386i	$-0.541810\pi$
$359$	−4.96315	−0.261945	−0.130973	+	0.991386i	$0.541810\pi$
$360$	0	0
$361$	21.0690	1.10889
$362$	1.19663	0.0628935
$363$	0	0
$364$	−50.8266	−2.66404
$365$	7.66239	0.401068
$366$	0	0
$367$	−16.5439	−0.863584	−0.431792	−	0.901973i	$-0.642118\pi$
$367$	−16.5439	−0.863584	−0.431792	+	0.901973i	$0.642118\pi$
$368$	−5.02891	−0.262150
$369$	0	0
$370$	12.4176	0.645563
$371$	−33.4598	−1.73715
$372$	0	0
$373$	17.9263	0.928188	0.464094	−	0.885786i	$-0.346380\pi$
$373$	17.9263	0.928188	0.464094	+	0.885786i	$0.346380\pi$
$374$	36.3663	1.88046
$375$	0	0
$376$	10.0799	0.519833
$377$	1.93472	0.0996431
$378$	0	0
$379$	30.6717	1.57550	0.787749	−	0.615996i	$-0.211246\pi$
$379$	30.6717	1.57550	0.787749	+	0.615996i	$0.211246\pi$
$380$	−23.2599	−1.19321
$381$	0	0
$382$	53.5903	2.74192
$383$	32.2046	1.64558	0.822789	−	0.568347i	$-0.192417\pi$
$383$	32.2046	1.64558	0.822789	+	0.568347i	$0.192417\pi$
$384$	0	0
$385$	15.2728	0.778372
$386$	47.5805	2.42178
$387$	0	0
$388$	53.3027	2.70603
$389$	1.34752	0.0683217	0.0341609	−	0.999416i	$-0.489124\pi$
$389$	1.34752	0.0683217	0.0341609	+	0.999416i	$0.489124\pi$
$390$	0	0
$391$	−31.8499	−1.61072
$392$	−29.8170	−1.50599
$393$	0	0
$394$	42.9919	2.16590
$395$	−10.8900	−0.547937
$396$	0	0
$397$	−13.7564	−0.690416	−0.345208	−	0.938526i	$-0.612192\pi$
$397$	−13.7564	−0.690416	−0.345208	+	0.938526i	$0.612192\pi$
$398$	−54.9106	−2.75242
$399$	0	0
$400$	−2.70681	−0.135341
$401$	−3.66356	−0.182950	−0.0914748	−	0.995807i	$-0.529158\pi$
$401$	−3.66356	−0.182950	−0.0914748	+	0.995807i	$0.529158\pi$
$402$	0	0
$403$	18.4849	0.920800
$404$	−17.9790	−0.894489
$405$	0	0
$406$	4.86549	0.241470
$407$	−17.2416	−0.854636
$408$	0	0
$409$	−11.6297	−0.575053	−0.287526	−	0.957773i	$-0.592833\pi$
$409$	−11.6297	−0.575053	−0.287526	+	0.957773i	$0.592833\pi$
$410$	−8.52911	−0.421223
$411$	0	0
$412$	66.7309	3.28759
$413$	16.0888	0.791681
$414$	0	0
$415$	−14.5848	−0.715940
$416$	−17.8832	−0.876795
$417$	0	0
$418$	51.3547	2.51184
$419$	26.0654	1.27338	0.636690	−	0.771120i	$-0.280303\pi$
$419$	26.0654	1.27338	0.636690	+	0.771120i	$0.280303\pi$
$420$	0	0
$421$	31.5514	1.53772	0.768861	−	0.639416i	$-0.220824\pi$
$421$	31.5514	1.53772	0.768861	+	0.639416i	$0.220824\pi$
$422$	3.33212	0.162205
$423$	0	0
$424$	26.7691	1.30002
$425$	−17.1432	−0.831569
$426$	0	0
$427$	21.3994	1.03559
$428$	48.9971	2.36836
$429$	0	0
$430$	10.1873	0.491274
$431$	−12.6638	−0.609993	−0.304997	−	0.952353i	$-0.598655\pi$
$431$	−12.6638	−0.609993	−0.304997	+	0.952353i	$0.598655\pi$
$432$	0	0
$433$	−30.8909	−1.48452	−0.742262	−	0.670110i	$-0.766247\pi$
$433$	−30.8909	−1.48452	−0.742262	+	0.670110i	$0.766247\pi$
$434$	46.4864	2.23142
$435$	0	0
$436$	−5.70361	−0.273153
$437$	−44.9768	−2.15153
$438$	0	0
$439$	12.8373	0.612692	0.306346	−	0.951920i	$-0.400893\pi$
$439$	12.8373	0.612692	0.306346	+	0.951920i	$0.400893\pi$
$440$	−12.2188	−0.582507
$441$	0	0
$442$	38.7178	1.84162
$443$	26.6816	1.26768	0.633841	−	0.773464i	$-0.281478\pi$
$443$	26.6816	1.26768	0.633841	+	0.773464i	$0.281478\pi$
$444$	0	0
$445$	2.31978	0.109968
$446$	−2.32144	−0.109923
$447$	0	0
$448$	−50.6787	−2.39434
$449$	34.5147	1.62885	0.814425	−	0.580268i	$-0.197052\pi$
$449$	34.5147	1.62885	0.814425	+	0.580268i	$0.197052\pi$
$450$	0	0
$451$	11.8425	0.557641
$452$	62.0969	2.92079
$453$	0	0
$454$	−14.9213	−0.700293
$455$	16.2603	0.762295
$456$	0	0
$457$	38.2803	1.79067	0.895337	−	0.445389i	$-0.146934\pi$
$457$	38.2803	1.79067	0.895337	+	0.445389i	$0.146934\pi$
$458$	2.06924	0.0966891
$459$	0	0
$460$	26.1089	1.21733
$461$	−22.9465	−1.06873	−0.534363	−	0.845255i	$-0.679449\pi$
$461$	−22.9465	−1.06873	−0.534363	+	0.845255i	$0.679449\pi$
$462$	0	0
$463$	−29.8411	−1.38683	−0.693416	−	0.720537i	$-0.743895\pi$
$463$	−29.8411	−1.38683	−0.693416	+	0.720537i	$0.743895\pi$
$464$	−0.368027	−0.0170852
$465$	0	0
$466$	−68.5271	−3.17446
$467$	10.0029	0.462877	0.231439	−	0.972849i	$-0.425657\pi$
$467$	10.0029	0.462877	0.231439	+	0.972849i	$0.425657\pi$
$468$	0	0
$469$	−51.8239	−2.39301
$470$	−7.86768	−0.362909
$471$	0	0
$472$	−12.8717	−0.592467
$473$	−14.1448	−0.650379
$474$	0	0
$475$	−24.2088	−1.11078
$476$	61.2331	2.80662
$477$	0	0
$478$	−47.0352	−2.15134
$479$	−1.16823	−0.0533776	−0.0266888	−	0.999644i	$-0.508496\pi$
$479$	−1.16823	−0.0533776	−0.0266888	+	0.999644i	$0.508496\pi$
$480$	0	0
$481$	−18.3565	−0.836983
$482$	−61.9025	−2.81958
$483$	0	0
$484$	4.11221	0.186919
$485$	−17.0524	−0.774312
$486$	0	0
$487$	27.9657	1.26725	0.633625	−	0.773641i	$-0.281566\pi$
$487$	27.9657	1.26725	0.633625	+	0.773641i	$0.281566\pi$
$488$	−17.1203	−0.775000
$489$	0	0
$490$	23.2731	1.05137
$491$	19.2069	0.866794	0.433397	−	0.901203i	$-0.357315\pi$
$491$	19.2069	0.866794	0.433397	+	0.901203i	$0.357315\pi$
$492$	0	0
$493$	−2.33084	−0.104976
$494$	54.6752	2.45996
$495$	0	0
$496$	−3.51624	−0.157884
$497$	16.3475	0.733287
$498$	0	0
$499$	−9.93756	−0.444866	−0.222433	−	0.974948i	$-0.571400\pi$
$499$	−9.93756	−0.444866	−0.222433	+	0.974948i	$0.571400\pi$
$500$	32.4259	1.45013
$501$	0	0
$502$	−13.3575	−0.596175
$503$	−15.9285	−0.710218	−0.355109	−	0.934825i	$-0.615556\pi$
$503$	−15.9285	−0.710218	−0.355109	+	0.934825i	$0.615556\pi$
$504$	0	0
$505$	5.75179	0.255951
$506$	−57.6447	−2.56262
$507$	0	0
$508$	−45.0282	−1.99780
$509$	−7.49117	−0.332040	−0.166020	−	0.986122i	$-0.553092\pi$
$509$	−7.49117	−0.332040	−0.166020	+	0.986122i	$0.553092\pi$
$510$	0	0
$511$	−28.4854	−1.26012
$512$	7.96644	0.352070
$513$	0	0
$514$	39.6211	1.74761
$515$	−21.3483	−0.940721
$516$	0	0
$517$	10.9241	0.480442
$518$	−46.1634	−2.02830
$519$	0	0
$520$	−13.0088	−0.570476
$521$	23.7532	1.04065	0.520323	−	0.853970i	$-0.325812\pi$
$521$	23.7532	1.04065	0.520323	+	0.853970i	$0.325812\pi$
$522$	0	0
$523$	5.45067	0.238341	0.119171	−	0.992874i	$-0.461976\pi$
$523$	5.45067	0.238341	0.119171	+	0.992874i	$0.461976\pi$
$524$	−12.2092	−0.533362
$525$	0	0
$526$	8.74343	0.381232
$527$	−22.2696	−0.970080
$528$	0	0
$529$	27.4857	1.19503
$530$	−20.8941	−0.907580
$531$	0	0
$532$	86.4703	3.74896
$533$	12.6082	0.546123
$534$	0	0
$535$	−15.6750	−0.677690
$536$	41.4610	1.79084
$537$	0	0
$538$	−20.3496	−0.877331
$539$	−32.3141	−1.39187
$540$	0	0
$541$	4.55299	0.195748	0.0978741	−	0.995199i	$-0.468796\pi$
$541$	4.55299	0.195748	0.0978741	+	0.995199i	$0.468796\pi$
$542$	40.3340	1.73249
$543$	0	0
$544$	21.5447	0.923721
$545$	1.82468	0.0781608
$546$	0	0
$547$	−23.2903	−0.995820	−0.497910	−	0.867229i	$-0.665899\pi$
$547$	−23.2903	−0.995820	−0.497910	+	0.867229i	$0.665899\pi$
$548$	48.8257	2.08573
$549$	0	0
$550$	−31.0273	−1.32301
$551$	−3.29150	−0.140223
$552$	0	0
$553$	40.4844	1.72157
$554$	−33.1076	−1.40661
$555$	0	0
$556$	−36.2690	−1.53815
$557$	41.0731	1.74032	0.870162	−	0.492765i	$-0.164014\pi$
$557$	41.0731	1.74032	0.870162	+	0.492765i	$0.164014\pi$
$558$	0	0
$559$	−15.0594	−0.636945
$560$	−3.09307	−0.130706
$561$	0	0
$562$	−0.528648	−0.0222997
$563$	1.92263	0.0810293	0.0405147	−	0.999179i	$-0.487100\pi$
$563$	1.92263	0.0810293	0.0405147	+	0.999179i	$0.487100\pi$
$564$	0	0
$565$	−19.8659	−0.835763
$566$	−27.2487	−1.14535
$567$	0	0
$568$	−13.0786	−0.548767
$569$	38.8106	1.62702	0.813512	−	0.581549i	$-0.197553\pi$
$569$	38.8106	1.62702	0.813512	+	0.581549i	$0.197553\pi$
$570$	0	0
$571$	18.4396	0.771673	0.385836	−	0.922567i	$-0.373913\pi$
$571$	18.4396	0.771673	0.385836	+	0.922567i	$0.373913\pi$
$572$	44.0687	1.84260
$573$	0	0
$574$	31.7075	1.32345
$575$	27.1740	1.13323
$576$	0	0
$577$	35.0551	1.45936	0.729681	−	0.683788i	$-0.239669\pi$
$577$	35.0551	1.45936	0.729681	+	0.683788i	$0.239669\pi$
$578$	−7.18053	−0.298670
$579$	0	0
$580$	1.91071	0.0793377
$581$	54.2199	2.24942
$582$	0	0
$583$	29.0109	1.20151
$584$	22.7894	0.943031
$585$	0	0
$586$	38.1853	1.57742
$587$	25.5045	1.05268	0.526342	−	0.850273i	$-0.323563\pi$
$587$	25.5045	1.05268	0.526342	+	0.850273i	$0.323563\pi$
$588$	0	0
$589$	−31.4480	−1.29579
$590$	10.0467	0.413617
$591$	0	0
$592$	3.49181	0.143512
$593$	−4.81765	−0.197837	−0.0989186	−	0.995096i	$-0.531538\pi$
$593$	−4.81765	−0.197837	−0.0989186	+	0.995096i	$0.531538\pi$
$594$	0	0
$595$	−19.5895	−0.803092
$596$	72.4043	2.96580
$597$	0	0
$598$	−61.3720	−2.50969
$599$	−12.6818	−0.518165	−0.259083	−	0.965855i	$-0.583420\pi$
$599$	−12.6818	−0.518165	−0.259083	+	0.965855i	$0.583420\pi$
$600$	0	0
$601$	10.1567	0.414299	0.207150	−	0.978309i	$-0.433581\pi$
$601$	10.1567	0.414299	0.207150	+	0.978309i	$0.433581\pi$
$602$	−37.8718	−1.54354
$603$	0	0
$604$	−26.3157	−1.07077
$605$	−1.31557	−0.0534854
$606$	0	0
$607$	31.1120	1.26280	0.631399	−	0.775458i	$-0.282481\pi$
$607$	31.1120	1.26280	0.631399	+	0.775458i	$0.282481\pi$
$608$	30.4243	1.23387
$609$	0	0
$610$	13.3629	0.541049
$611$	11.6305	0.470518
$612$	0	0
$613$	26.3440	1.06402	0.532012	−	0.846737i	$-0.321436\pi$
$613$	26.3440	1.06402	0.532012	+	0.846737i	$0.321436\pi$
$614$	21.8424	0.881486
$615$	0	0
$616$	45.4241	1.83019
$617$	−18.6505	−0.750840	−0.375420	−	0.926855i	$-0.622501\pi$
$617$	−18.6505	−0.750840	−0.375420	+	0.926855i	$0.622501\pi$
$618$	0	0
$619$	−23.9820	−0.963919	−0.481960	−	0.876193i	$-0.660075\pi$
$619$	−23.9820	−0.963919	−0.481960	+	0.876193i	$0.660075\pi$
$620$	18.2555	0.733158
$621$	0	0
$622$	−5.14310	−0.206220
$623$	−8.62392	−0.345510
$624$	0	0
$625$	8.74864	0.349946
$626$	18.2786	0.730559
$627$	0	0
$628$	62.6250	2.49901
$629$	22.1149	0.881777
$630$	0	0
$631$	−21.8217	−0.868708	−0.434354	−	0.900742i	$-0.643023\pi$
$631$	−21.8217	−0.868708	−0.434354	+	0.900742i	$0.643023\pi$
$632$	−32.3890	−1.28837
$633$	0	0
$634$	−40.1597	−1.59495
$635$	14.4053	0.571656
$636$	0	0
$637$	−34.4036	−1.36312
$638$	−4.21857	−0.167015
$639$	0	0
$640$	−21.2241	−0.838955
$641$	5.71339	0.225665	0.112833	−	0.993614i	$-0.464008\pi$
$641$	5.71339	0.225665	0.112833	+	0.993614i	$0.464008\pi$
$642$	0	0
$643$	−8.88697	−0.350468	−0.175234	−	0.984527i	$-0.556068\pi$
$643$	−8.88697	−0.350468	−0.175234	+	0.984527i	$0.556068\pi$
$644$	−97.0614	−3.82476
$645$	0	0
$646$	−65.8697	−2.59161
$647$	17.2033	0.676332	0.338166	−	0.941086i	$-0.390193\pi$
$647$	17.2033	0.676332	0.338166	+	0.941086i	$0.390193\pi$
$648$	0	0
$649$	−13.9497	−0.547572
$650$	−33.0335	−1.29568
$651$	0	0
$652$	7.46688	0.292426
$653$	22.3402	0.874238	0.437119	−	0.899404i	$-0.355999\pi$
$653$	22.3402	0.874238	0.437119	+	0.899404i	$0.355999\pi$
$654$	0	0
$655$	3.90593	0.152617
$656$	−2.39836	−0.0936403
$657$	0	0
$658$	29.2486	1.14023
$659$	−3.26690	−0.127260	−0.0636301	−	0.997974i	$-0.520268\pi$
$659$	−3.26690	−0.127260	−0.0636301	+	0.997974i	$0.520268\pi$
$660$	0	0
$661$	32.4045	1.26039	0.630194	−	0.776438i	$-0.282975\pi$
$661$	32.4045	1.26039	0.630194	+	0.776438i	$0.282975\pi$
$662$	40.3214	1.56713
$663$	0	0
$664$	−43.3779	−1.68339
$665$	−27.6633	−1.07274
$666$	0	0
$667$	3.69465	0.143057
$668$	−47.7182	−1.84627
$669$	0	0
$670$	−32.3616	−1.25024
$671$	−18.5541	−0.716273
$672$	0	0
$673$	21.8787	0.843361	0.421681	−	0.906744i	$-0.361440\pi$
$673$	21.8787	0.843361	0.421681	+	0.906744i	$0.361440\pi$
$674$	13.3195	0.513049
$675$	0	0
$676$	2.85992	0.109997
$677$	−11.7411	−0.451248	−0.225624	−	0.974214i	$-0.572442\pi$
$677$	−11.7411	−0.451248	−0.225624	+	0.974214i	$0.572442\pi$
$678$	0	0
$679$	63.3935	2.43282
$680$	15.6723	0.601007
$681$	0	0
$682$	−40.3055	−1.54338
$683$	−4.73224	−0.181074	−0.0905370	−	0.995893i	$-0.528858\pi$
$683$	−4.73224	−0.181074	−0.0905370	+	0.995893i	$0.528858\pi$
$684$	0	0
$685$	−15.6202	−0.596816
$686$	−21.0200	−0.802549
$687$	0	0
$688$	2.86463	0.109213
$689$	30.8868	1.17669
$690$	0	0
$691$	−1.32161	−0.0502765	−0.0251382	−	0.999684i	$-0.508003\pi$
$691$	−1.32161	−0.0502765	−0.0251382	+	0.999684i	$0.508003\pi$
$692$	−53.6267	−2.03858
$693$	0	0
$694$	4.03723	0.153251
$695$	11.6031	0.440130
$696$	0	0
$697$	−15.1897	−0.575351
$698$	40.8120	1.54476
$699$	0	0
$700$	−52.2434	−1.97462
$701$	12.1934	0.460539	0.230270	−	0.973127i	$-0.426039\pi$
$701$	12.1934	0.460539	0.230270	+	0.973127i	$0.426039\pi$
$702$	0	0
$703$	31.2295	1.17784
$704$	43.9404	1.65607
$705$	0	0
$706$	36.7300	1.38235
$707$	−21.3826	−0.804177
$708$	0	0
$709$	34.2838	1.28756	0.643778	−	0.765212i	$-0.277366\pi$
$709$	34.2838	1.28756	0.643778	+	0.765212i	$0.277366\pi$
$710$	10.2083	0.383109
$711$	0	0
$712$	6.89946	0.258568
$713$	35.2999	1.32199
$714$	0	0
$715$	−14.0983	−0.527247
$716$	54.3872	2.03255
$717$	0	0
$718$	11.5217	0.429985
$719$	−13.3194	−0.496731	−0.248365	−	0.968666i	$-0.579893\pi$
$719$	−13.3194	−0.496731	−0.248365	+	0.968666i	$0.579893\pi$
$720$	0	0
$721$	79.3638	2.95566
$722$	−48.9105	−1.82026
$723$	0	0
$724$	−1.74697	−0.0649256
$725$	1.98865	0.0738566
$726$	0	0
$727$	−39.6461	−1.47039	−0.735196	−	0.677854i	$-0.762910\pi$
$727$	−39.6461	−1.47039	−0.735196	+	0.677854i	$0.762910\pi$
$728$	48.3612	1.79239
$729$	0	0
$730$	−17.7878	−0.658355
$731$	18.1427	0.671034
$732$	0	0
$733$	5.49184	0.202846	0.101423	−	0.994843i	$-0.467660\pi$
$733$	5.49184	0.202846	0.101423	+	0.994843i	$0.467660\pi$
$734$	38.4057	1.41758
$735$	0	0
$736$	−34.1508	−1.25881
$737$	44.9333	1.65514
$738$	0	0
$739$	31.5699	1.16132	0.580658	−	0.814147i	$-0.302795\pi$
$739$	31.5699	1.16132	0.580658	+	0.814147i	$0.302795\pi$
$740$	−18.1286	−0.666422
$741$	0	0
$742$	77.6750	2.85154
$743$	−30.8268	−1.13093	−0.565463	−	0.824774i	$-0.691303\pi$
$743$	−30.8268	−1.13093	−0.565463	+	0.824774i	$0.691303\pi$
$744$	0	0
$745$	−23.1634	−0.848640
$746$	−41.6148	−1.52363
$747$	0	0
$748$	−53.0915	−1.94122
$749$	58.2728	2.12924
$750$	0	0
$751$	−38.3563	−1.39964	−0.699822	−	0.714318i	$-0.746737\pi$
$751$	−38.3563	−1.39964	−0.699822	+	0.714318i	$0.746737\pi$
$752$	−2.21237	−0.0806769
$753$	0	0
$754$	−4.49134	−0.163565
$755$	8.41886	0.306394
$756$	0	0
$757$	−21.0275	−0.764257	−0.382128	−	0.924109i	$-0.624809\pi$
$757$	−21.0275	−0.764257	−0.382128	+	0.924109i	$0.624809\pi$
$758$	−71.2025	−2.58619
$759$	0	0
$760$	22.1317	0.802800
$761$	−0.947754	−0.0343560	−0.0171780	−	0.999852i	$-0.505468\pi$
$761$	−0.947754	−0.0343560	−0.0171780	+	0.999852i	$0.505468\pi$
$762$	0	0
$763$	−6.78337	−0.245575
$764$	−78.2370	−2.83051
$765$	0	0
$766$	−74.7611	−2.70123
$767$	−14.8516	−0.536262
$768$	0	0
$769$	37.9725	1.36932	0.684662	−	0.728860i	$-0.259950\pi$
$769$	37.9725	1.36932	0.684662	+	0.728860i	$0.259950\pi$
$770$	−35.4549	−1.27770
$771$	0	0
$772$	−69.4631	−2.50003
$773$	50.0561	1.80039	0.900197	−	0.435483i	$-0.143422\pi$
$773$	50.0561	1.80039	0.900197	+	0.435483i	$0.143422\pi$
$774$	0	0
$775$	19.0002	0.682507
$776$	−50.7172	−1.82064
$777$	0	0
$778$	−3.12818	−0.112151
$779$	−21.4501	−0.768530
$780$	0	0
$781$	−14.1739	−0.507183
$782$	73.9377	2.64401
$783$	0	0
$784$	6.54432	0.233726
$785$	−20.0348	−0.715073
$786$	0	0
$787$	−9.60818	−0.342495	−0.171247	−	0.985228i	$-0.554780\pi$
$787$	−9.60818	−0.342495	−0.171247	+	0.985228i	$0.554780\pi$
$788$	−62.7643	−2.23589
$789$	0	0
$790$	25.2806	0.899443
$791$	73.8526	2.62590
$792$	0	0
$793$	−19.7538	−0.701479
$794$	31.9348	1.13332
$795$	0	0
$796$	80.1645	2.84136
$797$	−9.83233	−0.348279	−0.174140	−	0.984721i	$-0.555714\pi$
$797$	−9.83233	−0.348279	−0.174140	+	0.984721i	$0.555714\pi$
$798$	0	0
$799$	−14.0117	−0.495700
$800$	−18.3817	−0.649890
$801$	0	0
$802$	8.50475	0.300313
$803$	24.6979	0.871571
$804$	0	0
$805$	31.0516	1.09443
$806$	−42.9117	−1.51150
$807$	0	0
$808$	17.1069	0.601819
$809$	−21.8923	−0.769692	−0.384846	−	0.922981i	$-0.625745\pi$
$809$	−21.8923	−0.769692	−0.384846	+	0.922981i	$0.625745\pi$
$810$	0	0
$811$	−20.4077	−0.716613	−0.358306	−	0.933604i	$-0.616646\pi$
$811$	−20.4077	−0.716613	−0.358306	+	0.933604i	$0.616646\pi$
$812$	−7.10317	−0.249272
$813$	0	0
$814$	40.0254	1.40289
$815$	−2.38878	−0.0836754
$816$	0	0
$817$	25.6203	0.896339
$818$	26.9977	0.943953
$819$	0	0
$820$	12.4517	0.434833
$821$	35.7395	1.24732	0.623658	−	0.781697i	$-0.285646\pi$
$821$	35.7395	1.24732	0.623658	+	0.781697i	$0.285646\pi$
$822$	0	0
$823$	−36.1998	−1.26185	−0.630923	−	0.775845i	$-0.717324\pi$
$823$	−36.1998	−1.26185	−0.630923	+	0.775845i	$0.717324\pi$
$824$	−63.4940	−2.21192
$825$	0	0
$826$	−37.3493	−1.29955
$827$	−27.5629	−0.958454	−0.479227	−	0.877691i	$-0.659083\pi$
$827$	−27.5629	−0.958454	−0.479227	+	0.877691i	$0.659083\pi$
$828$	0	0
$829$	36.4260	1.26513	0.632563	−	0.774509i	$-0.282003\pi$
$829$	36.4260	1.26513	0.632563	+	0.774509i	$0.282003\pi$
$830$	33.8578	1.17522
$831$	0	0
$832$	46.7816	1.62186
$833$	41.4475	1.43607
$834$	0	0
$835$	15.2659	0.528297
$836$	−74.9731	−2.59300
$837$	0	0
$838$	−60.5094	−2.09026
$839$	15.1247	0.522164	0.261082	−	0.965317i	$-0.415921\pi$
$839$	15.1247	0.522164	0.261082	+	0.965317i	$0.415921\pi$
$840$	0	0
$841$	−28.7296	−0.990676
$842$	−73.2448	−2.52418
$843$	0	0
$844$	−4.86459	−0.167446
$845$	−0.914938	−0.0314748
$846$	0	0
$847$	4.89070	0.168046
$848$	−5.87535	−0.201760
$849$	0	0
$850$	39.7970	1.36503
$851$	−35.0546	−1.20165
$852$	0	0
$853$	−48.3555	−1.65566	−0.827830	−	0.560979i	$-0.810425\pi$
$853$	−48.3555	−1.65566	−0.827830	+	0.560979i	$0.810425\pi$
$854$	−49.6775	−1.69993
$855$	0	0
$856$	−46.6204	−1.59345
$857$	5.39794	0.184390	0.0921951	−	0.995741i	$-0.470612\pi$
$857$	5.39794	0.184390	0.0921951	+	0.995741i	$0.470612\pi$
$858$	0	0
$859$	44.1670	1.50696	0.753479	−	0.657472i	$-0.228374\pi$
$859$	44.1670	1.50696	0.753479	+	0.657472i	$0.228374\pi$
$860$	−14.8725	−0.507148
$861$	0	0
$862$	29.3983	1.00131
$863$	17.8409	0.607313	0.303656	−	0.952782i	$-0.401792\pi$
$863$	17.8409	0.607313	0.303656	+	0.952782i	$0.401792\pi$
$864$	0	0
$865$	17.1561	0.583325
$866$	71.7115	2.43686
$867$	0	0
$868$	−67.8659	−2.30352
$869$	−35.1015	−1.19074
$870$	0	0
$871$	47.8387	1.62095
$872$	5.42695	0.183780
$873$	0	0
$874$	104.411	3.53175
$875$	38.5645	1.30372
$876$	0	0
$877$	−40.4706	−1.36660	−0.683298	−	0.730140i	$-0.739455\pi$
$877$	−40.4706	−1.36660	−0.683298	+	0.730140i	$0.739455\pi$
$878$	−29.8011	−1.00574
$879$	0	0
$880$	2.68181	0.0904039
$881$	37.1454	1.25146	0.625731	−	0.780039i	$-0.284801\pi$
$881$	37.1454	1.25146	0.625731	+	0.780039i	$0.284801\pi$
$882$	0	0
$883$	25.6965	0.864757	0.432378	−	0.901692i	$-0.357675\pi$
$883$	25.6965	0.864757	0.432378	+	0.901692i	$0.357675\pi$
$884$	−56.5244	−1.90112
$885$	0	0
$886$	−61.9398	−2.08091
$887$	50.6538	1.70079	0.850394	−	0.526146i	$-0.176363\pi$
$887$	50.6538	1.70079	0.850394	+	0.526146i	$0.176363\pi$
$888$	0	0
$889$	−53.5525	−1.79609
$890$	−5.38523	−0.180513
$891$	0	0
$892$	3.38909	0.113475
$893$	−19.7867	−0.662135
$894$	0	0
$895$	−17.3994	−0.581598
$896$	78.9019	2.63593
$897$	0	0
$898$	−80.1239	−2.67377
$899$	2.58332	0.0861586
$900$	0	0
$901$	−37.2107	−1.23967
$902$	−27.4916	−0.915371
$903$	0	0
$904$	−59.0848	−1.96513
$905$	0.558886	0.0185780
$906$	0	0
$907$	10.3424	0.343414	0.171707	−	0.985148i	$-0.445072\pi$
$907$	10.3424	0.343414	0.171707	+	0.985148i	$0.445072\pi$
$908$	21.7838	0.722920
$909$	0	0
$910$	−37.7474	−1.25131
$911$	−22.7930	−0.755166	−0.377583	−	0.925976i	$-0.623245\pi$
$911$	−22.7930	−0.755166	−0.377583	+	0.925976i	$0.623245\pi$
$912$	0	0
$913$	−47.0108	−1.55583
$914$	−88.8654	−2.93941
$915$	0	0
$916$	−3.02090	−0.0998132
$917$	−14.5206	−0.479511
$918$	0	0
$919$	39.8129	1.31331	0.656653	−	0.754193i	$-0.271972\pi$
$919$	39.8129	1.31331	0.656653	+	0.754193i	$0.271972\pi$
$920$	−24.8424	−0.819031
$921$	0	0
$922$	53.2691	1.75432
$923$	−15.0904	−0.496707
$924$	0	0
$925$	−18.8681	−0.620381
$926$	69.2743	2.27650
$927$	0	0
$928$	−2.49923	−0.0820412
$929$	36.4567	1.19611	0.598053	−	0.801457i	$-0.295941\pi$
$929$	36.4567	1.19611	0.598053	+	0.801457i	$0.295941\pi$
$930$	0	0
$931$	58.5301	1.91825
$932$	100.043	3.27703
$933$	0	0
$934$	−23.2211	−0.759817
$935$	16.9849	0.555465
$936$	0	0
$937$	−11.8773	−0.388015	−0.194008	−	0.981000i	$-0.562149\pi$
$937$	−11.8773	−0.388015	−0.194008	+	0.981000i	$0.562149\pi$
$938$	120.306	3.92814
$939$	0	0
$940$	11.4861	0.374635
$941$	31.1078	1.01409	0.507043	−	0.861921i	$-0.330739\pi$
$941$	31.1078	1.01409	0.507043	+	0.861921i	$0.330739\pi$
$942$	0	0
$943$	24.0774	0.784067
$944$	2.82511	0.0919495
$945$	0	0
$946$	32.8363	1.06760
$947$	44.1368	1.43425	0.717127	−	0.696942i	$-0.245457\pi$
$947$	44.1368	1.43425	0.717127	+	0.696942i	$0.245457\pi$
$948$	0	0
$949$	26.2949	0.853569
$950$	56.1993	1.82335
$951$	0	0
$952$	−58.2629	−1.88831
$953$	16.1285	0.522452	0.261226	−	0.965278i	$-0.415873\pi$
$953$	16.1285	0.522452	0.261226	+	0.965278i	$0.415873\pi$
$954$	0	0
$955$	25.0293	0.809931
$956$	68.6671	2.22085
$957$	0	0
$958$	2.71197	0.0876197
$959$	58.0690	1.87514
$960$	0	0
$961$	−6.31811	−0.203810
$962$	42.6135	1.37391
$963$	0	0
$964$	90.3720	2.91069
$965$	22.2224	0.715366
$966$	0	0
$967$	52.1993	1.67862	0.839308	−	0.543656i	$-0.182960\pi$
$967$	52.1993	1.67862	0.839308	+	0.543656i	$0.182960\pi$
$968$	−3.91274	−0.125760
$969$	0	0
$970$	39.5863	1.27104
$971$	−47.9620	−1.53917	−0.769587	−	0.638542i	$-0.779538\pi$
$971$	−47.9620	−1.53917	−0.769587	+	0.638542i	$0.779538\pi$
$972$	0	0
$973$	−43.1352	−1.38285
$974$	−64.9209	−2.08020
$975$	0	0
$976$	3.75761	0.120278
$977$	−0.708707	−0.0226736	−0.0113368	−	0.999936i	$-0.503609\pi$
$977$	−0.708707	−0.0226736	−0.0113368	+	0.999936i	$0.503609\pi$
$978$	0	0
$979$	7.47727	0.238975
$980$	−33.9766	−1.08534
$981$	0	0
$982$	−44.5877	−1.42285
$983$	4.83678	0.154269	0.0771347	−	0.997021i	$-0.475423\pi$
$983$	4.83678	0.154269	0.0771347	+	0.997021i	$0.475423\pi$
$984$	0	0
$985$	20.0794	0.639782
$986$	5.41092	0.172319
$987$	0	0
$988$	−79.8209	−2.53944
$989$	−28.7583	−0.914461
$990$	0	0
$991$	−4.92667	−0.156501	−0.0782505	−	0.996934i	$-0.524933\pi$
$991$	−4.92667	−0.156501	−0.0782505	+	0.996934i	$0.524933\pi$
$992$	−23.8784	−0.758141
$993$	0	0
$994$	−37.9498	−1.20370
$995$	−25.6460	−0.813033
$996$	0	0
$997$	−52.8965	−1.67525	−0.837624	−	0.546247i	$-0.816056\pi$
$997$	−52.8965	−1.67525	−0.837624	+	0.546247i	$0.816056\pi$
$998$	23.0695	0.730251
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6021.2.a.t.1.5	✓	40
3.2	odd	2	inner	6021.2.a.t.1.36	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6021.2.a.t.1.5	✓	40	1.1	even	1	trivial
6021.2.a.t.1.36	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)

\(f(q)\)	\(=\)	\(q-2.32144 q^{2} +3.38909 q^{4} -1.08423 q^{5} +4.03069 q^{7} -3.22470 q^{8} +O(q^{10})\)	\(q-2.32144 q^{2} +3.38909 q^{4} -1.08423 q^{5} +4.03069 q^{7} -3.22470 q^{8} +2.51697 q^{10} -3.49476 q^{11} -3.72073 q^{13} -9.35701 q^{14} +0.707766 q^{16} +4.48254 q^{17} +6.33001 q^{19} -3.67455 q^{20} +8.11289 q^{22} -7.10533 q^{23} -3.82445 q^{25} +8.63747 q^{26} +13.6604 q^{28} -0.519983 q^{29} -4.96809 q^{31} +4.80636 q^{32} -10.4059 q^{34} -4.37019 q^{35} +4.93356 q^{37} -14.6947 q^{38} +3.49631 q^{40} -3.38864 q^{41} +4.04743 q^{43} -11.8441 q^{44} +16.4946 q^{46} -3.12585 q^{47} +9.24645 q^{49} +8.87823 q^{50} -12.6099 q^{52} -8.30126 q^{53} +3.78912 q^{55} -12.9978 q^{56} +1.20711 q^{58} +3.99159 q^{59} +5.30912 q^{61} +11.5331 q^{62} -12.5732 q^{64} +4.03413 q^{65} -12.8573 q^{67} +15.1917 q^{68} +10.1451 q^{70} +4.05577 q^{71} -7.06713 q^{73} -11.4530 q^{74} +21.4530 q^{76} -14.0863 q^{77} +10.0440 q^{79} -0.767380 q^{80} +7.86652 q^{82} +13.4518 q^{83} -4.86009 q^{85} -9.39587 q^{86} +11.2696 q^{88} -2.13957 q^{89} -14.9971 q^{91} -24.0806 q^{92} +7.25648 q^{94} -6.86318 q^{95} +15.7277 q^{97} -21.4651 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 46 q^{4} + 16 q^{7}+O(q^{10}) \) 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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