LMFDB - Embedded newform 6021.2.a.t.1.1

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			1.1
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)

$f(q)$	$=$	$q-2.70299 q^{2} +5.30613 q^{4} -3.42111 q^{5} -1.82484 q^{7} -8.93642 q^{8} +O(q^{10})$	\(q-2.70299 q^{2} +5.30613 q^{4} -3.42111 q^{5} -1.82484 q^{7} -8.93642 q^{8} +9.24720 q^{10} +0.926183 q^{11} -4.26452 q^{13} +4.93251 q^{14} +13.5427 q^{16} -3.66366 q^{17} +4.08345 q^{19} -18.1528 q^{20} -2.50346 q^{22} -3.32976 q^{23} +6.70398 q^{25} +11.5269 q^{26} -9.68282 q^{28} -8.34929 q^{29} -3.46529 q^{31} -18.7330 q^{32} +9.90282 q^{34} +6.24297 q^{35} -9.91704 q^{37} -11.0375 q^{38} +30.5724 q^{40} +7.41932 q^{41} +4.79841 q^{43} +4.91444 q^{44} +9.00030 q^{46} +1.83510 q^{47} -3.66997 q^{49} -18.1208 q^{50} -22.6281 q^{52} -5.01297 q^{53} -3.16857 q^{55} +16.3075 q^{56} +22.5680 q^{58} -14.7173 q^{59} -1.21288 q^{61} +9.36663 q^{62} +23.5495 q^{64} +14.5894 q^{65} +0.101296 q^{67} -19.4399 q^{68} -16.8746 q^{70} +4.13411 q^{71} -4.05614 q^{73} +26.8056 q^{74} +21.6673 q^{76} -1.69013 q^{77} -12.1326 q^{79} -46.3312 q^{80} -20.0543 q^{82} -8.34471 q^{83} +12.5338 q^{85} -12.9700 q^{86} -8.27675 q^{88} -6.88022 q^{89} +7.78206 q^{91} -17.6682 q^{92} -4.96024 q^{94} -13.9699 q^{95} +7.92018 q^{97} +9.91987 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.70299	−1.91130	−0.955650	−	0.294506i	$-0.904845\pi$
$2$	−2.70299	−1.91130	−0.955650	+	0.294506i	$0.904845\pi$
$3$	0	0
$3$	0	0
$4$	5.30613	2.65306
$5$	−3.42111	−1.52997	−0.764983	−	0.644051i	$-0.777253\pi$
$5$	−3.42111	−1.52997	−0.764983	+	0.644051i	$0.777253\pi$
$6$	0	0
$7$	−1.82484	−0.689724	−0.344862	−	0.938653i	$-0.612074\pi$
$7$	−1.82484	−0.689724	−0.344862	+	0.938653i	$0.612074\pi$
$8$	−8.93642	−3.15950
$9$	0	0
$10$	9.24720	2.92422
$11$	0.926183	0.279255	0.139627	−	0.990204i	$-0.455410\pi$
$11$	0.926183	0.279255	0.139627	+	0.990204i	$0.455410\pi$
$12$	0	0
$13$	−4.26452	−1.18277	−0.591383	−	0.806391i	$-0.701418\pi$
$13$	−4.26452	−1.18277	−0.591383	+	0.806391i	$0.701418\pi$
$14$	4.93251	1.31827
$15$	0	0
$16$	13.5427	3.38569
$17$	−3.66366	−0.888568	−0.444284	−	0.895886i	$-0.646542\pi$
$17$	−3.66366	−0.888568	−0.444284	+	0.895886i	$0.646542\pi$
$18$	0	0
$19$	4.08345	0.936808	0.468404	−	0.883515i	$-0.344829\pi$
$19$	4.08345	0.936808	0.468404	+	0.883515i	$0.344829\pi$
$20$	−18.1528	−4.05910
$21$	0	0
$22$	−2.50346	−0.533739
$23$	−3.32976	−0.694304	−0.347152	−	0.937809i	$-0.612851\pi$
$23$	−3.32976	−0.694304	−0.347152	+	0.937809i	$0.612851\pi$
$24$	0	0
$25$	6.70398	1.34080
$26$	11.5269	2.26062
$27$	0	0
$28$	−9.68282	−1.82988
$29$	−8.34929	−1.55042	−0.775212	−	0.631701i	$-0.782357\pi$
$29$	−8.34929	−1.55042	−0.775212	+	0.631701i	$0.782357\pi$
$30$	0	0
$31$	−3.46529	−0.622384	−0.311192	−	0.950347i	$-0.600728\pi$
$31$	−3.46529	−0.622384	−0.311192	+	0.950347i	$0.600728\pi$
$32$	−18.7330	−3.31156
$33$	0	0
$34$	9.90282	1.69832
$35$	6.24297	1.05525
$36$	0	0
$37$	−9.91704	−1.63035	−0.815175	−	0.579214i	$-0.803359\pi$
$37$	−9.91704	−1.63035	−0.815175	+	0.579214i	$0.803359\pi$
$38$	−11.0375	−1.79052
$39$	0	0
$40$	30.5724	4.83393
$41$	7.41932	1.15870	0.579351	−	0.815078i	$-0.303306\pi$
$41$	7.41932	1.15870	0.579351	+	0.815078i	$0.303306\pi$
$42$	0	0
$43$	4.79841	0.731750	0.365875	−	0.930664i	$-0.380770\pi$
$43$	4.79841	0.731750	0.365875	+	0.930664i	$0.380770\pi$
$44$	4.91444	0.740880
$45$	0	0
$46$	9.00030	1.32702
$47$	1.83510	0.267676	0.133838	−	0.991003i	$-0.457270\pi$
$47$	1.83510	0.267676	0.133838	+	0.991003i	$0.457270\pi$
$48$	0	0
$49$	−3.66997	−0.524281
$50$	−18.1208	−2.56266
$51$	0	0
$52$	−22.6281	−3.13795
$53$	−5.01297	−0.688584	−0.344292	−	0.938863i	$-0.611881\pi$
$53$	−5.01297	−0.688584	−0.344292	+	0.938863i	$0.611881\pi$
$54$	0	0
$55$	−3.16857	−0.427250
$56$	16.3075	2.17918
$57$	0	0
$58$	22.5680	2.96332
$59$	−14.7173	−1.91603	−0.958014	−	0.286721i	$-0.907435\pi$
$59$	−14.7173	−1.91603	−0.958014	+	0.286721i	$0.907435\pi$
$60$	0	0
$61$	−1.21288	−0.155293	−0.0776464	−	0.996981i	$-0.524741\pi$
$61$	−1.21288	−0.155293	−0.0776464	+	0.996981i	$0.524741\pi$
$62$	9.36663	1.18956
$63$	0	0
$64$	23.5495	2.94369
$65$	14.5894	1.80959
$66$	0	0
$67$	0.101296	0.0123753	0.00618763	−	0.999981i	$-0.498030\pi$
$67$	0.101296	0.0123753	0.00618763	+	0.999981i	$0.498030\pi$
$68$	−19.4399	−2.35743
$69$	0	0
$70$	−16.8746	−2.01691
$71$	4.13411	0.490629	0.245314	−	0.969444i	$-0.421109\pi$
$71$	4.13411	0.490629	0.245314	+	0.969444i	$0.421109\pi$
$72$	0	0
$73$	−4.05614	−0.474735	−0.237367	−	0.971420i	$-0.576285\pi$
$73$	−4.05614	−0.474735	−0.237367	+	0.971420i	$0.576285\pi$
$74$	26.8056	3.11609
$75$	0	0
$76$	21.6673	2.48541
$77$	−1.69013	−0.192609
$78$	0	0
$79$	−12.1326	−1.36502	−0.682512	−	0.730874i	$-0.739113\pi$
$79$	−12.1326	−1.36502	−0.682512	+	0.730874i	$0.739113\pi$
$80$	−46.3312	−5.17998
$81$	0	0
$82$	−20.0543	−2.21463
$83$	−8.34471	−0.915951	−0.457976	−	0.888965i	$-0.651425\pi$
$83$	−8.34471	−0.915951	−0.457976	+	0.888965i	$0.651425\pi$
$84$	0	0
$85$	12.5338	1.35948
$86$	−12.9700	−1.39859
$87$	0	0
$88$	−8.27675	−0.882305
$89$	−6.88022	−0.729302	−0.364651	−	0.931144i	$-0.618812\pi$
$89$	−6.88022	−0.729302	−0.364651	+	0.931144i	$0.618812\pi$
$90$	0	0
$91$	7.78206	0.815781
$92$	−17.6682	−1.84203
$93$	0	0
$94$	−4.96024	−0.511610
$95$	−13.9699	−1.43328
$96$	0	0
$97$	7.92018	0.804173	0.402086	−	0.915602i	$-0.368285\pi$
$97$	7.92018	0.804173	0.402086	+	0.915602i	$0.368285\pi$
$98$	9.91987	1.00206
$99$	0	0
$100$	35.5722	3.55722
$101$	9.73578	0.968746	0.484373	−	0.874862i	$-0.339048\pi$
$101$	9.73578	0.968746	0.484373	+	0.874862i	$0.339048\pi$
$102$	0	0
$103$	−10.2777	−1.01269	−0.506347	−	0.862330i	$-0.669005\pi$
$103$	−10.2777	−1.01269	−0.506347	+	0.862330i	$0.669005\pi$
$104$	38.1095	3.73695
$105$	0	0
$106$	13.5500	1.31609
$107$	2.97097	0.287214	0.143607	−	0.989635i	$-0.454130\pi$
$107$	2.97097	0.287214	0.143607	+	0.989635i	$0.454130\pi$
$108$	0	0
$109$	−9.52144	−0.911988	−0.455994	−	0.889983i	$-0.650716\pi$
$109$	−9.52144	−0.911988	−0.455994	+	0.889983i	$0.650716\pi$
$110$	8.56460	0.816603
$111$	0	0
$112$	−24.7133	−2.33519
$113$	−14.1363	−1.32983	−0.664914	−	0.746920i	$-0.731532\pi$
$113$	−14.1363	−1.32983	−0.664914	+	0.746920i	$0.731532\pi$
$114$	0	0
$115$	11.3915	1.06226
$116$	−44.3024	−4.11337
$117$	0	0
$118$	39.7806	3.66210
$119$	6.68559	0.612867
$120$	0	0
$121$	−10.1422	−0.922017
$122$	3.27838	0.296811
$123$	0	0
$124$	−18.3873	−1.65123
$125$	−5.82950	−0.521406
$126$	0	0
$127$	−18.2953	−1.62344	−0.811721	−	0.584045i	$-0.801469\pi$
$127$	−18.2953	−1.62344	−0.811721	+	0.584045i	$0.801469\pi$
$128$	−26.1880	−2.31472
$129$	0	0
$130$	−39.4349	−3.45867
$131$	−22.0623	−1.92759	−0.963796	−	0.266641i	$-0.914086\pi$
$131$	−22.0623	−1.92759	−0.963796	+	0.266641i	$0.914086\pi$
$132$	0	0
$133$	−7.45163	−0.646138
$134$	−0.273801	−0.0236528
$135$	0	0
$136$	32.7400	2.80743
$137$	−4.52347	−0.386466	−0.193233	−	0.981153i	$-0.561897\pi$
$137$	−4.52347	−0.386466	−0.193233	+	0.981153i	$0.561897\pi$
$138$	0	0
$139$	−5.81645	−0.493345	−0.246673	−	0.969099i	$-0.579337\pi$
$139$	−5.81645	−0.493345	−0.246673	+	0.969099i	$0.579337\pi$
$140$	33.1260	2.79966
$141$	0	0
$142$	−11.1744	−0.937738
$143$	−3.94973	−0.330293
$144$	0	0
$145$	28.5638	2.37210
$146$	10.9637	0.907360
$147$	0	0
$148$	−52.6211	−4.32542
$149$	14.7647	1.20957	0.604785	−	0.796389i	$-0.293259\pi$
$149$	14.7647	1.20957	0.604785	+	0.796389i	$0.293259\pi$
$150$	0	0
$151$	−0.940739	−0.0765563	−0.0382781	−	0.999267i	$-0.512187\pi$
$151$	−0.940739	−0.0765563	−0.0382781	+	0.999267i	$0.512187\pi$
$152$	−36.4914	−2.95984
$153$	0	0
$154$	4.56840	0.368133
$155$	11.8551	0.952227
$156$	0	0
$157$	−16.5019	−1.31700	−0.658499	−	0.752582i	$-0.728808\pi$
$157$	−16.5019	−1.31700	−0.658499	+	0.752582i	$0.728808\pi$
$158$	32.7942	2.60897
$159$	0	0
$160$	64.0876	5.06657
$161$	6.07628	0.478878
$162$	0	0
$163$	−3.88007	−0.303911	−0.151955	−	0.988387i	$-0.548557\pi$
$163$	−3.88007	−0.303911	−0.151955	+	0.988387i	$0.548557\pi$
$164$	39.3679	3.07411
$165$	0	0
$166$	22.5556	1.75066
$167$	−19.1122	−1.47895	−0.739474	−	0.673185i	$-0.764926\pi$
$167$	−19.1122	−1.47895	−0.739474	+	0.673185i	$0.764926\pi$
$168$	0	0
$169$	5.18614	0.398934
$170$	−33.8786	−2.59837
$171$	0	0
$172$	25.4610	1.94138
$173$	−0.757492	−0.0575910	−0.0287955	−	0.999585i	$-0.509167\pi$
$173$	−0.757492	−0.0575910	−0.0287955	+	0.999585i	$0.509167\pi$
$174$	0	0
$175$	−12.2337	−0.924779
$176$	12.5431	0.945468
$177$	0	0
$178$	18.5971	1.39391
$179$	14.8622	1.11085	0.555427	−	0.831566i	$-0.312555\pi$
$179$	14.8622	1.11085	0.555427	+	0.831566i	$0.312555\pi$
$180$	0	0
$181$	21.2852	1.58212	0.791058	−	0.611741i	$-0.209530\pi$
$181$	21.2852	1.58212	0.791058	+	0.611741i	$0.209530\pi$
$182$	−21.0348	−1.55920
$183$	0	0
$184$	29.7561	2.19365
$185$	33.9272	2.49438
$186$	0	0
$187$	−3.39322	−0.248137
$188$	9.73726	0.710163
$189$	0	0
$190$	37.7605	2.73943
$191$	14.3339	1.03716	0.518582	−	0.855028i	$-0.326460\pi$
$191$	14.3339	1.03716	0.518582	+	0.855028i	$0.326460\pi$
$192$	0	0
$193$	18.7075	1.34659	0.673297	−	0.739372i	$-0.264878\pi$
$193$	18.7075	1.34659	0.673297	+	0.739372i	$0.264878\pi$
$194$	−21.4081	−1.53701
$195$	0	0
$196$	−19.4733	−1.39095
$197$	17.6992	1.26102	0.630509	−	0.776182i	$-0.282846\pi$
$197$	17.6992	1.26102	0.630509	+	0.776182i	$0.282846\pi$
$198$	0	0
$199$	4.87808	0.345798	0.172899	−	0.984940i	$-0.444687\pi$
$199$	4.87808	0.345798	0.172899	+	0.984940i	$0.444687\pi$
$200$	−59.9095	−4.23624
$201$	0	0
$202$	−26.3157	−1.85156
$203$	15.2361	1.06936
$204$	0	0
$205$	−25.3823	−1.77278
$206$	27.7805	1.93556
$207$	0	0
$208$	−57.7533	−4.00447
$209$	3.78202	0.261608
$210$	0	0
$211$	1.02364	0.0704703	0.0352351	−	0.999379i	$-0.488782\pi$
$211$	1.02364	0.0704703	0.0352351	+	0.999379i	$0.488782\pi$
$212$	−26.5995	−1.82686
$213$	0	0
$214$	−8.03048	−0.548952
$215$	−16.4159	−1.11955
$216$	0	0
$217$	6.32359	0.429273
$218$	25.7363	1.74308
$219$	0	0
$220$	−16.8128	−1.13352
$221$	15.6238	1.05097
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	34.1847	2.28406
$225$	0	0
$226$	38.2101	2.54170
$227$	9.12064	0.605358	0.302679	−	0.953093i	$-0.402119\pi$
$227$	9.12064	0.605358	0.302679	+	0.953093i	$0.402119\pi$
$228$	0	0
$229$	−19.8725	−1.31321	−0.656607	−	0.754233i	$-0.728009\pi$
$229$	−19.8725	−1.31321	−0.656607	+	0.754233i	$0.728009\pi$
$230$	−30.7910	−2.03030
$231$	0	0
$232$	74.6127	4.89857
$233$	−24.0845	−1.57783	−0.788914	−	0.614504i	$-0.789356\pi$
$233$	−24.0845	−1.57783	−0.788914	+	0.614504i	$0.789356\pi$
$234$	0	0
$235$	−6.27807	−0.409536
$236$	−78.0918	−5.08335
$237$	0	0
$238$	−18.0710	−1.17137
$239$	−21.6649	−1.40139	−0.700694	−	0.713462i	$-0.747126\pi$
$239$	−21.6649	−1.40139	−0.700694	+	0.713462i	$0.747126\pi$
$240$	0	0
$241$	3.12316	0.201180	0.100590	−	0.994928i	$-0.467927\pi$
$241$	3.12316	0.201180	0.100590	+	0.994928i	$0.467927\pi$
$242$	27.4142	1.76225
$243$	0	0
$244$	−6.43567	−0.412002
$245$	12.5554	0.802132
$246$	0	0
$247$	−17.4140	−1.10802
$248$	30.9673	1.96642
$249$	0	0
$250$	15.7570	0.996563
$251$	−2.63752	−0.166479	−0.0832393	−	0.996530i	$-0.526527\pi$
$251$	−2.63752	−0.166479	−0.0832393	+	0.996530i	$0.526527\pi$
$252$	0	0
$253$	−3.08397	−0.193887
$254$	49.4518	3.10288
$255$	0	0
$256$	23.6868	1.48043
$257$	−8.40950	−0.524570	−0.262285	−	0.964990i	$-0.584476\pi$
$257$	−8.40950	−0.524570	−0.262285	+	0.964990i	$0.584476\pi$
$258$	0	0
$259$	18.0970	1.12449
$260$	77.4132	4.80096
$261$	0	0
$262$	59.6341	3.68420
$263$	1.27811	0.0788115	0.0394057	−	0.999223i	$-0.487454\pi$
$263$	1.27811	0.0788115	0.0394057	+	0.999223i	$0.487454\pi$
$264$	0	0
$265$	17.1499	1.05351
$266$	20.1416	1.23496
$267$	0	0
$268$	0.537489	0.0328324
$269$	10.2354	0.624065	0.312033	−	0.950071i	$-0.398990\pi$
$269$	10.2354	0.624065	0.312033	+	0.950071i	$0.398990\pi$
$270$	0	0
$271$	−20.9314	−1.27149	−0.635745	−	0.771899i	$-0.719307\pi$
$271$	−20.9314	−1.27149	−0.635745	+	0.771899i	$0.719307\pi$
$272$	−49.6160	−3.00841
$273$	0	0
$274$	12.2269	0.738652
$275$	6.20911	0.374423
$276$	0	0
$277$	−24.1119	−1.44875	−0.724373	−	0.689409i	$-0.757870\pi$
$277$	−24.1119	−1.44875	−0.724373	+	0.689409i	$0.757870\pi$
$278$	15.7218	0.942930
$279$	0	0
$280$	−55.7897	−3.33407
$281$	20.1217	1.20036	0.600181	−	0.799864i	$-0.295095\pi$
$281$	20.1217	1.20036	0.600181	+	0.799864i	$0.295095\pi$
$282$	0	0
$283$	−11.1652	−0.663704	−0.331852	−	0.943332i	$-0.607673\pi$
$283$	−11.1652	−0.663704	−0.331852	+	0.943332i	$0.607673\pi$
$284$	21.9361	1.30167
$285$	0	0
$286$	10.6761	0.631288
$287$	−13.5390	−0.799185
$288$	0	0
$289$	−3.57759	−0.210446
$290$	−77.2076	−4.53378
$291$	0	0
$292$	−21.5224	−1.25950
$293$	6.31711	0.369050	0.184525	−	0.982828i	$-0.440925\pi$
$293$	6.31711	0.369050	0.184525	+	0.982828i	$0.440925\pi$
$294$	0	0
$295$	50.3494	2.93146
$296$	88.6228	5.15109
$297$	0	0
$298$	−39.9087	−2.31185
$299$	14.1998	0.821198
$300$	0	0
$301$	−8.75631	−0.504706
$302$	2.54280	0.146322
$303$	0	0
$304$	55.3011	3.17174
$305$	4.14938	0.237593
$306$	0	0
$307$	−0.582251	−0.0332308	−0.0166154	−	0.999862i	$-0.505289\pi$
$307$	−0.582251	−0.0332308	−0.0166154	+	0.999862i	$0.505289\pi$
$308$	−8.96806	−0.511003
$309$	0	0
$310$	−32.0442	−1.81999
$311$	13.5438	0.768001	0.384000	−	0.923333i	$-0.374546\pi$
$311$	13.5438	0.768001	0.384000	+	0.923333i	$0.374546\pi$
$312$	0	0
$313$	−31.8044	−1.79769	−0.898844	−	0.438268i	$-0.855592\pi$
$313$	−31.8044	−1.79769	−0.898844	+	0.438268i	$0.855592\pi$
$314$	44.6045	2.51718
$315$	0	0
$316$	−64.3771	−3.62150
$317$	34.4926	1.93729	0.968647	−	0.248440i	$-0.0799178\pi$
$317$	34.4926	1.93729	0.968647	+	0.248440i	$0.0799178\pi$
$318$	0	0
$319$	−7.73297	−0.432963
$320$	−80.5655	−4.50375
$321$	0	0
$322$	−16.4241	−0.915279
$323$	−14.9604	−0.832418
$324$	0	0
$325$	−28.5893	−1.58585
$326$	10.4878	0.580864
$327$	0	0
$328$	−66.3021	−3.66092
$329$	−3.34875	−0.184623
$330$	0	0
$331$	1.95746	0.107592	0.0537958	−	0.998552i	$-0.482868\pi$
$331$	1.95746	0.107592	0.0537958	+	0.998552i	$0.482868\pi$
$332$	−44.2781	−2.43008
$333$	0	0
$334$	51.6601	2.82671
$335$	−0.346544	−0.0189337
$336$	0	0
$337$	1.60414	0.0873833	0.0436916	−	0.999045i	$-0.486088\pi$
$337$	1.60414	0.0873833	0.0436916	+	0.999045i	$0.486088\pi$
$338$	−14.0181	−0.762482
$339$	0	0
$340$	66.5059	3.60679
$341$	−3.20949	−0.173804
$342$	0	0
$343$	19.4710	1.05133
$344$	−42.8806	−2.31197
$345$	0	0
$346$	2.04749	0.110074
$347$	−23.0878	−1.23942	−0.619708	−	0.784833i	$-0.712749\pi$
$347$	−23.0878	−1.23942	−0.619708	+	0.784833i	$0.712749\pi$
$348$	0	0
$349$	−13.4893	−0.722068	−0.361034	−	0.932553i	$-0.617576\pi$
$349$	−13.4893	−0.722068	−0.361034	+	0.932553i	$0.617576\pi$
$350$	33.0674	1.76753
$351$	0	0
$352$	−17.3502	−0.924768
$353$	21.5674	1.14792	0.573958	−	0.818885i	$-0.305407\pi$
$353$	21.5674	1.14792	0.573958	+	0.818885i	$0.305407\pi$
$354$	0	0
$355$	−14.1432	−0.750645
$356$	−36.5073	−1.93488
$357$	0	0
$358$	−40.1723	−2.12317
$359$	−19.4804	−1.02814	−0.514069	−	0.857749i	$-0.671862\pi$
$359$	−19.4804	−1.02814	−0.514069	+	0.857749i	$0.671862\pi$
$360$	0	0
$361$	−2.32544	−0.122392
$362$	−57.5336	−3.02390
$363$	0	0
$364$	41.2926	2.16432
$365$	13.8765	0.726328
$366$	0	0
$367$	−19.1797	−1.00117	−0.500585	−	0.865687i	$-0.666882\pi$
$367$	−19.1797	−1.00117	−0.500585	+	0.865687i	$0.666882\pi$
$368$	−45.0941	−2.35069
$369$	0	0
$370$	−91.7048	−4.76751
$371$	9.14785	0.474933
$372$	0	0
$373$	−16.7305	−0.866275	−0.433137	−	0.901328i	$-0.642594\pi$
$373$	−16.7305	−0.866275	−0.433137	+	0.901328i	$0.642594\pi$
$374$	9.17182	0.474264
$375$	0	0
$376$	−16.3992	−0.845724
$377$	35.6057	1.83379
$378$	0	0
$379$	−16.1439	−0.829255	−0.414628	−	0.909991i	$-0.636088\pi$
$379$	−16.1439	−0.829255	−0.414628	+	0.909991i	$0.636088\pi$
$380$	−74.1262	−3.80259
$381$	0	0
$382$	−38.7443	−1.98233
$383$	−18.5313	−0.946905	−0.473452	−	0.880819i	$-0.656992\pi$
$383$	−18.5313	−0.946905	−0.473452	+	0.880819i	$0.656992\pi$
$384$	0	0
$385$	5.78213	0.294685
$386$	−50.5660	−2.57374
$387$	0	0
$388$	42.0255	2.13352
$389$	−23.4674	−1.18984	−0.594921	−	0.803784i	$-0.702817\pi$
$389$	−23.4674	−1.18984	−0.594921	+	0.803784i	$0.702817\pi$
$390$	0	0
$391$	12.1991	0.616936
$392$	32.7964	1.65647
$393$	0	0
$394$	−47.8408	−2.41018
$395$	41.5069	2.08844
$396$	0	0
$397$	28.9725	1.45409	0.727045	−	0.686590i	$-0.240893\pi$
$397$	28.9725	1.45409	0.727045	+	0.686590i	$0.240893\pi$
$398$	−13.1854	−0.660924
$399$	0	0
$400$	90.7903	4.53951
$401$	−6.59305	−0.329241	−0.164621	−	0.986357i	$-0.552640\pi$
$401$	−6.59305	−0.329241	−0.164621	+	0.986357i	$0.552640\pi$
$402$	0	0
$403$	14.7778	0.736135
$404$	51.6593	2.57015
$405$	0	0
$406$	−41.1829	−2.04387
$407$	−9.18499	−0.455283
$408$	0	0
$409$	35.3669	1.74878	0.874389	−	0.485225i	$-0.161262\pi$
$409$	35.3669	1.74878	0.874389	+	0.485225i	$0.161262\pi$
$410$	68.6079	3.38830
$411$	0	0
$412$	−54.5349	−2.68674
$413$	26.8567	1.32153
$414$	0	0
$415$	28.5482	1.40137
$416$	79.8873	3.91680
$417$	0	0
$418$	−10.2227	−0.500011
$419$	28.6184	1.39810	0.699051	−	0.715072i	$-0.253606\pi$
$419$	28.6184	1.39810	0.699051	+	0.715072i	$0.253606\pi$
$420$	0	0
$421$	23.8880	1.16423	0.582115	−	0.813106i	$-0.302225\pi$
$421$	23.8880	1.16423	0.582115	+	0.813106i	$0.302225\pi$
$422$	−2.76688	−0.134690
$423$	0	0
$424$	44.7980	2.17558
$425$	−24.5611	−1.19139
$426$	0	0
$427$	2.21330	0.107109
$428$	15.7643	0.761998
$429$	0	0
$430$	44.3719	2.13980
$431$	−7.17369	−0.345544	−0.172772	−	0.984962i	$-0.555272\pi$
$431$	−7.17369	−0.345544	−0.172772	+	0.984962i	$0.555272\pi$
$432$	0	0
$433$	28.0597	1.34846	0.674231	−	0.738521i	$-0.264475\pi$
$433$	28.0597	1.34846	0.674231	+	0.738521i	$0.264475\pi$
$434$	−17.0926	−0.820470
$435$	0	0
$436$	−50.5220	−2.41956
$437$	−13.5969	−0.650429
$438$	0	0
$439$	9.37131	0.447268	0.223634	−	0.974673i	$-0.428208\pi$
$439$	9.37131	0.447268	0.223634	+	0.974673i	$0.428208\pi$
$440$	28.3157	1.34990
$441$	0	0
$442$	−42.2308	−2.00871
$443$	−16.1622	−0.767888	−0.383944	−	0.923356i	$-0.625434\pi$
$443$	−16.1622	−0.767888	−0.383944	+	0.923356i	$0.625434\pi$
$444$	0	0
$445$	23.5380	1.11581
$446$	−2.70299	−0.127990
$447$	0	0
$448$	−42.9741	−2.03033
$449$	−1.20229	−0.0567398	−0.0283699	−	0.999597i	$-0.509032\pi$
$449$	−1.20229	−0.0567398	−0.0283699	+	0.999597i	$0.509032\pi$
$450$	0	0
$451$	6.87164	0.323573
$452$	−75.0088	−3.52812
$453$	0	0
$454$	−24.6530	−1.15702
$455$	−26.6233	−1.24812
$456$	0	0
$457$	−18.6945	−0.874491	−0.437246	−	0.899342i	$-0.644046\pi$
$457$	−18.6945	−0.874491	−0.437246	+	0.899342i	$0.644046\pi$
$458$	53.7151	2.50994
$459$	0	0
$460$	60.4447	2.81825
$461$	15.5178	0.722738	0.361369	−	0.932423i	$-0.382310\pi$
$461$	15.5178	0.722738	0.361369	+	0.932423i	$0.382310\pi$
$462$	0	0
$463$	−4.45173	−0.206889	−0.103445	−	0.994635i	$-0.532986\pi$
$463$	−4.45173	−0.206889	−0.103445	+	0.994635i	$0.532986\pi$
$464$	−113.072	−5.24925
$465$	0	0
$466$	65.1000	3.01570
$467$	17.0496	0.788961	0.394480	−	0.918904i	$-0.370925\pi$
$467$	17.0496	0.788961	0.394480	+	0.918904i	$0.370925\pi$
$468$	0	0
$469$	−0.184849	−0.00853551
$470$	16.9695	0.782746
$471$	0	0
$472$	131.520	6.05369
$473$	4.44420	0.204345
$474$	0	0
$475$	27.3754	1.25607
$476$	35.4746	1.62597
$477$	0	0
$478$	58.5600	2.67847
$479$	−14.6537	−0.669546	−0.334773	−	0.942299i	$-0.608660\pi$
$479$	−14.6537	−0.669546	−0.334773	+	0.942299i	$0.608660\pi$
$480$	0	0
$481$	42.2914	1.92832
$482$	−8.44185	−0.384516
$483$	0	0
$484$	−53.8157	−2.44617
$485$	−27.0958	−1.23036
$486$	0	0
$487$	23.2385	1.05304	0.526518	−	0.850164i	$-0.323497\pi$
$487$	23.2385	1.05304	0.526518	+	0.850164i	$0.323497\pi$
$488$	10.8388	0.490648
$489$	0	0
$490$	−33.9369	−1.53311
$491$	−40.6501	−1.83452	−0.917258	−	0.398294i	$-0.869602\pi$
$491$	−40.6501	−1.83452	−0.917258	+	0.398294i	$0.869602\pi$
$492$	0	0
$493$	30.5890	1.37766
$494$	47.0697	2.11776
$495$	0	0
$496$	−46.9295	−2.10720
$497$	−7.54408	−0.338398
$498$	0	0
$499$	43.0515	1.92725	0.963626	−	0.267255i	$-0.0861166\pi$
$499$	43.0515	1.92725	0.963626	+	0.267255i	$0.0861166\pi$
$500$	−30.9321	−1.38332
$501$	0	0
$502$	7.12917	0.318191
$503$	−8.79538	−0.392166	−0.196083	−	0.980587i	$-0.562822\pi$
$503$	−8.79538	−0.392166	−0.196083	+	0.980587i	$0.562822\pi$
$504$	0	0
$505$	−33.3071	−1.48215
$506$	8.33592	0.370577
$507$	0	0
$508$	−97.0770	−4.30710
$509$	−41.6789	−1.84739	−0.923693	−	0.383133i	$-0.874845\pi$
$509$	−41.6789	−1.84739	−0.923693	+	0.383133i	$0.874845\pi$
$510$	0	0
$511$	7.40179	0.327436
$512$	−11.6490	−0.514819
$513$	0	0
$514$	22.7308	1.00261
$515$	35.1612	1.54939
$516$	0	0
$517$	1.69964	0.0747499
$518$	−48.9159	−2.14924
$519$	0	0
$520$	−130.377	−5.71740
$521$	−37.7856	−1.65542	−0.827709	−	0.561157i	$-0.810356\pi$
$521$	−37.7856	−1.65542	−0.827709	+	0.561157i	$0.810356\pi$
$522$	0	0
$523$	22.8031	0.997109	0.498555	−	0.866858i	$-0.333864\pi$
$523$	22.8031	0.997109	0.498555	+	0.866858i	$0.333864\pi$
$524$	−117.065	−5.11402
$525$	0	0
$526$	−3.45470	−0.150632
$527$	12.6956	0.553031
$528$	0	0
$529$	−11.9127	−0.517942
$530$	−46.3559	−2.01357
$531$	0	0
$532$	−39.5393	−1.71425
$533$	−31.6398	−1.37047
$534$	0	0
$535$	−10.1640	−0.439428
$536$	−0.905222	−0.0390996
$537$	0	0
$538$	−27.6662	−1.19278
$539$	−3.39906	−0.146408
$540$	0	0
$541$	−17.6158	−0.757361	−0.378681	−	0.925527i	$-0.623622\pi$
$541$	−17.6158	−0.757361	−0.378681	+	0.925527i	$0.623622\pi$
$542$	56.5772	2.43020
$543$	0	0
$544$	68.6314	2.94255
$545$	32.5739	1.39531
$546$	0	0
$547$	−4.99410	−0.213532	−0.106766	−	0.994284i	$-0.534050\pi$
$547$	−4.99410	−0.213532	−0.106766	+	0.994284i	$0.534050\pi$
$548$	−24.0021	−1.02532
$549$	0	0
$550$	−16.7831	−0.715635
$551$	−34.0939	−1.45245
$552$	0	0
$553$	22.1400	0.941489
$554$	65.1742	2.76899
$555$	0	0
$556$	−30.8629	−1.30888
$557$	40.7972	1.72863	0.864316	−	0.502949i	$-0.167752\pi$
$557$	40.7972	1.72863	0.864316	+	0.502949i	$0.167752\pi$
$558$	0	0
$559$	−20.4629	−0.865489
$560$	84.5469	3.57276
$561$	0	0
$562$	−54.3888	−2.29425
$563$	−7.39042	−0.311469	−0.155735	−	0.987799i	$-0.549774\pi$
$563$	−7.39042	−0.311469	−0.155735	+	0.987799i	$0.549774\pi$
$564$	0	0
$565$	48.3617	2.03459
$566$	30.1794	1.26854
$567$	0	0
$568$	−36.9441	−1.55014
$569$	33.4642	1.40289	0.701446	−	0.712723i	$-0.252538\pi$
$569$	33.4642	1.40289	0.701446	+	0.712723i	$0.252538\pi$
$570$	0	0
$571$	40.5963	1.69890	0.849452	−	0.527666i	$-0.176933\pi$
$571$	40.5963	1.69890	0.849452	+	0.527666i	$0.176933\pi$
$572$	−20.9578	−0.876288
$573$	0	0
$574$	36.5958	1.52748
$575$	−22.3227	−0.930919
$576$	0	0
$577$	−4.79841	−0.199760	−0.0998801	−	0.994999i	$-0.531846\pi$
$577$	−4.79841	−0.199760	−0.0998801	+	0.994999i	$0.531846\pi$
$578$	9.67016	0.402226
$579$	0	0
$580$	151.563	6.29332
$581$	15.2277	0.631753
$582$	0	0
$583$	−4.64293	−0.192290
$584$	36.2473	1.49992
$585$	0	0
$586$	−17.0751	−0.705364
$587$	−15.4217	−0.636520	−0.318260	−	0.948003i	$-0.603099\pi$
$587$	−15.4217	−0.636520	−0.318260	+	0.948003i	$0.603099\pi$
$588$	0	0
$589$	−14.1503	−0.583054
$590$	−136.094	−5.60289
$591$	0	0
$592$	−134.304	−5.51985
$593$	14.0942	0.578779	0.289389	−	0.957211i	$-0.406548\pi$
$593$	14.0942	0.578779	0.289389	+	0.957211i	$0.406548\pi$
$594$	0	0
$595$	−22.8721	−0.937665
$596$	78.3433	3.20907
$597$	0	0
$598$	−38.3820	−1.56956
$599$	12.4429	0.508401	0.254201	−	0.967152i	$-0.418188\pi$
$599$	12.4429	0.508401	0.254201	+	0.967152i	$0.418188\pi$
$600$	0	0
$601$	28.5343	1.16394	0.581969	−	0.813211i	$-0.302282\pi$
$601$	28.5343	1.16394	0.581969	+	0.813211i	$0.302282\pi$
$602$	23.6682	0.964643
$603$	0	0
$604$	−4.99168	−0.203109
$605$	34.6975	1.41065
$606$	0	0
$607$	32.1105	1.30332	0.651662	−	0.758509i	$-0.274072\pi$
$607$	32.1105	1.30332	0.651662	+	0.758509i	$0.274072\pi$
$608$	−76.4953	−3.10229
$609$	0	0
$610$	−11.2157	−0.454111
$611$	−7.82581	−0.316598
$612$	0	0
$613$	16.4047	0.662581	0.331291	−	0.943529i	$-0.392516\pi$
$613$	16.4047	0.662581	0.331291	+	0.943529i	$0.392516\pi$
$614$	1.57382	0.0635141
$615$	0	0
$616$	15.1037	0.608547
$617$	−20.1751	−0.812218	−0.406109	−	0.913825i	$-0.633115\pi$
$617$	−20.1751	−0.812218	−0.406109	+	0.913825i	$0.633115\pi$
$618$	0	0
$619$	−11.2700	−0.452980	−0.226490	−	0.974013i	$-0.572725\pi$
$619$	−11.2700	−0.452980	−0.226490	+	0.974013i	$0.572725\pi$
$620$	62.9049	2.52632
$621$	0	0
$622$	−36.6088	−1.46788
$623$	12.5553	0.503017
$624$	0	0
$625$	−13.5766	−0.543062
$626$	85.9667	3.43592
$627$	0	0
$628$	−87.5614	−3.49408
$629$	36.3327	1.44868
$630$	0	0
$631$	−33.7370	−1.34305	−0.671524	−	0.740982i	$-0.734360\pi$
$631$	−33.7370	−1.34305	−0.671524	+	0.740982i	$0.734360\pi$
$632$	108.422	4.31279
$633$	0	0
$634$	−93.2329	−3.70275
$635$	62.5901	2.48381
$636$	0	0
$637$	15.6507	0.620102
$638$	20.9021	0.827522
$639$	0	0
$640$	89.5921	3.54144
$641$	14.7444	0.582370	0.291185	−	0.956667i	$-0.405951\pi$
$641$	14.7444	0.582370	0.291185	+	0.956667i	$0.405951\pi$
$642$	0	0
$643$	31.5345	1.24360	0.621800	−	0.783176i	$-0.286402\pi$
$643$	31.5345	1.24360	0.621800	+	0.783176i	$0.286402\pi$
$644$	32.2415	1.27049
$645$	0	0
$646$	40.4377	1.59100
$647$	37.2457	1.46428	0.732140	−	0.681154i	$-0.238521\pi$
$647$	37.2457	1.46428	0.732140	+	0.681154i	$0.238521\pi$
$648$	0	0
$649$	−13.6309	−0.535060
$650$	77.2763	3.03103
$651$	0	0
$652$	−20.5882	−0.806295
$653$	−15.3110	−0.599165	−0.299582	−	0.954070i	$-0.596847\pi$
$653$	−15.3110	−0.599165	−0.299582	+	0.954070i	$0.596847\pi$
$654$	0	0
$655$	75.4775	2.94915
$656$	100.478	3.92300
$657$	0	0
$658$	9.05163	0.352869
$659$	32.9877	1.28502	0.642508	−	0.766279i	$-0.277894\pi$
$659$	32.9877	1.28502	0.642508	+	0.766279i	$0.277894\pi$
$660$	0	0
$661$	−45.5428	−1.77141	−0.885704	−	0.464250i	$-0.846324\pi$
$661$	−45.5428	−1.77141	−0.885704	+	0.464250i	$0.846324\pi$
$662$	−5.29098	−0.205640
$663$	0	0
$664$	74.5718	2.89395
$665$	25.4928	0.988570
$666$	0	0
$667$	27.8012	1.07647
$668$	−101.412	−3.92375
$669$	0	0
$670$	0.936704	0.0361880
$671$	−1.12334	−0.0433662
$672$	0	0
$673$	−15.0271	−0.579254	−0.289627	−	0.957140i	$-0.593531\pi$
$673$	−15.0271	−0.579254	−0.289627	+	0.957140i	$0.593531\pi$
$674$	−4.33598	−0.167016
$675$	0	0
$676$	27.5183	1.05840
$677$	33.4828	1.28685	0.643425	−	0.765509i	$-0.277513\pi$
$677$	33.4828	1.28685	0.643425	+	0.765509i	$0.277513\pi$
$678$	0	0
$679$	−14.4530	−0.554657
$680$	−112.007	−4.29528
$681$	0	0
$682$	8.67521	0.332191
$683$	16.3036	0.623840	0.311920	−	0.950108i	$-0.399028\pi$
$683$	16.3036	0.623840	0.311920	+	0.950108i	$0.399028\pi$
$684$	0	0
$685$	15.4753	0.591279
$686$	−52.6297	−2.00941
$687$	0	0
$688$	64.9836	2.47748
$689$	21.3779	0.814434
$690$	0	0
$691$	17.2210	0.655119	0.327560	−	0.944831i	$-0.393774\pi$
$691$	17.2210	0.655119	0.327560	+	0.944831i	$0.393774\pi$
$692$	−4.01935	−0.152793
$693$	0	0
$694$	62.4059	2.36889
$695$	19.8987	0.754801
$696$	0	0
$697$	−27.1819	−1.02959
$698$	36.4615	1.38009
$699$	0	0
$700$	−64.9134	−2.45350
$701$	−25.9892	−0.981598	−0.490799	−	0.871273i	$-0.663295\pi$
$701$	−25.9892	−0.981598	−0.490799	+	0.871273i	$0.663295\pi$
$702$	0	0
$703$	−40.4957	−1.52732
$704$	21.8112	0.822039
$705$	0	0
$706$	−58.2963	−2.19401
$707$	−17.7662	−0.668167
$708$	0	0
$709$	−1.21579	−0.0456599	−0.0228299	−	0.999739i	$-0.507268\pi$
$709$	−1.21579	−0.0456599	−0.0228299	+	0.999739i	$0.507268\pi$
$710$	38.2290	1.43471
$711$	0	0
$712$	61.4845	2.30423
$713$	11.5386	0.432124
$714$	0	0
$715$	13.5124	0.505337
$716$	78.8608	2.94717
$717$	0	0
$718$	52.6553	1.96508
$719$	−4.68598	−0.174758	−0.0873788	−	0.996175i	$-0.527849\pi$
$719$	−4.68598	−0.174758	−0.0873788	+	0.996175i	$0.527849\pi$
$720$	0	0
$721$	18.7552	0.698479
$722$	6.28564	0.233927
$723$	0	0
$724$	112.942	4.19746
$725$	−55.9735	−2.07880
$726$	0	0
$727$	−25.3770	−0.941181	−0.470590	−	0.882352i	$-0.655959\pi$
$727$	−25.3770	−0.941181	−0.470590	+	0.882352i	$0.655959\pi$
$728$	−69.5437	−2.57746
$729$	0	0
$730$	−37.5079	−1.38823
$731$	−17.5797	−0.650210
$732$	0	0
$733$	−8.19740	−0.302778	−0.151389	−	0.988474i	$-0.548375\pi$
$733$	−8.19740	−0.302778	−0.151389	+	0.988474i	$0.548375\pi$
$734$	51.8424	1.91354
$735$	0	0
$736$	62.3765	2.29923
$737$	0.0938185	0.00345585
$738$	0	0
$739$	45.5022	1.67382	0.836912	−	0.547337i	$-0.184358\pi$
$739$	45.5022	1.67382	0.836912	+	0.547337i	$0.184358\pi$
$740$	180.022	6.61775
$741$	0	0
$742$	−24.7265	−0.907739
$743$	−8.12015	−0.297899	−0.148950	−	0.988845i	$-0.547589\pi$
$743$	−8.12015	−0.297899	−0.148950	+	0.988845i	$0.547589\pi$
$744$	0	0
$745$	−50.5116	−1.85060
$746$	45.2224	1.65571
$747$	0	0
$748$	−18.0049	−0.658323
$749$	−5.42153	−0.198098
$750$	0	0
$751$	1.84739	0.0674124	0.0337062	−	0.999432i	$-0.489269\pi$
$751$	1.84739	0.0674124	0.0337062	+	0.999432i	$0.489269\pi$
$752$	24.8523	0.906268
$753$	0	0
$754$	−96.2417	−3.50492
$755$	3.21837	0.117128
$756$	0	0
$757$	16.6634	0.605641	0.302820	−	0.953048i	$-0.402072\pi$
$757$	16.6634	0.605641	0.302820	+	0.953048i	$0.402072\pi$
$758$	43.6367	1.58496
$759$	0	0
$760$	124.841	4.52846
$761$	30.1240	1.09199	0.545996	−	0.837787i	$-0.316151\pi$
$761$	30.1240	1.09199	0.545996	+	0.837787i	$0.316151\pi$
$762$	0	0
$763$	17.3751	0.629020
$764$	76.0575	2.75166
$765$	0	0
$766$	50.0898	1.80982
$767$	62.7622	2.26621
$768$	0	0
$769$	22.4316	0.808903	0.404452	−	0.914559i	$-0.367462\pi$
$769$	22.4316	0.808903	0.404452	+	0.914559i	$0.367462\pi$
$770$	−15.6290	−0.563230
$771$	0	0
$772$	99.2643	3.57260
$773$	−29.0355	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$773$	−29.0355	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$774$	0	0
$775$	−23.2312	−0.834491
$776$	−70.7780	−2.54078
$777$	0	0
$778$	63.4319	2.27414
$779$	30.2964	1.08548
$780$	0	0
$781$	3.82894	0.137010
$782$	−32.9741	−1.17915
$783$	0	0
$784$	−49.7014	−1.77505
$785$	56.4549	2.01496
$786$	0	0
$787$	38.7706	1.38202	0.691011	−	0.722844i	$-0.257166\pi$
$787$	38.7706	1.38202	0.691011	+	0.722844i	$0.257166\pi$
$788$	93.9144	3.34556
$789$	0	0
$790$	−112.193	−3.99163
$791$	25.7964	0.917214
$792$	0	0
$793$	5.17233	0.183675
$794$	−78.3123	−2.77920
$795$	0	0
$796$	25.8837	0.917425
$797$	33.3549	1.18149	0.590746	−	0.806857i	$-0.298833\pi$
$797$	33.3549	1.18149	0.590746	+	0.806857i	$0.298833\pi$
$798$	0	0
$799$	−6.72318	−0.237849
$800$	−125.586	−4.44012
$801$	0	0
$802$	17.8209	0.629279
$803$	−3.75672	−0.132572
$804$	0	0
$805$	−20.7876	−0.732667
$806$	−39.9442	−1.40697
$807$	0	0
$808$	−87.0030	−3.06075
$809$	−42.5115	−1.49462	−0.747312	−	0.664473i	$-0.768656\pi$
$809$	−42.5115	−1.49462	−0.747312	+	0.664473i	$0.768656\pi$
$810$	0	0
$811$	38.0311	1.33545	0.667726	−	0.744408i	$-0.267268\pi$
$811$	38.0311	1.33545	0.667726	+	0.744408i	$0.267268\pi$
$812$	80.8447	2.83709
$813$	0	0
$814$	24.8269	0.870182
$815$	13.2741	0.464973
$816$	0	0
$817$	19.5941	0.685509
$818$	−95.5961	−3.34244
$819$	0	0
$820$	−134.682	−4.70329
$821$	8.25781	0.288199	0.144100	−	0.989563i	$-0.453971\pi$
$821$	8.25781	0.288199	0.144100	+	0.989563i	$0.453971\pi$
$822$	0	0
$823$	48.2929	1.68338	0.841692	−	0.539958i	$-0.181560\pi$
$823$	48.2929	1.68338	0.841692	+	0.539958i	$0.181560\pi$
$824$	91.8460	3.19961
$825$	0	0
$826$	−72.5932	−2.52584
$827$	49.3222	1.71510	0.857550	−	0.514401i	$-0.171986\pi$
$827$	49.3222	1.71510	0.857550	+	0.514401i	$0.171986\pi$
$828$	0	0
$829$	28.9334	1.00490	0.502449	−	0.864607i	$-0.332432\pi$
$829$	28.9334	1.00490	0.502449	+	0.864607i	$0.332432\pi$
$830$	−77.1652	−2.67844
$831$	0	0
$832$	−100.427	−3.48170
$833$	13.4455	0.465860
$834$	0	0
$835$	65.3850	2.26274
$836$	20.0679	0.694062
$837$	0	0
$838$	−77.3552	−2.67219
$839$	−49.6135	−1.71285	−0.856424	−	0.516274i	$-0.827319\pi$
$839$	−49.6135	−1.71285	−0.856424	+	0.516274i	$0.827319\pi$
$840$	0	0
$841$	40.7106	1.40381
$842$	−64.5689	−2.22519
$843$	0	0
$844$	5.43156	0.186962
$845$	−17.7423	−0.610355
$846$	0	0
$847$	18.5078	0.635937
$848$	−67.8893	−2.33133
$849$	0	0
$850$	66.3883	2.27710
$851$	33.0214	1.13196
$852$	0	0
$853$	−39.3666	−1.34789	−0.673943	−	0.738783i	$-0.735401\pi$
$853$	−39.3666	−1.34789	−0.673943	+	0.738783i	$0.735401\pi$
$854$	−5.98252	−0.204718
$855$	0	0
$856$	−26.5498	−0.907453
$857$	42.0438	1.43619	0.718095	−	0.695945i	$-0.245014\pi$
$857$	42.0438	1.43619	0.718095	+	0.695945i	$0.245014\pi$
$858$	0	0
$859$	−7.58227	−0.258704	−0.129352	−	0.991599i	$-0.541290\pi$
$859$	−7.58227	−0.258704	−0.129352	+	0.991599i	$0.541290\pi$
$860$	−87.1047	−2.97025
$861$	0	0
$862$	19.3904	0.660439
$863$	−32.1538	−1.09453	−0.547265	−	0.836960i	$-0.684331\pi$
$863$	−32.1538	−1.09453	−0.547265	+	0.836960i	$0.684331\pi$
$864$	0	0
$865$	2.59146	0.0881123
$866$	−75.8448	−2.57731
$867$	0	0
$868$	33.5538	1.13889
$869$	−11.2370	−0.381189
$870$	0	0
$871$	−0.431978	−0.0146370
$872$	85.0875	2.88143
$873$	0	0
$874$	36.7523	1.24316
$875$	10.6379	0.359626
$876$	0	0
$877$	17.2395	0.582135	0.291068	−	0.956702i	$-0.405990\pi$
$877$	17.2395	0.582135	0.291068	+	0.956702i	$0.405990\pi$
$878$	−25.3305	−0.854863
$879$	0	0
$880$	−42.9111	−1.44653
$881$	−38.2313	−1.28805	−0.644023	−	0.765006i	$-0.722736\pi$
$881$	−38.2313	−1.28805	−0.644023	+	0.765006i	$0.722736\pi$
$882$	0	0
$883$	−31.1066	−1.04682	−0.523411	−	0.852080i	$-0.675341\pi$
$883$	−31.1066	−1.04682	−0.523411	+	0.852080i	$0.675341\pi$
$884$	82.9017	2.78829
$885$	0	0
$886$	43.6861	1.46766
$887$	16.0884	0.540195	0.270098	−	0.962833i	$-0.412944\pi$
$887$	16.0884	0.540195	0.270098	+	0.962833i	$0.412944\pi$
$888$	0	0
$889$	33.3859	1.11973
$890$	−63.6228	−2.13264
$891$	0	0
$892$	5.30613	0.177662
$893$	7.49353	0.250761
$894$	0	0
$895$	−50.8452	−1.69957
$896$	47.7889	1.59652
$897$	0	0
$898$	3.24978	0.108447
$899$	28.9327	0.964960
$900$	0	0
$901$	18.3658	0.611854
$902$	−18.5740	−0.618445
$903$	0	0
$904$	126.328	4.20159
$905$	−72.8190	−2.42058
$906$	0	0
$907$	−32.8974	−1.09234	−0.546170	−	0.837674i	$-0.683915\pi$
$907$	−32.8974	−1.09234	−0.546170	+	0.837674i	$0.683915\pi$
$908$	48.3953	1.60605
$909$	0	0
$910$	71.9623	2.38553
$911$	8.68821	0.287853	0.143927	−	0.989588i	$-0.454027\pi$
$911$	8.68821	0.287853	0.143927	+	0.989588i	$0.454027\pi$
$912$	0	0
$913$	−7.72873	−0.255784
$914$	50.5309	1.67141
$915$	0	0
$916$	−105.446	−3.48404
$917$	40.2601	1.32951
$918$	0	0
$919$	22.9073	0.755642	0.377821	−	0.925879i	$-0.376674\pi$
$919$	22.9073	0.755642	0.377821	+	0.925879i	$0.376674\pi$
$920$	−101.799	−3.35621
$921$	0	0
$922$	−41.9445	−1.38137
$923$	−17.6300	−0.580299
$924$	0	0
$925$	−66.4836	−2.18597
$926$	12.0330	0.395428
$927$	0	0
$928$	156.407	5.13432
$929$	−3.68103	−0.120771	−0.0603853	−	0.998175i	$-0.519233\pi$
$929$	−3.68103	−0.120771	−0.0603853	+	0.998175i	$0.519233\pi$
$930$	0	0
$931$	−14.9861	−0.491151
$932$	−127.795	−4.18608
$933$	0	0
$934$	−46.0848	−1.50794
$935$	11.6086	0.379641
$936$	0	0
$937$	21.0149	0.686528	0.343264	−	0.939239i	$-0.388467\pi$
$937$	21.0149	0.686528	0.343264	+	0.939239i	$0.388467\pi$
$938$	0.499643	0.0163139
$939$	0	0
$940$	−33.3122	−1.08652
$941$	41.4275	1.35050	0.675250	−	0.737589i	$-0.264036\pi$
$941$	41.4275	1.35050	0.675250	+	0.737589i	$0.264036\pi$
$942$	0	0
$943$	−24.7046	−0.804492
$944$	−199.312	−6.48707
$945$	0	0
$946$	−12.0126	−0.390564
$947$	14.4794	0.470519	0.235259	−	0.971933i	$-0.424406\pi$
$947$	14.4794	0.470519	0.235259	+	0.971933i	$0.424406\pi$
$948$	0	0
$949$	17.2975	0.561500
$950$	−73.9952	−2.40072
$951$	0	0
$952$	−59.7452	−1.93635
$953$	23.2549	0.753301	0.376651	−	0.926355i	$-0.377076\pi$
$953$	23.2549	0.753301	0.376651	+	0.926355i	$0.377076\pi$
$954$	0	0
$955$	−49.0378	−1.58683
$956$	−114.957	−3.71797
$957$	0	0
$958$	39.6088	1.27970
$959$	8.25459	0.266555
$960$	0	0
$961$	−18.9918	−0.612638
$962$	−114.313	−3.68560
$963$	0	0
$964$	16.5719	0.533745
$965$	−64.0003	−2.06024
$966$	0	0
$967$	−21.9681	−0.706447	−0.353224	−	0.935539i	$-0.614915\pi$
$967$	−21.9681	−0.706447	−0.353224	+	0.935539i	$0.614915\pi$
$968$	90.6348	2.91311
$969$	0	0
$970$	73.2395	2.35158
$971$	−0.312009	−0.0100128	−0.00500642	−	0.999987i	$-0.501594\pi$
$971$	−0.312009	−0.0100128	−0.00500642	+	0.999987i	$0.501594\pi$
$972$	0	0
$973$	10.6141	0.340272
$974$	−62.8133	−2.01267
$975$	0	0
$976$	−16.4257	−0.525772
$977$	52.9958	1.69549	0.847743	−	0.530407i	$-0.177961\pi$
$977$	52.9958	1.69549	0.847743	+	0.530407i	$0.177961\pi$
$978$	0	0
$979$	−6.37234	−0.203661
$980$	66.6203	2.12811
$981$	0	0
$982$	109.877	3.50631
$983$	6.53441	0.208415	0.104208	−	0.994556i	$-0.466769\pi$
$983$	6.53441	0.208415	0.104208	+	0.994556i	$0.466769\pi$
$984$	0	0
$985$	−60.5510	−1.92931
$986$	−82.6815	−2.63312
$987$	0	0
$988$	−92.4007	−2.93966
$989$	−15.9776	−0.508057
$990$	0	0
$991$	−44.0955	−1.40074	−0.700370	−	0.713780i	$-0.746982\pi$
$991$	−44.0955	−1.40074	−0.700370	+	0.713780i	$0.746982\pi$
$992$	64.9153	2.06106
$993$	0	0
$994$	20.3915	0.646780
$995$	−16.6884	−0.529059
$996$	0	0
$997$	4.02146	0.127361	0.0636805	−	0.997970i	$-0.479716\pi$
$997$	4.02146	0.127361	0.0636805	+	0.997970i	$0.479716\pi$
$998$	−116.368	−3.68355
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6021.2.a.t.1.1	✓	40
3.2	odd	2	inner	6021.2.a.t.1.40	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6021.2.a.t.1.1	✓	40	1.1	even	1	trivial
6021.2.a.t.1.40	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)

\(f(q)\)	\(=\)	\(q-2.70299 q^{2} +5.30613 q^{4} -3.42111 q^{5} -1.82484 q^{7} -8.93642 q^{8} +O(q^{10})\)	\(q-2.70299 q^{2} +5.30613 q^{4} -3.42111 q^{5} -1.82484 q^{7} -8.93642 q^{8} +9.24720 q^{10} +0.926183 q^{11} -4.26452 q^{13} +4.93251 q^{14} +13.5427 q^{16} -3.66366 q^{17} +4.08345 q^{19} -18.1528 q^{20} -2.50346 q^{22} -3.32976 q^{23} +6.70398 q^{25} +11.5269 q^{26} -9.68282 q^{28} -8.34929 q^{29} -3.46529 q^{31} -18.7330 q^{32} +9.90282 q^{34} +6.24297 q^{35} -9.91704 q^{37} -11.0375 q^{38} +30.5724 q^{40} +7.41932 q^{41} +4.79841 q^{43} +4.91444 q^{44} +9.00030 q^{46} +1.83510 q^{47} -3.66997 q^{49} -18.1208 q^{50} -22.6281 q^{52} -5.01297 q^{53} -3.16857 q^{55} +16.3075 q^{56} +22.5680 q^{58} -14.7173 q^{59} -1.21288 q^{61} +9.36663 q^{62} +23.5495 q^{64} +14.5894 q^{65} +0.101296 q^{67} -19.4399 q^{68} -16.8746 q^{70} +4.13411 q^{71} -4.05614 q^{73} +26.8056 q^{74} +21.6673 q^{76} -1.69013 q^{77} -12.1326 q^{79} -46.3312 q^{80} -20.0543 q^{82} -8.34471 q^{83} +12.5338 q^{85} -12.9700 q^{86} -8.27675 q^{88} -6.88022 q^{89} +7.78206 q^{91} -17.6682 q^{92} -4.96024 q^{94} -13.9699 q^{95} +7.92018 q^{97} +9.91987 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 46 q^{4} + 16 q^{7}+O(q^{10}) \) 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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