LMFDB - Embedded newform 6021.2.a.t.1.11

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			1.11
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)

$f(q)$	$=$	$q-1.47084 q^{2} +0.163365 q^{4} -1.43270 q^{5} -5.26244 q^{7} +2.70139 q^{8} +O(q^{10})$	\(q-1.47084 q^{2} +0.163365 q^{4} -1.43270 q^{5} -5.26244 q^{7} +2.70139 q^{8} +2.10726 q^{10} +4.03771 q^{11} -5.88770 q^{13} +7.74020 q^{14} -4.30004 q^{16} +0.570315 q^{17} +7.70915 q^{19} -0.234053 q^{20} -5.93881 q^{22} -4.80802 q^{23} -2.94738 q^{25} +8.65985 q^{26} -0.859701 q^{28} +0.820324 q^{29} +2.77291 q^{31} +0.921882 q^{32} -0.838841 q^{34} +7.53948 q^{35} -0.0199806 q^{37} -11.3389 q^{38} -3.87027 q^{40} -10.0913 q^{41} -8.24247 q^{43} +0.659621 q^{44} +7.07182 q^{46} -6.63072 q^{47} +20.6933 q^{49} +4.33512 q^{50} -0.961846 q^{52} -1.10132 q^{53} -5.78481 q^{55} -14.2159 q^{56} -1.20656 q^{58} +8.41609 q^{59} -1.09674 q^{61} -4.07851 q^{62} +7.24415 q^{64} +8.43528 q^{65} -6.95836 q^{67} +0.0931697 q^{68} -11.0894 q^{70} -15.2643 q^{71} -1.95440 q^{73} +0.0293882 q^{74} +1.25941 q^{76} -21.2482 q^{77} -2.91865 q^{79} +6.16065 q^{80} +14.8427 q^{82} -3.60707 q^{83} -0.817088 q^{85} +12.1233 q^{86} +10.9074 q^{88} -12.4582 q^{89} +30.9837 q^{91} -0.785465 q^{92} +9.75272 q^{94} -11.0449 q^{95} -7.07069 q^{97} -30.4365 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.47084	−1.04004	−0.520020	−	0.854154i	$-0.674075\pi$
$2$	−1.47084	−1.04004	−0.520020	+	0.854154i	$0.674075\pi$
$3$	0	0
$3$	0	0
$4$	0.163365	0.0816827
$5$	−1.43270	−0.640721	−0.320361	−	0.947296i	$-0.603804\pi$
$5$	−1.43270	−0.640721	−0.320361	+	0.947296i	$0.603804\pi$
$6$	0	0
$7$	−5.26244	−1.98902	−0.994508	−	0.104663i	$-0.966624\pi$
$7$	−5.26244	−1.98902	−0.994508	+	0.104663i	$0.966624\pi$
$8$	2.70139	0.955087
$9$	0	0
$10$	2.10726	0.666376
$11$	4.03771	1.21741	0.608707	−	0.793395i	$-0.291689\pi$
$11$	4.03771	1.21741	0.608707	+	0.793395i	$0.291689\pi$
$12$	0	0
$13$	−5.88770	−1.63295	−0.816477	−	0.577378i	$-0.804076\pi$
$13$	−5.88770	−1.63295	−0.816477	+	0.577378i	$0.804076\pi$
$14$	7.74020	2.06866
$15$	0	0
$16$	−4.30004	−1.07501
$17$	0.570315	0.138322	0.0691608	−	0.997606i	$-0.477968\pi$
$17$	0.570315	0.138322	0.0691608	+	0.997606i	$0.477968\pi$
$18$	0	0
$19$	7.70915	1.76860	0.884300	−	0.466918i	$-0.154636\pi$
$19$	7.70915	1.76860	0.884300	+	0.466918i	$0.154636\pi$
$20$	−0.234053	−0.0523358
$21$	0	0
$22$	−5.93881	−1.26616
$23$	−4.80802	−1.00254	−0.501271	−	0.865290i	$-0.667134\pi$
$23$	−4.80802	−1.00254	−0.501271	+	0.865290i	$0.667134\pi$
$24$	0	0
$25$	−2.94738	−0.589476
$26$	8.65985	1.69834
$27$	0	0
$28$	−0.859701	−0.162468
$29$	0.820324	0.152330	0.0761652	−	0.997095i	$-0.475732\pi$
$29$	0.820324	0.152330	0.0761652	+	0.997095i	$0.475732\pi$
$30$	0	0
$31$	2.77291	0.498030	0.249015	−	0.968500i	$-0.419893\pi$
$31$	2.77291	0.498030	0.249015	+	0.968500i	$0.419893\pi$
$32$	0.921882	0.162967
$33$	0	0
$34$	−0.838841	−0.143860
$35$	7.53948	1.27440
$36$	0	0
$37$	−0.0199806	−0.00328479	−0.00164239	−	0.999999i	$-0.500523\pi$
$37$	−0.0199806	−0.00328479	−0.00164239	+	0.999999i	$0.500523\pi$
$38$	−11.3389	−1.83942
$39$	0	0
$40$	−3.87027	−0.611944
$41$	−10.0913	−1.57599	−0.787997	−	0.615679i	$-0.788882\pi$
$41$	−10.0913	−1.57599	−0.787997	+	0.615679i	$0.788882\pi$
$42$	0	0
$43$	−8.24247	−1.25696	−0.628482	−	0.777824i	$-0.716324\pi$
$43$	−8.24247	−1.25696	−0.628482	+	0.777824i	$0.716324\pi$
$44$	0.659621	0.0994417
$45$	0	0
$46$	7.07182	1.04268
$47$	−6.63072	−0.967190	−0.483595	−	0.875292i	$-0.660669\pi$
$47$	−6.63072	−0.967190	−0.483595	+	0.875292i	$0.660669\pi$
$48$	0	0
$49$	20.6933	2.95618
$50$	4.33512	0.613079
$51$	0	0
$52$	−0.961846	−0.133384
$53$	−1.10132	−0.151278	−0.0756390	−	0.997135i	$-0.524100\pi$
$53$	−1.10132	−0.151278	−0.0756390	+	0.997135i	$0.524100\pi$
$54$	0	0
$55$	−5.78481	−0.780023
$56$	−14.2159	−1.89968
$57$	0	0
$58$	−1.20656	−0.158430
$59$	8.41609	1.09568	0.547841	−	0.836583i	$-0.315450\pi$
$59$	8.41609	1.09568	0.547841	+	0.836583i	$0.315450\pi$
$60$	0	0
$61$	−1.09674	−0.140423	−0.0702114	−	0.997532i	$-0.522367\pi$
$61$	−1.09674	−0.140423	−0.0702114	+	0.997532i	$0.522367\pi$
$62$	−4.07851	−0.517971
$63$	0	0
$64$	7.24415	0.905518
$65$	8.43528	1.04627
$66$	0	0
$67$	−6.95836	−0.850099	−0.425050	−	0.905170i	$-0.639743\pi$
$67$	−6.95836	−0.850099	−0.425050	+	0.905170i	$0.639743\pi$
$68$	0.0931697	0.0112985
$69$	0	0
$70$	−11.0894	−1.32543
$71$	−15.2643	−1.81154	−0.905771	−	0.423768i	$-0.860707\pi$
$71$	−15.2643	−1.81154	−0.905771	+	0.423768i	$0.860707\pi$
$72$	0	0
$73$	−1.95440	−0.228745	−0.114372	−	0.993438i	$-0.536486\pi$
$73$	−1.95440	−0.228745	−0.114372	+	0.993438i	$0.536486\pi$
$74$	0.0293882	0.00341631
$75$	0	0
$76$	1.25941	0.144464
$77$	−21.2482	−2.42146
$78$	0	0
$79$	−2.91865	−0.328374	−0.164187	−	0.986429i	$-0.552500\pi$
$79$	−2.91865	−0.328374	−0.164187	+	0.986429i	$0.552500\pi$
$80$	6.16065	0.688782
$81$	0	0
$82$	14.8427	1.63910
$83$	−3.60707	−0.395928	−0.197964	−	0.980209i	$-0.563433\pi$
$83$	−3.60707	−0.395928	−0.197964	+	0.980209i	$0.563433\pi$
$84$	0	0
$85$	−0.817088	−0.0886256
$86$	12.1233	1.30729
$87$	0	0
$88$	10.9074	1.16274
$89$	−12.4582	−1.32057	−0.660284	−	0.751016i	$-0.729564\pi$
$89$	−12.4582	−1.32057	−0.660284	+	0.751016i	$0.729564\pi$
$90$	0	0
$91$	30.9837	3.24797
$92$	−0.785465	−0.0818903
$93$	0	0
$94$	9.75272	1.00592
$95$	−11.0449	−1.13318
$96$	0	0
$97$	−7.07069	−0.717920	−0.358960	−	0.933353i	$-0.616869\pi$
$97$	−7.07069	−0.717920	−0.358960	+	0.933353i	$0.616869\pi$
$98$	−30.4365	−3.07455
$99$	0	0
$100$	−0.481500	−0.0481500
$101$	5.52632	0.549889	0.274944	−	0.961460i	$-0.411341\pi$
$101$	5.52632	0.549889	0.274944	+	0.961460i	$0.411341\pi$
$102$	0	0
$103$	−8.67140	−0.854418	−0.427209	−	0.904153i	$-0.640503\pi$
$103$	−8.67140	−0.854418	−0.427209	+	0.904153i	$0.640503\pi$
$104$	−15.9050	−1.55961
$105$	0	0
$106$	1.61986	0.157335
$107$	−15.2601	−1.47525	−0.737623	−	0.675213i	$-0.764052\pi$
$107$	−15.2601	−1.47525	−0.737623	+	0.675213i	$0.764052\pi$
$108$	0	0
$109$	3.03277	0.290486	0.145243	−	0.989396i	$-0.453604\pi$
$109$	3.03277	0.290486	0.145243	+	0.989396i	$0.453604\pi$
$110$	8.50851	0.811255
$111$	0	0
$112$	22.6287	2.13821
$113$	9.81256	0.923088	0.461544	−	0.887117i	$-0.347296\pi$
$113$	9.81256	0.923088	0.461544	+	0.887117i	$0.347296\pi$
$114$	0	0
$115$	6.88844	0.642350
$116$	0.134013	0.0124428
$117$	0	0
$118$	−12.3787	−1.13955
$119$	−3.00125	−0.275124
$120$	0	0
$121$	5.30306	0.482097
$122$	1.61312	0.146045
$123$	0	0
$124$	0.452998	0.0406804
$125$	11.3862	1.01841
$126$	0	0
$127$	5.66532	0.502716	0.251358	−	0.967894i	$-0.419123\pi$
$127$	5.66532	0.502716	0.251358	+	0.967894i	$0.419123\pi$
$128$	−12.4987	−1.10474
$129$	0	0
$130$	−12.4069	−1.08816
$131$	1.82607	0.159544	0.0797722	−	0.996813i	$-0.474581\pi$
$131$	1.82607	0.159544	0.0797722	+	0.996813i	$0.474581\pi$
$132$	0	0
$133$	−40.5690	−3.51777
$134$	10.2346	0.884137
$135$	0	0
$136$	1.54064	0.132109
$137$	17.0698	1.45837	0.729184	−	0.684318i	$-0.239900\pi$
$137$	17.0698	1.45837	0.729184	+	0.684318i	$0.239900\pi$
$138$	0	0
$139$	−9.11075	−0.772764	−0.386382	−	0.922339i	$-0.626275\pi$
$139$	−9.11075	−0.772764	−0.386382	+	0.922339i	$0.626275\pi$
$140$	1.23169	0.104097
$141$	0	0
$142$	22.4514	1.88408
$143$	−23.7728	−1.98798
$144$	0	0
$145$	−1.17527	−0.0976013
$146$	2.87460	0.237904
$147$	0	0
$148$	−0.00326414	−0.000268310 0
$149$	−20.4810	−1.67786	−0.838932	−	0.544236i	$-0.816820\pi$
$149$	−20.4810	−1.67786	−0.838932	+	0.544236i	$0.816820\pi$
$150$	0	0
$151$	14.2182	1.15706	0.578531	−	0.815660i	$-0.303626\pi$
$151$	14.2182	1.15706	0.578531	+	0.815660i	$0.303626\pi$
$152$	20.8254	1.68917
$153$	0	0
$154$	31.2526	2.51841
$155$	−3.97274	−0.319098
$156$	0	0
$157$	−4.52306	−0.360979	−0.180490	−	0.983577i	$-0.557768\pi$
$157$	−4.52306	−0.360979	−0.180490	+	0.983577i	$0.557768\pi$
$158$	4.29286	0.341522
$159$	0	0
$160$	−1.32078	−0.104417
$161$	25.3019	1.99407
$162$	0	0
$163$	−7.88180	−0.617350	−0.308675	−	0.951168i	$-0.599886\pi$
$163$	−7.88180	−0.617350	−0.308675	+	0.951168i	$0.599886\pi$
$164$	−1.64857	−0.128731
$165$	0	0
$166$	5.30542	0.411780
$167$	−12.0879	−0.935387	−0.467693	−	0.883891i	$-0.654915\pi$
$167$	−12.0879	−0.935387	−0.467693	+	0.883891i	$0.654915\pi$
$168$	0	0
$169$	21.6650	1.66654
$170$	1.20180	0.0921742
$171$	0	0
$172$	−1.34653	−0.102672
$173$	−22.7540	−1.72996	−0.864978	−	0.501809i	$-0.832668\pi$
$173$	−22.7540	−1.72996	−0.864978	+	0.501809i	$0.832668\pi$
$174$	0	0
$175$	15.5104	1.17248
$176$	−17.3623	−1.30873
$177$	0	0
$178$	18.3240	1.37344
$179$	−17.2198	−1.28707	−0.643534	−	0.765417i	$-0.722533\pi$
$179$	−17.2198	−1.28707	−0.643534	+	0.765417i	$0.722533\pi$
$180$	0	0
$181$	17.1179	1.27236	0.636182	−	0.771539i	$-0.280513\pi$
$181$	17.1179	1.27236	0.636182	+	0.771539i	$0.280513\pi$
$182$	−45.5720	−3.37802
$183$	0	0
$184$	−12.9884	−0.957514
$185$	0.0286261	0.00210463
$186$	0	0
$187$	2.30276	0.168395
$188$	−1.08323	−0.0790027
$189$	0	0
$190$	16.2452	1.17855
$191$	5.16732	0.373894	0.186947	−	0.982370i	$-0.440141\pi$
$191$	5.16732	0.373894	0.186947	+	0.982370i	$0.440141\pi$
$192$	0	0
$193$	−0.989477	−0.0712241	−0.0356121	−	0.999366i	$-0.511338\pi$
$193$	−0.989477	−0.0712241	−0.0356121	+	0.999366i	$0.511338\pi$
$194$	10.3998	0.746665
$195$	0	0
$196$	3.38057	0.241469
$197$	23.3889	1.66639	0.833193	−	0.552982i	$-0.186510\pi$
$197$	23.3889	1.66639	0.833193	+	0.552982i	$0.186510\pi$
$198$	0	0
$199$	−10.3003	−0.730167	−0.365083	−	0.930975i	$-0.618959\pi$
$199$	−10.3003	−0.730167	−0.365083	+	0.930975i	$0.618959\pi$
$200$	−7.96203	−0.563001
$201$	0	0
$202$	−8.12832	−0.571906
$203$	−4.31691	−0.302987
$204$	0	0
$205$	14.4578	1.00977
$206$	12.7542	0.888629
$207$	0	0
$208$	25.3174	1.75544
$209$	31.1273	2.15312
$210$	0	0
$211$	7.44357	0.512436	0.256218	−	0.966619i	$-0.417523\pi$
$211$	7.44357	0.512436	0.256218	+	0.966619i	$0.417523\pi$
$212$	−0.179918	−0.0123568
$213$	0	0
$214$	22.4451	1.53431
$215$	11.8090	0.805364
$216$	0	0
$217$	−14.5923	−0.990589
$218$	−4.46071	−0.302117
$219$	0	0
$220$	−0.945037	−0.0637144
$221$	−3.35784	−0.225873
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	−4.85135	−0.324144
$225$	0	0
$226$	−14.4327	−0.960049
$227$	25.5887	1.69838	0.849191	−	0.528087i	$-0.177090\pi$
$227$	25.5887	1.69838	0.849191	+	0.528087i	$0.177090\pi$
$228$	0	0
$229$	−2.86392	−0.189253	−0.0946267	−	0.995513i	$-0.530166\pi$
$229$	−2.86392	−0.189253	−0.0946267	+	0.995513i	$0.530166\pi$
$230$	−10.1318	−0.668070
$231$	0	0
$232$	2.21602	0.145489
$233$	16.3388	1.07039	0.535195	−	0.844729i	$-0.320238\pi$
$233$	16.3388	1.07039	0.535195	+	0.844729i	$0.320238\pi$
$234$	0	0
$235$	9.49981	0.619699
$236$	1.37490	0.0894982
$237$	0	0
$238$	4.41435	0.286140
$239$	−1.78996	−0.115783	−0.0578915	−	0.998323i	$-0.518438\pi$
$239$	−1.78996	−0.115783	−0.0578915	+	0.998323i	$0.518438\pi$
$240$	0	0
$241$	4.75981	0.306606	0.153303	−	0.988179i	$-0.451009\pi$
$241$	4.75981	0.306606	0.153303	+	0.988179i	$0.451009\pi$
$242$	−7.79995	−0.501400
$243$	0	0
$244$	−0.179169	−0.0114701
$245$	−29.6472	−1.89409
$246$	0	0
$247$	−45.3892	−2.88804
$248$	7.49073	0.475662
$249$	0	0
$250$	−16.7472	−1.05919
$251$	−11.1884	−0.706208	−0.353104	−	0.935584i	$-0.614874\pi$
$251$	−11.1884	−0.706208	−0.353104	+	0.935584i	$0.614874\pi$
$252$	0	0
$253$	−19.4134	−1.22051
$254$	−8.33277	−0.522845
$255$	0	0
$256$	3.89532	0.243458
$257$	8.32837	0.519510	0.259755	−	0.965675i	$-0.416358\pi$
$257$	8.32837	0.519510	0.259755	+	0.965675i	$0.416358\pi$
$258$	0	0
$259$	0.105147	0.00653349
$260$	1.37803	0.0854620
$261$	0	0
$262$	−2.68585	−0.165932
$263$	−2.50629	−0.154544	−0.0772721	−	0.997010i	$-0.524621\pi$
$263$	−2.50629	−0.154544	−0.0772721	+	0.997010i	$0.524621\pi$
$264$	0	0
$265$	1.57786	0.0969270
$266$	59.6704	3.65863
$267$	0	0
$268$	−1.13676	−0.0694384
$269$	8.56659	0.522314	0.261157	−	0.965296i	$-0.415896\pi$
$269$	8.56659	0.522314	0.261157	+	0.965296i	$0.415896\pi$
$270$	0	0
$271$	−3.90472	−0.237195	−0.118597	−	0.992942i	$-0.537840\pi$
$271$	−3.90472	−0.237195	−0.118597	+	0.992942i	$0.537840\pi$
$272$	−2.45238	−0.148697
$273$	0	0
$274$	−25.1069	−1.51676
$275$	−11.9007	−0.717637
$276$	0	0
$277$	16.2506	0.976406	0.488203	−	0.872730i	$-0.337653\pi$
$277$	16.2506	0.976406	0.488203	+	0.872730i	$0.337653\pi$
$278$	13.4004	0.803705
$279$	0	0
$280$	20.3671	1.21717
$281$	−17.6315	−1.05181	−0.525904	−	0.850544i	$-0.676273\pi$
$281$	−17.6315	−1.05181	−0.525904	+	0.850544i	$0.676273\pi$
$282$	0	0
$283$	27.3617	1.62648	0.813241	−	0.581927i	$-0.197701\pi$
$283$	27.3617	1.62648	0.813241	+	0.581927i	$0.197701\pi$
$284$	−2.49366	−0.147972
$285$	0	0
$286$	34.9659	2.06758
$287$	53.1048	3.13468
$288$	0	0
$289$	−16.6747	−0.980867
$290$	1.72864	0.101509
$291$	0	0
$292$	−0.319281	−0.0186845
$293$	15.4459	0.902359	0.451180	−	0.892433i	$-0.351003\pi$
$293$	15.4459	0.902359	0.451180	+	0.892433i	$0.351003\pi$
$294$	0	0
$295$	−12.0577	−0.702026
$296$	−0.0539754	−0.00313726
$297$	0	0
$298$	30.1242	1.74505
$299$	28.3082	1.63710
$300$	0	0
$301$	43.3755	2.50012
$302$	−20.9127	−1.20339
$303$	0	0
$304$	−33.1497	−1.90126
$305$	1.57129	0.0899719
$306$	0	0
$307$	7.27856	0.415409	0.207705	−	0.978192i	$-0.433401\pi$
$307$	7.27856	0.415409	0.207705	+	0.978192i	$0.433401\pi$
$308$	−3.47122	−0.197791
$309$	0	0
$310$	5.84326	0.331875
$311$	34.0635	1.93156	0.965781	−	0.259359i	$-0.0835113\pi$
$311$	34.0635	1.93156	0.965781	+	0.259359i	$0.0835113\pi$
$312$	0	0
$313$	−12.4897	−0.705960	−0.352980	−	0.935631i	$-0.614832\pi$
$313$	−12.4897	−0.705960	−0.352980	+	0.935631i	$0.614832\pi$
$314$	6.65269	0.375433
$315$	0	0
$316$	−0.476807	−0.0268225
$317$	−21.7058	−1.21912	−0.609559	−	0.792741i	$-0.708653\pi$
$317$	−21.7058	−1.21912	−0.609559	+	0.792741i	$0.708653\pi$
$318$	0	0
$319$	3.31223	0.185449
$320$	−10.3787	−0.580185
$321$	0	0
$322$	−37.2151	−2.07391
$323$	4.39664	0.244636
$324$	0	0
$325$	17.3533	0.962587
$326$	11.5929	0.642069
$327$	0	0
$328$	−27.2605	−1.50521
$329$	34.8938	1.92376
$330$	0	0
$331$	20.4802	1.12569	0.562847	−	0.826561i	$-0.309706\pi$
$331$	20.4802	1.12569	0.562847	+	0.826561i	$0.309706\pi$
$332$	−0.589271	−0.0323404
$333$	0	0
$334$	17.7793	0.972840
$335$	9.96922	0.544677
$336$	0	0
$337$	8.22770	0.448191	0.224096	−	0.974567i	$-0.428057\pi$
$337$	8.22770	0.448191	0.224096	+	0.974567i	$0.428057\pi$
$338$	−31.8657	−1.73327
$339$	0	0
$340$	−0.133484	−0.00723918
$341$	11.1962	0.606308
$342$	0	0
$343$	−72.0601	−3.89088
$344$	−22.2661	−1.20051
$345$	0	0
$346$	33.4675	1.79922
$347$	−9.12688	−0.489956	−0.244978	−	0.969529i	$-0.578781\pi$
$347$	−9.12688	−0.489956	−0.244978	+	0.969529i	$0.578781\pi$
$348$	0	0
$349$	−27.2472	−1.45851	−0.729255	−	0.684243i	$-0.760133\pi$
$349$	−27.2472	−1.45851	−0.729255	+	0.684243i	$0.760133\pi$
$350$	−22.8133	−1.21942
$351$	0	0
$352$	3.72229	0.198399
$353$	14.4818	0.770791	0.385396	−	0.922751i	$-0.374065\pi$
$353$	14.4818	0.770791	0.385396	+	0.922751i	$0.374065\pi$
$354$	0	0
$355$	21.8691	1.16069
$356$	−2.03524	−0.107868
$357$	0	0
$358$	25.3276	1.33860
$359$	16.1661	0.853212	0.426606	−	0.904438i	$-0.359709\pi$
$359$	16.1661	0.853212	0.426606	+	0.904438i	$0.359709\pi$
$360$	0	0
$361$	40.4310	2.12795
$362$	−25.1777	−1.32331
$363$	0	0
$364$	5.06166	0.265303
$365$	2.80006	0.146562
$366$	0	0
$367$	29.6525	1.54785	0.773923	−	0.633279i	$-0.218292\pi$
$367$	29.6525	1.54785	0.773923	+	0.633279i	$0.218292\pi$
$368$	20.6747	1.07774
$369$	0	0
$370$	−0.0421044	−0.00218890
$371$	5.79563	0.300894
$372$	0	0
$373$	35.0580	1.81523	0.907617	−	0.419798i	$-0.137899\pi$
$373$	35.0580	1.81523	0.907617	+	0.419798i	$0.137899\pi$
$374$	−3.38699	−0.175137
$375$	0	0
$376$	−17.9122	−0.923750
$377$	−4.82982	−0.248748
$378$	0	0
$379$	15.2400	0.782826	0.391413	−	0.920215i	$-0.371986\pi$
$379$	15.2400	0.782826	0.391413	+	0.920215i	$0.371986\pi$
$380$	−1.80435	−0.0925612
$381$	0	0
$382$	−7.60029	−0.388865
$383$	4.42511	0.226113	0.113056	−	0.993589i	$-0.463936\pi$
$383$	4.42511	0.226113	0.113056	+	0.993589i	$0.463936\pi$
$384$	0	0
$385$	30.4422	1.55148
$386$	1.45536	0.0740759
$387$	0	0
$388$	−1.15511	−0.0586416
$389$	−25.0483	−1.27000	−0.635000	−	0.772512i	$-0.719000\pi$
$389$	−25.0483	−1.27000	−0.635000	+	0.772512i	$0.719000\pi$
$390$	0	0
$391$	−2.74209	−0.138673
$392$	55.9007	2.82341
$393$	0	0
$394$	−34.4012	−1.73311
$395$	4.18154	0.210396
$396$	0	0
$397$	−32.8300	−1.64769	−0.823846	−	0.566813i	$-0.808176\pi$
$397$	−32.8300	−1.64769	−0.823846	+	0.566813i	$0.808176\pi$
$398$	15.1500	0.759402
$399$	0	0
$400$	12.6739	0.633693
$401$	3.82600	0.191061	0.0955306	−	0.995426i	$-0.469545\pi$
$401$	3.82600	0.191061	0.0955306	+	0.995426i	$0.469545\pi$
$402$	0	0
$403$	−16.3261	−0.813260
$404$	0.902809	0.0449164
$405$	0	0
$406$	6.34947	0.315119
$407$	−0.0806757	−0.00399895
$408$	0	0
$409$	−2.94367	−0.145555	−0.0727776	−	0.997348i	$-0.523186\pi$
$409$	−2.94367	−0.145555	−0.0727776	+	0.997348i	$0.523186\pi$
$410$	−21.2650	−1.05020
$411$	0	0
$412$	−1.41661	−0.0697912
$413$	−44.2892	−2.17933
$414$	0	0
$415$	5.16784	0.253679
$416$	−5.42776	−0.266118
$417$	0	0
$418$	−45.7832	−2.23933
$419$	−25.9771	−1.26906	−0.634532	−	0.772897i	$-0.718807\pi$
$419$	−25.9771	−1.26906	−0.634532	+	0.772897i	$0.718807\pi$
$420$	0	0
$421$	−15.9318	−0.776468	−0.388234	−	0.921561i	$-0.626915\pi$
$421$	−15.9318	−0.776468	−0.388234	+	0.921561i	$0.626915\pi$
$422$	−10.9483	−0.532954
$423$	0	0
$424$	−2.97510	−0.144484
$425$	−1.68094	−0.0815374
$426$	0	0
$427$	5.77151	0.279303
$428$	−2.49297	−0.120502
$429$	0	0
$430$	−17.3691	−0.837611
$431$	−16.3078	−0.785521	−0.392761	−	0.919641i	$-0.628480\pi$
$431$	−16.3078	−0.785521	−0.392761	+	0.919641i	$0.628480\pi$
$432$	0	0
$433$	13.7387	0.660241	0.330120	−	0.943939i	$-0.392911\pi$
$433$	13.7387	0.660241	0.330120	+	0.943939i	$0.392911\pi$
$434$	21.4629	1.03025
$435$	0	0
$436$	0.495449	0.0237277
$437$	−37.0658	−1.77310
$438$	0	0
$439$	33.0319	1.57652	0.788262	−	0.615339i	$-0.210981\pi$
$439$	33.0319	1.57652	0.788262	+	0.615339i	$0.210981\pi$
$440$	−15.6270	−0.744989
$441$	0	0
$442$	4.93884	0.234917
$443$	1.25791	0.0597653	0.0298827	−	0.999553i	$-0.490487\pi$
$443$	1.25791	0.0597653	0.0298827	+	0.999553i	$0.490487\pi$
$444$	0	0
$445$	17.8488	0.846116
$446$	−1.47084	−0.0696462
$447$	0	0
$448$	−38.1219	−1.80109
$449$	33.0424	1.55937	0.779684	−	0.626173i	$-0.215380\pi$
$449$	33.0424	1.55937	0.779684	+	0.626173i	$0.215380\pi$
$450$	0	0
$451$	−40.7457	−1.91864
$452$	1.60303	0.0754004
$453$	0	0
$454$	−37.6368	−1.76638
$455$	−44.3902	−2.08104
$456$	0	0
$457$	34.9623	1.63547	0.817734	−	0.575596i	$-0.195230\pi$
$457$	34.9623	1.63547	0.817734	+	0.575596i	$0.195230\pi$
$458$	4.21237	0.196831
$459$	0	0
$460$	1.12533	0.0524689
$461$	−5.13575	−0.239196	−0.119598	−	0.992822i	$-0.538161\pi$
$461$	−5.13575	−0.239196	−0.119598	+	0.992822i	$0.538161\pi$
$462$	0	0
$463$	−16.4031	−0.762315	−0.381157	−	0.924510i	$-0.624474\pi$
$463$	−16.4031	−0.762315	−0.381157	+	0.924510i	$0.624474\pi$
$464$	−3.52743	−0.163757
$465$	0	0
$466$	−24.0317	−1.11325
$467$	−39.3575	−1.82125	−0.910624	−	0.413235i	$-0.864399\pi$
$467$	−39.3575	−1.82125	−0.910624	+	0.413235i	$0.864399\pi$
$468$	0	0
$469$	36.6180	1.69086
$470$	−13.9727	−0.644512
$471$	0	0
$472$	22.7352	1.04647
$473$	−33.2807	−1.53025
$474$	0	0
$475$	−22.7218	−1.04255
$476$	−0.490300	−0.0224729
$477$	0	0
$478$	2.63275	0.120419
$479$	−28.5495	−1.30446	−0.652229	−	0.758022i	$-0.726166\pi$
$479$	−28.5495	−1.30446	−0.652229	+	0.758022i	$0.726166\pi$
$480$	0	0
$481$	0.117640	0.00536391
$482$	−7.00091	−0.318883
$483$	0	0
$484$	0.866337	0.0393790
$485$	10.1302	0.459986
$486$	0	0
$487$	−41.2074	−1.86729	−0.933643	−	0.358205i	$-0.883389\pi$
$487$	−41.2074	−1.86729	−0.933643	+	0.358205i	$0.883389\pi$
$488$	−2.96272	−0.134116
$489$	0	0
$490$	43.6062	1.96993
$491$	−26.4181	−1.19223	−0.596116	−	0.802898i	$-0.703290\pi$
$491$	−26.4181	−1.19223	−0.596116	+	0.802898i	$0.703290\pi$
$492$	0	0
$493$	0.467843	0.0210706
$494$	66.7601	3.00368
$495$	0	0
$496$	−11.9236	−0.535387
$497$	80.3276	3.60318
$498$	0	0
$499$	0.546366	0.0244587	0.0122294	−	0.999925i	$-0.496107\pi$
$499$	0.546366	0.0244587	0.0122294	+	0.999925i	$0.496107\pi$
$500$	1.86011	0.0831866
$501$	0	0
$502$	16.4564	0.734484
$503$	−5.94432	−0.265044	−0.132522	−	0.991180i	$-0.542308\pi$
$503$	−5.94432	−0.265044	−0.132522	+	0.991180i	$0.542308\pi$
$504$	0	0
$505$	−7.91753	−0.352326
$506$	28.5539	1.26938
$507$	0	0
$508$	0.925518	0.0410632
$509$	13.4766	0.597340	0.298670	−	0.954357i	$-0.403457\pi$
$509$	13.4766	0.597340	0.298670	+	0.954357i	$0.403457\pi$
$510$	0	0
$511$	10.2849	0.454977
$512$	19.2681	0.851537
$513$	0	0
$514$	−12.2497	−0.540311
$515$	12.4235	0.547444
$516$	0	0
$517$	−26.7729	−1.17747
$518$	−0.154654	−0.00679509
$519$	0	0
$520$	22.7870	0.999276
$521$	−21.6526	−0.948617	−0.474309	−	0.880359i	$-0.657302\pi$
$521$	−21.6526	−0.948617	−0.474309	+	0.880359i	$0.657302\pi$
$522$	0	0
$523$	20.2226	0.884271	0.442135	−	0.896948i	$-0.354221\pi$
$523$	20.2226	0.884271	0.442135	+	0.896948i	$0.354221\pi$
$524$	0.298316	0.0130320
$525$	0	0
$526$	3.68634	0.160732
$527$	1.58143	0.0688883
$528$	0	0
$529$	0.117083	0.00509056
$530$	−2.32077	−0.100808
$531$	0	0
$532$	−6.62756	−0.287341
$533$	59.4145	2.57353
$534$	0	0
$535$	21.8630	0.945222
$536$	−18.7973	−0.811918
$537$	0	0
$538$	−12.6001	−0.543227
$539$	83.5534	3.59890
$540$	0	0
$541$	24.7922	1.06590	0.532949	−	0.846147i	$-0.321084\pi$
$541$	24.7922	1.06590	0.532949	+	0.846147i	$0.321084\pi$
$542$	5.74321	0.246692
$543$	0	0
$544$	0.525763	0.0225419
$545$	−4.34503	−0.186121
$546$	0	0
$547$	8.54834	0.365501	0.182750	−	0.983159i	$-0.441500\pi$
$547$	8.54834	0.365501	0.182750	+	0.983159i	$0.441500\pi$
$548$	2.78861	0.119123
$549$	0	0
$550$	17.5039	0.746371
$551$	6.32400	0.269412
$552$	0	0
$553$	15.3592	0.653141
$554$	−23.9021	−1.01550
$555$	0	0
$556$	−1.48838	−0.0631214
$557$	−26.5240	−1.12386	−0.561930	−	0.827185i	$-0.689941\pi$
$557$	−26.5240	−1.12386	−0.561930	+	0.827185i	$0.689941\pi$
$558$	0	0
$559$	48.5292	2.05257
$560$	−32.4201	−1.37000
$561$	0	0
$562$	25.9331	1.09392
$563$	7.82455	0.329765	0.164883	−	0.986313i	$-0.447275\pi$
$563$	7.82455	0.329765	0.164883	+	0.986313i	$0.447275\pi$
$564$	0	0
$565$	−14.0584	−0.591442
$566$	−40.2446	−1.69161
$567$	0	0
$568$	−41.2349	−1.73018
$569$	29.0988	1.21989	0.609943	−	0.792445i	$-0.291192\pi$
$569$	29.0988	1.21989	0.609943	+	0.792445i	$0.291192\pi$
$570$	0	0
$571$	18.7352	0.784045	0.392023	−	0.919956i	$-0.371776\pi$
$571$	18.7352	0.784045	0.392023	+	0.919956i	$0.371776\pi$
$572$	−3.88365	−0.162384
$573$	0	0
$574$	−78.1086	−3.26019
$575$	14.1711	0.590975
$576$	0	0
$577$	40.0937	1.66912	0.834561	−	0.550915i	$-0.185721\pi$
$577$	40.0937	1.66912	0.834561	+	0.550915i	$0.185721\pi$
$578$	24.5258	1.02014
$579$	0	0
$580$	−0.191999	−0.00797233
$581$	18.9820	0.787506
$582$	0	0
$583$	−4.44681	−0.184168
$584$	−5.27959	−0.218471
$585$	0	0
$586$	−22.7184	−0.938489
$587$	−27.7691	−1.14615	−0.573076	−	0.819502i	$-0.694250\pi$
$587$	−27.7691	−1.14615	−0.573076	+	0.819502i	$0.694250\pi$
$588$	0	0
$589$	21.3768	0.880816
$590$	17.7349	0.730135
$591$	0	0
$592$	0.0859173	0.00353118
$593$	−20.9019	−0.858336	−0.429168	−	0.903225i	$-0.641193\pi$
$593$	−20.9019	−0.858336	−0.429168	+	0.903225i	$0.641193\pi$
$594$	0	0
$595$	4.29988	0.176278
$596$	−3.34588	−0.137053
$597$	0	0
$598$	−41.6368	−1.70265
$599$	−4.21754	−0.172324	−0.0861620	−	0.996281i	$-0.527460\pi$
$599$	−4.21754	−0.172324	−0.0861620	+	0.996281i	$0.527460\pi$
$600$	0	0
$601$	−28.1187	−1.14699	−0.573493	−	0.819210i	$-0.694412\pi$
$601$	−28.1187	−1.14699	−0.573493	+	0.819210i	$0.694412\pi$
$602$	−63.7984	−2.60023
$603$	0	0
$604$	2.32277	0.0945120
$605$	−7.59768	−0.308890
$606$	0	0
$607$	16.4558	0.667921	0.333961	−	0.942587i	$-0.391615\pi$
$607$	16.4558	0.667921	0.333961	+	0.942587i	$0.391615\pi$
$608$	7.10693	0.288224
$609$	0	0
$610$	−2.31112	−0.0935743
$611$	39.0397	1.57938
$612$	0	0
$613$	13.2029	0.533258	0.266629	−	0.963799i	$-0.414090\pi$
$613$	13.2029	0.533258	0.266629	+	0.963799i	$0.414090\pi$
$614$	−10.7056	−0.432042
$615$	0	0
$616$	−57.3997	−2.31270
$617$	15.5483	0.625951	0.312975	−	0.949761i	$-0.398674\pi$
$617$	15.5483	0.625951	0.312975	+	0.949761i	$0.398674\pi$
$618$	0	0
$619$	36.7472	1.47700	0.738498	−	0.674256i	$-0.235536\pi$
$619$	36.7472	1.47700	0.738498	+	0.674256i	$0.235536\pi$
$620$	−0.649008	−0.0260648
$621$	0	0
$622$	−50.1019	−2.00890
$623$	65.5606	2.62663
$624$	0	0
$625$	−1.57603	−0.0630414
$626$	18.3703	0.734226
$627$	0	0
$628$	−0.738911	−0.0294858
$629$	−0.0113952	−0.000454357 0
$630$	0	0
$631$	41.3985	1.64805	0.824024	−	0.566555i	$-0.191724\pi$
$631$	41.3985	1.64805	0.824024	+	0.566555i	$0.191724\pi$
$632$	−7.88442	−0.313626
$633$	0	0
$634$	31.9257	1.26793
$635$	−8.11669	−0.322101
$636$	0	0
$637$	−121.836	−4.82731
$638$	−4.87175	−0.192874
$639$	0	0
$640$	17.9069	0.707832
$641$	−3.85367	−0.152211	−0.0761054	−	0.997100i	$-0.524249\pi$
$641$	−3.85367	−0.152211	−0.0761054	+	0.997100i	$0.524249\pi$
$642$	0	0
$643$	16.5477	0.652577	0.326289	−	0.945270i	$-0.394202\pi$
$643$	16.5477	0.652577	0.326289	+	0.945270i	$0.394202\pi$
$644$	4.13346	0.162881
$645$	0	0
$646$	−6.46675	−0.254431
$647$	17.0638	0.670848	0.335424	−	0.942067i	$-0.391120\pi$
$647$	17.0638	0.670848	0.335424	+	0.942067i	$0.391120\pi$
$648$	0	0
$649$	33.9817	1.33390
$650$	−25.5239	−1.00113
$651$	0	0
$652$	−1.28761	−0.0504268
$653$	−30.5896	−1.19706	−0.598531	−	0.801099i	$-0.704249\pi$
$653$	−30.5896	−1.19706	−0.598531	+	0.801099i	$0.704249\pi$
$654$	0	0
$655$	−2.61620	−0.102223
$656$	43.3930	1.69421
$657$	0	0
$658$	−51.3231	−2.00078
$659$	−20.9723	−0.816966	−0.408483	−	0.912766i	$-0.633942\pi$
$659$	−20.9723	−0.816966	−0.408483	+	0.912766i	$0.633942\pi$
$660$	0	0
$661$	−21.8835	−0.851169	−0.425584	−	0.904919i	$-0.639931\pi$
$661$	−21.8835	−0.851169	−0.425584	+	0.904919i	$0.639931\pi$
$662$	−30.1231	−1.17077
$663$	0	0
$664$	−9.74412	−0.378145
$665$	58.1230	2.25391
$666$	0	0
$667$	−3.94414	−0.152718
$668$	−1.97474	−0.0764049
$669$	0	0
$670$	−14.6631	−0.566485
$671$	−4.42830	−0.170953
$672$	0	0
$673$	15.7888	0.608614	0.304307	−	0.952574i	$-0.401575\pi$
$673$	15.7888	0.608614	0.304307	+	0.952574i	$0.401575\pi$
$674$	−12.1016	−0.466137
$675$	0	0
$676$	3.53931	0.136127
$677$	−10.8124	−0.415555	−0.207778	−	0.978176i	$-0.566623\pi$
$677$	−10.8124	−0.415555	−0.207778	+	0.978176i	$0.566623\pi$
$678$	0	0
$679$	37.2091	1.42795
$680$	−2.20728	−0.0846452
$681$	0	0
$682$	−16.4678	−0.630585
$683$	12.1583	0.465225	0.232613	−	0.972569i	$-0.425273\pi$
$683$	12.1583	0.465225	0.232613	+	0.972569i	$0.425273\pi$
$684$	0	0
$685$	−24.4558	−0.934407
$686$	105.989	4.04667
$687$	0	0
$688$	35.4430	1.35125
$689$	6.48424	0.247030
$690$	0	0
$691$	−2.20570	−0.0839086	−0.0419543	−	0.999120i	$-0.513358\pi$
$691$	−2.20570	−0.0839086	−0.0419543	+	0.999120i	$0.513358\pi$
$692$	−3.71722	−0.141307
$693$	0	0
$694$	13.4242	0.509574
$695$	13.0529	0.495126
$696$	0	0
$697$	−5.75521	−0.217994
$698$	40.0762	1.51691
$699$	0	0
$700$	2.53387	0.0957711
$701$	33.5028	1.26538	0.632691	−	0.774404i	$-0.281950\pi$
$701$	33.5028	1.26538	0.632691	+	0.774404i	$0.281950\pi$
$702$	0	0
$703$	−0.154033	−0.00580948
$704$	29.2497	1.10239
$705$	0	0
$706$	−21.3005	−0.801653
$707$	−29.0819	−1.09374
$708$	0	0
$709$	−15.6258	−0.586840	−0.293420	−	0.955984i	$-0.594793\pi$
$709$	−15.6258	−0.586840	−0.293420	+	0.955984i	$0.594793\pi$
$710$	−32.1660	−1.20717
$711$	0	0
$712$	−33.6545	−1.26126
$713$	−13.3322	−0.499296
$714$	0	0
$715$	34.0592	1.27374
$716$	−2.81312	−0.105131
$717$	0	0
$718$	−23.7777	−0.887374
$719$	17.8004	0.663844	0.331922	−	0.943307i	$-0.392303\pi$
$719$	17.8004	0.663844	0.331922	+	0.943307i	$0.392303\pi$
$720$	0	0
$721$	45.6327	1.69945
$722$	−59.4675	−2.21315
$723$	0	0
$724$	2.79647	0.103930
$725$	−2.41781	−0.0897951
$726$	0	0
$727$	42.1570	1.56352	0.781759	−	0.623581i	$-0.214323\pi$
$727$	42.1570	1.56352	0.781759	+	0.623581i	$0.214323\pi$
$728$	83.6990	3.10209
$729$	0	0
$730$	−4.11843	−0.152430
$731$	−4.70080	−0.173866
$732$	0	0
$733$	20.5510	0.759067	0.379534	−	0.925178i	$-0.376084\pi$
$733$	20.5510	0.759067	0.379534	+	0.925178i	$0.376084\pi$
$734$	−43.6140	−1.60982
$735$	0	0
$736$	−4.43243	−0.163382
$737$	−28.0958	−1.03492
$738$	0	0
$739$	33.0308	1.21506	0.607528	−	0.794298i	$-0.292161\pi$
$739$	33.0308	1.21506	0.607528	+	0.794298i	$0.292161\pi$
$740$	0.00467651	0.000171912 0
$741$	0	0
$742$	−8.52444	−0.312942
$743$	28.4349	1.04318	0.521588	−	0.853198i	$-0.325340\pi$
$743$	28.4349	1.04318	0.521588	+	0.853198i	$0.325340\pi$
$744$	0	0
$745$	29.3430	1.07504
$746$	−51.5647	−1.88792
$747$	0	0
$748$	0.376192	0.0137549
$749$	80.3052	2.93429
$750$	0	0
$751$	−15.6782	−0.572106	−0.286053	−	0.958214i	$-0.592343\pi$
$751$	−15.6782	−0.572106	−0.286053	+	0.958214i	$0.592343\pi$
$752$	28.5124	1.03974
$753$	0	0
$754$	7.10388	0.258708
$755$	−20.3704	−0.741355
$756$	0	0
$757$	4.61750	0.167826	0.0839129	−	0.996473i	$-0.473258\pi$
$757$	4.61750	0.167826	0.0839129	+	0.996473i	$0.473258\pi$
$758$	−22.4156	−0.814170
$759$	0	0
$760$	−29.8365	−1.08228
$761$	−9.72691	−0.352600	−0.176300	−	0.984336i	$-0.556413\pi$
$761$	−9.72691	−0.352600	−0.176300	+	0.984336i	$0.556413\pi$
$762$	0	0
$763$	−15.9597	−0.577782
$764$	0.844161	0.0305407
$765$	0	0
$766$	−6.50863	−0.235166
$767$	−49.5514	−1.78920
$768$	0	0
$769$	−10.7066	−0.386088	−0.193044	−	0.981190i	$-0.561836\pi$
$769$	−10.7066	−0.386088	−0.193044	+	0.981190i	$0.561836\pi$
$770$	−44.7755	−1.61360
$771$	0	0
$772$	−0.161646	−0.00581778
$773$	26.4937	0.952912	0.476456	−	0.879198i	$-0.341921\pi$
$773$	26.4937	0.952912	0.476456	+	0.879198i	$0.341921\pi$
$774$	0	0
$775$	−8.17283	−0.293577
$776$	−19.1007	−0.685676
$777$	0	0
$778$	36.8420	1.32085
$779$	−77.7953	−2.78731
$780$	0	0
$781$	−61.6328	−2.20540
$782$	4.03317	0.144226
$783$	0	0
$784$	−88.9820	−3.17793
$785$	6.48017	0.231287
$786$	0	0
$787$	19.8030	0.705901	0.352950	−	0.935642i	$-0.385178\pi$
$787$	19.8030	0.705901	0.352950	+	0.935642i	$0.385178\pi$
$788$	3.82093	0.136115
$789$	0	0
$790$	−6.15037	−0.218820
$791$	−51.6380	−1.83604
$792$	0	0
$793$	6.45726	0.229304
$794$	48.2877	1.71367
$795$	0	0
$796$	−1.68271	−0.0596420
$797$	42.9271	1.52056	0.760278	−	0.649598i	$-0.225063\pi$
$797$	42.9271	1.52056	0.760278	+	0.649598i	$0.225063\pi$
$798$	0	0
$799$	−3.78160	−0.133783
$800$	−2.71714	−0.0960654
$801$	0	0
$802$	−5.62742	−0.198711
$803$	−7.89127	−0.278477
$804$	0	0
$805$	−36.2500	−1.27764
$806$	24.0130	0.845822
$807$	0	0
$808$	14.9287	0.525192
$809$	−28.6911	−1.00873	−0.504363	−	0.863492i	$-0.668273\pi$
$809$	−28.6911	−1.00873	−0.504363	+	0.863492i	$0.668273\pi$
$810$	0	0
$811$	0.337468	0.0118501	0.00592505	−	0.999982i	$-0.498114\pi$
$811$	0.337468	0.0118501	0.00592505	+	0.999982i	$0.498114\pi$
$812$	−0.705233	−0.0247488
$813$	0	0
$814$	0.118661	0.00415906
$815$	11.2922	0.395549
$816$	0	0
$817$	−63.5425	−2.22307
$818$	4.32967	0.151383
$819$	0	0
$820$	2.36190	0.0824810
$821$	24.6415	0.859995	0.429998	−	0.902830i	$-0.358514\pi$
$821$	24.6415	0.859995	0.429998	+	0.902830i	$0.358514\pi$
$822$	0	0
$823$	53.2790	1.85719	0.928595	−	0.371095i	$-0.121018\pi$
$823$	53.2790	1.85719	0.928595	+	0.371095i	$0.121018\pi$
$824$	−23.4248	−0.816043
$825$	0	0
$826$	65.1422	2.26659
$827$	11.8391	0.411685	0.205842	−	0.978585i	$-0.434007\pi$
$827$	11.8391	0.411685	0.205842	+	0.978585i	$0.434007\pi$
$828$	0	0
$829$	−32.5268	−1.12970	−0.564851	−	0.825193i	$-0.691066\pi$
$829$	−32.5268	−1.12970	−0.564851	+	0.825193i	$0.691066\pi$
$830$	−7.60105	−0.263836
$831$	0	0
$832$	−42.6513	−1.47867
$833$	11.8017	0.408904
$834$	0	0
$835$	17.3182	0.599322
$836$	5.08512	0.175873
$837$	0	0
$838$	38.2081	1.31988
$839$	37.0903	1.28050	0.640249	−	0.768167i	$-0.278831\pi$
$839$	37.0903	1.28050	0.640249	+	0.768167i	$0.278831\pi$
$840$	0	0
$841$	−28.3271	−0.976795
$842$	23.4331	0.807558
$843$	0	0
$844$	1.21602	0.0418572
$845$	−31.0393	−1.06779
$846$	0	0
$847$	−27.9071	−0.958898
$848$	4.73572	0.162625
$849$	0	0
$850$	2.47238	0.0848021
$851$	0.0960671	0.00329314
$852$	0	0
$853$	16.7993	0.575197	0.287599	−	0.957751i	$-0.407143\pi$
$853$	16.7993	0.575197	0.287599	+	0.957751i	$0.407143\pi$
$854$	−8.48896	−0.290486
$855$	0	0
$856$	−41.2234	−1.40899
$857$	−18.5819	−0.634746	−0.317373	−	0.948301i	$-0.602801\pi$
$857$	−18.5819	−0.634746	−0.317373	+	0.948301i	$0.602801\pi$
$858$	0	0
$859$	27.6877	0.944691	0.472346	−	0.881413i	$-0.343407\pi$
$859$	27.6877	0.944691	0.472346	+	0.881413i	$0.343407\pi$
$860$	1.92917	0.0657843
$861$	0	0
$862$	23.9862	0.816973
$863$	−53.8587	−1.83337	−0.916686	−	0.399609i	$-0.869146\pi$
$863$	−53.8587	−1.83337	−0.916686	+	0.399609i	$0.869146\pi$
$864$	0	0
$865$	32.5996	1.10842
$866$	−20.2074	−0.686677
$867$	0	0
$868$	−2.38387	−0.0809140
$869$	−11.7847	−0.399767
$870$	0	0
$871$	40.9687	1.38817
$872$	8.19269	0.277439
$873$	0	0
$874$	54.5178	1.84409
$875$	−59.9191	−2.02564
$876$	0	0
$877$	27.4939	0.928404	0.464202	−	0.885729i	$-0.346341\pi$
$877$	27.4939	0.928404	0.464202	+	0.885729i	$0.346341\pi$
$878$	−48.5845	−1.63965
$879$	0	0
$880$	24.8749	0.838533
$881$	20.6512	0.695758	0.347879	−	0.937540i	$-0.386902\pi$
$881$	20.6512	0.695758	0.347879	+	0.937540i	$0.386902\pi$
$882$	0	0
$883$	−1.86974	−0.0629218	−0.0314609	−	0.999505i	$-0.510016\pi$
$883$	−1.86974	−0.0629218	−0.0314609	+	0.999505i	$0.510016\pi$
$884$	−0.548555	−0.0184499
$885$	0	0
$886$	−1.85019	−0.0621583
$887$	58.1259	1.95168	0.975838	−	0.218496i	$-0.0701151\pi$
$887$	58.1259	1.95168	0.975838	+	0.218496i	$0.0701151\pi$
$888$	0	0
$889$	−29.8134	−0.999910
$890$	−26.2527	−0.879994
$891$	0	0
$892$	0.163365	0.00546988
$893$	−51.1172	−1.71057
$894$	0	0
$895$	24.6708	0.824652
$896$	65.7738	2.19735
$897$	0	0
$898$	−48.6001	−1.62181
$899$	2.27469	0.0758650
$900$	0	0
$901$	−0.628099	−0.0209250
$902$	59.9303	1.99546
$903$	0	0
$904$	26.5076	0.881629
$905$	−24.5248	−0.815231
$906$	0	0
$907$	46.2983	1.53731	0.768655	−	0.639663i	$-0.220926\pi$
$907$	46.2983	1.53731	0.768655	+	0.639663i	$0.220926\pi$
$908$	4.18031	0.138728
$909$	0	0
$910$	65.2908	2.16437
$911$	16.2934	0.539823	0.269911	−	0.962885i	$-0.413006\pi$
$911$	16.2934	0.539823	0.269911	+	0.962885i	$0.413006\pi$
$912$	0	0
$913$	−14.5643	−0.482008
$914$	−51.4239	−1.70095
$915$	0	0
$916$	−0.467866	−0.0154587
$917$	−9.60958	−0.317336
$918$	0	0
$919$	4.26210	0.140594	0.0702969	−	0.997526i	$-0.477605\pi$
$919$	4.26210	0.140594	0.0702969	+	0.997526i	$0.477605\pi$
$920$	18.6084	0.613500
$921$	0	0
$922$	7.55386	0.248773
$923$	89.8717	2.95816
$924$	0	0
$925$	0.0588904	0.00193630
$926$	24.1262	0.792838
$927$	0	0
$928$	0.756242	0.0248249
$929$	12.3863	0.406381	0.203190	−	0.979139i	$-0.434869\pi$
$929$	12.3863	0.406381	0.203190	+	0.979139i	$0.434869\pi$
$930$	0	0
$931$	159.528	5.22831
$932$	2.66919	0.0874323
$933$	0	0
$934$	57.8885	1.89417
$935$	−3.29916	−0.107894
$936$	0	0
$937$	−29.8834	−0.976249	−0.488124	−	0.872774i	$-0.662319\pi$
$937$	−29.8834	−0.976249	−0.488124	+	0.872774i	$0.662319\pi$
$938$	−53.8591	−1.75856
$939$	0	0
$940$	1.55194	0.0506187
$941$	−23.1183	−0.753635	−0.376818	−	0.926287i	$-0.622982\pi$
$941$	−23.1183	−0.753635	−0.376818	+	0.926287i	$0.622982\pi$
$942$	0	0
$943$	48.5191	1.58000
$944$	−36.1895	−1.17787
$945$	0	0
$946$	48.9505	1.59152
$947$	53.0187	1.72288	0.861439	−	0.507861i	$-0.169564\pi$
$947$	53.0187	1.72288	0.861439	+	0.507861i	$0.169564\pi$
$948$	0	0
$949$	11.5069	0.373530
$950$	33.4201	1.08429
$951$	0	0
$952$	−8.10755	−0.262767
$953$	12.1624	0.393978	0.196989	−	0.980406i	$-0.436884\pi$
$953$	12.1624	0.393978	0.196989	+	0.980406i	$0.436884\pi$
$954$	0	0
$955$	−7.40320	−0.239562
$956$	−0.292418	−0.00945747
$957$	0	0
$958$	41.9916	1.35669
$959$	−89.8286	−2.90072
$960$	0	0
$961$	−23.3110	−0.751966
$962$	−0.173029	−0.00557867
$963$	0	0
$964$	0.777588	0.0250444
$965$	1.41762	0.0456348
$966$	0	0
$967$	44.1833	1.42084	0.710420	−	0.703778i	$-0.248505\pi$
$967$	44.1833	1.42084	0.710420	+	0.703778i	$0.248505\pi$
$968$	14.3257	0.460444
$969$	0	0
$970$	−14.8998	−0.478404
$971$	17.0574	0.547398	0.273699	−	0.961815i	$-0.411753\pi$
$971$	17.0574	0.547398	0.273699	+	0.961815i	$0.411753\pi$
$972$	0	0
$973$	47.9448	1.53704
$974$	60.6094	1.94205
$975$	0	0
$976$	4.71602	0.150956
$977$	−18.9481	−0.606204	−0.303102	−	0.952958i	$-0.598022\pi$
$977$	−18.9481	−0.606204	−0.303102	+	0.952958i	$0.598022\pi$
$978$	0	0
$979$	−50.3026	−1.60768
$980$	−4.84332	−0.154714
$981$	0	0
$982$	38.8568	1.23997
$983$	11.4907	0.366497	0.183248	−	0.983067i	$-0.441339\pi$
$983$	11.4907	0.366497	0.183248	+	0.983067i	$0.441339\pi$
$984$	0	0
$985$	−33.5091	−1.06769
$986$	−0.688121	−0.0219142
$987$	0	0
$988$	−7.41502	−0.235903
$989$	39.6300	1.26016
$990$	0	0
$991$	26.8062	0.851527	0.425764	−	0.904834i	$-0.360005\pi$
$991$	26.8062	0.851527	0.425764	+	0.904834i	$0.360005\pi$
$992$	2.55630	0.0811626
$993$	0	0
$994$	−118.149	−3.74746
$995$	14.7572	0.467833
$996$	0	0
$997$	−32.9097	−1.04226	−0.521130	−	0.853477i	$-0.674489\pi$
$997$	−32.9097	−1.04226	−0.521130	+	0.853477i	$0.674489\pi$
$998$	−0.803616	−0.0254380
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6021.2.a.t.1.11	✓	40
3.2	odd	2	inner	6021.2.a.t.1.30	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6021.2.a.t.1.11	✓	40	1.1	even	1	trivial
6021.2.a.t.1.30	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)

\(f(q)\)	\(=\)	\(q-1.47084 q^{2} +0.163365 q^{4} -1.43270 q^{5} -5.26244 q^{7} +2.70139 q^{8} +O(q^{10})\)	\(q-1.47084 q^{2} +0.163365 q^{4} -1.43270 q^{5} -5.26244 q^{7} +2.70139 q^{8} +2.10726 q^{10} +4.03771 q^{11} -5.88770 q^{13} +7.74020 q^{14} -4.30004 q^{16} +0.570315 q^{17} +7.70915 q^{19} -0.234053 q^{20} -5.93881 q^{22} -4.80802 q^{23} -2.94738 q^{25} +8.65985 q^{26} -0.859701 q^{28} +0.820324 q^{29} +2.77291 q^{31} +0.921882 q^{32} -0.838841 q^{34} +7.53948 q^{35} -0.0199806 q^{37} -11.3389 q^{38} -3.87027 q^{40} -10.0913 q^{41} -8.24247 q^{43} +0.659621 q^{44} +7.07182 q^{46} -6.63072 q^{47} +20.6933 q^{49} +4.33512 q^{50} -0.961846 q^{52} -1.10132 q^{53} -5.78481 q^{55} -14.2159 q^{56} -1.20656 q^{58} +8.41609 q^{59} -1.09674 q^{61} -4.07851 q^{62} +7.24415 q^{64} +8.43528 q^{65} -6.95836 q^{67} +0.0931697 q^{68} -11.0894 q^{70} -15.2643 q^{71} -1.95440 q^{73} +0.0293882 q^{74} +1.25941 q^{76} -21.2482 q^{77} -2.91865 q^{79} +6.16065 q^{80} +14.8427 q^{82} -3.60707 q^{83} -0.817088 q^{85} +12.1233 q^{86} +10.9074 q^{88} -12.4582 q^{89} +30.9837 q^{91} -0.785465 q^{92} +9.75272 q^{94} -11.0449 q^{95} -7.07069 q^{97} -30.4365 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 46 q^{4} + 16 q^{7}+O(q^{10}) \) 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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