LMFDB - Embedded newform 6028.2.a.e.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.62144 q^{3} -3.05441 q^{5} -4.42036 q^{7} -0.370921 q^{9} +O(q^{10})$	$q+1.62144 q^{3} -3.05441 q^{5} -4.42036 q^{7} -0.370921 q^{9} -1.00000 q^{11} -4.00341 q^{13} -4.95255 q^{15} +2.41906 q^{17} -2.70637 q^{19} -7.16736 q^{21} -0.219030 q^{23} +4.32941 q^{25} -5.46576 q^{27} -10.1142 q^{29} +0.535126 q^{31} -1.62144 q^{33} +13.5016 q^{35} -9.09201 q^{37} -6.49131 q^{39} +7.47262 q^{41} +4.20289 q^{43} +1.13295 q^{45} -10.7792 q^{47} +12.5395 q^{49} +3.92236 q^{51} +9.45692 q^{53} +3.05441 q^{55} -4.38823 q^{57} +12.4143 q^{59} +2.23899 q^{61} +1.63960 q^{63} +12.2281 q^{65} -7.75168 q^{67} -0.355145 q^{69} +0.710324 q^{71} +9.44921 q^{73} +7.01990 q^{75} +4.42036 q^{77} +13.6597 q^{79} -7.74965 q^{81} -12.5321 q^{83} -7.38879 q^{85} -16.3996 q^{87} -17.2404 q^{89} +17.6965 q^{91} +0.867677 q^{93} +8.26636 q^{95} +13.0067 q^{97} +0.370921 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) $ 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9	$ 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9} - 27 q^{11} + 2 q^{13} + 10 q^{15} + 17 q^{17} + 4 q^{19} - 4 q^{21} + 25 q^{23} + 21 q^{25} + 32 q^{27} - q^{29} + 8 q^{31} - 5 q^{33} + 30 q^{35} - 24 q^{37} + 10 q^{39} + 31 q^{41} + 17 q^{43} - 15 q^{45} + 27 q^{47} + 40 q^{49} + 30 q^{51} + 2 q^{55} + 13 q^{57} + 31 q^{59} + 2 q^{61} + 61 q^{63} + 2 q^{67} - 15 q^{69} + 40 q^{71} + 17 q^{73} + 30 q^{75} - 7 q^{77} + 33 q^{79} + 31 q^{81} + 66 q^{83} + 26 q^{85} + 43 q^{87} + 22 q^{89} + 18 q^{91} - 9 q^{93} + 95 q^{95} - 21 q^{97} - 28 q^{99}+O(q^{100}) $ 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9 - 27 * q^11 + 2 * q^13 + 10 * q^15 + 17 * q^17 + 4 * q^19 - 4 * q^21 + 25 * q^23 + 21 * q^25 + 32 * q^27 - q^29 + 8 * q^31 - 5 * q^33 + 30 * q^35 - 24 * q^37 + 10 * q^39 + 31 * q^41 + 17 * q^43 - 15 * q^45 + 27 * q^47 + 40 * q^49 + 30 * q^51 + 2 * q^55 + 13 * q^57 + 31 * q^59 + 2 * q^61 + 61 * q^63 + 2 * q^67 - 15 * q^69 + 40 * q^71 + 17 * q^73 + 30 * q^75 - 7 * q^77 + 33 * q^79 + 31 * q^81 + 66 * q^83 + 26 * q^85 + 43 * q^87 + 22 * q^89 + 18 * q^91 - 9 * q^93 + 95 * q^95 - 21 * q^97 - 28 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.62144	0.936141	0.468070	−	0.883691i	$-0.344949\pi$
$3$	1.62144	0.936141	0.468070	+	0.883691i	$0.344949\pi$
$4$	0	0
$5$	−3.05441	−1.36597	−0.682987	−	0.730431i	$-0.739319\pi$
$5$	−3.05441	−1.36597	−0.682987	+	0.730431i	$0.739319\pi$
$6$	0	0
$7$	−4.42036	−1.67074	−0.835369	−	0.549690i	$-0.814746\pi$
$7$	−4.42036	−1.67074	−0.835369	+	0.549690i	$0.814746\pi$
$8$	0	0
$9$	−0.370921	−0.123640
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−4.00341	−1.11035	−0.555173	−	0.831735i	$-0.687348\pi$
$13$	−4.00341	−1.11035	−0.555173	+	0.831735i	$0.687348\pi$
$14$	0	0
$15$	−4.95255	−1.27874
$16$	0	0
$17$	2.41906	0.586707	0.293354	−	0.956004i	$-0.405229\pi$
$17$	2.41906	0.586707	0.293354	+	0.956004i	$0.405229\pi$
$18$	0	0
$19$	−2.70637	−0.620884	−0.310442	−	0.950592i	$-0.600477\pi$
$19$	−2.70637	−0.620884	−0.310442	+	0.950592i	$0.600477\pi$
$20$	0	0
$21$	−7.16736	−1.56405
$22$	0	0
$23$	−0.219030	−0.0456710	−0.0228355	−	0.999739i	$-0.507269\pi$
$23$	−0.219030	−0.0456710	−0.0228355	+	0.999739i	$0.507269\pi$
$24$	0	0
$25$	4.32941	0.865883
$26$	0	0
$27$	−5.46576	−1.05189
$28$	0	0
$29$	−10.1142	−1.87816	−0.939081	−	0.343696i	$-0.888321\pi$
$29$	−10.1142	−1.87816	−0.939081	+	0.343696i	$0.888321\pi$
$30$	0	0
$31$	0.535126	0.0961116	0.0480558	−	0.998845i	$-0.484697\pi$
$31$	0.535126	0.0961116	0.0480558	+	0.998845i	$0.484697\pi$
$32$	0	0
$33$	−1.62144	−0.282257
$34$	0	0
$35$	13.5016	2.28218
$36$	0	0
$37$	−9.09201	−1.49472	−0.747359	−	0.664421i	$-0.768678\pi$
$37$	−9.09201	−1.49472	−0.747359	+	0.664421i	$0.768678\pi$
$38$	0	0
$39$	−6.49131	−1.03944
$40$	0	0
$41$	7.47262	1.16703	0.583513	−	0.812104i	$-0.301678\pi$
$41$	7.47262	1.16703	0.583513	+	0.812104i	$0.301678\pi$
$42$	0	0
$43$	4.20289	0.640935	0.320467	−	0.947260i	$-0.396160\pi$
$43$	4.20289	0.640935	0.320467	+	0.947260i	$0.396160\pi$
$44$	0	0
$45$	1.13295	0.168889
$46$	0	0
$47$	−10.7792	−1.57231	−0.786155	−	0.618030i	$-0.787931\pi$
$47$	−10.7792	−1.57231	−0.786155	+	0.618030i	$0.787931\pi$
$48$	0	0
$49$	12.5395	1.79136
$50$	0	0
$51$	3.92236	0.549241
$52$	0	0
$53$	9.45692	1.29901	0.649504	−	0.760358i	$-0.274976\pi$
$53$	9.45692	1.29901	0.649504	+	0.760358i	$0.274976\pi$
$54$	0	0
$55$	3.05441	0.411856
$56$	0	0
$57$	−4.38823	−0.581235
$58$	0	0
$59$	12.4143	1.61621	0.808105	−	0.589039i	$-0.200493\pi$
$59$	12.4143	1.61621	0.808105	+	0.589039i	$0.200493\pi$
$60$	0	0
$61$	2.23899	0.286674	0.143337	−	0.989674i	$-0.454217\pi$
$61$	2.23899	0.286674	0.143337	+	0.989674i	$0.454217\pi$
$62$	0	0
$63$	1.63960	0.206571
$64$	0	0
$65$	12.2281	1.51670
$66$	0	0
$67$	−7.75168	−0.947018	−0.473509	−	0.880789i	$-0.657013\pi$
$67$	−7.75168	−0.947018	−0.473509	+	0.880789i	$0.657013\pi$
$68$	0	0
$69$	−0.355145	−0.0427545
$70$	0	0
$71$	0.710324	0.0843000	0.0421500	−	0.999111i	$-0.486579\pi$
$71$	0.710324	0.0843000	0.0421500	+	0.999111i	$0.486579\pi$
$72$	0	0
$73$	9.44921	1.10595	0.552973	−	0.833199i	$-0.313493\pi$
$73$	9.44921	1.10595	0.552973	+	0.833199i	$0.313493\pi$
$74$	0	0
$75$	7.01990	0.810588
$76$	0	0
$77$	4.42036	0.503746
$78$	0	0
$79$	13.6597	1.53684	0.768419	−	0.639947i	$-0.221044\pi$
$79$	13.6597	1.53684	0.768419	+	0.639947i	$0.221044\pi$
$80$	0	0
$81$	−7.74965	−0.861073
$82$	0	0
$83$	−12.5321	−1.37558	−0.687789	−	0.725911i	$-0.741419\pi$
$83$	−12.5321	−1.37558	−0.687789	+	0.725911i	$0.741419\pi$
$84$	0	0
$85$	−7.38879	−0.801427
$86$	0	0
$87$	−16.3996	−1.75822
$88$	0	0
$89$	−17.2404	−1.82748	−0.913741	−	0.406298i	$-0.866819\pi$
$89$	−17.2404	−1.82748	−0.913741	+	0.406298i	$0.866819\pi$
$90$	0	0
$91$	17.6965	1.85510
$92$	0	0
$93$	0.867677	0.0899739
$94$	0	0
$95$	8.26636	0.848111
$96$	0	0
$97$	13.0067	1.32063	0.660313	−	0.750990i	$-0.270423\pi$
$97$	13.0067	1.32063	0.660313	+	0.750990i	$0.270423\pi$
$98$	0	0
$99$	0.370921	0.0372790
$100$	0	0
$101$	−14.6123	−1.45398	−0.726989	−	0.686649i	$-0.759081\pi$
$101$	−14.6123	−1.45398	−0.726989	+	0.686649i	$0.759081\pi$
$102$	0	0
$103$	5.44709	0.536717	0.268359	−	0.963319i	$-0.413519\pi$
$103$	5.44709	0.536717	0.268359	+	0.963319i	$0.413519\pi$
$104$	0	0
$105$	21.8920	2.13644
$106$	0	0
$107$	11.1043	1.07350	0.536749	−	0.843742i	$-0.319652\pi$
$107$	11.1043	1.07350	0.536749	+	0.843742i	$0.319652\pi$
$108$	0	0
$109$	4.55354	0.436150	0.218075	−	0.975932i	$-0.430022\pi$
$109$	4.55354	0.436150	0.218075	+	0.975932i	$0.430022\pi$
$110$	0	0
$111$	−14.7422	−1.39927
$112$	0	0
$113$	−6.89549	−0.648673	−0.324337	−	0.945942i	$-0.605141\pi$
$113$	−6.89549	−0.648673	−0.324337	+	0.945942i	$0.605141\pi$
$114$	0	0
$115$	0.669008	0.0623854
$116$	0	0
$117$	1.48495	0.137284
$118$	0	0
$119$	−10.6931	−0.980234
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	12.1164	1.09250
$124$	0	0
$125$	2.04825	0.183201
$126$	0	0
$127$	0.900162	0.0798765	0.0399382	−	0.999202i	$-0.487284\pi$
$127$	0.900162	0.0798765	0.0399382	+	0.999202i	$0.487284\pi$
$128$	0	0
$129$	6.81475	0.600005
$130$	0	0
$131$	−12.6955	−1.10921	−0.554607	−	0.832112i	$-0.687131\pi$
$131$	−12.6955	−1.10921	−0.554607	+	0.832112i	$0.687131\pi$
$132$	0	0
$133$	11.9631	1.03733
$134$	0	0
$135$	16.6947	1.43685
$136$	0	0
$137$	−1.00000	−0.0854358
$137$	−1.00000	−0.0854358
$138$	0	0
$139$	−0.788021	−0.0668390	−0.0334195	−	0.999441i	$-0.510640\pi$
$139$	−0.788021	−0.0668390	−0.0334195	+	0.999441i	$0.510640\pi$
$140$	0	0
$141$	−17.4779	−1.47190
$142$	0	0
$143$	4.00341	0.334782
$144$	0	0
$145$	30.8929	2.56552
$146$	0	0
$147$	20.3322	1.67697
$148$	0	0
$149$	−11.7281	−0.960805	−0.480403	−	0.877048i	$-0.659509\pi$
$149$	−11.7281	−0.960805	−0.480403	+	0.877048i	$0.659509\pi$
$150$	0	0
$151$	9.65662	0.785845	0.392922	−	0.919572i	$-0.371464\pi$
$151$	9.65662	0.785845	0.392922	+	0.919572i	$0.371464\pi$
$152$	0	0
$153$	−0.897280	−0.0725407
$154$	0	0
$155$	−1.63450	−0.131286
$156$	0	0
$157$	4.46974	0.356724	0.178362	−	0.983965i	$-0.442920\pi$
$157$	4.46974	0.356724	0.178362	+	0.983965i	$0.442920\pi$
$158$	0	0
$159$	15.3339	1.21605
$160$	0	0
$161$	0.968192	0.0763042
$162$	0	0
$163$	−21.8140	−1.70860	−0.854302	−	0.519777i	$-0.826015\pi$
$163$	−21.8140	−1.70860	−0.854302	+	0.519777i	$0.826015\pi$
$164$	0	0
$165$	4.95255	0.385556
$166$	0	0
$167$	19.2022	1.48591	0.742957	−	0.669339i	$-0.233423\pi$
$167$	19.2022	1.48591	0.742957	+	0.669339i	$0.233423\pi$
$168$	0	0
$169$	3.02731	0.232870
$170$	0	0
$171$	1.00385	0.0767664
$172$	0	0
$173$	10.6299	0.808176	0.404088	−	0.914720i	$-0.367589\pi$
$173$	10.6299	0.808176	0.404088	+	0.914720i	$0.367589\pi$
$174$	0	0
$175$	−19.1375	−1.44666
$176$	0	0
$177$	20.1292	1.51300
$178$	0	0
$179$	11.8049	0.882342	0.441171	−	0.897423i	$-0.354563\pi$
$179$	11.8049	0.882342	0.441171	+	0.897423i	$0.354563\pi$
$180$	0	0
$181$	−11.0581	−0.821945	−0.410972	−	0.911648i	$-0.634811\pi$
$181$	−11.0581	−0.821945	−0.410972	+	0.911648i	$0.634811\pi$
$182$	0	0
$183$	3.63040	0.268367
$184$	0	0
$185$	27.7707	2.04174
$186$	0	0
$187$	−2.41906	−0.176899
$188$	0	0
$189$	24.1606	1.75742
$190$	0	0
$191$	−13.9625	−1.01029	−0.505145	−	0.863034i	$-0.668561\pi$
$191$	−13.9625	−1.01029	−0.505145	+	0.863034i	$0.668561\pi$
$192$	0	0
$193$	13.4599	0.968863	0.484432	−	0.874829i	$-0.339026\pi$
$193$	13.4599	0.968863	0.484432	+	0.874829i	$0.339026\pi$
$194$	0	0
$195$	19.8271	1.41985
$196$	0	0
$197$	−23.0332	−1.64105	−0.820524	−	0.571613i	$-0.806318\pi$
$197$	−23.0332	−1.64105	−0.820524	+	0.571613i	$0.806318\pi$
$198$	0	0
$199$	−22.1306	−1.56880	−0.784399	−	0.620256i	$-0.787029\pi$
$199$	−22.1306	−1.56880	−0.784399	+	0.620256i	$0.787029\pi$
$200$	0	0
$201$	−12.5689	−0.886542
$202$	0	0
$203$	44.7084	3.13791
$204$	0	0
$205$	−22.8244	−1.59413
$206$	0	0
$207$	0.0812430	0.00564678
$208$	0	0
$209$	2.70637	0.187204
$210$	0	0
$211$	−4.81390	−0.331403	−0.165701	−	0.986176i	$-0.552989\pi$
$211$	−4.81390	−0.331403	−0.165701	+	0.986176i	$0.552989\pi$
$212$	0	0
$213$	1.15175	0.0789166
$214$	0	0
$215$	−12.8373	−0.875500
$216$	0	0
$217$	−2.36545	−0.160577
$218$	0	0
$219$	15.3214	1.03532
$220$	0	0
$221$	−9.68448	−0.651449
$222$	0	0
$223$	0.426988	0.0285932	0.0142966	−	0.999898i	$-0.495449\pi$
$223$	0.426988	0.0285932	0.0142966	+	0.999898i	$0.495449\pi$
$224$	0	0
$225$	−1.60587	−0.107058
$226$	0	0
$227$	−12.6294	−0.838245	−0.419122	−	0.907930i	$-0.637662\pi$
$227$	−12.6294	−0.838245	−0.419122	+	0.907930i	$0.637662\pi$
$228$	0	0
$229$	−18.6955	−1.23543	−0.617716	−	0.786401i	$-0.711942\pi$
$229$	−18.6955	−1.23543	−0.617716	+	0.786401i	$0.711942\pi$
$230$	0	0
$231$	7.16736	0.471577
$232$	0	0
$233$	3.97286	0.260271	0.130135	−	0.991496i	$-0.458459\pi$
$233$	3.97286	0.260271	0.130135	+	0.991496i	$0.458459\pi$
$234$	0	0
$235$	32.9241	2.14773
$236$	0	0
$237$	22.1485	1.43870
$238$	0	0
$239$	28.9934	1.87543	0.937713	−	0.347410i	$-0.112939\pi$
$239$	28.9934	1.87543	0.937713	+	0.347410i	$0.112939\pi$
$240$	0	0
$241$	3.99078	0.257069	0.128534	−	0.991705i	$-0.458973\pi$
$241$	3.99078	0.257069	0.128534	+	0.991705i	$0.458973\pi$
$242$	0	0
$243$	3.83165	0.245800
$244$	0	0
$245$	−38.3009	−2.44695
$246$	0	0
$247$	10.8347	0.689397
$248$	0	0
$249$	−20.3201	−1.28773
$250$	0	0
$251$	−3.17840	−0.200619	−0.100309	−	0.994956i	$-0.531983\pi$
$251$	−3.17840	−0.200619	−0.100309	+	0.994956i	$0.531983\pi$
$252$	0	0
$253$	0.219030	0.0137703
$254$	0	0
$255$	−11.9805	−0.750248
$256$	0	0
$257$	19.9676	1.24554	0.622772	−	0.782404i	$-0.286007\pi$
$257$	19.9676	1.24554	0.622772	+	0.782404i	$0.286007\pi$
$258$	0	0
$259$	40.1899	2.49728
$260$	0	0
$261$	3.75158	0.232217
$262$	0	0
$263$	12.2133	0.753105	0.376553	−	0.926395i	$-0.377109\pi$
$263$	12.2133	0.753105	0.376553	+	0.926395i	$0.377109\pi$
$264$	0	0
$265$	−28.8853	−1.77441
$266$	0	0
$267$	−27.9544	−1.71078
$268$	0	0
$269$	12.1605	0.741438	0.370719	−	0.928745i	$-0.379111\pi$
$269$	12.1605	0.741438	0.370719	+	0.928745i	$0.379111\pi$
$270$	0	0
$271$	−15.2454	−0.926093	−0.463046	−	0.886334i	$-0.653244\pi$
$271$	−15.2454	−0.926093	−0.463046	+	0.886334i	$0.653244\pi$
$272$	0	0
$273$	28.6939	1.73663
$274$	0	0
$275$	−4.32941	−0.261073
$276$	0	0
$277$	24.4869	1.47128	0.735639	−	0.677374i	$-0.236882\pi$
$277$	24.4869	1.47128	0.735639	+	0.677374i	$0.236882\pi$
$278$	0	0
$279$	−0.198490	−0.0118833
$280$	0	0
$281$	19.8383	1.18345	0.591727	−	0.806139i	$-0.298446\pi$
$281$	19.8383	1.18345	0.591727	+	0.806139i	$0.298446\pi$
$282$	0	0
$283$	14.9399	0.888082	0.444041	−	0.896006i	$-0.353544\pi$
$283$	14.9399	0.888082	0.444041	+	0.896006i	$0.353544\pi$
$284$	0	0
$285$	13.4034	0.793951
$286$	0	0
$287$	−33.0316	−1.94979
$288$	0	0
$289$	−11.1482	−0.655774
$290$	0	0
$291$	21.0896	1.23629
$292$	0	0
$293$	32.1276	1.87692	0.938458	−	0.345394i	$-0.112255\pi$
$293$	32.1276	1.87692	0.938458	+	0.345394i	$0.112255\pi$
$294$	0	0
$295$	−37.9185	−2.20770
$296$	0	0
$297$	5.46576	0.317155
$298$	0	0
$299$	0.876869	0.0507106
$300$	0	0
$301$	−18.5783	−1.07083
$302$	0	0
$303$	−23.6930	−1.36113
$304$	0	0
$305$	−6.83880	−0.391588
$306$	0	0
$307$	−12.6093	−0.719651	−0.359826	−	0.933020i	$-0.617164\pi$
$307$	−12.6093	−0.719651	−0.359826	+	0.933020i	$0.617164\pi$
$308$	0	0
$309$	8.83214	0.502443
$310$	0	0
$311$	20.5931	1.16773	0.583864	−	0.811852i	$-0.301540\pi$
$311$	20.5931	1.16773	0.583864	+	0.811852i	$0.301540\pi$
$312$	0	0
$313$	25.5756	1.44562	0.722808	−	0.691049i	$-0.242851\pi$
$313$	25.5756	1.44562	0.722808	+	0.691049i	$0.242851\pi$
$314$	0	0
$315$	−5.00802	−0.282170
$316$	0	0
$317$	−15.3398	−0.861567	−0.430784	−	0.902455i	$-0.641763\pi$
$317$	−15.3398	−0.861567	−0.430784	+	0.902455i	$0.641763\pi$
$318$	0	0
$319$	10.1142	0.566287
$320$	0	0
$321$	18.0051	1.00495
$322$	0	0
$323$	−6.54686	−0.364277
$324$	0	0
$325$	−17.3324	−0.961430
$326$	0	0
$327$	7.38330	0.408297
$328$	0	0
$329$	47.6479	2.62692
$330$	0	0
$331$	−9.18889	−0.505067	−0.252533	−	0.967588i	$-0.581264\pi$
$331$	−9.18889	−0.505067	−0.252533	+	0.967588i	$0.581264\pi$
$332$	0	0
$333$	3.37242	0.184807
$334$	0	0
$335$	23.6768	1.29360
$336$	0	0
$337$	−6.27461	−0.341800	−0.170900	−	0.985288i	$-0.554667\pi$
$337$	−6.27461	−0.341800	−0.170900	+	0.985288i	$0.554667\pi$
$338$	0	0
$339$	−11.1806	−0.607249
$340$	0	0
$341$	−0.535126	−0.0289787
$342$	0	0
$343$	−24.4868	−1.32216
$344$	0	0
$345$	1.08476	0.0584015
$346$	0	0
$347$	−27.8772	−1.49653	−0.748263	−	0.663403i	$-0.769112\pi$
$347$	−27.8772	−1.49653	−0.748263	+	0.663403i	$0.769112\pi$
$348$	0	0
$349$	−28.9843	−1.55150	−0.775748	−	0.631043i	$-0.782627\pi$
$349$	−28.9843	−1.55150	−0.775748	+	0.631043i	$0.782627\pi$
$350$	0	0
$351$	21.8817	1.16796
$352$	0	0
$353$	11.9002	0.633386	0.316693	−	0.948528i	$-0.397428\pi$
$353$	11.9002	0.633386	0.316693	+	0.948528i	$0.397428\pi$
$354$	0	0
$355$	−2.16962	−0.115151
$356$	0	0
$357$	−17.3382	−0.917637
$358$	0	0
$359$	7.86823	0.415270	0.207635	−	0.978206i	$-0.433423\pi$
$359$	7.86823	0.415270	0.207635	+	0.978206i	$0.433423\pi$
$360$	0	0
$361$	−11.6756	−0.614503
$362$	0	0
$363$	1.62144	0.0851037
$364$	0	0
$365$	−28.8617	−1.51069
$366$	0	0
$367$	19.8824	1.03785	0.518927	−	0.854819i	$-0.326332\pi$
$367$	19.8824	1.03785	0.518927	+	0.854819i	$0.326332\pi$
$368$	0	0
$369$	−2.77175	−0.144292
$370$	0	0
$371$	−41.8030	−2.17030
$372$	0	0
$373$	12.6604	0.655531	0.327766	−	0.944759i	$-0.393704\pi$
$373$	12.6604	0.655531	0.327766	+	0.944759i	$0.393704\pi$
$374$	0	0
$375$	3.32111	0.171502
$376$	0	0
$377$	40.4914	2.08541
$378$	0	0
$379$	−18.9122	−0.971455	−0.485727	−	0.874110i	$-0.661445\pi$
$379$	−18.9122	−0.971455	−0.485727	+	0.874110i	$0.661445\pi$
$380$	0	0
$381$	1.45956	0.0747756
$382$	0	0
$383$	2.59379	0.132536	0.0662681	−	0.997802i	$-0.478891\pi$
$383$	2.59379	0.132536	0.0662681	+	0.997802i	$0.478891\pi$
$384$	0	0
$385$	−13.5016	−0.688104
$386$	0	0
$387$	−1.55894	−0.0792454
$388$	0	0
$389$	11.7986	0.598212	0.299106	−	0.954220i	$-0.403312\pi$
$389$	11.7986	0.598212	0.299106	+	0.954220i	$0.403312\pi$
$390$	0	0
$391$	−0.529847	−0.0267955
$392$	0	0
$393$	−20.5851	−1.03838
$394$	0	0
$395$	−41.7223	−2.09928
$396$	0	0
$397$	−13.9119	−0.698219	−0.349110	−	0.937082i	$-0.613516\pi$
$397$	−13.9119	−0.698219	−0.349110	+	0.937082i	$0.613516\pi$
$398$	0	0
$399$	19.3975	0.971091
$400$	0	0
$401$	6.50997	0.325092	0.162546	−	0.986701i	$-0.448029\pi$
$401$	6.50997	0.325092	0.162546	+	0.986701i	$0.448029\pi$
$402$	0	0
$403$	−2.14233	−0.106717
$404$	0	0
$405$	23.6706	1.17620
$406$	0	0
$407$	9.09201	0.450674
$408$	0	0
$409$	13.2130	0.653339	0.326670	−	0.945139i	$-0.394074\pi$
$409$	13.2130	0.653339	0.326670	+	0.945139i	$0.394074\pi$
$410$	0	0
$411$	−1.62144	−0.0799799
$412$	0	0
$413$	−54.8758	−2.70026
$414$	0	0
$415$	38.2782	1.87900
$416$	0	0
$417$	−1.27773	−0.0625708
$418$	0	0
$419$	−9.60899	−0.469430	−0.234715	−	0.972064i	$-0.575416\pi$
$419$	−9.60899	−0.469430	−0.234715	+	0.972064i	$0.575416\pi$
$420$	0	0
$421$	10.8152	0.527100	0.263550	−	0.964646i	$-0.415107\pi$
$421$	10.8152	0.527100	0.263550	+	0.964646i	$0.415107\pi$
$422$	0	0
$423$	3.99824	0.194401
$424$	0	0
$425$	10.4731	0.508020
$426$	0	0
$427$	−9.89714	−0.478956
$428$	0	0
$429$	6.49131	0.313403
$430$	0	0
$431$	0.609099	0.0293393	0.0146696	−	0.999892i	$-0.495330\pi$
$431$	0.609099	0.0293393	0.0146696	+	0.999892i	$0.495330\pi$
$432$	0	0
$433$	−20.0787	−0.964922	−0.482461	−	0.875917i	$-0.660257\pi$
$433$	−20.0787	−0.964922	−0.482461	+	0.875917i	$0.660257\pi$
$434$	0	0
$435$	50.0911	2.40169
$436$	0	0
$437$	0.592777	0.0283564
$438$	0	0
$439$	5.27512	0.251768	0.125884	−	0.992045i	$-0.459823\pi$
$439$	5.27512	0.251768	0.125884	+	0.992045i	$0.459823\pi$
$440$	0	0
$441$	−4.65118	−0.221485
$442$	0	0
$443$	−0.676134	−0.0321241	−0.0160621	−	0.999871i	$-0.505113\pi$
$443$	−0.676134	−0.0321241	−0.0160621	+	0.999871i	$0.505113\pi$
$444$	0	0
$445$	52.6593	2.49629
$446$	0	0
$447$	−19.0165	−0.899449
$448$	0	0
$449$	−22.9141	−1.08139	−0.540693	−	0.841220i	$-0.681838\pi$
$449$	−22.9141	−1.08139	−0.540693	+	0.841220i	$0.681838\pi$
$450$	0	0
$451$	−7.47262	−0.351872
$452$	0	0
$453$	15.6577	0.735661
$454$	0	0
$455$	−54.0524	−2.53401
$456$	0	0
$457$	17.8879	0.836760	0.418380	−	0.908272i	$-0.362598\pi$
$457$	17.8879	0.836760	0.418380	+	0.908272i	$0.362598\pi$
$458$	0	0
$459$	−13.2220	−0.617149
$460$	0	0
$461$	18.0272	0.839609	0.419804	−	0.907615i	$-0.362099\pi$
$461$	18.0272	0.839609	0.419804	+	0.907615i	$0.362099\pi$
$462$	0	0
$463$	−21.8084	−1.01352	−0.506761	−	0.862087i	$-0.669157\pi$
$463$	−21.8084	−1.01352	−0.506761	+	0.862087i	$0.669157\pi$
$464$	0	0
$465$	−2.65024	−0.122902
$466$	0	0
$467$	26.2255	1.21357	0.606785	−	0.794866i	$-0.292459\pi$
$467$	26.2255	1.21357	0.606785	+	0.794866i	$0.292459\pi$
$468$	0	0
$469$	34.2652	1.58222
$470$	0	0
$471$	7.24743	0.333944
$472$	0	0
$473$	−4.20289	−0.193249
$474$	0	0
$475$	−11.7170	−0.537613
$476$	0	0
$477$	−3.50777	−0.160610
$478$	0	0
$479$	−33.1742	−1.51577	−0.757884	−	0.652389i	$-0.773767\pi$
$479$	−33.1742	−1.51577	−0.757884	+	0.652389i	$0.773767\pi$
$480$	0	0
$481$	36.3991	1.65965
$482$	0	0
$483$	1.56987	0.0714315
$484$	0	0
$485$	−39.7277	−1.80394
$486$	0	0
$487$	−12.5897	−0.570493	−0.285247	−	0.958454i	$-0.592076\pi$
$487$	−12.5897	−0.570493	−0.285247	+	0.958454i	$0.592076\pi$
$488$	0	0
$489$	−35.3702	−1.59949
$490$	0	0
$491$	12.6680	0.571699	0.285850	−	0.958275i	$-0.407724\pi$
$491$	12.6680	0.571699	0.285850	+	0.958275i	$0.407724\pi$
$492$	0	0
$493$	−24.4668	−1.10193
$494$	0	0
$495$	−1.13295	−0.0509221
$496$	0	0
$497$	−3.13989	−0.140843
$498$	0	0
$499$	13.6113	0.609326	0.304663	−	0.952460i	$-0.401456\pi$
$499$	13.6113	0.609326	0.304663	+	0.952460i	$0.401456\pi$
$500$	0	0
$501$	31.1354	1.39103
$502$	0	0
$503$	−30.8519	−1.37562	−0.687809	−	0.725892i	$-0.741427\pi$
$503$	−30.8519	−1.37562	−0.687809	+	0.725892i	$0.741427\pi$
$504$	0	0
$505$	44.6319	1.98610
$506$	0	0
$507$	4.90861	0.217999
$508$	0	0
$509$	−39.7267	−1.76085	−0.880427	−	0.474181i	$-0.842744\pi$
$509$	−39.7267	−1.76085	−0.880427	+	0.474181i	$0.842744\pi$
$510$	0	0
$511$	−41.7689	−1.84775
$512$	0	0
$513$	14.7924	0.653099
$514$	0	0
$515$	−16.6376	−0.733142
$516$	0	0
$517$	10.7792	0.474069
$518$	0	0
$519$	17.2358	0.756567
$520$	0	0
$521$	−19.3662	−0.848451	−0.424225	−	0.905557i	$-0.639454\pi$
$521$	−19.3662	−0.848451	−0.424225	+	0.905557i	$0.639454\pi$
$522$	0	0
$523$	−21.6905	−0.948460	−0.474230	−	0.880401i	$-0.657273\pi$
$523$	−21.6905	−0.948460	−0.474230	+	0.880401i	$0.657273\pi$
$524$	0	0
$525$	−31.0304	−1.35428
$526$	0	0
$527$	1.29450	0.0563894
$528$	0	0
$529$	−22.9520	−0.997914
$530$	0	0
$531$	−4.60474	−0.199829
$532$	0	0
$533$	−29.9160	−1.29580
$534$	0	0
$535$	−33.9172	−1.46637
$536$	0	0
$537$	19.1410	0.825996
$538$	0	0
$539$	−12.5395	−0.540116
$540$	0	0
$541$	30.8381	1.32583	0.662917	−	0.748693i	$-0.269318\pi$
$541$	30.8381	1.32583	0.662917	+	0.748693i	$0.269318\pi$
$542$	0	0
$543$	−17.9301	−0.769456
$544$	0	0
$545$	−13.9084	−0.595769
$546$	0	0
$547$	−24.5882	−1.05131	−0.525657	−	0.850697i	$-0.676180\pi$
$547$	−24.5882	−1.05131	−0.525657	+	0.850697i	$0.676180\pi$
$548$	0	0
$549$	−0.830490	−0.0354444
$550$	0	0
$551$	27.3728	1.16612
$552$	0	0
$553$	−60.3808	−2.56765
$554$	0	0
$555$	45.0286	1.91136
$556$	0	0
$557$	22.0641	0.934888	0.467444	−	0.884023i	$-0.345175\pi$
$557$	22.0641	0.934888	0.467444	+	0.884023i	$0.345175\pi$
$558$	0	0
$559$	−16.8259	−0.711660
$560$	0	0
$561$	−3.92236	−0.165602
$562$	0	0
$563$	17.4502	0.735440	0.367720	−	0.929937i	$-0.380139\pi$
$563$	17.4502	0.735440	0.367720	+	0.929937i	$0.380139\pi$
$564$	0	0
$565$	21.0616	0.886070
$566$	0	0
$567$	34.2562	1.43863
$568$	0	0
$569$	35.1805	1.47484	0.737421	−	0.675434i	$-0.236043\pi$
$569$	35.1805	1.47484	0.737421	+	0.675434i	$0.236043\pi$
$570$	0	0
$571$	−3.67864	−0.153946	−0.0769731	−	0.997033i	$-0.524526\pi$
$571$	−3.67864	−0.153946	−0.0769731	+	0.997033i	$0.524526\pi$
$572$	0	0
$573$	−22.6394	−0.945774
$574$	0	0
$575$	−0.948273	−0.0395457
$576$	0	0
$577$	−46.3020	−1.92758	−0.963789	−	0.266665i	$-0.914078\pi$
$577$	−46.3020	−1.92758	−0.963789	+	0.266665i	$0.914078\pi$
$578$	0	0
$579$	21.8244	0.906993
$580$	0	0
$581$	55.3964	2.29823
$582$	0	0
$583$	−9.45692	−0.391666
$584$	0	0
$585$	−4.53565	−0.187526
$586$	0	0
$587$	19.3147	0.797201	0.398601	−	0.917125i	$-0.369496\pi$
$587$	19.3147	0.797201	0.398601	+	0.917125i	$0.369496\pi$
$588$	0	0
$589$	−1.44825	−0.0596741
$590$	0	0
$591$	−37.3470	−1.53625
$592$	0	0
$593$	−43.1357	−1.77137	−0.885685	−	0.464286i	$-0.846311\pi$
$593$	−43.1357	−1.77137	−0.885685	+	0.464286i	$0.846311\pi$
$594$	0	0
$595$	32.6611	1.33897
$596$	0	0
$597$	−35.8836	−1.46862
$598$	0	0
$599$	28.6762	1.17168	0.585840	−	0.810427i	$-0.300765\pi$
$599$	28.6762	1.17168	0.585840	+	0.810427i	$0.300765\pi$
$600$	0	0
$601$	−31.2282	−1.27382	−0.636912	−	0.770937i	$-0.719788\pi$
$601$	−31.2282	−1.27382	−0.636912	+	0.770937i	$0.719788\pi$
$602$	0	0
$603$	2.87526	0.117090
$604$	0	0
$605$	−3.05441	−0.124179
$606$	0	0
$607$	−15.0477	−0.610765	−0.305383	−	0.952230i	$-0.598784\pi$
$607$	−15.0477	−0.610765	−0.305383	+	0.952230i	$0.598784\pi$
$608$	0	0
$609$	72.4921	2.93753
$610$	0	0
$611$	43.1536	1.74581
$612$	0	0
$613$	11.6608	0.470977	0.235489	−	0.971877i	$-0.424331\pi$
$613$	11.6608	0.470977	0.235489	+	0.971877i	$0.424331\pi$
$614$	0	0
$615$	−37.0085	−1.49233
$616$	0	0
$617$	−19.7004	−0.793109	−0.396555	−	0.918011i	$-0.629794\pi$
$617$	−19.7004	−0.793109	−0.396555	+	0.918011i	$0.629794\pi$
$618$	0	0
$619$	11.2522	0.452263	0.226131	−	0.974097i	$-0.427392\pi$
$619$	11.2522	0.452263	0.226131	+	0.974097i	$0.427392\pi$
$620$	0	0
$621$	1.19717	0.0480407
$622$	0	0
$623$	76.2088	3.05324
$624$	0	0
$625$	−27.9032	−1.11613
$626$	0	0
$627$	4.38823	0.175249
$628$	0	0
$629$	−21.9941	−0.876962
$630$	0	0
$631$	−7.01883	−0.279415	−0.139708	−	0.990193i	$-0.544616\pi$
$631$	−7.01883	−0.279415	−0.139708	+	0.990193i	$0.544616\pi$
$632$	0	0
$633$	−7.80547	−0.310240
$634$	0	0
$635$	−2.74946	−0.109109
$636$	0	0
$637$	−50.2010	−1.98903
$638$	0	0
$639$	−0.263474	−0.0104229
$640$	0	0
$641$	−14.4630	−0.571254	−0.285627	−	0.958341i	$-0.592202\pi$
$641$	−14.4630	−0.571254	−0.285627	+	0.958341i	$0.592202\pi$
$642$	0	0
$643$	−11.4767	−0.452596	−0.226298	−	0.974058i	$-0.572662\pi$
$643$	−11.4767	−0.452596	−0.226298	+	0.974058i	$0.572662\pi$
$644$	0	0
$645$	−20.8150	−0.819591
$646$	0	0
$647$	39.5759	1.55589	0.777945	−	0.628333i	$-0.216262\pi$
$647$	39.5759	1.55589	0.777945	+	0.628333i	$0.216262\pi$
$648$	0	0
$649$	−12.4143	−0.487306
$650$	0	0
$651$	−3.83544	−0.150323
$652$	0	0
$653$	23.9970	0.939076	0.469538	−	0.882912i	$-0.344421\pi$
$653$	23.9970	0.939076	0.469538	+	0.882912i	$0.344421\pi$
$654$	0	0
$655$	38.7773	1.51516
$656$	0	0
$657$	−3.50491	−0.136740
$658$	0	0
$659$	−10.4410	−0.406725	−0.203363	−	0.979103i	$-0.565187\pi$
$659$	−10.4410	−0.406725	−0.203363	+	0.979103i	$0.565187\pi$
$660$	0	0
$661$	6.04313	0.235051	0.117525	−	0.993070i	$-0.462504\pi$
$661$	6.04313	0.235051	0.117525	+	0.993070i	$0.462504\pi$
$662$	0	0
$663$	−15.7028	−0.609848
$664$	0	0
$665$	−36.5403	−1.41697
$666$	0	0
$667$	2.21532	0.0857775
$668$	0	0
$669$	0.692336	0.0267673
$670$	0	0
$671$	−2.23899	−0.0864353
$672$	0	0
$673$	−34.6137	−1.33426	−0.667130	−	0.744941i	$-0.732477\pi$
$673$	−34.6137	−1.33426	−0.667130	+	0.744941i	$0.732477\pi$
$674$	0	0
$675$	−23.6635	−0.910810
$676$	0	0
$677$	−17.2520	−0.663048	−0.331524	−	0.943447i	$-0.607563\pi$
$677$	−17.2520	−0.663048	−0.331524	+	0.943447i	$0.607563\pi$
$678$	0	0
$679$	−57.4941	−2.20642
$680$	0	0
$681$	−20.4779	−0.784715
$682$	0	0
$683$	10.1539	0.388530	0.194265	−	0.980949i	$-0.437768\pi$
$683$	10.1539	0.388530	0.194265	+	0.980949i	$0.437768\pi$
$684$	0	0
$685$	3.05441	0.116703
$686$	0	0
$687$	−30.3137	−1.15654
$688$	0	0
$689$	−37.8600	−1.44235
$690$	0	0
$691$	18.6763	0.710480	0.355240	−	0.934775i	$-0.384399\pi$
$691$	18.6763	0.710480	0.355240	+	0.934775i	$0.384399\pi$
$692$	0	0
$693$	−1.63960	−0.0622834
$694$	0	0
$695$	2.40694	0.0913003
$696$	0	0
$697$	18.0767	0.684703
$698$	0	0
$699$	6.44176	0.243650
$700$	0	0
$701$	−47.3353	−1.78783	−0.893915	−	0.448237i	$-0.852052\pi$
$701$	−47.3353	−1.78783	−0.893915	+	0.448237i	$0.852052\pi$
$702$	0	0
$703$	24.6063	0.928046
$704$	0	0
$705$	53.3846	2.01058
$706$	0	0
$707$	64.5916	2.42922
$708$	0	0
$709$	14.8649	0.558265	0.279132	−	0.960253i	$-0.409953\pi$
$709$	14.8649	0.558265	0.279132	+	0.960253i	$0.409953\pi$
$710$	0	0
$711$	−5.06668	−0.190015
$712$	0	0
$713$	−0.117209	−0.00438951
$714$	0	0
$715$	−12.2281	−0.457303
$716$	0	0
$717$	47.0111	1.75566
$718$	0	0
$719$	24.2419	0.904070	0.452035	−	0.892000i	$-0.350698\pi$
$719$	24.2419	0.904070	0.452035	+	0.892000i	$0.350698\pi$
$720$	0	0
$721$	−24.0781	−0.896714
$722$	0	0
$723$	6.47082	0.240652
$724$	0	0
$725$	−43.7886	−1.62627
$726$	0	0
$727$	−42.4835	−1.57563	−0.787813	−	0.615914i	$-0.788787\pi$
$727$	−42.4835	−1.57563	−0.787813	+	0.615914i	$0.788787\pi$
$728$	0	0
$729$	29.4618	1.09118
$730$	0	0
$731$	10.1670	0.376041
$732$	0	0
$733$	−18.8071	−0.694655	−0.347328	−	0.937744i	$-0.612911\pi$
$733$	−18.8071	−0.694655	−0.347328	+	0.937744i	$0.612911\pi$
$734$	0	0
$735$	−62.1027	−2.29069
$736$	0	0
$737$	7.75168	0.285537
$738$	0	0
$739$	15.1422	0.557016	0.278508	−	0.960434i	$-0.410160\pi$
$739$	15.1422	0.557016	0.278508	+	0.960434i	$0.410160\pi$
$740$	0	0
$741$	17.5679	0.645372
$742$	0	0
$743$	29.5314	1.08340	0.541701	−	0.840571i	$-0.317781\pi$
$743$	29.5314	1.08340	0.541701	+	0.840571i	$0.317781\pi$
$744$	0	0
$745$	35.8225	1.31243
$746$	0	0
$747$	4.64843	0.170077
$748$	0	0
$749$	−49.0852	−1.79353
$750$	0	0
$751$	−1.93493	−0.0706067	−0.0353034	−	0.999377i	$-0.511240\pi$
$751$	−1.93493	−0.0706067	−0.0353034	+	0.999377i	$0.511240\pi$
$752$	0	0
$753$	−5.15359	−0.187807
$754$	0	0
$755$	−29.4953	−1.07344
$756$	0	0
$757$	−28.3668	−1.03101	−0.515504	−	0.856887i	$-0.672395\pi$
$757$	−28.3668	−1.03101	−0.515504	+	0.856887i	$0.672395\pi$
$758$	0	0
$759$	0.355145	0.0128910
$760$	0	0
$761$	−17.7136	−0.642119	−0.321059	−	0.947059i	$-0.604039\pi$
$761$	−17.7136	−0.642119	−0.321059	+	0.947059i	$0.604039\pi$
$762$	0	0
$763$	−20.1282	−0.728691
$764$	0	0
$765$	2.74066	0.0990887
$766$	0	0
$767$	−49.6997	−1.79455
$768$	0	0
$769$	−26.1746	−0.943880	−0.471940	−	0.881631i	$-0.656446\pi$
$769$	−26.1746	−0.943880	−0.471940	+	0.881631i	$0.656446\pi$
$770$	0	0
$771$	32.3763	1.16600
$772$	0	0
$773$	24.0104	0.863593	0.431797	−	0.901971i	$-0.357880\pi$
$773$	24.0104	0.863593	0.431797	+	0.901971i	$0.357880\pi$
$774$	0	0
$775$	2.31678	0.0832213
$776$	0	0
$777$	65.1657	2.33781
$778$	0	0
$779$	−20.2237	−0.724588
$780$	0	0
$781$	−0.710324	−0.0254174
$782$	0	0
$783$	55.2818	1.97561
$784$	0	0
$785$	−13.6524	−0.487275
$786$	0	0
$787$	−40.6566	−1.44925	−0.724625	−	0.689143i	$-0.757987\pi$
$787$	−40.6566	−1.44925	−0.724625	+	0.689143i	$0.757987\pi$
$788$	0	0
$789$	19.8032	0.705013
$790$	0	0
$791$	30.4805	1.08376
$792$	0	0
$793$	−8.96361	−0.318307
$794$	0	0
$795$	−46.8359	−1.66110
$796$	0	0
$797$	46.8584	1.65981	0.829906	−	0.557904i	$-0.188394\pi$
$797$	46.8584	1.65981	0.829906	+	0.557904i	$0.188394\pi$
$798$	0	0
$799$	−26.0755	−0.922486
$800$	0	0
$801$	6.39484	0.225951
$802$	0	0
$803$	−9.44921	−0.333455
$804$	0	0
$805$	−2.95725	−0.104230
$806$	0	0
$807$	19.7176	0.694091
$808$	0	0
$809$	0.706526	0.0248401	0.0124201	−	0.999923i	$-0.496046\pi$
$809$	0.706526	0.0248401	0.0124201	+	0.999923i	$0.496046\pi$
$810$	0	0
$811$	15.9660	0.560643	0.280321	−	0.959906i	$-0.409559\pi$
$811$	15.9660	0.560643	0.280321	+	0.959906i	$0.409559\pi$
$812$	0	0
$813$	−24.7196	−0.866953
$814$	0	0
$815$	66.6289	2.33391
$816$	0	0
$817$	−11.3746	−0.397946
$818$	0	0
$819$	−6.56401	−0.229365
$820$	0	0
$821$	48.2648	1.68445	0.842227	−	0.539123i	$-0.181244\pi$
$821$	48.2648	1.68445	0.842227	+	0.539123i	$0.181244\pi$
$822$	0	0
$823$	11.3684	0.396277	0.198138	−	0.980174i	$-0.436510\pi$
$823$	11.3684	0.396277	0.198138	+	0.980174i	$0.436510\pi$
$824$	0	0
$825$	−7.01990	−0.244402
$826$	0	0
$827$	36.1476	1.25698	0.628488	−	0.777819i	$-0.283674\pi$
$827$	36.1476	1.25698	0.628488	+	0.777819i	$0.283674\pi$
$828$	0	0
$829$	29.8227	1.03579	0.517893	−	0.855445i	$-0.326716\pi$
$829$	29.8227	1.03579	0.517893	+	0.855445i	$0.326716\pi$
$830$	0	0
$831$	39.7042	1.37732
$832$	0	0
$833$	30.3339	1.05101
$834$	0	0
$835$	−58.6515	−2.02972
$836$	0	0
$837$	−2.92487	−0.101098
$838$	0	0
$839$	39.4935	1.36347	0.681733	−	0.731601i	$-0.261227\pi$
$839$	39.4935	1.36347	0.681733	+	0.731601i	$0.261227\pi$
$840$	0	0
$841$	73.2972	2.52749
$842$	0	0
$843$	32.1667	1.10788
$844$	0	0
$845$	−9.24664	−0.318094
$846$	0	0
$847$	−4.42036	−0.151885
$848$	0	0
$849$	24.2241	0.831370
$850$	0	0
$851$	1.99143	0.0682652
$852$	0	0
$853$	52.6645	1.80320	0.901599	−	0.432572i	$-0.142394\pi$
$853$	52.6645	1.80320	0.901599	+	0.432572i	$0.142394\pi$
$854$	0	0
$855$	−3.06617	−0.104861
$856$	0	0
$857$	14.4230	0.492680	0.246340	−	0.969184i	$-0.420772\pi$
$857$	14.4230	0.492680	0.246340	+	0.969184i	$0.420772\pi$
$858$	0	0
$859$	55.9066	1.90751	0.953754	−	0.300589i	$-0.0971833\pi$
$859$	55.9066	1.90751	0.953754	+	0.300589i	$0.0971833\pi$
$860$	0	0
$861$	−53.5589	−1.82528
$862$	0	0
$863$	−26.6334	−0.906611	−0.453305	−	0.891355i	$-0.649755\pi$
$863$	−26.6334	−0.906611	−0.453305	+	0.891355i	$0.649755\pi$
$864$	0	0
$865$	−32.4681	−1.10395
$866$	0	0
$867$	−18.0761	−0.613897
$868$	0	0
$869$	−13.6597	−0.463374
$870$	0	0
$871$	31.0332	1.05152
$872$	0	0
$873$	−4.82445	−0.163283
$874$	0	0
$875$	−9.05397	−0.306080
$876$	0	0
$877$	−36.0786	−1.21829	−0.609143	−	0.793060i	$-0.708487\pi$
$877$	−36.0786	−1.21829	−0.609143	+	0.793060i	$0.708487\pi$
$878$	0	0
$879$	52.0931	1.75706
$880$	0	0
$881$	−44.1981	−1.48907	−0.744536	−	0.667582i	$-0.767329\pi$
$881$	−44.1981	−1.48907	−0.744536	+	0.667582i	$0.767329\pi$
$882$	0	0
$883$	39.9090	1.34304	0.671522	−	0.740985i	$-0.265641\pi$
$883$	39.9090	1.34304	0.671522	+	0.740985i	$0.265641\pi$
$884$	0	0
$885$	−61.4827	−2.06672
$886$	0	0
$887$	33.9009	1.13828	0.569140	−	0.822240i	$-0.307276\pi$
$887$	33.9009	1.13828	0.569140	+	0.822240i	$0.307276\pi$
$888$	0	0
$889$	−3.97904	−0.133453
$890$	0	0
$891$	7.74965	0.259623
$892$	0	0
$893$	29.1725	0.976222
$894$	0	0
$895$	−36.0571	−1.20526
$896$	0	0
$897$	1.42179	0.0474723
$898$	0	0
$899$	−5.41238	−0.180513
$900$	0	0
$901$	22.8768	0.762138
$902$	0	0
$903$	−30.1236	−1.00245
$904$	0	0
$905$	33.7761	1.12275
$906$	0	0
$907$	45.4485	1.50909	0.754546	−	0.656247i	$-0.227857\pi$
$907$	45.4485	1.50909	0.754546	+	0.656247i	$0.227857\pi$
$908$	0	0
$909$	5.42001	0.179770
$910$	0	0
$911$	−30.6382	−1.01509	−0.507544	−	0.861626i	$-0.669446\pi$
$911$	−30.6382	−1.01509	−0.507544	+	0.861626i	$0.669446\pi$
$912$	0	0
$913$	12.5321	0.414752
$914$	0	0
$915$	−11.0887	−0.366582
$916$	0	0
$917$	56.1188	1.85320
$918$	0	0
$919$	−0.125354	−0.00413504	−0.00206752	−	0.999998i	$-0.500658\pi$
$919$	−0.125354	−0.00413504	−0.00206752	+	0.999998i	$0.500658\pi$
$920$	0	0
$921$	−20.4453	−0.673695
$922$	0	0
$923$	−2.84372	−0.0936022
$924$	0	0
$925$	−39.3631	−1.29425
$926$	0	0
$927$	−2.02044	−0.0663600
$928$	0	0
$929$	−23.5516	−0.772702	−0.386351	−	0.922352i	$-0.626265\pi$
$929$	−23.5516	−0.772702	−0.386351	+	0.922352i	$0.626265\pi$
$930$	0	0
$931$	−33.9367	−1.11223
$932$	0	0
$933$	33.3905	1.09316
$934$	0	0
$935$	7.38879	0.241639
$936$	0	0
$937$	10.6987	0.349512	0.174756	−	0.984612i	$-0.444086\pi$
$937$	10.6987	0.349512	0.174756	+	0.984612i	$0.444086\pi$
$938$	0	0
$939$	41.4693	1.35330
$940$	0	0
$941$	−37.7138	−1.22944	−0.614718	−	0.788747i	$-0.710730\pi$
$941$	−37.7138	−1.22944	−0.614718	+	0.788747i	$0.710730\pi$
$942$	0	0
$943$	−1.63673	−0.0532993
$944$	0	0
$945$	−73.7963	−2.40059
$946$	0	0
$947$	−35.9601	−1.16855	−0.584273	−	0.811557i	$-0.698620\pi$
$947$	−35.9601	−1.16855	−0.584273	+	0.811557i	$0.698620\pi$
$948$	0	0
$949$	−37.8291	−1.22798
$950$	0	0
$951$	−24.8726	−0.806548
$952$	0	0
$953$	−13.2836	−0.430297	−0.215148	−	0.976581i	$-0.569023\pi$
$953$	−13.2836	−0.430297	−0.215148	+	0.976581i	$0.569023\pi$
$954$	0	0
$955$	42.6472	1.38003
$956$	0	0
$957$	16.3996	0.530124
$958$	0	0
$959$	4.42036	0.142741
$960$	0	0
$961$	−30.7136	−0.990763
$962$	0	0
$963$	−4.11884	−0.132728
$964$	0	0
$965$	−41.1120	−1.32344
$966$	0	0
$967$	4.18948	0.134725	0.0673624	−	0.997729i	$-0.478542\pi$
$967$	4.18948	0.134725	0.0673624	+	0.997729i	$0.478542\pi$
$968$	0	0
$969$	−10.6154	−0.341015
$970$	0	0
$971$	21.0785	0.676440	0.338220	−	0.941067i	$-0.390175\pi$
$971$	21.0785	0.676440	0.338220	+	0.941067i	$0.390175\pi$
$972$	0	0
$973$	3.48333	0.111670
$974$	0	0
$975$	−28.1035	−0.900034
$976$	0	0
$977$	54.3488	1.73877	0.869387	−	0.494133i	$-0.164514\pi$
$977$	54.3488	1.73877	0.869387	+	0.494133i	$0.164514\pi$
$978$	0	0
$979$	17.2404	0.551006
$980$	0	0
$981$	−1.68900	−0.0539257
$982$	0	0
$983$	−16.8324	−0.536869	−0.268435	−	0.963298i	$-0.586506\pi$
$983$	−16.8324	−0.536869	−0.268435	+	0.963298i	$0.586506\pi$
$984$	0	0
$985$	70.3528	2.24163
$986$	0	0
$987$	77.2584	2.45916
$988$	0	0
$989$	−0.920561	−0.0292721
$990$	0	0
$991$	−47.3506	−1.50414	−0.752071	−	0.659082i	$-0.770945\pi$
$991$	−47.3506	−1.50414	−0.752071	+	0.659082i	$0.770945\pi$
$992$	0	0
$993$	−14.8993	−0.472814
$994$	0	0
$995$	67.5960	2.14294
$996$	0	0
$997$	−17.1669	−0.543682	−0.271841	−	0.962342i	$-0.587633\pi$
$997$	−17.1669	−0.543682	−0.271841	+	0.962342i	$0.587633\pi$
$998$	0	0
$999$	49.6947	1.57227

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.e.1.20	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.e.1.20	✓	27	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q+1.62144 q^{3} -3.05441 q^{5} -4.42036 q^{7} -0.370921 q^{9} +O(q^{10})\)	\(q+1.62144 q^{3} -3.05441 q^{5} -4.42036 q^{7} -0.370921 q^{9} -1.00000 q^{11} -4.00341 q^{13} -4.95255 q^{15} +2.41906 q^{17} -2.70637 q^{19} -7.16736 q^{21} -0.219030 q^{23} +4.32941 q^{25} -5.46576 q^{27} -10.1142 q^{29} +0.535126 q^{31} -1.62144 q^{33} +13.5016 q^{35} -9.09201 q^{37} -6.49131 q^{39} +7.47262 q^{41} +4.20289 q^{43} +1.13295 q^{45} -10.7792 q^{47} +12.5395 q^{49} +3.92236 q^{51} +9.45692 q^{53} +3.05441 q^{55} -4.38823 q^{57} +12.4143 q^{59} +2.23899 q^{61} +1.63960 q^{63} +12.2281 q^{65} -7.75168 q^{67} -0.355145 q^{69} +0.710324 q^{71} +9.44921 q^{73} +7.01990 q^{75} +4.42036 q^{77} +13.6597 q^{79} -7.74965 q^{81} -12.5321 q^{83} -7.38879 q^{85} -16.3996 q^{87} -17.2404 q^{89} +17.6965 q^{91} +0.867677 q^{93} +8.26636 q^{95} +13.0067 q^{97} +0.370921 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) \) 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9	\( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9} - 27 q^{11} + 2 q^{13} + 10 q^{15} + 17 q^{17} + 4 q^{19} - 4 q^{21} + 25 q^{23} + 21 q^{25} + 32 q^{27} - q^{29} + 8 q^{31} - 5 q^{33} + 30 q^{35} - 24 q^{37} + 10 q^{39} + 31 q^{41} + 17 q^{43} - 15 q^{45} + 27 q^{47} + 40 q^{49} + 30 q^{51} + 2 q^{55} + 13 q^{57} + 31 q^{59} + 2 q^{61} + 61 q^{63} + 2 q^{67} - 15 q^{69} + 40 q^{71} + 17 q^{73} + 30 q^{75} - 7 q^{77} + 33 q^{79} + 31 q^{81} + 66 q^{83} + 26 q^{85} + 43 q^{87} + 22 q^{89} + 18 q^{91} - 9 q^{93} + 95 q^{95} - 21 q^{97} - 28 q^{99}+O(q^{100}) \) 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9 - 27 * q^11 + 2 * q^13 + 10 * q^15 + 17 * q^17 + 4 * q^19 - 4 * q^21 + 25 * q^23 + 21 * q^25 + 32 * q^27 - q^29 + 8 * q^31 - 5 * q^33 + 30 * q^35 - 24 * q^37 + 10 * q^39 + 31 * q^41 + 17 * q^43 - 15 * q^45 + 27 * q^47 + 40 * q^49 + 30 * q^51 + 2 * q^55 + 13 * q^57 + 31 * q^59 + 2 * q^61 + 61 * q^63 + 2 * q^67 - 15 * q^69 + 40 * q^71 + 17 * q^73 + 30 * q^75 - 7 * q^77 + 33 * q^79 + 31 * q^81 + 66 * q^83 + 26 * q^85 + 43 * q^87 + 22 * q^89 + 18 * q^91 - 9 * q^93 + 95 * q^95 - 21 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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