LMFDB - Embedded newform 6028.2.a.e.1.10

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			1.10
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)

$f(q)$	$=$	$q-0.836719 q^{3} +1.09288 q^{5} -3.78282 q^{7} -2.29990 q^{9} +O(q^{10})$	$q-0.836719 q^{3} +1.09288 q^{5} -3.78282 q^{7} -2.29990 q^{9} -1.00000 q^{11} -2.30113 q^{13} -0.914437 q^{15} +3.68067 q^{17} +3.95497 q^{19} +3.16516 q^{21} -2.21016 q^{23} -3.80561 q^{25} +4.43453 q^{27} -0.515297 q^{29} +7.73364 q^{31} +0.836719 q^{33} -4.13419 q^{35} -8.97663 q^{37} +1.92540 q^{39} -8.84408 q^{41} -9.80228 q^{43} -2.51352 q^{45} +7.60642 q^{47} +7.30976 q^{49} -3.07969 q^{51} +7.02988 q^{53} -1.09288 q^{55} -3.30920 q^{57} -11.6890 q^{59} -11.1442 q^{61} +8.70012 q^{63} -2.51487 q^{65} -4.62936 q^{67} +1.84928 q^{69} -12.1288 q^{71} +12.9449 q^{73} +3.18422 q^{75} +3.78282 q^{77} +4.71973 q^{79} +3.18925 q^{81} -8.11210 q^{83} +4.02254 q^{85} +0.431159 q^{87} -4.40279 q^{89} +8.70478 q^{91} -6.47089 q^{93} +4.32233 q^{95} -17.3741 q^{97} +2.29990 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.836719	−0.483080	−0.241540	−	0.970391i	$-0.577653\pi$
$3$	−0.836719	−0.483080	−0.241540	+	0.970391i	$0.577653\pi$
$4$	0	0
$5$	1.09288	0.488752	0.244376	−	0.969681i	$-0.421417\pi$
$5$	1.09288	0.488752	0.244376	+	0.969681i	$0.421417\pi$
$6$	0	0
$7$	−3.78282	−1.42977	−0.714887	−	0.699240i	$-0.753522\pi$
$7$	−3.78282	−1.42977	−0.714887	+	0.699240i	$0.753522\pi$
$8$	0	0
$9$	−2.29990	−0.766634
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.30113	−0.638219	−0.319110	−	0.947718i	$-0.603384\pi$
$13$	−2.30113	−0.638219	−0.319110	+	0.947718i	$0.603384\pi$
$14$	0	0
$15$	−0.914437	−0.236106
$16$	0	0
$17$	3.68067	0.892694	0.446347	−	0.894860i	$-0.352725\pi$
$17$	3.68067	0.892694	0.446347	+	0.894860i	$0.352725\pi$
$18$	0	0
$19$	3.95497	0.907333	0.453667	−	0.891171i	$-0.350116\pi$
$19$	3.95497	0.907333	0.453667	+	0.891171i	$0.350116\pi$
$20$	0	0
$21$	3.16516	0.690695
$22$	0	0
$23$	−2.21016	−0.460850	−0.230425	−	0.973090i	$-0.574012\pi$
$23$	−2.21016	−0.460850	−0.230425	+	0.973090i	$0.574012\pi$
$24$	0	0
$25$	−3.80561	−0.761121
$26$	0	0
$27$	4.43453	0.853425
$28$	0	0
$29$	−0.515297	−0.0956882	−0.0478441	−	0.998855i	$-0.515235\pi$
$29$	−0.515297	−0.0956882	−0.0478441	+	0.998855i	$0.515235\pi$
$30$	0	0
$31$	7.73364	1.38900	0.694501	−	0.719491i	$-0.255625\pi$
$31$	7.73364	1.38900	0.694501	+	0.719491i	$0.255625\pi$
$32$	0	0
$33$	0.836719	0.145654
$34$	0	0
$35$	−4.13419	−0.698805
$36$	0	0
$37$	−8.97663	−1.47575	−0.737874	−	0.674938i	$-0.764170\pi$
$37$	−8.97663	−1.47575	−0.737874	+	0.674938i	$0.764170\pi$
$38$	0	0
$39$	1.92540	0.308311
$40$	0	0
$41$	−8.84408	−1.38121	−0.690607	−	0.723231i	$-0.742656\pi$
$41$	−8.84408	−1.38121	−0.690607	+	0.723231i	$0.742656\pi$
$42$	0	0
$43$	−9.80228	−1.49483	−0.747417	−	0.664355i	$-0.768706\pi$
$43$	−9.80228	−1.49483	−0.747417	+	0.664355i	$0.768706\pi$
$44$	0	0
$45$	−2.51352	−0.374694
$46$	0	0
$47$	7.60642	1.10951	0.554755	−	0.832014i	$-0.312812\pi$
$47$	7.60642	1.10951	0.554755	+	0.832014i	$0.312812\pi$
$48$	0	0
$49$	7.30976	1.04425
$50$	0	0
$51$	−3.07969	−0.431242
$52$	0	0
$53$	7.02988	0.965628	0.482814	−	0.875723i	$-0.339615\pi$
$53$	7.02988	0.965628	0.482814	+	0.875723i	$0.339615\pi$
$54$	0	0
$55$	−1.09288	−0.147364
$56$	0	0
$57$	−3.30920	−0.438315
$58$	0	0
$59$	−11.6890	−1.52177	−0.760886	−	0.648886i	$-0.775235\pi$
$59$	−11.6890	−1.52177	−0.760886	+	0.648886i	$0.775235\pi$
$60$	0	0
$61$	−11.1442	−1.42687	−0.713437	−	0.700719i	$-0.752863\pi$
$61$	−11.1442	−1.42687	−0.713437	+	0.700719i	$0.752863\pi$
$62$	0	0
$63$	8.70012	1.09611
$64$	0	0
$65$	−2.51487	−0.311931
$66$	0	0
$67$	−4.62936	−0.565566	−0.282783	−	0.959184i	$-0.591258\pi$
$67$	−4.62936	−0.565566	−0.282783	+	0.959184i	$0.591258\pi$
$68$	0	0
$69$	1.84928	0.222628
$70$	0	0
$71$	−12.1288	−1.43943	−0.719714	−	0.694271i	$-0.755727\pi$
$71$	−12.1288	−1.43943	−0.719714	+	0.694271i	$0.755727\pi$
$72$	0	0
$73$	12.9449	1.51509	0.757545	−	0.652782i	$-0.226398\pi$
$73$	12.9449	1.51509	0.757545	+	0.652782i	$0.226398\pi$
$74$	0	0
$75$	3.18422	0.367682
$76$	0	0
$77$	3.78282	0.431093
$78$	0	0
$79$	4.71973	0.531011	0.265505	−	0.964109i	$-0.414461\pi$
$79$	4.71973	0.531011	0.265505	+	0.964109i	$0.414461\pi$
$80$	0	0
$81$	3.18925	0.354361
$82$	0	0
$83$	−8.11210	−0.890418	−0.445209	−	0.895427i	$-0.646871\pi$
$83$	−8.11210	−0.890418	−0.445209	+	0.895427i	$0.646871\pi$
$84$	0	0
$85$	4.02254	0.436306
$86$	0	0
$87$	0.431159	0.0462250
$88$	0	0
$89$	−4.40279	−0.466694	−0.233347	−	0.972393i	$-0.574968\pi$
$89$	−4.40279	−0.466694	−0.233347	+	0.972393i	$0.574968\pi$
$90$	0	0
$91$	8.70478	0.912508
$92$	0	0
$93$	−6.47089	−0.671000
$94$	0	0
$95$	4.32233	0.443461
$96$	0	0
$97$	−17.3741	−1.76408	−0.882038	−	0.471177i	$-0.843829\pi$
$97$	−17.3741	−1.76408	−0.882038	+	0.471177i	$0.843829\pi$
$98$	0	0
$99$	2.29990	0.231149
$100$	0	0
$101$	9.37315	0.932663	0.466332	−	0.884610i	$-0.345575\pi$
$101$	9.37315	0.932663	0.466332	+	0.884610i	$0.345575\pi$
$102$	0	0
$103$	17.5618	1.73042	0.865209	−	0.501412i	$-0.167186\pi$
$103$	17.5618	1.73042	0.865209	+	0.501412i	$0.167186\pi$
$104$	0	0
$105$	3.45915	0.337579
$106$	0	0
$107$	13.9202	1.34572	0.672860	−	0.739770i	$-0.265065\pi$
$107$	13.9202	1.34572	0.672860	+	0.739770i	$0.265065\pi$
$108$	0	0
$109$	3.06620	0.293689	0.146844	−	0.989160i	$-0.453088\pi$
$109$	3.06620	0.293689	0.146844	+	0.989160i	$0.453088\pi$
$110$	0	0
$111$	7.51092	0.712905
$112$	0	0
$113$	12.7035	1.19504	0.597520	−	0.801854i	$-0.296153\pi$
$113$	12.7035	1.19504	0.597520	+	0.801854i	$0.296153\pi$
$114$	0	0
$115$	−2.41545	−0.225242
$116$	0	0
$117$	5.29237	0.489280
$118$	0	0
$119$	−13.9233	−1.27635
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	7.40001	0.667237
$124$	0	0
$125$	−9.62350	−0.860752
$126$	0	0
$127$	9.45063	0.838608	0.419304	−	0.907846i	$-0.362274\pi$
$127$	9.45063	0.838608	0.419304	+	0.907846i	$0.362274\pi$
$128$	0	0
$129$	8.20176	0.722124
$130$	0	0
$131$	5.83383	0.509704	0.254852	−	0.966980i	$-0.417973\pi$
$131$	5.83383	0.509704	0.254852	+	0.966980i	$0.417973\pi$
$132$	0	0
$133$	−14.9610	−1.29728
$134$	0	0
$135$	4.84642	0.417114
$136$	0	0
$137$	−1.00000	−0.0854358
$137$	−1.00000	−0.0854358
$138$	0	0
$139$	19.9469	1.69188	0.845939	−	0.533280i	$-0.179041\pi$
$139$	19.9469	1.69188	0.845939	+	0.533280i	$0.179041\pi$
$140$	0	0
$141$	−6.36444	−0.535982
$142$	0	0
$143$	2.30113	0.192430
$144$	0	0
$145$	−0.563159	−0.0467678
$146$	0	0
$147$	−6.11621	−0.504457
$148$	0	0
$149$	9.41396	0.771222	0.385611	−	0.922662i	$-0.373991\pi$
$149$	9.41396	0.771222	0.385611	+	0.922662i	$0.373991\pi$
$150$	0	0
$151$	18.3868	1.49630	0.748150	−	0.663530i	$-0.230942\pi$
$151$	18.3868	1.49630	0.748150	+	0.663530i	$0.230942\pi$
$152$	0	0
$153$	−8.46518	−0.684369
$154$	0	0
$155$	8.45197	0.678878
$156$	0	0
$157$	18.5594	1.48120	0.740601	−	0.671945i	$-0.234541\pi$
$157$	18.5594	1.48120	0.740601	+	0.671945i	$0.234541\pi$
$158$	0	0
$159$	−5.88203	−0.466475
$160$	0	0
$161$	8.36065	0.658911
$162$	0	0
$163$	21.5264	1.68607	0.843037	−	0.537855i	$-0.180765\pi$
$163$	21.5264	1.68607	0.843037	+	0.537855i	$0.180765\pi$
$164$	0	0
$165$	0.914437	0.0711888
$166$	0	0
$167$	−1.41762	−0.109699	−0.0548495	−	0.998495i	$-0.517468\pi$
$167$	−1.41762	−0.109699	−0.0548495	+	0.998495i	$0.517468\pi$
$168$	0	0
$169$	−7.70479	−0.592676
$170$	0	0
$171$	−9.09605	−0.695592
$172$	0	0
$173$	−4.59012	−0.348981	−0.174490	−	0.984659i	$-0.555828\pi$
$173$	−4.59012	−0.348981	−0.174490	+	0.984659i	$0.555828\pi$
$174$	0	0
$175$	14.3959	1.08823
$176$	0	0
$177$	9.78037	0.735138
$178$	0	0
$179$	12.8571	0.960985	0.480493	−	0.876999i	$-0.340458\pi$
$179$	12.8571	0.960985	0.480493	+	0.876999i	$0.340458\pi$
$180$	0	0
$181$	14.0817	1.04669	0.523343	−	0.852122i	$-0.324685\pi$
$181$	14.0817	1.04669	0.523343	+	0.852122i	$0.324685\pi$
$182$	0	0
$183$	9.32460	0.689295
$184$	0	0
$185$	−9.81041	−0.721275
$186$	0	0
$187$	−3.68067	−0.269157
$188$	0	0
$189$	−16.7750	−1.22020
$190$	0	0
$191$	19.5161	1.41213	0.706067	−	0.708145i	$-0.250468\pi$
$191$	19.5161	1.41213	0.706067	+	0.708145i	$0.250468\pi$
$192$	0	0
$193$	−0.670072	−0.0482328	−0.0241164	−	0.999709i	$-0.507677\pi$
$193$	−0.670072	−0.0482328	−0.0241164	+	0.999709i	$0.507677\pi$
$194$	0	0
$195$	2.10424	0.150688
$196$	0	0
$197$	−4.23867	−0.301992	−0.150996	−	0.988534i	$-0.548248\pi$
$197$	−4.23867	−0.301992	−0.150996	+	0.988534i	$0.548248\pi$
$198$	0	0
$199$	−4.22890	−0.299779	−0.149889	−	0.988703i	$-0.547892\pi$
$199$	−4.22890	−0.299779	−0.149889	+	0.988703i	$0.547892\pi$
$200$	0	0
$201$	3.87347	0.273214
$202$	0	0
$203$	1.94928	0.136812
$204$	0	0
$205$	−9.66555	−0.675071
$206$	0	0
$207$	5.08315	0.353303
$208$	0	0
$209$	−3.95497	−0.273571
$210$	0	0
$211$	20.5993	1.41811	0.709056	−	0.705152i	$-0.249121\pi$
$211$	20.5993	1.41811	0.709056	+	0.705152i	$0.249121\pi$
$212$	0	0
$213$	10.1484	0.695359
$214$	0	0
$215$	−10.7128	−0.730604
$216$	0	0
$217$	−29.2550	−1.98596
$218$	0	0
$219$	−10.8313	−0.731910
$220$	0	0
$221$	−8.46971	−0.569734
$222$	0	0
$223$	3.61814	0.242289	0.121144	−	0.992635i	$-0.461344\pi$
$223$	3.61814	0.242289	0.121144	+	0.992635i	$0.461344\pi$
$224$	0	0
$225$	8.75252	0.583501
$226$	0	0
$227$	20.9618	1.39128	0.695642	−	0.718388i	$-0.255120\pi$
$227$	20.9618	1.39128	0.695642	+	0.718388i	$0.255120\pi$
$228$	0	0
$229$	−6.68003	−0.441429	−0.220714	−	0.975338i	$-0.570839\pi$
$229$	−6.68003	−0.441429	−0.220714	+	0.975338i	$0.570839\pi$
$230$	0	0
$231$	−3.16516	−0.208252
$232$	0	0
$233$	−7.75414	−0.507990	−0.253995	−	0.967205i	$-0.581745\pi$
$233$	−7.75414	−0.507990	−0.253995	+	0.967205i	$0.581745\pi$
$234$	0	0
$235$	8.31293	0.542276
$236$	0	0
$237$	−3.94909	−0.256521
$238$	0	0
$239$	13.2983	0.860197	0.430099	−	0.902782i	$-0.358479\pi$
$239$	13.2983	0.860197	0.430099	+	0.902782i	$0.358479\pi$
$240$	0	0
$241$	−17.5276	−1.12905	−0.564525	−	0.825416i	$-0.690941\pi$
$241$	−17.5276	−1.12905	−0.564525	+	0.825416i	$0.690941\pi$
$242$	0	0
$243$	−15.9721	−1.02461
$244$	0	0
$245$	7.98871	0.510380
$246$	0	0
$247$	−9.10091	−0.579077
$248$	0	0
$249$	6.78755	0.430143
$250$	0	0
$251$	−21.2777	−1.34304	−0.671518	−	0.740988i	$-0.734358\pi$
$251$	−21.2777	−1.34304	−0.671518	+	0.740988i	$0.734358\pi$
$252$	0	0
$253$	2.21016	0.138952
$254$	0	0
$255$	−3.36574	−0.210771
$256$	0	0
$257$	26.6044	1.65954	0.829770	−	0.558106i	$-0.188472\pi$
$257$	26.6044	1.65954	0.829770	+	0.558106i	$0.188472\pi$
$258$	0	0
$259$	33.9570	2.10999
$260$	0	0
$261$	1.18513	0.0733578
$262$	0	0
$263$	−14.7346	−0.908577	−0.454288	−	0.890855i	$-0.650106\pi$
$263$	−14.7346	−0.908577	−0.454288	+	0.890855i	$0.650106\pi$
$264$	0	0
$265$	7.68283	0.471953
$266$	0	0
$267$	3.68389	0.225451
$268$	0	0
$269$	−6.97135	−0.425051	−0.212525	−	0.977156i	$-0.568169\pi$
$269$	−6.97135	−0.425051	−0.212525	+	0.977156i	$0.568169\pi$
$270$	0	0
$271$	−27.7549	−1.68599	−0.842996	−	0.537919i	$-0.819211\pi$
$271$	−27.7549	−1.68599	−0.842996	+	0.537919i	$0.819211\pi$
$272$	0	0
$273$	−7.28345	−0.440815
$274$	0	0
$275$	3.80561	0.229487
$276$	0	0
$277$	3.67486	0.220801	0.110401	−	0.993887i	$-0.464787\pi$
$277$	3.67486	0.220801	0.110401	+	0.993887i	$0.464787\pi$
$278$	0	0
$279$	−17.7866	−1.06486
$280$	0	0
$281$	8.50763	0.507522	0.253761	−	0.967267i	$-0.418332\pi$
$281$	8.50763	0.507522	0.253761	+	0.967267i	$0.418332\pi$
$282$	0	0
$283$	8.75447	0.520399	0.260200	−	0.965555i	$-0.416212\pi$
$283$	8.75447	0.520399	0.260200	+	0.965555i	$0.416212\pi$
$284$	0	0
$285$	−3.61657	−0.214227
$286$	0	0
$287$	33.4556	1.97482
$288$	0	0
$289$	−3.45267	−0.203098
$290$	0	0
$291$	14.5373	0.852190
$292$	0	0
$293$	−28.6435	−1.67337	−0.836685	−	0.547685i	$-0.815509\pi$
$293$	−28.6435	−1.67337	−0.836685	+	0.547685i	$0.815509\pi$
$294$	0	0
$295$	−12.7747	−0.743770
$296$	0	0
$297$	−4.43453	−0.257317
$298$	0	0
$299$	5.08587	0.294123
$300$	0	0
$301$	37.0803	2.13727
$302$	0	0
$303$	−7.84270	−0.450551
$304$	0	0
$305$	−12.1794	−0.697388
$306$	0	0
$307$	−4.95593	−0.282850	−0.141425	−	0.989949i	$-0.545168\pi$
$307$	−4.95593	−0.282850	−0.141425	+	0.989949i	$0.545168\pi$
$308$	0	0
$309$	−14.6943	−0.835930
$310$	0	0
$311$	−4.37834	−0.248273	−0.124136	−	0.992265i	$-0.539616\pi$
$311$	−4.37834	−0.248273	−0.124136	+	0.992265i	$0.539616\pi$
$312$	0	0
$313$	9.75937	0.551632	0.275816	−	0.961210i	$-0.411052\pi$
$313$	9.75937	0.551632	0.275816	+	0.961210i	$0.411052\pi$
$314$	0	0
$315$	9.50822	0.535727
$316$	0	0
$317$	−15.9250	−0.894437	−0.447219	−	0.894425i	$-0.647585\pi$
$317$	−15.9250	−0.894437	−0.447219	+	0.894425i	$0.647585\pi$
$318$	0	0
$319$	0.515297	0.0288511
$320$	0	0
$321$	−11.6473	−0.650091
$322$	0	0
$323$	14.5570	0.809971
$324$	0	0
$325$	8.75720	0.485762
$326$	0	0
$327$	−2.56555	−0.141875
$328$	0	0
$329$	−28.7737	−1.58635
$330$	0	0
$331$	11.0805	0.609041	0.304521	−	0.952506i	$-0.401504\pi$
$331$	11.0805	0.609041	0.304521	+	0.952506i	$0.401504\pi$
$332$	0	0
$333$	20.6454	1.13136
$334$	0	0
$335$	−5.05935	−0.276422
$336$	0	0
$337$	21.3732	1.16427	0.582136	−	0.813092i	$-0.302217\pi$
$337$	21.3732	1.16427	0.582136	+	0.813092i	$0.302217\pi$
$338$	0	0
$339$	−10.6292	−0.577300
$340$	0	0
$341$	−7.73364	−0.418800
$342$	0	0
$343$	−1.17176	−0.0632689
$344$	0	0
$345$	2.02105	0.108810
$346$	0	0
$347$	12.9088	0.692980	0.346490	−	0.938054i	$-0.387374\pi$
$347$	12.9088	0.692980	0.346490	+	0.938054i	$0.387374\pi$
$348$	0	0
$349$	−16.0989	−0.861753	−0.430877	−	0.902411i	$-0.641796\pi$
$349$	−16.0989	−0.861753	−0.430877	+	0.902411i	$0.641796\pi$
$350$	0	0
$351$	−10.2044	−0.544672
$352$	0	0
$353$	18.1805	0.967650	0.483825	−	0.875165i	$-0.339247\pi$
$353$	18.1805	0.967650	0.483825	+	0.875165i	$0.339247\pi$
$354$	0	0
$355$	−13.2554	−0.703524
$356$	0	0
$357$	11.6499	0.616579
$358$	0	0
$359$	−7.27946	−0.384195	−0.192098	−	0.981376i	$-0.561529\pi$
$359$	−7.27946	−0.384195	−0.192098	+	0.981376i	$0.561529\pi$
$360$	0	0
$361$	−3.35818	−0.176746
$362$	0	0
$363$	−0.836719	−0.0439164
$364$	0	0
$365$	14.1473	0.740504
$366$	0	0
$367$	12.1737	0.635460	0.317730	−	0.948181i	$-0.397079\pi$
$367$	12.1737	0.635460	0.317730	+	0.948181i	$0.397079\pi$
$368$	0	0
$369$	20.3405	1.05888
$370$	0	0
$371$	−26.5928	−1.38063
$372$	0	0
$373$	14.1179	0.730998	0.365499	−	0.930812i	$-0.380898\pi$
$373$	14.1179	0.730998	0.365499	+	0.930812i	$0.380898\pi$
$374$	0	0
$375$	8.05217	0.415812
$376$	0	0
$377$	1.18577	0.0610700
$378$	0	0
$379$	−8.42871	−0.432954	−0.216477	−	0.976288i	$-0.569457\pi$
$379$	−8.42871	−0.432954	−0.216477	+	0.976288i	$0.569457\pi$
$380$	0	0
$381$	−7.90752	−0.405115
$382$	0	0
$383$	−25.9163	−1.32426	−0.662131	−	0.749388i	$-0.730348\pi$
$383$	−25.9163	−1.32426	−0.662131	+	0.749388i	$0.730348\pi$
$384$	0	0
$385$	4.13419	0.210698
$386$	0	0
$387$	22.5443	1.14599
$388$	0	0
$389$	21.8947	1.11011	0.555053	−	0.831815i	$-0.312698\pi$
$389$	21.8947	1.11011	0.555053	+	0.831815i	$0.312698\pi$
$390$	0	0
$391$	−8.13487	−0.411398
$392$	0	0
$393$	−4.88128	−0.246228
$394$	0	0
$395$	5.15811	0.259533
$396$	0	0
$397$	17.9501	0.900889	0.450444	−	0.892804i	$-0.351266\pi$
$397$	17.9501	0.900889	0.450444	+	0.892804i	$0.351266\pi$
$398$	0	0
$399$	12.5181	0.626690
$400$	0	0
$401$	11.6307	0.580808	0.290404	−	0.956904i	$-0.406210\pi$
$401$	11.6307	0.580808	0.290404	+	0.956904i	$0.406210\pi$
$402$	0	0
$403$	−17.7961	−0.886488
$404$	0	0
$405$	3.48548	0.173195
$406$	0	0
$407$	8.97663	0.444955
$408$	0	0
$409$	6.11617	0.302425	0.151213	−	0.988501i	$-0.451682\pi$
$409$	6.11617	0.302425	0.151213	+	0.988501i	$0.451682\pi$
$410$	0	0
$411$	0.836719	0.0412723
$412$	0	0
$413$	44.2173	2.17579
$414$	0	0
$415$	−8.86558	−0.435194
$416$	0	0
$417$	−16.6900	−0.817312
$418$	0	0
$419$	−4.69406	−0.229320	−0.114660	−	0.993405i	$-0.536578\pi$
$419$	−4.69406	−0.229320	−0.114660	+	0.993405i	$0.536578\pi$
$420$	0	0
$421$	−28.9799	−1.41239	−0.706197	−	0.708016i	$-0.749591\pi$
$421$	−28.9799	−1.41239	−0.706197	+	0.708016i	$0.749591\pi$
$422$	0	0
$423$	−17.4940	−0.850588
$424$	0	0
$425$	−14.0072	−0.679448
$426$	0	0
$427$	42.1567	2.04011
$428$	0	0
$429$	−1.92540	−0.0929592
$430$	0	0
$431$	15.3278	0.738312	0.369156	−	0.929367i	$-0.379647\pi$
$431$	15.3278	0.738312	0.369156	+	0.929367i	$0.379647\pi$
$432$	0	0
$433$	−18.9421	−0.910299	−0.455149	−	0.890415i	$-0.650414\pi$
$433$	−18.9421	−0.910299	−0.455149	+	0.890415i	$0.650414\pi$
$434$	0	0
$435$	0.471206	0.0225926
$436$	0	0
$437$	−8.74113	−0.418145
$438$	0	0
$439$	19.2787	0.920121	0.460060	−	0.887888i	$-0.347828\pi$
$439$	19.2787	0.920121	0.460060	+	0.887888i	$0.347828\pi$
$440$	0	0
$441$	−16.8117	−0.800558
$442$	0	0
$443$	9.19135	0.436694	0.218347	−	0.975871i	$-0.429934\pi$
$443$	9.19135	0.436694	0.218347	+	0.975871i	$0.429934\pi$
$444$	0	0
$445$	−4.81173	−0.228098
$446$	0	0
$447$	−7.87684	−0.372562
$448$	0	0
$449$	−17.2934	−0.816128	−0.408064	−	0.912953i	$-0.633796\pi$
$449$	−17.2934	−0.816128	−0.408064	+	0.912953i	$0.633796\pi$
$450$	0	0
$451$	8.84408	0.416451
$452$	0	0
$453$	−15.3846	−0.722832
$454$	0	0
$455$	9.51330	0.445991
$456$	0	0
$457$	−11.8217	−0.552997	−0.276498	−	0.961014i	$-0.589174\pi$
$457$	−11.8217	−0.552997	−0.276498	+	0.961014i	$0.589174\pi$
$458$	0	0
$459$	16.3220	0.761848
$460$	0	0
$461$	−41.9298	−1.95287	−0.976433	−	0.215822i	$-0.930757\pi$
$461$	−41.9298	−1.95287	−0.976433	+	0.215822i	$0.930757\pi$
$462$	0	0
$463$	16.0627	0.746495	0.373247	−	0.927732i	$-0.378244\pi$
$463$	16.0627	0.746495	0.373247	+	0.927732i	$0.378244\pi$
$464$	0	0
$465$	−7.07192	−0.327953
$466$	0	0
$467$	3.02463	0.139963	0.0699815	−	0.997548i	$-0.477706\pi$
$467$	3.02463	0.139963	0.0699815	+	0.997548i	$0.477706\pi$
$468$	0	0
$469$	17.5120	0.808631
$470$	0	0
$471$	−15.5290	−0.715539
$472$	0	0
$473$	9.80228	0.450709
$474$	0	0
$475$	−15.0511	−0.690591
$476$	0	0
$477$	−16.1680	−0.740283
$478$	0	0
$479$	−18.0371	−0.824135	−0.412068	−	0.911153i	$-0.635193\pi$
$479$	−18.0371	−0.824135	−0.412068	+	0.911153i	$0.635193\pi$
$480$	0	0
$481$	20.6564	0.941851
$482$	0	0
$483$	−6.99551	−0.318307
$484$	0	0
$485$	−18.9879	−0.862197
$486$	0	0
$487$	−29.3662	−1.33071	−0.665356	−	0.746526i	$-0.731720\pi$
$487$	−29.3662	−1.33071	−0.665356	+	0.746526i	$0.731720\pi$
$488$	0	0
$489$	−18.0115	−0.814509
$490$	0	0
$491$	21.3185	0.962093	0.481046	−	0.876695i	$-0.340257\pi$
$491$	21.3185	0.962093	0.481046	+	0.876695i	$0.340257\pi$
$492$	0	0
$493$	−1.89664	−0.0854202
$494$	0	0
$495$	2.51352	0.112974
$496$	0	0
$497$	45.8813	2.05806
$498$	0	0
$499$	5.84359	0.261595	0.130798	−	0.991409i	$-0.458246\pi$
$499$	5.84359	0.261595	0.130798	+	0.991409i	$0.458246\pi$
$500$	0	0
$501$	1.18615	0.0529934
$502$	0	0
$503$	−14.0166	−0.624967	−0.312484	−	0.949923i	$-0.601161\pi$
$503$	−14.0166	−0.624967	−0.312484	+	0.949923i	$0.601161\pi$
$504$	0	0
$505$	10.2438	0.455841
$506$	0	0
$507$	6.44675	0.286310
$508$	0	0
$509$	−27.1631	−1.20398	−0.601992	−	0.798502i	$-0.705626\pi$
$509$	−27.1631	−1.20398	−0.601992	+	0.798502i	$0.705626\pi$
$510$	0	0
$511$	−48.9684	−2.16624
$512$	0	0
$513$	17.5384	0.774341
$514$	0	0
$515$	19.1930	0.845746
$516$	0	0
$517$	−7.60642	−0.334530
$518$	0	0
$519$	3.84064	0.168586
$520$	0	0
$521$	−15.6365	−0.685049	−0.342525	−	0.939509i	$-0.611282\pi$
$521$	−15.6365	−0.685049	−0.342525	+	0.939509i	$0.611282\pi$
$522$	0	0
$523$	43.8652	1.91809	0.959046	−	0.283250i	$-0.0914125\pi$
$523$	43.8652	1.91809	0.959046	+	0.283250i	$0.0914125\pi$
$524$	0	0
$525$	−12.0454	−0.525702
$526$	0	0
$527$	28.4650	1.23995
$528$	0	0
$529$	−18.1152	−0.787617
$530$	0	0
$531$	26.8834	1.16664
$532$	0	0
$533$	20.3514	0.881517
$534$	0	0
$535$	15.2132	0.657724
$536$	0	0
$537$	−10.7578	−0.464233
$538$	0	0
$539$	−7.30976	−0.314854
$540$	0	0
$541$	0.442032	0.0190044	0.00950222	−	0.999955i	$-0.496975\pi$
$541$	0.442032	0.0190044	0.00950222	+	0.999955i	$0.496975\pi$
$542$	0	0
$543$	−11.7824	−0.505633
$544$	0	0
$545$	3.35100	0.143541
$546$	0	0
$547$	30.6537	1.31066	0.655328	−	0.755344i	$-0.272530\pi$
$547$	30.6537	1.31066	0.655328	+	0.755344i	$0.272530\pi$
$548$	0	0
$549$	25.6307	1.09389
$550$	0	0
$551$	−2.03798	−0.0868211
$552$	0	0
$553$	−17.8539	−0.759225
$554$	0	0
$555$	8.20856	0.348434
$556$	0	0
$557$	25.3602	1.07455	0.537273	−	0.843408i	$-0.319455\pi$
$557$	25.3602	1.07455	0.537273	+	0.843408i	$0.319455\pi$
$558$	0	0
$559$	22.5563	0.954032
$560$	0	0
$561$	3.07969	0.130025
$562$	0	0
$563$	14.2446	0.600336	0.300168	−	0.953886i	$-0.402957\pi$
$563$	14.2446	0.600336	0.300168	+	0.953886i	$0.402957\pi$
$564$	0	0
$565$	13.8834	0.584079
$566$	0	0
$567$	−12.0644	−0.506656
$568$	0	0
$569$	41.7388	1.74978	0.874892	−	0.484319i	$-0.160932\pi$
$569$	41.7388	1.74978	0.874892	+	0.484319i	$0.160932\pi$
$570$	0	0
$571$	−1.38133	−0.0578070	−0.0289035	−	0.999582i	$-0.509202\pi$
$571$	−1.38133	−0.0578070	−0.0289035	+	0.999582i	$0.509202\pi$
$572$	0	0
$573$	−16.3295	−0.682173
$574$	0	0
$575$	8.41100	0.350763
$576$	0	0
$577$	16.4947	0.686682	0.343341	−	0.939211i	$-0.388441\pi$
$577$	16.4947	0.686682	0.343341	+	0.939211i	$0.388441\pi$
$578$	0	0
$579$	0.560662	0.0233003
$580$	0	0
$581$	30.6866	1.27310
$582$	0	0
$583$	−7.02988	−0.291148
$584$	0	0
$585$	5.78395	0.239137
$586$	0	0
$587$	36.9119	1.52352	0.761759	−	0.647860i	$-0.224336\pi$
$587$	36.9119	1.52352	0.761759	+	0.647860i	$0.224336\pi$
$588$	0	0
$589$	30.5863	1.26029
$590$	0	0
$591$	3.54657	0.145887
$592$	0	0
$593$	0.0763263	0.00313435	0.00156717	−	0.999999i	$-0.499501\pi$
$593$	0.0763263	0.00313435	0.00156717	+	0.999999i	$0.499501\pi$
$594$	0	0
$595$	−15.2166	−0.623819
$596$	0	0
$597$	3.53840	0.144817
$598$	0	0
$599$	−4.44272	−0.181525	−0.0907623	−	0.995873i	$-0.528930\pi$
$599$	−4.44272	−0.181525	−0.0907623	+	0.995873i	$0.528930\pi$
$600$	0	0
$601$	19.2754	0.786259	0.393130	−	0.919483i	$-0.371392\pi$
$601$	19.2754	0.786259	0.393130	+	0.919483i	$0.371392\pi$
$602$	0	0
$603$	10.6471	0.433582
$604$	0	0
$605$	1.09288	0.0444320
$606$	0	0
$607$	9.31210	0.377966	0.188983	−	0.981980i	$-0.439481\pi$
$607$	9.31210	0.377966	0.188983	+	0.981980i	$0.439481\pi$
$608$	0	0
$609$	−1.63100	−0.0660913
$610$	0	0
$611$	−17.5034	−0.708111
$612$	0	0
$613$	21.7104	0.876876	0.438438	−	0.898761i	$-0.355532\pi$
$613$	21.7104	0.876876	0.438438	+	0.898761i	$0.355532\pi$
$614$	0	0
$615$	8.08735	0.326113
$616$	0	0
$617$	26.8963	1.08280	0.541401	−	0.840764i	$-0.317894\pi$
$617$	26.8963	1.08280	0.541401	+	0.840764i	$0.317894\pi$
$618$	0	0
$619$	−3.92993	−0.157957	−0.0789787	−	0.996876i	$-0.525166\pi$
$619$	−3.92993	−0.157957	−0.0789787	+	0.996876i	$0.525166\pi$
$620$	0	0
$621$	−9.80102	−0.393301
$622$	0	0
$623$	16.6550	0.667267
$624$	0	0
$625$	8.51067	0.340427
$626$	0	0
$627$	3.30920	0.132157
$628$	0	0
$629$	−33.0400	−1.31739
$630$	0	0
$631$	1.32286	0.0526623	0.0263311	−	0.999653i	$-0.491618\pi$
$631$	1.32286	0.0526623	0.0263311	+	0.999653i	$0.491618\pi$
$632$	0	0
$633$	−17.2358	−0.685062
$634$	0	0
$635$	10.3284	0.409871
$636$	0	0
$637$	−16.8207	−0.666461
$638$	0	0
$639$	27.8951	1.10351
$640$	0	0
$641$	−14.6749	−0.579624	−0.289812	−	0.957084i	$-0.593593\pi$
$641$	−14.6749	−0.579624	−0.289812	+	0.957084i	$0.593593\pi$
$642$	0	0
$643$	28.0123	1.10470	0.552349	−	0.833613i	$-0.313732\pi$
$643$	28.0123	1.10470	0.552349	+	0.833613i	$0.313732\pi$
$644$	0	0
$645$	8.96356	0.352940
$646$	0	0
$647$	−26.9141	−1.05810	−0.529050	−	0.848590i	$-0.677452\pi$
$647$	−26.9141	−1.05810	−0.529050	+	0.848590i	$0.677452\pi$
$648$	0	0
$649$	11.6890	0.458832
$650$	0	0
$651$	24.4782	0.959377
$652$	0	0
$653$	−38.0491	−1.48898	−0.744488	−	0.667636i	$-0.767306\pi$
$653$	−38.0491	−1.48898	−0.744488	+	0.667636i	$0.767306\pi$
$654$	0	0
$655$	6.37570	0.249119
$656$	0	0
$657$	−29.7721	−1.16152
$658$	0	0
$659$	−44.0232	−1.71490	−0.857449	−	0.514568i	$-0.827952\pi$
$659$	−44.0232	−1.71490	−0.857449	+	0.514568i	$0.827952\pi$
$660$	0	0
$661$	−33.0172	−1.28422	−0.642110	−	0.766613i	$-0.721941\pi$
$661$	−33.0172	−1.28422	−0.642110	+	0.766613i	$0.721941\pi$
$662$	0	0
$663$	7.08677	0.275227
$664$	0	0
$665$	−16.3506	−0.634049
$666$	0	0
$667$	1.13889	0.0440979
$668$	0	0
$669$	−3.02737	−0.117045
$670$	0	0
$671$	11.1442	0.430219
$672$	0	0
$673$	−1.48633	−0.0572939	−0.0286470	−	0.999590i	$-0.509120\pi$
$673$	−1.48633	−0.0572939	−0.0286470	+	0.999590i	$0.509120\pi$
$674$	0	0
$675$	−16.8761	−0.649560
$676$	0	0
$677$	22.6318	0.869812	0.434906	−	0.900476i	$-0.356782\pi$
$677$	22.6318	0.869812	0.434906	+	0.900476i	$0.356782\pi$
$678$	0	0
$679$	65.7233	2.52223
$680$	0	0
$681$	−17.5392	−0.672102
$682$	0	0
$683$	−40.6060	−1.55375	−0.776873	−	0.629657i	$-0.783196\pi$
$683$	−40.6060	−1.55375	−0.776873	+	0.629657i	$0.783196\pi$
$684$	0	0
$685$	−1.09288	−0.0417569
$686$	0	0
$687$	5.58931	0.213246
$688$	0	0
$689$	−16.1767	−0.616282
$690$	0	0
$691$	−4.76287	−0.181188	−0.0905941	−	0.995888i	$-0.528877\pi$
$691$	−4.76287	−0.181188	−0.0905941	+	0.995888i	$0.528877\pi$
$692$	0	0
$693$	−8.70012	−0.330490
$694$	0	0
$695$	21.7997	0.826909
$696$	0	0
$697$	−32.5521	−1.23300
$698$	0	0
$699$	6.48803	0.245400
$700$	0	0
$701$	−2.79865	−0.105704	−0.0528519	−	0.998602i	$-0.516831\pi$
$701$	−2.79865	−0.105704	−0.0528519	+	0.998602i	$0.516831\pi$
$702$	0	0
$703$	−35.5023	−1.33900
$704$	0	0
$705$	−6.95559	−0.261963
$706$	0	0
$707$	−35.4570	−1.33350
$708$	0	0
$709$	14.4932	0.544303	0.272151	−	0.962254i	$-0.412265\pi$
$709$	14.4932	0.544303	0.272151	+	0.962254i	$0.412265\pi$
$710$	0	0
$711$	−10.8549	−0.407091
$712$	0	0
$713$	−17.0926	−0.640122
$714$	0	0
$715$	2.51487	0.0940508
$716$	0	0
$717$	−11.1270	−0.415544
$718$	0	0
$719$	−14.8570	−0.554071	−0.277035	−	0.960860i	$-0.589352\pi$
$719$	−14.8570	−0.554071	−0.277035	+	0.960860i	$0.589352\pi$
$720$	0	0
$721$	−66.4333	−2.47410
$722$	0	0
$723$	14.6657	0.545422
$724$	0	0
$725$	1.96102	0.0728303
$726$	0	0
$727$	28.3745	1.05235	0.526177	−	0.850375i	$-0.323625\pi$
$727$	28.3745	1.05235	0.526177	+	0.850375i	$0.323625\pi$
$728$	0	0
$729$	3.79641	0.140608
$730$	0	0
$731$	−36.0790	−1.33443
$732$	0	0
$733$	−11.2634	−0.416025	−0.208012	−	0.978126i	$-0.566699\pi$
$733$	−11.2634	−0.416025	−0.208012	+	0.978126i	$0.566699\pi$
$734$	0	0
$735$	−6.68431	−0.246554
$736$	0	0
$737$	4.62936	0.170524
$738$	0	0
$739$	−43.6446	−1.60549	−0.802747	−	0.596320i	$-0.796629\pi$
$739$	−43.6446	−1.60549	−0.802747	+	0.596320i	$0.796629\pi$
$740$	0	0
$741$	7.61491	0.279741
$742$	0	0
$743$	−12.8358	−0.470899	−0.235449	−	0.971887i	$-0.575656\pi$
$743$	−12.8358	−0.470899	−0.235449	+	0.971887i	$0.575656\pi$
$744$	0	0
$745$	10.2884	0.376936
$746$	0	0
$747$	18.6570	0.682625
$748$	0	0
$749$	−52.6578	−1.92408
$750$	0	0
$751$	10.3360	0.377165	0.188583	−	0.982057i	$-0.439611\pi$
$751$	10.3360	0.377165	0.188583	+	0.982057i	$0.439611\pi$
$752$	0	0
$753$	17.8035	0.648794
$754$	0	0
$755$	20.0947	0.731320
$756$	0	0
$757$	27.9283	1.01507	0.507535	−	0.861631i	$-0.330557\pi$
$757$	27.9283	1.01507	0.507535	+	0.861631i	$0.330557\pi$
$758$	0	0
$759$	−1.84928	−0.0671247
$760$	0	0
$761$	21.3723	0.774744	0.387372	−	0.921923i	$-0.373383\pi$
$761$	21.3723	0.774744	0.387372	+	0.921923i	$0.373383\pi$
$762$	0	0
$763$	−11.5989	−0.419908
$764$	0	0
$765$	−9.25145	−0.334487
$766$	0	0
$767$	26.8978	0.971224
$768$	0	0
$769$	−11.2073	−0.404146	−0.202073	−	0.979370i	$-0.564768\pi$
$769$	−11.2073	−0.404146	−0.202073	+	0.979370i	$0.564768\pi$
$770$	0	0
$771$	−22.2605	−0.801691
$772$	0	0
$773$	25.4297	0.914644	0.457322	−	0.889301i	$-0.348809\pi$
$773$	25.4297	0.914644	0.457322	+	0.889301i	$0.348809\pi$
$774$	0	0
$775$	−29.4312	−1.05720
$776$	0	0
$777$	−28.4125	−1.01929
$778$	0	0
$779$	−34.9781	−1.25322
$780$	0	0
$781$	12.1288	0.434004
$782$	0	0
$783$	−2.28510	−0.0816627
$784$	0	0
$785$	20.2833	0.723941
$786$	0	0
$787$	−25.3693	−0.904317	−0.452158	−	0.891938i	$-0.649346\pi$
$787$	−25.3693	−0.904317	−0.452158	+	0.891938i	$0.649346\pi$
$788$	0	0
$789$	12.3288	0.438915
$790$	0	0
$791$	−48.0550	−1.70864
$792$	0	0
$793$	25.6444	0.910659
$794$	0	0
$795$	−6.42837	−0.227991
$796$	0	0
$797$	−17.3907	−0.616012	−0.308006	−	0.951384i	$-0.599662\pi$
$797$	−17.3907	−0.616012	−0.308006	+	0.951384i	$0.599662\pi$
$798$	0	0
$799$	27.9967	0.990453
$800$	0	0
$801$	10.1260	0.357784
$802$	0	0
$803$	−12.9449	−0.456817
$804$	0	0
$805$	9.13721	0.322044
$806$	0	0
$807$	5.83306	0.205333
$808$	0	0
$809$	41.5871	1.46213	0.731063	−	0.682310i	$-0.239025\pi$
$809$	41.5871	1.46213	0.731063	+	0.682310i	$0.239025\pi$
$810$	0	0
$811$	27.4300	0.963198	0.481599	−	0.876392i	$-0.340056\pi$
$811$	27.4300	0.963198	0.481599	+	0.876392i	$0.340056\pi$
$812$	0	0
$813$	23.2231	0.814469
$814$	0	0
$815$	23.5258	0.824073
$816$	0	0
$817$	−38.7678	−1.35631
$818$	0	0
$819$	−20.0201	−0.699560
$820$	0	0
$821$	−34.1171	−1.19069	−0.595347	−	0.803469i	$-0.702985\pi$
$821$	−34.1171	−1.19069	−0.595347	+	0.803469i	$0.702985\pi$
$822$	0	0
$823$	−47.3129	−1.64923	−0.824613	−	0.565698i	$-0.808607\pi$
$823$	−47.3129	−1.64923	−0.824613	+	0.565698i	$0.808607\pi$
$824$	0	0
$825$	−3.18422	−0.110860
$826$	0	0
$827$	15.0641	0.523830	0.261915	−	0.965091i	$-0.415646\pi$
$827$	15.0641	0.523830	0.261915	+	0.965091i	$0.415646\pi$
$828$	0	0
$829$	23.6532	0.821508	0.410754	−	0.911746i	$-0.365265\pi$
$829$	23.6532	0.821508	0.410754	+	0.911746i	$0.365265\pi$
$830$	0	0
$831$	−3.07483	−0.106665
$832$	0	0
$833$	26.9048	0.932196
$834$	0	0
$835$	−1.54930	−0.0536157
$836$	0	0
$837$	34.2951	1.18541
$838$	0	0
$839$	51.7352	1.78610	0.893048	−	0.449961i	$-0.148562\pi$
$839$	51.7352	1.78610	0.893048	+	0.449961i	$0.148562\pi$
$840$	0	0
$841$	−28.7345	−0.990844
$842$	0	0
$843$	−7.11849	−0.245174
$844$	0	0
$845$	−8.42044	−0.289672
$846$	0	0
$847$	−3.78282	−0.129979
$848$	0	0
$849$	−7.32503	−0.251394
$850$	0	0
$851$	19.8398	0.680099
$852$	0	0
$853$	−28.9216	−0.990258	−0.495129	−	0.868819i	$-0.664879\pi$
$853$	−28.9216	−0.990258	−0.495129	+	0.868819i	$0.664879\pi$
$854$	0	0
$855$	−9.94092	−0.339972
$856$	0	0
$857$	28.4100	0.970467	0.485234	−	0.874385i	$-0.338735\pi$
$857$	28.4100	0.970467	0.485234	+	0.874385i	$0.338735\pi$
$858$	0	0
$859$	−27.2300	−0.929075	−0.464538	−	0.885553i	$-0.653779\pi$
$859$	−27.2300	−0.929075	−0.464538	+	0.885553i	$0.653779\pi$
$860$	0	0
$861$	−27.9929	−0.953997
$862$	0	0
$863$	−7.89034	−0.268590	−0.134295	−	0.990941i	$-0.542877\pi$
$863$	−7.89034	−0.268590	−0.134295	+	0.990941i	$0.542877\pi$
$864$	0	0
$865$	−5.01647	−0.170565
$866$	0	0
$867$	2.88891	0.0981126
$868$	0	0
$869$	−4.71973	−0.160106
$870$	0	0
$871$	10.6528	0.360955
$872$	0	0
$873$	39.9588	1.35240
$874$	0	0
$875$	36.4040	1.23068
$876$	0	0
$877$	−28.7947	−0.972327	−0.486163	−	0.873868i	$-0.661604\pi$
$877$	−28.7947	−0.972327	−0.486163	+	0.873868i	$0.661604\pi$
$878$	0	0
$879$	23.9665	0.808371
$880$	0	0
$881$	−20.3752	−0.686458	−0.343229	−	0.939252i	$-0.611521\pi$
$881$	−20.3752	−0.686458	−0.343229	+	0.939252i	$0.611521\pi$
$882$	0	0
$883$	9.99142	0.336238	0.168119	−	0.985767i	$-0.446231\pi$
$883$	9.99142	0.336238	0.168119	+	0.985767i	$0.446231\pi$
$884$	0	0
$885$	10.6888	0.359300
$886$	0	0
$887$	12.9092	0.433448	0.216724	−	0.976233i	$-0.430463\pi$
$887$	12.9092	0.433448	0.216724	+	0.976233i	$0.430463\pi$
$888$	0	0
$889$	−35.7501	−1.19902
$890$	0	0
$891$	−3.18925	−0.106844
$892$	0	0
$893$	30.0832	1.00670
$894$	0	0
$895$	14.0513	0.469684
$896$	0	0
$897$	−4.25544	−0.142085
$898$	0	0
$899$	−3.98512	−0.132911
$900$	0	0
$901$	25.8747	0.862010
$902$	0	0
$903$	−31.0258	−1.03247
$904$	0	0
$905$	15.3897	0.511570
$906$	0	0
$907$	−40.7192	−1.35206	−0.676030	−	0.736874i	$-0.736301\pi$
$907$	−40.7192	−1.35206	−0.676030	+	0.736874i	$0.736301\pi$
$908$	0	0
$909$	−21.5573	−0.715011
$910$	0	0
$911$	32.7682	1.08566	0.542830	−	0.839843i	$-0.317353\pi$
$911$	32.7682	1.08566	0.542830	+	0.839843i	$0.317353\pi$
$912$	0	0
$913$	8.11210	0.268471
$914$	0	0
$915$	10.1907	0.336894
$916$	0	0
$917$	−22.0684	−0.728762
$918$	0	0
$919$	54.2325	1.78897	0.894483	−	0.447102i	$-0.147544\pi$
$919$	54.2325	1.78897	0.894483	+	0.447102i	$0.147544\pi$
$920$	0	0
$921$	4.14672	0.136639
$922$	0	0
$923$	27.9101	0.918671
$924$	0	0
$925$	34.1615	1.12322
$926$	0	0
$927$	−40.3904	−1.32660
$928$	0	0
$929$	25.3498	0.831699	0.415849	−	0.909433i	$-0.363484\pi$
$929$	25.3498	0.831699	0.415849	+	0.909433i	$0.363484\pi$
$930$	0	0
$931$	28.9099	0.947484
$932$	0	0
$933$	3.66344	0.119936
$934$	0	0
$935$	−4.02254	−0.131551
$936$	0	0
$937$	−39.8450	−1.30168	−0.650840	−	0.759215i	$-0.725583\pi$
$937$	−39.8450	−1.30168	−0.650840	+	0.759215i	$0.725583\pi$
$938$	0	0
$939$	−8.16585	−0.266482
$940$	0	0
$941$	−47.8440	−1.55967	−0.779835	−	0.625985i	$-0.784697\pi$
$941$	−47.8440	−1.55967	−0.779835	+	0.625985i	$0.784697\pi$
$942$	0	0
$943$	19.5468	0.636532
$944$	0	0
$945$	−18.3332	−0.596378
$946$	0	0
$947$	22.0874	0.717745	0.358873	−	0.933387i	$-0.383161\pi$
$947$	22.0874	0.717745	0.358873	+	0.933387i	$0.383161\pi$
$948$	0	0
$949$	−29.7880	−0.966960
$950$	0	0
$951$	13.3248	0.432085
$952$	0	0
$953$	53.9308	1.74699	0.873495	−	0.486833i	$-0.161848\pi$
$953$	53.9308	1.74699	0.873495	+	0.486833i	$0.161848\pi$
$954$	0	0
$955$	21.3288	0.690183
$956$	0	0
$957$	−0.431159	−0.0139374
$958$	0	0
$959$	3.78282	0.122154
$960$	0	0
$961$	28.8092	0.929329
$962$	0	0
$963$	−32.0152	−1.03167
$964$	0	0
$965$	−0.732310	−0.0235739
$966$	0	0
$967$	−45.9124	−1.47644	−0.738222	−	0.674558i	$-0.764334\pi$
$967$	−45.9124	−1.47644	−0.738222	+	0.674558i	$0.764334\pi$
$968$	0	0
$969$	−12.1801	−0.391281
$970$	0	0
$971$	−6.07189	−0.194856	−0.0974280	−	0.995243i	$-0.531062\pi$
$971$	−6.07189	−0.194856	−0.0974280	+	0.995243i	$0.531062\pi$
$972$	0	0
$973$	−75.4558	−2.41900
$974$	0	0
$975$	−7.32732	−0.234662
$976$	0	0
$977$	28.5080	0.912052	0.456026	−	0.889966i	$-0.349272\pi$
$977$	28.5080	0.912052	0.456026	+	0.889966i	$0.349272\pi$
$978$	0	0
$979$	4.40279	0.140714
$980$	0	0
$981$	−7.05196	−0.225152
$982$	0	0
$983$	4.88136	0.155691	0.0778457	−	0.996965i	$-0.475196\pi$
$983$	4.88136	0.155691	0.0778457	+	0.996965i	$0.475196\pi$
$984$	0	0
$985$	−4.63237	−0.147599
$986$	0	0
$987$	24.0755	0.766333
$988$	0	0
$989$	21.6646	0.688895
$990$	0	0
$991$	20.0655	0.637402	0.318701	−	0.947855i	$-0.396753\pi$
$991$	20.0655	0.637402	0.318701	+	0.947855i	$0.396753\pi$
$992$	0	0
$993$	−9.27130	−0.294216
$994$	0	0
$995$	−4.62169	−0.146518
$996$	0	0
$997$	34.4488	1.09100	0.545501	−	0.838110i	$-0.316339\pi$
$997$	34.4488	1.09100	0.545501	+	0.838110i	$0.316339\pi$
$998$	0	0
$999$	−39.8071	−1.25944

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.e.1.10	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.e.1.10	✓	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)

\(f(q)\)	\(=\)	\(q-0.836719 q^{3} +1.09288 q^{5} -3.78282 q^{7} -2.29990 q^{9} +O(q^{10})\)	\(q-0.836719 q^{3} +1.09288 q^{5} -3.78282 q^{7} -2.29990 q^{9} -1.00000 q^{11} -2.30113 q^{13} -0.914437 q^{15} +3.68067 q^{17} +3.95497 q^{19} +3.16516 q^{21} -2.21016 q^{23} -3.80561 q^{25} +4.43453 q^{27} -0.515297 q^{29} +7.73364 q^{31} +0.836719 q^{33} -4.13419 q^{35} -8.97663 q^{37} +1.92540 q^{39} -8.84408 q^{41} -9.80228 q^{43} -2.51352 q^{45} +7.60642 q^{47} +7.30976 q^{49} -3.07969 q^{51} +7.02988 q^{53} -1.09288 q^{55} -3.30920 q^{57} -11.6890 q^{59} -11.1442 q^{61} +8.70012 q^{63} -2.51487 q^{65} -4.62936 q^{67} +1.84928 q^{69} -12.1288 q^{71} +12.9449 q^{73} +3.18422 q^{75} +3.78282 q^{77} +4.71973 q^{79} +3.18925 q^{81} -8.11210 q^{83} +4.02254 q^{85} +0.431159 q^{87} -4.40279 q^{89} +8.70478 q^{91} -6.47089 q^{93} +4.32233 q^{95} -17.3741 q^{97} +2.29990 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) \) 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

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Database

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