LMFDB - Embedded newform 8001.2.a.y.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.02489 q^{2} +2.10018 q^{4} +2.65037 q^{5} -1.00000 q^{7} -0.202857 q^{8} +O(q^{10})$	\(q-2.02489 q^{2} +2.10018 q^{4} +2.65037 q^{5} -1.00000 q^{7} -0.202857 q^{8} -5.36671 q^{10} +3.06138 q^{11} +0.621555 q^{13} +2.02489 q^{14} -3.78960 q^{16} +1.54932 q^{17} +7.42020 q^{19} +5.56626 q^{20} -6.19896 q^{22} +3.41609 q^{23} +2.02447 q^{25} -1.25858 q^{26} -2.10018 q^{28} -5.77234 q^{29} -8.60915 q^{31} +8.07924 q^{32} -3.13720 q^{34} -2.65037 q^{35} +8.63894 q^{37} -15.0251 q^{38} -0.537647 q^{40} +0.418206 q^{41} +4.93371 q^{43} +6.42946 q^{44} -6.91721 q^{46} +3.63973 q^{47} +1.00000 q^{49} -4.09932 q^{50} +1.30538 q^{52} +2.34131 q^{53} +8.11380 q^{55} +0.202857 q^{56} +11.6884 q^{58} +2.80139 q^{59} +8.10802 q^{61} +17.4326 q^{62} -8.78038 q^{64} +1.64735 q^{65} -1.88861 q^{67} +3.25385 q^{68} +5.36671 q^{70} -5.59109 q^{71} +1.70644 q^{73} -17.4929 q^{74} +15.5838 q^{76} -3.06138 q^{77} +1.89406 q^{79} -10.0438 q^{80} -0.846822 q^{82} -10.5424 q^{83} +4.10627 q^{85} -9.99023 q^{86} -0.621023 q^{88} +4.11133 q^{89} -0.621555 q^{91} +7.17441 q^{92} -7.37005 q^{94} +19.6663 q^{95} +12.6502 q^{97} -2.02489 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) $ 28 * q + 30 * q^4 - 28 * q^7	$ 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) $ 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * q^94 - 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.02489	−1.43181	−0.715907	−	0.698196i	$-0.753987\pi$
$2$	−2.02489	−1.43181	−0.715907	+	0.698196i	$0.753987\pi$
$3$	0	0
$3$	0	0
$4$	2.10018	1.05009
$5$	2.65037	1.18528	0.592641	−	0.805467i	$-0.298085\pi$
$5$	2.65037	1.18528	0.592641	+	0.805467i	$0.298085\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−0.202857	−0.0717208
$9$	0	0
$10$	−5.36671	−1.69710
$11$	3.06138	0.923041	0.461521	−	0.887129i	$-0.347304\pi$
$11$	3.06138	0.923041	0.461521	+	0.887129i	$0.347304\pi$
$12$	0	0
$13$	0.621555	0.172388	0.0861942	−	0.996278i	$-0.472529\pi$
$13$	0.621555	0.172388	0.0861942	+	0.996278i	$0.472529\pi$
$14$	2.02489	0.541175
$15$	0	0
$16$	−3.78960	−0.947400
$17$	1.54932	0.375765	0.187883	−	0.982192i	$-0.439838\pi$
$17$	1.54932	0.375765	0.187883	+	0.982192i	$0.439838\pi$
$18$	0	0
$19$	7.42020	1.70231	0.851155	−	0.524914i	$-0.175903\pi$
$19$	7.42020	1.70231	0.851155	+	0.524914i	$0.175903\pi$
$20$	5.56626	1.24465
$21$	0	0
$22$	−6.19896	−1.32162
$23$	3.41609	0.712304	0.356152	−	0.934428i	$-0.384089\pi$
$23$	3.41609	0.712304	0.356152	+	0.934428i	$0.384089\pi$
$24$	0	0
$25$	2.02447	0.404893
$26$	−1.25858	−0.246828
$27$	0	0
$28$	−2.10018	−0.396897
$29$	−5.77234	−1.07190	−0.535948	−	0.844251i	$-0.680046\pi$
$29$	−5.77234	−1.07190	−0.535948	+	0.844251i	$0.680046\pi$
$30$	0	0
$31$	−8.60915	−1.54625	−0.773124	−	0.634255i	$-0.781307\pi$
$31$	−8.60915	−1.54625	−0.773124	+	0.634255i	$0.781307\pi$
$32$	8.07924	1.42822
$33$	0	0
$34$	−3.13720	−0.538026
$35$	−2.65037	−0.447994
$36$	0	0
$37$	8.63894	1.42023	0.710117	−	0.704084i	$-0.248642\pi$
$37$	8.63894	1.42023	0.710117	+	0.704084i	$0.248642\pi$
$38$	−15.0251	−2.43739
$39$	0	0
$40$	−0.537647	−0.0850094
$41$	0.418206	0.0653129	0.0326564	−	0.999467i	$-0.489603\pi$
$41$	0.418206	0.0653129	0.0326564	+	0.999467i	$0.489603\pi$
$42$	0	0
$43$	4.93371	0.752384	0.376192	−	0.926542i	$-0.377233\pi$
$43$	4.93371	0.752384	0.376192	+	0.926542i	$0.377233\pi$
$44$	6.42946	0.969277
$45$	0	0
$46$	−6.91721	−1.01989
$47$	3.63973	0.530909	0.265454	−	0.964123i	$-0.414478\pi$
$47$	3.63973	0.530909	0.265454	+	0.964123i	$0.414478\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−4.09932	−0.579732
$51$	0	0
$52$	1.30538	0.181023
$53$	2.34131	0.321604	0.160802	−	0.986987i	$-0.448592\pi$
$53$	2.34131	0.321604	0.160802	+	0.986987i	$0.448592\pi$
$54$	0	0
$55$	8.11380	1.09406
$56$	0.202857	0.0271079
$57$	0	0
$58$	11.6884	1.53476
$59$	2.80139	0.364710	0.182355	−	0.983233i	$-0.441628\pi$
$59$	2.80139	0.364710	0.182355	+	0.983233i	$0.441628\pi$
$60$	0	0
$61$	8.10802	1.03812	0.519062	−	0.854736i	$-0.326281\pi$
$61$	8.10802	1.03812	0.519062	+	0.854736i	$0.326281\pi$
$62$	17.4326	2.21394
$63$	0	0
$64$	−8.78038	−1.09755
$65$	1.64735	0.204329
$66$	0	0
$67$	−1.88861	−0.230730	−0.115365	−	0.993323i	$-0.536804\pi$
$67$	−1.88861	−0.230730	−0.115365	+	0.993323i	$0.536804\pi$
$68$	3.25385	0.394587
$69$	0	0
$70$	5.36671	0.641445
$71$	−5.59109	−0.663541	−0.331770	−	0.943360i	$-0.607646\pi$
$71$	−5.59109	−0.663541	−0.331770	+	0.943360i	$0.607646\pi$
$72$	0	0
$73$	1.70644	0.199724	0.0998620	−	0.995001i	$-0.468160\pi$
$73$	1.70644	0.199724	0.0998620	+	0.995001i	$0.468160\pi$
$74$	−17.4929	−2.03351
$75$	0	0
$76$	15.5838	1.78758
$77$	−3.06138	−0.348877
$78$	0	0
$79$	1.89406	0.213099	0.106549	−	0.994307i	$-0.466020\pi$
$79$	1.89406	0.213099	0.106549	+	0.994307i	$0.466020\pi$
$80$	−10.0438	−1.12294
$81$	0	0
$82$	−0.846822	−0.0935159
$83$	−10.5424	−1.15718	−0.578592	−	0.815617i	$-0.696398\pi$
$83$	−10.5424	−1.15718	−0.578592	+	0.815617i	$0.696398\pi$
$84$	0	0
$85$	4.10627	0.445388
$86$	−9.99023	−1.07727
$87$	0	0
$88$	−0.621023	−0.0662013
$89$	4.11133	0.435800	0.217900	−	0.975971i	$-0.430079\pi$
$89$	4.11133	0.435800	0.217900	+	0.975971i	$0.430079\pi$
$90$	0	0
$91$	−0.621555	−0.0651567
$92$	7.17441	0.747984
$93$	0	0
$94$	−7.37005	−0.760163
$95$	19.6663	2.01772
$96$	0	0
$97$	12.6502	1.28444	0.642218	−	0.766522i	$-0.278014\pi$
$97$	12.6502	1.28444	0.642218	+	0.766522i	$0.278014\pi$
$98$	−2.02489	−0.204545
$99$	0	0
$100$	4.25175	0.425175
$101$	−7.10620	−0.707093	−0.353547	−	0.935417i	$-0.615024\pi$
$101$	−7.10620	−0.707093	−0.353547	+	0.935417i	$0.615024\pi$
$102$	0	0
$103$	4.36048	0.429651	0.214826	−	0.976652i	$-0.431082\pi$
$103$	4.36048	0.429651	0.214826	+	0.976652i	$0.431082\pi$
$104$	−0.126087	−0.0123638
$105$	0	0
$106$	−4.74089	−0.460476
$107$	−9.95118	−0.962016	−0.481008	−	0.876716i	$-0.659729\pi$
$107$	−9.95118	−0.962016	−0.481008	+	0.876716i	$0.659729\pi$
$108$	0	0
$109$	−0.756070	−0.0724183	−0.0362092	−	0.999344i	$-0.511528\pi$
$109$	−0.756070	−0.0724183	−0.0362092	+	0.999344i	$0.511528\pi$
$110$	−16.4296	−1.56650
$111$	0	0
$112$	3.78960	0.358084
$113$	3.60452	0.339084	0.169542	−	0.985523i	$-0.445771\pi$
$113$	3.60452	0.339084	0.169542	+	0.985523i	$0.445771\pi$
$114$	0	0
$115$	9.05390	0.844281
$116$	−12.1230	−1.12559
$117$	0	0
$118$	−5.67251	−0.522197
$119$	−1.54932	−0.142026
$120$	0	0
$121$	−1.62794	−0.147994
$122$	−16.4178	−1.48640
$123$	0	0
$124$	−18.0808	−1.62370
$125$	−7.88627	−0.705369
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	1.62082	0.143262
$129$	0	0
$130$	−3.33571	−0.292561
$131$	−14.0914	−1.23117	−0.615587	−	0.788069i	$-0.711081\pi$
$131$	−14.0914	−1.23117	−0.615587	+	0.788069i	$0.711081\pi$
$132$	0	0
$133$	−7.42020	−0.643413
$134$	3.82423	0.330363
$135$	0	0
$136$	−0.314291	−0.0269502
$137$	11.8267	1.01043	0.505213	−	0.862995i	$-0.331414\pi$
$137$	11.8267	1.01043	0.505213	+	0.862995i	$0.331414\pi$
$138$	0	0
$139$	21.6741	1.83837	0.919185	−	0.393825i	$-0.128849\pi$
$139$	21.6741	1.83837	0.919185	+	0.393825i	$0.128849\pi$
$140$	−5.56626	−0.470435
$141$	0	0
$142$	11.3213	0.950067
$143$	1.90282	0.159122
$144$	0	0
$145$	−15.2988	−1.27050
$146$	−3.45536	−0.285968
$147$	0	0
$148$	18.1434	1.49137
$149$	−3.57908	−0.293209	−0.146605	−	0.989195i	$-0.546835\pi$
$149$	−3.57908	−0.293209	−0.146605	+	0.989195i	$0.546835\pi$
$150$	0	0
$151$	21.7670	1.77138	0.885688	−	0.464281i	$-0.153687\pi$
$151$	21.7670	1.77138	0.885688	+	0.464281i	$0.153687\pi$
$152$	−1.50524	−0.122091
$153$	0	0
$154$	6.19896	0.499527
$155$	−22.8174	−1.83274
$156$	0	0
$157$	−4.43123	−0.353650	−0.176825	−	0.984242i	$-0.556583\pi$
$157$	−4.43123	−0.353650	−0.176825	+	0.984242i	$0.556583\pi$
$158$	−3.83527	−0.305118
$159$	0	0
$160$	21.4130	1.69284
$161$	−3.41609	−0.269226
$162$	0	0
$163$	20.0404	1.56969	0.784843	−	0.619694i	$-0.212743\pi$
$163$	20.0404	1.56969	0.784843	+	0.619694i	$0.212743\pi$
$164$	0.878310	0.0685845
$165$	0	0
$166$	21.3473	1.65687
$167$	9.72069	0.752210	0.376105	−	0.926577i	$-0.377263\pi$
$167$	9.72069	0.752210	0.376105	+	0.926577i	$0.377263\pi$
$168$	0	0
$169$	−12.6137	−0.970282
$170$	−8.31475	−0.637712
$171$	0	0
$172$	10.3617	0.790072
$173$	−6.82755	−0.519089	−0.259545	−	0.965731i	$-0.583572\pi$
$173$	−6.82755	−0.519089	−0.259545	+	0.965731i	$0.583572\pi$
$174$	0	0
$175$	−2.02447	−0.153035
$176$	−11.6014	−0.874489
$177$	0	0
$178$	−8.32499	−0.623984
$179$	16.0789	1.20179	0.600897	−	0.799327i	$-0.294810\pi$
$179$	16.0789	1.20179	0.600897	+	0.799327i	$0.294810\pi$
$180$	0	0
$181$	6.30601	0.468722	0.234361	−	0.972150i	$-0.424700\pi$
$181$	6.30601	0.468722	0.234361	+	0.972150i	$0.424700\pi$
$182$	1.25858	0.0932922
$183$	0	0
$184$	−0.692978	−0.0510870
$185$	22.8964	1.68338
$186$	0	0
$187$	4.74306	0.346847
$188$	7.64409	0.557502
$189$	0	0
$190$	−39.8221	−2.88900
$191$	−9.49179	−0.686802	−0.343401	−	0.939189i	$-0.611579\pi$
$191$	−9.49179	−0.686802	−0.343401	+	0.939189i	$0.611579\pi$
$192$	0	0
$193$	−12.1817	−0.876855	−0.438427	−	0.898767i	$-0.644464\pi$
$193$	−12.1817	−0.876855	−0.438427	+	0.898767i	$0.644464\pi$
$194$	−25.6153	−1.83907
$195$	0	0
$196$	2.10018	0.150013
$197$	3.39743	0.242057	0.121028	−	0.992649i	$-0.461381\pi$
$197$	3.39743	0.242057	0.121028	+	0.992649i	$0.461381\pi$
$198$	0	0
$199$	11.7385	0.832123	0.416062	−	0.909336i	$-0.363410\pi$
$199$	11.7385	0.832123	0.416062	+	0.909336i	$0.363410\pi$
$200$	−0.410678	−0.0290393
$201$	0	0
$202$	14.3893	1.01243
$203$	5.77234	0.405139
$204$	0	0
$205$	1.10840	0.0774142
$206$	−8.82950	−0.615180
$207$	0	0
$208$	−2.35545	−0.163321
$209$	22.7161	1.57130
$210$	0	0
$211$	−9.38616	−0.646170	−0.323085	−	0.946370i	$-0.604720\pi$
$211$	−9.38616	−0.646170	−0.323085	+	0.946370i	$0.604720\pi$
$212$	4.91717	0.337713
$213$	0	0
$214$	20.1500	1.37743
$215$	13.0762	0.891788
$216$	0	0
$217$	8.60915	0.584427
$218$	1.53096	0.103690
$219$	0	0
$220$	17.0405	1.14887
$221$	0.962987	0.0647775
$222$	0	0
$223$	−4.18718	−0.280395	−0.140197	−	0.990124i	$-0.544774\pi$
$223$	−4.18718	−0.280395	−0.140197	+	0.990124i	$0.544774\pi$
$224$	−8.07924	−0.539817
$225$	0	0
$226$	−7.29875	−0.485506
$227$	−7.79004	−0.517043	−0.258521	−	0.966006i	$-0.583235\pi$
$227$	−7.79004	−0.517043	−0.258521	+	0.966006i	$0.583235\pi$
$228$	0	0
$229$	0.564354	0.0372935	0.0186468	−	0.999826i	$-0.494064\pi$
$229$	0.564354	0.0372935	0.0186468	+	0.999826i	$0.494064\pi$
$230$	−18.3332	−1.20885
$231$	0	0
$232$	1.17096	0.0768773
$233$	28.2715	1.85213	0.926064	−	0.377366i	$-0.123170\pi$
$233$	28.2715	1.85213	0.926064	+	0.377366i	$0.123170\pi$
$234$	0	0
$235$	9.64663	0.629277
$236$	5.88343	0.382979
$237$	0	0
$238$	3.13720	0.203355
$239$	7.52800	0.486946	0.243473	−	0.969908i	$-0.421713\pi$
$239$	7.52800	0.486946	0.243473	+	0.969908i	$0.421713\pi$
$240$	0	0
$241$	19.5854	1.26161	0.630803	−	0.775943i	$-0.282726\pi$
$241$	19.5854	1.26161	0.630803	+	0.775943i	$0.282726\pi$
$242$	3.29640	0.211900
$243$	0	0
$244$	17.0283	1.09013
$245$	2.65037	0.169326
$246$	0	0
$247$	4.61206	0.293458
$248$	1.74643	0.110898
$249$	0	0
$250$	15.9688	1.00996
$251$	−12.8883	−0.813502	−0.406751	−	0.913539i	$-0.633338\pi$
$251$	−12.8883	−0.813502	−0.406751	+	0.913539i	$0.633338\pi$
$252$	0	0
$253$	10.4580	0.657486
$254$	−2.02489	−0.127053
$255$	0	0
$256$	14.2788	0.892423
$257$	−20.5522	−1.28201	−0.641005	−	0.767537i	$-0.721482\pi$
$257$	−20.5522	−1.28201	−0.641005	+	0.767537i	$0.721482\pi$
$258$	0	0
$259$	−8.63894	−0.536798
$260$	3.45974	0.214564
$261$	0	0
$262$	28.5336	1.76281
$263$	−19.6963	−1.21453	−0.607264	−	0.794500i	$-0.707733\pi$
$263$	−19.6963	−1.21453	−0.607264	+	0.794500i	$0.707733\pi$
$264$	0	0
$265$	6.20534	0.381191
$266$	15.0251	0.921247
$267$	0	0
$268$	−3.96642	−0.242288
$269$	25.4291	1.55044	0.775221	−	0.631690i	$-0.217639\pi$
$269$	25.4291	1.55044	0.775221	+	0.631690i	$0.217639\pi$
$270$	0	0
$271$	7.98850	0.485267	0.242633	−	0.970118i	$-0.421989\pi$
$271$	7.98850	0.485267	0.242633	+	0.970118i	$0.421989\pi$
$272$	−5.87130	−0.356000
$273$	0	0
$274$	−23.9478	−1.44674
$275$	6.19767	0.373733
$276$	0	0
$277$	−23.0567	−1.38534	−0.692670	−	0.721254i	$-0.743566\pi$
$277$	−23.0567	−1.38534	−0.692670	+	0.721254i	$0.743566\pi$
$278$	−43.8876	−2.63220
$279$	0	0
$280$	0.537647	0.0321305
$281$	−7.53946	−0.449766	−0.224883	−	0.974386i	$-0.572200\pi$
$281$	−7.53946	−0.449766	−0.224883	+	0.974386i	$0.572200\pi$
$282$	0	0
$283$	3.07735	0.182929	0.0914646	−	0.995808i	$-0.470845\pi$
$283$	3.07735	0.182929	0.0914646	+	0.995808i	$0.470845\pi$
$284$	−11.7423	−0.696778
$285$	0	0
$286$	−3.85300	−0.227833
$287$	−0.418206	−0.0246859
$288$	0	0
$289$	−14.5996	−0.858801
$290$	30.9785	1.81912
$291$	0	0
$292$	3.58384	0.209728
$293$	−30.9333	−1.80714	−0.903571	−	0.428439i	$-0.859064\pi$
$293$	−30.9333	−1.80714	−0.903571	+	0.428439i	$0.859064\pi$
$294$	0	0
$295$	7.42473	0.432284
$296$	−1.75247	−0.101860
$297$	0	0
$298$	7.24724	0.419821
$299$	2.12329	0.122793
$300$	0	0
$301$	−4.93371	−0.284375
$302$	−44.0759	−2.53628
$303$	0	0
$304$	−28.1196	−1.61277
$305$	21.4892	1.23047
$306$	0	0
$307$	−18.7615	−1.07077	−0.535387	−	0.844607i	$-0.679834\pi$
$307$	−18.7615	−1.07077	−0.535387	+	0.844607i	$0.679834\pi$
$308$	−6.42946	−0.366352
$309$	0	0
$310$	46.2028	2.62414
$311$	11.4690	0.650347	0.325173	−	0.945654i	$-0.394577\pi$
$311$	11.4690	0.650347	0.325173	+	0.945654i	$0.394577\pi$
$312$	0	0
$313$	−32.7099	−1.84887	−0.924435	−	0.381339i	$-0.875463\pi$
$313$	−32.7099	−1.84887	−0.924435	+	0.381339i	$0.875463\pi$
$314$	8.97275	0.506362
$315$	0	0
$316$	3.97788	0.223773
$317$	−28.2242	−1.58523	−0.792615	−	0.609722i	$-0.791281\pi$
$317$	−28.2242	−1.58523	−0.792615	+	0.609722i	$0.791281\pi$
$318$	0	0
$319$	−17.6713	−0.989405
$320$	−23.2713	−1.30090
$321$	0	0
$322$	6.91721	0.385481
$323$	11.4963	0.639669
$324$	0	0
$325$	1.25832	0.0697989
$326$	−40.5797	−2.24750
$327$	0	0
$328$	−0.0848362	−0.00468430
$329$	−3.63973	−0.200665
$330$	0	0
$331$	−24.1593	−1.32791	−0.663956	−	0.747771i	$-0.731124\pi$
$331$	−24.1593	−1.32791	−0.663956	+	0.747771i	$0.731124\pi$
$332$	−22.1411	−1.21515
$333$	0	0
$334$	−19.6833	−1.07702
$335$	−5.00552	−0.273481
$336$	0	0
$337$	−6.66154	−0.362877	−0.181439	−	0.983402i	$-0.558075\pi$
$337$	−6.66154	−0.362877	−0.181439	+	0.983402i	$0.558075\pi$
$338$	25.5413	1.38926
$339$	0	0
$340$	8.62391	0.467697
$341$	−26.3559	−1.42725
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−1.00084	−0.0539616
$345$	0	0
$346$	13.8250	0.743239
$347$	−6.60513	−0.354582	−0.177291	−	0.984158i	$-0.556733\pi$
$347$	−6.60513	−0.354582	−0.177291	+	0.984158i	$0.556733\pi$
$348$	0	0
$349$	18.8375	1.00835	0.504174	−	0.863602i	$-0.331797\pi$
$349$	18.8375	1.00835	0.504174	+	0.863602i	$0.331797\pi$
$350$	4.09932	0.219118
$351$	0	0
$352$	24.7336	1.31831
$353$	−29.8705	−1.58984	−0.794922	−	0.606712i	$-0.792488\pi$
$353$	−29.8705	−1.58984	−0.794922	+	0.606712i	$0.792488\pi$
$354$	0	0
$355$	−14.8185	−0.786483
$356$	8.63454	0.457630
$357$	0	0
$358$	−32.5580	−1.72074
$359$	27.9511	1.47520	0.737601	−	0.675237i	$-0.235959\pi$
$359$	27.9511	1.47520	0.737601	+	0.675237i	$0.235959\pi$
$360$	0	0
$361$	36.0594	1.89786
$362$	−12.7690	−0.671123
$363$	0	0
$364$	−1.30538	−0.0684204
$365$	4.52271	0.236729
$366$	0	0
$367$	−16.2031	−0.845797	−0.422898	−	0.906177i	$-0.638987\pi$
$367$	−16.2031	−0.845797	−0.422898	+	0.906177i	$0.638987\pi$
$368$	−12.9456	−0.674837
$369$	0	0
$370$	−46.3627	−2.41028
$371$	−2.34131	−0.121555
$372$	0	0
$373$	23.2966	1.20625	0.603126	−	0.797646i	$-0.293922\pi$
$373$	23.2966	1.20625	0.603126	+	0.797646i	$0.293922\pi$
$374$	−9.60417	−0.496620
$375$	0	0
$376$	−0.738345	−0.0380772
$377$	−3.58783	−0.184782
$378$	0	0
$379$	5.17118	0.265625	0.132813	−	0.991141i	$-0.457599\pi$
$379$	5.17118	0.265625	0.132813	+	0.991141i	$0.457599\pi$
$380$	41.3028	2.11879
$381$	0	0
$382$	19.2198	0.983372
$383$	5.61571	0.286949	0.143475	−	0.989654i	$-0.454172\pi$
$383$	5.61571	0.286949	0.143475	+	0.989654i	$0.454172\pi$
$384$	0	0
$385$	−8.11380	−0.413517
$386$	24.6665	1.25549
$387$	0	0
$388$	26.5678	1.34878
$389$	29.7379	1.50777	0.753885	−	0.657007i	$-0.228178\pi$
$389$	29.7379	1.50777	0.753885	+	0.657007i	$0.228178\pi$
$390$	0	0
$391$	5.29261	0.267659
$392$	−0.202857	−0.0102458
$393$	0	0
$394$	−6.87942	−0.346580
$395$	5.01997	0.252582
$396$	0	0
$397$	28.0723	1.40891	0.704455	−	0.709749i	$-0.251192\pi$
$397$	28.0723	1.40891	0.704455	+	0.709749i	$0.251192\pi$
$398$	−23.7693	−1.19145
$399$	0	0
$400$	−7.67192	−0.383596
$401$	−1.13573	−0.0567158	−0.0283579	−	0.999598i	$-0.509028\pi$
$401$	−1.13573	−0.0567158	−0.0283579	+	0.999598i	$0.509028\pi$
$402$	0	0
$403$	−5.35106	−0.266555
$404$	−14.9243	−0.742512
$405$	0	0
$406$	−11.6884	−0.580083
$407$	26.4471	1.31093
$408$	0	0
$409$	−18.8415	−0.931652	−0.465826	−	0.884876i	$-0.654243\pi$
$409$	−18.8415	−0.931652	−0.465826	+	0.884876i	$0.654243\pi$
$410$	−2.24439	−0.110843
$411$	0	0
$412$	9.15781	0.451173
$413$	−2.80139	−0.137848
$414$	0	0
$415$	−27.9414	−1.37159
$416$	5.02169	0.246209
$417$	0	0
$418$	−45.9975	−2.24981
$419$	4.06047	0.198367	0.0991836	−	0.995069i	$-0.468377\pi$
$419$	4.06047	0.198367	0.0991836	+	0.995069i	$0.468377\pi$
$420$	0	0
$421$	17.1377	0.835240	0.417620	−	0.908622i	$-0.362864\pi$
$421$	17.1377	0.835240	0.417620	+	0.908622i	$0.362864\pi$
$422$	19.0060	0.925195
$423$	0	0
$424$	−0.474951	−0.0230657
$425$	3.13655	0.152145
$426$	0	0
$427$	−8.10802	−0.392374
$428$	−20.8993	−1.01020
$429$	0	0
$430$	−26.4778	−1.27687
$431$	15.0193	0.723452	0.361726	−	0.932284i	$-0.382188\pi$
$431$	15.0193	0.723452	0.361726	+	0.932284i	$0.382188\pi$
$432$	0	0
$433$	−12.7800	−0.614169	−0.307084	−	0.951682i	$-0.599353\pi$
$433$	−12.7800	−0.614169	−0.307084	+	0.951682i	$0.599353\pi$
$434$	−17.4326	−0.836791
$435$	0	0
$436$	−1.58788	−0.0760458
$437$	25.3481	1.21256
$438$	0	0
$439$	23.0761	1.10136	0.550682	−	0.834715i	$-0.314368\pi$
$439$	23.0761	1.10136	0.550682	+	0.834715i	$0.314368\pi$
$440$	−1.64594	−0.0784672
$441$	0	0
$442$	−1.94994	−0.0927494
$443$	5.42372	0.257689	0.128844	−	0.991665i	$-0.458873\pi$
$443$	5.42372	0.257689	0.128844	+	0.991665i	$0.458873\pi$
$444$	0	0
$445$	10.8965	0.516546
$446$	8.47859	0.401473
$447$	0	0
$448$	8.78038	0.414834
$449$	−14.7937	−0.698158	−0.349079	−	0.937093i	$-0.613505\pi$
$449$	−14.7937	−0.698158	−0.349079	+	0.937093i	$0.613505\pi$
$450$	0	0
$451$	1.28029	0.0602865
$452$	7.57014	0.356069
$453$	0	0
$454$	15.7740	0.740309
$455$	−1.64735	−0.0772290
$456$	0	0
$457$	−28.0431	−1.31180	−0.655900	−	0.754848i	$-0.727711\pi$
$457$	−28.0431	−1.31180	−0.655900	+	0.754848i	$0.727711\pi$
$458$	−1.14275	−0.0533974
$459$	0	0
$460$	19.0148	0.886572
$461$	33.1736	1.54505	0.772523	−	0.634987i	$-0.218994\pi$
$461$	33.1736	1.54505	0.772523	+	0.634987i	$0.218994\pi$
$462$	0	0
$463$	−35.9991	−1.67302	−0.836510	−	0.547952i	$-0.815408\pi$
$463$	−35.9991	−1.67302	−0.836510	+	0.547952i	$0.815408\pi$
$464$	21.8749	1.01551
$465$	0	0
$466$	−57.2467	−2.65190
$467$	−15.3014	−0.708065	−0.354032	−	0.935233i	$-0.615190\pi$
$467$	−15.3014	−0.708065	−0.354032	+	0.935233i	$0.615190\pi$
$468$	0	0
$469$	1.88861	0.0872079
$470$	−19.5334	−0.901007
$471$	0	0
$472$	−0.568283	−0.0261573
$473$	15.1040	0.694482
$474$	0	0
$475$	15.0219	0.689254
$476$	−3.25385	−0.149140
$477$	0	0
$478$	−15.2434	−0.697215
$479$	−37.6363	−1.71964	−0.859822	−	0.510594i	$-0.829426\pi$
$479$	−37.6363	−1.71964	−0.859822	+	0.510594i	$0.829426\pi$
$480$	0	0
$481$	5.36958	0.244832
$482$	−39.6583	−1.80639
$483$	0	0
$484$	−3.41897	−0.155408
$485$	33.5278	1.52242
$486$	0	0
$487$	1.22801	0.0556465	0.0278233	−	0.999613i	$-0.491142\pi$
$487$	1.22801	0.0556465	0.0278233	+	0.999613i	$0.491142\pi$
$488$	−1.64477	−0.0744552
$489$	0	0
$490$	−5.36671	−0.242443
$491$	34.0980	1.53882	0.769411	−	0.638754i	$-0.220550\pi$
$491$	34.0980	1.53882	0.769411	+	0.638754i	$0.220550\pi$
$492$	0	0
$493$	−8.94319	−0.402781
$494$	−9.33892	−0.420178
$495$	0	0
$496$	32.6252	1.46492
$497$	5.59109	0.250795
$498$	0	0
$499$	−29.9594	−1.34117	−0.670584	−	0.741834i	$-0.733956\pi$
$499$	−29.9594	−1.34117	−0.670584	+	0.741834i	$0.733956\pi$
$500$	−16.5626	−0.740702
$501$	0	0
$502$	26.0974	1.16478
$503$	−0.673125	−0.0300131	−0.0150066	−	0.999887i	$-0.504777\pi$
$503$	−0.673125	−0.0300131	−0.0150066	+	0.999887i	$0.504777\pi$
$504$	0	0
$505$	−18.8341	−0.838105
$506$	−21.1762	−0.941397
$507$	0	0
$508$	2.10018	0.0931805
$509$	−34.4777	−1.52820	−0.764098	−	0.645100i	$-0.776816\pi$
$509$	−34.4777	−1.52820	−0.764098	+	0.645100i	$0.776816\pi$
$510$	0	0
$511$	−1.70644	−0.0754886
$512$	−32.1546	−1.42105
$513$	0	0
$514$	41.6159	1.83560
$515$	11.5569	0.509258
$516$	0	0
$517$	11.1426	0.490051
$518$	17.4929	0.768595
$519$	0	0
$520$	−0.334177	−0.0146546
$521$	−7.39889	−0.324151	−0.162076	−	0.986778i	$-0.551819\pi$
$521$	−7.39889	−0.324151	−0.162076	+	0.986778i	$0.551819\pi$
$522$	0	0
$523$	16.6161	0.726572	0.363286	−	0.931678i	$-0.381655\pi$
$523$	16.6161	0.726572	0.363286	+	0.931678i	$0.381655\pi$
$524$	−29.5946	−1.29284
$525$	0	0
$526$	39.8829	1.73898
$527$	−13.3383	−0.581026
$528$	0	0
$529$	−11.3303	−0.492623
$530$	−12.5651	−0.545794
$531$	0	0
$532$	−15.5838	−0.675642
$533$	0.259938	0.0112592
$534$	0	0
$535$	−26.3743	−1.14026
$536$	0.383118	0.0165482
$537$	0	0
$538$	−51.4912	−2.21994
$539$	3.06138	0.131863
$540$	0	0
$541$	19.9445	0.857480	0.428740	−	0.903428i	$-0.358958\pi$
$541$	19.9445	0.857480	0.428740	+	0.903428i	$0.358958\pi$
$542$	−16.1758	−0.694812
$543$	0	0
$544$	12.5173	0.536676
$545$	−2.00386	−0.0858361
$546$	0	0
$547$	16.4449	0.703131	0.351566	−	0.936163i	$-0.385649\pi$
$547$	16.4449	0.703131	0.351566	+	0.936163i	$0.385649\pi$
$548$	24.8383	1.06104
$549$	0	0
$550$	−12.5496	−0.535117
$551$	−42.8319	−1.82470
$552$	0	0
$553$	−1.89406	−0.0805438
$554$	46.6872	1.98355
$555$	0	0
$556$	45.5195	1.93046
$557$	19.4869	0.825688	0.412844	−	0.910802i	$-0.364535\pi$
$557$	19.4869	0.825688	0.412844	+	0.910802i	$0.364535\pi$
$558$	0	0
$559$	3.06658	0.129702
$560$	10.0438	0.424430
$561$	0	0
$562$	15.2666	0.643982
$563$	7.18834	0.302952	0.151476	−	0.988461i	$-0.451597\pi$
$563$	7.18834	0.302952	0.151476	+	0.988461i	$0.451597\pi$
$564$	0	0
$565$	9.55331	0.401911
$566$	−6.23129	−0.261921
$567$	0	0
$568$	1.13419	0.0475897
$569$	9.25158	0.387847	0.193923	−	0.981017i	$-0.437879\pi$
$569$	9.25158	0.387847	0.193923	+	0.981017i	$0.437879\pi$
$570$	0	0
$571$	33.4512	1.39989	0.699944	−	0.714198i	$-0.253208\pi$
$571$	33.4512	1.39989	0.699944	+	0.714198i	$0.253208\pi$
$572$	3.99626	0.167092
$573$	0	0
$574$	0.846822	0.0353457
$575$	6.91576	0.288407
$576$	0	0
$577$	38.6292	1.60815	0.804077	−	0.594525i	$-0.202660\pi$
$577$	38.6292	1.60815	0.804077	+	0.594525i	$0.202660\pi$
$578$	29.5626	1.22964
$579$	0	0
$580$	−32.1303	−1.33414
$581$	10.5424	0.437374
$582$	0	0
$583$	7.16764	0.296853
$584$	−0.346164	−0.0143244
$585$	0	0
$586$	62.6365	2.58749
$587$	44.6155	1.84148	0.920740	−	0.390177i	$-0.127586\pi$
$587$	44.6155	1.84148	0.920740	+	0.390177i	$0.127586\pi$
$588$	0	0
$589$	−63.8816	−2.63219
$590$	−15.0343	−0.618951
$591$	0	0
$592$	−32.7381	−1.34553
$593$	9.75368	0.400536	0.200268	−	0.979741i	$-0.435819\pi$
$593$	9.75368	0.400536	0.200268	+	0.979741i	$0.435819\pi$
$594$	0	0
$595$	−4.10627	−0.168341
$596$	−7.51671	−0.307896
$597$	0	0
$598$	−4.29942	−0.175817
$599$	−16.0380	−0.655294	−0.327647	−	0.944800i	$-0.606256\pi$
$599$	−16.0380	−0.655294	−0.327647	+	0.944800i	$0.606256\pi$
$600$	0	0
$601$	−6.54845	−0.267117	−0.133558	−	0.991041i	$-0.542640\pi$
$601$	−6.54845	−0.267117	−0.133558	+	0.991041i	$0.542640\pi$
$602$	9.99023	0.407171
$603$	0	0
$604$	45.7147	1.86011
$605$	−4.31464	−0.175415
$606$	0	0
$607$	−2.01172	−0.0816531	−0.0408266	−	0.999166i	$-0.512999\pi$
$607$	−2.01172	−0.0816531	−0.0408266	+	0.999166i	$0.512999\pi$
$608$	59.9496	2.43128
$609$	0	0
$610$	−43.5134	−1.76181
$611$	2.26229	0.0915225
$612$	0	0
$613$	−5.56186	−0.224641	−0.112321	−	0.993672i	$-0.535828\pi$
$613$	−5.56186	−0.224641	−0.112321	+	0.993672i	$0.535828\pi$
$614$	37.9899	1.53315
$615$	0	0
$616$	0.621023	0.0250217
$617$	20.6351	0.830740	0.415370	−	0.909653i	$-0.363652\pi$
$617$	20.6351	0.830740	0.415370	+	0.909653i	$0.363652\pi$
$618$	0	0
$619$	35.9581	1.44528	0.722639	−	0.691225i	$-0.242929\pi$
$619$	35.9581	1.44528	0.722639	+	0.691225i	$0.242929\pi$
$620$	−47.9208	−1.92454
$621$	0	0
$622$	−23.2235	−0.931176
$623$	−4.11133	−0.164717
$624$	0	0
$625$	−31.0239	−1.24095
$626$	66.2339	2.64724
$627$	0	0
$628$	−9.30638	−0.371365
$629$	13.3845	0.533674
$630$	0	0
$631$	0.410872	0.0163566	0.00817828	−	0.999967i	$-0.497397\pi$
$631$	0.410872	0.0163566	0.00817828	+	0.999967i	$0.497397\pi$
$632$	−0.384224	−0.0152836
$633$	0	0
$634$	57.1510	2.26976
$635$	2.65037	0.105177
$636$	0	0
$637$	0.621555	0.0246269
$638$	35.7825	1.41664
$639$	0	0
$640$	4.29578	0.169806
$641$	12.5079	0.494033	0.247016	−	0.969011i	$-0.420550\pi$
$641$	12.5079	0.494033	0.247016	+	0.969011i	$0.420550\pi$
$642$	0	0
$643$	−8.38699	−0.330751	−0.165375	−	0.986231i	$-0.552884\pi$
$643$	−8.38699	−0.330751	−0.165375	+	0.986231i	$0.552884\pi$
$644$	−7.17441	−0.282711
$645$	0	0
$646$	−23.2787	−0.915887
$647$	−16.0235	−0.629949	−0.314974	−	0.949100i	$-0.601996\pi$
$647$	−16.0235	−0.629949	−0.314974	+	0.949100i	$0.601996\pi$
$648$	0	0
$649$	8.57613	0.336643
$650$	−2.54796	−0.0999390
$651$	0	0
$652$	42.0885	1.64831
$653$	14.4221	0.564380	0.282190	−	0.959359i	$-0.408939\pi$
$653$	14.4221	0.564380	0.282190	+	0.959359i	$0.408939\pi$
$654$	0	0
$655$	−37.3475	−1.45929
$656$	−1.58484	−0.0618774
$657$	0	0
$658$	7.37005	0.287314
$659$	−7.11613	−0.277205	−0.138602	−	0.990348i	$-0.544261\pi$
$659$	−7.11613	−0.277205	−0.138602	+	0.990348i	$0.544261\pi$
$660$	0	0
$661$	31.4207	1.22213	0.611063	−	0.791582i	$-0.290742\pi$
$661$	31.4207	1.22213	0.611063	+	0.791582i	$0.290742\pi$
$662$	48.9198	1.90132
$663$	0	0
$664$	2.13861	0.0829942
$665$	−19.6663	−0.762626
$666$	0	0
$667$	−19.7188	−0.763516
$668$	20.4152	0.789889
$669$	0	0
$670$	10.1356	0.391573
$671$	24.8217	0.958232
$672$	0	0
$673$	18.6759	0.719902	0.359951	−	0.932971i	$-0.382793\pi$
$673$	18.6759	0.719902	0.359951	+	0.932971i	$0.382793\pi$
$674$	13.4889	0.519573
$675$	0	0
$676$	−26.4910	−1.01888
$677$	−24.7489	−0.951177	−0.475588	−	0.879668i	$-0.657765\pi$
$677$	−24.7489	−0.951177	−0.475588	+	0.879668i	$0.657765\pi$
$678$	0	0
$679$	−12.6502	−0.485471
$680$	−0.832987	−0.0319436
$681$	0	0
$682$	53.3678	2.04356
$683$	−18.5006	−0.707908	−0.353954	−	0.935263i	$-0.615163\pi$
$683$	−18.5006	−0.707908	−0.353954	+	0.935263i	$0.615163\pi$
$684$	0	0
$685$	31.3452	1.19764
$686$	2.02489	0.0773107
$687$	0	0
$688$	−18.6968	−0.712809
$689$	1.45525	0.0554407
$690$	0	0
$691$	−22.8004	−0.867367	−0.433684	−	0.901065i	$-0.642786\pi$
$691$	−22.8004	−0.867367	−0.433684	+	0.901065i	$0.642786\pi$
$692$	−14.3391	−0.545091
$693$	0	0
$694$	13.3747	0.507695
$695$	57.4443	2.17899
$696$	0	0
$697$	0.647935	0.0245423
$698$	−38.1439	−1.44377
$699$	0	0
$700$	−4.25175	−0.160701
$701$	−41.3712	−1.56257	−0.781285	−	0.624175i	$-0.785435\pi$
$701$	−41.3712	−1.56257	−0.781285	+	0.624175i	$0.785435\pi$
$702$	0	0
$703$	64.1027	2.41768
$704$	−26.8801	−1.01308
$705$	0	0
$706$	60.4844	2.27636
$707$	7.10620	0.267256
$708$	0	0
$709$	2.50912	0.0942320	0.0471160	−	0.998889i	$-0.484997\pi$
$709$	2.50912	0.0942320	0.0471160	+	0.998889i	$0.484997\pi$
$710$	30.0058	1.12610
$711$	0	0
$712$	−0.834013	−0.0312559
$713$	−29.4096	−1.10140
$714$	0	0
$715$	5.04317	0.188604
$716$	33.7686	1.26199
$717$	0	0
$718$	−56.5979	−2.11221
$719$	18.6223	0.694495	0.347248	−	0.937773i	$-0.387116\pi$
$719$	18.6223	0.694495	0.347248	+	0.937773i	$0.387116\pi$
$720$	0	0
$721$	−4.36048	−0.162393
$722$	−73.0162	−2.71738
$723$	0	0
$724$	13.2438	0.492201
$725$	−11.6859	−0.434004
$726$	0	0
$727$	11.5400	0.427993	0.213997	−	0.976834i	$-0.431352\pi$
$727$	11.5400	0.427993	0.213997	+	0.976834i	$0.431352\pi$
$728$	0.126087	0.00467309
$729$	0	0
$730$	−9.15799	−0.338952
$731$	7.64390	0.282720
$732$	0	0
$733$	−1.19505	−0.0441403	−0.0220701	−	0.999756i	$-0.507026\pi$
$733$	−1.19505	−0.0441403	−0.0220701	+	0.999756i	$0.507026\pi$
$734$	32.8096	1.21102
$735$	0	0
$736$	27.5994	1.01733
$737$	−5.78176	−0.212974
$738$	0	0
$739$	31.1176	1.14468	0.572340	−	0.820016i	$-0.306036\pi$
$739$	31.1176	1.14468	0.572340	+	0.820016i	$0.306036\pi$
$740$	48.0866	1.76770
$741$	0	0
$742$	4.74089	0.174044
$743$	−46.0989	−1.69120	−0.845602	−	0.533813i	$-0.820758\pi$
$743$	−46.0989	−1.69120	−0.845602	+	0.533813i	$0.820758\pi$
$744$	0	0
$745$	−9.48588	−0.347536
$746$	−47.1730	−1.72713
$747$	0	0
$748$	9.96128	0.364221
$749$	9.95118	0.363608
$750$	0	0
$751$	3.64940	0.133169	0.0665843	−	0.997781i	$-0.478790\pi$
$751$	3.64940	0.133169	0.0665843	+	0.997781i	$0.478790\pi$
$752$	−13.7931	−0.502983
$753$	0	0
$754$	7.26495	0.264574
$755$	57.6907	2.09958
$756$	0	0
$757$	13.6127	0.494763	0.247381	−	0.968918i	$-0.420430\pi$
$757$	13.6127	0.494763	0.247381	+	0.968918i	$0.420430\pi$
$758$	−10.4711	−0.380326
$759$	0	0
$760$	−3.98945	−0.144712
$761$	23.5233	0.852720	0.426360	−	0.904554i	$-0.359796\pi$
$761$	23.5233	0.852720	0.426360	+	0.904554i	$0.359796\pi$
$762$	0	0
$763$	0.756070	0.0273716
$764$	−19.9345	−0.721204
$765$	0	0
$766$	−11.3712	−0.410858
$767$	1.74122	0.0628718
$768$	0	0
$769$	−19.3910	−0.699257	−0.349628	−	0.936888i	$-0.613692\pi$
$769$	−19.3910	−0.699257	−0.349628	+	0.936888i	$0.613692\pi$
$770$	16.4296	0.592080
$771$	0	0
$772$	−25.5837	−0.920777
$773$	−5.89800	−0.212136	−0.106068	−	0.994359i	$-0.533826\pi$
$773$	−5.89800	−0.212136	−0.106068	+	0.994359i	$0.533826\pi$
$774$	0	0
$775$	−17.4289	−0.626066
$776$	−2.56619	−0.0921209
$777$	0	0
$778$	−60.2159	−2.15884
$779$	3.10318	0.111183
$780$	0	0
$781$	−17.1165	−0.612475
$782$	−10.7170	−0.383238
$783$	0	0
$784$	−3.78960	−0.135343
$785$	−11.7444	−0.419175
$786$	0	0
$787$	32.9716	1.17531	0.587656	−	0.809111i	$-0.300051\pi$
$787$	32.9716	1.17531	0.587656	+	0.809111i	$0.300051\pi$
$788$	7.13521	0.254182
$789$	0	0
$790$	−10.1649	−0.361651
$791$	−3.60452	−0.128162
$792$	0	0
$793$	5.03958	0.178961
$794$	−56.8434	−2.01730
$795$	0	0
$796$	24.6531	0.873805
$797$	14.2064	0.503215	0.251608	−	0.967829i	$-0.419041\pi$
$797$	14.2064	0.503215	0.251608	+	0.967829i	$0.419041\pi$
$798$	0	0
$799$	5.63910	0.199497
$800$	16.3562	0.578277
$801$	0	0
$802$	2.29973	0.0812064
$803$	5.22407	0.184354
$804$	0	0
$805$	−9.05390	−0.319108
$806$	10.8353	0.381657
$807$	0	0
$808$	1.44154	0.0507133
$809$	−21.0165	−0.738899	−0.369450	−	0.929251i	$-0.620454\pi$
$809$	−21.0165	−0.738899	−0.369450	+	0.929251i	$0.620454\pi$
$810$	0	0
$811$	−3.71115	−0.130316	−0.0651581	−	0.997875i	$-0.520755\pi$
$811$	−3.71115	−0.130316	−0.0651581	+	0.997875i	$0.520755\pi$
$812$	12.1230	0.425432
$813$	0	0
$814$	−53.5525	−1.87701
$815$	53.1145	1.86052
$816$	0	0
$817$	36.6091	1.28079
$818$	38.1520	1.33395
$819$	0	0
$820$	2.32785	0.0812919
$821$	39.8574	1.39103	0.695517	−	0.718509i	$-0.255175\pi$
$821$	39.8574	1.39103	0.695517	+	0.718509i	$0.255175\pi$
$822$	0	0
$823$	37.0554	1.29167	0.645836	−	0.763477i	$-0.276509\pi$
$823$	37.0554	1.29167	0.645836	+	0.763477i	$0.276509\pi$
$824$	−0.884555	−0.0308149
$825$	0	0
$826$	5.67251	0.197372
$827$	9.60653	0.334052	0.167026	−	0.985953i	$-0.446584\pi$
$827$	9.60653	0.334052	0.167026	+	0.985953i	$0.446584\pi$
$828$	0	0
$829$	18.7934	0.652723	0.326361	−	0.945245i	$-0.394177\pi$
$829$	18.7934	0.652723	0.326361	+	0.945245i	$0.394177\pi$
$830$	56.5783	1.96386
$831$	0	0
$832$	−5.45749	−0.189204
$833$	1.54932	0.0536807
$834$	0	0
$835$	25.7634	0.891581
$836$	47.7079	1.65001
$837$	0	0
$838$	−8.22202	−0.284025
$839$	1.09849	0.0379239	0.0189620	−	0.999820i	$-0.493964\pi$
$839$	1.09849	0.0379239	0.0189620	+	0.999820i	$0.493964\pi$
$840$	0	0
$841$	4.31988	0.148961
$842$	−34.7020	−1.19591
$843$	0	0
$844$	−19.7127	−0.678537
$845$	−33.4309	−1.15006
$846$	0	0
$847$	1.62794	0.0559366
$848$	−8.87262	−0.304687
$849$	0	0
$850$	−6.35116	−0.217843
$851$	29.5114	1.01164
$852$	0	0
$853$	−8.52189	−0.291784	−0.145892	−	0.989301i	$-0.546605\pi$
$853$	−8.52189	−0.291784	−0.145892	+	0.989301i	$0.546605\pi$
$854$	16.4178	0.561807
$855$	0	0
$856$	2.01867	0.0689966
$857$	15.3597	0.524676	0.262338	−	0.964976i	$-0.415506\pi$
$857$	15.3597	0.524676	0.262338	+	0.964976i	$0.415506\pi$
$858$	0	0
$859$	41.1885	1.40533	0.702666	−	0.711520i	$-0.251993\pi$
$859$	41.1885	1.40533	0.702666	+	0.711520i	$0.251993\pi$
$860$	27.4623	0.936458
$861$	0	0
$862$	−30.4124	−1.03585
$863$	7.96200	0.271030	0.135515	−	0.990775i	$-0.456731\pi$
$863$	7.96200	0.271030	0.135515	+	0.990775i	$0.456731\pi$
$864$	0	0
$865$	−18.0955	−0.615267
$866$	25.8782	0.879376
$867$	0	0
$868$	18.0808	0.613701
$869$	5.79845	0.196699
$870$	0	0
$871$	−1.17387	−0.0397752
$872$	0.153374	0.00519390
$873$	0	0
$874$	−51.3270	−1.73616
$875$	7.88627	0.266604
$876$	0	0
$877$	−26.3922	−0.891201	−0.445601	−	0.895232i	$-0.647010\pi$
$877$	−26.3922	−0.891201	−0.445601	+	0.895232i	$0.647010\pi$
$878$	−46.7266	−1.57695
$879$	0	0
$880$	−30.7481	−1.03652
$881$	10.4735	0.352860	0.176430	−	0.984313i	$-0.443545\pi$
$881$	10.4735	0.352860	0.176430	+	0.984313i	$0.443545\pi$
$882$	0	0
$883$	−8.27980	−0.278638	−0.139319	−	0.990248i	$-0.544491\pi$
$883$	−8.27980	−0.278638	−0.139319	+	0.990248i	$0.544491\pi$
$884$	2.02245	0.0680223
$885$	0	0
$886$	−10.9824	−0.368963
$887$	−20.3450	−0.683117	−0.341558	−	0.939861i	$-0.610955\pi$
$887$	−20.3450	−0.683117	−0.341558	+	0.939861i	$0.610955\pi$
$888$	0	0
$889$	−1.00000	−0.0335389
$890$	−22.0643	−0.739597
$891$	0	0
$892$	−8.79385	−0.294440
$893$	27.0075	0.903772
$894$	0	0
$895$	42.6150	1.42446
$896$	−1.62082	−0.0541478
$897$	0	0
$898$	29.9556	0.999632
$899$	49.6949	1.65742
$900$	0	0
$901$	3.62743	0.120847
$902$	−2.59245	−0.0863190
$903$	0	0
$904$	−0.731202	−0.0243194
$905$	16.7133	0.555568
$906$	0	0
$907$	38.4703	1.27738	0.638692	−	0.769462i	$-0.279476\pi$
$907$	38.4703	1.27738	0.638692	+	0.769462i	$0.279476\pi$
$908$	−16.3605	−0.542942
$909$	0	0
$910$	3.33571	0.110578
$911$	10.7180	0.355104	0.177552	−	0.984111i	$-0.443182\pi$
$911$	10.7180	0.355104	0.177552	+	0.984111i	$0.443182\pi$
$912$	0	0
$913$	−32.2745	−1.06813
$914$	56.7842	1.87825
$915$	0	0
$916$	1.18525	0.0391616
$917$	14.0914	0.465340
$918$	0	0
$919$	9.93297	0.327658	0.163829	−	0.986489i	$-0.447615\pi$
$919$	9.93297	0.327658	0.163829	+	0.986489i	$0.447615\pi$
$920$	−1.83665	−0.0605525
$921$	0	0
$922$	−67.1728	−2.21222
$923$	−3.47517	−0.114387
$924$	0	0
$925$	17.4893	0.575043
$926$	72.8942	2.39545
$927$	0	0
$928$	−46.6361	−1.53090
$929$	−59.2995	−1.94555	−0.972777	−	0.231743i	$-0.925557\pi$
$929$	−59.2995	−1.94555	−0.972777	+	0.231743i	$0.925557\pi$
$930$	0	0
$931$	7.42020	0.243187
$932$	59.3753	1.94490
$933$	0	0
$934$	30.9837	1.01382
$935$	12.5709	0.411111
$936$	0	0
$937$	−5.58991	−0.182614	−0.0913072	−	0.995823i	$-0.529105\pi$
$937$	−5.58991	−0.182614	−0.0913072	+	0.995823i	$0.529105\pi$
$938$	−3.82423	−0.124865
$939$	0	0
$940$	20.2597	0.660798
$941$	24.0290	0.783323	0.391661	−	0.920109i	$-0.371901\pi$
$941$	24.0290	0.783323	0.391661	+	0.920109i	$0.371901\pi$
$942$	0	0
$943$	1.42863	0.0465226
$944$	−10.6162	−0.345526
$945$	0	0
$946$	−30.5839	−0.994369
$947$	−33.2864	−1.08166	−0.540831	−	0.841131i	$-0.681890\pi$
$947$	−33.2864	−1.08166	−0.540831	+	0.841131i	$0.681890\pi$
$948$	0	0
$949$	1.06065	0.0344301
$950$	−30.4178	−0.986884
$951$	0	0
$952$	0.314291	0.0101862
$953$	−11.5468	−0.374036	−0.187018	−	0.982356i	$-0.559882\pi$
$953$	−11.5468	−0.374036	−0.187018	+	0.982356i	$0.559882\pi$
$954$	0	0
$955$	−25.1568	−0.814054
$956$	15.8102	0.511337
$957$	0	0
$958$	76.2093	2.46221
$959$	−11.8267	−0.381905
$960$	0	0
$961$	43.1174	1.39088
$962$	−10.8728	−0.350553
$963$	0	0
$964$	41.1329	1.32480
$965$	−32.2859	−1.03932
$966$	0	0
$967$	7.98233	0.256694	0.128347	−	0.991729i	$-0.459033\pi$
$967$	7.98233	0.256694	0.128347	+	0.991729i	$0.459033\pi$
$968$	0.330239	0.0106143
$969$	0	0
$970$	−67.8901	−2.17982
$971$	−0.517059	−0.0165932	−0.00829661	−	0.999966i	$-0.502641\pi$
$971$	−0.517059	−0.0165932	−0.00829661	+	0.999966i	$0.502641\pi$
$972$	0	0
$973$	−21.6741	−0.694839
$974$	−2.48659	−0.0796754
$975$	0	0
$976$	−30.7261	−0.983520
$977$	45.7360	1.46322	0.731612	−	0.681721i	$-0.238768\pi$
$977$	45.7360	1.46322	0.731612	+	0.681721i	$0.238768\pi$
$978$	0	0
$979$	12.5863	0.402261
$980$	5.56626	0.177808
$981$	0	0
$982$	−69.0448	−2.20331
$983$	29.8753	0.952875	0.476437	−	0.879208i	$-0.341928\pi$
$983$	29.8753	0.952875	0.476437	+	0.879208i	$0.341928\pi$
$984$	0	0
$985$	9.00444	0.286905
$986$	18.1090	0.576708
$987$	0	0
$988$	9.68617	0.308158
$989$	16.8540	0.535926
$990$	0	0
$991$	55.2582	1.75533	0.877667	−	0.479270i	$-0.159099\pi$
$991$	55.2582	1.75533	0.877667	+	0.479270i	$0.159099\pi$
$992$	−69.5554	−2.20838
$993$	0	0
$994$	−11.3213	−0.359091
$995$	31.1115	0.986301
$996$	0	0
$997$	−37.7020	−1.19403	−0.597017	−	0.802228i	$-0.703648\pi$
$997$	−37.7020	−1.19403	−0.597017	+	0.802228i	$0.703648\pi$
$998$	60.6645	1.92030
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.y.1.5	✓	28
3.2	odd	2	inner	8001.2.a.y.1.24	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8001.2.a.y.1.5	✓	28	1.1	even	1	trivial
8001.2.a.y.1.24	yes	28	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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q-2.02489 q^{2} +2.10018 q^{4} +2.65037 q^{5} -1.00000 q^{7} -0.202857 q^{8} +O(q^{10})\)	\(q-2.02489 q^{2} +2.10018 q^{4} +2.65037 q^{5} -1.00000 q^{7} -0.202857 q^{8} -5.36671 q^{10} +3.06138 q^{11} +0.621555 q^{13} +2.02489 q^{14} -3.78960 q^{16} +1.54932 q^{17} +7.42020 q^{19} +5.56626 q^{20} -6.19896 q^{22} +3.41609 q^{23} +2.02447 q^{25} -1.25858 q^{26} -2.10018 q^{28} -5.77234 q^{29} -8.60915 q^{31} +8.07924 q^{32} -3.13720 q^{34} -2.65037 q^{35} +8.63894 q^{37} -15.0251 q^{38} -0.537647 q^{40} +0.418206 q^{41} +4.93371 q^{43} +6.42946 q^{44} -6.91721 q^{46} +3.63973 q^{47} +1.00000 q^{49} -4.09932 q^{50} +1.30538 q^{52} +2.34131 q^{53} +8.11380 q^{55} +0.202857 q^{56} +11.6884 q^{58} +2.80139 q^{59} +8.10802 q^{61} +17.4326 q^{62} -8.78038 q^{64} +1.64735 q^{65} -1.88861 q^{67} +3.25385 q^{68} +5.36671 q^{70} -5.59109 q^{71} +1.70644 q^{73} -17.4929 q^{74} +15.5838 q^{76} -3.06138 q^{77} +1.89406 q^{79} -10.0438 q^{80} -0.846822 q^{82} -10.5424 q^{83} +4.10627 q^{85} -9.99023 q^{86} -0.621023 q^{88} +4.11133 q^{89} -0.621555 q^{91} +7.17441 q^{92} -7.37005 q^{94} +19.6663 q^{95} +12.6502 q^{97} -2.02489 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 30 q^{4} - 28 q^{7}+O(q^{10}) \) 28 * q + 30 * q^4 - 28 * q^7	\( 28 q + 30 q^{4} - 28 q^{7} + 4 q^{10} + 8 q^{13} + 42 q^{16} + 34 q^{19} - 10 q^{22} + 14 q^{25} - 30 q^{28} + 56 q^{31} - 6 q^{37} + 38 q^{40} + 18 q^{43} + 16 q^{46} + 28 q^{49} + 18 q^{52} + 48 q^{55} + 2 q^{58} + 36 q^{61} + 76 q^{64} - 4 q^{70} + 50 q^{73} + 132 q^{76} + 66 q^{79} - 36 q^{82} + 20 q^{85} + 6 q^{88} - 8 q^{91} + 54 q^{94} - 8 q^{97}+O(q^{100}) \) 28 * q + 30 * q^4 - 28 * q^7 + 4 * q^10 + 8 * q^13 + 42 * q^16 + 34 * q^19 - 10 * q^22 + 14 * q^25 - 30 * q^28 + 56 * q^31 - 6 * q^37 + 38 * q^40 + 18 * q^43 + 16 * q^46 + 28 * q^49 + 18 * q^52 + 48 * q^55 + 2 * q^58 + 36 * q^61 + 76 * q^64 - 4 * q^70 + 50 * q^73 + 132 * q^76 + 66 * q^79 - 36 * q^82 + 20 * q^85 + 6 * q^88 - 8 * q^91 + 54 * q^94 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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