LMFDB - Embedded newform 1331.2.a.f.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1331,2,Mod(1,1331)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1331, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1331.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1331 = 11^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1331.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:	$10.6280885090$
Analytic rank:	$0$
Dimension:	$30$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	1331.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.911222 q^{2} -0.648300 q^{3} -1.16967 q^{4} -1.23652 q^{5} -0.590746 q^{6} -1.76009 q^{7} -2.88828 q^{8} -2.57971 q^{9} +O(q^{10})$	\(q+0.911222 q^{2} -0.648300 q^{3} -1.16967 q^{4} -1.23652 q^{5} -0.590746 q^{6} -1.76009 q^{7} -2.88828 q^{8} -2.57971 q^{9} -1.12674 q^{10} +0.758300 q^{12} -4.35652 q^{13} -1.60383 q^{14} +0.801633 q^{15} -0.292515 q^{16} +7.66611 q^{17} -2.35069 q^{18} +4.10713 q^{19} +1.44632 q^{20} +1.14107 q^{21} +8.58230 q^{23} +1.87247 q^{24} -3.47103 q^{25} -3.96976 q^{26} +3.61733 q^{27} +2.05873 q^{28} +8.70739 q^{29} +0.730466 q^{30} -0.393275 q^{31} +5.51001 q^{32} +6.98553 q^{34} +2.17638 q^{35} +3.01742 q^{36} +0.904758 q^{37} +3.74251 q^{38} +2.82433 q^{39} +3.57140 q^{40} -4.27179 q^{41} +1.03976 q^{42} -9.86460 q^{43} +3.18985 q^{45} +7.82038 q^{46} +2.45376 q^{47} +0.189637 q^{48} -3.90209 q^{49} -3.16288 q^{50} -4.96994 q^{51} +5.09571 q^{52} +1.96031 q^{53} +3.29619 q^{54} +5.08363 q^{56} -2.66265 q^{57} +7.93436 q^{58} +0.899251 q^{59} -0.937649 q^{60} -8.22053 q^{61} -0.358361 q^{62} +4.54051 q^{63} +5.60587 q^{64} +5.38690 q^{65} +11.2903 q^{67} -8.96685 q^{68} -5.56391 q^{69} +1.98316 q^{70} -5.52569 q^{71} +7.45091 q^{72} -7.67482 q^{73} +0.824436 q^{74} +2.25027 q^{75} -4.80400 q^{76} +2.57360 q^{78} -4.87756 q^{79} +0.361699 q^{80} +5.39401 q^{81} -3.89255 q^{82} +1.25202 q^{83} -1.33468 q^{84} -9.47926 q^{85} -8.98885 q^{86} -5.64500 q^{87} +9.43597 q^{89} +2.90666 q^{90} +7.66787 q^{91} -10.0385 q^{92} +0.254960 q^{93} +2.23592 q^{94} -5.07853 q^{95} -3.57214 q^{96} +8.36224 q^{97} -3.55567 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q + 6 q^{3} + 58 q^{4} + 14 q^{5} + 32 q^{9}+O(q^{10}) $ 30 * q + 6 * q^3 + 58 * q^4 + 14 * q^5 + 32 * q^9	$ 30 q + 6 q^{3} + 58 q^{4} + 14 q^{5} + 32 q^{9} + 42 q^{12} + 18 q^{14} + 30 q^{15} + 62 q^{16} + 56 q^{20} - 14 q^{23} + 32 q^{25} + 20 q^{26} + 36 q^{27} + 40 q^{31} + 68 q^{34} + 90 q^{36} + 72 q^{37} - 44 q^{38} - 44 q^{42} + 66 q^{45} + 8 q^{47} - 16 q^{48} + 136 q^{49} + 20 q^{53} + 6 q^{56} - 22 q^{58} + 26 q^{59} - 32 q^{60} + 116 q^{64} - 8 q^{67} + 86 q^{69} - 36 q^{70} + 18 q^{71} - 10 q^{75} - 134 q^{78} - 16 q^{80} - 34 q^{81} - 38 q^{82} - 78 q^{86} - 8 q^{89} + 74 q^{91} - 98 q^{92} + 26 q^{93} + 98 q^{97}+O(q^{100}) $ 30 * q + 6 * q^3 + 58 * q^4 + 14 * q^5 + 32 * q^9 + 42 * q^12 + 18 * q^14 + 30 * q^15 + 62 * q^16 + 56 * q^20 - 14 * q^23 + 32 * q^25 + 20 * q^26 + 36 * q^27 + 40 * q^31 + 68 * q^34 + 90 * q^36 + 72 * q^37 - 44 * q^38 - 44 * q^42 + 66 * q^45 + 8 * q^47 - 16 * q^48 + 136 * q^49 + 20 * q^53 + 6 * q^56 - 22 * q^58 + 26 * q^59 - 32 * q^60 + 116 * q^64 - 8 * q^67 + 86 * q^69 - 36 * q^70 + 18 * q^71 - 10 * q^75 - 134 * q^78 - 16 * q^80 - 34 * q^81 - 38 * q^82 - 78 * q^86 - 8 * q^89 + 74 * q^91 - 98 * q^92 + 26 * q^93 + 98 * 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 $

$2$	0.911222	0.644331	0.322166	−	0.946683i	$-0.395589\pi$
$2$	0.911222	0.644331	0.322166	+	0.946683i	$0.395589\pi$
$3$	−0.648300	−0.374296	−0.187148	−	0.982332i	$-0.559924\pi$
$3$	−0.648300	−0.374296	−0.187148	+	0.982332i	$0.559924\pi$
$4$	−1.16967	−0.584837
$5$	−1.23652	−0.552986	−0.276493	−	0.961016i	$-0.589172\pi$
$5$	−1.23652	−0.552986	−0.276493	+	0.961016i	$0.589172\pi$
$6$	−0.590746	−0.241171
$7$	−1.76009	−0.665251	−0.332626	−	0.943059i	$-0.607935\pi$
$7$	−1.76009	−0.665251	−0.332626	+	0.943059i	$0.607935\pi$
$8$	−2.88828	−1.02116
$9$	−2.57971	−0.859902
$10$	−1.12674	−0.356307
$11$	0	0
$11$	0	0
$12$	0.758300	0.218902
$13$	−4.35652	−1.20828	−0.604141	−	0.796878i	$-0.706484\pi$
$13$	−4.35652	−1.20828	−0.604141	+	0.796878i	$0.706484\pi$
$14$	−1.60383	−0.428642
$15$	0.801633	0.206981
$16$	−0.292515	−0.0731287
$17$	7.66611	1.85930	0.929652	−	0.368438i	$-0.120107\pi$
$17$	7.66611	1.85930	0.929652	+	0.368438i	$0.120107\pi$
$18$	−2.35069	−0.554062
$19$	4.10713	0.942240	0.471120	−	0.882069i	$-0.343850\pi$
$19$	4.10713	0.942240	0.471120	+	0.882069i	$0.343850\pi$
$20$	1.44632	0.323407
$21$	1.14107	0.249001
$22$	0	0
$23$	8.58230	1.78953	0.894766	−	0.446534i	$-0.147342\pi$
$23$	8.58230	1.78953	0.894766	+	0.446534i	$0.147342\pi$
$24$	1.87247	0.382217
$25$	−3.47103	−0.694206
$26$	−3.96976	−0.778534
$27$	3.61733	0.696155
$28$	2.05873	0.389064
$29$	8.70739	1.61692	0.808461	−	0.588550i	$-0.200301\pi$
$29$	8.70739	1.61692	0.808461	+	0.588550i	$0.200301\pi$
$30$	0.730466	0.133364
$31$	−0.393275	−0.0706342	−0.0353171	−	0.999376i	$-0.511244\pi$
$31$	−0.393275	−0.0706342	−0.0353171	+	0.999376i	$0.511244\pi$
$32$	5.51001	0.974041
$33$	0	0
$34$	6.98553	1.19801
$35$	2.17638	0.367875
$36$	3.01742	0.502903
$37$	0.904758	0.148741	0.0743707	−	0.997231i	$-0.476305\pi$
$37$	0.904758	0.148741	0.0743707	+	0.997231i	$0.476305\pi$
$38$	3.74251	0.607115
$39$	2.82433	0.452255
$40$	3.57140	0.564688
$41$	−4.27179	−0.667142	−0.333571	−	0.942725i	$-0.608254\pi$
$41$	−4.27179	−0.667142	−0.333571	+	0.942725i	$0.608254\pi$
$42$	1.03976	0.160439
$43$	−9.86460	−1.50434	−0.752169	−	0.658970i	$-0.770992\pi$
$43$	−9.86460	−1.50434	−0.752169	+	0.658970i	$0.770992\pi$
$44$	0	0
$45$	3.18985	0.475514
$46$	7.82038	1.15305
$47$	2.45376	0.357918	0.178959	−	0.983857i	$-0.442727\pi$
$47$	2.45376	0.357918	0.178959	+	0.983857i	$0.442727\pi$
$48$	0.189637	0.0273718
$49$	−3.90209	−0.557441
$50$	−3.16288	−0.447299
$51$	−4.96994	−0.695931
$52$	5.09571	0.706648
$53$	1.96031	0.269269	0.134634	−	0.990895i	$-0.457014\pi$
$53$	1.96031	0.269269	0.134634	+	0.990895i	$0.457014\pi$
$54$	3.29619	0.448554
$55$	0	0
$56$	5.08363	0.679328
$57$	−2.66265	−0.352677
$58$	7.93436	1.04183
$59$	0.899251	0.117072	0.0585362	−	0.998285i	$-0.481357\pi$
$59$	0.899251	0.117072	0.0585362	+	0.998285i	$0.481357\pi$
$60$	−0.937649	−0.121050
$61$	−8.22053	−1.05253	−0.526265	−	0.850320i	$-0.676408\pi$
$61$	−8.22053	−1.05253	−0.526265	+	0.850320i	$0.676408\pi$
$62$	−0.358361	−0.0455118
$63$	4.54051	0.572051
$64$	5.60587	0.700734
$65$	5.38690	0.668163
$66$	0	0
$67$	11.2903	1.37933	0.689663	−	0.724130i	$-0.257759\pi$
$67$	11.2903	1.37933	0.689663	+	0.724130i	$0.257759\pi$
$68$	−8.96685	−1.08739
$69$	−5.56391	−0.669816
$70$	1.98316	0.237033
$71$	−5.52569	−0.655779	−0.327890	−	0.944716i	$-0.606337\pi$
$71$	−5.52569	−0.655779	−0.327890	+	0.944716i	$0.606337\pi$
$72$	7.45091	0.878098
$73$	−7.67482	−0.898269	−0.449135	−	0.893464i	$-0.648268\pi$
$73$	−7.67482	−0.898269	−0.449135	+	0.893464i	$0.648268\pi$
$74$	0.824436	0.0958387
$75$	2.25027	0.259839
$76$	−4.80400	−0.551057
$77$	0	0
$78$	2.57360	0.291402
$79$	−4.87756	−0.548768	−0.274384	−	0.961620i	$-0.588474\pi$
$79$	−4.87756	−0.548768	−0.274384	+	0.961620i	$0.588474\pi$
$80$	0.361699	0.0404392
$81$	5.39401	0.599334
$82$	−3.89255	−0.429860
$83$	1.25202	0.137427	0.0687133	−	0.997636i	$-0.478111\pi$
$83$	1.25202	0.137427	0.0687133	+	0.997636i	$0.478111\pi$
$84$	−1.33468	−0.145625
$85$	−9.47926	−1.02817
$86$	−8.98885	−0.969292
$87$	−5.64500	−0.605208
$88$	0	0
$89$	9.43597	1.00021	0.500105	−	0.865965i	$-0.333295\pi$
$89$	9.43597	1.00021	0.500105	+	0.865965i	$0.333295\pi$
$90$	2.90666	0.306389
$91$	7.66787	0.803811
$92$	−10.0385	−1.04659
$93$	0.254960	0.0264381
$94$	2.23592	0.230618
$95$	−5.07853	−0.521046
$96$	−3.57214	−0.364580
$97$	8.36224	0.849057	0.424528	−	0.905415i	$-0.360440\pi$
$97$	8.36224	0.849057	0.424528	+	0.905415i	$0.360440\pi$
$98$	−3.55567	−0.359177
$99$	0	0
$100$	4.05997	0.405997
$101$	5.33009	0.530363	0.265182	−	0.964198i	$-0.414568\pi$
$101$	5.33009	0.530363	0.265182	+	0.964198i	$0.414568\pi$
$102$	−4.52872	−0.448410
$103$	19.1234	1.88428	0.942141	−	0.335218i	$-0.108810\pi$
$103$	19.1234	1.88428	0.942141	+	0.335218i	$0.108810\pi$
$104$	12.5828	1.23385
$105$	−1.41095	−0.137694
$106$	1.78628	0.173498
$107$	−8.93178	−0.863468	−0.431734	−	0.902001i	$-0.642098\pi$
$107$	−8.93178	−0.863468	−0.431734	+	0.902001i	$0.642098\pi$
$108$	−4.23109	−0.407137
$109$	14.6833	1.40640	0.703202	−	0.710990i	$-0.251753\pi$
$109$	14.6833	1.40640	0.703202	+	0.710990i	$0.251753\pi$
$110$	0	0
$111$	−0.586555	−0.0556733
$112$	0.514852	0.0486490
$113$	4.92925	0.463705	0.231852	−	0.972751i	$-0.425521\pi$
$113$	4.92925	0.463705	0.231852	+	0.972751i	$0.425521\pi$
$114$	−2.42627	−0.227241
$115$	−10.6121	−0.989587
$116$	−10.1848	−0.945635
$117$	11.2385	1.03900
$118$	0.819417	0.0754335
$119$	−13.4930	−1.23690
$120$	−2.31534	−0.211361
$121$	0	0
$122$	−7.49073	−0.678179
$123$	2.76940	0.249709
$124$	0.460003	0.0413095
$125$	10.4746	0.936873
$126$	4.13742	0.368590
$127$	−7.70775	−0.683952	−0.341976	−	0.939709i	$-0.611096\pi$
$127$	−7.70775	−0.683952	−0.341976	+	0.939709i	$0.611096\pi$
$128$	−5.91182	−0.522536
$129$	6.39523	0.563068
$130$	4.90867	0.430519
$131$	10.8283	0.946073	0.473036	−	0.881043i	$-0.343158\pi$
$131$	10.8283	0.946073	0.473036	+	0.881043i	$0.343158\pi$
$132$	0	0
$133$	−7.22891	−0.626826
$134$	10.2879	0.888743
$135$	−4.47288	−0.384964
$136$	−22.1419	−1.89865
$137$	12.3607	1.05605	0.528025	−	0.849229i	$-0.322933\pi$
$137$	12.3607	1.05605	0.528025	+	0.849229i	$0.322933\pi$
$138$	−5.06995	−0.431583
$139$	10.4469	0.886093	0.443047	−	0.896499i	$-0.353898\pi$
$139$	10.4469	0.886093	0.443047	+	0.896499i	$0.353898\pi$
$140$	−2.54565	−0.215147
$141$	−1.59077	−0.133967
$142$	−5.03514	−0.422539
$143$	0	0
$144$	0.754602	0.0628835
$145$	−10.7668	−0.894135
$146$	−6.99346	−0.578783
$147$	2.52972	0.208648
$148$	−1.05827	−0.0869894
$149$	−9.72563	−0.796755	−0.398378	−	0.917222i	$-0.630427\pi$
$149$	−9.72563	−0.796755	−0.398378	+	0.917222i	$0.630427\pi$
$150$	2.05050	0.167422
$151$	15.3809	1.25168	0.625838	−	0.779953i	$-0.284757\pi$
$151$	15.3809	1.25168	0.625838	+	0.779953i	$0.284757\pi$
$152$	−11.8625	−0.962178
$153$	−19.7763	−1.59882
$154$	0	0
$155$	0.486290	0.0390598
$156$	−3.30355	−0.264496
$157$	−11.2907	−0.901099	−0.450549	−	0.892751i	$-0.648772\pi$
$157$	−11.2907	−0.901099	−0.450549	+	0.892751i	$0.648772\pi$
$158$	−4.44454	−0.353588
$159$	−1.27087	−0.100786
$160$	−6.81321	−0.538632
$161$	−15.1056	−1.19049
$162$	4.91514	0.386170
$163$	16.6683	1.30556	0.652779	−	0.757548i	$-0.273603\pi$
$163$	16.6683	1.30556	0.652779	+	0.757548i	$0.273603\pi$
$164$	4.99660	0.390169
$165$	0	0
$166$	1.14087	0.0885483
$167$	−18.1651	−1.40566	−0.702830	−	0.711358i	$-0.748080\pi$
$167$	−18.1651	−1.40566	−0.702830	+	0.711358i	$0.748080\pi$
$168$	−3.29572	−0.254270
$169$	5.97928	0.459944
$170$	−8.63771	−0.662482
$171$	−10.5952	−0.810234
$172$	11.5384	0.879793
$173$	9.72494	0.739373	0.369687	−	0.929157i	$-0.379465\pi$
$173$	9.72494	0.739373	0.369687	+	0.929157i	$0.379465\pi$
$174$	−5.14385	−0.389954
$175$	6.10932	0.461821
$176$	0	0
$177$	−0.582984	−0.0438198
$178$	8.59827	0.644467
$179$	9.89238	0.739391	0.369696	−	0.929153i	$-0.379462\pi$
$179$	9.89238	0.739391	0.369696	+	0.929153i	$0.379462\pi$
$180$	−3.73108	−0.278098
$181$	15.2227	1.13149	0.565747	−	0.824579i	$-0.308588\pi$
$181$	15.2227	1.13149	0.565747	+	0.824579i	$0.308588\pi$
$182$	6.98713	0.517921
$183$	5.32937	0.393958
$184$	−24.7881	−1.82740
$185$	−1.11875	−0.0822519
$186$	0.232325	0.0170349
$187$	0	0
$188$	−2.87010	−0.209324
$189$	−6.36682	−0.463118
$190$	−4.62767	−0.335726
$191$	−2.04171	−0.147733	−0.0738665	−	0.997268i	$-0.523534\pi$
$191$	−2.04171	−0.147733	−0.0738665	+	0.997268i	$0.523534\pi$
$192$	−3.63429	−0.262282
$193$	−2.41424	−0.173781	−0.0868904	−	0.996218i	$-0.527693\pi$
$193$	−2.41424	−0.173781	−0.0868904	+	0.996218i	$0.527693\pi$
$194$	7.61986	0.547074
$195$	−3.49233	−0.250091
$196$	4.56417	0.326012
$197$	2.76526	0.197017	0.0985084	−	0.995136i	$-0.468593\pi$
$197$	2.76526	0.197017	0.0985084	+	0.995136i	$0.468593\pi$
$198$	0	0
$199$	−10.7735	−0.763711	−0.381856	−	0.924222i	$-0.624715\pi$
$199$	−10.7735	−0.763711	−0.381856	+	0.924222i	$0.624715\pi$
$200$	10.0253	0.708896
$201$	−7.31949	−0.516277
$202$	4.85689	0.341730
$203$	−15.3258	−1.07566
$204$	5.81321	0.407006
$205$	5.28213	0.368920
$206$	17.4256	1.21410
$207$	−22.1398	−1.53882
$208$	1.27435	0.0883601
$209$	0	0
$210$	−1.28569	−0.0887207
$211$	4.79889	0.330369	0.165185	−	0.986263i	$-0.447178\pi$
$211$	4.79889	0.330369	0.165185	+	0.986263i	$0.447178\pi$
$212$	−2.29292	−0.157478
$213$	3.58231	0.245456
$214$	−8.13884	−0.556360
$215$	12.1977	0.831879
$216$	−10.4478	−0.710885
$217$	0.692199	0.0469895
$218$	13.3797	0.906190
$219$	4.97559	0.336219
$220$	0	0
$221$	−33.3976	−2.24656
$222$	−0.534482	−0.0358721
$223$	2.89043	0.193558	0.0967788	−	0.995306i	$-0.469146\pi$
$223$	2.89043	0.193558	0.0967788	+	0.995306i	$0.469146\pi$
$224$	−9.69811	−0.647982
$225$	8.95424	0.596949
$226$	4.49164	0.298779
$227$	−7.52787	−0.499642	−0.249821	−	0.968292i	$-0.580372\pi$
$227$	−7.52787	−0.499642	−0.249821	+	0.968292i	$0.580372\pi$
$228$	3.11444	0.206259
$229$	23.0327	1.52204	0.761021	−	0.648727i	$-0.224698\pi$
$229$	23.0327	1.52204	0.761021	+	0.648727i	$0.224698\pi$
$230$	−9.67002	−0.637622
$231$	0	0
$232$	−25.1493	−1.65114
$233$	19.6305	1.28603	0.643017	−	0.765852i	$-0.277682\pi$
$233$	19.6305	1.28603	0.643017	+	0.765852i	$0.277682\pi$
$234$	10.2408	0.669463
$235$	−3.03411	−0.197924
$236$	−1.05183	−0.0684683
$237$	3.16212	0.205402
$238$	−12.2952	−0.796977
$239$	−3.84209	−0.248524	−0.124262	−	0.992249i	$-0.539656\pi$
$239$	−3.84209	−0.248524	−0.124262	+	0.992249i	$0.539656\pi$
$240$	−0.234490	−0.0151362
$241$	−20.2441	−1.30404	−0.652018	−	0.758204i	$-0.726077\pi$
$241$	−20.2441	−1.30404	−0.652018	+	0.758204i	$0.726077\pi$
$242$	0	0
$243$	−14.3489	−0.920483
$244$	9.61534	0.615559
$245$	4.82499	0.308257
$246$	2.52354	0.160895
$247$	−17.8928	−1.13849
$248$	1.13589	0.0721289
$249$	−0.811683	−0.0514383
$250$	9.54465	0.603657
$251$	−1.36458	−0.0861317	−0.0430659	−	0.999072i	$-0.513713\pi$
$251$	−1.36458	−0.0861317	−0.0430659	+	0.999072i	$0.513713\pi$
$252$	−5.31092	−0.334557
$253$	0	0
$254$	−7.02347	−0.440692
$255$	6.14541	0.384840
$256$	−16.5987	−1.03742
$257$	−8.25614	−0.515004	−0.257502	−	0.966278i	$-0.582899\pi$
$257$	−8.25614	−0.515004	−0.257502	+	0.966278i	$0.582899\pi$
$258$	5.82747	0.362803
$259$	−1.59245	−0.0989503
$260$	−6.30092	−0.390767
$261$	−22.4625	−1.39039
$262$	9.86698	0.609584
$263$	23.7493	1.46444	0.732222	−	0.681066i	$-0.238483\pi$
$263$	23.7493	1.46444	0.732222	+	0.681066i	$0.238483\pi$
$264$	0	0
$265$	−2.42395	−0.148902
$266$	−6.58715	−0.403884
$267$	−6.11734	−0.374375
$268$	−13.2059	−0.806681
$269$	11.5433	0.703807	0.351903	−	0.936036i	$-0.385535\pi$
$269$	11.5433	0.703807	0.351903	+	0.936036i	$0.385535\pi$
$270$	−4.07579	−0.248044
$271$	−19.5924	−1.19015	−0.595075	−	0.803670i	$-0.702878\pi$
$271$	−19.5924	−1.19015	−0.595075	+	0.803670i	$0.702878\pi$
$272$	−2.24245	−0.135969
$273$	−4.97108	−0.300863
$274$	11.2634	0.680446
$275$	0	0
$276$	6.50796	0.391733
$277$	−8.40472	−0.504991	−0.252495	−	0.967598i	$-0.581251\pi$
$277$	−8.40472	−0.504991	−0.252495	+	0.967598i	$0.581251\pi$
$278$	9.51944	0.570938
$279$	1.01453	0.0607385
$280$	−6.28598	−0.375659
$281$	13.2679	0.791494	0.395747	−	0.918360i	$-0.370486\pi$
$281$	13.2679	0.791494	0.395747	+	0.918360i	$0.370486\pi$
$282$	−1.44955	−0.0863194
$283$	−21.5391	−1.28036	−0.640182	−	0.768223i	$-0.721141\pi$
$283$	−21.5391	−1.28036	−0.640182	+	0.768223i	$0.721141\pi$
$284$	6.46326	0.383524
$285$	3.29241	0.195026
$286$	0	0
$287$	7.51873	0.443817
$288$	−14.2142	−0.837580
$289$	41.7692	2.45701
$290$	−9.81096	−0.576120
$291$	−5.42124	−0.317799
$292$	8.97703	0.525341
$293$	19.9782	1.16714	0.583570	−	0.812063i	$-0.301655\pi$
$293$	19.9782	1.16714	0.583570	+	0.812063i	$0.301655\pi$
$294$	2.30514	0.134438
$295$	−1.11194	−0.0647395
$296$	−2.61319	−0.151889
$297$	0	0
$298$	−8.86221	−0.513374
$299$	−37.3890	−2.16226
$300$	−2.63208	−0.151963
$301$	17.3626	1.00076
$302$	14.0154	0.806494
$303$	−3.45550	−0.198513
$304$	−1.20140	−0.0689048
$305$	10.1648	0.582035
$306$	−18.0206	−1.03017
$307$	−1.80464	−0.102996	−0.0514981	−	0.998673i	$-0.516400\pi$
$307$	−1.80464	−0.102996	−0.0514981	+	0.998673i	$0.516400\pi$
$308$	0	0
$309$	−12.3977	−0.705280
$310$	0.443118	0.0251674
$311$	−5.00343	−0.283718	−0.141859	−	0.989887i	$-0.545308\pi$
$311$	−5.00343	−0.283718	−0.141859	+	0.989887i	$0.545308\pi$
$312$	−8.15746	−0.461825
$313$	28.8622	1.63139	0.815694	−	0.578484i	$-0.196355\pi$
$313$	28.8622	1.63139	0.815694	+	0.578484i	$0.196355\pi$
$314$	−10.2884	−0.580606
$315$	−5.61441	−0.316336
$316$	5.70515	0.320940
$317$	−5.72136	−0.321343	−0.160672	−	0.987008i	$-0.551366\pi$
$317$	−5.72136	−0.321343	−0.160672	+	0.987008i	$0.551366\pi$
$318$	−1.15804	−0.0649398
$319$	0	0
$320$	−6.93175	−0.387496
$321$	5.79048	0.323193
$322$	−13.7646	−0.767069
$323$	31.4857	1.75191
$324$	−6.30923	−0.350513
$325$	15.1216	0.838796
$326$	15.1885	0.841212
$327$	−9.51918	−0.526412
$328$	12.3381	0.681258
$329$	−4.31884	−0.238105
$330$	0	0
$331$	−18.5418	−1.01915	−0.509575	−	0.860426i	$-0.670197\pi$
$331$	−18.5418	−1.01915	−0.509575	+	0.860426i	$0.670197\pi$
$332$	−1.46445	−0.0803722
$333$	−2.33401	−0.127903
$334$	−16.5525	−0.905710
$335$	−13.9606	−0.762749
$336$	−0.333779	−0.0182091
$337$	9.24115	0.503398	0.251699	−	0.967806i	$-0.419011\pi$
$337$	9.24115	0.503398	0.251699	+	0.967806i	$0.419011\pi$
$338$	5.44845	0.296357
$339$	−3.19563	−0.173563
$340$	11.0876	0.601312
$341$	0	0
$342$	−9.65457	−0.522060
$343$	19.1886	1.03609
$344$	28.4917	1.53617
$345$	6.87985	0.370399
$346$	8.86158	0.476401
$347$	−13.9912	−0.751086	−0.375543	−	0.926805i	$-0.622544\pi$
$347$	−13.9912	−0.751086	−0.375543	+	0.926805i	$0.622544\pi$
$348$	6.60281	0.353948
$349$	26.7805	1.43353	0.716765	−	0.697315i	$-0.245622\pi$
$349$	26.7805	1.43353	0.716765	+	0.697315i	$0.245622\pi$
$350$	5.56695	0.297566
$351$	−15.7590	−0.841151
$352$	0	0
$353$	−28.2967	−1.50608	−0.753042	−	0.657973i	$-0.771414\pi$
$353$	−28.2967	−1.50608	−0.753042	+	0.657973i	$0.771414\pi$
$354$	−0.531228	−0.0282345
$355$	6.83260	0.362637
$356$	−11.0370	−0.584960
$357$	8.74754	0.462969
$358$	9.01416	0.476413
$359$	20.3602	1.07457	0.537286	−	0.843400i	$-0.319450\pi$
$359$	20.3602	1.07457	0.537286	+	0.843400i	$0.319450\pi$
$360$	−9.21316	−0.485576
$361$	−2.13149	−0.112184
$362$	13.8713	0.729057
$363$	0	0
$364$	−8.96890	−0.470098
$365$	9.49003	0.496731
$366$	4.85624	0.253840
$367$	9.30064	0.485489	0.242745	−	0.970090i	$-0.421952\pi$
$367$	9.30064	0.485489	0.242745	+	0.970090i	$0.421952\pi$
$368$	−2.51045	−0.130866
$369$	11.0200	0.573677
$370$	−1.01943	−0.0529975
$371$	−3.45032	−0.179132
$372$	−0.298220	−0.0154620
$373$	−19.5758	−1.01360	−0.506798	−	0.862065i	$-0.669171\pi$
$373$	−19.5758	−1.01360	−0.506798	+	0.862065i	$0.669171\pi$
$374$	0	0
$375$	−6.79066	−0.350668
$376$	−7.08714	−0.365492
$377$	−37.9339	−1.95370
$378$	−5.80158	−0.298401
$379$	1.95880	0.100617	0.0503083	−	0.998734i	$-0.483980\pi$
$379$	1.95880	0.100617	0.0503083	+	0.998734i	$0.483980\pi$
$380$	5.94022	0.304727
$381$	4.99693	0.256001
$382$	−1.86045	−0.0951891
$383$	−13.5977	−0.694809	−0.347404	−	0.937715i	$-0.612937\pi$
$383$	−13.5977	−0.694809	−0.347404	+	0.937715i	$0.612937\pi$
$384$	3.83264	0.195583
$385$	0	0
$386$	−2.19991	−0.111972
$387$	25.4478	1.29358
$388$	−9.78109	−0.496560
$389$	36.2978	1.84037	0.920186	−	0.391481i	$-0.128037\pi$
$389$	36.2978	1.84037	0.920186	+	0.391481i	$0.128037\pi$
$390$	−3.18229	−0.161142
$391$	65.7928	3.32729
$392$	11.2703	0.569236
$393$	−7.01999	−0.354111
$394$	2.51977	0.126944
$395$	6.03117	0.303461
$396$	0	0
$397$	5.93615	0.297927	0.148963	−	0.988843i	$-0.452406\pi$
$397$	5.93615	0.297927	0.148963	+	0.988843i	$0.452406\pi$
$398$	−9.81703	−0.492083
$399$	4.68651	0.234619
$400$	1.01533	0.0507664
$401$	−10.2104	−0.509882	−0.254941	−	0.966957i	$-0.582056\pi$
$401$	−10.2104	−0.509882	−0.254941	+	0.966957i	$0.582056\pi$
$402$	−6.66968	−0.332653
$403$	1.71331	0.0853460
$404$	−6.23446	−0.310176
$405$	−6.66977	−0.331424
$406$	−13.9652	−0.693081
$407$	0	0
$408$	14.3546	0.710657
$409$	3.46953	0.171557	0.0857785	−	0.996314i	$-0.472662\pi$
$409$	3.46953	0.171557	0.0857785	+	0.996314i	$0.472662\pi$
$410$	4.81320	0.237707
$411$	−8.01347	−0.395275
$412$	−22.3681	−1.10200
$413$	−1.58276	−0.0778826
$414$	−20.1743	−0.991512
$415$	−1.54814	−0.0759951
$416$	−24.0045	−1.17692
$417$	−6.77272	−0.331662
$418$	0	0
$419$	23.4202	1.14415	0.572075	−	0.820201i	$-0.306139\pi$
$419$	23.4202	1.14415	0.572075	+	0.820201i	$0.306139\pi$
$420$	1.65035	0.0805287
$421$	23.6349	1.15189	0.575946	−	0.817488i	$-0.304634\pi$
$421$	23.6349	1.15189	0.575946	+	0.817488i	$0.304634\pi$
$422$	4.37286	0.212867
$423$	−6.32998	−0.307774
$424$	−5.66191	−0.274967
$425$	−26.6093	−1.29074
$426$	3.26428	0.158155
$427$	14.4689	0.700197
$428$	10.4473	0.504988
$429$	0	0
$430$	11.1148	0.536005
$431$	27.4826	1.32379	0.661895	−	0.749596i	$-0.269752\pi$
$431$	27.4826	1.32379	0.661895	+	0.749596i	$0.269752\pi$
$432$	−1.05812	−0.0509089
$433$	−15.6838	−0.753714	−0.376857	−	0.926271i	$-0.622995\pi$
$433$	−15.6838	−0.753714	−0.376857	+	0.926271i	$0.622995\pi$
$434$	0.630747	0.0302768
$435$	6.98013	0.334672
$436$	−17.1747	−0.822517
$437$	35.2486	1.68617
$438$	4.53386	0.216636
$439$	−15.2516	−0.727919	−0.363959	−	0.931415i	$-0.618575\pi$
$439$	−15.2516	−0.727919	−0.363959	+	0.931415i	$0.618575\pi$
$440$	0	0
$441$	10.0662	0.479345
$442$	−30.4326	−1.44753
$443$	−23.2299	−1.10369	−0.551843	−	0.833948i	$-0.686075\pi$
$443$	−23.2299	−1.10369	−0.551843	+	0.833948i	$0.686075\pi$
$444$	0.686078	0.0325598
$445$	−11.6677	−0.553103
$446$	2.63383	0.124715
$447$	6.30513	0.298222
$448$	−9.86684	−0.466164
$449$	−33.1905	−1.56636	−0.783178	−	0.621798i	$-0.786403\pi$
$449$	−33.1905	−1.56636	−0.783178	+	0.621798i	$0.786403\pi$
$450$	8.15930	0.384633
$451$	0	0
$452$	−5.76561	−0.271192
$453$	−9.97141	−0.468498
$454$	−6.85956	−0.321935
$455$	−9.48143	−0.444496
$456$	7.69048	0.360140
$457$	−30.0421	−1.40531	−0.702656	−	0.711530i	$-0.748002\pi$
$457$	−30.0421	−1.40531	−0.702656	+	0.711530i	$0.748002\pi$
$458$	20.9879	0.980700
$459$	27.7308	1.29436
$460$	12.4127	0.578747
$461$	30.2629	1.40949	0.704743	−	0.709463i	$-0.251062\pi$
$461$	30.2629	1.40949	0.704743	+	0.709463i	$0.251062\pi$
$462$	0	0
$463$	−20.9120	−0.971865	−0.485933	−	0.873996i	$-0.661520\pi$
$463$	−20.9120	−0.971865	−0.485933	+	0.873996i	$0.661520\pi$
$464$	−2.54704	−0.118243
$465$	−0.315262	−0.0146199
$466$	17.8877	0.828632
$467$	20.1463	0.932261	0.466130	−	0.884716i	$-0.345648\pi$
$467$	20.1463	0.932261	0.466130	+	0.884716i	$0.345648\pi$
$468$	−13.1454	−0.607648
$469$	−19.8719	−0.917599
$470$	−2.76475	−0.127528
$471$	7.31979	0.337278
$472$	−2.59729	−0.119550
$473$	0	0
$474$	2.88139	0.132347
$475$	−14.2560	−0.654109
$476$	15.7825	0.723388
$477$	−5.05702	−0.231545
$478$	−3.50099	−0.160132
$479$	−22.1847	−1.01364	−0.506822	−	0.862051i	$-0.669180\pi$
$479$	−22.1847	−1.01364	−0.506822	+	0.862051i	$0.669180\pi$
$480$	4.41701	0.201608
$481$	−3.94160	−0.179721
$482$	−18.4469	−0.840231
$483$	9.79297	0.445596
$484$	0	0
$485$	−10.3400	−0.469517
$486$	−13.0750	−0.593096
$487$	−22.0243	−0.998017	−0.499008	−	0.866597i	$-0.666302\pi$
$487$	−22.0243	−0.998017	−0.499008	+	0.866597i	$0.666302\pi$
$488$	23.7432	1.07480
$489$	−10.8060	−0.488666
$490$	4.39664	0.198620
$491$	−20.9390	−0.944964	−0.472482	−	0.881340i	$-0.656642\pi$
$491$	−20.9390	−0.944964	−0.472482	+	0.881340i	$0.656642\pi$
$492$	−3.23930	−0.146039
$493$	66.7518	3.00635
$494$	−16.3043	−0.733566
$495$	0	0
$496$	0.115039	0.00516539
$497$	9.72571	0.436258
$498$	−0.739623	−0.0331433
$499$	−9.82320	−0.439747	−0.219873	−	0.975528i	$-0.570564\pi$
$499$	−9.82320	−0.439747	−0.219873	+	0.975528i	$0.570564\pi$
$500$	−12.2518	−0.547918
$501$	11.7765	0.526133
$502$	−1.24344	−0.0554974
$503$	5.72421	0.255230	0.127615	−	0.991824i	$-0.459268\pi$
$503$	5.72421	0.255230	0.127615	+	0.991824i	$0.459268\pi$
$504$	−13.1143	−0.584156
$505$	−6.59073	−0.293284
$506$	0	0
$507$	−3.87637	−0.172155
$508$	9.01555	0.400000
$509$	−2.68111	−0.118838	−0.0594191	−	0.998233i	$-0.518925\pi$
$509$	−2.68111	−0.118838	−0.0594191	+	0.998233i	$0.518925\pi$
$510$	5.59983	0.247965
$511$	13.5084	0.597575
$512$	−3.30149	−0.145907
$513$	14.8568	0.655945
$514$	−7.52318	−0.331833
$515$	−23.6463	−1.04198
$516$	−7.48033	−0.329303
$517$	0	0
$518$	−1.45108	−0.0637568
$519$	−6.30468	−0.276745
$520$	−15.5589	−0.682302
$521$	−11.4228	−0.500441	−0.250220	−	0.968189i	$-0.580503\pi$
$521$	−11.4228	−0.500441	−0.250220	+	0.968189i	$0.580503\pi$
$522$	−20.4683	−0.895875
$523$	−27.7334	−1.21269	−0.606347	−	0.795200i	$-0.707366\pi$
$523$	−27.7334	−1.21269	−0.606347	+	0.795200i	$0.707366\pi$
$524$	−12.6656	−0.553298
$525$	−3.96068	−0.172858
$526$	21.6409	0.943588
$527$	−3.01489	−0.131331
$528$	0	0
$529$	50.6558	2.20243
$530$	−2.20876	−0.0959423
$531$	−2.31980	−0.100671
$532$	8.45547	0.366591
$533$	18.6101	0.806095
$534$	−5.57426	−0.241222
$535$	11.0443	0.477486
$536$	−32.6094	−1.40851
$537$	−6.41323	−0.276751
$538$	10.5185	0.453485
$539$	0	0
$540$	5.23181	0.225141
$541$	17.6130	0.757244	0.378622	−	0.925551i	$-0.376398\pi$
$541$	17.6130	0.757244	0.378622	+	0.925551i	$0.376398\pi$
$542$	−17.8530	−0.766852
$543$	−9.86888	−0.423514
$544$	42.2403	1.81104
$545$	−18.1561	−0.777722
$546$	−4.52976	−0.193856
$547$	−26.2452	−1.12216	−0.561082	−	0.827760i	$-0.689615\pi$
$547$	−26.2452	−1.12216	−0.561082	+	0.827760i	$0.689615\pi$
$548$	−14.4580	−0.617617
$549$	21.2066	0.905074
$550$	0	0
$551$	35.7624	1.52353
$552$	16.0701	0.683989
$553$	8.58493	0.365068
$554$	−7.65857	−0.325382
$555$	0.725284	0.0307866
$556$	−12.2195	−0.518220
$557$	−6.37990	−0.270325	−0.135162	−	0.990823i	$-0.543156\pi$
$557$	−6.37990	−0.270325	−0.135162	+	0.990823i	$0.543156\pi$
$558$	0.924465	0.0391357
$559$	42.9754	1.81766
$560$	−0.636623	−0.0269022
$561$	0	0
$562$	12.0900	0.509984
$563$	2.46832	0.104027	0.0520137	−	0.998646i	$-0.483436\pi$
$563$	2.46832	0.104027	0.0520137	+	0.998646i	$0.483436\pi$
$564$	1.86069	0.0783491
$565$	−6.09509	−0.256422
$566$	−19.6269	−0.824979
$567$	−9.49393	−0.398708
$568$	15.9597	0.669656
$569$	−25.6500	−1.07530	−0.537652	−	0.843167i	$-0.680689\pi$
$569$	−25.6500	−1.07530	−0.537652	+	0.843167i	$0.680689\pi$
$570$	3.00012	0.125661
$571$	−19.6474	−0.822217	−0.411108	−	0.911586i	$-0.634858\pi$
$571$	−19.6474	−0.822217	−0.411108	+	0.911586i	$0.634858\pi$
$572$	0	0
$573$	1.32364	0.0552960
$574$	6.85124	0.285965
$575$	−29.7894	−1.24230
$576$	−14.4615	−0.602563
$577$	26.7327	1.11290	0.556449	−	0.830882i	$-0.312164\pi$
$577$	26.7327	1.11290	0.556449	+	0.830882i	$0.312164\pi$
$578$	38.0611	1.58313
$579$	1.56515	0.0650455
$580$	12.5937	0.522923
$581$	−2.20366	−0.0914233
$582$	−4.93996	−0.204768
$583$	0	0
$584$	22.1670	0.917277
$585$	−13.8966	−0.574555
$586$	18.2046	0.752025
$587$	25.0629	1.03446	0.517228	−	0.855847i	$-0.326964\pi$
$587$	25.0629	1.03446	0.517228	+	0.855847i	$0.326964\pi$
$588$	−2.95895	−0.122025
$589$	−1.61523	−0.0665544
$590$	−1.01322	−0.0417137
$591$	−1.79272	−0.0737426
$592$	−0.264655	−0.0108773
$593$	18.2192	0.748173	0.374087	−	0.927394i	$-0.377956\pi$
$593$	18.2192	0.748173	0.374087	+	0.927394i	$0.377956\pi$
$594$	0	0
$595$	16.6843	0.683992
$596$	11.3758	0.465972
$597$	6.98444	0.285854
$598$	−34.0697	−1.39321
$599$	−37.0590	−1.51419	−0.757095	−	0.653305i	$-0.773382\pi$
$599$	−37.0590	−1.51419	−0.757095	+	0.653305i	$0.773382\pi$
$600$	−6.49940	−0.265337
$601$	−6.61709	−0.269917	−0.134958	−	0.990851i	$-0.543090\pi$
$601$	−6.61709	−0.269917	−0.134958	+	0.990851i	$0.543090\pi$
$602$	15.8212	0.644823
$603$	−29.1256	−1.18609
$604$	−17.9906	−0.732026
$605$	0	0
$606$	−3.14873	−0.127908
$607$	13.2874	0.539320	0.269660	−	0.962956i	$-0.413089\pi$
$607$	13.2874	0.539320	0.269660	+	0.962956i	$0.413089\pi$
$608$	22.6303	0.917781
$609$	9.93570	0.402615
$610$	9.26240	0.375024
$611$	−10.6899	−0.432466
$612$	23.1318	0.935049
$613$	−4.04426	−0.163346	−0.0816730	−	0.996659i	$-0.526026\pi$
$613$	−4.04426	−0.163346	−0.0816730	+	0.996659i	$0.526026\pi$
$614$	−1.64443	−0.0663636
$615$	−3.42441	−0.138085
$616$	0	0
$617$	16.2380	0.653719	0.326859	−	0.945073i	$-0.394010\pi$
$617$	16.2380	0.653719	0.326859	+	0.945073i	$0.394010\pi$
$618$	−11.2970	−0.454434
$619$	17.2559	0.693572	0.346786	−	0.937944i	$-0.387273\pi$
$619$	17.2559	0.693572	0.346786	+	0.937944i	$0.387273\pi$
$620$	−0.568801	−0.0228436
$621$	31.0450	1.24579
$622$	−4.55924	−0.182809
$623$	−16.6082	−0.665392
$624$	−0.826159	−0.0330728
$625$	4.40320	0.176128
$626$	26.2999	1.05115
$627$	0	0
$628$	13.2065	0.526996
$629$	6.93598	0.276555
$630$	−5.11598	−0.203826
$631$	31.8528	1.26804	0.634020	−	0.773317i	$-0.281404\pi$
$631$	31.8528	1.26804	0.634020	+	0.773317i	$0.281404\pi$
$632$	14.0877	0.560380
$633$	−3.11112	−0.123656
$634$	−5.21343	−0.207052
$635$	9.53074	0.378216
$636$	1.48650	0.0589436
$637$	16.9995	0.673545
$638$	0	0
$639$	14.2547	0.563906
$640$	7.31006	0.288955
$641$	13.6181	0.537881	0.268941	−	0.963157i	$-0.413326\pi$
$641$	13.6181	0.537881	0.268941	+	0.963157i	$0.413326\pi$
$642$	5.27641	0.208243
$643$	−33.8320	−1.33420	−0.667102	−	0.744967i	$-0.732465\pi$
$643$	−33.8320	−1.33420	−0.667102	+	0.744967i	$0.732465\pi$
$644$	17.6686	0.696242
$645$	−7.90779	−0.311369
$646$	28.6905	1.12881
$647$	35.6321	1.40084	0.700421	−	0.713730i	$-0.252996\pi$
$647$	35.6321	1.40084	0.700421	+	0.713730i	$0.252996\pi$
$648$	−15.5794	−0.612016
$649$	0	0
$650$	13.7792	0.540463
$651$	−0.448752	−0.0175880
$652$	−19.4964	−0.763539
$653$	−23.5118	−0.920089	−0.460045	−	0.887896i	$-0.652167\pi$
$653$	−23.5118	−0.920089	−0.460045	+	0.887896i	$0.652167\pi$
$654$	−8.67408	−0.339184
$655$	−13.3894	−0.523165
$656$	1.24956	0.0487872
$657$	19.7988	0.772424
$658$	−3.93542	−0.153419
$659$	−29.5078	−1.14946	−0.574730	−	0.818343i	$-0.694893\pi$
$659$	−29.5078	−1.14946	−0.574730	+	0.818343i	$0.694893\pi$
$660$	0	0
$661$	29.3035	1.13977	0.569887	−	0.821723i	$-0.306987\pi$
$661$	29.3035	1.13977	0.569887	+	0.821723i	$0.306987\pi$
$662$	−16.8957	−0.656670
$663$	21.6517	0.840881
$664$	−3.61617	−0.140335
$665$	8.93866	0.346626
$666$	−2.12680	−0.0824119
$667$	74.7294	2.89353
$668$	21.2473	0.822082
$669$	−1.87387	−0.0724479
$670$	−12.7212	−0.491463
$671$	0	0
$672$	6.28729	0.242537
$673$	−49.6188	−1.91266	−0.956331	−	0.292285i	$-0.905585\pi$
$673$	−49.6188	−1.91266	−0.956331	+	0.292285i	$0.905585\pi$
$674$	8.42074	0.324355
$675$	−12.5558	−0.483275
$676$	−6.99380	−0.268992
$677$	−3.16016	−0.121455	−0.0607274	−	0.998154i	$-0.519342\pi$
$677$	−3.16016	−0.121455	−0.0607274	+	0.998154i	$0.519342\pi$
$678$	−2.91193	−0.111832
$679$	−14.7183	−0.564836
$680$	27.3787	1.04993
$681$	4.88032	0.187014
$682$	0	0
$683$	10.2494	0.392181	0.196091	−	0.980586i	$-0.437175\pi$
$683$	10.2494	0.392181	0.196091	+	0.980586i	$0.437175\pi$
$684$	12.3929	0.473855
$685$	−15.2842	−0.583981
$686$	17.4851	0.667585
$687$	−14.9321	−0.569695
$688$	2.88554	0.110010
$689$	−8.54012	−0.325353
$690$	6.26908	0.238660
$691$	−23.5290	−0.895085	−0.447543	−	0.894263i	$-0.647701\pi$
$691$	−23.5290	−0.895085	−0.447543	+	0.894263i	$0.647701\pi$
$692$	−11.3750	−0.432413
$693$	0	0
$694$	−12.7491	−0.483948
$695$	−12.9177	−0.489998
$696$	16.3043	0.618014
$697$	−32.7480	−1.24042
$698$	24.4030	0.923668
$699$	−12.7264	−0.481358
$700$	−7.14592	−0.270090
$701$	−40.1722	−1.51728	−0.758641	−	0.651508i	$-0.774136\pi$
$701$	−40.1722	−1.51728	−0.758641	+	0.651508i	$0.774136\pi$
$702$	−14.3599	−0.541980
$703$	3.71596	0.140150
$704$	0	0
$705$	1.96702	0.0740821
$706$	−25.7846	−0.970417
$707$	−9.38143	−0.352825
$708$	0.681902	0.0256274
$709$	23.4787	0.881760	0.440880	−	0.897566i	$-0.354666\pi$
$709$	23.4787	0.881760	0.440880	+	0.897566i	$0.354666\pi$
$710$	6.22602	0.233658
$711$	12.5827	0.471887
$712$	−27.2537	−1.02138
$713$	−3.37520	−0.126402
$714$	7.97095	0.298305
$715$	0	0
$716$	−11.5709	−0.432423
$717$	2.49083	0.0930216
$718$	18.5527	0.692380
$719$	−41.2951	−1.54005	−0.770024	−	0.638015i	$-0.779756\pi$
$719$	−41.2951	−1.54005	−0.770024	+	0.638015i	$0.779756\pi$
$720$	−0.933077	−0.0347737
$721$	−33.6588	−1.25352
$722$	−1.94226	−0.0722834
$723$	13.1242	0.488096
$724$	−17.8056	−0.661739
$725$	−30.2236	−1.12248
$726$	0	0
$727$	2.34426	0.0869439	0.0434719	−	0.999055i	$-0.486158\pi$
$727$	2.34426	0.0869439	0.0434719	+	0.999055i	$0.486158\pi$
$728$	−22.1469	−0.820820
$729$	−6.87962	−0.254801
$730$	8.64752	0.320059
$731$	−75.6231	−2.79702
$732$	−6.23363	−0.230401
$733$	−2.15334	−0.0795356	−0.0397678	−	0.999209i	$-0.512662\pi$
$733$	−2.15334	−0.0795356	−0.0397678	+	0.999209i	$0.512662\pi$
$734$	8.47495	0.312816
$735$	−3.12804	−0.115380
$736$	47.2885	1.74308
$737$	0	0
$738$	10.0416	0.369638
$739$	27.6717	1.01792	0.508959	−	0.860791i	$-0.330030\pi$
$739$	27.6717	1.01792	0.508959	+	0.860791i	$0.330030\pi$
$740$	1.30857	0.0481040
$741$	11.5999	0.426133
$742$	−3.14401	−0.115420
$743$	−15.9710	−0.585918	−0.292959	−	0.956125i	$-0.594640\pi$
$743$	−15.9710	−0.585918	−0.292959	+	0.956125i	$0.594640\pi$
$744$	−0.736395	−0.0269976
$745$	12.0259	0.440595
$746$	−17.8379	−0.653092
$747$	−3.22984	−0.118174
$748$	0	0
$749$	15.7207	0.574423
$750$	−6.18780	−0.225946
$751$	−22.3878	−0.816944	−0.408472	−	0.912771i	$-0.633938\pi$
$751$	−22.3878	−0.816944	−0.408472	+	0.912771i	$0.633938\pi$
$752$	−0.717762	−0.0261741
$753$	0.884660	0.0322388
$754$	−34.5662	−1.25883
$755$	−19.0187	−0.692160
$756$	7.44710	0.270848
$757$	8.73068	0.317322	0.158661	−	0.987333i	$-0.449282\pi$
$757$	8.73068	0.317322	0.158661	+	0.987333i	$0.449282\pi$
$758$	1.78490	0.0648305
$759$	0	0
$760$	14.6682	0.532071
$761$	14.6810	0.532187	0.266093	−	0.963947i	$-0.414267\pi$
$761$	14.6810	0.532187	0.266093	+	0.963947i	$0.414267\pi$
$762$	4.55332	0.164949
$763$	−25.8439	−0.935612
$764$	2.38814	0.0863998
$765$	24.4537	0.884126
$766$	−12.3905	−0.447687
$767$	−3.91760	−0.141456
$768$	10.7610	0.388303
$769$	31.1633	1.12378	0.561889	−	0.827213i	$-0.310075\pi$
$769$	31.1633	1.12378	0.561889	+	0.827213i	$0.310075\pi$
$770$	0	0
$771$	5.35246	0.192764
$772$	2.82387	0.101633
$773$	48.3814	1.74016	0.870079	−	0.492912i	$-0.164067\pi$
$773$	48.3814	1.74016	0.870079	+	0.492912i	$0.164067\pi$
$774$	23.1886	0.833497
$775$	1.36507	0.0490347
$776$	−24.1525	−0.867023
$777$	1.03239	0.0370367
$778$	33.0754	1.18581
$779$	−17.5448	−0.628608
$780$	4.08489	0.146263
$781$	0	0
$782$	59.9519	2.14388
$783$	31.4974	1.12563
$784$	1.14142	0.0407649
$785$	13.9612	0.498295
$786$	−6.39677	−0.228165
$787$	33.5759	1.19685	0.598426	−	0.801178i	$-0.295793\pi$
$787$	33.5759	1.19685	0.598426	+	0.801178i	$0.295793\pi$
$788$	−3.23445	−0.115223
$789$	−15.3967	−0.548136
$790$	5.49574	0.195530
$791$	−8.67592	−0.308480
$792$	0	0
$793$	35.8129	1.27175
$794$	5.40915	0.191964
$795$	1.57145	0.0557335
$796$	12.6014	0.446647
$797$	30.5323	1.08151	0.540755	−	0.841180i	$-0.318139\pi$
$797$	30.5323	1.08151	0.540755	+	0.841180i	$0.318139\pi$
$798$	4.27045	0.151172
$799$	18.8108	0.665478
$800$	−19.1254	−0.676185
$801$	−24.3420	−0.860084
$802$	−9.30393	−0.328533
$803$	0	0
$804$	8.56141	0.301938
$805$	18.6783	0.658324
$806$	1.56121	0.0549911
$807$	−7.48352	−0.263432
$808$	−15.3948	−0.541586
$809$	18.3141	0.643890	0.321945	−	0.946758i	$-0.395663\pi$
$809$	18.3141	0.643890	0.321945	+	0.946758i	$0.395663\pi$
$810$	−6.07765	−0.213547
$811$	55.3859	1.94486	0.972430	−	0.233194i	$-0.0749177\pi$
$811$	55.3859	1.94486	0.972430	+	0.233194i	$0.0749177\pi$
$812$	17.9262	0.629085
$813$	12.7017	0.445469
$814$	0	0
$815$	−20.6106	−0.721956
$816$	1.45378	0.0508925
$817$	−40.5152	−1.41745
$818$	3.16151	0.110540
$819$	−19.7808	−0.691199
$820$	−6.17837	−0.215758
$821$	−45.0552	−1.57244	−0.786219	−	0.617948i	$-0.787964\pi$
$821$	−45.0552	−1.57244	−0.786219	+	0.617948i	$0.787964\pi$
$822$	−7.30205	−0.254688
$823$	−9.59401	−0.334426	−0.167213	−	0.985921i	$-0.553477\pi$
$823$	−9.59401	−0.334426	−0.167213	+	0.985921i	$0.553477\pi$
$824$	−55.2336	−1.92415
$825$	0	0
$826$	−1.44225	−0.0501822
$827$	23.7350	0.825346	0.412673	−	0.910879i	$-0.364595\pi$
$827$	23.7350	0.825346	0.412673	+	0.910879i	$0.364595\pi$
$828$	25.8964	0.899961
$829$	13.2255	0.459342	0.229671	−	0.973268i	$-0.426235\pi$
$829$	13.2255	0.459342	0.229671	+	0.973268i	$0.426235\pi$
$830$	−1.41070	−0.0489660
$831$	5.44878	0.189016
$832$	−24.4221	−0.846684
$833$	−29.9138	−1.03645
$834$	−6.17145	−0.213700
$835$	22.4614	0.777310
$836$	0	0
$837$	−1.42260	−0.0491723
$838$	21.3410	0.737212
$839$	24.1741	0.834584	0.417292	−	0.908773i	$-0.362979\pi$
$839$	24.1741	0.834584	0.417292	+	0.908773i	$0.362979\pi$
$840$	4.07520	0.140608
$841$	46.8186	1.61443
$842$	21.5366	0.742201
$843$	−8.60155	−0.296253
$844$	−5.61314	−0.193212
$845$	−7.39347	−0.254343
$846$	−5.76802	−0.198309
$847$	0	0
$848$	−0.573419	−0.0196913
$849$	13.9638	0.479236
$850$	−24.2470	−0.831665
$851$	7.76490	0.266177
$852$	−4.19013	−0.143552
$853$	9.79676	0.335435	0.167717	−	0.985835i	$-0.446360\pi$
$853$	9.79676	0.335435	0.167717	+	0.985835i	$0.446360\pi$
$854$	13.1844	0.451159
$855$	13.1011	0.448049
$856$	25.7975	0.881739
$857$	12.7483	0.435472	0.217736	−	0.976008i	$-0.430133\pi$
$857$	12.7483	0.435472	0.217736	+	0.976008i	$0.430133\pi$
$858$	0	0
$859$	−12.6179	−0.430516	−0.215258	−	0.976557i	$-0.569059\pi$
$859$	−12.6179	−0.430516	−0.215258	+	0.976557i	$0.569059\pi$
$860$	−14.2674	−0.486513
$861$	−4.87440	−0.166119
$862$	25.0428	0.852960
$863$	−17.5109	−0.596076	−0.298038	−	0.954554i	$-0.596332\pi$
$863$	−17.5109	−0.596076	−0.298038	+	0.954554i	$0.596332\pi$
$864$	19.9315	0.678083
$865$	−12.0250	−0.408863
$866$	−14.2914	−0.485642
$867$	−27.0790	−0.919651
$868$	−0.809647	−0.0274812
$869$	0	0
$870$	6.36045	0.215639
$871$	−49.1863	−1.66661
$872$	−42.4094	−1.43616
$873$	−21.5721	−0.730106
$874$	32.1193	1.08645
$875$	−18.4362	−0.623256
$876$	−5.81981	−0.196633
$877$	8.97097	0.302928	0.151464	−	0.988463i	$-0.451601\pi$
$877$	8.97097	0.302928	0.151464	+	0.988463i	$0.451601\pi$
$878$	−13.8976	−0.469021
$879$	−12.9519	−0.436856
$880$	0	0
$881$	37.1472	1.25152	0.625761	−	0.780015i	$-0.284789\pi$
$881$	37.1472	1.25152	0.625761	+	0.780015i	$0.284789\pi$
$882$	9.17258	0.308857
$883$	55.6971	1.87436	0.937178	−	0.348850i	$-0.113428\pi$
$883$	55.6971	1.87436	0.937178	+	0.348850i	$0.113428\pi$
$884$	39.0643	1.31387
$885$	0.720869	0.0242317
$886$	−21.1676	−0.711140
$887$	14.3846	0.482986	0.241493	−	0.970403i	$-0.422363\pi$
$887$	14.3846	0.482986	0.241493	+	0.970403i	$0.422363\pi$
$888$	1.69413	0.0568514
$889$	13.5663	0.455000
$890$	−10.6319	−0.356382
$891$	0	0
$892$	−3.38086	−0.113200
$893$	10.0779	0.337245
$894$	5.74538	0.192154
$895$	−12.2321	−0.408873
$896$	10.4053	0.347618
$897$	24.2393	0.809326
$898$	−30.2439	−1.00925
$899$	−3.42439	−0.114210
$900$	−10.4735	−0.349118
$901$	15.0279	0.500653
$902$	0	0
$903$	−11.2562	−0.374582
$904$	−14.2370	−0.473517
$905$	−18.8231	−0.625701
$906$	−9.08617	−0.301868
$907$	−25.5035	−0.846829	−0.423414	−	0.905936i	$-0.639169\pi$
$907$	−25.5035	−0.846829	−0.423414	+	0.905936i	$0.639169\pi$
$908$	8.80515	0.292209
$909$	−13.7501	−0.456061
$910$	−8.63969	−0.286403
$911$	19.4116	0.643134	0.321567	−	0.946887i	$-0.395790\pi$
$911$	19.4116	0.643134	0.321567	+	0.946887i	$0.395790\pi$
$912$	0.778865	0.0257908
$913$	0	0
$914$	−27.3750	−0.905486
$915$	−6.58985	−0.217854
$916$	−26.9407	−0.890147
$917$	−19.0588	−0.629376
$918$	25.2689	0.833999
$919$	59.3556	1.95796	0.978980	−	0.203957i	$-0.0653802\pi$
$919$	59.3556	1.95796	0.978980	+	0.203957i	$0.0653802\pi$
$920$	30.6508	1.01053
$921$	1.16995	0.0385511
$922$	27.5763	0.908176
$923$	24.0728	0.792366
$924$	0	0
$925$	−3.14044	−0.103257
$926$	−19.0555	−0.626203
$927$	−49.3327	−1.62030
$928$	47.9778	1.57495
$929$	29.8277	0.978614	0.489307	−	0.872112i	$-0.337250\pi$
$929$	29.8277	0.978614	0.489307	+	0.872112i	$0.337250\pi$
$930$	−0.287274	−0.00942008
$931$	−16.0264	−0.525243
$932$	−22.9612	−0.752121
$933$	3.24372	0.106195
$934$	18.3578	0.600685
$935$	0	0
$936$	−32.4600	−1.06099
$937$	21.4289	0.700052	0.350026	−	0.936740i	$-0.386173\pi$
$937$	21.4289	0.700052	0.350026	+	0.936740i	$0.386173\pi$
$938$	−18.1077	−0.591238
$939$	−18.7114	−0.610623
$940$	3.54892	0.115753
$941$	15.1292	0.493198	0.246599	−	0.969118i	$-0.420687\pi$
$941$	15.1292	0.493198	0.246599	+	0.969118i	$0.420687\pi$
$942$	6.66995	0.217319
$943$	−36.6618	−1.19387
$944$	−0.263044	−0.00856136
$945$	7.87266	0.256098
$946$	0	0
$947$	−29.6808	−0.964497	−0.482248	−	0.876035i	$-0.660180\pi$
$947$	−29.6808	−0.964497	−0.482248	+	0.876035i	$0.660180\pi$
$948$	−3.69865	−0.120127
$949$	33.4355	1.08536
$950$	−12.9904	−0.421463
$951$	3.70916	0.120278
$952$	38.9716	1.26308
$953$	53.8119	1.74314	0.871569	−	0.490273i	$-0.163103\pi$
$953$	53.8119	1.74314	0.871569	+	0.490273i	$0.163103\pi$
$954$	−4.60807	−0.149192
$955$	2.52461	0.0816944
$956$	4.49399	0.145346
$957$	0	0
$958$	−20.2152	−0.653123
$959$	−21.7560	−0.702538
$960$	4.49385	0.145038
$961$	−30.8453	−0.995011
$962$	−3.59167	−0.115800
$963$	23.0414	0.742498
$964$	23.6790	0.762648
$965$	2.98524	0.0960984
$966$	8.92357	0.287111
$967$	30.5564	0.982626	0.491313	−	0.870983i	$-0.336517\pi$
$967$	30.5564	0.982626	0.491313	+	0.870983i	$0.336517\pi$
$968$	0	0
$969$	−20.4122	−0.655734
$970$	−9.42207	−0.302524
$971$	5.67439	0.182100	0.0910500	−	0.995846i	$-0.470978\pi$
$971$	5.67439	0.182100	0.0910500	+	0.995846i	$0.470978\pi$
$972$	16.7835	0.538333
$973$	−18.3875	−0.589475
$974$	−20.0690	−0.643054
$975$	−9.80335	−0.313958
$976$	2.40463	0.0769702
$977$	−30.8177	−0.985946	−0.492973	−	0.870045i	$-0.664090\pi$
$977$	−30.8177	−0.985946	−0.492973	+	0.870045i	$0.664090\pi$
$978$	−9.84670	−0.314863
$979$	0	0
$980$	−5.64366	−0.180280
$981$	−37.8786	−1.20937
$982$	−19.0801	−0.608870
$983$	−15.8060	−0.504135	−0.252067	−	0.967710i	$-0.581110\pi$
$983$	−15.8060	−0.504135	−0.252067	+	0.967710i	$0.581110\pi$
$984$	−7.99880	−0.254993
$985$	−3.41929	−0.108948
$986$	60.8257	1.93709
$987$	2.79990	0.0891219
$988$	20.9287	0.665832
$989$	−84.6610	−2.69206
$990$	0	0
$991$	−1.55511	−0.0493996	−0.0246998	−	0.999695i	$-0.507863\pi$
$991$	−1.55511	−0.0493996	−0.0246998	+	0.999695i	$0.507863\pi$
$992$	−2.16695	−0.0688006
$993$	12.0207	0.381464
$994$	8.86229	0.281095
$995$	13.3216	0.422322
$996$	0.949404	0.0300830
$997$	15.0959	0.478092	0.239046	−	0.971008i	$-0.423165\pi$
$997$	15.0959	0.478092	0.239046	+	0.971008i	$0.423165\pi$
$998$	−8.95112	−0.283343
$999$	3.27280	0.103547

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1331.2.a.f.1.17	yes	30
11.10	odd	2	inner	1331.2.a.f.1.14	✓	30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1331.2.a.f.1.14	✓	30	11.10	odd	2	inner
1331.2.a.f.1.17	yes	30	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:	\( N \)	\(=\)	\( 1331 = 11^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1331.a (trivial)

\(f(q)\)	\(=\)	\(q+0.911222 q^{2} -0.648300 q^{3} -1.16967 q^{4} -1.23652 q^{5} -0.590746 q^{6} -1.76009 q^{7} -2.88828 q^{8} -2.57971 q^{9} +O(q^{10})\)	\(q+0.911222 q^{2} -0.648300 q^{3} -1.16967 q^{4} -1.23652 q^{5} -0.590746 q^{6} -1.76009 q^{7} -2.88828 q^{8} -2.57971 q^{9} -1.12674 q^{10} +0.758300 q^{12} -4.35652 q^{13} -1.60383 q^{14} +0.801633 q^{15} -0.292515 q^{16} +7.66611 q^{17} -2.35069 q^{18} +4.10713 q^{19} +1.44632 q^{20} +1.14107 q^{21} +8.58230 q^{23} +1.87247 q^{24} -3.47103 q^{25} -3.96976 q^{26} +3.61733 q^{27} +2.05873 q^{28} +8.70739 q^{29} +0.730466 q^{30} -0.393275 q^{31} +5.51001 q^{32} +6.98553 q^{34} +2.17638 q^{35} +3.01742 q^{36} +0.904758 q^{37} +3.74251 q^{38} +2.82433 q^{39} +3.57140 q^{40} -4.27179 q^{41} +1.03976 q^{42} -9.86460 q^{43} +3.18985 q^{45} +7.82038 q^{46} +2.45376 q^{47} +0.189637 q^{48} -3.90209 q^{49} -3.16288 q^{50} -4.96994 q^{51} +5.09571 q^{52} +1.96031 q^{53} +3.29619 q^{54} +5.08363 q^{56} -2.66265 q^{57} +7.93436 q^{58} +0.899251 q^{59} -0.937649 q^{60} -8.22053 q^{61} -0.358361 q^{62} +4.54051 q^{63} +5.60587 q^{64} +5.38690 q^{65} +11.2903 q^{67} -8.96685 q^{68} -5.56391 q^{69} +1.98316 q^{70} -5.52569 q^{71} +7.45091 q^{72} -7.67482 q^{73} +0.824436 q^{74} +2.25027 q^{75} -4.80400 q^{76} +2.57360 q^{78} -4.87756 q^{79} +0.361699 q^{80} +5.39401 q^{81} -3.89255 q^{82} +1.25202 q^{83} -1.33468 q^{84} -9.47926 q^{85} -8.98885 q^{86} -5.64500 q^{87} +9.43597 q^{89} +2.90666 q^{90} +7.66787 q^{91} -10.0385 q^{92} +0.254960 q^{93} +2.23592 q^{94} -5.07853 q^{95} -3.57214 q^{96} +8.36224 q^{97} -3.55567 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{3} + 58 q^{4} + 14 q^{5} + 32 q^{9}+O(q^{10}) \) 30 * q + 6 * q^3 + 58 * q^4 + 14 * q^5 + 32 * q^9	\( 30 q + 6 q^{3} + 58 q^{4} + 14 q^{5} + 32 q^{9} + 42 q^{12} + 18 q^{14} + 30 q^{15} + 62 q^{16} + 56 q^{20} - 14 q^{23} + 32 q^{25} + 20 q^{26} + 36 q^{27} + 40 q^{31} + 68 q^{34} + 90 q^{36} + 72 q^{37} - 44 q^{38} - 44 q^{42} + 66 q^{45} + 8 q^{47} - 16 q^{48} + 136 q^{49} + 20 q^{53} + 6 q^{56} - 22 q^{58} + 26 q^{59} - 32 q^{60} + 116 q^{64} - 8 q^{67} + 86 q^{69} - 36 q^{70} + 18 q^{71} - 10 q^{75} - 134 q^{78} - 16 q^{80} - 34 q^{81} - 38 q^{82} - 78 q^{86} - 8 q^{89} + 74 q^{91} - 98 q^{92} + 26 q^{93} + 98 q^{97}+O(q^{100}) \) 30 * q + 6 * q^3 + 58 * q^4 + 14 * q^5 + 32 * q^9 + 42 * q^12 + 18 * q^14 + 30 * q^15 + 62 * q^16 + 56 * q^20 - 14 * q^23 + 32 * q^25 + 20 * q^26 + 36 * q^27 + 40 * q^31 + 68 * q^34 + 90 * q^36 + 72 * q^37 - 44 * q^38 - 44 * q^42 + 66 * q^45 + 8 * q^47 - 16 * q^48 + 136 * q^49 + 20 * q^53 + 6 * q^56 - 22 * q^58 + 26 * q^59 - 32 * q^60 + 116 * q^64 - 8 * q^67 + 86 * q^69 - 36 * q^70 + 18 * q^71 - 10 * q^75 - 134 * q^78 - 16 * q^80 - 34 * q^81 - 38 * q^82 - 78 * q^86 - 8 * q^89 + 74 * q^91 - 98 * q^92 + 26 * q^93 + 98 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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