LMFDB - Embedded newform 1458.3.b.c.1457.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1458,3,Mod(1457,1458)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1458, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1458.1457");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1458 = 2 \cdot 3^{6} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1458.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$39.7276225437$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.4
Character	$\chi$	$=$	1458.1457
Dual form			1458.3.b.c.1457.33

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} -2.00000 q^{4} -5.99439i q^{5} +7.30820 q^{7} +2.82843i q^{8} -8.47735 q^{10} -20.6977i q^{11} -6.81957 q^{13} -10.3353i q^{14} +4.00000 q^{16} +2.26910i q^{17} +24.3267 q^{19} +11.9888i q^{20} -29.2710 q^{22} -24.1635i q^{23} -10.9327 q^{25} +9.64433i q^{26} -14.6164 q^{28} -18.5012i q^{29} +21.7310 q^{31} -5.65685i q^{32} +3.20899 q^{34} -43.8082i q^{35} -44.7664 q^{37} -34.4032i q^{38} +16.9547 q^{40} +51.2829i q^{41} +36.9408 q^{43} +41.3954i q^{44} -34.1723 q^{46} -12.6884i q^{47} +4.40972 q^{49} +15.4612i q^{50} +13.6391 q^{52} +17.7730i q^{53} -124.070 q^{55} +20.6707i q^{56} -26.1646 q^{58} -65.9062i q^{59} +42.2218 q^{61} -30.7323i q^{62} -8.00000 q^{64} +40.8792i q^{65} +118.326 q^{67} -4.53820i q^{68} -61.9541 q^{70} -53.3127i q^{71} -67.7138 q^{73} +63.3092i q^{74} -48.6534 q^{76} -151.263i q^{77} -111.324 q^{79} -23.9776i q^{80} +72.5250 q^{82} -25.2483i q^{83} +13.6019 q^{85} -52.2421i q^{86} +58.5419 q^{88} +40.9280i q^{89} -49.8388 q^{91} +48.3269i q^{92} -17.9441 q^{94} -145.824i q^{95} +40.4271 q^{97} -6.23628i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 72 q^{4} + 144 q^{16} - 180 q^{25} + 252 q^{49} - 36 q^{61} - 288 q^{64} + 180 q^{67} - 252 q^{73} + 396 q^{91}+O(q^{100}) $ 36 * q - 72 * q^4 + 144 * q^16 - 180 * q^25 + 252 * q^49 - 36 * q^61 - 288 * q^64 + 180 * q^67 - 252 * q^73 + 396 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1458\mathbb{Z}\right)^\times$.

$n$	$731$
$\chi(n)$	$-1$

Coefficient data

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

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

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

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	− 5.99439i	− 1.19888i	−0.800420	−	0.599439i	$-0.795390\pi$
$5$	− 5.99439i	− 1.19888i	0.800420	−	0.599439i	$-0.204610\pi$
$6$	0	0
$7$	7.30820	1.04403	0.522014	−	0.852937i	$-0.325181\pi$
$7$	7.30820	1.04403	0.522014	+	0.852937i	$0.325181\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	−8.47735	−0.847735
$11$	− 20.6977i	− 1.88161i	−0.338951	−	0.940804i	$-0.610072\pi$
$11$	− 20.6977i	− 1.88161i	0.338951	−	0.940804i	$-0.389928\pi$
$12$	0	0
$13$	−6.81957	−0.524582	−0.262291	−	0.964989i	$-0.584478\pi$
$13$	−6.81957	−0.524582	−0.262291	+	0.964989i	$0.584478\pi$
$14$	− 10.3353i	− 0.738239i
$15$	0	0
$16$	4.00000	0.250000
$17$	2.26910i	0.133476i	0.997771	+	0.0667382i	$0.0212592\pi$
$17$	2.26910i	0.133476i	−0.997771	+	0.0667382i	$0.978741\pi$
$18$	0	0
$19$	24.3267	1.28035	0.640176	−	0.768228i	$-0.278861\pi$
$19$	24.3267	1.28035	0.640176	+	0.768228i	$0.278861\pi$
$20$	11.9888i	0.599439i
$21$	0	0
$22$	−29.2710	−1.33050
$23$	− 24.1635i	− 1.05059i	−0.850922	−	0.525293i	$-0.823956\pi$
$23$	− 24.1635i	− 1.05059i	0.850922	−	0.525293i	$-0.176044\pi$
$24$	0	0
$25$	−10.9327	−0.437308
$26$	9.64433i	0.370936i
$27$	0	0
$28$	−14.6164	−0.522014
$29$	− 18.5012i	− 0.637971i	−0.947760	−	0.318986i	$-0.896658\pi$
$29$	− 18.5012i	− 0.637971i	0.947760	−	0.318986i	$-0.103342\pi$
$30$	0	0
$31$	21.7310	0.701001	0.350501	−	0.936562i	$-0.386011\pi$
$31$	21.7310	0.701001	0.350501	+	0.936562i	$0.386011\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	3.20899	0.0943821
$35$	− 43.8082i	− 1.25166i
$36$	0	0
$37$	−44.7664	−1.20990	−0.604951	−	0.796263i	$-0.706807\pi$
$37$	−44.7664	−1.20990	−0.604951	+	0.796263i	$0.706807\pi$
$38$	− 34.4032i	− 0.905346i
$39$	0	0
$40$	16.9547	0.423867
$41$	51.2829i	1.25080i	0.780303	+	0.625402i	$0.215065\pi$
$41$	51.2829i	1.25080i	−0.780303	+	0.625402i	$0.784935\pi$
$42$	0	0
$43$	36.9408	0.859088	0.429544	−	0.903046i	$-0.358674\pi$
$43$	36.9408	0.859088	0.429544	+	0.903046i	$0.358674\pi$
$44$	41.3954i	0.940804i
$45$	0	0
$46$	−34.1723	−0.742876
$47$	− 12.6884i	− 0.269966i	−0.990848	−	0.134983i	$-0.956902\pi$
$47$	− 12.6884i	− 0.269966i	0.990848	−	0.134983i	$-0.0430980\pi$
$48$	0	0
$49$	4.40972	0.0899943
$50$	15.4612i	0.309224i
$51$	0	0
$52$	13.6391	0.262291
$53$	17.7730i	0.335339i	0.985843	+	0.167670i	$0.0536242\pi$
$53$	17.7730i	0.335339i	−0.985843	+	0.167670i	$0.946376\pi$
$54$	0	0
$55$	−124.070	−2.25582
$56$	20.6707i	0.369120i
$57$	0	0
$58$	−26.1646	−0.451114
$59$	− 65.9062i	− 1.11705i	−0.829486	−	0.558527i	$-0.811367\pi$
$59$	− 65.9062i	− 1.11705i	0.829486	−	0.558527i	$-0.188633\pi$
$60$	0	0
$61$	42.2218	0.692160	0.346080	−	0.938205i	$-0.387513\pi$
$61$	42.2218	0.692160	0.346080	+	0.938205i	$0.387513\pi$
$62$	− 30.7323i	− 0.495683i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	40.8792i	0.628910i
$66$	0	0
$67$	118.326	1.76607	0.883033	−	0.469311i	$-0.155498\pi$
$67$	118.326	1.76607	0.883033	+	0.469311i	$0.155498\pi$
$68$	− 4.53820i	− 0.0667382i
$69$	0	0
$70$	−61.9541	−0.885059
$71$	− 53.3127i	− 0.750883i	−0.926846	−	0.375442i	$-0.877491\pi$
$71$	− 53.3127i	− 0.750883i	0.926846	−	0.375442i	$-0.122509\pi$
$72$	0	0
$73$	−67.7138	−0.927587	−0.463793	−	0.885943i	$-0.653512\pi$
$73$	−67.7138	−0.927587	−0.463793	+	0.885943i	$0.653512\pi$
$74$	63.3092i	0.855529i
$75$	0	0
$76$	−48.6534	−0.640176
$77$	− 151.263i	− 1.96445i
$78$	0	0
$79$	−111.324	−1.40916	−0.704582	−	0.709623i	$-0.748865\pi$
$79$	−111.324	−1.40916	−0.704582	+	0.709623i	$0.748865\pi$
$80$	− 23.9776i	− 0.299719i
$81$	0	0
$82$	72.5250	0.884452
$83$	− 25.2483i	− 0.304196i	−0.988365	−	0.152098i	$-0.951397\pi$
$83$	− 25.2483i	− 0.304196i	0.988365	−	0.152098i	$-0.0486030\pi$
$84$	0	0
$85$	13.6019	0.160022
$86$	− 52.2421i	− 0.607467i
$87$	0	0
$88$	58.5419	0.665249
$89$	40.9280i	0.459865i	0.973207	+	0.229932i	$0.0738505\pi$
$89$	40.9280i	0.459865i	−0.973207	+	0.229932i	$0.926150\pi$
$90$	0	0
$91$	−49.8388	−0.547679
$92$	48.3269i	0.525293i
$93$	0	0
$94$	−17.9441	−0.190895
$95$	− 145.824i	− 1.53499i
$96$	0	0
$97$	40.4271	0.416775	0.208387	−	0.978046i	$-0.433179\pi$
$97$	40.4271	0.416775	0.208387	+	0.978046i	$0.433179\pi$
$98$	− 6.23628i	− 0.0636356i
$99$	0	0
$100$	21.8654	0.218654
$101$	− 11.6008i	− 0.114860i	−0.998350	−	0.0574299i	$-0.981709\pi$
$101$	− 11.6008i	− 0.114860i	0.998350	−	0.0574299i	$-0.0182906\pi$
$102$	0	0
$103$	124.649	1.21019	0.605094	−	0.796154i	$-0.293136\pi$
$103$	124.649	1.21019	0.605094	+	0.796154i	$0.293136\pi$
$104$	− 19.2887i	− 0.185468i
$105$	0	0
$106$	25.1348	0.237121
$107$	142.548i	1.33223i	0.745850	+	0.666113i	$0.232043\pi$
$107$	142.548i	1.33223i	−0.745850	+	0.666113i	$0.767957\pi$
$108$	0	0
$109$	−119.276	−1.09427	−0.547137	−	0.837043i	$-0.684282\pi$
$109$	−119.276	−1.09427	−0.547137	+	0.837043i	$0.684282\pi$
$110$	175.461i	1.59510i
$111$	0	0
$112$	29.2328	0.261007
$113$	103.667i	0.917408i	0.888589	+	0.458704i	$0.151686\pi$
$113$	103.667i	0.917408i	−0.888589	+	0.458704i	$0.848314\pi$
$114$	0	0
$115$	−144.845	−1.25952
$116$	37.0023i	0.318986i
$117$	0	0
$118$	−93.2054	−0.789876
$119$	16.5830i	0.139353i
$120$	0	0
$121$	−307.394	−2.54045
$122$	− 59.7106i	− 0.489431i
$123$	0	0
$124$	−43.4621	−0.350501
$125$	− 84.3248i	− 0.674599i
$126$	0	0
$127$	−219.539	−1.72866	−0.864329	−	0.502927i	$-0.832256\pi$
$127$	−219.539	−1.72866	−0.864329	+	0.502927i	$0.832256\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	57.8119	0.444707
$131$	130.641i	0.997261i	0.866815	+	0.498631i	$0.166164\pi$
$131$	130.641i	0.997261i	−0.866815	+	0.498631i	$0.833836\pi$
$132$	0	0
$133$	177.784	1.33672
$134$	− 167.339i	− 1.24880i
$135$	0	0
$136$	−6.41798	−0.0471911
$137$	− 77.2169i	− 0.563627i	−0.959469	−	0.281814i	$-0.909064\pi$
$137$	− 77.2169i	− 0.563627i	0.959469	−	0.281814i	$-0.0909360\pi$
$138$	0	0
$139$	−106.850	−0.768704	−0.384352	−	0.923187i	$-0.625575\pi$
$139$	−106.850	−0.768704	−0.384352	+	0.923187i	$0.625575\pi$
$140$	87.6163i	0.625831i
$141$	0	0
$142$	−75.3956	−0.530955
$143$	141.149i	0.987059i
$144$	0	0
$145$	−110.903	−0.764849
$146$	95.7618i	0.655903i
$147$	0	0
$148$	89.5327	0.604951
$149$	155.115i	1.04104i	0.853850	+	0.520520i	$0.174262\pi$
$149$	155.115i	1.04104i	−0.853850	+	0.520520i	$0.825738\pi$
$150$	0	0
$151$	−155.055	−1.02685	−0.513427	−	0.858133i	$-0.671624\pi$
$151$	−155.055	−1.02685	−0.513427	+	0.858133i	$0.671624\pi$
$152$	68.8063i	0.452673i
$153$	0	0
$154$	−213.918	−1.38908
$155$	− 130.264i	− 0.840415i
$156$	0	0
$157$	−121.480	−0.773758	−0.386879	−	0.922131i	$-0.626447\pi$
$157$	−121.480	−0.773758	−0.386879	+	0.922131i	$0.626447\pi$
$158$	157.436i	0.996429i
$159$	0	0
$160$	−33.9094	−0.211934
$161$	− 176.591i	− 1.09684i
$162$	0	0
$163$	−66.9033	−0.410450	−0.205225	−	0.978715i	$-0.565793\pi$
$163$	−66.9033	−0.410450	−0.205225	+	0.978715i	$0.565793\pi$
$164$	− 102.566i	− 0.625402i
$165$	0	0
$166$	−35.7065	−0.215099
$167$	7.25256i	0.0434285i	0.999764	+	0.0217143i	$0.00691240\pi$
$167$	7.25256i	0.0434285i	−0.999764	+	0.0217143i	$0.993088\pi$
$168$	0	0
$169$	−122.493	−0.724813
$170$	− 19.2360i	− 0.113153i
$171$	0	0
$172$	−73.8816	−0.429544
$173$	− 72.5019i	− 0.419086i	−0.977799	−	0.209543i	$-0.932802\pi$
$173$	− 72.5019i	− 0.419086i	0.977799	−	0.209543i	$-0.0671976\pi$
$174$	0	0
$175$	−79.8983	−0.456562
$176$	− 82.7908i	− 0.470402i
$177$	0	0
$178$	57.8809	0.325173
$179$	253.302i	1.41509i	0.706667	+	0.707547i	$0.250198\pi$
$179$	253.302i	1.41509i	−0.706667	+	0.707547i	$0.749802\pi$
$180$	0	0
$181$	−38.8301	−0.214531	−0.107265	−	0.994230i	$-0.534209\pi$
$181$	−38.8301	−0.214531	−0.107265	+	0.994230i	$0.534209\pi$
$182$	70.4827i	0.387267i
$183$	0	0
$184$	68.3446	0.371438
$185$	268.347i	1.45052i
$186$	0	0
$187$	46.9651	0.251150
$188$	25.3768i	0.134983i
$189$	0	0
$190$	−206.226	−1.08540
$191$	− 198.458i	− 1.03905i	−0.854456	−	0.519524i	$-0.826109\pi$
$191$	− 198.458i	− 1.03905i	0.854456	−	0.519524i	$-0.173891\pi$
$192$	0	0
$193$	56.9955	0.295314	0.147657	−	0.989039i	$-0.452827\pi$
$193$	56.9955	0.295314	0.147657	+	0.989039i	$0.452827\pi$
$194$	− 57.1726i	− 0.294704i
$195$	0	0
$196$	−8.81944	−0.0449971
$197$	74.8536i	0.379967i	0.981787	+	0.189984i	$0.0608435\pi$
$197$	74.8536i	0.379967i	−0.981787	+	0.189984i	$0.939157\pi$
$198$	0	0
$199$	130.432	0.655435	0.327717	−	0.944776i	$-0.393721\pi$
$199$	130.432	0.655435	0.327717	+	0.944776i	$0.393721\pi$
$200$	− 30.9224i	− 0.154612i
$201$	0	0
$202$	−16.4061	−0.0812182
$203$	− 135.210i	− 0.666060i
$204$	0	0
$205$	307.410	1.49956
$206$	− 176.281i	− 0.855732i
$207$	0	0
$208$	−27.2783	−0.131146
$209$	− 503.507i	− 2.40912i
$210$	0	0
$211$	31.6175	0.149846	0.0749230	−	0.997189i	$-0.476129\pi$
$211$	31.6175	0.149846	0.0749230	+	0.997189i	$0.476129\pi$
$212$	− 35.5459i	− 0.167670i
$213$	0	0
$214$	201.594	0.942027
$215$	− 221.437i	− 1.02994i
$216$	0	0
$217$	158.815	0.731865
$218$	168.682i	0.773769i
$219$	0	0
$220$	248.140	1.12791
$221$	− 15.4743i	− 0.0700194i
$222$	0	0
$223$	102.053	0.457636	0.228818	−	0.973469i	$-0.426514\pi$
$223$	102.053	0.457636	0.228818	+	0.973469i	$0.426514\pi$
$224$	− 41.3414i	− 0.184560i
$225$	0	0
$226$	146.607	0.648706
$227$	− 257.217i	− 1.13311i	−0.824023	−	0.566557i	$-0.808275\pi$
$227$	− 257.217i	− 1.13311i	0.824023	−	0.566557i	$-0.191725\pi$
$228$	0	0
$229$	274.974	1.20076	0.600381	−	0.799714i	$-0.295016\pi$
$229$	274.974	1.20076	0.600381	+	0.799714i	$0.295016\pi$
$230$	204.842i	0.890618i
$231$	0	0
$232$	52.3292	0.225557
$233$	− 257.857i	− 1.10668i	−0.832955	−	0.553341i	$-0.813353\pi$
$233$	− 257.857i	− 1.10668i	0.832955	−	0.553341i	$-0.186647\pi$
$234$	0	0
$235$	−76.0593	−0.323657
$236$	131.812i	0.558527i
$237$	0	0
$238$	23.4519	0.0985376
$239$	− 124.546i	− 0.521113i	−0.965459	−	0.260557i	$-0.916094\pi$
$239$	− 124.546i	− 0.521113i	0.965459	−	0.260557i	$-0.0839060\pi$
$240$	0	0
$241$	65.4473	0.271566	0.135783	−	0.990739i	$-0.456645\pi$
$241$	65.4473	0.271566	0.135783	+	0.990739i	$0.456645\pi$
$242$	434.721i	1.79637i
$243$	0	0
$244$	−84.4436	−0.346080
$245$	− 26.4336i	− 0.107892i
$246$	0	0
$247$	−165.898	−0.671651
$248$	61.4647i	0.247841i
$249$	0	0
$250$	−119.253	−0.477013
$251$	− 233.859i	− 0.931710i	−0.884861	−	0.465855i	$-0.845747\pi$
$251$	− 233.859i	− 0.931710i	0.884861	−	0.465855i	$-0.154253\pi$
$252$	0	0
$253$	−500.128	−1.97679
$254$	310.476i	1.22235i
$255$	0	0
$256$	16.0000	0.0625000
$257$	75.1638i	0.292466i	0.989250	+	0.146233i	$0.0467149\pi$
$257$	75.1638i	0.292466i	−0.989250	+	0.146233i	$0.953285\pi$
$258$	0	0
$259$	−327.161	−1.26317
$260$	− 81.7583i	− 0.314455i
$261$	0	0
$262$	184.755	0.705170
$263$	151.467i	0.575919i	0.957643	+	0.287960i	$0.0929769\pi$
$263$	151.467i	0.575919i	−0.957643	+	0.287960i	$0.907023\pi$
$264$	0	0
$265$	106.538	0.402031
$266$	− 251.425i	− 0.945207i
$267$	0	0
$268$	−236.653	−0.883033
$269$	170.868i	0.635198i	0.948225	+	0.317599i	$0.102877\pi$
$269$	170.868i	0.635198i	−0.948225	+	0.317599i	$0.897123\pi$
$270$	0	0
$271$	−8.46530	−0.0312373	−0.0156186	−	0.999878i	$-0.504972\pi$
$271$	−8.46530	−0.0312373	−0.0156186	+	0.999878i	$0.504972\pi$
$272$	9.07640i	0.0333691i
$273$	0	0
$274$	−109.201	−0.398545
$275$	226.282i	0.822843i
$276$	0	0
$277$	499.814	1.80438	0.902192	−	0.431335i	$-0.141957\pi$
$277$	499.814	1.80438	0.902192	+	0.431335i	$0.141957\pi$
$278$	151.108i	0.543556i
$279$	0	0
$280$	123.908	0.442529
$281$	− 70.0190i	− 0.249178i	−0.992208	−	0.124589i	$-0.960239\pi$
$281$	− 70.0190i	− 0.249178i	0.992208	−	0.124589i	$-0.0397612\pi$
$282$	0	0
$283$	−35.8457	−0.126663	−0.0633317	−	0.997993i	$-0.520173\pi$
$283$	−35.8457	−0.126663	−0.0633317	+	0.997993i	$0.520173\pi$
$284$	106.625i	0.375442i
$285$	0	0
$286$	199.615	0.697956
$287$	374.786i	1.30587i
$288$	0	0
$289$	283.851	0.982184
$290$	156.841i	0.540830i
$291$	0	0
$292$	135.428	0.463793
$293$	− 34.1314i	− 0.116489i	−0.998302	−	0.0582447i	$-0.981450\pi$
$293$	− 34.1314i	− 0.116489i	0.998302	−	0.0582447i	$-0.0185503\pi$
$294$	0	0
$295$	−395.067	−1.33921
$296$	− 126.618i	− 0.427765i
$297$	0	0
$298$	219.366	0.736126
$299$	164.784i	0.551119i
$300$	0	0
$301$	269.970	0.896912
$302$	219.281i	0.726096i
$303$	0	0
$304$	97.3068	0.320088
$305$	− 253.094i	− 0.829816i
$306$	0	0
$307$	330.610	1.07690	0.538452	−	0.842656i	$-0.319009\pi$
$307$	330.610	1.07690	0.538452	+	0.842656i	$0.319009\pi$
$308$	302.526i	0.982226i
$309$	0	0
$310$	−184.222	−0.594263
$311$	241.634i	0.776958i	0.921458	+	0.388479i	$0.126999\pi$
$311$	241.634i	0.776958i	−0.921458	+	0.388479i	$0.873001\pi$
$312$	0	0
$313$	−313.281	−1.00090	−0.500448	−	0.865766i	$-0.666831\pi$
$313$	−313.281	−1.00090	−0.500448	+	0.865766i	$0.666831\pi$
$314$	171.799i	0.547129i
$315$	0	0
$316$	222.648	0.704582
$317$	493.706i	1.55743i	0.627376	+	0.778717i	$0.284129\pi$
$317$	493.706i	1.55743i	−0.627376	+	0.778717i	$0.715871\pi$
$318$	0	0
$319$	−382.931	−1.20041
$320$	47.9551i	0.149860i
$321$	0	0
$322$	−249.738	−0.775583
$323$	55.1997i	0.170897i
$324$	0	0
$325$	74.5564	0.229404
$326$	94.6156i	0.290232i
$327$	0	0
$328$	−145.050	−0.442226
$329$	− 92.7294i	− 0.281852i
$330$	0	0
$331$	259.182	0.783026	0.391513	−	0.920172i	$-0.371952\pi$
$331$	259.182	0.783026	0.391513	+	0.920172i	$0.371952\pi$
$332$	50.4966i	0.152098i
$333$	0	0
$334$	10.2567	0.0307086
$335$	− 709.295i	− 2.11730i
$336$	0	0
$337$	458.840	1.36154	0.680771	−	0.732496i	$-0.261645\pi$
$337$	458.840	1.36154	0.680771	+	0.732496i	$0.261645\pi$
$338$	173.232i	0.512520i
$339$	0	0
$340$	−27.2037	−0.0800110
$341$	− 449.782i	− 1.31901i
$342$	0	0
$343$	−325.874	−0.950071
$344$	104.484i	0.303733i
$345$	0	0
$346$	−102.533	−0.296339
$347$	270.459i	0.779421i	0.920937	+	0.389711i	$0.127425\pi$
$347$	270.459i	0.779421i	−0.920937	+	0.389711i	$0.872575\pi$
$348$	0	0
$349$	334.313	0.957917	0.478959	−	0.877837i	$-0.341014\pi$
$349$	334.313	0.957917	0.478959	+	0.877837i	$0.341014\pi$
$350$	112.993i	0.322838i
$351$	0	0
$352$	−117.084	−0.332624
$353$	− 523.598i	− 1.48328i	−0.670798	−	0.741640i	$-0.734048\pi$
$353$	− 523.598i	− 1.48328i	0.670798	−	0.741640i	$-0.265952\pi$
$354$	0	0
$355$	−319.577	−0.900217
$356$	− 81.8559i	− 0.229932i
$357$	0	0
$358$	358.223	1.00062
$359$	53.1770i	0.148125i	0.997254	+	0.0740627i	$0.0235965\pi$
$359$	53.1770i	0.148125i	−0.997254	+	0.0740627i	$0.976404\pi$
$360$	0	0
$361$	230.789	0.639304
$362$	54.9140i	0.151696i
$363$	0	0
$364$	99.6775	0.273839
$365$	405.903i	1.11206i
$366$	0	0
$367$	439.214	1.19677	0.598384	−	0.801210i	$-0.295810\pi$
$367$	439.214	1.19677	0.598384	+	0.801210i	$0.295810\pi$
$368$	− 96.6539i	− 0.262646i
$369$	0	0
$370$	379.500	1.02568
$371$	129.888i	0.350103i
$372$	0	0
$373$	−591.646	−1.58618	−0.793091	−	0.609103i	$-0.791529\pi$
$373$	−591.646	−1.58618	−0.793091	+	0.609103i	$0.791529\pi$
$374$	− 66.4187i	− 0.177590i
$375$	0	0
$376$	35.8882	0.0954475
$377$	126.170i	0.334668i
$378$	0	0
$379$	198.665	0.524183	0.262092	−	0.965043i	$-0.415588\pi$
$379$	198.665	0.524183	0.262092	+	0.965043i	$0.415588\pi$
$380$	291.647i	0.767493i
$381$	0	0
$382$	−280.662	−0.734718
$383$	− 32.3369i	− 0.0844306i	−0.999109	−	0.0422153i	$-0.986558\pi$
$383$	− 32.3369i	− 0.0844306i	0.999109	−	0.0422153i	$-0.0134415\pi$
$384$	0	0
$385$	−906.728	−2.35514
$386$	− 80.6038i	− 0.208818i
$387$	0	0
$388$	−80.8543	−0.208387
$389$	218.700i	0.562212i	0.959677	+	0.281106i	$0.0907013\pi$
$389$	218.700i	0.562212i	−0.959677	+	0.281106i	$0.909299\pi$
$390$	0	0
$391$	54.8293	0.140228
$392$	12.4726i	0.0318178i
$393$	0	0
$394$	105.859	0.268677
$395$	667.319i	1.68941i
$396$	0	0
$397$	529.397	1.33349	0.666746	−	0.745285i	$-0.267686\pi$
$397$	529.397	1.33349	0.666746	+	0.745285i	$0.267686\pi$
$398$	− 184.458i	− 0.463462i
$399$	0	0
$400$	−43.7308	−0.109327
$401$	− 261.096i	− 0.651111i	−0.945523	−	0.325556i	$-0.894449\pi$
$401$	− 261.096i	− 0.651111i	0.945523	−	0.325556i	$-0.105551\pi$
$402$	0	0
$403$	−148.196	−0.367733
$404$	23.2017i	0.0574299i
$405$	0	0
$406$	−191.216	−0.470975
$407$	926.560i	2.27656i
$408$	0	0
$409$	639.558	1.56371	0.781856	−	0.623459i	$-0.214273\pi$
$409$	639.558	1.56371	0.781856	+	0.623459i	$0.214273\pi$
$410$	− 434.743i	− 1.06035i
$411$	0	0
$412$	−249.299	−0.605094
$413$	− 481.655i	− 1.16624i
$414$	0	0
$415$	−151.348	−0.364694
$416$	38.5773i	0.0927340i
$417$	0	0
$418$	−712.066	−1.70351
$419$	− 714.100i	− 1.70430i	−0.523301	−	0.852148i	$-0.675300\pi$
$419$	− 714.100i	− 1.70430i	0.523301	−	0.852148i	$-0.324700\pi$
$420$	0	0
$421$	−70.0167	−0.166310	−0.0831552	−	0.996537i	$-0.526500\pi$
$421$	−70.0167	−0.166310	−0.0831552	+	0.996537i	$0.526500\pi$
$422$	− 44.7139i	− 0.105957i
$423$	0	0
$424$	−50.2696	−0.118560
$425$	− 24.8074i	− 0.0583704i
$426$	0	0
$427$	308.565	0.722635
$428$	− 285.097i	− 0.666113i
$429$	0	0
$430$	−313.160	−0.728279
$431$	− 76.1177i	− 0.176607i	−0.996094	−	0.0883036i	$-0.971855\pi$
$431$	− 76.1177i	− 0.176607i	0.996094	−	0.0883036i	$-0.0281446\pi$
$432$	0	0
$433$	109.713	0.253380	0.126690	−	0.991942i	$-0.459565\pi$
$433$	109.713	0.253380	0.126690	+	0.991942i	$0.459565\pi$
$434$	− 224.598i	− 0.517507i
$435$	0	0
$436$	238.552	0.547137
$437$	− 587.817i	− 1.34512i
$438$	0	0
$439$	113.996	0.259672	0.129836	−	0.991536i	$-0.458555\pi$
$439$	113.996	0.259672	0.129836	+	0.991536i	$0.458555\pi$
$440$	− 350.923i	− 0.797552i
$441$	0	0
$442$	−21.8840	−0.0495112
$443$	172.711i	0.389867i	0.980816	+	0.194933i	$0.0624491\pi$
$443$	172.711i	0.389867i	−0.980816	+	0.194933i	$0.937551\pi$
$444$	0	0
$445$	245.338	0.551322
$446$	− 144.324i	− 0.323597i
$447$	0	0
$448$	−58.4656	−0.130503
$449$	190.458i	0.424182i	0.977250	+	0.212091i	$0.0680273\pi$
$449$	190.458i	0.424182i	−0.977250	+	0.212091i	$0.931973\pi$
$450$	0	0
$451$	1061.44	2.35352
$452$	− 207.334i	− 0.458704i
$453$	0	0
$454$	−363.759	−0.801232
$455$	298.753i	0.656600i
$456$	0	0
$457$	143.739	0.314528	0.157264	−	0.987557i	$-0.449733\pi$
$457$	143.739	0.314528	0.157264	+	0.987557i	$0.449733\pi$
$458$	− 388.872i	− 0.849066i
$459$	0	0
$460$	289.690	0.629762
$461$	− 653.134i	− 1.41678i	−0.705823	−	0.708388i	$-0.749423\pi$
$461$	− 653.134i	− 1.41678i	0.705823	−	0.708388i	$-0.250577\pi$
$462$	0	0
$463$	22.4229	0.0484296	0.0242148	−	0.999707i	$-0.492291\pi$
$463$	22.4229	0.0484296	0.0242148	+	0.999707i	$0.492291\pi$
$464$	− 74.0047i	− 0.159493i
$465$	0	0
$466$	−364.665	−0.782542
$467$	477.267i	1.02199i	0.859585	+	0.510993i	$0.170722\pi$
$467$	477.267i	1.02199i	−0.859585	+	0.510993i	$0.829278\pi$
$468$	0	0
$469$	864.753	1.84382
$470$	107.564i	0.228860i
$471$	0	0
$472$	186.411	0.394938
$473$	− 764.589i	− 1.61647i
$474$	0	0
$475$	−265.957	−0.559909
$476$	− 33.1661i	− 0.0696766i
$477$	0	0
$478$	−176.135	−0.368483
$479$	− 597.164i	− 1.24669i	−0.781948	−	0.623344i	$-0.785774\pi$
$479$	− 597.164i	− 1.24669i	0.781948	−	0.623344i	$-0.214226\pi$
$480$	0	0
$481$	305.287	0.634693
$482$	− 92.5565i	− 0.192026i
$483$	0	0
$484$	614.789	1.27022
$485$	− 242.336i	− 0.499662i
$486$	0	0
$487$	758.948	1.55842	0.779208	−	0.626766i	$-0.215622\pi$
$487$	758.948	1.55842	0.779208	+	0.626766i	$0.215622\pi$
$488$	119.421i	0.244716i
$489$	0	0
$490$	−37.3827	−0.0762913
$491$	− 262.641i	− 0.534910i	−0.963570	−	0.267455i	$-0.913817\pi$
$491$	− 262.641i	− 0.534910i	0.963570	−	0.267455i	$-0.0861826\pi$
$492$	0	0
$493$	41.9810	0.0851541
$494$	234.615i	0.474929i
$495$	0	0
$496$	86.9242	0.175250
$497$	− 389.620i	− 0.783943i
$498$	0	0
$499$	−90.9231	−0.182211	−0.0911054	−	0.995841i	$-0.529040\pi$
$499$	−90.9231	−0.182211	−0.0911054	+	0.995841i	$0.529040\pi$
$500$	168.650i	0.337299i
$501$	0	0
$502$	−330.727	−0.658818
$503$	− 425.234i	− 0.845395i	−0.906271	−	0.422698i	$-0.861083\pi$
$503$	− 425.234i	− 0.845395i	0.906271	−	0.422698i	$-0.138917\pi$
$504$	0	0
$505$	−69.5400	−0.137703
$506$	707.288i	1.39780i
$507$	0	0
$508$	439.079	0.864329
$509$	− 675.840i	− 1.32778i	−0.747830	−	0.663890i	$-0.768904\pi$
$509$	− 675.840i	− 1.32778i	0.747830	−	0.663890i	$-0.231096\pi$
$510$	0	0
$511$	−494.866	−0.968427
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	106.298	0.206805
$515$	− 747.196i	− 1.45087i
$516$	0	0
$517$	−262.621	−0.507971
$518$	462.676i	0.893197i
$519$	0	0
$520$	−115.624	−0.222353
$521$	782.422i	1.50177i	0.660433	+	0.750885i	$0.270373\pi$
$521$	782.422i	1.50177i	−0.660433	+	0.750885i	$0.729627\pi$
$522$	0	0
$523$	685.818	1.31132	0.655658	−	0.755058i	$-0.272392\pi$
$523$	685.818	1.31132	0.655658	+	0.755058i	$0.272392\pi$
$524$	− 261.282i	− 0.498631i
$525$	0	0
$526$	214.206	0.407236
$527$	49.3099i	0.0935672i
$528$	0	0
$529$	−54.8729	−0.103730
$530$	− 150.668i	− 0.284279i
$531$	0	0
$532$	−355.569	−0.668362
$533$	− 349.728i	− 0.656150i
$534$	0	0
$535$	854.490	1.59718
$536$	334.678i	0.624399i
$537$	0	0
$538$	241.644	0.449153
$539$	− 91.2710i	− 0.169334i
$540$	0	0
$541$	884.789	1.63547	0.817735	−	0.575595i	$-0.195229\pi$
$541$	884.789	1.63547	0.817735	+	0.575595i	$0.195229\pi$
$542$	11.9717i	0.0220881i
$543$	0	0
$544$	12.8360	0.0235955
$545$	714.986i	1.31190i
$546$	0	0
$547$	344.105	0.629077	0.314538	−	0.949245i	$-0.398150\pi$
$547$	344.105	0.629077	0.314538	+	0.949245i	$0.398150\pi$
$548$	154.434i	0.281814i
$549$	0	0
$550$	320.011	0.581838
$551$	− 450.072i	− 0.816828i
$552$	0	0
$553$	−813.577	−1.47121
$554$	− 706.844i	− 1.27589i
$555$	0	0
$556$	213.700	0.384352
$557$	− 317.052i	− 0.569213i	−0.958644	−	0.284606i	$-0.908137\pi$
$557$	− 317.052i	− 0.569213i	0.958644	−	0.284606i	$-0.0918629\pi$
$558$	0	0
$559$	−251.920	−0.450662
$560$	− 175.233i	− 0.312915i
$561$	0	0
$562$	−99.0218	−0.176195
$563$	203.004i	0.360575i	0.983614	+	0.180287i	$0.0577028\pi$
$563$	203.004i	0.360575i	−0.983614	+	0.180287i	$0.942297\pi$
$564$	0	0
$565$	621.421	1.09986
$566$	50.6935i	0.0895646i
$567$	0	0
$568$	150.791	0.265477
$569$	235.365i	0.413646i	0.978378	+	0.206823i	$0.0663124\pi$
$569$	235.365i	0.413646i	−0.978378	+	0.206823i	$0.933688\pi$
$570$	0	0
$571$	65.6422	0.114960	0.0574800	−	0.998347i	$-0.481693\pi$
$571$	65.6422	0.114960	0.0574800	+	0.998347i	$0.481693\pi$
$572$	− 282.299i	− 0.493529i
$573$	0	0
$574$	530.027	0.923392
$575$	264.172i	0.459430i
$576$	0	0
$577$	−144.256	−0.250011	−0.125005	−	0.992156i	$-0.539895\pi$
$577$	−144.256	−0.250011	−0.125005	+	0.992156i	$0.539895\pi$
$578$	− 401.426i	− 0.694509i
$579$	0	0
$580$	221.806	0.382425
$581$	− 184.519i	− 0.317589i
$582$	0	0
$583$	367.859	0.630977
$584$	− 191.524i	− 0.327951i
$585$	0	0
$586$	−48.2690	−0.0823704
$587$	− 567.446i	− 0.966687i	−0.875431	−	0.483344i	$-0.839422\pi$
$587$	− 567.446i	− 0.966687i	0.875431	−	0.483344i	$-0.160578\pi$
$588$	0	0
$589$	528.645	0.897529
$590$	558.710i	0.946965i
$591$	0	0
$592$	−179.065	−0.302475
$593$	− 174.635i	− 0.294494i	−0.989100	−	0.147247i	$-0.952959\pi$
$593$	− 174.635i	− 0.294494i	0.989100	−	0.147247i	$-0.0470412\pi$
$594$	0	0
$595$	99.4051	0.167067
$596$	− 310.230i	− 0.520520i
$597$	0	0
$598$	233.040	0.389700
$599$	431.459i	0.720300i	0.932894	+	0.360150i	$0.117274\pi$
$599$	431.459i	0.720300i	−0.932894	+	0.360150i	$0.882726\pi$
$600$	0	0
$601$	−830.822	−1.38240	−0.691200	−	0.722664i	$-0.742917\pi$
$601$	−830.822	−1.38240	−0.691200	+	0.722664i	$0.742917\pi$
$602$	− 381.796i	− 0.634212i
$603$	0	0
$604$	310.110	0.513427
$605$	1842.64i	3.04569i
$606$	0	0
$607$	−375.433	−0.618506	−0.309253	−	0.950980i	$-0.600079\pi$
$607$	−375.433	−0.618506	−0.309253	+	0.950980i	$0.600079\pi$
$608$	− 137.613i	− 0.226337i
$609$	0	0
$610$	−357.929	−0.586768
$611$	86.5295i	0.141620i
$612$	0	0
$613$	−384.933	−0.627949	−0.313974	−	0.949431i	$-0.601661\pi$
$613$	−384.933	−0.627949	−0.313974	+	0.949431i	$0.601661\pi$
$614$	− 467.553i	− 0.761487i
$615$	0	0
$616$	427.836	0.694538
$617$	− 165.163i	− 0.267688i	−0.991002	−	0.133844i	$-0.957268\pi$
$617$	− 165.163i	− 0.267688i	0.991002	−	0.133844i	$-0.0427321\pi$
$618$	0	0
$619$	478.604	0.773189	0.386594	−	0.922250i	$-0.373651\pi$
$619$	478.604	0.773189	0.386594	+	0.922250i	$0.373651\pi$
$620$	260.529i	0.420208i
$621$	0	0
$622$	341.722	0.549392
$623$	299.110i	0.480112i
$624$	0	0
$625$	−778.794	−1.24607
$626$	443.046i	0.707741i
$627$	0	0
$628$	242.960	0.386879
$629$	− 101.579i	− 0.161493i
$630$	0	0
$631$	−249.236	−0.394985	−0.197493	−	0.980304i	$-0.563280\pi$
$631$	−249.236	−0.394985	−0.197493	+	0.980304i	$0.563280\pi$
$632$	− 314.872i	− 0.498214i
$633$	0	0
$634$	698.206	1.10127
$635$	1316.01i	2.07245i
$636$	0	0
$637$	−30.0724	−0.0472094
$638$	541.547i	0.848819i
$639$	0	0
$640$	67.8188	0.105967
$641$	− 224.655i	− 0.350476i	−0.984526	−	0.175238i	$-0.943931\pi$
$641$	− 224.655i	− 0.350476i	0.984526	−	0.175238i	$-0.0560695\pi$
$642$	0	0
$643$	−364.144	−0.566320	−0.283160	−	0.959073i	$-0.591383\pi$
$643$	−364.144	−0.566320	−0.283160	+	0.959073i	$0.591383\pi$
$644$	353.183i	0.548420i
$645$	0	0
$646$	78.0642	0.120842
$647$	1200.78i	1.85593i	0.372671	+	0.927964i	$0.378442\pi$
$647$	1200.78i	1.85593i	−0.372671	+	0.927964i	$0.621558\pi$
$648$	0	0
$649$	−1364.11	−2.10186
$650$	− 105.439i	− 0.162213i
$651$	0	0
$652$	133.807	0.205225
$653$	− 635.854i	− 0.973742i	−0.873474	−	0.486871i	$-0.838138\pi$
$653$	− 635.854i	− 0.973742i	0.873474	−	0.486871i	$-0.161862\pi$
$654$	0	0
$655$	783.114	1.19559
$656$	205.132i	0.312701i
$657$	0	0
$658$	−131.139	−0.199300
$659$	1165.55i	1.76866i	0.466862	+	0.884330i	$0.345384\pi$
$659$	1165.55i	1.76866i	−0.466862	+	0.884330i	$0.654616\pi$
$660$	0	0
$661$	−962.967	−1.45683	−0.728417	−	0.685134i	$-0.759743\pi$
$661$	−962.967	−1.45683	−0.728417	+	0.685134i	$0.759743\pi$
$662$	− 366.538i	− 0.553683i
$663$	0	0
$664$	71.4129	0.107550
$665$	− 1065.71i	− 1.60257i
$666$	0	0
$667$	−447.052	−0.670243
$668$	− 14.5051i	− 0.0217143i
$669$	0	0
$670$	−1003.09	−1.49716
$671$	− 873.893i	− 1.30237i
$672$	0	0
$673$	−652.015	−0.968819	−0.484409	−	0.874841i	$-0.660965\pi$
$673$	−652.015	−0.968819	−0.484409	+	0.874841i	$0.660965\pi$
$674$	− 648.897i	− 0.962755i
$675$	0	0
$676$	244.987	0.362407
$677$	754.458i	1.11441i	0.830374	+	0.557207i	$0.188127\pi$
$677$	754.458i	1.11441i	−0.830374	+	0.557207i	$0.811873\pi$
$678$	0	0
$679$	295.449	0.435124
$680$	38.4719i	0.0565763i
$681$	0	0
$682$	−636.088	−0.932681
$683$	468.538i	0.686000i	0.939335	+	0.343000i	$0.111443\pi$
$683$	468.538i	0.686000i	−0.939335	+	0.343000i	$0.888557\pi$
$684$	0	0
$685$	−462.868	−0.675720
$686$	460.856i	0.671802i
$687$	0	0
$688$	147.763	0.214772
$689$	− 121.204i	− 0.175913i
$690$	0	0
$691$	215.048	0.311213	0.155607	−	0.987819i	$-0.450267\pi$
$691$	215.048	0.311213	0.155607	+	0.987819i	$0.450267\pi$
$692$	145.004i	0.209543i
$693$	0	0
$694$	382.487	0.551134
$695$	640.500i	0.921582i
$696$	0	0
$697$	−116.366	−0.166953
$698$	− 472.790i	− 0.677350i
$699$	0	0
$700$	159.797	0.228281
$701$	− 848.227i	− 1.21002i	−0.796216	−	0.605012i	$-0.793168\pi$
$701$	− 848.227i	− 1.21002i	0.796216	−	0.605012i	$-0.206832\pi$
$702$	0	0
$703$	−1089.02	−1.54910
$704$	165.582i	0.235201i
$705$	0	0
$706$	−740.479	−1.04884
$707$	− 84.7813i	− 0.119917i
$708$	0	0
$709$	−83.4604	−0.117716	−0.0588578	−	0.998266i	$-0.518746\pi$
$709$	−83.4604	−0.117716	−0.0588578	+	0.998266i	$0.518746\pi$
$710$	451.950i	0.636550i
$711$	0	0
$712$	−115.762	−0.162587
$713$	− 525.097i	− 0.736462i
$714$	0	0
$715$	846.104	1.18336
$716$	− 506.603i	− 0.707547i
$717$	0	0
$718$	75.2037	0.104740
$719$	1191.73i	1.65749i	0.559628	+	0.828744i	$0.310944\pi$
$719$	1191.73i	1.65749i	−0.559628	+	0.828744i	$0.689056\pi$
$720$	0	0
$721$	910.961	1.26347
$722$	− 326.384i	− 0.452056i
$723$	0	0
$724$	77.6602	0.107265
$725$	202.268i	0.278990i
$726$	0	0
$727$	674.507	0.927795	0.463898	−	0.885889i	$-0.346451\pi$
$727$	674.507	0.927795	0.463898	+	0.885889i	$0.346451\pi$
$728$	− 140.965i	− 0.193634i
$729$	0	0
$730$	574.034	0.786348
$731$	83.8223i	0.114668i
$732$	0	0
$733$	−195.294	−0.266431	−0.133215	−	0.991087i	$-0.542530\pi$
$733$	−195.294	−0.266431	−0.133215	+	0.991087i	$0.542530\pi$
$734$	− 621.142i	− 0.846242i
$735$	0	0
$736$	−136.689	−0.185719
$737$	− 2449.08i	− 3.32304i
$738$	0	0
$739$	1437.81	1.94561	0.972807	−	0.231616i	$-0.0744012\pi$
$739$	1437.81	1.94561	0.972807	+	0.231616i	$0.0744012\pi$
$740$	− 536.694i	− 0.725262i
$741$	0	0
$742$	183.690	0.247560
$743$	− 1001.88i	− 1.34842i	−0.738538	−	0.674212i	$-0.764484\pi$
$743$	− 1001.88i	− 1.34842i	0.738538	−	0.674212i	$-0.235516\pi$
$744$	0	0
$745$	929.819	1.24808
$746$	836.713i	1.12160i
$747$	0	0
$748$	−93.9303	−0.125575
$749$	1041.77i	1.39088i
$750$	0	0
$751$	−748.089	−0.996124	−0.498062	−	0.867142i	$-0.665955\pi$
$751$	−748.089	−0.996124	−0.498062	+	0.867142i	$0.665955\pi$
$752$	− 50.7536i	− 0.0674916i
$753$	0	0
$754$	178.431	0.236646
$755$	929.460i	1.23107i
$756$	0	0
$757$	788.946	1.04220	0.521100	−	0.853496i	$-0.325522\pi$
$757$	788.946	1.04220	0.521100	+	0.853496i	$0.325522\pi$
$758$	− 280.955i	− 0.370653i
$759$	0	0
$760$	412.452	0.542700
$761$	− 520.026i	− 0.683346i	−0.939819	−	0.341673i	$-0.889006\pi$
$761$	− 520.026i	− 0.683346i	0.939819	−	0.341673i	$-0.110994\pi$
$762$	0	0
$763$	−871.691	−1.14245
$764$	396.916i	0.519524i
$765$	0	0
$766$	−45.7313	−0.0597015
$767$	449.452i	0.585987i
$768$	0	0
$769$	−727.857	−0.946498	−0.473249	−	0.880929i	$-0.656919\pi$
$769$	−727.857	−0.946498	−0.473249	+	0.880929i	$0.656919\pi$
$770$	1282.31i	1.66533i
$771$	0	0
$772$	−113.991	−0.147657
$773$	− 1132.26i	− 1.46476i	−0.680894	−	0.732382i	$-0.738409\pi$
$773$	− 1132.26i	− 1.46476i	0.680894	−	0.732382i	$-0.261591\pi$
$774$	0	0
$775$	−237.579	−0.306554
$776$	114.345i	0.147352i
$777$	0	0
$778$	309.289	0.397544
$779$	1247.55i	1.60147i
$780$	0	0
$781$	−1103.45	−1.41287
$782$	− 77.5404i	− 0.0991565i
$783$	0	0
$784$	17.6389	0.0224986
$785$	728.198i	0.927641i
$786$	0	0
$787$	−772.596	−0.981698	−0.490849	−	0.871245i	$-0.663313\pi$
$787$	−772.596	−0.981698	−0.490849	+	0.871245i	$0.663313\pi$
$788$	− 149.707i	− 0.189984i
$789$	0	0
$790$	943.731	1.19460
$791$	757.620i	0.957800i
$792$	0	0
$793$	−287.934	−0.363095
$794$	− 748.680i	− 0.942922i
$795$	0	0
$796$	−260.863	−0.327717
$797$	1121.73i	1.40745i	0.710474	+	0.703723i	$0.248481\pi$
$797$	1121.73i	1.40745i	−0.710474	+	0.703723i	$0.751519\pi$
$798$	0	0
$799$	28.7913	0.0360341
$800$	61.8447i	0.0773059i
$801$	0	0
$802$	−369.245	−0.460405
$803$	1401.52i	1.74536i
$804$	0	0
$805$	−1058.56	−1.31498
$806$	209.581i	0.260027i
$807$	0	0
$808$	32.8122	0.0406091
$809$	− 1053.93i	− 1.30275i	−0.758754	−	0.651377i	$-0.774192\pi$
$809$	− 1053.93i	− 1.30275i	0.758754	−	0.651377i	$-0.225808\pi$
$810$	0	0
$811$	−440.011	−0.542554	−0.271277	−	0.962501i	$-0.587446\pi$
$811$	−440.011	−0.542554	−0.271277	+	0.962501i	$0.587446\pi$
$812$	270.420i	0.333030i
$813$	0	0
$814$	1310.35	1.60977
$815$	401.045i	0.492079i
$816$	0	0
$817$	898.647	1.09994
$818$	− 904.472i	− 1.10571i
$819$	0	0
$820$	−614.820	−0.749780
$821$	− 297.697i	− 0.362603i	−0.983428	−	0.181302i	$-0.941969\pi$
$821$	− 297.697i	− 0.362603i	0.983428	−	0.181302i	$-0.0580310\pi$
$822$	0	0
$823$	481.048	0.584505	0.292253	−	0.956341i	$-0.405595\pi$
$823$	481.048	0.584505	0.292253	+	0.956341i	$0.405595\pi$
$824$	352.561i	0.427866i
$825$	0	0
$826$	−681.163	−0.824653
$827$	− 313.336i	− 0.378882i	−0.981892	−	0.189441i	$-0.939332\pi$
$827$	− 313.336i	− 0.378882i	0.981892	−	0.189441i	$-0.0606676\pi$
$828$	0	0
$829$	−641.243	−0.773514	−0.386757	−	0.922182i	$-0.626405\pi$
$829$	−641.243	−0.773514	−0.386757	+	0.922182i	$0.626405\pi$
$830$	214.038i	0.257878i
$831$	0	0
$832$	54.5566	0.0655728
$833$	10.0061i	0.0120121i
$834$	0	0
$835$	43.4747	0.0520655
$836$	1007.01i	1.20456i
$837$	0	0
$838$	−1009.89	−1.20512
$839$	− 882.670i	− 1.05205i	−0.850469	−	0.526025i	$-0.823682\pi$
$839$	− 882.670i	− 1.05205i	0.850469	−	0.526025i	$-0.176318\pi$
$840$	0	0
$841$	498.707	0.592993
$842$	99.0185i	0.117599i
$843$	0	0
$844$	−63.2350	−0.0749230
$845$	734.273i	0.868963i
$846$	0	0
$847$	−2246.50	−2.65230
$848$	71.0919i	0.0838348i
$849$	0	0
$850$	−35.0830	−0.0412741
$851$	1081.71i	1.27110i
$852$	0	0
$853$	−1393.02	−1.63308	−0.816539	−	0.577291i	$-0.804110\pi$
$853$	−1393.02	−1.63308	−0.816539	+	0.577291i	$0.804110\pi$
$854$	− 436.377i	− 0.510980i
$855$	0	0
$856$	−403.187	−0.471013
$857$	126.277i	0.147348i	0.997282	+	0.0736741i	$0.0234724\pi$
$857$	126.277i	0.147348i	−0.997282	+	0.0736741i	$0.976528\pi$
$858$	0	0
$859$	1137.20	1.32386	0.661931	−	0.749565i	$-0.269737\pi$
$859$	1137.20	1.32386	0.661931	+	0.749565i	$0.269737\pi$
$860$	442.875i	0.514971i
$861$	0	0
$862$	−107.647	−0.124880
$863$	− 458.427i	− 0.531202i	−0.964083	−	0.265601i	$-0.914430\pi$
$863$	− 458.427i	− 0.531202i	0.964083	−	0.265601i	$-0.0855703\pi$
$864$	0	0
$865$	−434.605	−0.502433
$866$	− 155.158i	− 0.179167i
$867$	0	0
$868$	−317.629	−0.365933
$869$	2304.15i	2.65149i
$870$	0	0
$871$	−806.935	−0.926447
$872$	− 337.363i	− 0.386884i
$873$	0	0
$874$	−831.299	−0.951143
$875$	− 616.262i	− 0.704300i
$876$	0	0
$877$	1314.14	1.49845	0.749224	−	0.662316i	$-0.230426\pi$
$877$	1314.14	1.49845	0.749224	+	0.662316i	$0.230426\pi$
$878$	− 161.214i	− 0.183615i
$879$	0	0
$880$	−496.280	−0.563955
$881$	1517.25i	1.72219i	0.508442	+	0.861096i	$0.330222\pi$
$881$	1517.25i	1.72219i	−0.508442	+	0.861096i	$0.669778\pi$
$882$	0	0
$883$	1523.38	1.72524	0.862619	−	0.505855i	$-0.168823\pi$
$883$	1523.38	1.72524	0.862619	+	0.505855i	$0.168823\pi$
$884$	30.9486i	0.0350097i
$885$	0	0
$886$	244.250	0.275677
$887$	− 787.826i	− 0.888192i	−0.895979	−	0.444096i	$-0.853525\pi$
$887$	− 787.826i	− 0.888192i	0.895979	−	0.444096i	$-0.146475\pi$
$888$	0	0
$889$	−1604.44	−1.80477
$890$	− 346.960i	− 0.389843i
$891$	0	0
$892$	−204.105	−0.228818
$893$	− 308.667i	− 0.345652i
$894$	0	0
$895$	1518.39	1.69652
$896$	82.6828i	0.0922799i
$897$	0	0
$898$	269.348	0.299942
$899$	− 402.050i	− 0.447219i
$900$	0	0
$901$	−40.3287	−0.0447599
$902$	− 1501.10i	− 1.66419i
$903$	0	0
$904$	−293.215	−0.324353
$905$	232.763i	0.257196i
$906$	0	0
$907$	1700.38	1.87473	0.937366	−	0.348345i	$-0.113256\pi$
$907$	1700.38	1.87473	0.937366	+	0.348345i	$0.113256\pi$
$908$	514.433i	0.566557i
$909$	0	0
$910$	422.500	0.464286
$911$	303.682i	0.333350i	0.986012	+	0.166675i	$0.0533031\pi$
$911$	303.682i	0.333350i	−0.986012	+	0.166675i	$0.946697\pi$
$912$	0	0
$913$	−522.581	−0.572378
$914$	− 203.278i	− 0.222405i
$915$	0	0
$916$	−549.949	−0.600381
$917$	954.751i	1.04117i
$918$	0	0
$919$	−682.579	−0.742741	−0.371370	−	0.928485i	$-0.621112\pi$
$919$	−682.579	−0.742741	−0.371370	+	0.928485i	$0.621112\pi$
$920$	− 409.684i	− 0.445309i
$921$	0	0
$922$	−923.670	−1.00181
$923$	363.570i	0.393900i
$924$	0	0
$925$	489.417	0.529100
$926$	− 31.7108i	− 0.0342449i
$927$	0	0
$928$	−104.658	−0.112778
$929$	− 798.487i	− 0.859512i	−0.902945	−	0.429756i	$-0.858600\pi$
$929$	− 798.487i	− 0.859512i	0.902945	−	0.429756i	$-0.141400\pi$
$930$	0	0
$931$	107.274	0.115224
$932$	515.714i	0.553341i
$933$	0	0
$934$	674.958	0.722653
$935$	− 281.527i	− 0.301099i
$936$	0	0
$937$	773.853	0.825884	0.412942	−	0.910757i	$-0.364501\pi$
$937$	773.853	0.825884	0.412942	+	0.910757i	$0.364501\pi$
$938$	− 1222.94i	− 1.30378i
$939$	0	0
$940$	152.119	0.161828
$941$	1304.24i	1.38601i	0.720930	+	0.693007i	$0.243715\pi$
$941$	1304.24i	1.38601i	−0.720930	+	0.693007i	$0.756285\pi$
$942$	0	0
$943$	1239.17	1.31408
$944$	− 263.625i	− 0.279263i
$945$	0	0
$946$	−1081.29	−1.14301
$947$	825.166i	0.871347i	0.900105	+	0.435674i	$0.143490\pi$
$947$	825.166i	0.871347i	−0.900105	+	0.435674i	$0.856510\pi$
$948$	0	0
$949$	461.779	0.486596
$950$	376.120i	0.395915i
$951$	0	0
$952$	−46.9039	−0.0492688
$953$	− 1221.99i	− 1.28226i	−0.767434	−	0.641128i	$-0.778467\pi$
$953$	− 1221.99i	− 1.28226i	0.767434	−	0.641128i	$-0.221533\pi$
$954$	0	0
$955$	−1189.63	−1.24569
$956$	249.092i	0.260557i
$957$	0	0
$958$	−844.517	−0.881542
$959$	− 564.316i	− 0.588443i
$960$	0	0
$961$	−488.762	−0.508597
$962$	− 431.742i	− 0.448796i
$963$	0	0
$964$	−130.895	−0.135783
$965$	− 341.653i	− 0.354045i
$966$	0	0
$967$	−270.308	−0.279532	−0.139766	−	0.990185i	$-0.544635\pi$
$967$	−270.308	−0.279532	−0.139766	+	0.990185i	$0.544635\pi$
$968$	− 869.442i	− 0.898184i
$969$	0	0
$970$	−342.715	−0.353314
$971$	1817.80i	1.87209i	0.351883	+	0.936044i	$0.385542\pi$
$971$	1817.80i	1.87209i	−0.351883	+	0.936044i	$0.614458\pi$
$972$	0	0
$973$	−780.879	−0.802548
$974$	− 1073.31i	− 1.10197i
$975$	0	0
$976$	168.887	0.173040
$977$	− 1315.58i	− 1.34655i	−0.739391	−	0.673276i	$-0.764886\pi$
$977$	− 1315.58i	− 1.34655i	0.739391	−	0.673276i	$-0.235114\pi$
$978$	0	0
$979$	847.114	0.865285
$980$	52.8671i	0.0539461i
$981$	0	0
$982$	−371.430	−0.378238
$983$	633.484i	0.644439i	0.946665	+	0.322220i	$0.104429\pi$
$983$	633.484i	0.644439i	−0.946665	+	0.322220i	$0.895571\pi$
$984$	0	0
$985$	448.701	0.455534
$986$	− 59.3701i	− 0.0602131i
$987$	0	0
$988$	331.795	0.335825
$989$	− 892.617i	− 0.902545i
$990$	0	0
$991$	−1634.77	−1.64961	−0.824806	−	0.565416i	$-0.808716\pi$
$991$	−1634.77	−1.64961	−0.824806	+	0.565416i	$0.808716\pi$
$992$	− 122.929i	− 0.123921i
$993$	0	0
$994$	−551.006	−0.554332
$995$	− 781.857i	− 0.785786i
$996$	0	0
$997$	−439.024	−0.440345	−0.220173	−	0.975461i	$-0.570662\pi$
$997$	−439.024	−0.440345	−0.220173	+	0.975461i	$0.570662\pi$
$998$	128.585i	0.128842i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.3.b.c.1457.4		36
3.2	odd	2	inner	1458.3.b.c.1457.33		36
27.2	odd	18		54.3.f.a.23.5	✓	36
27.13	even	9		54.3.f.a.47.5	yes	36
27.14	odd	18		162.3.f.a.143.1		36
27.25	even	9		162.3.f.a.17.1		36
108.67	odd	18		432.3.bc.c.209.2		36
108.83	even	18		432.3.bc.c.401.2		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.3.f.a.23.5	✓	36	27.2	odd	18
54.3.f.a.47.5	yes	36	27.13	even	9
162.3.f.a.17.1		36	27.25	even	9
162.3.f.a.143.1		36	27.14	odd	18
432.3.bc.c.209.2		36	108.67	odd	18
432.3.bc.c.401.2		36	108.83	even	18
1458.3.b.c.1457.4		36	1.1	even	1	trivial
1458.3.b.c.1457.33		36	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:	\( N \)	\(=\)	\( 1458 = 2 \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1458.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -5.99439i q^{5} +7.30820 q^{7} +2.82843i q^{8} -8.47735 q^{10} -20.6977i q^{11} -6.81957 q^{13} -10.3353i q^{14} +4.00000 q^{16} +2.26910i q^{17} +24.3267 q^{19} +11.9888i q^{20} -29.2710 q^{22} -24.1635i q^{23} -10.9327 q^{25} +9.64433i q^{26} -14.6164 q^{28} -18.5012i q^{29} +21.7310 q^{31} -5.65685i q^{32} +3.20899 q^{34} -43.8082i q^{35} -44.7664 q^{37} -34.4032i q^{38} +16.9547 q^{40} +51.2829i q^{41} +36.9408 q^{43} +41.3954i q^{44} -34.1723 q^{46} -12.6884i q^{47} +4.40972 q^{49} +15.4612i q^{50} +13.6391 q^{52} +17.7730i q^{53} -124.070 q^{55} +20.6707i q^{56} -26.1646 q^{58} -65.9062i q^{59} +42.2218 q^{61} -30.7323i q^{62} -8.00000 q^{64} +40.8792i q^{65} +118.326 q^{67} -4.53820i q^{68} -61.9541 q^{70} -53.3127i q^{71} -67.7138 q^{73} +63.3092i q^{74} -48.6534 q^{76} -151.263i q^{77} -111.324 q^{79} -23.9776i q^{80} +72.5250 q^{82} -25.2483i q^{83} +13.6019 q^{85} -52.2421i q^{86} +58.5419 q^{88} +40.9280i q^{89} -49.8388 q^{91} +48.3269i q^{92} -17.9441 q^{94} -145.824i q^{95} +40.4271 q^{97} -6.23628i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 72 q^{4} + 144 q^{16} - 180 q^{25} + 252 q^{49} - 36 q^{61} - 288 q^{64} + 180 q^{67} - 252 q^{73} + 396 q^{91}+O(q^{100}) \) 36 * q - 72 * q^4 + 144 * q^16 - 180 * q^25 + 252 * q^49 - 36 * q^61 - 288 * q^64 + 180 * q^67 - 252 * q^73 + 396 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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