LMFDB - Embedded newform 1458.3.b.c.1457.13

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			1457.13
Character	$\chi$	$=$	1458.1457
Dual form			1458.3.b.c.1457.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} -2.00000 q^{4} +4.04819i q^{5} -13.2098 q^{7} +2.82843i q^{8} +5.72501 q^{10} -13.5794i q^{11} -18.5660 q^{13} +18.6814i q^{14} +4.00000 q^{16} -16.0261i q^{17} -0.654012 q^{19} -8.09638i q^{20} -19.2042 q^{22} -6.60266i q^{23} +8.61214 q^{25} +26.2563i q^{26} +26.4195 q^{28} +1.36420i q^{29} -18.6717 q^{31} -5.65685i q^{32} -22.6643 q^{34} -53.4756i q^{35} +16.8788 q^{37} +0.924912i q^{38} -11.4500 q^{40} +42.8656i q^{41} +0.0655728 q^{43} +27.1589i q^{44} -9.33756 q^{46} +46.8056i q^{47} +125.498 q^{49} -12.1794i q^{50} +37.1320 q^{52} +14.3109i q^{53} +54.9722 q^{55} -37.3628i q^{56} +1.92927 q^{58} -4.86429i q^{59} -15.7567 q^{61} +26.4057i q^{62} -8.00000 q^{64} -75.1588i q^{65} +39.5133 q^{67} +32.0522i q^{68} -75.6260 q^{70} +70.3335i q^{71} +68.3506 q^{73} -23.8702i q^{74} +1.30802 q^{76} +179.381i q^{77} -50.4483 q^{79} +16.1928i q^{80} +60.6211 q^{82} +146.909i q^{83} +64.8767 q^{85} -0.0927340i q^{86} +38.4085 q^{88} -19.9714i q^{89} +245.252 q^{91} +13.2053i q^{92} +66.1931 q^{94} -2.64757i q^{95} +105.729 q^{97} -177.480i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	4.04819i	0.809638i	0.914397	+	0.404819i	$0.132666\pi$
$5$	4.04819i	0.809638i	−0.914397	+	0.404819i	$0.867334\pi$
$6$	0	0
$7$	−13.2098	−1.88711	−0.943554	−	0.331219i	$-0.892540\pi$
$7$	−13.2098	−1.88711	−0.943554	+	0.331219i	$0.892540\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	5.72501	0.572501
$11$	− 13.5794i	− 1.23449i	−0.786769	−	0.617247i	$-0.788248\pi$
$11$	− 13.5794i	− 1.23449i	0.786769	−	0.617247i	$-0.211752\pi$
$12$	0	0
$13$	−18.5660	−1.42815	−0.714077	−	0.700067i	$-0.753153\pi$
$13$	−18.5660	−1.42815	−0.714077	+	0.700067i	$0.753153\pi$
$14$	18.6814i	1.33439i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 16.0261i	− 0.942711i	−0.881943	−	0.471355i	$-0.843765\pi$
$17$	− 16.0261i	− 0.942711i	0.881943	−	0.471355i	$-0.156235\pi$
$18$	0	0
$19$	−0.654012	−0.0344217	−0.0172108	−	0.999852i	$-0.505479\pi$
$19$	−0.654012	−0.0344217	−0.0172108	+	0.999852i	$0.505479\pi$
$20$	− 8.09638i	− 0.404819i
$21$	0	0
$22$	−19.2042	−0.872920
$23$	− 6.60266i	− 0.287072i	−0.989645	−	0.143536i	$-0.954153\pi$
$23$	− 6.60266i	− 0.287072i	0.989645	−	0.143536i	$-0.0458473\pi$
$24$	0	0
$25$	8.61214	0.344486
$26$	26.2563i	1.00986i
$27$	0	0
$28$	26.4195	0.943554
$29$	1.36420i	0.0470414i	0.999723	+	0.0235207i	$0.00748756\pi$
$29$	1.36420i	0.0470414i	−0.999723	+	0.0235207i	$0.992512\pi$
$30$	0	0
$31$	−18.6717	−0.602312	−0.301156	−	0.953575i	$-0.597372\pi$
$31$	−18.6717	−0.602312	−0.301156	+	0.953575i	$0.597372\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	−22.6643	−0.666597
$35$	− 53.4756i	− 1.52787i
$36$	0	0
$37$	16.8788	0.456184	0.228092	−	0.973640i	$-0.426751\pi$
$37$	16.8788	0.456184	0.228092	+	0.973640i	$0.426751\pi$
$38$	0.924912i	0.0243398i
$39$	0	0
$40$	−11.4500	−0.286250
$41$	42.8656i	1.04550i	0.852486	+	0.522751i	$0.175094\pi$
$41$	42.8656i	1.04550i	−0.852486	+	0.522751i	$0.824906\pi$
$42$	0	0
$43$	0.0655728	0.00152495	0.000762475	−	1.00000i	$-0.499757\pi$
$43$	0.0655728	0.00152495	0.000762475		1.00000i	$0.499757\pi$
$44$	27.1589i	0.617247i
$45$	0	0
$46$	−9.33756	−0.202991
$47$	46.8056i	0.995864i	0.867216	+	0.497932i	$0.165907\pi$
$47$	46.8056i	0.995864i	−0.867216	+	0.497932i	$0.834093\pi$
$48$	0	0
$49$	125.498	2.56118
$50$	− 12.1794i	− 0.243588i
$51$	0	0
$52$	37.1320	0.714077
$53$	14.3109i	0.270017i	0.990844	+	0.135008i	$0.0431061\pi$
$53$	14.3109i	0.270017i	−0.990844	+	0.135008i	$0.956894\pi$
$54$	0	0
$55$	54.9722	0.999494
$56$	− 37.3628i	− 0.667193i
$57$	0	0
$58$	1.92927	0.0332633
$59$	− 4.86429i	− 0.0824456i	−0.999150	−	0.0412228i	$-0.986875\pi$
$59$	− 4.86429i	− 0.0824456i	0.999150	−	0.0412228i	$-0.0131253\pi$
$60$	0	0
$61$	−15.7567	−0.258306	−0.129153	−	0.991625i	$-0.541226\pi$
$61$	−15.7567	−0.258306	−0.129153	+	0.991625i	$0.541226\pi$
$62$	26.4057i	0.425899i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 75.1588i	− 1.15629i
$66$	0	0
$67$	39.5133	0.589751	0.294875	−	0.955536i	$-0.404722\pi$
$67$	39.5133	0.589751	0.294875	+	0.955536i	$0.404722\pi$
$68$	32.0522i	0.471355i
$69$	0	0
$70$	−75.6260	−1.08037
$71$	70.3335i	0.990613i	0.868718	+	0.495306i	$0.164944\pi$
$71$	70.3335i	0.990613i	−0.868718	+	0.495306i	$0.835056\pi$
$72$	0	0
$73$	68.3506	0.936310	0.468155	−	0.883646i	$-0.344919\pi$
$73$	68.3506	0.936310	0.468155	+	0.883646i	$0.344919\pi$
$74$	− 23.8702i	− 0.322571i
$75$	0	0
$76$	1.30802	0.0172108
$77$	179.381i	2.32962i
$78$	0	0
$79$	−50.4483	−0.638586	−0.319293	−	0.947656i	$-0.603445\pi$
$79$	−50.4483	−0.638586	−0.319293	+	0.947656i	$0.603445\pi$
$80$	16.1928i	0.202410i
$81$	0	0
$82$	60.6211	0.739281
$83$	146.909i	1.76998i	0.465605	+	0.884992i	$0.345837\pi$
$83$	146.909i	1.76998i	−0.465605	+	0.884992i	$0.654163\pi$
$84$	0	0
$85$	64.8767	0.763255
$86$	− 0.0927340i	− 0.00107830i
$87$	0	0
$88$	38.4085	0.436460
$89$	− 19.9714i	− 0.224397i	−0.993686	−	0.112199i	$-0.964211\pi$
$89$	− 19.9714i	− 0.224397i	0.993686	−	0.112199i	$-0.0357893\pi$
$90$	0	0
$91$	245.252	2.69508
$92$	13.2053i	0.143536i
$93$	0	0
$94$	66.1931	0.704182
$95$	− 2.64757i	− 0.0278691i
$96$	0	0
$97$	105.729	1.08999	0.544993	−	0.838441i	$-0.316532\pi$
$97$	105.729	1.08999	0.544993	+	0.838441i	$0.316532\pi$
$98$	− 177.480i	− 1.81103i
$99$	0	0
$100$	−17.2243	−0.172243
$101$	60.2773i	0.596805i	0.954440	+	0.298402i	$0.0964538\pi$
$101$	60.2773i	0.596805i	−0.954440	+	0.298402i	$0.903546\pi$
$102$	0	0
$103$	−96.1261	−0.933263	−0.466631	−	0.884452i	$-0.654533\pi$
$103$	−96.1261	−0.933263	−0.466631	+	0.884452i	$0.654533\pi$
$104$	− 52.5126i	− 0.504929i
$105$	0	0
$106$	20.2386	0.190931
$107$	− 107.330i	− 1.00308i	−0.865133	−	0.501542i	$-0.832766\pi$
$107$	− 107.330i	− 1.00308i	0.865133	−	0.501542i	$-0.167234\pi$
$108$	0	0
$109$	85.1199	0.780917	0.390458	−	0.920621i	$-0.372317\pi$
$109$	85.1199	0.780917	0.390458	+	0.920621i	$0.372317\pi$
$110$	− 77.7424i	− 0.706749i
$111$	0	0
$112$	−52.8390	−0.471777
$113$	168.175i	1.48827i	0.668029	+	0.744135i	$0.267138\pi$
$113$	168.175i	1.48827i	−0.668029	+	0.744135i	$0.732862\pi$
$114$	0	0
$115$	26.7288	0.232424
$116$	− 2.72840i	− 0.0235207i
$117$	0	0
$118$	−6.87914	−0.0582978
$119$	211.701i	1.77900i
$120$	0	0
$121$	−63.4013	−0.523977
$122$	22.2833i	0.182650i
$123$	0	0
$124$	37.3433	0.301156
$125$	136.068i	1.08855i
$126$	0	0
$127$	171.665	1.35169	0.675845	−	0.737044i	$-0.263779\pi$
$127$	171.665	1.35169	0.675845	+	0.737044i	$0.263779\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	−106.291	−0.817620
$131$	− 175.072i	− 1.33643i	−0.743970	−	0.668213i	$-0.767060\pi$
$131$	− 175.072i	− 1.33643i	0.743970	−	0.668213i	$-0.232940\pi$
$132$	0	0
$133$	8.63934	0.0649574
$134$	− 55.8803i	− 0.417017i
$135$	0	0
$136$	45.3286	0.333299
$137$	49.0934i	0.358346i	0.983818	+	0.179173i	$0.0573421\pi$
$137$	49.0934i	0.358346i	−0.983818	+	0.179173i	$0.942658\pi$
$138$	0	0
$139$	−139.886	−1.00637	−0.503186	−	0.864178i	$-0.667839\pi$
$139$	−139.886	−1.00637	−0.503186	+	0.864178i	$0.667839\pi$
$140$	106.951i	0.763937i
$141$	0	0
$142$	99.4666	0.700469
$143$	252.116i	1.76305i
$144$	0	0
$145$	−5.52254	−0.0380865
$146$	− 96.6624i	− 0.662071i
$147$	0	0
$148$	−33.7576	−0.228092
$149$	− 240.264i	− 1.61251i	−0.591566	−	0.806257i	$-0.701490\pi$
$149$	− 240.264i	− 1.61251i	0.591566	−	0.806257i	$-0.298510\pi$
$150$	0	0
$151$	20.3771	0.134947	0.0674737	−	0.997721i	$-0.478506\pi$
$151$	20.3771	0.134947	0.0674737	+	0.997721i	$0.478506\pi$
$152$	− 1.84982i	− 0.0121699i
$153$	0	0
$154$	253.683	1.64729
$155$	− 75.5865i	− 0.487655i
$156$	0	0
$157$	−224.762	−1.43161	−0.715803	−	0.698303i	$-0.753939\pi$
$157$	−224.762	−1.43161	−0.715803	+	0.698303i	$0.753939\pi$
$158$	71.3446i	0.451548i
$159$	0	0
$160$	22.9000	0.143125
$161$	87.2195i	0.541736i
$162$	0	0
$163$	−155.441	−0.953626	−0.476813	−	0.879005i	$-0.658208\pi$
$163$	−155.441	−0.953626	−0.476813	+	0.879005i	$0.658208\pi$
$164$	− 85.7311i	− 0.522751i
$165$	0	0
$166$	207.760	1.25157
$167$	− 321.068i	− 1.92256i	−0.275565	−	0.961282i	$-0.588865\pi$
$167$	− 321.068i	− 1.92256i	0.275565	−	0.961282i	$-0.411135\pi$
$168$	0	0
$169$	175.697	1.03963
$170$	− 91.7495i	− 0.539703i
$171$	0	0
$172$	−0.131146	−0.000762475 0
$173$	142.485i	0.823613i	0.911271	+	0.411806i	$0.135102\pi$
$173$	142.485i	0.823613i	−0.911271	+	0.411806i	$0.864898\pi$
$174$	0	0
$175$	−113.764	−0.650082
$176$	− 54.3178i	− 0.308624i
$177$	0	0
$178$	−28.2438	−0.158673
$179$	− 214.891i	− 1.20051i	−0.799810	−	0.600253i	$-0.795066\pi$
$179$	− 214.891i	− 1.20051i	0.799810	−	0.600253i	$-0.204934\pi$
$180$	0	0
$181$	301.745	1.66710	0.833550	−	0.552444i	$-0.186305\pi$
$181$	301.745	1.66710	0.833550	+	0.552444i	$0.186305\pi$
$182$	− 346.839i	− 1.90571i
$183$	0	0
$184$	18.6751	0.101495
$185$	68.3286i	0.369344i
$186$	0	0
$187$	−217.625	−1.16377
$188$	− 93.6112i	− 0.497932i
$189$	0	0
$190$	−3.74422	−0.0197064
$191$	− 297.894i	− 1.55965i	−0.625996	−	0.779826i	$-0.715307\pi$
$191$	− 297.894i	− 1.55965i	0.625996	−	0.779826i	$-0.284693\pi$
$192$	0	0
$193$	125.871	0.652182	0.326091	−	0.945338i	$-0.394268\pi$
$193$	125.871	0.652182	0.326091	+	0.945338i	$0.394268\pi$
$194$	− 149.523i	− 0.770737i
$195$	0	0
$196$	−250.995	−1.28059
$197$	− 43.4026i	− 0.220318i	−0.993914	−	0.110159i	$-0.964864\pi$
$197$	− 43.4026i	− 0.220318i	0.993914	−	0.110159i	$-0.0351359\pi$
$198$	0	0
$199$	−75.6687	−0.380245	−0.190122	−	0.981760i	$-0.560888\pi$
$199$	−75.6687	−0.380245	−0.190122	+	0.981760i	$0.560888\pi$
$200$	24.3588i	0.121794i
$201$	0	0
$202$	85.2450	0.422005
$203$	− 18.0207i	− 0.0887721i
$204$	0	0
$205$	−173.528	−0.846478
$206$	135.943i	0.659917i
$207$	0	0
$208$	−74.2640	−0.357039
$209$	8.88112i	0.0424934i
$210$	0	0
$211$	215.427	1.02098	0.510490	−	0.859884i	$-0.329464\pi$
$211$	215.427	1.02098	0.510490	+	0.859884i	$0.329464\pi$
$212$	− 28.6218i	− 0.135008i
$213$	0	0
$214$	−151.788	−0.709287
$215$	0.265451i	0.00123466i
$216$	0	0
$217$	246.648	1.13663
$218$	− 120.378i	− 0.552191i
$219$	0	0
$220$	−109.944	−0.499747
$221$	297.540i	1.34634i
$222$	0	0
$223$	−83.2080	−0.373130	−0.186565	−	0.982443i	$-0.559736\pi$
$223$	−83.2080	−0.373130	−0.186565	+	0.982443i	$0.559736\pi$
$224$	74.7257i	0.333597i
$225$	0	0
$226$	237.835	1.05237
$227$	62.5222i	0.275428i	0.990472	+	0.137714i	$0.0439755\pi$
$227$	62.5222i	0.275428i	−0.990472	+	0.137714i	$0.956024\pi$
$228$	0	0
$229$	145.202	0.634071	0.317035	−	0.948414i	$-0.397313\pi$
$229$	145.202	0.634071	0.317035	+	0.948414i	$0.397313\pi$
$230$	− 37.8003i	− 0.164349i
$231$	0	0
$232$	−3.85854	−0.0166316
$233$	135.303i	0.580702i	0.956920	+	0.290351i	$0.0937720\pi$
$233$	135.303i	0.580702i	−0.956920	+	0.290351i	$0.906228\pi$
$234$	0	0
$235$	−189.478	−0.806289
$236$	9.72858i	0.0412228i
$237$	0	0
$238$	299.390	1.25794
$239$	− 146.553i	− 0.613193i	−0.951840	−	0.306597i	$-0.900810\pi$
$239$	− 146.553i	− 0.613193i	0.951840	−	0.306597i	$-0.0991903\pi$
$240$	0	0
$241$	317.139	1.31593	0.657966	−	0.753048i	$-0.271417\pi$
$241$	317.139	1.31593	0.657966	+	0.753048i	$0.271417\pi$
$242$	89.6629i	0.370508i
$243$	0	0
$244$	31.5133	0.129153
$245$	508.039i	2.07363i
$246$	0	0
$247$	12.1424	0.0491595
$248$	− 52.8114i	− 0.212949i
$249$	0	0
$250$	192.430	0.769719
$251$	− 159.268i	− 0.634534i	−0.948336	−	0.317267i	$-0.897235\pi$
$251$	− 159.268i	− 0.634534i	0.948336	−	0.317267i	$-0.102765\pi$
$252$	0	0
$253$	−89.6604	−0.354389
$254$	− 242.770i	− 0.955789i
$255$	0	0
$256$	16.0000	0.0625000
$257$	60.5675i	0.235671i	0.993033	+	0.117836i	$0.0375956\pi$
$257$	60.5675i	0.235671i	−0.993033	+	0.117836i	$0.962404\pi$
$258$	0	0
$259$	−222.965	−0.860868
$260$	150.318i	0.578144i
$261$	0	0
$262$	−247.589	−0.944996
$263$	337.834i	1.28454i	0.766479	+	0.642269i	$0.222007\pi$
$263$	337.834i	1.28454i	−0.766479	+	0.642269i	$0.777993\pi$
$264$	0	0
$265$	−57.9332	−0.218616
$266$	− 12.2179i	− 0.0459318i
$267$	0	0
$268$	−79.0266	−0.294875
$269$	− 39.0580i	− 0.145197i	−0.997361	−	0.0725986i	$-0.976871\pi$
$269$	− 39.0580i	− 0.145197i	0.997361	−	0.0725986i	$-0.0231292\pi$
$270$	0	0
$271$	247.005	0.911459	0.455730	−	0.890118i	$-0.349378\pi$
$271$	247.005	0.911459	0.455730	+	0.890118i	$0.349378\pi$
$272$	− 64.1043i	− 0.235678i
$273$	0	0
$274$	69.4285	0.253389
$275$	− 116.948i	− 0.425266i
$276$	0	0
$277$	−190.714	−0.688498	−0.344249	−	0.938878i	$-0.611866\pi$
$277$	−190.714	−0.688498	−0.344249	+	0.938878i	$0.611866\pi$
$278$	197.828i	0.711613i
$279$	0	0
$280$	151.252	0.540185
$281$	− 135.602i	− 0.482571i	−0.970454	−	0.241285i	$-0.922431\pi$
$281$	− 135.602i	− 0.482571i	0.970454	−	0.241285i	$-0.0775689\pi$
$282$	0	0
$283$	464.108	1.63996	0.819979	−	0.572394i	$-0.193985\pi$
$283$	464.108	1.63996	0.819979	+	0.572394i	$0.193985\pi$
$284$	− 140.667i	− 0.495306i
$285$	0	0
$286$	356.546	1.24666
$287$	− 566.244i	− 1.97297i
$288$	0	0
$289$	32.1647	0.111296
$290$	7.81005i	0.0269312i
$291$	0	0
$292$	−136.701	−0.468155
$293$	− 135.326i	− 0.461862i	−0.972970	−	0.230931i	$-0.925823\pi$
$293$	− 135.326i	− 0.461862i	0.972970	−	0.230931i	$-0.0741772\pi$
$294$	0	0
$295$	19.6916	0.0667511
$296$	47.7404i	0.161285i
$297$	0	0
$298$	−339.785	−1.14022
$299$	122.585i	0.409983i
$300$	0	0
$301$	−0.866201	−0.00287775
$302$	− 28.8175i	− 0.0954223i
$303$	0	0
$304$	−2.61605	−0.00860542
$305$	− 63.7860i	− 0.209134i
$306$	0	0
$307$	348.616	1.13556	0.567778	−	0.823182i	$-0.307803\pi$
$307$	348.616	1.13556	0.567778	+	0.823182i	$0.307803\pi$
$308$	− 358.762i	− 1.16481i
$309$	0	0
$310$	−106.895	−0.344824
$311$	219.028i	0.704270i	0.935949	+	0.352135i	$0.114544\pi$
$311$	219.028i	0.704270i	−0.935949	+	0.352135i	$0.885456\pi$
$312$	0	0
$313$	272.107	0.869350	0.434675	−	0.900587i	$-0.356863\pi$
$313$	272.107	0.869350	0.434675	+	0.900587i	$0.356863\pi$
$314$	317.861i	1.01230i
$315$	0	0
$316$	100.897	0.319293
$317$	467.002i	1.47319i	0.676333	+	0.736596i	$0.263568\pi$
$317$	467.002i	1.47319i	−0.676333	+	0.736596i	$0.736432\pi$
$318$	0	0
$319$	18.5251	0.0580723
$320$	− 32.3855i	− 0.101205i
$321$	0	0
$322$	123.347	0.383065
$323$	10.4812i	0.0324497i
$324$	0	0
$325$	−159.893	−0.491979
$326$	219.827i	0.674316i
$327$	0	0
$328$	−121.242	−0.369641
$329$	− 618.290i	− 1.87930i
$330$	0	0
$331$	103.885	0.313851	0.156925	−	0.987610i	$-0.449842\pi$
$331$	103.885	0.313851	0.156925	+	0.987610i	$0.449842\pi$
$332$	− 293.817i	− 0.884992i
$333$	0	0
$334$	−454.059	−1.35946
$335$	159.957i	0.477485i
$336$	0	0
$337$	−563.183	−1.67117	−0.835583	−	0.549364i	$-0.814870\pi$
$337$	−563.183	−1.67117	−0.835583	+	0.549364i	$0.814870\pi$
$338$	− 248.473i	− 0.735126i
$339$	0	0
$340$	−129.753	−0.381627
$341$	253.551i	0.743551i
$342$	0	0
$343$	−1010.52	−2.94611
$344$	0.185468i	0 0.000539151i
$345$	0	0
$346$	201.504	0.582382
$347$	− 271.814i	− 0.783326i	−0.920109	−	0.391663i	$-0.871900\pi$
$347$	− 271.814i	− 0.783326i	0.920109	−	0.391663i	$-0.128100\pi$
$348$	0	0
$349$	80.2151	0.229843	0.114921	−	0.993375i	$-0.463338\pi$
$349$	80.2151	0.229843	0.114921	+	0.993375i	$0.463338\pi$
$350$	160.887i	0.459677i
$351$	0	0
$352$	−76.8169	−0.218230
$353$	− 119.373i	− 0.338167i	−0.985602	−	0.169084i	$-0.945919\pi$
$353$	− 119.373i	− 0.338167i	0.985602	−	0.169084i	$-0.0540808\pi$
$354$	0	0
$355$	−284.723	−0.802038
$356$	39.9427i	0.112199i
$357$	0	0
$358$	−303.901	−0.848886
$359$	− 260.178i	− 0.724729i	−0.932037	−	0.362364i	$-0.881970\pi$
$359$	− 260.178i	− 0.724729i	0.932037	−	0.362364i	$-0.118030\pi$
$360$	0	0
$361$	−360.572	−0.998815
$362$	− 426.732i	− 1.17882i
$363$	0	0
$364$	−490.505	−1.34754
$365$	276.697i	0.758073i
$366$	0	0
$367$	−391.857	−1.06773	−0.533866	−	0.845569i	$-0.679261\pi$
$367$	−391.857	−1.06773	−0.533866	+	0.845569i	$0.679261\pi$
$368$	− 26.4106i	− 0.0717680i
$369$	0	0
$370$	96.6312	0.261166
$371$	− 189.043i	− 0.509550i
$372$	0	0
$373$	−111.052	−0.297726	−0.148863	−	0.988858i	$-0.547561\pi$
$373$	−111.052	−0.297726	−0.148863	+	0.988858i	$0.547561\pi$
$374$	307.769i	0.822911i
$375$	0	0
$376$	−132.386	−0.352091
$377$	− 25.3277i	− 0.0671823i
$378$	0	0
$379$	396.757	1.04685	0.523427	−	0.852071i	$-0.324653\pi$
$379$	396.757	1.04685	0.523427	+	0.852071i	$0.324653\pi$
$380$	5.29513i	0.0139346i
$381$	0	0
$382$	−421.285	−1.10284
$383$	226.074i	0.590273i	0.955455	+	0.295136i	$0.0953650\pi$
$383$	226.074i	0.590273i	−0.955455	+	0.295136i	$0.904635\pi$
$384$	0	0
$385$	−726.169	−1.88615
$386$	− 178.009i	− 0.461162i
$387$	0	0
$388$	−211.457	−0.544993
$389$	647.691i	1.66502i	0.554014	+	0.832508i	$0.313096\pi$
$389$	647.691i	1.66502i	−0.554014	+	0.832508i	$0.686904\pi$
$390$	0	0
$391$	−105.815	−0.270626
$392$	354.961i	0.905513i
$393$	0	0
$394$	−61.3805	−0.155788
$395$	− 204.224i	− 0.517023i
$396$	0	0
$397$	−360.160	−0.907204	−0.453602	−	0.891204i	$-0.649861\pi$
$397$	−360.160	−0.907204	−0.453602	+	0.891204i	$0.649861\pi$
$398$	107.012i	0.268873i
$399$	0	0
$400$	34.4486	0.0861214
$401$	− 246.471i	− 0.614641i	−0.951606	−	0.307321i	$-0.900568\pi$
$401$	− 246.471i	− 0.614641i	0.951606	−	0.307321i	$-0.0994324\pi$
$402$	0	0
$403$	346.658	0.860194
$404$	− 120.555i	− 0.298402i
$405$	0	0
$406$	−25.4852	−0.0627714
$407$	− 229.205i	− 0.563156i
$408$	0	0
$409$	44.0009	0.107582	0.0537909	−	0.998552i	$-0.482870\pi$
$409$	44.0009	0.107582	0.0537909	+	0.998552i	$0.482870\pi$
$410$	245.406i	0.598550i
$411$	0	0
$412$	192.252	0.466631
$413$	64.2561i	0.155584i
$414$	0	0
$415$	−594.715	−1.43305
$416$	105.025i	0.252464i
$417$	0	0
$418$	12.5598	0.0300474
$419$	299.934i	0.715834i	0.933753	+	0.357917i	$0.116513\pi$
$419$	299.934i	0.715834i	−0.933753	+	0.357917i	$0.883487\pi$
$420$	0	0
$421$	378.263	0.898487	0.449244	−	0.893409i	$-0.351693\pi$
$421$	378.263	0.898487	0.449244	+	0.893409i	$0.351693\pi$
$422$	− 304.660i	− 0.721942i
$423$	0	0
$424$	−40.4773	−0.0954653
$425$	− 138.019i	− 0.324750i
$426$	0	0
$427$	208.142	0.487451
$428$	214.660i	0.501542i
$429$	0	0
$430$	0.375405	0.000873035 0
$431$	739.788i	1.71645i	0.513278	+	0.858223i	$0.328431\pi$
$431$	739.788i	1.71645i	−0.513278	+	0.858223i	$0.671569\pi$
$432$	0	0
$433$	−856.804	−1.97876	−0.989381	−	0.145344i	$-0.953571\pi$
$433$	−856.804	−1.97876	−0.989381	+	0.145344i	$0.953571\pi$
$434$	− 348.813i	− 0.803717i
$435$	0	0
$436$	−170.240	−0.390458
$437$	4.31821i	0.00988150i
$438$	0	0
$439$	802.600	1.82825	0.914123	−	0.405438i	$-0.132881\pi$
$439$	802.600	1.82825	0.914123	+	0.405438i	$0.132881\pi$
$440$	155.485i	0.353375i
$441$	0	0
$442$	420.786	0.952004
$443$	305.701i	0.690071i	0.938590	+	0.345035i	$0.112133\pi$
$443$	305.701i	0.690071i	−0.938590	+	0.345035i	$0.887867\pi$
$444$	0	0
$445$	80.8479	0.181681
$446$	117.674i	0.263843i
$447$	0	0
$448$	105.678	0.235888
$449$	412.782i	0.919337i	0.888091	+	0.459668i	$0.152032\pi$
$449$	412.782i	0.919337i	−0.888091	+	0.459668i	$0.847968\pi$
$450$	0	0
$451$	582.090	1.29067
$452$	− 336.349i	− 0.744135i
$453$	0	0
$454$	88.4198	0.194757
$455$	992.829i	2.18204i
$456$	0	0
$457$	495.281	1.08377	0.541883	−	0.840454i	$-0.317711\pi$
$457$	495.281	1.08377	0.541883	+	0.840454i	$0.317711\pi$
$458$	− 205.347i	− 0.448356i
$459$	0	0
$460$	−53.4576	−0.116212
$461$	815.426i	1.76882i	0.466711	+	0.884410i	$0.345439\pi$
$461$	815.426i	1.76882i	−0.466711	+	0.884410i	$0.654561\pi$
$462$	0	0
$463$	416.269	0.899069	0.449534	−	0.893263i	$-0.351590\pi$
$463$	416.269	0.899069	0.449534	+	0.893263i	$0.351590\pi$
$464$	5.45680i	0.0117603i
$465$	0	0
$466$	191.348	0.410618
$467$	142.682i	0.305530i	0.988263	+	0.152765i	$0.0488178\pi$
$467$	142.682i	0.305530i	−0.988263	+	0.152765i	$0.951182\pi$
$468$	0	0
$469$	−521.961	−1.11292
$470$	267.962i	0.570133i
$471$	0	0
$472$	13.7583	0.0291489
$473$	− 0.890443i	− 0.00188254i
$474$	0	0
$475$	−5.63244	−0.0118578
$476$	− 423.401i	− 0.889499i
$477$	0	0
$478$	−207.257	−0.433593
$479$	103.522i	0.216121i	0.994144	+	0.108060i	$0.0344640\pi$
$479$	103.522i	0.216121i	−0.994144	+	0.108060i	$0.965536\pi$
$480$	0	0
$481$	−313.372	−0.651501
$482$	− 448.503i	− 0.930504i
$483$	0	0
$484$	126.803	0.261989
$485$	428.010i	0.882495i
$486$	0	0
$487$	−757.163	−1.55475	−0.777374	−	0.629038i	$-0.783449\pi$
$487$	−757.163	−1.55475	−0.777374	+	0.629038i	$0.783449\pi$
$488$	− 44.5666i	− 0.0913249i
$489$	0	0
$490$	718.475	1.46628
$491$	567.663i	1.15614i	0.815988	+	0.578069i	$0.196193\pi$
$491$	567.663i	1.15614i	−0.815988	+	0.578069i	$0.803807\pi$
$492$	0	0
$493$	21.8628	0.0443464
$494$	− 17.1719i	− 0.0347610i
$495$	0	0
$496$	−74.6866	−0.150578
$497$	− 929.088i	− 1.86939i
$498$	0	0
$499$	464.176	0.930212	0.465106	−	0.885255i	$-0.346016\pi$
$499$	464.176	0.930212	0.465106	+	0.885255i	$0.346016\pi$
$500$	− 272.137i	− 0.544274i
$501$	0	0
$502$	−225.239	−0.448683
$503$	7.86969i	0.0156455i	0.999969	+	0.00782275i	$0.00249009\pi$
$503$	7.86969i	0.0156455i	−0.999969	+	0.00782275i	$0.997510\pi$
$504$	0	0
$505$	−244.014	−0.483196
$506$	126.799i	0.250591i
$507$	0	0
$508$	−343.329	−0.675845
$509$	− 61.2790i	− 0.120391i	−0.998187	−	0.0601955i	$-0.980828\pi$
$509$	− 61.2790i	− 0.120391i	0.998187	−	0.0601955i	$-0.0191724\pi$
$510$	0	0
$511$	−902.895	−1.76692
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	85.6553	0.166645
$515$	− 389.137i	− 0.755605i
$516$	0	0
$517$	635.594	1.22939
$518$	315.320i	0.608726i
$519$	0	0
$520$	212.581	0.408810
$521$	967.977i	1.85792i	0.370179	+	0.928961i	$0.379296\pi$
$521$	967.977i	1.85792i	−0.370179	+	0.928961i	$0.620704\pi$
$522$	0	0
$523$	−365.117	−0.698121	−0.349061	−	0.937100i	$-0.613499\pi$
$523$	−365.117	−0.698121	−0.349061	+	0.937100i	$0.613499\pi$
$524$	350.144i	0.668213i
$525$	0	0
$526$	477.769	0.908306
$527$	299.234i	0.567806i
$528$	0	0
$529$	485.405	0.917590
$530$	81.9299i	0.154585i
$531$	0	0
$532$	−17.2787	−0.0324787
$533$	− 795.842i	− 1.49314i
$534$	0	0
$535$	434.492	0.812135
$536$	111.761i	0.208508i
$537$	0	0
$538$	−55.2364	−0.102670
$539$	− 1704.19i	− 3.16176i
$540$	0	0
$541$	1.90773	0.00352630	0.00176315	−	0.999998i	$-0.499439\pi$
$541$	1.90773	0.00352630	0.00176315	+	0.999998i	$0.499439\pi$
$542$	− 349.318i	− 0.644499i
$543$	0	0
$544$	−90.6572	−0.166649
$545$	344.582i	0.632260i
$546$	0	0
$547$	−0.721037	−0.00131817	−0.000659083	−	1.00000i	$-0.500210\pi$
$547$	−0.721037	−0.00131817	−0.000659083		1.00000i	$0.500210\pi$
$548$	− 98.1867i	− 0.179173i
$549$	0	0
$550$	−165.390	−0.300708
$551$	− 0.892203i	− 0.00161924i
$552$	0	0
$553$	666.409	1.20508
$554$	269.710i	0.486842i
$555$	0	0
$556$	279.771	0.503186
$557$	635.859i	1.14158i	0.821096	+	0.570789i	$0.193363\pi$
$557$	635.859i	1.14158i	−0.821096	+	0.570789i	$0.806637\pi$
$558$	0	0
$559$	−1.21743	−0.00217786
$560$	− 213.902i	− 0.381969i
$561$	0	0
$562$	−191.771	−0.341229
$563$	− 96.3409i	− 0.171121i	−0.996333	−	0.0855603i	$-0.972732\pi$
$563$	− 96.3409i	− 0.171121i	0.996333	−	0.0855603i	$-0.0272680\pi$
$564$	0	0
$565$	−680.803	−1.20496
$566$	− 656.348i	− 1.15962i
$567$	0	0
$568$	−198.933	−0.350234
$569$	− 1041.74i	− 1.83083i	−0.402507	−	0.915417i	$-0.631861\pi$
$569$	− 1041.74i	− 1.83083i	0.402507	−	0.915417i	$-0.368139\pi$
$570$	0	0
$571$	−260.344	−0.455944	−0.227972	−	0.973668i	$-0.573210\pi$
$571$	−260.344	−0.455944	−0.227972	+	0.973668i	$0.573210\pi$
$572$	− 504.232i	− 0.881525i
$573$	0	0
$574$	−800.789	−1.39510
$575$	− 56.8630i	− 0.0988922i
$576$	0	0
$577$	384.941	0.667142	0.333571	−	0.942725i	$-0.391746\pi$
$577$	384.941	0.667142	0.333571	+	0.942725i	$0.391746\pi$
$578$	− 45.4877i	− 0.0786984i
$579$	0	0
$580$	11.0451	0.0190432
$581$	− 1940.63i	− 3.34015i
$582$	0	0
$583$	194.334	0.333334
$584$	193.325i	0.331036i
$585$	0	0
$586$	−191.379	−0.326586
$587$	1038.58i	1.76930i	0.466259	+	0.884648i	$0.345601\pi$
$587$	1038.58i	1.76930i	−0.466259	+	0.884648i	$0.654399\pi$
$588$	0	0
$589$	12.2115	0.0207326
$590$	− 27.8481i	− 0.0472002i
$591$	0	0
$592$	67.5152	0.114046
$593$	464.135i	0.782690i	0.920244	+	0.391345i	$0.127990\pi$
$593$	464.135i	0.782690i	−0.920244	+	0.391345i	$0.872010\pi$
$594$	0	0
$595$	−857.005	−1.44034
$596$	480.529i	0.806257i
$597$	0	0
$598$	173.361	0.289902
$599$	816.768i	1.36355i	0.731561	+	0.681776i	$0.238792\pi$
$599$	816.768i	1.36355i	−0.731561	+	0.681776i	$0.761208\pi$
$600$	0	0
$601$	241.299	0.401495	0.200748	−	0.979643i	$-0.435663\pi$
$601$	241.299	0.401495	0.200748	+	0.979643i	$0.435663\pi$
$602$	1.22499i	0.00203487i
$603$	0	0
$604$	−40.7541	−0.0674737
$605$	− 256.660i	− 0.424232i
$606$	0	0
$607$	−524.042	−0.863331	−0.431665	−	0.902034i	$-0.642074\pi$
$607$	−524.042	−0.863331	−0.431665	+	0.902034i	$0.642074\pi$
$608$	3.69965i	0.00608495i
$609$	0	0
$610$	−90.2070	−0.147880
$611$	− 868.993i	− 1.42225i
$612$	0	0
$613$	−328.647	−0.536129	−0.268064	−	0.963401i	$-0.586384\pi$
$613$	−328.647	−0.536129	−0.268064	+	0.963401i	$0.586384\pi$
$614$	− 493.017i	− 0.802960i
$615$	0	0
$616$	−507.366	−0.823647
$617$	837.400i	1.35721i	0.734502	+	0.678606i	$0.237416\pi$
$617$	837.400i	1.35721i	−0.734502	+	0.678606i	$0.762584\pi$
$618$	0	0
$619$	−205.957	−0.332726	−0.166363	−	0.986065i	$-0.553202\pi$
$619$	−205.957	−0.332726	−0.166363	+	0.986065i	$0.553202\pi$
$620$	151.173i	0.243827i
$621$	0	0
$622$	309.752	0.497994
$623$	263.817i	0.423462i
$624$	0	0
$625$	−335.527	−0.536844
$626$	− 384.817i	− 0.614724i
$627$	0	0
$628$	449.524	0.715803
$629$	− 270.501i	− 0.430049i
$630$	0	0
$631$	379.363	0.601208	0.300604	−	0.953749i	$-0.402812\pi$
$631$	379.363	0.601208	0.300604	+	0.953749i	$0.402812\pi$
$632$	− 142.689i	− 0.225774i
$633$	0	0
$634$	660.440	1.04170
$635$	694.931i	1.09438i
$636$	0	0
$637$	−2329.99	−3.65776
$638$	− 26.1984i	− 0.0410633i
$639$	0	0
$640$	−45.8001	−0.0715626
$641$	− 883.358i	− 1.37809i	−0.724717	−	0.689047i	$-0.758029\pi$
$641$	− 883.358i	− 1.37809i	0.724717	−	0.689047i	$-0.241971\pi$
$642$	0	0
$643$	−967.318	−1.50438	−0.752192	−	0.658944i	$-0.771003\pi$
$643$	−967.318	−1.50438	−0.752192	+	0.658944i	$0.771003\pi$
$644$	− 174.439i	− 0.270868i
$645$	0	0
$646$	14.8227	0.0229454
$647$	− 199.904i	− 0.308971i	−0.987995	−	0.154485i	$-0.950628\pi$
$647$	− 199.904i	− 0.308971i	0.987995	−	0.154485i	$-0.0493719\pi$
$648$	0	0
$649$	−66.0543	−0.101779
$650$	226.123i	0.347882i
$651$	0	0
$652$	310.882	0.476813
$653$	− 167.043i	− 0.255809i	−0.991786	−	0.127904i	$-0.959175\pi$
$653$	− 167.043i	− 0.255809i	0.991786	−	0.127904i	$-0.0408251\pi$
$654$	0	0
$655$	708.724	1.08202
$656$	171.462i	0.261375i
$657$	0	0
$658$	−874.395	−1.32887
$659$	128.860i	0.195538i	0.995209	+	0.0977691i	$0.0311707\pi$
$659$	128.860i	0.195538i	−0.995209	+	0.0977691i	$0.968829\pi$
$660$	0	0
$661$	−290.981	−0.440214	−0.220107	−	0.975476i	$-0.570641\pi$
$661$	−290.981	−0.440214	−0.220107	+	0.975476i	$0.570641\pi$
$662$	− 146.915i	− 0.221926i
$663$	0	0
$664$	−415.521	−0.625784
$665$	34.9737i	0.0525920i
$666$	0	0
$667$	9.00734	0.0135043
$668$	642.137i	0.961282i
$669$	0	0
$670$	226.214	0.337633
$671$	213.967i	0.318877i
$672$	0	0
$673$	−449.428	−0.667799	−0.333899	−	0.942609i	$-0.608365\pi$
$673$	−449.428	−0.667799	−0.333899	+	0.942609i	$0.608365\pi$
$674$	796.461i	1.18169i
$675$	0	0
$676$	−351.393	−0.519813
$677$	1138.56i	1.68178i	0.541208	+	0.840889i	$0.317967\pi$
$677$	1138.56i	1.68178i	−0.541208	+	0.840889i	$0.682033\pi$
$678$	0	0
$679$	−1396.65	−2.05692
$680$	183.499i	0.269851i
$681$	0	0
$682$	358.575	0.525770
$683$	108.489i	0.158842i	0.996841	+	0.0794209i	$0.0253071\pi$
$683$	108.489i	0.158842i	−0.996841	+	0.0794209i	$0.974693\pi$
$684$	0	0
$685$	−198.739	−0.290130
$686$	1429.08i	2.08321i
$687$	0	0
$688$	0.262291	0.000381237 0
$689$	− 265.696i	− 0.385625i
$690$	0	0
$691$	1128.37	1.63296	0.816478	−	0.577377i	$-0.195924\pi$
$691$	1128.37	1.63296	0.816478	+	0.577377i	$0.195924\pi$
$692$	− 284.970i	− 0.411806i
$693$	0	0
$694$	−384.403	−0.553895
$695$	− 566.284i	− 0.814798i
$696$	0	0
$697$	686.967	0.985606
$698$	− 113.441i	− 0.162523i
$699$	0	0
$700$	227.529	0.325041
$701$	− 778.410i	− 1.11043i	−0.831707	−	0.555214i	$-0.812636\pi$
$701$	− 778.410i	− 1.11043i	0.831707	−	0.555214i	$-0.187364\pi$
$702$	0	0
$703$	−11.0389	−0.0157026
$704$	108.636i	0.154312i
$705$	0	0
$706$	−168.819	−0.239120
$707$	− 796.248i	− 1.12624i
$708$	0	0
$709$	−343.565	−0.484576	−0.242288	−	0.970204i	$-0.577898\pi$
$709$	−343.565	−0.484576	−0.242288	+	0.970204i	$0.577898\pi$
$710$	402.660i	0.567126i
$711$	0	0
$712$	56.4876	0.0793365
$713$	123.283i	0.172907i
$714$	0	0
$715$	−1020.61	−1.42743
$716$	429.781i	0.600253i
$717$	0	0
$718$	−367.947	−0.512461
$719$	610.207i	0.848689i	0.905501	+	0.424344i	$0.139495\pi$
$719$	610.207i	0.848689i	−0.905501	+	0.424344i	$0.860505\pi$
$720$	0	0
$721$	1269.80	1.76117
$722$	509.926i	0.706269i
$723$	0	0
$724$	−603.490	−0.833550
$725$	11.7487i	0.0162051i
$726$	0	0
$727$	208.437	0.286708	0.143354	−	0.989671i	$-0.454211\pi$
$727$	208.437	0.286708	0.143354	+	0.989671i	$0.454211\pi$
$728$	693.679i	0.952855i
$729$	0	0
$730$	391.308	0.536038
$731$	− 1.05088i	− 0.00143759i
$732$	0	0
$733$	−249.421	−0.340274	−0.170137	−	0.985420i	$-0.554421\pi$
$733$	−249.421	−0.340274	−0.170137	+	0.985420i	$0.554421\pi$
$734$	554.170i	0.755000i
$735$	0	0
$736$	−37.3503	−0.0507476
$737$	− 536.569i	− 0.728044i
$738$	0	0
$739$	175.446	0.237411	0.118705	−	0.992930i	$-0.462126\pi$
$739$	175.446	0.237411	0.118705	+	0.992930i	$0.462126\pi$
$740$	− 136.657i	− 0.184672i
$741$	0	0
$742$	−267.347	−0.360307
$743$	− 668.397i	− 0.899592i	−0.893131	−	0.449796i	$-0.851497\pi$
$743$	− 668.397i	− 0.899592i	0.893131	−	0.449796i	$-0.148503\pi$
$744$	0	0
$745$	972.637	1.30555
$746$	157.051i	0.210524i
$747$	0	0
$748$	435.251	0.581886
$749$	1417.80i	1.89293i
$750$	0	0
$751$	422.749	0.562915	0.281457	−	0.959574i	$-0.409182\pi$
$751$	422.749	0.562915	0.281457	+	0.959574i	$0.409182\pi$
$752$	187.222i	0.248966i
$753$	0	0
$754$	−35.8188	−0.0475051
$755$	82.4903i	0.109259i
$756$	0	0
$757$	938.792	1.24015	0.620074	−	0.784543i	$-0.287103\pi$
$757$	938.792	1.24015	0.620074	+	0.784543i	$0.287103\pi$
$758$	− 561.100i	− 0.740237i
$759$	0	0
$760$	7.48844	0.00985322
$761$	− 719.677i	− 0.945699i	−0.881143	−	0.472849i	$-0.843225\pi$
$761$	− 719.677i	− 0.945699i	0.881143	−	0.472849i	$-0.156775\pi$
$762$	0	0
$763$	−1124.41	−1.47367
$764$	595.787i	0.779826i
$765$	0	0
$766$	319.717	0.417386
$767$	90.3104i	0.117745i
$768$	0	0
$769$	−1264.85	−1.64480	−0.822402	−	0.568907i	$-0.807367\pi$
$769$	−1264.85	−1.64480	−0.822402	+	0.568907i	$0.807367\pi$
$770$	1026.96i	1.33371i
$771$	0	0
$772$	−251.742	−0.326091
$773$	947.031i	1.22514i	0.790418	+	0.612568i	$0.209864\pi$
$773$	947.031i	1.22514i	−0.790418	+	0.612568i	$0.790136\pi$
$774$	0	0
$775$	−160.803	−0.207488
$776$	299.046i	0.385368i
$777$	0	0
$778$	915.973	1.17734
$779$	− 28.0346i	− 0.0359879i
$780$	0	0
$781$	955.090	1.22291
$782$	149.645i	0.191361i
$783$	0	0
$784$	501.991	0.640294
$785$	− 909.880i	− 1.15908i
$786$	0	0
$787$	785.830	0.998513	0.499256	−	0.866454i	$-0.333607\pi$
$787$	785.830	0.998513	0.499256	+	0.866454i	$0.333607\pi$
$788$	86.8051i	0.110159i
$789$	0	0
$790$	−288.817	−0.365591
$791$	− 2221.55i	− 2.80853i
$792$	0	0
$793$	292.538	0.368901
$794$	509.343i	0.641490i
$795$	0	0
$796$	151.337	0.190122
$797$	706.554i	0.886517i	0.896394	+	0.443259i	$0.146178\pi$
$797$	706.554i	0.886517i	−0.896394	+	0.443259i	$0.853822\pi$
$798$	0	0
$799$	750.110	0.938811
$800$	− 48.7176i	− 0.0608970i
$801$	0	0
$802$	−348.563	−0.434617
$803$	− 928.164i	− 1.15587i
$804$	0	0
$805$	−353.081	−0.438610
$806$	− 490.249i	− 0.608249i
$807$	0	0
$808$	−170.490	−0.211002
$809$	604.777i	0.747562i	0.927517	+	0.373781i	$0.121939\pi$
$809$	604.777i	0.747562i	−0.927517	+	0.373781i	$0.878061\pi$
$810$	0	0
$811$	679.911	0.838362	0.419181	−	0.907903i	$-0.362317\pi$
$811$	679.911	0.838362	0.419181	+	0.907903i	$0.362317\pi$
$812$	36.0415i	0.0443861i
$813$	0	0
$814$	−324.144	−0.398212
$815$	− 629.255i	− 0.772093i
$816$	0	0
$817$	−0.0428854	−5.24913e−5 0
$818$	− 62.2267i	− 0.0760718i
$819$	0	0
$820$	347.056	0.423239
$821$	− 1256.17i	− 1.53004i	−0.644005	−	0.765021i	$-0.722728\pi$
$821$	− 1256.17i	− 1.53004i	0.644005	−	0.765021i	$-0.277272\pi$
$822$	0	0
$823$	643.659	0.782089	0.391044	−	0.920372i	$-0.372114\pi$
$823$	643.659	0.782089	0.391044	+	0.920372i	$0.372114\pi$
$824$	− 271.886i	− 0.329958i
$825$	0	0
$826$	90.8718	0.110014
$827$	− 1057.49i	− 1.27871i	−0.768914	−	0.639353i	$-0.779202\pi$
$827$	− 1057.49i	− 1.27871i	0.768914	−	0.639353i	$-0.220798\pi$
$828$	0	0
$829$	−99.0671	−0.119502	−0.0597510	−	0.998213i	$-0.519031\pi$
$829$	−99.0671	−0.119502	−0.0597510	+	0.998213i	$0.519031\pi$
$830$	841.054i	1.01332i
$831$	0	0
$832$	148.528	0.178519
$833$	− 2011.24i	− 2.41445i
$834$	0	0
$835$	1299.75	1.55658
$836$	− 17.7622i	− 0.0212467i
$837$	0	0
$838$	424.171	0.506171
$839$	− 288.672i	− 0.344067i	−0.985091	−	0.172034i	$-0.944966\pi$
$839$	− 288.672i	− 0.344067i	0.985091	−	0.172034i	$-0.0550338\pi$
$840$	0	0
$841$	839.139	0.997787
$842$	− 534.945i	− 0.635327i
$843$	0	0
$844$	−430.854	−0.510490
$845$	711.254i	0.841720i
$846$	0	0
$847$	837.515	0.988802
$848$	57.2435i	0.0675041i
$849$	0	0
$850$	−195.188	−0.229633
$851$	− 111.445i	− 0.130958i
$852$	0	0
$853$	721.783	0.846170	0.423085	−	0.906090i	$-0.360947\pi$
$853$	721.783	0.846170	0.423085	+	0.906090i	$0.360947\pi$
$854$	− 294.357i	− 0.344680i
$855$	0	0
$856$	303.575	0.354644
$857$	− 229.144i	− 0.267379i	−0.991023	−	0.133690i	$-0.957318\pi$
$857$	− 229.144i	− 0.267379i	0.991023	−	0.133690i	$-0.0426825\pi$
$858$	0	0
$859$	−635.243	−0.739514	−0.369757	−	0.929128i	$-0.620559\pi$
$859$	−635.243	−0.739514	−0.369757	+	0.929128i	$0.620559\pi$
$860$	− 0.530903i	0 0.000617329i
$861$	0	0
$862$	1046.22	1.21371
$863$	− 644.463i	− 0.746771i	−0.927676	−	0.373385i	$-0.878197\pi$
$863$	− 644.463i	− 0.746771i	0.927676	−	0.373385i	$-0.121803\pi$
$864$	0	0
$865$	−576.807	−0.666828
$866$	1211.70i	1.39920i
$867$	0	0
$868$	−493.296	−0.568313
$869$	685.059i	0.788331i
$870$	0	0
$871$	−733.604	−0.842255
$872$	240.755i	0.276096i
$873$	0	0
$874$	6.10688	0.00698727
$875$	− 1797.43i	− 2.05421i
$876$	0	0
$877$	1665.19	1.89873	0.949365	−	0.314176i	$-0.101728\pi$
$877$	1665.19	1.89873	0.949365	+	0.314176i	$0.101728\pi$
$878$	− 1135.05i	− 1.29276i
$879$	0	0
$880$	219.889	0.249874
$881$	− 398.999i	− 0.452893i	−0.974024	−	0.226447i	$-0.927289\pi$
$881$	− 398.999i	− 0.452893i	0.974024	−	0.226447i	$-0.0727109\pi$
$882$	0	0
$883$	13.9020	0.0157440	0.00787202	−	0.999969i	$-0.497494\pi$
$883$	13.9020	0.0157440	0.00787202	+	0.999969i	$0.497494\pi$
$884$	− 595.081i	− 0.673168i
$885$	0	0
$886$	432.327	0.487954
$887$	− 841.224i	− 0.948392i	−0.880419	−	0.474196i	$-0.842739\pi$
$887$	− 841.224i	− 0.948392i	0.880419	−	0.474196i	$-0.157261\pi$
$888$	0	0
$889$	−2267.65	−2.55078
$890$	− 114.336i	− 0.128468i
$891$	0	0
$892$	166.416	0.186565
$893$	− 30.6114i	− 0.0342793i
$894$	0	0
$895$	869.919	0.971976
$896$	− 149.451i	− 0.166798i
$897$	0	0
$898$	583.762	0.650069
$899$	− 25.4719i	− 0.0283336i
$900$	0	0
$901$	229.347	0.254548
$902$	− 823.200i	− 0.912639i
$903$	0	0
$904$	−475.670	−0.526183
$905$	1221.52i	1.34975i
$906$	0	0
$907$	−1074.23	−1.18437	−0.592187	−	0.805800i	$-0.701735\pi$
$907$	−1074.23	−1.18437	−0.592187	+	0.805800i	$0.701735\pi$
$908$	− 125.044i	− 0.137714i
$909$	0	0
$910$	1404.07	1.54294
$911$	− 680.759i	− 0.747265i	−0.927577	−	0.373633i	$-0.878112\pi$
$911$	− 680.759i	− 0.747265i	0.927577	−	0.373633i	$-0.121888\pi$
$912$	0	0
$913$	1994.94	2.18504
$914$	− 700.434i	− 0.766339i
$915$	0	0
$916$	−290.404	−0.317035
$917$	2312.66i	2.52198i
$918$	0	0
$919$	643.866	0.700616	0.350308	−	0.936635i	$-0.386077\pi$
$919$	643.866	0.700616	0.350308	+	0.936635i	$0.386077\pi$
$920$	75.6005i	0.0821745i
$921$	0	0
$922$	1153.19	1.25074
$923$	− 1305.81i	− 1.41475i
$924$	0	0
$925$	145.363	0.157149
$926$	− 588.693i	− 0.635738i
$927$	0	0
$928$	7.71708	0.00831582
$929$	822.844i	0.885731i	0.896588	+	0.442866i	$0.146038\pi$
$929$	822.844i	0.885731i	−0.896588	+	0.442866i	$0.853962\pi$
$930$	0	0
$931$	−82.0769	−0.0881600
$932$	− 270.607i	− 0.290351i
$933$	0	0
$934$	201.784	0.216042
$935$	− 880.989i	− 0.942234i
$936$	0	0
$937$	−320.345	−0.341884	−0.170942	−	0.985281i	$-0.554681\pi$
$937$	−320.345	−0.341884	−0.170942	+	0.985281i	$0.554681\pi$
$938$	738.165i	0.786956i
$939$	0	0
$940$	378.956	0.403145
$941$	343.761i	0.365314i	0.983177	+	0.182657i	$0.0584698\pi$
$941$	343.761i	0.365314i	−0.983177	+	0.182657i	$0.941530\pi$
$942$	0	0
$943$	283.027	0.300134
$944$	− 19.4572i	− 0.0206114i
$945$	0	0
$946$	−1.25928	−0.00133116
$947$	1025.04i	1.08240i	0.840893	+	0.541202i	$0.182031\pi$
$947$	1025.04i	1.08240i	−0.840893	+	0.541202i	$0.817969\pi$
$948$	0	0
$949$	−1269.00	−1.33720
$950$	7.96548i	0.00838471i
$951$	0	0
$952$	−598.780	−0.628970
$953$	1837.44i	1.92806i	0.265794	+	0.964030i	$0.414366\pi$
$953$	1837.44i	1.92806i	−0.265794	+	0.964030i	$0.585634\pi$
$954$	0	0
$955$	1205.93	1.26275
$956$	293.106i	0.306597i
$957$	0	0
$958$	146.402	0.152820
$959$	− 648.511i	− 0.676237i
$960$	0	0
$961$	−612.369	−0.637221
$962$	443.175i	0.460681i
$963$	0	0
$964$	−634.279	−0.657966
$965$	509.551i	0.528032i
$966$	0	0
$967$	530.916	0.549034	0.274517	−	0.961582i	$-0.411482\pi$
$967$	530.916	0.549034	0.274517	+	0.961582i	$0.411482\pi$
$968$	− 179.326i	− 0.185254i
$969$	0	0
$970$	605.297	0.624018
$971$	53.7560i	0.0553615i	0.999617	+	0.0276807i	$0.00881218\pi$
$971$	53.7560i	0.0553615i	−0.999617	+	0.0276807i	$0.991188\pi$
$972$	0	0
$973$	1847.86	1.89913
$974$	1070.79i	1.09937i
$975$	0	0
$976$	−63.0266	−0.0645765
$977$	− 808.421i	− 0.827452i	−0.910401	−	0.413726i	$-0.864227\pi$
$977$	− 808.421i	− 0.827452i	0.910401	−	0.413726i	$-0.135773\pi$
$978$	0	0
$979$	−271.200	−0.277017
$980$	− 1016.08i	− 1.03681i
$981$	0	0
$982$	802.797	0.817512
$983$	− 859.543i	− 0.874408i	−0.899362	−	0.437204i	$-0.855969\pi$
$983$	− 859.543i	− 0.874408i	0.899362	−	0.437204i	$-0.144031\pi$
$984$	0	0
$985$	175.702	0.178378
$986$	− 30.9186i	− 0.0313576i
$987$	0	0
$988$	−24.2848	−0.0245797
$989$	− 0.432955i	0 0.000437770i
$990$	0	0
$991$	−606.599	−0.612108	−0.306054	−	0.952014i	$-0.599009\pi$
$991$	−606.599	−0.612108	−0.306054	+	0.952014i	$0.599009\pi$
$992$	105.623i	0.106475i
$993$	0	0
$994$	−1313.93	−1.32186
$995$	− 306.321i	− 0.307861i
$996$	0	0
$997$	1244.36	1.24811	0.624054	−	0.781381i	$-0.285485\pi$
$997$	1244.36	1.24811	0.624054	+	0.781381i	$0.285485\pi$
$998$	− 656.444i	− 0.657759i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1458.3.b.c.1457.13		36
3.2	odd	2	inner	1458.3.b.c.1457.24		36
27.2	odd	18		162.3.f.a.17.5		36
27.13	even	9		162.3.f.a.143.5		36
27.14	odd	18		54.3.f.a.47.2	yes	36
27.25	even	9		54.3.f.a.23.2	✓	36
108.79	odd	18		432.3.bc.c.401.4		36
108.95	even	18		432.3.bc.c.209.4		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.3.f.a.23.2	✓	36	27.25	even	9
54.3.f.a.47.2	yes	36	27.14	odd	18
162.3.f.a.17.5		36	27.2	odd	18
162.3.f.a.143.5		36	27.13	even	9
432.3.bc.c.209.4		36	108.95	even	18
432.3.bc.c.401.4		36	108.79	odd	18
1458.3.b.c.1457.13		36	1.1	even	1	trivial
1458.3.b.c.1457.24		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)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} +4.04819i q^{5} -13.2098 q^{7} +2.82843i q^{8} +5.72501 q^{10} -13.5794i q^{11} -18.5660 q^{13} +18.6814i q^{14} +4.00000 q^{16} -16.0261i q^{17} -0.654012 q^{19} -8.09638i q^{20} -19.2042 q^{22} -6.60266i q^{23} +8.61214 q^{25} +26.2563i q^{26} +26.4195 q^{28} +1.36420i q^{29} -18.6717 q^{31} -5.65685i q^{32} -22.6643 q^{34} -53.4756i q^{35} +16.8788 q^{37} +0.924912i q^{38} -11.4500 q^{40} +42.8656i q^{41} +0.0655728 q^{43} +27.1589i q^{44} -9.33756 q^{46} +46.8056i q^{47} +125.498 q^{49} -12.1794i q^{50} +37.1320 q^{52} +14.3109i q^{53} +54.9722 q^{55} -37.3628i q^{56} +1.92927 q^{58} -4.86429i q^{59} -15.7567 q^{61} +26.4057i q^{62} -8.00000 q^{64} -75.1588i q^{65} +39.5133 q^{67} +32.0522i q^{68} -75.6260 q^{70} +70.3335i q^{71} +68.3506 q^{73} -23.8702i q^{74} +1.30802 q^{76} +179.381i q^{77} -50.4483 q^{79} +16.1928i q^{80} +60.6211 q^{82} +146.909i q^{83} +64.8767 q^{85} -0.0927340i q^{86} +38.4085 q^{88} -19.9714i q^{89} +245.252 q^{91} +13.2053i q^{92} +66.1931 q^{94} -2.64757i q^{95} +105.729 q^{97} -177.480i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 72 q^{4} + 144 q^{16} - 180 q^{25} + 252 q^{49} - 36 q^{61} - 288 q^{64} + 180 q^{67} - 252 q^{73} + 396 q^{91}+O(q^{100}) \) 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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