LMFDB - Embedded newform 1728.4.d.i.865.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(865,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1728.865");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.d (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:	$101.955300490$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 170x^{12} + 7609x^{8} + 59868x^{4} + 104976 $ x^16 + 170x^12 + 7609x^8 + 59868*x^4 + 104976
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			865.14
Root			$1.97781 + 1.97781i$ of defining polynomial
Character	$\chi$	$=$	1728.865
Dual form			1728.4.d.i.865.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.8148i q^{5} +24.7677 q^{7} +O(q^{10})$	$q+10.8148i q^{5} +24.7677 q^{7} +61.0491i q^{11} -57.9369i q^{13} +25.3300 q^{17} -97.0940i q^{19} +99.6991 q^{23} +8.04076 q^{25} -34.3255i q^{29} +68.0761 q^{31} +267.856i q^{35} -271.478i q^{37} +295.771 q^{41} -433.480i q^{43} +436.016 q^{47} +270.437 q^{49} -476.675i q^{53} -660.232 q^{55} -601.244i q^{59} +374.463i q^{61} +626.574 q^{65} +869.510i q^{67} -721.597 q^{71} +1129.80 q^{73} +1512.04i q^{77} -425.209 q^{79} -148.588i q^{83} +273.939i q^{85} -428.154 q^{89} -1434.96i q^{91} +1050.05 q^{95} -683.483 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 240 q^{17} - 304 q^{25} + 1008 q^{41} + 1616 q^{49} + 2736 q^{65} + 128 q^{73} + 5856 q^{89} - 2576 q^{97}+O(q^{100}) $ 16 * q - 240 * q^17 - 304 * q^25 + 1008 * q^41 + 1616 * q^49 + 2736 * q^65 + 128 * q^73 + 5856 * q^89 - 2576 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	10.8148i	0.967302i	0.875261	+	0.483651i	$0.160690\pi$
$5$	10.8148i	0.967302i	−0.875261	+	0.483651i	$0.839310\pi$
$6$	0	0
$7$	24.7677	1.33733	0.668664	−	0.743565i	$-0.266867\pi$
$7$	24.7677	1.33733	0.668664	+	0.743565i	$0.266867\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	61.0491i	1.67336i	0.547690	+	0.836682i	$0.315507\pi$
$11$	61.0491i	1.67336i	−0.547690	+	0.836682i	$0.684493\pi$
$12$	0	0
$13$	− 57.9369i	− 1.23606i	−0.786154	−	0.618031i	$-0.787931\pi$
$13$	− 57.9369i	− 1.23606i	0.786154	−	0.618031i	$-0.212069\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	25.3300	0.361379	0.180689	−	0.983540i	$-0.442167\pi$
$17$	25.3300	0.361379	0.180689	+	0.983540i	$0.442167\pi$
$18$	0	0
$19$	− 97.0940i	− 1.17236i	−0.810180	−	0.586181i	$-0.800631\pi$
$19$	− 97.0940i	− 1.17236i	0.810180	−	0.586181i	$-0.199369\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	99.6991	0.903856	0.451928	−	0.892054i	$-0.350736\pi$
$23$	99.6991	0.903856	0.451928	+	0.892054i	$0.350736\pi$
$24$	0	0
$25$	8.04076	0.0643261
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 34.3255i	− 0.219796i	−0.993943	−	0.109898i	$-0.964948\pi$
$29$	− 34.3255i	− 0.219796i	0.993943	−	0.109898i	$-0.0350525\pi$
$30$	0	0
$31$	68.0761	0.394414	0.197207	−	0.980362i	$-0.436813\pi$
$31$	68.0761	0.394414	0.197207	+	0.980362i	$0.436813\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	267.856i	1.29360i
$36$	0	0
$37$	− 271.478i	− 1.20624i	−0.797651	−	0.603119i	$-0.793925\pi$
$37$	− 271.478i	− 1.20624i	0.797651	−	0.603119i	$-0.206075\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	295.771	1.12663	0.563313	−	0.826244i	$-0.309527\pi$
$41$	295.771	1.12663	0.563313	+	0.826244i	$0.309527\pi$
$42$	0	0
$43$	− 433.480i	− 1.53733i	−0.639653	−	0.768664i	$-0.720922\pi$
$43$	− 433.480i	− 1.53733i	0.639653	−	0.768664i	$-0.279078\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	436.016	1.35318	0.676590	−	0.736360i	$-0.263457\pi$
$47$	436.016	1.35318	0.676590	+	0.736360i	$0.263457\pi$
$48$	0	0
$49$	270.437	0.788445
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 476.675i	− 1.23540i	−0.786413	−	0.617701i	$-0.788064\pi$
$53$	− 476.675i	− 1.23540i	0.786413	−	0.617701i	$-0.211936\pi$
$54$	0	0
$55$	−660.232	−1.61865
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 601.244i	− 1.32670i	−0.748309	−	0.663350i	$-0.769134\pi$
$59$	− 601.244i	− 1.32670i	0.748309	−	0.663350i	$-0.230866\pi$
$60$	0	0
$61$	374.463i	0.785985i	0.919542	+	0.392993i	$0.128560\pi$
$61$	374.463i	0.785985i	−0.919542	+	0.392993i	$0.871440\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	626.574	1.19565
$66$	0	0
$67$	869.510i	1.58549i	0.609556	+	0.792743i	$0.291348\pi$
$67$	869.510i	1.58549i	−0.609556	+	0.792743i	$0.708652\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−721.597	−1.20617	−0.603083	−	0.797678i	$-0.706061\pi$
$71$	−721.597	−1.20617	−0.603083	+	0.797678i	$0.706061\pi$
$72$	0	0
$73$	1129.80	1.81141	0.905705	−	0.423909i	$-0.139342\pi$
$73$	1129.80	1.81141	0.905705	+	0.423909i	$0.139342\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1512.04i	2.23783i
$78$	0	0
$79$	−425.209	−0.605567	−0.302783	−	0.953059i	$-0.597916\pi$
$79$	−425.209	−0.605567	−0.302783	+	0.953059i	$0.597916\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 148.588i	− 0.196502i	−0.995162	−	0.0982512i	$-0.968675\pi$
$83$	− 148.588i	− 0.196502i	0.995162	−	0.0982512i	$-0.0313249\pi$
$84$	0	0
$85$	273.939i	0.349562i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−428.154	−0.509935	−0.254967	−	0.966950i	$-0.582065\pi$
$89$	−428.154	−0.509935	−0.254967	+	0.966950i	$0.582065\pi$
$90$	0	0
$91$	− 1434.96i	− 1.65302i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1050.05	1.13403
$96$	0	0
$97$	−683.483	−0.715435	−0.357717	−	0.933830i	$-0.616445\pi$
$97$	−683.483	−0.715435	−0.357717	+	0.933830i	$0.616445\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	741.725i	0.730736i	0.930863	+	0.365368i	$0.119057\pi$
$101$	741.725i	0.730736i	−0.930863	+	0.365368i	$0.880943\pi$
$102$	0	0
$103$	838.961	0.802576	0.401288	−	0.915952i	$-0.368563\pi$
$103$	838.961	0.802576	0.401288	+	0.915952i	$0.368563\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 585.351i	− 0.528860i	−0.964405	−	0.264430i	$-0.914816\pi$
$107$	− 585.351i	− 0.528860i	0.964405	−	0.264430i	$-0.0851838\pi$
$108$	0	0
$109$	247.532i	0.217516i	0.994068	+	0.108758i	$0.0346874\pi$
$109$	247.532i	0.217516i	−0.994068	+	0.108758i	$0.965313\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2140.71	−1.78213	−0.891065	−	0.453875i	$-0.850041\pi$
$113$	−2140.71	−1.78213	−0.891065	+	0.453875i	$0.850041\pi$
$114$	0	0
$115$	1078.22i	0.874302i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	627.366	0.483282
$120$	0	0
$121$	−2395.99	−1.80014
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1438.81i	1.02953i
$126$	0	0
$127$	514.727	0.359643	0.179821	−	0.983699i	$-0.442448\pi$
$127$	514.727	0.359643	0.179821	+	0.983699i	$0.442448\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2772.36i	− 1.84902i	−0.381156	−	0.924511i	$-0.624474\pi$
$131$	− 2772.36i	− 1.84902i	0.381156	−	0.924511i	$-0.375526\pi$
$132$	0	0
$133$	− 2404.79i	− 1.56783i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1082.26	0.674920	0.337460	−	0.941340i	$-0.390432\pi$
$137$	1082.26	0.674920	0.337460	+	0.941340i	$0.390432\pi$
$138$	0	0
$139$	666.475i	0.406688i	0.979107	+	0.203344i	$0.0651810\pi$
$139$	666.475i	0.406688i	−0.979107	+	0.203344i	$0.934819\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3536.99	2.06838
$144$	0	0
$145$	371.223	0.212609
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2036.41i	1.11966i	0.828609	+	0.559828i	$0.189133\pi$
$149$	2036.41i	1.11966i	−0.828609	+	0.559828i	$0.810867\pi$
$150$	0	0
$151$	2809.54	1.51415	0.757077	−	0.653325i	$-0.226627\pi$
$151$	2809.54	1.51415	0.757077	+	0.653325i	$0.226627\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	736.228i	0.381518i
$156$	0	0
$157$	1937.49i	0.984896i	0.870342	+	0.492448i	$0.163898\pi$
$157$	1937.49i	0.984896i	−0.870342	+	0.492448i	$0.836102\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2469.31	1.20875
$162$	0	0
$163$	682.959i	0.328181i	0.986445	+	0.164091i	$0.0524689\pi$
$163$	682.959i	0.328181i	−0.986445	+	0.164091i	$0.947531\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2004.85	−0.928980	−0.464490	−	0.885578i	$-0.653762\pi$
$167$	−2004.85	−0.928980	−0.464490	+	0.885578i	$0.653762\pi$
$168$	0	0
$169$	−1159.68	−0.527848
$170$	0	0
$171$	0	0
$172$	0	0
$173$	411.426i	0.180810i	0.995905	+	0.0904050i	$0.0288162\pi$
$173$	411.426i	0.180810i	−0.995905	+	0.0904050i	$0.971184\pi$
$174$	0	0
$175$	199.151	0.0860250
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1622.26i	− 0.677391i	−0.940896	−	0.338696i	$-0.890014\pi$
$179$	− 1622.26i	− 0.677391i	0.940896	−	0.338696i	$-0.109986\pi$
$180$	0	0
$181$	− 746.020i	− 0.306360i	−0.988198	−	0.153180i	$-0.951049\pi$
$181$	− 746.020i	− 0.306360i	0.988198	−	0.153180i	$-0.0489515\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2935.98	1.16680
$186$	0	0
$187$	1546.38i	0.604718i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4553.58	1.72506	0.862528	−	0.506009i	$-0.168880\pi$
$191$	4553.58	1.72506	0.862528	+	0.506009i	$0.168880\pi$
$192$	0	0
$193$	−1608.97	−0.600083	−0.300042	−	0.953926i	$-0.597001\pi$
$193$	−1608.97	−0.600083	−0.300042	+	0.953926i	$0.597001\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4142.26i	1.49809i	0.662518	+	0.749046i	$0.269488\pi$
$197$	4142.26i	1.49809i	−0.662518	+	0.749046i	$0.730512\pi$
$198$	0	0
$199$	−3056.79	−1.08890	−0.544448	−	0.838795i	$-0.683261\pi$
$199$	−3056.79	−1.08890	−0.544448	+	0.838795i	$0.683261\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 850.163i	− 0.293940i
$204$	0	0
$205$	3198.69i	1.08979i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5927.50	1.96179
$210$	0	0
$211$	− 1426.29i	− 0.465355i	−0.972554	−	0.232678i	$-0.925251\pi$
$211$	− 1426.29i	− 0.465355i	0.972554	−	0.232678i	$-0.0747487\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4687.99	1.48706
$216$	0	0
$217$	1686.09	0.527461
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1467.54i	− 0.446686i
$222$	0	0
$223$	64.7099	0.0194318	0.00971590	−	0.999953i	$-0.496907\pi$
$223$	64.7099	0.0194318	0.00971590	+	0.999953i	$0.496907\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1678.16i	0.490676i	0.969438	+	0.245338i	$0.0788990\pi$
$227$	1678.16i	0.490676i	−0.969438	+	0.245338i	$0.921101\pi$
$228$	0	0
$229$	1483.99i	0.428230i	0.976808	+	0.214115i	$0.0686868\pi$
$229$	1483.99i	0.428230i	−0.976808	+	0.214115i	$0.931313\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6731.40	1.89265	0.946327	−	0.323210i	$-0.104762\pi$
$233$	6731.40	1.89265	0.946327	+	0.323210i	$0.104762\pi$
$234$	0	0
$235$	4715.41i	1.30893i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2789.46	0.754959	0.377479	−	0.926018i	$-0.376791\pi$
$239$	2789.46	0.754959	0.377479	+	0.926018i	$0.376791\pi$
$240$	0	0
$241$	3712.61	0.992324	0.496162	−	0.868230i	$-0.334742\pi$
$241$	3712.61	0.992324	0.496162	+	0.868230i	$0.334742\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2924.71i	0.762665i
$246$	0	0
$247$	−5625.33	−1.44911
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 1932.01i	− 0.485846i	−0.970046	−	0.242923i	$-0.921894\pi$
$251$	− 1932.01i	− 0.485846i	0.970046	−	0.242923i	$-0.0781062\pi$
$252$	0	0
$253$	6086.54i	1.51248i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4806.99	−1.16674	−0.583370	−	0.812207i	$-0.698266\pi$
$257$	−4806.99	−1.16674	−0.583370	+	0.812207i	$0.698266\pi$
$258$	0	0
$259$	− 6723.88i	− 1.61313i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5966.47	1.39889	0.699446	−	0.714686i	$-0.253430\pi$
$263$	5966.47	1.39889	0.699446	+	0.714686i	$0.253430\pi$
$264$	0	0
$265$	5155.13	1.19501
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5916.36i	1.34099i	0.741913	+	0.670497i	$0.233919\pi$
$269$	5916.36i	1.34099i	−0.741913	+	0.670497i	$0.766081\pi$
$270$	0	0
$271$	−990.081	−0.221930	−0.110965	−	0.993824i	$-0.535394\pi$
$271$	−990.081	−0.221930	−0.110965	+	0.993824i	$0.535394\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	490.881i	0.107641i
$276$	0	0
$277$	− 9190.01i	− 1.99341i	−0.0811261	−	0.996704i	$-0.525852\pi$
$277$	− 9190.01i	− 1.99341i	0.0811261	−	0.996704i	$-0.474148\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3335.49	−0.708110	−0.354055	−	0.935225i	$-0.615197\pi$
$281$	−3335.49	−0.708110	−0.354055	+	0.935225i	$0.615197\pi$
$282$	0	0
$283$	− 1226.71i	− 0.257668i	−0.991666	−	0.128834i	$-0.958877\pi$
$283$	− 1226.71i	− 0.257668i	0.991666	−	0.128834i	$-0.0411235\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7325.55	1.50667
$288$	0	0
$289$	−4271.39	−0.869406
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6104.03i	1.21707i	0.793527	+	0.608535i	$0.208242\pi$
$293$	6104.03i	1.21707i	−0.793527	+	0.608535i	$0.791758\pi$
$294$	0	0
$295$	6502.32	1.28332
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 5776.25i	− 1.11722i
$300$	0	0
$301$	− 10736.3i	− 2.05591i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4049.73	−0.760285
$306$	0	0
$307$	− 117.036i	− 0.0217577i	−0.999941	−	0.0108789i	$-0.996537\pi$
$307$	− 117.036i	− 0.0217577i	0.999941	−	0.0108789i	$-0.00346292\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6757.78	1.23215	0.616075	−	0.787688i	$-0.288722\pi$
$311$	6757.78	1.23215	0.616075	+	0.787688i	$0.288722\pi$
$312$	0	0
$313$	−3315.26	−0.598689	−0.299345	−	0.954145i	$-0.596768\pi$
$313$	−3315.26	−0.598689	−0.299345	+	0.954145i	$0.596768\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3363.65i	0.595966i	0.954571	+	0.297983i	$0.0963139\pi$
$317$	3363.65i	0.595966i	−0.954571	+	0.297983i	$0.903686\pi$
$318$	0	0
$319$	2095.54	0.367799
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 2459.40i	− 0.423667i
$324$	0	0
$325$	− 465.856i	− 0.0795110i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	10799.1	1.80964
$330$	0	0
$331$	− 6637.85i	− 1.10226i	−0.834418	−	0.551131i	$-0.814196\pi$
$331$	− 6637.85i	− 1.10226i	0.834418	−	0.551131i	$-0.185804\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9403.55	−1.53364
$336$	0	0
$337$	12118.2	1.95881	0.979407	−	0.201894i	$-0.0647097\pi$
$337$	12118.2	1.95881	0.979407	+	0.201894i	$0.0647097\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	4155.99i	0.659998i
$342$	0	0
$343$	−1797.22	−0.282918
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 5144.69i	− 0.795912i	−0.917404	−	0.397956i	$-0.869720\pi$
$347$	− 5144.69i	− 0.795912i	0.917404	−	0.397956i	$-0.130280\pi$
$348$	0	0
$349$	− 3398.88i	− 0.521313i	−0.965432	−	0.260656i	$-0.916061\pi$
$349$	− 3398.88i	− 0.521313i	0.965432	−	0.260656i	$-0.0839389\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1293.98	0.195104	0.0975521	−	0.995230i	$-0.468899\pi$
$353$	1293.98	0.195104	0.0975521	+	0.995230i	$0.468899\pi$
$354$	0	0
$355$	− 7803.90i	− 1.16673i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10123.9	−1.48836	−0.744179	−	0.667980i	$-0.767159\pi$
$359$	−10123.9	−1.48836	−0.744179	+	0.667980i	$0.767159\pi$
$360$	0	0
$361$	−2568.25	−0.374435
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12218.5i	1.75218i
$366$	0	0
$367$	−2329.96	−0.331397	−0.165699	−	0.986176i	$-0.552988\pi$
$367$	−2329.96	−0.331397	−0.165699	+	0.986176i	$0.552988\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 11806.1i	− 1.65214i
$372$	0	0
$373$	5489.47i	0.762021i	0.924571	+	0.381011i	$0.124424\pi$
$373$	5489.47i	0.762021i	−0.924571	+	0.381011i	$0.875576\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1988.71	−0.271682
$378$	0	0
$379$	6549.70i	0.887693i	0.896103	+	0.443847i	$0.146386\pi$
$379$	6549.70i	0.887693i	−0.896103	+	0.443847i	$0.853614\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3485.16	0.464969	0.232484	−	0.972600i	$-0.425315\pi$
$383$	3485.16	0.464969	0.232484	+	0.972600i	$0.425315\pi$
$384$	0	0
$385$	−16352.4	−2.16466
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5081.13i	0.662271i	0.943583	+	0.331136i	$0.107432\pi$
$389$	5081.13i	0.662271i	−0.943583	+	0.331136i	$0.892568\pi$
$390$	0	0
$391$	2525.38	0.326634
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 4598.54i	− 0.585766i
$396$	0	0
$397$	15181.0i	1.91918i	0.281402	+	0.959590i	$0.409201\pi$
$397$	15181.0i	1.91918i	−0.281402	+	0.959590i	$0.590799\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2280.54	−0.284002	−0.142001	−	0.989866i	$-0.545354\pi$
$401$	−2280.54	−0.284002	−0.142001	+	0.989866i	$0.545354\pi$
$402$	0	0
$403$	− 3944.12i	− 0.487520i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	16573.5	2.01847
$408$	0	0
$409$	4701.84	0.568438	0.284219	−	0.958759i	$-0.408266\pi$
$409$	4701.84	0.568438	0.284219	+	0.958759i	$0.408266\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 14891.4i	− 1.77423i
$414$	0	0
$415$	1606.95	0.190077
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 736.121i	− 0.0858278i	−0.999079	−	0.0429139i	$-0.986336\pi$
$419$	− 736.121i	− 0.0858278i	0.999079	−	0.0429139i	$-0.0136641\pi$
$420$	0	0
$421$	− 11067.7i	− 1.28125i	−0.767854	−	0.640625i	$-0.778675\pi$
$421$	− 11067.7i	− 1.28125i	0.767854	−	0.640625i	$-0.221325\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	203.673	0.0232461
$426$	0	0
$427$	9274.57i	1.05112i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6603.96	−0.738055	−0.369027	−	0.929419i	$-0.620309\pi$
$431$	−6603.96	−0.738055	−0.369027	+	0.929419i	$0.620309\pi$
$432$	0	0
$433$	13001.6	1.44299	0.721496	−	0.692419i	$-0.243455\pi$
$433$	13001.6	1.44299	0.721496	+	0.692419i	$0.243455\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 9680.18i	− 1.05965i
$438$	0	0
$439$	2996.10	0.325731	0.162866	−	0.986648i	$-0.447926\pi$
$439$	2996.10	0.325731	0.162866	+	0.986648i	$0.447926\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	83.1158i	0.00891411i	0.999990	+	0.00445706i	$0.00141873\pi$
$443$	83.1158i	0.00891411i	−0.999990	+	0.00445706i	$0.998581\pi$
$444$	0	0
$445$	− 4630.38i	− 0.493261i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4081.07	−0.428947	−0.214474	−	0.976730i	$-0.568804\pi$
$449$	−4081.07	−0.428947	−0.214474	+	0.976730i	$0.568804\pi$
$450$	0	0
$451$	18056.5i	1.88525i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	15518.8	1.59897
$456$	0	0
$457$	4429.17	0.453365	0.226682	−	0.973969i	$-0.427212\pi$
$457$	4429.17	0.453365	0.226682	+	0.973969i	$0.427212\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17727.5i	1.79101i	0.445056	+	0.895503i	$0.353184\pi$
$461$	17727.5i	1.79101i	−0.445056	+	0.895503i	$0.646816\pi$
$462$	0	0
$463$	−15118.0	−1.51748	−0.758740	−	0.651394i	$-0.774185\pi$
$463$	−15118.0	−1.51748	−0.758740	+	0.651394i	$0.774185\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9501.20i	0.941463i	0.882277	+	0.470731i	$0.156010\pi$
$467$	9501.20i	0.941463i	−0.882277	+	0.470731i	$0.843990\pi$
$468$	0	0
$469$	21535.7i	2.12031i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	26463.6	2.57251
$474$	0	0
$475$	− 780.710i	− 0.0754135i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15186.6	−1.44863	−0.724317	−	0.689467i	$-0.757845\pi$
$479$	−15186.6	−1.44863	−0.724317	+	0.689467i	$0.757845\pi$
$480$	0	0
$481$	−15728.6	−1.49098
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 7391.71i	− 0.692042i
$486$	0	0
$487$	−13111.1	−1.21996	−0.609981	−	0.792416i	$-0.708823\pi$
$487$	−13111.1	−1.21996	−0.609981	+	0.792416i	$0.708823\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	14207.9i	1.30589i	0.757405	+	0.652946i	$0.226467\pi$
$491$	14207.9i	1.30589i	−0.757405	+	0.652946i	$0.773533\pi$
$492$	0	0
$493$	− 869.467i	− 0.0794297i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−17872.3	−1.61304
$498$	0	0
$499$	14346.0i	1.28700i	0.765446	+	0.643500i	$0.222518\pi$
$499$	14346.0i	1.28700i	−0.765446	+	0.643500i	$0.777482\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8042.33	0.712902	0.356451	−	0.934314i	$-0.383987\pi$
$503$	8042.33	0.712902	0.356451	+	0.934314i	$0.383987\pi$
$504$	0	0
$505$	−8021.58	−0.706843
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 8465.24i	− 0.737161i	−0.929596	−	0.368581i	$-0.879844\pi$
$509$	− 8465.24i	− 0.737161i	0.929596	−	0.368581i	$-0.120156\pi$
$510$	0	0
$511$	27982.4	2.42245
$512$	0	0
$513$	0	0
$514$	0	0
$515$	9073.17i	0.776334i
$516$	0	0
$517$	26618.4i	2.26436i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1886.80	−0.158661	−0.0793305	−	0.996848i	$-0.525278\pi$
$521$	−1886.80	−0.158661	−0.0793305	+	0.996848i	$0.525278\pi$
$522$	0	0
$523$	− 11332.8i	− 0.947509i	−0.880657	−	0.473755i	$-0.842898\pi$
$523$	− 11332.8i	− 0.947509i	0.880657	−	0.473755i	$-0.157102\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1724.37	0.142533
$528$	0	0
$529$	−2227.10	−0.183044
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 17136.0i	− 1.39258i
$534$	0	0
$535$	6330.44	0.511567
$536$	0	0
$537$	0	0
$538$	0	0
$539$	16509.9i	1.31935i
$540$	0	0
$541$	− 15975.9i	− 1.26961i	−0.772673	−	0.634804i	$-0.781081\pi$
$541$	− 15975.9i	− 1.26961i	0.772673	−	0.634804i	$-0.218919\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2677.00	−0.210404
$546$	0	0
$547$	10110.2i	0.790280i	0.918621	+	0.395140i	$0.129304\pi$
$547$	10110.2i	0.790280i	−0.918621	+	0.395140i	$0.870696\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3332.80	−0.257681
$552$	0	0
$553$	−10531.4	−0.809841
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20452.5i	1.55583i	0.628367	+	0.777917i	$0.283724\pi$
$557$	20452.5i	1.55583i	−0.628367	+	0.777917i	$0.716276\pi$
$558$	0	0
$559$	−25114.5	−1.90023
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12012.2i	0.899207i	0.893228	+	0.449604i	$0.148435\pi$
$563$	12012.2i	0.899207i	−0.893228	+	0.449604i	$0.851565\pi$
$564$	0	0
$565$	− 23151.3i	− 1.72386i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16457.8	1.21256	0.606278	−	0.795252i	$-0.292662\pi$
$569$	16457.8	1.21256	0.606278	+	0.795252i	$0.292662\pi$
$570$	0	0
$571$	− 21065.9i	− 1.54392i	−0.635670	−	0.771961i	$-0.719276\pi$
$571$	− 21065.9i	− 1.54392i	0.635670	−	0.771961i	$-0.280724\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	801.656	0.0581415
$576$	0	0
$577$	14086.4	1.01633	0.508167	−	0.861258i	$-0.330323\pi$
$577$	14086.4	1.01633	0.508167	+	0.861258i	$0.330323\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 3680.19i	− 0.262788i
$582$	0	0
$583$	29100.6	2.06728
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5796.83i	0.407599i	0.979013	+	0.203800i	$0.0653291\pi$
$587$	5796.83i	0.407599i	−0.979013	+	0.203800i	$0.934671\pi$
$588$	0	0
$589$	− 6609.79i	− 0.462396i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1375.06	0.0952226	0.0476113	−	0.998866i	$-0.484839\pi$
$593$	1375.06	0.0952226	0.0476113	+	0.998866i	$0.484839\pi$
$594$	0	0
$595$	6784.81i	0.467479i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−26452.8	−1.80440	−0.902198	−	0.431322i	$-0.858047\pi$
$599$	−26452.8	−1.80440	−0.902198	+	0.431322i	$0.858047\pi$
$600$	0	0
$601$	−20708.6	−1.40553	−0.702763	−	0.711424i	$-0.748050\pi$
$601$	−20708.6	−1.40553	−0.702763	+	0.711424i	$0.748050\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 25912.1i	− 1.74128i
$606$	0	0
$607$	13567.6	0.907235	0.453618	−	0.891196i	$-0.350133\pi$
$607$	13567.6	0.907235	0.453618	+	0.891196i	$0.350133\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 25261.4i	− 1.67261i
$612$	0	0
$613$	− 78.8760i	− 0.00519702i	−0.999997	−	0.00259851i	$-0.999173\pi$
$613$	− 78.8760i	− 0.00519702i	0.999997	−	0.00259851i	$-0.000827132\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−14909.5	−0.972828	−0.486414	−	0.873729i	$-0.661695\pi$
$617$	−14909.5	−0.972828	−0.486414	+	0.873729i	$0.661695\pi$
$618$	0	0
$619$	− 19774.8i	− 1.28403i	−0.766691	−	0.642016i	$-0.778098\pi$
$619$	− 19774.8i	− 1.28403i	0.766691	−	0.642016i	$-0.221902\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10604.4	−0.681950
$624$	0	0
$625$	−14555.3	−0.931536
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 6876.56i	− 0.435908i
$630$	0	0
$631$	−12377.7	−0.780901	−0.390450	−	0.920624i	$-0.627681\pi$
$631$	−12377.7	−0.780901	−0.390450	+	0.920624i	$0.627681\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5566.65i	0.347883i
$636$	0	0
$637$	− 15668.3i	− 0.974567i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−28130.3	−1.73335	−0.866677	−	0.498869i	$-0.833749\pi$
$641$	−28130.3	−1.73335	−0.866677	+	0.498869i	$0.833749\pi$
$642$	0	0
$643$	5767.89i	0.353753i	0.984233	+	0.176877i	$0.0565994\pi$
$643$	5767.89i	0.353753i	−0.984233	+	0.176877i	$0.943401\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17564.0	−1.06725	−0.533626	−	0.845720i	$-0.679171\pi$
$647$	−17564.0	−1.06725	−0.533626	+	0.845720i	$0.679171\pi$
$648$	0	0
$649$	36705.4	2.22005
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 32099.1i	− 1.92364i	−0.273690	−	0.961818i	$-0.588244\pi$
$653$	− 32099.1i	− 1.92364i	0.273690	−	0.961818i	$-0.411756\pi$
$654$	0	0
$655$	29982.4	1.78856
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 17089.1i	− 1.01016i	−0.863071	−	0.505082i	$-0.831462\pi$
$659$	− 17089.1i	− 1.01016i	0.863071	−	0.505082i	$-0.168538\pi$
$660$	0	0
$661$	9116.04i	0.536419i	0.963361	+	0.268209i	$0.0864319\pi$
$661$	9116.04i	0.536419i	−0.963361	+	0.268209i	$0.913568\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	26007.3	1.51657
$666$	0	0
$667$	− 3422.22i	− 0.198664i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−22860.6	−1.31524
$672$	0	0
$673$	12124.6	0.694455	0.347228	−	0.937781i	$-0.387123\pi$
$673$	12124.6	0.694455	0.347228	+	0.937781i	$0.387123\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5739.16i	0.325811i	0.986642	+	0.162905i	$0.0520865\pi$
$677$	5739.16i	0.325811i	−0.986642	+	0.162905i	$0.947913\pi$
$678$	0	0
$679$	−16928.3	−0.956770
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 13514.1i	− 0.757105i	−0.925580	−	0.378552i	$-0.876422\pi$
$683$	− 13514.1i	− 0.757105i	0.925580	−	0.378552i	$-0.123578\pi$
$684$	0	0
$685$	11704.4i	0.652852i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−27617.1	−1.52703
$690$	0	0
$691$	− 27880.0i	− 1.53488i	−0.641120	−	0.767441i	$-0.721530\pi$
$691$	− 27880.0i	− 1.53488i	0.641120	−	0.767441i	$-0.278470\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7207.77	−0.393390
$696$	0	0
$697$	7491.88	0.407138
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 18202.6i	− 0.980745i	−0.871513	−	0.490372i	$-0.836861\pi$
$701$	− 18202.6i	− 0.980745i	0.871513	−	0.490372i	$-0.163139\pi$
$702$	0	0
$703$	−26358.9	−1.41415
$704$	0	0
$705$	0	0
$706$	0	0
$707$	18370.8i	0.977234i
$708$	0	0
$709$	− 7077.13i	− 0.374876i	−0.982276	−	0.187438i	$-0.939982\pi$
$709$	− 7077.13i	− 0.374876i	0.982276	−	0.187438i	$-0.0600184\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6787.13	0.356494
$714$	0	0
$715$	38251.8i	2.00075i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	20951.8	1.08674	0.543372	−	0.839492i	$-0.317147\pi$
$719$	20951.8	1.08674	0.543372	+	0.839492i	$0.317147\pi$
$720$	0	0
$721$	20779.1	1.07331
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 276.003i	− 0.0141386i
$726$	0	0
$727$	−36144.5	−1.84392	−0.921958	−	0.387291i	$-0.873411\pi$
$727$	−36144.5	−1.84392	−0.921958	+	0.387291i	$0.873411\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 10980.1i	− 0.555557i
$732$	0	0
$733$	− 8915.77i	− 0.449265i	−0.974443	−	0.224633i	$-0.927882\pi$
$733$	− 8915.77i	− 0.449265i	0.974443	−	0.224633i	$-0.0721182\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−53082.8	−2.65309
$738$	0	0
$739$	− 11324.4i	− 0.563700i	−0.959458	−	0.281850i	$-0.909052\pi$
$739$	− 11324.4i	− 0.563700i	0.959458	−	0.281850i	$-0.0909481\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−23750.6	−1.17271	−0.586357	−	0.810053i	$-0.699438\pi$
$743$	−23750.6	−1.17271	−0.586357	+	0.810053i	$0.699438\pi$
$744$	0	0
$745$	−22023.3	−1.08305
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 14497.8i	− 0.707259i
$750$	0	0
$751$	−11684.7	−0.567750	−0.283875	−	0.958861i	$-0.591620\pi$
$751$	−11684.7	−0.567750	−0.283875	+	0.958861i	$0.591620\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	30384.6i	1.46465i
$756$	0	0
$757$	31064.7i	1.49150i	0.666227	+	0.745749i	$0.267908\pi$
$757$	31064.7i	1.49150i	−0.666227	+	0.745749i	$0.732092\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−35894.4	−1.70982	−0.854909	−	0.518778i	$-0.826387\pi$
$761$	−35894.4	−1.70982	−0.854909	+	0.518778i	$0.826387\pi$
$762$	0	0
$763$	6130.79i	0.290890i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−34834.2	−1.63988
$768$	0	0
$769$	21025.7	0.985965	0.492982	−	0.870039i	$-0.335907\pi$
$769$	21025.7	0.985965	0.492982	+	0.870039i	$0.335907\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18259.0i	0.849587i	0.905290	+	0.424794i	$0.139653\pi$
$773$	18259.0i	0.849587i	−0.905290	+	0.424794i	$0.860347\pi$
$774$	0	0
$775$	547.384	0.0253711
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 28717.6i	− 1.32081i
$780$	0	0
$781$	− 44052.8i	− 2.01835i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−20953.5	−0.952692
$786$	0	0
$787$	− 21135.3i	− 0.957296i	−0.878007	−	0.478648i	$-0.841127\pi$
$787$	− 21135.3i	− 0.957296i	0.878007	−	0.478648i	$-0.158873\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−53020.3	−2.38329
$792$	0	0
$793$	21695.2	0.971526
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 260.755i	− 0.0115890i	−0.999983	−	0.00579450i	$-0.998156\pi$
$797$	− 260.755i	− 0.0115890i	0.999983	−	0.00579450i	$-0.00184446\pi$
$798$	0	0
$799$	11044.3	0.489010
$800$	0	0
$801$	0	0
$802$	0	0
$803$	68973.2i	3.03115i
$804$	0	0
$805$	26705.0i	1.16923i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	19659.9	0.854395	0.427197	−	0.904158i	$-0.359501\pi$
$809$	19659.9	0.854395	0.427197	+	0.904158i	$0.359501\pi$
$810$	0	0
$811$	− 12687.2i	− 0.549333i	−0.961539	−	0.274667i	$-0.911432\pi$
$811$	− 12687.2i	− 0.549333i	0.961539	−	0.274667i	$-0.0885675\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7386.05	−0.317450
$816$	0	0
$817$	−42088.3	−1.80231
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 3518.62i	− 0.149574i	−0.997200	−	0.0747872i	$-0.976172\pi$
$821$	− 3518.62i	− 0.149574i	0.997200	−	0.0747872i	$-0.0238278\pi$
$822$	0	0
$823$	8701.87	0.368564	0.184282	−	0.982873i	$-0.441004\pi$
$823$	8701.87	0.368564	0.184282	+	0.982873i	$0.441004\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	21017.1i	0.883721i	0.897084	+	0.441861i	$0.145681\pi$
$827$	21017.1i	0.883721i	−0.897084	+	0.441861i	$0.854319\pi$
$828$	0	0
$829$	38803.1i	1.62568i	0.582488	+	0.812840i	$0.302079\pi$
$829$	38803.1i	1.62568i	−0.582488	+	0.812840i	$0.697921\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6850.17	0.284927
$834$	0	0
$835$	− 21682.0i	− 0.898605i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	14442.2	0.594281	0.297140	−	0.954834i	$-0.403967\pi$
$839$	14442.2	0.594281	0.297140	+	0.954834i	$0.403967\pi$
$840$	0	0
$841$	23210.8	0.951690
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 12541.7i	− 0.510589i
$846$	0	0
$847$	−59343.1	−2.40738
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 27066.1i	− 1.09027i
$852$	0	0
$853$	− 17466.3i	− 0.701094i	−0.936545	−	0.350547i	$-0.885996\pi$
$853$	− 17466.3i	− 0.701094i	0.936545	−	0.350547i	$-0.114004\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	42467.6	1.69272	0.846362	−	0.532607i	$-0.178788\pi$
$857$	42467.6	1.69272	0.846362	+	0.532607i	$0.178788\pi$
$858$	0	0
$859$	36343.9i	1.44358i	0.692111	+	0.721791i	$0.256681\pi$
$859$	36343.9i	1.44358i	−0.692111	+	0.721791i	$0.743319\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24569.9	0.969140	0.484570	−	0.874753i	$-0.338976\pi$
$863$	24569.9	0.969140	0.484570	+	0.874753i	$0.338976\pi$
$864$	0	0
$865$	−4449.48	−0.174898
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 25958.6i	− 1.01333i
$870$	0	0
$871$	50376.7	1.95976
$872$	0	0
$873$	0	0
$874$	0	0
$875$	35635.8i	1.37681i
$876$	0	0
$877$	2294.97i	0.0883644i	0.999023	+	0.0441822i	$0.0140682\pi$
$877$	2294.97i	0.0883644i	−0.999023	+	0.0441822i	$0.985932\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−34279.4	−1.31090	−0.655450	−	0.755238i	$-0.727521\pi$
$881$	−34279.4	−1.31090	−0.655450	+	0.755238i	$0.727521\pi$
$882$	0	0
$883$	41156.3i	1.56854i	0.620421	+	0.784269i	$0.286962\pi$
$883$	41156.3i	1.56854i	−0.620421	+	0.784269i	$0.713038\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−26283.7	−0.994949	−0.497474	−	0.867479i	$-0.665739\pi$
$887$	−26283.7	−0.994949	−0.497474	+	0.867479i	$0.665739\pi$
$888$	0	0
$889$	12748.6	0.480960
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 42334.5i	− 1.58642i
$894$	0	0
$895$	17544.3	0.655242
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 2336.75i	− 0.0866907i
$900$	0	0
$901$	− 12074.2i	− 0.446448i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	8068.03	0.296343
$906$	0	0
$907$	− 17488.7i	− 0.640245i	−0.947376	−	0.320123i	$-0.896276\pi$
$907$	− 17488.7i	− 0.640245i	0.947376	−	0.320123i	$-0.103724\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24740.9	0.899782	0.449891	−	0.893084i	$-0.351463\pi$
$911$	24740.9	0.899782	0.449891	+	0.893084i	$0.351463\pi$
$912$	0	0
$913$	9071.19	0.328820
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 68664.7i	− 2.47275i
$918$	0	0
$919$	20067.1	0.720296	0.360148	−	0.932895i	$-0.382726\pi$
$919$	20067.1	0.720296	0.360148	+	0.932895i	$0.382726\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	41807.1i	1.49090i
$924$	0	0
$925$	− 2182.89i	− 0.0775925i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−45723.3	−1.61478	−0.807391	−	0.590016i	$-0.799121\pi$
$929$	−45723.3	−1.61478	−0.807391	+	0.590016i	$0.799121\pi$
$930$	0	0
$931$	− 26257.8i	− 0.924344i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−16723.7	−0.584945
$936$	0	0
$937$	−35381.0	−1.23356	−0.616780	−	0.787136i	$-0.711563\pi$
$937$	−35381.0	−1.23356	−0.616780	+	0.787136i	$0.711563\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	34719.7i	1.20280i	0.798950	+	0.601398i	$0.205389\pi$
$941$	34719.7i	1.20280i	−0.798950	+	0.601398i	$0.794611\pi$
$942$	0	0
$943$	29488.1	1.01831
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 18784.2i	− 0.644566i	−0.946643	−	0.322283i	$-0.895550\pi$
$947$	− 18784.2i	− 0.644566i	0.946643	−	0.322283i	$-0.104450\pi$
$948$	0	0
$949$	− 65457.0i	− 2.23901i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	31205.8	1.06071	0.530354	−	0.847776i	$-0.322059\pi$
$953$	31205.8	1.06071	0.530354	+	0.847776i	$0.322059\pi$
$954$	0	0
$955$	49246.0i	1.66865i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	26805.2	0.902590
$960$	0	0
$961$	−25156.6	−0.844438
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 17400.6i	− 0.580462i
$966$	0	0
$967$	5655.36	0.188070	0.0940352	−	0.995569i	$-0.470023\pi$
$967$	5655.36	0.188070	0.0940352	+	0.995569i	$0.470023\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10354.1i	0.342204i	0.985253	+	0.171102i	$0.0547327\pi$
$971$	10354.1i	0.342204i	−0.985253	+	0.171102i	$0.945267\pi$
$972$	0	0
$973$	16507.0i	0.543875i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9485.97	0.310628	0.155314	−	0.987865i	$-0.450361\pi$
$977$	9485.97	0.310628	0.155314	+	0.987865i	$0.450361\pi$
$978$	0	0
$979$	− 26138.4i	− 0.853306i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−32420.3	−1.05193	−0.525965	−	0.850506i	$-0.676296\pi$
$983$	−32420.3	−1.05193	−0.525965	+	0.850506i	$0.676296\pi$
$984$	0	0
$985$	−44797.6	−1.44911
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 43217.5i	− 1.38952i
$990$	0	0
$991$	53990.7	1.73065	0.865324	−	0.501213i	$-0.167113\pi$
$991$	53990.7	1.73065	0.865324	+	0.501213i	$0.167113\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 33058.5i	− 1.05329i
$996$	0	0
$997$	− 14501.8i	− 0.460658i	−0.973113	−	0.230329i	$-0.926020\pi$
$997$	− 14501.8i	− 0.460658i	0.973113	−	0.230329i	$-0.0739803\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.d.i.865.14	yes	16
3.2	odd	2		1728.4.d.k.865.4	yes	16
4.3	odd	2	inner	1728.4.d.i.865.13	yes	16
8.3	odd	2	inner	1728.4.d.i.865.3	✓	16
8.5	even	2	inner	1728.4.d.i.865.4	yes	16
12.11	even	2		1728.4.d.k.865.3	yes	16
24.5	odd	2		1728.4.d.k.865.14	yes	16
24.11	even	2		1728.4.d.k.865.13	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.4.d.i.865.3	✓	16	8.3	odd	2	inner
1728.4.d.i.865.4	yes	16	8.5	even	2	inner
1728.4.d.i.865.13	yes	16	4.3	odd	2	inner
1728.4.d.i.865.14	yes	16	1.1	even	1	trivial
1728.4.d.k.865.3	yes	16	12.11	even	2
1728.4.d.k.865.4	yes	16	3.2	odd	2
1728.4.d.k.865.13	yes	16	24.11	even	2
1728.4.d.k.865.14	yes	16	24.5	odd	2

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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+10.8148i q^{5} +24.7677 q^{7} +O(q^{10})\)	\(q+10.8148i q^{5} +24.7677 q^{7} +61.0491i q^{11} -57.9369i q^{13} +25.3300 q^{17} -97.0940i q^{19} +99.6991 q^{23} +8.04076 q^{25} -34.3255i q^{29} +68.0761 q^{31} +267.856i q^{35} -271.478i q^{37} +295.771 q^{41} -433.480i q^{43} +436.016 q^{47} +270.437 q^{49} -476.675i q^{53} -660.232 q^{55} -601.244i q^{59} +374.463i q^{61} +626.574 q^{65} +869.510i q^{67} -721.597 q^{71} +1129.80 q^{73} +1512.04i q^{77} -425.209 q^{79} -148.588i q^{83} +273.939i q^{85} -428.154 q^{89} -1434.96i q^{91} +1050.05 q^{95} -683.483 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 240 q^{17} - 304 q^{25} + 1008 q^{41} + 1616 q^{49} + 2736 q^{65} + 128 q^{73} + 5856 q^{89} - 2576 q^{97}+O(q^{100}) \) 16 * q - 240 * q^17 - 304 * q^25 + 1008 * q^41 + 1616 * q^49 + 2736 * q^65 + 128 * q^73 + 5856 * q^89 - 2576 * q^97

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