LMFDB - Embedded newform 2736.3.m.e.1711.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2736,3,Mod(1711,2736)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2736, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2736.1711");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2736 = 2^{4} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2736.m (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:	$74.5506003290$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 2 x^{10} - 28 x^{9} - 400 x^{8} - 520 x^{7} + 17067 x^{6} - 3250 x^{5} + \cdots + 2052928 $ x^12 - 2x^11 - 2x^10 - 28x^9 - 400x^8 - 520x^7 + 17067x^6 - 3250x^5 - 195494x^4 + 302996x^3 + 602332x^2 - 2263536*x + 2052928
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{16} $
Twist minimal:	no (minimal twist has level 912)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1711.10
Root			$1.30395 - 1.41343i$ of defining polynomial
Character	$\chi$	$=$	2736.1711
Dual form			2736.3.m.e.1711.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.08630 q^{5} +8.85163i q^{7} +O(q^{10})$	$q+6.08630 q^{5} +8.85163i q^{7} -6.64046i q^{11} -2.95680 q^{13} -21.7835 q^{17} +4.35890i q^{19} +30.4779i q^{23} +12.0431 q^{25} +17.0059 q^{29} -58.0298i q^{31} +53.8737i q^{35} -44.3601 q^{37} +5.33039 q^{41} +50.1443i q^{43} +74.0431i q^{47} -29.3513 q^{49} -51.5148 q^{53} -40.4159i q^{55} +18.5370i q^{59} -100.168 q^{61} -17.9960 q^{65} +31.1882i q^{67} +74.3566i q^{71} -50.4741 q^{73} +58.7789 q^{77} -6.81405i q^{79} +25.6400i q^{83} -132.581 q^{85} -3.61427 q^{89} -26.1725i q^{91} +26.5296i q^{95} +140.421 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 10 q^{5}+O(q^{10}) $ 12 * q + 10 * q^5	$ 12 q + 10 q^{5} + 36 q^{13} - 14 q^{17} + 10 q^{25} + 108 q^{29} - 16 q^{37} - 16 q^{41} - 118 q^{49} - 220 q^{53} + 366 q^{61} - 140 q^{65} - 158 q^{73} + 286 q^{77} + 98 q^{85} + 396 q^{89} + 224 q^{97}+O(q^{100}) $ 12 * q + 10 * q^5 + 36 * q^13 - 14 * q^17 + 10 * q^25 + 108 * q^29 - 16 * q^37 - 16 * q^41 - 118 * q^49 - 220 * q^53 + 366 * q^61 - 140 * q^65 - 158 * q^73 + 286 * q^77 + 98 * q^85 + 396 * q^89 + 224 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times$.

$n$	$1009$	$1217$	$1711$	$2053$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$	6.08630	1.21726	0.608630	−	0.793454i	$-0.291719\pi$
$5$	6.08630	1.21726	0.608630	+	0.793454i	$0.291719\pi$
$6$	0	0
$7$	8.85163i	1.26452i	0.774757	+	0.632259i	$0.217872\pi$
$7$	8.85163i	1.26452i	−0.774757	+	0.632259i	$0.782128\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 6.64046i	− 0.603678i	−0.953359	−	0.301839i	$-0.902399\pi$
$11$	− 6.64046i	− 0.603678i	0.953359	−	0.301839i	$-0.0976005\pi$
$12$	0	0
$13$	−2.95680	−0.227446	−0.113723	−	0.993512i	$-0.536278\pi$
$13$	−2.95680	−0.227446	−0.113723	+	0.993512i	$0.536278\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−21.7835	−1.28138	−0.640691	−	0.767799i	$-0.721352\pi$
$17$	−21.7835	−1.28138	−0.640691	+	0.767799i	$0.721352\pi$
$18$	0	0
$19$	4.35890i	0.229416i
$19$	4.35890i	0.229416i
$20$	0	0
$21$	0	0
$22$	0	0
$23$	30.4779i	1.32513i	0.749006	+	0.662563i	$0.230531\pi$
$23$	30.4779i	1.32513i	−0.749006	+	0.662563i	$0.769469\pi$
$24$	0	0
$25$	12.0431	0.481724
$26$	0	0
$27$	0	0
$28$	0	0
$29$	17.0059	0.586410	0.293205	−	0.956050i	$-0.405278\pi$
$29$	17.0059	0.586410	0.293205	+	0.956050i	$0.405278\pi$
$30$	0	0
$31$	− 58.0298i	− 1.87193i	−0.352094	−	0.935965i	$-0.614530\pi$
$31$	− 58.0298i	− 1.87193i	0.352094	−	0.935965i	$-0.385470\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	53.8737i	1.53925i
$36$	0	0
$37$	−44.3601	−1.19892	−0.599461	−	0.800404i	$-0.704618\pi$
$37$	−44.3601	−1.19892	−0.599461	+	0.800404i	$0.704618\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.33039	0.130009	0.0650047	−	0.997885i	$-0.479294\pi$
$41$	5.33039	0.130009	0.0650047	+	0.997885i	$0.479294\pi$
$42$	0	0
$43$	50.1443i	1.16615i	0.812419	+	0.583074i	$0.198150\pi$
$43$	50.1443i	1.16615i	−0.812419	+	0.583074i	$0.801850\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	74.0431i	1.57538i	0.616069	+	0.787692i	$0.288724\pi$
$47$	74.0431i	1.57538i	−0.616069	+	0.787692i	$0.711276\pi$
$48$	0	0
$49$	−29.3513	−0.599005
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−51.5148	−0.971977	−0.485988	−	0.873965i	$-0.661540\pi$
$53$	−51.5148	−0.971977	−0.485988	+	0.873965i	$0.661540\pi$
$54$	0	0
$55$	− 40.4159i	− 0.734834i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	18.5370i	0.314187i	0.987584	+	0.157093i	$0.0502124\pi$
$59$	18.5370i	0.314187i	−0.987584	+	0.157093i	$0.949788\pi$
$60$	0	0
$61$	−100.168	−1.64210	−0.821048	−	0.570859i	$-0.806610\pi$
$61$	−100.168	−1.64210	−0.821048	+	0.570859i	$0.806610\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−17.9960	−0.276861
$66$	0	0
$67$	31.1882i	0.465495i	0.972537	+	0.232747i	$0.0747715\pi$
$67$	31.1882i	0.465495i	−0.972537	+	0.232747i	$0.925228\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	74.3566i	1.04728i	0.851941	+	0.523638i	$0.175426\pi$
$71$	74.3566i	1.04728i	−0.851941	+	0.523638i	$0.824574\pi$
$72$	0	0
$73$	−50.4741	−0.691426	−0.345713	−	0.938340i	$-0.612363\pi$
$73$	−50.4741	−0.691426	−0.345713	+	0.938340i	$0.612363\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	58.7789	0.763362
$78$	0	0
$79$	− 6.81405i	− 0.0862538i	−0.999070	−	0.0431269i	$-0.986268\pi$
$79$	− 6.81405i	− 0.0862538i	0.999070	−	0.0431269i	$-0.0137320\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	25.6400i	0.308916i	0.987999	+	0.154458i	$0.0493632\pi$
$83$	25.6400i	0.308916i	−0.987999	+	0.154458i	$0.950637\pi$
$84$	0	0
$85$	−132.581	−1.55978
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.61427	−0.0406098	−0.0203049	−	0.999794i	$-0.506464\pi$
$89$	−3.61427	−0.0406098	−0.0203049	+	0.999794i	$0.506464\pi$
$90$	0	0
$91$	− 26.1725i	− 0.287610i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	26.5296i	0.279259i
$96$	0	0
$97$	140.421	1.44764	0.723819	−	0.689990i	$-0.242385\pi$
$97$	140.421	1.44764	0.723819	+	0.689990i	$0.242385\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−71.3972	−0.706903	−0.353451	−	0.935453i	$-0.614992\pi$
$101$	−71.3972	−0.706903	−0.353451	+	0.935453i	$0.614992\pi$
$102$	0	0
$103$	− 102.721i	− 0.997295i	−0.866805	−	0.498647i	$-0.833830\pi$
$103$	− 102.721i	− 0.997295i	0.866805	−	0.498647i	$-0.166170\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	64.9271i	0.606796i	0.952864	+	0.303398i	$0.0981211\pi$
$107$	64.9271i	0.606796i	−0.952864	+	0.303398i	$0.901879\pi$
$108$	0	0
$109$	−115.572	−1.06029	−0.530145	−	0.847907i	$-0.677863\pi$
$109$	−115.572	−1.06029	−0.530145	+	0.847907i	$0.677863\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	217.263	1.92268	0.961339	−	0.275366i	$-0.0887991\pi$
$113$	217.263	1.92268	0.961339	+	0.275366i	$0.0887991\pi$
$114$	0	0
$115$	185.498i	1.61302i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 192.819i	− 1.62033i
$120$	0	0
$121$	76.9043	0.635573
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−78.8596	−0.630877
$126$	0	0
$127$	− 17.3029i	− 0.136243i	−0.997677	−	0.0681217i	$-0.978299\pi$
$127$	− 17.3029i	− 0.136243i	0.997677	−	0.0681217i	$-0.0217006\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	72.8478i	0.556090i	0.960568	+	0.278045i	$0.0896864\pi$
$131$	72.8478i	0.556090i	−0.960568	+	0.278045i	$0.910314\pi$
$132$	0	0
$133$	−38.5833	−0.290100
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−166.375	−1.21441	−0.607207	−	0.794544i	$-0.707710\pi$
$137$	−166.375	−1.21441	−0.607207	+	0.794544i	$0.707710\pi$
$138$	0	0
$139$	− 197.109i	− 1.41805i	−0.705184	−	0.709024i	$-0.749136\pi$
$139$	− 197.109i	− 1.41805i	0.705184	−	0.709024i	$-0.250864\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	19.6345i	0.137304i
$144$	0	0
$145$	103.503	0.713814
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−240.667	−1.61522	−0.807608	−	0.589720i	$-0.799238\pi$
$149$	−240.667	−1.61522	−0.807608	+	0.589720i	$0.799238\pi$
$150$	0	0
$151$	− 123.722i	− 0.819354i	−0.912231	−	0.409677i	$-0.865641\pi$
$151$	− 123.722i	− 0.819354i	0.912231	−	0.409677i	$-0.134359\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 353.187i	− 2.27863i
$156$	0	0
$157$	114.012	0.726194	0.363097	−	0.931751i	$-0.381719\pi$
$157$	114.012	0.726194	0.363097	+	0.931751i	$0.381719\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−269.779	−1.67565
$162$	0	0
$163$	− 227.685i	− 1.39684i	−0.715689	−	0.698419i	$-0.753887\pi$
$163$	− 227.685i	− 1.39684i	0.715689	−	0.698419i	$-0.246113\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	96.0499i	0.575149i	0.957758	+	0.287575i	$0.0928489\pi$
$167$	96.0499i	0.575149i	−0.957758	+	0.287575i	$0.907151\pi$
$168$	0	0
$169$	−160.257	−0.948268
$170$	0	0
$171$	0	0
$172$	0	0
$173$	277.388	1.60340	0.801698	−	0.597729i	$-0.203930\pi$
$173$	277.388	1.60340	0.801698	+	0.597729i	$0.203930\pi$
$174$	0	0
$175$	106.601i	0.609149i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	103.569i	0.578600i	0.957239	+	0.289300i	$0.0934225\pi$
$179$	103.569i	0.578600i	−0.957239	+	0.289300i	$0.906578\pi$
$180$	0	0
$181$	−126.631	−0.699621	−0.349810	−	0.936821i	$-0.613754\pi$
$181$	−126.631	−0.699621	−0.349810	+	0.936821i	$0.613754\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−269.989	−1.45940
$186$	0	0
$187$	144.652i	0.773542i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	118.419i	0.619997i	0.950737	+	0.309998i	$0.100328\pi$
$191$	118.419i	0.619997i	−0.950737	+	0.309998i	$0.899672\pi$
$192$	0	0
$193$	−179.569	−0.930412	−0.465206	−	0.885203i	$-0.654020\pi$
$193$	−179.569	−0.930412	−0.465206	+	0.885203i	$0.654020\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−185.985	−0.944088	−0.472044	−	0.881575i	$-0.656484\pi$
$197$	−185.985	−0.944088	−0.472044	+	0.881575i	$0.656484\pi$
$198$	0	0
$199$	11.9251i	0.0599251i	0.999551	+	0.0299625i	$0.00953880\pi$
$199$	11.9251i	0.0599251i	−0.999551	+	0.0299625i	$0.990461\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	150.530i	0.741526i
$204$	0	0
$205$	32.4423	0.158255
$206$	0	0
$207$	0	0
$208$	0	0
$209$	28.9451	0.138493
$210$	0	0
$211$	54.6535i	0.259021i	0.991578	+	0.129511i	$0.0413406\pi$
$211$	54.6535i	0.259021i	−0.991578	+	0.129511i	$0.958659\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	305.194i	1.41951i
$216$	0	0
$217$	513.658	2.36709
$218$	0	0
$219$	0	0
$220$	0	0
$221$	64.4094	0.291445
$222$	0	0
$223$	125.844i	0.564321i	0.959367	+	0.282161i	$0.0910511\pi$
$223$	125.844i	0.564321i	−0.959367	+	0.282161i	$0.908949\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	372.784i	1.64222i	0.570769	+	0.821110i	$0.306645\pi$
$227$	372.784i	1.64222i	−0.570769	+	0.821110i	$0.693355\pi$
$228$	0	0
$229$	−14.2915	−0.0624082	−0.0312041	−	0.999513i	$-0.509934\pi$
$229$	−14.2915	−0.0624082	−0.0312041	+	0.999513i	$0.509934\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.2964	−0.0828170	−0.0414085	−	0.999142i	$-0.513184\pi$
$233$	−19.2964	−0.0828170	−0.0414085	+	0.999142i	$0.513184\pi$
$234$	0	0
$235$	450.649i	1.91765i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 147.097i	− 0.615470i	−0.951472	−	0.307735i	$-0.900429\pi$
$239$	− 147.097i	− 0.615470i	0.951472	−	0.307735i	$-0.0995710\pi$
$240$	0	0
$241$	−210.083	−0.871713	−0.435857	−	0.900016i	$-0.643555\pi$
$241$	−210.083	−0.871713	−0.435857	+	0.900016i	$0.643555\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−178.641	−0.729146
$246$	0	0
$247$	− 12.8884i	− 0.0521797i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 53.6081i	− 0.213578i	−0.994282	−	0.106789i	$-0.965943\pi$
$251$	− 53.6081i	− 0.213578i	0.994282	−	0.106789i	$-0.0340570\pi$
$252$	0	0
$253$	202.387	0.799950
$254$	0	0
$255$	0	0
$256$	0	0
$257$	346.789	1.34937	0.674686	−	0.738105i	$-0.264279\pi$
$257$	346.789	1.34937	0.674686	+	0.738105i	$0.264279\pi$
$258$	0	0
$259$	− 392.659i	− 1.51606i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 355.183i	− 1.35050i	−0.737587	−	0.675252i	$-0.764035\pi$
$263$	− 355.183i	− 1.35050i	0.737587	−	0.675252i	$-0.235965\pi$
$264$	0	0
$265$	−313.535	−1.18315
$266$	0	0
$267$	0	0
$268$	0	0
$269$	41.7600	0.155242	0.0776209	−	0.996983i	$-0.475268\pi$
$269$	41.7600	0.155242	0.0776209	+	0.996983i	$0.475268\pi$
$270$	0	0
$271$	433.198i	1.59852i	0.600987	+	0.799259i	$0.294774\pi$
$271$	433.198i	1.59852i	−0.600987	+	0.799259i	$0.705226\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 79.9717i	− 0.290806i
$276$	0	0
$277$	380.398	1.37328	0.686640	−	0.726998i	$-0.259085\pi$
$277$	380.398	1.37328	0.686640	+	0.726998i	$0.259085\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−185.121	−0.658792	−0.329396	−	0.944192i	$-0.606845\pi$
$281$	−185.121	−0.658792	−0.329396	+	0.944192i	$0.606845\pi$
$282$	0	0
$283$	397.379i	1.40417i	0.712096	+	0.702083i	$0.247746\pi$
$283$	397.379i	1.40417i	−0.712096	+	0.702083i	$0.752254\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	47.1826i	0.164399i
$288$	0	0
$289$	185.520	0.641939
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−489.006	−1.66896	−0.834480	−	0.551037i	$-0.814232\pi$
$293$	−489.006	−1.66896	−0.834480	+	0.551037i	$0.814232\pi$
$294$	0	0
$295$	112.822i	0.382447i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 90.1170i	− 0.301395i
$300$	0	0
$301$	−443.859	−1.47461
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−609.652	−1.99886
$306$	0	0
$307$	− 93.4725i	− 0.304471i	−0.988344	−	0.152235i	$-0.951353\pi$
$307$	− 93.4725i	− 0.304471i	0.988344	−	0.152235i	$-0.0486471\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	214.371i	0.689294i	0.938732	+	0.344647i	$0.112001\pi$
$311$	214.371i	0.689294i	−0.938732	+	0.344647i	$0.887999\pi$
$312$	0	0
$313$	148.994	0.476020	0.238010	−	0.971263i	$-0.423505\pi$
$313$	148.994	0.476020	0.238010	+	0.971263i	$0.423505\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	490.937	1.54870	0.774349	−	0.632759i	$-0.218077\pi$
$317$	490.937	1.54870	0.774349	+	0.632759i	$0.218077\pi$
$318$	0	0
$319$	− 112.927i	− 0.354003i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 94.9520i	− 0.293969i
$324$	0	0
$325$	−35.6090	−0.109566
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−655.401	−1.99210
$330$	0	0
$331$	489.131i	1.47774i	0.673850	+	0.738868i	$0.264639\pi$
$331$	489.131i	1.47774i	−0.673850	+	0.738868i	$0.735361\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	189.821i	0.566629i
$336$	0	0
$337$	3.82451	0.0113487	0.00567435	−	0.999984i	$-0.498194\pi$
$337$	3.82451	0.0113487	0.00567435	+	0.999984i	$0.498194\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−385.345	−1.13004
$342$	0	0
$343$	173.923i	0.507065i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 151.748i	− 0.437314i	−0.975802	−	0.218657i	$-0.929832\pi$
$347$	− 151.748i	− 0.437314i	0.975802	−	0.218657i	$-0.0701676\pi$
$348$	0	0
$349$	12.2852	0.0352011	0.0176005	−	0.999845i	$-0.494397\pi$
$349$	12.2852	0.0352011	0.0176005	+	0.999845i	$0.494397\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.69743	0.0104743	0.00523716	−	0.999986i	$-0.498333\pi$
$353$	3.69743	0.0104743	0.00523716	+	0.999986i	$0.498333\pi$
$354$	0	0
$355$	452.557i	1.27481i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 109.483i	− 0.304966i	−0.988306	−	0.152483i	$-0.951273\pi$
$359$	− 109.483i	− 0.304966i	0.988306	−	0.152483i	$-0.0487270\pi$
$360$	0	0
$361$	−19.0000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−307.201	−0.841646
$366$	0	0
$367$	611.742i	1.66687i	0.552615	+	0.833436i	$0.313630\pi$
$367$	611.742i	1.66687i	−0.552615	+	0.833436i	$0.686370\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 455.989i	− 1.22908i
$372$	0	0
$373$	−670.407	−1.79734	−0.898669	−	0.438628i	$-0.855465\pi$
$373$	−670.407	−1.79734	−0.898669	+	0.438628i	$0.855465\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−50.2830	−0.133377
$378$	0	0
$379$	187.724i	0.495315i	0.968848	+	0.247658i	$0.0796608\pi$
$379$	187.724i	0.495315i	−0.968848	+	0.247658i	$0.920339\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	190.583i	0.497605i	0.968554	+	0.248803i	$0.0800370\pi$
$383$	190.583i	0.497605i	−0.968554	+	0.248803i	$0.919963\pi$
$384$	0	0
$385$	357.746	0.929211
$386$	0	0
$387$	0	0
$388$	0	0
$389$	584.136	1.50164	0.750818	−	0.660510i	$-0.229660\pi$
$389$	584.136	1.50164	0.750818	+	0.660510i	$0.229660\pi$
$390$	0	0
$391$	− 663.915i	− 1.69799i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 41.4724i	− 0.104993i
$396$	0	0
$397$	−571.136	−1.43863	−0.719315	−	0.694684i	$-0.755544\pi$
$397$	−571.136	−1.43863	−0.719315	+	0.694684i	$0.755544\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−56.0922	−0.139881	−0.0699404	−	0.997551i	$-0.522281\pi$
$401$	−56.0922	−0.139881	−0.0699404	+	0.997551i	$0.522281\pi$
$402$	0	0
$403$	171.582i	0.425763i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	294.572i	0.723763i
$408$	0	0
$409$	−136.122	−0.332816	−0.166408	−	0.986057i	$-0.553217\pi$
$409$	−136.122	−0.332816	−0.166408	+	0.986057i	$0.553217\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−164.083	−0.397295
$414$	0	0
$415$	156.053i	0.376032i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 235.462i	− 0.561962i	−0.959713	−	0.280981i	$-0.909340\pi$
$419$	− 235.462i	− 0.561962i	0.959713	−	0.280981i	$-0.0906598\pi$
$420$	0	0
$421$	−568.408	−1.35014	−0.675068	−	0.737755i	$-0.735886\pi$
$421$	−568.408	−1.35014	−0.675068	+	0.737755i	$0.735886\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−262.341	−0.617272
$426$	0	0
$427$	− 886.649i	− 2.07646i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	40.3202i	0.0935504i	0.998905	+	0.0467752i	$0.0148944\pi$
$431$	40.3202i	0.0935504i	−0.998905	+	0.0467752i	$0.985106\pi$
$432$	0	0
$433$	122.057	0.281887	0.140943	−	0.990018i	$-0.454986\pi$
$433$	122.057	0.281887	0.140943	+	0.990018i	$0.454986\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−132.850	−0.304005
$438$	0	0
$439$	701.020i	1.59686i	0.602090	+	0.798428i	$0.294335\pi$
$439$	701.020i	1.59686i	−0.602090	+	0.798428i	$0.705665\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	693.845i	1.56624i	0.621870	+	0.783120i	$0.286373\pi$
$443$	693.845i	1.56624i	−0.621870	+	0.783120i	$0.713627\pi$
$444$	0	0
$445$	−21.9976	−0.0494327
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−383.490	−0.854098	−0.427049	−	0.904228i	$-0.640447\pi$
$449$	−383.490	−0.854098	−0.427049	+	0.904228i	$0.640447\pi$
$450$	0	0
$451$	− 35.3962i	− 0.0784838i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 159.294i	− 0.350096i
$456$	0	0
$457$	−191.679	−0.419429	−0.209715	−	0.977763i	$-0.567253\pi$
$457$	−191.679	−0.419429	−0.209715	+	0.977763i	$0.567253\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	674.072	1.46220	0.731098	−	0.682272i	$-0.239008\pi$
$461$	674.072	1.46220	0.731098	+	0.682272i	$0.239008\pi$
$462$	0	0
$463$	− 439.695i	− 0.949666i	−0.880076	−	0.474833i	$-0.842509\pi$
$463$	− 439.695i	− 0.949666i	0.880076	−	0.474833i	$-0.157491\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 221.787i	− 0.474920i	−0.971397	−	0.237460i	$-0.923685\pi$
$467$	− 221.787i	− 0.474920i	0.971397	−	0.237460i	$-0.0763148\pi$
$468$	0	0
$469$	−276.066	−0.588627
$470$	0	0
$471$	0	0
$472$	0	0
$473$	332.981	0.703978
$474$	0	0
$475$	52.4947i	0.110515i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 226.990i	− 0.473882i	−0.971524	−	0.236941i	$-0.923855\pi$
$479$	− 226.990i	− 0.473882i	0.971524	−	0.236941i	$-0.0761449\pi$
$480$	0	0
$481$	131.164	0.272690
$482$	0	0
$483$	0	0
$484$	0	0
$485$	854.644	1.76215
$486$	0	0
$487$	− 801.415i	− 1.64562i	−0.568319	−	0.822808i	$-0.692406\pi$
$487$	− 801.415i	− 1.64562i	0.568319	−	0.822808i	$-0.307594\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	64.6780i	0.131727i	0.997829	+	0.0658635i	$0.0209802\pi$
$491$	64.6780i	0.131727i	−0.997829	+	0.0658635i	$0.979020\pi$
$492$	0	0
$493$	−370.448	−0.751415
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−658.177	−1.32430
$498$	0	0
$499$	− 183.365i	− 0.367465i	−0.982976	−	0.183732i	$-0.941182\pi$
$499$	− 183.365i	− 0.367465i	0.982976	−	0.183732i	$-0.0588180\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 587.722i	− 1.16843i	−0.811598	−	0.584217i	$-0.801402\pi$
$503$	− 587.722i	− 1.16843i	0.811598	−	0.584217i	$-0.198598\pi$
$504$	0	0
$505$	−434.545	−0.860485
$506$	0	0
$507$	0	0
$508$	0	0
$509$	152.232	0.299080	0.149540	−	0.988756i	$-0.452221\pi$
$509$	152.232	0.299080	0.149540	+	0.988756i	$0.452221\pi$
$510$	0	0
$511$	− 446.778i	− 0.874321i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 625.193i	− 1.21397i
$516$	0	0
$517$	491.680	0.951025
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−705.378	−1.35389	−0.676946	−	0.736032i	$-0.736697\pi$
$521$	−705.378	−1.35389	−0.676946	+	0.736032i	$0.736697\pi$
$522$	0	0
$523$	− 579.461i	− 1.10796i	−0.832531	−	0.553978i	$-0.813109\pi$
$523$	− 579.461i	− 1.10796i	0.832531	−	0.553978i	$-0.186891\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1264.09i	2.39866i
$528$	0	0
$529$	−399.902	−0.755959
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−15.7609	−0.0295701
$534$	0	0
$535$	395.166i	0.738629i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	194.906i	0.361607i
$540$	0	0
$541$	275.470	0.509187	0.254593	−	0.967048i	$-0.418058\pi$
$541$	275.470	0.509187	0.254593	+	0.967048i	$0.418058\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−703.405	−1.29065
$546$	0	0
$547$	− 920.704i	− 1.68319i	−0.540111	−	0.841594i	$-0.681618\pi$
$547$	− 920.704i	− 1.68319i	0.540111	−	0.841594i	$-0.318382\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	74.1269i	0.134532i
$552$	0	0
$553$	60.3154	0.109069
$554$	0	0
$555$	0	0
$556$	0	0
$557$	154.313	0.277043	0.138522	−	0.990359i	$-0.455765\pi$
$557$	154.313	0.277043	0.138522	+	0.990359i	$0.455765\pi$
$558$	0	0
$559$	− 148.267i	− 0.265235i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 317.068i	− 0.563176i	−0.959535	−	0.281588i	$-0.909139\pi$
$563$	− 317.068i	− 0.563176i	0.959535	−	0.281588i	$-0.0908611\pi$
$564$	0	0
$565$	1322.33	2.34040
$566$	0	0
$567$	0	0
$568$	0	0
$569$	420.374	0.738794	0.369397	−	0.929272i	$-0.379564\pi$
$569$	420.374	0.738794	0.369397	+	0.929272i	$0.379564\pi$
$570$	0	0
$571$	− 426.174i	− 0.746364i	−0.927758	−	0.373182i	$-0.878267\pi$
$571$	− 426.174i	− 0.746364i	0.927758	−	0.373182i	$-0.121733\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	367.048i	0.638345i
$576$	0	0
$577$	865.702	1.50035	0.750175	−	0.661239i	$-0.229969\pi$
$577$	865.702	1.50035	0.750175	+	0.661239i	$0.229969\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−226.956	−0.390630
$582$	0	0
$583$	342.082i	0.586761i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	327.570i	0.558041i	0.960285	+	0.279021i	$0.0900098\pi$
$587$	327.570i	0.558041i	−0.960285	+	0.279021i	$0.909990\pi$
$588$	0	0
$589$	252.946	0.429450
$590$	0	0
$591$	0	0
$592$	0	0
$593$	250.431	0.422312	0.211156	−	0.977452i	$-0.432277\pi$
$593$	250.431	0.422312	0.211156	+	0.977452i	$0.432277\pi$
$594$	0	0
$595$	− 1173.56i	− 1.97236i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	369.741i	0.617264i	0.951182	+	0.308632i	$0.0998711\pi$
$599$	369.741i	0.617264i	−0.951182	+	0.308632i	$0.900129\pi$
$600$	0	0
$601$	1109.33	1.84581	0.922903	−	0.385034i	$-0.125810\pi$
$601$	1109.33	1.84581	0.922903	+	0.385034i	$0.125810\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	468.063	0.773658
$606$	0	0
$607$	1103.84i	1.81852i	0.416228	+	0.909260i	$0.363352\pi$
$607$	1103.84i	1.81852i	−0.416228	+	0.909260i	$0.636648\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 218.930i	− 0.358315i
$612$	0	0
$613$	808.834	1.31947	0.659734	−	0.751499i	$-0.270669\pi$
$613$	808.834	1.31947	0.659734	+	0.751499i	$0.270669\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	681.933	1.10524	0.552620	−	0.833433i	$-0.313628\pi$
$617$	681.933	1.10524	0.552620	+	0.833433i	$0.313628\pi$
$618$	0	0
$619$	203.788i	0.329222i	0.986359	+	0.164611i	$0.0526368\pi$
$619$	203.788i	0.329222i	−0.986359	+	0.164611i	$0.947363\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 31.9922i	− 0.0513518i
$624$	0	0
$625$	−781.041	−1.24967
$626$	0	0
$627$	0	0
$628$	0	0
$629$	966.318	1.53628
$630$	0	0
$631$	51.4201i	0.0814899i	0.999170	+	0.0407449i	$0.0129731\pi$
$631$	51.4201i	0.0814899i	−0.999170	+	0.0407449i	$0.987027\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 105.311i	− 0.165844i
$636$	0	0
$637$	86.7858	0.136241
$638$	0	0
$639$	0	0
$640$	0	0
$641$	652.952	1.01865	0.509323	−	0.860576i	$-0.329896\pi$
$641$	652.952	1.01865	0.509323	+	0.860576i	$0.329896\pi$
$642$	0	0
$643$	− 620.192i	− 0.964529i	−0.876026	−	0.482264i	$-0.839814\pi$
$643$	− 620.192i	− 0.964529i	0.876026	−	0.482264i	$-0.160186\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	510.756i	0.789421i	0.918805	+	0.394711i	$0.129155\pi$
$647$	510.756i	0.789421i	−0.918805	+	0.394711i	$0.870845\pi$
$648$	0	0
$649$	123.094	0.189668
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1227.41	1.87964	0.939821	−	0.341666i	$-0.110991\pi$
$653$	1227.41	1.87964	0.939821	+	0.341666i	$0.110991\pi$
$654$	0	0
$655$	443.374i	0.676906i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	335.934i	0.509763i	0.966972	+	0.254881i	$0.0820364\pi$
$659$	335.934i	0.509763i	−0.966972	+	0.254881i	$0.917964\pi$
$660$	0	0
$661$	170.169	0.257442	0.128721	−	0.991681i	$-0.458913\pi$
$661$	170.169	0.257442	0.128721	+	0.991681i	$0.458913\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−234.830	−0.353128
$666$	0	0
$667$	518.304i	0.777067i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	665.161i	0.991298i
$672$	0	0
$673$	−738.836	−1.09782	−0.548912	−	0.835880i	$-0.684958\pi$
$673$	−738.836	−1.09782	−0.548912	+	0.835880i	$0.684958\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−443.165	−0.654601	−0.327300	−	0.944920i	$-0.606139\pi$
$677$	−443.165	−0.654601	−0.327300	+	0.944920i	$0.606139\pi$
$678$	0	0
$679$	1242.95i	1.83056i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	335.857i	0.491738i	0.969303	+	0.245869i	$0.0790734\pi$
$683$	335.857i	0.491738i	−0.969303	+	0.245869i	$0.920927\pi$
$684$	0	0
$685$	−1012.61	−1.47826
$686$	0	0
$687$	0	0
$688$	0	0
$689$	152.319	0.221072
$690$	0	0
$691$	− 818.494i	− 1.18451i	−0.805752	−	0.592254i	$-0.798238\pi$
$691$	− 818.494i	− 1.18451i	0.805752	−	0.592254i	$-0.201762\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1199.66i	− 1.72613i
$696$	0	0
$697$	−116.114	−0.166592
$698$	0	0
$699$	0	0
$700$	0	0
$701$	601.994	0.858765	0.429383	−	0.903123i	$-0.358731\pi$
$701$	601.994	0.858765	0.429383	+	0.903123i	$0.358731\pi$
$702$	0	0
$703$	− 193.361i	− 0.275052i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 631.981i	− 0.893891i
$708$	0	0
$709$	720.442	1.01614	0.508069	−	0.861316i	$-0.330359\pi$
$709$	720.442	1.01614	0.508069	+	0.861316i	$0.330359\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1768.63	2.48054
$714$	0	0
$715$	119.502i	0.167135i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1106.86i	1.53944i	0.638379	+	0.769722i	$0.279605\pi$
$719$	1106.86i	1.53944i	−0.638379	+	0.769722i	$0.720395\pi$
$720$	0	0
$721$	909.251	1.26110
$722$	0	0
$723$	0	0
$724$	0	0
$725$	204.804	0.282488
$726$	0	0
$727$	− 108.860i	− 0.149739i	−0.997193	−	0.0748696i	$-0.976146\pi$
$727$	− 108.860i	− 0.149739i	0.997193	−	0.0748696i	$-0.0238541\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1092.32i	− 1.49428i
$732$	0	0
$733$	−1251.06	−1.70677	−0.853385	−	0.521280i	$-0.825455\pi$
$733$	−1251.06	−1.70677	−0.853385	+	0.521280i	$0.825455\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	207.104	0.281009
$738$	0	0
$739$	246.376i	0.333391i	0.986008	+	0.166696i	$0.0533098\pi$
$739$	246.376i	0.333391i	−0.986008	+	0.166696i	$0.946690\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	160.506i	0.216024i	0.994150	+	0.108012i	$0.0344486\pi$
$743$	160.506i	0.216024i	−0.994150	+	0.108012i	$0.965551\pi$
$744$	0	0
$745$	−1464.77	−1.96614
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−574.711	−0.767304
$750$	0	0
$751$	− 774.408i	− 1.03117i	−0.856839	−	0.515584i	$-0.827575\pi$
$751$	− 774.408i	− 1.03117i	0.856839	−	0.515584i	$-0.172425\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 753.012i	− 0.997367i
$756$	0	0
$757$	−159.858	−0.211174	−0.105587	−	0.994410i	$-0.533672\pi$
$757$	−159.858	−0.211174	−0.105587	+	0.994410i	$0.533672\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−66.4269	−0.0872889	−0.0436445	−	0.999047i	$-0.513897\pi$
$761$	−66.4269	−0.0872889	−0.0436445	+	0.999047i	$0.513897\pi$
$762$	0	0
$763$	− 1023.00i	− 1.34076i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 54.8102i	− 0.0714605i
$768$	0	0
$769$	1469.57	1.91102	0.955509	−	0.294960i	$-0.0953064\pi$
$769$	1469.57	1.91102	0.955509	+	0.294960i	$0.0953064\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−255.482	−0.330508	−0.165254	−	0.986251i	$-0.552844\pi$
$773$	−255.482	−0.330508	−0.165254	+	0.986251i	$0.552844\pi$
$774$	0	0
$775$	− 698.859i	− 0.901754i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	23.2346i	0.0298262i
$780$	0	0
$781$	493.762	0.632218
$782$	0	0
$783$	0	0
$784$	0	0
$785$	693.915	0.883968
$786$	0	0
$787$	− 640.558i	− 0.813923i	−0.913445	−	0.406962i	$-0.866588\pi$
$787$	− 640.558i	− 0.813923i	0.913445	−	0.406962i	$-0.133412\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1923.13i	2.43126i
$792$	0	0
$793$	296.176	0.373488
$794$	0	0
$795$	0	0
$796$	0	0
$797$	780.994	0.979917	0.489959	−	0.871746i	$-0.337012\pi$
$797$	780.994	0.979917	0.489959	+	0.871746i	$0.337012\pi$
$798$	0	0
$799$	− 1612.92i	− 2.01867i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	335.171i	0.417399i
$804$	0	0
$805$	−1641.96	−2.03970
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1091.04	1.34862	0.674312	−	0.738446i	$-0.264440\pi$
$809$	1091.04	1.34862	0.674312	+	0.738446i	$0.264440\pi$
$810$	0	0
$811$	− 587.805i	− 0.724790i	−0.932025	−	0.362395i	$-0.881959\pi$
$811$	− 587.805i	− 0.724790i	0.932025	−	0.362395i	$-0.118041\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1385.76i	− 1.70032i
$816$	0	0
$817$	−218.574	−0.267532
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−397.797	−0.484527	−0.242264	−	0.970210i	$-0.577890\pi$
$821$	−397.797	−0.484527	−0.242264	+	0.970210i	$0.577890\pi$
$822$	0	0
$823$	772.958i	0.939195i	0.882880	+	0.469598i	$0.155601\pi$
$823$	772.958i	0.939195i	−0.882880	+	0.469598i	$0.844399\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 604.655i	− 0.731143i	−0.930783	−	0.365572i	$-0.880874\pi$
$827$	− 604.655i	− 0.731143i	0.930783	−	0.365572i	$-0.119126\pi$
$828$	0	0
$829$	391.614	0.472393	0.236197	−	0.971705i	$-0.424099\pi$
$829$	391.614	0.472393	0.236197	+	0.971705i	$0.424099\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	639.373	0.767555
$834$	0	0
$835$	584.589i	0.700106i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1446.75i	1.72437i	0.506590	+	0.862187i	$0.330906\pi$
$839$	1446.75i	1.72437i	−0.506590	+	0.862187i	$0.669094\pi$
$840$	0	0
$841$	−551.800	−0.656123
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−975.375	−1.15429
$846$	0	0
$847$	680.728i	0.803693i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1352.00i	− 1.58872i
$852$	0	0
$853$	77.7668	0.0911686	0.0455843	−	0.998960i	$-0.485485\pi$
$853$	77.7668	0.0911686	0.0455843	+	0.998960i	$0.485485\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1213.91	1.41646	0.708230	−	0.705981i	$-0.249494\pi$
$857$	1213.91	1.41646	0.708230	+	0.705981i	$0.249494\pi$
$858$	0	0
$859$	1427.11i	1.66136i	0.556751	+	0.830679i	$0.312048\pi$
$859$	1427.11i	1.66136i	−0.556751	+	0.830679i	$0.687952\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1371.18i	1.58885i	0.607362	+	0.794425i	$0.292228\pi$
$863$	1371.18i	1.58885i	−0.607362	+	0.794425i	$0.707772\pi$
$864$	0	0
$865$	1688.27	1.95175
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−45.2484	−0.0520695
$870$	0	0
$871$	− 92.2171i	− 0.105875i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 698.036i	− 0.797755i
$876$	0	0
$877$	1329.39	1.51584	0.757920	−	0.652348i	$-0.226216\pi$
$877$	1329.39	1.51584	0.757920	+	0.652348i	$0.226216\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	887.373	1.00723	0.503617	−	0.863927i	$-0.332002\pi$
$881$	887.373	1.00723	0.503617	+	0.863927i	$0.332002\pi$
$882$	0	0
$883$	− 1238.75i	− 1.40288i	−0.712726	−	0.701442i	$-0.752540\pi$
$883$	− 1238.75i	− 1.40288i	0.712726	−	0.701442i	$-0.247460\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 717.716i	− 0.809150i	−0.914505	−	0.404575i	$-0.867419\pi$
$887$	− 717.716i	− 0.809150i	0.914505	−	0.404575i	$-0.132581\pi$
$888$	0	0
$889$	153.159	0.172282
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−322.746	−0.361418
$894$	0	0
$895$	630.355i	0.704307i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 986.848i	− 1.09772i
$900$	0	0
$901$	1122.17	1.24547
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−770.717	−0.851621
$906$	0	0
$907$	484.018i	0.533647i	0.963745	+	0.266824i	$0.0859741\pi$
$907$	484.018i	0.533647i	−0.963745	+	0.266824i	$0.914026\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1391.42i	− 1.52736i	−0.645595	−	0.763680i	$-0.723391\pi$
$911$	− 1391.42i	− 1.52736i	0.645595	−	0.763680i	$-0.276609\pi$
$912$	0	0
$913$	170.262	0.186486
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−644.821	−0.703186
$918$	0	0
$919$	50.4361i	0.0548815i	0.999623	+	0.0274407i	$0.00873576\pi$
$919$	50.4361i	0.0548815i	−0.999623	+	0.0274407i	$0.991264\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 219.857i	− 0.238199i
$924$	0	0
$925$	−534.233	−0.577550
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−783.700	−0.843595	−0.421798	−	0.906690i	$-0.638601\pi$
$929$	−783.700	−0.843595	−0.421798	+	0.906690i	$0.638601\pi$
$930$	0	0
$931$	− 127.939i	− 0.137421i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	880.398i	0.941603i
$936$	0	0
$937$	−50.8290	−0.0542465	−0.0271233	−	0.999632i	$-0.508635\pi$
$937$	−50.8290	−0.0542465	−0.0271233	+	0.999632i	$0.508635\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1572.18	−1.67076	−0.835380	−	0.549673i	$-0.814752\pi$
$941$	−1572.18	−1.67076	−0.835380	+	0.549673i	$0.814752\pi$
$942$	0	0
$943$	162.459i	0.172279i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 81.5277i	− 0.0860905i	−0.999073	−	0.0430453i	$-0.986294\pi$
$947$	− 81.5277i	− 0.0860905i	0.999073	−	0.0430453i	$-0.0137060\pi$
$948$	0	0
$949$	149.242	0.157262
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1153.28	1.21016	0.605079	−	0.796165i	$-0.293141\pi$
$953$	1153.28	1.21016	0.605079	+	0.796165i	$0.293141\pi$
$954$	0	0
$955$	720.736i	0.754698i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1472.69i	− 1.53565i
$960$	0	0
$961$	−2406.46	−2.50412
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1092.91	−1.13255
$966$	0	0
$967$	889.643i	0.920003i	0.887918	+	0.460001i	$0.152151\pi$
$967$	889.643i	0.920003i	−0.887918	+	0.460001i	$0.847849\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1664.80i	− 1.71452i	−0.514882	−	0.857261i	$-0.672164\pi$
$971$	− 1664.80i	− 1.71452i	0.514882	−	0.857261i	$-0.327836\pi$
$972$	0	0
$973$	1744.73	1.79315
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1362.00	1.39406	0.697030	−	0.717042i	$-0.254504\pi$
$977$	1362.00	1.39406	0.697030	+	0.717042i	$0.254504\pi$
$978$	0	0
$979$	24.0004i	0.0245152i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1173.24i	1.19353i	0.802415	+	0.596766i	$0.203548\pi$
$983$	1173.24i	1.19353i	−0.802415	+	0.596766i	$0.796452\pi$
$984$	0	0
$985$	−1131.96	−1.14920
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1528.29	−1.54529
$990$	0	0
$991$	− 797.134i	− 0.804373i	−0.915558	−	0.402186i	$-0.868250\pi$
$991$	− 797.134i	− 0.804373i	0.915558	−	0.402186i	$-0.131750\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	72.5797i	0.0729444i
$996$	0	0
$997$	−1534.22	−1.53884	−0.769419	−	0.638744i	$-0.779454\pi$
$997$	−1534.22	−1.53884	−0.769419	+	0.638744i	$0.779454\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2736.3.m.e.1711.10		12
3.2	odd	2		912.3.m.b.799.2	✓	12
4.3	odd	2	inner	2736.3.m.e.1711.9		12
12.11	even	2		912.3.m.b.799.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
912.3.m.b.799.2	✓	12	3.2	odd	2
912.3.m.b.799.8	yes	12	12.11	even	2
2736.3.m.e.1711.9		12	4.3	odd	2	inner
2736.3.m.e.1711.10		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2736.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.08630 q^{5} +8.85163i q^{7} +O(q^{10})\)	\(q+6.08630 q^{5} +8.85163i q^{7} -6.64046i q^{11} -2.95680 q^{13} -21.7835 q^{17} +4.35890i q^{19} +30.4779i q^{23} +12.0431 q^{25} +17.0059 q^{29} -58.0298i q^{31} +53.8737i q^{35} -44.3601 q^{37} +5.33039 q^{41} +50.1443i q^{43} +74.0431i q^{47} -29.3513 q^{49} -51.5148 q^{53} -40.4159i q^{55} +18.5370i q^{59} -100.168 q^{61} -17.9960 q^{65} +31.1882i q^{67} +74.3566i q^{71} -50.4741 q^{73} +58.7789 q^{77} -6.81405i q^{79} +25.6400i q^{83} -132.581 q^{85} -3.61427 q^{89} -26.1725i q^{91} +26.5296i q^{95} +140.421 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 10 q^{5}+O(q^{10}) \) 12 * q + 10 * q^5	\( 12 q + 10 q^{5} + 36 q^{13} - 14 q^{17} + 10 q^{25} + 108 q^{29} - 16 q^{37} - 16 q^{41} - 118 q^{49} - 220 q^{53} + 366 q^{61} - 140 q^{65} - 158 q^{73} + 286 q^{77} + 98 q^{85} + 396 q^{89} + 224 q^{97}+O(q^{100}) \) 12 * q + 10 * q^5 + 36 * q^13 - 14 * q^17 + 10 * q^25 + 108 * q^29 - 16 * q^37 - 16 * q^41 - 118 * q^49 - 220 * q^53 + 366 * q^61 - 140 * q^65 - 158 * q^73 + 286 * q^77 + 98 * q^85 + 396 * q^89 + 224 * q^97

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