LMFDB - Embedded newform 2916.3.c.b.1457.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2916,3,Mod(1457,2916)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2916, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2916.1457");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2916 = 2^{2} \cdot 3^{6} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2916.c (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:	$79.4552450875$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1457.16
Character	$\chi$	$=$	2916.1457
Dual form			2916.3.c.b.1457.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.52502i q^{5} -3.44709 q^{7} +O(q^{10})$	$q-1.52502i q^{5} -3.44709 q^{7} -4.99580i q^{11} -4.48502 q^{13} +4.12519i q^{17} -13.5127 q^{19} +16.9547i q^{23} +22.6743 q^{25} +47.8469i q^{29} +15.2644 q^{31} +5.25687i q^{35} +64.7672 q^{37} -56.2567i q^{41} +29.6664 q^{43} -20.5911i q^{47} -37.1176 q^{49} +19.8596i q^{53} -7.61868 q^{55} -99.0845i q^{59} -82.5257 q^{61} +6.83973i q^{65} +64.7834 q^{67} -126.761i q^{71} -36.0692 q^{73} +17.2210i q^{77} -83.3856 q^{79} +16.8059i q^{83} +6.29098 q^{85} +76.2593i q^{89} +15.4602 q^{91} +20.6072i q^{95} +107.254 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q+O(q^{10}) $ 36 * q	$ 36 q - 180 q^{25} + 252 q^{49} + 18 q^{61} - 90 q^{67} + 126 q^{73} - 198 q^{91}+O(q^{100}) $ 36 * q - 180 * q^25 + 252 * q^49 + 18 * q^61 - 90 * q^67 + 126 * q^73 - 198 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2916\mathbb{Z}\right)^\times$.

$n$	$1459$	$2189$
$\chi(n)$	$1$	$-1$

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
$5$	− 1.52502i	− 0.305003i	−0.988303	−	0.152502i	$-0.951267\pi$
$5$	− 1.52502i	− 0.305003i	0.988303	−	0.152502i	$-0.0487330\pi$
$6$	0	0
$7$	−3.44709	−0.492441	−0.246220	−	0.969214i	$-0.579189\pi$
$7$	−3.44709	−0.492441	−0.246220	+	0.969214i	$0.579189\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 4.99580i	− 0.454164i	−0.973876	−	0.227082i	$-0.927082\pi$
$11$	− 4.99580i	− 0.454164i	0.973876	−	0.227082i	$-0.0729185\pi$
$12$	0	0
$13$	−4.48502	−0.345001	−0.172501	−	0.985009i	$-0.555185\pi$
$13$	−4.48502	−0.345001	−0.172501	+	0.985009i	$0.555185\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.12519i	0.242658i	0.992612	+	0.121329i	$0.0387156\pi$
$17$	4.12519i	0.242658i	−0.992612	+	0.121329i	$0.961284\pi$
$18$	0	0
$19$	−13.5127	−0.711197	−0.355599	−	0.934639i	$-0.615723\pi$
$19$	−13.5127	−0.711197	−0.355599	+	0.934639i	$0.615723\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	16.9547i	0.737161i	0.929596	+	0.368580i	$0.120156\pi$
$23$	16.9547i	0.737161i	−0.929596	+	0.368580i	$0.879844\pi$
$24$	0	0
$25$	22.6743	0.906973
$26$	0	0
$27$	0	0
$28$	0	0
$29$	47.8469i	1.64989i	0.565210	+	0.824947i	$0.308795\pi$
$29$	47.8469i	1.64989i	−0.565210	+	0.824947i	$0.691205\pi$
$30$	0	0
$31$	15.2644	0.492402	0.246201	−	0.969219i	$-0.420818\pi$
$31$	15.2644	0.492402	0.246201	+	0.969219i	$0.420818\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.25687i	0.150196i
$36$	0	0
$37$	64.7672	1.75047	0.875233	−	0.483702i	$-0.160708\pi$
$37$	64.7672	1.75047	0.875233	+	0.483702i	$0.160708\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 56.2567i	− 1.37211i	−0.727548	−	0.686057i	$-0.759340\pi$
$41$	− 56.2567i	− 1.37211i	0.727548	−	0.686057i	$-0.240660\pi$
$42$	0	0
$43$	29.6664	0.689916	0.344958	−	0.938618i	$-0.387893\pi$
$43$	29.6664	0.689916	0.344958	+	0.938618i	$0.387893\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 20.5911i	− 0.438107i	−0.975713	−	0.219054i	$-0.929703\pi$
$47$	− 20.5911i	− 0.438107i	0.975713	−	0.219054i	$-0.0702970\pi$
$48$	0	0
$49$	−37.1176	−0.757502
$50$	0	0
$51$	0	0
$52$	0	0
$53$	19.8596i	0.374710i	0.982292	+	0.187355i	$0.0599915\pi$
$53$	19.8596i	0.374710i	−0.982292	+	0.187355i	$0.940009\pi$
$54$	0	0
$55$	−7.61868	−0.138522
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 99.0845i	− 1.67940i	−0.543052	−	0.839699i	$-0.682731\pi$
$59$	− 99.0845i	− 1.67940i	0.543052	−	0.839699i	$-0.317269\pi$
$60$	0	0
$61$	−82.5257	−1.35288	−0.676440	−	0.736497i	$-0.736478\pi$
$61$	−82.5257	−1.35288	−0.676440	+	0.736497i	$0.736478\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.83973i	0.105227i
$66$	0	0
$67$	64.7834	0.966916	0.483458	−	0.875368i	$-0.339381\pi$
$67$	64.7834	0.966916	0.483458	+	0.875368i	$0.339381\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 126.761i	− 1.78536i	−0.450692	−	0.892680i	$-0.648823\pi$
$71$	− 126.761i	− 1.78536i	0.450692	−	0.892680i	$-0.351177\pi$
$72$	0	0
$73$	−36.0692	−0.494099	−0.247049	−	0.969003i	$-0.579461\pi$
$73$	−36.0692	−0.494099	−0.247049	+	0.969003i	$0.579461\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	17.2210i	0.223649i
$78$	0	0
$79$	−83.3856	−1.05551	−0.527757	−	0.849395i	$-0.676967\pi$
$79$	−83.3856	−1.05551	−0.527757	+	0.849395i	$0.676967\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	16.8059i	0.202480i	0.994862	+	0.101240i	$0.0322811\pi$
$83$	16.8059i	0.202480i	−0.994862	+	0.101240i	$0.967719\pi$
$84$	0	0
$85$	6.29098	0.0740116
$86$	0	0
$87$	0	0
$88$	0	0
$89$	76.2593i	0.856846i	0.903578	+	0.428423i	$0.140931\pi$
$89$	76.2593i	0.856846i	−0.903578	+	0.428423i	$0.859069\pi$
$90$	0	0
$91$	15.4602	0.169893
$92$	0	0
$93$	0	0
$94$	0	0
$95$	20.6072i	0.216918i
$96$	0	0
$97$	107.254	1.10571	0.552855	−	0.833277i	$-0.313538\pi$
$97$	107.254	1.10571	0.552855	+	0.833277i	$0.313538\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 112.916i	− 1.11798i	−0.829174	−	0.558991i	$-0.811189\pi$
$101$	− 112.916i	− 1.11798i	0.829174	−	0.558991i	$-0.188811\pi$
$102$	0	0
$103$	−144.890	−1.40670	−0.703349	−	0.710845i	$-0.748313\pi$
$103$	−144.890	−1.40670	−0.703349	+	0.710845i	$0.748313\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	90.9102i	0.849628i	0.905281	+	0.424814i	$0.139661\pi$
$107$	90.9102i	0.849628i	−0.905281	+	0.424814i	$0.860339\pi$
$108$	0	0
$109$	−44.2294	−0.405774	−0.202887	−	0.979202i	$-0.565032\pi$
$109$	−44.2294	−0.405774	−0.202887	+	0.979202i	$0.565032\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 143.711i	− 1.27178i	−0.771780	−	0.635890i	$-0.780633\pi$
$113$	− 143.711i	− 1.27178i	0.771780	−	0.635890i	$-0.219367\pi$
$114$	0	0
$115$	25.8562	0.224837
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 14.2199i	− 0.119495i
$120$	0	0
$121$	96.0420	0.793735
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 72.7042i	− 0.581633i
$126$	0	0
$127$	37.6692	0.296608	0.148304	−	0.988942i	$-0.452619\pi$
$127$	37.6692	0.296608	0.148304	+	0.988942i	$0.452619\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 153.197i	− 1.16944i	−0.811234	−	0.584722i	$-0.801204\pi$
$131$	− 153.197i	− 1.16944i	0.811234	−	0.584722i	$-0.198796\pi$
$132$	0	0
$133$	46.5796	0.350223
$134$	0	0
$135$	0	0
$136$	0	0
$137$	59.3686i	0.433347i	0.976244	+	0.216674i	$0.0695207\pi$
$137$	59.3686i	0.433347i	−0.976244	+	0.216674i	$0.930479\pi$
$138$	0	0
$139$	60.0165	0.431773	0.215887	−	0.976418i	$-0.430736\pi$
$139$	60.0165	0.431773	0.215887	+	0.976418i	$0.430736\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	22.4063i	0.156687i
$144$	0	0
$145$	72.9674	0.503223
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 268.791i	− 1.80397i	−0.431771	−	0.901983i	$-0.642111\pi$
$149$	− 268.791i	− 1.80397i	0.431771	−	0.901983i	$-0.357889\pi$
$150$	0	0
$151$	12.2072	0.0808424	0.0404212	−	0.999183i	$-0.487130\pi$
$151$	12.2072	0.0808424	0.0404212	+	0.999183i	$0.487130\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 23.2786i	− 0.150184i
$156$	0	0
$157$	−45.1623	−0.287658	−0.143829	−	0.989603i	$-0.545941\pi$
$157$	−45.1623	−0.287658	−0.143829	+	0.989603i	$0.545941\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 58.4443i	− 0.363008i
$162$	0	0
$163$	5.88721	0.0361178	0.0180589	−	0.999837i	$-0.494251\pi$
$163$	5.88721	0.0361178	0.0180589	+	0.999837i	$0.494251\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 222.730i	− 1.33371i	−0.745186	−	0.666857i	$-0.767639\pi$
$167$	− 222.730i	− 1.33371i	0.745186	−	0.666857i	$-0.232361\pi$
$168$	0	0
$169$	−148.885	−0.880974
$170$	0	0
$171$	0	0
$172$	0	0
$173$	25.6607i	0.148328i	0.997246	+	0.0741640i	$0.0236288\pi$
$173$	25.6607i	0.148328i	−0.997246	+	0.0741640i	$0.976371\pi$
$174$	0	0
$175$	−78.1603	−0.446631
$176$	0	0
$177$	0	0
$178$	0	0
$179$	78.2689i	0.437256i	0.975808	+	0.218628i	$0.0701582\pi$
$179$	78.2689i	0.437256i	−0.975808	+	0.218628i	$0.929842\pi$
$180$	0	0
$181$	−144.327	−0.797386	−0.398693	−	0.917084i	$-0.630536\pi$
$181$	−144.327	−0.797386	−0.398693	+	0.917084i	$0.630536\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 98.7712i	− 0.533898i
$186$	0	0
$187$	20.6086	0.110207
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 111.341i	− 0.582937i	−0.956580	−	0.291468i	$-0.905856\pi$
$191$	− 111.341i	− 0.582937i	0.956580	−	0.291468i	$-0.0941438\pi$
$192$	0	0
$193$	−96.7390	−0.501238	−0.250619	−	0.968086i	$-0.580634\pi$
$193$	−96.7390	−0.501238	−0.250619	+	0.968086i	$0.580634\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.8923i	0.0705193i	0.999378	+	0.0352596i	$0.0112258\pi$
$197$	13.8923i	0.0705193i	−0.999378	+	0.0352596i	$0.988774\pi$
$198$	0	0
$199$	−155.549	−0.781654	−0.390827	−	0.920464i	$-0.627811\pi$
$199$	−155.549	−0.781654	−0.390827	+	0.920464i	$0.627811\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 164.933i	− 0.812475i
$204$	0	0
$205$	−85.7924	−0.418500
$206$	0	0
$207$	0	0
$208$	0	0
$209$	67.5070i	0.323000i
$210$	0	0
$211$	372.585	1.76581	0.882904	−	0.469554i	$-0.155585\pi$
$211$	372.585	1.76581	0.882904	+	0.469554i	$0.155585\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 45.2418i	− 0.210427i
$216$	0	0
$217$	−52.6179	−0.242479
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 18.5015i	− 0.0837174i
$222$	0	0
$223$	272.454	1.22177	0.610884	−	0.791720i	$-0.290814\pi$
$223$	272.454	1.22177	0.610884	+	0.791720i	$0.290814\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 289.896i	− 1.27707i	−0.769591	−	0.638537i	$-0.779540\pi$
$227$	− 289.896i	− 1.27707i	0.769591	−	0.638537i	$-0.220460\pi$
$228$	0	0
$229$	−143.351	−0.625988	−0.312994	−	0.949755i	$-0.601332\pi$
$229$	−143.351	−0.625988	−0.312994	+	0.949755i	$0.601332\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	281.846i	1.20964i	0.796363	+	0.604819i	$0.206755\pi$
$233$	281.846i	1.20964i	−0.796363	+	0.604819i	$0.793245\pi$
$234$	0	0
$235$	−31.4017	−0.133624
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 387.319i	− 1.62058i	−0.586029	−	0.810290i	$-0.699309\pi$
$239$	− 387.319i	− 1.62058i	0.586029	−	0.810290i	$-0.300691\pi$
$240$	0	0
$241$	191.297	0.793764	0.396882	−	0.917870i	$-0.370092\pi$
$241$	191.297	0.793764	0.396882	+	0.917870i	$0.370092\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	56.6050i	0.231041i
$246$	0	0
$247$	60.6049	0.245364
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 435.813i	− 1.73631i	−0.496297	−	0.868153i	$-0.665307\pi$
$251$	− 435.813i	− 1.73631i	0.496297	−	0.868153i	$-0.334693\pi$
$252$	0	0
$253$	84.7023	0.334792
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 418.861i	− 1.62981i	−0.579594	−	0.814906i	$-0.696789\pi$
$257$	− 418.861i	− 1.62981i	0.579594	−	0.814906i	$-0.303211\pi$
$258$	0	0
$259$	−223.258	−0.862001
$260$	0	0
$261$	0	0
$262$	0	0
$263$	330.839i	1.25794i	0.777428	+	0.628972i	$0.216524\pi$
$263$	330.839i	1.25794i	−0.777428	+	0.628972i	$0.783476\pi$
$264$	0	0
$265$	30.2863	0.114288
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 136.569i	− 0.507690i	−0.967245	−	0.253845i	$-0.918305\pi$
$269$	− 136.569i	− 0.507690i	0.967245	−	0.253845i	$-0.0816953\pi$
$270$	0	0
$271$	170.355	0.628615	0.314307	−	0.949321i	$-0.398228\pi$
$271$	170.355	0.628615	0.314307	+	0.949321i	$0.398228\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 113.276i	− 0.411914i
$276$	0	0
$277$	133.668	0.482557	0.241279	−	0.970456i	$-0.422433\pi$
$277$	133.668	0.482557	0.241279	+	0.970456i	$0.422433\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	454.024i	1.61574i	0.589358	+	0.807872i	$0.299381\pi$
$281$	454.024i	1.61574i	−0.589358	+	0.807872i	$0.700619\pi$
$282$	0	0
$283$	−389.539	−1.37646	−0.688232	−	0.725491i	$-0.741613\pi$
$283$	−389.539	−1.37646	−0.688232	+	0.725491i	$0.741613\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	193.922i	0.675685i
$288$	0	0
$289$	271.983	0.941117
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 340.058i	− 1.16061i	−0.814401	−	0.580303i	$-0.802934\pi$
$293$	− 340.058i	− 1.16061i	0.814401	−	0.580303i	$-0.197066\pi$
$294$	0	0
$295$	−151.106	−0.512222
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 76.0422i	− 0.254322i
$300$	0	0
$301$	−102.263	−0.339743
$302$	0	0
$303$	0	0
$304$	0	0
$305$	125.853i	0.412633i
$306$	0	0
$307$	−465.839	−1.51739	−0.758695	−	0.651446i	$-0.774163\pi$
$307$	−465.839	−1.51739	−0.758695	+	0.651446i	$0.774163\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	83.2808i	0.267784i	0.990996	+	0.133892i	$0.0427475\pi$
$311$	83.2808i	0.267784i	−0.990996	+	0.133892i	$0.957252\pi$
$312$	0	0
$313$	−278.391	−0.889430	−0.444715	−	0.895672i	$-0.646695\pi$
$313$	−278.391	−0.889430	−0.444715	+	0.895672i	$0.646695\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	165.540i	0.522209i	0.965311	+	0.261104i	$0.0840866\pi$
$317$	165.540i	0.522209i	−0.965311	+	0.261104i	$0.915913\pi$
$318$	0	0
$319$	239.034	0.749322
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 55.7426i	− 0.172578i
$324$	0	0
$325$	−101.695	−0.312907
$326$	0	0
$327$	0	0
$328$	0	0
$329$	70.9791i	0.215742i
$330$	0	0
$331$	11.4660	0.0346405	0.0173203	−	0.999850i	$-0.494487\pi$
$331$	11.4660	0.0346405	0.0173203	+	0.999850i	$0.494487\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 98.7958i	− 0.294913i
$336$	0	0
$337$	404.137	1.19922	0.599610	−	0.800292i	$-0.295322\pi$
$337$	404.137	1.19922	0.599610	+	0.800292i	$0.295322\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 76.2582i	− 0.223631i
$342$	0	0
$343$	296.855	0.865466
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 533.716i	− 1.53809i	−0.639196	−	0.769044i	$-0.720733\pi$
$347$	− 533.716i	− 1.53809i	0.639196	−	0.769044i	$-0.279267\pi$
$348$	0	0
$349$	−233.788	−0.669881	−0.334940	−	0.942239i	$-0.608716\pi$
$349$	−233.788	−0.669881	−0.334940	+	0.942239i	$0.608716\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 494.164i	− 1.39990i	−0.714193	−	0.699949i	$-0.753206\pi$
$353$	− 494.164i	− 1.39990i	0.714193	−	0.699949i	$-0.246794\pi$
$354$	0	0
$355$	−193.312	−0.544541
$356$	0	0
$357$	0	0
$358$	0	0
$359$	53.2393i	0.148299i	0.997247	+	0.0741494i	$0.0236242\pi$
$359$	53.2393i	0.148299i	−0.997247	+	0.0741494i	$0.976376\pi$
$360$	0	0
$361$	−178.406	−0.494198
$362$	0	0
$363$	0	0
$364$	0	0
$365$	55.0062i	0.150702i
$366$	0	0
$367$	−67.4089	−0.183676	−0.0918378	−	0.995774i	$-0.529274\pi$
$367$	−67.4089	−0.183676	−0.0918378	+	0.995774i	$0.529274\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 68.4579i	− 0.184523i
$372$	0	0
$373$	−180.582	−0.484134	−0.242067	−	0.970260i	$-0.577825\pi$
$373$	−180.582	−0.484134	−0.242067	+	0.970260i	$0.577825\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 214.594i	− 0.569216i
$378$	0	0
$379$	609.218	1.60743	0.803717	−	0.595011i	$-0.202852\pi$
$379$	609.218	1.60743	0.803717	+	0.595011i	$0.202852\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 343.791i	− 0.897626i	−0.893626	−	0.448813i	$-0.851847\pi$
$383$	− 343.791i	− 0.897626i	0.893626	−	0.448813i	$-0.148153\pi$
$384$	0	0
$385$	26.2623	0.0682137
$386$	0	0
$387$	0	0
$388$	0	0
$389$	488.684i	1.25626i	0.778109	+	0.628129i	$0.216179\pi$
$389$	488.684i	1.25626i	−0.778109	+	0.628129i	$0.783821\pi$
$390$	0	0
$391$	−69.9413	−0.178878
$392$	0	0
$393$	0	0
$394$	0	0
$395$	127.164i	0.321935i
$396$	0	0
$397$	370.209	0.932515	0.466258	−	0.884649i	$-0.345602\pi$
$397$	370.209	0.932515	0.466258	+	0.884649i	$0.345602\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 534.514i	− 1.33295i	−0.745526	−	0.666476i	$-0.767802\pi$
$401$	− 534.514i	− 1.33295i	0.745526	−	0.666476i	$-0.232198\pi$
$402$	0	0
$403$	−68.4613	−0.169879
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 323.564i	− 0.794998i
$408$	0	0
$409$	−146.558	−0.358333	−0.179166	−	0.983819i	$-0.557340\pi$
$409$	−146.558	−0.358333	−0.179166	+	0.983819i	$0.557340\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	341.553i	0.827004i
$414$	0	0
$415$	25.6293	0.0617572
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 200.186i	− 0.477771i	−0.971048	−	0.238885i	$-0.923218\pi$
$419$	− 200.186i	− 0.477771i	0.971048	−	0.238885i	$-0.0767820\pi$
$420$	0	0
$421$	441.208	1.04800	0.524000	−	0.851718i	$-0.324439\pi$
$421$	441.208	1.04800	0.524000	+	0.851718i	$0.324439\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	93.5359i	0.220084i
$426$	0	0
$427$	284.473	0.666214
$428$	0	0
$429$	0	0
$430$	0	0
$431$	474.920i	1.10190i	0.834537	+	0.550951i	$0.185735\pi$
$431$	474.920i	1.10190i	−0.834537	+	0.550951i	$0.814265\pi$
$432$	0	0
$433$	−540.928	−1.24926	−0.624628	−	0.780922i	$-0.714749\pi$
$433$	−540.928	−1.24926	−0.624628	+	0.780922i	$0.714749\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 229.105i	− 0.524267i
$438$	0	0
$439$	655.377	1.49289	0.746443	−	0.665449i	$-0.231760\pi$
$439$	655.377	1.49289	0.746443	+	0.665449i	$0.231760\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 717.516i	− 1.61967i	−0.586654	−	0.809837i	$-0.699555\pi$
$443$	− 717.516i	− 1.61967i	0.586654	−	0.809837i	$-0.300445\pi$
$444$	0	0
$445$	116.297	0.261341
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 136.567i	− 0.304157i	−0.988368	−	0.152079i	$-0.951403\pi$
$449$	− 136.567i	− 0.304157i	0.988368	−	0.152079i	$-0.0485967\pi$
$450$	0	0
$451$	−281.047	−0.623165
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 23.5771i	− 0.0518179i
$456$	0	0
$457$	250.871	0.548953	0.274476	−	0.961594i	$-0.411495\pi$
$457$	250.871	0.548953	0.274476	+	0.961594i	$0.411495\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	230.934i	0.500942i	0.968124	+	0.250471i	$0.0805855\pi$
$461$	230.934i	0.500942i	−0.968124	+	0.250471i	$0.919415\pi$
$462$	0	0
$463$	445.185	0.961522	0.480761	−	0.876852i	$-0.340360\pi$
$463$	445.185	0.961522	0.480761	+	0.876852i	$0.340360\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 595.736i	− 1.27567i	−0.770175	−	0.637833i	$-0.779831\pi$
$467$	− 595.736i	− 1.27567i	0.770175	−	0.637833i	$-0.220169\pi$
$468$	0	0
$469$	−223.314	−0.476149
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 148.207i	− 0.313335i
$474$	0	0
$475$	−306.392	−0.645037
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 327.569i	− 0.683860i	−0.939725	−	0.341930i	$-0.888919\pi$
$479$	− 327.569i	− 0.683860i	0.939725	−	0.341930i	$-0.111081\pi$
$480$	0	0
$481$	−290.482	−0.603913
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 163.564i	− 0.337245i
$486$	0	0
$487$	−39.4271	−0.0809592	−0.0404796	−	0.999180i	$-0.512889\pi$
$487$	−39.4271	−0.0809592	−0.0404796	+	0.999180i	$0.512889\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	326.558i	0.665088i	0.943088	+	0.332544i	$0.107907\pi$
$491$	326.558i	0.665088i	−0.943088	+	0.332544i	$0.892093\pi$
$492$	0	0
$493$	−197.378	−0.400360
$494$	0	0
$495$	0	0
$496$	0	0
$497$	436.954i	0.879184i
$498$	0	0
$499$	−443.983	−0.889746	−0.444873	−	0.895594i	$-0.646751\pi$
$499$	−443.983	−0.889746	−0.444873	+	0.895594i	$0.646751\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	287.052i	0.570679i	0.958426	+	0.285340i	$0.0921064\pi$
$503$	287.052i	0.570679i	−0.958426	+	0.285340i	$0.907894\pi$
$504$	0	0
$505$	−172.199	−0.340988
$506$	0	0
$507$	0	0
$508$	0	0
$509$	813.294i	1.59783i	0.601445	+	0.798914i	$0.294592\pi$
$509$	813.294i	1.59783i	−0.601445	+	0.798914i	$0.705408\pi$
$510$	0	0
$511$	124.334	0.243315
$512$	0	0
$513$	0	0
$514$	0	0
$515$	220.960i	0.429048i
$516$	0	0
$517$	−102.869	−0.198973
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 19.7617i	− 0.0379304i	−0.999820	−	0.0189652i	$-0.993963\pi$
$521$	− 19.7617i	− 0.0379304i	0.999820	−	0.0189652i	$-0.00603716\pi$
$522$	0	0
$523$	684.310	1.30843	0.654216	−	0.756308i	$-0.272999\pi$
$523$	684.310	1.30843	0.654216	+	0.756308i	$0.272999\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	62.9687i	0.119485i
$528$	0	0
$529$	241.538	0.456594
$530$	0	0
$531$	0	0
$532$	0	0
$533$	252.312i	0.473381i
$534$	0	0
$535$	138.640	0.259140
$536$	0	0
$537$	0	0
$538$	0	0
$539$	185.432i	0.344030i
$540$	0	0
$541$	283.876	0.524724	0.262362	−	0.964970i	$-0.415499\pi$
$541$	283.876	0.524724	0.262362	+	0.964970i	$0.415499\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	67.4506i	0.123763i
$546$	0	0
$547$	−724.992	−1.32540	−0.662698	−	0.748886i	$-0.730589\pi$
$547$	−724.992	−1.32540	−0.662698	+	0.748886i	$0.730589\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 646.544i	− 1.17340i
$552$	0	0
$553$	287.437	0.519778
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 735.862i	− 1.32112i	−0.750775	−	0.660558i	$-0.770320\pi$
$557$	− 735.862i	− 1.32112i	0.750775	−	0.660558i	$-0.229680\pi$
$558$	0	0
$559$	−133.054	−0.238022
$560$	0	0
$561$	0	0
$562$	0	0
$563$	774.685i	1.37600i	0.725713	+	0.687998i	$0.241510\pi$
$563$	774.685i	1.37600i	−0.725713	+	0.687998i	$0.758490\pi$
$564$	0	0
$565$	−219.162	−0.387897
$566$	0	0
$567$	0	0
$568$	0	0
$569$	659.378i	1.15884i	0.815030	+	0.579419i	$0.196720\pi$
$569$	659.378i	1.15884i	−0.815030	+	0.579419i	$0.803280\pi$
$570$	0	0
$571$	−102.479	−0.179473	−0.0897363	−	0.995966i	$-0.528602\pi$
$571$	−102.479	−0.179473	−0.0897363	+	0.995966i	$0.528602\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	384.436i	0.668585i
$576$	0	0
$577$	948.934	1.64460	0.822300	−	0.569055i	$-0.192691\pi$
$577$	948.934	1.64460	0.822300	+	0.569055i	$0.192691\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 57.9313i	− 0.0997096i
$582$	0	0
$583$	99.2148	0.170180
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 836.623i	− 1.42525i	−0.701544	−	0.712626i	$-0.747506\pi$
$587$	− 836.623i	− 1.42525i	0.701544	−	0.712626i	$-0.252494\pi$
$588$	0	0
$589$	−206.265	−0.350195
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 485.118i	− 0.818075i	−0.912518	−	0.409037i	$-0.865865\pi$
$593$	− 485.118i	− 0.818075i	0.912518	−	0.409037i	$-0.134135\pi$
$594$	0	0
$595$	−21.6856	−0.0364463
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 295.759i	− 0.493755i	−0.969047	−	0.246878i	$-0.920595\pi$
$599$	− 295.759i	− 0.493755i	0.969047	−	0.246878i	$-0.0794046\pi$
$600$	0	0
$601$	−683.580	−1.13740	−0.568702	−	0.822544i	$-0.692554\pi$
$601$	−683.580	−1.13740	−0.568702	+	0.822544i	$0.692554\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 146.466i	− 0.242092i
$606$	0	0
$607$	159.932	0.263479	0.131740	−	0.991284i	$-0.457944\pi$
$607$	159.932	0.263479	0.131740	+	0.991284i	$0.457944\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	92.3513i	0.151148i
$612$	0	0
$613$	−741.174	−1.20909	−0.604546	−	0.796570i	$-0.706646\pi$
$613$	−741.174	−1.20909	−0.604546	+	0.796570i	$0.706646\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	99.2535i	0.160865i	0.996760	+	0.0804323i	$0.0256301\pi$
$617$	99.2535i	0.160865i	−0.996760	+	0.0804323i	$0.974370\pi$
$618$	0	0
$619$	387.605	0.626179	0.313090	−	0.949724i	$-0.398636\pi$
$619$	387.605	0.626179	0.313090	+	0.949724i	$0.398636\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 262.872i	− 0.421946i
$624$	0	0
$625$	455.983	0.729573
$626$	0	0
$627$	0	0
$628$	0	0
$629$	267.177i	0.424765i
$630$	0	0
$631$	−232.514	−0.368485	−0.184243	−	0.982881i	$-0.558983\pi$
$631$	−232.514	−0.368485	−0.184243	+	0.982881i	$0.558983\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 57.4461i	− 0.0904664i
$636$	0	0
$637$	166.473	0.261339
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 68.7477i	− 0.107251i	−0.998561	−	0.0536254i	$-0.982922\pi$
$641$	− 68.7477i	− 0.107251i	0.998561	−	0.0536254i	$-0.0170777\pi$
$642$	0	0
$643$	−861.182	−1.33932	−0.669660	−	0.742668i	$-0.733560\pi$
$643$	−861.182	−1.33932	−0.669660	+	0.742668i	$0.733560\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	175.249i	0.270864i	0.990787	+	0.135432i	$0.0432422\pi$
$647$	175.249i	0.270864i	−0.990787	+	0.135432i	$0.956758\pi$
$648$	0	0
$649$	−495.006	−0.762722
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 359.848i	− 0.551068i	−0.961291	−	0.275534i	$-0.911145\pi$
$653$	− 359.848i	− 0.551068i	0.961291	−	0.275534i	$-0.0888547\pi$
$654$	0	0
$655$	−233.628	−0.356684
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 30.8838i	− 0.0468646i	−0.999725	−	0.0234323i	$-0.992541\pi$
$659$	− 30.8838i	− 0.0468646i	0.999725	−	0.0234323i	$-0.00745941\pi$
$660$	0	0
$661$	226.617	0.342840	0.171420	−	0.985198i	$-0.445165\pi$
$661$	226.617	0.342840	0.171420	+	0.985198i	$0.445165\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 71.0347i	− 0.106819i
$666$	0	0
$667$	−811.230	−1.21624
$668$	0	0
$669$	0	0
$670$	0	0
$671$	412.282i	0.614429i
$672$	0	0
$673$	583.199	0.866566	0.433283	−	0.901258i	$-0.357355\pi$
$673$	583.199	0.866566	0.433283	+	0.901258i	$0.357355\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1312.91i	1.93930i	0.244497	+	0.969650i	$0.421377\pi$
$677$	1312.91i	1.93930i	−0.244497	+	0.969650i	$0.578623\pi$
$678$	0	0
$679$	−369.713	−0.544497
$680$	0	0
$681$	0	0
$682$	0	0
$683$	233.018i	0.341168i	0.985343	+	0.170584i	$0.0545655\pi$
$683$	233.018i	0.341168i	−0.985343	+	0.170584i	$0.945435\pi$
$684$	0	0
$685$	90.5381	0.132172
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 89.0708i	− 0.129275i
$690$	0	0
$691$	837.760	1.21239	0.606194	−	0.795317i	$-0.292696\pi$
$691$	837.760	1.21239	0.606194	+	0.795317i	$0.292696\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 91.5261i	− 0.131692i
$696$	0	0
$697$	232.069	0.332955
$698$	0	0
$699$	0	0
$700$	0	0
$701$	855.750i	1.22076i	0.792111	+	0.610378i	$0.208982\pi$
$701$	855.750i	1.22076i	−0.792111	+	0.610378i	$0.791018\pi$
$702$	0	0
$703$	−875.183	−1.24493
$704$	0	0
$705$	0	0
$706$	0	0
$707$	389.232i	0.550540i
$708$	0	0
$709$	122.722	0.173092	0.0865461	−	0.996248i	$-0.472417\pi$
$709$	122.722	0.173092	0.0865461	+	0.996248i	$0.472417\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	258.804i	0.362979i
$714$	0	0
$715$	34.1699	0.0477901
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 780.498i	− 1.08553i	−0.839884	−	0.542766i	$-0.817377\pi$
$719$	− 780.498i	− 1.08553i	0.839884	−	0.542766i	$-0.182623\pi$
$720$	0	0
$721$	499.448	0.692716
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1084.90i	1.49641i
$726$	0	0
$727$	−786.007	−1.08116	−0.540582	−	0.841291i	$-0.681796\pi$
$727$	−786.007	−1.08116	−0.540582	+	0.841291i	$0.681796\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	122.379i	0.167414i
$732$	0	0
$733$	1127.68	1.53845	0.769225	−	0.638978i	$-0.220642\pi$
$733$	1127.68	1.53845	0.769225	+	0.638978i	$0.220642\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 323.645i	− 0.439138i
$738$	0	0
$739$	−114.302	−0.154671	−0.0773357	−	0.997005i	$-0.524641\pi$
$739$	−114.302	−0.154671	−0.0773357	+	0.997005i	$0.524641\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1153.85i	1.55296i	0.630145	+	0.776478i	$0.282995\pi$
$743$	1153.85i	1.55296i	−0.630145	+	0.776478i	$0.717005\pi$
$744$	0	0
$745$	−409.911	−0.550216
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 313.375i	− 0.418392i
$750$	0	0
$751$	−975.576	−1.29904	−0.649518	−	0.760346i	$-0.725029\pi$
$751$	−975.576	−1.29904	−0.649518	+	0.760346i	$0.725029\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 18.6162i	− 0.0246572i
$756$	0	0
$757$	886.690	1.17132	0.585661	−	0.810556i	$-0.300835\pi$
$757$	886.690	1.17132	0.585661	+	0.810556i	$0.300835\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	860.890i	1.13126i	0.824659	+	0.565631i	$0.191367\pi$
$761$	860.890i	1.13126i	−0.824659	+	0.565631i	$0.808633\pi$
$762$	0	0
$763$	152.463	0.199820
$764$	0	0
$765$	0	0
$766$	0	0
$767$	444.396i	0.579395i
$768$	0	0
$769$	254.019	0.330323	0.165162	−	0.986267i	$-0.447185\pi$
$769$	254.019	0.330323	0.165162	+	0.986267i	$0.447185\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 52.6116i	− 0.0680616i	−0.999421	−	0.0340308i	$-0.989166\pi$
$773$	− 52.6116i	− 0.0680616i	0.999421	−	0.0340308i	$-0.0108344\pi$
$774$	0	0
$775$	346.111	0.446595
$776$	0	0
$777$	0	0
$778$	0	0
$779$	760.183i	0.975844i
$780$	0	0
$781$	−633.270	−0.810845
$782$	0	0
$783$	0	0
$784$	0	0
$785$	68.8732i	0.0877366i
$786$	0	0
$787$	5.06217	0.00643224	0.00321612	−	0.999995i	$-0.498976\pi$
$787$	5.06217	0.00643224	0.00321612	+	0.999995i	$0.498976\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	495.385i	0.626276i
$792$	0	0
$793$	370.129	0.466746
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1030.57i	1.29306i	0.762888	+	0.646530i	$0.223781\pi$
$797$	1030.57i	1.29306i	−0.762888	+	0.646530i	$0.776219\pi$
$798$	0	0
$799$	84.9420	0.106310
$800$	0	0
$801$	0	0
$802$	0	0
$803$	180.195i	0.224402i
$804$	0	0
$805$	−89.1286	−0.110719
$806$	0	0
$807$	0	0
$808$	0	0
$809$	369.548i	0.456796i	0.973568	+	0.228398i	$0.0733488\pi$
$809$	369.548i	0.456796i	−0.973568	+	0.228398i	$0.926651\pi$
$810$	0	0
$811$	−366.571	−0.451998	−0.225999	−	0.974127i	$-0.572565\pi$
$811$	−366.571	−0.451998	−0.225999	+	0.974127i	$0.572565\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 8.97809i	− 0.0110161i
$816$	0	0
$817$	−400.875	−0.490667
$818$	0	0
$819$	0	0
$820$	0	0
$821$	227.417i	0.277000i	0.990362	+	0.138500i	$0.0442280\pi$
$821$	227.417i	0.277000i	−0.990362	+	0.138500i	$0.955772\pi$
$822$	0	0
$823$	1418.03	1.72300	0.861499	−	0.507759i	$-0.169526\pi$
$823$	1418.03	1.72300	0.861499	+	0.507759i	$0.169526\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1092.92i	− 1.32154i	−0.750587	−	0.660771i	$-0.770229\pi$
$827$	− 1092.92i	− 1.32154i	0.750587	−	0.660771i	$-0.229771\pi$
$828$	0	0
$829$	−391.526	−0.472287	−0.236144	−	0.971718i	$-0.575884\pi$
$829$	−391.526	−0.472287	−0.236144	+	0.971718i	$0.575884\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 153.117i	− 0.183814i
$834$	0	0
$835$	−339.667	−0.406787
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 888.734i	− 1.05928i	−0.848223	−	0.529639i	$-0.822327\pi$
$839$	− 888.734i	− 1.05928i	0.848223	−	0.529639i	$-0.177673\pi$
$840$	0	0
$841$	−1448.33	−1.72215
$842$	0	0
$843$	0	0
$844$	0	0
$845$	227.052i	0.268700i
$846$	0	0
$847$	−331.065	−0.390868
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1098.11i	1.29037i
$852$	0	0
$853$	765.613	0.897553	0.448777	−	0.893644i	$-0.351860\pi$
$853$	765.613	0.897553	0.448777	+	0.893644i	$0.351860\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	962.423i	1.12301i	0.827472	+	0.561507i	$0.189778\pi$
$857$	962.423i	1.12301i	−0.827472	+	0.561507i	$0.810222\pi$
$858$	0	0
$859$	520.247	0.605643	0.302822	−	0.953047i	$-0.402071\pi$
$859$	520.247	0.605643	0.302822	+	0.953047i	$0.402071\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 384.728i	− 0.445803i	−0.974841	−	0.222901i	$-0.928447\pi$
$863$	− 384.728i	− 0.445803i	0.974841	−	0.222901i	$-0.0715528\pi$
$864$	0	0
$865$	39.1331	0.0452406
$866$	0	0
$867$	0	0
$868$	0	0
$869$	416.578i	0.479376i
$870$	0	0
$871$	−290.555	−0.333587
$872$	0	0
$873$	0	0
$874$	0	0
$875$	250.618i	0.286420i
$876$	0	0
$877$	−1265.16	−1.44260	−0.721301	−	0.692622i	$-0.756455\pi$
$877$	−1265.16	−1.44260	−0.721301	+	0.692622i	$0.756455\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	767.742i	0.871443i	0.900081	+	0.435722i	$0.143507\pi$
$881$	767.742i	0.871443i	−0.900081	+	0.435722i	$0.856493\pi$
$882$	0	0
$883$	999.576	1.13202	0.566012	−	0.824397i	$-0.308486\pi$
$883$	999.576	1.13202	0.566012	+	0.824397i	$0.308486\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	49.1364i	0.0553961i	0.999616	+	0.0276981i	$0.00881770\pi$
$887$	49.1364i	0.0553961i	−0.999616	+	0.0276981i	$0.991182\pi$
$888$	0	0
$889$	−129.849	−0.146062
$890$	0	0
$891$	0	0
$892$	0	0
$893$	278.242i	0.311581i
$894$	0	0
$895$	119.361	0.133365
$896$	0	0
$897$	0	0
$898$	0	0
$899$	730.357i	0.812411i
$900$	0	0
$901$	−81.9247	−0.0909264
$902$	0	0
$903$	0	0
$904$	0	0
$905$	220.101i	0.243206i
$906$	0	0
$907$	386.029	0.425611	0.212806	−	0.977095i	$-0.431740\pi$
$907$	386.029	0.425611	0.212806	+	0.977095i	$0.431740\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 727.670i	− 0.798759i	−0.916786	−	0.399380i	$-0.869226\pi$
$911$	− 727.670i	− 0.798759i	0.916786	−	0.399380i	$-0.130774\pi$
$912$	0	0
$913$	83.9588	0.0919593
$914$	0	0
$915$	0	0
$916$	0	0
$917$	528.084i	0.575882i
$918$	0	0
$919$	407.999	0.443960	0.221980	−	0.975051i	$-0.428748\pi$
$919$	407.999	0.443960	0.221980	+	0.975051i	$0.428748\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	568.523i	0.615952i
$924$	0	0
$925$	1468.55	1.58763
$926$	0	0
$927$	0	0
$928$	0	0
$929$	480.207i	0.516907i	0.966024	+	0.258454i	$0.0832129\pi$
$929$	480.207i	0.516907i	−0.966024	+	0.258454i	$0.916787\pi$
$930$	0	0
$931$	501.561	0.538733
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 31.4285i	− 0.0336134i
$936$	0	0
$937$	−167.930	−0.179221	−0.0896105	−	0.995977i	$-0.528562\pi$
$937$	−167.930	−0.179221	−0.0896105	+	0.995977i	$0.528562\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1172.44i	1.24595i	0.782242	+	0.622975i	$0.214076\pi$
$941$	1172.44i	1.24595i	−0.782242	+	0.622975i	$0.785924\pi$
$942$	0	0
$943$	953.815	1.01147
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1243.41i	− 1.31300i	−0.754326	−	0.656500i	$-0.772036\pi$
$947$	− 1243.41i	− 1.31300i	0.754326	−	0.656500i	$-0.227964\pi$
$948$	0	0
$949$	161.771	0.170465
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1113.62i	− 1.16854i	−0.811559	−	0.584270i	$-0.801381\pi$
$953$	− 1113.62i	− 1.16854i	0.811559	−	0.584270i	$-0.198619\pi$
$954$	0	0
$955$	−169.797	−0.177798
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 204.649i	− 0.213398i
$960$	0	0
$961$	−727.997	−0.757541
$962$	0	0
$963$	0	0
$964$	0	0
$965$	147.529i	0.152879i
$966$	0	0
$967$	−427.758	−0.442356	−0.221178	−	0.975233i	$-0.570990\pi$
$967$	−427.758	−0.442356	−0.221178	+	0.975233i	$0.570990\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 713.622i	− 0.734935i	−0.930036	−	0.367467i	$-0.880225\pi$
$971$	− 713.622i	− 0.734935i	0.930036	−	0.367467i	$-0.119775\pi$
$972$	0	0
$973$	−206.882	−0.212623
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1309.53i	1.34035i	0.742201	+	0.670177i	$0.233782\pi$
$977$	1309.53i	1.34035i	−0.742201	+	0.670177i	$0.766218\pi$
$978$	0	0
$979$	380.976	0.389148
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1315.70i	− 1.33845i	−0.743060	−	0.669225i	$-0.766626\pi$
$983$	− 1315.70i	− 1.33845i	0.743060	−	0.669225i	$-0.233374\pi$
$984$	0	0
$985$	21.1860	0.0215086
$986$	0	0
$987$	0	0
$988$	0	0
$989$	502.985i	0.508579i
$990$	0	0
$991$	184.711	0.186389	0.0931944	−	0.995648i	$-0.470292\pi$
$991$	184.711	0.186389	0.0931944	+	0.995648i	$0.470292\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	237.215i	0.238407i
$996$	0	0
$997$	1042.33	1.04546	0.522731	−	0.852497i	$-0.324913\pi$
$997$	1042.33	1.04546	0.522731	+	0.852497i	$0.324913\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2916.3.c.b.1457.16		36
3.2	odd	2	inner	2916.3.c.b.1457.21		36
27.5	odd	18		108.3.k.a.29.6	✓	36
27.11	odd	18		324.3.k.a.233.3		36
27.16	even	9		108.3.k.a.41.6	yes	36
27.22	even	9		324.3.k.a.89.3		36
108.43	odd	18		432.3.bc.b.257.1		36
108.59	even	18		432.3.bc.b.353.1		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.3.k.a.29.6	✓	36	27.5	odd	18
108.3.k.a.41.6	yes	36	27.16	even	9
324.3.k.a.89.3		36	27.22	even	9
324.3.k.a.233.3		36	27.11	odd	18
432.3.bc.b.257.1		36	108.43	odd	18
432.3.bc.b.353.1		36	108.59	even	18
2916.3.c.b.1457.16		36	1.1	even	1	trivial
2916.3.c.b.1457.21		36	3.2	odd	2	inner

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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:	\( N \)	\(=\)	\( 2916 = 2^{2} \cdot 3^{6} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2916.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.52502i q^{5} -3.44709 q^{7} +O(q^{10})\)	\(q-1.52502i q^{5} -3.44709 q^{7} -4.99580i q^{11} -4.48502 q^{13} +4.12519i q^{17} -13.5127 q^{19} +16.9547i q^{23} +22.6743 q^{25} +47.8469i q^{29} +15.2644 q^{31} +5.25687i q^{35} +64.7672 q^{37} -56.2567i q^{41} +29.6664 q^{43} -20.5911i q^{47} -37.1176 q^{49} +19.8596i q^{53} -7.61868 q^{55} -99.0845i q^{59} -82.5257 q^{61} +6.83973i q^{65} +64.7834 q^{67} -126.761i q^{71} -36.0692 q^{73} +17.2210i q^{77} -83.3856 q^{79} +16.8059i q^{83} +6.29098 q^{85} +76.2593i q^{89} +15.4602 q^{91} +20.6072i q^{95} +107.254 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q+O(q^{10}) \) 36 * q	\( 36 q - 180 q^{25} + 252 q^{49} + 18 q^{61} - 90 q^{67} + 126 q^{73} - 198 q^{91}+O(q^{100}) \) 36 * q - 180 * q^25 + 252 * q^49 + 18 * q^61 - 90 * q^67 + 126 * q^73 - 198 * q^91

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