LMFDB - Embedded newform 1296.3.g.k.1135.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,3,Mod(1135,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1296.1135");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1296.g (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:	$35.3134422611$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.856615824.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 $ x^8 + 11x^6 + 36x^4 + 32*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1135.3
Root			$-0.385731i$ of defining polynomial
Character	$\chi$	$=$	1296.1135
Dual form			1296.3.g.k.1135.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.909226 q^{5} -7.05186i q^{7} +O(q^{10})$	$q-0.909226 q^{5} -7.05186i q^{7} +8.04435i q^{11} -6.71904 q^{13} +26.3462 q^{17} +20.5603i q^{19} -25.2076i q^{23} -24.1733 q^{25} +30.3387 q^{29} +0.138610i q^{31} +6.41173i q^{35} +69.7588 q^{37} -58.7588 q^{41} +2.83886i q^{43} -81.7046i q^{47} -0.728766 q^{49} +30.0259 q^{53} -7.31413i q^{55} -89.0785i q^{59} -48.1377 q^{61} +6.10913 q^{65} -50.9025i q^{67} -68.4355i q^{71} -22.1474 q^{73} +56.7277 q^{77} -39.7613i q^{79} +26.6506i q^{83} -23.9547 q^{85} +25.7926 q^{89} +47.3818i q^{91} -18.6939i q^{95} +104.739 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{5}+O(q^{10}) $ 8 * q + 6 * q^5	$ 8 q + 6 q^{5} - 10 q^{13} - 6 q^{17} + 46 q^{25} + 138 q^{29} - 20 q^{37} + 108 q^{41} - 82 q^{49} + 252 q^{53} - 14 q^{61} + 186 q^{65} + 74 q^{73} + 414 q^{77} + 60 q^{85} + 168 q^{89} + 20 q^{97}+O(q^{100}) $ 8 * q + 6 * q^5 - 10 * q^13 - 6 * q^17 + 46 * q^25 + 138 * q^29 - 20 * q^37 + 108 * q^41 - 82 * q^49 + 252 * q^53 - 14 * q^61 + 186 * q^65 + 74 * q^73 + 414 * q^77 + 60 * q^85 + 168 * q^89 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	−0.909226	−0.181845	−0.0909226	−	0.995858i	$-0.528982\pi$
$5$	−0.909226	−0.181845	−0.0909226	+	0.995858i	$0.528982\pi$
$6$	0	0
$7$	− 7.05186i	− 1.00741i	−0.863876	−	0.503704i	$-0.831970\pi$
$7$	− 7.05186i	− 1.00741i	0.863876	−	0.503704i	$-0.168030\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	8.04435i	0.731305i	0.930751	+	0.365652i	$0.119154\pi$
$11$	8.04435i	0.731305i	−0.930751	+	0.365652i	$0.880846\pi$
$12$	0	0
$13$	−6.71904	−0.516850	−0.258425	−	0.966031i	$-0.583203\pi$
$13$	−6.71904	−0.516850	−0.258425	+	0.966031i	$0.583203\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	26.3462	1.54978	0.774889	−	0.632097i	$-0.217806\pi$
$17$	26.3462	1.54978	0.774889	+	0.632097i	$0.217806\pi$
$18$	0	0
$19$	20.5603i	1.08212i	0.840984	+	0.541059i	$0.181977\pi$
$19$	20.5603i	1.08212i	−0.840984	+	0.541059i	$0.818023\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 25.2076i	− 1.09598i	−0.836484	−	0.547992i	$-0.815392\pi$
$23$	− 25.2076i	− 1.09598i	0.836484	−	0.547992i	$-0.184608\pi$
$24$	0	0
$25$	−24.1733	−0.966932
$26$	0	0
$27$	0	0
$28$	0	0
$29$	30.3387	1.04616	0.523081	−	0.852283i	$-0.324783\pi$
$29$	30.3387	1.04616	0.523081	+	0.852283i	$0.324783\pi$
$30$	0	0
$31$	0.138610i	0.00447129i	0.999998	+	0.00223565i	$0.000711629\pi$
$31$	0.138610i	0.00447129i	−0.999998	+	0.00223565i	$0.999288\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	6.41173i	0.183192i
$36$	0	0
$37$	69.7588	1.88537	0.942687	−	0.333679i	$-0.108290\pi$
$37$	69.7588	1.88537	0.942687	+	0.333679i	$0.108290\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−58.7588	−1.43314	−0.716571	−	0.697514i	$-0.754289\pi$
$41$	−58.7588	−1.43314	−0.716571	+	0.697514i	$0.754289\pi$
$42$	0	0
$43$	2.83886i	0.0660201i	0.999455	+	0.0330100i	$0.0105093\pi$
$43$	2.83886i	0.0660201i	−0.999455	+	0.0330100i	$0.989491\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 81.7046i	− 1.73840i	−0.494464	−	0.869198i	$-0.664636\pi$
$47$	− 81.7046i	− 1.73840i	0.494464	−	0.869198i	$-0.335364\pi$
$48$	0	0
$49$	−0.728766	−0.0148728
$50$	0	0
$51$	0	0
$52$	0	0
$53$	30.0259	0.566526	0.283263	−	0.959042i	$-0.408583\pi$
$53$	30.0259	0.566526	0.283263	+	0.959042i	$0.408583\pi$
$54$	0	0
$55$	− 7.31413i	− 0.132984i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 89.0785i	− 1.50980i	−0.655837	−	0.754902i	$-0.727684\pi$
$59$	− 89.0785i	− 1.50980i	0.655837	−	0.754902i	$-0.272316\pi$
$60$	0	0
$61$	−48.1377	−0.789142	−0.394571	−	0.918865i	$-0.629107\pi$
$61$	−48.1377	−0.789142	−0.394571	+	0.918865i	$0.629107\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.10913	0.0939866
$66$	0	0
$67$	− 50.9025i	− 0.759739i	−0.925040	−	0.379870i	$-0.875969\pi$
$67$	− 50.9025i	− 0.759739i	0.925040	−	0.379870i	$-0.124031\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 68.4355i	− 0.963881i	−0.876204	−	0.481940i	$-0.839932\pi$
$71$	− 68.4355i	− 0.963881i	0.876204	−	0.481940i	$-0.160068\pi$
$72$	0	0
$73$	−22.1474	−0.303389	−0.151695	−	0.988427i	$-0.548473\pi$
$73$	−22.1474	−0.303389	−0.151695	+	0.988427i	$0.548473\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	56.7277	0.736723
$78$	0	0
$79$	− 39.7613i	− 0.503308i	−0.967817	−	0.251654i	$-0.919026\pi$
$79$	− 39.7613i	− 0.503308i	0.967817	−	0.251654i	$-0.0809745\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	26.6506i	0.321091i	0.987028	+	0.160546i	$0.0513254\pi$
$83$	26.6506i	0.321091i	−0.987028	+	0.160546i	$0.948675\pi$
$84$	0	0
$85$	−23.9547	−0.281820
$86$	0	0
$87$	0	0
$88$	0	0
$89$	25.7926	0.289804	0.144902	−	0.989446i	$-0.453713\pi$
$89$	25.7926	0.289804	0.144902	+	0.989446i	$0.453713\pi$
$90$	0	0
$91$	47.3818i	0.520679i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 18.6939i	− 0.196778i
$96$	0	0
$97$	104.739	1.07979	0.539894	−	0.841733i	$-0.318464\pi$
$97$	104.739	1.07979	0.539894	+	0.841733i	$0.318464\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	40.9581	0.405525	0.202763	−	0.979228i	$-0.435008\pi$
$101$	40.9581	0.405525	0.202763	+	0.979228i	$0.435008\pi$
$102$	0	0
$103$	144.659i	1.40445i	0.711953	+	0.702227i	$0.247811\pi$
$103$	144.659i	1.40445i	−0.711953	+	0.702227i	$0.752189\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 177.858i	− 1.66222i	−0.556105	−	0.831112i	$-0.687705\pi$
$107$	− 177.858i	− 1.66222i	0.556105	−	0.831112i	$-0.312295\pi$
$108$	0	0
$109$	−142.616	−1.30840	−0.654200	−	0.756322i	$-0.726994\pi$
$109$	−142.616	−1.30840	−0.654200	+	0.756322i	$0.726994\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	200.293	1.77251	0.886254	−	0.463200i	$-0.153299\pi$
$113$	200.293	1.77251	0.886254	+	0.463200i	$0.153299\pi$
$114$	0	0
$115$	22.9194i	0.199299i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 185.790i	− 1.56126i
$120$	0	0
$121$	56.2884	0.465193
$122$	0	0
$123$	0	0
$124$	0	0
$125$	44.7096	0.357677
$126$	0	0
$127$	− 181.723i	− 1.43089i	−0.698670	−	0.715445i	$-0.746224\pi$
$127$	− 181.723i	− 1.43089i	0.698670	−	0.715445i	$-0.253776\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 61.1253i	− 0.466605i	−0.972404	−	0.233303i	$-0.925047\pi$
$131$	− 61.1253i	− 0.466605i	0.972404	−	0.233303i	$-0.0749533\pi$
$132$	0	0
$133$	144.988	1.09014
$134$	0	0
$135$	0	0
$136$	0	0
$137$	36.2263	0.264425	0.132213	−	0.991221i	$-0.457792\pi$
$137$	36.2263	0.264425	0.132213	+	0.991221i	$0.457792\pi$
$138$	0	0
$139$	− 178.577i	− 1.28473i	−0.766400	−	0.642363i	$-0.777954\pi$
$139$	− 178.577i	− 1.28473i	0.766400	−	0.642363i	$-0.222046\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 54.0504i	− 0.377975i
$144$	0	0
$145$	−27.5847	−0.190239
$146$	0	0
$147$	0	0
$148$	0	0
$149$	240.087	1.61132	0.805660	−	0.592378i	$-0.201811\pi$
$149$	240.087	1.61132	0.805660	+	0.592378i	$0.201811\pi$
$150$	0	0
$151$	113.321i	0.750473i	0.926929	+	0.375237i	$0.122439\pi$
$151$	113.321i	0.750473i	−0.926929	+	0.375237i	$0.877561\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 0.126028i	0 0.000813083i
$156$	0	0
$157$	120.721	0.768923	0.384461	−	0.923141i	$-0.374387\pi$
$157$	120.721	0.768923	0.384461	+	0.923141i	$0.374387\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−177.761	−1.10410
$162$	0	0
$163$	20.3498i	0.124845i	0.998050	+	0.0624226i	$0.0198827\pi$
$163$	20.3498i	0.124845i	−0.998050	+	0.0624226i	$0.980117\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	174.972i	1.04773i	0.851800	+	0.523867i	$0.175511\pi$
$167$	174.972i	1.04773i	−0.851800	+	0.523867i	$0.824489\pi$
$168$	0	0
$169$	−123.854	−0.732866
$170$	0	0
$171$	0	0
$172$	0	0
$173$	111.835	0.646445	0.323223	−	0.946323i	$-0.395234\pi$
$173$	111.835	0.646445	0.323223	+	0.946323i	$0.395234\pi$
$174$	0	0
$175$	170.467i	0.974096i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 18.6939i	− 0.104435i	−0.998636	−	0.0522176i	$-0.983371\pi$
$179$	− 18.6939i	− 0.104435i	0.998636	−	0.0522176i	$-0.0166289\pi$
$180$	0	0
$181$	−98.0536	−0.541733	−0.270866	−	0.962617i	$-0.587310\pi$
$181$	−98.0536	−0.541733	−0.270866	+	0.962617i	$0.587310\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−63.4265	−0.342846
$186$	0	0
$187$	211.938i	1.13336i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 277.951i	− 1.45524i	−0.685979	−	0.727621i	$-0.740626\pi$
$191$	− 277.951i	− 1.45524i	0.685979	−	0.727621i	$-0.259374\pi$
$192$	0	0
$193$	−162.257	−0.840710	−0.420355	−	0.907360i	$-0.638094\pi$
$193$	−162.257	−0.840710	−0.420355	+	0.907360i	$0.638094\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	106.182	0.538993	0.269496	−	0.963001i	$-0.413143\pi$
$197$	106.182	0.538993	0.269496	+	0.963001i	$0.413143\pi$
$198$	0	0
$199$	− 63.3880i	− 0.318532i	−0.987236	−	0.159266i	$-0.949087\pi$
$199$	− 63.3880i	− 0.318532i	0.987236	−	0.159266i	$-0.0509128\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 213.944i	− 1.05391i
$204$	0	0
$205$	53.4250	0.260610
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−165.394	−0.791359
$210$	0	0
$211$	5.75062i	0.0272541i	0.999907	+	0.0136271i	$0.00433777\pi$
$211$	5.75062i	0.0272541i	−0.999907	+	0.0136271i	$0.995662\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 2.58117i	− 0.0120054i
$216$	0	0
$217$	0.977459	0.00450442
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−177.021	−0.801002
$222$	0	0
$223$	− 207.599i	− 0.930937i	−0.885064	−	0.465469i	$-0.845886\pi$
$223$	− 207.599i	− 0.930937i	0.885064	−	0.465469i	$-0.154114\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	91.0931i	0.401291i	0.979664	+	0.200646i	$0.0643040\pi$
$227$	91.0931i	0.401291i	−0.979664	+	0.200646i	$0.935696\pi$
$228$	0	0
$229$	−4.09889	−0.0178991	−0.00894954	−	0.999960i	$-0.502849\pi$
$229$	−4.09889	−0.0178991	−0.00894954	+	0.999960i	$0.502849\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−171.761	−0.737170	−0.368585	−	0.929594i	$-0.620158\pi$
$233$	−171.761	−0.737170	−0.368585	+	0.929594i	$0.620158\pi$
$234$	0	0
$235$	74.2879i	0.316119i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 91.1137i	− 0.381229i	−0.981665	−	0.190614i	$-0.938952\pi$
$239$	− 91.1137i	− 0.381229i	0.981665	−	0.190614i	$-0.0610480\pi$
$240$	0	0
$241$	−74.4704	−0.309006	−0.154503	−	0.987992i	$-0.549378\pi$
$241$	−74.4704	−0.309006	−0.154503	+	0.987992i	$0.549378\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.662613	0.00270454
$246$	0	0
$247$	− 138.145i	− 0.559293i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 216.868i	− 0.864014i	−0.901870	−	0.432007i	$-0.857806\pi$
$251$	− 216.868i	− 0.864014i	0.901870	−	0.432007i	$-0.142194\pi$
$252$	0	0
$253$	202.779	0.801499
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−287.154	−1.11733	−0.558665	−	0.829394i	$-0.688686\pi$
$257$	−287.154	−1.11733	−0.558665	+	0.829394i	$0.688686\pi$
$258$	0	0
$259$	− 491.930i	− 1.89934i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	417.701i	1.58822i	0.607776	+	0.794108i	$0.292062\pi$
$263$	417.701i	1.58822i	−0.607776	+	0.794108i	$0.707938\pi$
$264$	0	0
$265$	−27.3003	−0.103020
$266$	0	0
$267$	0	0
$268$	0	0
$269$	489.868	1.82107	0.910535	−	0.413433i	$-0.135670\pi$
$269$	489.868	1.82107	0.910535	+	0.413433i	$0.135670\pi$
$270$	0	0
$271$	325.133i	1.19975i	0.800093	+	0.599876i	$0.204784\pi$
$271$	325.133i	1.19975i	−0.800093	+	0.599876i	$0.795216\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 194.459i	− 0.707122i
$276$	0	0
$277$	−172.176	−0.621576	−0.310788	−	0.950479i	$-0.600593\pi$
$277$	−172.176	−0.621576	−0.310788	+	0.950479i	$0.600593\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−120.204	−0.427772	−0.213886	−	0.976859i	$-0.568612\pi$
$281$	−120.204	−0.427772	−0.213886	+	0.976859i	$0.568612\pi$
$282$	0	0
$283$	− 110.725i	− 0.391256i	−0.980678	−	0.195628i	$-0.937326\pi$
$283$	− 110.725i	− 0.391256i	0.980678	−	0.195628i	$-0.0626745\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	414.359i	1.44376i
$288$	0	0
$289$	405.124	1.40181
$290$	0	0
$291$	0	0
$292$	0	0
$293$	84.6729	0.288986	0.144493	−	0.989506i	$-0.453845\pi$
$293$	84.6729	0.288986	0.144493	+	0.989506i	$0.453845\pi$
$294$	0	0
$295$	80.9924i	0.274551i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	169.371i	0.566459i
$300$	0	0
$301$	20.0193	0.0665092
$302$	0	0
$303$	0	0
$304$	0	0
$305$	43.7680	0.143502
$306$	0	0
$307$	220.477i	0.718167i	0.933306	+	0.359083i	$0.116911\pi$
$307$	220.477i	0.718167i	−0.933306	+	0.359083i	$0.883089\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	310.577i	0.998641i	0.866417	+	0.499321i	$0.166417\pi$
$311$	310.577i	0.998641i	−0.866417	+	0.499321i	$0.833583\pi$
$312$	0	0
$313$	131.157	0.419033	0.209516	−	0.977805i	$-0.432811\pi$
$313$	131.157	0.419033	0.209516	+	0.977805i	$0.432811\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	339.960	1.07243	0.536214	−	0.844082i	$-0.319854\pi$
$317$	339.960	1.07243	0.536214	+	0.844082i	$0.319854\pi$
$318$	0	0
$319$	244.055i	0.765063i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	541.685i	1.67704i
$324$	0	0
$325$	162.422	0.499759
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−576.170	−1.75128
$330$	0	0
$331$	455.468i	1.37604i	0.725694	+	0.688018i	$0.241519\pi$
$331$	455.468i	1.37604i	−0.725694	+	0.688018i	$0.758481\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	46.2819i	0.138155i
$336$	0	0
$337$	43.9085	0.130292	0.0651462	−	0.997876i	$-0.479249\pi$
$337$	43.9085	0.130292	0.0651462	+	0.997876i	$0.479249\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.11503	−0.00326988
$342$	0	0
$343$	− 340.402i	− 0.992426i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	392.925i	1.13235i	0.824286	+	0.566174i	$0.191577\pi$
$347$	392.925i	1.13235i	−0.824286	+	0.566174i	$0.808423\pi$
$348$	0	0
$349$	−543.959	−1.55862	−0.779310	−	0.626638i	$-0.784430\pi$
$349$	−543.959	−1.55862	−0.779310	+	0.626638i	$0.784430\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−127.592	−0.361451	−0.180725	−	0.983534i	$-0.557845\pi$
$353$	−127.592	−0.361451	−0.180725	+	0.983534i	$0.557845\pi$
$354$	0	0
$355$	62.2233i	0.175277i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	285.077i	0.794085i	0.917800	+	0.397043i	$0.129963\pi$
$359$	285.077i	0.794085i	−0.917800	+	0.397043i	$0.870037\pi$
$360$	0	0
$361$	−61.7239	−0.170980
$362$	0	0
$363$	0	0
$364$	0	0
$365$	20.1370	0.0551698
$366$	0	0
$367$	− 323.413i	− 0.881234i	−0.897695	−	0.440617i	$-0.854760\pi$
$367$	− 323.413i	− 0.881234i	0.897695	−	0.440617i	$-0.145240\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 211.738i	− 0.570724i
$372$	0	0
$373$	−515.481	−1.38199	−0.690993	−	0.722862i	$-0.742826\pi$
$373$	−515.481	−1.38199	−0.690993	+	0.722862i	$0.742826\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−203.847	−0.540708
$378$	0	0
$379$	− 21.2535i	− 0.0560779i	−0.999607	−	0.0280389i	$-0.991074\pi$
$379$	− 21.2535i	− 0.0560779i	0.999607	−	0.0280389i	$-0.00892624\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 107.863i	− 0.281628i	−0.990036	−	0.140814i	$-0.955028\pi$
$383$	− 107.863i	− 0.281628i	0.990036	−	0.140814i	$-0.0449719\pi$
$384$	0	0
$385$	−51.5783	−0.133969
$386$	0	0
$387$	0	0
$388$	0	0
$389$	150.295	0.386362	0.193181	−	0.981163i	$-0.438120\pi$
$389$	150.295	0.386362	0.193181	+	0.981163i	$0.438120\pi$
$390$	0	0
$391$	− 664.126i	− 1.69853i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	36.1520i	0.0915241i
$396$	0	0
$397$	137.203	0.345600	0.172800	−	0.984957i	$-0.444719\pi$
$397$	137.203	0.345600	0.172800	+	0.984957i	$0.444719\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−266.731	−0.665165	−0.332583	−	0.943074i	$-0.607920\pi$
$401$	−266.731	−0.665165	−0.332583	+	0.943074i	$0.607920\pi$
$402$	0	0
$403$	− 0.931327i	− 0.00231099i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	561.165i	1.37878i
$408$	0	0
$409$	682.808	1.66946	0.834729	−	0.550662i	$-0.185624\pi$
$409$	682.808	1.66946	0.834729	+	0.550662i	$0.185624\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−628.169	−1.52099
$414$	0	0
$415$	− 24.2314i	− 0.0583889i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	518.496i	1.23746i	0.785604	+	0.618730i	$0.212352\pi$
$419$	518.496i	1.23746i	−0.785604	+	0.618730i	$0.787648\pi$
$420$	0	0
$421$	341.516	0.811202	0.405601	−	0.914050i	$-0.367062\pi$
$421$	341.516	0.811202	0.405601	+	0.914050i	$0.367062\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−636.875	−1.49853
$426$	0	0
$427$	339.460i	0.794989i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 166.603i	− 0.386550i	−0.981145	−	0.193275i	$-0.938089\pi$
$431$	− 166.603i	− 0.386550i	0.981145	−	0.193275i	$-0.0619110\pi$
$432$	0	0
$433$	−677.766	−1.56528	−0.782640	−	0.622475i	$-0.786127\pi$
$433$	−677.766	−1.56528	−0.782640	+	0.622475i	$0.786127\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	518.275	1.18598
$438$	0	0
$439$	493.094i	1.12322i	0.827402	+	0.561610i	$0.189818\pi$
$439$	493.094i	1.12322i	−0.827402	+	0.561610i	$0.810182\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	328.032i	0.740478i	0.928937	+	0.370239i	$0.120724\pi$
$443$	328.032i	0.740478i	−0.928937	+	0.370239i	$0.879276\pi$
$444$	0	0
$445$	−23.4513	−0.0526994
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−19.0862	−0.0425082	−0.0212541	−	0.999774i	$-0.506766\pi$
$449$	−19.0862	−0.0425082	−0.0212541	+	0.999774i	$0.506766\pi$
$450$	0	0
$451$	− 472.677i	− 1.04806i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 43.0807i	− 0.0946829i
$456$	0	0
$457$	275.611	0.603088	0.301544	−	0.953452i	$-0.402498\pi$
$457$	275.611	0.603088	0.301544	+	0.953452i	$0.402498\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−398.933	−0.865365	−0.432683	−	0.901546i	$-0.642433\pi$
$461$	−398.933	−0.865365	−0.432683	+	0.901546i	$0.642433\pi$
$462$	0	0
$463$	− 18.2305i	− 0.0393747i	−0.999806	−	0.0196873i	$-0.993733\pi$
$463$	− 18.2305i	− 0.0393747i	0.999806	−	0.0196873i	$-0.00626708\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	499.275i	1.06911i	0.845133	+	0.534555i	$0.179521\pi$
$467$	499.275i	1.06911i	−0.845133	+	0.534555i	$0.820479\pi$
$468$	0	0
$469$	−358.958	−0.765368
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−22.8368	−0.0482808
$474$	0	0
$475$	− 497.009i	− 1.04634i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 60.5609i	− 0.126432i	−0.998000	−	0.0632160i	$-0.979864\pi$
$479$	− 60.5609i	− 0.126432i	0.998000	−	0.0632160i	$-0.0201357\pi$
$480$	0	0
$481$	−468.713	−0.974454
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−95.2317	−0.196354
$486$	0	0
$487$	852.354i	1.75021i	0.483931	+	0.875106i	$0.339209\pi$
$487$	852.354i	1.75021i	−0.483931	+	0.875106i	$0.660791\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	676.028i	1.37684i	0.725313	+	0.688419i	$0.241695\pi$
$491$	676.028i	1.37684i	−0.725313	+	0.688419i	$0.758305\pi$
$492$	0	0
$493$	799.310	1.62132
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−482.598	−0.971022
$498$	0	0
$499$	849.957i	1.70332i	0.524094	+	0.851660i	$0.324404\pi$
$499$	849.957i	1.70332i	−0.524094	+	0.851660i	$0.675596\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 945.179i	− 1.87908i	−0.342433	−	0.939542i	$-0.611251\pi$
$503$	− 945.179i	− 1.87908i	0.342433	−	0.939542i	$-0.388749\pi$
$504$	0	0
$505$	−37.2401	−0.0737428
$506$	0	0
$507$	0	0
$508$	0	0
$509$	251.862	0.494817	0.247409	−	0.968911i	$-0.420421\pi$
$509$	251.862	0.494817	0.247409	+	0.968911i	$0.420421\pi$
$510$	0	0
$511$	156.181i	0.305637i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 131.527i	− 0.255393i
$516$	0	0
$517$	657.261	1.27130
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−856.423	−1.64381	−0.821903	−	0.569628i	$-0.807087\pi$
$521$	−856.423	−1.64381	−0.821903	+	0.569628i	$0.807087\pi$
$522$	0	0
$523$	− 741.634i	− 1.41804i	−0.705189	−	0.709019i	$-0.749138\pi$
$523$	− 741.634i	− 1.41804i	0.705189	−	0.709019i	$-0.250862\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.65185i	0.00692951i
$528$	0	0
$529$	−106.425	−0.201181
$530$	0	0
$531$	0	0
$532$	0	0
$533$	394.803	0.740719
$534$	0	0
$535$	161.713i	0.302267i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 5.86245i	− 0.0108765i
$540$	0	0
$541$	−434.157	−0.802509	−0.401254	−	0.915967i	$-0.631426\pi$
$541$	−434.157	−0.802509	−0.401254	+	0.915967i	$0.631426\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	129.670	0.237926
$546$	0	0
$547$	271.665i	0.496645i	0.968677	+	0.248323i	$0.0798793\pi$
$547$	271.665i	0.496645i	−0.968677	+	0.248323i	$0.920121\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	623.771i	1.13207i
$552$	0	0
$553$	−280.391	−0.507037
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−41.7759	−0.0750016	−0.0375008	−	0.999297i	$-0.511940\pi$
$557$	−41.7759	−0.0750016	−0.0375008	+	0.999297i	$0.511940\pi$
$558$	0	0
$559$	− 19.0744i	− 0.0341224i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	448.490i	0.796607i	0.917254	+	0.398303i	$0.130401\pi$
$563$	448.490i	0.796607i	−0.917254	+	0.398303i	$0.869599\pi$
$564$	0	0
$565$	−182.112	−0.322322
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−360.416	−0.633421	−0.316710	−	0.948522i	$-0.602578\pi$
$569$	−360.416	−0.633421	−0.316710	+	0.948522i	$0.602578\pi$
$570$	0	0
$571$	− 768.782i	− 1.34638i	−0.739471	−	0.673189i	$-0.764924\pi$
$571$	− 768.782i	− 1.34638i	0.739471	−	0.673189i	$-0.235076\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	609.352i	1.05974i
$576$	0	0
$577$	−413.359	−0.716394	−0.358197	−	0.933646i	$-0.616608\pi$
$577$	−413.359	−0.716394	−0.358197	+	0.933646i	$0.616608\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	187.936	0.323470
$582$	0	0
$583$	241.539i	0.414303i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.63090i	− 0.00277837i	−0.999999	−	0.00138918i	$-0.999558\pi$
$587$	− 1.63090i	− 0.00277837i	0.999999	−	0.00138918i	$-0.000442191\pi$
$588$	0	0
$589$	−2.84986	−0.00483847
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−652.407	−1.10018	−0.550091	−	0.835105i	$-0.685407\pi$
$593$	−652.407	−1.10018	−0.550091	+	0.835105i	$0.685407\pi$
$594$	0	0
$595$	168.925i	0.283908i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 171.260i	− 0.285909i	−0.989729	−	0.142955i	$-0.954340\pi$
$599$	− 171.260i	− 0.285909i	0.989729	−	0.142955i	$-0.0456603\pi$
$600$	0	0
$601$	131.975	0.219592	0.109796	−	0.993954i	$-0.464980\pi$
$601$	131.975	0.219592	0.109796	+	0.993954i	$0.464980\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−51.1788	−0.0845931
$606$	0	0
$607$	− 594.397i	− 0.979238i	−0.871937	−	0.489619i	$-0.837136\pi$
$607$	− 594.397i	− 0.979238i	0.871937	−	0.489619i	$-0.162864\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	548.977i	0.898489i
$612$	0	0
$613$	367.632	0.599727	0.299863	−	0.953982i	$-0.403059\pi$
$613$	367.632	0.599727	0.299863	+	0.953982i	$0.403059\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	42.1611	0.0683324	0.0341662	−	0.999416i	$-0.489122\pi$
$617$	42.1611	0.0683324	0.0341662	+	0.999416i	$0.489122\pi$
$618$	0	0
$619$	247.450i	0.399757i	0.979821	+	0.199879i	$0.0640548\pi$
$619$	247.450i	0.399757i	−0.979821	+	0.199879i	$0.935945\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 181.886i	− 0.291951i
$624$	0	0
$625$	563.682	0.901891
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1837.88	2.92191
$630$	0	0
$631$	− 725.556i	− 1.14985i	−0.818206	−	0.574926i	$-0.805031\pi$
$631$	− 725.556i	− 1.14985i	0.818206	−	0.574926i	$-0.194969\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	165.227i	0.260200i
$636$	0	0
$637$	4.89661	0.00768699
$638$	0	0
$639$	0	0
$640$	0	0
$641$	916.013	1.42904	0.714518	−	0.699617i	$-0.246646\pi$
$641$	916.013	1.42904	0.714518	+	0.699617i	$0.246646\pi$
$642$	0	0
$643$	793.052i	1.23336i	0.787213	+	0.616681i	$0.211523\pi$
$643$	793.052i	1.23336i	−0.787213	+	0.616681i	$0.788477\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 78.3837i	− 0.121150i	−0.998164	−	0.0605748i	$-0.980707\pi$
$647$	− 78.3837i	− 0.121150i	0.998164	−	0.0605748i	$-0.0192934\pi$
$648$	0	0
$649$	716.579	1.10413
$650$	0	0
$651$	0	0
$652$	0	0
$653$	262.501	0.401992	0.200996	−	0.979592i	$-0.435582\pi$
$653$	262.501	0.401992	0.200996	+	0.979592i	$0.435582\pi$
$654$	0	0
$655$	55.5767i	0.0848499i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1288.68i	− 1.95551i	−0.209758	−	0.977753i	$-0.567268\pi$
$659$	− 1288.68i	− 1.95551i	0.209758	−	0.977753i	$-0.432732\pi$
$660$	0	0
$661$	368.859	0.558032	0.279016	−	0.960287i	$-0.409992\pi$
$661$	368.859	0.558032	0.279016	+	0.960287i	$0.409992\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−131.827	−0.198236
$666$	0	0
$667$	− 764.766i	− 1.14658i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 387.237i	− 0.577104i
$672$	0	0
$673$	−255.724	−0.379976	−0.189988	−	0.981786i	$-0.560845\pi$
$673$	−255.724	−0.379976	−0.189988	+	0.981786i	$0.560845\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1133.14	1.67377	0.836885	−	0.547379i	$-0.184374\pi$
$677$	1133.14	1.67377	0.836885	+	0.547379i	$0.184374\pi$
$678$	0	0
$679$	− 738.608i	− 1.08779i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 941.046i	− 1.37781i	−0.724850	−	0.688907i	$-0.758091\pi$
$683$	− 941.046i	− 1.37781i	0.724850	−	0.688907i	$-0.241909\pi$
$684$	0	0
$685$	−32.9379	−0.0480845
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−201.745	−0.292809
$690$	0	0
$691$	− 1343.85i	− 1.94479i	−0.233351	−	0.972393i	$-0.574969\pi$
$691$	− 1343.85i	− 1.94479i	0.233351	−	0.972393i	$-0.425031\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	162.367i	0.233621i
$696$	0	0
$697$	−1548.07	−2.22105
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.1783	0.0401973	0.0200986	−	0.999798i	$-0.493602\pi$
$701$	28.1783	0.0401973	0.0200986	+	0.999798i	$0.493602\pi$
$702$	0	0
$703$	1434.26i	2.04020i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 288.831i	− 0.408530i
$708$	0	0
$709$	−1249.32	−1.76209	−0.881043	−	0.473036i	$-0.843158\pi$
$709$	−1249.32	−1.76209	−0.881043	+	0.473036i	$0.843158\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.49403	0.00490046
$714$	0	0
$715$	49.1440i	0.0687328i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	738.132i	1.02661i	0.858206	+	0.513305i	$0.171579\pi$
$719$	738.132i	1.02661i	−0.858206	+	0.513305i	$0.828421\pi$
$720$	0	0
$721$	1020.11	1.41486
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−733.386	−1.01157
$726$	0	0
$727$	377.213i	0.518862i	0.965762	+	0.259431i	$0.0835350\pi$
$727$	377.213i	0.518862i	−0.965762	+	0.259431i	$0.916465\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	74.7933i	0.102316i
$732$	0	0
$733$	−561.698	−0.766300	−0.383150	−	0.923686i	$-0.625161\pi$
$733$	−561.698	−0.766300	−0.383150	+	0.923686i	$0.625161\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	409.478	0.555601
$738$	0	0
$739$	912.732i	1.23509i	0.786535	+	0.617545i	$0.211873\pi$
$739$	912.732i	1.23509i	−0.786535	+	0.617545i	$0.788127\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1054.25i	− 1.41891i	−0.704753	−	0.709453i	$-0.748942\pi$
$743$	− 1054.25i	− 1.41891i	0.704753	−	0.709453i	$-0.251058\pi$
$744$	0	0
$745$	−218.293	−0.293011
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−1254.23	−1.67454
$750$	0	0
$751$	1058.26i	1.40914i	0.709635	+	0.704569i	$0.248860\pi$
$751$	1058.26i	1.40914i	−0.709635	+	0.704569i	$0.751140\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 103.035i	− 0.136470i
$756$	0	0
$757$	−359.804	−0.475302	−0.237651	−	0.971351i	$-0.576377\pi$
$757$	−359.804	−0.475302	−0.237651	+	0.971351i	$0.576377\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−622.948	−0.818592	−0.409296	−	0.912402i	$-0.634226\pi$
$761$	−622.948	−0.818592	−0.409296	+	0.912402i	$0.634226\pi$
$762$	0	0
$763$	1005.71i	1.31809i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	598.522i	0.780342i
$768$	0	0
$769$	1068.91	1.38999	0.694997	−	0.719013i	$-0.255406\pi$
$769$	1068.91	1.38999	0.694997	+	0.719013i	$0.255406\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	512.261	0.662692	0.331346	−	0.943509i	$-0.392497\pi$
$773$	512.261	0.662692	0.331346	+	0.943509i	$0.392497\pi$
$774$	0	0
$775$	− 3.35066i	− 0.00432344i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1208.10i	− 1.55083i
$780$	0	0
$781$	550.520	0.704891
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−109.763	−0.139825
$786$	0	0
$787$	− 177.604i	− 0.225672i	−0.993614	−	0.112836i	$-0.964007\pi$
$787$	− 177.604i	− 0.225672i	0.993614	−	0.112836i	$-0.0359935\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1412.44i	− 1.78564i
$792$	0	0
$793$	323.439	0.407868
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−380.763	−0.477745	−0.238872	−	0.971051i	$-0.576778\pi$
$797$	−380.763	−0.477745	−0.238872	+	0.971051i	$0.576778\pi$
$798$	0	0
$799$	− 2152.61i	− 2.69413i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 178.162i	− 0.221870i
$804$	0	0
$805$	161.625	0.200776
$806$	0	0
$807$	0	0
$808$	0	0
$809$	720.850	0.891039	0.445519	−	0.895272i	$-0.353019\pi$
$809$	720.850	0.891039	0.445519	+	0.895272i	$0.353019\pi$
$810$	0	0
$811$	1048.82i	1.29324i	0.762813	+	0.646619i	$0.223818\pi$
$811$	1048.82i	1.29324i	−0.762813	+	0.646619i	$0.776182\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 18.5025i	− 0.0227025i
$816$	0	0
$817$	−58.3677	−0.0714415
$818$	0	0
$819$	0	0
$820$	0	0
$821$	723.332	0.881038	0.440519	−	0.897743i	$-0.354794\pi$
$821$	723.332	0.881038	0.440519	+	0.897743i	$0.354794\pi$
$822$	0	0
$823$	− 1241.19i	− 1.50813i	−0.656799	−	0.754066i	$-0.728090\pi$
$823$	− 1241.19i	− 1.50813i	0.656799	−	0.754066i	$-0.271910\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 348.111i	− 0.420932i	−0.977601	−	0.210466i	$-0.932502\pi$
$827$	− 348.111i	− 0.420932i	0.977601	−	0.210466i	$-0.0674981\pi$
$828$	0	0
$829$	731.151	0.881968	0.440984	−	0.897515i	$-0.354630\pi$
$829$	731.151	0.881968	0.440984	+	0.897515i	$0.354630\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−19.2002	−0.0230495
$834$	0	0
$835$	− 159.089i	− 0.190525i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 650.259i	− 0.775041i	−0.921861	−	0.387520i	$-0.873332\pi$
$839$	− 650.259i	− 0.775041i	0.921861	−	0.387520i	$-0.126668\pi$
$840$	0	0
$841$	79.4357	0.0944539
$842$	0	0
$843$	0	0
$844$	0	0
$845$	112.612	0.133268
$846$	0	0
$847$	− 396.938i	− 0.468640i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1758.45i	− 2.06634i
$852$	0	0
$853$	1328.55	1.55750	0.778750	−	0.627334i	$-0.215854\pi$
$853$	1328.55	1.55750	0.778750	+	0.627334i	$0.215854\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−646.676	−0.754581	−0.377290	−	0.926095i	$-0.623144\pi$
$857$	−646.676	−0.754581	−0.377290	+	0.926095i	$0.623144\pi$
$858$	0	0
$859$	609.083i	0.709061i	0.935044	+	0.354530i	$0.115359\pi$
$859$	609.083i	0.709061i	−0.935044	+	0.354530i	$0.884641\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 93.0849i	− 0.107862i	−0.998545	−	0.0539310i	$-0.982825\pi$
$863$	− 93.0849i	− 0.107862i	0.998545	−	0.0539310i	$-0.0171751\pi$
$864$	0	0
$865$	−101.683	−0.117553
$866$	0	0
$867$	0	0
$868$	0	0
$869$	319.854	0.368071
$870$	0	0
$871$	342.016i	0.392671i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 315.286i	− 0.360327i
$876$	0	0
$877$	−661.368	−0.754125	−0.377063	−	0.926188i	$-0.623066\pi$
$877$	−661.368	−0.754125	−0.377063	+	0.926188i	$0.623066\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	659.388	0.748455	0.374227	−	0.927337i	$-0.377908\pi$
$881$	659.388	0.748455	0.374227	+	0.927337i	$0.377908\pi$
$882$	0	0
$883$	− 776.943i	− 0.879890i	−0.898025	−	0.439945i	$-0.854998\pi$
$883$	− 776.943i	− 0.879890i	0.898025	−	0.439945i	$-0.145002\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	691.491i	0.779584i	0.920903	+	0.389792i	$0.127453\pi$
$887$	691.491i	0.779584i	−0.920903	+	0.389792i	$0.872547\pi$
$888$	0	0
$889$	−1281.49	−1.44149
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1679.87	1.88115
$894$	0	0
$895$	16.9970i	0.0189910i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.20525i	0.00467769i
$900$	0	0
$901$	791.069	0.877990
$902$	0	0
$903$	0	0
$904$	0	0
$905$	89.1529	0.0985115
$906$	0	0
$907$	− 94.0706i	− 0.103716i	−0.998654	−	0.0518581i	$-0.983486\pi$
$907$	− 94.0706i	− 0.103716i	0.998654	−	0.0518581i	$-0.0165144\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 945.098i	− 1.03743i	−0.854948	−	0.518714i	$-0.826411\pi$
$911$	− 945.098i	− 1.03743i	0.854948	−	0.518714i	$-0.173589\pi$
$912$	0	0
$913$	−214.387	−0.234816
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−431.047	−0.470062
$918$	0	0
$919$	− 29.1099i	− 0.0316757i	−0.999875	−	0.0158378i	$-0.994958\pi$
$919$	− 29.1099i	− 0.0316757i	0.999875	−	0.0158378i	$-0.00504155\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	459.821i	0.498181i
$924$	0	0
$925$	−1686.30	−1.82303
$926$	0	0
$927$	0	0
$928$	0	0
$929$	767.537	0.826197	0.413099	−	0.910686i	$-0.364446\pi$
$929$	767.537	0.826197	0.413099	+	0.910686i	$0.364446\pi$
$930$	0	0
$931$	− 14.9836i	− 0.0160941i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 192.700i	− 0.206096i
$936$	0	0
$937$	−736.778	−0.786316	−0.393158	−	0.919471i	$-0.628617\pi$
$937$	−736.778	−0.786316	−0.393158	+	0.919471i	$0.628617\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1062.70	1.12933	0.564667	−	0.825319i	$-0.309005\pi$
$941$	1062.70	1.12933	0.564667	+	0.825319i	$0.309005\pi$
$942$	0	0
$943$	1481.17i	1.57070i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 576.522i	− 0.608788i	−0.952546	−	0.304394i	$-0.901546\pi$
$947$	− 576.522i	− 0.608788i	0.952546	−	0.304394i	$-0.0984539\pi$
$948$	0	0
$949$	148.809	0.156807
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−677.961	−0.711397	−0.355698	−	0.934601i	$-0.615757\pi$
$953$	−677.961	−0.711397	−0.355698	+	0.934601i	$0.615757\pi$
$954$	0	0
$955$	252.720i	0.264629i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 255.463i	− 0.266385i
$960$	0	0
$961$	960.981	0.999980
$962$	0	0
$963$	0	0
$964$	0	0
$965$	147.528	0.152879
$966$	0	0
$967$	1331.66i	1.37710i	0.725187	+	0.688552i	$0.241753\pi$
$967$	1331.66i	1.37710i	−0.725187	+	0.688552i	$0.758247\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 385.062i	− 0.396563i	−0.980145	−	0.198281i	$-0.936464\pi$
$971$	− 385.062i	− 0.396563i	0.980145	−	0.198281i	$-0.0635360\pi$
$972$	0	0
$973$	−1259.30	−1.29424
$974$	0	0
$975$	0	0
$976$	0	0
$977$	285.101	0.291813	0.145906	−	0.989298i	$-0.453390\pi$
$977$	285.101	0.291813	0.145906	+	0.989298i	$0.453390\pi$
$978$	0	0
$979$	207.484i	0.211935i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1344.56i	− 1.36781i	−0.729570	−	0.683907i	$-0.760280\pi$
$983$	− 1344.56i	− 1.36781i	0.729570	−	0.683907i	$-0.239720\pi$
$984$	0	0
$985$	−96.5430	−0.0980132
$986$	0	0
$987$	0	0
$988$	0	0
$989$	71.5610	0.0723570
$990$	0	0
$991$	− 1457.00i	− 1.47023i	−0.677944	−	0.735114i	$-0.737129\pi$
$991$	− 1457.00i	− 1.47023i	0.677944	−	0.735114i	$-0.262871\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	57.6340i	0.0579236i
$996$	0	0
$997$	−321.379	−0.322346	−0.161173	−	0.986926i	$-0.551528\pi$
$997$	−321.379	−0.322346	−0.161173	+	0.986926i	$0.551528\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.g.k.1135.3		8
3.2	odd	2		1296.3.g.j.1135.5		8
4.3	odd	2	inner	1296.3.g.k.1135.4		8
9.2	odd	6		144.3.o.a.31.2	✓	8
9.4	even	3		432.3.o.a.127.3		8
9.5	odd	6		144.3.o.c.79.3	yes	8
9.7	even	3		432.3.o.b.415.3		8
12.11	even	2		1296.3.g.j.1135.6		8
36.7	odd	6		432.3.o.a.415.3		8
36.11	even	6		144.3.o.c.31.3	yes	8
36.23	even	6		144.3.o.a.79.2	yes	8
36.31	odd	6		432.3.o.b.127.3		8
72.5	odd	6		576.3.o.d.511.2		8
72.11	even	6		576.3.o.d.319.2		8
72.13	even	6		1728.3.o.e.127.2		8
72.29	odd	6		576.3.o.f.319.3		8
72.43	odd	6		1728.3.o.e.1279.2		8
72.59	even	6		576.3.o.f.511.3		8
72.61	even	6		1728.3.o.f.1279.2		8
72.67	odd	6		1728.3.o.f.127.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.o.a.31.2	✓	8	9.2	odd	6
144.3.o.a.79.2	yes	8	36.23	even	6
144.3.o.c.31.3	yes	8	36.11	even	6
144.3.o.c.79.3	yes	8	9.5	odd	6
432.3.o.a.127.3		8	9.4	even	3
432.3.o.a.415.3		8	36.7	odd	6
432.3.o.b.127.3		8	36.31	odd	6
432.3.o.b.415.3		8	9.7	even	3
576.3.o.d.319.2		8	72.11	even	6
576.3.o.d.511.2		8	72.5	odd	6
576.3.o.f.319.3		8	72.29	odd	6
576.3.o.f.511.3		8	72.59	even	6
1296.3.g.j.1135.5		8	3.2	odd	2
1296.3.g.j.1135.6		8	12.11	even	2
1296.3.g.k.1135.3		8	1.1	even	1	trivial
1296.3.g.k.1135.4		8	4.3	odd	2	inner
1728.3.o.e.127.2		8	72.13	even	6
1728.3.o.e.1279.2		8	72.43	odd	6
1728.3.o.f.127.2		8	72.67	odd	6
1728.3.o.f.1279.2		8	72.61	even	6

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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1296.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.909226 q^{5} -7.05186i q^{7} +O(q^{10})\)	\(q-0.909226 q^{5} -7.05186i q^{7} +8.04435i q^{11} -6.71904 q^{13} +26.3462 q^{17} +20.5603i q^{19} -25.2076i q^{23} -24.1733 q^{25} +30.3387 q^{29} +0.138610i q^{31} +6.41173i q^{35} +69.7588 q^{37} -58.7588 q^{41} +2.83886i q^{43} -81.7046i q^{47} -0.728766 q^{49} +30.0259 q^{53} -7.31413i q^{55} -89.0785i q^{59} -48.1377 q^{61} +6.10913 q^{65} -50.9025i q^{67} -68.4355i q^{71} -22.1474 q^{73} +56.7277 q^{77} -39.7613i q^{79} +26.6506i q^{83} -23.9547 q^{85} +25.7926 q^{89} +47.3818i q^{91} -18.6939i q^{95} +104.739 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{5}+O(q^{10}) \) 8 * q + 6 * q^5	\( 8 q + 6 q^{5} - 10 q^{13} - 6 q^{17} + 46 q^{25} + 138 q^{29} - 20 q^{37} + 108 q^{41} - 82 q^{49} + 252 q^{53} - 14 q^{61} + 186 q^{65} + 74 q^{73} + 414 q^{77} + 60 q^{85} + 168 q^{89} + 20 q^{97}+O(q^{100}) \) 8 * q + 6 * q^5 - 10 * q^13 - 6 * q^17 + 46 * q^25 + 138 * q^29 - 20 * q^37 + 108 * q^41 - 82 * q^49 + 252 * q^53 - 14 * q^61 + 186 * q^65 + 74 * q^73 + 414 * q^77 + 60 * q^85 + 168 * q^89 + 20 * q^97

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