LMFDB - Embedded newform 1296.3.g.c.1135.2

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:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1135.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1296.1135
Dual form			1296.3.g.c.1135.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.73205 q^{5} +9.46410i q^{7} +O(q^{10})$	$q-7.73205 q^{5} +9.46410i q^{7} -16.3923i q^{11} +17.3923 q^{13} -20.6603 q^{17} -33.4641i q^{19} +7.60770i q^{23} +34.7846 q^{25} +7.73205 q^{29} +32.7846i q^{31} -73.1769i q^{35} -26.1769 q^{37} +41.5692 q^{41} +37.1769i q^{43} -24.0000i q^{47} -40.5692 q^{49} +15.2154 q^{53} +126.746i q^{55} +96.0000i q^{59} -34.9615 q^{61} -134.478 q^{65} +40.8897i q^{67} +51.5307i q^{71} +132.138 q^{73} +155.138 q^{77} +49.6743i q^{79} +91.9230i q^{83} +159.746 q^{85} -77.6936 q^{89} +164.603i q^{91} +258.746i q^{95} -43.5692 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 24 q^{5}+O(q^{10}) $ 4 * q - 24 * q^5	$ 4 q - 24 q^{5} + 28 q^{13} - 48 q^{17} + 56 q^{25} + 24 q^{29} + 20 q^{37} + 4 q^{49} + 144 q^{53} + 68 q^{61} - 240 q^{65} + 196 q^{73} + 288 q^{77} + 348 q^{85} - 96 q^{89} - 8 q^{97}+O(q^{100}) $ 4 * q - 24 * q^5 + 28 * q^13 - 48 * q^17 + 56 * q^25 + 24 * q^29 + 20 * q^37 + 4 * q^49 + 144 * q^53 + 68 * q^61 - 240 * q^65 + 196 * q^73 + 288 * q^77 + 348 * q^85 - 96 * q^89 - 8 * 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
$5$	−7.73205	−1.54641	−0.773205	−	0.634156i	$-0.781348\pi$
$5$	−7.73205	−1.54641	−0.773205	+	0.634156i	$0.781348\pi$
$6$	0	0
$7$	9.46410i	1.35201i	0.736895	+	0.676007i	$0.236291\pi$
$7$	9.46410i	1.35201i	−0.736895	+	0.676007i	$0.763709\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 16.3923i	− 1.49021i	−0.666947	−	0.745105i	$-0.732399\pi$
$11$	− 16.3923i	− 1.49021i	0.666947	−	0.745105i	$-0.267601\pi$
$12$	0	0
$13$	17.3923	1.33787	0.668935	−	0.743321i	$-0.266751\pi$
$13$	17.3923	1.33787	0.668935	+	0.743321i	$0.266751\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−20.6603	−1.21531	−0.607655	−	0.794201i	$-0.707889\pi$
$17$	−20.6603	−1.21531	−0.607655	+	0.794201i	$0.707889\pi$
$18$	0	0
$19$	− 33.4641i	− 1.76127i	−0.473797	−	0.880634i	$-0.657117\pi$
$19$	− 33.4641i	− 1.76127i	0.473797	−	0.880634i	$-0.342883\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	7.60770i	0.330769i	0.986229	+	0.165385i	$0.0528866\pi$
$23$	7.60770i	0.330769i	−0.986229	+	0.165385i	$0.947113\pi$
$24$	0	0
$25$	34.7846	1.39138
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.73205	0.266622	0.133311	−	0.991074i	$-0.457439\pi$
$29$	7.73205	0.266622	0.133311	+	0.991074i	$0.457439\pi$
$30$	0	0
$31$	32.7846i	1.05757i	0.848756	+	0.528784i	$0.177352\pi$
$31$	32.7846i	1.05757i	−0.848756	+	0.528784i	$0.822648\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 73.1769i	− 2.09077i
$36$	0	0
$37$	−26.1769	−0.707484	−0.353742	−	0.935343i	$-0.615091\pi$
$37$	−26.1769	−0.707484	−0.353742	+	0.935343i	$0.615091\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	41.5692	1.01388	0.506942	−	0.861980i	$-0.330776\pi$
$41$	41.5692	1.01388	0.506942	+	0.861980i	$0.330776\pi$
$42$	0	0
$43$	37.1769i	0.864579i	0.901735	+	0.432290i	$0.142294\pi$
$43$	37.1769i	0.864579i	−0.901735	+	0.432290i	$0.857706\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 24.0000i	− 0.510638i	−0.966857	−	0.255319i	$-0.917819\pi$
$47$	− 24.0000i	− 0.510638i	0.966857	−	0.255319i	$-0.0821805\pi$
$48$	0	0
$49$	−40.5692	−0.827943
$50$	0	0
$51$	0	0
$52$	0	0
$53$	15.2154	0.287083	0.143541	−	0.989644i	$-0.454151\pi$
$53$	15.2154	0.287083	0.143541	+	0.989644i	$0.454151\pi$
$54$	0	0
$55$	126.746i	2.30448i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	96.0000i	1.62712i	0.581482	+	0.813559i	$0.302473\pi$
$59$	96.0000i	1.62712i	−0.581482	+	0.813559i	$0.697527\pi$
$60$	0	0
$61$	−34.9615	−0.573140	−0.286570	−	0.958059i	$-0.592515\pi$
$61$	−34.9615	−0.573140	−0.286570	+	0.958059i	$0.592515\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−134.478	−2.06890
$66$	0	0
$67$	40.8897i	0.610294i	0.952305	+	0.305147i	$0.0987057\pi$
$67$	40.8897i	0.610294i	−0.952305	+	0.305147i	$0.901294\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	51.5307i	0.725785i	0.931831	+	0.362893i	$0.118211\pi$
$71$	51.5307i	0.725785i	−0.931831	+	0.362893i	$0.881789\pi$
$72$	0	0
$73$	132.138	1.81012	0.905058	−	0.425289i	$-0.139827\pi$
$73$	132.138	1.81012	0.905058	+	0.425289i	$0.139827\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	155.138	2.01478
$78$	0	0
$79$	49.6743i	0.628789i	0.949292	+	0.314395i	$0.101801\pi$
$79$	49.6743i	0.628789i	−0.949292	+	0.314395i	$0.898199\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	91.9230i	1.10751i	0.832681	+	0.553753i	$0.186805\pi$
$83$	91.9230i	1.10751i	−0.832681	+	0.553753i	$0.813195\pi$
$84$	0	0
$85$	159.746	1.87937
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−77.6936	−0.872962	−0.436481	−	0.899714i	$-0.643775\pi$
$89$	−77.6936	−0.872962	−0.436481	+	0.899714i	$0.643775\pi$
$90$	0	0
$91$	164.603i	1.80882i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	258.746i	2.72364i
$96$	0	0
$97$	−43.5692	−0.449167	−0.224584	−	0.974455i	$-0.572102\pi$
$97$	−43.5692	−0.449167	−0.224584	+	0.974455i	$0.572102\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	170.354	1.68667	0.843336	−	0.537387i	$-0.180589\pi$
$101$	170.354	1.68667	0.843336	+	0.537387i	$0.180589\pi$
$102$	0	0
$103$	40.2102i	0.390391i	0.980764	+	0.195195i	$0.0625341\pi$
$103$	40.2102i	0.390391i	−0.980764	+	0.195195i	$0.937466\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 152.785i	− 1.42789i	−0.700200	−	0.713947i	$-0.746906\pi$
$107$	− 152.785i	− 1.42789i	0.700200	−	0.713947i	$-0.253094\pi$
$108$	0	0
$109$	80.6077	0.739520	0.369760	−	0.929127i	$-0.379440\pi$
$109$	80.6077	0.739520	0.369760	+	0.929127i	$0.379440\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−24.7372	−0.218913	−0.109457	−	0.993992i	$-0.534911\pi$
$113$	−24.7372	−0.218913	−0.109457	+	0.993992i	$0.534911\pi$
$114$	0	0
$115$	− 58.8231i	− 0.511505i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 195.531i	− 1.64312i
$120$	0	0
$121$	−147.708	−1.22072
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−75.6551	−0.605241
$126$	0	0
$127$	− 26.0385i	− 0.205027i	−0.994732	−	0.102514i	$-0.967311\pi$
$127$	− 26.0385i	− 0.205027i	0.994732	−	0.102514i	$-0.0326885\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	217.177i	1.65784i	0.559368	+	0.828920i	$0.311044\pi$
$131$	217.177i	1.65784i	−0.559368	+	0.828920i	$0.688956\pi$
$132$	0	0
$133$	316.708	2.38126
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.52182	0.0695023	0.0347512	−	0.999396i	$-0.488936\pi$
$137$	9.52182	0.0695023	0.0347512	+	0.999396i	$0.488936\pi$
$138$	0	0
$139$	22.6410i	0.162885i	0.996678	+	0.0814425i	$0.0259527\pi$
$139$	22.6410i	0.162885i	−0.996678	+	0.0814425i	$0.974047\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 285.100i	− 1.99371i
$144$	0	0
$145$	−59.7846	−0.412308
$146$	0	0
$147$	0	0
$148$	0	0
$149$	64.5167	0.432998	0.216499	−	0.976283i	$-0.430536\pi$
$149$	64.5167	0.432998	0.216499	+	0.976283i	$0.430536\pi$
$150$	0	0
$151$	198.067i	1.31170i	0.754891	+	0.655850i	$0.227690\pi$
$151$	198.067i	1.31170i	−0.754891	+	0.655850i	$0.772310\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 253.492i	− 1.63543i
$156$	0	0
$157$	284.100	1.80955	0.904777	−	0.425886i	$-0.140037\pi$
$157$	284.100	1.80955	0.904777	+	0.425886i	$0.140037\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−72.0000	−0.447205
$162$	0	0
$163$	− 23.0052i	− 0.141136i	−0.997507	−	0.0705680i	$-0.977519\pi$
$163$	− 23.0052i	− 0.141136i	0.997507	−	0.0705680i	$-0.0224812\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 217.808i	− 1.30424i	−0.758117	−	0.652119i	$-0.773880\pi$
$167$	− 217.808i	− 1.30424i	0.758117	−	0.652119i	$-0.226120\pi$
$168$	0	0
$169$	133.492	0.789895
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−109.914	−0.635342	−0.317671	−	0.948201i	$-0.602901\pi$
$173$	−109.914	−0.635342	−0.317671	+	0.948201i	$0.602901\pi$
$174$	0	0
$175$	329.205i	1.88117i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	190.277i	1.06300i	0.847059	+	0.531500i	$0.178371\pi$
$179$	190.277i	1.06300i	−0.847059	+	0.531500i	$0.821629\pi$
$180$	0	0
$181$	34.8616	0.192605	0.0963027	−	0.995352i	$-0.469298\pi$
$181$	34.8616	0.192605	0.0963027	+	0.995352i	$0.469298\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	202.401	1.09406
$186$	0	0
$187$	338.669i	1.81107i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	165.100i	0.864398i	0.901778	+	0.432199i	$0.142262\pi$
$191$	165.100i	0.864398i	−0.901778	+	0.432199i	$0.857738\pi$
$192$	0	0
$193$	42.9230	0.222399	0.111200	−	0.993798i	$-0.464531\pi$
$193$	42.9230	0.222399	0.111200	+	0.993798i	$0.464531\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−147.406	−0.748256	−0.374128	−	0.927377i	$-0.622058\pi$
$197$	−147.406	−0.748256	−0.374128	+	0.927377i	$0.622058\pi$
$198$	0	0
$199$	− 278.851i	− 1.40126i	−0.713524	−	0.700631i	$-0.752902\pi$
$199$	− 278.851i	− 1.40126i	0.713524	−	0.700631i	$-0.247098\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	73.1769i	0.360477i
$204$	0	0
$205$	−321.415	−1.56788
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−548.554	−2.62466
$210$	0	0
$211$	131.818i	0.624730i	0.949962	+	0.312365i	$0.101121\pi$
$211$	131.818i	0.624730i	−0.949962	+	0.312365i	$0.898879\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 287.454i	− 1.33699i
$216$	0	0
$217$	−310.277	−1.42985
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−359.329	−1.62593
$222$	0	0
$223$	34.8231i	0.156157i	0.996947	+	0.0780787i	$0.0248785\pi$
$223$	34.8231i	0.156157i	−0.996947	+	0.0780787i	$0.975121\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 42.7461i	− 0.188309i	−0.995558	−	0.0941545i	$-0.969985\pi$
$227$	− 42.7461i	− 0.188309i	0.995558	−	0.0941545i	$-0.0300148\pi$
$228$	0	0
$229$	−192.808	−0.841955	−0.420977	−	0.907071i	$-0.638313\pi$
$229$	−192.808	−0.841955	−0.420977	+	0.907071i	$0.638313\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	304.583	1.30722	0.653612	−	0.756830i	$-0.273253\pi$
$233$	304.583	1.30722	0.653612	+	0.756830i	$0.273253\pi$
$234$	0	0
$235$	185.569i	0.789656i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	240.631i	1.00682i	0.864047	+	0.503412i	$0.167922\pi$
$239$	240.631i	1.00682i	−0.864047	+	0.503412i	$0.832078\pi$
$240$	0	0
$241$	−294.061	−1.22017	−0.610086	−	0.792335i	$-0.708865\pi$
$241$	−294.061	−1.22017	−0.610086	+	0.792335i	$0.708865\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	313.683	1.28034
$246$	0	0
$247$	− 582.018i	− 2.35635i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 51.5307i	− 0.205302i	−0.994717	−	0.102651i	$-0.967268\pi$
$251$	− 51.5307i	− 0.205302i	0.994717	−	0.102651i	$-0.0327324\pi$
$252$	0	0
$253$	124.708	0.492916
$254$	0	0
$255$	0	0
$256$	0	0
$257$	459.722	1.78880	0.894400	−	0.447267i	$-0.147603\pi$
$257$	459.722	1.78880	0.894400	+	0.447267i	$0.147603\pi$
$258$	0	0
$259$	− 247.741i	− 0.956529i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	351.846i	1.33782i	0.743344	+	0.668909i	$0.233238\pi$
$263$	351.846i	1.33782i	−0.743344	+	0.668909i	$0.766762\pi$
$264$	0	0
$265$	−117.646	−0.443948
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−219.655	−0.816562	−0.408281	−	0.912856i	$-0.633872\pi$
$269$	−219.655	−0.816562	−0.408281	+	0.912856i	$0.633872\pi$
$270$	0	0
$271$	355.244i	1.31086i	0.755255	+	0.655431i	$0.227513\pi$
$271$	355.244i	1.31086i	−0.755255	+	0.655431i	$0.772487\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 570.200i	− 2.07345i
$276$	0	0
$277$	415.415	1.49969	0.749847	−	0.661611i	$-0.230127\pi$
$277$	415.415	1.49969	0.749847	+	0.661611i	$0.230127\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	240.986	0.857601	0.428801	−	0.903399i	$-0.358936\pi$
$281$	240.986	0.857601	0.428801	+	0.903399i	$0.358936\pi$
$282$	0	0
$283$	107.138i	0.378581i	0.981921	+	0.189291i	$0.0606188\pi$
$283$	107.138i	0.378581i	−0.981921	+	0.189291i	$0.939381\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	393.415i	1.37079i
$288$	0	0
$289$	137.846	0.476976
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−366.640	−1.25133	−0.625665	−	0.780092i	$-0.715172\pi$
$293$	−366.640	−1.25133	−0.625665	+	0.780092i	$0.715172\pi$
$294$	0	0
$295$	− 742.277i	− 2.51619i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	132.315i	0.442526i
$300$	0	0
$301$	−351.846	−1.16892
$302$	0	0
$303$	0	0
$304$	0	0
$305$	270.324	0.886309
$306$	0	0
$307$	− 173.338i	− 0.564620i	−0.959323	−	0.282310i	$-0.908899\pi$
$307$	− 173.338i	− 0.564620i	0.959323	−	0.282310i	$-0.0911007\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	107.769i	0.346525i	0.984876	+	0.173262i	$0.0554308\pi$
$311$	107.769i	0.346525i	−0.984876	+	0.173262i	$0.944569\pi$
$312$	0	0
$313$	−65.9385	−0.210666	−0.105333	−	0.994437i	$-0.533591\pi$
$313$	−65.9385	−0.210666	−0.105333	+	0.994437i	$0.533591\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	547.883	1.72834	0.864169	−	0.503202i	$-0.167845\pi$
$317$	547.883	1.72834	0.864169	+	0.503202i	$0.167845\pi$
$318$	0	0
$319$	− 126.746i	− 0.397323i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	691.377i	2.14049i
$324$	0	0
$325$	604.985	1.86149
$326$	0	0
$327$	0	0
$328$	0	0
$329$	227.138	0.690390
$330$	0	0
$331$	− 159.167i	− 0.480866i	−0.970666	−	0.240433i	$-0.922711\pi$
$331$	− 159.167i	− 0.480866i	0.970666	−	0.240433i	$-0.0772894\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 316.161i	− 0.943766i
$336$	0	0
$337$	634.400	1.88249	0.941246	−	0.337722i	$-0.109656\pi$
$337$	634.400	1.88249	0.941246	+	0.337722i	$0.109656\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	537.415	1.57600
$342$	0	0
$343$	79.7898i	0.232623i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 30.9770i	− 0.0892709i	−0.999003	−	0.0446354i	$-0.985787\pi$
$347$	− 30.9770i	− 0.0892709i	0.999003	−	0.0446354i	$-0.0142126\pi$
$348$	0	0
$349$	176.431	0.505532	0.252766	−	0.967527i	$-0.418660\pi$
$349$	176.431	0.505532	0.252766	+	0.967527i	$0.418660\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	38.5847	0.109305	0.0546525	−	0.998505i	$-0.482595\pi$
$353$	38.5847	0.109305	0.0546525	+	0.998505i	$0.482595\pi$
$354$	0	0
$355$	− 398.438i	− 1.12236i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	573.100i	1.59638i	0.602407	+	0.798189i	$0.294209\pi$
$359$	573.100i	1.59638i	−0.602407	+	0.798189i	$0.705791\pi$
$360$	0	0
$361$	−758.846	−2.10207
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1021.70	−2.79918
$366$	0	0
$367$	− 273.051i	− 0.744009i	−0.928231	−	0.372004i	$-0.878671\pi$
$367$	− 273.051i	− 0.744009i	0.928231	−	0.372004i	$-0.121329\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	144.000i	0.388140i
$372$	0	0
$373$	−210.554	−0.564487	−0.282244	−	0.959343i	$-0.591079\pi$
$373$	−210.554	−0.564487	−0.282244	+	0.959343i	$0.591079\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	134.478	0.356706
$378$	0	0
$379$	− 310.641i	− 0.819633i	−0.912168	−	0.409817i	$-0.865593\pi$
$379$	− 310.641i	− 0.819633i	0.912168	−	0.409817i	$-0.134407\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 465.962i	− 1.21661i	−0.793703	−	0.608305i	$-0.791850\pi$
$383$	− 465.962i	− 1.21661i	0.793703	−	0.608305i	$-0.208150\pi$
$384$	0	0
$385$	−1199.54	−3.11568
$386$	0	0
$387$	0	0
$388$	0	0
$389$	90.1999	0.231876	0.115938	−	0.993256i	$-0.463013\pi$
$389$	90.1999	0.231876	0.115938	+	0.993256i	$0.463013\pi$
$390$	0	0
$391$	− 157.177i	− 0.401987i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 384.084i	− 0.972366i
$396$	0	0
$397$	89.7461	0.226061	0.113030	−	0.993592i	$-0.463944\pi$
$397$	89.7461	0.226061	0.113030	+	0.993592i	$0.463944\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−588.755	−1.46822	−0.734109	−	0.679032i	$-0.762400\pi$
$401$	−588.755	−1.46822	−0.734109	+	0.679032i	$0.762400\pi$
$402$	0	0
$403$	570.200i	1.41489i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	429.100i	1.05430i
$408$	0	0
$409$	−721.354	−1.76370	−0.881851	−	0.471529i	$-0.843702\pi$
$409$	−721.354	−1.76370	−0.881851	+	0.471529i	$0.843702\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−908.554	−2.19989
$414$	0	0
$415$	− 710.754i	− 1.71266i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 792.631i	− 1.89172i	−0.324575	−	0.945860i	$-0.605221\pi$
$419$	− 792.631i	− 1.89172i	0.324575	−	0.945860i	$-0.394779\pi$
$420$	0	0
$421$	397.946	0.945240	0.472620	−	0.881266i	$-0.343308\pi$
$421$	397.946	0.945240	0.472620	+	0.881266i	$0.343308\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−718.659	−1.69096
$426$	0	0
$427$	− 330.879i	− 0.774893i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 48.6307i	− 0.112832i	−0.998407	−	0.0564161i	$-0.982033\pi$
$431$	− 48.6307i	− 0.112832i	0.998407	−	0.0564161i	$-0.0179674\pi$
$432$	0	0
$433$	−379.200	−0.875750	−0.437875	−	0.899036i	$-0.644269\pi$
$433$	−379.200	−0.875750	−0.437875	+	0.899036i	$0.644269\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	254.585	0.582574
$438$	0	0
$439$	224.154i	0.510601i	0.966862	+	0.255301i	$0.0821744\pi$
$439$	224.154i	0.510601i	−0.966862	+	0.255301i	$0.917826\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	299.138i	0.675256i	0.941280	+	0.337628i	$0.109625\pi$
$443$	299.138i	0.675256i	−0.941280	+	0.337628i	$0.890375\pi$
$444$	0	0
$445$	600.731	1.34996
$446$	0	0
$447$	0	0
$448$	0	0
$449$	396.400	0.882850	0.441425	−	0.897298i	$-0.354473\pi$
$449$	396.400	0.882850	0.441425	+	0.897298i	$0.354473\pi$
$450$	0	0
$451$	− 681.415i	− 1.51090i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 1272.72i	− 2.79718i
$456$	0	0
$457$	43.1999	0.0945294	0.0472647	−	0.998882i	$-0.484950\pi$
$457$	43.1999	0.0945294	0.0472647	+	0.998882i	$0.484950\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−31.5232	−0.0683800	−0.0341900	−	0.999415i	$-0.510885\pi$
$461$	−31.5232	−0.0683800	−0.0341900	+	0.999415i	$0.510885\pi$
$462$	0	0
$463$	6.16421i	0.0133136i	0.999978	+	0.00665682i	$0.00211895\pi$
$463$	6.16421i	0.0133136i	−0.999978	+	0.00665682i	$0.997881\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 330.200i	− 0.707066i	−0.935422	−	0.353533i	$-0.884980\pi$
$467$	− 330.200i	− 0.707066i	0.935422	−	0.353533i	$-0.115020\pi$
$468$	0	0
$469$	−386.985	−0.825127
$470$	0	0
$471$	0	0
$472$	0	0
$473$	609.415	1.28840
$474$	0	0
$475$	− 1164.04i	− 2.45060i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 66.7461i	− 0.139345i	−0.997570	−	0.0696724i	$-0.977805\pi$
$479$	− 66.7461i	− 0.139345i	0.997570	−	0.0696724i	$-0.0221954\pi$
$480$	0	0
$481$	−455.277	−0.946522
$482$	0	0
$483$	0	0
$484$	0	0
$485$	336.879	0.694597
$486$	0	0
$487$	− 313.359i	− 0.643448i	−0.946834	−	0.321724i	$-0.895738\pi$
$487$	− 313.359i	− 0.643448i	0.946834	−	0.321724i	$-0.104262\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	172.161i	0.350634i	0.984512	+	0.175317i	$0.0560951\pi$
$491$	172.161i	0.350634i	−0.984512	+	0.175317i	$0.943905\pi$
$492$	0	0
$493$	−159.746	−0.324029
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−487.692	−0.981272
$498$	0	0
$499$	20.2384i	0.0405579i	0.999794	+	0.0202790i	$0.00645544\pi$
$499$	20.2384i	0.0405579i	−0.999794	+	0.0202790i	$0.993545\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	368.785i	0.733170i	0.930385	+	0.366585i	$0.119473\pi$
$503$	368.785i	0.733170i	−0.930385	+	0.366585i	$0.880527\pi$
$504$	0	0
$505$	−1317.18	−2.60829
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−234.200	−0.460118	−0.230059	−	0.973177i	$-0.573892\pi$
$509$	−234.200	−0.460118	−0.230059	+	0.973177i	$0.573892\pi$
$510$	0	0
$511$	1250.57i	2.44730i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 310.908i	− 0.603704i
$516$	0	0
$517$	−393.415	−0.760958
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1010.98	1.94047	0.970235	−	0.242167i	$-0.0778581\pi$
$521$	1010.98	1.94047	0.970235	+	0.242167i	$0.0778581\pi$
$522$	0	0
$523$	− 525.864i	− 1.00548i	−0.864439	−	0.502738i	$-0.832326\pi$
$523$	− 525.864i	− 1.00548i	0.864439	−	0.502738i	$-0.167674\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 677.338i	− 1.28527i
$528$	0	0
$529$	471.123	0.890592
$530$	0	0
$531$	0	0
$532$	0	0
$533$	722.985	1.35644
$534$	0	0
$535$	1181.34i	2.20811i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	665.023i	1.23381i
$540$	0	0
$541$	412.531	0.762534	0.381267	−	0.924465i	$-0.375488\pi$
$541$	412.531	0.762534	0.381267	+	0.924465i	$0.375488\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−623.263	−1.14360
$546$	0	0
$547$	− 939.251i	− 1.71710i	−0.512734	−	0.858548i	$-0.671367\pi$
$547$	− 939.251i	− 1.71710i	0.512734	−	0.858548i	$-0.328633\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 258.746i	− 0.469594i
$552$	0	0
$553$	−470.123	−0.850132
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−631.022	−1.13289	−0.566447	−	0.824098i	$-0.691682\pi$
$557$	−631.022	−1.13289	−0.566447	+	0.824098i	$0.691682\pi$
$558$	0	0
$559$	646.592i	1.15669i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	94.2769i	0.167454i	0.996489	+	0.0837272i	$0.0266824\pi$
$563$	94.2769i	0.167454i	−0.996489	+	0.0837272i	$0.973318\pi$
$564$	0	0
$565$	191.269	0.338530
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−42.9371	−0.0754607	−0.0377303	−	0.999288i	$-0.512013\pi$
$569$	−42.9371	−0.0754607	−0.0377303	+	0.999288i	$0.512013\pi$
$570$	0	0
$571$	− 423.118i	− 0.741012i	−0.928830	−	0.370506i	$-0.879184\pi$
$571$	− 423.118i	− 0.741012i	0.928830	−	0.370506i	$-0.120816\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	264.631i	0.460227i
$576$	0	0
$577$	937.523	1.62482	0.812411	−	0.583085i	$-0.198154\pi$
$577$	937.523	1.62482	0.812411	+	0.583085i	$0.198154\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−869.969	−1.49737
$582$	0	0
$583$	− 249.415i	− 0.427814i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	604.161i	1.02924i	0.857420	+	0.514618i	$0.172066\pi$
$587$	604.161i	1.02924i	−0.857420	+	0.514618i	$0.827934\pi$
$588$	0	0
$589$	1097.11	1.86266
$590$	0	0
$591$	0	0
$592$	0	0
$593$	91.5678	0.154415	0.0772073	−	0.997015i	$-0.475400\pi$
$593$	91.5678	0.154415	0.0772073	+	0.997015i	$0.475400\pi$
$594$	0	0
$595$	1511.85i	2.54093i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	949.031i	1.58436i	0.610289	+	0.792179i	$0.291053\pi$
$599$	949.031i	1.58436i	−0.610289	+	0.792179i	$0.708947\pi$
$600$	0	0
$601$	−362.723	−0.603533	−0.301766	−	0.953382i	$-0.597576\pi$
$601$	−362.723	−0.603533	−0.301766	+	0.953382i	$0.597576\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1142.08	1.88774
$606$	0	0
$607$	− 1051.24i	− 1.73187i	−0.500159	−	0.865934i	$-0.666725\pi$
$607$	− 1051.24i	− 1.73187i	0.500159	−	0.865934i	$-0.333275\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 417.415i	− 0.683167i
$612$	0	0
$613$	208.585	0.340269	0.170134	−	0.985421i	$-0.445580\pi$
$613$	208.585	0.340269	0.170134	+	0.985421i	$0.445580\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	689.845	1.11806	0.559031	−	0.829147i	$-0.311173\pi$
$617$	689.845	1.11806	0.559031	+	0.829147i	$0.311173\pi$
$618$	0	0
$619$	885.451i	1.43045i	0.698892	+	0.715227i	$0.253677\pi$
$619$	885.451i	1.43045i	−0.698892	+	0.715227i	$0.746323\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 735.300i	− 1.18026i
$624$	0	0
$625$	−284.646	−0.455434
$626$	0	0
$627$	0	0
$628$	0	0
$629$	540.822	0.859812
$630$	0	0
$631$	− 790.179i	− 1.25227i	−0.779717	−	0.626133i	$-0.784637\pi$
$631$	− 790.179i	− 1.25227i	0.779717	−	0.626133i	$-0.215363\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	201.331i	0.317056i
$636$	0	0
$637$	−705.592	−1.10768
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−364.601	−0.568801	−0.284400	−	0.958706i	$-0.591794\pi$
$641$	−364.601	−0.568801	−0.284400	+	0.958706i	$0.591794\pi$
$642$	0	0
$643$	− 1189.49i	− 1.84991i	−0.380076	−	0.924955i	$-0.624102\pi$
$643$	− 1189.49i	− 1.84991i	0.380076	−	0.924955i	$-0.375898\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	68.0075i	0.105112i	0.998618	+	0.0525561i	$0.0167368\pi$
$647$	68.0075i	0.105112i	−0.998618	+	0.0525561i	$0.983263\pi$
$648$	0	0
$649$	1573.66	2.42475
$650$	0	0
$651$	0	0
$652$	0	0
$653$	416.785	0.638261	0.319131	−	0.947711i	$-0.396609\pi$
$653$	416.785	0.638261	0.319131	+	0.947711i	$0.396609\pi$
$654$	0	0
$655$	− 1679.22i	− 2.56370i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1204.33i	1.82751i	0.406263	+	0.913756i	$0.366832\pi$
$659$	1204.33i	1.82751i	−0.406263	+	0.913756i	$0.633168\pi$
$660$	0	0
$661$	1205.52	1.82378	0.911888	−	0.410440i	$-0.134625\pi$
$661$	1205.52	1.82378	0.911888	+	0.410440i	$0.134625\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2448.80	−3.68241
$666$	0	0
$667$	58.8231i	0.0881905i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	573.100i	0.854098i
$672$	0	0
$673$	41.6616	0.0619044	0.0309522	−	0.999521i	$-0.490146\pi$
$673$	41.6616	0.0619044	0.0309522	+	0.999521i	$0.490146\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15.2154	−0.0224747	−0.0112374	−	0.999937i	$-0.503577\pi$
$677$	−15.2154	−0.0224747	−0.0112374	+	0.999937i	$0.503577\pi$
$678$	0	0
$679$	− 412.344i	− 0.607281i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	951.384i	1.39295i	0.717581	+	0.696475i	$0.245249\pi$
$683$	951.384i	1.39295i	−0.717581	+	0.696475i	$0.754751\pi$
$684$	0	0
$685$	−73.6232	−0.107479
$686$	0	0
$687$	0	0
$688$	0	0
$689$	264.631	0.384079
$690$	0	0
$691$	− 131.454i	− 0.190237i	−0.995466	−	0.0951185i	$-0.969677\pi$
$691$	− 131.454i	− 0.190237i	0.995466	−	0.0951185i	$-0.0303230\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 175.061i	− 0.251887i
$696$	0	0
$697$	−858.831	−1.23218
$698$	0	0
$699$	0	0
$700$	0	0
$701$	466.930	0.666091	0.333045	−	0.942911i	$-0.391924\pi$
$701$	466.930	0.666091	0.333045	+	0.942911i	$0.391924\pi$
$702$	0	0
$703$	875.987i	1.24607i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1612.25i	2.28040i
$708$	0	0
$709$	327.038	0.461267	0.230634	−	0.973041i	$-0.425920\pi$
$709$	327.038	0.461267	0.230634	+	0.973041i	$0.425920\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−249.415	−0.349811
$714$	0	0
$715$	2204.41i	3.08309i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 670.361i	− 0.932352i	−0.884692	−	0.466176i	$-0.845631\pi$
$719$	− 670.361i	− 0.932352i	0.884692	−	0.466176i	$-0.154369\pi$
$720$	0	0
$721$	−380.554	−0.527814
$722$	0	0
$723$	0	0
$724$	0	0
$725$	268.956	0.370974
$726$	0	0
$727$	693.500i	0.953920i	0.878925	+	0.476960i	$0.158261\pi$
$727$	693.500i	0.953920i	−0.878925	+	0.476960i	$0.841739\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 768.084i	− 1.05073i
$732$	0	0
$733$	−314.000	−0.428377	−0.214188	−	0.976792i	$-0.568711\pi$
$733$	−314.000	−0.428377	−0.214188	+	0.976792i	$0.568711\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	670.277	0.909467
$738$	0	0
$739$	233.569i	0.316061i	0.987434	+	0.158031i	$0.0505145\pi$
$739$	233.569i	0.316061i	−0.987434	+	0.158031i	$0.949486\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	927.846i	1.24878i	0.781111	+	0.624392i	$0.214653\pi$
$743$	927.846i	1.24878i	−0.781111	+	0.624392i	$0.785347\pi$
$744$	0	0
$745$	−498.846	−0.669592
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1445.97	1.93053
$750$	0	0
$751$	428.505i	0.570579i	0.958441	+	0.285290i	$0.0920898\pi$
$751$	428.505i	0.570579i	−0.958441	+	0.285290i	$0.907910\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1531.46i	− 2.02843i
$756$	0	0
$757$	−342.246	−0.452108	−0.226054	−	0.974115i	$-0.572583\pi$
$757$	−342.246	−0.452108	−0.226054	+	0.974115i	$0.572583\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	469.768	0.617303	0.308652	−	0.951175i	$-0.400122\pi$
$761$	469.768	0.617303	0.308652	+	0.951175i	$0.400122\pi$
$762$	0	0
$763$	762.879i	0.999842i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1669.66i	2.17687i
$768$	0	0
$769$	272.246	0.354026	0.177013	−	0.984208i	$-0.443356\pi$
$769$	272.246	0.354026	0.177013	+	0.984208i	$0.443356\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1135.27	1.46866	0.734328	−	0.678795i	$-0.237498\pi$
$773$	1135.27	1.46866	0.734328	+	0.678795i	$0.237498\pi$
$774$	0	0
$775$	1140.40i	1.47148i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1391.08i	− 1.78572i
$780$	0	0
$781$	844.708	1.08157
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−2196.68	−2.79831
$786$	0	0
$787$	613.905i	0.780057i	0.920803	+	0.390029i	$0.127535\pi$
$787$	613.905i	0.780057i	−0.920803	+	0.390029i	$0.872465\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 234.115i	− 0.295974i
$792$	0	0
$793$	−608.061	−0.766786
$794$	0	0
$795$	0	0
$796$	0	0
$797$	596.763	0.748761	0.374381	−	0.927275i	$-0.377855\pi$
$797$	596.763	0.748761	0.374381	+	0.927275i	$0.377855\pi$
$798$	0	0
$799$	495.846i	0.620583i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 2166.05i	− 2.69745i
$804$	0	0
$805$	556.708	0.691562
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23.8935	−0.0295346	−0.0147673	−	0.999891i	$-0.504701\pi$
$809$	−23.8935	−0.0295346	−0.0147673	+	0.999891i	$0.504701\pi$
$810$	0	0
$811$	548.092i	0.675822i	0.941178	+	0.337911i	$0.109720\pi$
$811$	548.092i	0.675822i	−0.941178	+	0.337911i	$0.890280\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	177.877i	0.218254i
$816$	0	0
$817$	1244.09	1.52276
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−976.055	−1.18886	−0.594431	−	0.804147i	$-0.702622\pi$
$821$	−976.055	−1.18886	−0.594431	+	0.804147i	$0.702622\pi$
$822$	0	0
$823$	1359.65i	1.65207i	0.563621	+	0.826033i	$0.309408\pi$
$823$	1359.65i	1.65207i	−0.563621	+	0.826033i	$0.690592\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 663.762i	− 0.802614i	−0.915944	−	0.401307i	$-0.868556\pi$
$827$	− 663.762i	− 0.802614i	0.915944	−	0.401307i	$-0.131444\pi$
$828$	0	0
$829$	−403.108	−0.486258	−0.243129	−	0.969994i	$-0.578174\pi$
$829$	−403.108	−0.486258	−0.243129	+	0.969994i	$0.578174\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	838.170	1.00621
$834$	0	0
$835$	1684.10i	2.01689i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1029.73i	− 1.22733i	−0.789566	−	0.613665i	$-0.789694\pi$
$839$	− 1029.73i	− 1.22733i	0.789566	−	0.613665i	$-0.210306\pi$
$840$	0	0
$841$	−781.215	−0.928912
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1032.17	−1.22150
$846$	0	0
$847$	− 1397.92i	− 1.65044i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 199.146i	− 0.234014i
$852$	0	0
$853$	467.415	0.547966	0.273983	−	0.961734i	$-0.411659\pi$
$853$	467.415	0.547966	0.273983	+	0.961734i	$0.411659\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−480.355	−0.560508	−0.280254	−	0.959926i	$-0.590419\pi$
$857$	−480.355	−0.560508	−0.280254	+	0.959926i	$0.590419\pi$
$858$	0	0
$859$	342.964i	0.399260i	0.979871	+	0.199630i	$0.0639739\pi$
$859$	342.964i	0.399260i	−0.979871	+	0.199630i	$0.936026\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 888.084i	− 1.02907i	−0.857470	−	0.514533i	$-0.827965\pi$
$863$	− 888.084i	− 1.02907i	0.857470	−	0.514533i	$-0.172035\pi$
$864$	0	0
$865$	849.862	0.982499
$866$	0	0
$867$	0	0
$868$	0	0
$869$	814.277	0.937027
$870$	0	0
$871$	711.167i	0.816494i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 716.008i	− 0.818294i
$876$	0	0
$877$	−709.946	−0.809517	−0.404758	−	0.914424i	$-0.632644\pi$
$877$	−709.946	−0.809517	−0.404758	+	0.914424i	$0.632644\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−996.862	−1.13151	−0.565756	−	0.824573i	$-0.691415\pi$
$881$	−996.862	−1.13151	−0.565756	+	0.824573i	$0.691415\pi$
$882$	0	0
$883$	− 897.513i	− 1.01644i	−0.861228	−	0.508218i	$-0.830304\pi$
$883$	− 897.513i	− 1.01644i	0.861228	−	0.508218i	$-0.169696\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1268.47i	1.43007i	0.699090	+	0.715033i	$0.253589\pi$
$887$	1268.47i	1.43007i	−0.699090	+	0.715033i	$0.746411\pi$
$888$	0	0
$889$	246.431	0.277200
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−803.138	−0.899371
$894$	0	0
$895$	− 1471.23i	− 1.64383i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	253.492i	0.281971i
$900$	0	0
$901$	−314.354	−0.348894
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−269.551	−0.297847
$906$	0	0
$907$	52.8052i	0.0582197i	0.999576	+	0.0291098i	$0.00926726\pi$
$907$	52.8052i	0.0582197i	−0.999576	+	0.0291098i	$0.990733\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	629.423i	0.690914i	0.938435	+	0.345457i	$0.112276\pi$
$911$	629.423i	0.690914i	−0.938435	+	0.345457i	$0.887724\pi$
$912$	0	0
$913$	1506.83	1.65042
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2055.38	−2.24142
$918$	0	0
$919$	1112.18i	1.21020i	0.796148	+	0.605101i	$0.206867\pi$
$919$	1112.18i	1.21020i	−0.796148	+	0.605101i	$0.793133\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	896.238i	0.971006i
$924$	0	0
$925$	−910.554	−0.984382
$926$	0	0
$927$	0	0
$928$	0	0
$929$	374.940	0.403595	0.201798	−	0.979427i	$-0.435322\pi$
$929$	374.940	0.403595	0.201798	+	0.979427i	$0.435322\pi$
$930$	0	0
$931$	1357.61i	1.45823i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 2618.61i	− 2.80065i
$936$	0	0
$937$	555.246	0.592578	0.296289	−	0.955098i	$-0.404251\pi$
$937$	555.246	0.592578	0.296289	+	0.955098i	$0.404251\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	646.486	0.687020	0.343510	−	0.939149i	$-0.388384\pi$
$941$	646.486	0.687020	0.343510	+	0.939149i	$0.388384\pi$
$942$	0	0
$943$	316.246i	0.335362i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 255.931i	− 0.270254i	−0.990828	−	0.135127i	$-0.956856\pi$
$947$	− 255.931i	− 0.270254i	0.990828	−	0.135127i	$-0.0431443\pi$
$948$	0	0
$949$	2298.19	2.42170
$950$	0	0
$951$	0	0
$952$	0	0
$953$	45.1220	0.0473473	0.0236736	−	0.999720i	$-0.492464\pi$
$953$	45.1220	0.0473473	0.0236736	+	0.999720i	$0.492464\pi$
$954$	0	0
$955$	− 1276.56i	− 1.33671i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	90.1154i	0.0939681i
$960$	0	0
$961$	−113.831	−0.118450
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−331.883	−0.343920
$966$	0	0
$967$	1029.60i	1.06473i	0.846514	+	0.532367i	$0.178697\pi$
$967$	1029.60i	1.06473i	−0.846514	+	0.532367i	$0.821303\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	405.731i	0.417848i	0.977932	+	0.208924i	$0.0669962\pi$
$971$	405.731i	0.417848i	−0.977932	+	0.208924i	$0.933004\pi$
$972$	0	0
$973$	−214.277	−0.220223
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1094.12	−1.11988	−0.559940	−	0.828533i	$-0.689176\pi$
$977$	−1094.12	−1.11988	−0.559940	+	0.828533i	$0.689176\pi$
$978$	0	0
$979$	1273.58i	1.30090i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	405.184i	0.412192i	0.978532	+	0.206096i	$0.0660759\pi$
$983$	405.184i	0.412192i	−0.978532	+	0.206096i	$0.933924\pi$
$984$	0	0
$985$	1139.75	1.15711
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−282.831	−0.285976
$990$	0	0
$991$	− 181.905i	− 0.183557i	−0.995779	−	0.0917786i	$-0.970745\pi$
$991$	− 181.905i	− 0.183557i	0.995779	−	0.0917786i	$-0.0292552\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2156.09i	2.16693i
$996$	0	0
$997$	877.038	0.879678	0.439839	−	0.898077i	$-0.355036\pi$
$997$	877.038	0.879678	0.439839	+	0.898077i	$0.355036\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.3.g.c.1135.2	yes	4
3.2	odd	2		1296.3.g.i.1135.4	yes	4
4.3	odd	2	inner	1296.3.g.c.1135.1	✓	4
9.2	odd	6		1296.3.o.r.271.1		4
9.4	even	3		1296.3.o.be.703.2		4
9.5	odd	6		1296.3.o.q.703.1		4
9.7	even	3		1296.3.o.bf.271.2		4
12.11	even	2		1296.3.g.i.1135.3	yes	4
36.7	odd	6		1296.3.o.be.271.2		4
36.11	even	6		1296.3.o.q.271.1		4
36.23	even	6		1296.3.o.r.703.1		4
36.31	odd	6		1296.3.o.bf.703.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1296.3.g.c.1135.1	✓	4	4.3	odd	2	inner
1296.3.g.c.1135.2	yes	4	1.1	even	1	trivial
1296.3.g.i.1135.3	yes	4	12.11	even	2
1296.3.g.i.1135.4	yes	4	3.2	odd	2
1296.3.o.q.271.1		4	36.11	even	6
1296.3.o.q.703.1		4	9.5	odd	6
1296.3.o.r.271.1		4	9.2	odd	6
1296.3.o.r.703.1		4	36.23	even	6
1296.3.o.be.271.2		4	36.7	odd	6
1296.3.o.be.703.2		4	9.4	even	3
1296.3.o.bf.271.2		4	9.7	even	3
1296.3.o.bf.703.2		4	36.31	odd	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)

\(f(q)\)	\(=\)	\(q-7.73205 q^{5} +9.46410i q^{7} +O(q^{10})\)	\(q-7.73205 q^{5} +9.46410i q^{7} -16.3923i q^{11} +17.3923 q^{13} -20.6603 q^{17} -33.4641i q^{19} +7.60770i q^{23} +34.7846 q^{25} +7.73205 q^{29} +32.7846i q^{31} -73.1769i q^{35} -26.1769 q^{37} +41.5692 q^{41} +37.1769i q^{43} -24.0000i q^{47} -40.5692 q^{49} +15.2154 q^{53} +126.746i q^{55} +96.0000i q^{59} -34.9615 q^{61} -134.478 q^{65} +40.8897i q^{67} +51.5307i q^{71} +132.138 q^{73} +155.138 q^{77} +49.6743i q^{79} +91.9230i q^{83} +159.746 q^{85} -77.6936 q^{89} +164.603i q^{91} +258.746i q^{95} -43.5692 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 24 q^{5}+O(q^{10}) \) 4 * q - 24 * q^5	\( 4 q - 24 q^{5} + 28 q^{13} - 48 q^{17} + 56 q^{25} + 24 q^{29} + 20 q^{37} + 4 q^{49} + 144 q^{53} + 68 q^{61} - 240 q^{65} + 196 q^{73} + 288 q^{77} + 348 q^{85} - 96 q^{89} - 8 q^{97}+O(q^{100}) \) 4 * q - 24 * q^5 + 28 * q^13 - 48 * q^17 + 56 * q^25 + 24 * q^29 + 20 * q^37 + 4 * q^49 + 144 * q^53 + 68 * q^61 - 240 * q^65 + 196 * q^73 + 288 * q^77 + 348 * q^85 - 96 * q^89 - 8 * q^97

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