LMFDB - Embedded newform 1296.3.g.k.1135.5

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			1135.5
Root			$2.33086i$ of defining polynomial
Character	$\chi$	$=$	1296.1135
Dual form			1296.3.g.k.1135.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.710609 q^{5} -3.12324i q^{7} +O(q^{10})$	$q+0.710609 q^{5} -3.12324i q^{7} +16.6072i q^{11} +18.3549 q^{13} +9.69321 q^{17} +8.20686i q^{19} -2.24953i q^{23} -24.4950 q^{25} -41.6434 q^{29} -24.9759i q^{31} -2.21940i q^{35} -40.3888 q^{37} +51.3888 q^{41} +65.4277i q^{43} +33.8205i q^{47} +39.2454 q^{49} +90.6691 q^{53} +11.8012i q^{55} +76.4693i q^{59} -2.71643 q^{61} +13.0431 q^{65} -39.8859i q^{67} +102.923i q^{71} +38.1741 q^{73} +51.8682 q^{77} -109.119i q^{79} +131.062i q^{83} +6.88808 q^{85} -38.0903 q^{89} -57.3266i q^{91} +5.83187i q^{95} +24.3923 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	0.710609	0.142122	0.0710609	−	0.997472i	$-0.477362\pi$
$5$	0.710609	0.142122	0.0710609	+	0.997472i	$0.477362\pi$
$6$	0	0
$7$	− 3.12324i	− 0.446177i	−0.974798	−	0.223088i	$-0.928386\pi$
$7$	− 3.12324i	− 0.446177i	0.974798	−	0.223088i	$-0.0716139\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	16.6072i	1.50974i	0.655873	+	0.754872i	$0.272301\pi$
$11$	16.6072i	1.50974i	−0.655873	+	0.754872i	$0.727699\pi$
$12$	0	0
$13$	18.3549	1.41191	0.705956	−	0.708255i	$-0.250517\pi$
$13$	18.3549	1.41191	0.705956	+	0.708255i	$0.250517\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	9.69321	0.570189	0.285095	−	0.958499i	$-0.407975\pi$
$17$	9.69321	0.570189	0.285095	+	0.958499i	$0.407975\pi$
$18$	0	0
$19$	8.20686i	0.431940i	0.976400	+	0.215970i	$0.0692913\pi$
$19$	8.20686i	0.431940i	−0.976400	+	0.215970i	$0.930709\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 2.24953i	− 0.0978057i	−0.998804	−	0.0489028i	$-0.984428\pi$
$23$	− 2.24953i	− 0.0978057i	0.998804	−	0.0489028i	$-0.0155725\pi$
$24$	0	0
$25$	−24.4950	−0.979801
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−41.6434	−1.43598	−0.717989	−	0.696054i	$-0.754937\pi$
$29$	−41.6434	−1.43598	−0.717989	+	0.696054i	$0.754937\pi$
$30$	0	0
$31$	− 24.9759i	− 0.805674i	−0.915272	−	0.402837i	$-0.868024\pi$
$31$	− 24.9759i	− 0.805674i	0.915272	−	0.402837i	$-0.131976\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.21940i	− 0.0634115i
$36$	0	0
$37$	−40.3888	−1.09159	−0.545794	−	0.837919i	$-0.683772\pi$
$37$	−40.3888	−1.09159	−0.545794	+	0.837919i	$0.683772\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	51.3888	1.25338	0.626692	−	0.779267i	$-0.284408\pi$
$41$	51.3888	1.25338	0.626692	+	0.779267i	$0.284408\pi$
$42$	0	0
$43$	65.4277i	1.52157i	0.649001	+	0.760787i	$0.275187\pi$
$43$	65.4277i	1.52157i	−0.649001	+	0.760787i	$0.724813\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	33.8205i	0.719585i	0.933032	+	0.359793i	$0.117153\pi$
$47$	33.8205i	0.719585i	−0.933032	+	0.359793i	$0.882847\pi$
$48$	0	0
$49$	39.2454	0.800926
$50$	0	0
$51$	0	0
$52$	0	0
$53$	90.6691	1.71074	0.855369	−	0.518019i	$-0.173330\pi$
$53$	90.6691	1.71074	0.855369	+	0.518019i	$0.173330\pi$
$54$	0	0
$55$	11.8012i	0.214567i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	76.4693i	1.29609i	0.761602	+	0.648045i	$0.224413\pi$
$59$	76.4693i	1.29609i	−0.761602	+	0.648045i	$0.775587\pi$
$60$	0	0
$61$	−2.71643	−0.0445317	−0.0222659	−	0.999752i	$-0.507088\pi$
$61$	−2.71643	−0.0445317	−0.0222659	+	0.999752i	$0.507088\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	13.0431	0.200664
$66$	0	0
$67$	− 39.8859i	− 0.595312i	−0.954673	−	0.297656i	$-0.903795\pi$
$67$	− 39.8859i	− 0.595312i	0.954673	−	0.297656i	$-0.0962048\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	102.923i	1.44962i	0.688950	+	0.724809i	$0.258072\pi$
$71$	102.923i	1.44962i	−0.688950	+	0.724809i	$0.741928\pi$
$72$	0	0
$73$	38.1741	0.522933	0.261466	−	0.965213i	$-0.415794\pi$
$73$	38.1741	0.522933	0.261466	+	0.965213i	$0.415794\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	51.8682	0.673613
$78$	0	0
$79$	− 109.119i	− 1.38125i	−0.723214	−	0.690624i	$-0.757336\pi$
$79$	− 109.119i	− 1.38125i	0.723214	−	0.690624i	$-0.242664\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	131.062i	1.57906i	0.613710	+	0.789531i	$0.289676\pi$
$83$	131.062i	1.57906i	−0.613710	+	0.789531i	$0.710324\pi$
$84$	0	0
$85$	6.88808	0.0810363
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−38.0903	−0.427981	−0.213991	−	0.976836i	$-0.568646\pi$
$89$	−38.0903	−0.427981	−0.213991	+	0.976836i	$0.568646\pi$
$90$	0	0
$91$	− 57.3266i	− 0.629963i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.83187i	0.0613881i
$96$	0	0
$97$	24.3923	0.251467	0.125733	−	0.992064i	$-0.459872\pi$
$97$	24.3923	0.251467	0.125733	+	0.992064i	$0.459872\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	196.261	1.94318	0.971589	−	0.236674i	$-0.0760572\pi$
$101$	196.261	1.94318	0.971589	+	0.236674i	$0.0760572\pi$
$102$	0	0
$103$	120.634i	1.17121i	0.810597	+	0.585604i	$0.199143\pi$
$103$	120.634i	1.17121i	−0.810597	+	0.585604i	$0.800857\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 6.52440i	− 0.0609757i	−0.999535	−	0.0304878i	$-0.990294\pi$
$107$	− 6.52440i	− 0.0609757i	0.999535	−	0.0304878i	$-0.00970609\pi$
$108$	0	0
$109$	38.0272	0.348873	0.174437	−	0.984668i	$-0.444190\pi$
$109$	38.0272	0.348873	0.174437	+	0.984668i	$0.444190\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−107.017	−0.947054	−0.473527	−	0.880779i	$-0.657019\pi$
$113$	−107.017	−0.947054	−0.473527	+	0.880779i	$0.657019\pi$
$114$	0	0
$115$	− 1.59854i	− 0.0139003i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 30.2742i	− 0.254405i
$120$	0	0
$121$	−154.798	−1.27932
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−35.1716	−0.281373
$126$	0	0
$127$	101.437i	0.798713i	0.916796	+	0.399357i	$0.130766\pi$
$127$	101.437i	0.798713i	−0.916796	+	0.399357i	$0.869234\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	188.008i	1.43518i	0.696467	+	0.717589i	$0.254754\pi$
$131$	188.008i	1.43518i	−0.696467	+	0.717589i	$0.745246\pi$
$132$	0	0
$133$	25.6320	0.192722
$134$	0	0
$135$	0	0
$136$	0	0
$137$	62.6542	0.457330	0.228665	−	0.973505i	$-0.426564\pi$
$137$	62.6542	0.457330	0.228665	+	0.973505i	$0.426564\pi$
$138$	0	0
$139$	46.8579i	0.337107i	0.985693	+	0.168553i	$0.0539096\pi$
$139$	46.8579i	0.337107i	−0.985693	+	0.168553i	$0.946090\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	304.822i	2.13163i
$144$	0	0
$145$	−29.5922	−0.204084
$146$	0	0
$147$	0	0
$148$	0	0
$149$	107.972	0.724644	0.362322	−	0.932053i	$-0.381984\pi$
$149$	107.972	0.724644	0.362322	+	0.932053i	$0.381984\pi$
$150$	0	0
$151$	3.17819i	0.0210476i	0.999945	+	0.0105238i	$0.00334990\pi$
$151$	3.17819i	0.0210476i	−0.999945	+	0.0105238i	$0.996650\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 17.7481i	− 0.114504i
$156$	0	0
$157$	−256.430	−1.63331	−0.816656	−	0.577125i	$-0.804175\pi$
$157$	−256.430	−1.63331	−0.816656	+	0.577125i	$0.804175\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−7.02582	−0.0436386
$162$	0	0
$163$	− 201.100i	− 1.23374i	−0.787065	−	0.616870i	$-0.788400\pi$
$163$	− 201.100i	− 1.23374i	0.787065	−	0.616870i	$-0.211600\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	127.812i	0.765344i	0.923884	+	0.382672i	$0.124996\pi$
$167$	127.812i	0.765344i	−0.923884	+	0.382672i	$0.875004\pi$
$168$	0	0
$169$	167.901	0.993498
$170$	0	0
$171$	0	0
$172$	0	0
$173$	201.435	1.16437	0.582183	−	0.813058i	$-0.302199\pi$
$173$	201.435	1.16437	0.582183	+	0.813058i	$0.302199\pi$
$174$	0	0
$175$	76.5038i	0.437165i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.83187i	0.0325803i	0.999867	+	0.0162901i	$0.00518554\pi$
$179$	5.83187i	0.0325803i	−0.999867	+	0.0162901i	$0.994814\pi$
$180$	0	0
$181$	132.737	0.733353	0.366677	−	0.930348i	$-0.380496\pi$
$181$	132.737	0.733353	0.366677	+	0.930348i	$0.380496\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−28.7006	−0.155139
$186$	0	0
$187$	160.977i	0.860839i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 41.6666i	− 0.218150i	−0.994034	−	0.109075i	$-0.965211\pi$
$191$	− 41.6666i	− 0.218150i	0.994034	−	0.109075i	$-0.0347888\pi$
$192$	0	0
$193$	138.385	0.717022	0.358511	−	0.933525i	$-0.383285\pi$
$193$	138.385	0.717022	0.358511	+	0.933525i	$0.383285\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	109.421	0.555438	0.277719	−	0.960662i	$-0.410422\pi$
$197$	109.421	0.555438	0.277719	+	0.960662i	$0.410422\pi$
$198$	0	0
$199$	87.0243i	0.437308i	0.975802	+	0.218654i	$0.0701666\pi$
$199$	87.0243i	0.437308i	−0.975802	+	0.218654i	$0.929833\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	130.062i	0.640701i
$204$	0	0
$205$	36.5173	0.178133
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−136.293	−0.652118
$210$	0	0
$211$	− 282.189i	− 1.33739i	−0.743536	−	0.668695i	$-0.766853\pi$
$211$	− 282.189i	− 1.33739i	0.743536	−	0.668695i	$-0.233147\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	46.4935i	0.216249i
$216$	0	0
$217$	−78.0057	−0.359473
$218$	0	0
$219$	0	0
$220$	0	0
$221$	177.918	0.805057
$222$	0	0
$223$	− 294.864i	− 1.32226i	−0.750272	−	0.661129i	$-0.770077\pi$
$223$	− 294.864i	− 1.32226i	0.750272	−	0.661129i	$-0.229923\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	143.633i	0.632743i	0.948635	+	0.316372i	$0.102465\pi$
$227$	143.633i	0.632743i	−0.948635	+	0.316372i	$0.897535\pi$
$228$	0	0
$229$	282.853	1.23517	0.617583	−	0.786506i	$-0.288112\pi$
$229$	282.853	1.23517	0.617583	+	0.786506i	$0.288112\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	25.9127	0.111213	0.0556067	−	0.998453i	$-0.482291\pi$
$233$	25.9127	0.111213	0.0556067	+	0.998453i	$0.482291\pi$
$234$	0	0
$235$	24.0331i	0.102269i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 358.889i	− 1.50163i	−0.660514	−	0.750813i	$-0.729662\pi$
$239$	− 358.889i	− 1.50163i	0.660514	−	0.750813i	$-0.270338\pi$
$240$	0	0
$241$	−175.410	−0.727840	−0.363920	−	0.931430i	$-0.618562\pi$
$241$	−175.410	−0.727840	−0.363920	+	0.931430i	$0.618562\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	27.8881	0.113829
$246$	0	0
$247$	150.636i	0.609861i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 410.044i	− 1.63364i	−0.576891	−	0.816821i	$-0.695734\pi$
$251$	− 410.044i	− 1.63364i	0.576891	−	0.816821i	$-0.304266\pi$
$252$	0	0
$253$	37.3583	0.147661
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−172.856	−0.672591	−0.336296	−	0.941756i	$-0.609174\pi$
$257$	−172.856	−0.672591	−0.336296	+	0.941756i	$0.609174\pi$
$258$	0	0
$259$	126.144i	0.487042i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	153.224i	0.582600i	0.956632	+	0.291300i	$0.0940877\pi$
$263$	153.224i	0.582600i	−0.956632	+	0.291300i	$0.905912\pi$
$264$	0	0
$265$	64.4303	0.243133
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−12.6752	−0.0471198	−0.0235599	−	0.999722i	$-0.507500\pi$
$269$	−12.6752	−0.0471198	−0.0235599	+	0.999722i	$0.507500\pi$
$270$	0	0
$271$	40.7101i	0.150222i	0.997175	+	0.0751108i	$0.0239311\pi$
$271$	40.7101i	0.150222i	−0.997175	+	0.0751108i	$0.976069\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 406.793i	− 1.47925i
$276$	0	0
$277$	−368.286	−1.32955	−0.664776	−	0.747043i	$-0.731473\pi$
$277$	−368.286	−1.32955	−0.664776	+	0.747043i	$0.731473\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−476.620	−1.69616	−0.848078	−	0.529872i	$-0.822240\pi$
$281$	−476.620	−1.69616	−0.848078	+	0.529872i	$0.822240\pi$
$282$	0	0
$283$	173.265i	0.612242i	0.951993	+	0.306121i	$0.0990313\pi$
$283$	173.265i	0.612242i	−0.951993	+	0.306121i	$0.900969\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 160.499i	− 0.559231i
$288$	0	0
$289$	−195.042	−0.674884
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5.83666	−0.0199203	−0.00996017	−	0.999950i	$-0.503170\pi$
$293$	−5.83666	−0.0199203	−0.00996017	+	0.999950i	$0.503170\pi$
$294$	0	0
$295$	54.3397i	0.184203i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 41.2898i	− 0.138093i
$300$	0	0
$301$	204.346	0.678891
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.93032	−0.00632893
$306$	0	0
$307$	371.717i	1.21080i	0.795920	+	0.605402i	$0.206988\pi$
$307$	371.717i	1.21080i	−0.795920	+	0.605402i	$0.793012\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	223.971i	0.720163i	0.932921	+	0.360081i	$0.117251\pi$
$311$	223.971i	0.720163i	−0.932921	+	0.360081i	$0.882749\pi$
$312$	0	0
$313$	−158.192	−0.505406	−0.252703	−	0.967544i	$-0.581319\pi$
$313$	−158.192	−0.505406	−0.252703	+	0.967544i	$0.581319\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−412.855	−1.30238	−0.651191	−	0.758914i	$-0.725730\pi$
$317$	−412.855	−1.30238	−0.651191	+	0.758914i	$0.725730\pi$
$318$	0	0
$319$	− 691.579i	− 2.16796i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	79.5508i	0.246287i
$324$	0	0
$325$	−449.603	−1.38339
$326$	0	0
$327$	0	0
$328$	0	0
$329$	105.629	0.321062
$330$	0	0
$331$	146.574i	0.442822i	0.975181	+	0.221411i	$0.0710663\pi$
$331$	146.574i	0.442822i	−0.975181	+	0.221411i	$0.928934\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 28.3433i	− 0.0846068i
$336$	0	0
$337$	94.6998	0.281008	0.140504	−	0.990080i	$-0.455128\pi$
$337$	94.6998	0.281008	0.140504	+	0.990080i	$0.455128\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	414.779	1.21636
$342$	0	0
$343$	− 275.611i	− 0.803532i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	93.9277i	0.270685i	0.990799	+	0.135343i	$0.0432135\pi$
$347$	93.9277i	0.270685i	−0.990799	+	0.135343i	$0.956787\pi$
$348$	0	0
$349$	231.158	0.662343	0.331171	−	0.943571i	$-0.392556\pi$
$349$	231.158	0.662343	0.331171	+	0.943571i	$0.392556\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−97.9246	−0.277407	−0.138703	−	0.990334i	$-0.544293\pi$
$353$	−97.9246	−0.277407	−0.138703	+	0.990334i	$0.544293\pi$
$354$	0	0
$355$	73.1379i	0.206022i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 244.287i	− 0.680465i	−0.940341	−	0.340233i	$-0.889494\pi$
$359$	− 244.287i	− 0.680465i	0.940341	−	0.340233i	$-0.110506\pi$
$360$	0	0
$361$	293.647	0.813428
$362$	0	0
$363$	0	0
$364$	0	0
$365$	27.1268	0.0743201
$366$	0	0
$367$	425.308i	1.15888i	0.815016	+	0.579438i	$0.196728\pi$
$367$	425.308i	1.15888i	−0.815016	+	0.579438i	$0.803272\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 283.181i	− 0.763292i
$372$	0	0
$373$	−89.7135	−0.240519	−0.120259	−	0.992743i	$-0.538373\pi$
$373$	−89.7135	−0.240519	−0.120259	+	0.992743i	$0.538373\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−764.359	−2.02748
$378$	0	0
$379$	− 406.140i	− 1.07161i	−0.844342	−	0.535805i	$-0.820008\pi$
$379$	− 406.140i	− 1.07161i	0.844342	−	0.535805i	$-0.179992\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 324.357i	− 0.846884i	−0.905923	−	0.423442i	$-0.860822\pi$
$383$	− 324.357i	− 0.846884i	0.905923	−	0.423442i	$-0.139178\pi$
$384$	0	0
$385$	36.8580	0.0957350
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−57.3523	−0.147435	−0.0737176	−	0.997279i	$-0.523486\pi$
$389$	−57.3523	−0.147435	−0.0737176	+	0.997279i	$0.523486\pi$
$390$	0	0
$391$	− 21.8052i	− 0.0557677i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 77.5406i	− 0.196305i
$396$	0	0
$397$	194.475	0.489861	0.244931	−	0.969541i	$-0.421235\pi$
$397$	194.475	0.489861	0.244931	+	0.969541i	$0.421235\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−327.886	−0.817671	−0.408836	−	0.912608i	$-0.634065\pi$
$401$	−327.886	−0.817671	−0.408836	+	0.912608i	$0.634065\pi$
$402$	0	0
$403$	− 458.429i	− 1.13754i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 670.744i	− 1.64802i
$408$	0	0
$409$	73.3572	0.179357	0.0896787	−	0.995971i	$-0.471416\pi$
$409$	73.3572	0.179357	0.0896787	+	0.995971i	$0.471416\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	238.832	0.578285
$414$	0	0
$415$	93.1340i	0.224419i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 643.619i	− 1.53608i	−0.640400	−	0.768041i	$-0.721232\pi$
$419$	− 643.619i	− 1.53608i	0.640400	−	0.768041i	$-0.278768\pi$
$420$	0	0
$421$	560.302	1.33088	0.665441	−	0.746450i	$-0.268243\pi$
$421$	560.302	1.33088	0.665441	+	0.746450i	$0.268243\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−237.436	−0.558672
$426$	0	0
$427$	8.48407i	0.0198690i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 536.437i	− 1.24463i	−0.782765	−	0.622317i	$-0.786191\pi$
$431$	− 536.437i	− 1.24463i	0.782765	−	0.622317i	$-0.213809\pi$
$432$	0	0
$433$	281.999	0.651268	0.325634	−	0.945496i	$-0.394422\pi$
$433$	281.999	0.651268	0.325634	+	0.945496i	$0.394422\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	18.4616	0.0422462
$438$	0	0
$439$	− 100.760i	− 0.229521i	−0.993393	−	0.114760i	$-0.963390\pi$
$439$	− 100.760i	− 0.229521i	0.993393	−	0.114760i	$-0.0366100\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 600.212i	− 1.35488i	−0.735578	−	0.677440i	$-0.763089\pi$
$443$	− 600.212i	− 1.35488i	0.735578	−	0.677440i	$-0.236911\pi$
$444$	0	0
$445$	−27.0673	−0.0608255
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−639.843	−1.42504	−0.712520	−	0.701651i	$-0.752446\pi$
$449$	−639.843	−1.42504	−0.712520	+	0.701651i	$0.752446\pi$
$450$	0	0
$451$	853.422i	1.89229i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 40.7368i	− 0.0895315i
$456$	0	0
$457$	−173.544	−0.379747	−0.189873	−	0.981809i	$-0.560808\pi$
$457$	−173.544	−0.379747	−0.189873	+	0.981809i	$0.560808\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	723.311	1.56900	0.784502	−	0.620126i	$-0.212919\pi$
$461$	723.311	1.56900	0.784502	+	0.620126i	$0.212919\pi$
$462$	0	0
$463$	743.488i	1.60581i	0.596110	+	0.802903i	$0.296712\pi$
$463$	743.488i	1.60581i	−0.596110	+	0.802903i	$0.703288\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	98.0700i	0.210000i	0.994472	+	0.105000i	$0.0334842\pi$
$467$	98.0700i	0.210000i	−0.994472	+	0.105000i	$0.966516\pi$
$468$	0	0
$469$	−124.573	−0.265614
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1086.57	−2.29719
$474$	0	0
$475$	− 201.027i	− 0.423215i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	339.075i	0.707882i	0.935268	+	0.353941i	$0.115159\pi$
$479$	339.075i	0.707882i	−0.935268	+	0.353941i	$0.884841\pi$
$480$	0	0
$481$	−741.331	−1.54123
$482$	0	0
$483$	0	0
$484$	0	0
$485$	17.3334	0.0357389
$486$	0	0
$487$	− 777.718i	− 1.59696i	−0.602023	−	0.798479i	$-0.705638\pi$
$487$	− 777.718i	− 1.59696i	0.602023	−	0.798479i	$-0.294362\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	60.0507i	0.122303i	0.998128	+	0.0611514i	$0.0194773\pi$
$491$	60.0507i	0.122303i	−0.998128	+	0.0611514i	$0.980523\pi$
$492$	0	0
$493$	−403.658	−0.818779
$494$	0	0
$495$	0	0
$496$	0	0
$497$	321.453	0.646786
$498$	0	0
$499$	288.802i	0.578761i	0.957214	+	0.289381i	$0.0934493\pi$
$499$	288.802i	0.578761i	−0.957214	+	0.289381i	$0.906551\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 567.389i	− 1.12801i	−0.825772	−	0.564005i	$-0.809260\pi$
$503$	− 567.389i	− 1.12801i	0.825772	−	0.564005i	$-0.190740\pi$
$504$	0	0
$505$	139.465	0.276168
$506$	0	0
$507$	0	0
$508$	0	0
$509$	316.567	0.621939	0.310970	−	0.950420i	$-0.399346\pi$
$509$	316.567	0.621939	0.310970	+	0.950420i	$0.399346\pi$
$510$	0	0
$511$	− 119.227i	− 0.233320i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	85.7239i	0.166454i
$516$	0	0
$517$	−561.663	−1.08639
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−597.419	−1.14668	−0.573339	−	0.819318i	$-0.694352\pi$
$521$	−597.419	−1.14668	−0.573339	+	0.819318i	$0.694352\pi$
$522$	0	0
$523$	− 630.846i	− 1.20621i	−0.797663	−	0.603103i	$-0.793931\pi$
$523$	− 630.846i	− 1.20621i	0.797663	−	0.603103i	$-0.206069\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 242.097i	− 0.459386i
$528$	0	0
$529$	523.940	0.990434
$530$	0	0
$531$	0	0
$532$	0	0
$533$	943.234	1.76967
$534$	0	0
$535$	− 4.63630i	− 0.00866597i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	651.755i	1.20919i
$540$	0	0
$541$	−144.808	−0.267667	−0.133834	−	0.991004i	$-0.542729\pi$
$541$	−144.808	−0.267667	−0.133834	+	0.991004i	$0.542729\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	27.0225	0.0495825
$546$	0	0
$547$	− 784.161i	− 1.43357i	−0.697296	−	0.716784i	$-0.745613\pi$
$547$	− 784.161i	− 1.43357i	0.697296	−	0.716784i	$-0.254387\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 341.761i	− 0.620256i
$552$	0	0
$553$	−340.803	−0.616281
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−92.1884	−0.165509	−0.0827544	−	0.996570i	$-0.526372\pi$
$557$	−92.1884	−0.165509	−0.0827544	+	0.996570i	$0.526372\pi$
$558$	0	0
$559$	1200.92i	2.14833i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 163.897i	− 0.291114i	−0.989350	−	0.145557i	$-0.953503\pi$
$563$	− 163.897i	− 0.291114i	0.989350	−	0.145557i	$-0.0464974\pi$
$564$	0	0
$565$	−76.0473	−0.134597
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−704.438	−1.23803	−0.619014	−	0.785380i	$-0.712468\pi$
$569$	−704.438	−1.23803	−0.619014	+	0.785380i	$0.712468\pi$
$570$	0	0
$571$	477.835i	0.836838i	0.908254	+	0.418419i	$0.137416\pi$
$571$	477.835i	0.836838i	−0.908254	+	0.418419i	$0.862584\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	55.1023i	0.0958301i
$576$	0	0
$577$	−21.5525	−0.0373527	−0.0186764	−	0.999826i	$-0.505945\pi$
$577$	−21.5525	−0.0373527	−0.0186764	+	0.999826i	$0.505945\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	409.338	0.704541
$582$	0	0
$583$	1505.76i	2.58278i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 792.329i	− 1.34979i	−0.737912	−	0.674897i	$-0.764188\pi$
$587$	− 792.329i	− 1.34979i	0.737912	−	0.674897i	$-0.235812\pi$
$588$	0	0
$589$	204.974	0.348003
$590$	0	0
$591$	0	0
$592$	0	0
$593$	571.777	0.964211	0.482105	−	0.876113i	$-0.339872\pi$
$593$	571.777	0.964211	0.482105	+	0.876113i	$0.339872\pi$
$594$	0	0
$595$	− 21.5131i	− 0.0361565i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	927.989i	1.54923i	0.632433	+	0.774615i	$0.282056\pi$
$599$	927.989i	1.54923i	−0.632433	+	0.774615i	$0.717944\pi$
$600$	0	0
$601$	132.863	0.221070	0.110535	−	0.993872i	$-0.464744\pi$
$601$	132.863	0.221070	0.110535	+	0.993872i	$0.464744\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−110.001	−0.181820
$606$	0	0
$607$	− 554.374i	− 0.913301i	−0.889646	−	0.456651i	$-0.849049\pi$
$607$	− 554.374i	− 0.913301i	0.889646	−	0.456651i	$-0.150951\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	620.771i	1.01599i
$612$	0	0
$613$	1096.88	1.78937	0.894684	−	0.446700i	$-0.147401\pi$
$613$	1096.88	1.78937	0.894684	+	0.446700i	$0.147401\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−623.272	−1.01017	−0.505083	−	0.863071i	$-0.668538\pi$
$617$	−623.272	−1.01017	−0.505083	+	0.863071i	$0.668538\pi$
$618$	0	0
$619$	− 708.330i	− 1.14431i	−0.820144	−	0.572157i	$-0.806107\pi$
$619$	− 708.330i	− 1.14431i	0.820144	−	0.572157i	$-0.193893\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	118.965i	0.190955i
$624$	0	0
$625$	587.383	0.939812
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−391.497	−0.622412
$630$	0	0
$631$	− 1142.86i	− 1.81119i	−0.424139	−	0.905597i	$-0.639423\pi$
$631$	− 1142.86i	− 1.81119i	0.424139	−	0.905597i	$-0.360577\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	72.0818i	0.113515i
$636$	0	0
$637$	720.344	1.13084
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−277.083	−0.432267	−0.216134	−	0.976364i	$-0.569345\pi$
$641$	−277.083	−0.432267	−0.216134	+	0.976364i	$0.569345\pi$
$642$	0	0
$643$	− 851.287i	− 1.32393i	−0.749535	−	0.661965i	$-0.769723\pi$
$643$	− 851.287i	− 1.32393i	0.749535	−	0.661965i	$-0.230277\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 706.622i	− 1.09215i	−0.837736	−	0.546076i	$-0.816121\pi$
$647$	− 706.622i	− 1.09215i	0.837736	−	0.546076i	$-0.183879\pi$
$648$	0	0
$649$	−1269.94	−1.95676
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−780.683	−1.19553	−0.597767	−	0.801670i	$-0.703945\pi$
$653$	−780.683	−1.19553	−0.597767	+	0.801670i	$0.703945\pi$
$654$	0	0
$655$	133.600i	0.203970i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	567.329i	0.860895i	0.902616	+	0.430447i	$0.141644\pi$
$659$	567.329i	0.860895i	−0.902616	+	0.430447i	$0.858356\pi$
$660$	0	0
$661$	−93.8327	−0.141956	−0.0709778	−	0.997478i	$-0.522612\pi$
$661$	−93.8327	−0.141956	−0.0709778	+	0.997478i	$0.522612\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	18.2143	0.0273899
$666$	0	0
$667$	93.6780i	0.140447i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 45.1123i	− 0.0672315i
$672$	0	0
$673$	−201.484	−0.299382	−0.149691	−	0.988733i	$-0.547828\pi$
$673$	−201.484	−0.299382	−0.149691	+	0.988733i	$0.547828\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−673.976	−0.995533	−0.497766	−	0.867311i	$-0.665846\pi$
$677$	−673.976	−0.995533	−0.497766	+	0.867311i	$0.665846\pi$
$678$	0	0
$679$	− 76.1828i	− 0.112199i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 155.633i	− 0.227867i	−0.993488	−	0.113933i	$-0.963655\pi$
$683$	− 155.633i	− 0.227867i	0.993488	−	0.113933i	$-0.0363450\pi$
$684$	0	0
$685$	44.5226	0.0649965
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1664.22	2.41541
$690$	0	0
$691$	800.511i	1.15848i	0.815157	+	0.579241i	$0.196651\pi$
$691$	800.511i	1.15848i	−0.815157	+	0.579241i	$0.803349\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	33.2976i	0.0479102i
$696$	0	0
$697$	498.122	0.714666
$698$	0	0
$699$	0	0
$700$	0	0
$701$	488.317	0.696601	0.348300	−	0.937383i	$-0.386759\pi$
$701$	488.317	0.696601	0.348300	+	0.937383i	$0.386759\pi$
$702$	0	0
$703$	− 331.465i	− 0.471501i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 612.970i	− 0.867001i
$708$	0	0
$709$	−717.265	−1.01166	−0.505829	−	0.862634i	$-0.668813\pi$
$709$	−717.265	−1.01166	−0.505829	+	0.862634i	$0.668813\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−56.1840	−0.0787995
$714$	0	0
$715$	216.610i	0.302950i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	512.219i	0.712404i	0.934409	+	0.356202i	$0.115929\pi$
$719$	512.219i	0.712404i	−0.934409	+	0.356202i	$0.884071\pi$
$720$	0	0
$721$	376.770	0.522566
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1020.06	1.40697
$726$	0	0
$727$	− 663.374i	− 0.912481i	−0.889856	−	0.456241i	$-0.849196\pi$
$727$	− 663.374i	− 0.912481i	0.889856	−	0.456241i	$-0.150804\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	634.205i	0.867585i
$732$	0	0
$733$	293.765	0.400771	0.200386	−	0.979717i	$-0.435781\pi$
$733$	293.765	0.400771	0.200386	+	0.979717i	$0.435781\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	662.392	0.898768
$738$	0	0
$739$	1335.27i	1.80686i	0.428736	+	0.903430i	$0.358959\pi$
$739$	1335.27i	1.80686i	−0.428736	+	0.903430i	$0.641041\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	563.711i	0.758696i	0.925254	+	0.379348i	$0.123852\pi$
$743$	563.711i	0.758696i	−0.925254	+	0.379348i	$0.876148\pi$
$744$	0	0
$745$	76.7258	0.102988
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−20.3773	−0.0272059
$750$	0	0
$751$	− 682.898i	− 0.909319i	−0.890665	−	0.454659i	$-0.849761\pi$
$751$	− 682.898i	− 0.909319i	0.890665	−	0.454659i	$-0.150239\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.25845i	0.00299133i
$756$	0	0
$757$	−534.746	−0.706401	−0.353201	−	0.935548i	$-0.614907\pi$
$757$	−534.746	−0.706401	−0.353201	+	0.935548i	$0.614907\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−832.381	−1.09380	−0.546900	−	0.837198i	$-0.684192\pi$
$761$	−832.381	−1.09380	−0.546900	+	0.837198i	$0.684192\pi$
$762$	0	0
$763$	− 118.768i	− 0.155659i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1403.58i	1.82996i
$768$	0	0
$769$	702.041	0.912927	0.456464	−	0.889742i	$-0.349116\pi$
$769$	702.041	0.912927	0.456464	+	0.889742i	$0.349116\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	472.477	0.611225	0.305612	−	0.952156i	$-0.401139\pi$
$773$	472.477	0.611225	0.305612	+	0.952156i	$0.401139\pi$
$774$	0	0
$775$	611.785i	0.789400i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	421.740i	0.541387i
$780$	0	0
$781$	−1709.26	−2.18855
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−182.221	−0.232129
$786$	0	0
$787$	642.229i	0.816046i	0.912972	+	0.408023i	$0.133782\pi$
$787$	642.229i	0.816046i	−0.912972	+	0.408023i	$0.866218\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	334.240i	0.422554i
$792$	0	0
$793$	−49.8598	−0.0628749
$794$	0	0
$795$	0	0
$796$	0	0
$797$	84.6334	0.106190	0.0530950	−	0.998589i	$-0.483091\pi$
$797$	84.6334	0.106190	0.0530950	+	0.998589i	$0.483091\pi$
$798$	0	0
$799$	327.829i	0.410300i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	633.964i	0.789494i
$804$	0	0
$805$	−4.99261	−0.00620200
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1071.63	1.32464	0.662319	−	0.749222i	$-0.269573\pi$
$809$	1071.63	1.32464	0.662319	+	0.749222i	$0.269573\pi$
$810$	0	0
$811$	− 745.523i	− 0.919264i	−0.888110	−	0.459632i	$-0.847981\pi$
$811$	− 745.523i	− 0.919264i	0.888110	−	0.459632i	$-0.152019\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 142.903i	− 0.175341i
$816$	0	0
$817$	−536.956	−0.657229
$818$	0	0
$819$	0	0
$820$	0	0
$821$	835.099	1.01717	0.508587	−	0.861011i	$-0.330168\pi$
$821$	835.099	1.01717	0.508587	+	0.861011i	$0.330168\pi$
$822$	0	0
$823$	426.982i	0.518812i	0.965768	+	0.259406i	$0.0835267\pi$
$823$	426.982i	0.518812i	−0.965768	+	0.259406i	$0.916473\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 235.457i	− 0.284713i	−0.989815	−	0.142356i	$-0.954532\pi$
$827$	− 235.457i	− 0.284713i	0.989815	−	0.142356i	$-0.0454679\pi$
$828$	0	0
$829$	−1336.05	−1.61164	−0.805819	−	0.592161i	$-0.798275\pi$
$829$	−1336.05	−1.61164	−0.805819	+	0.592161i	$0.798275\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	380.414	0.456679
$834$	0	0
$835$	90.8247i	0.108772i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1518.90i	1.81037i	0.425021	+	0.905183i	$0.360267\pi$
$839$	1518.90i	1.81037i	−0.425021	+	0.905183i	$0.639733\pi$
$840$	0	0
$841$	893.171	1.06203
$842$	0	0
$843$	0	0
$844$	0	0
$845$	119.312	0.141198
$846$	0	0
$847$	483.472i	0.570805i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	90.8558i	0.106764i
$852$	0	0
$853$	439.269	0.514970	0.257485	−	0.966282i	$-0.417106\pi$
$853$	439.269	0.514970	0.257485	+	0.966282i	$0.417106\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	85.9349	0.100274	0.0501370	−	0.998742i	$-0.484034\pi$
$857$	85.9349	0.100274	0.0501370	+	0.998742i	$0.484034\pi$
$858$	0	0
$859$	− 866.477i	− 1.00870i	−0.863498	−	0.504352i	$-0.831731\pi$
$859$	− 866.477i	− 1.00870i	0.863498	−	0.504352i	$-0.168269\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	651.682i	0.755136i	0.925982	+	0.377568i	$0.123240\pi$
$863$	651.682i	0.755136i	−0.925982	+	0.377568i	$0.876760\pi$
$864$	0	0
$865$	143.142	0.165482
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1812.15	2.08533
$870$	0	0
$871$	− 732.100i	− 0.840528i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	109.849i	0.125542i
$876$	0	0
$877$	718.005	0.818706	0.409353	−	0.912376i	$-0.365754\pi$
$877$	718.005	0.818706	0.409353	+	0.912376i	$0.365754\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−292.378	−0.331870	−0.165935	−	0.986137i	$-0.553064\pi$
$881$	−292.378	−0.331870	−0.165935	+	0.986137i	$0.553064\pi$
$882$	0	0
$883$	507.123i	0.574318i	0.957883	+	0.287159i	$0.0927109\pi$
$883$	507.123i	0.574318i	−0.957883	+	0.287159i	$0.907289\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	256.712i	0.289417i	0.989474	+	0.144708i	$0.0462244\pi$
$887$	256.712i	0.289417i	−0.989474	+	0.144708i	$0.953776\pi$
$888$	0	0
$889$	316.811	0.356367
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−277.560	−0.310818
$894$	0	0
$895$	4.14418i	0.00463037i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1040.08i	1.15693i
$900$	0	0
$901$	878.875	0.975444
$902$	0	0
$903$	0	0
$904$	0	0
$905$	94.3241	0.104225
$906$	0	0
$907$	1548.73i	1.70753i	0.520657	+	0.853766i	$0.325687\pi$
$907$	1548.73i	1.70753i	−0.520657	+	0.853766i	$0.674313\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 537.876i	− 0.590424i	−0.955432	−	0.295212i	$-0.904610\pi$
$911$	− 537.876i	− 0.590424i	0.955432	−	0.295212i	$-0.0953903\pi$
$912$	0	0
$913$	−2176.57	−2.38398
$914$	0	0
$915$	0	0
$916$	0	0
$917$	587.195	0.640343
$918$	0	0
$919$	− 1099.66i	− 1.19659i	−0.801277	−	0.598294i	$-0.795845\pi$
$919$	− 1099.66i	− 1.19659i	0.801277	−	0.598294i	$-0.204155\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1889.14i	2.04673i
$924$	0	0
$925$	989.325	1.06954
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−195.311	−0.210238	−0.105119	−	0.994460i	$-0.533522\pi$
$929$	−195.311	−0.210238	−0.105119	+	0.994460i	$0.533522\pi$
$930$	0	0
$931$	322.081i	0.345952i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	114.392i	0.122344i
$936$	0	0
$937$	1286.97	1.37350	0.686752	−	0.726892i	$-0.259036\pi$
$937$	1286.97	1.37350	0.686752	+	0.726892i	$0.259036\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1011.37	−1.07478	−0.537391	−	0.843333i	$-0.680590\pi$
$941$	−1011.37	−1.07478	−0.537391	+	0.843333i	$0.680590\pi$
$942$	0	0
$943$	− 115.601i	− 0.122588i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 691.107i	− 0.729785i	−0.931050	−	0.364893i	$-0.881106\pi$
$947$	− 691.107i	− 0.729785i	0.931050	−	0.364893i	$-0.118894\pi$
$948$	0	0
$949$	700.680	0.738335
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−659.310	−0.691826	−0.345913	−	0.938267i	$-0.612431\pi$
$953$	−659.310	−0.691826	−0.345913	+	0.938267i	$0.612431\pi$
$954$	0	0
$955$	− 29.6086i	− 0.0310038i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 195.684i	− 0.204050i
$960$	0	0
$961$	337.205	0.350890
$962$	0	0
$963$	0	0
$964$	0	0
$965$	98.3378	0.101904
$966$	0	0
$967$	− 1132.55i	− 1.17120i	−0.810600	−	0.585600i	$-0.800859\pi$
$967$	− 1132.55i	− 1.17120i	0.810600	−	0.585600i	$-0.199141\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 277.255i	− 0.285536i	−0.989756	−	0.142768i	$-0.954400\pi$
$971$	− 277.255i	− 0.285536i	0.989756	−	0.142768i	$-0.0456002\pi$
$972$	0	0
$973$	146.348	0.150409
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−333.092	−0.340934	−0.170467	−	0.985363i	$-0.554528\pi$
$977$	−333.092	−0.340934	−0.170467	+	0.985363i	$0.554528\pi$
$978$	0	0
$979$	− 632.573i	− 0.646142i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1486.24i	− 1.51195i	−0.654603	−	0.755973i	$-0.727164\pi$
$983$	− 1486.24i	− 1.51195i	0.654603	−	0.755973i	$-0.272836\pi$
$984$	0	0
$985$	77.7557	0.0789398
$986$	0	0
$987$	0	0
$988$	0	0
$989$	147.182	0.148819
$990$	0	0
$991$	− 17.1782i	− 0.0173343i	−0.999962	−	0.00866713i	$-0.997241\pi$
$991$	− 17.1782i	− 0.0173343i	0.999962	−	0.00866713i	$-0.00275887\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	61.8402i	0.0621510i
$996$	0	0
$997$	1475.31	1.47975	0.739874	−	0.672745i	$-0.234885\pi$
$997$	1475.31	1.47975	0.739874	+	0.672745i	$0.234885\pi$
$998$	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.1135.5		8
3.2	odd	2		1296.3.g.j.1135.3		8
4.3	odd	2	inner	1296.3.g.k.1135.6		8
9.2	odd	6		144.3.o.c.31.1	yes	8
9.4	even	3		432.3.o.b.127.2		8
9.5	odd	6		144.3.o.a.79.4	yes	8
9.7	even	3		432.3.o.a.415.2		8
12.11	even	2		1296.3.g.j.1135.4		8
36.7	odd	6		432.3.o.b.415.2		8
36.11	even	6		144.3.o.a.31.4	✓	8
36.23	even	6		144.3.o.c.79.1	yes	8
36.31	odd	6		432.3.o.a.127.2		8
72.5	odd	6		576.3.o.f.511.1		8
72.11	even	6		576.3.o.f.319.1		8
72.13	even	6		1728.3.o.f.127.3		8
72.29	odd	6		576.3.o.d.319.4		8
72.43	odd	6		1728.3.o.f.1279.3		8
72.59	even	6		576.3.o.d.511.4		8
72.61	even	6		1728.3.o.e.1279.3		8
72.67	odd	6		1728.3.o.e.127.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.o.a.31.4	✓	8	36.11	even	6
144.3.o.a.79.4	yes	8	9.5	odd	6
144.3.o.c.31.1	yes	8	9.2	odd	6
144.3.o.c.79.1	yes	8	36.23	even	6
432.3.o.a.127.2		8	36.31	odd	6
432.3.o.a.415.2		8	9.7	even	3
432.3.o.b.127.2		8	9.4	even	3
432.3.o.b.415.2		8	36.7	odd	6
576.3.o.d.319.4		8	72.29	odd	6
576.3.o.d.511.4		8	72.59	even	6
576.3.o.f.319.1		8	72.11	even	6
576.3.o.f.511.1		8	72.5	odd	6
1296.3.g.j.1135.3		8	3.2	odd	2
1296.3.g.j.1135.4		8	12.11	even	2
1296.3.g.k.1135.5		8	1.1	even	1	trivial
1296.3.g.k.1135.6		8	4.3	odd	2	inner
1728.3.o.e.127.3		8	72.67	odd	6
1728.3.o.e.1279.3		8	72.61	even	6
1728.3.o.f.127.3		8	72.13	even	6
1728.3.o.f.1279.3		8	72.43	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)

\(f(q)\)	\(=\)	\(q+0.710609 q^{5} -3.12324i q^{7} +O(q^{10})\)	\(q+0.710609 q^{5} -3.12324i q^{7} +16.6072i q^{11} +18.3549 q^{13} +9.69321 q^{17} +8.20686i q^{19} -2.24953i q^{23} -24.4950 q^{25} -41.6434 q^{29} -24.9759i q^{31} -2.21940i q^{35} -40.3888 q^{37} +51.3888 q^{41} +65.4277i q^{43} +33.8205i q^{47} +39.2454 q^{49} +90.6691 q^{53} +11.8012i q^{55} +76.4693i q^{59} -2.71643 q^{61} +13.0431 q^{65} -39.8859i q^{67} +102.923i q^{71} +38.1741 q^{73} +51.8682 q^{77} -109.119i q^{79} +131.062i q^{83} +6.88808 q^{85} -38.0903 q^{89} -57.3266i q^{91} +5.83187i q^{95} +24.3923 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{5}+O(q^{10}) \) 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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