LMFDB - Embedded newform 1296.3.g.k.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:	$8$
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			1135.2
Root			$-1.07834i$ of defining polynomial
Character	$\chi$	$=$	1296.1135
Dual form			1296.3.g.k.1135.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.03459 q^{5} +11.8163i q^{7} +O(q^{10})$	$q-6.03459 q^{5} +11.8163i q^{7} -6.10206i q^{11} -14.8974 q^{13} -26.6919 q^{17} +9.45610i q^{19} -19.9386i q^{23} +11.4163 q^{25} +44.6229 q^{29} -6.26262i q^{31} -71.3065i q^{35} -6.65707 q^{37} +17.6571 q^{41} +23.4053i q^{43} +42.0612i q^{47} -90.6249 q^{49} +51.6192 q^{53} +36.8234i q^{55} -37.9924i q^{59} +90.7631 q^{61} +89.8994 q^{65} -61.7276i q^{67} +39.5232i q^{71} +35.0355 q^{73} +72.1038 q^{77} -90.0210i q^{79} -118.191i q^{83} +161.074 q^{85} +14.4499 q^{89} -176.032i q^{91} -57.0637i q^{95} -135.112 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−6.03459	−1.20692	−0.603459	−	0.797394i	$-0.706211\pi$
$5$	−6.03459	−1.20692	−0.603459	+	0.797394i	$0.706211\pi$
$6$	0	0
$7$	11.8163i	1.68804i	0.536310	+	0.844021i	$0.319818\pi$
$7$	11.8163i	1.68804i	−0.536310	+	0.844021i	$0.680182\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 6.10206i	− 0.554733i	−0.960764	−	0.277366i	$-0.910538\pi$
$11$	− 6.10206i	− 0.554733i	0.960764	−	0.277366i	$-0.0894616\pi$
$12$	0	0
$13$	−14.8974	−1.14595	−0.572975	−	0.819573i	$-0.694211\pi$
$13$	−14.8974	−1.14595	−0.572975	+	0.819573i	$0.694211\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−26.6919	−1.57011	−0.785055	−	0.619427i	$-0.787365\pi$
$17$	−26.6919	−1.57011	−0.785055	+	0.619427i	$0.787365\pi$
$18$	0	0
$19$	9.45610i	0.497689i	0.968543	+	0.248845i	$0.0800509\pi$
$19$	9.45610i	0.497689i	−0.968543	+	0.248845i	$0.919949\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 19.9386i	− 0.866896i	−0.901179	−	0.433448i	$-0.857297\pi$
$23$	− 19.9386i	− 0.866896i	0.901179	−	0.433448i	$-0.142703\pi$
$24$	0	0
$25$	11.4163	0.456650
$26$	0	0
$27$	0	0
$28$	0	0
$29$	44.6229	1.53872	0.769360	−	0.638816i	$-0.220575\pi$
$29$	44.6229	1.53872	0.769360	+	0.638816i	$0.220575\pi$
$30$	0	0
$31$	− 6.26262i	− 0.202020i	−0.994885	−	0.101010i	$-0.967793\pi$
$31$	− 6.26262i	− 0.202020i	0.994885	−	0.101010i	$-0.0322074\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 71.3065i	− 2.03733i
$36$	0	0
$37$	−6.65707	−0.179921	−0.0899604	−	0.995945i	$-0.528674\pi$
$37$	−6.65707	−0.179921	−0.0899604	+	0.995945i	$0.528674\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	17.6571	0.430660	0.215330	−	0.976541i	$-0.430917\pi$
$41$	17.6571	0.430660	0.215330	+	0.976541i	$0.430917\pi$
$42$	0	0
$43$	23.4053i	0.544310i	0.962253	+	0.272155i	$0.0877364\pi$
$43$	23.4053i	0.544310i	−0.962253	+	0.272155i	$0.912264\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	42.0612i	0.894920i	0.894304	+	0.447460i	$0.147671\pi$
$47$	42.0612i	0.894920i	−0.894304	+	0.447460i	$0.852329\pi$
$48$	0	0
$49$	−90.6249	−1.84949
$50$	0	0
$51$	0	0
$52$	0	0
$53$	51.6192	0.973948	0.486974	−	0.873416i	$-0.338101\pi$
$53$	51.6192	0.973948	0.486974	+	0.873416i	$0.338101\pi$
$54$	0	0
$55$	36.8234i	0.669517i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 37.9924i	− 0.643939i	−0.946750	−	0.321970i	$-0.895655\pi$
$59$	− 37.9924i	− 0.643939i	0.946750	−	0.321970i	$-0.104345\pi$
$60$	0	0
$61$	90.7631	1.48792	0.743960	−	0.668225i	$-0.232945\pi$
$61$	90.7631	1.48792	0.743960	+	0.668225i	$0.232945\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	89.8994	1.38307
$66$	0	0
$67$	− 61.7276i	− 0.921308i	−0.887580	−	0.460654i	$-0.847615\pi$
$67$	− 61.7276i	− 0.921308i	0.887580	−	0.460654i	$-0.152385\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	39.5232i	0.556665i	0.960485	+	0.278333i	$0.0897817\pi$
$71$	39.5232i	0.556665i	−0.960485	+	0.278333i	$0.910218\pi$
$72$	0	0
$73$	35.0355	0.479938	0.239969	−	0.970780i	$-0.422863\pi$
$73$	35.0355	0.479938	0.239969	+	0.970780i	$0.422863\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	72.1038	0.936413
$78$	0	0
$79$	− 90.0210i	− 1.13951i	−0.821816	−	0.569753i	$-0.807039\pi$
$79$	− 90.0210i	− 1.13951i	0.821816	−	0.569753i	$-0.192961\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 118.191i	− 1.42399i	−0.702183	−	0.711996i	$-0.747791\pi$
$83$	− 118.191i	− 1.42399i	0.702183	−	0.711996i	$-0.252209\pi$
$84$	0	0
$85$	161.074	1.89499
$86$	0	0
$87$	0	0
$88$	0	0
$89$	14.4499	0.162359	0.0811794	−	0.996700i	$-0.474131\pi$
$89$	14.4499	0.162359	0.0811794	+	0.996700i	$0.474131\pi$
$90$	0	0
$91$	− 176.032i	− 1.93441i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 57.0637i	− 0.600670i
$96$	0	0
$97$	−135.112	−1.39291	−0.696455	−	0.717601i	$-0.745240\pi$
$97$	−135.112	−1.39291	−0.696455	+	0.717601i	$0.745240\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−23.4879	−0.232553	−0.116277	−	0.993217i	$-0.537096\pi$
$101$	−23.4879	−0.232553	−0.116277	+	0.993217i	$0.537096\pi$
$102$	0	0
$103$	31.5062i	0.305885i	0.988235	+	0.152943i	$0.0488750\pi$
$103$	31.5062i	0.305885i	−0.988235	+	0.152943i	$0.951125\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 208.386i	− 1.94753i	−0.227558	−	0.973765i	$-0.573074\pi$
$107$	− 208.386i	− 1.94753i	0.227558	−	0.973765i	$-0.426926\pi$
$108$	0	0
$109$	64.5228	0.591952	0.295976	−	0.955195i	$-0.404355\pi$
$109$	64.5228	0.591952	0.295976	+	0.955195i	$0.404355\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.58230	0.0317017	0.0158509	−	0.999874i	$-0.494954\pi$
$113$	3.58230	0.0317017	0.0158509	+	0.999874i	$0.494954\pi$
$114$	0	0
$115$	120.321i	1.04627i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 315.399i	− 2.65041i
$120$	0	0
$121$	83.7649	0.692272
$122$	0	0
$123$	0	0
$124$	0	0
$125$	81.9723	0.655778
$126$	0	0
$127$	92.5083i	0.728412i	0.931319	+	0.364206i	$0.118660\pi$
$127$	92.5083i	0.728412i	−0.931319	+	0.364206i	$0.881340\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 121.550i	− 0.927862i	−0.885871	−	0.463931i	$-0.846439\pi$
$131$	− 121.550i	− 0.927862i	0.885871	−	0.463931i	$-0.153561\pi$
$132$	0	0
$133$	−111.736	−0.840121
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−256.840	−1.87474	−0.937372	−	0.348330i	$-0.886749\pi$
$137$	−256.840	−1.87474	−0.937372	+	0.348330i	$0.886749\pi$
$138$	0	0
$139$	128.352i	0.923396i	0.887037	+	0.461698i	$0.152760\pi$
$139$	128.352i	0.923396i	−0.887037	+	0.461698i	$0.847240\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	90.9045i	0.635696i
$144$	0	0
$145$	−269.281	−1.85711
$146$	0	0
$147$	0	0
$148$	0	0
$149$	21.7171	0.145752	0.0728762	−	0.997341i	$-0.476782\pi$
$149$	21.7171	0.145752	0.0728762	+	0.997341i	$0.476782\pi$
$150$	0	0
$151$	− 280.520i	− 1.85775i	−0.370396	−	0.928874i	$-0.620778\pi$
$151$	− 280.520i	− 1.85775i	0.370396	−	0.928874i	$-0.379222\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	37.7924i	0.243822i
$156$	0	0
$157$	−105.069	−0.669231	−0.334615	−	0.942355i	$-0.608606\pi$
$157$	−105.069	−0.669231	−0.334615	+	0.942355i	$0.608606\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	235.601	1.46336
$162$	0	0
$163$	145.690i	0.893804i	0.894583	+	0.446902i	$0.147473\pi$
$163$	145.690i	0.893804i	−0.894583	+	0.446902i	$0.852527\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 245.695i	− 1.47123i	−0.677402	−	0.735613i	$-0.736894\pi$
$167$	− 245.695i	− 1.47123i	0.677402	−	0.735613i	$-0.263106\pi$
$168$	0	0
$169$	52.9311	0.313202
$170$	0	0
$171$	0	0
$172$	0	0
$173$	43.8202	0.253296	0.126648	−	0.991948i	$-0.459578\pi$
$173$	43.8202	0.253296	0.126648	+	0.991948i	$0.459578\pi$
$174$	0	0
$175$	134.898i	0.770845i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 57.0637i	− 0.318791i	−0.987215	−	0.159396i	$-0.949045\pi$
$179$	− 57.0637i	− 0.318791i	0.987215	−	0.159396i	$-0.0509546\pi$
$180$	0	0
$181$	92.7281	0.512310	0.256155	−	0.966636i	$-0.417544\pi$
$181$	92.7281	0.512310	0.256155	+	0.966636i	$0.417544\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	40.1727	0.217150
$186$	0	0
$187$	162.875i	0.870991i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	89.4508i	0.468329i	0.972197	+	0.234164i	$0.0752354\pi$
$191$	89.4508i	0.468329i	−0.972197	+	0.234164i	$0.924765\pi$
$192$	0	0
$193$	230.426	1.19392	0.596959	−	0.802271i	$-0.296375\pi$
$193$	230.426	1.19392	0.596959	+	0.802271i	$0.296375\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	95.9308	0.486958	0.243479	−	0.969906i	$-0.421711\pi$
$197$	95.9308	0.486958	0.243479	+	0.969906i	$0.421711\pi$
$198$	0	0
$199$	50.1566i	0.252043i	0.992028	+	0.126021i	$0.0402208\pi$
$199$	50.1566i	0.252043i	−0.992028	+	0.126021i	$0.959779\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	527.277i	2.59742i
$204$	0	0
$205$	−106.553	−0.519771
$206$	0	0
$207$	0	0
$208$	0	0
$209$	57.7017	0.276085
$210$	0	0
$211$	2.63305i	0.0124789i	0.999981	+	0.00623946i	$0.00198610\pi$
$211$	2.63305i	0.0124789i	−0.999981	+	0.00623946i	$0.998014\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 141.242i	− 0.656938i
$216$	0	0
$217$	74.0010	0.341019
$218$	0	0
$219$	0	0
$220$	0	0
$221$	397.638	1.79927
$222$	0	0
$223$	176.388i	0.790978i	0.918471	+	0.395489i	$0.129425\pi$
$223$	176.388i	0.790978i	−0.918471	+	0.395489i	$0.870575\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.14511i	0.00944984i	0.999989	+	0.00472492i	$0.00150399\pi$
$227$	2.14511i	0.00944984i	−0.999989	+	0.00472492i	$0.998496\pi$
$228$	0	0
$229$	−127.289	−0.555849	−0.277925	−	0.960603i	$-0.589647\pi$
$229$	−127.289	−0.555849	−0.277925	+	0.960603i	$0.589647\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−262.526	−1.12672	−0.563360	−	0.826211i	$-0.690492\pi$
$233$	−262.526	−1.12672	−0.563360	+	0.826211i	$0.690492\pi$
$234$	0	0
$235$	− 253.822i	− 1.08009i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 312.413i	− 1.30717i	−0.756854	−	0.653584i	$-0.773265\pi$
$239$	− 312.413i	− 1.30717i	0.756854	−	0.653584i	$-0.226735\pi$
$240$	0	0
$241$	29.4219	0.122083	0.0610413	−	0.998135i	$-0.480558\pi$
$241$	29.4219	0.122083	0.0610413	+	0.998135i	$0.480558\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	546.884	2.23218
$246$	0	0
$247$	− 140.871i	− 0.570327i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	195.326i	0.778192i	0.921197	+	0.389096i	$0.127213\pi$
$251$	195.326i	0.778192i	−0.921197	+	0.389096i	$0.872787\pi$
$252$	0	0
$253$	−121.667	−0.480896
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−437.346	−1.70173	−0.850867	−	0.525381i	$-0.823923\pi$
$257$	−437.346	−1.70173	−0.850867	+	0.525381i	$0.823923\pi$
$258$	0	0
$259$	− 78.6619i	− 0.303714i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 74.8405i	− 0.284565i	−0.989826	−	0.142282i	$-0.954556\pi$
$263$	− 74.8405i	− 0.284565i	0.989826	−	0.142282i	$-0.0454441\pi$
$264$	0	0
$265$	−311.501	−1.17548
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−40.6759	−0.151212	−0.0756058	−	0.997138i	$-0.524089\pi$
$269$	−40.6759	−0.151212	−0.0756058	+	0.997138i	$0.524089\pi$
$270$	0	0
$271$	130.442i	0.481337i	0.970607	+	0.240668i	$0.0773666\pi$
$271$	130.442i	0.481337i	−0.970607	+	0.240668i	$0.922633\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 69.6627i	− 0.253319i
$276$	0	0
$277$	228.816	0.826049	0.413025	−	0.910720i	$-0.364472\pi$
$277$	228.816	0.826049	0.413025	+	0.910720i	$0.364472\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	47.9065	0.170486	0.0852429	−	0.996360i	$-0.472833\pi$
$281$	47.9065	0.170486	0.0852429	+	0.996360i	$0.472833\pi$
$282$	0	0
$283$	− 354.782i	− 1.25365i	−0.779161	−	0.626824i	$-0.784355\pi$
$283$	− 354.782i	− 1.25365i	0.779161	−	0.626824i	$-0.215645\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	208.641i	0.726973i
$288$	0	0
$289$	423.455	1.46524
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−55.8957	−0.190770	−0.0953851	−	0.995440i	$-0.530408\pi$
$293$	−55.8957	−0.190770	−0.0953851	+	0.995440i	$0.530408\pi$
$294$	0	0
$295$	229.269i	0.777182i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	297.032i	0.993420i
$300$	0	0
$301$	−276.564	−0.918819
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−547.718	−1.79580
$306$	0	0
$307$	− 57.3939i	− 0.186951i	−0.995622	−	0.0934754i	$-0.970202\pi$
$307$	− 57.3939i	− 0.186951i	0.995622	−	0.0934754i	$-0.0297976\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	248.836i	0.800115i	0.916490	+	0.400058i	$0.131010\pi$
$311$	248.836i	0.800115i	−0.916490	+	0.400058i	$0.868990\pi$
$312$	0	0
$313$	615.712	1.96713	0.983565	−	0.180555i	$-0.0577893\pi$
$313$	615.712	1.96713	0.983565	+	0.180555i	$0.0577893\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−256.627	−0.809549	−0.404775	−	0.914417i	$-0.632650\pi$
$317$	−256.627	−0.809549	−0.404775	+	0.914417i	$0.632650\pi$
$318$	0	0
$319$	− 272.291i	− 0.853578i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 252.401i	− 0.781427i
$324$	0	0
$325$	−170.072	−0.523299
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−497.008	−1.51066
$330$	0	0
$331$	− 145.196i	− 0.438658i	−0.975651	−	0.219329i	$-0.929613\pi$
$331$	− 145.196i	− 0.438658i	0.975651	−	0.219329i	$-0.0703868\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	372.501i	1.11194i
$336$	0	0
$337$	−43.6272	−0.129458	−0.0647288	−	0.997903i	$-0.520618\pi$
$337$	−43.6272	−0.129458	−0.0647288	+	0.997903i	$0.520618\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−38.2149	−0.112067
$342$	0	0
$343$	− 491.852i	− 1.43397i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	180.267i	0.519502i	0.965676	+	0.259751i	$0.0836406\pi$
$347$	180.267i	0.519502i	−0.965676	+	0.259751i	$0.916359\pi$
$348$	0	0
$349$	−93.1258	−0.266836	−0.133418	−	0.991060i	$-0.542595\pi$
$349$	−93.1258	−0.266836	−0.133418	+	0.991060i	$0.542595\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−430.909	−1.22071	−0.610353	−	0.792130i	$-0.708972\pi$
$353$	−430.909	−1.22071	−0.610353	+	0.792130i	$0.708972\pi$
$354$	0	0
$355$	− 238.506i	− 0.671849i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 316.054i	− 0.880374i	−0.897906	−	0.440187i	$-0.854912\pi$
$359$	− 316.054i	− 0.880374i	0.897906	−	0.440187i	$-0.145088\pi$
$360$	0	0
$361$	271.582	0.752305
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−211.425	−0.579246
$366$	0	0
$367$	− 523.947i	− 1.42765i	−0.700325	−	0.713824i	$-0.746962\pi$
$367$	− 523.947i	− 1.42765i	0.700325	−	0.713824i	$-0.253038\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	609.948i	1.64407i
$372$	0	0
$373$	327.134	0.877035	0.438518	−	0.898723i	$-0.355504\pi$
$373$	327.134	0.877035	0.438518	+	0.898723i	$0.355504\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−664.763	−1.76330
$378$	0	0
$379$	443.580i	1.17040i	0.810890	+	0.585198i	$0.198983\pi$
$379$	443.580i	1.17040i	−0.810890	+	0.585198i	$0.801017\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	453.806i	1.18487i	0.805617	+	0.592436i	$0.201834\pi$
$383$	453.806i	1.18487i	−0.805617	+	0.592436i	$0.798166\pi$
$384$	0	0
$385$	−435.117	−1.13017
$386$	0	0
$387$	0	0
$388$	0	0
$389$	164.872	0.423836	0.211918	−	0.977287i	$-0.432029\pi$
$389$	164.872	0.423836	0.211918	+	0.977287i	$0.432029\pi$
$390$	0	0
$391$	532.199i	1.36112i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	543.240i	1.37529i
$396$	0	0
$397$	−395.775	−0.996914	−0.498457	−	0.866915i	$-0.666100\pi$
$397$	−395.775	−0.996914	−0.498457	+	0.866915i	$0.666100\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	347.676	0.867023	0.433512	−	0.901148i	$-0.357274\pi$
$401$	347.676	0.867023	0.433512	+	0.901148i	$0.357274\pi$
$402$	0	0
$403$	93.2965i	0.231505i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	40.6218i	0.0998079i
$408$	0	0
$409$	65.5978	0.160386	0.0801930	−	0.996779i	$-0.474446\pi$
$409$	65.5978	0.160386	0.0801930	+	0.996779i	$0.474446\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	448.930	1.08700
$414$	0	0
$415$	713.236i	1.71864i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 688.113i	− 1.64227i	−0.570731	−	0.821137i	$-0.693340\pi$
$419$	− 688.113i	− 1.64227i	0.570731	−	0.821137i	$-0.306660\pi$
$420$	0	0
$421$	293.709	0.697647	0.348823	−	0.937188i	$-0.386581\pi$
$421$	293.709	0.697647	0.348823	+	0.937188i	$0.386581\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−304.721	−0.716991
$426$	0	0
$427$	1072.48i	2.51167i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	555.264i	1.28832i	0.764893	+	0.644158i	$0.222792\pi$
$431$	555.264i	1.28832i	−0.764893	+	0.644158i	$0.777208\pi$
$432$	0	0
$433$	−559.107	−1.29124	−0.645620	−	0.763659i	$-0.723401\pi$
$433$	−559.107	−1.29124	−0.645620	+	0.763659i	$0.723401\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	188.541	0.431445
$438$	0	0
$439$	598.785i	1.36398i	0.731364	+	0.681988i	$0.238884\pi$
$439$	598.785i	1.36398i	−0.731364	+	0.681988i	$0.761116\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 610.200i	− 1.37743i	−0.725034	−	0.688713i	$-0.758176\pi$
$443$	− 610.200i	− 1.37743i	0.725034	−	0.688713i	$-0.241824\pi$
$444$	0	0
$445$	−87.1994	−0.195954
$446$	0	0
$447$	0	0
$448$	0	0
$449$	342.989	0.763896	0.381948	−	0.924184i	$-0.375253\pi$
$449$	342.989	0.763896	0.381948	+	0.924184i	$0.375253\pi$
$450$	0	0
$451$	− 107.744i	− 0.238901i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1062.28i	2.33468i
$456$	0	0
$457$	−281.540	−0.616061	−0.308030	−	0.951377i	$-0.599670\pi$
$457$	−281.540	−0.616061	−0.308030	+	0.951377i	$0.599670\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	98.1287	0.212860	0.106430	−	0.994320i	$-0.466058\pi$
$461$	98.1287	0.212860	0.106430	+	0.994320i	$0.466058\pi$
$462$	0	0
$463$	− 722.026i	− 1.55945i	−0.626121	−	0.779726i	$-0.715358\pi$
$463$	− 722.026i	− 1.55945i	0.626121	−	0.779726i	$-0.284642\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	723.722i	1.54972i	0.632130	+	0.774862i	$0.282181\pi$
$467$	723.722i	1.54972i	−0.632130	+	0.774862i	$0.717819\pi$
$468$	0	0
$469$	729.392	1.55521
$470$	0	0
$471$	0	0
$472$	0	0
$473$	142.821	0.301947
$474$	0	0
$475$	107.953i	0.227270i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	350.649i	0.732045i	0.930606	+	0.366022i	$0.119281\pi$
$479$	350.649i	0.732045i	−0.930606	+	0.366022i	$0.880719\pi$
$480$	0	0
$481$	99.1727	0.206180
$482$	0	0
$483$	0	0
$484$	0	0
$485$	815.347	1.68113
$486$	0	0
$487$	− 693.565i	− 1.42416i	−0.702099	−	0.712079i	$-0.747754\pi$
$487$	− 693.565i	− 1.42416i	0.702099	−	0.712079i	$-0.252246\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	520.597i	1.06028i	0.847911	+	0.530139i	$0.177860\pi$
$491$	520.597i	1.06028i	−0.847911	+	0.530139i	$0.822140\pi$
$492$	0	0
$493$	−1191.07	−2.41596
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−467.018	−0.939675
$498$	0	0
$499$	− 415.528i	− 0.832721i	−0.909200	−	0.416361i	$-0.863305\pi$
$499$	− 415.528i	− 0.832721i	0.909200	−	0.416361i	$-0.136695\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	153.470i	0.305109i	0.988295	+	0.152555i	$0.0487500\pi$
$503$	153.470i	0.305109i	−0.988295	+	0.152555i	$0.951250\pi$
$504$	0	0
$505$	141.740	0.280673
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−581.209	−1.14186	−0.570932	−	0.820998i	$-0.693418\pi$
$509$	−581.209	−1.14186	−0.570932	+	0.820998i	$0.693418\pi$
$510$	0	0
$511$	413.990i	0.810157i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 190.127i	− 0.369179i
$516$	0	0
$517$	256.660	0.496441
$518$	0	0
$519$	0	0
$520$	0	0
$521$	119.457	0.229284	0.114642	−	0.993407i	$-0.463428\pi$
$521$	119.457	0.229284	0.114642	+	0.993407i	$0.463428\pi$
$522$	0	0
$523$	− 291.527i	− 0.557413i	−0.960376	−	0.278706i	$-0.910094\pi$
$523$	− 291.527i	− 0.557413i	0.960376	−	0.278706i	$-0.0899056\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	167.161i	0.317194i
$528$	0	0
$529$	131.452	0.248491
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−263.044	−0.493515
$534$	0	0
$535$	1257.52i	2.35051i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	552.999i	1.02597i
$540$	0	0
$541$	−918.712	−1.69817	−0.849087	−	0.528254i	$-0.822847\pi$
$541$	−918.712	−1.69817	−0.849087	+	0.528254i	$0.822847\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−389.368	−0.714438
$546$	0	0
$547$	− 999.655i	− 1.82752i	−0.406252	−	0.913761i	$-0.633164\pi$
$547$	− 999.655i	− 1.82752i	0.406252	−	0.913761i	$-0.366836\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	421.958i	0.765804i
$552$	0	0
$553$	1063.72	1.92354
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−730.122	−1.31081	−0.655406	−	0.755277i	$-0.727503\pi$
$557$	−730.122	−1.31081	−0.655406	+	0.755277i	$0.727503\pi$
$558$	0	0
$559$	− 348.678i	− 0.623752i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 387.852i	− 0.688901i	−0.938804	−	0.344451i	$-0.888065\pi$
$563$	− 387.852i	− 0.688901i	0.938804	−	0.344451i	$-0.111935\pi$
$564$	0	0
$565$	−21.6177	−0.0382614
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−556.778	−0.978520	−0.489260	−	0.872138i	$-0.662733\pi$
$569$	−556.778	−0.978520	−0.489260	+	0.872138i	$0.662733\pi$
$570$	0	0
$571$	− 32.3859i	− 0.0567179i	−0.999598	−	0.0283589i	$-0.990972\pi$
$571$	− 32.3859i	− 0.0567179i	0.999598	−	0.0283589i	$-0.00902814\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 227.624i	− 0.395869i
$576$	0	0
$577$	221.536	0.383945	0.191972	−	0.981400i	$-0.438512\pi$
$577$	221.536	0.383945	0.191972	+	0.981400i	$0.438512\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1396.58	2.40376
$582$	0	0
$583$	− 314.984i	− 0.540281i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 754.882i	− 1.28600i	−0.765866	−	0.643000i	$-0.777689\pi$
$587$	− 754.882i	− 1.28600i	0.765866	−	0.643000i	$-0.222311\pi$
$588$	0	0
$589$	59.2200	0.100543
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−697.420	−1.17609	−0.588044	−	0.808829i	$-0.700102\pi$
$593$	−697.420	−1.17609	−0.588044	+	0.808829i	$0.700102\pi$
$594$	0	0
$595$	1903.30i	3.19883i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	93.2741i	0.155716i	0.996964	+	0.0778581i	$0.0248081\pi$
$599$	93.2741i	0.155716i	−0.996964	+	0.0778581i	$0.975192\pi$
$600$	0	0
$601$	−430.027	−0.715519	−0.357760	−	0.933814i	$-0.616459\pi$
$601$	−430.027	−0.715519	−0.357760	+	0.933814i	$0.616459\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−505.486	−0.835515
$606$	0	0
$607$	− 547.538i	− 0.902039i	−0.892514	−	0.451019i	$-0.851060\pi$
$607$	− 547.538i	− 0.902039i	0.892514	−	0.451019i	$-0.148940\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 626.601i	− 1.02553i
$612$	0	0
$613$	−6.42863	−0.0104872	−0.00524358	−	0.999986i	$-0.501669\pi$
$613$	−6.42863	−0.0104872	−0.00524358	+	0.999986i	$0.501669\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1034.31	−1.67635	−0.838177	−	0.545398i	$-0.816379\pi$
$617$	−1034.31	−1.67635	−0.838177	+	0.545398i	$0.816379\pi$
$618$	0	0
$619$	− 740.791i	− 1.19675i	−0.801215	−	0.598377i	$-0.795812\pi$
$619$	− 740.791i	− 1.19675i	0.801215	−	0.598377i	$-0.204188\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	170.745i	0.274069i
$624$	0	0
$625$	−780.076	−1.24812
$626$	0	0
$627$	0	0
$628$	0	0
$629$	177.689	0.282495
$630$	0	0
$631$	− 693.165i	− 1.09852i	−0.835652	−	0.549259i	$-0.814910\pi$
$631$	− 693.165i	− 1.09852i	0.835652	−	0.549259i	$-0.185090\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 558.249i	− 0.879133i
$636$	0	0
$637$	1350.07	2.11942
$638$	0	0
$639$	0	0
$640$	0	0
$641$	230.291	0.359268	0.179634	−	0.983734i	$-0.442509\pi$
$641$	230.291	0.359268	0.179634	+	0.983734i	$0.442509\pi$
$642$	0	0
$643$	765.470i	1.19047i	0.803553	+	0.595233i	$0.202940\pi$
$643$	765.470i	1.19047i	−0.803553	+	0.595233i	$0.797060\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 339.078i	− 0.524078i	−0.965057	−	0.262039i	$-0.915605\pi$
$647$	− 339.078i	− 0.524078i	0.965057	−	0.262039i	$-0.0843949\pi$
$648$	0	0
$649$	−231.832	−0.357214
$650$	0	0
$651$	0	0
$652$	0	0
$653$	147.649	0.226108	0.113054	−	0.993589i	$-0.463937\pi$
$653$	147.649	0.226108	0.113054	+	0.993589i	$0.463937\pi$
$654$	0	0
$655$	733.504i	1.11985i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	918.482i	1.39375i	0.717192	+	0.696875i	$0.245427\pi$
$659$	918.482i	1.39375i	−0.717192	+	0.696875i	$0.754573\pi$
$660$	0	0
$661$	206.301	0.312104	0.156052	−	0.987749i	$-0.450123\pi$
$661$	206.301	0.312104	0.156052	+	0.987749i	$0.450123\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	674.281	1.01396
$666$	0	0
$667$	− 889.718i	− 1.33391i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 553.842i	− 0.825397i
$672$	0	0
$673$	666.545	0.990409	0.495204	−	0.868777i	$-0.335093\pi$
$673$	666.545	0.990409	0.495204	+	0.868777i	$0.335093\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	470.994	0.695707	0.347854	−	0.937549i	$-0.386911\pi$
$677$	470.994	0.695707	0.347854	+	0.937549i	$0.386911\pi$
$678$	0	0
$679$	− 1596.53i	− 2.35129i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 909.494i	− 1.33162i	−0.746123	−	0.665808i	$-0.768087\pi$
$683$	− 909.494i	− 1.33162i	0.746123	−	0.665808i	$-0.231913\pi$
$684$	0	0
$685$	1549.92	2.26266
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−768.990	−1.11610
$690$	0	0
$691$	− 423.529i	− 0.612921i	−0.951883	−	0.306461i	$-0.900855\pi$
$691$	− 423.529i	− 0.612921i	0.951883	−	0.306461i	$-0.0991448\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 774.552i	− 1.11446i
$696$	0	0
$697$	−471.300	−0.676183
$698$	0	0
$699$	0	0
$700$	0	0
$701$	66.5738	0.0949697	0.0474849	−	0.998872i	$-0.484879\pi$
$701$	66.5738	0.0949697	0.0474849	+	0.998872i	$0.484879\pi$
$702$	0	0
$703$	− 62.9499i	− 0.0895446i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 277.540i	− 0.392560i
$708$	0	0
$709$	96.7863	0.136511	0.0682555	−	0.997668i	$-0.478257\pi$
$709$	96.7863	0.136511	0.0682555	+	0.997668i	$0.478257\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−124.868	−0.175130
$714$	0	0
$715$	− 548.571i	− 0.767233i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1184.39i	− 1.64727i	−0.567120	−	0.823635i	$-0.691942\pi$
$719$	− 1184.39i	− 1.64727i	0.567120	−	0.823635i	$-0.308058\pi$
$720$	0	0
$721$	−372.287	−0.516348
$722$	0	0
$723$	0	0
$724$	0	0
$725$	509.426	0.702657
$726$	0	0
$727$	794.088i	1.09228i	0.837694	+	0.546140i	$0.183903\pi$
$727$	794.088i	1.09228i	−0.837694	+	0.546140i	$0.816097\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 624.732i	− 0.854626i
$732$	0	0
$733$	128.986	0.175971	0.0879853	−	0.996122i	$-0.471957\pi$
$733$	128.986	0.175971	0.0879853	+	0.996122i	$0.471957\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−376.666	−0.511080
$738$	0	0
$739$	17.7228i	0.0239822i	0.999928	+	0.0119911i	$0.00381697\pi$
$739$	17.7228i	0.0239822i	−0.999928	+	0.0119911i	$0.996183\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	46.1150i	0.0620659i	0.999518	+	0.0310330i	$0.00987968\pi$
$743$	46.1150i	0.0620659i	−0.999518	+	0.0310330i	$0.990120\pi$
$744$	0	0
$745$	−131.054	−0.175911
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2462.35	3.28751
$750$	0	0
$751$	236.855i	0.315386i	0.987488	+	0.157693i	$0.0504056\pi$
$751$	236.855i	0.315386i	−0.987488	+	0.157693i	$0.949594\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1692.82i	2.24215i
$756$	0	0
$757$	193.736	0.255925	0.127963	−	0.991779i	$-0.459156\pi$
$757$	193.736	0.255925	0.127963	+	0.991779i	$0.459156\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−82.1296	−0.107923	−0.0539616	−	0.998543i	$-0.517185\pi$
$761$	−82.1296	−0.107923	−0.0539616	+	0.998543i	$0.517185\pi$
$762$	0	0
$763$	762.421i	0.999241i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	565.987i	0.737922i
$768$	0	0
$769$	−1318.45	−1.71451	−0.857253	−	0.514896i	$-0.827831\pi$
$769$	−1318.45	−1.71451	−0.857253	+	0.514896i	$0.827831\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.74554	0.00225814	0.00112907	−	0.999999i	$-0.499641\pi$
$773$	1.74554	0.00225814	0.00112907	+	0.999999i	$0.499641\pi$
$774$	0	0
$775$	− 71.4958i	− 0.0922526i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	166.967i	0.214335i
$780$	0	0
$781$	241.173	0.308800
$782$	0	0
$783$	0	0
$784$	0	0
$785$	634.050	0.807706
$786$	0	0
$787$	− 1184.97i	− 1.50568i	−0.658205	−	0.752838i	$-0.728684\pi$
$787$	− 1184.97i	− 1.50568i	0.658205	−	0.752838i	$-0.271316\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	42.3295i	0.0535139i
$792$	0	0
$793$	−1352.13	−1.70508
$794$	0	0
$795$	0	0
$796$	0	0
$797$	486.841	0.610842	0.305421	−	0.952217i	$-0.401203\pi$
$797$	486.841	0.610842	0.305421	+	0.952217i	$0.401203\pi$
$798$	0	0
$799$	− 1122.69i	− 1.40512i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 213.789i	− 0.266238i
$804$	0	0
$805$	−1421.75	−1.76615
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1104.99	−1.36587	−0.682936	−	0.730478i	$-0.739297\pi$
$809$	−1104.99	−1.36587	−0.682936	+	0.730478i	$0.739297\pi$
$810$	0	0
$811$	321.548i	0.396483i	0.980153	+	0.198241i	$0.0635230\pi$
$811$	321.548i	0.396483i	−0.980153	+	0.198241i	$0.936477\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 879.180i	− 1.07875i
$816$	0	0
$817$	−221.323	−0.270897
$818$	0	0
$819$	0	0
$820$	0	0
$821$	794.622	0.967871	0.483936	−	0.875104i	$-0.339207\pi$
$821$	794.622	0.967871	0.483936	+	0.875104i	$0.339207\pi$
$822$	0	0
$823$	1178.13i	1.43150i	0.698356	+	0.715750i	$0.253915\pi$
$823$	1178.13i	1.43150i	−0.698356	+	0.715750i	$0.746085\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 722.570i	− 0.873725i	−0.899528	−	0.436862i	$-0.856090\pi$
$827$	− 722.570i	− 0.873725i	0.899528	−	0.436862i	$-0.143910\pi$
$828$	0	0
$829$	1435.00	1.73100	0.865501	−	0.500906i	$-0.167000\pi$
$829$	1435.00	1.73100	0.865501	+	0.500906i	$0.167000\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2418.95	2.90390
$834$	0	0
$835$	1482.67i	1.77565i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 589.167i	− 0.702226i	−0.936333	−	0.351113i	$-0.885803\pi$
$839$	− 589.167i	− 0.702226i	0.936333	−	0.351113i	$-0.114197\pi$
$840$	0	0
$841$	1150.20	1.36766
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−319.417	−0.378009
$846$	0	0
$847$	989.791i	1.16858i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	132.733i	0.155973i
$852$	0	0
$853$	317.907	0.372693	0.186347	−	0.982484i	$-0.440335\pi$
$853$	317.907	0.372693	0.186347	+	0.982484i	$0.440335\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	540.476	0.630661	0.315330	−	0.948982i	$-0.397885\pi$
$857$	540.476	0.630661	0.315330	+	0.948982i	$0.397885\pi$
$858$	0	0
$859$	1014.85i	1.18144i	0.806878	+	0.590718i	$0.201156\pi$
$859$	1014.85i	1.18144i	−0.806878	+	0.590718i	$0.798844\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 894.091i	− 1.03603i	−0.855373	−	0.518013i	$-0.826672\pi$
$863$	− 894.091i	− 1.03603i	0.855373	−	0.518013i	$-0.173328\pi$
$864$	0	0
$865$	−264.437	−0.305707
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−549.314	−0.632122
$870$	0	0
$871$	919.578i	1.05577i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	968.609i	1.10698i
$876$	0	0
$877$	709.759	0.809303	0.404652	−	0.914471i	$-0.367393\pi$
$877$	709.759	0.809303	0.404652	+	0.914471i	$0.367393\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1318.63	1.49674	0.748371	−	0.663281i	$-0.230837\pi$
$881$	1318.63	1.49674	0.748371	+	0.663281i	$0.230837\pi$
$882$	0	0
$883$	− 1035.56i	− 1.17277i	−0.810032	−	0.586386i	$-0.800550\pi$
$883$	− 1035.56i	− 1.17277i	0.810032	−	0.586386i	$-0.199450\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	832.842i	0.938943i	0.882948	+	0.469472i	$0.155556\pi$
$887$	832.842i	0.938943i	−0.882948	+	0.469472i	$0.844444\pi$
$888$	0	0
$889$	−1093.11	−1.22959
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−397.735	−0.445392
$894$	0	0
$895$	344.356i	0.384755i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 279.456i	− 0.310852i
$900$	0	0
$901$	−1377.81	−1.52920
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−559.576	−0.618316
$906$	0	0
$907$	9.92593i	0.0109437i	0.999985	+	0.00547185i	$0.00174175\pi$
$907$	9.92593i	0.0109437i	−0.999985	+	0.00547185i	$0.998258\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1668.11i	− 1.83108i	−0.402228	−	0.915540i	$-0.631764\pi$
$911$	− 1668.11i	− 1.83108i	0.402228	−	0.915540i	$-0.368236\pi$
$912$	0	0
$913$	−721.211	−0.789935
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1436.27	1.56627
$918$	0	0
$919$	1787.42i	1.94496i	0.232988	+	0.972480i	$0.425150\pi$
$919$	1787.42i	1.94496i	−0.232988	+	0.972480i	$0.574850\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 588.791i	− 0.637911i
$924$	0	0
$925$	−75.9988	−0.0821609
$926$	0	0
$927$	0	0
$928$	0	0
$929$	719.714	0.774719	0.387360	−	0.921929i	$-0.373387\pi$
$929$	719.714	0.774719	0.387360	+	0.921929i	$0.373387\pi$
$930$	0	0
$931$	− 856.958i	− 0.920470i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 982.886i	− 1.05121i
$936$	0	0
$937$	1352.51	1.44345	0.721724	−	0.692181i	$-0.243350\pi$
$937$	1352.51	1.44345	0.721724	+	0.692181i	$0.243350\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1511.82	−1.60661	−0.803303	−	0.595570i	$-0.796926\pi$
$941$	−1511.82	−1.60661	−0.803303	+	0.595570i	$0.796926\pi$
$942$	0	0
$943$	− 352.057i	− 0.373338i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	846.884i	0.894281i	0.894464	+	0.447140i	$0.147558\pi$
$947$	846.884i	0.894281i	−0.894464	+	0.447140i	$0.852442\pi$
$948$	0	0
$949$	−521.936	−0.549986
$950$	0	0
$951$	0	0
$952$	0	0
$953$	198.475	0.208263	0.104131	−	0.994564i	$-0.466794\pi$
$953$	198.475	0.208263	0.104131	+	0.994564i	$0.466794\pi$
$954$	0	0
$955$	− 539.799i	− 0.565234i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 3034.90i	− 3.16465i
$960$	0	0
$961$	921.780	0.959188
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1390.53	−1.44096
$966$	0	0
$967$	− 205.958i	− 0.212987i	−0.994313	−	0.106493i	$-0.966038\pi$
$967$	− 205.958i	− 0.212987i	0.994313	−	0.106493i	$-0.0339623\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1364.48i	− 1.40523i	−0.711568	−	0.702617i	$-0.752015\pi$
$971$	− 1364.48i	− 1.40523i	0.711568	−	0.702617i	$-0.247985\pi$
$972$	0	0
$973$	−1516.65	−1.55873
$974$	0	0
$975$	0	0
$976$	0	0
$977$	944.161	0.966388	0.483194	−	0.875513i	$-0.339477\pi$
$977$	944.161	0.966388	0.483194	+	0.875513i	$0.339477\pi$
$978$	0	0
$979$	− 88.1744i	− 0.0900657i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1526.67i	− 1.55307i	−0.630072	−	0.776537i	$-0.716975\pi$
$983$	− 1526.67i	− 1.55307i	0.630072	−	0.776537i	$-0.283025\pi$
$984$	0	0
$985$	−578.903	−0.587719
$986$	0	0
$987$	0	0
$988$	0	0
$989$	466.670	0.471860
$990$	0	0
$991$	− 599.630i	− 0.605075i	−0.953137	−	0.302538i	$-0.902166\pi$
$991$	− 599.630i	− 0.605075i	0.953137	−	0.302538i	$-0.0978338\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 302.674i	− 0.304195i
$996$	0	0
$997$	−594.234	−0.596022	−0.298011	−	0.954562i	$-0.596323\pi$
$997$	−594.234	−0.596022	−0.298011	+	0.954562i	$0.596323\pi$
$998$	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.1135.2		8
3.2	odd	2		1296.3.g.j.1135.8		8
4.3	odd	2	inner	1296.3.g.k.1135.1		8
9.2	odd	6		144.3.o.a.31.1	✓	8
9.4	even	3		432.3.o.a.127.4		8
9.5	odd	6		144.3.o.c.79.4	yes	8
9.7	even	3		432.3.o.b.415.4		8
12.11	even	2		1296.3.g.j.1135.7		8
36.7	odd	6		432.3.o.a.415.4		8
36.11	even	6		144.3.o.c.31.4	yes	8
36.23	even	6		144.3.o.a.79.1	yes	8
36.31	odd	6		432.3.o.b.127.4		8
72.5	odd	6		576.3.o.d.511.1		8
72.11	even	6		576.3.o.d.319.1		8
72.13	even	6		1728.3.o.e.127.1		8
72.29	odd	6		576.3.o.f.319.4		8
72.43	odd	6		1728.3.o.e.1279.1		8
72.59	even	6		576.3.o.f.511.4		8
72.61	even	6		1728.3.o.f.1279.1		8
72.67	odd	6		1728.3.o.f.127.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.3.o.a.31.1	✓	8	9.2	odd	6
144.3.o.a.79.1	yes	8	36.23	even	6
144.3.o.c.31.4	yes	8	36.11	even	6
144.3.o.c.79.4	yes	8	9.5	odd	6
432.3.o.a.127.4		8	9.4	even	3
432.3.o.a.415.4		8	36.7	odd	6
432.3.o.b.127.4		8	36.31	odd	6
432.3.o.b.415.4		8	9.7	even	3
576.3.o.d.319.1		8	72.11	even	6
576.3.o.d.511.1		8	72.5	odd	6
576.3.o.f.319.4		8	72.29	odd	6
576.3.o.f.511.4		8	72.59	even	6
1296.3.g.j.1135.7		8	12.11	even	2
1296.3.g.j.1135.8		8	3.2	odd	2
1296.3.g.k.1135.1		8	4.3	odd	2	inner
1296.3.g.k.1135.2		8	1.1	even	1	trivial
1728.3.o.e.127.1		8	72.13	even	6
1728.3.o.e.1279.1		8	72.43	odd	6
1728.3.o.f.127.1		8	72.67	odd	6
1728.3.o.f.1279.1		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)

\(f(q)\)	\(=\)	\(q-6.03459 q^{5} +11.8163i q^{7} +O(q^{10})\)	\(q-6.03459 q^{5} +11.8163i q^{7} -6.10206i q^{11} -14.8974 q^{13} -26.6919 q^{17} +9.45610i q^{19} -19.9386i q^{23} +11.4163 q^{25} +44.6229 q^{29} -6.26262i q^{31} -71.3065i q^{35} -6.65707 q^{37} +17.6571 q^{41} +23.4053i q^{43} +42.0612i q^{47} -90.6249 q^{49} +51.6192 q^{53} +36.8234i q^{55} -37.9924i q^{59} +90.7631 q^{61} +89.8994 q^{65} -61.7276i q^{67} +39.5232i q^{71} +35.0355 q^{73} +72.1038 q^{77} -90.0210i q^{79} -118.191i q^{83} +161.074 q^{85} +14.4499 q^{89} -176.032i q^{91} -57.0637i q^{95} -135.112 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{5}+O(q^{10}) \) 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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