LMFDB - Embedded newform 864.2.f.b.431.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(431,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("864.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	864.f (of order $2$, degree $1$, not 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:	$6.89907473464$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.170772624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 5x^{6} - 6x^{5} + 6x^{4} - 12x^{3} + 20x^{2} - 24x + 16 $ x^8 - 3x^7 + 5x^6 - 6x^5 + 6x^4 - 12x^3 + 20x^2 - 24*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.6
Root			$-1.02187 + 0.977642i$ of defining polynomial
Character	$\chi$	$=$	864.431
Dual form			864.2.f.b.431.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.505408 q^{5} +3.42703i q^{7} +O(q^{10})$	$q+0.505408 q^{5} +3.42703i q^{7} +3.31662i q^{11} +2.55164i q^{13} -5.04868i q^{17} -4.74456 q^{19} -6.44121 q^{23} -4.74456 q^{25} +5.43039 q^{29} +5.97868i q^{31} +1.73205i q^{35} +11.1565i q^{37} +1.87953i q^{41} +4.00000 q^{43} +10.8608 q^{47} -4.74456 q^{49} -5.93580 q^{53} +1.67625i q^{55} -6.63325i q^{59} +5.10328i q^{61} +1.28962i q^{65} +4.00000 q^{67} -4.41957 q^{71} +7.74456 q^{73} -11.3662 q^{77} +4.30243i q^{79} -3.61158i q^{83} -2.55164i q^{85} +17.0256i q^{89} -8.74456 q^{91} -2.39794 q^{95} -1.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{19} + 8 q^{25} + 32 q^{43} + 8 q^{49} + 32 q^{67} + 16 q^{73} - 24 q^{91} - 8 q^{97}+O(q^{100}) $ 8 * q + 8 * q^19 + 8 * q^25 + 32 * q^43 + 8 * q^49 + 32 * q^67 + 16 * q^73 - 24 * q^91 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$-1$	$-1$

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
$5$	0.505408	0.226025	0.113013	−	0.993594i	$-0.463950\pi$
$5$	0.505408	0.226025	0.113013	+	0.993594i	$0.463950\pi$
$6$	0	0
$7$	3.42703i	1.29530i	0.761939	+	0.647649i	$0.224247\pi$
$7$	3.42703i	1.29530i	−0.761939	+	0.647649i	$0.775753\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.31662i	1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$11$	3.31662i	1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$12$	0	0
$13$	2.55164i	0.707698i	0.935303	+	0.353849i	$0.115127\pi$
$13$	2.55164i	0.707698i	−0.935303	+	0.353849i	$0.884873\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.04868i	− 1.22448i	−0.790671	−	0.612242i	$-0.790268\pi$
$17$	− 5.04868i	− 1.22448i	0.790671	−	0.612242i	$-0.209732\pi$
$18$	0	0
$19$	−4.74456	−1.08848	−0.544239	−	0.838930i	$-0.683181\pi$
$19$	−4.74456	−1.08848	−0.544239	+	0.838930i	$0.683181\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.44121	−1.34308	−0.671542	−	0.740966i	$-0.734368\pi$
$23$	−6.44121	−1.34308	−0.671542	+	0.740966i	$0.734368\pi$
$24$	0	0
$25$	−4.74456	−0.948913
$26$	0	0
$27$	0	0
$28$	0	0
$29$	5.43039	1.00840	0.504199	−	0.863588i	$-0.331788\pi$
$29$	5.43039	1.00840	0.504199	+	0.863588i	$0.331788\pi$
$30$	0	0
$31$	5.97868i	1.07380i	0.843645	+	0.536901i	$0.180405\pi$
$31$	5.97868i	1.07380i	−0.843645	+	0.536901i	$0.819595\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.73205i	0.292770i
$36$	0	0
$37$	11.1565i	1.83412i	0.398753	+	0.917058i	$0.369443\pi$
$37$	11.1565i	1.83412i	−0.398753	+	0.917058i	$0.630557\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.87953i	0.293533i	0.989171	+	0.146766i	$0.0468866\pi$
$41$	1.87953i	0.293533i	−0.989171	+	0.146766i	$0.953113\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.8608	1.58421	0.792104	−	0.610387i	$-0.208986\pi$
$47$	10.8608	1.58421	0.792104	+	0.610387i	$0.208986\pi$
$48$	0	0
$49$	−4.74456	−0.677795
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.93580	−0.815344	−0.407672	−	0.913128i	$-0.633659\pi$
$53$	−5.93580	−0.815344	−0.407672	+	0.913128i	$0.633659\pi$
$54$	0	0
$55$	1.67625i	0.226025i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 6.63325i	− 0.863576i	−0.901975	−	0.431788i	$-0.857883\pi$
$59$	− 6.63325i	− 0.863576i	0.901975	−	0.431788i	$-0.142117\pi$
$60$	0	0
$61$	5.10328i	0.653408i	0.945127	+	0.326704i	$0.105938\pi$
$61$	5.10328i	0.653408i	−0.945127	+	0.326704i	$0.894062\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.28962i	0.159958i
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.41957	−0.524507	−0.262253	−	0.964999i	$-0.584466\pi$
$71$	−4.41957	−0.524507	−0.262253	+	0.964999i	$0.584466\pi$
$72$	0	0
$73$	7.74456	0.906432	0.453216	−	0.891401i	$-0.350277\pi$
$73$	7.74456	0.906432	0.453216	+	0.891401i	$0.350277\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−11.3662	−1.29530
$78$	0	0
$79$	4.30243i	0.484061i	0.970269	+	0.242030i	$0.0778134\pi$
$79$	4.30243i	0.484061i	−0.970269	+	0.242030i	$0.922187\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 3.61158i	− 0.396422i	−0.980159	−	0.198211i	$-0.936487\pi$
$83$	− 3.61158i	− 0.396422i	0.980159	−	0.198211i	$-0.0635132\pi$
$84$	0	0
$85$	− 2.55164i	− 0.276764i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	17.0256i	1.80471i	0.430999	+	0.902353i	$0.358161\pi$
$89$	17.0256i	1.80471i	−0.430999	+	0.902353i	$0.641839\pi$
$90$	0	0
$91$	−8.74456	−0.916679
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.39794	−0.246023
$96$	0	0
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.7858	1.57074	0.785371	−	0.619026i	$-0.212472\pi$
$101$	15.7858	1.57074	0.785371	+	0.619026i	$0.212472\pi$
$102$	0	0
$103$	− 0.800857i	− 0.0789107i	−0.999221	−	0.0394554i	$-0.987438\pi$
$103$	− 0.800857i	− 0.0789107i	0.999221	−	0.0394554i	$-0.0125623\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.90120i	− 0.473817i	−0.971532	−	0.236908i	$-0.923866\pi$
$107$	− 4.90120i	− 0.473817i	0.971532	−	0.236908i	$-0.0761341\pi$
$108$	0	0
$109$	− 11.1565i	− 1.06860i	−0.845295	−	0.534299i	$-0.820576\pi$
$109$	− 11.1565i	− 1.06860i	0.845295	−	0.534299i	$-0.179424\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 11.9769i	− 1.12669i	−0.826222	−	0.563345i	$-0.809514\pi$
$113$	− 11.9769i	− 1.12669i	0.826222	−	0.563345i	$-0.190486\pi$
$114$	0	0
$115$	−3.25544	−0.303571
$116$	0	0
$117$	0	0
$118$	0	0
$119$	17.3020	1.58607
$120$	0	0
$121$	0	0
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−4.92498	−0.440504
$126$	0	0
$127$	− 10.2811i	− 0.912300i	−0.889903	−	0.456150i	$-0.849228\pi$
$127$	− 10.2811i	− 0.912300i	0.889903	−	0.456150i	$-0.150772\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.02700i	0.177100i	0.996072	+	0.0885501i	$0.0282233\pi$
$131$	2.02700i	0.177100i	−0.996072	+	0.0885501i	$0.971777\pi$
$132$	0	0
$133$	− 16.2598i	− 1.40990i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.0256i	1.45459i	0.686324	+	0.727296i	$0.259223\pi$
$137$	17.0256i	1.45459i	−0.686324	+	0.727296i	$0.740777\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.46284	−0.707698
$144$	0	0
$145$	2.74456	0.227924
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.8182	−1.54165	−0.770824	−	0.637048i	$-0.780155\pi$
$149$	−18.8182	−1.54165	−0.770824	+	0.637048i	$0.780155\pi$
$150$	0	0
$151$	− 7.72946i	− 0.629015i	−0.949255	−	0.314507i	$-0.898161\pi$
$151$	− 7.72946i	− 0.629015i	0.949255	−	0.314507i	$-0.101839\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.02167i	0.242706i
$156$	0	0
$157$	− 5.10328i	− 0.407286i	−0.979045	−	0.203643i	$-0.934722\pi$
$157$	− 5.10328i	− 0.407286i	0.979045	−	0.203643i	$-0.0652782\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 22.0742i	− 1.73969i
$162$	0	0
$163$	18.2337	1.42817	0.714086	−	0.700058i	$-0.246842\pi$
$163$	18.2337	1.42817	0.714086	+	0.700058i	$0.246842\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.02163	−0.156439	−0.0782193	−	0.996936i	$-0.524923\pi$
$167$	−2.02163	−0.156439	−0.0782193	+	0.996936i	$0.524923\pi$
$168$	0	0
$169$	6.48913	0.499163
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.3662	0.864155	0.432078	−	0.901836i	$-0.357781\pi$
$173$	11.3662	0.864155	0.432078	+	0.901836i	$0.357781\pi$
$174$	0	0
$175$	− 16.2598i	− 1.22912i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 11.8294i	− 0.884171i	−0.896973	−	0.442086i	$-0.854239\pi$
$179$	− 11.8294i	− 0.884171i	0.896973	−	0.442086i	$-0.145761\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.63858i	0.414557i
$186$	0	0
$187$	16.7446	1.22448
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.46284	−0.612349	−0.306175	−	0.951975i	$-0.599049\pi$
$191$	−8.46284	−0.612349	−0.306175	+	0.951975i	$0.599049\pi$
$192$	0	0
$193$	13.2337	0.952582	0.476291	−	0.879288i	$-0.341981\pi$
$193$	13.2337	0.952582	0.476291	+	0.879288i	$0.341981\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.3878	0.953843	0.476921	−	0.878946i	$-0.341753\pi$
$197$	13.3878	0.953843	0.476921	+	0.878946i	$0.341753\pi$
$198$	0	0
$199$	− 6.77953i	− 0.480588i	−0.970700	−	0.240294i	$-0.922756\pi$
$199$	− 6.77953i	− 0.480588i	0.970700	−	0.240294i	$-0.0772439\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	18.6101i	1.30617i
$204$	0	0
$205$	0.949929i	0.0663459i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 15.7359i	− 1.08848i
$210$	0	0
$211$	12.7446	0.877372	0.438686	−	0.898640i	$-0.355444\pi$
$211$	12.7446	0.877372	0.438686	+	0.898640i	$0.355444\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.02163	0.137874
$216$	0	0
$217$	−20.4891	−1.39089
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.8824	0.866565
$222$	0	0
$223$	− 18.0106i	− 1.20608i	−0.797712	−	0.603038i	$-0.793957\pi$
$223$	− 18.0106i	− 1.20608i	0.797712	−	0.603038i	$-0.206043\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.6588i	1.57029i	0.619312	+	0.785145i	$0.287412\pi$
$227$	23.6588i	1.57029i	−0.619312	+	0.785145i	$0.712588\pi$
$228$	0	0
$229$	− 8.60485i	− 0.568625i	−0.958732	−	0.284312i	$-0.908235\pi$
$229$	− 8.60485i	− 0.568625i	0.958732	−	0.284312i	$-0.0917653\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.589907i	0.0386461i	0.999813	+	0.0193231i	$0.00615110\pi$
$233$	0.589907i	0.0386461i	−0.999813	+	0.0193231i	$0.993849\pi$
$234$	0	0
$235$	5.48913	0.358071
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.2804	−0.988404	−0.494202	−	0.869347i	$-0.664540\pi$
$239$	−15.2804	−0.988404	−0.494202	+	0.869347i	$0.664540\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.39794	−0.153199
$246$	0	0
$247$	− 12.1064i	− 0.770313i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	9.80240i	0.618722i	0.950945	+	0.309361i	$0.100115\pi$
$251$	9.80240i	0.618722i	−0.950945	+	0.309361i	$0.899885\pi$
$252$	0	0
$253$	− 21.3631i	− 1.34308i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 18.9051i	− 1.17927i	−0.807671	−	0.589633i	$-0.799272\pi$
$257$	− 18.9051i	− 1.17927i	0.807671	−	0.589633i	$-0.200728\pi$
$258$	0	0
$259$	−38.2337	−2.37573
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.44121	0.397182	0.198591	−	0.980082i	$-0.436364\pi$
$263$	6.44121	0.397182	0.198591	+	0.980082i	$0.436364\pi$
$264$	0	0
$265$	−3.00000	−0.184289
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.45202	−0.454358	−0.227179	−	0.973853i	$-0.572950\pi$
$269$	−7.45202	−0.454358	−0.227179	+	0.973853i	$0.572950\pi$
$270$	0	0
$271$	23.1884i	1.40859i	0.709905	+	0.704297i	$0.248738\pi$
$271$	23.1884i	1.40859i	−0.709905	+	0.704297i	$0.751262\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 15.7359i	− 0.948913i
$276$	0	0
$277$	5.10328i	0.306627i	0.988178	+	0.153313i	$0.0489944\pi$
$277$	5.10328i	0.306627i	−0.988178	+	0.153313i	$0.951006\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.80773i	0.525425i	0.964874	+	0.262713i	$0.0846171\pi$
$281$	8.80773i	0.525425i	−0.964874	+	0.262713i	$0.915383\pi$
$282$	0	0
$283$	−11.2554	−0.669066	−0.334533	−	0.942384i	$-0.608579\pi$
$283$	−11.2554	−0.669066	−0.334533	+	0.942384i	$0.608579\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.44121	−0.380212
$288$	0	0
$289$	−8.48913	−0.499360
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.47365	0.553457	0.276728	−	0.960948i	$-0.410750\pi$
$293$	9.47365	0.553457	0.276728	+	0.960948i	$0.410750\pi$
$294$	0	0
$295$	− 3.35250i	− 0.195190i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 16.4356i	− 0.950498i
$300$	0	0
$301$	13.7081i	0.790124i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2.57924i	0.147687i
$306$	0	0
$307$	7.25544	0.414090	0.207045	−	0.978331i	$-0.433615\pi$
$307$	7.25544	0.414090	0.207045	+	0.978331i	$0.433615\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.39794	0.135975	0.0679874	−	0.997686i	$-0.478342\pi$
$311$	2.39794	0.135975	0.0679874	+	0.997686i	$0.478342\pi$
$312$	0	0
$313$	25.2337	1.42629	0.713146	−	0.701015i	$-0.247270\pi$
$313$	25.2337	1.42629	0.713146	+	0.701015i	$0.247270\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.95743	−0.446934	−0.223467	−	0.974712i	$-0.571737\pi$
$317$	−7.95743	−0.446934	−0.223467	+	0.974712i	$0.571737\pi$
$318$	0	0
$319$	18.0106i	1.00840i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	23.9538i	1.33282i
$324$	0	0
$325$	− 12.1064i	− 0.671544i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	37.2203i	2.05202i
$330$	0	0
$331$	10.5109	0.577730	0.288865	−	0.957370i	$-0.406722\pi$
$331$	10.5109	0.577730	0.288865	+	0.957370i	$0.406722\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	2.02163	0.110454
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−19.8290	−1.07380
$342$	0	0
$343$	7.72946i	0.417352i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 26.9754i	− 1.44812i	−0.689739	−	0.724058i	$-0.742275\pi$
$347$	− 26.9754i	− 1.44812i	0.689739	−	0.724058i	$-0.257725\pi$
$348$	0	0
$349$	16.2598i	0.870366i	0.900342	+	0.435183i	$0.143316\pi$
$349$	16.2598i	0.870366i	−0.900342	+	0.435183i	$0.856684\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 28.4125i	− 1.51225i	−0.654429	−	0.756123i	$-0.727091\pi$
$353$	− 28.4125i	− 1.51225i	0.654429	−	0.756123i	$-0.272909\pi$
$354$	0	0
$355$	−2.23369	−0.118552
$356$	0	0
$357$	0	0
$358$	0	0
$359$	23.7432	1.25312	0.626559	−	0.779374i	$-0.284463\pi$
$359$	23.7432	1.25312	0.626559	+	0.779374i	$0.284463\pi$
$360$	0	0
$361$	3.51087	0.184783
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.91416	0.204877
$366$	0	0
$367$	11.0820i	0.578474i	0.957258	+	0.289237i	$0.0934015\pi$
$367$	11.0820i	0.578474i	−0.957258	+	0.289237i	$0.906599\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 20.3422i	− 1.05611i
$372$	0	0
$373$	2.55164i	0.132119i	0.997816	+	0.0660595i	$0.0210427\pi$
$373$	2.55164i	0.132119i	−0.997816	+	0.0660595i	$0.978957\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	13.8564i	0.713641i
$378$	0	0
$379$	9.48913	0.487424	0.243712	−	0.969848i	$-0.421635\pi$
$379$	9.48913	0.487424	0.243712	+	0.969848i	$0.421635\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	26.1411	1.33575	0.667875	−	0.744274i	$-0.267204\pi$
$383$	26.1411	1.33575	0.667875	+	0.744274i	$0.267204\pi$
$384$	0	0
$385$	−5.74456	−0.292770
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−22.8615	−1.15912	−0.579561	−	0.814929i	$-0.696776\pi$
$389$	−22.8615	−1.15912	−0.579561	+	0.814929i	$0.696776\pi$
$390$	0	0
$391$	32.5196i	1.64458i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.17448i	0.109410i
$396$	0	0
$397$	29.9679i	1.50405i	0.659137	+	0.752023i	$0.270922\pi$
$397$	29.9679i	1.50405i	−0.659137	+	0.752023i	$0.729078\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 3.75906i	− 0.187718i	−0.995585	−	0.0938591i	$-0.970080\pi$
$401$	− 3.75906i	− 0.187718i	0.995585	−	0.0938591i	$-0.0299203\pi$
$402$	0	0
$403$	−15.2554	−0.759927
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−37.0019	−1.83412
$408$	0	0
$409$	11.0000	0.543915	0.271957	−	0.962309i	$-0.412329\pi$
$409$	11.0000	0.543915	0.271957	+	0.962309i	$0.412329\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	22.7324	1.11859
$414$	0	0
$415$	− 1.82532i	− 0.0896015i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.22316i	0.352874i	0.984312	+	0.176437i	$0.0564572\pi$
$419$	7.22316i	0.352874i	−0.984312	+	0.176437i	$0.943543\pi$
$420$	0	0
$421$	27.4163i	1.33619i	0.744077	+	0.668094i	$0.232889\pi$
$421$	27.4163i	1.33619i	−0.744077	+	0.668094i	$0.767111\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	23.9538i	1.16193i
$426$	0	0
$427$	−17.4891	−0.846358
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.9040	0.717902	0.358951	−	0.933356i	$-0.383134\pi$
$431$	14.9040	0.717902	0.358951	+	0.933356i	$0.383134\pi$
$432$	0	0
$433$	−18.4891	−0.888531	−0.444265	−	0.895895i	$-0.646535\pi$
$433$	−18.4891	−0.888531	−0.444265	+	0.895895i	$0.646535\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	30.5607	1.46192
$438$	0	0
$439$	18.7369i	0.894263i	0.894468	+	0.447131i	$0.147554\pi$
$439$	18.7369i	0.894263i	−0.894468	+	0.447131i	$0.852446\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 6.63325i	− 0.315155i	−0.987507	−	0.157578i	$-0.949632\pi$
$443$	− 6.63325i	− 0.315155i	0.987507	−	0.157578i	$-0.0503684\pi$
$444$	0	0
$445$	8.60485i	0.407909i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.45877i	0.210422i	0.994450	+	0.105211i	$0.0335518\pi$
$449$	4.45877i	0.210422i	−0.994450	+	0.105211i	$0.966448\pi$
$450$	0	0
$451$	−6.23369	−0.293533
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.41957	−0.207193
$456$	0	0
$457$	−41.4674	−1.93976	−0.969881	−	0.243579i	$-0.921678\pi$
$457$	−41.4674	−1.93976	−0.969881	+	0.243579i	$0.921678\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−34.4749	−1.60565	−0.802827	−	0.596212i	$-0.796672\pi$
$461$	−34.4749	−1.60565	−0.802827	+	0.596212i	$0.796672\pi$
$462$	0	0
$463$	− 31.6442i	− 1.47063i	−0.677726	−	0.735314i	$-0.737034\pi$
$463$	− 31.6442i	− 1.47063i	0.677726	−	0.735314i	$-0.262966\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 31.3244i	− 1.44952i	−0.689001	−	0.724760i	$-0.741951\pi$
$467$	− 31.3244i	− 1.44952i	0.689001	−	0.724760i	$-0.258049\pi$
$468$	0	0
$469$	13.7081i	0.632983i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	13.2665i	0.609994i
$474$	0	0
$475$	22.5109	1.03287
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.02163	0.0923707	0.0461854	−	0.998933i	$-0.485294\pi$
$479$	2.02163	0.0923707	0.0461854	+	0.998933i	$0.485294\pi$
$480$	0	0
$481$	−28.4674	−1.29800
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.505408	−0.0229494
$486$	0	0
$487$	9.40571i	0.426213i	0.977029	+	0.213107i	$0.0683582\pi$
$487$	9.40571i	0.426213i	−0.977029	+	0.213107i	$0.931642\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	27.9701i	1.26227i	0.775672	+	0.631136i	$0.217411\pi$
$491$	27.9701i	1.26227i	−0.775672	+	0.631136i	$0.782589\pi$
$492$	0	0
$493$	− 27.4163i	− 1.23477i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 15.1460i	− 0.679392i
$498$	0	0
$499$	28.0000	1.25345	0.626726	−	0.779240i	$-0.284395\pi$
$499$	28.0000	1.25345	0.626726	+	0.779240i	$0.284395\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−19.3236	−0.861597	−0.430799	−	0.902448i	$-0.641768\pi$
$503$	−19.3236	−0.861597	−0.430799	+	0.902448i	$0.641768\pi$
$504$	0	0
$505$	7.97825	0.355027
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−23.2378	−1.03000	−0.514998	−	0.857191i	$-0.672207\pi$
$509$	−23.2378	−1.03000	−0.514998	+	0.857191i	$0.672207\pi$
$510$	0	0
$511$	26.5409i	1.17410i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 0.404759i	− 0.0178358i
$516$	0	0
$517$	36.0211i	1.58421i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 5.04868i	− 0.221186i	−0.993866	−	0.110593i	$-0.964725\pi$
$521$	− 5.04868i	− 0.221186i	0.993866	−	0.110593i	$-0.0352751\pi$
$522$	0	0
$523$	−34.2337	−1.49693	−0.748467	−	0.663172i	$-0.769210\pi$
$523$	−34.2337	−1.49693	−0.748467	+	0.663172i	$0.769210\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	30.1844	1.31485
$528$	0	0
$529$	18.4891	0.803875
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.79588	−0.207733
$534$	0	0
$535$	− 2.47711i	− 0.107095i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 15.7359i	− 0.677795i
$540$	0	0
$541$	37.6228i	1.61753i	0.588130	+	0.808766i	$0.299864\pi$
$541$	37.6228i	1.61753i	−0.588130	+	0.808766i	$0.700136\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 5.63858i	− 0.241530i
$546$	0	0
$547$	−5.76631	−0.246550	−0.123275	−	0.992373i	$-0.539340\pi$
$547$	−5.76631	−0.246550	−0.123275	+	0.992373i	$0.539340\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−25.7648	−1.09762
$552$	0	0
$553$	−14.7446	−0.627003
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.96825	0.379997	0.189998	−	0.981784i	$-0.439152\pi$
$557$	8.96825	0.379997	0.189998	+	0.981784i	$0.439152\pi$
$558$	0	0
$559$	10.2066i	0.431692i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.8241i	0.540470i	0.962794	+	0.270235i	$0.0871014\pi$
$563$	12.8241i	0.540470i	−0.962794	+	0.270235i	$0.912899\pi$
$564$	0	0
$565$	− 6.05321i	− 0.254661i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.51811i	0.315176i	0.987505	+	0.157588i	$0.0503717\pi$
$569$	7.51811i	0.315176i	−0.987505	+	0.157588i	$0.949628\pi$
$570$	0	0
$571$	−34.2337	−1.43264	−0.716318	−	0.697774i	$-0.754174\pi$
$571$	−34.2337	−1.43264	−0.716318	+	0.697774i	$0.754174\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	30.5607	1.27447
$576$	0	0
$577$	36.9783	1.53942	0.769712	−	0.638391i	$-0.220400\pi$
$577$	36.9783	1.53942	0.769712	+	0.638391i	$0.220400\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	12.3770	0.513485
$582$	0	0
$583$	− 19.6868i	− 0.815344i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	39.2473i	1.61991i	0.586494	+	0.809954i	$0.300508\pi$
$587$	39.2473i	1.61991i	−0.586494	+	0.809954i	$0.699492\pi$
$588$	0	0
$589$	− 28.3662i	− 1.16881i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 13.2665i	− 0.544790i	−0.962186	−	0.272395i	$-0.912184\pi$
$593$	− 13.2665i	− 0.544790i	0.962186	−	0.272395i	$-0.0878157\pi$
$594$	0	0
$595$	8.74456	0.358492
$596$	0	0
$597$	0	0
$598$	0	0
$599$	27.7865	1.13532	0.567662	−	0.823262i	$-0.307848\pi$
$599$	27.7865	1.13532	0.567662	+	0.823262i	$0.307848\pi$
$600$	0	0
$601$	−3.23369	−0.131905	−0.0659524	−	0.997823i	$-0.521009\pi$
$601$	−3.23369	−0.131905	−0.0659524	+	0.997823i	$0.521009\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	36.8220i	1.49456i	0.664510	+	0.747279i	$0.268640\pi$
$607$	36.8220i	1.49456i	−0.664510	+	0.747279i	$0.731360\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.7128i	1.12114i
$612$	0	0
$613$	− 34.1213i	− 1.37815i	−0.724692	−	0.689073i	$-0.758018\pi$
$613$	− 34.1213i	− 1.37815i	0.724692	−	0.689073i	$-0.241982\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 28.4125i	− 1.14385i	−0.820308	−	0.571923i	$-0.806198\pi$
$617$	− 28.4125i	− 1.14385i	0.820308	−	0.571923i	$-0.193802\pi$
$618$	0	0
$619$	42.2337	1.69752	0.848758	−	0.528782i	$-0.177351\pi$
$619$	42.2337	1.69752	0.848758	+	0.528782i	$0.177351\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−58.3472	−2.33763
$624$	0	0
$625$	21.2337	0.849348
$626$	0	0
$627$	0	0
$628$	0	0
$629$	56.3255	2.24585
$630$	0	0
$631$	− 13.7827i	− 0.548680i	−0.961633	−	0.274340i	$-0.911541\pi$
$631$	− 13.7827i	− 0.548680i	0.961633	−	0.274340i	$-0.0884593\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 5.19615i	− 0.206203i
$636$	0	0
$637$	− 12.1064i	− 0.479674i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.87953i	0.0742369i	0.999311	+	0.0371184i	$0.0118179\pi$
$641$	1.87953i	0.0742369i	−0.999311	+	0.0371184i	$0.988182\pi$
$642$	0	0
$643$	44.4674	1.75362	0.876811	−	0.480835i	$-0.159666\pi$
$643$	44.4674	1.75362	0.876811	+	0.480835i	$0.159666\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−29.8081	−1.17188	−0.585938	−	0.810356i	$-0.699274\pi$
$647$	−29.8081	−1.17188	−0.585938	+	0.810356i	$0.699274\pi$
$648$	0	0
$649$	22.0000	0.863576
$650$	0	0
$651$	0	0
$652$	0	0
$653$	41.9269	1.64073	0.820363	−	0.571843i	$-0.193771\pi$
$653$	41.9269	1.64073	0.820363	+	0.571843i	$0.193771\pi$
$654$	0	0
$655$	1.02446i	0.0400291i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 39.5422i	− 1.54035i	−0.637835	−	0.770173i	$-0.720170\pi$
$659$	− 39.5422i	− 1.54035i	0.637835	−	0.770173i	$-0.279830\pi$
$660$	0	0
$661$	− 16.2598i	− 0.632432i	−0.948687	−	0.316216i	$-0.897588\pi$
$661$	− 16.2598i	− 0.632432i	0.948687	−	0.316216i	$-0.102412\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 8.21782i	− 0.318674i
$666$	0	0
$667$	−34.9783	−1.35436
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−16.9257	−0.653408
$672$	0	0
$673$	−7.51087	−0.289523	−0.144761	−	0.989467i	$-0.546241\pi$
$673$	−7.51087	−0.289523	−0.144761	+	0.989467i	$0.546241\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−42.0560	−1.61634	−0.808171	−	0.588947i	$-0.799543\pi$
$677$	−42.0560	−1.61634	−0.808171	+	0.588947i	$0.799543\pi$
$678$	0	0
$679$	− 3.42703i	− 0.131517i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.80240i	0.375078i	0.982257	+	0.187539i	$0.0600512\pi$
$683$	9.80240i	0.375078i	−0.982257	+	0.187539i	$0.939949\pi$
$684$	0	0
$685$	8.60485i	0.328775i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 15.1460i	− 0.577018i
$690$	0	0
$691$	−49.4891	−1.88266	−0.941328	−	0.337494i	$-0.890421\pi$
$691$	−49.4891	−1.88266	−0.941328	+	0.337494i	$0.890421\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.04326	−0.153370
$696$	0	0
$697$	9.48913	0.359426
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−10.3554	−0.391117	−0.195558	−	0.980692i	$-0.562652\pi$
$701$	−10.3554	−0.391117	−0.195558	+	0.980692i	$0.562652\pi$
$702$	0	0
$703$	− 52.9327i	− 1.99639i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	54.0983i	2.03458i
$708$	0	0
$709$	12.1064i	0.454666i	0.973817	+	0.227333i	$0.0730006\pi$
$709$	12.1064i	0.454666i	−0.973817	+	0.227333i	$0.926999\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 38.5099i	− 1.44221i
$714$	0	0
$715$	−4.27719	−0.159958
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.83915	−0.329645	−0.164822	−	0.986323i	$-0.552705\pi$
$719$	−8.83915	−0.329645	−0.164822	+	0.986323i	$0.552705\pi$
$720$	0	0
$721$	2.74456	0.102213
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−25.7648	−0.956881
$726$	0	0
$727$	10.4302i	0.386834i	0.981117	+	0.193417i	$0.0619570\pi$
$727$	10.4302i	0.386834i	−0.981117	+	0.193417i	$0.938043\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 20.1947i	− 0.746928i
$732$	0	0
$733$	− 38.5728i	− 1.42472i	−0.701815	−	0.712359i	$-0.747627\pi$
$733$	− 38.5728i	− 1.42472i	0.701815	−	0.712359i	$-0.252373\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.2665i	0.488678i
$738$	0	0
$739$	7.25544	0.266896	0.133448	−	0.991056i	$-0.457395\pi$
$739$	7.25544	0.266896	0.133448	+	0.991056i	$0.457395\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17.3020	0.634748	0.317374	−	0.948300i	$-0.397199\pi$
$743$	17.3020	0.634748	0.317374	+	0.948300i	$0.397199\pi$
$744$	0	0
$745$	−9.51087	−0.348451
$746$	0	0
$747$	0	0
$748$	0	0
$749$	16.7966	0.613734
$750$	0	0
$751$	− 1.02446i	− 0.0373832i	−0.999825	−	0.0186916i	$-0.994050\pi$
$751$	− 1.02446i	− 0.0373832i	0.999825	−	0.0186916i	$-0.00595007\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 3.90653i	− 0.142173i
$756$	0	0
$757$	− 34.1213i	− 1.24016i	−0.784539	−	0.620079i	$-0.787100\pi$
$757$	− 34.1213i	− 1.24016i	0.784539	−	0.620079i	$-0.212900\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12.6766i	0.459526i	0.973247	+	0.229763i	$0.0737951\pi$
$761$	12.6766i	0.459526i	−0.973247	+	0.229763i	$0.926205\pi$
$762$	0	0
$763$	38.2337	1.38415
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16.9257	0.611151
$768$	0	0
$769$	−21.7446	−0.784129	−0.392064	−	0.919938i	$-0.628239\pi$
$769$	−21.7446	−0.784129	−0.392064	+	0.919938i	$0.628239\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−33.9695	−1.22180	−0.610898	−	0.791709i	$-0.709192\pi$
$773$	−33.9695	−1.22180	−0.610898	+	0.791709i	$0.709192\pi$
$774$	0	0
$775$	− 28.3662i	− 1.01894i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 8.91754i	− 0.319504i
$780$	0	0
$781$	− 14.6581i	− 0.524507i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 2.57924i	− 0.0920570i
$786$	0	0
$787$	−1.48913	−0.0530816	−0.0265408	−	0.999648i	$-0.508449\pi$
$787$	−1.48913	−0.0530816	−0.0265408	+	0.999648i	$0.508449\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	41.0452	1.45940
$792$	0	0
$793$	−13.0217	−0.462416
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.4095	0.545831	0.272915	−	0.962038i	$-0.412012\pi$
$797$	15.4095	0.545831	0.272915	+	0.962038i	$0.412012\pi$
$798$	0	0
$799$	− 54.8325i	− 1.93984i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	25.6858i	0.906432i
$804$	0	0
$805$	− 11.1565i	− 0.393215i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	49.8968i	1.75428i	0.480235	+	0.877140i	$0.340551\pi$
$809$	49.8968i	1.75428i	−0.480235	+	0.877140i	$0.659449\pi$
$810$	0	0
$811$	−4.74456	−0.166604	−0.0833021	−	0.996524i	$-0.526547\pi$
$811$	−4.74456	−0.166604	−0.0833021	+	0.996524i	$0.526547\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9.21545	0.322803
$816$	0	0
$817$	−18.9783	−0.663965
$818$	0	0
$819$	0	0
$820$	0	0
$821$	35.2385	1.22983	0.614916	−	0.788593i	$-0.289190\pi$
$821$	35.2385	1.22983	0.614916	+	0.788593i	$0.289190\pi$
$822$	0	0
$823$	− 50.4556i	− 1.75877i	−0.476110	−	0.879386i	$-0.657954\pi$
$823$	− 50.4556i	− 1.75877i	0.476110	−	0.879386i	$-0.342046\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 6.63325i	− 0.230661i	−0.993327	−	0.115330i	$-0.963207\pi$
$827$	− 6.63325i	− 0.230661i	0.993327	−	0.115330i	$-0.0367927\pi$
$828$	0	0
$829$	3.50157i	0.121615i	0.998150	+	0.0608073i	$0.0193675\pi$
$829$	3.50157i	0.121615i	−0.998150	+	0.0608073i	$0.980632\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	23.9538i	0.829949i
$834$	0	0
$835$	−1.02175	−0.0353591
$836$	0	0
$837$	0	0
$838$	0	0
$839$	12.5061	0.431759	0.215879	−	0.976420i	$-0.430738\pi$
$839$	12.5061	0.431759	0.215879	+	0.976420i	$0.430738\pi$
$840$	0	0
$841$	0.489125	0.0168664
$842$	0	0
$843$	0	0
$844$	0	0
$845$	3.27966	0.112824
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 71.8613i	− 2.46337i
$852$	0	0
$853$	− 6.70500i	− 0.229575i	−0.993390	−	0.114787i	$-0.963381\pi$
$853$	− 6.70500i	− 0.229575i	0.993390	−	0.114787i	$-0.0366187\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.46943i	− 0.0843543i	−0.999110	−	0.0421771i	$-0.986571\pi$
$857$	− 2.46943i	− 0.0843543i	0.999110	−	0.0421771i	$-0.0134294\pi$
$858$	0	0
$859$	20.4674	0.698338	0.349169	−	0.937060i	$-0.386464\pi$
$859$	20.4674	0.698338	0.349169	+	0.937060i	$0.386464\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−26.5174	−0.902664	−0.451332	−	0.892356i	$-0.649051\pi$
$863$	−26.5174	−0.902664	−0.451332	+	0.892356i	$0.649051\pi$
$864$	0	0
$865$	5.74456	0.195321
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−14.2695	−0.484061
$870$	0	0
$871$	10.2066i	0.345836i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 16.8781i	− 0.570583i
$876$	0	0
$877$	13.7081i	0.462891i	0.972848	+	0.231445i	$0.0743455\pi$
$877$	13.7081i	0.462891i	−0.972848	+	0.231445i	$0.925655\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 7.62792i	− 0.256991i	−0.991710	−	0.128496i	$-0.958985\pi$
$881$	− 7.62792i	− 0.256991i	0.991710	−	0.128496i	$-0.0410148\pi$
$882$	0	0
$883$	23.7228	0.798336	0.399168	−	0.916878i	$-0.369299\pi$
$883$	23.7228	0.798336	0.399168	+	0.916878i	$0.369299\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.21545	−0.309425	−0.154712	−	0.987960i	$-0.549445\pi$
$887$	−9.21545	−0.309425	−0.154712	+	0.987960i	$0.549445\pi$
$888$	0	0
$889$	35.2337	1.18170
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−51.5296	−1.72437
$894$	0	0
$895$	− 5.97868i	− 0.199845i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	32.4665i	1.08282i
$900$	0	0
$901$	29.9679i	0.998376i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	54.2337	1.80080	0.900400	−	0.435063i	$-0.143274\pi$
$907$	54.2337	1.80080	0.900400	+	0.435063i	$0.143274\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10.8608	0.359834	0.179917	−	0.983682i	$-0.442417\pi$
$911$	10.8608	0.359834	0.179917	+	0.983682i	$0.442417\pi$
$912$	0	0
$913$	11.9783	0.396422
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.94661	−0.229397
$918$	0	0
$919$	26.6900i	0.880420i	0.897895	+	0.440210i	$0.145096\pi$
$919$	26.6900i	0.880420i	−0.897895	+	0.440210i	$0.854904\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 11.2772i	− 0.371192i
$924$	0	0
$925$	− 52.9327i	− 1.74042i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	42.9686i	1.40976i	0.709329	+	0.704878i	$0.248998\pi$
$929$	42.9686i	1.40976i	−0.709329	+	0.704878i	$0.751002\pi$
$930$	0	0
$931$	22.5109	0.737764
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.46284	0.276764
$936$	0	0
$937$	35.0000	1.14340	0.571700	−	0.820463i	$-0.306284\pi$
$937$	35.0000	1.14340	0.571700	+	0.820463i	$0.306284\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.91416	−0.127598	−0.0637991	−	0.997963i	$-0.520322\pi$
$941$	−3.91416	−0.127598	−0.0637991	+	0.997963i	$0.520322\pi$
$942$	0	0
$943$	− 12.1064i	− 0.394239i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 11.8294i	− 0.384404i	−0.981355	−	0.192202i	$-0.938437\pi$
$947$	− 11.8294i	− 0.384404i	0.981355	−	0.192202i	$-0.0615629\pi$
$948$	0	0
$949$	19.7613i	0.641481i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 31.4719i	− 1.01947i	−0.860330	−	0.509737i	$-0.829743\pi$
$953$	− 31.4719i	− 1.01947i	0.860330	−	0.509737i	$-0.170257\pi$
$954$	0	0
$955$	−4.27719	−0.138407
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−58.3472	−1.88413
$960$	0	0
$961$	−4.74456	−0.153050
$962$	0	0
$963$	0	0
$964$	0	0
$965$	6.68841	0.215308
$966$	0	0
$967$	− 12.8327i	− 0.412673i	−0.978481	−	0.206337i	$-0.933846\pi$
$967$	− 12.8327i	− 0.412673i	0.978481	−	0.206337i	$-0.0661542\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	14.5938i	0.468337i	0.972196	+	0.234169i	$0.0752367\pi$
$971$	14.5938i	0.468337i	−0.972196	+	0.234169i	$0.924763\pi$
$972$	0	0
$973$	− 27.4163i	− 0.878925i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.3176i	1.51382i	0.653517	+	0.756912i	$0.273293\pi$
$977$	47.3176i	1.51382i	−0.653517	+	0.756912i	$0.726707\pi$
$978$	0	0
$979$	−56.4674	−1.80471
$980$	0	0
$981$	0	0
$982$	0	0
$983$	28.5391	0.910255	0.455127	−	0.890426i	$-0.349594\pi$
$983$	28.5391	0.910255	0.455127	+	0.890426i	$0.349594\pi$
$984$	0	0
$985$	6.76631	0.215593
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−25.7648	−0.819274
$990$	0	0
$991$	12.0319i	0.382205i	0.981570	+	0.191103i	$0.0612064\pi$
$991$	12.0319i	0.382205i	−0.981570	+	0.191103i	$0.938794\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 3.42643i	− 0.108625i
$996$	0	0
$997$	− 31.8678i	− 1.00926i	−0.863335	−	0.504631i	$-0.831628\pi$
$997$	− 31.8678i	− 1.00926i	0.863335	−	0.504631i	$-0.168372\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.f.b.431.6		8
3.2	odd	2	inner	864.2.f.b.431.4		8
4.3	odd	2		216.2.f.b.107.2	yes	8
8.3	odd	2	inner	864.2.f.b.431.3		8
8.5	even	2		216.2.f.b.107.8	yes	8
9.2	odd	6		2592.2.p.e.2159.3		8
9.4	even	3		2592.2.p.e.431.2		8
9.5	odd	6		2592.2.p.d.431.3		8
9.7	even	3		2592.2.p.d.2159.2		8
12.11	even	2		216.2.f.b.107.7	yes	8
24.5	odd	2		216.2.f.b.107.1	✓	8
24.11	even	2	inner	864.2.f.b.431.5		8
36.7	odd	6		648.2.l.d.539.4		8
36.11	even	6		648.2.l.e.539.1		8
36.23	even	6		648.2.l.d.107.3		8
36.31	odd	6		648.2.l.e.107.2		8
72.5	odd	6		648.2.l.d.107.4		8
72.11	even	6		2592.2.p.e.2159.2		8
72.13	even	6		648.2.l.e.107.1		8
72.29	odd	6		648.2.l.e.539.2		8
72.43	odd	6		2592.2.p.d.2159.3		8
72.59	even	6		2592.2.p.d.431.2		8
72.61	even	6		648.2.l.d.539.3		8
72.67	odd	6		2592.2.p.e.431.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.f.b.107.1	✓	8	24.5	odd	2
216.2.f.b.107.2	yes	8	4.3	odd	2
216.2.f.b.107.7	yes	8	12.11	even	2
216.2.f.b.107.8	yes	8	8.5	even	2
648.2.l.d.107.3		8	36.23	even	6
648.2.l.d.107.4		8	72.5	odd	6
648.2.l.d.539.3		8	72.61	even	6
648.2.l.d.539.4		8	36.7	odd	6
648.2.l.e.107.1		8	72.13	even	6
648.2.l.e.107.2		8	36.31	odd	6
648.2.l.e.539.1		8	36.11	even	6
648.2.l.e.539.2		8	72.29	odd	6
864.2.f.b.431.3		8	8.3	odd	2	inner
864.2.f.b.431.4		8	3.2	odd	2	inner
864.2.f.b.431.5		8	24.11	even	2	inner
864.2.f.b.431.6		8	1.1	even	1	trivial
2592.2.p.d.431.2		8	72.59	even	6
2592.2.p.d.431.3		8	9.5	odd	6
2592.2.p.d.2159.2		8	9.7	even	3
2592.2.p.d.2159.3		8	72.43	odd	6
2592.2.p.e.431.2		8	9.4	even	3
2592.2.p.e.431.3		8	72.67	odd	6
2592.2.p.e.2159.2		8	72.11	even	6
2592.2.p.e.2159.3		8	9.2	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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+0.505408 q^{5} +3.42703i q^{7} +O(q^{10})\)	\(q+0.505408 q^{5} +3.42703i q^{7} +3.31662i q^{11} +2.55164i q^{13} -5.04868i q^{17} -4.74456 q^{19} -6.44121 q^{23} -4.74456 q^{25} +5.43039 q^{29} +5.97868i q^{31} +1.73205i q^{35} +11.1565i q^{37} +1.87953i q^{41} +4.00000 q^{43} +10.8608 q^{47} -4.74456 q^{49} -5.93580 q^{53} +1.67625i q^{55} -6.63325i q^{59} +5.10328i q^{61} +1.28962i q^{65} +4.00000 q^{67} -4.41957 q^{71} +7.74456 q^{73} -11.3662 q^{77} +4.30243i q^{79} -3.61158i q^{83} -2.55164i q^{85} +17.0256i q^{89} -8.74456 q^{91} -2.39794 q^{95} -1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{19} + 8 q^{25} + 32 q^{43} + 8 q^{49} + 32 q^{67} + 16 q^{73} - 24 q^{91} - 8 q^{97}+O(q^{100}) \) 8 * q + 8 * q^19 + 8 * q^25 + 32 * q^43 + 8 * q^49 + 32 * q^67 + 16 * q^73 - 24 * q^91 - 8 * q^97

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