LMFDB - Embedded newform 864.3.g.b.703.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,3,Mod(703,864)]

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, 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("864.703");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	864.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:	$23.5422948407$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.56070144.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 $ x^8 - 4x^7 + 16x^6 - 34x^5 + 63x^4 - 74x^3 + 70x^2 - 38*x + 13
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.3
Root			$0.500000 - 2.19293i$ of defining polynomial
Character	$\chi$	$=$	864.703
Dual form			864.3.g.b.703.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.13244 q^{5} -4.02516i q^{7} +O(q^{10})$	$q-5.13244 q^{5} -4.02516i q^{7} +4.30344i q^{11} -18.4862 q^{13} +23.5751 q^{17} -21.7259i q^{19} +30.7461i q^{23} +1.34199 q^{25} +12.6840 q^{29} +24.5298i q^{31} +20.6589i q^{35} +18.2314 q^{37} +38.0990 q^{41} +34.9724i q^{43} -29.6295i q^{47} +32.7981 q^{49} +39.3043 q^{53} -22.0872i q^{55} +65.3429i q^{59} -29.8541 q^{61} +94.8794 q^{65} +11.8634i q^{67} +140.651i q^{71} +119.285 q^{73} +17.3220 q^{77} -9.18859i q^{79} +113.180i q^{83} -120.998 q^{85} +7.88346 q^{89} +74.4099i q^{91} +111.507i q^{95} +55.5080 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^5	$ 8 q - 8 q^{5} + 8 q^{13} - 24 q^{17} + 24 q^{25} + 128 q^{29} + 24 q^{37} - 160 q^{41} - 144 q^{49} + 48 q^{53} - 136 q^{61} + 280 q^{65} + 72 q^{73} - 520 q^{77} + 96 q^{85} + 168 q^{89} + 104 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 + 8 * q^13 - 24 * q^17 + 24 * q^25 + 128 * q^29 + 24 * q^37 - 160 * q^41 - 144 * q^49 + 48 * q^53 - 136 * q^61 + 280 * q^65 + 72 * q^73 - 520 * q^77 + 96 * q^85 + 168 * q^89 + 104 * 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
$5$	−5.13244	−1.02649	−0.513244	−	0.858242i	$-0.671557\pi$
$5$	−5.13244	−1.02649	−0.513244	+	0.858242i	$0.671557\pi$
$6$	0	0
$7$	− 4.02516i	− 0.575023i	−0.957777	−	0.287511i	$-0.907172\pi$
$7$	− 4.02516i	− 0.575023i	0.957777	−	0.287511i	$-0.0928279\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.30344i	0.391222i	0.980682	+	0.195611i	$0.0626689\pi$
$11$	4.30344i	0.391222i	−0.980682	+	0.195611i	$0.937331\pi$
$12$	0	0
$13$	−18.4862	−1.42202	−0.711008	−	0.703184i	$-0.751761\pi$
$13$	−18.4862	−1.42202	−0.711008	+	0.703184i	$0.751761\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	23.5751	1.38677	0.693384	−	0.720568i	$-0.256119\pi$
$17$	23.5751	1.38677	0.693384	+	0.720568i	$0.256119\pi$
$18$	0	0
$19$	− 21.7259i	− 1.14347i	−0.820438	−	0.571735i	$-0.806271\pi$
$19$	− 21.7259i	− 1.14347i	0.820438	−	0.571735i	$-0.193729\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	30.7461i	1.33679i	0.743809	+	0.668393i	$0.233017\pi$
$23$	30.7461i	1.33679i	−0.743809	+	0.668393i	$0.766983\pi$
$24$	0	0
$25$	1.34199	0.0536794
$26$	0	0
$27$	0	0
$28$	0	0
$29$	12.6840	0.437378	0.218689	−	0.975795i	$-0.429822\pi$
$29$	12.6840	0.437378	0.218689	+	0.975795i	$0.429822\pi$
$30$	0	0
$31$	24.5298i	0.791283i	0.918405	+	0.395642i	$0.129478\pi$
$31$	24.5298i	0.791283i	−0.918405	+	0.395642i	$0.870522\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	20.6589i	0.590254i
$36$	0	0
$37$	18.2314	0.492740	0.246370	−	0.969176i	$-0.420762\pi$
$37$	18.2314	0.492740	0.246370	+	0.969176i	$0.420762\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	38.0990	0.929244	0.464622	−	0.885509i	$-0.346190\pi$
$41$	38.0990	0.929244	0.464622	+	0.885509i	$0.346190\pi$
$42$	0	0
$43$	34.9724i	0.813312i	0.913581	+	0.406656i	$0.133305\pi$
$43$	34.9724i	0.813312i	−0.913581	+	0.406656i	$0.866695\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 29.6295i	− 0.630415i	−0.949023	−	0.315208i	$-0.897926\pi$
$47$	− 29.6295i	− 0.630415i	0.949023	−	0.315208i	$-0.102074\pi$
$48$	0	0
$49$	32.7981	0.669349
$50$	0	0
$51$	0	0
$52$	0	0
$53$	39.3043	0.741591	0.370796	−	0.928715i	$-0.379085\pi$
$53$	39.3043	0.741591	0.370796	+	0.928715i	$0.379085\pi$
$54$	0	0
$55$	− 22.0872i	− 0.401585i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	65.3429i	1.10751i	0.832681	+	0.553753i	$0.186805\pi$
$59$	65.3429i	1.10751i	−0.832681	+	0.553753i	$0.813195\pi$
$60$	0	0
$61$	−29.8541	−0.489412	−0.244706	−	0.969597i	$-0.578691\pi$
$61$	−29.8541	−0.489412	−0.244706	+	0.969597i	$0.578691\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	94.8794	1.45968
$66$	0	0
$67$	11.8634i	0.177066i	0.996073	+	0.0885329i	$0.0282178\pi$
$67$	11.8634i	0.177066i	−0.996073	+	0.0885329i	$0.971782\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	140.651i	1.98100i	0.137529	+	0.990498i	$0.456084\pi$
$71$	140.651i	1.98100i	−0.137529	+	0.990498i	$0.543916\pi$
$72$	0	0
$73$	119.285	1.63404	0.817022	−	0.576607i	$-0.195624\pi$
$73$	119.285	1.63404	0.817022	+	0.576607i	$0.195624\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	17.3220	0.224961
$78$	0	0
$79$	− 9.18859i	− 0.116311i	−0.998308	−	0.0581556i	$-0.981478\pi$
$79$	− 9.18859i	− 0.116311i	0.998308	−	0.0581556i	$-0.0185220\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	113.180i	1.36362i	0.731529	+	0.681810i	$0.238807\pi$
$83$	113.180i	1.36362i	−0.731529	+	0.681810i	$0.761193\pi$
$84$	0	0
$85$	−120.998	−1.42350
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.88346	0.0885782	0.0442891	−	0.999019i	$-0.485898\pi$
$89$	7.88346	0.0885782	0.0442891	+	0.999019i	$0.485898\pi$
$90$	0	0
$91$	74.4099i	0.817691i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	111.507i	1.17376i
$96$	0	0
$97$	55.5080	0.572248	0.286124	−	0.958193i	$-0.407633\pi$
$97$	55.5080	0.572248	0.286124	+	0.958193i	$0.407633\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	195.186	1.93253	0.966265	−	0.257550i	$-0.0829152\pi$
$101$	195.186	1.93253	0.966265	+	0.257550i	$0.0829152\pi$
$102$	0	0
$103$	29.3579i	0.285028i	0.989793	+	0.142514i	$0.0455186\pi$
$103$	29.3579i	0.285028i	−0.989793	+	0.142514i	$0.954481\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 133.256i	− 1.24538i	−0.782468	−	0.622691i	$-0.786039\pi$
$107$	− 133.256i	− 1.24538i	0.782468	−	0.622691i	$-0.213961\pi$
$108$	0	0
$109$	−198.485	−1.82096	−0.910481	−	0.413552i	$-0.864288\pi$
$109$	−198.485	−1.82096	−0.910481	+	0.413552i	$0.864288\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−178.745	−1.58182	−0.790908	−	0.611934i	$-0.790392\pi$
$113$	−178.745	−1.58182	−0.790908	+	0.611934i	$0.790392\pi$
$114$	0	0
$115$	− 157.802i	− 1.37220i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 94.8934i	− 0.797424i
$120$	0	0
$121$	102.480	0.846946
$122$	0	0
$123$	0	0
$124$	0	0
$125$	121.423	0.971388
$126$	0	0
$127$	− 172.213i	− 1.35601i	−0.735058	−	0.678004i	$-0.762845\pi$
$127$	− 172.213i	− 1.35601i	0.735058	−	0.678004i	$-0.237155\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	206.300i	1.57481i	0.616435	+	0.787406i	$0.288576\pi$
$131$	206.300i	1.57481i	−0.616435	+	0.787406i	$0.711424\pi$
$132$	0	0
$133$	−87.4503	−0.657521
$134$	0	0
$135$	0	0
$136$	0	0
$137$	85.1089	0.621233	0.310616	−	0.950535i	$-0.399465\pi$
$137$	85.1089	0.621233	0.310616	+	0.950535i	$0.399465\pi$
$138$	0	0
$139$	− 150.326i	− 1.08148i	−0.841189	−	0.540742i	$-0.818144\pi$
$139$	− 150.326i	− 1.08148i	0.841189	−	0.540742i	$-0.181856\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 79.5542i	− 0.556323i
$144$	0	0
$145$	−65.0998	−0.448964
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−111.133	−0.745861	−0.372931	−	0.927859i	$-0.621647\pi$
$149$	−111.133	−0.745861	−0.372931	+	0.927859i	$0.621647\pi$
$150$	0	0
$151$	166.295i	1.10129i	0.834739	+	0.550646i	$0.185619\pi$
$151$	166.295i	1.10129i	−0.834739	+	0.550646i	$0.814381\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 125.898i	− 0.812243i
$156$	0	0
$157$	50.1560	0.319465	0.159732	−	0.987160i	$-0.448937\pi$
$157$	50.1560	0.319465	0.159732	+	0.987160i	$0.448937\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	123.758	0.768682
$162$	0	0
$163$	265.994i	1.63187i	0.578145	+	0.815934i	$0.303777\pi$
$163$	265.994i	1.63187i	−0.578145	+	0.815934i	$0.696223\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	223.651i	1.33923i	0.742708	+	0.669615i	$0.233541\pi$
$167$	223.651i	1.33923i	−0.742708	+	0.669615i	$0.766459\pi$
$168$	0	0
$169$	172.740	1.02213
$170$	0	0
$171$	0	0
$172$	0	0
$173$	55.4735	0.320656	0.160328	−	0.987064i	$-0.448745\pi$
$173$	55.4735	0.320656	0.160328	+	0.987064i	$0.448745\pi$
$174$	0	0
$175$	− 5.40171i	− 0.0308669i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.7562i	0.0600907i	0.999549	+	0.0300453i	$0.00956517\pi$
$179$	10.7562i	0.0600907i	−0.999549	+	0.0300453i	$0.990435\pi$
$180$	0	0
$181$	−36.3638	−0.200905	−0.100453	−	0.994942i	$-0.532029\pi$
$181$	−36.3638	−0.200905	−0.100453	+	0.994942i	$0.532029\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−93.5715	−0.505792
$186$	0	0
$187$	101.454i	0.542534i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 214.445i	− 1.12275i	−0.827563	−	0.561373i	$-0.810273\pi$
$191$	− 214.445i	− 1.12275i	0.827563	−	0.561373i	$-0.189727\pi$
$192$	0	0
$193$	280.031	1.45094	0.725470	−	0.688254i	$-0.241622\pi$
$193$	280.031	1.45094	0.725470	+	0.688254i	$0.241622\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−254.078	−1.28974	−0.644869	−	0.764293i	$-0.723088\pi$
$197$	−254.078	−1.28974	−0.644869	+	0.764293i	$0.723088\pi$
$198$	0	0
$199$	150.197i	0.754757i	0.926059	+	0.377379i	$0.123174\pi$
$199$	150.197i	0.754757i	−0.926059	+	0.377379i	$0.876826\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 51.0550i	− 0.251503i
$204$	0	0
$205$	−195.541	−0.953858
$206$	0	0
$207$	0	0
$208$	0	0
$209$	93.4962	0.447350
$210$	0	0
$211$	− 294.457i	− 1.39553i	−0.716326	−	0.697765i	$-0.754178\pi$
$211$	− 294.457i	− 1.39553i	0.716326	−	0.697765i	$-0.245822\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 179.494i	− 0.834855i
$216$	0	0
$217$	98.7363	0.455006
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−435.813	−1.97201
$222$	0	0
$223$	− 90.5784i	− 0.406181i	−0.979160	−	0.203091i	$-0.934901\pi$
$223$	− 90.5784i	− 0.406181i	0.979160	−	0.203091i	$-0.0650986\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 401.033i	− 1.76667i	−0.468746	−	0.883333i	$-0.655294\pi$
$227$	− 401.033i	− 1.76667i	0.468746	−	0.883333i	$-0.344706\pi$
$228$	0	0
$229$	−155.539	−0.679209	−0.339605	−	0.940568i	$-0.610293\pi$
$229$	−155.539	−0.679209	−0.339605	+	0.940568i	$0.610293\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−238.595	−1.02401	−0.512007	−	0.858981i	$-0.671098\pi$
$233$	−238.595	−1.02401	−0.512007	+	0.858981i	$0.671098\pi$
$234$	0	0
$235$	152.072i	0.647114i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 61.3314i	− 0.256617i	−0.991734	−	0.128308i	$-0.959045\pi$
$239$	− 61.3314i	− 0.256617i	0.991734	−	0.128308i	$-0.0409547\pi$
$240$	0	0
$241$	270.776	1.12355	0.561776	−	0.827290i	$-0.310118\pi$
$241$	270.776	1.12355	0.561776	+	0.827290i	$0.310118\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−168.334	−0.687079
$246$	0	0
$247$	401.630i	1.62603i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	229.191i	0.913110i	0.889695	+	0.456555i	$0.150917\pi$
$251$	229.191i	0.913110i	−0.889695	+	0.456555i	$0.849083\pi$
$252$	0	0
$253$	−132.314	−0.522979
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−366.617	−1.42653	−0.713263	−	0.700897i	$-0.752783\pi$
$257$	−366.617	−1.42653	−0.713263	+	0.700897i	$0.752783\pi$
$258$	0	0
$259$	− 73.3842i	− 0.283337i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 159.531i	− 0.606581i	−0.952898	−	0.303290i	$-0.901915\pi$
$263$	− 159.531i	− 0.606581i	0.952898	−	0.303290i	$-0.0980852\pi$
$264$	0	0
$265$	−201.727	−0.761235
$266$	0	0
$267$	0	0
$268$	0	0
$269$	91.2480	0.339212	0.169606	−	0.985512i	$-0.445751\pi$
$269$	91.2480	0.339212	0.169606	+	0.985512i	$0.445751\pi$
$270$	0	0
$271$	− 506.145i	− 1.86769i	−0.357675	−	0.933846i	$-0.616430\pi$
$271$	− 506.145i	− 1.86769i	0.357675	−	0.933846i	$-0.383570\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.77515i	0.0210006i
$276$	0	0
$277$	−28.7143	−0.103662	−0.0518308	−	0.998656i	$-0.516506\pi$
$277$	−28.7143	−0.103662	−0.0518308	+	0.998656i	$0.516506\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	194.297	0.691447	0.345724	−	0.938336i	$-0.387633\pi$
$281$	194.297	0.691447	0.345724	+	0.938336i	$0.387633\pi$
$282$	0	0
$283$	212.089i	0.749432i	0.927140	+	0.374716i	$0.122260\pi$
$283$	212.089i	0.749432i	−0.927140	+	0.374716i	$0.877740\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 153.355i	− 0.534336i
$288$	0	0
$289$	266.784	0.923127
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−118.777	−0.405382	−0.202691	−	0.979243i	$-0.564969\pi$
$293$	−118.777	−0.405382	−0.202691	+	0.979243i	$0.564969\pi$
$294$	0	0
$295$	− 335.369i	− 1.13684i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 568.378i	− 1.90093i
$300$	0	0
$301$	140.769	0.467673
$302$	0	0
$303$	0	0
$304$	0	0
$305$	153.225	0.502376
$306$	0	0
$307$	− 118.198i	− 0.385011i	−0.981296	−	0.192506i	$-0.938339\pi$
$307$	− 118.198i	− 0.385011i	0.981296	−	0.192506i	$-0.0616614\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	57.5386i	0.185012i	0.995712	+	0.0925059i	$0.0294877\pi$
$311$	57.5386i	0.185012i	−0.995712	+	0.0925059i	$0.970512\pi$
$312$	0	0
$313$	−401.012	−1.28119	−0.640594	−	0.767879i	$-0.721312\pi$
$313$	−401.012	−1.28119	−0.640594	+	0.767879i	$0.721312\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.34020	0.0263098	0.0131549	−	0.999913i	$-0.495813\pi$
$317$	8.34020	0.0263098	0.0131549	+	0.999913i	$0.495813\pi$
$318$	0	0
$319$	54.5847i	0.171112i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 512.190i	− 1.58573i
$324$	0	0
$325$	−24.8082	−0.0763330
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−119.264	−0.362503
$330$	0	0
$331$	253.990i	0.767340i	0.923470	+	0.383670i	$0.125340\pi$
$331$	253.990i	0.767340i	−0.923470	+	0.383670i	$0.874660\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 60.8883i	− 0.181756i
$336$	0	0
$337$	−620.332	−1.84075	−0.920374	−	0.391040i	$-0.872115\pi$
$337$	−620.332	−1.84075	−0.920374	+	0.391040i	$0.872115\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−105.562	−0.309567
$342$	0	0
$343$	− 329.250i	− 0.959914i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	496.172i	1.42989i	0.699181	+	0.714945i	$0.253548\pi$
$347$	496.172i	1.42989i	−0.699181	+	0.714945i	$0.746452\pi$
$348$	0	0
$349$	63.4267	0.181738	0.0908692	−	0.995863i	$-0.471035\pi$
$349$	63.4267	0.181738	0.0908692	+	0.995863i	$0.471035\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−439.982	−1.24641	−0.623204	−	0.782059i	$-0.714170\pi$
$353$	−439.982	−1.24641	−0.623204	+	0.782059i	$0.714170\pi$
$354$	0	0
$355$	− 721.882i	− 2.03347i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 489.550i	− 1.36365i	−0.731516	−	0.681824i	$-0.761187\pi$
$359$	− 489.550i	− 1.36365i	0.731516	−	0.681824i	$-0.238813\pi$
$360$	0	0
$361$	−111.016	−0.307524
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−612.224	−1.67733
$366$	0	0
$367$	393.063i	1.07102i	0.844530	+	0.535509i	$0.179880\pi$
$367$	393.063i	1.07102i	−0.844530	+	0.535509i	$0.820120\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 158.206i	− 0.426432i
$372$	0	0
$373$	11.4757	0.0307658	0.0153829	−	0.999882i	$-0.495103\pi$
$373$	11.4757	0.0307658	0.0153829	+	0.999882i	$0.495103\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−234.478	−0.621959
$378$	0	0
$379$	661.804i	1.74618i	0.487556	+	0.873092i	$0.337888\pi$
$379$	661.804i	1.74618i	−0.487556	+	0.873092i	$0.662112\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	95.8558i	0.250276i	0.992139	+	0.125138i	$0.0399374\pi$
$383$	95.8558i	0.250276i	−0.992139	+	0.125138i	$0.960063\pi$
$384$	0	0
$385$	−88.9043	−0.230920
$386$	0	0
$387$	0	0
$388$	0	0
$389$	471.609	1.21236	0.606182	−	0.795326i	$-0.292700\pi$
$389$	471.609	1.21236	0.606182	+	0.795326i	$0.292700\pi$
$390$	0	0
$391$	724.840i	1.85381i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	47.1599i	0.119392i
$396$	0	0
$397$	718.221	1.80912	0.904560	−	0.426346i	$-0.140199\pi$
$397$	718.221	1.80912	0.904560	+	0.426346i	$0.140199\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−225.675	−0.562780	−0.281390	−	0.959593i	$-0.590796\pi$
$401$	−225.675	−0.562780	−0.281390	+	0.959593i	$0.590796\pi$
$402$	0	0
$403$	− 453.462i	− 1.12522i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	78.4576i	0.192770i
$408$	0	0
$409$	588.917	1.43990	0.719948	−	0.694028i	$-0.244166\pi$
$409$	588.917	1.43990	0.719948	+	0.694028i	$0.244166\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	263.015	0.636841
$414$	0	0
$415$	− 580.892i	− 1.39974i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	121.906i	0.290944i	0.989362	+	0.145472i	$0.0464701\pi$
$419$	121.906i	0.290944i	−0.989362	+	0.145472i	$0.953530\pi$
$420$	0	0
$421$	289.293	0.687157	0.343578	−	0.939124i	$-0.388361\pi$
$421$	289.293	0.687157	0.343578	+	0.939124i	$0.388361\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	31.6374	0.0744410
$426$	0	0
$427$	120.168i	0.281423i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	337.402i	0.782835i	0.920213	+	0.391418i	$0.128015\pi$
$431$	337.402i	0.782835i	−0.920213	+	0.391418i	$0.871985\pi$
$432$	0	0
$433$	−7.18171	−0.0165859	−0.00829297	−	0.999966i	$-0.502640\pi$
$433$	−7.18171	−0.0165859	−0.00829297	+	0.999966i	$0.502640\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	667.987	1.52857
$438$	0	0
$439$	780.145i	1.77710i	0.458783	+	0.888548i	$0.348285\pi$
$439$	780.145i	1.77710i	−0.458783	+	0.888548i	$0.651715\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	478.844i	1.08091i	0.841372	+	0.540456i	$0.181748\pi$
$443$	478.844i	1.08091i	−0.841372	+	0.540456i	$0.818252\pi$
$444$	0	0
$445$	−40.4614	−0.0909245
$446$	0	0
$447$	0	0
$448$	0	0
$449$	55.5835	0.123794	0.0618970	−	0.998083i	$-0.480285\pi$
$449$	55.5835	0.123794	0.0618970	+	0.998083i	$0.480285\pi$
$450$	0	0
$451$	163.957i	0.363540i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 381.905i	− 0.839351i
$456$	0	0
$457$	31.6281	0.0692080	0.0346040	−	0.999401i	$-0.488983\pi$
$457$	31.6281	0.0692080	0.0346040	+	0.999401i	$0.488983\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	498.508	1.08136	0.540681	−	0.841228i	$-0.318167\pi$
$461$	498.508	1.08136	0.540681	+	0.841228i	$0.318167\pi$
$462$	0	0
$463$	− 63.5797i	− 0.137321i	−0.997640	−	0.0686606i	$-0.978127\pi$
$463$	− 63.5797i	− 0.137321i	0.997640	−	0.0686606i	$-0.0218726\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	550.457i	1.17871i	0.807875	+	0.589354i	$0.200618\pi$
$467$	550.457i	1.17871i	−0.807875	+	0.589354i	$0.799382\pi$
$468$	0	0
$469$	47.7521	0.101817
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−150.502	−0.318185
$474$	0	0
$475$	− 29.1559i	− 0.0613808i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	760.357i	1.58738i	0.608320	+	0.793692i	$0.291844\pi$
$479$	760.357i	1.58738i	−0.608320	+	0.793692i	$0.708156\pi$
$480$	0	0
$481$	−337.029	−0.700683
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−284.892	−0.587406
$486$	0	0
$487$	− 340.198i	− 0.698558i	−0.937019	−	0.349279i	$-0.886427\pi$
$487$	− 340.198i	− 0.698558i	0.937019	−	0.349279i	$-0.113573\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 488.166i	− 0.994229i	−0.867685	−	0.497114i	$-0.834393\pi$
$491$	− 488.166i	− 0.994229i	0.867685	−	0.497114i	$-0.165607\pi$
$492$	0	0
$493$	299.026	0.606543
$494$	0	0
$495$	0	0
$496$	0	0
$497$	566.141	1.13912
$498$	0	0
$499$	− 195.282i	− 0.391347i	−0.980669	−	0.195674i	$-0.937311\pi$
$499$	− 195.282i	− 0.391347i	0.980669	−	0.195674i	$-0.0626893\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	167.239i	0.332483i	0.986085	+	0.166241i	$0.0531631\pi$
$503$	167.239i	0.332483i	−0.986085	+	0.166241i	$0.946837\pi$
$504$	0	0
$505$	−1001.78	−1.98372
$506$	0	0
$507$	0	0
$508$	0	0
$509$	374.972	0.736683	0.368342	−	0.929691i	$-0.379926\pi$
$509$	374.972	0.736683	0.368342	+	0.929691i	$0.379926\pi$
$510$	0	0
$511$	− 480.142i	− 0.939612i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 150.678i	− 0.292578i
$516$	0	0
$517$	127.509	0.246632
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−49.7644	−0.0955170	−0.0477585	−	0.998859i	$-0.515208\pi$
$521$	−49.7644	−0.0955170	−0.0477585	+	0.998859i	$0.515208\pi$
$522$	0	0
$523$	878.700i	1.68012i	0.542497	+	0.840058i	$0.317479\pi$
$523$	878.700i	1.68012i	−0.542497	+	0.840058i	$0.682521\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	578.291i	1.09733i
$528$	0	0
$529$	−416.320	−0.786995
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−704.306	−1.32140
$534$	0	0
$535$	683.928i	1.27837i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	141.145i	0.261864i
$540$	0	0
$541$	183.963	0.340043	0.170022	−	0.985440i	$-0.445616\pi$
$541$	183.963	0.340043	0.170022	+	0.985440i	$0.445616\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1018.71	1.86920
$546$	0	0
$547$	− 642.639i	− 1.17484i	−0.809281	−	0.587421i	$-0.800143\pi$
$547$	− 642.639i	− 1.17484i	0.809281	−	0.587421i	$-0.199857\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 275.571i	− 0.500129i
$552$	0	0
$553$	−36.9855	−0.0668816
$554$	0	0
$555$	0	0
$556$	0	0
$557$	231.980	0.416480	0.208240	−	0.978078i	$-0.433226\pi$
$557$	231.980	0.416480	0.208240	+	0.978078i	$0.433226\pi$
$558$	0	0
$559$	− 646.507i	− 1.15654i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 356.868i	− 0.633869i	−0.948447	−	0.316935i	$-0.897346\pi$
$563$	− 356.868i	− 0.633869i	0.948447	−	0.316935i	$-0.102654\pi$
$564$	0	0
$565$	917.400	1.62372
$566$	0	0
$567$	0	0
$568$	0	0
$569$	440.786	0.774669	0.387334	−	0.921939i	$-0.373396\pi$
$569$	440.786	0.774669	0.387334	+	0.921939i	$0.373396\pi$
$570$	0	0
$571$	− 712.649i	− 1.24807i	−0.781396	−	0.624036i	$-0.785492\pi$
$571$	− 712.649i	− 1.24807i	0.781396	−	0.624036i	$-0.214508\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	41.2608i	0.0717579i
$576$	0	0
$577$	−253.401	−0.439170	−0.219585	−	0.975593i	$-0.570470\pi$
$577$	−253.401	−0.439170	−0.219585	+	0.975593i	$0.570470\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	455.569	0.784112
$582$	0	0
$583$	169.144i	0.290126i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	800.800i	1.36422i	0.731248	+	0.682112i	$0.238938\pi$
$587$	800.800i	1.36422i	−0.731248	+	0.682112i	$0.761062\pi$
$588$	0	0
$589$	532.932	0.904809
$590$	0	0
$591$	0	0
$592$	0	0
$593$	618.665	1.04328	0.521640	−	0.853166i	$-0.325321\pi$
$593$	618.665	1.04328	0.521640	+	0.853166i	$0.325321\pi$
$594$	0	0
$595$	487.035i	0.818546i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	249.660i	0.416795i	0.978044	+	0.208398i	$0.0668248\pi$
$599$	249.660i	0.416795i	−0.978044	+	0.208398i	$0.933175\pi$
$600$	0	0
$601$	−1137.02	−1.89188	−0.945941	−	0.324339i	$-0.894858\pi$
$601$	−1137.02	−1.89188	−0.945941	+	0.324339i	$0.894858\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−525.975	−0.869380
$606$	0	0
$607$	− 520.853i	− 0.858077i	−0.903286	−	0.429038i	$-0.858852\pi$
$607$	− 520.853i	− 0.858077i	0.903286	−	0.429038i	$-0.141148\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	547.737i	0.896460i
$612$	0	0
$613$	148.439	0.242152	0.121076	−	0.992643i	$-0.461365\pi$
$613$	148.439	0.242152	0.121076	+	0.992643i	$0.461365\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1081.44	1.75273	0.876366	−	0.481646i	$-0.159961\pi$
$617$	1081.44	1.75273	0.876366	+	0.481646i	$0.159961\pi$
$618$	0	0
$619$	306.359i	0.494925i	0.968897	+	0.247463i	$0.0795967\pi$
$619$	306.359i	0.494925i	−0.968897	+	0.247463i	$0.920403\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 31.7322i	− 0.0509345i
$624$	0	0
$625$	−656.749	−1.05080
$626$	0	0
$627$	0	0
$628$	0	0
$629$	429.806	0.683316
$630$	0	0
$631$	807.507i	1.27973i	0.768489	+	0.639863i	$0.221009\pi$
$631$	807.507i	1.27973i	−0.768489	+	0.639863i	$0.778991\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	883.874i	1.39193i
$636$	0	0
$637$	−606.312	−0.951824
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−387.743	−0.604904	−0.302452	−	0.953165i	$-0.597805\pi$
$641$	−387.743	−0.604904	−0.302452	+	0.953165i	$0.597805\pi$
$642$	0	0
$643$	− 978.913i	− 1.52242i	−0.648508	−	0.761208i	$-0.724607\pi$
$643$	− 978.913i	− 1.52242i	0.648508	−	0.761208i	$-0.275393\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	786.067i	1.21494i	0.794342	+	0.607470i	$0.207816\pi$
$647$	786.067i	1.21494i	−0.794342	+	0.607470i	$0.792184\pi$
$648$	0	0
$649$	−281.199	−0.433280
$650$	0	0
$651$	0	0
$652$	0	0
$653$	251.026	0.384419	0.192209	−	0.981354i	$-0.438435\pi$
$653$	251.026	0.384419	0.192209	+	0.981354i	$0.438435\pi$
$654$	0	0
$655$	− 1058.82i	− 1.61653i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	933.805i	1.41700i	0.705709	+	0.708501i	$0.250628\pi$
$659$	933.805i	1.41700i	−0.705709	+	0.708501i	$0.749372\pi$
$660$	0	0
$661$	473.319	0.716065	0.358033	−	0.933709i	$-0.383448\pi$
$661$	473.319	0.716065	0.358033	+	0.933709i	$0.383448\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	448.834	0.674938
$666$	0	0
$667$	389.982i	0.584681i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 128.475i	− 0.191469i
$672$	0	0
$673$	254.846	0.378671	0.189336	−	0.981912i	$-0.439367\pi$
$673$	254.846	0.378671	0.189336	+	0.981912i	$0.439367\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	104.575	0.154468	0.0772338	−	0.997013i	$-0.475391\pi$
$677$	104.575	0.154468	0.0772338	+	0.997013i	$0.475391\pi$
$678$	0	0
$679$	− 223.429i	− 0.329055i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 623.576i	− 0.912995i	−0.889725	−	0.456497i	$-0.849104\pi$
$683$	− 623.576i	− 0.912995i	0.889725	−	0.456497i	$-0.150896\pi$
$684$	0	0
$685$	−436.817	−0.637689
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−726.588	−1.05455
$690$	0	0
$691$	544.623i	0.788167i	0.919075	+	0.394083i	$0.128938\pi$
$691$	544.623i	0.788167i	−0.919075	+	0.394083i	$0.871062\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	771.541i	1.11013i
$696$	0	0
$697$	898.186	1.28865
$698$	0	0
$699$	0	0
$700$	0	0
$701$	411.842	0.587507	0.293753	−	0.955881i	$-0.405096\pi$
$701$	411.842	0.587507	0.293753	+	0.955881i	$0.405096\pi$
$702$	0	0
$703$	− 396.093i	− 0.563433i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 785.653i	− 1.11125i
$708$	0	0
$709$	679.236	0.958020	0.479010	−	0.877809i	$-0.340996\pi$
$709$	679.236	0.958020	0.479010	+	0.877809i	$0.340996\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−754.194	−1.05778
$714$	0	0
$715$	408.308i	0.571060i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	619.766i	0.861983i	0.902356	+	0.430991i	$0.141836\pi$
$719$	619.766i	0.861983i	−0.902356	+	0.430991i	$0.858164\pi$
$720$	0	0
$721$	118.170	0.163898
$722$	0	0
$723$	0	0
$724$	0	0
$725$	17.0217	0.0234782
$726$	0	0
$727$	1224.73i	1.68463i	0.538982	+	0.842317i	$0.318809\pi$
$727$	1224.73i	1.68463i	−0.538982	+	0.842317i	$0.681191\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	824.477i	1.12788i
$732$	0	0
$733$	797.758	1.08835	0.544173	−	0.838973i	$-0.316844\pi$
$733$	797.758	1.08835	0.544173	+	0.838973i	$0.316844\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−51.0534	−0.0692719
$738$	0	0
$739$	− 422.067i	− 0.571133i	−0.958359	−	0.285567i	$-0.907818\pi$
$739$	− 422.067i	− 0.571133i	0.958359	−	0.285567i	$-0.0921818\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 805.671i	− 1.08435i	−0.840266	−	0.542174i	$-0.817601\pi$
$743$	− 805.671i	− 1.08435i	0.840266	−	0.542174i	$-0.182399\pi$
$744$	0	0
$745$	570.385	0.765618
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−536.376	−0.716123
$750$	0	0
$751$	1014.96i	1.35148i	0.737138	+	0.675742i	$0.236177\pi$
$751$	1014.96i	1.35148i	−0.737138	+	0.675742i	$0.763823\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 853.501i	− 1.13046i
$756$	0	0
$757$	−14.8738	−0.0196483	−0.00982416	−	0.999952i	$-0.503127\pi$
$757$	−14.8738	−0.0196483	−0.00982416	+	0.999952i	$0.503127\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	884.373	1.16212	0.581060	−	0.813861i	$-0.302638\pi$
$761$	884.373	1.16212	0.581060	+	0.813861i	$0.302638\pi$
$762$	0	0
$763$	798.933i	1.04709i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1207.94i	− 1.57489i
$768$	0	0
$769$	−775.760	−1.00879	−0.504395	−	0.863473i	$-0.668285\pi$
$769$	−775.760	−1.00879	−0.504395	+	0.863473i	$0.668285\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−794.115	−1.02732	−0.513658	−	0.857995i	$-0.671710\pi$
$773$	−794.115	−1.02732	−0.513658	+	0.857995i	$0.671710\pi$
$774$	0	0
$775$	32.9186i	0.0424756i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 827.736i	− 1.06256i
$780$	0	0
$781$	−605.281	−0.775008
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−257.423	−0.327927
$786$	0	0
$787$	344.287i	0.437468i	0.975785	+	0.218734i	$0.0701927\pi$
$787$	344.287i	0.437468i	−0.975785	+	0.218734i	$0.929807\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	719.478i	0.909581i
$792$	0	0
$793$	551.890	0.695952
$794$	0	0
$795$	0	0
$796$	0	0
$797$	986.969	1.23836	0.619178	−	0.785251i	$-0.287466\pi$
$797$	986.969	1.23836	0.619178	+	0.785251i	$0.287466\pi$
$798$	0	0
$799$	− 698.518i	− 0.874240i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	513.336i	0.639273i
$804$	0	0
$805$	−635.180	−0.789043
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−33.7831	−0.0417591	−0.0208795	−	0.999782i	$-0.506647\pi$
$809$	−33.7831	−0.0417591	−0.0208795	+	0.999782i	$0.506647\pi$
$810$	0	0
$811$	174.359i	0.214993i	0.994205	+	0.107496i	$0.0342834\pi$
$811$	174.359i	0.214993i	−0.994205	+	0.107496i	$0.965717\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 1365.20i	− 1.67509i
$816$	0	0
$817$	759.808	0.929997
$818$	0	0
$819$	0	0
$820$	0	0
$821$	826.976	1.00728	0.503640	−	0.863914i	$-0.331994\pi$
$821$	826.976	1.00728	0.503640	+	0.863914i	$0.331994\pi$
$822$	0	0
$823$	789.862i	0.959735i	0.877341	+	0.479868i	$0.159315\pi$
$823$	789.862i	0.959735i	−0.877341	+	0.479868i	$0.840685\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	94.0797i	0.113760i	0.998381	+	0.0568801i	$0.0181153\pi$
$827$	94.0797i	0.113760i	−0.998381	+	0.0568801i	$0.981885\pi$
$828$	0	0
$829$	383.996	0.463204	0.231602	−	0.972811i	$-0.425603\pi$
$829$	383.996	0.463204	0.231602	+	0.972811i	$0.425603\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	773.217	0.928232
$834$	0	0
$835$	− 1147.88i	− 1.37470i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1148.75i	1.36919i	0.728924	+	0.684594i	$0.240021\pi$
$839$	1148.75i	1.36919i	−0.728924	+	0.684594i	$0.759979\pi$
$840$	0	0
$841$	−680.117	−0.808700
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−886.576	−1.04920
$846$	0	0
$847$	− 412.500i	− 0.487013i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	560.543i	0.658687i
$852$	0	0
$853$	165.565	0.194097	0.0970485	−	0.995280i	$-0.469060\pi$
$853$	165.565	0.194097	0.0970485	+	0.995280i	$0.469060\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1294.69	−1.51073	−0.755363	−	0.655307i	$-0.772539\pi$
$857$	−1294.69	−1.51073	−0.755363	+	0.655307i	$0.772539\pi$
$858$	0	0
$859$	− 14.0574i	− 0.0163648i	−0.999967	−	0.00818241i	$-0.997395\pi$
$859$	− 14.0574i	− 0.0163648i	0.999967	−	0.00818241i	$-0.00260457\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 409.653i	− 0.474685i	−0.971426	−	0.237343i	$-0.923724\pi$
$863$	− 409.653i	− 0.474685i	0.971426	−	0.237343i	$-0.0762764\pi$
$864$	0	0
$865$	−284.715	−0.329150
$866$	0	0
$867$	0	0
$868$	0	0
$869$	39.5425	0.0455035
$870$	0	0
$871$	− 219.309i	− 0.251790i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 488.749i	− 0.558570i
$876$	0	0
$877$	−10.0270	−0.0114333	−0.00571666	−	0.999984i	$-0.501820\pi$
$877$	−10.0270	−0.0114333	−0.00571666	+	0.999984i	$0.501820\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1388.38	−1.57591	−0.787956	−	0.615731i	$-0.788861\pi$
$881$	−1388.38	−1.57591	−0.787956	+	0.615731i	$0.788861\pi$
$882$	0	0
$883$	− 1445.91i	− 1.63750i	−0.574153	−	0.818748i	$-0.694668\pi$
$883$	− 1445.91i	− 1.63750i	0.574153	−	0.818748i	$-0.305332\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 661.924i	− 0.746250i	−0.927781	−	0.373125i	$-0.878286\pi$
$887$	− 661.924i	− 0.746250i	0.927781	−	0.373125i	$-0.121714\pi$
$888$	0	0
$889$	−693.185	−0.779736
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−643.729	−0.720861
$894$	0	0
$895$	− 55.2058i	− 0.0616824i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	311.135i	0.346090i
$900$	0	0
$901$	926.602	1.02842
$902$	0	0
$903$	0	0
$904$	0	0
$905$	186.635	0.206227
$906$	0	0
$907$	− 314.099i	− 0.346306i	−0.984895	−	0.173153i	$-0.944605\pi$
$907$	− 314.099i	− 0.346306i	0.984895	−	0.173153i	$-0.0553955\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	620.050i	0.680626i	0.940312	+	0.340313i	$0.110533\pi$
$911$	620.050i	0.680626i	−0.940312	+	0.340313i	$0.889467\pi$
$912$	0	0
$913$	−487.065	−0.533478
$914$	0	0
$915$	0	0
$916$	0	0
$917$	830.391	0.905552
$918$	0	0
$919$	− 259.390i	− 0.282253i	−0.989992	−	0.141126i	$-0.954928\pi$
$919$	− 259.390i	− 0.282253i	0.989992	−	0.141126i	$-0.0450724\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 2600.10i	− 2.81701i
$924$	0	0
$925$	24.4662	0.0264500
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−511.871	−0.550991	−0.275496	−	0.961302i	$-0.588842\pi$
$929$	−511.871	−0.550991	−0.275496	+	0.961302i	$0.588842\pi$
$930$	0	0
$931$	− 712.569i	− 0.765380i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 520.706i	− 0.556905i
$936$	0	0
$937$	531.153	0.566866	0.283433	−	0.958992i	$-0.408527\pi$
$937$	531.153	0.566866	0.283433	+	0.958992i	$0.408527\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1401.22	1.48908	0.744540	−	0.667578i	$-0.232669\pi$
$941$	1401.22	1.48908	0.744540	+	0.667578i	$0.232669\pi$
$942$	0	0
$943$	1171.39i	1.24220i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	246.077i	0.259849i	0.991524	+	0.129925i	$0.0414736\pi$
$947$	246.077i	0.259849i	−0.991524	+	0.129925i	$0.958526\pi$
$948$	0	0
$949$	−2205.13	−2.32363
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1711.93	−1.79636	−0.898178	−	0.439631i	$-0.855109\pi$
$953$	−1711.93	−1.79636	−0.898178	+	0.439631i	$0.855109\pi$
$954$	0	0
$955$	1100.62i	1.15249i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 342.577i	− 0.357223i
$960$	0	0
$961$	359.290	0.373871
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1437.25	−1.48937
$966$	0	0
$967$	− 801.048i	− 0.828385i	−0.910189	−	0.414192i	$-0.864064\pi$
$967$	− 801.048i	− 0.828385i	0.910189	−	0.414192i	$-0.135936\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 526.798i	− 0.542532i	−0.962504	−	0.271266i	$-0.912558\pi$
$971$	− 526.798i	− 0.542532i	0.962504	−	0.271266i	$-0.0874423\pi$
$972$	0	0
$973$	−605.087	−0.621877
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−630.886	−0.645738	−0.322869	−	0.946444i	$-0.604647\pi$
$977$	−630.886	−0.645738	−0.322869	+	0.946444i	$0.604647\pi$
$978$	0	0
$979$	33.9260i	0.0346537i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1553.23i	− 1.58009i	−0.613051	−	0.790043i	$-0.710058\pi$
$983$	− 1553.23i	− 1.58009i	0.613051	−	0.790043i	$-0.289942\pi$
$984$	0	0
$985$	1304.04	1.32390
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1075.26	−1.08722
$990$	0	0
$991$	− 487.053i	− 0.491476i	−0.969336	−	0.245738i	$-0.920970\pi$
$991$	− 487.053i	− 0.491476i	0.969336	−	0.245738i	$-0.0790303\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 770.876i	− 0.774750i
$996$	0	0
$997$	−1665.23	−1.67024	−0.835122	−	0.550065i	$-0.814603\pi$
$997$	−1665.23	−1.67024	−0.835122	+	0.550065i	$0.814603\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.3.g.b.703.3	✓	8
3.2	odd	2		864.3.g.d.703.5	yes	8
4.3	odd	2	inner	864.3.g.b.703.4	yes	8
8.3	odd	2		1728.3.g.m.703.6		8
8.5	even	2		1728.3.g.m.703.5		8
12.11	even	2		864.3.g.d.703.6	yes	8
24.5	odd	2		1728.3.g.j.703.3		8
24.11	even	2		1728.3.g.j.703.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.3.g.b.703.3	✓	8	1.1	even	1	trivial
864.3.g.b.703.4	yes	8	4.3	odd	2	inner
864.3.g.d.703.5	yes	8	3.2	odd	2
864.3.g.d.703.6	yes	8	12.11	even	2
1728.3.g.j.703.3		8	24.5	odd	2
1728.3.g.j.703.4		8	24.11	even	2
1728.3.g.m.703.5		8	8.5	even	2
1728.3.g.m.703.6		8	8.3	odd	2

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	864.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.13244 q^{5} -4.02516i q^{7} +O(q^{10})\)	\(q-5.13244 q^{5} -4.02516i q^{7} +4.30344i q^{11} -18.4862 q^{13} +23.5751 q^{17} -21.7259i q^{19} +30.7461i q^{23} +1.34199 q^{25} +12.6840 q^{29} +24.5298i q^{31} +20.6589i q^{35} +18.2314 q^{37} +38.0990 q^{41} +34.9724i q^{43} -29.6295i q^{47} +32.7981 q^{49} +39.3043 q^{53} -22.0872i q^{55} +65.3429i q^{59} -29.8541 q^{61} +94.8794 q^{65} +11.8634i q^{67} +140.651i q^{71} +119.285 q^{73} +17.3220 q^{77} -9.18859i q^{79} +113.180i q^{83} -120.998 q^{85} +7.88346 q^{89} +74.4099i q^{91} +111.507i q^{95} +55.5080 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^5	\( 8 q - 8 q^{5} + 8 q^{13} - 24 q^{17} + 24 q^{25} + 128 q^{29} + 24 q^{37} - 160 q^{41} - 144 q^{49} + 48 q^{53} - 136 q^{61} + 280 q^{65} + 72 q^{73} - 520 q^{77} + 96 q^{85} + 168 q^{89} + 104 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 + 8 * q^13 - 24 * q^17 + 24 * q^25 + 128 * q^29 + 24 * q^37 - 160 * q^41 - 144 * q^49 + 48 * q^53 - 136 * q^61 + 280 * q^65 + 72 * q^73 - 520 * q^77 + 96 * q^85 + 168 * q^89 + 104 * q^97

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