LMFDB - Embedded newform 864.3.g.a.703.5

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:	8.0.22581504.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 $ x^8 - 4x^7 + 5x^6 + 2x^5 - 11x^4 + 4x^3 + 20x^2 - 32*x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.5
Root			$1.40994 + 0.109843i$ of defining polynomial
Character	$\chi$	$=$	864.703
Dual form			864.3.g.a.703.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.22512 q^{5} -6.57221i q^{7} +O(q^{10})$	$q+3.22512 q^{5} -6.57221i q^{7} +13.8903i q^{11} -19.0865 q^{13} -23.0865 q^{17} +18.8425i q^{19} -13.5947i q^{23} -14.5986 q^{25} -0.752110 q^{29} +46.4433i q^{31} -21.1962i q^{35} -2.68178 q^{37} +34.1405 q^{41} +20.9127i q^{43} +15.4086i q^{47} +5.80609 q^{49} +46.8189 q^{53} +44.7980i q^{55} +40.4126i q^{59} -105.543 q^{61} -61.5562 q^{65} +27.9369i q^{67} -24.0130i q^{71} -120.117 q^{73} +91.2901 q^{77} +95.6394i q^{79} -115.446i q^{83} -74.4567 q^{85} -169.589 q^{89} +125.440i q^{91} +60.7694i q^{95} -93.1142 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^5	$ 8 q - 8 q^{5} - 16 q^{13} - 48 q^{17} + 48 q^{25} + 32 q^{29} - 96 q^{37} + 128 q^{41} - 168 q^{53} + 32 q^{61} + 112 q^{65} + 24 q^{73} + 440 q^{77} + 144 q^{85} - 624 q^{89} - 136 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 - 16 * q^13 - 48 * q^17 + 48 * q^25 + 32 * q^29 - 96 * q^37 + 128 * q^41 - 168 * q^53 + 32 * q^61 + 112 * q^65 + 24 * q^73 + 440 * q^77 + 144 * q^85 - 624 * q^89 - 136 * 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
$5$	3.22512	0.645024	0.322512	−	0.946565i	$-0.395473\pi$
$5$	3.22512	0.645024	0.322512	+	0.946565i	$0.395473\pi$
$6$	0	0
$7$	− 6.57221i	− 0.938887i	−0.882963	−	0.469443i	$-0.844455\pi$
$7$	− 6.57221i	− 0.938887i	0.882963	−	0.469443i	$-0.155545\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	13.8903i	1.26276i	0.775475	+	0.631379i	$0.217511\pi$
$11$	13.8903i	1.26276i	−0.775475	+	0.631379i	$0.782489\pi$
$12$	0	0
$13$	−19.0865	−1.46819	−0.734095	−	0.679046i	$-0.762394\pi$
$13$	−19.0865	−1.46819	−0.734095	+	0.679046i	$0.762394\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−23.0865	−1.35803	−0.679014	−	0.734125i	$-0.737593\pi$
$17$	−23.0865	−1.35803	−0.679014	+	0.734125i	$0.737593\pi$
$18$	0	0
$19$	18.8425i	0.991713i	0.868405	+	0.495856i	$0.165146\pi$
$19$	18.8425i	0.991713i	−0.868405	+	0.495856i	$0.834854\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 13.5947i	− 0.591072i	−0.955332	−	0.295536i	$-0.904502\pi$
$23$	− 13.5947i	− 0.591072i	0.955332	−	0.295536i	$-0.0954982\pi$
$24$	0	0
$25$	−14.5986	−0.583944
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.752110	−0.0259348	−0.0129674	−	0.999916i	$-0.504128\pi$
$29$	−0.752110	−0.0259348	−0.0129674	+	0.999916i	$0.504128\pi$
$30$	0	0
$31$	46.4433i	1.49817i	0.662473	+	0.749085i	$0.269507\pi$
$31$	46.4433i	1.49817i	−0.662473	+	0.749085i	$0.730493\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 21.1962i	− 0.605604i
$36$	0	0
$37$	−2.68178	−0.0724807	−0.0362403	−	0.999343i	$-0.511538\pi$
$37$	−2.68178	−0.0724807	−0.0362403	+	0.999343i	$0.511538\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	34.1405	0.832694	0.416347	−	0.909206i	$-0.363310\pi$
$41$	34.1405	0.832694	0.416347	+	0.909206i	$0.363310\pi$
$42$	0	0
$43$	20.9127i	0.486341i	0.969984	+	0.243171i	$0.0781875\pi$
$43$	20.9127i	0.486341i	−0.969984	+	0.243171i	$0.921812\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	15.4086i	0.327844i	0.986473	+	0.163922i	$0.0524145\pi$
$47$	15.4086i	0.327844i	−0.986473	+	0.163922i	$0.947586\pi$
$48$	0	0
$49$	5.80609	0.118492
$50$	0	0
$51$	0	0
$52$	0	0
$53$	46.8189	0.883376	0.441688	−	0.897169i	$-0.354380\pi$
$53$	46.8189	0.883376	0.441688	+	0.897169i	$0.354380\pi$
$54$	0	0
$55$	44.7980i	0.814508i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	40.4126i	0.684959i	0.939525	+	0.342480i	$0.111267\pi$
$59$	40.4126i	0.684959i	−0.939525	+	0.342480i	$0.888733\pi$
$60$	0	0
$61$	−105.543	−1.73022	−0.865108	−	0.501586i	$-0.832750\pi$
$61$	−105.543	−1.73022	−0.865108	+	0.501586i	$0.832750\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−61.5562	−0.947018
$66$	0	0
$67$	27.9369i	0.416969i	0.978026	+	0.208485i	$0.0668531\pi$
$67$	27.9369i	0.416969i	−0.978026	+	0.208485i	$0.933147\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 24.0130i	− 0.338212i	−0.985598	−	0.169106i	$-0.945912\pi$
$71$	− 24.0130i	− 0.338212i	0.985598	−	0.169106i	$-0.0540880\pi$
$72$	0	0
$73$	−120.117	−1.64545	−0.822723	−	0.568443i	$-0.807546\pi$
$73$	−120.117	−1.64545	−0.822723	+	0.568443i	$0.807546\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	91.2901	1.18559
$78$	0	0
$79$	95.6394i	1.21062i	0.795988	+	0.605312i	$0.206952\pi$
$79$	95.6394i	1.21062i	−0.795988	+	0.605312i	$0.793048\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 115.446i	− 1.39091i	−0.718568	−	0.695457i	$-0.755202\pi$
$83$	− 115.446i	− 1.39091i	0.718568	−	0.695457i	$-0.244798\pi$
$84$	0	0
$85$	−74.4567	−0.875961
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−169.589	−1.90549	−0.952745	−	0.303770i	$-0.901755\pi$
$89$	−169.589	−1.90549	−0.952745	+	0.303770i	$0.901755\pi$
$90$	0	0
$91$	125.440i	1.37846i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	60.7694i	0.639678i
$96$	0	0
$97$	−93.1142	−0.959940	−0.479970	−	0.877285i	$-0.659352\pi$
$97$	−93.1142	−0.959940	−0.479970	+	0.877285i	$0.659352\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	159.090	1.57514	0.787572	−	0.616222i	$-0.211338\pi$
$101$	159.090	1.57514	0.787572	+	0.616222i	$0.211338\pi$
$102$	0	0
$103$	− 75.3674i	− 0.731722i	−0.930669	−	0.365861i	$-0.880774\pi$
$103$	− 75.3674i	− 0.731722i	0.930669	−	0.365861i	$-0.119226\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 69.4279i	− 0.648859i	−0.945910	−	0.324429i	$-0.894828\pi$
$107$	− 69.4279i	− 0.648859i	0.945910	−	0.324429i	$-0.105172\pi$
$108$	0	0
$109$	60.4669	0.554742	0.277371	−	0.960763i	$-0.410537\pi$
$109$	60.4669	0.554742	0.277371	+	0.960763i	$0.410537\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−140.698	−1.24512	−0.622559	−	0.782573i	$-0.713907\pi$
$113$	−140.698	−1.24512	−0.622559	+	0.782573i	$0.713907\pi$
$114$	0	0
$115$	− 43.8444i	− 0.381255i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	151.729i	1.27503i
$120$	0	0
$121$	−71.9412	−0.594556
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−127.710	−1.02168
$126$	0	0
$127$	− 66.9726i	− 0.527343i	−0.964612	−	0.263672i	$-0.915066\pi$
$127$	− 66.9726i	− 0.527343i	0.964612	−	0.263672i	$-0.0849335\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	255.148i	1.94770i	0.227196	+	0.973849i	$0.427044\pi$
$131$	255.148i	1.94770i	−0.227196	+	0.973849i	$0.572956\pi$
$132$	0	0
$133$	123.837	0.931106
$134$	0	0
$135$	0	0
$136$	0	0
$137$	27.1946	0.198501	0.0992505	−	0.995062i	$-0.468355\pi$
$137$	27.1946	0.198501	0.0992505	+	0.995062i	$0.468355\pi$
$138$	0	0
$139$	203.956i	1.46731i	0.679521	+	0.733656i	$0.262188\pi$
$139$	203.956i	1.46731i	−0.679521	+	0.733656i	$0.737812\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 265.117i	− 1.85397i
$144$	0	0
$145$	−2.42564	−0.0167286
$146$	0	0
$147$	0	0
$148$	0	0
$149$	251.507	1.68797	0.843984	−	0.536369i	$-0.180204\pi$
$149$	251.507	1.68797	0.843984	+	0.536369i	$0.180204\pi$
$150$	0	0
$151$	250.251i	1.65729i	0.559774	+	0.828645i	$0.310888\pi$
$151$	250.251i	1.65729i	−0.559774	+	0.828645i	$0.689112\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	149.785i	0.966356i
$156$	0	0
$157$	18.1730	0.115751	0.0578757	−	0.998324i	$-0.481567\pi$
$157$	18.1730	0.115751	0.0578757	+	0.998324i	$0.481567\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−89.3469	−0.554950
$162$	0	0
$163$	− 50.0180i	− 0.306859i	−0.988160	−	0.153429i	$-0.950968\pi$
$163$	− 50.0180i	− 0.306859i	0.988160	−	0.153429i	$-0.0490318\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	84.8389i	0.508017i	0.967202	+	0.254009i	$0.0817492\pi$
$167$	84.8389i	0.508017i	−0.967202	+	0.254009i	$0.918251\pi$
$168$	0	0
$169$	195.294	1.15558
$170$	0	0
$171$	0	0
$172$	0	0
$173$	144.703	0.836433	0.418216	−	0.908347i	$-0.362655\pi$
$173$	144.703	0.836433	0.418216	+	0.908347i	$0.362655\pi$
$174$	0	0
$175$	95.9451i	0.548258i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 280.926i	− 1.56942i	−0.619864	−	0.784709i	$-0.712812\pi$
$179$	− 280.926i	− 1.56942i	0.619864	−	0.784709i	$-0.287188\pi$
$180$	0	0
$181$	−295.508	−1.63264	−0.816320	−	0.577600i	$-0.803989\pi$
$181$	−295.508	−1.63264	−0.816320	+	0.577600i	$0.803989\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.64907	−0.0467518
$186$	0	0
$187$	− 320.679i	− 1.71486i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 295.982i	− 1.54965i	−0.632179	−	0.774823i	$-0.717839\pi$
$191$	− 295.982i	− 1.54965i	0.632179	−	0.774823i	$-0.282161\pi$
$192$	0	0
$193$	−18.0202	−0.0933688	−0.0466844	−	0.998910i	$-0.514866\pi$
$193$	−18.0202	−0.0933688	−0.0466844	+	0.998910i	$0.514866\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−119.020	−0.604162	−0.302081	−	0.953282i	$-0.597681\pi$
$197$	−119.020	−0.604162	−0.302081	+	0.953282i	$0.597681\pi$
$198$	0	0
$199$	− 304.912i	− 1.53222i	−0.642708	−	0.766111i	$-0.722189\pi$
$199$	− 304.912i	− 1.53222i	0.642708	−	0.766111i	$-0.277811\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	4.94302i	0.0243499i
$204$	0	0
$205$	110.107	0.537108
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−261.729	−1.25229
$210$	0	0
$211$	− 282.143i	− 1.33717i	−0.743635	−	0.668585i	$-0.766900\pi$
$211$	− 282.143i	− 1.33717i	0.743635	−	0.668585i	$-0.233100\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	67.4459i	0.313702i
$216$	0	0
$217$	305.235	1.40661
$218$	0	0
$219$	0	0
$220$	0	0
$221$	440.640	1.99384
$222$	0	0
$223$	− 91.9070i	− 0.412139i	−0.978537	−	0.206070i	$-0.933933\pi$
$223$	− 91.9070i	− 0.412139i	0.978537	−	0.206070i	$-0.0660673\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	31.5125i	0.138822i	0.997588	+	0.0694108i	$0.0221119\pi$
$227$	31.5125i	0.138822i	−0.997588	+	0.0694108i	$0.977888\pi$
$228$	0	0
$229$	242.515	1.05902	0.529509	−	0.848304i	$-0.322376\pi$
$229$	242.515	1.05902	0.529509	+	0.848304i	$0.322376\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−402.297	−1.72660	−0.863298	−	0.504695i	$-0.831605\pi$
$233$	−402.297	−1.72660	−0.863298	+	0.504695i	$0.831605\pi$
$234$	0	0
$235$	49.6947i	0.211467i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	167.034i	0.698886i	0.936958	+	0.349443i	$0.113629\pi$
$239$	167.034i	0.698886i	−0.936958	+	0.349443i	$0.886371\pi$
$240$	0	0
$241$	−21.4458	−0.0889868	−0.0444934	−	0.999010i	$-0.514167\pi$
$241$	−21.4458	−0.0889868	−0.0444934	+	0.999010i	$0.514167\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	18.7253	0.0764299
$246$	0	0
$247$	− 359.638i	− 1.45602i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	22.4825i	0.0895718i	0.998997	+	0.0447859i	$0.0142606\pi$
$251$	22.4825i	0.0895718i	−0.998997	+	0.0447859i	$0.985739\pi$
$252$	0	0
$253$	188.834	0.746380
$254$	0	0
$255$	0	0
$256$	0	0
$257$	437.652	1.70293	0.851463	−	0.524415i	$-0.175716\pi$
$257$	437.652	1.70293	0.851463	+	0.524415i	$0.175716\pi$
$258$	0	0
$259$	17.6252i	0.0680511i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	266.194i	1.01214i	0.862491	+	0.506072i	$0.168903\pi$
$263$	266.194i	1.01214i	−0.862491	+	0.506072i	$0.831097\pi$
$264$	0	0
$265$	150.997	0.569799
$266$	0	0
$267$	0	0
$268$	0	0
$269$	220.671	0.820338	0.410169	−	0.912010i	$-0.365470\pi$
$269$	220.671	0.820338	0.410169	+	0.912010i	$0.365470\pi$
$270$	0	0
$271$	− 224.295i	− 0.827656i	−0.910355	−	0.413828i	$-0.864192\pi$
$271$	− 224.295i	− 0.827656i	0.910355	−	0.413828i	$-0.135808\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 202.779i	− 0.737380i
$276$	0	0
$277$	−51.4458	−0.185725	−0.0928625	−	0.995679i	$-0.529602\pi$
$277$	−51.4458	−0.185725	−0.0928625	+	0.995679i	$0.529602\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−40.1606	−0.142920	−0.0714602	−	0.997443i	$-0.522766\pi$
$281$	−40.1606	−0.142920	−0.0714602	+	0.997443i	$0.522766\pi$
$282$	0	0
$283$	− 190.774i	− 0.674112i	−0.941485	−	0.337056i	$-0.890569\pi$
$283$	− 190.774i	− 0.674112i	0.941485	−	0.337056i	$-0.109431\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 224.378i	− 0.781806i
$288$	0	0
$289$	243.986	0.844241
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−115.872	−0.395467	−0.197733	−	0.980256i	$-0.563358\pi$
$293$	−115.872	−0.395467	−0.197733	+	0.980256i	$0.563358\pi$
$294$	0	0
$295$	130.335i	0.441815i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	259.474i	0.867806i
$300$	0	0
$301$	137.442	0.456619
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−340.389	−1.11603
$306$	0	0
$307$	497.554i	1.62070i	0.585949	+	0.810348i	$0.300722\pi$
$307$	497.554i	1.62070i	−0.585949	+	0.810348i	$0.699278\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	358.447i	1.15256i	0.817252	+	0.576281i	$0.195497\pi$
$311$	358.447i	1.15256i	−0.817252	+	0.576281i	$0.804503\pi$
$312$	0	0
$313$	32.9519	0.105278	0.0526388	−	0.998614i	$-0.483237\pi$
$313$	32.9519	0.105278	0.0526388	+	0.998614i	$0.483237\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−216.101	−0.681707	−0.340853	−	0.940117i	$-0.610716\pi$
$317$	−216.101	−0.681707	−0.340853	+	0.940117i	$0.610716\pi$
$318$	0	0
$319$	− 10.4471i	− 0.0327494i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 435.008i	− 1.34677i
$324$	0	0
$325$	278.636	0.857342
$326$	0	0
$327$	0	0
$328$	0	0
$329$	101.269	0.307808
$330$	0	0
$331$	− 578.213i	− 1.74687i	−0.486944	−	0.873433i	$-0.661888\pi$
$331$	− 578.213i	− 1.74687i	0.486944	−	0.873433i	$-0.338112\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	90.0999i	0.268955i
$336$	0	0
$337$	358.117	1.06266	0.531331	−	0.847164i	$-0.321692\pi$
$337$	358.117	1.06266	0.531331	+	0.847164i	$0.321692\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−645.113	−1.89183
$342$	0	0
$343$	− 360.197i	− 1.05014i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	483.855i	1.39440i	0.716879	+	0.697198i	$0.245570\pi$
$347$	483.855i	1.39440i	−0.716879	+	0.697198i	$0.754430\pi$
$348$	0	0
$349$	22.1213	0.0633849	0.0316924	−	0.999498i	$-0.489910\pi$
$349$	22.1213	0.0633849	0.0316924	+	0.999498i	$0.489910\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	459.068	1.30048	0.650238	−	0.759731i	$-0.274669\pi$
$353$	459.068	1.30048	0.650238	+	0.759731i	$0.274669\pi$
$354$	0	0
$355$	− 77.4449i	− 0.218155i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 414.323i	− 1.15410i	−0.816708	−	0.577052i	$-0.804203\pi$
$359$	− 414.323i	− 1.15410i	0.816708	−	0.577052i	$-0.195797\pi$
$360$	0	0
$361$	5.95856	0.0165057
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−387.393	−1.06135
$366$	0	0
$367$	90.5390i	0.246700i	0.992363	+	0.123350i	$0.0393638\pi$
$367$	90.5390i	0.246700i	−0.992363	+	0.123350i	$0.960636\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 307.704i	− 0.829390i
$372$	0	0
$373$	223.011	0.597884	0.298942	−	0.954271i	$-0.403366\pi$
$373$	223.011	0.597884	0.298942	+	0.954271i	$0.403366\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	14.3551	0.0380773
$378$	0	0
$379$	− 118.162i	− 0.311774i	−0.987775	−	0.155887i	$-0.950176\pi$
$379$	− 118.162i	− 0.311774i	0.987775	−	0.155887i	$-0.0498236\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19.6418i	0.0512840i	0.999671	+	0.0256420i	$0.00816300\pi$
$383$	19.6418i	0.0512840i	−0.999671	+	0.0256420i	$0.991837\pi$
$384$	0	0
$385$	294.422	0.764731
$386$	0	0
$387$	0	0
$388$	0	0
$389$	352.392	0.905892	0.452946	−	0.891538i	$-0.350373\pi$
$389$	352.392	0.905892	0.452946	+	0.891538i	$0.350373\pi$
$390$	0	0
$391$	313.853i	0.802692i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	308.448i	0.780882i
$396$	0	0
$397$	−5.66047	−0.0142581	−0.00712906	−	0.999975i	$-0.502269\pi$
$397$	−5.66047	−0.0142581	−0.00712906	+	0.999975i	$0.502269\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	250.097	0.623683	0.311842	−	0.950134i	$-0.399054\pi$
$401$	250.097	0.623683	0.311842	+	0.950134i	$0.399054\pi$
$402$	0	0
$403$	− 886.439i	− 2.19960i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 37.2509i	− 0.0915255i
$408$	0	0
$409$	−643.885	−1.57429	−0.787145	−	0.616767i	$-0.788442\pi$
$409$	−643.885	−1.57429	−0.787145	+	0.616767i	$0.788442\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	265.600	0.643099
$414$	0	0
$415$	− 372.327i	− 0.897173i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 439.563i	− 1.04908i	−0.851387	−	0.524539i	$-0.824238\pi$
$419$	− 439.563i	− 1.04908i	0.851387	−	0.524539i	$-0.175762\pi$
$420$	0	0
$421$	65.4392	0.155438	0.0777188	−	0.996975i	$-0.475236\pi$
$421$	65.4392	0.155438	0.0777188	+	0.996975i	$0.475236\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	337.030	0.793013
$426$	0	0
$427$	693.651i	1.62448i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 299.824i	− 0.695646i	−0.937560	−	0.347823i	$-0.886921\pi$
$431$	− 299.824i	− 0.695646i	0.937560	−	0.347823i	$-0.113079\pi$
$432$	0	0
$433$	−699.217	−1.61482	−0.807410	−	0.589991i	$-0.799131\pi$
$433$	−699.217	−1.61482	−0.807410	+	0.589991i	$0.799131\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	256.158	0.586174
$438$	0	0
$439$	744.832i	1.69666i	0.529472	+	0.848328i	$0.322390\pi$
$439$	744.832i	1.69666i	−0.529472	+	0.848328i	$0.677610\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	320.933i	0.724455i	0.932090	+	0.362227i	$0.117984\pi$
$443$	320.933i	0.724455i	−0.932090	+	0.362227i	$0.882016\pi$
$444$	0	0
$445$	−546.944	−1.22909
$446$	0	0
$447$	0	0
$448$	0	0
$449$	439.651	0.979179	0.489589	−	0.871953i	$-0.337147\pi$
$449$	439.651	0.979179	0.489589	+	0.871953i	$0.337147\pi$
$450$	0	0
$451$	474.222i	1.05149i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	404.560i	0.889143i
$456$	0	0
$457$	−732.044	−1.60185	−0.800923	−	0.598767i	$-0.795658\pi$
$457$	−732.044	−1.60185	−0.800923	+	0.598767i	$0.795658\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−381.075	−0.826627	−0.413314	−	0.910589i	$-0.635629\pi$
$461$	−381.075	−0.826627	−0.413314	+	0.910589i	$0.635629\pi$
$462$	0	0
$463$	− 591.775i	− 1.27813i	−0.769152	−	0.639066i	$-0.779321\pi$
$463$	− 591.775i	− 1.27813i	0.769152	−	0.639066i	$-0.220679\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	50.8002i	0.108780i	0.998520	+	0.0543899i	$0.0173214\pi$
$467$	50.8002i	0.108780i	−0.998520	+	0.0543899i	$0.982679\pi$
$468$	0	0
$469$	183.607	0.391487
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−290.484	−0.614131
$474$	0	0
$475$	− 275.075i	− 0.579105i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 844.826i	− 1.76373i	−0.471503	−	0.881865i	$-0.656288\pi$
$479$	− 844.826i	− 1.76373i	0.471503	−	0.881865i	$-0.343712\pi$
$480$	0	0
$481$	51.1858	0.106415
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−300.304	−0.619184
$486$	0	0
$487$	328.276i	0.674077i	0.941491	+	0.337039i	$0.109425\pi$
$487$	328.276i	0.674077i	−0.941491	+	0.337039i	$0.890575\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	558.319i	1.13711i	0.822647	+	0.568553i	$0.192496\pi$
$491$	558.319i	1.13711i	−0.822647	+	0.568553i	$0.807504\pi$
$492$	0	0
$493$	17.3636	0.0352202
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−157.819	−0.317543
$498$	0	0
$499$	507.059i	1.01615i	0.861313	+	0.508075i	$0.169643\pi$
$499$	507.059i	1.01615i	−0.861313	+	0.508075i	$0.830357\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	884.817i	1.75908i	0.475825	+	0.879540i	$0.342150\pi$
$503$	884.817i	1.75908i	−0.475825	+	0.879540i	$0.657850\pi$
$504$	0	0
$505$	513.083	1.01601
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−398.001	−0.781928	−0.390964	−	0.920406i	$-0.627858\pi$
$509$	−398.001	−0.781928	−0.390964	+	0.920406i	$0.627858\pi$
$510$	0	0
$511$	789.437i	1.54489i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 243.069i	− 0.471978i
$516$	0	0
$517$	−214.031	−0.413987
$518$	0	0
$519$	0	0
$520$	0	0
$521$	716.672	1.37557	0.687785	−	0.725914i	$-0.258583\pi$
$521$	716.672	1.37557	0.687785	+	0.725914i	$0.258583\pi$
$522$	0	0
$523$	706.115i	1.35012i	0.737761	+	0.675062i	$0.235883\pi$
$523$	706.115i	1.35012i	−0.737761	+	0.675062i	$0.764117\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1072.21i	− 2.03456i
$528$	0	0
$529$	344.185	0.650634
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−651.621	−1.22255
$534$	0	0
$535$	− 223.913i	− 0.418529i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	80.6485i	0.149626i
$540$	0	0
$541$	−116.644	−0.215609	−0.107804	−	0.994172i	$-0.534382\pi$
$541$	−116.644	−0.215609	−0.107804	+	0.994172i	$0.534382\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	195.013	0.357822
$546$	0	0
$547$	422.278i	0.771989i	0.922501	+	0.385995i	$0.126142\pi$
$547$	422.278i	0.771989i	−0.922501	+	0.385995i	$0.873858\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 14.1717i	− 0.0257199i
$552$	0	0
$553$	628.562	1.13664
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−686.288	−1.23212	−0.616058	−	0.787701i	$-0.711271\pi$
$557$	−686.288	−1.23212	−0.616058	+	0.787701i	$0.711271\pi$
$558$	0	0
$559$	− 399.149i	− 0.714042i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 439.191i	− 0.780090i	−0.920796	−	0.390045i	$-0.872459\pi$
$563$	− 439.191i	− 0.780090i	0.920796	−	0.390045i	$-0.127541\pi$
$564$	0	0
$565$	−453.769	−0.803130
$566$	0	0
$567$	0	0
$568$	0	0
$569$	406.032	0.713589	0.356794	−	0.934183i	$-0.383870\pi$
$569$	406.032	0.713589	0.356794	+	0.934183i	$0.383870\pi$
$570$	0	0
$571$	− 166.816i	− 0.292148i	−0.989274	−	0.146074i	$-0.953336\pi$
$571$	− 166.816i	− 0.292148i	0.989274	−	0.146074i	$-0.0466637\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	198.463i	0.345153i
$576$	0	0
$577$	−897.895	−1.55614	−0.778072	−	0.628175i	$-0.783802\pi$
$577$	−897.895	−1.55614	−0.778072	+	0.628175i	$0.783802\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−758.734	−1.30591
$582$	0	0
$583$	650.330i	1.11549i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 152.207i	− 0.259296i	−0.991560	−	0.129648i	$-0.958615\pi$
$587$	− 152.207i	− 0.259296i	0.991560	−	0.129648i	$-0.0413847\pi$
$588$	0	0
$589$	−875.110	−1.48576
$590$	0	0
$591$	0	0
$592$	0	0
$593$	964.580	1.62661	0.813305	−	0.581838i	$-0.197666\pi$
$593$	964.580	1.62661	0.813305	+	0.581838i	$0.197666\pi$
$594$	0	0
$595$	489.345i	0.822428i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	983.578i	1.64203i	0.570905	+	0.821016i	$0.306593\pi$
$599$	983.578i	1.64203i	−0.570905	+	0.821016i	$0.693407\pi$
$600$	0	0
$601$	−443.392	−0.737757	−0.368878	−	0.929478i	$-0.620258\pi$
$601$	−443.392	−0.737757	−0.368878	+	0.929478i	$0.620258\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−232.019	−0.383503
$606$	0	0
$607$	487.520i	0.803163i	0.915823	+	0.401582i	$0.131539\pi$
$607$	487.520i	0.803163i	−0.915823	+	0.401582i	$0.868461\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 294.097i	− 0.481337i
$612$	0	0
$613$	−909.815	−1.48420	−0.742100	−	0.670289i	$-0.766170\pi$
$613$	−909.815	−1.48420	−0.742100	+	0.670289i	$0.766170\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	481.557	0.780481	0.390241	−	0.920713i	$-0.372392\pi$
$617$	481.557	0.780481	0.390241	+	0.920713i	$0.372392\pi$
$618$	0	0
$619$	350.103i	0.565594i	0.959180	+	0.282797i	$0.0912623\pi$
$619$	350.103i	0.565594i	−0.959180	+	0.282797i	$0.908738\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1114.57i	1.78904i
$624$	0	0
$625$	−46.9156	−0.0750649
$626$	0	0
$627$	0	0
$628$	0	0
$629$	61.9130	0.0984308
$630$	0	0
$631$	− 274.658i	− 0.435274i	−0.976030	−	0.217637i	$-0.930165\pi$
$631$	− 274.658i	− 0.435274i	0.976030	−	0.217637i	$-0.0698350\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 215.995i	− 0.340149i
$636$	0	0
$637$	−110.818	−0.173968
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−383.085	−0.597636	−0.298818	−	0.954310i	$-0.596592\pi$
$641$	−383.085	−0.597636	−0.298818	+	0.954310i	$0.596592\pi$
$642$	0	0
$643$	177.134i	0.275480i	0.990468	+	0.137740i	$0.0439838\pi$
$643$	177.134i	0.275480i	−0.990468	+	0.137740i	$0.956016\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	877.305i	1.35596i	0.735081	+	0.677979i	$0.237144\pi$
$647$	877.305i	1.35596i	−0.735081	+	0.677979i	$0.762856\pi$
$648$	0	0
$649$	−561.344	−0.864937
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−35.3934	−0.0542012	−0.0271006	−	0.999633i	$-0.508627\pi$
$653$	−35.3934	−0.0542012	−0.0271006	+	0.999633i	$0.508627\pi$
$654$	0	0
$655$	822.884i	1.25631i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	556.337i	0.844214i	0.906546	+	0.422107i	$0.138709\pi$
$659$	556.337i	0.844214i	−0.906546	+	0.422107i	$0.861291\pi$
$660$	0	0
$661$	318.875	0.482413	0.241206	−	0.970474i	$-0.422457\pi$
$661$	318.875	0.482413	0.241206	+	0.970474i	$0.422457\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	399.389	0.600586
$666$	0	0
$667$	10.2247i	0.0153293i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1466.03i	− 2.18484i
$672$	0	0
$673$	135.295	0.201033	0.100517	−	0.994935i	$-0.467950\pi$
$673$	135.295	0.201033	0.100517	+	0.994935i	$0.467950\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	123.989	0.183145	0.0915723	−	0.995798i	$-0.470811\pi$
$677$	123.989	0.183145	0.0915723	+	0.995798i	$0.470811\pi$
$678$	0	0
$679$	611.966i	0.901275i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	609.347i	0.892163i	0.894992	+	0.446081i	$0.147181\pi$
$683$	609.347i	0.892163i	−0.894992	+	0.446081i	$0.852819\pi$
$684$	0	0
$685$	87.7059	0.128038
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−893.609	−1.29696
$690$	0	0
$691$	− 353.682i	− 0.511840i	−0.966698	−	0.255920i	$-0.917622\pi$
$691$	− 353.682i	− 0.511840i	0.966698	−	0.255920i	$-0.0823784\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	657.784i	0.946452i
$696$	0	0
$697$	−788.183	−1.13082
$698$	0	0
$699$	0	0
$700$	0	0
$701$	85.2186	0.121567	0.0607836	−	0.998151i	$-0.480640\pi$
$701$	85.2186	0.121567	0.0607836	+	0.998151i	$0.480640\pi$
$702$	0	0
$703$	− 50.5316i	− 0.0718800i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1045.57i	− 1.47888i
$708$	0	0
$709$	−701.943	−0.990046	−0.495023	−	0.868880i	$-0.664840\pi$
$709$	−701.943	−0.990046	−0.495023	+	0.868880i	$0.664840\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	631.381	0.885527
$714$	0	0
$715$	− 855.036i	− 1.19585i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 993.471i	− 1.38174i	−0.722979	−	0.690870i	$-0.757228\pi$
$719$	− 993.471i	− 1.38174i	0.722979	−	0.690870i	$-0.242772\pi$
$720$	0	0
$721$	−495.330	−0.687004
$722$	0	0
$723$	0	0
$724$	0	0
$725$	10.9798	0.0151445
$726$	0	0
$727$	− 485.145i	− 0.667325i	−0.942693	−	0.333662i	$-0.891715\pi$
$727$	− 485.145i	− 0.667325i	0.942693	−	0.333662i	$-0.108285\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 482.800i	− 0.660465i
$732$	0	0
$733$	706.167	0.963394	0.481697	−	0.876338i	$-0.340021\pi$
$733$	706.167	0.963394	0.481697	+	0.876338i	$0.340021\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−388.053	−0.526531
$738$	0	0
$739$	− 414.250i	− 0.560554i	−0.959919	−	0.280277i	$-0.909574\pi$
$739$	− 414.250i	− 0.560554i	0.959919	−	0.280277i	$-0.0904264\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1386.38i	− 1.86592i	−0.359978	−	0.932961i	$-0.617216\pi$
$743$	− 1386.38i	− 1.86592i	0.359978	−	0.932961i	$-0.382784\pi$
$744$	0	0
$745$	811.141	1.08878
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−456.294	−0.609205
$750$	0	0
$751$	601.365i	0.800752i	0.916351	+	0.400376i	$0.131121\pi$
$751$	601.365i	0.800752i	−0.916351	+	0.400376i	$0.868879\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	807.089i	1.06899i
$756$	0	0
$757$	−254.294	−0.335924	−0.167962	−	0.985794i	$-0.553719\pi$
$757$	−254.294	−0.335924	−0.167962	+	0.985794i	$0.553719\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	466.017	0.612375	0.306187	−	0.951971i	$-0.400947\pi$
$761$	466.017	0.612375	0.306187	+	0.951971i	$0.400947\pi$
$762$	0	0
$763$	− 397.401i	− 0.520840i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 771.334i	− 1.00565i
$768$	0	0
$769$	−365.822	−0.475712	−0.237856	−	0.971300i	$-0.576445\pi$
$769$	−365.822	−0.475712	−0.237856	+	0.971300i	$0.576445\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1426.70	1.84566	0.922830	−	0.385207i	$-0.125870\pi$
$773$	1426.70	1.84566	0.922830	+	0.385207i	$0.125870\pi$
$774$	0	0
$775$	− 678.007i	− 0.874848i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	643.293i	0.825794i
$780$	0	0
$781$	333.549	0.427079
$782$	0	0
$783$	0	0
$784$	0	0
$785$	58.6100	0.0746624
$786$	0	0
$787$	410.337i	0.521394i	0.965421	+	0.260697i	$0.0839523\pi$
$787$	410.337i	0.521394i	−0.965421	+	0.260697i	$0.916048\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	924.698i	1.16902i
$792$	0	0
$793$	2014.45	2.54029
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−738.968	−0.927187	−0.463594	−	0.886048i	$-0.653440\pi$
$797$	−738.968	−0.927187	−0.463594	+	0.886048i	$0.653440\pi$
$798$	0	0
$799$	− 355.731i	− 0.445221i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1668.47i	− 2.07780i
$804$	0	0
$805$	−288.154	−0.357956
$806$	0	0
$807$	0	0
$808$	0	0
$809$	130.444	0.161241	0.0806204	−	0.996745i	$-0.474310\pi$
$809$	130.444	0.161241	0.0806204	+	0.996745i	$0.474310\pi$
$810$	0	0
$811$	1558.44i	1.92163i	0.277187	+	0.960816i	$0.410598\pi$
$811$	1558.44i	1.92163i	−0.277187	+	0.960816i	$0.589402\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 161.314i	− 0.197931i
$816$	0	0
$817$	−394.048	−0.482311
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−55.0345	−0.0670335	−0.0335168	−	0.999438i	$-0.510671\pi$
$821$	−55.0345	−0.0670335	−0.0335168	+	0.999438i	$0.510671\pi$
$822$	0	0
$823$	− 293.058i	− 0.356085i	−0.984023	−	0.178043i	$-0.943023\pi$
$823$	− 293.058i	− 0.356085i	0.984023	−	0.178043i	$-0.0569765\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1118.55i	1.35253i	0.736657	+	0.676267i	$0.236403\pi$
$827$	1118.55i	1.35253i	−0.736657	+	0.676267i	$0.763597\pi$
$828$	0	0
$829$	1069.30	1.28986	0.644932	−	0.764240i	$-0.276885\pi$
$829$	1069.30	1.28986	0.644932	+	0.764240i	$0.276885\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−134.042	−0.160915
$834$	0	0
$835$	273.616i	0.327683i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 41.6050i	− 0.0495888i	−0.999693	−	0.0247944i	$-0.992107\pi$
$839$	− 41.6050i	− 0.0495888i	0.999693	−	0.0247944i	$-0.00789311\pi$
$840$	0	0
$841$	−840.434	−0.999327
$842$	0	0
$843$	0	0
$844$	0	0
$845$	629.846	0.745379
$846$	0	0
$847$	472.813i	0.558220i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	36.4579i	0.0428413i
$852$	0	0
$853$	1302.63	1.52711	0.763555	−	0.645743i	$-0.223452\pi$
$853$	1302.63	1.52711	0.763555	+	0.645743i	$0.223452\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−734.683	−0.857273	−0.428636	−	0.903477i	$-0.641006\pi$
$857$	−734.683	−0.857273	−0.428636	+	0.903477i	$0.641006\pi$
$858$	0	0
$859$	445.296i	0.518388i	0.965825	+	0.259194i	$0.0834570\pi$
$859$	445.296i	0.518388i	−0.965825	+	0.259194i	$0.916543\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1338.23i	1.55067i	0.631550	+	0.775335i	$0.282419\pi$
$863$	1338.23i	1.55067i	−0.631550	+	0.775335i	$0.717581\pi$
$864$	0	0
$865$	466.684	0.539519
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1328.46	−1.52873
$870$	0	0
$871$	− 533.218i	− 0.612190i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	839.338i	0.959244i
$876$	0	0
$877$	−329.803	−0.376058	−0.188029	−	0.982163i	$-0.560210\pi$
$877$	−329.803	−0.376058	−0.188029	+	0.982163i	$0.560210\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1173.32	1.33181	0.665903	−	0.746038i	$-0.268046\pi$
$881$	1173.32	1.33181	0.665903	+	0.746038i	$0.268046\pi$
$882$	0	0
$883$	− 410.459i	− 0.464846i	−0.972615	−	0.232423i	$-0.925335\pi$
$883$	− 410.459i	− 0.464846i	0.972615	−	0.232423i	$-0.0746654\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	604.280i	0.681263i	0.940197	+	0.340631i	$0.110641\pi$
$887$	604.280i	0.681263i	−0.940197	+	0.340631i	$0.889359\pi$
$888$	0	0
$889$	−440.158	−0.495116
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−290.338	−0.325127
$894$	0	0
$895$	− 906.020i	− 1.01231i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 34.9305i	− 0.0388548i
$900$	0	0
$901$	−1080.88	−1.19965
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−953.048	−1.05309
$906$	0	0
$907$	− 735.914i	− 0.811371i	−0.914013	−	0.405686i	$-0.867033\pi$
$907$	− 735.914i	− 0.811371i	0.914013	−	0.405686i	$-0.132967\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 586.390i	− 0.643677i	−0.946795	−	0.321838i	$-0.895699\pi$
$911$	− 586.390i	− 0.643677i	0.946795	−	0.321838i	$-0.104301\pi$
$912$	0	0
$913$	1603.58	1.75639
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1676.89	1.82867
$918$	0	0
$919$	158.999i	0.173013i	0.996251	+	0.0865063i	$0.0275703\pi$
$919$	158.999i	0.173013i	−0.996251	+	0.0865063i	$0.972430\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	458.324i	0.496560i
$924$	0	0
$925$	39.1503	0.0423247
$926$	0	0
$927$	0	0
$928$	0	0
$929$	540.670	0.581992	0.290996	−	0.956724i	$-0.406013\pi$
$929$	540.670	0.581992	0.290996	+	0.956724i	$0.406013\pi$
$930$	0	0
$931$	109.402i	0.117510i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 1034.23i	− 1.10613i
$936$	0	0
$937$	384.947	0.410829	0.205415	−	0.978675i	$-0.434146\pi$
$937$	384.947	0.410829	0.205415	+	0.978675i	$0.434146\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−39.6287	−0.0421134	−0.0210567	−	0.999778i	$-0.506703\pi$
$941$	−39.6287	−0.0421134	−0.0210567	+	0.999778i	$0.506703\pi$
$942$	0	0
$943$	− 464.128i	− 0.492182i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 449.648i	− 0.474813i	−0.971410	−	0.237407i	$-0.923703\pi$
$947$	− 449.648i	− 0.474813i	0.971410	−	0.237407i	$-0.0762974\pi$
$948$	0	0
$949$	2292.62	2.41583
$950$	0	0
$951$	0	0
$952$	0	0
$953$	21.5070	0.0225676	0.0112838	−	0.999936i	$-0.496408\pi$
$953$	21.5070	0.0225676	0.0112838	+	0.999936i	$0.496408\pi$
$954$	0	0
$955$	− 954.578i	− 0.999558i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 178.729i	− 0.186370i
$960$	0	0
$961$	−1195.98	−1.24452
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−58.1172	−0.0602251
$966$	0	0
$967$	− 1355.49i	− 1.40175i	−0.713283	−	0.700876i	$-0.752793\pi$
$967$	− 1355.49i	− 1.40175i	0.713283	−	0.700876i	$-0.247207\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	938.488i	0.966517i	0.875478	+	0.483258i	$0.160547\pi$
$971$	938.488i	0.966517i	−0.875478	+	0.483258i	$0.839453\pi$
$972$	0	0
$973$	1340.44	1.37764
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1090.81	−1.11649	−0.558243	−	0.829678i	$-0.688524\pi$
$977$	−1090.81	−1.11649	−0.558243	+	0.829678i	$0.688524\pi$
$978$	0	0
$979$	− 2355.64i	− 2.40617i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 27.7492i	− 0.0282291i	−0.999900	−	0.0141146i	$-0.995507\pi$
$983$	− 27.7492i	− 0.0282291i	0.999900	−	0.0141146i	$-0.00449296\pi$
$984$	0	0
$985$	−383.853	−0.389699
$986$	0	0
$987$	0	0
$988$	0	0
$989$	284.301	0.287463
$990$	0	0
$991$	− 272.858i	− 0.275336i	−0.990478	−	0.137668i	$-0.956039\pi$
$991$	− 272.858i	− 0.275336i	0.990478	−	0.137668i	$-0.0439607\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 983.378i	− 0.988320i
$996$	0	0
$997$	1190.35	1.19393	0.596964	−	0.802268i	$-0.296374\pi$
$997$	1190.35	1.19393	0.596964	+	0.802268i	$0.296374\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.3.g.a.703.5	✓	8	1.1	even	1	trivial
864.3.g.a.703.6	yes	8	4.3	odd	2	inner
864.3.g.c.703.3	yes	8	3.2	odd	2
864.3.g.c.703.4	yes	8	12.11	even	2
1728.3.g.k.703.5		8	24.5	odd	2
1728.3.g.k.703.6		8	24.11	even	2
1728.3.g.n.703.3		8	8.5	even	2
1728.3.g.n.703.4		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)

\(f(q)\)	\(=\)	\(q+3.22512 q^{5} -6.57221i q^{7} +O(q^{10})\)	\(q+3.22512 q^{5} -6.57221i q^{7} +13.8903i q^{11} -19.0865 q^{13} -23.0865 q^{17} +18.8425i q^{19} -13.5947i q^{23} -14.5986 q^{25} -0.752110 q^{29} +46.4433i q^{31} -21.1962i q^{35} -2.68178 q^{37} +34.1405 q^{41} +20.9127i q^{43} +15.4086i q^{47} +5.80609 q^{49} +46.8189 q^{53} +44.7980i q^{55} +40.4126i q^{59} -105.543 q^{61} -61.5562 q^{65} +27.9369i q^{67} -24.0130i q^{71} -120.117 q^{73} +91.2901 q^{77} +95.6394i q^{79} -115.446i q^{83} -74.4567 q^{85} -169.589 q^{89} +125.440i q^{91} +60.7694i q^{95} -93.1142 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^5	\( 8 q - 8 q^{5} - 16 q^{13} - 48 q^{17} + 48 q^{25} + 32 q^{29} - 96 q^{37} + 128 q^{41} - 168 q^{53} + 32 q^{61} + 112 q^{65} + 24 q^{73} + 440 q^{77} + 144 q^{85} - 624 q^{89} - 136 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 - 16 * q^13 - 48 * q^17 + 48 * q^25 + 32 * q^29 - 96 * q^37 + 128 * q^41 - 168 * q^53 + 32 * q^61 + 112 * q^65 + 24 * q^73 + 440 * q^77 + 144 * q^85 - 624 * q^89 - 136 * q^97

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