LMFDB - Embedded newform 864.3.b.a.271.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,3,Mod(271,864)]

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, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("864.271");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	864.b (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:	$23.5422948407$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + x^{14} - 8x^{12} + 4x^{10} + 160x^{8} + 64x^{6} - 2048x^{4} + 4096x^{2} + 65536 $ x^16 + x^14 - 8x^12 + 4x^10 + 160x^8 + 64x^6 - 2048x^4 + 4096x^2 + 65536
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{10} $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.4
Root			$-1.95589 - 0.417734i$ of defining polynomial
Character	$\chi$	$=$	864.271
Dual form			864.3.b.a.271.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.14075i q^{5} +12.6525i q^{7} +O(q^{10})$	$q-5.14075i q^{5} +12.6525i q^{7} +1.46281 q^{11} +5.68201i q^{13} -14.2764 q^{17} -26.6211 q^{19} -36.7058i q^{23} -1.42727 q^{25} +19.4918i q^{29} -16.1884i q^{31} +65.0432 q^{35} -37.2185i q^{37} -58.8886 q^{41} -61.2705 q^{43} +61.6746i q^{47} -111.085 q^{49} +42.0953i q^{53} -7.51995i q^{55} -2.74134 q^{59} +71.6574i q^{61} +29.2098 q^{65} +10.3192 q^{67} -70.6220i q^{71} -104.912 q^{73} +18.5082i q^{77} +82.0064i q^{79} -68.7766 q^{83} +73.3915i q^{85} +65.6313 q^{89} -71.8916 q^{91} +136.852i q^{95} +5.31919 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q - 32 q^{19} - 80 q^{25} - 128 q^{43} - 80 q^{49} - 128 q^{67} - 160 q^{73} + 96 q^{91} - 208 q^{97}+O(q^{100}) $ 16 * q - 32 * q^19 - 80 * q^25 - 128 * q^43 - 80 * q^49 - 128 * q^67 - 160 * q^73 + 96 * q^91 - 208 * 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
$5$	− 5.14075i	− 1.02815i	−0.857745	−	0.514075i	$-0.828135\pi$
$5$	− 5.14075i	− 1.02815i	0.857745	−	0.514075i	$-0.171865\pi$
$6$	0	0
$7$	12.6525i	1.80750i	0.428063	+	0.903749i	$0.359196\pi$
$7$	12.6525i	1.80750i	−0.428063	+	0.903749i	$0.640804\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.46281	0.132983	0.0664915	−	0.997787i	$-0.478819\pi$
$11$	1.46281	0.132983	0.0664915	+	0.997787i	$0.478819\pi$
$12$	0	0
$13$	5.68201i	0.437078i	0.975828	+	0.218539i	$0.0701291\pi$
$13$	5.68201i	0.437078i	−0.975828	+	0.218539i	$0.929871\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−14.2764	−0.839790	−0.419895	−	0.907573i	$-0.637933\pi$
$17$	−14.2764	−0.839790	−0.419895	+	0.907573i	$0.637933\pi$
$18$	0	0
$19$	−26.6211	−1.40111	−0.700556	−	0.713598i	$-0.747064\pi$
$19$	−26.6211	−1.40111	−0.700556	+	0.713598i	$0.747064\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 36.7058i	− 1.59590i	−0.602721	−	0.797952i	$-0.705917\pi$
$23$	− 36.7058i	− 1.59590i	0.602721	−	0.797952i	$-0.294083\pi$
$24$	0	0
$25$	−1.42727	−0.0570909
$26$	0	0
$27$	0	0
$28$	0	0
$29$	19.4918i	0.672133i	0.941838	+	0.336066i	$0.109097\pi$
$29$	19.4918i	0.672133i	−0.941838	+	0.336066i	$0.890903\pi$
$30$	0	0
$31$	− 16.1884i	− 0.522206i	−0.965311	−	0.261103i	$-0.915914\pi$
$31$	− 16.1884i	− 0.522206i	0.965311	−	0.261103i	$-0.0840863\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	65.0432	1.85838
$36$	0	0
$37$	− 37.2185i	− 1.00590i	−0.864314	−	0.502952i	$-0.832247\pi$
$37$	− 37.2185i	− 1.00590i	0.864314	−	0.502952i	$-0.167753\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−58.8886	−1.43631	−0.718153	−	0.695885i	$-0.755012\pi$
$41$	−58.8886	−1.43631	−0.718153	+	0.695885i	$0.755012\pi$
$42$	0	0
$43$	−61.2705	−1.42490	−0.712448	−	0.701725i	$-0.752413\pi$
$43$	−61.2705	−1.42490	−0.712448	+	0.701725i	$0.752413\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	61.6746i	1.31223i	0.754662	+	0.656113i	$0.227801\pi$
$47$	61.6746i	1.31223i	−0.754662	+	0.656113i	$0.772199\pi$
$48$	0	0
$49$	−111.085	−2.26705
$50$	0	0
$51$	0	0
$52$	0	0
$53$	42.0953i	0.794251i	0.917764	+	0.397125i	$0.129992\pi$
$53$	42.0953i	0.794251i	−0.917764	+	0.397125i	$0.870008\pi$
$54$	0	0
$55$	− 7.51995i	− 0.136726i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.74134	−0.0464635	−0.0232317	−	0.999730i	$-0.507396\pi$
$59$	−2.74134	−0.0464635	−0.0232317	+	0.999730i	$0.507396\pi$
$60$	0	0
$61$	71.6574i	1.17471i	0.809329	+	0.587355i	$0.199831\pi$
$61$	71.6574i	1.17471i	−0.809329	+	0.587355i	$0.800169\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	29.2098	0.449381
$66$	0	0
$67$	10.3192	0.154018	0.0770088	−	0.997030i	$-0.475463\pi$
$67$	10.3192	0.154018	0.0770088	+	0.997030i	$0.475463\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 70.6220i	− 0.994676i	−0.867557	−	0.497338i	$-0.834311\pi$
$71$	− 70.6220i	− 0.994676i	0.867557	−	0.497338i	$-0.165689\pi$
$72$	0	0
$73$	−104.912	−1.43715	−0.718573	−	0.695451i	$-0.755205\pi$
$73$	−104.912	−1.43715	−0.718573	+	0.695451i	$0.755205\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.5082i	0.240367i
$78$	0	0
$79$	82.0064i	1.03806i	0.854757	+	0.519028i	$0.173706\pi$
$79$	82.0064i	1.03806i	−0.854757	+	0.519028i	$0.826294\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−68.7766	−0.828633	−0.414317	−	0.910133i	$-0.635979\pi$
$83$	−68.7766	−0.828633	−0.414317	+	0.910133i	$0.635979\pi$
$84$	0	0
$85$	73.3915i	0.863430i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	65.6313	0.737431	0.368715	−	0.929542i	$-0.379798\pi$
$89$	65.6313	0.737431	0.368715	+	0.929542i	$0.379798\pi$
$90$	0	0
$91$	−71.8916	−0.790018
$92$	0	0
$93$	0	0
$94$	0	0
$95$	136.852i	1.44055i
$96$	0	0
$97$	5.31919	0.0548370	0.0274185	−	0.999624i	$-0.491271\pi$
$97$	5.31919	0.0548370	0.0274185	+	0.999624i	$0.491271\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 132.472i	− 1.31161i	−0.754932	−	0.655803i	$-0.772330\pi$
$101$	− 132.472i	− 1.31161i	0.754932	−	0.655803i	$-0.227670\pi$
$102$	0	0
$103$	− 7.79346i	− 0.0756647i	−0.999284	−	0.0378323i	$-0.987955\pi$
$103$	− 7.79346i	− 0.0756647i	0.999284	−	0.0378323i	$-0.0120453\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	70.3416	0.657398	0.328699	−	0.944435i	$-0.393390\pi$
$107$	70.3416	0.657398	0.328699	+	0.944435i	$0.393390\pi$
$108$	0	0
$109$	131.604i	1.20738i	0.797221	+	0.603688i	$0.206303\pi$
$109$	131.604i	1.20738i	−0.797221	+	0.603688i	$0.793697\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−13.0165	−0.115191	−0.0575953	−	0.998340i	$-0.518343\pi$
$113$	−13.0165	−0.115191	−0.0575953	+	0.998340i	$0.518343\pi$
$114$	0	0
$115$	−188.695	−1.64083
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 180.632i	− 1.51792i
$120$	0	0
$121$	−118.860	−0.982316
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 121.181i	− 0.969451i
$126$	0	0
$127$	− 2.36611i	− 0.0186308i	−0.999957	−	0.00931539i	$-0.997035\pi$
$127$	− 2.36611i	− 0.0186308i	0.999957	−	0.00931539i	$-0.00296522\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−236.933	−1.80865	−0.904326	−	0.426842i	$-0.859626\pi$
$131$	−236.933	−1.80865	−0.904326	+	0.426842i	$0.859626\pi$
$132$	0	0
$133$	− 336.823i	− 2.53251i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−60.8593	−0.444229	−0.222114	−	0.975021i	$-0.571296\pi$
$137$	−60.8593	−0.444229	−0.222114	+	0.975021i	$0.571296\pi$
$138$	0	0
$139$	39.5897	0.284818	0.142409	−	0.989808i	$-0.454515\pi$
$139$	39.5897	0.284818	0.142409	+	0.989808i	$0.454515\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.31172i	0.0581239i
$144$	0	0
$145$	100.203	0.691053
$146$	0	0
$147$	0	0
$148$	0	0
$149$	146.973i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$149$	146.973i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$150$	0	0
$151$	− 62.7781i	− 0.415749i	−0.978155	−	0.207875i	$-0.933345\pi$
$151$	− 62.7781i	− 0.415749i	0.978155	−	0.207875i	$-0.0666546\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−83.2204	−0.536906
$156$	0	0
$157$	− 82.5467i	− 0.525775i	−0.964826	−	0.262888i	$-0.915325\pi$
$157$	− 82.5467i	− 0.525775i	0.964826	−	0.262888i	$-0.0846749\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	464.420	2.88459
$162$	0	0
$163$	−29.4072	−0.180412	−0.0902060	−	0.995923i	$-0.528753\pi$
$163$	−29.4072	−0.180412	−0.0902060	+	0.995923i	$0.528753\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	92.9654i	0.556679i	0.960483	+	0.278339i	$0.0897840\pi$
$167$	92.9654i	0.556679i	−0.960483	+	0.278339i	$0.910216\pi$
$168$	0	0
$169$	136.715	0.808963
$170$	0	0
$171$	0	0
$172$	0	0
$173$	48.3804i	0.279655i	0.990176	+	0.139828i	$0.0446549\pi$
$173$	48.3804i	0.279655i	−0.990176	+	0.139828i	$0.955345\pi$
$174$	0	0
$175$	− 18.0585i	− 0.103192i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	148.467	0.829422	0.414711	−	0.909953i	$-0.363883\pi$
$179$	148.467	0.829422	0.414711	+	0.909953i	$0.363883\pi$
$180$	0	0
$181$	− 94.3854i	− 0.521466i	−0.965411	−	0.260733i	$-0.916036\pi$
$181$	− 94.3854i	− 0.521466i	0.965411	−	0.260733i	$-0.0839643\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−191.331	−1.03422
$186$	0	0
$187$	−20.8838	−0.111678
$188$	0	0
$189$	0	0
$190$	0	0
$191$	198.584i	1.03971i	0.854255	+	0.519854i	$0.174014\pi$
$191$	198.584i	1.03971i	−0.854255	+	0.519854i	$0.825986\pi$
$192$	0	0
$193$	−2.71819	−0.0140839	−0.00704193	−	0.999975i	$-0.502242\pi$
$193$	−2.71819	−0.0140839	−0.00704193	+	0.999975i	$0.502242\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 197.266i	− 1.00135i	−0.865635	−	0.500675i	$-0.833085\pi$
$197$	− 197.266i	− 1.00135i	0.865635	−	0.500675i	$-0.166915\pi$
$198$	0	0
$199$	275.188i	1.38285i	0.722446	+	0.691427i	$0.243018\pi$
$199$	275.188i	1.38285i	−0.722446	+	0.691427i	$0.756982\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−246.620	−1.21488
$204$	0	0
$205$	302.731i	1.47674i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−38.9417	−0.186324
$210$	0	0
$211$	301.299	1.42796	0.713979	−	0.700167i	$-0.246891\pi$
$211$	301.299	1.42796	0.713979	+	0.700167i	$0.246891\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	314.976i	1.46500i
$216$	0	0
$217$	204.823	0.943887
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 81.1189i	− 0.367054i
$222$	0	0
$223$	− 171.918i	− 0.770934i	−0.922722	−	0.385467i	$-0.874040\pi$
$223$	− 171.918i	− 0.770934i	0.922722	−	0.385467i	$-0.125960\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	240.769	1.06066	0.530329	−	0.847792i	$-0.322069\pi$
$227$	240.769	1.06066	0.530329	+	0.847792i	$0.322069\pi$
$228$	0	0
$229$	416.468i	1.81864i	0.416102	+	0.909318i	$0.363396\pi$
$229$	416.468i	1.81864i	−0.416102	+	0.909318i	$0.636604\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	39.4647	0.169376	0.0846881	−	0.996408i	$-0.473011\pi$
$233$	39.4647	0.169376	0.0846881	+	0.996408i	$0.473011\pi$
$234$	0	0
$235$	317.054	1.34916
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 390.425i	− 1.63358i	−0.576937	−	0.816789i	$-0.695752\pi$
$239$	− 390.425i	− 1.63358i	0.576937	−	0.816789i	$-0.304248\pi$
$240$	0	0
$241$	107.903	0.447729	0.223864	−	0.974620i	$-0.428133\pi$
$241$	107.903	0.447729	0.223864	+	0.974620i	$0.428133\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	571.062i	2.33087i
$246$	0	0
$247$	− 151.262i	− 0.612395i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	239.656	0.954805	0.477403	−	0.878685i	$-0.341578\pi$
$251$	239.656	0.954805	0.477403	+	0.878685i	$0.341578\pi$
$252$	0	0
$253$	− 53.6937i	− 0.212228i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−219.964	−0.855893	−0.427946	−	0.903804i	$-0.640763\pi$
$257$	−219.964	−0.855893	−0.427946	+	0.903804i	$0.640763\pi$
$258$	0	0
$259$	470.906	1.81817
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 59.9658i	− 0.228007i	−0.993480	−	0.114003i	$-0.963633\pi$
$263$	− 59.9658i	− 0.228007i	0.993480	−	0.114003i	$-0.0363675\pi$
$264$	0	0
$265$	216.401	0.816608
$266$	0	0
$267$	0	0
$268$	0	0
$269$	216.414i	0.804515i	0.915527	+	0.402257i	$0.131774\pi$
$269$	216.414i	0.804515i	−0.915527	+	0.402257i	$0.868226\pi$
$270$	0	0
$271$	− 153.616i	− 0.566849i	−0.958995	−	0.283424i	$-0.908530\pi$
$271$	− 153.616i	− 0.566849i	0.958995	−	0.283424i	$-0.0914705\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.08783	−0.00759212
$276$	0	0
$277$	− 267.370i	− 0.965234i	−0.875832	−	0.482617i	$-0.839686\pi$
$277$	− 267.370i	− 0.965234i	0.875832	−	0.482617i	$-0.160314\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−485.895	−1.72916	−0.864582	−	0.502492i	$-0.832416\pi$
$281$	−485.895	−1.72916	−0.864582	+	0.502492i	$0.832416\pi$
$282$	0	0
$283$	74.3478	0.262713	0.131357	−	0.991335i	$-0.458067\pi$
$283$	74.3478	0.262713	0.131357	+	0.991335i	$0.458067\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 745.087i	− 2.59612i
$288$	0	0
$289$	−85.1835	−0.294752
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 468.531i	− 1.59908i	−0.600612	−	0.799540i	$-0.705076\pi$
$293$	− 468.531i	− 1.59908i	0.600612	−	0.799540i	$-0.294924\pi$
$294$	0	0
$295$	14.0926i	0.0477714i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	208.563	0.697535
$300$	0	0
$301$	− 775.224i	− 2.57550i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	368.372	1.20778
$306$	0	0
$307$	−58.3993	−0.190226	−0.0951129	−	0.995466i	$-0.530321\pi$
$307$	−58.3993	−0.190226	−0.0951129	+	0.995466i	$0.530321\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	268.018i	0.861796i	0.902401	+	0.430898i	$0.141803\pi$
$311$	268.018i	0.861796i	−0.902401	+	0.430898i	$0.858197\pi$
$312$	0	0
$313$	−78.0050	−0.249217	−0.124609	−	0.992206i	$-0.539768\pi$
$313$	−78.0050	−0.249217	−0.124609	+	0.992206i	$0.539768\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 262.045i	− 0.826639i	−0.910586	−	0.413320i	$-0.864369\pi$
$317$	− 262.045i	− 0.826639i	0.910586	−	0.413320i	$-0.135631\pi$
$318$	0	0
$319$	28.5129i	0.0893822i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	380.055	1.17664
$324$	0	0
$325$	− 8.10978i	− 0.0249532i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−780.338	−2.37185
$330$	0	0
$331$	279.482	0.844355	0.422178	−	0.906513i	$-0.361266\pi$
$331$	279.482	0.844355	0.422178	+	0.906513i	$0.361266\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 53.0483i	− 0.158353i
$336$	0	0
$337$	552.770	1.64027	0.820133	−	0.572173i	$-0.193899\pi$
$337$	552.770	1.64027	0.820133	+	0.572173i	$0.193899\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 23.6806i	− 0.0694446i
$342$	0	0
$343$	− 785.535i	− 2.29019i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−210.871	−0.607697	−0.303848	−	0.952720i	$-0.598272\pi$
$347$	−210.871	−0.607697	−0.303848	+	0.952720i	$0.598272\pi$
$348$	0	0
$349$	306.200i	0.877363i	0.898643	+	0.438681i	$0.144554\pi$
$349$	306.200i	0.877363i	−0.898643	+	0.438681i	$0.855446\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	359.324	1.01791	0.508957	−	0.860792i	$-0.330031\pi$
$353$	359.324	1.01791	0.508957	+	0.860792i	$0.330031\pi$
$354$	0	0
$355$	−363.050	−1.02267
$356$	0	0
$357$	0	0
$358$	0	0
$359$	336.189i	0.936459i	0.883607	+	0.468230i	$0.155108\pi$
$359$	336.189i	0.936459i	−0.883607	+	0.468230i	$0.844892\pi$
$360$	0	0
$361$	347.684	0.963113
$362$	0	0
$363$	0	0
$364$	0	0
$365$	539.325i	1.47760i
$366$	0	0
$367$	462.021i	1.25891i	0.777036	+	0.629456i	$0.216722\pi$
$367$	462.021i	1.25891i	−0.777036	+	0.629456i	$0.783278\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−532.610	−1.43561
$372$	0	0
$373$	201.874i	0.541217i	0.962689	+	0.270608i	$0.0872249\pi$
$373$	201.874i	0.541217i	−0.962689	+	0.270608i	$0.912775\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−110.753	−0.293774
$378$	0	0
$379$	328.194	0.865946	0.432973	−	0.901407i	$-0.357465\pi$
$379$	328.194	0.865946	0.432973	+	0.901407i	$0.357465\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	57.5543i	0.150272i	0.997173	+	0.0751362i	$0.0239391\pi$
$383$	57.5543i	0.150272i	−0.997173	+	0.0751362i	$0.976061\pi$
$384$	0	0
$385$	95.1461	0.247133
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 528.085i	− 1.35755i	−0.734349	−	0.678773i	$-0.762512\pi$
$389$	− 528.085i	− 1.35755i	0.734349	−	0.678773i	$-0.237488\pi$
$390$	0	0
$391$	524.028i	1.34023i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	421.574	1.06728
$396$	0	0
$397$	− 151.307i	− 0.381125i	−0.981675	−	0.190562i	$-0.938969\pi$
$397$	− 151.307i	− 0.381125i	0.981675	−	0.190562i	$-0.0610312\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	730.954	1.82283	0.911413	−	0.411492i	$-0.134992\pi$
$401$	730.954	1.82283	0.911413	+	0.411492i	$0.134992\pi$
$402$	0	0
$403$	91.9827	0.228245
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 54.4437i	− 0.133768i
$408$	0	0
$409$	314.720	0.769487	0.384744	−	0.923023i	$-0.374290\pi$
$409$	314.720	0.769487	0.384744	+	0.923023i	$0.374290\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 34.6848i	− 0.0839826i
$414$	0	0
$415$	353.563i	0.851959i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	114.634	0.273589	0.136794	−	0.990599i	$-0.456320\pi$
$419$	114.634	0.273589	0.136794	+	0.990599i	$0.456320\pi$
$420$	0	0
$421$	530.807i	1.26082i	0.776261	+	0.630412i	$0.217114\pi$
$421$	530.807i	1.26082i	−0.776261	+	0.630412i	$0.782886\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	20.3763	0.0479443
$426$	0	0
$427$	−906.644	−2.12329
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 91.4566i	− 0.212196i	−0.994356	−	0.106098i	$-0.966164\pi$
$431$	− 91.4566i	− 0.212196i	0.994356	−	0.106098i	$-0.0338358\pi$
$432$	0	0
$433$	−449.530	−1.03817	−0.519087	−	0.854721i	$-0.673728\pi$
$433$	−449.530	−1.03817	−0.519087	+	0.854721i	$0.673728\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	977.149i	2.23604i
$438$	0	0
$439$	59.9309i	0.136517i	0.997668	+	0.0682585i	$0.0217442\pi$
$439$	59.9309i	0.136517i	−0.997668	+	0.0682585i	$0.978256\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−332.627	−0.750851	−0.375425	−	0.926853i	$-0.622503\pi$
$443$	−332.627	−0.750851	−0.375425	+	0.926853i	$0.622503\pi$
$444$	0	0
$445$	− 337.394i	− 0.758189i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−204.039	−0.454431	−0.227215	−	0.973845i	$-0.572962\pi$
$449$	−204.039	−0.454431	−0.227215	+	0.973845i	$0.572962\pi$
$450$	0	0
$451$	−86.1430	−0.191004
$452$	0	0
$453$	0	0
$454$	0	0
$455$	369.577i	0.812256i
$456$	0	0
$457$	−483.610	−1.05823	−0.529114	−	0.848551i	$-0.677476\pi$
$457$	−483.610	−1.05823	−0.529114	+	0.848551i	$0.677476\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 75.8364i	− 0.164504i	−0.996612	−	0.0822521i	$-0.973789\pi$
$461$	− 75.8364i	− 0.164504i	0.996612	−	0.0822521i	$-0.0262113\pi$
$462$	0	0
$463$	55.9734i	0.120893i	0.998171	+	0.0604464i	$0.0192524\pi$
$463$	55.9734i	0.120893i	−0.998171	+	0.0604464i	$0.980748\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−519.757	−1.11297	−0.556485	−	0.830857i	$-0.687850\pi$
$467$	−519.757	−1.11297	−0.556485	+	0.830857i	$0.687850\pi$
$468$	0	0
$469$	130.563i	0.278387i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−89.6273	−0.189487
$474$	0	0
$475$	37.9956	0.0799907
$476$	0	0
$477$	0	0
$478$	0	0
$479$	381.514i	0.796479i	0.917281	+	0.398240i	$0.130379\pi$
$479$	381.514i	0.796479i	−0.917281	+	0.398240i	$0.869621\pi$
$480$	0	0
$481$	211.476	0.439659
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 27.3446i	− 0.0563806i
$486$	0	0
$487$	593.446i	1.21858i	0.792949	+	0.609288i	$0.208544\pi$
$487$	593.446i	1.21858i	−0.792949	+	0.609288i	$0.791456\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	125.688	0.255984	0.127992	−	0.991775i	$-0.459147\pi$
$491$	125.688	0.255984	0.127992	+	0.991775i	$0.459147\pi$
$492$	0	0
$493$	− 278.274i	− 0.564450i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	893.544	1.79787
$498$	0	0
$499$	470.104	0.942093	0.471046	−	0.882108i	$-0.343876\pi$
$499$	470.104	0.942093	0.471046	+	0.882108i	$0.343876\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	21.6483i	0.0430384i	0.999768	+	0.0215192i	$0.00685030\pi$
$503$	21.6483i	0.0430384i	−0.999768	+	0.0215192i	$0.993150\pi$
$504$	0	0
$505$	−681.006	−1.34853
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 612.338i	− 1.20302i	−0.798865	−	0.601510i	$-0.794566\pi$
$509$	− 612.338i	− 1.20302i	0.798865	−	0.601510i	$-0.205434\pi$
$510$	0	0
$511$	− 1327.39i	− 2.59764i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−40.0642	−0.0777946
$516$	0	0
$517$	90.2185i	0.174504i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−449.648	−0.863047	−0.431524	−	0.902102i	$-0.642024\pi$
$521$	−449.648	−0.863047	−0.431524	+	0.902102i	$0.642024\pi$
$522$	0	0
$523$	−314.249	−0.600859	−0.300429	−	0.953804i	$-0.597130\pi$
$523$	−314.249	−0.600859	−0.300429	+	0.953804i	$0.597130\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	231.113i	0.438544i
$528$	0	0
$529$	−818.316	−1.54691
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 334.606i	− 0.627778i
$534$	0	0
$535$	− 361.609i	− 0.675904i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−162.497	−0.301479
$540$	0	0
$541$	− 0.255704i	0 0.000472651i	−1.00000		0.000236326i	$-0.999925\pi$
$541$	− 0.255704i	0 0.000472651i	1.00000		0.000236326i	$-7.52247e-5\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	676.542	1.24136
$546$	0	0
$547$	−341.864	−0.624980	−0.312490	−	0.949921i	$-0.601163\pi$
$547$	−341.864	−0.624980	−0.312490	+	0.949921i	$0.601163\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 518.895i	− 0.941733i
$552$	0	0
$553$	−1037.59	−1.87628
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 261.620i	− 0.469694i	−0.972032	−	0.234847i	$-0.924541\pi$
$557$	− 261.620i	− 0.469694i	0.972032	−	0.234847i	$-0.0754589\pi$
$558$	0	0
$559$	− 348.140i	− 0.622790i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−354.965	−0.630489	−0.315244	−	0.949011i	$-0.602087\pi$
$563$	−354.965	−0.630489	−0.315244	+	0.949011i	$0.602087\pi$
$564$	0	0
$565$	66.9147i	0.118433i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−528.724	−0.929216	−0.464608	−	0.885517i	$-0.653805\pi$
$569$	−528.724	−0.929216	−0.464608	+	0.885517i	$0.653805\pi$
$570$	0	0
$571$	−500.724	−0.876924	−0.438462	−	0.898750i	$-0.644477\pi$
$571$	−500.724	−0.876924	−0.438462	+	0.898750i	$0.644477\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	52.3892i	0.0911116i
$576$	0	0
$577$	308.270	0.534263	0.267131	−	0.963660i	$-0.413924\pi$
$577$	308.270	0.534263	0.267131	+	0.963660i	$0.413924\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 870.195i	− 1.49775i
$582$	0	0
$583$	61.5775i	0.105622i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−434.777	−0.740676	−0.370338	−	0.928897i	$-0.620758\pi$
$587$	−434.777	−0.740676	−0.370338	+	0.928897i	$0.620758\pi$
$588$	0	0
$589$	430.953i	0.731669i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1007.42	1.69885	0.849427	−	0.527707i	$-0.176948\pi$
$593$	1007.42	1.69885	0.849427	+	0.527707i	$0.176948\pi$
$594$	0	0
$595$	−928.585	−1.56065
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1057.10i	− 1.76477i	−0.470525	−	0.882387i	$-0.655935\pi$
$599$	− 1057.10i	− 1.76477i	0.470525	−	0.882387i	$-0.344065\pi$
$600$	0	0
$601$	−122.611	−0.204011	−0.102006	−	0.994784i	$-0.532526\pi$
$601$	−122.611	−0.204011	−0.102006	+	0.994784i	$0.532526\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	611.030i	1.00997i
$606$	0	0
$607$	204.786i	0.337374i	0.985670	+	0.168687i	$0.0539527\pi$
$607$	204.786i	0.337374i	−0.985670	+	0.168687i	$0.946047\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−350.436	−0.573545
$612$	0	0
$613$	608.733i	0.993040i	0.868025	+	0.496520i	$0.165389\pi$
$613$	608.733i	0.993040i	−0.868025	+	0.496520i	$0.834611\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	17.0108	0.0275701	0.0137851	−	0.999905i	$-0.495612\pi$
$617$	17.0108	0.0275701	0.0137851	+	0.999905i	$0.495612\pi$
$618$	0	0
$619$	−175.916	−0.284193	−0.142097	−	0.989853i	$-0.545384\pi$
$619$	−175.916	−0.284193	−0.142097	+	0.989853i	$0.545384\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	830.400i	1.33290i
$624$	0	0
$625$	−658.645	−1.05383
$626$	0	0
$627$	0	0
$628$	0	0
$629$	531.347i	0.844749i
$630$	0	0
$631$	160.940i	0.255055i	0.991835	+	0.127528i	$0.0407041\pi$
$631$	160.940i	0.255055i	−0.991835	+	0.127528i	$0.959296\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−12.1636	−0.0191552
$636$	0	0
$637$	− 631.189i	− 0.990878i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	301.976	0.471101	0.235551	−	0.971862i	$-0.424311\pi$
$641$	301.976	0.471101	0.235551	+	0.971862i	$0.424311\pi$
$642$	0	0
$643$	−356.979	−0.555177	−0.277589	−	0.960700i	$-0.589535\pi$
$643$	−356.979	−0.555177	−0.277589	+	0.960700i	$0.589535\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 170.568i	− 0.263630i	−0.991274	−	0.131815i	$-0.957920\pi$
$647$	− 170.568i	− 0.263630i	0.991274	−	0.131815i	$-0.0420804\pi$
$648$	0	0
$649$	−4.01008	−0.00617885
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 509.245i	− 0.779855i	−0.920846	−	0.389927i	$-0.872500\pi$
$653$	− 509.245i	− 0.779855i	0.920846	−	0.389927i	$-0.127500\pi$
$654$	0	0
$655$	1218.01i	1.85956i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1224.45	1.85804	0.929019	−	0.370033i	$-0.120653\pi$
$659$	1224.45	1.85804	0.929019	+	0.370033i	$0.120653\pi$
$660$	0	0
$661$	257.526i	0.389600i	0.980843	+	0.194800i	$0.0624059\pi$
$661$	257.526i	0.389600i	−0.980843	+	0.194800i	$0.937594\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1731.52	−2.60379
$666$	0	0
$667$	715.464	1.07266
$668$	0	0
$669$	0	0
$670$	0	0
$671$	104.821i	0.156217i
$672$	0	0
$673$	−432.596	−0.642787	−0.321394	−	0.946946i	$-0.604151\pi$
$673$	−432.596	−0.642787	−0.321394	+	0.946946i	$0.604151\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	235.171i	0.347372i	0.984801	+	0.173686i	$0.0555678\pi$
$677$	235.171i	0.347372i	−0.984801	+	0.173686i	$0.944432\pi$
$678$	0	0
$679$	67.3009i	0.0991177i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1040.38	−1.52326	−0.761628	−	0.648015i	$-0.775599\pi$
$683$	−1040.38	−1.52326	−0.761628	+	0.648015i	$0.775599\pi$
$684$	0	0
$685$	312.862i	0.456734i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−239.186	−0.347149
$690$	0	0
$691$	−121.607	−0.175987	−0.0879937	−	0.996121i	$-0.528046\pi$
$691$	−121.607	−0.175987	−0.0879937	+	0.996121i	$0.528046\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 203.521i	− 0.292835i
$696$	0	0
$697$	840.719	1.20620
$698$	0	0
$699$	0	0
$700$	0	0
$701$	28.6350i	0.0408488i	0.999791	+	0.0204244i	$0.00650174\pi$
$701$	28.6350i	0.0408488i	−0.999791	+	0.0204244i	$0.993498\pi$
$702$	0	0
$703$	990.797i	1.40938i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1676.10	2.37072
$708$	0	0
$709$	− 260.652i	− 0.367634i	−0.982961	−	0.183817i	$-0.941155\pi$
$709$	− 260.652i	− 0.367634i	0.982961	−	0.183817i	$-0.0588453\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−594.208	−0.833391
$714$	0	0
$715$	42.7285	0.0597601
$716$	0	0
$717$	0	0
$718$	0	0
$719$	911.912i	1.26831i	0.773208	+	0.634153i	$0.218651\pi$
$719$	911.912i	1.26831i	−0.773208	+	0.634153i	$0.781349\pi$
$720$	0	0
$721$	98.6067	0.136764
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 27.8202i	− 0.0383726i
$726$	0	0
$727$	− 126.575i	− 0.174105i	−0.996204	−	0.0870527i	$-0.972255\pi$
$727$	− 126.575i	− 0.174105i	0.996204	−	0.0870527i	$-0.0277449\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	874.724	1.19661
$732$	0	0
$733$	825.221i	1.12581i	0.826521	+	0.562906i	$0.190317\pi$
$733$	825.221i	1.12581i	−0.826521	+	0.562906i	$0.809683\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	15.0950	0.0204817
$738$	0	0
$739$	−1098.07	−1.48588	−0.742942	−	0.669356i	$-0.766570\pi$
$739$	−1098.07	−1.48588	−0.742942	+	0.669356i	$0.766570\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 862.493i	− 1.16082i	−0.814323	−	0.580412i	$-0.802891\pi$
$743$	− 862.493i	− 1.16082i	0.814323	−	0.580412i	$-0.197109\pi$
$744$	0	0
$745$	755.549	1.01416
$746$	0	0
$747$	0	0
$748$	0	0
$749$	889.997i	1.18825i
$750$	0	0
$751$	307.835i	0.409900i	0.978772	+	0.204950i	$0.0657032\pi$
$751$	307.835i	0.409900i	−0.978772	+	0.204950i	$0.934297\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−322.726	−0.427452
$756$	0	0
$757$	− 383.571i	− 0.506698i	−0.967375	−	0.253349i	$-0.918468\pi$
$757$	− 383.571i	− 0.506698i	0.967375	−	0.253349i	$-0.0815321\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	424.861	0.558294	0.279147	−	0.960248i	$-0.409948\pi$
$761$	424.861	0.558294	0.279147	+	0.960248i	$0.409948\pi$
$762$	0	0
$763$	−1665.12	−2.18233
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 15.5764i	− 0.0203082i
$768$	0	0
$769$	−1172.64	−1.52489	−0.762443	−	0.647055i	$-0.776000\pi$
$769$	−1172.64	−1.52489	−0.762443	+	0.647055i	$0.776000\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1210.65i	1.56617i	0.621914	+	0.783085i	$0.286355\pi$
$773$	1210.65i	1.56617i	−0.621914	+	0.783085i	$0.713645\pi$
$774$	0	0
$775$	23.1052i	0.0298132i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1567.68	2.01243
$780$	0	0
$781$	− 103.307i	− 0.132275i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−424.352	−0.540576
$786$	0	0
$787$	949.469	1.20644	0.603220	−	0.797574i	$-0.293884\pi$
$787$	949.469	1.20644	0.603220	+	0.797574i	$0.293884\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 164.692i	− 0.208207i
$792$	0	0
$793$	−407.158	−0.513440
$794$	0	0
$795$	0	0
$796$	0	0
$797$	668.187i	0.838378i	0.907899	+	0.419189i	$0.137686\pi$
$797$	668.187i	0.838378i	−0.907899	+	0.419189i	$0.862314\pi$
$798$	0	0
$799$	− 880.494i	− 1.10199i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−153.466	−0.191116
$804$	0	0
$805$	− 2387.46i	− 2.96579i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−896.533	−1.10820	−0.554099	−	0.832451i	$-0.686937\pi$
$809$	−896.533	−1.10820	−0.554099	+	0.832451i	$0.686937\pi$
$810$	0	0
$811$	1083.73	1.33628	0.668141	−	0.744034i	$-0.267090\pi$
$811$	1083.73	1.33628	0.668141	+	0.744034i	$0.267090\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	151.175i	0.185490i
$816$	0	0
$817$	1631.09	1.99644
$818$	0	0
$819$	0	0
$820$	0	0
$821$	464.546i	0.565830i	0.959145	+	0.282915i	$0.0913014\pi$
$821$	464.546i	0.565830i	−0.959145	+	0.282915i	$0.908699\pi$
$822$	0	0
$823$	− 609.423i	− 0.740490i	−0.928934	−	0.370245i	$-0.879274\pi$
$823$	− 609.423i	− 0.740490i	0.928934	−	0.370245i	$-0.120726\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−1179.52	−1.42627	−0.713133	−	0.701029i	$-0.752725\pi$
$827$	−1179.52	−1.42627	−0.713133	+	0.701029i	$0.752725\pi$
$828$	0	0
$829$	− 645.824i	− 0.779040i	−0.921018	−	0.389520i	$-0.872641\pi$
$829$	− 645.824i	− 0.779040i	0.921018	−	0.389520i	$-0.127359\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1585.90	1.90385
$834$	0	0
$835$	477.911	0.572349
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1050.96i	− 1.25263i	−0.779571	−	0.626314i	$-0.784563\pi$
$839$	− 1050.96i	− 1.25263i	0.779571	−	0.626314i	$-0.215437\pi$
$840$	0	0
$841$	461.068	0.548238
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 702.816i	− 0.831735i
$846$	0	0
$847$	− 1503.88i	− 1.77553i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1366.13	−1.60533
$852$	0	0
$853$	917.334i	1.07542i	0.843130	+	0.537710i	$0.180711\pi$
$853$	917.334i	1.07542i	−0.843130	+	0.537710i	$0.819289\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−714.451	−0.833665	−0.416832	−	0.908983i	$-0.636860\pi$
$857$	−714.451	−0.833665	−0.416832	+	0.908983i	$0.636860\pi$
$858$	0	0
$859$	−151.374	−0.176221	−0.0881104	−	0.996111i	$-0.528083\pi$
$859$	−151.374	−0.176221	−0.0881104	+	0.996111i	$0.528083\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	647.037i	0.749753i	0.927075	+	0.374877i	$0.122315\pi$
$863$	647.037i	0.749753i	−0.927075	+	0.374877i	$0.877685\pi$
$864$	0	0
$865$	248.711	0.287527
$866$	0	0
$867$	0	0
$868$	0	0
$869$	119.960i	0.138044i
$870$	0	0
$871$	58.6338i	0.0673177i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1533.25	1.75228
$876$	0	0
$877$	1622.75i	1.85034i	0.379549	+	0.925172i	$0.376079\pi$
$877$	1622.75i	1.85034i	−0.379549	+	0.925172i	$0.623921\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1008.69	−1.14494	−0.572471	−	0.819925i	$-0.694015\pi$
$881$	−1008.69	−1.14494	−0.572471	+	0.819925i	$0.694015\pi$
$882$	0	0
$883$	−76.0921	−0.0861745	−0.0430873	−	0.999071i	$-0.513719\pi$
$883$	−76.0921	−0.0861745	−0.0430873	+	0.999071i	$0.513719\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1463.05i	1.64943i	0.565545	+	0.824717i	$0.308666\pi$
$887$	1463.05i	1.64943i	−0.565545	+	0.824717i	$0.691334\pi$
$888$	0	0
$889$	29.9372	0.0336751
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1641.85i	− 1.83858i
$894$	0	0
$895$	− 763.229i	− 0.852770i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	315.542	0.350992
$900$	0	0
$901$	− 600.970i	− 0.667004i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−485.212	−0.536145
$906$	0	0
$907$	535.897	0.590846	0.295423	−	0.955367i	$-0.404539\pi$
$907$	535.897	0.590846	0.295423	+	0.955367i	$0.404539\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1025.19i	− 1.12535i	−0.826680	−	0.562673i	$-0.809773\pi$
$911$	− 1025.19i	− 1.12535i	0.826680	−	0.562673i	$-0.190227\pi$
$912$	0	0
$913$	−100.607	−0.110194
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2997.80i	− 3.26914i
$918$	0	0
$919$	− 1192.03i	− 1.29709i	−0.761176	−	0.648545i	$-0.775378\pi$
$919$	− 1192.03i	− 1.29709i	0.761176	−	0.648545i	$-0.224622\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	401.275	0.434751
$924$	0	0
$925$	53.1209i	0.0574280i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1133.29	1.21991	0.609954	−	0.792437i	$-0.291188\pi$
$929$	1133.29	1.21991	0.609954	+	0.792437i	$0.291188\pi$
$930$	0	0
$931$	2957.22	3.17639
$932$	0	0
$933$	0	0
$934$	0	0
$935$	107.358i	0.114821i
$936$	0	0
$937$	−389.378	−0.415558	−0.207779	−	0.978176i	$-0.566624\pi$
$937$	−389.378	−0.415558	−0.207779	+	0.978176i	$0.566624\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	768.324i	0.816497i	0.912871	+	0.408249i	$0.133860\pi$
$941$	768.324i	0.816497i	−0.912871	+	0.408249i	$0.866140\pi$
$942$	0	0
$943$	2161.55i	2.29221i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1592.82	−1.68196	−0.840980	−	0.541066i	$-0.818021\pi$
$947$	−1592.82	−1.68196	−0.840980	+	0.541066i	$0.818021\pi$
$948$	0	0
$949$	− 596.110i	− 0.628145i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−204.697	−0.214792	−0.107396	−	0.994216i	$-0.534251\pi$
$953$	−204.697	−0.214792	−0.107396	+	0.994216i	$0.534251\pi$
$954$	0	0
$955$	1020.87	1.06898
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 770.022i	− 0.802943i
$960$	0	0
$961$	698.936	0.727301
$962$	0	0
$963$	0	0
$964$	0	0
$965$	13.9735i	0.0144803i
$966$	0	0
$967$	732.916i	0.757928i	0.925411	+	0.378964i	$0.123720\pi$
$967$	732.916i	0.757928i	−0.925411	+	0.378964i	$0.876280\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−101.413	−0.104442	−0.0522210	−	0.998636i	$-0.516630\pi$
$971$	−101.413	−0.104442	−0.0522210	+	0.998636i	$0.516630\pi$
$972$	0	0
$973$	500.908i	0.514808i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1439.84	1.47374	0.736868	−	0.676037i	$-0.236304\pi$
$977$	1439.84	1.47374	0.736868	+	0.676037i	$0.236304\pi$
$978$	0	0
$979$	96.0064	0.0980658
$980$	0	0
$981$	0	0
$982$	0	0
$983$	720.894i	0.733361i	0.930347	+	0.366681i	$0.119506\pi$
$983$	720.894i	0.733361i	−0.930347	+	0.366681i	$0.880494\pi$
$984$	0	0
$985$	−1014.09	−1.02954
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2248.98i	2.27400i
$990$	0	0
$991$	861.974i	0.869802i	0.900478	+	0.434901i	$0.143217\pi$
$991$	861.974i	0.869802i	−0.900478	+	0.434901i	$0.856783\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1414.67	1.42178
$996$	0	0
$997$	464.804i	0.466203i	0.972452	+	0.233102i	$0.0748875\pi$
$997$	464.804i	0.466203i	−0.972452	+	0.233102i	$0.925113\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.3.b.a.271.4		16
3.2	odd	2	inner	864.3.b.a.271.14		16
4.3	odd	2		216.3.b.a.163.16	yes	16
8.3	odd	2	inner	864.3.b.a.271.13		16
8.5	even	2		216.3.b.a.163.15	yes	16
12.11	even	2		216.3.b.a.163.1	✓	16
24.5	odd	2		216.3.b.a.163.2	yes	16
24.11	even	2	inner	864.3.b.a.271.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.b.a.163.1	✓	16	12.11	even	2
216.3.b.a.163.2	yes	16	24.5	odd	2
216.3.b.a.163.15	yes	16	8.5	even	2
216.3.b.a.163.16	yes	16	4.3	odd	2
864.3.b.a.271.3		16	24.11	even	2	inner
864.3.b.a.271.4		16	1.1	even	1	trivial
864.3.b.a.271.13		16	8.3	odd	2	inner
864.3.b.a.271.14		16	3.2	odd	2	inner

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.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-5.14075i q^{5} +12.6525i q^{7} +O(q^{10})\)	\(q-5.14075i q^{5} +12.6525i q^{7} +1.46281 q^{11} +5.68201i q^{13} -14.2764 q^{17} -26.6211 q^{19} -36.7058i q^{23} -1.42727 q^{25} +19.4918i q^{29} -16.1884i q^{31} +65.0432 q^{35} -37.2185i q^{37} -58.8886 q^{41} -61.2705 q^{43} +61.6746i q^{47} -111.085 q^{49} +42.0953i q^{53} -7.51995i q^{55} -2.74134 q^{59} +71.6574i q^{61} +29.2098 q^{65} +10.3192 q^{67} -70.6220i q^{71} -104.912 q^{73} +18.5082i q^{77} +82.0064i q^{79} -68.7766 q^{83} +73.3915i q^{85} +65.6313 q^{89} -71.8916 q^{91} +136.852i q^{95} +5.31919 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q - 32 q^{19} - 80 q^{25} - 128 q^{43} - 80 q^{49} - 128 q^{67} - 160 q^{73} + 96 q^{91} - 208 q^{97}+O(q^{100}) \) 16 * q - 32 * q^19 - 80 * q^25 - 128 * q^43 - 80 * q^49 - 128 * q^67 - 160 * q^73 + 96 * q^91 - 208 * q^97

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