LMFDB - Embedded newform 864.3.b.a.271.5

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			271.5
Root			$0.316912 - 1.97473i$ of defining polynomial
Character	$\chi$	$=$	864.271
Dual form			864.3.b.a.271.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.41296i q^{5} -3.59985i q^{7} +O(q^{10})$	$q-4.41296i q^{5} -3.59985i q^{7} -18.7798 q^{11} +19.9181i q^{13} +8.00578 q^{17} -16.1354 q^{19} +18.3926i q^{23} +5.52575 q^{25} +17.9287i q^{29} +29.7323i q^{31} -15.8860 q^{35} +26.7200i q^{37} +40.2437 q^{41} +71.8376 q^{43} -23.3952i q^{47} +36.0411 q^{49} -90.7227i q^{53} +82.8745i q^{55} -10.9410 q^{59} +90.9260i q^{61} +87.8980 q^{65} -74.0054 q^{67} +13.9001i q^{71} -56.0158 q^{73} +67.6044i q^{77} +118.415i q^{79} -8.08995 q^{83} -35.3292i q^{85} -83.9483 q^{89} +71.7022 q^{91} +71.2049i q^{95} -79.0054 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	− 4.41296i	− 0.882593i	−0.897361	−	0.441296i	$-0.854519\pi$
$5$	− 4.41296i	− 0.882593i	0.897361	−	0.441296i	$-0.145481\pi$
$6$	0	0
$7$	− 3.59985i	− 0.514264i	−0.966376	−	0.257132i	$-0.917223\pi$
$7$	− 3.59985i	− 0.514264i	0.966376	−	0.257132i	$-0.0827775\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−18.7798	−1.70725	−0.853627	−	0.520885i	$-0.825602\pi$
$11$	−18.7798	−1.70725	−0.853627	+	0.520885i	$0.825602\pi$
$12$	0	0
$13$	19.9181i	1.53216i	0.642743	+	0.766082i	$0.277796\pi$
$13$	19.9181i	1.53216i	−0.642743	+	0.766082i	$0.722204\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	8.00578	0.470928	0.235464	−	0.971883i	$-0.424339\pi$
$17$	8.00578	0.470928	0.235464	+	0.971883i	$0.424339\pi$
$18$	0	0
$19$	−16.1354	−0.849231	−0.424616	−	0.905374i	$-0.639591\pi$
$19$	−16.1354	−0.849231	−0.424616	+	0.905374i	$0.639591\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	18.3926i	0.799680i	0.916585	+	0.399840i	$0.130934\pi$
$23$	18.3926i	0.799680i	−0.916585	+	0.399840i	$0.869066\pi$
$24$	0	0
$25$	5.52575	0.221030
$26$	0	0
$27$	0	0
$28$	0	0
$29$	17.9287i	0.618232i	0.951024	+	0.309116i	$0.100033\pi$
$29$	17.9287i	0.618232i	−0.951024	+	0.309116i	$0.899967\pi$
$30$	0	0
$31$	29.7323i	0.959106i	0.877513	+	0.479553i	$0.159201\pi$
$31$	29.7323i	0.959106i	−0.877513	+	0.479553i	$0.840799\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−15.8860	−0.453886
$36$	0	0
$37$	26.7200i	0.722162i	0.932535	+	0.361081i	$0.117592\pi$
$37$	26.7200i	0.722162i	−0.932535	+	0.361081i	$0.882408\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	40.2437	0.981553	0.490776	−	0.871286i	$-0.336713\pi$
$41$	40.2437	0.981553	0.490776	+	0.871286i	$0.336713\pi$
$42$	0	0
$43$	71.8376	1.67064	0.835321	−	0.549762i	$-0.185282\pi$
$43$	71.8376	1.67064	0.835321	+	0.549762i	$0.185282\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 23.3952i	− 0.497770i	−0.968533	−	0.248885i	$-0.919936\pi$
$47$	− 23.3952i	− 0.497770i	0.968533	−	0.248885i	$-0.0800641\pi$
$48$	0	0
$49$	36.0411	0.735533
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 90.7227i	− 1.71175i	−0.517183	−	0.855875i	$-0.673019\pi$
$53$	− 90.7227i	− 1.71175i	0.517183	−	0.855875i	$-0.326981\pi$
$54$	0	0
$55$	82.8745i	1.50681i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.9410	−0.185440	−0.0927200	−	0.995692i	$-0.529556\pi$
$59$	−10.9410	−0.185440	−0.0927200	+	0.995692i	$0.529556\pi$
$60$	0	0
$61$	90.9260i	1.49059i	0.666735	+	0.745295i	$0.267691\pi$
$61$	90.9260i	1.49059i	−0.666735	+	0.745295i	$0.732309\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	87.8980	1.35228
$66$	0	0
$67$	−74.0054	−1.10456	−0.552279	−	0.833659i	$-0.686242\pi$
$67$	−74.0054	−1.10456	−0.552279	+	0.833659i	$0.686242\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.9001i	0.195776i	0.995197	+	0.0978882i	$0.0312088\pi$
$71$	13.9001i	0.195776i	−0.995197	+	0.0978882i	$0.968791\pi$
$72$	0	0
$73$	−56.0158	−0.767340	−0.383670	−	0.923470i	$-0.625340\pi$
$73$	−56.0158	−0.767340	−0.383670	+	0.923470i	$0.625340\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	67.6044i	0.877979i
$78$	0	0
$79$	118.415i	1.49893i	0.662044	+	0.749465i	$0.269689\pi$
$79$	118.415i	1.49893i	−0.662044	+	0.749465i	$0.730311\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.08995	−0.0974693	−0.0487347	−	0.998812i	$-0.515519\pi$
$83$	−8.08995	−0.0974693	−0.0487347	+	0.998812i	$0.515519\pi$
$84$	0	0
$85$	− 35.3292i	− 0.415638i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−83.9483	−0.943239	−0.471620	−	0.881802i	$-0.656330\pi$
$89$	−83.9483	−0.943239	−0.471620	+	0.881802i	$0.656330\pi$
$90$	0	0
$91$	71.7022	0.787937
$92$	0	0
$93$	0	0
$94$	0	0
$95$	71.2049i	0.749525i
$96$	0	0
$97$	−79.0054	−0.814489	−0.407244	−	0.913319i	$-0.633510\pi$
$97$	−79.0054	−0.814489	−0.407244	+	0.913319i	$0.633510\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	87.0864i	0.862241i	0.902294	+	0.431121i	$0.141882\pi$
$101$	87.0864i	0.862241i	−0.902294	+	0.431121i	$0.858118\pi$
$102$	0	0
$103$	138.658i	1.34619i	0.739556	+	0.673095i	$0.235035\pi$
$103$	138.658i	1.34619i	−0.739556	+	0.673095i	$0.764965\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	54.4965	0.509313	0.254656	−	0.967032i	$-0.418038\pi$
$107$	54.4965	0.509313	0.254656	+	0.967032i	$0.418038\pi$
$108$	0	0
$109$	143.879i	1.31999i	0.751271	+	0.659994i	$0.229441\pi$
$109$	143.879i	1.31999i	−0.751271	+	0.659994i	$0.770559\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−128.630	−1.13832	−0.569161	−	0.822226i	$-0.692732\pi$
$113$	−128.630	−1.13832	−0.569161	+	0.822226i	$0.692732\pi$
$114$	0	0
$115$	81.1661	0.705792
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 28.8196i	− 0.242181i
$120$	0	0
$121$	231.681	1.91472
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 134.709i	− 1.07767i
$126$	0	0
$127$	− 90.8699i	− 0.715511i	−0.933815	−	0.357756i	$-0.883542\pi$
$127$	− 90.8699i	− 0.715511i	0.933815	−	0.357756i	$-0.116458\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	78.5838	0.599876	0.299938	−	0.953959i	$-0.403034\pi$
$131$	78.5838	0.599876	0.299938	+	0.953959i	$0.403034\pi$
$132$	0	0
$133$	58.0850i	0.436729i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−271.485	−1.98164	−0.990821	−	0.135178i	$-0.956839\pi$
$137$	−271.485	−1.98164	−0.990821	+	0.135178i	$0.956839\pi$
$138$	0	0
$139$	−177.843	−1.27945	−0.639723	−	0.768605i	$-0.720951\pi$
$139$	−177.843	−1.27945	−0.639723	+	0.768605i	$0.720951\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 374.059i	− 2.61579i
$144$	0	0
$145$	79.1188	0.545647
$146$	0	0
$147$	0	0
$148$	0	0
$149$	206.173i	1.38371i	0.722037	+	0.691854i	$0.243206\pi$
$149$	206.173i	1.38371i	−0.722037	+	0.691854i	$0.756794\pi$
$150$	0	0
$151$	− 215.526i	− 1.42732i	−0.700490	−	0.713662i	$-0.747035\pi$
$151$	− 215.526i	− 1.42732i	0.700490	−	0.713662i	$-0.252965\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	131.207	0.846500
$156$	0	0
$157$	163.503i	1.04142i	0.853734	+	0.520709i	$0.174332\pi$
$157$	163.503i	1.04142i	−0.853734	+	0.520709i	$0.825668\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	66.2107	0.411247
$162$	0	0
$163$	72.2438	0.443214	0.221607	−	0.975136i	$-0.428870\pi$
$163$	72.2438	0.443214	0.221607	+	0.975136i	$0.428870\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	211.488i	1.26639i	0.773990	+	0.633197i	$0.218258\pi$
$167$	211.488i	1.26639i	−0.773990	+	0.633197i	$0.781742\pi$
$168$	0	0
$169$	−227.732	−1.34753
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 113.539i	− 0.656292i	−0.944627	−	0.328146i	$-0.893576\pi$
$173$	− 113.539i	− 0.656292i	0.944627	−	0.328146i	$-0.106424\pi$
$174$	0	0
$175$	− 19.8919i	− 0.113668i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−29.2370	−0.163335	−0.0816677	−	0.996660i	$-0.526025\pi$
$179$	−29.2370	−0.163335	−0.0816677	+	0.996660i	$0.526025\pi$
$180$	0	0
$181$	− 170.599i	− 0.942534i	−0.881991	−	0.471267i	$-0.843797\pi$
$181$	− 170.599i	− 0.942534i	0.881991	−	0.471267i	$-0.156203\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	117.914	0.637375
$186$	0	0
$187$	−150.347	−0.803994
$188$	0	0
$189$	0	0
$190$	0	0
$191$	353.213i	1.84928i	0.380841	+	0.924641i	$0.375635\pi$
$191$	353.213i	1.84928i	−0.380841	+	0.924641i	$0.624365\pi$
$192$	0	0
$193$	−23.5773	−0.122162	−0.0610810	−	0.998133i	$-0.519455\pi$
$193$	−23.5773	−0.122162	−0.0610810	+	0.998133i	$0.519455\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 21.5269i	− 0.109273i	−0.998506	−	0.0546367i	$-0.982600\pi$
$197$	− 21.5269i	− 0.109273i	0.998506	−	0.0546367i	$-0.0174001\pi$
$198$	0	0
$199$	34.8638i	0.175195i	0.996156	+	0.0875974i	$0.0279189\pi$
$199$	34.8638i	0.175195i	−0.996156	+	0.0875974i	$0.972081\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	64.5407	0.317934
$204$	0	0
$205$	− 177.594i	− 0.866311i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	303.019	1.44985
$210$	0	0
$211$	179.116	0.848889	0.424444	−	0.905454i	$-0.360469\pi$
$211$	179.116	0.848889	0.424444	+	0.905454i	$0.360469\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 317.017i	− 1.47450i
$216$	0	0
$217$	107.032	0.493233
$218$	0	0
$219$	0	0
$220$	0	0
$221$	159.460i	0.721539i
$222$	0	0
$223$	− 1.38743i	− 0.00622167i	−0.999995	−	0.00311084i	$-0.999010\pi$
$223$	− 1.38743i	− 0.00622167i	0.999995	−	0.00311084i	$-0.000990211\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	111.238	0.490033	0.245017	−	0.969519i	$-0.421207\pi$
$227$	111.238	0.490033	0.245017	+	0.969519i	$0.421207\pi$
$228$	0	0
$229$	− 385.872i	− 1.68503i	−0.538673	−	0.842515i	$-0.681074\pi$
$229$	− 385.872i	− 1.68503i	0.538673	−	0.842515i	$-0.318926\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−174.604	−0.749372	−0.374686	−	0.927152i	$-0.622250\pi$
$233$	−174.604	−0.749372	−0.374686	+	0.927152i	$0.622250\pi$
$234$	0	0
$235$	−103.242	−0.439328
$236$	0	0
$237$	0	0
$238$	0	0
$239$	112.809i	0.472004i	0.971753	+	0.236002i	$0.0758373\pi$
$239$	112.809i	0.472004i	−0.971753	+	0.236002i	$0.924163\pi$
$240$	0	0
$241$	10.3355	0.0428861	0.0214430	−	0.999770i	$-0.493174\pi$
$241$	10.3355	0.0428861	0.0214430	+	0.999770i	$0.493174\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 159.048i	− 0.649176i
$246$	0	0
$247$	− 321.387i	− 1.30116i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−334.659	−1.33330	−0.666651	−	0.745370i	$-0.732273\pi$
$251$	−334.659	−1.33330	−0.666651	+	0.745370i	$0.732273\pi$
$252$	0	0
$253$	− 345.410i	− 1.36526i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	218.689	0.850929	0.425464	−	0.904975i	$-0.360111\pi$
$257$	218.689	0.850929	0.425464	+	0.904975i	$0.360111\pi$
$258$	0	0
$259$	96.1879	0.371382
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 150.622i	− 0.572709i	−0.958124	−	0.286354i	$-0.907557\pi$
$263$	− 150.622i	− 0.572709i	0.958124	−	0.286354i	$-0.0924435\pi$
$264$	0	0
$265$	−400.356	−1.51078
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 340.377i	− 1.26534i	−0.774420	−	0.632672i	$-0.781958\pi$
$269$	− 340.377i	− 1.26534i	0.774420	−	0.632672i	$-0.218042\pi$
$270$	0	0
$271$	− 137.486i	− 0.507328i	−0.967292	−	0.253664i	$-0.918364\pi$
$271$	− 137.486i	− 0.507328i	0.967292	−	0.253664i	$-0.0816358\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−103.773	−0.377355
$276$	0	0
$277$	− 31.5211i	− 0.113795i	−0.998380	−	0.0568973i	$-0.981879\pi$
$277$	− 31.5211i	− 0.113795i	0.998380	−	0.0568973i	$-0.0181208\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−285.437	−1.01579	−0.507896	−	0.861419i	$-0.669576\pi$
$281$	−285.437	−1.01579	−0.507896	+	0.861419i	$0.669576\pi$
$282$	0	0
$283$	0.947759	0.00334897	0.00167449	−	0.999999i	$-0.499467\pi$
$283$	0.947759	0.00334897	0.00167449	+	0.999999i	$0.499467\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 144.871i	− 0.504777i
$288$	0	0
$289$	−224.908	−0.778227
$290$	0	0
$291$	0	0
$292$	0	0
$293$	204.998i	0.699652i	0.936815	+	0.349826i	$0.113759\pi$
$293$	204.998i	0.699652i	−0.936815	+	0.349826i	$0.886241\pi$
$294$	0	0
$295$	48.2821i	0.163668i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−366.347	−1.22524
$300$	0	0
$301$	− 258.605i	− 0.859151i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	401.253	1.31558
$306$	0	0
$307$	−229.805	−0.748551	−0.374276	−	0.927317i	$-0.622109\pi$
$307$	−229.805	−0.748551	−0.374276	+	0.927317i	$0.622109\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	286.078i	0.919864i	0.887954	+	0.459932i	$0.152126\pi$
$311$	286.078i	0.919864i	−0.887954	+	0.459932i	$0.847874\pi$
$312$	0	0
$313$	447.913	1.43103	0.715517	−	0.698596i	$-0.246191\pi$
$313$	447.913	1.43103	0.715517	+	0.698596i	$0.246191\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	387.037i	1.22094i	0.792040	+	0.610469i	$0.209019\pi$
$317$	387.037i	1.22094i	−0.792040	+	0.610469i	$0.790981\pi$
$318$	0	0
$319$	− 336.698i	− 1.05548i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−129.176	−0.399927
$324$	0	0
$325$	110.063i	0.338655i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−84.2191	−0.255985
$330$	0	0
$331$	41.5163	0.125427	0.0627134	−	0.998032i	$-0.480025\pi$
$331$	41.5163	0.125427	0.0627134	+	0.998032i	$0.480025\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	326.583i	0.974875i
$336$	0	0
$337$	146.594	0.434997	0.217499	−	0.976061i	$-0.430210\pi$
$337$	146.594	0.434997	0.217499	+	0.976061i	$0.430210\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 558.366i	− 1.63744i
$342$	0	0
$343$	− 306.135i	− 0.892522i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−109.781	−0.316373	−0.158186	−	0.987409i	$-0.550565\pi$
$347$	−109.781	−0.316373	−0.158186	+	0.987409i	$0.550565\pi$
$348$	0	0
$349$	− 430.104i	− 1.23239i	−0.787593	−	0.616195i	$-0.788673\pi$
$349$	− 430.104i	− 1.23239i	0.787593	−	0.616195i	$-0.211327\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	324.763	0.920008	0.460004	−	0.887917i	$-0.347848\pi$
$353$	324.763	0.920008	0.460004	+	0.887917i	$0.347848\pi$
$354$	0	0
$355$	61.3407	0.172791
$356$	0	0
$357$	0	0
$358$	0	0
$359$	160.539i	0.447185i	0.974683	+	0.223592i	$0.0717784\pi$
$359$	160.539i	0.447185i	−0.974683	+	0.223592i	$0.928222\pi$
$360$	0	0
$361$	−100.649	−0.278806
$362$	0	0
$363$	0	0
$364$	0	0
$365$	247.196i	0.677249i
$366$	0	0
$367$	− 373.416i	− 1.01748i	−0.860919	−	0.508742i	$-0.830111\pi$
$367$	− 373.416i	− 1.01748i	0.860919	−	0.508742i	$-0.169889\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−326.588	−0.880291
$372$	0	0
$373$	359.197i	0.962995i	0.876448	+	0.481497i	$0.159907\pi$
$373$	359.197i	0.962995i	−0.876448	+	0.481497i	$0.840093\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−357.107	−0.947233
$378$	0	0
$379$	258.439	0.681896	0.340948	−	0.940082i	$-0.389252\pi$
$379$	258.439	0.681896	0.340948	+	0.940082i	$0.389252\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	185.218i	0.483597i	0.970326	+	0.241799i	$0.0777373\pi$
$383$	185.218i	0.483597i	−0.970326	+	0.241799i	$0.922263\pi$
$384$	0	0
$385$	298.336	0.774898
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 107.100i	− 0.275321i	−0.990479	−	0.137660i	$-0.956042\pi$
$389$	− 107.100i	− 0.275321i	0.990479	−	0.137660i	$-0.0439583\pi$
$390$	0	0
$391$	147.247i	0.376592i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	522.563	1.32294
$396$	0	0
$397$	132.442i	0.333607i	0.985990	+	0.166803i	$0.0533446\pi$
$397$	132.442i	0.333607i	−0.985990	+	0.166803i	$0.946655\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	471.890	1.17678	0.588392	−	0.808576i	$-0.299761\pi$
$401$	471.890	1.17678	0.588392	+	0.808576i	$0.299761\pi$
$402$	0	0
$403$	−592.212	−1.46951
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 501.796i	− 1.23291i
$408$	0	0
$409$	−386.361	−0.944649	−0.472324	−	0.881425i	$-0.656585\pi$
$409$	−386.361	−0.944649	−0.472324	+	0.881425i	$0.656585\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	39.3858i	0.0953651i
$414$	0	0
$415$	35.7007i	0.0860257i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	41.3087	0.0985888	0.0492944	−	0.998784i	$-0.484303\pi$
$419$	41.3087	0.0985888	0.0492944	+	0.998784i	$0.484303\pi$
$420$	0	0
$421$	− 3.75696i	− 0.00892390i	−0.999990	−	0.00446195i	$-0.998580\pi$
$421$	− 3.75696i	− 0.00892390i	0.999990	−	0.00446195i	$-0.00142029\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	44.2379	0.104089
$426$	0	0
$427$	327.320	0.766557
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 451.946i	− 1.04860i	−0.851534	−	0.524299i	$-0.824327\pi$
$431$	− 451.946i	− 1.04860i	0.851534	−	0.524299i	$-0.175673\pi$
$432$	0	0
$433$	−127.903	−0.295389	−0.147694	−	0.989033i	$-0.547185\pi$
$433$	−127.903	−0.295389	−0.147694	+	0.989033i	$0.547185\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 296.773i	− 0.679114i
$438$	0	0
$439$	133.514i	0.304132i	0.988370	+	0.152066i	$0.0485926\pi$
$439$	133.514i	0.304132i	−0.988370	+	0.152066i	$0.951407\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	631.208	1.42485	0.712424	−	0.701749i	$-0.247597\pi$
$443$	631.208	1.42485	0.712424	+	0.701749i	$0.247597\pi$
$444$	0	0
$445$	370.461i	0.832496i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	84.8691	0.189018	0.0945091	−	0.995524i	$-0.469872\pi$
$449$	84.8691	0.189018	0.0945091	+	0.995524i	$0.469872\pi$
$450$	0	0
$451$	−755.768	−1.67576
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 316.419i	− 0.695427i
$456$	0	0
$457$	301.582	0.659917	0.329959	−	0.943995i	$-0.392965\pi$
$457$	301.582	0.659917	0.329959	+	0.943995i	$0.392965\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	185.028i	0.401362i	0.979657	+	0.200681i	$0.0643154\pi$
$461$	185.028i	0.401362i	−0.979657	+	0.200681i	$0.935685\pi$
$462$	0	0
$463$	515.131i	1.11259i	0.830984	+	0.556297i	$0.187778\pi$
$463$	515.131i	1.11259i	−0.830984	+	0.556297i	$0.812222\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	301.784	0.646218	0.323109	−	0.946362i	$-0.395272\pi$
$467$	301.784	0.646218	0.323109	+	0.946362i	$0.395272\pi$
$468$	0	0
$469$	266.408i	0.568034i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1349.10	−2.85221
$474$	0	0
$475$	−89.1602	−0.187706
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 263.369i	− 0.549831i	−0.961468	−	0.274915i	$-0.911350\pi$
$479$	− 263.369i	− 0.549831i	0.961468	−	0.274915i	$-0.0886499\pi$
$480$	0	0
$481$	−532.212	−1.10647
$482$	0	0
$483$	0	0
$484$	0	0
$485$	348.648i	0.718862i
$486$	0	0
$487$	100.621i	0.206615i	0.994649	+	0.103307i	$0.0329425\pi$
$487$	100.621i	0.206615i	−0.994649	+	0.103307i	$0.967057\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	506.967	1.03252	0.516260	−	0.856432i	$-0.327324\pi$
$491$	506.967	1.03252	0.516260	+	0.856432i	$0.327324\pi$
$492$	0	0
$493$	143.533i	0.291143i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	50.0383	0.100681
$498$	0	0
$499$	609.604	1.22165	0.610825	−	0.791765i	$-0.290838\pi$
$499$	609.604	1.22165	0.610825	+	0.791765i	$0.290838\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	794.669i	1.57986i	0.613197	+	0.789930i	$0.289883\pi$
$503$	794.669i	1.57986i	−0.613197	+	0.789930i	$0.710117\pi$
$504$	0	0
$505$	384.309	0.761008
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 155.999i	− 0.306481i	−0.988189	−	0.153241i	$-0.951029\pi$
$509$	− 155.999i	− 0.306481i	0.988189	−	0.153241i	$-0.0489709\pi$
$510$	0	0
$511$	201.648i	0.394615i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	611.891	1.18814
$516$	0	0
$517$	439.357i	0.849820i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−719.474	−1.38095	−0.690474	−	0.723357i	$-0.742598\pi$
$521$	−719.474	−1.38095	−0.690474	+	0.723357i	$0.742598\pi$
$522$	0	0
$523$	963.965	1.84314	0.921572	−	0.388206i	$-0.126905\pi$
$523$	963.965	1.84314	0.921572	+	0.388206i	$0.126905\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	238.030i	0.451670i
$528$	0	0
$529$	190.710	0.360511
$530$	0	0
$531$	0	0
$532$	0	0
$533$	801.579i	1.50390i
$534$	0	0
$535$	− 240.491i	− 0.449516i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−676.844	−1.25574
$540$	0	0
$541$	− 592.256i	− 1.09474i	−0.836890	−	0.547371i	$-0.815629\pi$
$541$	− 592.256i	− 1.09474i	0.836890	−	0.547371i	$-0.184371\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	634.931	1.16501
$546$	0	0
$547$	285.828	0.522538	0.261269	−	0.965266i	$-0.415859\pi$
$547$	285.828	0.522538	0.261269	+	0.965266i	$0.415859\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 289.287i	− 0.525022i
$552$	0	0
$553$	426.277	0.770845
$554$	0	0
$555$	0	0
$556$	0	0
$557$	148.974i	0.267459i	0.991018	+	0.133729i	$0.0426953\pi$
$557$	148.974i	0.267459i	−0.991018	+	0.133729i	$0.957305\pi$
$558$	0	0
$559$	1430.87i	2.55970i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−107.185	−0.190382	−0.0951908	−	0.995459i	$-0.530346\pi$
$563$	−107.185	−0.190382	−0.0951908	+	0.995459i	$0.530346\pi$
$564$	0	0
$565$	567.641i	1.00467i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−119.321	−0.209703	−0.104851	−	0.994488i	$-0.533437\pi$
$569$	−119.321	−0.209703	−0.104851	+	0.994488i	$0.533437\pi$
$570$	0	0
$571$	−118.725	−0.207926	−0.103963	−	0.994581i	$-0.533152\pi$
$571$	−118.725	−0.207926	−0.103963	+	0.994581i	$0.533152\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	101.633i	0.176753i
$576$	0	0
$577$	490.151	0.849481	0.424741	−	0.905315i	$-0.360365\pi$
$577$	490.151	0.849481	0.424741	+	0.905315i	$0.360365\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	29.1226i	0.0501250i
$582$	0	0
$583$	1703.75i	2.92239i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−686.259	−1.16909	−0.584547	−	0.811360i	$-0.698728\pi$
$587$	−686.259	−1.16909	−0.584547	+	0.811360i	$0.698728\pi$
$588$	0	0
$589$	− 479.742i	− 0.814503i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	177.526	0.299369	0.149684	−	0.988734i	$-0.452174\pi$
$593$	177.526	0.299369	0.149684	+	0.988734i	$0.452174\pi$
$594$	0	0
$595$	−127.180	−0.213747
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 192.768i	− 0.321817i	−0.986969	−	0.160908i	$-0.948558\pi$
$599$	− 192.768i	− 0.321817i	0.986969	−	0.160908i	$-0.0514423\pi$
$600$	0	0
$601$	−1097.74	−1.82652	−0.913261	−	0.407375i	$-0.866444\pi$
$601$	−1097.74	−1.82652	−0.913261	+	0.407375i	$0.866444\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 1022.40i	− 1.68991i
$606$	0	0
$607$	− 4.30901i	− 0.00709886i	−0.999994	−	0.00354943i	$-0.998870\pi$
$607$	− 4.30901i	− 0.00709886i	0.999994	−	0.00354943i	$-0.00112982\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	465.989	0.762666
$612$	0	0
$613$	618.106i	1.00833i	0.863608	+	0.504165i	$0.168200\pi$
$613$	618.106i	1.00833i	−0.863608	+	0.504165i	$0.831800\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−793.183	−1.28555	−0.642774	−	0.766056i	$-0.722217\pi$
$617$	−793.183	−1.28555	−0.642774	+	0.766056i	$0.722217\pi$
$618$	0	0
$619$	−635.387	−1.02647	−0.513237	−	0.858247i	$-0.671554\pi$
$619$	−635.387	−1.02647	−0.513237	+	0.858247i	$0.671554\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	302.201i	0.485074i
$624$	0	0
$625$	−456.322	−0.730116
$626$	0	0
$627$	0	0
$628$	0	0
$629$	213.914i	0.340086i
$630$	0	0
$631$	− 307.959i	− 0.488049i	−0.969769	−	0.244024i	$-0.921532\pi$
$631$	− 307.959i	− 0.488049i	0.969769	−	0.244024i	$-0.0784677\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−401.006	−0.631505
$636$	0	0
$637$	717.872i	1.12696i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−449.718	−0.701588	−0.350794	−	0.936453i	$-0.614088\pi$
$641$	−449.718	−0.701588	−0.350794	+	0.936453i	$0.614088\pi$
$642$	0	0
$643$	665.066	1.03432	0.517158	−	0.855890i	$-0.326990\pi$
$643$	665.066	1.03432	0.517158	+	0.855890i	$0.326990\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 9.25693i	− 0.0143075i	−0.999974	−	0.00715373i	$-0.997723\pi$
$647$	− 9.25693i	− 0.0143075i	0.999974	−	0.00715373i	$-0.00227712\pi$
$648$	0	0
$649$	205.469	0.316593
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 692.725i	− 1.06083i	−0.847737	−	0.530417i	$-0.822035\pi$
$653$	− 692.725i	− 1.06083i	0.847737	−	0.530417i	$-0.177965\pi$
$654$	0	0
$655$	− 346.787i	− 0.529446i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−1008.98	−1.53108	−0.765538	−	0.643391i	$-0.777527\pi$
$659$	−1008.98	−1.53108	−0.765538	+	0.643391i	$0.777527\pi$
$660$	0	0
$661$	150.898i	0.228287i	0.993464	+	0.114144i	$0.0364124\pi$
$661$	150.898i	0.228287i	−0.993464	+	0.114144i	$0.963588\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	256.327	0.385454
$666$	0	0
$667$	−329.757	−0.494388
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1707.57i	− 2.54482i
$672$	0	0
$673$	−124.651	−0.185217	−0.0926084	−	0.995703i	$-0.529520\pi$
$673$	−124.651	−0.185217	−0.0926084	+	0.995703i	$0.529520\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 200.927i	− 0.296791i	−0.988928	−	0.148395i	$-0.952589\pi$
$677$	− 200.927i	− 0.296791i	0.988928	−	0.148395i	$-0.0474108\pi$
$678$	0	0
$679$	284.407i	0.418862i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	764.228	1.11893	0.559464	−	0.828855i	$-0.311007\pi$
$683$	764.228	1.11893	0.559464	+	0.828855i	$0.311007\pi$
$684$	0	0
$685$	1198.05i	1.74898i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1807.03	2.62268
$690$	0	0
$691$	130.928	0.189476	0.0947378	−	0.995502i	$-0.469799\pi$
$691$	130.928	0.189476	0.0947378	+	0.995502i	$0.469799\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	784.815i	1.12923i
$696$	0	0
$697$	322.182	0.462241
$698$	0	0
$699$	0	0
$700$	0	0
$701$	721.278i	1.02893i	0.857512	+	0.514464i	$0.172009\pi$
$701$	721.278i	1.02893i	−0.857512	+	0.514464i	$0.827991\pi$
$702$	0	0
$703$	− 431.137i	− 0.613282i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	313.498	0.443420
$708$	0	0
$709$	− 4.66911i	− 0.00658548i	−0.999995	−	0.00329274i	$-0.998952\pi$
$709$	− 4.66911i	− 0.00658548i	0.999995	−	0.00329274i	$-0.00104811\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−546.855	−0.766978
$714$	0	0
$715$	−1650.71	−2.30868
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1307.39i	1.81835i	0.416417	+	0.909174i	$0.363286\pi$
$719$	1307.39i	1.81835i	−0.416417	+	0.909174i	$0.636714\pi$
$720$	0	0
$721$	499.146	0.692297
$722$	0	0
$723$	0	0
$724$	0	0
$725$	99.0698i	0.136648i
$726$	0	0
$727$	− 838.418i	− 1.15326i	−0.817006	−	0.576629i	$-0.804368\pi$
$727$	− 838.418i	− 1.15326i	0.817006	−	0.576629i	$-0.195632\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	575.116	0.786753
$732$	0	0
$733$	1062.62i	1.44969i	0.688914	+	0.724843i	$0.258088\pi$
$733$	1062.62i	1.44969i	−0.688914	+	0.724843i	$0.741912\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1389.81	1.88576
$738$	0	0
$739$	518.771	0.701991	0.350996	−	0.936377i	$-0.385843\pi$
$739$	518.771	0.701991	0.350996	+	0.936377i	$0.385843\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	130.095i	0.175094i	0.996160	+	0.0875471i	$0.0279028\pi$
$743$	130.095i	0.175094i	−0.996160	+	0.0875471i	$0.972097\pi$
$744$	0	0
$745$	909.832	1.22125
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 196.179i	− 0.261921i
$750$	0	0
$751$	144.284i	0.192122i	0.995375	+	0.0960609i	$0.0306244\pi$
$751$	144.284i	0.192122i	−0.995375	+	0.0960609i	$0.969376\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−951.108	−1.25975
$756$	0	0
$757$	− 700.450i	− 0.925297i	−0.886542	−	0.462648i	$-0.846899\pi$
$757$	− 700.450i	− 0.925297i	0.886542	−	0.462648i	$-0.153101\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−508.775	−0.668561	−0.334281	−	0.942474i	$-0.608493\pi$
$761$	−508.775	−0.668561	−0.334281	+	0.942474i	$0.608493\pi$
$762$	0	0
$763$	517.941	0.678822
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 217.924i	− 0.284125i
$768$	0	0
$769$	1485.01	1.93110	0.965548	−	0.260226i	$-0.0837971\pi$
$769$	1485.01	1.93110	0.965548	+	0.260226i	$0.0837971\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1419.38i	1.83619i	0.396359	+	0.918096i	$0.370274\pi$
$773$	1419.38i	1.83619i	−0.396359	+	0.918096i	$0.629726\pi$
$774$	0	0
$775$	164.293i	0.211991i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−649.347	−0.833565
$780$	0	0
$781$	− 261.042i	− 0.334240i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	721.531	0.919148
$786$	0	0
$787$	277.527	0.352639	0.176319	−	0.984333i	$-0.443581\pi$
$787$	277.527	0.352639	0.176319	+	0.984333i	$0.443581\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	463.050i	0.585398i
$792$	0	0
$793$	−1811.08	−2.28383
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 599.355i	− 0.752013i	−0.926617	−	0.376007i	$-0.877297\pi$
$797$	− 599.355i	− 0.752013i	0.926617	−	0.376007i	$-0.122703\pi$
$798$	0	0
$799$	− 187.297i	− 0.234414i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1051.97	1.31004
$804$	0	0
$805$	− 292.186i	− 0.362963i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	297.174	0.367336	0.183668	−	0.982988i	$-0.441203\pi$
$809$	297.174	0.367336	0.183668	+	0.982988i	$0.441203\pi$
$810$	0	0
$811$	632.860	0.780346	0.390173	−	0.920742i	$-0.372415\pi$
$811$	632.860	0.780346	0.390173	+	0.920742i	$0.372415\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 318.809i	− 0.391177i
$816$	0	0
$817$	−1159.13	−1.41876
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1343.47i	− 1.63638i	−0.574947	−	0.818190i	$-0.694978\pi$
$821$	− 1343.47i	− 1.63638i	0.574947	−	0.818190i	$-0.305022\pi$
$822$	0	0
$823$	1078.41i	1.31034i	0.755480	+	0.655172i	$0.227404\pi$
$823$	1078.41i	1.31034i	−0.755480	+	0.655172i	$0.772596\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−922.881	−1.11594	−0.557969	−	0.829862i	$-0.688419\pi$
$827$	−922.881	−1.11594	−0.557969	+	0.829862i	$0.688419\pi$
$828$	0	0
$829$	− 295.258i	− 0.356162i	−0.984016	−	0.178081i	$-0.943011\pi$
$829$	− 295.258i	− 0.356162i	0.984016	−	0.178081i	$-0.0569889\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	288.537	0.346383
$834$	0	0
$835$	933.288	1.11771
$836$	0	0
$837$	0	0
$838$	0	0
$839$	101.848i	0.121392i	0.998156	+	0.0606960i	$0.0193320\pi$
$839$	101.848i	0.121392i	−0.998156	+	0.0606960i	$0.980668\pi$
$840$	0	0
$841$	519.561	0.617789
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1004.97i	1.18932i
$846$	0	0
$847$	− 834.015i	− 0.984670i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−491.451	−0.577499
$852$	0	0
$853$	− 740.329i	− 0.867912i	−0.900934	−	0.433956i	$-0.857117\pi$
$853$	− 740.329i	− 0.867912i	0.900934	−	0.433956i	$-0.142883\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	553.996	0.646436	0.323218	−	0.946325i	$-0.395235\pi$
$857$	553.996	0.646436	0.323218	+	0.946325i	$0.395235\pi$
$858$	0	0
$859$	−1350.24	−1.57187	−0.785934	−	0.618310i	$-0.787818\pi$
$859$	−1350.24	−1.57187	−0.785934	+	0.618310i	$0.787818\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 656.568i	− 0.760798i	−0.924823	−	0.380399i	$-0.875787\pi$
$863$	− 656.568i	− 0.760798i	0.924823	−	0.380399i	$-0.124213\pi$
$864$	0	0
$865$	−501.042	−0.579239
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 2223.82i	− 2.55905i
$870$	0	0
$871$	− 1474.05i	− 1.69237i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−484.932	−0.554208
$876$	0	0
$877$	− 1238.69i	− 1.41242i	−0.708001	−	0.706211i	$-0.750403\pi$
$877$	− 1238.69i	− 1.41242i	0.708001	−	0.706211i	$-0.249597\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−122.310	−0.138831	−0.0694154	−	0.997588i	$-0.522113\pi$
$881$	−122.310	−0.138831	−0.0694154	+	0.997588i	$0.522113\pi$
$882$	0	0
$883$	−1362.04	−1.54251	−0.771254	−	0.636527i	$-0.780370\pi$
$883$	−1362.04	−1.54251	−0.771254	+	0.636527i	$0.780370\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 519.056i	− 0.585182i	−0.956238	−	0.292591i	$-0.905483\pi$
$887$	− 519.056i	− 0.585182i	0.956238	−	0.292591i	$-0.0945174\pi$
$888$	0	0
$889$	−327.118	−0.367962
$890$	0	0
$891$	0	0
$892$	0	0
$893$	377.491i	0.422722i
$894$	0	0
$895$	129.022i	0.144159i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−533.062	−0.592950
$900$	0	0
$901$	− 726.306i	− 0.806111i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−752.845	−0.831873
$906$	0	0
$907$	169.347	0.186711	0.0933553	−	0.995633i	$-0.470241\pi$
$907$	169.347	0.186711	0.0933553	+	0.995633i	$0.470241\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	266.499i	0.292535i	0.989245	+	0.146267i	$0.0467260\pi$
$911$	266.499i	0.292535i	−0.989245	+	0.146267i	$0.953274\pi$
$912$	0	0
$913$	151.928	0.166405
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 282.890i	− 0.308495i
$918$	0	0
$919$	10.0832i	0.0109720i	0.999985	+	0.00548598i	$0.00174625\pi$
$919$	10.0832i	0.0109720i	−0.999985	+	0.00548598i	$0.998254\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−276.865	−0.299962
$924$	0	0
$925$	147.648i	0.159620i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	832.048	0.895639	0.447819	−	0.894124i	$-0.352201\pi$
$929$	832.048	0.895639	0.447819	+	0.894124i	$0.352201\pi$
$930$	0	0
$931$	−581.537	−0.624637
$932$	0	0
$933$	0	0
$934$	0	0
$935$	663.475i	0.709599i
$936$	0	0
$937$	846.801	0.903736	0.451868	−	0.892085i	$-0.350758\pi$
$937$	846.801	0.903736	0.451868	+	0.892085i	$0.350758\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	239.064i	0.254053i	0.991899	+	0.127027i	$0.0405433\pi$
$941$	239.064i	0.254053i	−0.991899	+	0.127027i	$0.959457\pi$
$942$	0	0
$943$	740.187i	0.784928i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−629.352	−0.664575	−0.332287	−	0.943178i	$-0.607820\pi$
$947$	−629.352	−0.664575	−0.332287	+	0.943178i	$0.607820\pi$
$948$	0	0
$949$	− 1115.73i	− 1.17569i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−1221.93	−1.28220	−0.641098	−	0.767459i	$-0.721521\pi$
$953$	−1221.93	−1.28220	−0.641098	+	0.767459i	$0.721521\pi$
$954$	0	0
$955$	1558.71	1.63216
$956$	0	0
$957$	0	0
$958$	0	0
$959$	977.305i	1.01909i
$960$	0	0
$961$	76.9918	0.0801163
$962$	0	0
$963$	0	0
$964$	0	0
$965$	104.046i	0.107819i
$966$	0	0
$967$	− 1138.01i	− 1.17685i	−0.808553	−	0.588423i	$-0.799749\pi$
$967$	− 1138.01i	− 1.17685i	0.808553	−	0.588423i	$-0.200251\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1822.54	1.87697	0.938484	−	0.345322i	$-0.112230\pi$
$971$	1822.54	1.87697	0.938484	+	0.345322i	$0.112230\pi$
$972$	0	0
$973$	640.208i	0.657973i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1695.80	−1.73572	−0.867862	−	0.496806i	$-0.834506\pi$
$977$	−1695.80	−1.73572	−0.867862	+	0.496806i	$0.834506\pi$
$978$	0	0
$979$	1576.53	1.61035
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 925.775i	− 0.941785i	−0.882191	−	0.470893i	$-0.843932\pi$
$983$	− 925.775i	− 0.941785i	0.882191	−	0.470893i	$-0.156068\pi$
$984$	0	0
$985$	−94.9973	−0.0964440
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1321.28i	1.33598i
$990$	0	0
$991$	− 820.777i	− 0.828231i	−0.910224	−	0.414115i	$-0.864091\pi$
$991$	− 820.777i	− 0.828231i	0.910224	−	0.414115i	$-0.135909\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	153.853	0.154626
$996$	0	0
$997$	− 990.056i	− 0.993035i	−0.868027	−	0.496517i	$-0.834612\pi$
$997$	− 990.056i	− 0.993035i	0.868027	−	0.496517i	$-0.165388\pi$
$998$	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.271.5		16
3.2	odd	2	inner	864.3.b.a.271.11		16
4.3	odd	2		216.3.b.a.163.8	yes	16
8.3	odd	2	inner	864.3.b.a.271.12		16
8.5	even	2		216.3.b.a.163.7	✓	16
12.11	even	2		216.3.b.a.163.9	yes	16
24.5	odd	2		216.3.b.a.163.10	yes	16
24.11	even	2	inner	864.3.b.a.271.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.3.b.a.163.7	✓	16	8.5	even	2
216.3.b.a.163.8	yes	16	4.3	odd	2
216.3.b.a.163.9	yes	16	12.11	even	2
216.3.b.a.163.10	yes	16	24.5	odd	2
864.3.b.a.271.5		16	1.1	even	1	trivial
864.3.b.a.271.6		16	24.11	even	2	inner
864.3.b.a.271.11		16	3.2	odd	2	inner
864.3.b.a.271.12		16	8.3	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)

\(f(q)\)	\(=\)	\(q-4.41296i q^{5} -3.59985i q^{7} +O(q^{10})\)	\(q-4.41296i q^{5} -3.59985i q^{7} -18.7798 q^{11} +19.9181i q^{13} +8.00578 q^{17} -16.1354 q^{19} +18.3926i q^{23} +5.52575 q^{25} +17.9287i q^{29} +29.7323i q^{31} -15.8860 q^{35} +26.7200i q^{37} +40.2437 q^{41} +71.8376 q^{43} -23.3952i q^{47} +36.0411 q^{49} -90.7227i q^{53} +82.8745i q^{55} -10.9410 q^{59} +90.9260i q^{61} +87.8980 q^{65} -74.0054 q^{67} +13.9001i q^{71} -56.0158 q^{73} +67.6044i q^{77} +118.415i q^{79} -8.08995 q^{83} -35.3292i q^{85} -83.9483 q^{89} +71.7022 q^{91} +71.2049i q^{95} -79.0054 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 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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