LMFDB - Embedded newform 864.3.e.a.161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,3,Mod(161,864)]

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([0, 0, 1]))

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

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

chi := DirichletCharacter("864.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	864.e (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:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$-1.58114 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	864.161
Dual form			864.3.e.a.161.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.47214i q^{5} -9.32456 q^{7} +O(q^{10})$	$q-4.47214i q^{5} -9.32456 q^{7} -19.0733i q^{11} -23.9737 q^{13} +30.3870i q^{17} +21.6491 q^{19} +13.4164i q^{23} +5.00000 q^{25} +42.8854i q^{29} +19.2982 q^{31} +41.7007i q^{35} +25.9737 q^{37} -10.7802i q^{41} -32.5964 q^{43} +28.9355i q^{47} +37.9473 q^{49} +58.9380i q^{53} -85.2982 q^{55} -68.5335i q^{59} -2.02633 q^{61} +107.213i q^{65} +70.2982 q^{67} +2.90291i q^{71} -70.9473 q^{73} +177.850i q^{77} -32.6754 q^{79} -41.0494i q^{83} +135.895 q^{85} +116.158i q^{89} +223.544 q^{91} -96.8178i q^{95} +5.05267 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{7}+O(q^{10}) $ 4 * q - 12 * q^7	$ 4 q - 12 q^{7} - 20 q^{13} + 36 q^{19} + 20 q^{25} - 24 q^{31} + 28 q^{37} + 72 q^{43} - 240 q^{55} - 84 q^{61} + 180 q^{67} - 132 q^{73} - 156 q^{79} + 240 q^{85} + 540 q^{91} + 172 q^{97}+O(q^{100}) $ 4 * q - 12 * q^7 - 20 * q^13 + 36 * q^19 + 20 * q^25 - 24 * q^31 + 28 * q^37 + 72 * q^43 - 240 * q^55 - 84 * q^61 + 180 * q^67 - 132 * q^73 - 156 * q^79 + 240 * q^85 + 540 * q^91 + 172 * 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
$5$	− 4.47214i	− 0.894427i	−0.894427	−	0.447214i	$-0.852416\pi$
$5$	− 4.47214i	− 0.894427i	0.894427	−	0.447214i	$-0.147584\pi$
$6$	0	0
$7$	−9.32456	−1.33208	−0.666040	−	0.745916i	$-0.732012\pi$
$7$	−9.32456	−1.33208	−0.666040	+	0.745916i	$0.732012\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 19.0733i	− 1.73393i	−0.498366	−	0.866966i	$-0.666067\pi$
$11$	− 19.0733i	− 1.73393i	0.498366	−	0.866966i	$-0.333933\pi$
$12$	0	0
$13$	−23.9737	−1.84413	−0.922064	−	0.387037i	$-0.873498\pi$
$13$	−23.9737	−1.84413	−0.922064	+	0.387037i	$0.873498\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	30.3870i	1.78747i	0.448596	+	0.893734i	$0.351924\pi$
$17$	30.3870i	1.78747i	−0.448596	+	0.893734i	$0.648076\pi$
$18$	0	0
$19$	21.6491	1.13943	0.569713	−	0.821843i	$-0.307054\pi$
$19$	21.6491	1.13943	0.569713	+	0.821843i	$0.307054\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	13.4164i	0.583322i	0.956522	+	0.291661i	$0.0942079\pi$
$23$	13.4164i	0.583322i	−0.956522	+	0.291661i	$0.905792\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	42.8854i	1.47881i	0.673263	+	0.739403i	$0.264892\pi$
$29$	42.8854i	1.47881i	−0.673263	+	0.739403i	$0.735108\pi$
$30$	0	0
$31$	19.2982	0.622523	0.311262	−	0.950324i	$-0.399248\pi$
$31$	19.2982	0.622523	0.311262	+	0.950324i	$0.399248\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	41.7007i	1.19145i
$36$	0	0
$37$	25.9737	0.701991	0.350995	−	0.936377i	$-0.385843\pi$
$37$	25.9737	0.701991	0.350995	+	0.936377i	$0.385843\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 10.7802i	− 0.262933i	−0.991321	−	0.131466i	$-0.958031\pi$
$41$	− 10.7802i	− 0.262933i	0.991321	−	0.131466i	$-0.0419685\pi$
$42$	0	0
$43$	−32.5964	−0.758057	−0.379028	−	0.925385i	$-0.623742\pi$
$43$	−32.5964	−0.758057	−0.379028	+	0.925385i	$0.623742\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	28.9355i	0.615649i	0.951443	+	0.307825i	$0.0996010\pi$
$47$	28.9355i	0.615649i	−0.951443	+	0.307825i	$0.900399\pi$
$48$	0	0
$49$	37.9473	0.774435
$50$	0	0
$51$	0	0
$52$	0	0
$53$	58.9380i	1.11204i	0.831170	+	0.556019i	$0.187672\pi$
$53$	58.9380i	1.11204i	−0.831170	+	0.556019i	$0.812328\pi$
$54$	0	0
$55$	−85.2982	−1.55088
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 68.5335i	− 1.16158i	−0.814052	−	0.580792i	$-0.802743\pi$
$59$	− 68.5335i	− 1.16158i	0.814052	−	0.580792i	$-0.197257\pi$
$60$	0	0
$61$	−2.02633	−0.0332186	−0.0166093	−	0.999862i	$-0.505287\pi$
$61$	−2.02633	−0.0332186	−0.0166093	+	0.999862i	$0.505287\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	107.213i	1.64944i
$66$	0	0
$67$	70.2982	1.04923	0.524614	−	0.851340i	$-0.324210\pi$
$67$	70.2982	1.04923	0.524614	+	0.851340i	$0.324210\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.90291i	0.0408861i	0.999791	+	0.0204430i	$0.00650767\pi$
$71$	2.90291i	0.0408861i	−0.999791	+	0.0204430i	$0.993492\pi$
$72$	0	0
$73$	−70.9473	−0.971881	−0.485941	−	0.873992i	$-0.661523\pi$
$73$	−70.9473	−0.971881	−0.485941	+	0.873992i	$0.661523\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	177.850i	2.30974i
$78$	0	0
$79$	−32.6754	−0.413613	−0.206807	−	0.978382i	$-0.566307\pi$
$79$	−32.6754	−0.413613	−0.206807	+	0.978382i	$0.566307\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 41.0494i	− 0.494572i	−0.968943	−	0.247286i	$-0.920461\pi$
$83$	− 41.0494i	− 0.494572i	0.968943	−	0.247286i	$-0.0795386\pi$
$84$	0	0
$85$	135.895	1.59876
$86$	0	0
$87$	0	0
$88$	0	0
$89$	116.158i	1.30514i	0.757727	+	0.652572i	$0.226310\pi$
$89$	116.158i	1.30514i	−0.757727	+	0.652572i	$0.773690\pi$
$90$	0	0
$91$	223.544	2.45652
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 96.8178i	− 1.01913i
$96$	0	0
$97$	5.05267	0.0520894	0.0260447	−	0.999661i	$-0.491709\pi$
$97$	5.05267	0.0520894	0.0260447	+	0.999661i	$0.491709\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 73.3901i	− 0.726635i	−0.931665	−	0.363318i	$-0.881644\pi$
$101$	− 73.3901i	− 0.726635i	0.931665	−	0.363318i	$-0.118356\pi$
$102$	0	0
$103$	−141.974	−1.37839	−0.689193	−	0.724578i	$-0.742035\pi$
$103$	−141.974	−1.37839	−0.689193	+	0.724578i	$0.742035\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	154.838i	1.44708i	0.690281	+	0.723541i	$0.257487\pi$
$107$	154.838i	1.44708i	−0.690281	+	0.723541i	$0.742513\pi$
$108$	0	0
$109$	−61.8947	−0.567841	−0.283920	−	0.958848i	$-0.591635\pi$
$109$	−61.8947	−0.567841	−0.283920	+	0.958848i	$0.591635\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	80.3807i	0.711333i	0.934613	+	0.355667i	$0.115746\pi$
$113$	80.3807i	0.711333i	−0.934613	+	0.355667i	$0.884254\pi$
$114$	0	0
$115$	60.0000	0.521739
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 283.345i	− 2.38105i
$120$	0	0
$121$	−242.789	−2.00652
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 134.164i	− 1.07331i
$126$	0	0
$127$	−140.596	−1.10706	−0.553529	−	0.832830i	$-0.686719\pi$
$127$	−140.596	−1.10706	−0.553529	+	0.832830i	$0.686719\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	165.202i	1.26109i	0.776154	+	0.630543i	$0.217168\pi$
$131$	165.202i	1.26109i	−0.776154	+	0.630543i	$0.782832\pi$
$132$	0	0
$133$	−201.868	−1.51781
$134$	0	0
$135$	0	0
$136$	0	0
$137$	57.2198i	0.417663i	0.977952	+	0.208831i	$0.0669660\pi$
$137$	57.2198i	0.417663i	−0.977952	+	0.208831i	$0.933034\pi$
$138$	0	0
$139$	2.50889	0.0180496	0.00902480	−	0.999959i	$-0.497127\pi$
$139$	2.50889	0.0180496	0.00902480	+	0.999959i	$0.497127\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	457.256i	3.19759i
$144$	0	0
$145$	191.789	1.32269
$146$	0	0
$147$	0	0
$148$	0	0
$149$	194.938i	1.30831i	0.756361	+	0.654154i	$0.226975\pi$
$149$	194.938i	1.30831i	−0.756361	+	0.654154i	$0.773025\pi$
$150$	0	0
$151$	−26.1843	−0.173406	−0.0867031	−	0.996234i	$-0.527633\pi$
$151$	−26.1843	−0.173406	−0.0867031	+	0.996234i	$0.527633\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 86.3043i	− 0.556802i
$156$	0	0
$157$	−37.8947	−0.241367	−0.120684	−	0.992691i	$-0.538509\pi$
$157$	−37.8947	−0.241367	−0.120684	+	0.992691i	$0.538509\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 125.102i	− 0.777031i
$162$	0	0
$163$	−230.035	−1.41126	−0.705628	−	0.708582i	$-0.749335\pi$
$163$	−230.035	−1.41126	−0.705628	+	0.708582i	$0.749335\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 31.8384i	− 0.190649i	−0.995446	−	0.0953246i	$-0.969611\pi$
$167$	− 31.8384i	− 0.190649i	0.995446	−	0.0953246i	$-0.0303889\pi$
$168$	0	0
$169$	405.737	2.40081
$170$	0	0
$171$	0	0
$172$	0	0
$173$	134.164i	0.775515i	0.921761	+	0.387757i	$0.126750\pi$
$173$	134.164i	0.775515i	−0.921761	+	0.387757i	$0.873250\pi$
$174$	0	0
$175$	−46.6228	−0.266416
$176$	0	0
$177$	0	0
$178$	0	0
$179$	8.70873i	0.0486521i	0.999704	+	0.0243261i	$0.00774399\pi$
$179$	8.70873i	0.0486521i	−0.999704	+	0.0243261i	$0.992256\pi$
$180$	0	0
$181$	91.9210	0.507851	0.253925	−	0.967224i	$-0.418278\pi$
$181$	91.9210	0.507851	0.253925	+	0.967224i	$0.418278\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 116.158i	− 0.627880i
$186$	0	0
$187$	579.579	3.09935
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 136.565i	− 0.714999i	−0.933913	−	0.357499i	$-0.883629\pi$
$191$	− 136.565i	− 0.714999i	0.933913	−	0.357499i	$-0.116371\pi$
$192$	0	0
$193$	−41.1053	−0.212981	−0.106491	−	0.994314i	$-0.533961\pi$
$193$	−41.1053	−0.212981	−0.106491	+	0.994314i	$0.533961\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	88.6425i	0.449962i	0.974363	+	0.224981i	$0.0722320\pi$
$197$	88.6425i	0.449962i	−0.974363	+	0.224981i	$0.927768\pi$
$198$	0	0
$199$	35.1139	0.176452	0.0882258	−	0.996100i	$-0.471880\pi$
$199$	35.1139	0.176452	0.0882258	+	0.996100i	$0.471880\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 399.887i	− 1.96989i
$204$	0	0
$205$	−48.2107	−0.235174
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 412.919i	− 1.97569i
$210$	0	0
$211$	183.491	0.869626	0.434813	−	0.900521i	$-0.356814\pi$
$211$	183.491	0.869626	0.434813	+	0.900521i	$0.356814\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	145.776i	0.678027i
$216$	0	0
$217$	−179.947	−0.829250
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 728.487i	− 3.29632i
$222$	0	0
$223$	−255.088	−1.14389	−0.571945	−	0.820292i	$-0.693811\pi$
$223$	−255.088	−1.14389	−0.571945	+	0.820292i	$0.693811\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 169.706i	− 0.747602i	−0.927509	−	0.373801i	$-0.878054\pi$
$227$	− 169.706i	− 0.747602i	0.927509	−	0.373801i	$-0.121946\pi$
$228$	0	0
$229$	146.000	0.637555	0.318777	−	0.947830i	$-0.396728\pi$
$229$	146.000	0.637555	0.318777	+	0.947830i	$0.396728\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 48.1577i	− 0.206686i	−0.994646	−	0.103343i	$-0.967046\pi$
$233$	− 48.1577i	− 0.206686i	0.994646	−	0.103343i	$-0.0329539\pi$
$234$	0	0
$235$	129.404	0.550653
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 257.312i	− 1.07662i	−0.842747	−	0.538310i	$-0.819063\pi$
$239$	− 257.312i	− 1.07662i	0.842747	−	0.538310i	$-0.180937\pi$
$240$	0	0
$241$	−112.684	−0.467568	−0.233784	−	0.972289i	$-0.575111\pi$
$241$	−112.684	−0.467568	−0.233784	+	0.972289i	$0.575111\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 169.706i	− 0.692676i
$246$	0	0
$247$	−519.009	−2.10125
$248$	0	0
$249$	0	0
$250$	0	0
$251$	400.185i	1.59436i	0.603740	+	0.797182i	$0.293677\pi$
$251$	400.185i	1.59436i	−0.603740	+	0.797182i	$0.706323\pi$
$252$	0	0
$253$	255.895	1.01144
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 121.312i	− 0.472032i	−0.971749	−	0.236016i	$-0.924158\pi$
$257$	− 121.312i	− 0.472032i	0.971749	−	0.236016i	$-0.0758419\pi$
$258$	0	0
$259$	−242.193	−0.935108
$260$	0	0
$261$	0	0
$262$	0	0
$263$	200.446i	0.762152i	0.924544	+	0.381076i	$0.124446\pi$
$263$	200.446i	0.762152i	−0.924544	+	0.381076i	$0.875554\pi$
$264$	0	0
$265$	263.579	0.994636
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 48.9580i	− 0.182000i	−0.995851	−	0.0909999i	$-0.970994\pi$
$269$	− 48.9580i	− 0.182000i	0.995851	−	0.0909999i	$-0.0290063\pi$
$270$	0	0
$271$	207.974	0.767431	0.383715	−	0.923451i	$-0.374644\pi$
$271$	207.974	0.767431	0.383715	+	0.923451i	$0.374644\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 95.3663i	− 0.346787i
$276$	0	0
$277$	−293.895	−1.06099	−0.530496	−	0.847688i	$-0.677994\pi$
$277$	−293.895	−1.06099	−0.530496	+	0.847688i	$0.677994\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	159.396i	0.567247i	0.958936	+	0.283624i	$0.0915366\pi$
$281$	159.396i	0.567247i	−0.958936	+	0.283624i	$0.908463\pi$
$282$	0	0
$283$	87.4036	0.308846	0.154423	−	0.988005i	$-0.450648\pi$
$283$	87.4036	0.308846	0.154423	+	0.988005i	$0.450648\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	100.521i	0.350247i
$288$	0	0
$289$	−634.368	−2.19504
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 317.286i	− 1.08289i	−0.840737	−	0.541444i	$-0.817878\pi$
$293$	− 317.286i	− 1.08289i	0.840737	−	0.541444i	$-0.182122\pi$
$294$	0	0
$295$	−306.491	−1.03895
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 321.640i	− 1.07572i
$300$	0	0
$301$	303.947	1.00979
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.06204i	0.0297116i
$306$	0	0
$307$	−70.2107	−0.228699	−0.114350	−	0.993441i	$-0.536478\pi$
$307$	−70.2107	−0.228699	−0.114350	+	0.993441i	$0.536478\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	342.519i	1.10135i	0.834721	+	0.550673i	$0.185629\pi$
$311$	342.519i	1.10135i	−0.834721	+	0.550673i	$0.814371\pi$
$312$	0	0
$313$	127.316	0.406760	0.203380	−	0.979100i	$-0.434807\pi$
$313$	127.316	0.406760	0.203380	+	0.979100i	$0.434807\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 157.089i	− 0.495550i	−0.968818	−	0.247775i	$-0.920301\pi$
$317$	− 157.089i	− 0.495550i	0.968818	−	0.247775i	$-0.0796994\pi$
$318$	0	0
$319$	817.964	2.56415
$320$	0	0
$321$	0	0
$322$	0	0
$323$	657.851i	2.03669i
$324$	0	0
$325$	−119.868	−0.368826
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 269.811i	− 0.820094i
$330$	0	0
$331$	−225.175	−0.680287	−0.340144	−	0.940373i	$-0.610476\pi$
$331$	−225.175	−0.680287	−0.340144	+	0.940373i	$0.610476\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 314.383i	− 0.938457i
$336$	0	0
$337$	−490.737	−1.45619	−0.728096	−	0.685475i	$-0.759594\pi$
$337$	−490.737	−1.45619	−0.728096	+	0.685475i	$0.759594\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 368.080i	− 1.07941i
$342$	0	0
$343$	103.061	0.300470
$344$	0	0
$345$	0	0
$346$	0	0
$347$	155.191i	0.447237i	0.974677	+	0.223618i	$0.0717869\pi$
$347$	155.191i	0.447237i	−0.974677	+	0.223618i	$0.928213\pi$
$348$	0	0
$349$	−419.974	−1.20336	−0.601681	−	0.798736i	$-0.705502\pi$
$349$	−419.974	−1.20336	−0.601681	+	0.798736i	$0.705502\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 304.105i	− 0.861488i	−0.902474	−	0.430744i	$-0.858251\pi$
$353$	− 304.105i	− 0.861488i	0.902474	−	0.430744i	$-0.141749\pi$
$354$	0	0
$355$	12.9822	0.0365696
$356$	0	0
$357$	0	0
$358$	0	0
$359$	181.820i	0.506461i	0.967406	+	0.253231i	$0.0814931\pi$
$359$	181.820i	0.506461i	−0.967406	+	0.253231i	$0.918507\pi$
$360$	0	0
$361$	107.684	0.298294
$362$	0	0
$363$	0	0
$364$	0	0
$365$	317.286i	0.869277i
$366$	0	0
$367$	679.868	1.85250	0.926251	−	0.376907i	$-0.123012\pi$
$367$	679.868	1.85250	0.926251	+	0.376907i	$0.123012\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 549.570i	− 1.48132i
$372$	0	0
$373$	−117.605	−0.315295	−0.157647	−	0.987495i	$-0.550391\pi$
$373$	−117.605	−0.315295	−0.157647	+	0.987495i	$0.550391\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 1028.12i	− 2.72711i
$378$	0	0
$379$	−94.1402	−0.248391	−0.124196	−	0.992258i	$-0.539635\pi$
$379$	−94.1402	−0.248391	−0.124196	+	0.992258i	$0.539635\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	17.1195i	0.0446985i	0.999750	+	0.0223493i	$0.00711458\pi$
$383$	17.1195i	0.0446985i	−0.999750	+	0.0223493i	$0.992885\pi$
$384$	0	0
$385$	795.368	2.06589
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 208.354i	− 0.535615i	−0.963472	−	0.267808i	$-0.913701\pi$
$389$	− 208.354i	− 0.535615i	0.963472	−	0.267808i	$-0.0862992\pi$
$390$	0	0
$391$	−407.684	−1.04267
$392$	0	0
$393$	0	0
$394$	0	0
$395$	146.129i	0.369947i
$396$	0	0
$397$	609.684	1.53573	0.767864	−	0.640613i	$-0.221320\pi$
$397$	609.684	1.53573	0.767864	+	0.640613i	$0.221320\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	500.879i	1.24908i	0.780995	+	0.624538i	$0.214713\pi$
$401$	500.879i	1.24908i	−0.780995	+	0.624538i	$0.785287\pi$
$402$	0	0
$403$	−462.649	−1.14801
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 495.403i	− 1.21721i
$408$	0	0
$409$	−506.737	−1.23896	−0.619482	−	0.785010i	$-0.712658\pi$
$409$	−506.737	−1.23896	−0.619482	+	0.785010i	$0.712658\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	639.044i	1.54732i
$414$	0	0
$415$	−183.579	−0.442358
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 235.336i	− 0.561662i	−0.959757	−	0.280831i	$-0.909390\pi$
$419$	− 235.336i	− 0.561662i	0.959757	−	0.280831i	$-0.0906100\pi$
$420$	0	0
$421$	−42.3423	−0.100576	−0.0502878	−	0.998735i	$-0.516014\pi$
$421$	−42.3423	−0.100576	−0.0502878	+	0.998735i	$0.516014\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	151.935i	0.357494i
$426$	0	0
$427$	18.8947	0.0442498
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 524.840i	− 1.21773i	−0.793275	−	0.608863i	$-0.791626\pi$
$431$	− 524.840i	− 1.21773i	0.793275	−	0.608863i	$-0.208374\pi$
$432$	0	0
$433$	845.157	1.95186	0.975932	−	0.218074i	$-0.0699774\pi$
$433$	845.157	1.95186	0.975932	+	0.218074i	$0.0699774\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	290.453i	0.664653i
$438$	0	0
$439$	−454.877	−1.03617	−0.518083	−	0.855330i	$-0.673354\pi$
$439$	−454.877	−1.03617	−0.518083	+	0.855330i	$0.673354\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	679.529i	1.53393i	0.641691	+	0.766963i	$0.278233\pi$
$443$	679.529i	1.53393i	−0.641691	+	0.766963i	$0.721767\pi$
$444$	0	0
$445$	519.473	1.16736
$446$	0	0
$447$	0	0
$448$	0	0
$449$	480.801i	1.07083i	0.844590	+	0.535414i	$0.179844\pi$
$449$	480.801i	1.07083i	−0.844590	+	0.535414i	$0.820156\pi$
$450$	0	0
$451$	−205.614	−0.455907
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 999.718i	− 2.19718i
$456$	0	0
$457$	741.789	1.62317	0.811586	−	0.584233i	$-0.198605\pi$
$457$	741.789	1.62317	0.811586	+	0.584233i	$0.198605\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	254.912i	0.552954i	0.961021	+	0.276477i	$0.0891669\pi$
$461$	254.912i	0.552954i	−0.961021	+	0.276477i	$0.910833\pi$
$462$	0	0
$463$	683.430	1.47609	0.738045	−	0.674751i	$-0.235749\pi$
$463$	683.430	1.47609	0.738045	+	0.674751i	$0.235749\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 441.290i	− 0.944946i	−0.881345	−	0.472473i	$-0.843361\pi$
$467$	− 441.290i	− 0.944946i	0.881345	−	0.472473i	$-0.156639\pi$
$468$	0	0
$469$	−655.500	−1.39765
$470$	0	0
$471$	0	0
$472$	0	0
$473$	621.720i	1.31442i
$474$	0	0
$475$	108.246	0.227885
$476$	0	0
$477$	0	0
$478$	0	0
$479$	419.910i	0.876638i	0.898819	+	0.438319i	$0.144426\pi$
$479$	419.910i	0.876638i	−0.898819	+	0.438319i	$0.855574\pi$
$480$	0	0
$481$	−622.684	−1.29456
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 22.5962i	− 0.0465901i
$486$	0	0
$487$	−29.1139	−0.0597821	−0.0298911	−	0.999553i	$-0.509516\pi$
$487$	−29.1139	−0.0597821	−0.0298911	+	0.999553i	$0.509516\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	710.214i	1.44646i	0.690605	+	0.723232i	$0.257344\pi$
$491$	710.214i	1.44646i	−0.690605	+	0.723232i	$0.742656\pi$
$492$	0	0
$493$	−1303.16	−2.64332
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 27.0684i	− 0.0544635i
$498$	0	0
$499$	−969.579	−1.94304	−0.971522	−	0.236951i	$-0.923852\pi$
$499$	−969.579	−1.94304	−0.971522	+	0.236951i	$0.923852\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	636.079i	1.26457i	0.774736	+	0.632285i	$0.217883\pi$
$503$	636.079i	1.26457i	−0.774736	+	0.632285i	$0.782117\pi$
$504$	0	0
$505$	−328.211	−0.649922
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 112.039i	− 0.220116i	−0.993925	−	0.110058i	$-0.964896\pi$
$509$	− 112.039i	− 0.220116i	0.993925	−	0.110058i	$-0.0351036\pi$
$510$	0	0
$511$	661.552	1.29462
$512$	0	0
$513$	0	0
$514$	0	0
$515$	634.926i	1.23287i
$516$	0	0
$517$	551.895	1.06749
$518$	0	0
$519$	0	0
$520$	0	0
$521$	402.139i	0.771860i	0.922528	+	0.385930i	$0.126119\pi$
$521$	402.139i	0.771860i	−0.922528	+	0.385930i	$0.873881\pi$
$522$	0	0
$523$	993.999	1.90057	0.950286	−	0.311378i	$-0.100791\pi$
$523$	993.999	1.90057	0.950286	+	0.311378i	$0.100791\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	586.414i	1.11274i
$528$	0	0
$529$	349.000	0.659735
$530$	0	0
$531$	0	0
$532$	0	0
$533$	258.442i	0.484881i
$534$	0	0
$535$	692.456	1.29431
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 723.779i	− 1.34282i
$540$	0	0
$541$	192.342	0.355531	0.177766	−	0.984073i	$-0.443113\pi$
$541$	192.342	0.355531	0.177766	+	0.984073i	$0.443113\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	276.801i	0.507892i
$546$	0	0
$547$	−760.947	−1.39113	−0.695564	−	0.718464i	$-0.744846\pi$
$547$	−760.947	−1.39113	−0.695564	+	0.718464i	$0.744846\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	928.431i	1.68499i
$552$	0	0
$553$	304.684	0.550966
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 546.636i	− 0.981394i	−0.871330	−	0.490697i	$-0.836742\pi$
$557$	− 546.636i	− 0.981394i	0.871330	−	0.490697i	$-0.163258\pi$
$558$	0	0
$559$	781.456	1.39795
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 894.192i	− 1.58826i	−0.607746	−	0.794131i	$-0.707926\pi$
$563$	− 894.192i	− 1.58826i	0.607746	−	0.794131i	$-0.292074\pi$
$564$	0	0
$565$	359.473	0.636236
$566$	0	0
$567$	0	0
$568$	0	0
$569$	330.585i	0.580993i	0.956876	+	0.290496i	$0.0938204\pi$
$569$	330.585i	0.580993i	−0.956876	+	0.290496i	$0.906180\pi$
$570$	0	0
$571$	−623.456	−1.09187	−0.545933	−	0.837829i	$-0.683825\pi$
$571$	−623.456	−1.09187	−0.545933	+	0.837829i	$0.683825\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	67.0820i	0.116664i
$576$	0	0
$577$	−809.789	−1.40345	−0.701723	−	0.712450i	$-0.747586\pi$
$577$	−809.789	−1.40345	−0.701723	+	0.712450i	$0.747586\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	382.768i	0.658808i
$582$	0	0
$583$	1124.14	1.92820
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 140.025i	− 0.238544i	−0.992862	−	0.119272i	$-0.961944\pi$
$587$	− 140.025i	− 0.238544i	0.992862	−	0.119272i	$-0.0380560\pi$
$588$	0	0
$589$	417.789	0.709320
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1027.65i	1.73297i	0.499206	+	0.866483i	$0.333625\pi$
$593$	1027.65i	1.73297i	−0.499206	+	0.866483i	$0.666375\pi$
$594$	0	0
$595$	−1267.16	−2.12968
$596$	0	0
$597$	0	0
$598$	0	0
$599$	691.141i	1.15382i	0.816806	+	0.576912i	$0.195743\pi$
$599$	691.141i	1.15382i	−0.816806	+	0.576912i	$0.804257\pi$
$600$	0	0
$601$	−744.736	−1.23916	−0.619581	−	0.784933i	$-0.712697\pi$
$601$	−744.736	−1.23916	−0.619581	+	0.784933i	$0.712697\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1085.79i	1.79469i
$606$	0	0
$607$	−515.622	−0.849460	−0.424730	−	0.905320i	$-0.639631\pi$
$607$	−515.622	−0.849460	−0.424730	+	0.905320i	$0.639631\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 693.690i	− 1.13534i
$612$	0	0
$613$	347.605	0.567055	0.283528	−	0.958964i	$-0.408495\pi$
$613$	347.605	0.567055	0.283528	+	0.958964i	$0.408495\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 161.350i	− 0.261508i	−0.991415	−	0.130754i	$-0.958260\pi$
$617$	− 161.350i	− 0.261508i	0.991415	−	0.130754i	$-0.0417397\pi$
$618$	0	0
$619$	252.579	0.408043	0.204022	−	0.978966i	$-0.434599\pi$
$619$	252.579	0.408043	0.204022	+	0.978966i	$0.434599\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1083.12i	− 1.73855i
$624$	0	0
$625$	−475.000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	789.261i	1.25479i
$630$	0	0
$631$	−1064.04	−1.68628	−0.843141	−	0.537693i	$-0.819296\pi$
$631$	−1064.04	−1.68628	−0.843141	+	0.537693i	$0.819296\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	628.766i	0.990183i
$636$	0	0
$637$	−909.737	−1.42816
$638$	0	0
$639$	0	0
$640$	0	0
$641$	616.919i	0.962432i	0.876602	+	0.481216i	$0.159805\pi$
$641$	616.919i	0.962432i	−0.876602	+	0.481216i	$0.840195\pi$
$642$	0	0
$643$	−781.789	−1.21585	−0.607923	−	0.793996i	$-0.707997\pi$
$643$	−781.789	−1.21585	−0.607923	+	0.793996i	$0.707997\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 490.993i	− 0.758876i	−0.925217	−	0.379438i	$-0.876117\pi$
$647$	− 490.993i	− 0.758876i	0.925217	−	0.379438i	$-0.123883\pi$
$648$	0	0
$649$	−1307.16	−2.01411
$650$	0	0
$651$	0	0
$652$	0	0
$653$	446.507i	0.683778i	0.939740	+	0.341889i	$0.111067\pi$
$653$	446.507i	0.683778i	−0.939740	+	0.341889i	$0.888933\pi$
$654$	0	0
$655$	738.807	1.12795
$656$	0	0
$657$	0	0
$658$	0	0
$659$	471.268i	0.715126i	0.933889	+	0.357563i	$0.116392\pi$
$659$	471.268i	0.715126i	−0.933889	+	0.357563i	$0.883608\pi$
$660$	0	0
$661$	487.921	0.738156	0.369078	−	0.929398i	$-0.379674\pi$
$661$	487.921	0.738156	0.369078	+	0.929398i	$0.379674\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	902.783i	1.35757i
$666$	0	0
$667$	−575.368	−0.862621
$668$	0	0
$669$	0	0
$670$	0	0
$671$	38.6488i	0.0575988i
$672$	0	0
$673$	600.315	0.891999	0.445999	−	0.895033i	$-0.352848\pi$
$673$	600.315	0.891999	0.445999	+	0.895033i	$0.352848\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	481.955i	0.711898i	0.934505	+	0.355949i	$0.115842\pi$
$677$	481.955i	0.711898i	−0.934505	+	0.355949i	$0.884158\pi$
$678$	0	0
$679$	−47.1139	−0.0693872
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 720.579i	− 1.05502i	−0.849549	−	0.527510i	$-0.823126\pi$
$683$	− 720.579i	− 1.05502i	0.849549	−	0.527510i	$-0.176874\pi$
$684$	0	0
$685$	255.895	0.373569
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1412.96i	− 2.05074i
$690$	0	0
$691$	−187.614	−0.271511	−0.135756	−	0.990742i	$-0.543346\pi$
$691$	−187.614	−0.271511	−0.135756	+	0.990742i	$0.543346\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 11.2201i	− 0.0161440i
$696$	0	0
$697$	327.579	0.469984
$698$	0	0
$699$	0	0
$700$	0	0
$701$	451.450i	0.644009i	0.946738	+	0.322004i	$0.104357\pi$
$701$	451.450i	0.644009i	−0.946738	+	0.322004i	$0.895643\pi$
$702$	0	0
$703$	562.307	0.799867
$704$	0	0
$705$	0	0
$706$	0	0
$707$	684.330i	0.967935i
$708$	0	0
$709$	729.447	1.02884	0.514420	−	0.857539i	$-0.328007\pi$
$709$	729.447	1.02884	0.514420	+	0.857539i	$0.328007\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	258.913i	0.363132i
$714$	0	0
$715$	2044.91	2.86002
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1070.60i	− 1.48901i	−0.667618	−	0.744504i	$-0.732686\pi$
$719$	− 1070.60i	− 1.48901i	0.667618	−	0.744504i	$-0.267314\pi$
$720$	0	0
$721$	1323.84	1.83612
$722$	0	0
$723$	0	0
$724$	0	0
$725$	214.427i	0.295761i
$726$	0	0
$727$	−242.105	−0.333020	−0.166510	−	0.986040i	$-0.553250\pi$
$727$	−242.105	−0.333020	−0.166510	+	0.986040i	$0.553250\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 990.507i	− 1.35500i
$732$	0	0
$733$	−1204.84	−1.64371	−0.821856	−	0.569695i	$-0.807061\pi$
$733$	−1204.84	−1.64371	−0.821856	+	0.569695i	$0.807061\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1340.82i	− 1.81929i
$738$	0	0
$739$	163.964	0.221873	0.110937	−	0.993827i	$-0.464615\pi$
$739$	163.964	0.221873	0.110937	+	0.993827i	$0.464615\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 219.370i	− 0.295249i	−0.989043	−	0.147625i	$-0.952837\pi$
$743$	− 219.370i	− 0.295249i	0.989043	−	0.147625i	$-0.0471628\pi$
$744$	0	0
$745$	871.789	1.17019
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1443.79i	− 1.92763i
$750$	0	0
$751$	283.535	0.377544	0.188772	−	0.982021i	$-0.439549\pi$
$751$	283.535	0.377544	0.188772	+	0.982021i	$0.439549\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	117.100i	0.155099i
$756$	0	0
$757$	616.815	0.814815	0.407408	−	0.913246i	$-0.366433\pi$
$757$	616.815	0.814815	0.407408	+	0.913246i	$0.366433\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 721.826i	− 0.948523i	−0.880384	−	0.474261i	$-0.842715\pi$
$761$	− 721.826i	− 0.948523i	0.880384	−	0.474261i	$-0.157285\pi$
$762$	0	0
$763$	577.140	0.756409
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1643.00i	2.14211i
$768$	0	0
$769$	813.947	1.05845	0.529224	−	0.848482i	$-0.322483\pi$
$769$	813.947	1.05845	0.529224	+	0.848482i	$0.322483\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	807.479i	1.04460i	0.852761	+	0.522302i	$0.174927\pi$
$773$	807.479i	1.04460i	−0.852761	+	0.522302i	$0.825073\pi$
$774$	0	0
$775$	96.4911	0.124505
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 233.382i	− 0.299592i
$780$	0	0
$781$	55.3680	0.0708937
$782$	0	0
$783$	0	0
$784$	0	0
$785$	169.470i	0.215885i
$786$	0	0
$787$	−808.631	−1.02749	−0.513743	−	0.857944i	$-0.671741\pi$
$787$	−808.631	−1.02749	−0.513743	+	0.857944i	$0.671741\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 749.514i	− 0.947553i
$792$	0	0
$793$	48.5787	0.0612593
$794$	0	0
$795$	0	0
$796$	0	0
$797$	188.536i	0.236557i	0.992980	+	0.118279i	$0.0377376\pi$
$797$	188.536i	0.236557i	−0.992980	+	0.118279i	$0.962262\pi$
$798$	0	0
$799$	−879.263	−1.10045
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1353.20i	1.68518i
$804$	0	0
$805$	−559.473	−0.694998
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1260.91i	1.55860i	0.626651	+	0.779300i	$0.284425\pi$
$809$	1260.91i	1.55860i	−0.626651	+	0.779300i	$0.715575\pi$
$810$	0	0
$811$	221.018	0.272525	0.136263	−	0.990673i	$-0.456491\pi$
$811$	221.018	0.272525	0.136263	+	0.990673i	$0.456491\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1028.75i	1.26227i
$816$	0	0
$817$	−705.684	−0.863750
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1539.64i	− 1.87532i	−0.347554	−	0.937660i	$-0.612988\pi$
$821$	− 1539.64i	− 1.87532i	0.347554	−	0.937660i	$-0.387012\pi$
$822$	0	0
$823$	−523.377	−0.635938	−0.317969	−	0.948101i	$-0.603001\pi$
$823$	−523.377	−0.635938	−0.317969	+	0.948101i	$0.603001\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 479.326i	− 0.579596i	−0.957088	−	0.289798i	$-0.906412\pi$
$827$	− 479.326i	− 0.579596i	0.957088	−	0.289798i	$-0.0935881\pi$
$828$	0	0
$829$	778.184	0.938702	0.469351	−	0.883012i	$-0.344488\pi$
$829$	778.184	0.938702	0.469351	+	0.883012i	$0.344488\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1153.10i	1.38428i
$834$	0	0
$835$	−142.386	−0.170522
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1451.17i	1.72964i	0.502082	+	0.864820i	$0.332568\pi$
$839$	1451.17i	1.72964i	−0.502082	+	0.864820i	$0.667432\pi$
$840$	0	0
$841$	−998.157	−1.18687
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1814.51i	− 2.14735i
$846$	0	0
$847$	2263.90	2.67285
$848$	0	0
$849$	0	0
$850$	0	0
$851$	348.473i	0.409487i
$852$	0	0
$853$	−853.078	−1.00009	−0.500046	−	0.865999i	$-0.666684\pi$
$853$	−853.078	−1.00009	−0.500046	+	0.865999i	$0.666684\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	975.161i	1.13788i	0.822380	+	0.568939i	$0.192646\pi$
$857$	975.161i	1.13788i	−0.822380	+	0.568939i	$0.807354\pi$
$858$	0	0
$859$	−594.912	−0.692563	−0.346282	−	0.938131i	$-0.612556\pi$
$859$	−594.912	−0.692563	−0.346282	+	0.938131i	$0.612556\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 1579.62i	− 1.83038i	−0.403020	−	0.915191i	$-0.632039\pi$
$863$	− 1579.62i	− 1.83038i	0.403020	−	0.915191i	$-0.367961\pi$
$864$	0	0
$865$	600.000	0.693642
$866$	0	0
$867$	0	0
$868$	0	0
$869$	623.227i	0.717178i
$870$	0	0
$871$	−1685.31	−1.93491
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1251.02i	1.42974i
$876$	0	0
$877$	−519.552	−0.592420	−0.296210	−	0.955123i	$-0.595723\pi$
$877$	−519.552	−0.592420	−0.296210	+	0.955123i	$0.595723\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1368.31i	− 1.55313i	−0.630037	−	0.776565i	$-0.716960\pi$
$881$	− 1368.31i	− 1.55313i	0.630037	−	0.776565i	$-0.283040\pi$
$882$	0	0
$883$	−431.438	−0.488605	−0.244303	−	0.969699i	$-0.578559\pi$
$883$	−431.438	−0.488605	−0.244303	+	0.969699i	$0.578559\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 425.418i	− 0.479614i	−0.970821	−	0.239807i	$-0.922916\pi$
$887$	− 425.418i	− 0.479614i	0.970821	−	0.239807i	$-0.0770842\pi$
$888$	0	0
$889$	1311.00	1.47469
$890$	0	0
$891$	0	0
$892$	0	0
$893$	626.428i	0.701487i
$894$	0	0
$895$	38.9466	0.0435158
$896$	0	0
$897$	0	0
$898$	0	0
$899$	827.612i	0.920592i
$900$	0	0
$901$	−1790.95	−1.98773
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 411.083i	− 0.454236i
$906$	0	0
$907$	−1364.56	−1.50448	−0.752238	−	0.658891i	$-0.771026\pi$
$907$	−1364.56	−1.50448	−0.752238	+	0.658891i	$0.771026\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1109.75i	1.21816i	0.793107	+	0.609082i	$0.208462\pi$
$911$	1109.75i	1.21816i	−0.793107	+	0.609082i	$0.791538\pi$
$912$	0	0
$913$	−782.947	−0.857554
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1540.44i	− 1.67987i
$918$	0	0
$919$	−498.316	−0.542237	−0.271119	−	0.962546i	$-0.587394\pi$
$919$	−498.316	−0.542237	−0.271119	+	0.962546i	$0.587394\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 69.5934i	− 0.0753992i
$924$	0	0
$925$	129.868	0.140398
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1125.57i	− 1.21159i	−0.795622	−	0.605794i	$-0.792856\pi$
$929$	− 1125.57i	− 1.21159i	0.795622	−	0.605794i	$-0.207144\pi$
$930$	0	0
$931$	821.526	0.882412
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 2591.95i	− 2.77214i
$936$	0	0
$937$	−1028.79	−1.09796	−0.548980	−	0.835835i	$-0.684984\pi$
$937$	−1028.79	−1.09796	−0.548980	+	0.835835i	$0.684984\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 863.781i	− 0.917939i	−0.888452	−	0.458969i	$-0.848219\pi$
$941$	− 863.781i	− 0.917939i	0.888452	−	0.458969i	$-0.151781\pi$
$942$	0	0
$943$	144.632	0.153374
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 478.730i	− 0.505523i	−0.967529	−	0.252761i	$-0.918661\pi$
$947$	− 478.730i	− 0.505523i	0.967529	−	0.252761i	$-0.0813387\pi$
$948$	0	0
$949$	1700.87	1.79227
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1302.55i	− 1.36678i	−0.730052	−	0.683392i	$-0.760504\pi$
$953$	− 1302.55i	− 1.36678i	0.730052	−	0.683392i	$-0.239496\pi$
$954$	0	0
$955$	−610.736	−0.639514
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 533.549i	− 0.556360i
$960$	0	0
$961$	−588.579	−0.612465
$962$	0	0
$963$	0	0
$964$	0	0
$965$	183.829i	0.190496i
$966$	0	0
$967$	727.236	0.752054	0.376027	−	0.926609i	$-0.377290\pi$
$967$	727.236	0.752054	0.376027	+	0.926609i	$0.377290\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1262.99i	− 1.30071i	−0.759632	−	0.650353i	$-0.774621\pi$
$971$	− 1262.99i	− 1.30071i	0.759632	−	0.650353i	$-0.225379\pi$
$972$	0	0
$973$	−23.3943	−0.0240435
$974$	0	0
$975$	0	0
$976$	0	0
$977$	635.514i	0.650475i	0.945632	+	0.325238i	$0.105444\pi$
$977$	635.514i	0.650475i	−0.945632	+	0.325238i	$0.894556\pi$
$978$	0	0
$979$	2215.51	2.26303
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1726.81i	− 1.75667i	−0.478043	−	0.878336i	$-0.658654\pi$
$983$	− 1726.81i	− 1.75667i	0.478043	−	0.878336i	$-0.341346\pi$
$984$	0	0
$985$	396.421	0.402458
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 437.327i	− 0.442191i
$990$	0	0
$991$	−495.465	−0.499965	−0.249983	−	0.968250i	$-0.580425\pi$
$991$	−495.465	−0.499965	−0.249983	+	0.968250i	$0.580425\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 157.034i	− 0.157823i
$996$	0	0
$997$	−508.420	−0.509950	−0.254975	−	0.966948i	$-0.582067\pi$
$997$	−508.420	−0.509950	−0.254975	+	0.966948i	$0.582067\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.3.e.a.161.1	✓	4
3.2	odd	2	inner	864.3.e.a.161.3	yes	4
4.3	odd	2		864.3.e.c.161.2	yes	4
8.3	odd	2		1728.3.e.t.1025.4		4
8.5	even	2		1728.3.e.q.1025.3		4
12.11	even	2		864.3.e.c.161.4	yes	4
24.5	odd	2		1728.3.e.q.1025.1		4
24.11	even	2		1728.3.e.t.1025.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.3.e.a.161.1	✓	4	1.1	even	1	trivial
864.3.e.a.161.3	yes	4	3.2	odd	2	inner
864.3.e.c.161.2	yes	4	4.3	odd	2
864.3.e.c.161.4	yes	4	12.11	even	2
1728.3.e.q.1025.1		4	24.5	odd	2
1728.3.e.q.1025.3		4	8.5	even	2
1728.3.e.t.1025.2		4	24.11	even	2
1728.3.e.t.1025.4		4	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.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.47214i q^{5} -9.32456 q^{7} +O(q^{10})\)	\(q-4.47214i q^{5} -9.32456 q^{7} -19.0733i q^{11} -23.9737 q^{13} +30.3870i q^{17} +21.6491 q^{19} +13.4164i q^{23} +5.00000 q^{25} +42.8854i q^{29} +19.2982 q^{31} +41.7007i q^{35} +25.9737 q^{37} -10.7802i q^{41} -32.5964 q^{43} +28.9355i q^{47} +37.9473 q^{49} +58.9380i q^{53} -85.2982 q^{55} -68.5335i q^{59} -2.02633 q^{61} +107.213i q^{65} +70.2982 q^{67} +2.90291i q^{71} -70.9473 q^{73} +177.850i q^{77} -32.6754 q^{79} -41.0494i q^{83} +135.895 q^{85} +116.158i q^{89} +223.544 q^{91} -96.8178i q^{95} +5.05267 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{7}+O(q^{10}) \) 4 * q - 12 * q^7	\( 4 q - 12 q^{7} - 20 q^{13} + 36 q^{19} + 20 q^{25} - 24 q^{31} + 28 q^{37} + 72 q^{43} - 240 q^{55} - 84 q^{61} + 180 q^{67} - 132 q^{73} - 156 q^{79} + 240 q^{85} + 540 q^{91} + 172 q^{97}+O(q^{100}) \) 4 * q - 12 * q^7 - 20 * q^13 + 36 * q^19 + 20 * q^25 - 24 * q^31 + 28 * q^37 + 72 * q^43 - 240 * q^55 - 84 * q^61 + 180 * q^67 - 132 * q^73 - 156 * q^79 + 240 * q^85 + 540 * q^91 + 172 * q^97

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