LMFDB - Embedded newform 1728.3.h.e.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(161,1728)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("1728.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.h (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:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	1728.161
Dual form			1728.3.h.e.161.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.19615 q^{5} +5.19615 q^{7} +O(q^{10})$	$q+5.19615 q^{5} +5.19615 q^{7} +3.00000 q^{11} +10.3923i q^{13} +6.00000i q^{17} +2.00000i q^{19} +10.3923i q^{23} +2.00000 q^{25} +20.7846 q^{29} +36.3731 q^{31} +27.0000 q^{35} +51.9615i q^{37} +42.0000i q^{41} +4.00000i q^{43} +41.5692i q^{47} -22.0000 q^{49} -67.5500 q^{53} +15.5885 q^{55} +66.0000 q^{59} -62.3538i q^{61} +54.0000i q^{65} -44.0000i q^{67} -135.100i q^{71} +29.0000 q^{73} +15.5885 q^{77} -83.1384 q^{79} +99.0000 q^{83} +31.1769i q^{85} +144.000i q^{89} +54.0000i q^{91} +10.3923i q^{95} +31.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 12 q^{11} + 8 q^{25} + 108 q^{35} - 88 q^{49} + 264 q^{59} + 116 q^{73} + 396 q^{83} + 124 q^{97}+O(q^{100}) $ 4 * q + 12 * q^11 + 8 * q^25 + 108 * q^35 - 88 * q^49 + 264 * q^59 + 116 * q^73 + 396 * q^83 + 124 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$1$	$-1$

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
$5$	5.19615	1.03923	0.519615	−	0.854400i	$-0.326075\pi$
$5$	5.19615	1.03923	0.519615	+	0.854400i	$0.326075\pi$
$6$	0	0
$7$	5.19615	0.742307	0.371154	−	0.928571i	$-0.378962\pi$
$7$	5.19615	0.742307	0.371154	+	0.928571i	$0.378962\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.00000	0.272727	0.136364	−	0.990659i	$-0.456458\pi$
$11$	3.00000	0.272727	0.136364	+	0.990659i	$0.456458\pi$
$12$	0	0
$13$	10.3923i	0.799408i	0.916644	+	0.399704i	$0.130887\pi$
$13$	10.3923i	0.799408i	−0.916644	+	0.399704i	$0.869113\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000i	0.352941i	0.984306	+	0.176471i	$0.0564680\pi$
$17$	6.00000i	0.352941i	−0.984306	+	0.176471i	$0.943532\pi$
$18$	0	0
$19$	2.00000i	0.105263i	0.998614	+	0.0526316i	$0.0167609\pi$
$19$	2.00000i	0.105263i	−0.998614	+	0.0526316i	$0.983239\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	10.3923i	0.451839i	0.974146	+	0.225920i	$0.0725387\pi$
$23$	10.3923i	0.451839i	−0.974146	+	0.225920i	$0.927461\pi$
$24$	0	0
$25$	2.00000	0.0800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	20.7846	0.716711	0.358355	−	0.933585i	$-0.383338\pi$
$29$	20.7846	0.716711	0.358355	+	0.933585i	$0.383338\pi$
$30$	0	0
$31$	36.3731	1.17332	0.586662	−	0.809832i	$-0.300442\pi$
$31$	36.3731	1.17332	0.586662	+	0.809832i	$0.300442\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	27.0000	0.771429
$36$	0	0
$37$	51.9615i	1.40437i	0.711997	+	0.702183i	$0.247791\pi$
$37$	51.9615i	1.40437i	−0.711997	+	0.702183i	$0.752209\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	42.0000i	1.02439i	0.858869	+	0.512195i	$0.171168\pi$
$41$	42.0000i	1.02439i	−0.858869	+	0.512195i	$0.828832\pi$
$42$	0	0
$43$	4.00000i	0.0930233i	0.998918	+	0.0465116i	$0.0148105\pi$
$43$	4.00000i	0.0930233i	−0.998918	+	0.0465116i	$0.985190\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	41.5692i	0.884451i	0.896904	+	0.442226i	$0.145811\pi$
$47$	41.5692i	0.884451i	−0.896904	+	0.442226i	$0.854189\pi$
$48$	0	0
$49$	−22.0000	−0.448980
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−67.5500	−1.27453	−0.637264	−	0.770646i	$-0.719934\pi$
$53$	−67.5500	−1.27453	−0.637264	+	0.770646i	$0.719934\pi$
$54$	0	0
$55$	15.5885	0.283426
$56$	0	0
$57$	0	0
$58$	0	0
$59$	66.0000	1.11864	0.559322	−	0.828950i	$-0.311062\pi$
$59$	66.0000	1.11864	0.559322	+	0.828950i	$0.311062\pi$
$60$	0	0
$61$	− 62.3538i	− 1.02219i	−0.859523	−	0.511097i	$-0.829239\pi$
$61$	− 62.3538i	− 1.02219i	0.859523	−	0.511097i	$-0.170761\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	54.0000i	0.830769i
$66$	0	0
$67$	− 44.0000i	− 0.656716i	−0.944553	−	0.328358i	$-0.893505\pi$
$67$	− 44.0000i	− 0.656716i	0.944553	−	0.328358i	$-0.106495\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 135.100i	− 1.90282i	−0.307933	−	0.951408i	$-0.599637\pi$
$71$	− 135.100i	− 1.90282i	0.307933	−	0.951408i	$-0.400363\pi$
$72$	0	0
$73$	29.0000	0.397260	0.198630	−	0.980075i	$-0.436351\pi$
$73$	29.0000	0.397260	0.198630	+	0.980075i	$0.436351\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.5885	0.202447
$78$	0	0
$79$	−83.1384	−1.05239	−0.526193	−	0.850365i	$-0.676381\pi$
$79$	−83.1384	−1.05239	−0.526193	+	0.850365i	$0.676381\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	99.0000	1.19277	0.596386	−	0.802698i	$-0.296603\pi$
$83$	99.0000	1.19277	0.596386	+	0.802698i	$0.296603\pi$
$84$	0	0
$85$	31.1769i	0.366787i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	144.000i	1.61798i	0.587824	+	0.808989i	$0.299985\pi$
$89$	144.000i	1.61798i	−0.587824	+	0.808989i	$0.700015\pi$
$90$	0	0
$91$	54.0000i	0.593407i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10.3923i	0.109393i
$96$	0	0
$97$	31.0000	0.319588	0.159794	−	0.987150i	$-0.448917\pi$
$97$	31.0000	0.319588	0.159794	+	0.987150i	$0.448917\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	57.1577	0.565918	0.282959	−	0.959132i	$-0.408684\pi$
$101$	57.1577	0.565918	0.282959	+	0.959132i	$0.408684\pi$
$102$	0	0
$103$	−62.3538	−0.605377	−0.302688	−	0.953090i	$-0.597884\pi$
$103$	−62.3538	−0.605377	−0.302688	+	0.953090i	$0.597884\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	129.000	1.20561	0.602804	−	0.797889i	$-0.294050\pi$
$107$	129.000	1.20561	0.602804	+	0.797889i	$0.294050\pi$
$108$	0	0
$109$	− 176.669i	− 1.62082i	−0.585864	−	0.810409i	$-0.699245\pi$
$109$	− 176.669i	− 1.62082i	0.585864	−	0.810409i	$-0.300755\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	90.0000i	0.796460i	0.917286	+	0.398230i	$0.130375\pi$
$113$	90.0000i	0.796460i	−0.917286	+	0.398230i	$0.869625\pi$
$114$	0	0
$115$	54.0000i	0.469565i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	31.1769i	0.261991i
$120$	0	0
$121$	−112.000	−0.925620
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−119.512	−0.956092
$126$	0	0
$127$	−36.3731	−0.286402	−0.143201	−	0.989694i	$-0.545740\pi$
$127$	−36.3731	−0.286402	−0.143201	+	0.989694i	$0.545740\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	153.000	1.16794	0.583969	−	0.811776i	$-0.301499\pi$
$131$	153.000	1.16794	0.583969	+	0.811776i	$0.301499\pi$
$132$	0	0
$133$	10.3923i	0.0781376i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	240.000i	1.75182i	0.482470	+	0.875912i	$0.339740\pi$
$137$	240.000i	1.75182i	−0.482470	+	0.875912i	$0.660260\pi$
$138$	0	0
$139$	− 136.000i	− 0.978417i	−0.872167	−	0.489209i	$-0.837286\pi$
$139$	− 136.000i	− 0.978417i	0.872167	−	0.489209i	$-0.162714\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	31.1769i	0.218020i
$144$	0	0
$145$	108.000	0.744828
$146$	0	0
$147$	0	0
$148$	0	0
$149$	77.9423	0.523103	0.261551	−	0.965190i	$-0.415766\pi$
$149$	77.9423	0.523103	0.261551	+	0.965190i	$0.415766\pi$
$150$	0	0
$151$	−46.7654	−0.309704	−0.154852	−	0.987938i	$-0.549490\pi$
$151$	−46.7654	−0.309704	−0.154852	+	0.987938i	$0.549490\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	189.000	1.21935
$156$	0	0
$157$	− 103.923i	− 0.661930i	−0.943643	−	0.330965i	$-0.892626\pi$
$157$	− 103.923i	− 0.661930i	0.943643	−	0.330965i	$-0.107374\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	54.0000i	0.335404i
$162$	0	0
$163$	110.000i	0.674847i	0.941353	+	0.337423i	$0.109555\pi$
$163$	110.000i	0.674847i	−0.941353	+	0.337423i	$0.890445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	145.492i	0.871211i	0.900138	+	0.435606i	$0.143466\pi$
$167$	145.492i	0.871211i	−0.900138	+	0.435606i	$0.856534\pi$
$168$	0	0
$169$	61.0000	0.360947
$170$	0	0
$171$	0	0
$172$	0	0
$173$	67.5500	0.390462	0.195231	−	0.980757i	$-0.437454\pi$
$173$	67.5500	0.390462	0.195231	+	0.980757i	$0.437454\pi$
$174$	0	0
$175$	10.3923	0.0593846
$176$	0	0
$177$	0	0
$178$	0	0
$179$	153.000	0.854749	0.427374	−	0.904075i	$-0.359439\pi$
$179$	153.000	0.854749	0.427374	+	0.904075i	$0.359439\pi$
$180$	0	0
$181$	166.277i	0.918657i	0.888267	+	0.459328i	$0.151910\pi$
$181$	166.277i	0.918657i	−0.888267	+	0.459328i	$0.848090\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	270.000i	1.45946i
$186$	0	0
$187$	18.0000i	0.0962567i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	114.315i	0.598510i	0.954173	+	0.299255i	$0.0967381\pi$
$191$	114.315i	0.598510i	−0.954173	+	0.299255i	$0.903262\pi$
$192$	0	0
$193$	−271.000	−1.40415	−0.702073	−	0.712105i	$-0.747742\pi$
$193$	−271.000	−1.40415	−0.702073	+	0.712105i	$0.747742\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	109.119	0.553905	0.276952	−	0.960884i	$-0.410676\pi$
$197$	109.119	0.553905	0.276952	+	0.960884i	$0.410676\pi$
$198$	0	0
$199$	129.904	0.652783	0.326391	−	0.945235i	$-0.394167\pi$
$199$	129.904	0.652783	0.326391	+	0.945235i	$0.394167\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	108.000	0.532020
$204$	0	0
$205$	218.238i	1.06458i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.00000i	0.0287081i
$210$	0	0
$211$	− 314.000i	− 1.48815i	−0.668095	−	0.744076i	$-0.732890\pi$
$211$	− 314.000i	− 1.48815i	0.668095	−	0.744076i	$-0.267110\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	20.7846i	0.0966726i
$216$	0	0
$217$	189.000	0.870968
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−62.3538	−0.282144
$222$	0	0
$223$	353.338	1.58448	0.792238	−	0.610212i	$-0.208916\pi$
$223$	353.338	1.58448	0.792238	+	0.610212i	$0.208916\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	150.000	0.660793	0.330396	−	0.943842i	$-0.392818\pi$
$227$	150.000	0.660793	0.330396	+	0.943842i	$0.392818\pi$
$228$	0	0
$229$	83.1384i	0.363050i	0.983386	+	0.181525i	$0.0581033\pi$
$229$	83.1384i	0.363050i	−0.983386	+	0.181525i	$0.941897\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	300.000i	1.28755i	0.765213	+	0.643777i	$0.222633\pi$
$233$	300.000i	1.28755i	−0.765213	+	0.643777i	$0.777367\pi$
$234$	0	0
$235$	216.000i	0.919149i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 384.515i	− 1.60885i	−0.594054	−	0.804425i	$-0.702473\pi$
$239$	− 384.515i	− 1.60885i	0.594054	−	0.804425i	$-0.297527\pi$
$240$	0	0
$241$	122.000	0.506224	0.253112	−	0.967437i	$-0.418546\pi$
$241$	122.000	0.506224	0.253112	+	0.967437i	$0.418546\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−114.315	−0.466593
$246$	0	0
$247$	−20.7846	−0.0841482
$248$	0	0
$249$	0	0
$250$	0	0
$251$	102.000	0.406375	0.203187	−	0.979140i	$-0.434870\pi$
$251$	102.000	0.406375	0.203187	+	0.979140i	$0.434870\pi$
$252$	0	0
$253$	31.1769i	0.123229i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 114.000i	− 0.443580i	−0.975094	−	0.221790i	$-0.928810\pi$
$257$	− 114.000i	− 0.443580i	0.975094	−	0.221790i	$-0.0711899\pi$
$258$	0	0
$259$	270.000i	1.04247i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 342.946i	− 1.30398i	−0.758229	−	0.651989i	$-0.773935\pi$
$263$	− 342.946i	− 1.30398i	0.758229	−	0.651989i	$-0.226065\pi$
$264$	0	0
$265$	−351.000	−1.32453
$266$	0	0
$267$	0	0
$268$	0	0
$269$	249.415	0.927194	0.463597	−	0.886046i	$-0.346558\pi$
$269$	249.415	0.927194	0.463597	+	0.886046i	$0.346558\pi$
$270$	0	0
$271$	−171.473	−0.632742	−0.316371	−	0.948636i	$-0.602464\pi$
$271$	−171.473	−0.632742	−0.316371	+	0.948636i	$0.602464\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.00000	0.0218182
$276$	0	0
$277$	20.7846i	0.0750347i	0.999296	+	0.0375173i	$0.0119449\pi$
$277$	20.7846i	0.0750347i	−0.999296	+	0.0375173i	$0.988055\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	198.000i	0.704626i	0.935882	+	0.352313i	$0.114605\pi$
$281$	198.000i	0.704626i	−0.935882	+	0.352313i	$0.885395\pi$
$282$	0	0
$283$	− 298.000i	− 1.05300i	−0.850174	−	0.526502i	$-0.823503\pi$
$283$	− 298.000i	− 1.05300i	0.850174	−	0.526502i	$-0.176497\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	218.238i	0.760413i
$288$	0	0
$289$	253.000	0.875433
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−270.200	−0.922184	−0.461092	−	0.887352i	$-0.652542\pi$
$293$	−270.200	−0.922184	−0.461092	+	0.887352i	$0.652542\pi$
$294$	0	0
$295$	342.946	1.16253
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−108.000	−0.361204
$300$	0	0
$301$	20.7846i	0.0690519i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 324.000i	− 1.06230i
$306$	0	0
$307$	286.000i	0.931596i	0.884891	+	0.465798i	$0.154233\pi$
$307$	286.000i	0.931596i	−0.884891	+	0.465798i	$0.845767\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	363.731i	1.16955i	0.811195	+	0.584776i	$0.198818\pi$
$311$	363.731i	1.16955i	−0.811195	+	0.584776i	$0.801182\pi$
$312$	0	0
$313$	193.000	0.616613	0.308307	−	0.951287i	$-0.400238\pi$
$313$	193.000	0.616613	0.308307	+	0.951287i	$0.400238\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−483.242	−1.52442	−0.762212	−	0.647328i	$-0.775886\pi$
$317$	−483.242	−1.52442	−0.762212	+	0.647328i	$0.775886\pi$
$318$	0	0
$319$	62.3538	0.195467
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−12.0000	−0.0371517
$324$	0	0
$325$	20.7846i	0.0639526i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	216.000i	0.656535i
$330$	0	0
$331$	− 40.0000i	− 0.120846i	−0.998173	−	0.0604230i	$-0.980755\pi$
$331$	− 40.0000i	− 0.120846i	0.998173	−	0.0604230i	$-0.0192449\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 228.631i	− 0.682480i
$336$	0	0
$337$	−422.000	−1.25223	−0.626113	−	0.779733i	$-0.715355\pi$
$337$	−422.000	−1.25223	−0.626113	+	0.779733i	$0.715355\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	109.119	0.319998
$342$	0	0
$343$	−368.927	−1.07559
$344$	0	0
$345$	0	0
$346$	0	0
$347$	39.0000	0.112392	0.0561960	−	0.998420i	$-0.482103\pi$
$347$	39.0000	0.112392	0.0561960	+	0.998420i	$0.482103\pi$
$348$	0	0
$349$	114.315i	0.327551i	0.986498	+	0.163776i	$0.0523673\pi$
$349$	114.315i	0.327551i	−0.986498	+	0.163776i	$0.947633\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 474.000i	− 1.34278i	−0.741106	−	0.671388i	$-0.765698\pi$
$353$	− 474.000i	− 1.34278i	0.741106	−	0.671388i	$-0.234302\pi$
$354$	0	0
$355$	− 702.000i	− 1.97746i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	270.200i	0.752646i	0.926489	+	0.376323i	$0.122812\pi$
$359$	270.200i	0.752646i	−0.926489	+	0.376323i	$0.877188\pi$
$360$	0	0
$361$	357.000	0.988920
$362$	0	0
$363$	0	0
$364$	0	0
$365$	150.688	0.412845
$366$	0	0
$367$	−607.950	−1.65654	−0.828270	−	0.560330i	$-0.810674\pi$
$367$	−607.950	−1.65654	−0.828270	+	0.560330i	$0.810674\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−351.000	−0.946092
$372$	0	0
$373$	− 613.146i	− 1.64382i	−0.569615	−	0.821912i	$-0.692908\pi$
$373$	− 613.146i	− 1.64382i	0.569615	−	0.821912i	$-0.307092\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	216.000i	0.572944i
$378$	0	0
$379$	412.000i	1.08707i	0.839386	+	0.543536i	$0.182915\pi$
$379$	412.000i	1.08707i	−0.839386	+	0.543536i	$0.817085\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 571.577i	− 1.49237i	−0.665740	−	0.746184i	$-0.731884\pi$
$383$	− 571.577i	− 1.49237i	0.665740	−	0.746184i	$-0.268116\pi$
$384$	0	0
$385$	81.0000	0.210390
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.19615	−0.0133577	−0.00667886	−	0.999978i	$-0.502126\pi$
$389$	−5.19615	−0.0133577	−0.00667886	+	0.999978i	$0.502126\pi$
$390$	0	0
$391$	−62.3538	−0.159473
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−432.000	−1.09367
$396$	0	0
$397$	− 197.454i	− 0.497365i	−0.968585	−	0.248682i	$-0.920002\pi$
$397$	− 197.454i	− 0.497365i	0.968585	−	0.248682i	$-0.0799975\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 660.000i	− 1.64589i	−0.568124	−	0.822943i	$-0.692331\pi$
$401$	− 660.000i	− 1.64589i	0.568124	−	0.822943i	$-0.307669\pi$
$402$	0	0
$403$	378.000i	0.937965i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	155.885i	0.383009i
$408$	0	0
$409$	−649.000	−1.58680	−0.793399	−	0.608703i	$-0.791690\pi$
$409$	−649.000	−1.58680	−0.793399	+	0.608703i	$0.791690\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	342.946	0.830378
$414$	0	0
$415$	514.419	1.23956
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−198.000	−0.472554	−0.236277	−	0.971686i	$-0.575927\pi$
$419$	−198.000	−0.472554	−0.236277	+	0.971686i	$0.575927\pi$
$420$	0	0
$421$	− 789.815i	− 1.87605i	−0.346574	−	0.938023i	$-0.612655\pi$
$421$	− 789.815i	− 1.87605i	0.346574	−	0.938023i	$-0.387345\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	12.0000i	0.0282353i
$426$	0	0
$427$	− 324.000i	− 0.758782i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	581.969i	1.35028i	0.737691	+	0.675138i	$0.235916\pi$
$431$	581.969i	1.35028i	−0.737691	+	0.675138i	$0.764084\pi$
$432$	0	0
$433$	331.000	0.764434	0.382217	−	0.924073i	$-0.375161\pi$
$433$	331.000	0.764434	0.382217	+	0.924073i	$0.375161\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−20.7846	−0.0475620
$438$	0	0
$439$	−514.419	−1.17180	−0.585899	−	0.810384i	$-0.699258\pi$
$439$	−514.419	−1.17180	−0.585899	+	0.810384i	$0.699258\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−102.000	−0.230248	−0.115124	−	0.993351i	$-0.536727\pi$
$443$	−102.000	−0.230248	−0.115124	+	0.993351i	$0.536727\pi$
$444$	0	0
$445$	748.246i	1.68145i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 690.000i	− 1.53675i	−0.640001	−	0.768374i	$-0.721066\pi$
$449$	− 690.000i	− 1.53675i	0.640001	−	0.768374i	$-0.278934\pi$
$450$	0	0
$451$	126.000i	0.279379i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	280.592i	0.616686i
$456$	0	0
$457$	203.000	0.444201	0.222101	−	0.975024i	$-0.428709\pi$
$457$	203.000	0.444201	0.222101	+	0.975024i	$0.428709\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	701.481	1.52165	0.760825	−	0.648957i	$-0.224795\pi$
$461$	701.481	1.52165	0.760825	+	0.648957i	$0.224795\pi$
$462$	0	0
$463$	−504.027	−1.08861	−0.544305	−	0.838887i	$-0.683207\pi$
$463$	−504.027	−1.08861	−0.544305	+	0.838887i	$0.683207\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−381.000	−0.815846	−0.407923	−	0.913016i	$-0.633747\pi$
$467$	−381.000	−0.815846	−0.407923	+	0.913016i	$0.633747\pi$
$468$	0	0
$469$	− 228.631i	− 0.487486i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	12.0000i	0.0253700i
$474$	0	0
$475$	4.00000i	0.00842105i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 62.3538i	− 0.130175i	−0.997880	−	0.0650875i	$-0.979267\pi$
$479$	− 62.3538i	− 0.130175i	0.997880	−	0.0650875i	$-0.0207327\pi$
$480$	0	0
$481$	−540.000	−1.12266
$482$	0	0
$483$	0	0
$484$	0	0
$485$	161.081	0.332125
$486$	0	0
$487$	228.631	0.469468	0.234734	−	0.972060i	$-0.424578\pi$
$487$	228.631	0.469468	0.234734	+	0.972060i	$0.424578\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−759.000	−1.54582	−0.772912	−	0.634513i	$-0.781201\pi$
$491$	−759.000	−1.54582	−0.772912	+	0.634513i	$0.781201\pi$
$492$	0	0
$493$	124.708i	0.252957i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 702.000i	− 1.41247i
$498$	0	0
$499$	− 692.000i	− 1.38677i	−0.720566	−	0.693387i	$-0.756118\pi$
$499$	− 692.000i	− 1.38677i	0.720566	−	0.693387i	$-0.243882\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 800.207i	− 1.59087i	−0.606039	−	0.795435i	$-0.707243\pi$
$503$	− 800.207i	− 1.59087i	0.606039	−	0.795435i	$-0.292757\pi$
$504$	0	0
$505$	297.000	0.588119
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−161.081	−0.316465	−0.158233	−	0.987402i	$-0.550580\pi$
$509$	−161.081	−0.316465	−0.158233	+	0.987402i	$0.550580\pi$
$510$	0	0
$511$	150.688	0.294889
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−324.000	−0.629126
$516$	0	0
$517$	124.708i	0.241214i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 210.000i	− 0.403071i	−0.979481	−	0.201536i	$-0.935407\pi$
$521$	− 210.000i	− 0.403071i	0.979481	−	0.201536i	$-0.0645931\pi$
$522$	0	0
$523$	682.000i	1.30402i	0.758212	+	0.652008i	$0.226073\pi$
$523$	682.000i	1.30402i	−0.758212	+	0.652008i	$0.773927\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	218.238i	0.414115i
$528$	0	0
$529$	421.000	0.795841
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−436.477	−0.818906
$534$	0	0
$535$	670.304	1.25290
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−66.0000	−0.122449
$540$	0	0
$541$	83.1384i	0.153675i	0.997044	+	0.0768377i	$0.0244823\pi$
$541$	83.1384i	0.153675i	−0.997044	+	0.0768377i	$0.975518\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 918.000i	− 1.68440i
$546$	0	0
$547$	518.000i	0.946984i	0.880798	+	0.473492i	$0.157007\pi$
$547$	518.000i	0.946984i	−0.880798	+	0.473492i	$0.842993\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	41.5692i	0.0754432i
$552$	0	0
$553$	−432.000	−0.781193
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−795.011	−1.42731	−0.713655	−	0.700498i	$-0.752961\pi$
$557$	−795.011	−1.42731	−0.713655	+	0.700498i	$0.752961\pi$
$558$	0	0
$559$	−41.5692	−0.0743635
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−561.000	−0.996448	−0.498224	−	0.867048i	$-0.666014\pi$
$563$	−561.000	−0.996448	−0.498224	+	0.867048i	$0.666014\pi$
$564$	0	0
$565$	467.654i	0.827706i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	264.000i	0.463972i	0.972719	+	0.231986i	$0.0745223\pi$
$569$	264.000i	0.463972i	−0.972719	+	0.231986i	$0.925478\pi$
$570$	0	0
$571$	− 406.000i	− 0.711033i	−0.934670	−	0.355517i	$-0.884305\pi$
$571$	− 406.000i	− 0.711033i	0.934670	−	0.355517i	$-0.115695\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	20.7846i	0.0361471i
$576$	0	0
$577$	610.000	1.05719	0.528596	−	0.848873i	$-0.322719\pi$
$577$	610.000	1.05719	0.528596	+	0.848873i	$0.322719\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	514.419	0.885403
$582$	0	0
$583$	−202.650	−0.347599
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−483.000	−0.822828	−0.411414	−	0.911449i	$-0.634965\pi$
$587$	−483.000	−0.822828	−0.411414	+	0.911449i	$0.634965\pi$
$588$	0	0
$589$	72.7461i	0.123508i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 828.000i	− 1.39629i	−0.715956	−	0.698145i	$-0.754009\pi$
$593$	− 828.000i	− 1.39629i	0.715956	−	0.698145i	$-0.245991\pi$
$594$	0	0
$595$	162.000i	0.272269i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 20.7846i	− 0.0346988i	−0.999849	−	0.0173494i	$-0.994477\pi$
$599$	− 20.7846i	− 0.0346988i	0.999849	−	0.0173494i	$-0.00552277\pi$
$600$	0	0
$601$	5.00000	0.00831947	0.00415973	−	0.999991i	$-0.498676\pi$
$601$	5.00000	0.00831947	0.00415973	+	0.999991i	$0.498676\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−581.969	−0.961932
$606$	0	0
$607$	769.031	1.26694	0.633468	−	0.773769i	$-0.281631\pi$
$607$	769.031	1.26694	0.633468	+	0.773769i	$0.281631\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−432.000	−0.707038
$612$	0	0
$613$	311.769i	0.508596i	0.967126	+	0.254298i	$0.0818443\pi$
$613$	311.769i	0.508596i	−0.967126	+	0.254298i	$0.918156\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	318.000i	0.515397i	0.966225	+	0.257699i	$0.0829641\pi$
$617$	318.000i	0.515397i	−0.966225	+	0.257699i	$0.917036\pi$
$618$	0	0
$619$	− 418.000i	− 0.675283i	−0.941275	−	0.337641i	$-0.890371\pi$
$619$	− 418.000i	− 0.675283i	0.941275	−	0.337641i	$-0.109629\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	748.246i	1.20104i
$624$	0	0
$625$	−671.000	−1.07360
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−311.769	−0.495658
$630$	0	0
$631$	−36.3731	−0.0576435	−0.0288218	−	0.999585i	$-0.509176\pi$
$631$	−36.3731	−0.0576435	−0.0288218	+	0.999585i	$0.509176\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−189.000	−0.297638
$636$	0	0
$637$	− 228.631i	− 0.358918i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1086.00i	− 1.69423i	−0.531411	−	0.847114i	$-0.678338\pi$
$641$	− 1086.00i	− 1.69423i	0.531411	−	0.847114i	$-0.321662\pi$
$642$	0	0
$643$	− 748.000i	− 1.16330i	−0.813440	−	0.581649i	$-0.802408\pi$
$643$	− 748.000i	− 1.16330i	0.813440	−	0.581649i	$-0.197592\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 665.108i	− 1.02799i	−0.857794	−	0.513993i	$-0.828166\pi$
$647$	− 665.108i	− 1.02799i	0.857794	−	0.513993i	$-0.171834\pi$
$648$	0	0
$649$	198.000	0.305085
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−452.065	−0.692290	−0.346145	−	0.938181i	$-0.612509\pi$
$653$	−452.065	−0.692290	−0.346145	+	0.938181i	$0.612509\pi$
$654$	0	0
$655$	795.011	1.21376
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−279.000	−0.423369	−0.211684	−	0.977338i	$-0.567895\pi$
$659$	−279.000	−0.423369	−0.211684	+	0.977338i	$0.567895\pi$
$660$	0	0
$661$	342.946i	0.518829i	0.965766	+	0.259415i	$0.0835296\pi$
$661$	342.946i	0.518829i	−0.965766	+	0.259415i	$0.916470\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	54.0000i	0.0812030i
$666$	0	0
$667$	216.000i	0.323838i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 187.061i	− 0.278780i
$672$	0	0
$673$	−77.0000	−0.114413	−0.0572065	−	0.998362i	$-0.518219\pi$
$673$	−77.0000	−0.114413	−0.0572065	+	0.998362i	$0.518219\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1143.15	1.68856	0.844279	−	0.535904i	$-0.180029\pi$
$677$	1143.15	1.68856	0.844279	+	0.535904i	$0.180029\pi$
$678$	0	0
$679$	161.081	0.237232
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−714.000	−1.04539	−0.522694	−	0.852520i	$-0.675073\pi$
$683$	−714.000	−1.04539	−0.522694	+	0.852520i	$0.675073\pi$
$684$	0	0
$685$	1247.08i	1.82055i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 702.000i	− 1.01887i
$690$	0	0
$691$	1004.00i	1.45297i	0.687184	+	0.726483i	$0.258847\pi$
$691$	1004.00i	1.45297i	−0.687184	+	0.726483i	$0.741153\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 706.677i	− 1.01680i
$696$	0	0
$697$	−252.000	−0.361549
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−140.296	−0.200137	−0.100069	−	0.994981i	$-0.531906\pi$
$701$	−140.296	−0.200137	−0.100069	+	0.994981i	$0.531906\pi$
$702$	0	0
$703$	−103.923	−0.147828
$704$	0	0
$705$	0	0
$706$	0	0
$707$	297.000	0.420085
$708$	0	0
$709$	− 20.7846i	− 0.0293154i	−0.999893	−	0.0146577i	$-0.995334\pi$
$709$	− 20.7846i	− 0.0293154i	0.999893	−	0.0146577i	$-0.00466586\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	378.000i	0.530154i
$714$	0	0
$715$	162.000i	0.226573i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 935.307i	− 1.30084i	−0.759573	−	0.650422i	$-0.774592\pi$
$719$	− 935.307i	− 1.30084i	0.759573	−	0.650422i	$-0.225408\pi$
$720$	0	0
$721$	−324.000	−0.449376
$722$	0	0
$723$	0	0
$724$	0	0
$725$	41.5692	0.0573369
$726$	0	0
$727$	171.473	0.235864	0.117932	−	0.993022i	$-0.462374\pi$
$727$	171.473	0.235864	0.117932	+	0.993022i	$0.462374\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−24.0000	−0.0328317
$732$	0	0
$733$	− 800.207i	− 1.09169i	−0.837887	−	0.545844i	$-0.816209\pi$
$733$	− 800.207i	− 1.09169i	0.837887	−	0.545844i	$-0.183791\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 132.000i	− 0.179104i
$738$	0	0
$739$	286.000i	0.387009i	0.981099	+	0.193505i	$0.0619855\pi$
$739$	286.000i	0.387009i	−0.981099	+	0.193505i	$0.938015\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	363.731i	0.489543i	0.969581	+	0.244772i	$0.0787130\pi$
$743$	363.731i	0.489543i	−0.969581	+	0.244772i	$0.921287\pi$
$744$	0	0
$745$	405.000	0.543624
$746$	0	0
$747$	0	0
$748$	0	0
$749$	670.304	0.894931
$750$	0	0
$751$	−1460.12	−1.94423	−0.972116	−	0.234499i	$-0.924655\pi$
$751$	−1460.12	−1.94423	−0.972116	+	0.234499i	$0.924655\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−243.000	−0.321854
$756$	0	0
$757$	− 685.892i	− 0.906066i	−0.891494	−	0.453033i	$-0.850342\pi$
$757$	− 685.892i	− 0.906066i	0.891494	−	0.453033i	$-0.149658\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0	0	1.00000			$0$
$761$	0	0	−1.00000			$\pi$
$762$	0	0
$763$	− 918.000i	− 1.20315i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	685.892i	0.894253i
$768$	0	0
$769$	−55.0000	−0.0715215	−0.0357607	−	0.999360i	$-0.511385\pi$
$769$	−55.0000	−0.0715215	−0.0357607	+	0.999360i	$0.511385\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−1267.86	−1.64018	−0.820091	−	0.572233i	$-0.806077\pi$
$773$	−1267.86	−1.64018	−0.820091	+	0.572233i	$0.806077\pi$
$774$	0	0
$775$	72.7461	0.0938660
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−84.0000	−0.107831
$780$	0	0
$781$	− 405.300i	− 0.518950i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 540.000i	− 0.687898i
$786$	0	0
$787$	− 964.000i	− 1.22490i	−0.790508	−	0.612452i	$-0.790183\pi$
$787$	− 964.000i	− 1.22490i	0.790508	−	0.612452i	$-0.209817\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	467.654i	0.591218i
$792$	0	0
$793$	648.000	0.817150
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1314.63	1.64947	0.824734	−	0.565520i	$-0.191325\pi$
$797$	1314.63	1.64947	0.824734	+	0.565520i	$0.191325\pi$
$798$	0	0
$799$	−249.415	−0.312159
$800$	0	0
$801$	0	0
$802$	0	0
$803$	87.0000	0.108344
$804$	0	0
$805$	280.592i	0.348562i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	600.000i	0.741656i	0.928701	+	0.370828i	$0.120926\pi$
$809$	600.000i	0.741656i	−0.928701	+	0.370828i	$0.879074\pi$
$810$	0	0
$811$	250.000i	0.308261i	0.988050	+	0.154131i	$0.0492577\pi$
$811$	250.000i	0.308261i	−0.988050	+	0.154131i	$0.950742\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	571.577i	0.701321i
$816$	0	0
$817$	−8.00000	−0.00979192
$818$	0	0
$819$	0	0
$820$	0	0
$821$	270.200	0.329111	0.164555	−	0.986368i	$-0.447381\pi$
$821$	270.200	0.329111	0.164555	+	0.986368i	$0.447381\pi$
$822$	0	0
$823$	−275.396	−0.334625	−0.167312	−	0.985904i	$-0.553509\pi$
$823$	−275.396	−0.334625	−0.167312	+	0.985904i	$0.553509\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1026.00	1.24063	0.620314	−	0.784353i	$-0.287005\pi$
$827$	1026.00	1.24063	0.620314	+	0.784353i	$0.287005\pi$
$828$	0	0
$829$	− 852.169i	− 1.02795i	−0.857806	−	0.513974i	$-0.828173\pi$
$829$	− 852.169i	− 1.02795i	0.857806	−	0.513974i	$-0.171827\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 132.000i	− 0.158463i
$834$	0	0
$835$	756.000i	0.905389i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	1132.76i	1.35013i	0.737757	+	0.675066i	$0.235885\pi$
$839$	1132.76i	1.35013i	−0.737757	+	0.675066i	$0.764115\pi$
$840$	0	0
$841$	−409.000	−0.486326
$842$	0	0
$843$	0	0
$844$	0	0
$845$	316.965	0.375107
$846$	0	0
$847$	−581.969	−0.687095
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−540.000	−0.634548
$852$	0	0
$853$	685.892i	0.804094i	0.915619	+	0.402047i	$0.131701\pi$
$853$	685.892i	0.804094i	−0.915619	+	0.402047i	$0.868299\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	810.000i	0.945158i	0.881288	+	0.472579i	$0.156677\pi$
$857$	810.000i	0.945158i	−0.881288	+	0.472579i	$0.843323\pi$
$858$	0	0
$859$	748.000i	0.870780i	0.900242	+	0.435390i	$0.143390\pi$
$859$	748.000i	0.870780i	−0.900242	+	0.435390i	$0.856610\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.7846i	0.0240841i	0.999927	+	0.0120421i	$0.00383320\pi$
$863$	20.7846i	0.0240841i	−0.999927	+	0.0120421i	$0.996167\pi$
$864$	0	0
$865$	351.000	0.405780
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−249.415	−0.287014
$870$	0	0
$871$	457.261	0.524984
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−621.000	−0.709714
$876$	0	0
$877$	− 1475.71i	− 1.68268i	−0.540509	−	0.841338i	$-0.681768\pi$
$877$	− 1475.71i	− 1.68268i	0.540509	−	0.841338i	$-0.318232\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 342.000i	− 0.388195i	−0.980982	−	0.194098i	$-0.937822\pi$
$881$	− 342.000i	− 0.388195i	0.980982	−	0.194098i	$-0.0621779\pi$
$882$	0	0
$883$	− 538.000i	− 0.609287i	−0.952467	−	0.304643i	$-0.901463\pi$
$883$	− 538.000i	− 0.609287i	0.952467	−	0.304643i	$-0.0985372\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 779.423i	− 0.878718i	−0.898312	−	0.439359i	$-0.855206\pi$
$887$	− 779.423i	− 0.878718i	0.898312	−	0.439359i	$-0.144794\pi$
$888$	0	0
$889$	−189.000	−0.212598
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−83.1384	−0.0931002
$894$	0	0
$895$	795.011	0.888281
$896$	0	0
$897$	0	0
$898$	0	0
$899$	756.000	0.840934
$900$	0	0
$901$	− 405.300i	− 0.449833i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	864.000i	0.954696i
$906$	0	0
$907$	− 374.000i	− 0.412348i	−0.978515	−	0.206174i	$-0.933899\pi$
$907$	− 374.000i	− 0.412348i	0.978515	−	0.206174i	$-0.0661014\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 457.261i	− 0.501933i	−0.967996	−	0.250967i	$-0.919252\pi$
$911$	− 457.261i	− 0.501933i	0.967996	−	0.250967i	$-0.0807485\pi$
$912$	0	0
$913$	297.000	0.325301
$914$	0	0
$915$	0	0
$916$	0	0
$917$	795.011	0.866970
$918$	0	0
$919$	−1293.84	−1.40788	−0.703940	−	0.710259i	$-0.748578\pi$
$919$	−1293.84	−1.40788	−0.703940	+	0.710259i	$0.748578\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1404.00	1.52113
$924$	0	0
$925$	103.923i	0.112349i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	108.000i	0.116254i	0.998309	+	0.0581270i	$0.0185128\pi$
$929$	108.000i	0.116254i	−0.998309	+	0.0581270i	$0.981487\pi$
$930$	0	0
$931$	− 44.0000i	− 0.0472610i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	93.5307i	0.100033i
$936$	0	0
$937$	−713.000	−0.760939	−0.380470	−	0.924793i	$-0.624238\pi$
$937$	−713.000	−0.760939	−0.380470	+	0.924793i	$0.624238\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1148.35	−1.22035	−0.610175	−	0.792267i	$-0.708901\pi$
$941$	−1148.35	−1.22035	−0.610175	+	0.792267i	$0.708901\pi$
$942$	0	0
$943$	−436.477	−0.462860
$944$	0	0
$945$	0	0
$946$	0	0
$947$	825.000	0.871172	0.435586	−	0.900147i	$-0.356541\pi$
$947$	825.000	0.871172	0.435586	+	0.900147i	$0.356541\pi$
$948$	0	0
$949$	301.377i	0.317573i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	876.000i	0.919203i	0.888125	+	0.459601i	$0.152008\pi$
$953$	876.000i	0.919203i	−0.888125	+	0.459601i	$0.847992\pi$
$954$	0	0
$955$	594.000i	0.621990i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1247.08i	1.30039i
$960$	0	0
$961$	362.000	0.376691
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1408.16	−1.45923
$966$	0	0
$967$	−1730.32	−1.78937	−0.894684	−	0.446700i	$-0.852599\pi$
$967$	−1730.32	−1.78937	−0.894684	+	0.446700i	$0.852599\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−729.000	−0.750772	−0.375386	−	0.926868i	$-0.622490\pi$
$971$	−729.000	−0.750772	−0.375386	+	0.926868i	$0.622490\pi$
$972$	0	0
$973$	− 706.677i	− 0.726286i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	636.000i	0.650972i	0.945547	+	0.325486i	$0.105528\pi$
$977$	636.000i	0.650972i	−0.945547	+	0.325486i	$0.894472\pi$
$978$	0	0
$979$	432.000i	0.441267i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 249.415i	− 0.253729i	−0.991920	−	0.126864i	$-0.959509\pi$
$983$	− 249.415i	− 0.253729i	0.991920	−	0.126864i	$-0.0404913\pi$
$984$	0	0
$985$	567.000	0.575635
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−41.5692	−0.0420316
$990$	0	0
$991$	358.535	0.361791	0.180895	−	0.983502i	$-0.442100\pi$
$991$	358.535	0.361791	0.180895	+	0.983502i	$0.442100\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	675.000	0.678392
$996$	0	0
$997$	1215.90i	1.21956i	0.792571	+	0.609779i	$0.208742\pi$
$997$	1215.90i	1.21956i	−0.792571	+	0.609779i	$0.791258\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.h.e.161.4	yes	4
3.2	odd	2		1728.3.h.b.161.2	yes	4
4.3	odd	2		1728.3.h.b.161.4	yes	4
8.3	odd	2	inner	1728.3.h.e.161.1	yes	4
8.5	even	2		1728.3.h.b.161.1	✓	4
12.11	even	2	inner	1728.3.h.e.161.2	yes	4
24.5	odd	2	inner	1728.3.h.e.161.3	yes	4
24.11	even	2		1728.3.h.b.161.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.3.h.b.161.1	✓	4	8.5	even	2
1728.3.h.b.161.2	yes	4	3.2	odd	2
1728.3.h.b.161.3	yes	4	24.11	even	2
1728.3.h.b.161.4	yes	4	4.3	odd	2
1728.3.h.e.161.1	yes	4	8.3	odd	2	inner
1728.3.h.e.161.2	yes	4	12.11	even	2	inner
1728.3.h.e.161.3	yes	4	24.5	odd	2	inner
1728.3.h.e.161.4	yes	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.19615 q^{5} +5.19615 q^{7} +O(q^{10})\)	\(q+5.19615 q^{5} +5.19615 q^{7} +3.00000 q^{11} +10.3923i q^{13} +6.00000i q^{17} +2.00000i q^{19} +10.3923i q^{23} +2.00000 q^{25} +20.7846 q^{29} +36.3731 q^{31} +27.0000 q^{35} +51.9615i q^{37} +42.0000i q^{41} +4.00000i q^{43} +41.5692i q^{47} -22.0000 q^{49} -67.5500 q^{53} +15.5885 q^{55} +66.0000 q^{59} -62.3538i q^{61} +54.0000i q^{65} -44.0000i q^{67} -135.100i q^{71} +29.0000 q^{73} +15.5885 q^{77} -83.1384 q^{79} +99.0000 q^{83} +31.1769i q^{85} +144.000i q^{89} +54.0000i q^{91} +10.3923i q^{95} +31.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 12 q^{11} + 8 q^{25} + 108 q^{35} - 88 q^{49} + 264 q^{59} + 116 q^{73} + 396 q^{83} + 124 q^{97}+O(q^{100}) \) 4 * q + 12 * q^11 + 8 * q^25 + 108 * q^35 - 88 * q^49 + 264 * q^59 + 116 * q^73 + 396 * q^83 + 124 * q^97

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