LMFDB - Embedded newform 1728.4.f.g.863.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(863,1728)]

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([1, 1, 1]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1728.863");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.f (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:	$101.955300490$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.58594980096.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 21x^{6} + 341x^{4} - 2100x^{2} + 10000 $ x^8 - 21x^6 + 341x^4 - 2100*x^2 + 10000
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			863.4
Root			$3.20565 + 1.85078i$ of defining polynomial
Character	$\chi$	$=$	1728.863
Dual form			1728.4.f.g.863.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-9.35849 q^{5} +21.2094i q^{7} +O(q^{10})$	$q-9.35849 q^{5} +21.2094i q^{7} +50.9277i q^{11} -33.2716i q^{13} -120.628i q^{17} +82.1317 q^{19} +95.3719 q^{23} -37.4187 q^{25} -83.1384 q^{29} +36.4187i q^{31} -198.488i q^{35} -201.724i q^{37} -291.769i q^{41} +457.424 q^{43} -628.397 q^{47} -106.837 q^{49} +659.013 q^{53} -476.606i q^{55} -44.8330i q^{59} +149.600i q^{61} +311.372i q^{65} -134.256 q^{67} -291.769 q^{71} +888.212 q^{73} -1080.14 q^{77} +714.653i q^{79} +827.466i q^{83} +1128.90i q^{85} +19.6032i q^{89} +705.670 q^{91} -768.628 q^{95} -805.469 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 1224 q^{23} + 8 q^{25} - 1800 q^{47} - 240 q^{49} + 432 q^{71} + 344 q^{73} - 5688 q^{95} + 1240 q^{97}+O(q^{100}) $ 8 * q + 1224 * q^23 + 8 * q^25 - 1800 * q^47 - 240 * q^49 + 432 * q^71 + 344 * q^73 - 5688 * q^95 + 1240 * 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
$5$	−9.35849	−0.837048	−0.418524	−	0.908206i	$-0.637452\pi$
$5$	−9.35849	−0.837048	−0.418524	+	0.908206i	$0.637452\pi$
$6$	0	0
$7$	21.2094i	1.14520i	0.819835	+	0.572599i	$0.194065\pi$
$7$	21.2094i	1.14520i	−0.819835	+	0.572599i	$0.805935\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	50.9277i	1.39593i	0.716130	+	0.697967i	$0.245912\pi$
$11$	50.9277i	1.39593i	−0.716130	+	0.697967i	$0.754088\pi$
$12$	0	0
$13$	− 33.2716i	− 0.709837i	−0.934897	−	0.354919i	$-0.884509\pi$
$13$	− 33.2716i	− 0.709837i	0.934897	−	0.354919i	$-0.115491\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 120.628i	− 1.72098i	−0.509470	−	0.860489i	$-0.670158\pi$
$17$	− 120.628i	− 1.72098i	0.509470	−	0.860489i	$-0.329842\pi$
$18$	0	0
$19$	82.1317	0.991700	0.495850	−	0.868408i	$-0.334857\pi$
$19$	82.1317	0.991700	0.495850	+	0.868408i	$0.334857\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	95.3719	0.864627	0.432313	−	0.901723i	$-0.357697\pi$
$23$	95.3719	0.864627	0.432313	+	0.901723i	$0.357697\pi$
$24$	0	0
$25$	−37.4187	−0.299350
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−83.1384	−0.532359	−0.266180	−	0.963923i	$-0.585761\pi$
$29$	−83.1384	−0.532359	−0.266180	+	0.963923i	$0.585761\pi$
$30$	0	0
$31$	36.4187i	0.211000i	0.994419	+	0.105500i	$0.0336443\pi$
$31$	36.4187i	0.211000i	−0.994419	+	0.105500i	$0.966356\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 198.488i	− 0.958587i
$36$	0	0
$37$	− 201.724i	− 0.896305i	−0.893957	−	0.448152i	$-0.852082\pi$
$37$	− 201.724i	− 0.896305i	0.893957	−	0.448152i	$-0.147918\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 291.769i	− 1.11138i	−0.831389	−	0.555690i	$-0.812454\pi$
$41$	− 291.769i	− 1.11138i	0.831389	−	0.555690i	$-0.187546\pi$
$42$	0	0
$43$	457.424	1.62224	0.811122	−	0.584877i	$-0.198857\pi$
$43$	457.424	1.62224	0.811122	+	0.584877i	$0.198857\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−628.397	−1.95024	−0.975118	−	0.221686i	$-0.928844\pi$
$47$	−628.397	−1.95024	−0.975118	+	0.221686i	$0.928844\pi$
$48$	0	0
$49$	−106.837	−0.311480
$50$	0	0
$51$	0	0
$52$	0	0
$53$	659.013	1.70797	0.853985	−	0.520298i	$-0.174179\pi$
$53$	659.013	1.70797	0.853985	+	0.520298i	$0.174179\pi$
$54$	0	0
$55$	− 476.606i	− 1.16846i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 44.8330i	− 0.0989282i	−0.998776	−	0.0494641i	$-0.984249\pi$
$59$	− 44.8330i	− 0.0989282i	0.998776	−	0.0494641i	$-0.0157513\pi$
$60$	0	0
$61$	149.600i	0.314006i	0.987598	+	0.157003i	$0.0501833\pi$
$61$	149.600i	0.314006i	−0.987598	+	0.157003i	$0.949817\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	311.372i	0.594168i
$66$	0	0
$67$	−134.256	−0.244805	−0.122402	−	0.992481i	$-0.539060\pi$
$67$	−134.256	−0.244805	−0.122402	+	0.992481i	$0.539060\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−291.769	−0.487698	−0.243849	−	0.969813i	$-0.578410\pi$
$71$	−291.769	−0.487698	−0.243849	+	0.969813i	$0.578410\pi$
$72$	0	0
$73$	888.212	1.42407	0.712037	−	0.702142i	$-0.247773\pi$
$73$	888.212	1.42407	0.712037	+	0.702142i	$0.247773\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1080.14	−1.59862
$78$	0	0
$79$	714.653i	1.01778i	0.860831	+	0.508891i	$0.169944\pi$
$79$	714.653i	1.01778i	−0.860831	+	0.508891i	$0.830056\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	827.466i	1.09429i	0.837038	+	0.547145i	$0.184286\pi$
$83$	827.466i	1.09429i	−0.837038	+	0.547145i	$0.815714\pi$
$84$	0	0
$85$	1128.90i	1.44054i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	19.6032i	0.0233476i	0.999932	+	0.0116738i	$0.00371596\pi$
$89$	19.6032i	0.0233476i	−0.999932	+	0.0116738i	$0.996284\pi$
$90$	0	0
$91$	705.670	0.812905
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−768.628	−0.830101
$96$	0	0
$97$	−805.469	−0.843123	−0.421562	−	0.906800i	$-0.638518\pi$
$97$	−805.469	−0.843123	−0.421562	+	0.906800i	$0.638518\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−421.787	−0.415538	−0.207769	−	0.978178i	$-0.566620\pi$
$101$	−421.787	−0.415538	−0.207769	+	0.978178i	$0.566620\pi$
$102$	0	0
$103$	372.653i	0.356491i	0.983986	+	0.178246i	$0.0570422\pi$
$103$	372.653i	0.356491i	−0.983986	+	0.178246i	$0.942958\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	40.9142i	0.0369656i	0.999829	+	0.0184828i	$0.00588360\pi$
$107$	40.9142i	0.0369656i	−0.999829	+	0.0184828i	$0.994116\pi$
$108$	0	0
$109$	2080.96i	1.82862i	0.405011	+	0.914312i	$0.367268\pi$
$109$	2080.96i	1.82862i	−0.405011	+	0.914312i	$0.632732\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	614.447i	0.511525i	0.966740	+	0.255762i	$0.0823265\pi$
$113$	614.447i	0.511525i	−0.966740	+	0.255762i	$0.917674\pi$
$114$	0	0
$115$	−892.536	−0.723734
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2558.45	1.97086
$120$	0	0
$121$	−1262.63	−0.948633
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1519.99	1.08762
$126$	0	0
$127$	− 1682.24i	− 1.17539i	−0.809083	−	0.587695i	$-0.800036\pi$
$127$	− 1682.24i	− 1.17539i	0.809083	−	0.587695i	$-0.199964\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2008.81i	− 1.33978i	−0.742462	−	0.669888i	$-0.766342\pi$
$131$	− 2008.81i	− 1.33978i	0.742462	−	0.669888i	$-0.233658\pi$
$132$	0	0
$133$	1741.96i	1.13569i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2140.40i	− 1.33479i	−0.744703	−	0.667396i	$-0.767409\pi$
$137$	− 2140.40i	− 1.33479i	0.744703	−	0.667396i	$-0.232591\pi$
$138$	0	0
$139$	2626.56	1.60275	0.801373	−	0.598165i	$-0.204103\pi$
$139$	2626.56	1.60275	0.801373	+	0.598165i	$0.204103\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1694.45	0.990886
$144$	0	0
$145$	778.050	0.445611
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−260.728	−0.143353	−0.0716766	−	0.997428i	$-0.522835\pi$
$149$	−260.728	−0.143353	−0.0716766	+	0.997428i	$0.522835\pi$
$150$	0	0
$151$	1825.26i	0.983692i	0.870682	+	0.491846i	$0.163678\pi$
$151$	1825.26i	0.983692i	−0.870682	+	0.491846i	$0.836322\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 340.824i	− 0.176617i
$156$	0	0
$157$	1731.00i	0.879929i	0.898015	+	0.439965i	$0.145009\pi$
$157$	1731.00i	0.879929i	−0.898015	+	0.439965i	$0.854991\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2022.78i	0.990169i
$162$	0	0
$163$	1066.74	0.512597	0.256299	−	0.966598i	$-0.417497\pi$
$163$	1066.74	0.512597	0.256299	+	0.966598i	$0.417497\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1643.93	0.761745	0.380873	−	0.924628i	$-0.375624\pi$
$167$	1643.93	0.761745	0.380873	+	0.924628i	$0.375624\pi$
$168$	0	0
$169$	1090.00	0.496131
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3182.76	1.39873	0.699367	−	0.714763i	$-0.253465\pi$
$173$	3182.76	1.39873	0.699367	+	0.714763i	$0.253465\pi$
$174$	0	0
$175$	− 793.628i	− 0.342815i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3669.40i	1.53220i	0.642720	+	0.766101i	$0.277806\pi$
$179$	3669.40i	1.53220i	−0.642720	+	0.766101i	$0.722194\pi$
$180$	0	0
$181$	1363.16i	0.559796i	0.960030	+	0.279898i	$0.0903006\pi$
$181$	1363.16i	0.559796i	−0.960030	+	0.279898i	$0.909699\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1887.83i	0.750251i
$186$	0	0
$187$	6143.31	2.40237
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−807.834	−0.306036	−0.153018	−	0.988223i	$-0.548899\pi$
$191$	−807.834	−0.306036	−0.153018	+	0.988223i	$0.548899\pi$
$192$	0	0
$193$	898.681	0.335173	0.167587	−	0.985857i	$-0.446403\pi$
$193$	898.681	0.335173	0.167587	+	0.985857i	$0.446403\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4394.80	−1.58943	−0.794713	−	0.606986i	$-0.792378\pi$
$197$	−4394.80	−1.58943	−0.794713	+	0.606986i	$0.792378\pi$
$198$	0	0
$199$	− 878.234i	− 0.312846i	−0.987690	−	0.156423i	$-0.950004\pi$
$199$	− 878.234i	− 0.312846i	0.987690	−	0.156423i	$-0.0499963\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1763.31i	− 0.609657i
$204$	0	0
$205$	2730.51i	0.930280i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4182.78i	1.38435i
$210$	0	0
$211$	5015.10	1.63627	0.818137	−	0.575024i	$-0.195007\pi$
$211$	5015.10	1.63627	0.818137	+	0.575024i	$0.195007\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−4280.79	−1.35790
$216$	0	0
$217$	−772.419	−0.241637
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4013.49	−1.22161
$222$	0	0
$223$	1304.69i	0.391788i	0.980625	+	0.195894i	$0.0627609\pi$
$223$	1304.69i	0.391788i	−0.980625	+	0.195894i	$0.937239\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1393.77i	− 0.407523i	−0.979021	−	0.203762i	$-0.934683\pi$
$227$	− 1393.77i	− 0.407523i	0.979021	−	0.203762i	$-0.0653168\pi$
$228$	0	0
$229$	− 1153.16i	− 0.332763i	−0.986061	−	0.166381i	$-0.946792\pi$
$229$	− 1153.16i	− 0.332763i	0.986061	−	0.166381i	$-0.0532083\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 3540.43i	− 0.995456i	−0.867333	−	0.497728i	$-0.834168\pi$
$233$	− 3540.43i	− 0.995456i	0.867333	−	0.497728i	$-0.165832\pi$
$234$	0	0
$235$	5880.84	1.63244
$236$	0	0
$237$	0	0
$238$	0	0
$239$	78.4127	0.0212222	0.0106111	−	0.999944i	$-0.496622\pi$
$239$	78.4127	0.0212222	0.0106111	+	0.999944i	$0.496622\pi$
$240$	0	0
$241$	3906.31	1.04410	0.522048	−	0.852916i	$-0.325168\pi$
$241$	3906.31	1.04410	0.522048	+	0.852916i	$0.325168\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	999.837	0.260723
$246$	0	0
$247$	− 2732.65i	− 0.703946i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	5007.89i	1.25934i	0.776861	+	0.629672i	$0.216811\pi$
$251$	5007.89i	1.25934i	−0.776861	+	0.629672i	$0.783189\pi$
$252$	0	0
$253$	4857.07i	1.20696i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2019.77i	− 0.490232i	−0.969494	−	0.245116i	$-0.921174\pi$
$257$	− 2019.77i	− 0.490232i	0.969494	−	0.245116i	$-0.0788261\pi$
$258$	0	0
$259$	4278.45	1.02645
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6575.74	−1.54174	−0.770869	−	0.636993i	$-0.780178\pi$
$263$	−6575.74	−1.54174	−0.770869	+	0.636993i	$0.780178\pi$
$264$	0	0
$265$	−6167.36	−1.42965
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3661.36	0.829876	0.414938	−	0.909850i	$-0.363803\pi$
$269$	3661.36	0.829876	0.414938	+	0.909850i	$0.363803\pi$
$270$	0	0
$271$	3645.23i	0.817092i	0.912738	+	0.408546i	$0.133964\pi$
$271$	3645.23i	0.817092i	−0.912738	+	0.408546i	$0.866036\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1905.65i	− 0.417873i
$276$	0	0
$277$	5103.76i	1.10706i	0.832830	+	0.553529i	$0.186719\pi$
$277$	5103.76i	1.10706i	−0.832830	+	0.553529i	$0.813281\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	487.800i	0.103558i	0.998659	+	0.0517789i	$0.0164891\pi$
$281$	487.800i	0.103558i	−0.998659	+	0.0517789i	$0.983511\pi$
$282$	0	0
$283$	339.147	0.0712374	0.0356187	−	0.999365i	$-0.488660\pi$
$283$	339.147	0.0712374	0.0356187	+	0.999365i	$0.488660\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6188.23	1.27275
$288$	0	0
$289$	−9638.14	−1.96176
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9014.85	1.79745	0.898726	−	0.438511i	$-0.144494\pi$
$293$	9014.85	1.79745	0.898726	+	0.438511i	$0.144494\pi$
$294$	0	0
$295$	419.569i	0.0828077i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 3173.18i	− 0.613744i
$300$	0	0
$301$	9701.67i	1.85779i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1400.03i	− 0.262838i
$306$	0	0
$307$	4127.92	0.767404	0.383702	−	0.923457i	$-0.374649\pi$
$307$	4127.92	0.767404	0.383702	+	0.923457i	$0.374649\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2597.65	−0.473631	−0.236816	−	0.971555i	$-0.576104\pi$
$311$	−2597.65	−0.473631	−0.236816	+	0.971555i	$0.576104\pi$
$312$	0	0
$313$	8571.57	1.54790	0.773952	−	0.633245i	$-0.218277\pi$
$313$	8571.57	1.54790	0.773952	+	0.633245i	$0.218277\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−3321.62	−0.588520	−0.294260	−	0.955725i	$-0.595073\pi$
$317$	−3321.62	−0.588520	−0.294260	+	0.955725i	$0.595073\pi$
$318$	0	0
$319$	− 4234.05i	− 0.743139i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 9907.39i	− 1.70669i
$324$	0	0
$325$	1244.98i	0.212490i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 13327.9i	− 2.23341i
$330$	0	0
$331$	184.431	0.0306262	0.0153131	−	0.999883i	$-0.495126\pi$
$331$	184.431	0.0306262	0.0153131	+	0.999883i	$0.495126\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1256.43	0.204914
$336$	0	0
$337$	6322.76	1.02202	0.511012	−	0.859573i	$-0.329271\pi$
$337$	6322.76	1.02202	0.511012	+	0.859573i	$0.329271\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1854.72	−0.294542
$342$	0	0
$343$	5008.86i	0.788493i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6318.50i	0.977507i	0.872422	+	0.488753i	$0.162548\pi$
$347$	6318.50i	0.977507i	−0.872422	+	0.488753i	$0.837452\pi$
$348$	0	0
$349$	8503.65i	1.30427i	0.758103	+	0.652135i	$0.226126\pi$
$349$	8503.65i	1.30427i	−0.758103	+	0.652135i	$0.773874\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 7697.59i	− 1.16063i	−0.814393	−	0.580313i	$-0.802930\pi$
$353$	− 7697.59i	− 1.16063i	0.814393	−	0.580313i	$-0.197070\pi$
$354$	0	0
$355$	2730.51	0.408227
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−903.571	−0.132838	−0.0664188	−	0.997792i	$-0.521157\pi$
$359$	−903.571	−0.132838	−0.0664188	+	0.997792i	$0.521157\pi$
$360$	0	0
$361$	−113.388	−0.0165312
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8312.32	−1.19202
$366$	0	0
$367$	6619.95i	0.941576i	0.882246	+	0.470788i	$0.156030\pi$
$367$	6619.95i	0.941576i	−0.882246	+	0.470788i	$0.843970\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	13977.2i	1.95596i
$372$	0	0
$373$	11830.0i	1.64219i	0.570794	+	0.821093i	$0.306636\pi$
$373$	11830.0i	1.64219i	−0.570794	+	0.821093i	$0.693364\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2766.15i	0.377888i
$378$	0	0
$379$	−3711.81	−0.503068	−0.251534	−	0.967849i	$-0.580935\pi$
$379$	−3711.81	−0.503068	−0.251534	+	0.967849i	$0.580935\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1761.92	0.235065	0.117532	−	0.993069i	$-0.462502\pi$
$383$	1761.92	0.235065	0.117532	+	0.993069i	$0.462502\pi$
$384$	0	0
$385$	10108.5	1.33812
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4849.66	−0.632102	−0.316051	−	0.948742i	$-0.602357\pi$
$389$	−4849.66	−0.632102	−0.316051	+	0.948742i	$0.602357\pi$
$390$	0	0
$391$	− 11504.5i	− 1.48800i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 6688.07i	− 0.851933i
$396$	0	0
$397$	8066.93i	1.01982i	0.860228	+	0.509909i	$0.170321\pi$
$397$	8066.93i	1.01982i	−0.860228	+	0.509909i	$0.829679\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14632.2i	1.82219i	0.412199	+	0.911094i	$0.364761\pi$
$401$	14632.2i	1.82219i	−0.412199	+	0.911094i	$0.635239\pi$
$402$	0	0
$403$	1211.71	0.149776
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10273.4	1.25118
$408$	0	0
$409$	−12812.9	−1.54904	−0.774519	−	0.632551i	$-0.782008\pi$
$409$	−12812.9	−1.54904	−0.774519	+	0.632551i	$0.782008\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	950.880	0.113292
$414$	0	0
$415$	− 7743.82i	− 0.915974i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9701.95i	1.13120i	0.824681	+	0.565598i	$0.191355\pi$
$419$	9701.95i	1.13120i	−0.824681	+	0.565598i	$0.808645\pi$
$420$	0	0
$421$	− 16265.0i	− 1.88292i	−0.337123	−	0.941460i	$-0.609454\pi$
$421$	− 16265.0i	− 1.88292i	0.337123	−	0.941460i	$-0.390546\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4513.75i	0.515175i
$426$	0	0
$427$	−3172.93	−0.359599
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16472.2	−1.84092	−0.920460	−	0.390837i	$-0.872185\pi$
$431$	−16472.2	−1.84092	−0.920460	+	0.390837i	$0.872185\pi$
$432$	0	0
$433$	−14696.1	−1.63106	−0.815532	−	0.578712i	$-0.803556\pi$
$433$	−14696.1	−1.63106	−0.815532	+	0.578712i	$0.803556\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7833.05	0.857450
$438$	0	0
$439$	7192.32i	0.781938i	0.920404	+	0.390969i	$0.127860\pi$
$439$	7192.32i	0.781938i	−0.920404	+	0.390969i	$0.872140\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 12096.0i	− 1.29728i	−0.761094	−	0.648642i	$-0.775337\pi$
$443$	− 12096.0i	− 1.29728i	0.761094	−	0.648642i	$-0.224663\pi$
$444$	0	0
$445$	− 183.456i	− 0.0195430i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2061.98i	0.216728i	0.994111	+	0.108364i	$0.0345613\pi$
$449$	2061.98i	0.216728i	−0.994111	+	0.108364i	$0.965439\pi$
$450$	0	0
$451$	14859.1	1.55142
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6604.00	−0.680440
$456$	0	0
$457$	18568.8	1.90068	0.950340	−	0.311215i	$-0.100736\pi$
$457$	18568.8	1.90068	0.950340	+	0.311215i	$0.100736\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2231.88	−0.225486	−0.112743	−	0.993624i	$-0.535964\pi$
$461$	−2231.88	−0.225486	−0.112743	+	0.993624i	$0.535964\pi$
$462$	0	0
$463$	− 3414.82i	− 0.342765i	−0.985205	−	0.171383i	$-0.945177\pi$
$463$	− 3414.82i	− 0.342765i	0.985205	−	0.171383i	$-0.0548234\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8313.82i	− 0.823806i	−0.911228	−	0.411903i	$-0.864864\pi$
$467$	− 8313.82i	− 0.823806i	0.911228	−	0.411903i	$-0.135136\pi$
$468$	0	0
$469$	− 2847.48i	− 0.280350i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	23295.5i	2.26455i
$474$	0	0
$475$	−3073.26	−0.296865
$476$	0	0
$477$	0	0
$478$	0	0
$479$	3888.73	0.370941	0.185470	−	0.982650i	$-0.440619\pi$
$479$	3888.73	0.370941	0.185470	+	0.982650i	$0.440619\pi$
$480$	0	0
$481$	−6711.69	−0.636231
$482$	0	0
$483$	0	0
$484$	0	0
$485$	7537.97	0.705735
$486$	0	0
$487$	− 1477.35i	− 0.137464i	−0.997635	−	0.0687320i	$-0.978105\pi$
$487$	− 1477.35i	− 0.137464i	0.997635	−	0.0687320i	$-0.0218953\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	11772.5i	1.08205i	0.841006	+	0.541026i	$0.181964\pi$
$491$	11772.5i	1.08205i	−0.841006	+	0.541026i	$0.818036\pi$
$492$	0	0
$493$	10028.8i	0.916178i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6188.23i	− 0.558511i
$498$	0	0
$499$	9378.01	0.841318	0.420659	−	0.907219i	$-0.361799\pi$
$499$	9378.01	0.841318	0.420659	+	0.907219i	$0.361799\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	22078.3	1.95710	0.978552	−	0.205998i	$-0.0660442\pi$
$503$	22078.3	1.95710	0.978552	+	0.205998i	$0.0660442\pi$
$504$	0	0
$505$	3947.29	0.347826
$506$	0	0
$507$	0	0
$508$	0	0
$509$	18081.3	1.57453	0.787267	−	0.616612i	$-0.211495\pi$
$509$	18081.3	1.57453	0.787267	+	0.616612i	$0.211495\pi$
$510$	0	0
$511$	18838.4i	1.63085i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 3487.47i	− 0.298400i
$516$	0	0
$517$	− 32002.8i	− 2.72240i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8655.05i	0.727801i	0.931438	+	0.363901i	$0.118555\pi$
$521$	8655.05i	0.727801i	−0.931438	+	0.363901i	$0.881445\pi$
$522$	0	0
$523$	−9920.20	−0.829407	−0.414704	−	0.909956i	$-0.636115\pi$
$523$	−9920.20	−0.829407	−0.414704	+	0.909956i	$0.636115\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4393.12	0.363126
$528$	0	0
$529$	−3071.20	−0.252421
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9707.61	−0.788900
$534$	0	0
$535$	− 382.895i	− 0.0309420i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 5440.99i	− 0.434805i
$540$	0	0
$541$	− 12337.7i	− 0.980481i	−0.871587	−	0.490241i	$-0.836909\pi$
$541$	− 12337.7i	− 0.980481i	0.871587	−	0.490241i	$-0.163091\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 19474.6i	− 1.53065i
$546$	0	0
$547$	7446.60	0.582073	0.291036	−	0.956712i	$-0.406000\pi$
$547$	7446.60	0.582073	0.291036	+	0.956712i	$0.406000\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6828.30	−0.527941
$552$	0	0
$553$	−15157.3	−1.16556
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10322.9	−0.785268	−0.392634	−	0.919695i	$-0.628436\pi$
$557$	−10322.9	−0.785268	−0.392634	+	0.919695i	$0.628436\pi$
$558$	0	0
$559$	− 15219.2i	− 1.15153i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 18218.6i	− 1.36381i	−0.731442	−	0.681904i	$-0.761152\pi$
$563$	− 18218.6i	− 1.36381i	0.731442	−	0.681904i	$-0.238848\pi$
$564$	0	0
$565$	− 5750.29i	− 0.428171i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6399.31i	0.471481i	0.971816	+	0.235741i	$0.0757516\pi$
$569$	6399.31i	0.471481i	−0.971816	+	0.235741i	$0.924248\pi$
$570$	0	0
$571$	1999.51	0.146545	0.0732723	−	0.997312i	$-0.476656\pi$
$571$	1999.51	0.146545	0.0732723	+	0.997312i	$0.476656\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3568.70	−0.258826
$576$	0	0
$577$	10659.3	0.769067	0.384534	−	0.923111i	$-0.374362\pi$
$577$	10659.3	0.769067	0.384534	+	0.923111i	$0.374362\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−17550.0	−1.25318
$582$	0	0
$583$	33562.0i	2.38421i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8379.15i	0.589173i	0.955625	+	0.294586i	$0.0951819\pi$
$587$	8379.15i	0.589173i	−0.955625	+	0.294586i	$0.904818\pi$
$588$	0	0
$589$	2991.13i	0.209249i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 7052.96i	− 0.488416i	−0.969723	−	0.244208i	$-0.921472\pi$
$593$	− 7052.96i	− 0.488416i	0.969723	−	0.244208i	$-0.0785279\pi$
$594$	0	0
$595$	−23943.2	−1.64971
$596$	0	0
$597$	0	0
$598$	0	0
$599$	21730.4	1.48227	0.741135	−	0.671356i	$-0.234288\pi$
$599$	21730.4	1.48227	0.741135	+	0.671356i	$0.234288\pi$
$600$	0	0
$601$	18068.1	1.22631	0.613156	−	0.789962i	$-0.289900\pi$
$601$	18068.1	1.22631	0.613156	+	0.789962i	$0.289900\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	11816.3	0.794052
$606$	0	0
$607$	− 21545.7i	− 1.44071i	−0.693603	−	0.720357i	$-0.743978\pi$
$607$	− 21545.7i	− 1.44071i	0.693603	−	0.720357i	$-0.256022\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	20907.8i	1.38435i
$612$	0	0
$613$	− 19035.8i	− 1.25424i	−0.778923	−	0.627120i	$-0.784234\pi$
$613$	− 19035.8i	− 1.25424i	0.778923	−	0.627120i	$-0.215766\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 17984.2i	− 1.17344i	−0.809788	−	0.586722i	$-0.800418\pi$
$617$	− 17984.2i	− 1.17344i	0.809788	−	0.586722i	$-0.199582\pi$
$618$	0	0
$619$	−10517.1	−0.682907	−0.341454	−	0.939899i	$-0.610919\pi$
$619$	−10517.1	−0.682907	−0.341454	+	0.939899i	$0.610919\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−415.771	−0.0267376
$624$	0	0
$625$	−9547.49	−0.611040
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−24333.6	−1.54252
$630$	0	0
$631$	− 23831.9i	− 1.50354i	−0.659425	−	0.751771i	$-0.729200\pi$
$631$	− 23831.9i	− 1.50354i	0.659425	−	0.751771i	$-0.270800\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	15743.2i	0.983858i
$636$	0	0
$637$	3554.66i	0.221100i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15647.0i	0.964149i	0.876130	+	0.482075i	$0.160117\pi$
$641$	15647.0i	0.964149i	−0.876130	+	0.482075i	$0.839883\pi$
$642$	0	0
$643$	−29777.2	−1.82628	−0.913139	−	0.407648i	$-0.866349\pi$
$643$	−29777.2	−1.82628	−0.913139	+	0.407648i	$0.866349\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9149.68	0.555968	0.277984	−	0.960586i	$-0.410334\pi$
$647$	9149.68	0.555968	0.277984	+	0.960586i	$0.410334\pi$
$648$	0	0
$649$	2283.24	0.138097
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19011.7	1.13933	0.569667	−	0.821875i	$-0.307072\pi$
$653$	19011.7	1.13933	0.569667	+	0.821875i	$0.307072\pi$
$654$	0	0
$655$	18799.4i	1.12146i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 299.938i	− 0.0177298i	−0.999961	−	0.00886490i	$-0.997178\pi$
$659$	− 299.938i	− 0.0177298i	0.999961	−	0.00886490i	$-0.00282182\pi$
$660$	0	0
$661$	− 23512.2i	− 1.38354i	−0.722118	−	0.691769i	$-0.756831\pi$
$661$	− 23512.2i	− 1.38354i	0.722118	−	0.691769i	$-0.243169\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 16302.1i	− 0.950630i
$666$	0	0
$667$	−7929.07	−0.460292
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−7618.81	−0.438332
$672$	0	0
$673$	22913.5	1.31241	0.656204	−	0.754583i	$-0.272161\pi$
$673$	22913.5	1.31241	0.656204	+	0.754583i	$0.272161\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16220.1	0.920809	0.460405	−	0.887709i	$-0.347704\pi$
$677$	16220.1	0.920809	0.460405	+	0.887709i	$0.347704\pi$
$678$	0	0
$679$	− 17083.5i	− 0.965543i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	18445.4i	1.03337i	0.856175	+	0.516686i	$0.172835\pi$
$683$	18445.4i	1.03337i	−0.856175	+	0.516686i	$0.827165\pi$
$684$	0	0
$685$	20030.9i	1.11729i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 21926.4i	− 1.21238i
$690$	0	0
$691$	−18971.8	−1.04446	−0.522231	−	0.852804i	$-0.674900\pi$
$691$	−18971.8	−1.04446	−0.522231	+	0.852804i	$0.674900\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−24580.6	−1.34158
$696$	0	0
$697$	−35195.5	−1.91266
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−9303.46	−0.501265	−0.250632	−	0.968082i	$-0.580639\pi$
$701$	−9303.46	−0.501265	−0.250632	+	0.968082i	$0.580639\pi$
$702$	0	0
$703$	− 16568.0i	− 0.888865i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 8945.84i	− 0.475874i
$708$	0	0
$709$	− 33456.9i	− 1.77221i	−0.463481	−	0.886107i	$-0.653400\pi$
$709$	− 33456.9i	− 1.77221i	0.463481	−	0.886107i	$-0.346600\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3473.32i	0.182436i
$714$	0	0
$715$	−15857.5	−0.829420
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−19474.6	−1.01013	−0.505064	−	0.863082i	$-0.668531\pi$
$719$	−19474.6	−1.01013	−0.505064	+	0.863082i	$0.668531\pi$
$720$	0	0
$721$	−7903.74	−0.408253
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3110.94	0.159362
$726$	0	0
$727$	7641.59i	0.389836i	0.980820	+	0.194918i	$0.0624441\pi$
$727$	7641.59i	0.389836i	−0.980820	+	0.194918i	$0.937556\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 55178.2i	− 2.79184i
$732$	0	0
$733$	23153.2i	1.16669i	0.812224	+	0.583345i	$0.198257\pi$
$733$	23153.2i	1.16669i	−0.812224	+	0.583345i	$0.801743\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 6837.33i	− 0.341732i
$738$	0	0
$739$	−20627.7	−1.02680	−0.513398	−	0.858150i	$-0.671614\pi$
$739$	−20627.7	−1.02680	−0.513398	+	0.858150i	$0.671614\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−30112.9	−1.48686	−0.743429	−	0.668815i	$-0.766802\pi$
$743$	−30112.9	−1.48686	−0.743429	+	0.668815i	$0.766802\pi$
$744$	0	0
$745$	2440.02	0.119994
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−867.764	−0.0423330
$750$	0	0
$751$	13806.8i	0.670861i	0.942065	+	0.335431i	$0.108882\pi$
$751$	13806.8i	0.670861i	−0.942065	+	0.335431i	$0.891118\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 17081.7i	− 0.823398i
$756$	0	0
$757$	− 16505.6i	− 0.792478i	−0.918147	−	0.396239i	$-0.870315\pi$
$757$	− 16505.6i	− 0.792478i	0.918147	−	0.396239i	$-0.129685\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	35472.2i	1.68971i	0.534997	+	0.844854i	$0.320313\pi$
$761$	35472.2i	1.68971i	−0.534997	+	0.844854i	$0.679687\pi$
$762$	0	0
$763$	−44135.9	−2.09414
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1491.67	−0.0702229
$768$	0	0
$769$	2816.64	0.132081	0.0660406	−	0.997817i	$-0.478963\pi$
$769$	2816.64	0.132081	0.0660406	+	0.997817i	$0.478963\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	30378.5	1.41351	0.706753	−	0.707461i	$-0.250159\pi$
$773$	30378.5	1.41351	0.706753	+	0.707461i	$0.250159\pi$
$774$	0	0
$775$	− 1362.74i	− 0.0631628i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 23963.5i	− 1.10216i
$780$	0	0
$781$	− 14859.1i	− 0.680795i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 16199.5i	− 0.736543i
$786$	0	0
$787$	−26302.1	−1.19132	−0.595660	−	0.803237i	$-0.703109\pi$
$787$	−26302.1	−1.19132	−0.595660	+	0.803237i	$0.703109\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−13032.0	−0.585797
$792$	0	0
$793$	4977.45	0.222893
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27016.4	−1.20072	−0.600358	−	0.799731i	$-0.704975\pi$
$797$	−27016.4	−1.20072	−0.600358	+	0.799731i	$0.704975\pi$
$798$	0	0
$799$	75802.3i	3.35631i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	45234.6i	1.98791i
$804$	0	0
$805$	− 18930.1i	− 0.828819i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	33893.5i	1.47297i	0.676455	+	0.736484i	$0.263516\pi$
$809$	33893.5i	1.47297i	−0.676455	+	0.736484i	$0.736484\pi$
$810$	0	0
$811$	16820.8	0.728309	0.364155	−	0.931338i	$-0.381358\pi$
$811$	16820.8	0.728309	0.364155	+	0.931338i	$0.381358\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9983.05	−0.429069
$816$	0	0
$817$	37569.0	1.60878
$818$	0	0
$819$	0	0
$820$	0	0
$821$	27615.2	1.17391	0.586953	−	0.809621i	$-0.300327\pi$
$821$	27615.2	1.17391	0.586953	+	0.809621i	$0.300327\pi$
$822$	0	0
$823$	− 27645.0i	− 1.17089i	−0.810712	−	0.585445i	$-0.800920\pi$
$823$	− 27645.0i	− 1.17089i	0.810712	−	0.585445i	$-0.199080\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	33748.9i	1.41906i	0.704674	+	0.709531i	$0.251093\pi$
$827$	33748.9i	1.41906i	−0.704674	+	0.709531i	$0.748907\pi$
$828$	0	0
$829$	− 19289.5i	− 0.808145i	−0.914727	−	0.404072i	$-0.867594\pi$
$829$	− 19289.5i	− 0.808145i	0.914727	−	0.404072i	$-0.132406\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	12887.6i	0.536049i
$834$	0	0
$835$	−15384.7	−0.637618
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8017.26	0.329900	0.164950	−	0.986302i	$-0.447254\pi$
$839$	8017.26	0.329900	0.164950	+	0.986302i	$0.447254\pi$
$840$	0	0
$841$	−17477.0	−0.716594
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10200.7	−0.415286
$846$	0	0
$847$	− 26779.6i	− 1.08637i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 19238.8i	− 0.774969i
$852$	0	0
$853$	− 23777.8i	− 0.954439i	−0.878784	−	0.477220i	$-0.841645\pi$
$853$	− 23777.8i	− 0.954439i	0.878784	−	0.477220i	$-0.158355\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 40669.8i	− 1.62107i	−0.585692	−	0.810534i	$-0.699177\pi$
$857$	− 40669.8i	− 1.62107i	0.585692	−	0.810534i	$-0.300823\pi$
$858$	0	0
$859$	7166.21	0.284642	0.142321	−	0.989821i	$-0.454543\pi$
$859$	7166.21	0.284642	0.142321	+	0.989821i	$0.454543\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	30914.4	1.21939	0.609697	−	0.792635i	$-0.291291\pi$
$863$	30914.4	1.21939	0.609697	+	0.792635i	$0.291291\pi$
$864$	0	0
$865$	−29785.8	−1.17081
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−36395.6	−1.42076
$870$	0	0
$871$	4466.90i	0.173772i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	32238.1i	1.24554i
$876$	0	0
$877$	− 5414.60i	− 0.208481i	−0.994552	−	0.104241i	$-0.966759\pi$
$877$	− 5414.60i	− 0.208481i	0.994552	−	0.104241i	$-0.0332412\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 6604.46i	− 0.252565i	−0.991994	−	0.126283i	$-0.959695\pi$
$881$	− 6604.46i	− 0.252565i	0.991994	−	0.126283i	$-0.0403046\pi$
$882$	0	0
$883$	−35256.0	−1.34367	−0.671834	−	0.740701i	$-0.734493\pi$
$883$	−35256.0	−1.34367	−0.671834	+	0.740701i	$0.734493\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17276.2	0.653976	0.326988	−	0.945028i	$-0.393966\pi$
$887$	17276.2	0.653976	0.326988	+	0.945028i	$0.393966\pi$
$888$	0	0
$889$	35679.2	1.34605
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−51611.3	−1.93405
$894$	0	0
$895$	− 34340.1i	− 1.28253i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 3027.80i	− 0.112328i
$900$	0	0
$901$	− 79495.5i	− 2.93938i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 12757.1i	− 0.468576i
$906$	0	0
$907$	34491.5	1.26270	0.631351	−	0.775497i	$-0.282501\pi$
$907$	34491.5	1.26270	0.631351	+	0.775497i	$0.282501\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	38213.5	1.38976	0.694879	−	0.719127i	$-0.255458\pi$
$911$	38213.5	1.38976	0.694879	+	0.719127i	$0.255458\pi$
$912$	0	0
$913$	−42140.9	−1.52756
$914$	0	0
$915$	0	0
$916$	0	0
$917$	42605.6	1.53431
$918$	0	0
$919$	36899.1i	1.32447i	0.749295	+	0.662236i	$0.230392\pi$
$919$	36899.1i	1.32447i	−0.749295	+	0.662236i	$0.769608\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	9707.61i	0.346186i
$924$	0	0
$925$	7548.27i	0.268309i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 279.002i	− 0.00985335i	−0.999988	−	0.00492668i	$-0.998432\pi$
$929$	− 279.002i	− 0.00985335i	0.999988	−	0.00492668i	$-0.00156822\pi$
$930$	0	0
$931$	−8774.74	−0.308894
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−57492.1	−2.01090
$936$	0	0
$937$	−41546.4	−1.44852	−0.724259	−	0.689528i	$-0.757818\pi$
$937$	−41546.4	−1.44852	−0.724259	+	0.689528i	$0.757818\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	18490.9	0.640580	0.320290	−	0.947320i	$-0.396220\pi$
$941$	18490.9	0.640580	0.320290	+	0.947320i	$0.396220\pi$
$942$	0	0
$943$	− 27826.5i	− 0.960930i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7862.91i	0.269810i	0.990859	+	0.134905i	$0.0430730\pi$
$947$	7862.91i	0.269810i	−0.990859	+	0.134905i	$0.956927\pi$
$948$	0	0
$949$	− 29552.3i	− 1.01086i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 23650.7i	− 0.803904i	−0.915661	−	0.401952i	$-0.868332\pi$
$953$	− 23650.7i	− 0.803904i	0.915661	−	0.401952i	$-0.131668\pi$
$954$	0	0
$955$	7560.11	0.256167
$956$	0	0
$957$	0	0
$958$	0	0
$959$	45396.5	1.52860
$960$	0	0
$961$	28464.7	0.955479
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−8410.29	−0.280556
$966$	0	0
$967$	− 13515.5i	− 0.449460i	−0.974421	−	0.224730i	$-0.927850\pi$
$967$	− 13515.5i	− 0.449460i	0.974421	−	0.224730i	$-0.0721501\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 39505.1i	− 1.30564i	−0.757512	−	0.652821i	$-0.773585\pi$
$971$	− 39505.1i	− 1.30564i	0.757512	−	0.652821i	$-0.226415\pi$
$972$	0	0
$973$	55707.6i	1.83546i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	42131.6i	1.37964i	0.723981	+	0.689820i	$0.242310\pi$
$977$	42131.6i	1.37964i	−0.723981	+	0.689820i	$0.757690\pi$
$978$	0	0
$979$	−998.345	−0.0325917
$980$	0	0
$981$	0	0
$982$	0	0
$983$	48982.9	1.58933	0.794665	−	0.607049i	$-0.207647\pi$
$983$	48982.9	1.58933	0.794665	+	0.607049i	$0.207647\pi$
$984$	0	0
$985$	41128.7	1.33043
$986$	0	0
$987$	0	0
$988$	0	0
$989$	43625.4	1.40263
$990$	0	0
$991$	− 3555.12i	− 0.113958i	−0.998375	−	0.0569788i	$-0.981853\pi$
$991$	− 3555.12i	− 0.113958i	0.998375	−	0.0569788i	$-0.0181468\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8218.94i	0.261867i
$996$	0	0
$997$	− 43140.0i	− 1.37037i	−0.728370	−	0.685184i	$-0.759722\pi$
$997$	− 43140.0i	− 1.37037i	0.728370	−	0.685184i	$-0.240278\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.f.g.863.4	yes	8
3.2	odd	2		1728.4.f.c.863.6	yes	8
4.3	odd	2		1728.4.f.c.863.3	✓	8
8.3	odd	2		1728.4.f.c.863.5	yes	8
8.5	even	2	inner	1728.4.f.g.863.6	yes	8
12.11	even	2	inner	1728.4.f.g.863.5	yes	8
24.5	odd	2		1728.4.f.c.863.4	yes	8
24.11	even	2	inner	1728.4.f.g.863.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.4.f.c.863.3	✓	8	4.3	odd	2
1728.4.f.c.863.4	yes	8	24.5	odd	2
1728.4.f.c.863.5	yes	8	8.3	odd	2
1728.4.f.c.863.6	yes	8	3.2	odd	2
1728.4.f.g.863.3	yes	8	24.11	even	2	inner
1728.4.f.g.863.4	yes	8	1.1	even	1	trivial
1728.4.f.g.863.5	yes	8	12.11	even	2	inner
1728.4.f.g.863.6	yes	8	8.5	even	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-9.35849 q^{5} +21.2094i q^{7} +O(q^{10})\)	\(q-9.35849 q^{5} +21.2094i q^{7} +50.9277i q^{11} -33.2716i q^{13} -120.628i q^{17} +82.1317 q^{19} +95.3719 q^{23} -37.4187 q^{25} -83.1384 q^{29} +36.4187i q^{31} -198.488i q^{35} -201.724i q^{37} -291.769i q^{41} +457.424 q^{43} -628.397 q^{47} -106.837 q^{49} +659.013 q^{53} -476.606i q^{55} -44.8330i q^{59} +149.600i q^{61} +311.372i q^{65} -134.256 q^{67} -291.769 q^{71} +888.212 q^{73} -1080.14 q^{77} +714.653i q^{79} +827.466i q^{83} +1128.90i q^{85} +19.6032i q^{89} +705.670 q^{91} -768.628 q^{95} -805.469 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 1224 q^{23} + 8 q^{25} - 1800 q^{47} - 240 q^{49} + 432 q^{71} + 344 q^{73} - 5688 q^{95} + 1240 q^{97}+O(q^{100}) \) 8 * q + 1224 * q^23 + 8 * q^25 - 1800 * q^47 - 240 * q^49 + 432 * q^71 + 344 * q^73 - 5688 * q^95 + 1240 * q^97

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