LMFDB - Embedded newform 1728.4.f.d.863.8

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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			863.8
Root			$2.33962 + 1.35078i$ of defining polynomial
Character	$\chi$	$=$	1728.863
Dual form			1728.4.f.d.863.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{5} +13.0000i q^{7} +O(q^{10})$	$q+1.73205 q^{5} +13.0000i q^{7} +22.5167i q^{11} +66.5432i q^{13} +115.256i q^{17} -66.5432 q^{19} -115.256 q^{23} -122.000 q^{25} -207.846 q^{29} -13.0000i q^{31} +22.5167i q^{35} -332.716i q^{37} -345.769i q^{41} +266.173 q^{43} +230.512 q^{47} +174.000 q^{49} -112.583 q^{53} +39.0000i q^{55} +135.100i q^{59} +399.259i q^{61} +115.256i q^{65} +345.769 q^{71} +85.0000 q^{73} -292.717 q^{77} -844.000i q^{79} -1259.20i q^{83} +199.630i q^{85} -230.512i q^{89} -865.062 q^{91} -115.256 q^{95} -481.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 976 q^{25} + 1392 q^{49} + 680 q^{73} - 3848 q^{97}+O(q^{100}) $ 8 * q - 976 * q^25 + 1392 * q^49 + 680 * q^73 - 3848 * 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
$5$	1.73205	0.154919	0.0774597	−	0.996995i	$-0.475319\pi$
$5$	1.73205	0.154919	0.0774597	+	0.996995i	$0.475319\pi$
$6$	0	0
$7$	13.0000i	0.701934i	0.936388	+	0.350967i	$0.114147\pi$
$7$	13.0000i	0.701934i	−0.936388	+	0.350967i	$0.885853\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	22.5167i	0.617184i	0.951194	+	0.308592i	$0.0998578\pi$
$11$	22.5167i	0.617184i	−0.951194	+	0.308592i	$0.900142\pi$
$12$	0	0
$13$	66.5432i	1.41967i	0.704366	+	0.709837i	$0.251232\pi$
$13$	66.5432i	1.41967i	−0.704366	+	0.709837i	$0.748768\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	115.256i	1.64434i	0.569244	+	0.822169i	$0.307236\pi$
$17$	115.256i	1.64434i	−0.569244	+	0.822169i	$0.692764\pi$
$18$	0	0
$19$	−66.5432	−0.803477	−0.401738	−	0.915754i	$-0.631594\pi$
$19$	−66.5432	−0.803477	−0.401738	+	0.915754i	$0.631594\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−115.256	−1.04490	−0.522448	−	0.852671i	$-0.674981\pi$
$23$	−115.256	−1.04490	−0.522448	+	0.852671i	$0.674981\pi$
$24$	0	0
$25$	−122.000	−0.976000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−207.846	−1.33090	−0.665449	−	0.746443i	$-0.731760\pi$
$29$	−207.846	−1.33090	−0.665449	+	0.746443i	$0.731760\pi$
$30$	0	0
$31$	− 13.0000i	− 0.0753184i	−0.999291	−	0.0376592i	$-0.988010\pi$
$31$	− 13.0000i	− 0.0753184i	0.999291	−	0.0376592i	$-0.0119901\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	22.5167i	0.108743i
$36$	0	0
$37$	− 332.716i	− 1.47833i	−0.673525	−	0.739165i	$-0.735220\pi$
$37$	− 332.716i	− 1.47833i	0.673525	−	0.739165i	$-0.264780\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 345.769i	− 1.31707i	−0.752549	−	0.658537i	$-0.771176\pi$
$41$	− 345.769i	− 1.31707i	0.752549	−	0.658537i	$-0.228824\pi$
$42$	0	0
$43$	266.173	0.943976	0.471988	−	0.881605i	$-0.343536\pi$
$43$	266.173	0.943976	0.471988	+	0.881605i	$0.343536\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	230.512	0.715398	0.357699	−	0.933837i	$-0.383561\pi$
$47$	230.512	0.715398	0.357699	+	0.933837i	$0.383561\pi$
$48$	0	0
$49$	174.000	0.507289
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−112.583	−0.291783	−0.145892	−	0.989301i	$-0.546605\pi$
$53$	−112.583	−0.291783	−0.145892	+	0.989301i	$0.546605\pi$
$54$	0	0
$55$	39.0000i	0.0956138i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	135.100i	0.298110i	0.988829	+	0.149055i	$0.0476232\pi$
$59$	135.100i	0.298110i	−0.988829	+	0.149055i	$0.952377\pi$
$60$	0	0
$61$	399.259i	0.838031i	0.907979	+	0.419016i	$0.137625\pi$
$61$	399.259i	0.838031i	−0.907979	+	0.419016i	$0.862375\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	115.256i	0.219935i
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	345.769	0.577961	0.288980	−	0.957335i	$-0.406684\pi$
$71$	345.769	0.577961	0.288980	+	0.957335i	$0.406684\pi$
$72$	0	0
$73$	85.0000	0.136281	0.0681404	−	0.997676i	$-0.478293\pi$
$73$	85.0000	0.136281	0.0681404	+	0.997676i	$0.478293\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−292.717	−0.433223
$78$	0	0
$79$	− 844.000i	− 1.20199i	−0.799252	−	0.600996i	$-0.794771\pi$
$79$	− 844.000i	− 1.20199i	0.799252	−	0.600996i	$-0.205229\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 1259.20i	− 1.66524i	−0.553842	−	0.832622i	$-0.686839\pi$
$83$	− 1259.20i	− 1.66524i	0.553842	−	0.832622i	$-0.313161\pi$
$84$	0	0
$85$	199.630i	0.254740i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 230.512i	− 0.274542i	−0.990534	−	0.137271i	$-0.956167\pi$
$89$	− 230.512i	− 0.274542i	0.990534	−	0.137271i	$-0.0438332\pi$
$90$	0	0
$91$	−865.062	−0.996518
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−115.256	−0.124474
$96$	0	0
$97$	−481.000	−0.503486	−0.251743	−	0.967794i	$-0.581004\pi$
$97$	−481.000	−0.503486	−0.251743	+	0.967794i	$0.581004\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−694.552	−0.684263	−0.342131	−	0.939652i	$-0.611149\pi$
$101$	−694.552	−0.684263	−0.342131	+	0.939652i	$0.611149\pi$
$102$	0	0
$103$	632.000i	0.604590i	0.953214	+	0.302295i	$0.0977528\pi$
$103$	632.000i	0.604590i	−0.953214	+	0.302295i	$0.902247\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1311.16i	− 1.18462i	−0.805708	−	0.592312i	$-0.798215\pi$
$107$	− 1311.16i	− 1.18462i	0.805708	−	0.592312i	$-0.201785\pi$
$108$	0	0
$109$	66.5432i	0.0584742i	0.999573	+	0.0292371i	$0.00930778\pi$
$109$	66.5432i	0.0584742i	−0.999573	+	0.0292371i	$0.990692\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1498.33i	1.24736i	0.781682	+	0.623678i	$0.214362\pi$
$113$	1498.33i	1.24736i	−0.781682	+	0.623678i	$0.785638\pi$
$114$	0	0
$115$	−199.630	−0.161874
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1498.33	−1.15422
$120$	0	0
$121$	824.000	0.619083
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−427.817	−0.306121
$126$	0	0
$127$	1201.00i	0.839146i	0.907722	+	0.419573i	$0.137820\pi$
$127$	1201.00i	0.839146i	−0.907722	+	0.419573i	$0.862180\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2274.18i	− 1.51677i	−0.651810	−	0.758383i	$-0.725990\pi$
$131$	− 2274.18i	− 1.51677i	0.651810	−	0.758383i	$-0.274010\pi$
$132$	0	0
$133$	− 865.062i	− 0.563988i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 230.512i	− 0.143752i	−0.997414	−	0.0718759i	$-0.977101\pi$
$137$	− 230.512i	− 0.143752i	0.997414	−	0.0718759i	$-0.0228986\pi$
$138$	0	0
$139$	1330.86	0.812104	0.406052	−	0.913850i	$-0.366905\pi$
$139$	1330.86	0.812104	0.406052	+	0.913850i	$0.366905\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1498.33	−0.876201
$144$	0	0
$145$	−360.000	−0.206182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1352.73	−0.743759	−0.371880	−	0.928281i	$-0.621287\pi$
$149$	−1352.73	−0.743759	−0.371880	+	0.928281i	$0.621287\pi$
$150$	0	0
$151$	319.000i	0.171920i	0.996299	+	0.0859598i	$0.0273957\pi$
$151$	319.000i	0.171920i	−0.996299	+	0.0859598i	$0.972604\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 22.5167i	− 0.0116683i
$156$	0	0
$157$	− 1730.12i	− 0.879483i	−0.898124	−	0.439742i	$-0.855070\pi$
$157$	− 1730.12i	− 0.879483i	0.898124	−	0.439742i	$-0.144930\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 1498.33i	− 0.733447i
$162$	0	0
$163$	3127.53	1.50287	0.751433	−	0.659809i	$-0.229363\pi$
$163$	3127.53	1.50287	0.751433	+	0.659809i	$0.229363\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3227.17	1.49537	0.747683	−	0.664055i	$-0.231166\pi$
$167$	3227.17	1.49537	0.747683	+	0.664055i	$0.231166\pi$
$168$	0	0
$169$	−2231.00	−1.01548
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3628.65	−1.59469	−0.797344	−	0.603526i	$-0.793762\pi$
$173$	−3628.65	−1.59469	−0.797344	+	0.603526i	$0.793762\pi$
$174$	0	0
$175$	− 1586.00i	− 0.685088i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 3576.68i	− 1.49349i	−0.665113	−	0.746743i	$-0.731617\pi$
$179$	− 3576.68i	− 1.49349i	0.665113	−	0.746743i	$-0.268383\pi$
$180$	0	0
$181$	3460.25i	1.42098i	0.703705	+	0.710492i	$0.251528\pi$
$181$	3460.25i	1.42098i	−0.703705	+	0.710492i	$0.748472\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 576.281i	− 0.229022i
$186$	0	0
$187$	−2595.19	−1.01486
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3572.94	1.35356	0.676778	−	0.736187i	$-0.263376\pi$
$191$	3572.94	1.35356	0.676778	+	0.736187i	$0.263376\pi$
$192$	0	0
$193$	3101.00	1.15655	0.578277	−	0.815841i	$-0.303725\pi$
$193$	3101.00	1.15655	0.578277	+	0.815841i	$0.303725\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4480.82	−1.62053	−0.810266	−	0.586062i	$-0.800677\pi$
$197$	−4480.82	−1.62053	−0.810266	+	0.586062i	$0.800677\pi$
$198$	0	0
$199$	5.00000i	0.00178111i	1.00000		0.000890554i	$0.000283472\pi$
$199$	5.00000i	0.00178111i	−1.00000		0.000890554i	$0.999717\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2702.00i	− 0.934203i
$204$	0	0
$205$	− 598.889i	− 0.204040i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 1498.33i	− 0.495893i
$210$	0	0
$211$	−6055.43	−1.97570	−0.987851	−	0.155403i	$-0.950332\pi$
$211$	−6055.43	−1.97570	−0.987851	+	0.155403i	$0.950332\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	461.025	0.146240
$216$	0	0
$217$	169.000	0.0528685
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7669.52	−2.33442
$222$	0	0
$223$	2032.00i	0.610192i	0.952322	+	0.305096i	$0.0986885\pi$
$223$	2032.00i	0.610192i	−0.952322	+	0.305096i	$0.901311\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2546.11i	− 0.744456i	−0.928141	−	0.372228i	$-0.878594\pi$
$227$	− 2546.11i	− 0.744456i	0.928141	−	0.372228i	$-0.121406\pi$
$228$	0	0
$229$	1730.12i	0.499257i	0.968342	+	0.249628i	$0.0803085\pi$
$229$	1730.12i	0.499257i	−0.968342	+	0.249628i	$0.919692\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4610.25i	1.29626i	0.761532	+	0.648128i	$0.224448\pi$
$233$	4610.25i	1.29626i	−0.761532	+	0.648128i	$0.775552\pi$
$234$	0	0
$235$	399.259	0.110829
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6339.09	−1.71566	−0.857828	−	0.513937i	$-0.828187\pi$
$239$	−6339.09	−1.71566	−0.857828	+	0.513937i	$0.828187\pi$
$240$	0	0
$241$	2626.00	0.701890	0.350945	−	0.936396i	$-0.385860\pi$
$241$	2626.00	0.701890	0.350945	+	0.936396i	$0.385860\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	301.377	0.0785888
$246$	0	0
$247$	− 4428.00i	− 1.14068i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 1714.73i	− 0.431206i	−0.976481	−	0.215603i	$-0.930828\pi$
$251$	− 1714.73i	− 0.431206i	0.976481	−	0.215603i	$-0.0691718\pi$
$252$	0	0
$253$	− 2595.19i	− 0.644893i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4494.99i	− 1.09101i	−0.838107	−	0.545506i	$-0.816338\pi$
$257$	− 4494.99i	− 1.09101i	0.838107	−	0.545506i	$-0.183662\pi$
$258$	0	0
$259$	4325.31	1.03769
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3572.94	−0.837708	−0.418854	−	0.908054i	$-0.637568\pi$
$263$	−3572.94	−0.837708	−0.418854	+	0.908054i	$0.637568\pi$
$264$	0	0
$265$	−195.000	−0.0452028
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5133.80	−1.16362	−0.581809	−	0.813325i	$-0.697655\pi$
$269$	−5133.80	−1.16362	−0.581809	+	0.813325i	$0.697655\pi$
$270$	0	0
$271$	1723.00i	0.386217i	0.981177	+	0.193108i	$0.0618569\pi$
$271$	1723.00i	0.386217i	−0.981177	+	0.193108i	$0.938143\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2747.03i	− 0.602372i
$276$	0	0
$277$	− 1463.95i	− 0.317546i	−0.987315	−	0.158773i	$-0.949246\pi$
$277$	− 1463.95i	− 0.317546i	0.987315	−	0.158773i	$-0.0507538\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4264.48i	0.905330i	0.891681	+	0.452665i	$0.149527\pi$
$281$	4264.48i	0.905330i	−0.891681	+	0.452665i	$0.850473\pi$
$282$	0	0
$283$	7519.38	1.57944	0.789719	−	0.613469i	$-0.210226\pi$
$283$	7519.38	1.57944	0.789719	+	0.613469i	$0.210226\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4494.99	0.924498
$288$	0	0
$289$	−8371.00	−1.70385
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1621.20	−0.323247	−0.161624	−	0.986852i	$-0.551673\pi$
$293$	−1621.20	−0.323247	−0.161624	+	0.986852i	$0.551673\pi$
$294$	0	0
$295$	234.000i	0.0461831i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 7669.52i	− 1.48341i
$300$	0	0
$301$	3460.25i	0.662609i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	691.537i	0.129827i
$306$	0	0
$307$	−3792.96	−0.705133	−0.352567	−	0.935787i	$-0.614691\pi$
$307$	−3792.96	−0.705133	−0.352567	+	0.935787i	$0.614691\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3111.92	−0.567398	−0.283699	−	0.958913i	$-0.591562\pi$
$311$	−3111.92	−0.567398	−0.283699	+	0.958913i	$0.591562\pi$
$312$	0	0
$313$	625.000	0.112866	0.0564330	−	0.998406i	$-0.482027\pi$
$313$	625.000	0.112866	0.0564330	+	0.998406i	$0.482027\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7588.11	−1.34445	−0.672226	−	0.740346i	$-0.734661\pi$
$317$	−7588.11	−1.34445	−0.672226	+	0.740346i	$0.734661\pi$
$318$	0	0
$319$	− 4680.00i	− 0.821410i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 7669.52i	− 1.32119i
$324$	0	0
$325$	− 8118.27i	− 1.38560i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2996.66i	0.502162i
$330$	0	0
$331$	−931.605	−0.154700	−0.0773499	−	0.997004i	$-0.524646\pi$
$331$	−931.605	−0.154700	−0.0773499	+	0.997004i	$0.524646\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2738.00	0.442577	0.221288	−	0.975208i	$-0.428974\pi$
$337$	2738.00	0.442577	0.221288	+	0.975208i	$0.428974\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	292.717	0.0464853
$342$	0	0
$343$	6721.00i	1.05802i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 9614.61i	− 1.48743i	−0.668495	−	0.743717i	$-0.733061\pi$
$347$	− 9614.61i	− 1.48743i	0.668495	−	0.743717i	$-0.266939\pi$
$348$	0	0
$349$	7253.21i	1.11248i	0.831022	+	0.556240i	$0.187756\pi$
$349$	7253.21i	1.11248i	−0.831022	+	0.556240i	$0.812244\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2189.87i	0.330184i	0.986278	+	0.165092i	$0.0527921\pi$
$353$	2189.87i	0.330184i	−0.986278	+	0.165092i	$0.947208\pi$
$354$	0	0
$355$	598.889	0.0895373
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−2431.00	−0.354425
$362$	0	0
$363$	0	0
$364$	0	0
$365$	147.224	0.0211125
$366$	0	0
$367$	10895.0i	1.54963i	0.632188	+	0.774815i	$0.282157\pi$
$367$	10895.0i	1.54963i	−0.632188	+	0.774815i	$0.717843\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1463.58i	− 0.204813i
$372$	0	0
$373$	− 6055.43i	− 0.840586i	−0.907389	−	0.420293i	$-0.861927\pi$
$373$	− 6055.43i	− 0.840586i	0.907389	−	0.420293i	$-0.138073\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 13830.7i	− 1.88944i
$378$	0	0
$379$	−10247.7	−1.38888	−0.694442	−	0.719549i	$-0.744349\pi$
$379$	−10247.7	−1.38888	−0.694442	+	0.719549i	$0.744349\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	13485.0	1.79909	0.899543	−	0.436831i	$-0.143899\pi$
$383$	13485.0	1.79909	0.899543	+	0.436831i	$0.143899\pi$
$384$	0	0
$385$	−507.000	−0.0671146
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2682.95	−0.349694	−0.174847	−	0.984596i	$-0.555943\pi$
$389$	−2682.95	−0.349694	−0.174847	+	0.984596i	$0.555943\pi$
$390$	0	0
$391$	− 13284.0i	− 1.71816i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1461.85i	− 0.186212i
$396$	0	0
$397$	7253.21i	0.916948i	0.888708	+	0.458474i	$0.151604\pi$
$397$	7253.21i	0.916948i	−0.888708	+	0.458474i	$0.848396\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2996.66i	0.373182i	0.982438	+	0.186591i	$0.0597440\pi$
$401$	2996.66i	0.373182i	−0.982438	+	0.186591i	$0.940256\pi$
$402$	0	0
$403$	865.062	0.106928
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7491.66	0.912402
$408$	0	0
$409$	10435.0	1.26156	0.630779	−	0.775962i	$-0.282735\pi$
$409$	10435.0	1.26156	0.630779	+	0.775962i	$0.282735\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1756.30	−0.209254
$414$	0	0
$415$	− 2181.00i	− 0.257979i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4250.45i	0.495580i	0.968814	+	0.247790i	$0.0797044\pi$
$419$	4250.45i	0.495580i	−0.968814	+	0.247790i	$0.920296\pi$
$420$	0	0
$421$	7719.01i	0.893591i	0.894636	+	0.446795i	$0.147435\pi$
$421$	7719.01i	0.893591i	−0.894636	+	0.446795i	$0.852565\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 14061.3i	− 1.60487i
$426$	0	0
$427$	−5190.37	−0.588243
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10834.1	−1.21081	−0.605406	−	0.795917i	$-0.706989\pi$
$431$	−10834.1	−1.21081	−0.605406	+	0.795917i	$0.706989\pi$
$432$	0	0
$433$	−7493.00	−0.831618	−0.415809	−	0.909452i	$-0.636502\pi$
$433$	−7493.00	−0.831618	−0.415809	+	0.909452i	$0.636502\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	7669.52	0.839549
$438$	0	0
$439$	4217.00i	0.458466i	0.973372	+	0.229233i	$0.0736217\pi$
$439$	4217.00i	0.458466i	−0.973372	+	0.229233i	$0.926378\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 6765.39i	− 0.725583i	−0.931870	−	0.362792i	$-0.881824\pi$
$443$	− 6765.39i	− 0.725583i	0.931870	−	0.362792i	$-0.118176\pi$
$444$	0	0
$445$	− 399.259i	− 0.0425319i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	10488.3i	1.10239i	0.834376	+	0.551196i	$0.185828\pi$
$449$	10488.3i	1.10239i	−0.834376	+	0.551196i	$0.814172\pi$
$450$	0	0
$451$	7785.56	0.812877
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1498.33	−0.154380
$456$	0	0
$457$	−10049.0	−1.02860	−0.514302	−	0.857609i	$-0.671949\pi$
$457$	−10049.0	−1.02860	−0.514302	+	0.857609i	$0.671949\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−8509.57	−0.859718	−0.429859	−	0.902896i	$-0.641437\pi$
$461$	−8509.57	−0.859718	−0.429859	+	0.902896i	$0.641437\pi$
$462$	0	0
$463$	− 16061.0i	− 1.61213i	−0.591824	−	0.806067i	$-0.701592\pi$
$463$	− 16061.0i	− 1.61213i	0.591824	−	0.806067i	$-0.298408\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9881.35i	0.979131i	0.871966	+	0.489566i	$0.162845\pi$
$467$	9881.35i	0.979131i	−0.871966	+	0.489566i	$0.837155\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5993.32i	0.582607i
$474$	0	0
$475$	8118.27	0.784193
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17518.9	−1.67111	−0.835554	−	0.549408i	$-0.814853\pi$
$479$	−17518.9	−1.67111	−0.835554	+	0.549408i	$0.814853\pi$
$480$	0	0
$481$	22140.0	2.09875
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−833.116	−0.0779997
$486$	0	0
$487$	13984.0i	1.30118i	0.759428	+	0.650591i	$0.225479\pi$
$487$	13984.0i	1.30118i	−0.759428	+	0.650591i	$0.774521\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6060.45i	0.557035i	0.960431	+	0.278517i	$0.0898430\pi$
$491$	6060.45i	0.557035i	−0.960431	+	0.278517i	$0.910157\pi$
$492$	0	0
$493$	− 23955.6i	− 2.18845i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4494.99i	0.405690i
$498$	0	0
$499$	10380.7	0.931274	0.465637	−	0.884976i	$-0.345825\pi$
$499$	10380.7	0.931274	0.465637	+	0.884976i	$0.345825\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4494.99	−0.398453	−0.199226	−	0.979953i	$-0.563843\pi$
$503$	−4494.99	−0.398453	−0.199226	+	0.979953i	$0.563843\pi$
$504$	0	0
$505$	−1203.00	−0.106006
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−16189.5	−1.40980	−0.704898	−	0.709309i	$-0.749007\pi$
$509$	−16189.5	−1.40980	−0.704898	+	0.709309i	$0.749007\pi$
$510$	0	0
$511$	1105.00i	0.0956601i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1094.66i	0.0936627i
$516$	0	0
$517$	5190.37i	0.441532i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8413.71i	0.707507i	0.935339	+	0.353753i	$0.115095\pi$
$521$	8413.71i	0.707507i	−0.935339	+	0.353753i	$0.884905\pi$
$522$	0	0
$523$	12576.7	1.05151	0.525755	−	0.850636i	$-0.323783\pi$
$523$	12576.7	1.05151	0.525755	+	0.850636i	$0.323783\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1498.33	0.123849
$528$	0	0
$529$	1117.00	0.0918057
$530$	0	0
$531$	0	0
$532$	0	0
$533$	23008.6	1.86982
$534$	0	0
$535$	− 2271.00i	− 0.183521i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3917.90i	0.313091i
$540$	0	0
$541$	− 3460.25i	− 0.274986i	−0.990503	−	0.137493i	$-0.956095\pi$
$541$	− 3460.25i	− 0.274986i	0.990503	−	0.137493i	$-0.0439045\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	115.256i	0.00905878i
$546$	0	0
$547$	−22957.4	−1.79449	−0.897247	−	0.441529i	$-0.854436\pi$
$547$	−22957.4	−1.79449	−0.897247	+	0.441529i	$0.854436\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13830.7	1.06935
$552$	0	0
$553$	10972.0	0.843720
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10380.2	−0.789627	−0.394814	−	0.918761i	$-0.629191\pi$
$557$	−10380.2	−0.789627	−0.394814	+	0.918761i	$0.629191\pi$
$558$	0	0
$559$	17712.0i	1.34014i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10331.7i	0.773408i	0.922204	+	0.386704i	$0.126386\pi$
$563$	10331.7i	0.773408i	−0.922204	+	0.386704i	$0.873614\pi$
$564$	0	0
$565$	2595.19i	0.193239i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11986.6i	0.883140i	0.897227	+	0.441570i	$0.145578\pi$
$569$	11986.6i	0.883140i	−0.897227	+	0.441570i	$0.854422\pi$
$570$	0	0
$571$	−16702.3	−1.22412	−0.612059	−	0.790812i	$-0.709659\pi$
$571$	−16702.3	−1.22412	−0.612059	+	0.790812i	$0.709659\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	14061.3	1.01982
$576$	0	0
$577$	−24518.0	−1.76897	−0.884487	−	0.466565i	$-0.845491\pi$
$577$	−24518.0	−1.76897	−0.884487	+	0.466565i	$0.845491\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	16369.6	1.16889
$582$	0	0
$583$	− 2535.00i	− 0.180084i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	19161.7i	1.34734i	0.739033	+	0.673669i	$0.235283\pi$
$587$	19161.7i	1.34734i	−0.739033	+	0.673669i	$0.764717\pi$
$588$	0	0
$589$	865.062i	0.0605166i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 11986.6i	− 0.830072i	−0.909805	−	0.415036i	$-0.863769\pi$
$593$	− 11986.6i	− 0.830072i	0.909805	−	0.415036i	$-0.136231\pi$
$594$	0	0
$595$	−2595.19	−0.178810
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−12908.7	−0.880526	−0.440263	−	0.897869i	$-0.645115\pi$
$599$	−12908.7	−0.880526	−0.440263	+	0.897869i	$0.645115\pi$
$600$	0	0
$601$	−15835.0	−1.07475	−0.537374	−	0.843344i	$-0.680583\pi$
$601$	−15835.0	−1.07475	−0.537374	+	0.843344i	$0.680583\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1427.21	0.0959080
$606$	0	0
$607$	− 18344.0i	− 1.22662i	−0.789841	−	0.613311i	$-0.789837\pi$
$607$	− 18344.0i	− 1.22662i	0.789841	−	0.613311i	$-0.210163\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	15339.0i	1.01563i
$612$	0	0
$613$	21560.0i	1.42056i	0.703922	+	0.710278i	$0.251431\pi$
$613$	21560.0i	1.42056i	−0.703922	+	0.710278i	$0.748569\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 21322.4i	− 1.39126i	−0.718400	−	0.695630i	$-0.755125\pi$
$617$	− 21322.4i	− 1.39126i	0.718400	−	0.695630i	$-0.244875\pi$
$618$	0	0
$619$	−1397.41	−0.0907376	−0.0453688	−	0.998970i	$-0.514446\pi$
$619$	−1397.41	−0.0907376	−0.0453688	+	0.998970i	$0.514446\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2996.66	0.192711
$624$	0	0
$625$	14509.0	0.928576
$626$	0	0
$627$	0	0
$628$	0	0
$629$	38347.6	2.43087
$630$	0	0
$631$	11245.0i	0.709440i	0.934973	+	0.354720i	$0.115424\pi$
$631$	11245.0i	0.709440i	−0.934973	+	0.354720i	$0.884576\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2080.19i	0.130000i
$636$	0	0
$637$	11578.5i	0.720185i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 4494.99i	− 0.276976i	−0.990364	−	0.138488i	$-0.955776\pi$
$641$	− 4494.99i	− 0.276976i	0.990364	−	0.138488i	$-0.0442242\pi$
$642$	0	0
$643$	−5988.89	−0.367308	−0.183654	−	0.982991i	$-0.558793\pi$
$643$	−5988.89	−0.367308	−0.183654	+	0.982991i	$0.558793\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5993.32	0.364176	0.182088	−	0.983282i	$-0.441714\pi$
$647$	5993.32	0.364176	0.182088	+	0.983282i	$0.441714\pi$
$648$	0	0
$649$	−3042.00	−0.183989
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10858.2	0.650712	0.325356	−	0.945592i	$-0.394516\pi$
$653$	10858.2	0.650712	0.325356	+	0.945592i	$0.394516\pi$
$654$	0	0
$655$	− 3939.00i	− 0.234976i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	13217.3i	0.781293i	0.920541	+	0.390647i	$0.127749\pi$
$659$	13217.3i	0.781293i	−0.920541	+	0.390647i	$0.872251\pi$
$660$	0	0
$661$	− 3792.96i	− 0.223191i	−0.993754	−	0.111595i	$-0.964404\pi$
$661$	− 3792.96i	− 0.223191i	0.993754	−	0.111595i	$-0.0355961\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1498.33i	− 0.0873726i
$666$	0	0
$667$	23955.6	1.39065
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8989.99	−0.517220
$672$	0	0
$673$	−21965.0	−1.25808	−0.629041	−	0.777373i	$-0.716552\pi$
$673$	−21965.0	−1.25808	−0.629041	+	0.777373i	$0.716552\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17292.8	0.981708	0.490854	−	0.871242i	$-0.336685\pi$
$677$	17292.8	0.981708	0.490854	+	0.871242i	$0.336685\pi$
$678$	0	0
$679$	− 6253.00i	− 0.353414i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 5372.82i	− 0.301003i	−0.988610	−	0.150502i	$-0.951911\pi$
$683$	− 5372.82i	− 0.301003i	0.988610	−	0.150502i	$-0.0480889\pi$
$684$	0	0
$685$	− 399.259i	− 0.0222699i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 7491.66i	− 0.414237i
$690$	0	0
$691$	19031.4	1.04774	0.523869	−	0.851799i	$-0.324488\pi$
$691$	19031.4	1.04774	0.523869	+	0.851799i	$0.324488\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	2305.12	0.125811
$696$	0	0
$697$	39852.0	2.16571
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−7272.88	−0.391859	−0.195929	−	0.980618i	$-0.562772\pi$
$701$	−7272.88	−0.391859	−0.195929	+	0.980618i	$0.562772\pi$
$702$	0	0
$703$	22140.0i	1.18780i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 9029.18i	− 0.480307i
$708$	0	0
$709$	− 16103.5i	− 0.853002i	−0.904487	−	0.426501i	$-0.859746\pi$
$709$	− 16103.5i	− 0.853002i	0.904487	−	0.426501i	$-0.140254\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	1498.33i	0.0786998i
$714$	0	0
$715$	−2595.19	−0.135740
$716$	0	0
$717$	0	0
$718$	0	0
$719$	19593.6	1.01630	0.508148	−	0.861270i	$-0.330330\pi$
$719$	19593.6	1.01630	0.508148	+	0.861270i	$0.330330\pi$
$720$	0	0
$721$	−8216.00	−0.424383
$722$	0	0
$723$	0	0
$724$	0	0
$725$	25357.2	1.29896
$726$	0	0
$727$	949.000i	0.0484133i	0.999707	+	0.0242066i	$0.00770597\pi$
$727$	949.000i	0.0484133i	−0.999707	+	0.0242066i	$0.992294\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	30678.1i	1.55222i
$732$	0	0
$733$	− 10314.2i	− 0.519732i	−0.965645	−	0.259866i	$-0.916322\pi$
$733$	− 10314.2i	− 0.519732i	0.965645	−	0.259866i	$-0.0836784\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−28014.7	−1.39450	−0.697251	−	0.716827i	$-0.745594\pi$
$739$	−28014.7	−1.39450	−0.697251	+	0.716827i	$0.745594\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1498.33	−0.0739817	−0.0369909	−	0.999316i	$-0.511777\pi$
$743$	−1498.33	−0.0739817	−0.0369909	+	0.999316i	$0.511777\pi$
$744$	0	0
$745$	−2343.00	−0.115223
$746$	0	0
$747$	0	0
$748$	0	0
$749$	17045.1	0.831528
$750$	0	0
$751$	− 40261.0i	− 1.95625i	−0.208015	−	0.978126i	$-0.566700\pi$
$751$	− 40261.0i	− 1.95625i	0.208015	−	0.978126i	$-0.433300\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	552.524i	0.0266337i
$756$	0	0
$757$	− 20761.5i	− 0.996815i	−0.866943	−	0.498408i	$-0.833918\pi$
$757$	− 20761.5i	− 0.996815i	0.866943	−	0.498408i	$-0.166082\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	14752.8i	0.702744i	0.936236	+	0.351372i	$0.114285\pi$
$761$	14752.8i	0.702744i	−0.936236	+	0.351372i	$0.885715\pi$
$762$	0	0
$763$	−865.062	−0.0410450
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8989.99	−0.423220
$768$	0	0
$769$	28877.0	1.35414	0.677068	−	0.735920i	$-0.263250\pi$
$769$	28877.0	1.35414	0.677068	+	0.735920i	$0.263250\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	38119.0	1.77367	0.886833	−	0.462090i	$-0.152900\pi$
$773$	38119.0	1.77367	0.886833	+	0.462090i	$0.152900\pi$
$774$	0	0
$775$	1586.00i	0.0735107i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	23008.6i	1.05824i
$780$	0	0
$781$	7785.56i	0.356708i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 2996.66i	− 0.136249i
$786$	0	0
$787$	40325.2	1.82648	0.913239	−	0.407425i	$-0.133573\pi$
$787$	40325.2	1.82648	0.913239	+	0.407425i	$0.133573\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−19478.3	−0.875561
$792$	0	0
$793$	−26568.0	−1.18973
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2315.75	0.102921	0.0514606	−	0.998675i	$-0.483612\pi$
$797$	2315.75	0.102921	0.0514606	+	0.998675i	$0.483612\pi$
$798$	0	0
$799$	26568.0i	1.17636i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1913.92i	0.0841104i
$804$	0	0
$805$	− 2595.19i	− 0.113625i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	38956.6i	1.69301i	0.532384	+	0.846503i	$0.321296\pi$
$809$	38956.6i	1.69301i	−0.532384	+	0.846503i	$0.678704\pi$
$810$	0	0
$811$	−25086.8	−1.08621	−0.543105	−	0.839665i	$-0.682751\pi$
$811$	−25086.8	−1.08621	−0.543105	+	0.839665i	$0.682751\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	5417.04	0.232823
$816$	0	0
$817$	−17712.0	−0.758463
$818$	0	0
$819$	0	0
$820$	0	0
$821$	32424.0	1.37833	0.689163	−	0.724607i	$-0.257979\pi$
$821$	32424.0	1.37833	0.689163	+	0.724607i	$0.257979\pi$
$822$	0	0
$823$	− 43277.0i	− 1.83298i	−0.400059	−	0.916489i	$-0.631010\pi$
$823$	− 43277.0i	− 1.83298i	0.400059	−	0.916489i	$-0.368990\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	42660.4i	1.79377i	0.442264	+	0.896885i	$0.354176\pi$
$827$	42660.4i	1.79377i	−0.442264	+	0.896885i	$0.645824\pi$
$828$	0	0
$829$	− 13841.0i	− 0.579876i	−0.957045	−	0.289938i	$-0.906365\pi$
$829$	− 13841.0i	− 0.579876i	0.957045	−	0.289938i	$-0.0936347\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	20054.6i	0.834154i
$834$	0	0
$835$	5589.63	0.231661
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−25702.1	−1.05761	−0.528806	−	0.848743i	$-0.677360\pi$
$839$	−25702.1	−1.05761	−0.528806	+	0.848743i	$0.677360\pi$
$840$	0	0
$841$	18811.0	0.771290
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−3864.21	−0.157317
$846$	0	0
$847$	10712.0i	0.434556i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	38347.6i	1.54470i
$852$	0	0
$853$	23157.0i	0.929522i	0.885436	+	0.464761i	$0.153860\pi$
$853$	23157.0i	0.929522i	−0.885436	+	0.464761i	$0.846140\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11871.4i	0.473184i	0.971609	+	0.236592i	$0.0760305\pi$
$857$	11871.4i	0.473184i	−0.971609	+	0.236592i	$0.923969\pi$
$858$	0	0
$859$	3060.99	0.121583	0.0607914	−	0.998150i	$-0.480638\pi$
$859$	3060.99	0.121583	0.0607914	+	0.998150i	$0.480638\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−42644.8	−1.68209	−0.841046	−	0.540963i	$-0.818060\pi$
$863$	−42644.8	−1.68209	−0.841046	+	0.540963i	$0.818060\pi$
$864$	0	0
$865$	−6285.00	−0.247048
$866$	0	0
$867$	0	0
$868$	0	0
$869$	19004.1	0.741851
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 5561.62i	− 0.214876i
$876$	0	0
$877$	34602.5i	1.33232i	0.745810	+	0.666159i	$0.232063\pi$
$877$	34602.5i	1.33232i	−0.745810	+	0.666159i	$0.767937\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 32848.0i	− 1.25616i	−0.778148	−	0.628081i	$-0.783841\pi$
$881$	− 32848.0i	− 1.25616i	0.778148	−	0.628081i	$-0.216159\pi$
$882$	0	0
$883$	−23090.5	−0.880019	−0.440010	−	0.897993i	$-0.645025\pi$
$883$	−23090.5	−0.880019	−0.440010	+	0.897993i	$0.645025\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	48522.9	1.83680	0.918398	−	0.395657i	$-0.129483\pi$
$887$	48522.9	1.83680	0.918398	+	0.395657i	$0.129483\pi$
$888$	0	0
$889$	−15613.0	−0.589025
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−15339.0	−0.574806
$894$	0	0
$895$	− 6195.00i	− 0.231370i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2702.00i	0.100241i
$900$	0	0
$901$	− 12975.9i	− 0.479790i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5993.32i	0.220138i
$906$	0	0
$907$	−7519.38	−0.275278	−0.137639	−	0.990482i	$-0.543951\pi$
$907$	−7519.38	−0.275278	−0.137639	+	0.990482i	$0.543951\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	44027.9	1.60122	0.800609	−	0.599188i	$-0.204510\pi$
$911$	44027.9	1.60122	0.800609	+	0.599188i	$0.204510\pi$
$912$	0	0
$913$	28353.0	1.02776
$914$	0	0
$915$	0	0
$916$	0	0
$917$	29564.4	1.06467
$918$	0	0
$919$	35671.0i	1.28039i	0.768213	+	0.640195i	$0.221146\pi$
$919$	35671.0i	1.28039i	−0.768213	+	0.640195i	$0.778854\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	23008.6i	0.820516i
$924$	0	0
$925$	40591.4i	1.44285i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 38956.6i	− 1.37581i	−0.725802	−	0.687904i	$-0.758531\pi$
$929$	− 38956.6i	− 1.37581i	0.725802	−	0.687904i	$-0.241469\pi$
$930$	0	0
$931$	−11578.5	−0.407595
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4494.99	−0.157221
$936$	0	0
$937$	−21605.0	−0.753260	−0.376630	−	0.926364i	$-0.622917\pi$
$937$	−21605.0	−0.753260	−0.376630	+	0.926364i	$0.622917\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−31476.6	−1.09044	−0.545221	−	0.838292i	$-0.683554\pi$
$941$	−31476.6	−1.09044	−0.545221	+	0.838292i	$0.683554\pi$
$942$	0	0
$943$	39852.0i	1.37620i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10349.0i	0.355119i	0.984110	+	0.177559i	$0.0568202\pi$
$947$	10349.0i	0.355119i	−0.984110	+	0.177559i	$0.943180\pi$
$948$	0	0
$949$	5656.17i	0.193474i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 50943.3i	− 1.73160i	−0.500391	−	0.865800i	$-0.666810\pi$
$953$	− 50943.3i	− 1.73160i	0.500391	−	0.865800i	$-0.333190\pi$
$954$	0	0
$955$	6188.52	0.209692
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2996.66	0.100904
$960$	0	0
$961$	29622.0	0.994327
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5371.09	0.179173
$966$	0	0
$967$	− 5629.00i	− 0.187194i	−0.995610	−	0.0935969i	$-0.970164\pi$
$967$	− 5629.00i	− 0.187194i	0.995610	−	0.0935969i	$-0.0298365\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	22241.3i	0.735073i	0.930009	+	0.367537i	$0.119799\pi$
$971$	22241.3i	0.735073i	−0.930009	+	0.367537i	$0.880201\pi$
$972$	0	0
$973$	17301.2i	0.570043i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1844.10i	0.0603869i	0.999544	+	0.0301934i	$0.00961233\pi$
$977$	1844.10i	0.0603869i	−0.999544	+	0.0301934i	$0.990388\pi$
$978$	0	0
$979$	5190.37	0.169443
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−5762.81	−0.186984	−0.0934919	−	0.995620i	$-0.529803\pi$
$983$	−5762.81	−0.186984	−0.0934919	+	0.995620i	$0.529803\pi$
$984$	0	0
$985$	−7761.00	−0.251052
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−30678.1	−0.986356
$990$	0	0
$991$	9293.00i	0.297883i	0.988846	+	0.148941i	$0.0475866\pi$
$991$	9293.00i	0.297883i	−0.988846	+	0.148941i	$0.952413\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.66025i	0 0.000275928i
$996$	0	0
$997$	− 54498.9i	− 1.73119i	−0.500744	−	0.865595i	$-0.666940\pi$
$997$	− 54498.9i	− 1.73119i	0.500744	−	0.865595i	$-0.333060\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.f.d.863.8	yes	8
3.2	odd	2	inner	1728.4.f.d.863.4	yes	8
4.3	odd	2	inner	1728.4.f.d.863.6	yes	8
8.3	odd	2	inner	1728.4.f.d.863.1	✓	8
8.5	even	2	inner	1728.4.f.d.863.3	yes	8
12.11	even	2	inner	1728.4.f.d.863.2	yes	8
24.5	odd	2	inner	1728.4.f.d.863.7	yes	8
24.11	even	2	inner	1728.4.f.d.863.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.4.f.d.863.1	✓	8	8.3	odd	2	inner
1728.4.f.d.863.2	yes	8	12.11	even	2	inner
1728.4.f.d.863.3	yes	8	8.5	even	2	inner
1728.4.f.d.863.4	yes	8	3.2	odd	2	inner
1728.4.f.d.863.5	yes	8	24.11	even	2	inner
1728.4.f.d.863.6	yes	8	4.3	odd	2	inner
1728.4.f.d.863.7	yes	8	24.5	odd	2	inner
1728.4.f.d.863.8	yes	8	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:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{5} +13.0000i q^{7} +O(q^{10})\)	\(q+1.73205 q^{5} +13.0000i q^{7} +22.5167i q^{11} +66.5432i q^{13} +115.256i q^{17} -66.5432 q^{19} -115.256 q^{23} -122.000 q^{25} -207.846 q^{29} -13.0000i q^{31} +22.5167i q^{35} -332.716i q^{37} -345.769i q^{41} +266.173 q^{43} +230.512 q^{47} +174.000 q^{49} -112.583 q^{53} +39.0000i q^{55} +135.100i q^{59} +399.259i q^{61} +115.256i q^{65} +345.769 q^{71} +85.0000 q^{73} -292.717 q^{77} -844.000i q^{79} -1259.20i q^{83} +199.630i q^{85} -230.512i q^{89} -865.062 q^{91} -115.256 q^{95} -481.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 976 q^{25} + 1392 q^{49} + 680 q^{73} - 3848 q^{97}+O(q^{100}) \) 8 * q - 976 * q^25 + 1392 * q^49 + 680 * q^73 - 3848 * q^97

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