LMFDB - Embedded newform 2160.4.h.c.431.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,4,Mod(431,2160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2160.431"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2160.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,-192] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$127.444125612$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 8 x^{10} - 100 x^{9} + 160 x^{8} - 522 x^{7} + 3906 x^{6} - 4698 x^{5} + \cdots + 531441 $ x^12 - 2x^11 + 8x^10 - 100x^9 + 160x^8 - 522x^7 + 3906x^6 - 4698x^5 + 12960x^4 - 72900x^3 + 52488x^2 - 118098*x + 531441
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.3
Root			$-0.671227 - 2.92394i$ of defining polynomial
Character	$\chi$	$=$	2160.431
Dual form			2160.4.h.c.431.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000i q^{5} -6.39707i q^{7} -55.5166 q^{11} +66.4777 q^{13} +5.75985i q^{17} -20.4831i q^{19} -23.9237 q^{23} -25.0000 q^{25} +264.356i q^{29} +54.2716i q^{31} -31.9853 q^{35} +273.756 q^{37} +169.715i q^{41} -180.798i q^{43} -124.124 q^{47} +302.077 q^{49} -174.189i q^{53} +277.583i q^{55} -19.6299 q^{59} -676.182 q^{61} -332.389i q^{65} -374.946i q^{67} +976.477 q^{71} -246.465 q^{73} +355.143i q^{77} +1153.63i q^{79} +59.4554 q^{83} +28.7992 q^{85} -474.905i q^{89} -425.263i q^{91} -102.415 q^{95} -54.9428 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 192 q^{13} - 300 q^{25} + 936 q^{37} + 2820 q^{49} - 492 q^{61} + 4488 q^{73} + 900 q^{85} + 2544 q^{97}+O(q^{100}) $ 12 * q - 192 * q^13 - 300 * q^25 + 936 * q^37 + 2820 * q^49 - 492 * q^61 + 4488 * q^73 + 900 * q^85 + 2544 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$-1$	$1$	$1$	$-1$

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
$5$	− 5.00000i	− 0.447214i
$5$	− 5.00000i	− 0.447214i
$6$	0	0
$7$	− 6.39707i	− 0.345409i	−0.984974	−	0.172705i	$-0.944749\pi$
$7$	− 6.39707i	− 0.345409i	0.984974	−	0.172705i	$-0.0552506\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−55.5166	−1.52172	−0.760858	−	0.648918i	$-0.775222\pi$
$11$	−55.5166	−1.52172	−0.760858	+	0.648918i	$0.775222\pi$
$12$	0	0
$13$	66.4777	1.41828	0.709139	−	0.705069i	$-0.249084\pi$
$13$	66.4777	1.41828	0.709139	+	0.705069i	$0.249084\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.75985i	0.0821746i	0.999156	+	0.0410873i	$0.0130822\pi$
$17$	5.75985i	0.0821746i	−0.999156	+	0.0410873i	$0.986918\pi$
$18$	0	0
$19$	− 20.4831i	− 0.247323i	−0.992324	−	0.123662i	$-0.960536\pi$
$19$	− 20.4831i	− 0.247323i	0.992324	−	0.123662i	$-0.0394637\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−23.9237	−0.216889	−0.108444	−	0.994103i	$-0.534587\pi$
$23$	−23.9237	−0.216889	−0.108444	+	0.994103i	$0.534587\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	264.356i	1.69275i	0.532591	+	0.846373i	$0.321219\pi$
$29$	264.356i	1.69275i	−0.532591	+	0.846373i	$0.678781\pi$
$30$	0	0
$31$	54.2716i	0.314435i	0.987564	+	0.157217i	$0.0502523\pi$
$31$	54.2716i	0.314435i	−0.987564	+	0.157217i	$0.949748\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−31.9853	−0.154472
$36$	0	0
$37$	273.756	1.21636	0.608178	−	0.793800i	$-0.291900\pi$
$37$	273.756	1.21636	0.608178	+	0.793800i	$0.291900\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	169.715i	0.646465i	0.946320	+	0.323233i	$0.104770\pi$
$41$	169.715i	0.646465i	−0.946320	+	0.323233i	$0.895230\pi$
$42$	0	0
$43$	− 180.798i	− 0.641196i	−0.947215	−	0.320598i	$-0.896116\pi$
$43$	− 180.798i	− 0.641196i	0.947215	−	0.320598i	$-0.103884\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−124.124	−0.385219	−0.192610	−	0.981275i	$-0.561695\pi$
$47$	−124.124	−0.385219	−0.192610	+	0.981275i	$0.561695\pi$
$48$	0	0
$49$	302.077	0.880692
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 174.189i	− 0.451448i	−0.974191	−	0.225724i	$-0.927525\pi$
$53$	− 174.189i	− 0.451448i	0.974191	−	0.225724i	$-0.0724747\pi$
$54$	0	0
$55$	277.583i	0.680532i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−19.6299	−0.0433151	−0.0216576	−	0.999765i	$-0.506894\pi$
$59$	−19.6299	−0.0433151	−0.0216576	+	0.999765i	$0.506894\pi$
$60$	0	0
$61$	−676.182	−1.41928	−0.709641	−	0.704564i	$-0.751143\pi$
$61$	−676.182	−1.41928	−0.709641	+	0.704564i	$0.751143\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 332.389i	− 0.634273i
$66$	0	0
$67$	− 374.946i	− 0.683686i	−0.939757	−	0.341843i	$-0.888949\pi$
$67$	− 374.946i	− 0.683686i	0.939757	−	0.341843i	$-0.111051\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	976.477	1.63220	0.816102	−	0.577907i	$-0.196130\pi$
$71$	976.477	1.63220	0.816102	+	0.577907i	$0.196130\pi$
$72$	0	0
$73$	−246.465	−0.395158	−0.197579	−	0.980287i	$-0.563308\pi$
$73$	−246.465	−0.395158	−0.197579	+	0.980287i	$0.563308\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	355.143i	0.525615i
$78$	0	0
$79$	1153.63i	1.64295i	0.570245	+	0.821475i	$0.306848\pi$
$79$	1153.63i	1.64295i	−0.570245	+	0.821475i	$0.693152\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	59.4554	0.0786275	0.0393137	−	0.999227i	$-0.487483\pi$
$83$	59.4554	0.0786275	0.0393137	+	0.999227i	$0.487483\pi$
$84$	0	0
$85$	28.7992	0.0367496
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 474.905i	− 0.565616i	−0.959177	−	0.282808i	$-0.908734\pi$
$89$	− 474.905i	− 0.565616i	0.959177	−	0.282808i	$-0.0912659\pi$
$90$	0	0
$91$	− 425.263i	− 0.489886i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−102.415	−0.110606
$96$	0	0
$97$	−54.9428	−0.0575113	−0.0287556	−	0.999586i	$-0.509154\pi$
$97$	−54.9428	−0.0575113	−0.0287556	+	0.999586i	$0.509154\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1348.85i	1.32887i	0.747346	+	0.664435i	$0.231328\pi$
$101$	1348.85i	1.32887i	−0.747346	+	0.664435i	$0.768672\pi$
$102$	0	0
$103$	− 1197.10i	− 1.14518i	−0.819842	−	0.572590i	$-0.805939\pi$
$103$	− 1197.10i	− 1.14518i	0.819842	−	0.572590i	$-0.194061\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1589.60	1.43619	0.718095	−	0.695945i	$-0.245014\pi$
$107$	1589.60	1.43619	0.718095	+	0.695945i	$0.245014\pi$
$108$	0	0
$109$	1686.52	1.48201	0.741007	−	0.671497i	$-0.234349\pi$
$109$	1686.52	1.48201	0.741007	+	0.671497i	$0.234349\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1501.59i	− 1.25007i	−0.780597	−	0.625034i	$-0.785085\pi$
$113$	− 1501.59i	− 1.25007i	0.780597	−	0.625034i	$-0.214915\pi$
$114$	0	0
$115$	119.619i	0.0969956i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	36.8462	0.0283839
$120$	0	0
$121$	1751.09	1.31562
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000i	0.0894427i
$126$	0	0
$127$	− 415.412i	− 0.290251i	−0.989413	−	0.145125i	$-0.953641\pi$
$127$	− 415.412i	− 0.290251i	0.989413	−	0.145125i	$-0.0463586\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	165.063	0.110089	0.0550445	−	0.998484i	$-0.482470\pi$
$131$	165.063	0.110089	0.0550445	+	0.998484i	$0.482470\pi$
$132$	0	0
$133$	−131.032	−0.0854278
$134$	0	0
$135$	0	0
$136$	0	0
$137$	213.152i	0.132926i	0.997789	+	0.0664629i	$0.0211714\pi$
$137$	213.152i	0.132926i	−0.997789	+	0.0664629i	$0.978829\pi$
$138$	0	0
$139$	− 817.013i	− 0.498548i	−0.968433	−	0.249274i	$-0.919808\pi$
$139$	− 817.013i	− 0.498548i	0.968433	−	0.249274i	$-0.0801919\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3690.62	−2.15822
$144$	0	0
$145$	1321.78	0.757019
$146$	0	0
$147$	0	0
$148$	0	0
$149$	371.997i	0.204532i	0.994757	+	0.102266i	$0.0326092\pi$
$149$	371.997i	0.204532i	−0.994757	+	0.102266i	$0.967391\pi$
$150$	0	0
$151$	− 3302.89i	− 1.78004i	−0.455925	−	0.890018i	$-0.650691\pi$
$151$	− 3302.89i	− 1.78004i	0.455925	−	0.890018i	$-0.349309\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	271.358	0.140619
$156$	0	0
$157$	464.066	0.235901	0.117951	−	0.993019i	$-0.462368\pi$
$157$	464.066	0.235901	0.117951	+	0.993019i	$0.462368\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	153.042i	0.0749154i
$162$	0	0
$163$	− 2398.33i	− 1.15246i	−0.817286	−	0.576232i	$-0.804523\pi$
$163$	− 2398.33i	− 1.15246i	0.817286	−	0.576232i	$-0.195477\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−605.878	−0.280744	−0.140372	−	0.990099i	$-0.544830\pi$
$167$	−605.878	−0.280744	−0.140372	+	0.990099i	$0.544830\pi$
$168$	0	0
$169$	2222.29	1.01151
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1271.79i	− 0.558914i	−0.960158	−	0.279457i	$-0.909845\pi$
$173$	− 1271.79i	− 0.558914i	0.960158	−	0.279457i	$-0.0901545\pi$
$174$	0	0
$175$	159.927i	0.0690819i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2858.61	−1.19365	−0.596823	−	0.802373i	$-0.703571\pi$
$179$	−2858.61	−1.19365	−0.596823	+	0.802373i	$0.703571\pi$
$180$	0	0
$181$	1736.90	0.713276	0.356638	−	0.934243i	$-0.383923\pi$
$181$	1736.90	0.713276	0.356638	+	0.934243i	$0.383923\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 1368.78i	− 0.543971i
$186$	0	0
$187$	− 319.767i	− 0.125046i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1097.42	−0.415743	−0.207871	−	0.978156i	$-0.566654\pi$
$191$	−1097.42	−0.415743	−0.207871	+	0.978156i	$0.566654\pi$
$192$	0	0
$193$	1273.79	0.475073	0.237537	−	0.971379i	$-0.423660\pi$
$193$	1273.79	0.475073	0.237537	+	0.971379i	$0.423660\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2170.76i	0.785077i	0.919736	+	0.392538i	$0.128403\pi$
$197$	2170.76i	0.785077i	−0.919736	+	0.392538i	$0.871597\pi$
$198$	0	0
$199$	5210.83i	1.85621i	0.372320	+	0.928104i	$0.378562\pi$
$199$	5210.83i	1.85621i	−0.372320	+	0.928104i	$0.621438\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1691.10	0.584690
$204$	0	0
$205$	848.577	0.289108
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1137.15i	0.376356i
$210$	0	0
$211$	− 183.077i	− 0.0597326i	−0.999554	−	0.0298663i	$-0.990492\pi$
$211$	− 183.077i	− 0.0597326i	0.999554	−	0.0298663i	$-0.00950815\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−903.989	−0.286751
$216$	0	0
$217$	347.180	0.108609
$218$	0	0
$219$	0	0
$220$	0	0
$221$	382.902i	0.116546i
$222$	0	0
$223$	− 6323.84i	− 1.89899i	−0.313778	−	0.949496i	$-0.601595\pi$
$223$	− 6323.84i	− 1.89899i	0.313778	−	0.949496i	$-0.398405\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	566.540	0.165650	0.0828251	−	0.996564i	$-0.473606\pi$
$227$	566.540	0.165650	0.0828251	+	0.996564i	$0.473606\pi$
$228$	0	0
$229$	463.926	0.133874	0.0669368	−	0.997757i	$-0.478677\pi$
$229$	463.926	0.133874	0.0669368	+	0.997757i	$0.478677\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2099.73i	0.590377i	0.955439	+	0.295188i	$0.0953824\pi$
$233$	2099.73i	0.590377i	−0.955439	+	0.295188i	$0.904618\pi$
$234$	0	0
$235$	620.618i	0.172275i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2481.10	0.671503	0.335752	−	0.941951i	$-0.391010\pi$
$239$	2481.10	0.671503	0.335752	+	0.941951i	$0.391010\pi$
$240$	0	0
$241$	3380.39	0.903526	0.451763	−	0.892138i	$-0.350795\pi$
$241$	3380.39	0.903526	0.451763	+	0.892138i	$0.350795\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1510.39i	− 0.393858i
$246$	0	0
$247$	− 1361.67i	− 0.350773i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−925.089	−0.232634	−0.116317	−	0.993212i	$-0.537109\pi$
$251$	−925.089	−0.232634	−0.116317	+	0.993212i	$0.537109\pi$
$252$	0	0
$253$	1328.16	0.330043
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6512.95i	− 1.58080i	−0.612588	−	0.790402i	$-0.709872\pi$
$257$	− 6512.95i	− 1.58080i	0.612588	−	0.790402i	$-0.290128\pi$
$258$	0	0
$259$	− 1751.24i	− 0.420141i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	4789.97	1.12305	0.561525	−	0.827460i	$-0.310215\pi$
$263$	4789.97	1.12305	0.561525	+	0.827460i	$0.310215\pi$
$264$	0	0
$265$	−870.946	−0.201894
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 4947.13i	− 1.12131i	−0.828050	−	0.560654i	$-0.810550\pi$
$269$	− 4947.13i	− 1.12131i	0.828050	−	0.560654i	$-0.189450\pi$
$270$	0	0
$271$	− 2410.24i	− 0.540264i	−0.962823	−	0.270132i	$-0.912933\pi$
$271$	− 2410.24i	− 0.540264i	0.962823	−	0.270132i	$-0.0870674\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1387.91	0.304343
$276$	0	0
$277$	7868.39	1.70674	0.853368	−	0.521309i	$-0.174556\pi$
$277$	7868.39	1.70674	0.853368	+	0.521309i	$0.174556\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	105.174i	0.0223281i	0.999938	+	0.0111640i	$0.00355369\pi$
$281$	105.174i	0.0223281i	−0.999938	+	0.0111640i	$0.996446\pi$
$282$	0	0
$283$	7571.83i	1.59045i	0.606312	+	0.795227i	$0.292648\pi$
$283$	7571.83i	1.59045i	−0.606312	+	0.795227i	$0.707352\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1085.68	0.223295
$288$	0	0
$289$	4879.82	0.993247
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3710.49i	0.739826i	0.929066	+	0.369913i	$0.120612\pi$
$293$	3710.49i	0.739826i	−0.929066	+	0.369913i	$0.879388\pi$
$294$	0	0
$295$	98.1494i	0.0193711i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1590.39	−0.307608
$300$	0	0
$301$	−1156.58	−0.221475
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3380.91i	0.634722i
$306$	0	0
$307$	− 2213.98i	− 0.411591i	−0.978595	−	0.205795i	$-0.934022\pi$
$307$	− 2213.98i	− 0.411591i	0.978595	−	0.205795i	$-0.0659781\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2363.06	0.430857	0.215429	−	0.976520i	$-0.430885\pi$
$311$	2363.06	0.430857	0.215429	+	0.976520i	$0.430885\pi$
$312$	0	0
$313$	6549.59	1.18276	0.591381	−	0.806392i	$-0.298583\pi$
$313$	6549.59	1.18276	0.591381	+	0.806392i	$0.298583\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 9439.49i	− 1.67248i	−0.548367	−	0.836238i	$-0.684750\pi$
$317$	− 9439.49i	− 1.67248i	0.548367	−	0.836238i	$-0.315250\pi$
$318$	0	0
$319$	− 14676.1i	− 2.57588i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	117.980	0.0203237
$324$	0	0
$325$	−1661.94	−0.283656
$326$	0	0
$327$	0	0
$328$	0	0
$329$	794.028i	0.133058i
$330$	0	0
$331$	5204.47i	0.864241i	0.901816	+	0.432120i	$0.142234\pi$
$331$	5204.47i	0.864241i	−0.901816	+	0.432120i	$0.857766\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1874.73	−0.305754
$336$	0	0
$337$	7094.31	1.14674	0.573370	−	0.819296i	$-0.305636\pi$
$337$	7094.31	1.14674	0.573370	+	0.819296i	$0.305636\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 3012.98i	− 0.478480i
$342$	0	0
$343$	− 4126.61i	− 0.649609i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7220.79	1.11710	0.558548	−	0.829472i	$-0.311359\pi$
$347$	7220.79	1.11710	0.558548	+	0.829472i	$0.311359\pi$
$348$	0	0
$349$	8494.10	1.30280	0.651402	−	0.758733i	$-0.274181\pi$
$349$	8494.10	1.30280	0.651402	+	0.758733i	$0.274181\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12828.2i	1.93421i	0.254385	+	0.967103i	$0.418127\pi$
$353$	12828.2i	1.93421i	−0.254385	+	0.967103i	$0.581873\pi$
$354$	0	0
$355$	− 4882.39i	− 0.729944i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7108.51	−1.04505	−0.522524	−	0.852624i	$-0.675010\pi$
$359$	−7108.51	−1.04505	−0.522524	+	0.852624i	$0.675010\pi$
$360$	0	0
$361$	6439.44	0.938831
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1232.32i	0.176720i
$366$	0	0
$367$	− 11036.4i	− 1.56975i	−0.619656	−	0.784873i	$-0.712728\pi$
$367$	− 11036.4i	− 1.56975i	0.619656	−	0.784873i	$-0.287272\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1114.30	−0.155934
$372$	0	0
$373$	12170.7	1.68948	0.844739	−	0.535178i	$-0.179755\pi$
$373$	12170.7	1.68948	0.844739	+	0.535178i	$0.179755\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	17573.8i	2.40078i
$378$	0	0
$379$	− 3342.88i	− 0.453066i	−0.974003	−	0.226533i	$-0.927261\pi$
$379$	− 3342.88i	− 0.453066i	0.974003	−	0.226533i	$-0.0727392\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5913.51	0.788946	0.394473	−	0.918907i	$-0.370927\pi$
$383$	5913.51	0.788946	0.394473	+	0.918907i	$0.370927\pi$
$384$	0	0
$385$	1775.72	0.235062
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5614.04i	0.731731i	0.930668	+	0.365865i	$0.119227\pi$
$389$	5614.04i	0.731731i	−0.930668	+	0.365865i	$0.880773\pi$
$390$	0	0
$391$	− 137.797i	− 0.0178227i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5768.13	0.734749
$396$	0	0
$397$	8698.42	1.09965	0.549825	−	0.835280i	$-0.314694\pi$
$397$	8698.42	1.09965	0.549825	+	0.835280i	$0.314694\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 6213.28i	− 0.773757i	−0.922131	−	0.386878i	$-0.873553\pi$
$401$	− 6213.28i	− 0.773757i	0.922131	−	0.386878i	$-0.126447\pi$
$402$	0	0
$403$	3607.86i	0.445956i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−15198.0	−1.85095
$408$	0	0
$409$	−7049.01	−0.852203	−0.426101	−	0.904675i	$-0.640113\pi$
$409$	−7049.01	−0.852203	−0.426101	+	0.904675i	$0.640113\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	125.574i	0.0149615i
$414$	0	0
$415$	− 297.277i	− 0.0351633i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1852.88	−0.216036	−0.108018	−	0.994149i	$-0.534450\pi$
$419$	−1852.88	−0.216036	−0.108018	+	0.994149i	$0.534450\pi$
$420$	0	0
$421$	−3719.81	−0.430623	−0.215312	−	0.976545i	$-0.569077\pi$
$421$	−3719.81	−0.430623	−0.215312	+	0.976545i	$0.569077\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 143.996i	− 0.0164349i
$426$	0	0
$427$	4325.58i	0.490233i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10575.5	−1.18191	−0.590953	−	0.806706i	$-0.701248\pi$
$431$	−10575.5	−1.18191	−0.590953	+	0.806706i	$0.701248\pi$
$432$	0	0
$433$	392.491	0.0435610	0.0217805	−	0.999763i	$-0.493067\pi$
$433$	392.491	0.0435610	0.0217805	+	0.999763i	$0.493067\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	490.032i	0.0536416i
$438$	0	0
$439$	− 5896.93i	− 0.641105i	−0.947231	−	0.320553i	$-0.896131\pi$
$439$	− 5896.93i	− 0.641105i	0.947231	−	0.320553i	$-0.103869\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9000.77	−0.965326	−0.482663	−	0.875806i	$-0.660330\pi$
$443$	−9000.77	−0.965326	−0.482663	+	0.875806i	$0.660330\pi$
$444$	0	0
$445$	−2374.52	−0.252951
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 16867.0i	− 1.77283i	−0.462892	−	0.886415i	$-0.653188\pi$
$449$	− 16867.0i	− 1.77283i	0.462892	−	0.886415i	$-0.346812\pi$
$450$	0	0
$451$	− 9422.02i	− 0.983737i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2126.31	−0.219084
$456$	0	0
$457$	−14093.6	−1.44260	−0.721302	−	0.692621i	$-0.756456\pi$
$457$	−14093.6	−1.44260	−0.721302	+	0.692621i	$0.756456\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 571.138i	− 0.0577018i	−0.999584	−	0.0288509i	$-0.990815\pi$
$461$	− 571.138i	− 0.0577018i	0.999584	−	0.0288509i	$-0.00918481\pi$
$462$	0	0
$463$	8105.80i	0.813625i	0.913512	+	0.406813i	$0.133360\pi$
$463$	8105.80i	0.813625i	−0.913512	+	0.406813i	$0.866640\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6979.97	0.691637	0.345818	−	0.938301i	$-0.387601\pi$
$467$	6979.97	0.691637	0.345818	+	0.938301i	$0.387601\pi$
$468$	0	0
$469$	−2398.56	−0.236151
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10037.3i	0.975718i
$474$	0	0
$475$	512.077i	0.0494647i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4346.35	0.414592	0.207296	−	0.978278i	$-0.433534\pi$
$479$	4346.35	0.414592	0.207296	+	0.978278i	$0.433534\pi$
$480$	0	0
$481$	18198.7	1.72513
$482$	0	0
$483$	0	0
$484$	0	0
$485$	274.714i	0.0257198i
$486$	0	0
$487$	7542.79i	0.701841i	0.936405	+	0.350921i	$0.114131\pi$
$487$	7542.79i	0.701841i	−0.936405	+	0.350921i	$0.885869\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9224.04	0.847811	0.423905	−	0.905706i	$-0.360659\pi$
$491$	9224.04	0.847811	0.423905	+	0.905706i	$0.360659\pi$
$492$	0	0
$493$	−1522.65	−0.139101
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 6246.59i	− 0.563779i
$498$	0	0
$499$	− 9222.35i	− 0.827353i	−0.910424	−	0.413676i	$-0.864245\pi$
$499$	− 9222.35i	− 0.827353i	0.910424	−	0.413676i	$-0.135755\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15756.4	1.39670	0.698351	−	0.715755i	$-0.253917\pi$
$503$	15756.4	1.39670	0.698351	+	0.715755i	$0.253917\pi$
$504$	0	0
$505$	6744.27	0.594289
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 13026.3i	− 1.13435i	−0.823599	−	0.567173i	$-0.808037\pi$
$509$	− 13026.3i	− 1.13435i	0.823599	−	0.567173i	$-0.191963\pi$
$510$	0	0
$511$	1576.65i	0.136491i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5985.49	−0.512140
$516$	0	0
$517$	6890.92	0.586194
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10658.0i	0.896233i	0.893975	+	0.448116i	$0.147905\pi$
$521$	10658.0i	0.896233i	−0.893975	+	0.448116i	$0.852095\pi$
$522$	0	0
$523$	− 2978.41i	− 0.249019i	−0.992218	−	0.124509i	$-0.960264\pi$
$523$	− 2978.41i	− 0.249019i	0.992218	−	0.124509i	$-0.0397357\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−312.597	−0.0258386
$528$	0	0
$529$	−11594.7	−0.952959
$530$	0	0
$531$	0	0
$532$	0	0
$533$	11282.3i	0.916868i
$534$	0	0
$535$	− 7947.99i	− 0.642283i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−16770.3	−1.34016
$540$	0	0
$541$	1745.08	0.138682	0.0693410	−	0.997593i	$-0.477910\pi$
$541$	1745.08	0.138682	0.0693410	+	0.997593i	$0.477910\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 8432.61i	− 0.662777i
$546$	0	0
$547$	7508.43i	0.586905i	0.955974	+	0.293453i	$0.0948043\pi$
$547$	7508.43i	0.586905i	−0.955974	+	0.293453i	$0.905196\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5414.82	0.418655
$552$	0	0
$553$	7379.82	0.567490
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10270.1i	0.781250i	0.920550	+	0.390625i	$0.127741\pi$
$557$	10270.1i	0.781250i	−0.920550	+	0.390625i	$0.872259\pi$
$558$	0	0
$559$	− 12019.0i	− 0.909394i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16226.8	1.21470	0.607352	−	0.794433i	$-0.292232\pi$
$563$	16226.8	1.21470	0.607352	+	0.794433i	$0.292232\pi$
$564$	0	0
$565$	−7507.95	−0.559048
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 5838.15i	− 0.430137i	−0.976599	−	0.215069i	$-0.931003\pi$
$569$	− 5838.15i	− 0.430137i	0.976599	−	0.215069i	$-0.0689975\pi$
$570$	0	0
$571$	6560.15i	0.480794i	0.970675	+	0.240397i	$0.0772777\pi$
$571$	6560.15i	0.480794i	−0.970675	+	0.240397i	$0.922722\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	598.093	0.0433777
$576$	0	0
$577$	−18667.7	−1.34688	−0.673438	−	0.739243i	$-0.735183\pi$
$577$	−18667.7	−1.34688	−0.673438	+	0.739243i	$0.735183\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 380.341i	− 0.0271587i
$582$	0	0
$583$	9670.39i	0.686975i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6635.47	0.466568	0.233284	−	0.972409i	$-0.425053\pi$
$587$	6635.47	0.466568	0.233284	+	0.972409i	$0.425053\pi$
$588$	0	0
$589$	1111.65	0.0777670
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 10842.9i	− 0.750870i	−0.926849	−	0.375435i	$-0.877493\pi$
$593$	− 10842.9i	− 0.750870i	0.926849	−	0.375435i	$-0.122507\pi$
$594$	0	0
$595$	− 184.231i	− 0.0126937i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	24193.1	1.65025	0.825127	−	0.564948i	$-0.191104\pi$
$599$	24193.1	1.65025	0.825127	+	0.564948i	$0.191104\pi$
$600$	0	0
$601$	15971.1	1.08399	0.541994	−	0.840383i	$-0.317670\pi$
$601$	15971.1	1.08399	0.541994	+	0.840383i	$0.317670\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 8755.46i	− 0.588364i
$606$	0	0
$607$	− 8024.95i	− 0.536611i	−0.963334	−	0.268305i	$-0.913536\pi$
$607$	− 8024.95i	− 0.536611i	0.963334	−	0.268305i	$-0.0864636\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8251.46	−0.546348
$612$	0	0
$613$	9725.48	0.640797	0.320398	−	0.947283i	$-0.396183\pi$
$613$	9725.48	0.640797	0.320398	+	0.947283i	$0.396183\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 24882.0i	− 1.62352i	−0.583990	−	0.811761i	$-0.698509\pi$
$617$	− 24882.0i	− 1.62352i	0.583990	−	0.811761i	$-0.301491\pi$
$618$	0	0
$619$	20644.9i	1.34053i	0.742120	+	0.670267i	$0.233820\pi$
$619$	20644.9i	1.34053i	−0.742120	+	0.670267i	$0.766180\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3038.00	−0.195369
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1576.79i	0.0999537i
$630$	0	0
$631$	− 7597.15i	− 0.479299i	−0.970859	−	0.239650i	$-0.922967\pi$
$631$	− 7597.15i	− 0.479299i	0.970859	−	0.239650i	$-0.0770325\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2077.06	−0.129804
$636$	0	0
$637$	20081.4	1.24907
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3812.73i	0.234936i	0.993077	+	0.117468i	$0.0374777\pi$
$641$	3812.73i	0.234936i	−0.993077	+	0.117468i	$0.962522\pi$
$642$	0	0
$643$	− 4046.71i	− 0.248191i	−0.992270	−	0.124095i	$-0.960397\pi$
$643$	− 4046.71i	− 0.248191i	0.992270	−	0.124095i	$-0.0396029\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28536.0	−1.73395	−0.866974	−	0.498353i	$-0.833938\pi$
$647$	−28536.0	−1.73395	−0.866974	+	0.498353i	$0.833938\pi$
$648$	0	0
$649$	1089.78	0.0659133
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 16067.6i	− 0.962899i	−0.876474	−	0.481449i	$-0.840110\pi$
$653$	− 16067.6i	− 0.962899i	0.876474	−	0.481449i	$-0.159890\pi$
$654$	0	0
$655$	− 825.317i	− 0.0492333i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−13800.3	−0.815756	−0.407878	−	0.913036i	$-0.633731\pi$
$659$	−13800.3	−0.815756	−0.407878	+	0.913036i	$0.633731\pi$
$660$	0	0
$661$	−14129.5	−0.831431	−0.415715	−	0.909495i	$-0.636469\pi$
$661$	−14129.5	−0.831431	−0.415715	+	0.909495i	$0.636469\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	655.159i	0.0382045i
$666$	0	0
$667$	− 6324.37i	− 0.367137i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	37539.3	2.15974
$672$	0	0
$673$	−5999.06	−0.343606	−0.171803	−	0.985131i	$-0.554959\pi$
$673$	−5999.06	−0.343606	−0.171803	+	0.985131i	$0.554959\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	10956.0i	0.621968i	0.950415	+	0.310984i	$0.100658\pi$
$677$	10956.0i	0.621968i	−0.950415	+	0.310984i	$0.899342\pi$
$678$	0	0
$679$	351.473i	0.0198649i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−33310.5	−1.86617	−0.933084	−	0.359659i	$-0.882893\pi$
$683$	−33310.5	−1.86617	−0.933084	+	0.359659i	$0.882893\pi$
$684$	0	0
$685$	1065.76	0.0594462
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 11579.7i	− 0.640278i
$690$	0	0
$691$	− 298.971i	− 0.0164593i	−0.999966	−	0.00822966i	$-0.997380\pi$
$691$	− 298.971i	− 0.0164593i	0.999966	−	0.00822966i	$-0.00261961\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4085.06	−0.222957
$696$	0	0
$697$	−977.535	−0.0531231
$698$	0	0
$699$	0	0
$700$	0	0
$701$	29219.4i	1.57432i	0.616747	+	0.787161i	$0.288450\pi$
$701$	29219.4i	1.57432i	−0.616747	+	0.787161i	$0.711550\pi$
$702$	0	0
$703$	− 5607.37i	− 0.300833i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8628.71	0.459004
$708$	0	0
$709$	17485.6	0.926214	0.463107	−	0.886302i	$-0.346735\pi$
$709$	17485.6	0.926214	0.463107	+	0.886302i	$0.346735\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 1298.38i	− 0.0681973i
$714$	0	0
$715$	18453.1i	0.965184i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−23281.8	−1.20760	−0.603801	−	0.797135i	$-0.706348\pi$
$719$	−23281.8	−1.20760	−0.603801	+	0.797135i	$0.706348\pi$
$720$	0	0
$721$	−7657.91	−0.395556
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 6608.89i	− 0.338549i
$726$	0	0
$727$	1403.69i	0.0716093i	0.999359	+	0.0358046i	$0.0113994\pi$
$727$	1403.69i	0.0716093i	−0.999359	+	0.0358046i	$0.988601\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1041.37	0.0526900
$732$	0	0
$733$	−30220.4	−1.52280	−0.761402	−	0.648280i	$-0.775489\pi$
$733$	−30220.4	−1.52280	−0.761402	+	0.648280i	$0.775489\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	20815.7i	1.04038i
$738$	0	0
$739$	34125.5i	1.69868i	0.527843	+	0.849342i	$0.323001\pi$
$739$	34125.5i	1.69868i	−0.527843	+	0.849342i	$0.676999\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	12683.6	0.626268	0.313134	−	0.949709i	$-0.398621\pi$
$743$	12683.6	0.626268	0.313134	+	0.949709i	$0.398621\pi$
$744$	0	0
$745$	1859.99	0.0914693
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 10168.8i	− 0.496073i
$750$	0	0
$751$	15777.5i	0.766616i	0.923621	+	0.383308i	$0.125215\pi$
$751$	15777.5i	0.766616i	−0.923621	+	0.383308i	$0.874785\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−16514.5	−0.796057
$756$	0	0
$757$	−13079.9	−0.628003	−0.314001	−	0.949423i	$-0.601670\pi$
$757$	−13079.9	−0.628003	−0.314001	+	0.949423i	$0.601670\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	7814.60i	0.372246i	0.982526	+	0.186123i	$0.0595922\pi$
$761$	7814.60i	0.372246i	−0.982526	+	0.186123i	$0.940408\pi$
$762$	0	0
$763$	− 10788.8i	− 0.511901i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1304.95	−0.0614329
$768$	0	0
$769$	1314.68	0.0616495	0.0308248	−	0.999525i	$-0.490187\pi$
$769$	1314.68	0.0616495	0.0308248	+	0.999525i	$0.490187\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	28082.8i	1.30669i	0.757061	+	0.653344i	$0.226634\pi$
$773$	28082.8i	1.30669i	−0.757061	+	0.653344i	$0.773366\pi$
$774$	0	0
$775$	− 1356.79i	− 0.0628869i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3476.29	0.159886
$780$	0	0
$781$	−54210.7	−2.48375
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 2320.33i	− 0.105498i
$786$	0	0
$787$	23777.6i	1.07698i	0.842633	+	0.538489i	$0.181005\pi$
$787$	23777.6i	1.07698i	−0.842633	+	0.538489i	$0.818995\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9605.78	−0.431785
$792$	0	0
$793$	−44951.0	−2.01294
$794$	0	0
$795$	0	0
$796$	0	0
$797$	21940.2i	0.975111i	0.873092	+	0.487555i	$0.162111\pi$
$797$	21940.2i	0.975111i	−0.873092	+	0.487555i	$0.837889\pi$
$798$	0	0
$799$	− 714.934i	− 0.0316552i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13682.9	0.601318
$804$	0	0
$805$	765.208	0.0335032
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 22835.5i	− 0.992402i	−0.868208	−	0.496201i	$-0.834728\pi$
$809$	− 22835.5i	− 0.992402i	0.868208	−	0.496201i	$-0.165272\pi$
$810$	0	0
$811$	29505.2i	1.27752i	0.769407	+	0.638759i	$0.220552\pi$
$811$	29505.2i	1.27752i	−0.769407	+	0.638759i	$0.779448\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11991.6	−0.515397
$816$	0	0
$817$	−3703.30	−0.158583
$818$	0	0
$819$	0	0
$820$	0	0
$821$	35909.6i	1.52650i	0.646105	+	0.763248i	$0.276397\pi$
$821$	35909.6i	1.52650i	−0.646105	+	0.763248i	$0.723603\pi$
$822$	0	0
$823$	− 29714.2i	− 1.25853i	−0.777189	−	0.629267i	$-0.783355\pi$
$823$	− 29714.2i	− 1.25853i	0.777189	−	0.629267i	$-0.216645\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	14542.2	0.611465	0.305732	−	0.952118i	$-0.401099\pi$
$827$	14542.2	0.611465	0.305732	+	0.952118i	$0.401099\pi$
$828$	0	0
$829$	−6765.34	−0.283438	−0.141719	−	0.989907i	$-0.545263\pi$
$829$	−6765.34	−0.283438	−0.141719	+	0.989907i	$0.545263\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1739.92i	0.0723706i
$834$	0	0
$835$	3029.39i	0.125553i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9817.26	−0.403968	−0.201984	−	0.979389i	$-0.564739\pi$
$839$	−9817.26	−0.403968	−0.201984	+	0.979389i	$0.564739\pi$
$840$	0	0
$841$	−45495.0	−1.86539
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 11111.5i	− 0.452362i
$846$	0	0
$847$	− 11201.9i	− 0.454428i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6549.26	−0.263814
$852$	0	0
$853$	5322.48	0.213644	0.106822	−	0.994278i	$-0.465933\pi$
$853$	5322.48	0.213644	0.106822	+	0.994278i	$0.465933\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	20964.5i	0.835630i	0.908532	+	0.417815i	$0.137204\pi$
$857$	20964.5i	0.835630i	−0.908532	+	0.417815i	$0.862796\pi$
$858$	0	0
$859$	41995.8i	1.66808i	0.551705	+	0.834039i	$0.313977\pi$
$859$	41995.8i	1.66808i	−0.551705	+	0.834039i	$0.686023\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−14325.6	−0.565065	−0.282532	−	0.959258i	$-0.591174\pi$
$863$	−14325.6	−0.565065	−0.282532	+	0.959258i	$0.591174\pi$
$864$	0	0
$865$	−6358.94	−0.249954
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 64045.4i	− 2.50010i
$870$	0	0
$871$	− 24925.6i	− 0.969656i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	799.634	0.0308943
$876$	0	0
$877$	−30888.3	−1.18931	−0.594654	−	0.803981i	$-0.702711\pi$
$877$	−30888.3	−1.18931	−0.594654	+	0.803981i	$0.702711\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 13371.4i	− 0.511344i	−0.966764	−	0.255672i	$-0.917703\pi$
$881$	− 13371.4i	− 0.511344i	0.966764	−	0.255672i	$-0.0822968\pi$
$882$	0	0
$883$	32657.0i	1.24462i	0.782773	+	0.622308i	$0.213805\pi$
$883$	32657.0i	1.24462i	−0.782773	+	0.622308i	$0.786195\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45855.4	1.73582	0.867911	−	0.496719i	$-0.165462\pi$
$887$	45855.4	1.73582	0.867911	+	0.496719i	$0.165462\pi$
$888$	0	0
$889$	−2657.42	−0.100255
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2542.44i	0.0952736i
$894$	0	0
$895$	14293.1i	0.533815i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−14347.0	−0.532258
$900$	0	0
$901$	1003.30	0.0370976
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 8684.52i	− 0.318987i
$906$	0	0
$907$	− 9944.06i	− 0.364043i	−0.983295	−	0.182022i	$-0.941736\pi$
$907$	− 9944.06i	− 0.364043i	0.983295	−	0.182022i	$-0.0582641\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2143.92	−0.0779707	−0.0389854	−	0.999240i	$-0.512413\pi$
$911$	−2143.92	−0.0779707	−0.0389854	+	0.999240i	$0.512413\pi$
$912$	0	0
$913$	−3300.76	−0.119649
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 1055.92i	− 0.0380258i
$918$	0	0
$919$	− 14171.4i	− 0.508673i	−0.967116	−	0.254336i	$-0.918143\pi$
$919$	− 14171.4i	− 0.508673i	0.967116	−	0.254336i	$-0.0818570\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	64914.0	2.31492
$924$	0	0
$925$	−6843.90	−0.243271
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 6611.21i	− 0.233484i	−0.993162	−	0.116742i	$-0.962755\pi$
$929$	− 6611.21i	− 0.233484i	0.993162	−	0.116742i	$-0.0372451\pi$
$930$	0	0
$931$	− 6187.48i	− 0.217816i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1598.84	−0.0559225
$936$	0	0
$937$	4874.63	0.169954	0.0849771	−	0.996383i	$-0.472918\pi$
$937$	4874.63	0.169954	0.0849771	+	0.996383i	$0.472918\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 9360.15i	− 0.324264i	−0.986769	−	0.162132i	$-0.948163\pi$
$941$	− 9360.15i	− 0.324264i	0.986769	−	0.162132i	$-0.0518370\pi$
$942$	0	0
$943$	− 4060.22i	− 0.140211i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−33182.4	−1.13863	−0.569315	−	0.822119i	$-0.692792\pi$
$947$	−33182.4	−1.13863	−0.569315	+	0.822119i	$0.692792\pi$
$948$	0	0
$949$	−16384.4	−0.560444
$950$	0	0
$951$	0	0
$952$	0	0
$953$	50051.9i	1.70130i	0.525730	+	0.850651i	$0.323792\pi$
$953$	50051.9i	1.70130i	−0.525730	+	0.850651i	$0.676208\pi$
$954$	0	0
$955$	5487.12i	0.185926i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1363.55	0.0459138
$960$	0	0
$961$	26845.6	0.901131
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 6368.93i	− 0.212459i
$966$	0	0
$967$	− 12066.8i	− 0.401283i	−0.979665	−	0.200641i	$-0.935697\pi$
$967$	− 12066.8i	− 0.401283i	0.979665	−	0.200641i	$-0.0643026\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	38464.1	1.27124	0.635619	−	0.772003i	$-0.280745\pi$
$971$	38464.1	1.27124	0.635619	+	0.772003i	$0.280745\pi$
$972$	0	0
$973$	−5226.49	−0.172203
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 9271.79i	− 0.303614i	−0.988410	−	0.151807i	$-0.951491\pi$
$977$	− 9271.79i	− 0.303614i	0.988410	−	0.151807i	$-0.0485092\pi$
$978$	0	0
$979$	26365.1i	0.860707i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	27407.2	0.889270	0.444635	−	0.895712i	$-0.353333\pi$
$983$	27407.2	0.889270	0.444635	+	0.895712i	$0.353333\pi$
$984$	0	0
$985$	10853.8	0.351097
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4325.36i	0.139068i
$990$	0	0
$991$	− 21099.0i	− 0.676320i	−0.941089	−	0.338160i	$-0.890196\pi$
$991$	− 21099.0i	− 0.676320i	0.941089	−	0.338160i	$-0.109804\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	26054.1	0.830122
$996$	0	0
$997$	−12537.7	−0.398269	−0.199135	−	0.979972i	$-0.563813\pi$
$997$	−12537.7	−0.398269	−0.199135	+	0.979972i	$0.563813\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.h.c.431.3	✓	12
3.2	odd	2	inner	2160.4.h.c.431.9	yes	12
4.3	odd	2	inner	2160.4.h.c.431.4	yes	12
12.11	even	2	inner	2160.4.h.c.431.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2160.4.h.c.431.3	✓	12	1.1	even	1	trivial
2160.4.h.c.431.4	yes	12	4.3	odd	2	inner
2160.4.h.c.431.9	yes	12	3.2	odd	2	inner
2160.4.h.c.431.10	yes	12	12.11	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2160.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.00000i q^{5} -6.39707i q^{7} -55.5166 q^{11} +66.4777 q^{13} +5.75985i q^{17} -20.4831i q^{19} -23.9237 q^{23} -25.0000 q^{25} +264.356i q^{29} +54.2716i q^{31} -31.9853 q^{35} +273.756 q^{37} +169.715i q^{41} -180.798i q^{43} -124.124 q^{47} +302.077 q^{49} -174.189i q^{53} +277.583i q^{55} -19.6299 q^{59} -676.182 q^{61} -332.389i q^{65} -374.946i q^{67} +976.477 q^{71} -246.465 q^{73} +355.143i q^{77} +1153.63i q^{79} +59.4554 q^{83} +28.7992 q^{85} -474.905i q^{89} -425.263i q^{91} -102.415 q^{95} -54.9428 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 192 q^{13} - 300 q^{25} + 936 q^{37} + 2820 q^{49} - 492 q^{61} + 4488 q^{73} + 900 q^{85} + 2544 q^{97}+O(q^{100}) \) 12 * q - 192 * q^13 - 300 * q^25 + 936 * q^37 + 2820 * q^49 - 492 * q^61 + 4488 * q^73 + 900 * q^85 + 2544 * q^97

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