LMFDB - Embedded newform 1728.2.r.c.289.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,Mod(289,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 4]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1728.289");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1728.r (of order $6$, degree $2$, not 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:	$13.7981494693$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			289.4
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1728.289
Dual form			1728.2.r.c.1441.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+O(q^{10})$	$q+(4.71940 + 2.72474i) q^{11} +1.89898 q^{17} -8.34847i q^{19} +(-2.50000 + 4.33013i) q^{25} +(6.39898 + 11.0834i) q^{41} +(-2.03383 - 1.17423i) q^{43} +(3.50000 + 6.06218i) q^{49} +(8.00853 - 4.62372i) q^{59} +(12.4261 - 7.17423i) q^{67} +13.6969 q^{73} +(15.5885 + 9.00000i) q^{83} -18.0000 q^{89} +(9.84847 - 17.0580i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 24 q^{17} - 20 q^{25} + 12 q^{41} + 28 q^{49} - 8 q^{73} - 144 q^{89} + 20 q^{97}+O(q^{100}) $ 8 * q - 24 * q^17 - 20 * q^25 + 12 * q^41 + 28 * q^49 - 8 * q^73 - 144 * q^89 + 20 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$7$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.71940	+	2.72474i	1.42295	+	0.821541i	0.996550	−	0.0829925i	$-0.0264478\pi$
$11$	4.71940	+	2.72474i	1.42295	+	0.821541i	0.426401	+	0.904534i	$0.359781\pi$
$12$		0			0
$13$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$13$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.89898			0.460570			0.230285	−	0.973123i	$-0.426034\pi$
$17$	1.89898			0.460570			0.230285	+	0.973123i	$0.426034\pi$
$18$		0			0
$19$		−	8.34847i		−	1.91527i	−0.287984	−	0.957635i	$-0.592985\pi$
$19$		−	8.34847i		−	1.91527i	0.287984	−	0.957635i	$-0.407015\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	−2.50000	+	4.33013i	−0.500000	+	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	6.39898	+	11.0834i	0.999353	+	1.73093i	0.530831	+	0.847477i	$0.321880\pi$
$41$	6.39898	+	11.0834i	0.999353	+	1.73093i	0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$	−2.03383	−	1.17423i	−0.310157	−	0.179069i	0.336840	−	0.941562i	$-0.390642\pi$
$43$	−2.03383	−	1.17423i	−0.310157	−	0.179069i	−0.646997	+	0.762493i	$0.723975\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		0			0
$49$	3.50000	+	6.06218i	0.500000	+	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.00853	−	4.62372i	1.04262	−	0.601958i	0.122047	−	0.992524i	$-0.461054\pi$
$59$	8.00853	−	4.62372i	1.04262	−	0.601958i	0.920575	+	0.390567i	$0.127721\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	12.4261	−	7.17423i	1.51809	−	0.876472i	0.518321	−	0.855186i	$-0.326557\pi$
$67$	12.4261	−	7.17423i	1.51809	−	0.876472i	0.999773	−	0.0212861i	$-0.00677610\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	13.6969			1.60311			0.801553	−	0.597924i	$-0.204008\pi$
$73$	13.6969			1.60311			0.801553	+	0.597924i	$0.204008\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$79$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	15.5885	+	9.00000i	1.71106	+	0.987878i	0.933143	+	0.359506i	$0.117055\pi$
$83$	15.5885	+	9.00000i	1.71106	+	0.987878i	0.777913	+	0.628372i	$0.216279\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−18.0000			−1.90800			−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000			−1.90800			−0.953998	+	0.299813i	$0.903076\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	9.84847	−	17.0580i	0.999961	−	1.73198i	0.492287	−	0.870433i	$-0.336161\pi$
$97$	9.84847	−	17.0580i	0.999961	−	1.73198i	0.507673	−	0.861550i	$-0.330506\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	20.1464i		−	1.94763i	−0.227345	−	0.973814i	$-0.573004\pi$
$107$		−	20.1464i		−	1.94763i	0.227345	−	0.973814i	$-0.426996\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.00000	+	15.5885i	0.846649	+	1.46644i	0.884182	+	0.467143i	$0.154717\pi$
$113$	9.00000	+	15.5885i	0.846649	+	1.46644i	−0.0375328	+	0.999295i	$0.511950\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	9.34847	+	16.1920i	0.849861	+	1.47200i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−15.5885	+	9.00000i	−1.36197	+	0.786334i	−0.989886	−	0.141865i	$-0.954690\pi$
$131$	−15.5885	+	9.00000i	−1.36197	+	0.786334i	−0.372084	+	0.928199i	$0.621357\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−8.29796	+	14.3725i	−0.708942	+	1.22792i	0.256307	+	0.966595i	$0.417494\pi$
$137$	−8.29796	+	14.3725i	−0.708942	+	1.22792i	−0.965250	+	0.261329i	$0.915839\pi$
$138$		0			0
$139$	3.16232	−	1.82577i	0.268224	−	0.154859i	−0.359856	−	0.933008i	$-0.617174\pi$
$139$	3.16232	−	1.82577i	0.268224	−	0.154859i	0.628080	+	0.778148i	$0.283841\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$151$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$157$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$			2.00000i			0.156652i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$			2.00000i			0.156652i	−0.996928	+	0.0783260i	$0.975042\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	−6.50000	+	11.2583i	−0.500000	+	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	18.0000i		−	1.34538i	−0.739923	−	0.672692i	$-0.765138\pi$
$179$		−	18.0000i		−	1.34538i	0.739923	−	0.672692i	$-0.234862\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	8.96204	+	5.17423i	0.655369	+	0.378378i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$191$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$192$		0			0
$193$	−12.8485	−	22.2542i	−0.924853	−	1.60189i	−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−12.8485	−	22.2542i	−0.924853	−	1.60189i	−0.133056	−	0.991109i	$-0.542479\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	22.7474	−	39.3997i	1.57347	−	2.72534i
$210$		0			0
$211$	−12.1244	+	7.00000i	−0.834675	+	0.481900i	−0.855451	−	0.517884i	$-0.826720\pi$
$211$	−12.1244	+	7.00000i	−0.834675	+	0.481900i	0.0207756	+	0.999784i	$0.493386\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$223$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	10.8691	+	6.27526i	0.721405	+	0.416503i	0.815270	−	0.579082i	$-0.196589\pi$
$227$	10.8691	+	6.27526i	0.721405	+	0.416503i	−0.0938647	+	0.995585i	$0.529922\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−19.8990			−1.30363			−0.651813	−	0.758380i	$-0.725991\pi$
$233$	−19.8990			−1.30363			−0.651813	+	0.758380i	$0.725991\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$239$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$240$		0			0
$241$	−0.848469	+	1.46959i	−0.0546547	+	0.0946647i	−0.892058	−	0.451920i	$-0.850739\pi$
$241$	−0.848469	+	1.46959i	−0.0546547	+	0.0946647i	0.837404	+	0.546585i	$0.184072\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$			23.9444i			1.51136i	0.654943	+	0.755678i	$0.272693\pi$
$251$			23.9444i			1.51136i	−0.654943	+	0.755678i	$0.727307\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.60102	+	4.50510i	0.162247	+	0.281020i	0.935674	−	0.352865i	$-0.114792\pi$
$257$	2.60102	+	4.50510i	0.162247	+	0.281020i	−0.773427	+	0.633885i	$0.781459\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$263$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−23.5970	+	13.6237i	−1.42295	+	0.821541i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.00000	−	15.5885i	0.536895	−	0.929929i	−0.462174	−	0.886789i	$-0.652930\pi$
$281$	9.00000	−	15.5885i	0.536895	−	0.929929i	0.999069	−	0.0431402i	$-0.0137362\pi$
$282$		0			0
$283$	−19.0526	+	11.0000i	−1.13256	+	0.653882i	−0.944577	−	0.328291i	$-0.893527\pi$
$283$	−19.0526	+	11.0000i	−1.13256	+	0.653882i	−0.187980	+	0.982173i	$0.560194\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.3939			−0.787875
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	9.65153i		−	0.550842i	−0.961324	−	0.275421i	$-0.911183\pi$
$307$		−	9.65153i		−	0.550842i	0.961324	−	0.275421i	$-0.0888172\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−12.1969	+	21.1257i	−0.689412	+	1.19410i	0.282617	+	0.959233i	$0.408798\pi$
$313$	−12.1969	+	21.1257i	−0.689412	+	1.19410i	−0.972028	+	0.234863i	$0.924536\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	15.8536i		−	0.882116i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−22.5167	−	13.0000i	−1.23763	−	0.714545i	−0.269019	−	0.963135i	$-0.586699\pi$
$331$	−22.5167	−	13.0000i	−1.23763	−	0.714545i	−0.968609	+	0.248590i	$0.920033\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−18.1969	−	31.5180i	−0.991250	−	1.71690i	−0.609936	−	0.792451i	$-0.708805\pi$
$337$	−18.1969	−	31.5180i	−0.991250	−	1.71690i	−0.381314	−	0.924445i	$-0.624528\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−30.1752	+	17.4217i	−1.61989	+	0.935245i	−0.632945	+	0.774197i	$0.718154\pi$
$347$	−30.1752	+	17.4217i	−1.61989	+	0.935245i	−0.986947	+	0.161048i	$0.948512\pi$
$348$		0			0
$349$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$349$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	17.2980	−	29.9609i	0.920677	−	1.59466i	0.122308	−	0.992492i	$-0.460970\pi$
$353$	17.2980	−	29.9609i	0.920677	−	1.59466i	0.798369	−	0.602168i	$-0.205696\pi$
$354$		0			0
$355$		0			0
$356$		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
$361$	−50.6969			−2.66826
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$367$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$			26.3485i			1.35343i	0.736245	+	0.676715i	$0.236597\pi$
$379$			26.3485i			1.35343i	−0.736245	+	0.676715i	$0.763403\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−15.6464	−	27.1004i	−0.781345	−	1.35333i	−0.931158	−	0.364615i	$-0.881200\pi$
$401$	−15.6464	−	27.1004i	−0.781345	−	1.35333i	0.149813	−	0.988714i	$-0.452133\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	9.19694	+	15.9296i	0.454759	+	0.787666i	0.998674	−	0.0514740i	$-0.0163919\pi$
$409$	9.19694	+	15.9296i	0.454759	+	0.787666i	−0.543915	+	0.839140i	$0.683059\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−15.5885	+	9.00000i	−0.761546	+	0.439679i	−0.829851	−	0.557986i	$-0.811574\pi$
$419$	−15.5885	+	9.00000i	−0.761546	+	0.439679i	0.0683046	+	0.997665i	$0.478241\pi$
$420$		0			0
$421$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$421$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−4.74745	+	8.22282i	−0.230285	+	0.398865i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	4.30306			0.206792			0.103396	−	0.994640i	$-0.467029\pi$
$433$	4.30306			0.206792			0.103396	+	0.994640i	$0.467029\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$439$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−20.3079	−	11.7247i	−0.964855	−	0.557059i	−0.0671913	−	0.997740i	$-0.521404\pi$
$443$	−20.3079	−	11.7247i	−0.964855	−	0.557059i	−0.897664	+	0.440681i	$0.854737\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	16.1010			0.759854			0.379927	−	0.925016i	$-0.375949\pi$
$449$	16.1010			0.759854			0.379927	+	0.925016i	$0.375949\pi$
$450$		0			0
$451$			69.7423i			3.28404i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	21.1969	−	36.7142i	0.991551	−	1.71742i	0.383437	−	0.923567i	$-0.374740\pi$
$457$	21.1969	−	36.7142i	0.991551	−	1.71742i	0.608114	−	0.793849i	$-0.291926\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$461$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$462$		0			0
$463$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$463$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	41.9444i		−	1.94095i	−0.241192	−	0.970477i	$-0.577538\pi$
$467$		−	41.9444i		−	1.94095i	0.241192	−	0.970477i	$-0.422462\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−6.39898	−	11.0834i	−0.294225	−	0.509613i
$474$		0			0
$475$	36.1499	+	20.8712i	1.65867	+	0.957635i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	7.57993	−	4.37628i	0.342078	−	0.197499i	−0.319113	−	0.947717i	$-0.603385\pi$
$491$	7.57993	−	4.37628i	0.342078	−	0.197499i	0.661190	+	0.750218i	$0.270052\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−25.7576	+	14.8712i	−1.15307	+	0.665725i	−0.949633	−	0.313363i	$-0.898544\pi$
$499$	−25.7576	+	14.8712i	−1.15307	+	0.665725i	−0.203436	+	0.979088i	$0.565211\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−42.1918			−1.84846			−0.924229	−	0.381839i	$-0.875291\pi$
$521$	−42.1918			−1.84846			−0.924229	+	0.381839i	$0.875291\pi$
$522$		0			0
$523$		−	38.0000i		−	1.66162i	−0.556553	−	0.830812i	$-0.687876\pi$
$523$		−	38.0000i		−	1.66162i	0.556553	−	0.830812i	$-0.312124\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$			38.1464i			1.64308i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−13.5546	−	7.82577i	−0.579554	−	0.334606i	0.181402	−	0.983409i	$-0.441936\pi$
$547$	−13.5546	−	7.82577i	−0.579554	−	0.334606i	−0.760956	+	0.648803i	$0.775270\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	14.5868	−	8.42168i	0.614760	−	0.354932i	−0.160066	−	0.987106i	$-0.551171\pi$
$563$	14.5868	−	8.42168i	0.614760	−	0.354932i	0.774826	+	0.632175i	$0.217837\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.702041	+	1.21597i	−0.0294311	+	0.0509761i	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−0.702041	+	1.21597i	−0.0294311	+	0.0509761i	0.850935	+	0.525271i	$0.176036\pi$
$570$		0			0
$571$	41.3461	−	23.8712i	1.73028	−	0.998978i	0.842535	−	0.538642i	$-0.181062\pi$
$571$	41.3461	−	23.8712i	1.73028	−	0.998978i	0.887745	−	0.460336i	$-0.152271\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	12.3939			0.515964			0.257982	−	0.966150i	$-0.416942\pi$
$577$	12.3939			0.515964			0.257982	+	0.966150i	$0.416942\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−33.4644	−	19.3207i	−1.38122	−	0.797449i	−0.388918	−	0.921272i	$-0.627151\pi$
$587$	−33.4644	−	19.3207i	−1.38122	−	0.797449i	−0.992304	+	0.123823i	$0.960484\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−18.0000			−0.739171			−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000			−0.739171			−0.369586	+	0.929197i	$0.620500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$599$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$600$		0			0
$601$	−18.8485	+	32.6465i	−0.768845	+	1.33168i	0.169344	+	0.985557i	$0.445835\pi$
$601$	−18.8485	+	32.6465i	−0.768845	+	1.33168i	−0.938190	+	0.346122i	$0.887498\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$607$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	24.6464	+	42.6889i	0.992228	+	1.71859i	0.603877	+	0.797077i	$0.293622\pi$
$617$	24.6464	+	42.6889i	0.992228	+	1.71859i	0.388351	+	0.921512i	$0.373045\pi$
$618$		0			0
$619$	−20.5615	−	11.8712i	−0.826435	−	0.477143i	0.0261952	−	0.999657i	$-0.491661\pi$
$619$	−20.5615	−	11.8712i	−0.826435	−	0.477143i	−0.852631	+	0.522514i	$0.824994\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−22.7474	+	39.3997i	−0.898470	+	1.55620i	−0.0690201	+	0.997615i	$0.521987\pi$
$641$	−22.7474	+	39.3997i	−0.898470	+	1.55620i	−0.829450	+	0.558581i	$0.811346\pi$
$642$		0			0
$643$	−28.0146	+	16.1742i	−1.10479	+	0.637850i	−0.937474	−	0.348054i	$-0.886843\pi$
$643$	−28.0146	+	16.1742i	−1.10479	+	0.637850i	−0.167313	+	0.985904i	$0.553509\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	50.3939			1.97813
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	15.5885	+	9.00000i	0.607240	+	0.350590i	0.771885	−	0.635763i	$-0.219314\pi$
$659$	15.5885	+	9.00000i	0.607240	+	0.350590i	−0.164644	+	0.986353i	$0.552648\pi$
$660$		0			0
$661$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$661$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−5.00000	+	8.66025i	−0.192736	+	0.333828i	−0.946156	−	0.323711i	$-0.895069\pi$
$673$	−5.00000	+	8.66025i	−0.192736	+	0.333828i	0.753420	+	0.657539i	$0.228403\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	5.94439i		−	0.227456i	−0.993512	−	0.113728i	$-0.963721\pi$
$683$		−	5.94439i		−	0.227456i	0.993512	−	0.113728i	$-0.0362792\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	39.8372	+	23.0000i	1.51548	+	0.874961i	0.999835	+	0.0181572i	$0.00577993\pi$
$691$	39.8372	+	23.0000i	1.51548	+	0.874961i	0.515642	+	0.856804i	$0.327553\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	12.1515	+	21.0471i	0.460272	+	0.797215i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$709$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$727$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3.86221	−	2.22985i	−0.142849	−	0.0824739i
$732$		0			0
$733$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$733$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	78.1918			2.88023
$738$		0			0
$739$		−	53.7423i		−	1.97694i	−0.151403	−	0.988472i	$-0.548379\pi$
$739$		−	53.7423i		−	1.97694i	0.151403	−	0.988472i	$-0.451621\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$743$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$751$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−27.0000	−	46.7654i	−0.978749	−	1.69524i	−0.666962	−	0.745091i	$-0.732406\pi$
$761$	−27.0000	−	46.7654i	−0.978749	−	1.69524i	−0.311787	−	0.950152i	$-0.600927\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	11.0000	+	19.0526i	0.396670	+	0.687053i	0.993313	−	0.115454i	$-0.0368323\pi$
$769$	11.0000	+	19.0526i	0.396670	+	0.687053i	−0.596643	+	0.802507i	$0.703499\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	92.5291	−	53.4217i	3.31520	−	1.91403i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	43.3013	−	25.0000i	1.54352	−	0.891154i	0.544911	−	0.838494i	$-0.316563\pi$
$787$	43.3013	−	25.0000i	1.54352	−	0.891154i	0.998613	−	0.0526599i	$-0.0167699\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	64.6413	+	37.3207i	2.28114	+	1.31702i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	45.9898			1.61692			0.808458	−	0.588555i	$-0.200303\pi$
$809$	45.9898			1.61692			0.808458	+	0.588555i	$0.200303\pi$
$810$		0			0
$811$		−	17.7423i		−	0.623018i	−0.950243	−	0.311509i	$-0.899166\pi$
$811$		−	17.7423i		−	0.623018i	0.950243	−	0.311509i	$-0.100834\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−9.80306	+	16.9794i	−0.342966	+	0.594034i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$823$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			54.0000i			1.87776i	0.344239	+	0.938882i	$0.388137\pi$
$827$			54.0000i			1.87776i	−0.344239	+	0.938882i	$0.611863\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6.64643	+	11.5120i	0.230285	+	0.398865i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	−14.5000	−	25.1147i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$853$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−27.0000	+	46.7654i	−0.922302	+	1.59747i	−0.126459	+	0.991972i	$0.540361\pi$
$857$	−27.0000	+	46.7654i	−0.922302	+	1.59747i	−0.795843	+	0.605503i	$0.792972\pi$
$858$		0			0
$859$	−18.7508	+	10.8258i	−0.639768	+	0.369370i	−0.784525	−	0.620097i	$-0.787093\pi$
$859$	−18.7508	+	10.8258i	−0.639768	+	0.369370i	0.144757	+	0.989467i	$0.453760\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−18.0000			−0.606435			−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000			−0.606435			−0.303218	+	0.952921i	$0.598061\pi$
$882$		0			0
$883$		−	52.4393i		−	1.76472i	−0.470573	−	0.882361i	$-0.655953\pi$
$883$		−	52.4393i		−	1.76472i	0.470573	−	0.882361i	$-0.344047\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−40.2176	−	23.2196i	−1.33540	−	0.770996i	−0.349281	−	0.937018i	$-0.613574\pi$
$907$	−40.2176	−	23.2196i	−1.33540	−	0.770996i	−0.986122	+	0.166022i	$0.946908\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$911$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$912$		0			0
$913$	49.0454	+	84.9491i	1.62317	+	2.81141i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−27.0000	+	46.7654i	−0.885841	+	1.53432i	−0.0410949	+	0.999155i	$0.513085\pi$
$929$	−27.0000	+	46.7654i	−0.885841	+	1.53432i	−0.844746	+	0.535167i	$0.820249\pi$
$930$		0			0
$931$	50.6099	−	29.2196i	1.65867	−	0.957635i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	34.0000			1.11073			0.555366	−	0.831606i	$-0.312578\pi$
$937$	34.0000			1.11073			0.555366	+	0.831606i	$0.312578\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−27.3147	−	15.7702i	−0.887609	−	0.512461i	−0.0144491	−	0.999896i	$-0.504599\pi$
$947$	−27.3147	−	15.7702i	−0.887609	−	0.512461i	−0.873160	+	0.487435i	$0.837933\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	60.1918			1.94980			0.974902	−	0.222633i	$-0.0714650\pi$
$953$	60.1918			1.94980			0.974902	+	0.222633i	$0.0714650\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.5000	−	26.8468i	0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$967$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			54.0000i			1.73294i	0.499227	+	0.866471i	$0.333617\pi$
$971$			54.0000i			1.73294i	−0.499227	+	0.866471i	$0.666383\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	28.4444	+	49.2671i	0.910017	+	1.57619i	0.814038	+	0.580812i	$0.197265\pi$
$977$	28.4444	+	49.2671i	0.910017	+	1.57619i	0.0959785	+	0.995383i	$0.469402\pi$
$978$		0			0
$979$	−84.9491	−	49.0454i	−2.71499	−	1.56750i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$997$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.r.c.289.4		8
3.2	odd	2		576.2.r.c.97.2	✓	8
4.3	odd	2	inner	1728.2.r.c.289.1		8
8.3	odd	2	CM	1728.2.r.c.289.4		8
8.5	even	2	inner	1728.2.r.c.289.1		8
9.2	odd	6		5184.2.d.l.2593.4		4
9.4	even	3	inner	1728.2.r.c.1441.1		8
9.5	odd	6		576.2.r.c.481.3	yes	8
9.7	even	3		5184.2.d.e.2593.1		4
12.11	even	2		576.2.r.c.97.3	yes	8
24.5	odd	2		576.2.r.c.97.3	yes	8
24.11	even	2		576.2.r.c.97.2	✓	8
36.7	odd	6		5184.2.d.e.2593.4		4
36.11	even	6		5184.2.d.l.2593.1		4
36.23	even	6		576.2.r.c.481.2	yes	8
36.31	odd	6	inner	1728.2.r.c.1441.4		8
72.5	odd	6		576.2.r.c.481.2	yes	8
72.11	even	6		5184.2.d.l.2593.4		4
72.13	even	6	inner	1728.2.r.c.1441.4		8
72.29	odd	6		5184.2.d.l.2593.1		4
72.43	odd	6		5184.2.d.e.2593.1		4
72.59	even	6		576.2.r.c.481.3	yes	8
72.61	even	6		5184.2.d.e.2593.4		4
72.67	odd	6	inner	1728.2.r.c.1441.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.2.r.c.97.2	✓	8	3.2	odd	2
576.2.r.c.97.2	✓	8	24.11	even	2
576.2.r.c.97.3	yes	8	12.11	even	2
576.2.r.c.97.3	yes	8	24.5	odd	2
576.2.r.c.481.2	yes	8	36.23	even	6
576.2.r.c.481.2	yes	8	72.5	odd	6
576.2.r.c.481.3	yes	8	9.5	odd	6
576.2.r.c.481.3	yes	8	72.59	even	6
1728.2.r.c.289.1		8	4.3	odd	2	inner
1728.2.r.c.289.1		8	8.5	even	2	inner
1728.2.r.c.289.4		8	1.1	even	1	trivial
1728.2.r.c.289.4		8	8.3	odd	2	CM
1728.2.r.c.1441.1		8	9.4	even	3	inner
1728.2.r.c.1441.1		8	72.67	odd	6	inner
1728.2.r.c.1441.4		8	36.31	odd	6	inner
1728.2.r.c.1441.4		8	72.13	even	6	inner
5184.2.d.e.2593.1		4	9.7	even	3
5184.2.d.e.2593.1		4	72.43	odd	6
5184.2.d.e.2593.4		4	36.7	odd	6
5184.2.d.e.2593.4		4	72.61	even	6
5184.2.d.l.2593.1		4	36.11	even	6
5184.2.d.l.2593.1		4	72.29	odd	6
5184.2.d.l.2593.4		4	9.2	odd	6
5184.2.d.l.2593.4		4	72.11	even	6

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:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.r (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+O(q^{10})\)	\(q+(4.71940 + 2.72474i) q^{11} +1.89898 q^{17} -8.34847i q^{19} +(-2.50000 + 4.33013i) q^{25} +(6.39898 + 11.0834i) q^{41} +(-2.03383 - 1.17423i) q^{43} +(3.50000 + 6.06218i) q^{49} +(8.00853 - 4.62372i) q^{59} +(12.4261 - 7.17423i) q^{67} +13.6969 q^{73} +(15.5885 + 9.00000i) q^{83} -18.0000 q^{89} +(9.84847 - 17.0580i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 24 q^{17} - 20 q^{25} + 12 q^{41} + 28 q^{49} - 8 q^{73} - 144 q^{89} + 20 q^{97}+O(q^{100}) \) 8 * q - 24 * q^17 - 20 * q^25 + 12 * q^41 + 28 * q^49 - 8 * q^73 - 144 * q^89 + 20 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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