LMFDB - Embedded newform 1728.3.n.a.737.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(737,1728)]

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, 1]))

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

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

chi := DirichletCharacter("1728.737");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.n (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:	$47.0845896815$
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:	$\Z[a_1, \ldots, a_{41}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 576)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			737.4
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1728.737
Dual form			1728.3.n.a.1313.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+O(q^{10})$	$q+(10.3048 - 17.8485i) q^{11} -15.2385i q^{17} +31.6969i q^{19} +(12.5000 - 21.6506i) q^{25} +(-5.10612 + 2.94802i) q^{41} +(-69.7018 - 40.2423i) q^{43} +(24.5000 + 42.4352i) q^{49} +(-56.7202 - 98.2423i) q^{59} +(-62.2487 + 35.9393i) q^{67} +41.6061 q^{73} +(25.4558 - 44.0908i) q^{83} -101.823i q^{89} +(96.9847 - 167.982i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 100 q^{25} - 276 q^{41} + 196 q^{49} + 568 q^{73} + 188 q^{97}+O(q^{100}) $ 8 * q + 100 * q^25 - 276 * q^41 + 196 * q^49 + 568 * q^73 + 188 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{6}\right)$

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		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		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
$11$	10.3048	−	17.8485i	0.936802	−	1.62259i	0.165412	−	0.986224i	$-0.447104\pi$
$11$	10.3048	−	17.8485i	0.936802	−	1.62259i	0.771389	−	0.636364i	$-0.219562\pi$
$12$		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
$17$		−	15.2385i		−	0.896383i	−0.893938	−	0.448192i	$-0.852068\pi$
$17$		−	15.2385i		−	0.896383i	0.893938	−	0.448192i	$-0.147932\pi$
$18$		0			0
$19$			31.6969i			1.66826i	0.551568	+	0.834130i	$0.314030\pi$
$19$			31.6969i			1.66826i	−0.551568	+	0.834130i	$0.685970\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	12.5000	−	21.6506i	0.500000	−	0.866025i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$29$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$30$		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
$41$	−5.10612	+	2.94802i	−0.124540	+	0.0719030i	−0.560976	−	0.827832i	$-0.689574\pi$
$41$	−5.10612	+	2.94802i	−0.124540	+	0.0719030i	0.436436	+	0.899735i	$0.356241\pi$
$42$		0			0
$43$	−69.7018	−	40.2423i	−1.62097	−	0.935869i	−0.986661	−	0.162791i	$-0.947950\pi$
$43$	−69.7018	−	40.2423i	−1.62097	−	0.935869i	−0.634311	−	0.773078i	$-0.718716\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		0			0
$49$	24.5000	+	42.4352i	0.500000	+	0.866025i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−56.7202	−	98.2423i	−0.961360	−	1.66512i	−0.719092	−	0.694915i	$-0.755442\pi$
$59$	−56.7202	−	98.2423i	−0.961360	−	1.66512i	−0.242268	−	0.970209i	$-0.577891\pi$
$60$		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
$67$	−62.2487	+	35.9393i	−0.929085	+	0.536407i	−0.886522	−	0.462687i	$-0.846886\pi$
$67$	−62.2487	+	35.9393i	−0.929085	+	0.536407i	−0.0425626	+	0.999094i	$0.513552\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	41.6061			0.569947			0.284973	−	0.958535i	$-0.408015\pi$
$73$	41.6061			0.569947			0.284973	+	0.958535i	$0.408015\pi$
$74$		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
$83$	25.4558	−	44.0908i	0.306697	−	0.531215i	−0.670941	−	0.741511i	$-0.734110\pi$
$83$	25.4558	−	44.0908i	0.306697	−	0.531215i	0.977638	+	0.210296i	$0.0674429\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	101.823i		−	1.14408i	−0.820225	−	0.572041i	$-0.806152\pi$
$89$		−	101.823i		−	1.14408i	0.820225	−	0.572041i	$-0.193848\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	96.9847	−	167.982i	0.999842	−	1.73178i	0.484536	−	0.874771i	$-0.338988\pi$
$97$	96.9847	−	167.982i	0.999842	−	1.73178i	0.515306	−	0.857006i	$-0.327678\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		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
$107$	94.7556			0.885566			0.442783	−	0.896629i	$-0.353991\pi$
$107$	94.7556			0.885566			0.442783	+	0.896629i	$0.353991\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	176.363	−	101.823i	1.56074	−	0.901092i	0.563554	−	0.826079i	$-0.309434\pi$
$113$	176.363	−	101.823i	1.56074	−	0.901092i	0.997183	−	0.0750128i	$-0.0238998\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−151.879	−	263.061i	−1.25519	−	2.17406i
$122$		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
$131$	−127.279	−	220.454i	−0.971597	−	1.68286i	−0.690736	−	0.723107i	$-0.742713\pi$
$131$	−127.279	−	220.454i	−0.971597	−	1.68286i	−0.280861	−	0.959748i	$-0.590620\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	237.288	+	136.998i	1.73203	+	0.999986i	0.868613	+	0.495491i	$0.165012\pi$
$137$	237.288	+	136.998i	1.73203	+	0.999986i	0.863414	+	0.504496i	$0.168322\pi$
$138$		0			0
$139$	−229.208	+	132.333i	−1.64898	+	0.952037i	−0.671497	+	0.741007i	$0.734348\pi$
$139$	−229.208	+	132.333i	−1.64898	+	0.952037i	−0.977480	+	0.211030i	$0.932318\pi$
$140$		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		0.866025	−	0.500000i	$-0.166667\pi$
$149$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$150$		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
$163$		−	322.000i		−	1.97546i	−0.156171	−	0.987730i	$-0.549915\pi$
$163$		−	322.000i		−	1.97546i	0.156171	−	0.987730i	$-0.450085\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$167$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$168$		0			0
$169$	−84.5000	+	146.358i	−0.500000	+	0.866025i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−356.382			−1.99096			−0.995480	−	0.0949721i	$-0.969724\pi$
$179$	−356.382			−1.99096			−0.995480	+	0.0949721i	$0.969724\pi$
$180$		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
$187$	−271.984	−	157.030i	−1.45446	−	0.839733i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	−137.166	−	237.579i	−0.710706	−	1.23098i	−0.964592	−	0.263745i	$-0.915042\pi$
$193$	−137.166	−	237.579i	−0.710706	−	1.23098i	0.253886	−	0.967234i	$-0.418291\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$		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
$209$	565.742	+	326.631i	2.70690	+	1.56283i
$210$		0			0
$211$	195.722	−	113.000i	0.927591	−	0.535545i	0.0415423	−	0.999137i	$-0.486773\pi$
$211$	195.722	−	113.000i	0.927591	−	0.535545i	0.886049	+	0.463592i	$0.153440\pi$
$212$		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
$227$	−171.910	+	297.758i	−0.757315	+	1.31171i	0.186900	+	0.982379i	$0.440156\pi$
$227$	−171.910	+	297.758i	−0.757315	+	1.31171i	−0.944215	+	0.329329i	$0.893178\pi$
$228$		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
$233$		−	460.708i		−	1.97729i	−0.150280	−	0.988643i	$-0.548018\pi$
$233$		−	460.708i		−	1.97729i	0.150280	−	0.988643i	$-0.451982\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$239$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$240$		0			0
$241$	239.560	−	414.930i	0.994026	−	1.72170i	0.402490	−	0.915425i	$-0.368145\pi$
$241$	239.560	−	414.930i	0.994026	−	1.72170i	0.591536	−	0.806279i	$-0.298522\pi$
$242$		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
$251$	496.906			1.97970			0.989852	−	0.142099i	$-0.0453851\pi$
$251$	496.906			1.97970			0.989852	+	0.142099i	$0.0453851\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−142.531	+	82.2901i	−0.554594	+	0.320195i	−0.750973	−	0.660333i	$-0.770415\pi$
$257$	−142.531	+	82.2901i	−0.554594	+	0.320195i	0.196379	+	0.980528i	$0.437082\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		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
$275$	−257.620	−	446.212i	−0.936802	−	1.62259i
$276$		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
$281$	−440.908	−	254.558i	−1.56907	−	0.905902i	−0.996278	−	0.0862000i	$-0.972528\pi$
$281$	−440.908	−	254.558i	−1.56907	−	0.905902i	−0.572790	−	0.819702i	$-0.694139\pi$
$282$		0			0
$283$	−71.0141	+	41.0000i	−0.250933	+	0.144876i	−0.620191	−	0.784450i	$-0.712945\pi$
$283$	−71.0141	+	41.0000i	−0.250933	+	0.144876i	0.369258	+	0.929327i	$0.379612\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	56.7878			0.196497
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$293$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$294$		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
$307$			520.848i			1.69657i	0.529537	+	0.848287i	$0.322366\pi$
$307$			520.848i			1.69657i	−0.529537	+	0.848287i	$0.677634\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	−15.4694	+	26.7938i	−0.0494230	+	0.0856031i	−0.889679	−	0.456587i	$-0.849072\pi$
$313$	−15.4694	+	26.7938i	−0.0494230	+	0.0856031i	0.840256	+	0.542191i	$0.182405\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$317$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	483.014			1.49540
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−12.1244	−	7.00000i	−0.0366295	−	0.0211480i	0.481573	−	0.876406i	$-0.340065\pi$
$331$	−12.1244	−	7.00000i	−0.0366295	−	0.0211480i	−0.518203	+	0.855258i	$0.673399\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	325.257	+	563.362i	0.965155	+	1.67170i	0.709199	+	0.705009i	$0.249057\pi$
$337$	325.257	+	563.362i	0.965155	+	1.67170i	0.255956	+	0.966688i	$0.417610\pi$
$338$		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
$347$	−229.768	−	397.970i	−0.662156	−	1.14689i	−0.980048	−	0.198761i	$-0.936308\pi$
$347$	−229.768	−	397.970i	−0.662156	−	1.14689i	0.317892	−	0.948127i	$-0.397025\pi$
$348$		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
$353$	439.439	+	253.710i	1.24487	+	0.718725i	0.970081	−	0.242779i	$-0.0780591\pi$
$353$	439.439	+	253.710i	1.24487	+	0.718725i	0.274788	+	0.961505i	$0.411392\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−643.696			−1.78309
$362$		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
$379$			63.7582i			0.168227i	0.996456	+	0.0841137i	$0.0268059\pi$
$379$			63.7582i			0.168227i	−0.996456	+	0.0841137i	$0.973194\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$389$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$390$		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
$401$	−677.379	+	391.085i	−1.68922	+	0.975274i	−0.734111	+	0.679029i	$0.762401\pi$
$401$	−677.379	+	391.085i	−1.68922	+	0.975274i	−0.955112	+	0.296244i	$0.904266\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	239.833	+	415.402i	0.586388	+	1.01565i	0.994701	+	0.102812i	$0.0327839\pi$
$409$	239.833	+	415.402i	0.586388	+	1.01565i	−0.408313	+	0.912842i	$0.633883\pi$
$410$		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
$419$	330.926	+	573.181i	0.789799	+	1.36797i	0.926090	+	0.377304i	$0.123149\pi$
$419$	330.926	+	573.181i	0.789799	+	1.36797i	−0.136290	+	0.990669i	$0.543518\pi$
$420$		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
$425$	−329.923	−	190.481i	−0.776290	−	0.448192i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$	847.484			1.95724			0.978619	−	0.205684i	$-0.0659418\pi$
$433$	847.484			1.95724			0.978619	+	0.205684i	$0.0659418\pi$
$434$		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
$443$	−409.884	+	709.939i	−0.925245	+	1.60257i	−0.134079	+	0.990971i	$0.542808\pi$
$443$	−409.884	+	709.939i	−0.925245	+	1.60257i	−0.791166	+	0.611601i	$0.790526\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	631.184i		−	1.40576i	−0.711311	−	0.702878i	$-0.751898\pi$
$449$		−	631.184i		−	1.40576i	0.711311	−	0.702878i	$-0.248102\pi$
$450$		0			0
$451$			121.515i			0.269435i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−441.620	+	764.909i	−0.966347	+	1.67376i	−0.260394	+	0.965502i	$0.583852\pi$
$457$	−441.620	+	764.909i	−0.966347	+	1.67376i	−0.705953	+	0.708259i	$0.749481\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		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
$467$	437.246			0.936286			0.468143	−	0.883653i	$-0.344923\pi$
$467$	437.246			0.936286			0.468143	+	0.883653i	$0.344923\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1436.53	+	829.380i	−3.03706	+	1.75345i
$474$		0			0
$475$	686.259	+	396.212i	1.44476	+	0.834130i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		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
$491$	−190.124	−	329.304i	−0.387217	−	0.670679i	0.604857	−	0.796334i	$-0.293230\pi$
$491$	−190.124	−	329.304i	−0.387217	−	0.670679i	−0.992074	+	0.125655i	$0.959897\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	98.2011	−	56.6964i	0.196796	−	0.113620i	−0.398364	−	0.917227i	$-0.630422\pi$
$499$	98.2011	−	56.6964i	0.196796	−	0.113620i	0.595160	+	0.803607i	$0.297089\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		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
$521$		−	735.457i		−	1.41163i	−0.708398	−	0.705813i	$-0.750582\pi$
$521$		−	735.457i		−	1.41163i	0.708398	−	0.705813i	$-0.249418\pi$
$522$		0			0
$523$		−	398.000i		−	0.760994i	−0.924782	−	0.380497i	$-0.875753\pi$
$523$		−	398.000i		−	0.760994i	0.924782	−	0.380497i	$-0.124247\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−264.500	+	458.127i	−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
$539$	1009.87			1.87360
$540$		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
$547$	735.281	+	424.515i	1.34421	+	0.776078i	0.987422	−	0.158108i	$-0.0505395\pi$
$547$	735.281	+	424.515i	1.34421	+	0.776078i	0.356785	+	0.934186i	$0.383873\pi$
$548$		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			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	373.633	+	647.150i	0.663646	+	1.14947i	0.979651	+	0.200710i	$0.0643251\pi$
$563$	373.633	+	647.150i	0.663646	+	1.14947i	−0.316005	+	0.948758i	$0.602342\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	57.9857	+	33.4781i	0.101908	+	0.0588367i	0.550088	−	0.835107i	$-0.314594\pi$
$569$	57.9857	+	33.4781i	0.101908	+	0.0588367i	−0.448180	+	0.893943i	$0.647928\pi$
$570$		0			0
$571$	984.958	−	568.666i	1.72497	−	0.995912i	0.817321	−	0.576182i	$-0.195458\pi$
$571$	984.958	−	568.666i	1.72497	−	0.995912i	0.907649	−	0.419730i	$-0.137875\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	1000.39			1.73378			0.866891	−	0.498498i	$-0.166115\pi$
$577$	1000.39			1.73378			0.866891	+	0.498498i	$0.166115\pi$
$578$		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
$587$	−564.893	+	978.424i	−0.962340	+	1.66682i	−0.245741	+	0.969336i	$0.579031\pi$
$587$	−564.893	+	978.424i	−0.962340	+	1.66682i	−0.716599	+	0.697485i	$0.754302\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			814.587i			1.37367i	0.726813	+	0.686836i	$0.241001\pi$
$593$			814.587i			1.37367i	−0.726813	+	0.686836i	$0.758999\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$	−109.530	+	189.711i	−0.182246	+	0.315659i	−0.942645	−	0.333797i	$-0.891670\pi$
$601$	−109.530	+	189.711i	−0.182246	+	0.315659i	0.760399	+	0.649456i	$0.225003\pi$
$602$		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
$617$	−263.893	+	152.359i	−0.427703	+	0.246935i	−0.698368	−	0.715739i	$-0.746090\pi$
$617$	−263.893	+	152.359i	−0.427703	+	0.246935i	0.270665	+	0.962674i	$0.412757\pi$
$618$		0			0
$619$	583.962	+	337.150i	0.943395	+	0.544670i	0.891023	−	0.453958i	$-0.149988\pi$
$619$	583.962	+	337.150i	0.943395	+	0.544670i	0.0523724	+	0.998628i	$0.483322\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−312.500	−	541.266i	−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
$641$	875.893	+	505.697i	1.36645	+	0.788919i	0.990472	−	0.137711i	$-0.0439745\pi$
$641$	875.893	+	505.697i	1.36645	+	0.788919i	0.375975	+	0.926630i	$0.377308\pi$
$642$		0			0
$643$	−207.479	+	119.788i	−0.322674	+	0.186296i	−0.652584	−	0.757717i	$-0.726315\pi$
$643$	−207.479	+	119.788i	−0.322674	+	0.186296i	0.329910	+	0.944012i	$0.392982\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	−2337.97			−3.60241
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$653$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−432.749	+	749.544i	−0.656676	+	1.13740i	0.324795	+	0.945784i	$0.394705\pi$
$659$	−432.749	+	749.544i	−0.656676	+	1.13740i	−0.981471	+	0.191611i	$0.938629\pi$
$660$		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
$673$	−623.000	+	1079.07i	−0.925706	+	1.60337i	−0.135284	+	0.990807i	$0.543195\pi$
$673$	−623.000	+	1079.07i	−0.925706	+	1.60337i	−0.790422	+	0.612563i	$0.790139\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−308.689			−0.451960			−0.225980	−	0.974132i	$-0.572558\pi$
$683$	−308.689			−0.451960			−0.225980	+	0.974132i	$0.572558\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−635.663	−	367.000i	−0.919917	−	0.531114i	−0.0363084	−	0.999341i	$-0.511560\pi$
$691$	−635.663	−	367.000i	−0.919917	−	0.531114i	−0.883609	+	0.468226i	$0.844893\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	44.9235	+	77.8097i	0.0644526	+	0.111635i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		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		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		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
$731$	−613.233	+	1062.15i	−0.838897	+	1.45301i
$732$		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
$737$			1481.39i			2.01003i
$738$		0			0
$739$		−	1410.24i		−	1.90831i	−0.299316	−	0.954154i	$-0.596758\pi$
$739$		−	1410.24i		−	1.90831i	0.299316	−	0.954154i	$-0.403242\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$743$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$744$		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
$761$	−529.090	+	305.470i	−0.695256	+	0.401406i	−0.805578	−	0.592490i	$-0.798145\pi$
$761$	−529.090	+	305.470i	−0.695256	+	0.401406i	0.110322	+	0.993896i	$0.464812\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	527.000	+	912.791i	0.685306	+	1.18698i	0.973341	+	0.229364i	$0.0736647\pi$
$769$	527.000	+	912.791i	0.685306	+	1.18698i	−0.288035	+	0.957620i	$0.593002\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−93.4433	−	161.848i	−0.119953	−	0.207764i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	801.940	−	463.000i	1.01898	−	0.588310i	0.105174	−	0.994454i	$-0.466460\pi$
$787$	801.940	−	463.000i	1.01898	−	0.588310i	0.913809	+	0.406144i	$0.133127\pi$
$788$		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		0.866025	−	0.500000i	$-0.166667\pi$
$797$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$	428.744	−	742.606i	0.533927	−	0.924789i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1539.76i		−	1.90329i	−0.307207	−	0.951643i	$-0.599394\pi$
$809$		−	1539.76i		−	1.90329i	0.307207	−	0.951643i	$-0.400606\pi$
$810$		0			0
$811$			1307.21i			1.61185i	0.592019	+	0.805924i	$0.298331\pi$
$811$			1307.21i			1.61185i	−0.592019	+	0.805924i	$0.701669\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1275.56	−	2209.33i	1.56127	−	2.70420i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$821$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$822$		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
$827$	1069.15			1.29280			0.646400	−	0.762999i	$-0.276274\pi$
$827$	1069.15			1.29280			0.646400	+	0.762999i	$0.276274\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	646.650	−	373.344i	0.776290	−	0.448192i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$839$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$840$		0			0
$841$	420.500	+	728.327i	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
$857$	−1058.18	−	610.940i	−1.23475	−	0.712882i	−0.266733	−	0.963771i	$-0.585944\pi$
$857$	−1058.18	−	610.940i	−1.23475	−	0.712882i	−0.968016	+	0.250888i	$0.919277\pi$
$858$		0			0
$859$	1081.85	−	624.606i	1.25943	−	0.727131i	0.286465	−	0.958091i	$-0.407520\pi$
$859$	1081.85	−	624.606i	1.25943	−	0.727131i	0.972963	+	0.230960i	$0.0741866\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		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
$881$		−	1018.23i		−	1.15577i	−0.816118	−	0.577885i	$-0.803878\pi$
$881$		−	1018.23i		−	1.15577i	0.816118	−	0.577885i	$-0.196122\pi$
$882$		0			0
$883$			983.879i			1.11425i	0.830430	+	0.557123i	$0.188095\pi$
$883$			983.879i			1.11425i	−0.830430	+	0.557123i	$0.811905\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		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
$907$	296.706	+	171.304i	0.327130	+	0.188868i	0.654566	−	0.756005i	$-0.272851\pi$
$907$	296.706	+	171.304i	0.327130	+	0.188868i	−0.327436	+	0.944873i	$0.606185\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$911$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$912$		0			0
$913$	−524.636	−	908.696i	−0.574628	−	0.995286i
$914$		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
$929$	1322.72	+	763.675i	1.42382	+	0.822040i	0.996622	−	0.0821203i	$-0.0261692\pi$
$929$	1322.72	+	763.675i	1.42382	+	0.822040i	0.427193	+	0.904160i	$0.359503\pi$
$930$		0			0
$931$	−1345.07	+	776.575i	−1.44476	+	0.834130i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	718.000			0.766275			0.383138	−	0.923691i	$-0.374843\pi$
$937$	718.000			0.766275			0.383138	+	0.923691i	$0.374843\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	833.465	−	1443.60i	0.880111	−	1.52440i	0.0288952	−	0.999582i	$-0.490801\pi$
$947$	833.465	−	1443.60i	0.880111	−	1.52440i	0.851216	−	0.524815i	$-0.175866\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	827.376i		−	0.868180i	−0.900869	−	0.434090i	$-0.857070\pi$
$953$		−	827.376i		−	0.868180i	0.900869	−	0.434090i	$-0.142930\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	480.500	−	832.250i	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
$971$	−1680.09			−1.73026			−0.865132	−	0.501545i	$-0.832765\pi$
$971$	−1680.09			−1.73026			−0.865132	+	0.501545i	$0.832765\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1276.83	+	737.180i	−1.30689	+	0.754535i	−0.981576	−	0.191071i	$-0.938804\pi$
$977$	−1276.83	+	737.180i	−1.30689	+	0.754535i	−0.325316	+	0.945605i	$0.605471\pi$
$978$		0			0
$979$	−1817.39	−	1049.27i	−1.85638	−	1.07178i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		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	1728.3.n.a.737.4		8
3.2	odd	2		576.3.n.a.353.3	yes	8
4.3	odd	2	inner	1728.3.n.a.737.1		8
8.3	odd	2	CM	1728.3.n.a.737.4		8
8.5	even	2	inner	1728.3.n.a.737.1		8
9.4	even	3		576.3.n.a.545.2	yes	8
9.5	odd	6	inner	1728.3.n.a.1313.1		8
12.11	even	2		576.3.n.a.353.2	✓	8
24.5	odd	2		576.3.n.a.353.2	✓	8
24.11	even	2		576.3.n.a.353.3	yes	8
36.23	even	6	inner	1728.3.n.a.1313.4		8
36.31	odd	6		576.3.n.a.545.3	yes	8
72.5	odd	6	inner	1728.3.n.a.1313.4		8
72.13	even	6		576.3.n.a.545.3	yes	8
72.59	even	6	inner	1728.3.n.a.1313.1		8
72.67	odd	6		576.3.n.a.545.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.3.n.a.353.2	✓	8	12.11	even	2
576.3.n.a.353.2	✓	8	24.5	odd	2
576.3.n.a.353.3	yes	8	3.2	odd	2
576.3.n.a.353.3	yes	8	24.11	even	2
576.3.n.a.545.2	yes	8	9.4	even	3
576.3.n.a.545.2	yes	8	72.67	odd	6
576.3.n.a.545.3	yes	8	36.31	odd	6
576.3.n.a.545.3	yes	8	72.13	even	6
1728.3.n.a.737.1		8	4.3	odd	2	inner
1728.3.n.a.737.1		8	8.5	even	2	inner
1728.3.n.a.737.4		8	1.1	even	1	trivial
1728.3.n.a.737.4		8	8.3	odd	2	CM
1728.3.n.a.1313.1		8	9.5	odd	6	inner
1728.3.n.a.1313.1		8	72.59	even	6	inner
1728.3.n.a.1313.4		8	36.23	even	6	inner
1728.3.n.a.1313.4		8	72.5	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.n (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+O(q^{10})\)	\(q+(10.3048 - 17.8485i) q^{11} -15.2385i q^{17} +31.6969i q^{19} +(12.5000 - 21.6506i) q^{25} +(-5.10612 + 2.94802i) q^{41} +(-69.7018 - 40.2423i) q^{43} +(24.5000 + 42.4352i) q^{49} +(-56.7202 - 98.2423i) q^{59} +(-62.2487 + 35.9393i) q^{67} +41.6061 q^{73} +(25.4558 - 44.0908i) q^{83} -101.823i q^{89} +(96.9847 - 167.982i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 100 q^{25} - 276 q^{41} + 196 q^{49} + 568 q^{73} + 188 q^{97}+O(q^{100}) \) 8 * q + 100 * q^25 - 276 * q^41 + 196 * q^49 + 568 * q^73 + 188 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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