LMFDB - Embedded newform 1728.3.n.a.737.2

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			737.2
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1728.737
Dual form			1728.3.n.a.1313.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+O(q^{10})$	$q+(-1.81954 + 3.15153i) q^{11} +18.7026i q^{17} -2.30306i q^{19} +(12.5000 - 21.6506i) q^{25} +(-63.8939 + 36.8891i) q^{41} +(-57.5774 - 33.2423i) q^{43} +(24.5000 + 42.4352i) q^{49} +(14.2938 + 24.7577i) q^{59} +(-115.942 + 66.9393i) q^{67} +100.394 q^{73} +(25.4558 - 44.0908i) q^{83} +101.823i q^{89} +(-49.9847 + 86.5760i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$11$	−1.81954	+	3.15153i	−0.165412	+	0.286503i	−0.936802	−	0.349861i	$-0.886229\pi$
$11$	−1.81954	+	3.15153i	−0.165412	+	0.286503i	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
$17$			18.7026i			1.10015i	0.835114	+	0.550077i	$0.185402\pi$
$17$			18.7026i			1.10015i	−0.835114	+	0.550077i	$0.814598\pi$
$18$		0			0
$19$		−	2.30306i		−	0.121214i	−0.998162	−	0.0606069i	$-0.980696\pi$
$19$		−	2.30306i		−	0.121214i	0.998162	−	0.0606069i	$-0.0193036\pi$
$20$		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
$41$	−63.8939	+	36.8891i	−1.55839	+	0.899735i	−0.560976	+	0.827832i	$0.689574\pi$
$41$	−63.8939	+	36.8891i	−1.55839	+	0.899735i	−0.997412	+	0.0719030i	$0.977093\pi$
$42$		0			0
$43$	−57.5774	−	33.2423i	−1.33901	−	0.773078i	−0.352349	−	0.935869i	$-0.614617\pi$
$43$	−57.5774	−	33.2423i	−1.33901	−	0.773078i	−0.986661	+	0.162791i	$0.947950\pi$
$44$		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
$59$	14.2938	+	24.7577i	0.242268	+	0.419621i	0.961360	−	0.275294i	$-0.0887752\pi$
$59$	14.2938	+	24.7577i	0.242268	+	0.419621i	−0.719092	+	0.694915i	$0.755442\pi$
$60$		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
$67$	−115.942	+	66.9393i	−1.73048	+	0.999094i	−0.843959	+	0.536407i	$0.819781\pi$
$67$	−115.942	+	66.9393i	−1.73048	+	0.999094i	−0.886522	+	0.462687i	$0.846886\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	100.394			1.37526			0.687629	−	0.726062i	$-0.258651\pi$
$73$	100.394			1.37526			0.687629	+	0.726062i	$0.258651\pi$
$74$		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	$0.193848\pi$
$89$			101.823i			1.14408i	−0.820225	+	0.572041i	$0.806152\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−49.9847	+	86.5760i	−0.515306	+	0.892536i	0.484536	+	0.874771i	$0.338988\pi$
$97$	−49.9847	+	86.5760i	−0.515306	+	0.892536i	−0.999842	+	0.0177651i	$0.994345\pi$
$98$		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
$107$	−213.549			−1.99579			−0.997895	−	0.0648531i	$-0.979342\pi$
$107$	−213.549			−1.99579			−0.997895	+	0.0648531i	$0.979342\pi$
$108$		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	$0.690566\pi$
$113$	−176.363	+	101.823i	−1.56074	+	0.901092i	−0.997183	+	0.0750128i	$0.976100\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	53.8786	+	93.3204i	0.445277	+	0.771243i
$122$		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
$137$	119.712	+	69.1159i	0.873812	+	0.504496i	0.868613	−	0.495491i	$-0.165012\pi$
$137$	119.712	+	69.1159i	0.873812	+	0.504496i	0.00519888	+	0.999986i	$0.498345\pi$
$138$		0			0
$139$	−50.8065	+	29.3332i	−0.365515	+	0.211030i	−0.671497	−	0.741007i	$-0.734348\pi$
$139$	−50.8065	+	29.3332i	−0.365515	+	0.211030i	0.305983	+	0.952037i	$0.401015\pi$
$140$		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	$0.450085\pi$
$163$			322.000i			1.97546i	−0.156171	+	0.987730i	$0.549915\pi$
$164$		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
$187$	−58.9419	−	34.0301i	−0.315197	−	0.181979i
$188$		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
$193$	186.166	+	322.450i	0.964592	+	1.67072i	0.710706	+	0.703489i	$0.248375\pi$
$193$	186.166	+	322.450i	0.964592	+	1.67072i	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
$209$	7.25817	+	4.19051i	0.0347281	+	0.0200503i
$210$		0			0
$211$	−195.722	+	113.000i	−0.927591	+	0.535545i	−0.886049	−	0.463592i	$-0.846560\pi$
$211$	−195.722	+	113.000i	−0.927591	+	0.535545i	−0.0415423	+	0.999137i	$0.513227\pi$
$212$		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
$227$	214.337	−	371.242i	0.944215	−	1.63543i	0.186900	−	0.982379i	$-0.440156\pi$
$227$	214.337	−	371.242i	0.944215	−	1.63543i	0.757315	−	0.653050i	$-0.226511\pi$
$228$		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
$233$		−	291.002i		−	1.24894i	−0.781050	−	0.624468i	$-0.785316\pi$
$233$		−	291.002i		−	1.24894i	0.781050	−	0.624468i	$-0.214684\pi$
$234$		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
$241$	−142.560	+	246.922i	−0.591536	+	1.02457i	0.402490	+	0.915425i	$0.368145\pi$
$241$	−142.560	+	246.922i	−0.591536	+	1.02457i	−0.994026	+	0.109146i	$0.965188\pi$
$242$		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
$251$	−310.230			−1.23598			−0.617988	−	0.786188i	$-0.712052\pi$
$251$	−310.230			−1.23598			−0.617988	+	0.786188i	$0.712052\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−436.469	+	251.996i	−1.69832	+	0.980528i	−0.750973	+	0.660333i	$0.770415\pi$
$257$	−436.469	+	251.996i	−1.69832	+	0.980528i	−0.947352	+	0.320195i	$0.896252\pi$
$258$		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
$275$	45.4884	+	78.7883i	0.165412	+	0.286503i
$276$		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	$0.0274724\pi$
$281$	440.908	+	254.558i	1.56907	+	0.905902i	0.572790	+	0.819702i	$0.305861\pi$
$282$		0			0
$283$	71.0141	−	41.0000i	0.250933	−	0.144876i	−0.369258	−	0.929327i	$-0.620388\pi$
$283$	71.0141	−	41.0000i	0.250933	−	0.144876i	0.620191	+	0.784450i	$0.287055\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−60.7878			−0.210338
$290$		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
$307$		−	21.1520i		−	0.0688992i	−0.999406	−	0.0344496i	$-0.989032\pi$
$307$		−	21.1520i		−	0.0688992i	0.999406	−	0.0344496i	$-0.0109678\pi$
$308$		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
$313$	278.469	−	482.323i	0.889679	−	1.54097i	0.0494230	−	0.998778i	$-0.484262\pi$
$313$	278.469	−	482.323i	0.889679	−	1.54097i	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
$323$	43.0733			0.133354
$324$		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	0.518203	−	0.855258i	$-0.326601\pi$
$331$	12.1244	+	7.00000i	0.0366295	+	0.0211480i	−0.481573	+	0.876406i	$0.659935\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−86.2571	−	149.402i	−0.255956	−	0.443329i	0.709199	−	0.705009i	$-0.249057\pi$
$337$	−86.2571	−	149.402i	−0.255956	−	0.443329i	−0.965155	+	0.261680i	$0.915724\pi$
$338$		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
$347$	340.077	+	589.030i	0.980048	+	1.69749i	0.662156	+	0.749366i	$0.269642\pi$
$347$	340.077	+	589.030i	0.980048	+	1.69749i	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
$353$	−148.439	−	85.7012i	−0.420506	−	0.242779i	0.274788	−	0.961505i	$-0.411392\pi$
$353$	−148.439	−	85.7012i	−0.420506	−	0.242779i	−0.695294	+	0.718725i	$0.744726\pi$
$354$		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
$361$	355.696			0.985307
$362$		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
$379$		−	622.242i		−	1.64180i	−0.571073	−	0.820900i	$-0.693473\pi$
$379$		−	622.242i		−	1.64180i	0.571073	−	0.820900i	$-0.306527\pi$
$380$		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
$401$	−471.621	+	272.291i	−1.17611	+	0.679029i	−0.955112	−	0.296244i	$-0.904266\pi$
$401$	−471.621	+	272.291i	−1.17611	+	0.679029i	−0.221001	+	0.975274i	$0.570932\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−406.833	−	704.655i	−0.994701	−	1.72287i	−0.586388	−	0.810030i	$-0.699451\pi$
$409$	−406.833	−	704.655i	−0.994701	−	1.72287i	−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
$425$	404.923	+	233.783i	0.952761	+	0.550077i
$426$		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
$433$	−269.484			−0.622364			−0.311182	−	0.950350i	$-0.600725\pi$
$433$	−269.484			−0.622364			−0.311182	+	0.950350i	$0.600725\pi$
$434$		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
$443$	350.487	−	607.061i	0.791166	−	1.37034i	−0.134079	−	0.990971i	$-0.542808\pi$
$443$	350.487	−	607.061i	0.791166	−	1.37034i	0.925245	−	0.379370i	$-0.123859\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	868.772i		−	1.93490i	−0.253054	−	0.967452i	$-0.581435\pi$
$449$		−	868.772i		−	1.93490i	0.253054	−	0.967452i	$-0.418565\pi$
$450$		0			0
$451$		−	268.485i		−	0.595310i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	322.620	−	558.795i	0.705953	−	1.22275i	−0.260394	−	0.965502i	$-0.583852\pi$
$457$	322.620	−	558.795i	0.705953	−	1.22275i	0.966347	−	0.257244i	$-0.0828143\pi$
$458$		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
$467$	496.135			1.06239			0.531194	−	0.847250i	$-0.321743\pi$
$467$	496.135			1.06239			0.531194	+	0.847250i	$0.321743\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	209.529	−	120.971i	0.442978	−	0.255753i
$474$		0			0
$475$	−49.8627	−	28.7883i	−0.104974	−	0.0606069i
$476$		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
$491$	487.108	+	843.696i	0.992074	+	1.71832i	0.604857	+	0.796334i	$0.293230\pi$
$491$	487.108	+	843.696i	0.992074	+	1.71832i	0.387217	+	0.921989i	$0.373436\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	792.753	−	457.696i	1.58868	−	0.917227i	0.595160	−	0.803607i	$-0.297089\pi$
$499$	792.753	−	457.696i	1.58868	−	0.917227i	0.993524	−	0.113620i	$-0.0362447\pi$
$500$		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
$521$		−	1006.99i		−	1.93279i	−0.257053	−	0.966397i	$-0.582751\pi$
$521$		−	1006.99i		−	1.93279i	0.257053	−	0.966397i	$-0.417249\pi$
$522$		0			0
$523$			398.000i			0.760994i	0.924782	+	0.380497i	$0.124247\pi$
$523$			398.000i			0.760994i	−0.924782	+	0.380497i	$0.875753\pi$
$524$		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
$539$	−178.315			−0.330825
$540$		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
$547$	−149.797	−	86.4852i	−0.273852	−	0.158108i	0.356785	−	0.934186i	$-0.383873\pi$
$547$	−149.797	−	86.4852i	−0.273852	−	0.158108i	−0.630637	+	0.776078i	$0.717206\pi$
$548$		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
$563$	177.911	+	308.150i	0.316005	+	0.547337i	0.979651	−	0.200710i	$-0.0643251\pi$
$563$	177.911	+	308.150i	0.316005	+	0.547337i	−0.663646	+	0.748047i	$0.730992\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	881.014	+	508.654i	1.54836	+	0.893943i	0.998268	+	0.0588367i	$0.0187391\pi$
$569$	881.014	+	508.654i	1.54836	+	0.893943i	0.550088	+	0.835107i	$0.314594\pi$
$570$		0			0
$571$	415.113	−	239.666i	0.726994	−	0.419730i	−0.0903277	−	0.995912i	$-0.528791\pi$
$571$	415.113	−	239.666i	0.726994	−	0.419730i	0.817321	+	0.576182i	$0.195458\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−998.392			−1.73032			−0.865158	−	0.501500i	$-0.832782\pi$
$577$	−998.392			−1.73032			−0.865158	+	0.501500i	$0.832782\pi$
$578$		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
$587$	420.644	−	728.576i	0.716599	−	1.24119i	−0.245741	−	0.969336i	$-0.579031\pi$
$587$	420.644	−	728.576i	0.716599	−	1.24119i	0.962340	−	0.271850i	$-0.0876355\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	814.587i		−	1.37367i	−0.726813	−	0.686836i	$-0.758999\pi$
$593$		−	814.587i		−	1.37367i	0.726813	−	0.686836i	$-0.241001\pi$
$594$		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
$601$	566.530	−	981.258i	0.942645	−	1.63271i	0.182246	−	0.983253i	$-0.441663\pi$
$601$	566.530	−	981.258i	0.942645	−	1.63271i	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
$617$	764.893	−	441.611i	1.23970	−	0.715739i	0.270665	−	0.962674i	$-0.412757\pi$
$617$	764.893	−	441.611i	1.23970	−	0.715739i	0.969032	+	0.246935i	$0.0794233\pi$
$618$		0			0
$619$	1070.67	+	618.150i	1.72967	+	0.998628i	0.891023	+	0.453958i	$0.149988\pi$
$619$	1070.67	+	618.150i	1.72967	+	0.998628i	0.838651	+	0.544670i	$0.183345\pi$
$620$		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
$641$	−152.893	−	88.2727i	−0.238522	−	0.137711i	0.375975	−	0.926630i	$-0.377308\pi$
$641$	−152.893	−	88.2727i	−0.238522	−	0.137711i	−0.614497	+	0.788919i	$0.710641\pi$
$642$		0			0
$643$	843.875	−	487.212i	1.31240	−	0.757717i	0.329910	−	0.944012i	$-0.392982\pi$
$643$	843.875	−	487.212i	1.31240	−	0.757717i	0.982494	+	0.186296i	$0.0596483\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$		0			0
$649$	−104.033			−0.160297
$650$		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
$683$	−998.045			−1.46127			−0.730633	−	0.682770i	$-0.760775\pi$
$683$	−998.045			−1.46127			−0.730633	+	0.682770i	$0.760775\pi$
$684$		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	0.883609	−	0.468226i	$-0.155107\pi$
$691$	635.663	+	367.000i	0.919917	+	0.531114i	0.0363084	+	0.999341i	$0.488440\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−689.923	−	1194.98i	−0.989847	−	1.71447i
$698$		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
$731$	621.719	−	1076.85i	0.850504	−	1.47312i
$732$		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
$737$		−	487.194i		−	0.661050i
$738$		0			0
$739$		−	1088.24i		−	1.47258i	−0.676664	−	0.736292i	$-0.736575\pi$
$739$		−	1088.24i		−	1.47258i	0.676664	−	0.736292i	$-0.263425\pi$
$740$		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	−0.110322	−	0.993896i	$-0.535188\pi$
$761$	529.090	−	305.470i	0.695256	−	0.401406i	0.805578	+	0.592490i	$0.201855\pi$
$762$		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
$779$	84.9580	+	147.152i	0.109060	+	0.188898i
$780$		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	−0.913809	−	0.406144i	$-0.866873\pi$
$787$	−801.940	+	463.000i	−1.01898	+	0.588310i	−0.105174	+	0.994454i	$0.533540\pi$
$788$		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
$803$	−182.670	+	316.394i	−0.227485	+	0.394015i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1200.35i		−	1.48374i	−0.670543	−	0.741871i	$-0.733939\pi$
$809$		−	1200.35i		−	1.48374i	0.670543	−	0.741871i	$-0.266061\pi$
$810$		0			0
$811$			1485.21i			1.83133i	0.401942	+	0.915665i	$0.368335\pi$
$811$			1485.21i			1.83133i	−0.401942	+	0.915665i	$0.631665\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−76.5592	+	132.604i	−0.0937077	+	0.162306i
$818$		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
$833$	−793.650	+	458.214i	−0.952761	+	0.550077i
$834$		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	0.968016	−	0.250888i	$-0.0807226\pi$
$857$	1058.18	+	610.940i	1.23475	+	0.712882i	0.266733	+	0.963771i	$0.414056\pi$
$858$		0			0
$859$	−343.629	+	198.394i	−0.400034	+	0.230960i	−0.686499	−	0.727131i	$-0.740853\pi$
$859$	−343.629	+	198.394i	−0.400034	+	0.230960i	0.286465	+	0.958091i	$0.407520\pi$
$860$		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	$0.196122\pi$
$881$			1018.23i			1.15577i	−0.816118	+	0.577885i	$0.803878\pi$
$882$		0			0
$883$		−	778.121i		−	0.881225i	−0.897697	−	0.440612i	$-0.854761\pi$
$883$		−	778.121i		−	0.881225i	0.897697	−	0.440612i	$-0.145239\pi$
$884$		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
$907$	−1187.66	−	685.696i	−1.30944	−	0.756005i	−0.327436	−	0.944873i	$-0.606185\pi$
$907$	−1187.66	−	685.696i	−1.30944	−	0.756005i	−0.982002	+	0.188868i	$0.939518\pi$
$908$		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
$913$	92.6357	+	160.450i	0.101463	+	0.175739i
$914$		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	−0.427193	−	0.904160i	$-0.640497\pi$
$929$	−1322.72	−	763.675i	−1.42382	−	0.822040i	−0.996622	+	0.0821203i	$0.973831\pi$
$930$		0			0
$931$	97.7310	−	56.4250i	0.104974	−	0.0606069i
$932$		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
$947$	−27.3638	+	47.3954i	−0.0288952	+	0.0500480i	−0.880111	−	0.474767i	$-0.842532\pi$
$947$	−27.3638	+	47.3954i	−0.0288952	+	0.0500480i	0.851216	+	0.524815i	$0.175866\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$			1073.33i			1.12626i	0.826368	+	0.563131i	$0.190403\pi$
$953$			1073.33i			1.12626i	−0.826368	+	0.563131i	$0.809597\pi$
$954$		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
$977$	−1600.17	+	923.856i	−1.63784	+	0.945605i	−0.656260	+	0.754535i	$0.727863\pi$
$977$	−1600.17	+	923.856i	−1.63784	+	0.945605i	−0.981576	+	0.191071i	$0.938804\pi$
$978$		0			0
$979$	−320.900	−	185.271i	−0.327783	−	0.189246i
$980$		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.737.2		8
3.2	odd	2		576.3.n.a.353.4	yes	8
4.3	odd	2	inner	1728.3.n.a.737.3		8
8.3	odd	2	CM	1728.3.n.a.737.2		8
8.5	even	2	inner	1728.3.n.a.737.3		8
9.4	even	3		576.3.n.a.545.1	yes	8
9.5	odd	6	inner	1728.3.n.a.1313.3		8
12.11	even	2		576.3.n.a.353.1	✓	8
24.5	odd	2		576.3.n.a.353.1	✓	8
24.11	even	2		576.3.n.a.353.4	yes	8
36.23	even	6	inner	1728.3.n.a.1313.2		8
36.31	odd	6		576.3.n.a.545.4	yes	8
72.5	odd	6	inner	1728.3.n.a.1313.2		8
72.13	even	6		576.3.n.a.545.4	yes	8
72.59	even	6	inner	1728.3.n.a.1313.3		8
72.67	odd	6		576.3.n.a.545.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
576.3.n.a.353.1	✓	8	12.11	even	2
576.3.n.a.353.1	✓	8	24.5	odd	2
576.3.n.a.353.4	yes	8	3.2	odd	2
576.3.n.a.353.4	yes	8	24.11	even	2
576.3.n.a.545.1	yes	8	9.4	even	3
576.3.n.a.545.1	yes	8	72.67	odd	6
576.3.n.a.545.4	yes	8	36.31	odd	6
576.3.n.a.545.4	yes	8	72.13	even	6
1728.3.n.a.737.2		8	1.1	even	1	trivial
1728.3.n.a.737.2		8	8.3	odd	2	CM
1728.3.n.a.737.3		8	4.3	odd	2	inner
1728.3.n.a.737.3		8	8.5	even	2	inner
1728.3.n.a.1313.2		8	36.23	even	6	inner
1728.3.n.a.1313.2		8	72.5	odd	6	inner
1728.3.n.a.1313.3		8	9.5	odd	6	inner
1728.3.n.a.1313.3		8	72.59	even	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)

\(f(q)\)	\(=\)	\(q+O(q^{10})\)	\(q+(-1.81954 + 3.15153i) q^{11} +18.7026i q^{17} -2.30306i q^{19} +(12.5000 - 21.6506i) q^{25} +(-63.8939 + 36.8891i) q^{41} +(-57.5774 - 33.2423i) q^{43} +(24.5000 + 42.4352i) q^{49} +(14.2938 + 24.7577i) q^{59} +(-115.942 + 66.9393i) q^{67} +100.394 q^{73} +(25.4558 - 44.0908i) q^{83} +101.823i q^{89} +(-49.9847 + 86.5760i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 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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