LMFDB - Embedded newform 756.3.bk.c.53.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,3,Mod(53,756)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("756.53"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	756.bk (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,11,0,0,0,0,0,-44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$20.5995079856$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			53.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	756.53
Dual form			756.3.bk.c.485.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.50000 + 4.33013i) q^{7} -22.0000 q^{13} +(18.5000 + 32.0429i) q^{19} +(-12.5000 + 21.6506i) q^{25} +(6.50000 - 11.2583i) q^{31} +(-13.0000 - 22.5167i) q^{37} -61.0000 q^{43} +(11.5000 + 47.6314i) q^{49} +(60.5000 + 104.789i) q^{61} +(-61.0000 + 105.655i) q^{67} +(-71.5000 + 123.842i) q^{73} +(71.0000 + 122.976i) q^{79} +(-121.000 - 95.2628i) q^{91} +167.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 11 q^{7} - 44 q^{13} + 37 q^{19} - 25 q^{25} + 13 q^{31} - 26 q^{37} - 122 q^{43} + 23 q^{49} + 121 q^{61} - 122 q^{67} - 143 q^{73} + 142 q^{79} - 242 q^{91} + 334 q^{97}+O(q^{100}) $ 2 * q + 11 * q^7 - 44 * q^13 + 37 * q^19 - 25 * q^25 + 13 * q^31 - 26 * q^37 - 122 * q^43 + 23 * q^49 + 121 * q^61 - 122 * q^67 - 143 * q^73 + 142 * q^79 - 242 * q^91 + 334 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$	5.50000	+	4.33013i	0.785714	+	0.618590i
$7$	5.50000	+	4.33013i	0.785714	+	0.618590i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$		0			0
$13$	−22.0000			−1.69231			−0.846154	−	0.532939i	$-0.821088\pi$
$13$	−22.0000			−1.69231			−0.846154	+	0.532939i	$0.821088\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$	18.5000	+	32.0429i	0.973684	+	1.68647i	0.684211	+	0.729285i	$0.260147\pi$
$19$	18.5000	+	32.0429i	0.973684	+	1.68647i	0.289474	+	0.957186i	$0.406520\pi$
$20$		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		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	6.50000	−	11.2583i	0.209677	−	0.363172i	−0.741935	−	0.670471i	$-0.766092\pi$
$31$	6.50000	−	11.2583i	0.209677	−	0.363172i	0.951613	+	0.307299i	$0.0994253\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−13.0000	−	22.5167i	−0.351351	−	0.608558i	0.635135	−	0.772401i	$-0.280944\pi$
$37$	−13.0000	−	22.5167i	−0.351351	−	0.608558i	−0.986486	+	0.163843i	$0.947611\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−61.0000			−1.41860			−0.709302	−	0.704904i	$-0.750990\pi$
$43$	−61.0000			−1.41860			−0.709302	+	0.704904i	$0.750990\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$		0			0
$49$	11.5000	+	47.6314i	0.234694	+	0.972069i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	60.5000	+	104.789i	0.991803	+	1.71785i	0.606557	+	0.795040i	$0.292550\pi$
$61$	60.5000	+	104.789i	0.991803	+	1.71785i	0.385246	+	0.922814i	$0.374117\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−61.0000	+	105.655i	−0.910448	+	1.57694i	−0.0970149	+	0.995283i	$0.530929\pi$
$67$	−61.0000	+	105.655i	−0.910448	+	1.57694i	−0.813433	+	0.581659i	$0.802404\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−71.5000	+	123.842i	−0.979452	+	1.69646i	−0.315068	+	0.949069i	$0.602027\pi$
$73$	−71.5000	+	123.842i	−0.979452	+	1.69646i	−0.664384	+	0.747392i	$0.731306\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	71.0000	+	122.976i	0.898734	+	1.55665i	0.829114	+	0.559080i	$0.188845\pi$
$79$	71.0000	+	122.976i	0.898734	+	1.55665i	0.0696203	+	0.997574i	$0.477821\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$		0			0
$91$	−121.000	−	95.2628i	−1.32967	−	1.04684i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	167.000			1.72165			0.860825	−	0.508902i	$-0.169948\pi$
$97$	167.000			1.72165			0.860825	+	0.508902i	$0.169948\pi$
$98$		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
$103$	−97.0000	−	168.009i	−0.941748	−	1.63115i	−0.762136	−	0.647417i	$-0.775849\pi$
$103$	−97.0000	−	168.009i	−0.941748	−	1.63115i	−0.179612	−	0.983738i	$-0.557484\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$		0			0
$109$	−35.5000	+	61.4878i	−0.325688	+	0.564108i	−0.981651	−	0.190684i	$-0.938929\pi$
$109$	−35.5000	+	61.4878i	−0.325688	+	0.564108i	0.655963	+	0.754793i	$0.272263\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−60.5000	−	104.789i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	107.000			0.842520			0.421260	−	0.906940i	$-0.361588\pi$
$127$	107.000			0.842520			0.421260	+	0.906940i	$0.361588\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$	−37.0000	+	256.344i	−0.278195	+	1.92739i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$	−22.0000			−0.158273			−0.0791367	−	0.996864i	$-0.525216\pi$
$139$	−22.0000			−0.158273			−0.0791367	+	0.996864i	$0.525216\pi$
$140$		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
$151$	−29.5000	+	51.0955i	−0.195364	+	0.338381i	−0.947020	−	0.321175i	$-0.895922\pi$
$151$	−29.5000	+	51.0955i	−0.195364	+	0.338381i	0.751656	+	0.659556i	$0.229256\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	59.0000	−	102.191i	0.375796	−	0.650898i	−0.614650	−	0.788800i	$-0.710703\pi$
$157$	59.0000	−	102.191i	0.375796	−	0.650898i	0.990446	+	0.137902i	$0.0440359\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−149.500	−	258.942i	−0.917178	−	1.58860i	−0.803681	−	0.595060i	$-0.797128\pi$
$163$	−149.500	−	258.942i	−0.917178	−	1.58860i	−0.113497	−	0.993538i	$-0.536205\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	315.000			1.86391
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	−162.500	+	64.9519i	−0.928571	+	0.371154i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$	−1.00000			−0.00552486			−0.00276243	−	0.999996i	$-0.500879\pi$
$181$	−1.00000			−0.00552486			−0.00276243	+	0.999996i	$0.500879\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$		0			0
$193$	191.000	−	330.822i	0.989637	−	1.71410i	0.370466	−	0.928846i	$-0.379198\pi$
$193$	191.000	−	330.822i	0.989637	−	1.71410i	0.619171	−	0.785256i	$-0.287469\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	138.500	−	239.889i	0.695980	−	1.20547i	−0.273869	−	0.961767i	$-0.588304\pi$
$199$	138.500	−	239.889i	0.695980	−	1.20547i	0.969849	−	0.243706i	$-0.0783631\pi$
$200$		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
$209$		0			0
$210$		0			0
$211$	−253.000			−1.19905			−0.599526	−	0.800355i	$-0.704644\pi$
$211$	−253.000			−1.19905			−0.599526	+	0.800355i	$0.704644\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	84.5000	−	33.7750i	0.389401	−	0.155645i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	338.000			1.51570			0.757848	−	0.652432i	$-0.226251\pi$
$223$	338.000			1.51570			0.757848	+	0.652432i	$0.226251\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$		0			0
$229$	204.500	+	354.204i	0.893013	+	1.54674i	0.836245	+	0.548357i	$0.184746\pi$
$229$	204.500	+	354.204i	0.893013	+	1.54674i	0.0567686	+	0.998387i	$0.481920\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	96.5000	−	167.143i	0.400415	−	0.693539i	−0.593361	−	0.804936i	$-0.702199\pi$
$241$	96.5000	−	167.143i	0.400415	−	0.693539i	0.993776	+	0.111397i	$0.0355327\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−407.000	−	704.945i	−1.64777	−	2.85403i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$	26.0000	−	180.133i	0.100386	−	0.695495i
$260$		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		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	−149.500	−	258.942i	−0.551661	−	0.955504i	−0.998155	−	0.0607176i	$-0.980661\pi$
$271$	−149.500	−	258.942i	−0.551661	−	0.955504i	0.446494	−	0.894786i	$-0.352672\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−203.500	+	352.472i	−0.734657	+	1.27246i	0.220217	+	0.975451i	$0.429324\pi$
$277$	−203.500	+	352.472i	−0.734657	+	1.27246i	−0.954874	+	0.297012i	$0.904010\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	258.500	−	447.735i	0.913428	−	1.58210i	0.104240	−	0.994552i	$-0.466759\pi$
$283$	258.500	−	447.735i	0.913428	−	1.58210i	0.809187	−	0.587551i	$-0.199908\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−144.500	−	250.281i	−0.500000	−	0.866025i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−335.500	−	264.138i	−1.11462	−	0.877534i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	611.000			1.99023			0.995114	−	0.0987325i	$-0.0314788\pi$
$307$	611.000			1.99023			0.995114	+	0.0987325i	$0.0314788\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	228.500	+	395.774i	0.730032	+	1.26445i	0.956869	+	0.290520i	$0.0938282\pi$
$313$	228.500	+	395.774i	0.730032	+	1.26445i	−0.226837	+	0.973933i	$0.572838\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	275.000	−	476.314i	0.846154	−	1.46558i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−149.500	−	258.942i	−0.451662	−	0.782301i	0.546828	−	0.837245i	$-0.315835\pi$
$331$	−149.500	−	258.942i	−0.451662	−	0.782301i	−0.998489	+	0.0549442i	$0.982502\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	167.000			0.495549			0.247774	−	0.968818i	$-0.420301\pi$
$337$	167.000			0.495549			0.247774	+	0.968818i	$0.420301\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−143.000	+	311.769i	−0.416910	+	0.908948i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	671.000			1.92264			0.961318	−	0.275441i	$-0.0888238\pi$
$349$	671.000			1.92264			0.961318	+	0.275441i	$0.0888238\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	−504.000	+	872.954i	−1.39612	+	2.41815i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−113.500	+	196.588i	−0.309264	+	0.535661i	−0.978202	−	0.207657i	$-0.933416\pi$
$367$	−113.500	+	196.588i	−0.309264	+	0.535661i	0.668937	+	0.743319i	$0.266749\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	60.5000	+	104.789i	0.162198	+	0.280936i	0.935657	−	0.352911i	$-0.114808\pi$
$373$	60.5000	+	104.789i	0.162198	+	0.280936i	−0.773458	+	0.633847i	$0.781475\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−694.000			−1.83113			−0.915567	−	0.402165i	$-0.868258\pi$
$379$	−694.000			−1.83113			−0.915567	+	0.402165i	$0.868258\pi$
$380$		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		0.500000	−	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
$397$	−215.500	−	373.257i	−0.542821	−	0.940194i	−0.998741	−	0.0501728i	$-0.984023\pi$
$397$	−215.500	−	373.257i	−0.542821	−	0.940194i	0.455919	−	0.890021i	$-0.349311\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	−143.000	+	247.683i	−0.354839	+	0.614599i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−313.000	+	542.132i	−0.765281	+	1.32551i	0.174817	+	0.984601i	$0.444067\pi$
$409$	−313.000	+	542.132i	−0.765281	+	1.32551i	−0.940098	+	0.340905i	$0.889267\pi$
$410$		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
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	839.000			1.99287			0.996437	−	0.0843398i	$-0.0268781\pi$
$421$	839.000			1.99287			0.996437	+	0.0843398i	$0.0268781\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−121.000	+	838.313i	−0.283372	+	1.96326i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$		0			0
$433$	359.000			0.829099			0.414550	−	0.910027i	$-0.363939\pi$
$433$	359.000			0.829099			0.414550	+	0.910027i	$0.363939\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	47.0000	+	81.4064i	0.107062	+	0.185436i	0.914579	−	0.404408i	$-0.132522\pi$
$439$	47.0000	+	81.4064i	0.107062	+	0.185436i	−0.807517	+	0.589844i	$0.799189\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−383.500	−	664.241i	−0.839168	−	1.45348i	−0.890591	−	0.454805i	$-0.849709\pi$
$457$	−383.500	−	664.241i	−0.839168	−	1.45348i	0.0514223	−	0.998677i	$-0.483625\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	923.000			1.99352			0.996760	−	0.0804300i	$-0.0256293\pi$
$463$	923.000			1.99352			0.996760	+	0.0804300i	$0.0256293\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$		0			0
$469$	−793.000	+	316.965i	−1.69083	+	0.675832i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−925.000			−1.94737
$476$		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
$481$	286.000	+	495.367i	0.594595	+	1.02987i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	306.500	−	530.874i	0.629363	−	1.09009i	−0.358316	−	0.933600i	$-0.616649\pi$
$487$	306.500	−	530.874i	0.629363	−	1.09009i	0.987680	−	0.156489i	$-0.0500176\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−425.500	−	736.988i	−0.852705	−	1.47693i	−0.878758	−	0.477269i	$-0.841627\pi$
$499$	−425.500	−	736.988i	−0.852705	−	1.47693i	0.0260521	−	0.999661i	$-0.491706\pi$
$500$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$	−929.500	+	371.525i	−1.81898	+	0.727055i
$512$		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
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	491.000	+	850.437i	0.938815	+	1.62607i	0.767686	+	0.640826i	$0.221408\pi$
$523$	491.000	+	850.437i	0.938815	+	1.62607i	0.171128	+	0.985249i	$0.445259\pi$
$524$		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
$539$		0			0
$540$		0			0
$541$	396.500	+	686.758i	0.732902	+	1.26942i	0.955638	+	0.294545i	$0.0951680\pi$
$541$	396.500	+	686.758i	0.732902	+	1.26942i	−0.222736	+	0.974879i	$0.571499\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1093.00			−1.99817			−0.999086	−	0.0427471i	$-0.986389\pi$
$547$	−1093.00			−1.99817			−0.999086	+	0.0427471i	$0.986389\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−142.000	+	983.805i	−0.256781	+	1.77903i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$	1342.00			2.40072
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	−533.500	+	924.049i	−0.934326	+	1.61830i	−0.158494	+	0.987360i	$0.550664\pi$
$571$	−533.500	+	924.049i	−0.934326	+	1.61830i	−0.775832	+	0.630940i	$0.782670\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−481.000	+	833.116i	−0.833622	+	1.44388i	0.0615251	+	0.998106i	$0.480404\pi$
$577$	−481.000	+	833.116i	−0.833622	+	1.44388i	−0.895147	+	0.445770i	$0.852930\pi$
$578$		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
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$		0			0
$589$	481.000			0.816638
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$	1199.00			1.99501			0.997504	−	0.0706077i	$-0.0224939\pi$
$601$	1199.00			1.99501			0.997504	+	0.0706077i	$0.0224939\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	186.500	+	323.027i	0.307249	+	0.532170i	0.977759	−	0.209729i	$-0.0672583\pi$
$607$	186.500	+	323.027i	0.307249	+	0.532170i	−0.670511	+	0.741900i	$0.733925\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−491.500	+	851.303i	−0.801794	+	1.38875i	0.116639	+	0.993174i	$0.462788\pi$
$613$	−491.500	+	851.303i	−0.801794	+	1.38875i	−0.918434	+	0.395574i	$0.870546\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	107.000	−	185.329i	0.172859	−	0.299401i	−0.766559	−	0.642174i	$-0.778033\pi$
$619$	107.000	−	185.329i	0.172859	−	0.299401i	0.939418	+	0.342773i	$0.111366\pi$
$620$		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
$631$	587.000			0.930269			0.465135	−	0.885240i	$-0.346006\pi$
$631$	587.000			0.930269			0.465135	+	0.885240i	$0.346006\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−253.000	−	1047.89i	−0.397174	−	1.64504i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	−1237.00			−1.92379			−0.961897	−	0.273411i	$-0.911848\pi$
$643$	−1237.00			−1.92379			−0.961897	+	0.273411i	$0.911848\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$		0			0
$649$		0			0
$650$		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
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	600.500	−	1040.10i	0.908472	−	1.57352i	0.0922844	−	0.995733i	$-0.470583\pi$
$661$	600.500	−	1040.10i	0.908472	−	1.57352i	0.816188	−	0.577787i	$-0.196084\pi$
$662$		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
$673$	−1177.00			−1.74889			−0.874443	−	0.485129i	$-0.838773\pi$
$673$	−1177.00			−1.74889			−0.874443	+	0.485129i	$0.838773\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	918.500	+	723.131i	1.35272	+	1.06499i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−509.500	−	882.480i	−0.737337	−	1.27711i	−0.953690	−	0.300790i	$-0.902750\pi$
$691$	−509.500	−	882.480i	−0.737337	−	1.27711i	0.216353	−	0.976315i	$-0.430584\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	481.000	−	833.116i	0.684211	−	1.18509i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−695.500	−	1204.64i	−0.980959	−	1.69907i	−0.658674	−	0.752428i	$-0.728882\pi$
$709$	−695.500	−	1204.64i	−0.980959	−	1.69907i	−0.322285	−	0.946643i	$-0.604451\pi$
$710$		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		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$	194.000	−	1344.07i	0.269071	−	1.86418i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	947.000			1.30261			0.651307	−	0.758815i	$-0.274221\pi$
$727$	947.000			1.30261			0.651307	+	0.758815i	$0.274221\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	708.500	+	1227.16i	0.966576	+	1.67416i	0.705321	+	0.708888i	$0.250803\pi$
$733$	708.500	+	1227.16i	0.966576	+	1.67416i	0.261255	+	0.965270i	$0.415864\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−665.500	+	1152.68i	−0.900541	+	1.55978i	−0.0737483	+	0.997277i	$0.523496\pi$
$739$	−665.500	+	1152.68i	−0.900541	+	1.55978i	−0.826793	+	0.562506i	$0.809837\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	690.500	+	1195.98i	0.919441	+	1.59252i	0.800266	+	0.599645i	$0.204691\pi$
$751$	690.500	+	1195.98i	0.919441	+	1.59252i	0.119174	+	0.992873i	$0.461975\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−673.000			−0.889036			−0.444518	−	0.895770i	$-0.646625\pi$
$757$	−673.000			−0.889036			−0.444518	+	0.895770i	$0.646625\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$	−461.500	+	184.463i	−0.604849	+	0.241761i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	671.000			0.872562			0.436281	−	0.899811i	$-0.356295\pi$
$769$	671.000			0.872562			0.436281	+	0.899811i	$0.356295\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$	162.500	+	281.458i	0.209677	+	0.363172i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	306.500	−	530.874i	0.389454	−	0.674553i	−0.602922	−	0.797800i	$-0.705997\pi$
$787$	306.500	−	530.874i	0.389454	−	0.674553i	0.992376	+	0.123246i	$0.0393305\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1331.00	−	2305.36i	−1.67844	−	2.90714i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	1514.00			1.86683			0.933416	−	0.358797i	$-0.116813\pi$
$811$	1514.00			1.86683			0.933416	+	0.358797i	$0.116813\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1128.50	−	1954.62i	−1.38127	−	2.39243i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	810.500	−	1403.83i	0.984812	−	1.70574i	0.342041	−	0.939685i	$-0.388882\pi$
$823$	810.500	−	1403.83i	0.984812	−	1.70574i	0.642770	−	0.766059i	$-0.277785\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−575.500	+	996.795i	−0.694210	+	1.20241i	0.276236	+	0.961090i	$0.410913\pi$
$829$	−575.500	+	996.795i	−0.694210	+	1.20241i	−0.970446	+	0.241317i	$0.922421\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	841.000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	121.000	−	838.313i	0.142857	−	0.989743i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1177.00			−1.37984			−0.689918	−	0.723888i	$-0.742353\pi$
$853$	−1177.00			−1.37984			−0.689918	+	0.723888i	$0.742353\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$	774.500	+	1341.47i	0.901630	+	1.56167i	0.825378	+	0.564580i	$0.190962\pi$
$859$	774.500	+	1341.47i	0.901630	+	1.56167i	0.0762515	+	0.997089i	$0.475705\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	1342.00	−	2324.41i	1.54076	−	2.66867i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−863.500	−	1495.63i	−0.984607	−	1.70539i	−0.643672	−	0.765302i	$-0.722590\pi$
$877$	−863.500	−	1495.63i	−0.984607	−	1.70539i	−0.340935	−	0.940087i	$-0.610744\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	1259.00			1.42582			0.712911	−	0.701255i	$-0.247377\pi$
$883$	1259.00			1.42582			0.712911	+	0.701255i	$0.247377\pi$
$884$		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
$889$	588.500	+	463.324i	0.661980	+	0.521174i
$890$		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
$907$	107.000	−	185.329i	0.117971	−	0.204332i	−0.800992	−	0.598675i	$-0.795694\pi$
$907$	107.000	−	185.329i	0.117971	−	0.204332i	0.918964	+	0.394342i	$0.129028\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	918.500	+	1590.89i	0.999456	+	1.73111i	0.528292	+	0.849063i	$0.322833\pi$
$919$	918.500	+	1590.89i	0.999456	+	1.73111i	0.471164	+	0.882045i	$0.343834\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	650.000			0.702703
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$	−1313.50	+	1249.67i	−1.41085	+	1.34229i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1198.00			−1.27855			−0.639274	−	0.768979i	$-0.720765\pi$
$937$	−1198.00			−1.27855			−0.639274	+	0.768979i	$0.720765\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$		0			0
$949$	1573.00	−	2724.52i	1.65753	−	2.87093i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	396.000	+	685.892i	0.412071	+	0.713727i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−1534.00			−1.58635			−0.793175	−	0.608994i	$-0.791573\pi$
$967$	−1534.00			−1.58635			−0.793175	+	0.608994i	$0.791573\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$		0			0
$973$	−121.000	−	95.2628i	−0.124358	−	0.0979063i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		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
$991$	23.0000	−	39.8372i	0.0232089	−	0.0401990i	−0.854188	−	0.519965i	$-0.825945\pi$
$991$	23.0000	−	39.8372i	0.0232089	−	0.0401990i	0.877397	+	0.479766i	$0.159278\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−203.500	+	352.472i	−0.204112	+	0.353533i	−0.949850	−	0.312707i	$-0.898764\pi$
$997$	−203.500	+	352.472i	−0.204112	+	0.353533i	0.745737	+	0.666240i	$0.232097\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.3.bk.c.53.1	✓	2
3.2	odd	2	CM	756.3.bk.c.53.1	✓	2
7.2	even	3	inner	756.3.bk.c.485.1	yes	2
21.2	odd	6	inner	756.3.bk.c.485.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.3.bk.c.53.1	✓	2	1.1	even	1	trivial
756.3.bk.c.53.1	✓	2	3.2	odd	2	CM
756.3.bk.c.485.1	yes	2	7.2	even	3	inner
756.3.bk.c.485.1	yes	2	21.2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	756.bk (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(5.50000 + 4.33013i) q^{7} -22.0000 q^{13} +(18.5000 + 32.0429i) q^{19} +(-12.5000 + 21.6506i) q^{25} +(6.50000 - 11.2583i) q^{31} +(-13.0000 - 22.5167i) q^{37} -61.0000 q^{43} +(11.5000 + 47.6314i) q^{49} +(60.5000 + 104.789i) q^{61} +(-61.0000 + 105.655i) q^{67} +(-71.5000 + 123.842i) q^{73} +(71.0000 + 122.976i) q^{79} +(-121.000 - 95.2628i) q^{91} +167.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 11 q^{7} - 44 q^{13} + 37 q^{19} - 25 q^{25} + 13 q^{31} - 26 q^{37} - 122 q^{43} + 23 q^{49} + 121 q^{61} - 122 q^{67} - 143 q^{73} + 142 q^{79} - 242 q^{91} + 334 q^{97}+O(q^{100}) \) 2 * q + 11 * q^7 - 44 * q^13 + 37 * q^19 - 25 * q^25 + 13 * q^31 - 26 * q^37 - 122 * q^43 + 23 * q^49 + 121 * q^61 - 122 * q^67 - 143 * q^73 + 142 * q^79 - 242 * q^91 + 334 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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