LMFDB - Embedded newform 756.3.bk.a.485.1

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 = [2,0,0,0,0,0,2,0,0,0,0,0,-2] 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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			485.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	756.485
Dual form			756.3.bk.a.53.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 + 6.92820i) q^{7} -1.00000 q^{13} +(-13.0000 + 22.5167i) q^{19} +(-12.5000 - 21.6506i) q^{25} +(6.50000 + 11.2583i) q^{31} +(-23.5000 + 40.7032i) q^{37} -61.0000 q^{43} +(-47.0000 + 13.8564i) q^{49} +(-23.5000 + 40.7032i) q^{61} +(54.5000 + 94.3968i) q^{67} +(23.0000 + 39.8372i) q^{73} +(-65.5000 + 113.449i) q^{79} +(-1.00000 - 6.92820i) q^{91} -169.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{7} - 2 q^{13} - 26 q^{19} - 25 q^{25} + 13 q^{31} - 47 q^{37} - 122 q^{43} - 94 q^{49} - 47 q^{61} + 109 q^{67} + 46 q^{73} - 131 q^{79} - 2 q^{91} - 338 q^{97}+O(q^{100}) $ 2 * q + 2 * q^7 - 2 * q^13 - 26 * q^19 - 25 * q^25 + 13 * q^31 - 47 * q^37 - 122 * q^43 - 94 * q^49 - 47 * q^61 + 109 * q^67 + 46 * q^73 - 131 * q^79 - 2 * q^91 - 338 * q^97

Character values

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

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

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	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$	1.00000	+	6.92820i	0.142857	+	0.989743i
$7$	1.00000	+	6.92820i	0.142857	+	0.989743i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$		0			0
$13$	−1.00000			−0.0769231			−0.0384615	−	0.999260i	$-0.512246\pi$
$13$	−1.00000			−0.0769231			−0.0384615	+	0.999260i	$0.512246\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	−13.0000	+	22.5167i	−0.684211	+	1.18509i	0.289474	+	0.957186i	$0.406520\pi$
$19$	−13.0000	+	22.5167i	−0.684211	+	1.18509i	−0.973684	+	0.227901i	$0.926814\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		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	0.951613	−	0.307299i	$-0.0994253\pi$
$31$	6.50000	+	11.2583i	0.209677	+	0.363172i	−0.741935	+	0.670471i	$0.766092\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−23.5000	+	40.7032i	−0.635135	+	1.10009i	0.351351	+	0.936244i	$0.385722\pi$
$37$	−23.5000	+	40.7032i	−0.635135	+	1.10009i	−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	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$		0			0
$49$	−47.0000	+	13.8564i	−0.959184	+	0.282784i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	−23.5000	+	40.7032i	−0.385246	+	0.667265i	−0.991803	−	0.127774i	$-0.959217\pi$
$61$	−23.5000	+	40.7032i	−0.385246	+	0.667265i	0.606557	+	0.795040i	$0.292550\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	54.5000	+	94.3968i	0.813433	+	1.40891i	0.910448	+	0.413624i	$0.135737\pi$
$67$	54.5000	+	94.3968i	0.813433	+	1.40891i	−0.0970149	+	0.995283i	$0.530929\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	23.0000	+	39.8372i	0.315068	+	0.545715i	0.979452	−	0.201677i	$-0.0646392\pi$
$73$	23.0000	+	39.8372i	0.315068	+	0.545715i	−0.664384	+	0.747392i	$0.731306\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−65.5000	+	113.449i	−0.829114	+	1.43607i	0.0696203	+	0.997574i	$0.477821\pi$
$79$	−65.5000	+	113.449i	−0.829114	+	1.43607i	−0.898734	+	0.438494i	$0.855512\pi$
$80$		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	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$	−1.00000	−	6.92820i	−0.0109890	−	0.0761341i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−169.000			−1.74227			−0.871134	−	0.491045i	$-0.836615\pi$
$97$	−169.000			−1.74227			−0.871134	+	0.491045i	$0.836615\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$	18.5000	−	32.0429i	0.179612	−	0.311097i	−0.762136	−	0.647417i	$-0.775849\pi$
$103$	18.5000	−	32.0429i	0.179612	−	0.311097i	0.941748	+	0.336321i	$0.109183\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$	−35.5000	−	61.4878i	−0.325688	−	0.564108i	0.655963	−	0.754793i	$-0.272263\pi$
$109$	−35.5000	−	61.4878i	−0.325688	−	0.564108i	−0.981651	+	0.190684i	$0.938929\pi$
$110$		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	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	−169.000	−	67.5500i	−1.27068	−	0.507895i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$	251.000			1.80576			0.902878	−	0.429898i	$-0.141450\pi$
$139$	251.000			1.80576			0.902878	+	0.429898i	$0.141450\pi$
$140$		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	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	−113.500	−	196.588i	−0.751656	−	1.30191i	−0.947020	−	0.321175i	$-0.895922\pi$
$151$	−113.500	−	196.588i	−0.751656	−	1.30191i	0.195364	−	0.980731i	$-0.437411\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	59.0000	+	102.191i	0.375796	+	0.650898i	0.990446	−	0.137902i	$-0.0440359\pi$
$157$	59.0000	+	102.191i	0.375796	+	0.650898i	−0.614650	+	0.788800i	$0.710703\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	18.5000	−	32.0429i	0.113497	−	0.196582i	−0.803681	−	0.595060i	$-0.797128\pi$
$163$	18.5000	−	32.0429i	0.113497	−	0.196582i	0.917178	+	0.398478i	$0.130461\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−168.000			−0.994083
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$	137.500	−	108.253i	0.785714	−	0.618590i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$	314.000			1.73481			0.867403	−	0.497606i	$-0.165787\pi$
$181$	314.000			1.73481			0.867403	+	0.497606i	$0.165787\pi$
$182$		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	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$		0			0
$193$	−71.5000	−	123.842i	−0.370466	−	0.641666i	0.619171	−	0.785256i	$-0.287469\pi$
$193$	−71.5000	−	123.842i	−0.370466	−	0.641666i	−0.989637	+	0.143590i	$0.954135\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	54.5000	+	94.3968i	0.273869	+	0.474356i	0.969849	−	0.243706i	$-0.0783631\pi$
$199$	54.5000	+	94.3968i	0.273869	+	0.474356i	−0.695980	+	0.718061i	$0.745030\pi$
$200$		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
$211$	419.000			1.98578			0.992891	−	0.119027i	$-0.0379776\pi$
$211$	419.000			1.98578			0.992891	+	0.119027i	$0.0379776\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−71.5000	+	56.2917i	−0.329493	+	0.259409i
$218$		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	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$		0			0
$229$	204.500	−	354.204i	0.893013	−	1.54674i	0.0567686	−	0.998387i	$-0.481920\pi$
$229$	204.500	−	354.204i	0.893013	−	1.54674i	0.836245	−	0.548357i	$-0.184746\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		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
$241$	−239.500	−	414.826i	−0.993776	−	1.72127i	−0.593361	−	0.804936i	$-0.702199\pi$
$241$	−239.500	−	414.826i	−0.993776	−	1.72127i	−0.400415	−	0.916334i	$-0.631134\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	13.0000	−	22.5167i	0.0526316	−	0.0911606i
$248$		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	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$	−305.500	−	122.110i	−1.17954	−	0.471466i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$	270.500	−	468.520i	0.998155	−	1.72886i	0.446494	−	0.894786i	$-0.352672\pi$
$271$	270.500	−	468.520i	0.998155	−	1.72886i	0.551661	−	0.834069i	$-0.313994\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−203.500	−	352.472i	−0.734657	−	1.27246i	−0.954874	−	0.297012i	$-0.904010\pi$
$277$	−203.500	−	352.472i	−0.734657	−	1.27246i	0.220217	−	0.975451i	$-0.429324\pi$
$278$		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	0.809187	+	0.587551i	$0.199908\pi$
$283$	258.500	+	447.735i	0.913428	+	1.58210i	0.104240	+	0.994552i	$0.466759\pi$
$284$		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
$301$	−61.0000	−	422.620i	−0.202658	−	1.40405i
$302$		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	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	71.0000	−	122.976i	0.226837	−	0.392893i	−0.730032	−	0.683413i	$-0.760495\pi$
$313$	71.0000	−	122.976i	0.226837	−	0.392893i	0.956869	+	0.290520i	$0.0938282\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	12.5000	+	21.6506i	0.0384615	+	0.0666173i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−181.000	+	313.501i	−0.546828	+	0.947134i	0.451662	+	0.892189i	$0.350831\pi$
$331$	−181.000	+	313.501i	−0.546828	+	0.947134i	−0.998489	+	0.0549442i	$0.982502\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	482.000			1.43027			0.715134	−	0.698988i	$-0.246366\pi$
$337$	482.000			1.43027			0.715134	+	0.698988i	$0.246366\pi$
$338$		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	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	−169.000			−0.484241			−0.242120	−	0.970246i	$-0.577843\pi$
$349$	−169.000			−0.484241			−0.242120	+	0.970246i	$0.577843\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−157.500	−	272.798i	−0.436288	−	0.755673i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	359.000	+	621.806i	0.978202	+	1.69429i	0.668937	+	0.743319i	$0.266749\pi$
$367$	359.000	+	621.806i	0.978202	+	1.69429i	0.309264	+	0.950976i	$0.399917\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−349.000	+	604.486i	−0.935657	+	1.62061i	−0.162198	+	0.986758i	$0.551858\pi$
$373$	−349.000	+	604.486i	−0.935657	+	1.62061i	−0.773458	+	0.633847i	$0.781475\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	83.0000			0.218997			0.109499	−	0.993987i	$-0.465075\pi$
$379$	83.0000			0.218997			0.109499	+	0.993987i	$0.465075\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		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	0.455919	+	0.890021i	$0.349311\pi$
$397$	−215.500	+	373.257i	−0.542821	+	0.940194i	−0.998741	+	0.0501728i	$0.984023\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	−6.50000	−	11.2583i	−0.0161290	−	0.0279363i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−71.5000	−	123.842i	−0.174817	−	0.302791i	0.765281	−	0.643696i	$-0.222600\pi$
$409$	−71.5000	−	123.842i	−0.174817	−	0.302791i	−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
$421$	−358.000			−0.850356			−0.425178	−	0.905110i	$-0.639789\pi$
$421$	−358.000			−0.850356			−0.425178	+	0.905110i	$0.639789\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−305.500	−	122.110i	−0.715457	−	0.285971i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$		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	−0.807517	−	0.589844i	$-0.799189\pi$
$439$	47.0000	−	81.4064i	0.107062	−	0.185436i	0.914579	+	0.404408i	$0.132522\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		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	0.0514223	+	0.998677i	$0.483625\pi$
$457$	−383.500	+	664.241i	−0.839168	+	1.45348i	−0.890591	+	0.454805i	$0.849709\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−526.000			−1.13607			−0.568035	−	0.823005i	$-0.692296\pi$
$463$	−526.000			−1.13607			−0.568035	+	0.823005i	$0.692296\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$		0			0
$469$	−599.500	+	471.984i	−1.27825	+	1.00636i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	650.000			1.36842
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$	23.5000	−	40.7032i	0.0488565	−	0.0846220i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−481.000	−	833.116i	−0.987680	−	1.71071i	−0.629363	−	0.777111i	$-0.716684\pi$
$487$	−481.000	−	833.116i	−0.987680	−	1.71071i	−0.358316	−	0.933600i	$-0.616649\pi$
$488$		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	0.0260521	+	0.999661i	$0.491706\pi$
$499$	−425.500	+	736.988i	−0.852705	+	1.47693i	−0.878758	+	0.477269i	$0.841627\pi$
$500$		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	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	−253.000	+	199.186i	−0.495108	+	0.389796i
$512$		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	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	−401.500	+	695.418i	−0.767686	+	1.32967i	0.171128	+	0.985249i	$0.445259\pi$
$523$	−401.500	+	695.418i	−0.767686	+	1.32967i	−0.938815	+	0.344423i	$0.888075\pi$
$524$		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
$541$	−517.000	+	895.470i	−0.955638	+	1.65521i	−0.222736	+	0.974879i	$0.571499\pi$
$541$	−517.000	+	895.470i	−0.955638	+	1.65521i	−0.732902	+	0.680334i	$0.761835\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	587.000			1.07313			0.536563	−	0.843860i	$-0.319722\pi$
$547$	587.000			1.07313			0.536563	+	0.843860i	$0.319722\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−851.500	−	340.348i	−1.53978	−	0.615457i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$	61.0000			0.109123
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	443.000	+	767.299i	0.775832	+	1.34378i	0.934326	+	0.356420i	$0.116003\pi$
$571$	443.000	+	767.299i	0.775832	+	1.34378i	−0.158494	+	0.987360i	$0.550664\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	516.500	+	894.604i	0.895147	+	1.55044i	0.833622	+	0.552335i	$0.186263\pi$
$577$	516.500	+	894.604i	0.895147	+	1.55044i	0.0615251	+	0.998106i	$0.480404\pi$
$578$		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
$589$	−338.000			−0.573854
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$		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
$601$	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
$607$	407.000	−	704.945i	0.670511	−	1.16136i	−0.307249	−	0.951629i	$-0.599408\pi$
$607$	407.000	−	704.945i	0.670511	−	1.16136i	0.977759	−	0.209729i	$-0.0672583\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−71.5000	−	123.842i	−0.116639	−	0.202026i	0.801794	−	0.597600i	$-0.203879\pi$
$613$	−71.5000	−	123.842i	−0.116639	−	0.202026i	−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
$619$	474.500	+	821.858i	0.766559	+	1.32772i	0.939418	+	0.342773i	$0.111366\pi$
$619$	474.500	+	821.858i	0.766559	+	1.32772i	−0.172859	+	0.984947i	$0.555301\pi$
$620$		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
$631$	−1261.00			−1.99842			−0.999208	−	0.0398015i	$-0.987327\pi$
$631$	−1261.00			−1.99842			−0.999208	+	0.0398015i	$0.987327\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	47.0000	−	13.8564i	0.0737834	−	0.0217526i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		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	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		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
$661$	−61.0000	−	105.655i	−0.0922844	−	0.159841i	0.816188	−	0.577787i	$-0.196084\pi$
$661$	−61.0000	−	105.655i	−0.0922844	−	0.159841i	−0.908472	+	0.417946i	$0.862750\pi$
$662$		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
$673$	1154.00			1.71471			0.857355	−	0.514725i	$-0.172106\pi$
$673$	1154.00			1.71471			0.857355	+	0.514725i	$0.172106\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	−169.000	−	1170.87i	−0.248895	−	1.72440i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$		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	0.216353	+	0.976315i	$0.430584\pi$
$691$	−509.500	+	882.480i	−0.737337	+	1.27711i	−0.953690	+	0.300790i	$0.902750\pi$
$692$		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
$703$	−611.000	−	1058.28i	−0.869132	−	1.50538i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	228.500	−	395.774i	0.322285	−	0.558214i	−0.658674	−	0.752428i	$-0.728882\pi$
$709$	228.500	−	395.774i	0.322285	−	0.558214i	0.980959	+	0.194214i	$0.0622158\pi$
$710$		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	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$	240.500	+	96.1288i	0.333564	+	0.133327i
$722$		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	0.261255	−	0.965270i	$-0.415864\pi$
$733$	708.500	−	1227.16i	0.966576	−	1.67416i	0.705321	−	0.708888i	$-0.250803\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−665.500	−	1152.68i	−0.900541	−	1.55978i	−0.826793	−	0.562506i	$-0.809837\pi$
$739$	−665.500	−	1152.68i	−0.900541	−	1.55978i	−0.0737483	−	0.997277i	$-0.523496\pi$
$740$		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
$751$	−601.000	+	1040.96i	−0.800266	+	1.38610i	0.119174	+	0.992873i	$0.461975\pi$
$751$	−601.000	+	1040.96i	−0.800266	+	1.38610i	−0.919441	+	0.393229i	$0.871358\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1511.00			1.99604			0.998018	−	0.0629213i	$-0.0200417\pi$
$757$	1511.00			1.99604			0.998018	+	0.0629213i	$0.0200417\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	390.500	−	307.439i	0.511796	−	0.402934i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1534.00			−1.99480			−0.997399	−	0.0720749i	$-0.977038\pi$
$769$	−1534.00			−1.99480			−0.997399	+	0.0720749i	$0.977038\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		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
$787$	474.500	+	821.858i	0.602922	+	1.04429i	0.992376	+	0.123246i	$0.0393305\pi$
$787$	474.500	+	821.858i	0.602922	+	1.04429i	−0.389454	+	0.921046i	$0.627336\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	23.5000	−	40.7032i	0.0296343	−	0.0513281i
$794$		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	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		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
$817$	793.000	−	1373.52i	0.970624	−	1.68117i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	−281.500	−	487.572i	−0.342041	−	0.592433i	0.642770	−	0.766059i	$-0.277785\pi$
$823$	−281.500	−	487.572i	−0.342041	−	0.592433i	−0.984812	+	0.173626i	$0.944452\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−229.000	−	396.640i	−0.276236	−	0.478456i	0.694210	−	0.719773i	$-0.255754\pi$
$829$	−229.000	−	396.640i	−0.276236	−	0.478456i	−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
$847$	−786.500	−	314.367i	−0.928571	−	0.371154i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1658.00			1.94373			0.971864	−	0.235543i	$-0.0756867\pi$
$853$	1658.00			1.94373			0.971864	+	0.235543i	$0.0756867\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	−65.5000	+	113.449i	−0.0762515	+	0.132071i	−0.901630	−	0.432509i	$-0.857629\pi$
$859$	−65.5000	+	113.449i	−0.0762515	+	0.132071i	0.825378	+	0.564580i	$0.190962\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−54.5000	−	94.3968i	−0.0625718	−	0.108377i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	564.500	−	977.743i	0.643672	−	1.11487i	−0.340935	−	0.940087i	$-0.610744\pi$
$877$	564.500	−	977.743i	0.643672	−	1.11487i	0.984607	−	0.174785i	$-0.0559231\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−1702.00			−1.92752			−0.963760	−	0.266771i	$-0.914043\pi$
$883$	−1702.00			−1.92752			−0.963760	+	0.266771i	$0.914043\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	107.000	+	741.318i	0.120360	+	0.833878i
$890$		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
$907$	726.500	+	1258.33i	0.800992	+	1.38736i	0.918964	+	0.394342i	$0.129028\pi$
$907$	726.500	+	1258.33i	0.800992	+	1.38736i	−0.117971	+	0.993017i	$0.537639\pi$
$908$		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	0.471164	−	0.882045i	$-0.343834\pi$
$919$	918.500	−	1590.89i	0.999456	−	1.73111i	0.528292	−	0.849063i	$-0.322833\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1175.00			1.27027
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$	299.000	−	1238.42i	0.321160	−	1.33020i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1847.00			1.97118			0.985592	−	0.169138i	$-0.0540985\pi$
$937$	1847.00			1.97118			0.985592	+	0.169138i	$0.0540985\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$		0			0
$949$	−23.0000	−	39.8372i	−0.0242360	−	0.0419780i
$950$		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
$967$	−253.000			−0.261634			−0.130817	−	0.991407i	$-0.541760\pi$
$967$	−253.000			−0.261634			−0.130817	+	0.991407i	$0.541760\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$		0			0
$973$	251.000	+	1738.98i	0.257965	+	1.78723i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−869.500	−	1506.02i	−0.877397	−	1.51970i	−0.854188	−	0.519965i	$-0.825945\pi$
$991$	−869.500	−	1506.02i	−0.877397	−	1.51970i	−0.0232089	−	0.999731i	$-0.507388\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−203.500	−	352.472i	−0.204112	−	0.353533i	0.745737	−	0.666240i	$-0.232097\pi$
$997$	−203.500	−	352.472i	−0.204112	−	0.353533i	−0.949850	+	0.312707i	$0.898764\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.3.bk.a.53.1	✓	2	7.4	even	3	inner
756.3.bk.a.53.1	✓	2	21.11	odd	6	inner
756.3.bk.a.485.1	yes	2	1.1	even	1	trivial
756.3.bk.a.485.1	yes	2	3.2	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(1.00000 + 6.92820i) q^{7} -1.00000 q^{13} +(-13.0000 + 22.5167i) q^{19} +(-12.5000 - 21.6506i) q^{25} +(6.50000 + 11.2583i) q^{31} +(-23.5000 + 40.7032i) q^{37} -61.0000 q^{43} +(-47.0000 + 13.8564i) q^{49} +(-23.5000 + 40.7032i) q^{61} +(54.5000 + 94.3968i) q^{67} +(23.0000 + 39.8372i) q^{73} +(-65.5000 + 113.449i) q^{79} +(-1.00000 - 6.92820i) q^{91} -169.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{7} - 2 q^{13} - 26 q^{19} - 25 q^{25} + 13 q^{31} - 47 q^{37} - 122 q^{43} - 94 q^{49} - 47 q^{61} + 109 q^{67} + 46 q^{73} - 131 q^{79} - 2 q^{91} - 338 q^{97}+O(q^{100}) \) 2 * q + 2 * q^7 - 2 * q^13 - 26 * q^19 - 25 * q^25 + 13 * q^31 - 47 * q^37 - 122 * q^43 - 94 * q^49 - 47 * q^61 + 109 * q^67 + 46 * q^73 - 131 * q^79 - 2 * q^91 - 338 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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