Embedded newform 432.2.bj.a.131.10

• Label: 432.2.bj.a.131.10
• Level: $432$
• Weight: $2$
• Character: 432.131
• Analytic conductor: $3.450$
• Analytic rank: $0$
• Dimension: $840$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	432.bj (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.44953736732\)
Analytic rank:	\(0\)
Dimension:	\(840\)
Relative dimension:	\(70\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			131.10
Character	\(\chi\)	\(=\)	432.131
Dual form			432.2.bj.a.155.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28525 + 0.590035i) q^{2} +(-1.66392 + 0.481027i) q^{3} +(1.30372 - 1.51668i) q^{4} +(1.14406 - 0.801082i) q^{5} +(1.85472 - 1.60001i) q^{6} +(-2.09620 + 1.75892i) q^{7} +(-0.780703 + 2.71855i) q^{8} +(2.53723 - 1.60078i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/432\mathbb{Z}\right)^\times\).

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{11}{18}\right)\)

Coefficient data

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

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28525	+	0.590035i	−0.908807	+	0.417218i
							\(3\)	−1.66392	+	0.481027i	−0.960662	+	0.277721i
\(5\)	1.14406	−	0.801082i	0.511641	−	0.358255i								−0.289068	−	0.957309i	\(-0.593345\pi\)
							0.800709	+	0.599054i	\(0.204457\pi\)
\(7\)	−2.09620	+	1.75892i	−0.792289	+	0.664810i	−0.946311	−	0.323258i	\(-0.895222\pi\)
							0.154022	+	0.988067i	\(0.450777\pi\)
\(11\)	−3.15290	−	2.20768i	−0.950634	−	0.665641i	−0.00817624	−	0.999967i	\(-0.502603\pi\)
							−0.942458	+	0.334326i	\(0.891491\pi\)
\(13\)	2.37214	+	1.10615i	0.657912	+	0.306790i	0.722753	−	0.691106i	\(-0.242876\pi\)
							−0.0648410	+	0.997896i	\(0.520654\pi\)
\(17\)	1.58287	+	0.913869i	0.383902	+	0.221646i	0.679515	−	0.733662i	\(-0.262190\pi\)
							−0.295613	+	0.955308i	\(0.595524\pi\)
\(19\)	7.61198	+	2.03962i	1.74631	+	0.467922i	0.983832	−	0.179094i	\(-0.0573167\pi\)
							0.762476	+	0.647016i	\(0.223983\pi\)
\(23\)	−3.16285	+	3.76934i	−0.659500	+	0.785961i	−0.987314	−	0.158781i	\(-0.949244\pi\)
							0.327814	+	0.944742i	\(0.393688\pi\)
\(29\)	−1.30285	+	0.607531i	−0.241934	+	0.112816i	−0.539804	−	0.841791i	\(-0.681502\pi\)
							0.297870	+	0.954606i	\(0.403724\pi\)
\(31\)	−1.00592	+	1.19881i	−0.180669	+	0.215313i	−0.848776	−	0.528752i	\(-0.822660\pi\)
							0.668108	+	0.744065i	\(0.267105\pi\)
\(37\)	2.83034	+	10.5630i	0.465305	+	1.73654i	0.655876	+	0.754869i	\(0.272299\pi\)
							−0.190571	+	0.981673i	\(0.561034\pi\)
\(41\)	−11.3019	+	4.11355i	−1.76506	+	0.642429i	−0.999999	−	0.00153167i	\(-0.999512\pi\)
							−0.765059	+	0.643960i	\(0.777290\pi\)
\(43\)	1.01376	−	1.44780i	0.154597	−	0.220788i	−0.734381	−	0.678738i	\(-0.762527\pi\)
							0.888978	+	0.457950i	\(0.151416\pi\)
\(47\)	1.94607	−	1.63295i	0.283864	−	0.238190i	−0.489726	−	0.871876i	\(-0.662903\pi\)
							0.773590	+	0.633686i	\(0.218459\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bj.a.131.10	✓	840
16.11	odd	4	inner	432.2.bj.a.347.45	yes	840
27.20	odd	18	inner	432.2.bj.a.371.45	yes	840
432.155	even	36	inner	432.2.bj.a.155.10	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bj.a.131.10	✓	840	1.1	even	1	trivial
432.2.bj.a.155.10	yes	840	432.155	even	36	inner
432.2.bj.a.347.45	yes	840	16.11	odd	4	inner
432.2.bj.a.371.45	yes	840	27.20	odd	18	inner