Embedded newform 432.2.bj.a.155.11

• Label: 432.2.bj.a.155.11
• Level: $432$
• Weight: $2$
• Character: 432.155
• 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			155.11
Character	\(\chi\)	\(=\)	432.155
Dual form			432.2.bj.a.131.11

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.28349 + 0.593851i) q^{2} +(-1.49854 - 0.868551i) q^{3} +(1.29468 - 1.52440i) q^{4} +(2.70467 + 1.89383i) q^{5} +(2.43915 + 0.224867i) q^{6} +(-0.207189 - 0.173852i) q^{7} +(-0.756443 + 2.72540i) q^{8} +(1.49124 + 2.60312i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{7}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.28349	+	0.593851i	−0.907563	+	0.419916i
							\(3\)	−1.49854	−	0.868551i	−0.865182	−	0.501458i
\(5\)	2.70467	+	1.89383i	1.20957	+	0.846948i								0.991642	−	0.129017i	\(-0.0411823\pi\)
							0.217925	+	0.975966i	\(0.430071\pi\)
\(7\)	−0.207189	−	0.173852i	−0.0783100	−	0.0657099i	0.602792	−	0.797898i	\(-0.294055\pi\)
							−0.681102	+	0.732188i	\(0.738499\pi\)
\(11\)	1.45001	−	1.01531i	0.437194	−	0.306127i	−0.334179	−	0.942510i	\(-0.608459\pi\)
							0.771373	+	0.636383i	\(0.219570\pi\)
\(13\)	−1.92185	+	0.896172i	−0.533025	+	0.248554i	−0.670445	−	0.741959i	\(-0.733897\pi\)
							0.137420	+	0.990513i	\(0.456119\pi\)
\(17\)	−1.54027	+	0.889273i	−0.373570	+	0.215680i	−0.675017	−	0.737802i	\(-0.735864\pi\)
							0.301447	+	0.953483i	\(0.402530\pi\)
\(19\)	0.494595	−	0.132526i	0.113468	−	0.0304036i	−0.201638	−	0.979460i	\(-0.564627\pi\)
							0.315106	+	0.949056i	\(0.397960\pi\)
\(23\)	3.09758	+	3.69156i	0.645891	+	0.769743i	0.985288	−	0.170900i	\(-0.0546675\pi\)
							−0.339397	+	0.940643i	\(0.610223\pi\)
\(29\)	8.64511	+	4.03128i	1.60536	+	0.748590i	0.998916	−	0.0465386i	\(-0.0148190\pi\)
							0.606441	+	0.795129i	\(0.292597\pi\)
\(31\)	−2.29288	−	2.73255i	−0.411813	−	0.490780i	0.519771	−	0.854306i	\(-0.326017\pi\)
							−0.931584	+	0.363526i	\(0.881573\pi\)
\(37\)	−2.76817	+	10.3309i	−0.455084	+	1.69840i	0.232757	+	0.972535i	\(0.425225\pi\)
							−0.687841	+	0.725862i	\(0.741441\pi\)
\(41\)	1.67449	+	0.609463i	0.261511	+	0.0951821i	0.469448	−	0.882960i	\(-0.344453\pi\)
							−0.207937	+	0.978142i	\(0.566675\pi\)
\(43\)	5.11178	+	7.30038i	0.779540	+	1.11330i	0.990729	+	0.135853i	\(0.0433775\pi\)
							−0.211189	+	0.977445i	\(0.567734\pi\)
\(47\)	4.52032	+	3.79300i	0.659357	+	0.553266i	0.909894	−	0.414841i	\(-0.136163\pi\)
							−0.250537	+	0.968107i	\(0.580607\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bj.a.155.11	yes	840
16.3	odd	4	inner	432.2.bj.a.371.65	yes	840
27.23	odd	18	inner	432.2.bj.a.347.65	yes	840
432.131	even	36	inner	432.2.bj.a.131.11	✓	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bj.a.131.11	✓	840	432.131	even	36	inner
432.2.bj.a.155.11	yes	840	1.1	even	1	trivial
432.2.bj.a.347.65	yes	840	27.23	odd	18	inner
432.2.bj.a.371.65	yes	840	16.3	odd	4	inner