Embedded newform 432.2.bj.a.11.15

• Label: 432.2.bj.a.11.15
• Level: $432$
• Weight: $2$
• Character: 432.11
• 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			11.15
Character	\(\chi\)	\(=\)	432.11
Dual form			432.2.bj.a.275.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.11141 + 0.874510i) q^{2} +(-1.49998 - 0.866058i) q^{3} +(0.470465 - 1.94388i) q^{4} +(2.31389 - 1.07899i) q^{5} +(2.42447 - 0.349202i) q^{6} +(-0.254246 + 1.44190i) q^{7} +(1.17706 + 2.57187i) q^{8} +(1.49989 + 2.59814i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 18 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{1}{4}\right)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.11141	+	0.874510i	−0.785886	+	0.618372i
							\(3\)	−1.49998	−	0.866058i	−0.866015	−	0.500019i
\(5\)	2.31389	−	1.07899i	1.03480	−	0.482537i								0.170422	−	0.985371i	\(-0.445487\pi\)
							0.864383	+	0.502834i	\(0.167709\pi\)
\(7\)	−0.254246	+	1.44190i	−0.0960961	+	0.544988i	0.898310	+	0.439363i	\(0.144796\pi\)
							−0.994406	+	0.105626i	\(0.966315\pi\)
\(11\)	−0.899375	−	0.419385i	−0.271172	−	0.126449i	0.282276	−	0.959333i	\(-0.408910\pi\)
							−0.553448	+	0.832884i	\(0.686688\pi\)
\(13\)	0.00392828	−	0.0449004i	0.00108951	−	0.0124531i	−0.995630	−	0.0933806i	\(-0.970233\pi\)
							0.996720	+	0.0809275i	\(0.0257882\pi\)
\(17\)	5.76966	−	3.33111i	1.39935	−	0.807914i	0.405024	−	0.914306i	\(-0.367263\pi\)
							0.994324	+	0.106392i	\(0.0339299\pi\)
\(19\)	−2.77916	+	0.744673i	−0.637583	+	0.170840i	−0.563108	−	0.826383i	\(-0.690395\pi\)
							−0.0744744	+	0.997223i	\(0.523728\pi\)
\(23\)	7.11802	−	1.25510i	1.48421	−	0.261706i	0.627951	−	0.778253i	\(-0.283894\pi\)
							0.856259	+	0.516547i	\(0.172783\pi\)
\(29\)	−0.121038	−	1.38347i	−0.0224762	−	0.256904i	−0.999117	−	0.0420238i	\(-0.986619\pi\)
							0.976640	−	0.214880i	\(-0.0689361\pi\)
\(31\)	3.22103	−	0.567954i	0.578514	−	0.102008i	0.123268	−	0.992373i	\(-0.460663\pi\)
							0.455246	+	0.890366i	\(0.349551\pi\)
\(37\)	0.379245	−	1.41536i	0.0623475	−	0.232684i	−0.927720	−	0.373277i	\(-0.878234\pi\)
							0.990068	+	0.140593i	\(0.0449008\pi\)
\(41\)	1.10395	−	0.926321i	0.172407	−	0.144667i	−0.552501	−	0.833512i	\(-0.686326\pi\)
							0.724908	+	0.688845i	\(0.241882\pi\)
\(43\)	0.659649	−	1.41462i	0.100596	−	0.215728i	−0.849568	−	0.527478i	\(-0.823138\pi\)
							0.950164	+	0.311751i	\(0.100915\pi\)
\(47\)	−0.211782	+	1.20108i	−0.0308916	+	0.175195i	−0.996350	−	0.0853624i	\(-0.972795\pi\)
							0.965458	+	0.260557i	\(0.0839063\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bj.a.11.15	✓	840
16.3	odd	4	inner	432.2.bj.a.227.27	yes	840
27.5	odd	18	inner	432.2.bj.a.59.27	yes	840
432.275	even	36	inner	432.2.bj.a.275.15	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bj.a.11.15	✓	840	1.1	even	1	trivial
432.2.bj.a.59.27	yes	840	27.5	odd	18	inner
432.2.bj.a.227.27	yes	840	16.3	odd	4	inner
432.2.bj.a.275.15	yes	840	432.275	even	36	inner