Embedded newform 432.2.bj.a.11.2

• Label: 432.2.bj.a.11.2
• 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			11.2
Character	\(\chi\)	\(=\)	432.11
Dual form			432.2.bj.a.275.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.41262 + 0.0671088i) q^{2} +(-1.64576 - 0.539886i) q^{3} +(1.99099 - 0.189599i) q^{4} +(-2.14457 + 1.00003i) q^{5} +(2.36106 + 0.652210i) q^{6} +(-0.446923 + 2.53462i) q^{7} +(-2.79979 + 0.401444i) q^{8} +(2.41705 + 1.77705i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 24 q^{7} - 18 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.41262	+	0.0671088i	−0.998873	+	0.0474531i
							\(3\)	−1.64576	−	0.539886i	−0.950179	−	0.311704i
\(5\)	−2.14457	+	1.00003i	−0.959082	+	0.447227i								−0.838154	−	0.545434i	\(-0.816365\pi\)
							−0.120928	+	0.992661i	\(0.538587\pi\)
\(7\)	−0.446923	+	2.53462i	−0.168921	+	0.957998i	0.776008	+	0.630723i	\(0.217241\pi\)
							−0.944929	+	0.327275i	\(0.893870\pi\)
\(11\)	0.966049	+	0.450476i	0.291275	+	0.135824i	0.562763	−	0.826618i	\(-0.309738\pi\)
							−0.271489	+	0.962442i	\(0.587516\pi\)
\(13\)	0.350561	−	4.00693i	0.0972281	−	1.11132i	−0.778214	−	0.627999i	\(-0.783874\pi\)
							0.875442	−	0.483323i	\(-0.160570\pi\)
\(17\)	−0.332822	+	0.192155i	−0.0807211	+	0.0466044i	−0.539817	−	0.841782i	\(-0.681507\pi\)
							0.459096	+	0.888387i	\(0.348173\pi\)
\(19\)	1.21289	−	0.324992i	0.278256	−	0.0745584i	−0.116992	−	0.993133i	\(-0.537325\pi\)
							0.395248	+	0.918574i	\(0.370659\pi\)
\(23\)	−6.52207	+	1.15002i	−1.35995	+	0.239795i	−0.805585	−	0.592481i	\(-0.798149\pi\)
							−0.554362	+	0.832276i	\(0.687038\pi\)
\(29\)	−0.632540	−	7.22997i	−0.117460	−	1.34257i	−0.794787	−	0.606889i	\(-0.792417\pi\)
							0.677327	−	0.735682i	\(-0.263138\pi\)
\(31\)	−7.03605	+	1.24065i	−1.26371	+	0.222826i	−0.765050	−	0.643970i	\(-0.777286\pi\)
							−0.498661	+	0.866797i	\(0.666175\pi\)
\(37\)	3.12306	−	11.6554i	0.513429	−	1.91614i	0.133781	−	0.991011i	\(-0.457288\pi\)
							0.379648	−	0.925131i	\(-0.376045\pi\)
\(41\)	1.15377	−	0.968132i	0.180189	−	0.151197i	−0.548232	−	0.836326i	\(-0.684699\pi\)
							0.728421	+	0.685130i	\(0.240254\pi\)
\(43\)	−1.49203	+	3.19966i	−0.227532	+	0.487944i	−0.986619	−	0.163046i	\(-0.947868\pi\)
							0.759086	+	0.650990i	\(0.225646\pi\)
\(47\)	2.25403	−	12.7833i	0.328785	−	1.86463i	−0.152838	−	0.988251i	\(-0.548841\pi\)
							0.481623	−	0.876379i	\(-0.340047\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bj.a.11.2	✓	840
16.3	odd	4	inner	432.2.bj.a.227.37	yes	840
27.5	odd	18	inner	432.2.bj.a.59.37	yes	840
432.275	even	36	inner	432.2.bj.a.275.2	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bj.a.11.2	✓	840	1.1	even	1	trivial
432.2.bj.a.59.37	yes	840	27.5	odd	18	inner
432.2.bj.a.227.37	yes	840	16.3	odd	4	inner
432.2.bj.a.275.2	yes	840	432.275	even	36	inner