Embedded newform 432.2.bg.a.13.65

• Label: 432.2.bg.a.13.65
• Level: $432$
• Weight: $2$
• Character: 432.13
• 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.bg (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			13.65
Character	\(\chi\)	\(=\)	432.13
Dual form			432.2.bg.a.133.65

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.36013 - 0.387357i) q^{2} +(1.13120 + 1.31163i) q^{3} +(1.69991 - 1.05371i) q^{4} +(-0.110154 + 0.00963718i) q^{5} +(2.04665 + 1.34581i) q^{6} +(0.959174 - 2.63531i) q^{7} +(1.90393 - 2.09166i) q^{8} +(-0.440761 + 2.96745i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(271\)	\(325\)	\(353\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{3}{4}\right)\)	\(e\left(\frac{4}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.36013	−	0.387357i	0.961757	−	0.273903i
							\(3\)	1.13120	+	1.31163i	0.653100	+	0.757272i
\(5\)	−0.110154	+	0.00963718i	−0.0492621	+	0.00430988i								−0.111760	−	0.993735i	\(-0.535649\pi\)
							0.0624980	+	0.998045i	\(0.480093\pi\)
\(7\)	0.959174	−	2.63531i	0.362534	−	0.996054i	−0.615597	−	0.788061i	\(-0.711085\pi\)
							0.978130	−	0.207992i	\(-0.0666929\pi\)
\(11\)	0.413396	−	4.72513i	0.124643	−	1.42468i	−0.634080	−	0.773268i	\(-0.718621\pi\)
							0.758723	−	0.651413i	\(-0.225824\pi\)
\(13\)	−2.83779	+	1.98704i	−0.787061	+	0.551106i	−0.896582	−	0.442878i	\(-0.853957\pi\)
							0.109520	+	0.993985i	\(0.465069\pi\)
\(17\)	−1.45042	+	2.51220i	−0.351778	+	0.609298i	−0.986561	−	0.163393i	\(-0.947756\pi\)
							0.634783	+	0.772691i	\(0.281090\pi\)
\(19\)	−1.75304	+	6.54242i	−0.402174	+	1.50093i	0.407034	+	0.913413i	\(0.366563\pi\)
							−0.809208	+	0.587522i	\(0.800104\pi\)
\(23\)	2.71564	+	7.46117i	0.566251	+	1.55576i	0.810310	+	0.586001i	\(0.199299\pi\)
							−0.244059	+	0.969760i	\(0.578479\pi\)
\(29\)	4.33763	−	6.19478i	0.805478	−	1.15034i	−0.180517	−	0.983572i	\(-0.557777\pi\)
							0.985996	−	0.166770i	\(-0.0533339\pi\)
\(31\)	−2.07111	+	0.753823i	−0.371983	+	0.135391i	−0.521245	−	0.853407i	\(-0.674532\pi\)
							0.149263	+	0.988798i	\(0.452310\pi\)
\(37\)	−0.852349	−	3.18101i	−0.140125	−	0.522955i	−0.999924	−	0.0123218i	\(-0.996078\pi\)
							0.859799	−	0.510633i	\(-0.170589\pi\)
\(41\)	−9.01073	−	1.58883i	−1.40724	−	0.248134i	−0.582124	−	0.813100i	\(-0.697778\pi\)
							−0.825115	+	0.564965i	\(0.808889\pi\)
\(43\)	−0.748784	+	8.55864i	−0.114189	+	1.30518i	0.695587	+	0.718441i	\(0.255144\pi\)
							−0.809776	+	0.586739i	\(0.800411\pi\)
\(47\)	−2.65909	−	0.967828i	−0.387868	−	0.141172i	0.140723	−	0.990049i	\(-0.455057\pi\)
							−0.528591	+	0.848877i	\(0.677279\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bg.a.13.65	✓	840
16.5	even	4	inner	432.2.bg.a.229.2	yes	840
27.25	even	9	inner	432.2.bg.a.349.2	yes	840
432.133	even	36	inner	432.2.bg.a.133.65	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bg.a.13.65	✓	840	1.1	even	1	trivial
432.2.bg.a.133.65	yes	840	432.133	even	36	inner
432.2.bg.a.229.2	yes	840	16.5	even	4	inner
432.2.bg.a.349.2	yes	840	27.25	even	9	inner