Embedded newform 432.2.bg.a.13.4

• Label: 432.2.bg.a.13.4
• 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			13.4
Character	\(\chi\)	\(=\)	432.13
Dual form			432.2.bg.a.133.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.39069 - 0.256893i) q^{2} +(-0.791539 - 1.54061i) q^{3} +(1.86801 + 0.714515i) q^{4} +(-2.39012 + 0.209108i) q^{5} +(0.705011 + 2.34584i) q^{6} +(0.307660 - 0.845290i) q^{7} +(-2.41426 - 1.47355i) q^{8} +(-1.74693 + 2.43890i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.39069	−	0.256893i	−0.983363	−	0.181651i
							\(3\)	−0.791539	−	1.54061i	−0.456995	−	0.889469i
\(5\)	−2.39012	+	0.209108i	−1.06889	+	0.0935161i								−0.607998	−	0.793938i	\(-0.708027\pi\)
							−0.460896	+	0.887454i	\(0.652472\pi\)
\(7\)	0.307660	−	0.845290i	0.116285	−	0.319489i	−0.867873	−	0.496787i	\(-0.834513\pi\)
							0.984157	+	0.177297i	\(0.0567354\pi\)
\(11\)	−0.422373	+	4.82775i	−0.127350	+	1.45562i	0.616542	+	0.787322i	\(0.288533\pi\)
							−0.743892	+	0.668299i	\(0.767023\pi\)
\(13\)	1.13954	−	0.797914i	0.316051	−	0.221301i	−0.404777	−	0.914416i	\(-0.632651\pi\)
							0.720828	+	0.693114i	\(0.243762\pi\)
\(17\)	1.53003	−	2.65010i	0.371088	−	0.642743i	−0.618645	−	0.785670i	\(-0.712318\pi\)
							0.989733	+	0.142928i	\(0.0456516\pi\)
\(19\)	−0.517049	+	1.92965i	−0.118619	+	0.442692i	−0.999532	−	0.0305857i	\(-0.990263\pi\)
							0.880913	+	0.473278i	\(0.156929\pi\)
\(23\)	1.62446	+	4.46318i	0.338724	+	0.930637i	0.985757	+	0.168175i	\(0.0537873\pi\)
							−0.647033	+	0.762462i	\(0.723990\pi\)
\(29\)	0.0256813	−	0.0366767i	0.00476889	−	0.00681069i	−0.816761	−	0.576977i	\(-0.804232\pi\)
							0.821529	+	0.570166i	\(0.193121\pi\)
\(31\)	4.31451	−	1.57035i	0.774909	−	0.282044i	0.0758611	−	0.997118i	\(-0.475829\pi\)
							0.699048	+	0.715074i	\(0.253607\pi\)
\(37\)	0.234299	+	0.874416i	0.0385185	+	0.143753i	0.982507	−	0.186225i	\(-0.0596253\pi\)
							−0.943989	+	0.329978i	\(0.892959\pi\)
\(41\)	6.11909	+	1.07896i	0.955642	+	0.168505i	0.629660	−	0.776871i	\(-0.283194\pi\)
							0.325982	+	0.945376i	\(0.394305\pi\)
\(43\)	−0.547919	+	6.26274i	−0.0835568	+	0.955059i	0.832504	+	0.554019i	\(0.186907\pi\)
							−0.916061	+	0.401040i	\(0.868649\pi\)
\(47\)	11.8015	+	4.29539i	1.72142	+	0.626547i	0.997962	−	0.0638183i	\(-0.0203278\pi\)
							0.723461	+	0.690365i	\(0.242550\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bg.a.13.4	✓	840
16.5	even	4	inner	432.2.bg.a.229.60	yes	840
27.25	even	9	inner	432.2.bg.a.349.60	yes	840
432.133	even	36	inner	432.2.bg.a.133.4	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bg.a.13.4	✓	840	1.1	even	1	trivial
432.2.bg.a.133.4	yes	840	432.133	even	36	inner
432.2.bg.a.229.60	yes	840	16.5	even	4	inner
432.2.bg.a.349.60	yes	840	27.25	even	9	inner