Embedded newform 432.2.bg.a.13.15

• Label: 432.2.bg.a.13.15
• 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			13.15
Character	\(\chi\)	\(=\)	432.13
Dual form			432.2.bg.a.133.15

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.09334 - 0.896994i) q^{2} +(1.02688 - 1.39482i) q^{3} +(0.390802 + 1.96145i) q^{4} +(-3.34554 + 0.292697i) q^{5} +(-2.37388 + 0.603909i) q^{6} +(-0.981004 + 2.69529i) q^{7} +(1.33213 - 2.49508i) q^{8} +(-0.891032 - 2.86462i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 840 q - 12 q^{2} - 12 q^{3} - 12 q^{4} - 12 q^{5} - 12 q^{6} - 6 q^{8}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.09334	−	0.896994i	−0.773111	−	0.634271i
							\(3\)	1.02688	−	1.39482i	0.592870	−	0.805298i
\(5\)	−3.34554	+	0.292697i	−1.49617	+	0.130898i								−0.805640	−	0.592406i	\(-0.798178\pi\)
							−0.690530	+	0.723304i	\(0.742623\pi\)
\(7\)	−0.981004	+	2.69529i	−0.370785	+	1.01872i	0.604274	+	0.796777i	\(0.293463\pi\)
							−0.975059	+	0.221946i	\(0.928759\pi\)
\(11\)	0.270188	−	3.08826i	0.0814646	−	0.931145i	−0.839975	−	0.542624i	\(-0.817431\pi\)
							0.921440	−	0.388521i	\(-0.127014\pi\)
\(13\)	−0.321705	+	0.225261i	−0.0892251	+	0.0624761i	−0.617338	−	0.786698i	\(-0.711789\pi\)
							0.528113	+	0.849174i	\(0.322900\pi\)
\(17\)	−2.70356	+	4.68270i	−0.655710	+	1.13572i	0.326006	+	0.945368i	\(0.394297\pi\)
							−0.981715	+	0.190355i	\(0.939036\pi\)
\(19\)	−1.93495	+	7.22134i	−0.443908	+	1.65669i	0.274894	+	0.961474i	\(0.411357\pi\)
							−0.718803	+	0.695214i	\(0.755309\pi\)
\(23\)	2.87061	+	7.88693i	0.598563	+	1.64454i	0.754139	+	0.656714i	\(0.228054\pi\)
							−0.155576	+	0.987824i	\(0.549723\pi\)
\(29\)	−0.685206	+	0.978576i	−0.127240	+	0.181717i	−0.877717	−	0.479179i	\(-0.840935\pi\)
							0.750478	+	0.660896i	\(0.229823\pi\)
\(31\)	−3.31553	+	1.20676i	−0.595487	+	0.216740i	−0.622141	−	0.782905i	\(-0.713737\pi\)
							0.0266538	+	0.999645i	\(0.491515\pi\)
\(37\)	0.818016	+	3.05288i	0.134481	+	0.501890i	0.999999	+	0.00101283i	\(0.000322393\pi\)
							−0.865519	+	0.500877i	\(0.833011\pi\)
\(41\)	−2.24101	−	0.395150i	−0.349987	−	0.0617121i	−0.00410871	−	0.999992i	\(-0.501308\pi\)
							−0.345878	+	0.938279i	\(0.612419\pi\)
\(43\)	1.13177	−	12.9362i	0.172593	−	1.97275i	−0.0532268	−	0.998582i	\(-0.516951\pi\)
							0.225820	−	0.974169i	\(-0.427494\pi\)
\(47\)	−8.26261	−	3.00734i	−1.20523	−	0.438666i	−0.340180	−	0.940360i	\(-0.610488\pi\)
							−0.865045	+	0.501694i	\(0.832710\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.bg.a.13.15	✓	840
16.5	even	4	inner	432.2.bg.a.229.48	yes	840
27.25	even	9	inner	432.2.bg.a.349.48	yes	840
432.133	even	36	inner	432.2.bg.a.133.15	yes	840

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.bg.a.13.15	✓	840	1.1	even	1	trivial
432.2.bg.a.133.15	yes	840	432.133	even	36	inner
432.2.bg.a.229.48	yes	840	16.5	even	4	inner
432.2.bg.a.349.48	yes	840	27.25	even	9	inner