Embedded newform 429.2.be.a.89.8

• Label: 429.2.be.a.89.8
• Level: $429$
• Weight: $2$
• Character: 429.89
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.be (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(46\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			89.8
Character	\(\chi\)	\(=\)	429.89
Dual form			429.2.be.a.188.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.99225 - 0.533822i) q^{2} +(-1.33180 + 1.10738i) q^{3} +(1.95204 + 1.12701i) q^{4} +(-0.191278 + 0.191278i) q^{5} +(3.24443 - 1.49524i) q^{6} +(-0.125579 - 0.468666i) q^{7} +(-0.370477 - 0.370477i) q^{8} +(0.547407 - 2.94963i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 184 q - 12 q^{6} + 12 q^{7}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.99225	−	0.533822i	−1.40873	−	0.377469i	−0.527260	−	0.849704i	\(-0.676781\pi\)
							−0.881473	+	0.472235i	\(0.843447\pi\)
\(3\)	−1.33180	+	1.10738i	−0.768918	+	0.639348i
							\(5\)	−0.191278	+	0.191278i	−0.0855421	+	0.0855421i	−0.748583	−	0.663041i	\(-0.769266\pi\)
0.663041	+	0.748583i	\(0.269266\pi\)
\(7\)	−0.125579	−	0.468666i	−0.0474643	−	0.177139i	0.938124	−	0.346298i	\(-0.112561\pi\)
							−0.985589	+	0.169159i	\(0.945895\pi\)
\(11\)	0.258819	−	0.965926i	0.0780369	−	0.291238i
							\(13\)	−3.18841	+	1.68346i	−0.884306	+	0.466907i
\(17\)	2.16195	−	3.74461i	0.524350	−	0.908201i								−0.475248	−	0.879852i	\(-0.657642\pi\)
							0.999598	−	0.0283490i	\(-0.00902499\pi\)
\(19\)	0.193576	−	0.0518684i	0.0444093	−	0.0118994i	−0.236546	−	0.971620i	\(-0.576015\pi\)
							0.280955	+	0.959721i	\(0.409349\pi\)
\(23\)	0.377783	+	0.654340i	0.0787732	+	0.136439i	0.902721	−	0.430227i	\(-0.141566\pi\)
							−0.823948	+	0.566666i	\(0.808233\pi\)
\(29\)	2.02304	−	1.16801i	0.375670	−	0.216893i	−0.300263	−	0.953857i	\(-0.597074\pi\)
							0.675933	+	0.736963i	\(0.263741\pi\)
\(31\)	3.60502	+	3.60502i	0.647480	+	0.647480i	0.952383	−	0.304903i	\(-0.0986241\pi\)
							−0.304903	+	0.952383i	\(0.598624\pi\)
\(37\)	5.71277	+	1.53073i	0.939174	+	0.251651i	0.695762	−	0.718272i	\(-0.255067\pi\)
							0.243412	+	0.969923i	\(0.421733\pi\)
\(41\)	7.84503	+	2.10207i	1.22519	+	0.328288i	0.812704	−	0.582677i	\(-0.197995\pi\)
							0.412484	+	0.910965i	\(0.364661\pi\)
\(43\)	−1.76101	−	1.01672i	−0.268551	−	0.155048i	0.359678	−	0.933077i	\(-0.382887\pi\)
							−0.628229	+	0.778028i	\(0.716220\pi\)
\(47\)	4.94020	+	4.94020i	0.720602	+	0.720602i	0.968728	−	0.248126i	\(-0.0798147\pi\)
							−0.248126	+	0.968728i	\(0.579815\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.be.a.89.8	✓	184
3.2	odd	2	inner	429.2.be.a.89.39	yes	184
13.6	odd	12	inner	429.2.be.a.188.39	yes	184
39.32	even	12	inner	429.2.be.a.188.8	yes	184

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.be.a.89.8	✓	184	1.1	even	1	trivial
429.2.be.a.89.39	yes	184	3.2	odd	2	inner
429.2.be.a.188.8	yes	184	39.32	even	12	inner
429.2.be.a.188.39	yes	184	13.6	odd	12	inner