Embedded newform 429.2.p.b.230.13

• Label: 429.2.p.b.230.13
• Level: $429$
• Weight: $2$
• Character: 429.230
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $96$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(96\)
Relative dimension:	\(48\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			230.13
Character	\(\chi\)	\(=\)	429.230
Dual form			429.2.p.b.263.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.790522 + 1.36922i) q^{2} +(-1.05747 - 1.37177i) q^{3} +(-0.249851 - 0.432755i) q^{4} +3.89310i q^{5} +(2.71422 - 0.363499i) q^{6} +(-2.15619 + 1.24488i) q^{7} -2.37204 q^{8} +(-0.763513 + 2.90121i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 2 q^{3} - 60 q^{4} - 2 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(-1\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.790522	+	1.36922i	−0.558984	+	0.968188i	0.438598	+	0.898683i	\(0.355475\pi\)
							−0.997582	+	0.0695048i	\(0.977858\pi\)
\(3\)	−1.05747	−	1.37177i	−0.610531	−	0.791993i
							\(5\)			3.89310i			1.74105i	0.492128	+	0.870523i	\(0.336219\pi\)
−0.492128	+	0.870523i	\(0.663781\pi\)
\(7\)	−2.15619	+	1.24488i	−0.814965	+	0.470520i	−0.848677	−	0.528912i	\(-0.822600\pi\)
							0.0337124	+	0.999432i	\(0.489267\pi\)
\(11\)	3.29661	−	0.363858i	0.993964	−	0.109707i
							\(13\)	2.65036	−	2.44450i	0.735078	−	0.677982i
\(17\)	−0.828012	−	1.43416i	−0.200822	−	0.347835i								0.747971	−	0.663731i	\(-0.231028\pi\)
							−0.948794	+	0.315897i	\(0.897695\pi\)
\(19\)	−6.49576	+	3.75033i	−1.49023	+	0.860385i	−0.999938	−	0.0111733i	\(-0.996443\pi\)
							−0.490292	+	0.871558i	\(0.663110\pi\)
\(23\)	−5.90260	−	3.40787i	−1.23078	−	0.710589i	−0.263585	−	0.964636i	\(-0.584905\pi\)
							−0.967192	+	0.254047i	\(0.918238\pi\)
\(29\)	−0.214377	+	0.371312i	−0.0398088	+	0.0689508i	−0.885243	−	0.465128i	\(-0.846008\pi\)
							0.845435	+	0.534079i	\(0.179342\pi\)
\(31\)	0.540608			0.0970960			0.0485480	−	0.998821i	\(-0.484541\pi\)
							0.0485480	+	0.998821i	\(0.484541\pi\)
\(37\)	−2.10493	+	3.64585i	−0.346048	+	0.599373i	−0.985544	−	0.169422i	\(-0.945810\pi\)
							0.639495	+	0.768795i	\(0.279143\pi\)
\(41\)	−3.49499	+	6.05349i	−0.545825	+	0.945397i	0.452729	+	0.891648i	\(0.350450\pi\)
							−0.998554	+	0.0537490i	\(0.982883\pi\)
\(43\)	4.38219	−	2.53006i	0.668278	−	0.385831i	−0.127146	−	0.991884i	\(-0.540582\pi\)
							0.795424	+	0.606053i	\(0.207248\pi\)
\(47\)		−	6.66133i		−	0.971655i	−0.874055	−	0.485827i	\(-0.838518\pi\)
							0.874055	−	0.485827i	\(-0.161482\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.p.b.230.13	✓	96
3.2	odd	2	inner	429.2.p.b.230.36	yes	96
11.10	odd	2	inner	429.2.p.b.230.35	yes	96
13.3	even	3	inner	429.2.p.b.263.14	yes	96
33.32	even	2	inner	429.2.p.b.230.14	yes	96
39.29	odd	6	inner	429.2.p.b.263.35	yes	96
143.120	odd	6	inner	429.2.p.b.263.36	yes	96
429.263	even	6	inner	429.2.p.b.263.13	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.p.b.230.13	✓	96	1.1	even	1	trivial
429.2.p.b.230.14	yes	96	33.32	even	2	inner
429.2.p.b.230.35	yes	96	11.10	odd	2	inner
429.2.p.b.230.36	yes	96	3.2	odd	2	inner
429.2.p.b.263.13	yes	96	429.263	even	6	inner
429.2.p.b.263.14	yes	96	13.3	even	3	inner
429.2.p.b.263.35	yes	96	39.29	odd	6	inner
429.2.p.b.263.36	yes	96	143.120	odd	6	inner