Embedded newform 429.2.p.b.230.17

• Label: 429.2.p.b.230.17
• 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			230.17
Character	\(\chi\)	\(=\)	429.230
Dual form			429.2.p.b.263.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.570464 + 0.988072i) q^{2} +(0.0427205 - 1.73152i) q^{3} +(0.349142 + 0.604732i) q^{4} +0.927976i q^{5} +(1.68650 + 1.02998i) q^{6} +(-1.26517 + 0.730447i) q^{7} -3.07855 q^{8} +(-2.99635 - 0.147943i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 2 q^{3} - 60 q^{4} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.570464	+	0.988072i	−0.403379	+	0.698672i	−0.994131	−	0.108180i	\(-0.965498\pi\)
							0.590753	+	0.806853i	\(0.298831\pi\)
\(3\)	0.0427205	−	1.73152i	0.0246647	−	0.999696i
							\(5\)			0.927976i			0.415003i	0.978235	+	0.207502i	\(0.0665332\pi\)
−0.978235	+	0.207502i	\(0.933467\pi\)
\(7\)	−1.26517	+	0.730447i	−0.478190	+	0.276083i	−0.719662	−	0.694325i	\(-0.755703\pi\)
							0.241472	+	0.970408i	\(0.422370\pi\)
\(11\)	−3.26288	+	0.594635i	−0.983796	+	0.179289i
							\(13\)	−3.28660	+	1.48265i	−0.911539	+	0.411214i
\(17\)	1.09224	+	1.89182i	0.264908	+	0.458834i								0.967540	−	0.252720i	\(-0.0813250\pi\)
							−0.702631	+	0.711554i	\(0.747992\pi\)
\(19\)	−0.853358	+	0.492686i	−0.195774	+	0.113030i	−0.594683	−	0.803961i	\(-0.702722\pi\)
							0.398909	+	0.916991i	\(0.369389\pi\)
\(23\)	0.289443	+	0.167110i	0.0603529	+	0.0348448i	0.529873	−	0.848077i	\(-0.322240\pi\)
							−0.469520	+	0.882922i	\(0.655573\pi\)
\(29\)	0.566826	−	0.981772i	0.105257	−	0.182310i	−0.808586	−	0.588378i	\(-0.799767\pi\)
							0.913843	+	0.406067i	\(0.133100\pi\)
\(31\)	−6.43584			−1.15591			−0.577955	−	0.816069i	\(-0.696149\pi\)
							−0.577955	+	0.816069i	\(0.696149\pi\)
\(37\)	−3.87545	+	6.71248i	−0.637121	+	1.10353i	0.348941	+	0.937145i	\(0.386541\pi\)
							−0.986062	+	0.166380i	\(0.946792\pi\)
\(41\)	1.63404	−	2.83023i	0.255194	−	0.442008i	−0.709754	−	0.704449i	\(-0.751194\pi\)
							0.964948	+	0.262441i	\(0.0845275\pi\)
\(43\)	−3.51723	+	2.03067i	−0.536373	+	0.309675i	−0.743608	−	0.668616i	\(-0.766887\pi\)
							0.207235	+	0.978291i	\(0.433554\pi\)
\(47\)		−	0.882849i		−	0.128777i	−0.997925	−	0.0643884i	\(-0.979490\pi\)
							0.997925	−	0.0643884i	\(-0.0205096\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.p.b.230.17	✓	96
3.2	odd	2	inner	429.2.p.b.230.32	yes	96
11.10	odd	2	inner	429.2.p.b.230.31	yes	96
13.3	even	3	inner	429.2.p.b.263.18	yes	96
33.32	even	2	inner	429.2.p.b.230.18	yes	96
39.29	odd	6	inner	429.2.p.b.263.31	yes	96
143.120	odd	6	inner	429.2.p.b.263.32	yes	96
429.263	even	6	inner	429.2.p.b.263.17	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.p.b.230.17	✓	96	1.1	even	1	trivial
429.2.p.b.230.18	yes	96	33.32	even	2	inner
429.2.p.b.230.31	yes	96	11.10	odd	2	inner
429.2.p.b.230.32	yes	96	3.2	odd	2	inner
429.2.p.b.263.17	yes	96	429.263	even	6	inner
429.2.p.b.263.18	yes	96	13.3	even	3	inner
429.2.p.b.263.31	yes	96	39.29	odd	6	inner
429.2.p.b.263.32	yes	96	143.120	odd	6	inner