Embedded newform 429.2.x.a.248.8

• Label: 429.2.x.a.248.8
• Level: $429$
• Weight: $2$
• Character: 429.248
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.x (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(48\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			248.8
Character	\(\chi\)	\(=\)	429.248
Dual form			429.2.x.a.365.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.641276 - 1.97365i) q^{2} +(1.56505 - 0.742034i) q^{3} +(-1.86601 + 1.35573i) q^{4} +(2.08634 + 0.677894i) q^{5} +(-2.46814 - 2.61301i) q^{6} +(-1.98628 - 2.73388i) q^{7} +(0.514597 + 0.373877i) q^{8} +(1.89877 - 2.32264i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 192 q + 6 q^{3} - 48 q^{4} - 10 q^{6} - 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{9}{10}\right)\)	\(-1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.641276	−	1.97365i	−0.453451	−	1.39558i	−0.872944	−	0.487820i	\(-0.837792\pi\)
							0.419494	−	0.907758i	\(-0.362208\pi\)
\(3\)	1.56505	−	0.742034i	0.903583	−	0.428414i
							\(5\)	2.08634	+	0.677894i	0.933041	+	0.303163i	0.735806	−	0.677193i	\(-0.236804\pi\)
0.197235	+	0.980356i	\(0.436804\pi\)
\(7\)	−1.98628	−	2.73388i	−0.750742	−	1.03331i	−0.997928	−	0.0643399i	\(-0.979506\pi\)
							0.247186	−	0.968968i	\(-0.420494\pi\)
\(11\)	2.72539	+	1.89004i	0.821736	+	0.569868i
							\(13\)	−0.951057	+	0.309017i	−0.263776	+	0.0857059i
\(17\)	1.41628	−	4.35887i	0.343499	−	1.05718i								−0.618884	−	0.785483i	\(-0.712415\pi\)
							0.962383	−	0.271698i	\(-0.0875852\pi\)
\(19\)	−1.57999	+	2.17467i	−0.362475	+	0.498904i	−0.950836	−	0.309694i	\(-0.899773\pi\)
							0.588361	+	0.808598i	\(0.299773\pi\)
\(23\)		−	5.73689i		−	1.19622i	−0.801413	−	0.598112i	\(-0.795918\pi\)
							0.801413	−	0.598112i	\(-0.204082\pi\)
\(29\)	0.109085	−	0.0792549i	0.0202566	−	0.0147173i	−0.577611	−	0.816312i	\(-0.696015\pi\)
							0.597867	+	0.801595i	\(0.296015\pi\)
\(31\)	2.48761	+	7.65608i	0.446788	+	1.37507i	0.880511	+	0.474026i	\(0.157200\pi\)
							−0.433723	+	0.901046i	\(0.642800\pi\)
\(37\)	−2.00924	+	1.45980i	−0.330316	+	0.239989i	−0.740565	−	0.671985i	\(-0.765442\pi\)
							0.410248	+	0.911974i	\(0.365442\pi\)
\(41\)	0.529534	+	0.384729i	0.0826993	+	0.0600846i	0.628367	−	0.777917i	\(-0.283724\pi\)
							−0.545667	+	0.838002i	\(0.683724\pi\)
\(43\)			3.88198i			0.591996i	0.955189	+	0.295998i	\(0.0956522\pi\)
							−0.955189	+	0.295998i	\(0.904348\pi\)
\(47\)	−6.23676	+	8.58417i	−0.909725	+	1.25213i	0.0575351	+	0.998343i	\(0.481676\pi\)
							−0.967260	+	0.253786i	\(0.918324\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.x.a.248.8	✓	192
3.2	odd	2	inner	429.2.x.a.248.41	yes	192
11.2	odd	10	inner	429.2.x.a.365.41	yes	192
33.2	even	10	inner	429.2.x.a.365.8	yes	192

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.x.a.248.8	✓	192	1.1	even	1	trivial
429.2.x.a.248.41	yes	192	3.2	odd	2	inner
429.2.x.a.365.8	yes	192	33.2	even	10	inner
429.2.x.a.365.41	yes	192	11.2	odd	10	inner