Embedded newform 429.2.p.b.230.8

• Label: 429.2.p.b.230.8
• 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			230.8
Character	\(\chi\)	\(=\)	429.230
Dual form			429.2.p.b.263.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.15197 + 1.99526i) q^{2} +(1.72583 + 0.146661i) q^{3} +(-1.65405 - 2.86490i) q^{4} +3.64772i q^{5} +(-2.28072 + 3.27454i) q^{6} +(-1.03838 + 0.599507i) q^{7} +3.01377 q^{8} +(2.95698 + 0.506224i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 96 q + 2 q^{3} - 60 q^{4} - 2 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.15197	+	1.99526i	−0.814563	+	1.41086i	0.0950786	+	0.995470i	\(0.469690\pi\)
							−0.909641	+	0.415394i	\(0.863644\pi\)
\(3\)	1.72583	+	0.146661i	0.996409	+	0.0846748i
							\(5\)			3.64772i			1.63131i	0.578538	+	0.815656i	\(0.303624\pi\)
−0.578538	+	0.815656i	\(0.696376\pi\)
\(7\)	−1.03838	+	0.599507i	−0.392470	+	0.226592i	−0.683230	−	0.730204i	\(-0.739425\pi\)
							0.290760	+	0.956796i	\(0.406092\pi\)
\(11\)	−3.19124	+	0.903309i	−0.962196	+	0.272358i
							\(13\)	2.76793	+	2.31053i	0.767685	+	0.640827i
\(17\)	−1.77057	−	3.06672i	−0.429427	−	0.743790i								0.567395	−	0.823446i	\(-0.307951\pi\)
							−0.996822	+	0.0796557i	\(0.974618\pi\)
\(19\)	1.82948	−	1.05625i	0.419712	−	0.242321i	−0.275242	−	0.961375i	\(-0.588758\pi\)
							0.694954	+	0.719054i	\(0.255425\pi\)
\(23\)	1.24693	+	0.719917i	0.260003	+	0.150113i	0.624336	−	0.781156i	\(-0.285370\pi\)
							−0.364333	+	0.931269i	\(0.618703\pi\)
\(29\)	−2.05794	+	3.56446i	−0.382150	+	0.661904i	−0.991369	−	0.131098i	\(-0.958150\pi\)
							0.609219	+	0.793002i	\(0.291483\pi\)
\(31\)	−6.43553			−1.15585			−0.577927	−	0.816088i	\(-0.696138\pi\)
							−0.577927	+	0.816088i	\(0.696138\pi\)
\(37\)	−2.72708	+	4.72344i	−0.448329	+	0.776529i	−0.998277	−	0.0586698i	\(-0.981314\pi\)
							0.549948	+	0.835199i	\(0.314647\pi\)
\(41\)	4.45109	−	7.70952i	0.695144	−	1.20402i	−0.274989	−	0.961447i	\(-0.588674\pi\)
							0.970132	−	0.242577i	\(-0.0779926\pi\)
\(43\)	4.64547	−	2.68207i	0.708428	−	0.409011i	−0.102051	−	0.994779i	\(-0.532540\pi\)
							0.810479	+	0.585768i	\(0.199207\pi\)
\(47\)			0.223378i			0.0325831i	0.999867	+	0.0162915i	\(0.00518598\pi\)
							−0.999867	+	0.0162915i	\(0.994814\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.p.b.230.8	yes	96
3.2	odd	2	inner	429.2.p.b.230.41	yes	96
11.10	odd	2	inner	429.2.p.b.230.42	yes	96
13.3	even	3	inner	429.2.p.b.263.7	yes	96
33.32	even	2	inner	429.2.p.b.230.7	✓	96
39.29	odd	6	inner	429.2.p.b.263.42	yes	96
143.120	odd	6	inner	429.2.p.b.263.41	yes	96
429.263	even	6	inner	429.2.p.b.263.8	yes	96

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.p.b.230.7	✓	96	33.32	even	2	inner
429.2.p.b.230.8	yes	96	1.1	even	1	trivial
429.2.p.b.230.41	yes	96	3.2	odd	2	inner
429.2.p.b.230.42	yes	96	11.10	odd	2	inner
429.2.p.b.263.7	yes	96	13.3	even	3	inner
429.2.p.b.263.8	yes	96	429.263	even	6	inner
429.2.p.b.263.41	yes	96	143.120	odd	6	inner
429.2.p.b.263.42	yes	96	39.29	odd	6	inner