Embedded newform 429.2.y.b.116.16

• Label: 429.2.y.b.116.16
• Level: $429$
• Weight: $2$
• Character: 429.116
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $176$
• CM: no
• Inner twists: $8$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(176\)
Relative dimension:	\(44\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			116.16
Character	\(\chi\)	\(=\)	429.116
Dual form			429.2.y.b.233.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.743149 + 0.241464i) q^{2} +(0.418569 - 1.68071i) q^{3} +(-1.12407 + 0.816683i) q^{4} +(0.877069 - 2.69934i) q^{5} +(0.0947722 + 1.35009i) q^{6} +(1.52431 - 1.10747i) q^{7} +(1.55673 - 2.14266i) q^{8} +(-2.64960 - 1.40699i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q - 6 q^{3} + 20 q^{4} + 6 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.743149	+	0.241464i	−0.525486	+	0.170741i	−0.559734	−	0.828673i	\(-0.689097\pi\)
							0.0342476	+	0.999413i	\(0.489097\pi\)
\(3\)	0.418569	−	1.68071i	0.241661	−	0.970361i
							\(5\)	0.877069	−	2.69934i	0.392237	−	1.20718i	−0.538855	−	0.842398i	\(-0.681143\pi\)
0.931092	−	0.364783i	\(-0.118857\pi\)
\(7\)	1.52431	−	1.10747i	0.576134	−	0.418586i	−0.261194	−	0.965286i	\(-0.584116\pi\)
							0.837328	+	0.546700i	\(0.184116\pi\)
\(11\)	2.99867	+	1.41704i	0.904132	+	0.427254i
							\(13\)	−3.55668	−	0.591660i	−0.986444	−	0.164097i
\(17\)	−1.42322	+	4.38022i	−0.345182	+	1.06236i								0.616305	+	0.787508i	\(0.288629\pi\)
							−0.961486	+	0.274853i	\(0.911371\pi\)
\(19\)	−2.60261	−	1.89090i	−0.597079	−	0.433803i	0.247762	−	0.968821i	\(-0.420305\pi\)
							−0.844841	+	0.535018i	\(0.820305\pi\)
\(23\)		−	8.59808i		−	1.79282i	−0.443221	−	0.896412i	\(-0.646164\pi\)
							0.443221	−	0.896412i	\(-0.353836\pi\)
\(29\)	3.66292	−	2.66127i	0.680187	−	0.494185i	−0.193233	−	0.981153i	\(-0.561897\pi\)
							0.873420	+	0.486968i	\(0.161897\pi\)
\(31\)	−1.60292	+	0.520819i	−0.287892	+	0.0935419i	−0.449403	−	0.893329i	\(-0.648363\pi\)
							0.161511	+	0.986871i	\(0.448363\pi\)
\(37\)	−6.05447	−	8.33326i	−0.995348	−	1.36998i	−0.928136	−	0.372240i	\(-0.878590\pi\)
							−0.0672116	−	0.997739i	\(-0.521410\pi\)
\(41\)	2.22232	−	3.05876i	0.347068	−	0.477698i	−0.599421	−	0.800434i	\(-0.704602\pi\)
							0.946489	+	0.322736i	\(0.104602\pi\)
\(43\)			4.46942i			0.681580i	0.940139	+	0.340790i	\(0.110695\pi\)
							−0.940139	+	0.340790i	\(0.889305\pi\)
\(47\)	−0.150355	−	0.109240i	−0.0219316	−	0.0159342i	0.576765	−	0.816910i	\(-0.304315\pi\)
							−0.598697	+	0.800976i	\(0.704315\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.y.b.116.16	yes	176
3.2	odd	2	inner	429.2.y.b.116.29	yes	176
11.2	odd	10	inner	429.2.y.b.233.15	yes	176
13.12	even	2	inner	429.2.y.b.116.30	yes	176
33.2	even	10	inner	429.2.y.b.233.30	yes	176
39.38	odd	2	inner	429.2.y.b.116.15	✓	176
143.90	odd	10	inner	429.2.y.b.233.29	yes	176
429.233	even	10	inner	429.2.y.b.233.16	yes	176

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.y.b.116.15	✓	176	39.38	odd	2	inner
429.2.y.b.116.16	yes	176	1.1	even	1	trivial
429.2.y.b.116.29	yes	176	3.2	odd	2	inner
429.2.y.b.116.30	yes	176	13.12	even	2	inner
429.2.y.b.233.15	yes	176	11.2	odd	10	inner
429.2.y.b.233.16	yes	176	429.233	even	10	inner
429.2.y.b.233.29	yes	176	143.90	odd	10	inner
429.2.y.b.233.30	yes	176	33.2	even	10	inner