Embedded newform 429.2.bn.a.49.6

• Label: 429.2.bn.a.49.6
• Level: $429$
• Weight: $2$
• Character: 429.49
• Analytic conductor: $3.426$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 429 = 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	429.bn (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.42558224671\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(14\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			49.6
Character	\(\chi\)	\(=\)	429.49
Dual form			429.2.bn.a.394.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.280336 - 0.252415i) q^{2} +(0.104528 - 0.994522i) q^{3} +(-0.194182 - 1.84752i) q^{4} +(-2.49942 - 0.812112i) q^{5} +(-0.280336 + 0.252415i) q^{6} +(-1.73582 + 0.182442i) q^{7} +(-0.855366 + 1.17731i) q^{8} +(-0.978148 - 0.207912i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 112 q - 14 q^{3} - 20 q^{4} + 14 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(67\)	\(79\)	\(287\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{5}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.280336	−	0.252415i	−0.198227	−	0.178485i	0.564039	−	0.825748i	\(-0.309247\pi\)
							−0.762266	+	0.647263i	\(0.775913\pi\)
\(3\)	0.104528	−	0.994522i	0.0603495	−	0.574187i
							\(5\)	−2.49942	−	0.812112i	−1.11778	−	0.363188i	−0.308858	−	0.951108i	\(-0.599947\pi\)
−0.808919	+	0.587921i	\(0.799947\pi\)
\(7\)	−1.73582	+	0.182442i	−0.656077	+	0.0689565i	−0.426717	−	0.904385i	\(-0.640330\pi\)
							−0.229360	+	0.973342i	\(0.573663\pi\)
\(11\)	2.46457	+	2.21944i	0.743095	+	0.669186i
							\(13\)	−3.32953	−	1.38355i	−0.923446	−	0.383728i
\(17\)	3.49809	+	3.88502i	0.848411	+	0.942256i								0.998925	−	0.0463516i	\(-0.0147594\pi\)
							−0.150514	+	0.988608i	\(0.548093\pi\)
\(19\)	−0.529392	−	1.18903i	−0.121451	−	0.272783i	0.842581	−	0.538569i	\(-0.181035\pi\)
							−0.964032	+	0.265787i	\(0.914368\pi\)
\(23\)	−0.314029	−	0.543914i	−0.0654796	−	0.113414i	0.831427	−	0.555634i	\(-0.187524\pi\)
							−0.896907	+	0.442220i	\(0.854191\pi\)
\(29\)	−0.814948	−	0.362838i	−0.151332	−	0.0673774i	0.329673	−	0.944095i	\(-0.393061\pi\)
							−0.481005	+	0.876718i	\(0.659728\pi\)
\(31\)	−7.19549	+	2.33796i	−1.29235	+	0.419909i	−0.872913	−	0.487876i	\(-0.837772\pi\)
							−0.419435	+	0.907786i	\(0.637772\pi\)
\(37\)	1.96389	−	4.41096i	0.322861	−	0.725158i	−0.677081	−	0.735908i	\(-0.736755\pi\)
							0.999942	+	0.0107507i	\(0.00342211\pi\)
\(41\)	−7.99106	−	0.839894i	−1.24799	−	0.131169i	−0.542603	−	0.839989i	\(-0.682561\pi\)
							−0.705390	+	0.708820i	\(0.749228\pi\)
\(43\)	4.20790	−	7.28829i	0.641698	−	1.11145i	−0.343355	−	0.939206i	\(-0.611563\pi\)
							0.985054	−	0.172249i	\(-0.0551032\pi\)
\(47\)	−2.57441	+	3.54337i	−0.375516	+	0.516853i	−0.954390	−	0.298564i	\(-0.903492\pi\)
							0.578874	+	0.815417i	\(0.303492\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	429.2.bn.a.49.6	✓	112
11.9	even	5	inner	429.2.bn.a.361.9	yes	112
13.4	even	6	inner	429.2.bn.a.82.9	yes	112
143.108	even	30	inner	429.2.bn.a.394.6	yes	112

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.bn.a.49.6	✓	112	1.1	even	1	trivial
429.2.bn.a.82.9	yes	112	13.4	even	6	inner
429.2.bn.a.361.9	yes	112	11.9	even	5	inner
429.2.bn.a.394.6	yes	112	143.108	even	30	inner