Embedded newform 403.2.bz.a.38.2

• Label: 403.2.bz.a.38.2
• Level: $403$
• Weight: $2$
• Character: 403.38
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bz (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(36\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			38.2
Character	\(\chi\)	\(=\)	403.38
Dual form			403.2.bz.a.350.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.44210 + 0.793486i) q^{2} +(0.151621 - 0.0322281i) q^{3} +(3.71619 - 2.69997i) q^{4} +(-1.21524 - 0.701620i) q^{5} +(-0.344702 + 0.199014i) q^{6} +(0.456028 - 1.02426i) q^{7} +(-3.91432 + 5.38760i) q^{8} +(-2.71869 + 1.21044i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 16 q^{3} + 60 q^{4} + 16 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/403\mathbb{Z}\right)^\times\).

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{14}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.44210	+	0.793486i	−1.72682	+	0.561079i	−0.992985	−	0.118243i	\(-0.962274\pi\)
							−0.733840	+	0.679322i	\(0.762274\pi\)
\(3\)	0.151621	−	0.0322281i	0.0875387	−	0.0186069i	−0.163934	−	0.986471i	\(-0.552418\pi\)
							0.251473	+	0.967864i	\(0.419085\pi\)
\(5\)	−1.21524	−	0.701620i	−0.543472	−	0.313774i	0.203013	−	0.979176i	\(-0.434927\pi\)
							−0.746485	+	0.665402i	\(0.768260\pi\)
\(7\)	0.456028	−	1.02426i	0.172362	−	0.387132i	−0.806622	−	0.591068i	\(-0.798707\pi\)
							0.978984	+	0.203936i	\(0.0653733\pi\)
\(11\)	−1.10421	+	0.116057i	−0.332933	+	0.0349926i	−0.269520	−	0.962995i	\(-0.586865\pi\)
							−0.0634126	+	0.997987i	\(0.520198\pi\)
\(13\)	−1.92703	−	3.04738i	−0.534463	−	0.845192i
							\(17\)	−0.544149	+	5.17723i	−0.131975	+	1.25566i	0.705311	+	0.708898i	\(0.250807\pi\)
−0.837286	+	0.546765i	\(0.815859\pi\)
\(19\)	5.99259	+	5.39575i	1.37479	+	1.23787i	0.941671	+	0.336535i	\(0.109255\pi\)
							0.433123	+	0.901335i	\(0.357412\pi\)
\(23\)	5.43028	+	3.94533i	1.13229	+	0.822659i	0.986027	−	0.166587i	\(-0.0532745\pi\)
							0.146266	+	0.989245i	\(0.453275\pi\)
\(29\)	0.140846	+	0.433478i	0.0261544	+	0.0804949i	0.963282	−	0.268493i	\(-0.0865255\pi\)
							−0.937127	+	0.348988i	\(0.886526\pi\)
\(31\)	−0.759323	+	5.51574i	−0.136378	+	0.990657i
							\(37\)	1.19781	−	0.691559i	0.196920	−	0.113692i	−0.398298	−	0.917256i	\(-0.630399\pi\)
0.595218	+	0.803564i	\(0.297066\pi\)
\(41\)	−1.72611	+	8.12070i	−0.269573	+	1.26824i	0.609972	+	0.792423i	\(0.291181\pi\)
							−0.879544	+	0.475817i	\(0.842153\pi\)
\(43\)	−6.05063	+	6.71991i	−0.922713	+	1.02478i	0.0769025	+	0.997039i	\(0.475497\pi\)
							−0.999615	+	0.0277376i	\(0.991170\pi\)
\(47\)	−7.60087	−	2.46967i	−1.10870	−	0.360239i	−0.303255	−	0.952909i	\(-0.598074\pi\)
							−0.805445	+	0.592671i	\(0.798074\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bz.a.38.2	✓	288
13.12	even	2	inner	403.2.bz.a.38.35	yes	288
31.9	even	15	inner	403.2.bz.a.350.35	yes	288
403.350	even	30	inner	403.2.bz.a.350.2	yes	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bz.a.38.2	✓	288	1.1	even	1	trivial
403.2.bz.a.38.35	yes	288	13.12	even	2	inner
403.2.bz.a.350.2	yes	288	403.350	even	30	inner
403.2.bz.a.350.35	yes	288	31.9	even	15	inner