Embedded newform 403.2.bt.a.82.13

• Label: 403.2.bt.a.82.13
• Level: $403$
• Weight: $2$
• Character: 403.82
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $280$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(35\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			82.13
Character	\(\chi\)	\(=\)	403.82
Dual form			403.2.bt.a.231.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.190379 + 0.895663i) q^{2} +(-3.22061 + 0.684561i) q^{3} +(1.06112 + 0.472442i) q^{4} +(0.521441 - 0.301054i) q^{5} -3.01491i q^{6} +(0.972860 + 1.33903i) q^{7} +(-1.70160 + 2.34205i) q^{8} +(7.16305 - 3.18920i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 9 q^{2} - 3 q^{3} - 35 q^{4} - 15 q^{7} + 45 q^{8} + 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{4}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.190379	+	0.895663i	−0.134618	+	0.633329i	0.858162	+	0.513380i	\(0.171607\pi\)
							−0.992780	+	0.119950i	\(0.961727\pi\)
\(3\)	−3.22061	+	0.684561i	−1.85942	+	0.395232i	−0.994326	−	0.106380i	\(-0.966074\pi\)
							−0.865093	+	0.501611i	\(0.832741\pi\)
\(5\)	0.521441	−	0.301054i	0.233196	−	0.134636i	−0.378850	−	0.925458i	\(-0.623680\pi\)
							0.612045	+	0.790823i	\(0.290347\pi\)
\(7\)	0.972860	+	1.33903i	0.367706	+	0.506104i	0.952276	−	0.305239i	\(-0.0987365\pi\)
							−0.584569	+	0.811344i	\(0.698736\pi\)
\(11\)	2.73153	+	3.75963i	0.823588	+	1.13357i	0.989083	+	0.147361i	\(0.0470779\pi\)
							−0.165495	+	0.986211i	\(0.552922\pi\)
\(13\)	−3.28932	+	1.47662i	−0.912292	+	0.409541i
							\(17\)	−2.13174	−	1.54880i	−0.517022	−	0.375639i	0.298459	−	0.954423i	\(-0.403528\pi\)
−0.815481	+	0.578784i	\(0.803528\pi\)
\(19\)	4.03958	−	1.31254i	0.926744	−	0.301117i	0.193514	−	0.981098i	\(-0.438012\pi\)
							0.733231	+	0.679980i	\(0.238012\pi\)
\(23\)	−0.953229	+	0.424405i	−0.198762	+	0.0884945i	−0.503704	−	0.863876i	\(-0.668030\pi\)
							0.304942	+	0.952371i	\(0.401363\pi\)
\(29\)	−8.81777	−	1.87427i	−1.63742	−	0.348044i	−0.704939	−	0.709268i	\(-0.749026\pi\)
							−0.932479	+	0.361224i	\(0.882359\pi\)
\(31\)	−5.53154	+	0.634078i	−0.993494	+	0.113884i
							\(37\)	8.92952	−	5.15546i	1.46800	−	0.847553i	0.468647	−	0.883386i	\(-0.344742\pi\)
0.999358	+	0.0358327i	\(0.0114083\pi\)
\(41\)	−5.14592	+	1.67201i	−0.803658	+	0.261124i	−0.681909	−	0.731437i	\(-0.738850\pi\)
							−0.121749	+	0.992561i	\(0.538850\pi\)
\(43\)	1.39898	+	4.30562i	0.213343	+	0.656601i	0.999267	+	0.0382786i	\(0.0121874\pi\)
							−0.785924	+	0.618323i	\(0.787813\pi\)
\(47\)	0.806935	+	0.262189i	0.117704	+	0.0382442i	0.367277	−	0.930112i	\(-0.380290\pi\)
							−0.249573	+	0.968356i	\(0.580290\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bt.a.82.13	yes	280
13.10	even	6		403.2.bp.a.361.23	yes	280
31.14	even	15		403.2.bp.a.355.23	✓	280
403.231	even	30	inner	403.2.bt.a.231.13	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bp.a.355.23	✓	280	31.14	even	15
403.2.bp.a.361.23	yes	280	13.10	even	6
403.2.bt.a.82.13	yes	280	1.1	even	1	trivial
403.2.bt.a.231.13	yes	280	403.231	even	30	inner