Embedded newform 403.2.bj.a.100.6

• Label: 403.2.bj.a.100.6
• Level: $403$
• Weight: $2$
• Character: 403.100
• 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.bj (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(280\)
Relative dimension:	\(35\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			100.6
Character	\(\chi\)	\(=\)	403.100
Dual form			403.2.bj.a.133.6

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.14008 - 0.454887i) q^{2} +(-0.792588 + 2.43933i) q^{3} +(2.54591 + 1.13351i) q^{4} +(1.99812 - 3.46084i) q^{5} +(2.80582 - 4.85982i) q^{6} +(-0.425169 - 4.04521i) q^{7} +(-1.39275 - 1.01189i) q^{8} +(-2.89510 - 2.10342i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 3 q^{2} + 3 q^{3} + 29 q^{4} - 8 q^{5} + 5 q^{6} - q^{7} - 29 q^{8} - 63 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{14}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.14008	−	0.454887i	−1.51326	−	0.321654i	−0.624866	−	0.780732i	\(-0.714847\pi\)
							−0.888396	+	0.459078i	\(0.848180\pi\)
\(3\)	−0.792588	+	2.43933i	−0.457601	+	1.40835i	0.410454	+	0.911881i	\(0.365370\pi\)
							−0.868055	+	0.496469i	\(0.834630\pi\)
\(5\)	1.99812	−	3.46084i	0.893584	−	1.54773i	0.0580375	−	0.998314i	\(-0.481516\pi\)
							0.835547	−	0.549419i	\(-0.185151\pi\)
\(7\)	−0.425169	−	4.04521i	−0.160699	−	1.52895i	−0.716474	−	0.697614i	\(-0.754245\pi\)
							0.555775	−	0.831333i	\(-0.312422\pi\)
\(11\)	−0.672915	−	0.299601i	−0.202892	−	0.0903331i	0.302775	−	0.953062i	\(-0.402087\pi\)
							−0.505667	+	0.862729i	\(0.668753\pi\)
\(13\)	−2.11235	+	2.92198i	−0.585862	+	0.810411i
							\(17\)	−1.80228	+	0.802426i	−0.437117	+	0.194617i	−0.613483	−	0.789708i	\(-0.710232\pi\)
0.176367	+	0.984325i	\(0.443566\pi\)
\(19\)	−6.58853	−	1.40043i	−1.51151	−	0.321282i	−0.623764	−	0.781613i	\(-0.714397\pi\)
							−0.887747	+	0.460331i	\(0.847731\pi\)
\(23\)	−4.96438	+	2.21028i	−1.03514	+	0.460876i	−0.852733	−	0.522346i	\(-0.825057\pi\)
							−0.182411	+	0.983222i	\(0.558390\pi\)
\(29\)	4.69323	+	0.997578i	0.871512	+	0.185246i	0.621896	−	0.783100i	\(-0.286362\pi\)
							0.249615	+	0.968345i	\(0.419696\pi\)
\(31\)	−4.41243	−	3.39565i	−0.792496	−	0.609876i
							\(37\)	−2.86796			−0.471490			−0.235745	−	0.971815i	\(-0.575753\pi\)
−0.235745	+	0.971815i	\(0.575753\pi\)
\(41\)	4.69228	−	5.21130i	0.732810	−	0.813868i	−0.255422	−	0.966830i	\(-0.582214\pi\)
							0.988232	+	0.152961i	\(0.0488810\pi\)
\(43\)	4.57309	+	0.972041i	0.697390	+	0.148235i	0.542947	−	0.839767i	\(-0.317308\pi\)
							0.154443	+	0.988002i	\(0.450642\pi\)
\(47\)	−2.25941	+	6.95374i	−0.329568	+	1.01431i	0.639768	+	0.768568i	\(0.279030\pi\)
							−0.969336	+	0.245739i	\(0.920970\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bj.a.100.6	✓	280
13.3	even	3		403.2.bk.a.224.30	yes	280
31.9	even	15		403.2.bk.a.9.30	yes	280
403.133	even	15	inner	403.2.bj.a.133.6	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bj.a.100.6	✓	280	1.1	even	1	trivial
403.2.bj.a.133.6	yes	280	403.133	even	15	inner
403.2.bk.a.9.30	yes	280	31.9	even	15
403.2.bk.a.224.30	yes	280	13.3	even	3