Embedded newform 403.2.bk.a.9.8

• Label: 403.2.bk.a.9.8
• Level: $403$
• Weight: $2$
• Character: 403.9
• 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.bk (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			9.8
Character	\(\chi\)	\(=\)	403.9
Dual form			403.2.bk.a.224.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.24715 - 1.38510i) q^{2} +(-0.607940 + 0.675186i) q^{3} +(-0.154065 + 1.46583i) q^{4} +(1.19707 + 2.07339i) q^{5} +1.69340 q^{6} +(-2.18489 - 1.58741i) q^{7} +(-0.793286 + 0.576356i) q^{8} +(0.227301 + 2.16262i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 3 q^{2} - 9 q^{3} + 29 q^{4} - 8 q^{5} - 10 q^{6} - 13 q^{7} - 29 q^{8} + 24 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{1}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.24715	−	1.38510i	−0.881871	−	0.979417i	0.118037	−	0.993009i	\(-0.462340\pi\)
							−0.999908	+	0.0135925i	\(0.995673\pi\)
\(3\)	−0.607940	+	0.675186i	−0.350994	+	0.389819i	−0.892627	−	0.450797i	\(-0.851140\pi\)
							0.541632	+	0.840616i	\(0.317807\pi\)
\(5\)	1.19707	+	2.07339i	0.535346	+	0.927247i	0.999147	+	0.0413069i	\(0.0131521\pi\)
							−0.463800	+	0.885940i	\(0.653515\pi\)
\(7\)	−2.18489	−	1.58741i	−0.825810	−	0.599986i	0.0925607	−	0.995707i	\(-0.470495\pi\)
							−0.918371	+	0.395721i	\(0.870495\pi\)
\(11\)	−4.25150	−	3.08890i	−1.28188	−	0.931338i	−0.282268	−	0.959335i	\(-0.591087\pi\)
							−0.999608	+	0.0279978i	\(0.991087\pi\)
\(13\)	1.71232	−	3.17301i	0.474911	−	0.880034i
							\(17\)	4.86569	−	3.53513i	1.18010	−	0.857395i	0.187919	−	0.982184i	\(-0.439826\pi\)
0.992183	+	0.124790i	\(0.0398257\pi\)
\(19\)	1.62281	−	4.99451i	0.372299	−	1.14582i	−0.572983	−	0.819567i	\(-0.694214\pi\)
							0.945283	−	0.326252i	\(-0.105786\pi\)
\(23\)	−0.605147	−	5.75759i	−0.126182	−	1.20054i	−0.856028	−	0.516929i	\(-0.827075\pi\)
							0.729846	−	0.683612i	\(-0.239592\pi\)
\(29\)	2.19539	+	2.43823i	0.407674	+	0.452768i	0.911661	−	0.410943i	\(-0.134801\pi\)
							−0.503987	+	0.863711i	\(0.668134\pi\)
\(31\)	−5.17844	−	2.04543i	−0.930075	−	0.367371i
							\(37\)	−0.184825	−	0.320127i	−0.0303851	−	0.0526285i	0.850433	−	0.526083i	\(-0.176340\pi\)
−0.880818	+	0.473455i	\(0.843007\pi\)
\(41\)	−1.27093	+	3.91152i	−0.198486	+	0.610877i	0.801432	+	0.598086i	\(0.204072\pi\)
							−0.999918	+	0.0127914i	\(0.995928\pi\)
\(43\)	0.629032	−	1.93596i	0.0959264	−	0.295231i	−0.891568	−	0.452888i	\(-0.850394\pi\)
							0.987494	+	0.157656i	\(0.0503939\pi\)
\(47\)	1.52333	+	4.68831i	0.222200	+	0.683861i	0.998564	+	0.0535758i	\(0.0170619\pi\)
							−0.776364	+	0.630285i	\(0.782938\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bk.a.9.8	yes	280
13.3	even	3		403.2.bj.a.133.28	yes	280
31.7	even	15		403.2.bj.a.100.28	✓	280
403.224	even	15	inner	403.2.bk.a.224.8	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bj.a.100.28	✓	280	31.7	even	15
403.2.bj.a.133.28	yes	280	13.3	even	3
403.2.bk.a.9.8	yes	280	1.1	even	1	trivial
403.2.bk.a.224.8	yes	280	403.224	even	15	inner