Embedded newform 403.2.bi.a.14.13

• Label: 403.2.bi.a.14.13
• Level: $403$
• Weight: $2$
• Character: 403.14
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bi (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(15\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

Embedding invariants

Embedding label			14.13
Character	\(\chi\)	\(=\)	403.14
Dual form			403.2.bi.a.144.13

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.523105 - 1.60995i) q^{2} +(0.991294 - 1.10094i) q^{3} +(-0.700275 - 0.508780i) q^{4} +(-1.03169 + 1.78693i) q^{5} +(-1.25392 - 2.17185i) q^{6} +(0.334356 - 3.18118i) q^{7} +(1.55359 - 1.12875i) q^{8} +(0.0841724 + 0.800847i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} + 2 q^{3} - 22 q^{4} + 13 q^{5} - 10 q^{6} - 16 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{11}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.523105	−	1.60995i	0.369891	−	1.13841i	−0.576970	−	0.816766i	\(-0.695765\pi\)
							0.946861	−	0.321643i	\(-0.104235\pi\)
\(3\)	0.991294	−	1.10094i	0.572324	−	0.635630i	−0.385595	−	0.922668i	\(-0.626004\pi\)
							0.957919	+	0.287038i	\(0.0926705\pi\)
\(5\)	−1.03169	+	1.78693i	−0.461384	+	0.799140i	−0.999030	−	0.0440302i	\(-0.985980\pi\)
							0.537646	+	0.843170i	\(0.319314\pi\)
\(7\)	0.334356	−	3.18118i	0.126375	−	1.20237i	−0.729055	−	0.684455i	\(-0.760040\pi\)
							0.855430	−	0.517919i	\(-0.173293\pi\)
\(11\)	4.45676	−	1.98428i	1.34376	−	0.598282i	0.396292	−	0.918124i	\(-0.370297\pi\)
							0.947471	+	0.319843i	\(0.103630\pi\)
\(13\)	0.978148	+	0.207912i	0.271289	+	0.0576643i
							\(17\)	−5.81467	−	2.58886i	−1.41026	−	0.627890i	−0.446533	−	0.894767i	\(-0.647342\pi\)
−0.963731	+	0.266877i	\(0.914008\pi\)
\(19\)	−6.12349	+	1.30159i	−1.40482	+	0.298605i	−0.847105	−	0.531425i	\(-0.821657\pi\)
							−0.557719	+	0.830030i	\(0.688323\pi\)
\(23\)	−0.870353	+	0.632349i	−0.181481	+	0.131854i	−0.674817	−	0.737985i	\(-0.735777\pi\)
							0.493336	+	0.869839i	\(0.335777\pi\)
\(29\)	−2.23105	+	6.86647i	−0.414296	+	1.27507i	0.498584	+	0.866842i	\(0.333854\pi\)
							−0.912879	+	0.408230i	\(0.866146\pi\)
\(31\)	5.56349	+	0.218143i	0.999232	+	0.0391797i
							\(37\)	−0.699066	−	1.21082i	−0.114926	−	0.199057i	0.802824	−	0.596216i	\(-0.203330\pi\)
−0.917750	+	0.397158i	\(0.869996\pi\)
\(41\)	4.37001	+	4.85338i	0.682480	+	0.757971i	0.980485	−	0.196593i	\(-0.0629878\pi\)
							−0.298005	+	0.954564i	\(0.596321\pi\)
\(43\)	4.71235	−	1.00164i	0.718626	−	0.152749i	0.165937	−	0.986136i	\(-0.446935\pi\)
							0.552689	+	0.833388i	\(0.313602\pi\)
\(47\)	0.624049	+	1.92063i	0.0910269	+	0.280152i	0.986198	−	0.165571i	\(-0.0529467\pi\)
							−0.895171	+	0.445723i	\(0.852947\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bi.a.14.13	✓	120
31.20	even	15	inner	403.2.bi.a.144.13	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bi.a.14.13	✓	120	1.1	even	1	trivial
403.2.bi.a.144.13	yes	120	31.20	even	15	inner