Embedded newform 403.2.bi.a.14.10

• Label: 403.2.bi.a.14.10
• 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			14.10
Character	\(\chi\)	\(=\)	403.14
Dual form			403.2.bi.a.144.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.138780 - 0.427121i) q^{2} +(-2.06376 + 2.29204i) q^{3} +(1.45486 + 1.05702i) q^{4} +(0.873401 - 1.51278i) q^{5} +(0.692568 + 1.19956i) q^{6} +(0.471332 - 4.48443i) q^{7} +(1.38004 - 1.00266i) q^{8} +(-0.680746 - 6.47687i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 120 q + 6 q^{2} + 2 q^{3} - 22 q^{4} + 13 q^{5} - 10 q^{6} - 16 q^{7} - 6 q^{8} + 13 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.138780	−	0.427121i	0.0981322	−	0.302020i	−0.889925	−	0.456107i	\(-0.849244\pi\)
							0.988057	+	0.154087i	\(0.0492435\pi\)
\(3\)	−2.06376	+	2.29204i	−1.19151	+	1.32331i	−0.257411	+	0.966302i	\(0.582869\pi\)
							−0.934102	+	0.357007i	\(0.883797\pi\)
\(5\)	0.873401	−	1.51278i	0.390597	−	0.676534i	−0.601931	−	0.798548i	\(-0.705602\pi\)
							0.992528	+	0.122014i	\(0.0389353\pi\)
\(7\)	0.471332	−	4.48443i	0.178147	−	1.69495i	−0.431365	−	0.902177i	\(-0.641968\pi\)
							0.609512	−	0.792777i	\(-0.291365\pi\)
\(11\)	2.38887	−	1.06359i	0.720270	−	0.320685i	−0.0136630	−	0.999907i	\(-0.504349\pi\)
							0.733933	+	0.679222i	\(0.237683\pi\)
\(13\)	0.978148	+	0.207912i	0.271289	+	0.0576643i
							\(17\)	−6.03205	−	2.68564i	−1.46299	−	0.651364i	−0.487842	−	0.872932i	\(-0.662216\pi\)
−0.975145	+	0.221568i	\(0.928883\pi\)
\(19\)	3.52239	−	0.748708i	0.808093	−	0.171765i	0.214702	−	0.976680i	\(-0.431122\pi\)
							0.593391	+	0.804914i	\(0.297789\pi\)
\(23\)	3.47037	−	2.52137i	0.723622	−	0.525742i	−0.163917	−	0.986474i	\(-0.552413\pi\)
							0.887539	+	0.460732i	\(0.152413\pi\)
\(29\)	−2.52209	+	7.76221i	−0.468341	+	1.44141i	0.386391	+	0.922335i	\(0.373722\pi\)
							−0.854732	+	0.519070i	\(0.826278\pi\)
\(31\)	−0.824047	−	5.50645i	−0.148003	−	0.988987i
							\(37\)	2.76169	+	4.78338i	0.454018	+	0.786383i	0.998631	−	0.0523046i	\(-0.0166567\pi\)
−0.544613	+	0.838688i	\(0.683323\pi\)
\(41\)	0.174691	+	0.194014i	0.0272822	+	0.0302999i	0.756631	−	0.653842i	\(-0.226844\pi\)
							−0.729349	+	0.684142i	\(0.760177\pi\)
\(43\)	−5.43750	+	1.15578i	−0.829211	+	0.176254i	−0.602916	−	0.797805i	\(-0.705995\pi\)
							−0.226295	+	0.974059i	\(0.572661\pi\)
\(47\)	−2.15475	−	6.63163i	−0.314302	−	0.967323i	−0.976041	−	0.217588i	\(-0.930181\pi\)
							0.661739	−	0.749735i	\(-0.269819\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bi.a.14.10	✓	120
31.20	even	15	inner	403.2.bi.a.144.10	yes	120

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bi.a.14.10	✓	120	1.1	even	1	trivial
403.2.bi.a.144.10	yes	120	31.20	even	15	inner