Embedded newform 403.2.bt.a.82.10

• Label: 403.2.bt.a.82.10
• 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			82.10
Character	\(\chi\)	\(=\)	403.82
Dual form			403.2.bt.a.231.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.285763 + 1.34441i) q^{2} +(1.01586 - 0.215927i) q^{3} +(0.101320 + 0.0451105i) q^{4} +(-1.34641 + 0.777351i) q^{5} +1.42743i q^{6} +(2.50350 + 3.44577i) q^{7} +(-1.70535 + 2.34722i) q^{8} +(-1.75529 + 0.781506i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 9 q^{2} - 3 q^{3} - 35 q^{4} - 15 q^{7} + 45 q^{8} + 24 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.285763	+	1.34441i	−0.202065	+	0.950640i	0.753862	+	0.657033i	\(0.228189\pi\)
							−0.955926	+	0.293606i	\(0.905144\pi\)
\(3\)	1.01586	−	0.215927i	0.586506	−	0.124666i	0.0949053	−	0.995486i	\(-0.469745\pi\)
							0.491601	+	0.870821i	\(0.336412\pi\)
\(5\)	−1.34641	+	0.777351i	−0.602134	+	0.347642i	−0.769881	−	0.638188i	\(-0.779684\pi\)
							0.167747	+	0.985830i	\(0.446351\pi\)
\(7\)	2.50350	+	3.44577i	0.946232	+	1.30238i	0.953181	+	0.302401i	\(0.0977883\pi\)
							−0.00694819	+	0.999976i	\(0.502212\pi\)
\(11\)	−0.971854	−	1.33764i	−0.293025	−	0.403314i	0.636969	−	0.770890i	\(-0.280188\pi\)
							−0.929994	+	0.367575i	\(0.880188\pi\)
\(13\)	−0.323154	−	3.59104i	−0.0896267	−	0.995975i
							\(17\)	1.59347	+	1.15772i	0.386473	+	0.280789i	0.764009	−	0.645206i	\(-0.223228\pi\)
−0.377536	+	0.925995i	\(0.623228\pi\)
\(19\)	6.65096	−	2.16103i	1.52583	−	0.495773i	0.578408	−	0.815748i	\(-0.303674\pi\)
							0.947426	+	0.319974i	\(0.103674\pi\)
\(23\)	−6.35442	+	2.82917i	−1.32499	+	0.589922i	−0.942552	−	0.334060i	\(-0.891581\pi\)
							−0.382435	+	0.923982i	\(0.624915\pi\)
\(29\)	4.05562	+	0.862049i	0.753110	+	0.160078i	0.568443	−	0.822723i	\(-0.307546\pi\)
							0.184667	+	0.982801i	\(0.440879\pi\)
\(31\)	−0.473907	+	5.54756i	−0.0851162	+	0.996371i
							\(37\)	6.65024	−	3.83952i	1.09329	−	0.631212i	0.158841	−	0.987304i	\(-0.449224\pi\)
0.934451	+	0.356092i	\(0.115891\pi\)
\(41\)	9.16941	−	2.97932i	1.43202	−	0.465292i	0.512621	−	0.858615i	\(-0.328675\pi\)
							0.919400	+	0.393323i	\(0.128675\pi\)
\(43\)	−2.16695	−	6.66919i	−0.330457	−	1.01704i	−0.968917	−	0.247387i	\(-0.920428\pi\)
							0.638460	−	0.769655i	\(-0.279572\pi\)
\(47\)	2.17925	+	0.708081i	0.317876	+	0.103284i	0.463609	−	0.886040i	\(-0.346554\pi\)
							−0.145733	+	0.989324i	\(0.546554\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bt.a.82.10	yes	280
13.10	even	6		403.2.bp.a.361.26	yes	280
31.14	even	15		403.2.bp.a.355.26	✓	280
403.231	even	30	inner	403.2.bt.a.231.10	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bp.a.355.26	✓	280	31.14	even	15
403.2.bp.a.361.26	yes	280	13.10	even	6
403.2.bt.a.82.10	yes	280	1.1	even	1	trivial
403.2.bt.a.231.10	yes	280	403.231	even	30	inner