Embedded newform 403.2.bs.a.101.10

• Label: 403.2.bs.a.101.10
• Level: $403$
• Weight: $2$
• Character: 403.101
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bs (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(36\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

Embedding invariants

Embedding label			101.10
Character	\(\chi\)	\(=\)	403.101
Dual form			403.2.bs.a.4.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.323679 - 1.52279i) q^{2} +(0.859963 - 0.955086i) q^{3} +(-0.387033 + 0.172318i) q^{4} -2.26589i q^{5} +(-1.73275 - 1.00040i) q^{6} +(-1.51411 - 3.40074i) q^{7} +(-1.44246 - 1.98538i) q^{8} +(0.140933 + 1.34089i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 288 q - 9 q^{2} - q^{3} - 39 q^{4} - 42 q^{6} - 15 q^{7} + 31 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{2}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.323679	−	1.52279i	−0.228876	−	1.07678i	−0.931084	−	0.364804i	\(-0.881136\pi\)
							0.702208	−	0.711971i	\(-0.252197\pi\)
\(3\)	0.859963	−	0.955086i	0.496500	−	0.551419i	−0.441858	−	0.897085i	\(-0.645680\pi\)
							0.938358	+	0.345666i	\(0.112347\pi\)
\(5\)		−	2.26589i		−	1.01334i	−0.862141	−	0.506669i	\(-0.830877\pi\)
							0.862141	−	0.506669i	\(-0.169123\pi\)
\(7\)	−1.51411	−	3.40074i	−0.572279	−	1.28536i	−0.935392	−	0.353614i	\(-0.884953\pi\)
							0.363113	−	0.931745i	\(-0.381714\pi\)
\(11\)	4.01759	+	0.422266i	1.21135	+	0.127318i	0.688582	−	0.725158i	\(-0.258233\pi\)
							0.522766	+	0.852476i	\(0.324900\pi\)
\(13\)	1.09913	+	3.43393i	0.304845	+	0.952402i
							\(17\)	0.487089	+	4.63434i	0.118136	+	1.12399i	0.879576	+	0.475758i	\(0.157826\pi\)
−0.761440	+	0.648236i	\(0.775507\pi\)
\(19\)	−0.0685638	+	0.0617351i	−0.0157296	+	0.0141630i	−0.676958	−	0.736022i	\(-0.736702\pi\)
							0.661228	+	0.750185i	\(0.270035\pi\)
\(23\)	−4.76474	−	2.12140i	−0.993516	−	0.442342i	−0.155411	−	0.987850i	\(-0.549670\pi\)
							−0.838106	+	0.545508i	\(0.816337\pi\)
\(29\)	9.74892	−	2.07220i	1.81033	−	0.384797i	0.826361	−	0.563141i	\(-0.190407\pi\)
							0.983968	+	0.178344i	\(0.0570739\pi\)
\(31\)	−3.29769	−	4.48612i	−0.592282	−	0.805731i
							\(37\)	−6.10868	+	3.52685i	−1.00426	+	0.579810i	−0.909506	−	0.415691i	\(-0.863540\pi\)
−0.0947543	+	0.995501i	\(0.530207\pi\)
\(41\)	−0.271676	−	1.27814i	−0.0424287	−	0.199611i	0.951828	−	0.306632i	\(-0.0992021\pi\)
							−0.994257	+	0.107021i	\(0.965869\pi\)
\(43\)	−5.54872	−	6.16248i	−0.846172	−	0.939769i	0.152651	−	0.988280i	\(-0.451219\pi\)
							−0.998823	+	0.0485110i	\(0.984552\pi\)
\(47\)	3.82893	−	1.24409i	0.558506	−	0.181470i	−0.0161426	−	0.999870i	\(-0.505139\pi\)
							0.574649	+	0.818400i	\(0.305139\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bs.a.101.10	yes	288
13.4	even	6	inner	403.2.bs.a.225.27	yes	288
31.4	even	5	inner	403.2.bs.a.283.27	yes	288
403.4	even	30	inner	403.2.bs.a.4.10	✓	288

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bs.a.4.10	✓	288	403.4	even	30	inner
403.2.bs.a.101.10	yes	288	1.1	even	1	trivial
403.2.bs.a.225.27	yes	288	13.4	even	6	inner
403.2.bs.a.283.27	yes	288	31.4	even	5	inner