Embedded newform 403.2.bp.a.10.3

• Label: 403.2.bp.a.10.3
• Level: $403$
• Weight: $2$
• Character: 403.10
• 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.bp (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			10.3
Character	\(\chi\)	\(=\)	403.10
Dual form			403.2.bp.a.121.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.60859 - 0.274174i) q^{2} +(2.47351 - 1.79711i) q^{3} +(4.77329 + 1.01459i) q^{4} +(0.288360 - 0.166485i) q^{5} +(-6.94509 + 4.00975i) q^{6} +(2.54091 + 2.28784i) q^{7} +(-7.18423 - 2.33430i) q^{8} +(1.96159 - 6.03714i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 280 q - 9 q^{2} - 9 q^{3} - 35 q^{4} - 21 q^{6} - 3 q^{7} - 45 q^{8} - 63 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(e\left(\frac{7}{15}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.60859	−	0.274174i	−1.84455	−	0.193870i	−0.883165	−	0.469063i	\(-0.844592\pi\)
							−0.961389	+	0.275192i	\(0.911258\pi\)
\(3\)	2.47351	−	1.79711i	1.42808	−	1.03756i	0.437709	−	0.899117i	\(-0.355790\pi\)
							0.990370	−	0.138443i	\(-0.0442099\pi\)
\(5\)	0.288360	−	0.166485i	0.128958	−	0.0744541i	−0.434133	−	0.900849i	\(-0.642945\pi\)
							0.563091	+	0.826395i	\(0.309612\pi\)
\(7\)	2.54091	+	2.28784i	0.960373	+	0.864724i	0.990871	−	0.134811i	\(-0.0430426\pi\)
							−0.0304980	+	0.999535i	\(0.509709\pi\)
\(11\)	−0.139129	+	0.654549i	−0.0419489	+	0.197354i	−0.994132	−	0.108174i	\(-0.965500\pi\)
							0.952183	+	0.305528i	\(0.0988330\pi\)
\(13\)	2.20777	+	2.85057i	0.612325	+	0.790606i
							\(17\)	5.81023	−	1.23500i	1.40919	−	0.299532i	0.560381	−	0.828235i	\(-0.310655\pi\)
0.848807	+	0.528703i	\(0.177322\pi\)
\(19\)	−0.510817	−	0.0536890i	−0.117189	−	0.0123171i	0.0457522	−	0.998953i	\(-0.485432\pi\)
							−0.162942	+	0.986636i	\(0.552098\pi\)
\(23\)	−5.56512	+	1.18290i	−1.16041	+	0.246652i	−0.747590	−	0.664161i	\(-0.768789\pi\)
							−0.412818	+	0.910813i	\(0.635456\pi\)
\(29\)	−0.0397337	+	0.378041i	−0.00737836	+	0.0702004i	−0.997591	−	0.0693631i	\(-0.977903\pi\)
							0.990213	+	0.139564i	\(0.0445700\pi\)
\(31\)	−3.75347	+	4.11236i	−0.674143	+	0.738601i
							\(37\)		−	7.04751i		−	1.15860i	−0.815113	−	0.579301i	\(-0.803325\pi\)
0.815113	−	0.579301i	\(-0.196675\pi\)
\(41\)	−0.610143	−	1.37040i	−0.0952883	−	0.214021i	0.859612	−	0.510947i	\(-0.170705\pi\)
							−0.954900	+	0.296926i	\(0.904039\pi\)
\(43\)	0.850575	−	8.09268i	0.129712	−	1.23412i	−0.715086	−	0.699036i	\(-0.753612\pi\)
							0.844798	−	0.535086i	\(-0.179721\pi\)
\(47\)	2.97707	+	4.09759i	0.434250	+	0.597694i	0.968922	−	0.247365i	\(-0.0795648\pi\)
							−0.534672	+	0.845060i	\(0.679565\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bp.a.10.3	✓	280
13.4	even	6		403.2.bt.a.134.3	yes	280
31.28	even	15		403.2.bt.a.400.3	yes	280
403.121	even	30	inner	403.2.bp.a.121.3	yes	280

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.bp.a.10.3	✓	280	1.1	even	1	trivial
403.2.bp.a.121.3	yes	280	403.121	even	30	inner
403.2.bt.a.134.3	yes	280	13.4	even	6
403.2.bt.a.400.3	yes	280	31.28	even	15