Embedded newform 403.2.e.a.191.7

• Label: 403.2.e.a.191.7
• Level: $403$
• Weight: $2$
• Character: 403.191
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $70$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			191.7
Character	\(\chi\)	\(=\)	403.191
Dual form			403.2.e.a.211.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.948089 + 1.64214i) q^{2} +(-0.203682 - 0.352788i) q^{3} +(-0.797747 - 1.38174i) q^{4} +(0.840173 + 1.45522i) q^{5} +0.772435 q^{6} -3.74155 q^{7} -0.767016 q^{8} +(1.41703 - 2.45436i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 2 q^{2} + 4 q^{3} - 34 q^{4} - 2 q^{5} + 8 q^{7} - 6 q^{8} - 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.948089	+	1.64214i	−0.670400	+	1.16117i	0.307390	+	0.951584i	\(0.400544\pi\)
							−0.977791	+	0.209584i	\(0.932789\pi\)
\(3\)	−0.203682	−	0.352788i	−0.117596	−	0.203682i	0.801219	−	0.598372i	\(-0.204185\pi\)
							−0.918814	+	0.394690i	\(0.870852\pi\)
\(5\)	0.840173	+	1.45522i	0.375737	+	0.650796i	0.990437	−	0.137965i	\(-0.0440563\pi\)
							−0.614700	+	0.788761i	\(0.710723\pi\)
\(7\)	−3.74155			−1.41417			−0.707086	−	0.707127i	\(-0.749991\pi\)
							−0.707086	+	0.707127i	\(0.749991\pi\)
\(11\)	−3.21871			−0.970477			−0.485238	−	0.874382i	\(-0.661267\pi\)
							−0.485238	+	0.874382i	\(0.661267\pi\)
\(13\)	3.09590	+	1.84808i	0.858648	+	0.512565i
							\(17\)	−5.54025			−1.34371			−0.671854	−	0.740684i	\(-0.734502\pi\)
−0.671854	+	0.740684i	\(0.734502\pi\)
\(19\)	−6.71616			−1.54079			−0.770397	−	0.637565i	\(-0.779942\pi\)
							−0.770397	+	0.637565i	\(0.779942\pi\)
\(23\)	−2.02107	+	3.50060i	−0.421423	+	0.729925i	−0.996079	−	0.0884695i	\(-0.971802\pi\)
							0.574656	+	0.818395i	\(0.305136\pi\)
\(29\)	1.34047	−	2.32175i	0.248918	−	0.431139i	−0.714308	−	0.699832i	\(-0.753258\pi\)
							0.963226	+	0.268693i	\(0.0865916\pi\)
\(31\)	0.484679	−	5.54663i	0.0870510	−	0.996204i
							\(37\)	1.14846	+	1.98919i	0.188806	+	0.327022i	0.944852	−	0.327496i	\(-0.106205\pi\)
−0.756046	+	0.654518i	\(0.772872\pi\)
\(41\)	−5.35339			−0.836059			−0.418029	−	0.908434i	\(-0.637279\pi\)
							−0.418029	+	0.908434i	\(0.637279\pi\)
\(43\)	0.134156			0.0204586			0.0102293	−	0.999948i	\(-0.496744\pi\)
							0.0102293	+	0.999948i	\(0.496744\pi\)
\(47\)	−2.34144			−0.341534			−0.170767	−	0.985311i	\(-0.554625\pi\)
							−0.170767	+	0.985311i	\(0.554625\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.e.a.191.7	✓	70
13.3	even	3		403.2.g.a.315.7	yes	70
31.25	even	3		403.2.g.a.87.7	yes	70
403.211	even	3	inner	403.2.e.a.211.7	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.e.a.191.7	✓	70	1.1	even	1	trivial
403.2.e.a.211.7	yes	70	403.211	even	3	inner
403.2.g.a.87.7	yes	70	31.25	even	3
403.2.g.a.315.7	yes	70	13.3	even	3