Embedded newform 403.2.s.a.160.7

• Label: 403.2.s.a.160.7
• Level: $403$
• Weight: $2$
• Character: 403.160
• 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.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(70\)
Relative dimension:	\(35\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			160.7
Character	\(\chi\)	\(=\)	403.160
Dual form			403.2.s.a.335.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.86211 - 1.07509i) q^{2} +(-1.53180 - 2.65315i) q^{3} +(1.31164 + 2.27183i) q^{4} +(2.43528 - 1.40601i) q^{5} +6.58728i q^{6} -2.16759i q^{7} -1.34017i q^{8} +(-3.19279 + 5.53008i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 70 q - 6 q^{2} - 2 q^{3} + 30 q^{4} - 29 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{1}{6}\right)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.86211	−	1.07509i	−1.31671	−	0.760204i	−0.333514	−	0.942745i	\(-0.608234\pi\)
							−0.983198	+	0.182541i	\(0.941568\pi\)
\(3\)	−1.53180	−	2.65315i	−0.884382	−	1.53180i	−0.846420	−	0.532516i	\(-0.821247\pi\)
							−0.0379625	−	0.999279i	\(-0.512087\pi\)
\(5\)	2.43528	−	1.40601i	1.08909	−	0.628786i	0.155755	−	0.987796i	\(-0.450219\pi\)
							0.933334	+	0.359010i	\(0.116886\pi\)
\(7\)		−	2.16759i		−	0.819270i	−0.912249	−	0.409635i	\(-0.865656\pi\)
							0.912249	−	0.409635i	\(-0.134344\pi\)
\(11\)			0.744756i			0.224552i	0.993677	+	0.112276i	\(0.0358141\pi\)
							−0.993677	+	0.112276i	\(0.964186\pi\)
\(13\)	−3.57942	+	0.433270i	−0.992754	+	0.120167i
							\(17\)	−7.43498			−1.80325			−0.901624	−	0.432521i	\(-0.857624\pi\)
−0.901624	+	0.432521i	\(0.857624\pi\)
\(19\)			1.93828i			0.444672i	0.974970	+	0.222336i	\(0.0713682\pi\)
							−0.974970	+	0.222336i	\(0.928632\pi\)
\(23\)	0.142675	−	0.247120i	0.0297497	−	0.0515281i	−0.850767	−	0.525543i	\(-0.823862\pi\)
							0.880517	+	0.474015i	\(0.157196\pi\)
\(29\)	−2.55345	+	4.42271i	−0.474164	+	0.821276i	−0.999562	−	0.0295802i	\(-0.990583\pi\)
							0.525398	+	0.850856i	\(0.323916\pi\)
\(31\)	−3.35789	−	4.44123i	−0.603096	−	0.797669i
							\(37\)	3.04644	−	1.75887i	0.500832	−	0.289156i	−0.228225	−	0.973608i	\(-0.573292\pi\)
0.729057	+	0.684453i	\(0.239959\pi\)
\(41\)			6.92192i			1.08102i	0.841337	+	0.540511i	\(0.181769\pi\)
							−0.841337	+	0.540511i	\(0.818231\pi\)
\(43\)	8.05953			1.22907			0.614533	−	0.788891i	\(-0.289344\pi\)
							0.614533	+	0.788891i	\(0.289344\pi\)
\(47\)		−	7.33889i		−	1.07049i	−0.844698	−	0.535243i	\(-0.820220\pi\)
							0.844698	−	0.535243i	\(-0.179780\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.s.a.160.7	✓	70
13.10	even	6		403.2.v.a.36.7	yes	70
31.25	even	3		403.2.v.a.56.7	yes	70
403.335	even	6	inner	403.2.s.a.335.7	yes	70

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.s.a.160.7	✓	70	1.1	even	1	trivial
403.2.s.a.335.7	yes	70	403.335	even	6	inner
403.2.v.a.36.7	yes	70	13.10	even	6
403.2.v.a.56.7	yes	70	31.25	even	3