Embedded newform 403.2.y.a.64.12

• Label: 403.2.y.a.64.12
• Level: $403$
• Weight: $2$
• Character: 403.64
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.y (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(34\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

Embedding invariants

Embedding label			64.12
Character	\(\chi\)	\(=\)	403.64
Dual form			403.2.y.a.233.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.618027 + 0.850641i) q^{2} +(2.69208 - 1.95591i) q^{3} +(0.276401 + 0.850674i) q^{4} +1.05598i q^{5} +3.49880i q^{6} +(0.656905 - 0.213441i) q^{7} +(-2.89442 - 0.940454i) q^{8} +(2.49466 - 7.67776i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 10 q^{3} + 22 q^{4} - 32 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(-1\)	\(e\left(\frac{4}{5}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.618027	+	0.850641i	−0.437011	+	0.601494i	−0.969545	−	0.244915i	\(-0.921240\pi\)
							0.532533	+	0.846409i	\(0.321240\pi\)
\(3\)	2.69208	−	1.95591i	1.55427	−	1.12925i	0.613757	−	0.789495i	\(-0.289657\pi\)
							0.940516	−	0.339751i	\(-0.110343\pi\)
\(5\)			1.05598i			0.472247i	0.971723	+	0.236124i	\(0.0758770\pi\)
							−0.971723	+	0.236124i	\(0.924123\pi\)
\(7\)	0.656905	−	0.213441i	0.248287	−	0.0806732i	−0.182230	−	0.983256i	\(-0.558331\pi\)
							0.430516	+	0.902583i	\(0.358331\pi\)
\(11\)	5.87560	−	1.90910i	1.77156	−	0.575614i	0.773269	−	0.634078i	\(-0.218620\pi\)
							0.998290	+	0.0584632i	\(0.0186200\pi\)
\(13\)	−3.56881	−	0.513386i	−0.989811	−	0.142388i
							\(17\)	−0.546630	+	1.68235i	−0.132577	+	0.408031i	−0.995205	−	0.0978078i	\(-0.968817\pi\)
0.862628	+	0.505839i	\(0.168817\pi\)
\(19\)	−1.31500	+	1.80995i	−0.301683	+	0.415230i	−0.932765	−	0.360486i	\(-0.882611\pi\)
							0.631082	+	0.775716i	\(0.282611\pi\)
\(23\)	−1.86926	+	5.75299i	−0.389768	+	1.19958i	0.543194	+	0.839607i	\(0.317215\pi\)
							−0.932962	+	0.359975i	\(0.882785\pi\)
\(29\)	0.776731	+	0.564328i	0.144235	+	0.104793i	0.657563	−	0.753400i	\(-0.271587\pi\)
							−0.513328	+	0.858193i	\(0.671587\pi\)
\(31\)	−3.21081	+	4.54870i	−0.576678	+	0.816971i
							\(37\)			6.06658i			0.997340i	0.866792	+	0.498670i	\(0.166178\pi\)
−0.866792	+	0.498670i	\(0.833822\pi\)
\(41\)	4.50935	−	6.20658i	0.704242	−	0.969305i	−0.295660	−	0.955293i	\(-0.595540\pi\)
							0.999902	−	0.0140122i	\(-0.00446038\pi\)
\(43\)	−8.23914	−	5.98608i	−1.25646	−	0.912869i	−0.257879	−	0.966177i	\(-0.583023\pi\)
							−0.998578	+	0.0533081i	\(0.983023\pi\)
\(47\)	−2.65074	−	3.64843i	−0.386650	−	0.532178i	0.570681	−	0.821172i	\(-0.306679\pi\)
							−0.957331	+	0.288994i	\(0.906679\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.y.a.64.12	✓	136
13.12	even	2	inner	403.2.y.a.64.23	yes	136
31.16	even	5	inner	403.2.y.a.233.23	yes	136
403.233	even	10	inner	403.2.y.a.233.12	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.y.a.64.12	✓	136	1.1	even	1	trivial
403.2.y.a.64.23	yes	136	13.12	even	2	inner
403.2.y.a.233.12	yes	136	403.233	even	10	inner
403.2.y.a.233.23	yes	136	31.16	even	5	inner