Embedded newform 403.2.y.a.64.3

• Label: 403.2.y.a.64.3
• 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			64.3
Character	\(\chi\)	\(=\)	403.64
Dual form			403.2.y.a.233.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.44841 + 1.99357i) q^{2} +(1.15476 - 0.838982i) q^{3} +(-1.25838 - 3.87289i) q^{4} -1.37198i q^{5} +3.51728i q^{6} +(-4.14268 + 1.34604i) q^{7} +(4.85636 + 1.57793i) q^{8} +(-0.297472 + 0.915526i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 136 q - 10 q^{3} + 22 q^{4} - 32 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.44841	+	1.99357i	−1.02418	+	1.40966i	−0.114950	+	0.993371i	\(0.536671\pi\)
							−0.909231	+	0.416293i	\(0.863329\pi\)
\(3\)	1.15476	−	0.838982i	0.666701	−	0.484386i	−0.202219	−	0.979340i	\(-0.564815\pi\)
							0.868919	+	0.494954i	\(0.164815\pi\)
\(5\)		−	1.37198i		−	0.613570i	−0.951779	−	0.306785i	\(-0.900747\pi\)
							0.951779	−	0.306785i	\(-0.0992533\pi\)
\(7\)	−4.14268	+	1.34604i	−1.56579	+	0.508755i	−0.958346	−	0.285610i	\(-0.907804\pi\)
							−0.607441	+	0.794365i	\(0.707804\pi\)
\(11\)	2.01656	−	0.655220i	0.608016	−	0.197556i	0.0112037	−	0.999937i	\(-0.496434\pi\)
							0.596812	+	0.802381i	\(0.296434\pi\)
\(13\)	0.185833	+	3.60076i	0.0515407	+	0.998671i
							\(17\)	−1.78086	+	5.48091i	−0.431921	+	1.32932i	0.464288	+	0.885684i	\(0.346310\pi\)
−0.896209	+	0.443632i	\(0.853690\pi\)
\(19\)	−0.849603	+	1.16938i	−0.194912	+	0.268274i	−0.895276	−	0.445513i	\(-0.853021\pi\)
							0.700363	+	0.713787i	\(0.253021\pi\)
\(23\)	−2.46523	+	7.58719i	−0.514036	+	1.58204i	0.270994	+	0.962581i	\(0.412648\pi\)
							−0.785030	+	0.619458i	\(0.787352\pi\)
\(29\)	−8.65140	−	6.28561i	−1.60652	−	1.16721i	−0.873245	−	0.487281i	\(-0.837989\pi\)
							−0.733280	−	0.679927i	\(-0.762011\pi\)
\(31\)	2.11857	+	5.14895i	0.380507	+	0.924778i
							\(37\)		−	2.41689i		−	0.397335i	−0.980067	−	0.198667i	\(-0.936339\pi\)
0.980067	−	0.198667i	\(-0.0636613\pi\)
\(41\)	−0.867022	+	1.19335i	−0.135406	+	0.186371i	−0.871335	−	0.490688i	\(-0.836746\pi\)
							0.735929	+	0.677058i	\(0.236746\pi\)
\(43\)	1.58653	+	1.15268i	0.241943	+	0.175782i	0.702148	−	0.712031i	\(-0.252224\pi\)
							−0.460205	+	0.887812i	\(0.652224\pi\)
\(47\)	−7.03168	−	9.67828i	−1.02568	−	1.41172i	−0.908147	−	0.418653i	\(-0.862503\pi\)
							−0.117529	−	0.993069i	\(-0.537497\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.y.a.64.3	✓	136
13.12	even	2	inner	403.2.y.a.64.32	yes	136
31.16	even	5	inner	403.2.y.a.233.32	yes	136
403.233	even	10	inner	403.2.y.a.233.3	yes	136

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.y.a.64.3	✓	136	1.1	even	1	trivial
403.2.y.a.64.32	yes	136	13.12	even	2	inner
403.2.y.a.233.3	yes	136	403.233	even	10	inner
403.2.y.a.233.32	yes	136	31.16	even	5	inner