Embedded newform 403.2.r.a.218.10

• Label: 403.2.r.a.218.10
• Level: $403$
• Weight: $2$
• Character: 403.218
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(34\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			218.10
Character	\(\chi\)	\(=\)	403.218
Dual form			403.2.r.a.342.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.25179 + 0.722722i) q^{2} +(-0.0126133 - 0.0218468i) q^{3} +(0.0446555 - 0.0773455i) q^{4} +1.27994i q^{5} +(0.0315784 + 0.0182318i) q^{6} +(-1.82792 - 1.05535i) q^{7} -2.76180i q^{8} +(1.49968 - 2.59753i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 68 q + 32 q^{4} - 34 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)	\(1\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.25179	+	0.722722i	−0.885151	+	0.511042i	−0.872353	−	0.488876i	\(-0.837407\pi\)
							−0.0127974	+	0.999918i	\(0.504074\pi\)
\(3\)	−0.0126133	−	0.0218468i	−0.00728227	−	0.0126133i	0.862361	−	0.506293i	\(-0.168985\pi\)
							−0.869644	+	0.493680i	\(0.835651\pi\)
\(5\)			1.27994i			0.572407i	0.958169	+	0.286203i	\(0.0923933\pi\)
							−0.958169	+	0.286203i	\(0.907607\pi\)
\(7\)	−1.82792	−	1.05535i	−0.690890	−	0.398886i	0.113055	−	0.993589i	\(-0.463936\pi\)
							−0.803945	+	0.594703i	\(0.797270\pi\)
\(11\)	4.05786	−	2.34281i	1.22349	−	0.706383i	0.257830	−	0.966190i	\(-0.416993\pi\)
							0.965660	+	0.259808i	\(0.0836592\pi\)
\(13\)	−1.96656	−	3.02203i	−0.545424	−	0.838160i
							\(17\)	−2.15100	+	3.72564i	−0.521694	+	0.903601i	0.477987	+	0.878367i	\(0.341367\pi\)
−0.999682	+	0.0252344i	\(0.991967\pi\)
\(19\)	7.01839	+	4.05207i	1.61013	+	0.929609i	0.989339	+	0.145633i	\(0.0465218\pi\)
							0.620791	+	0.783976i	\(0.286812\pi\)
\(23\)	−3.14791	−	5.45233i	−0.656384	−	1.13689i	−0.981545	−	0.191232i	\(-0.938752\pi\)
							0.325161	−	0.945659i	\(-0.394581\pi\)
\(29\)	0.829903	+	1.43743i	0.154109	+	0.266925i	0.932734	−	0.360565i	\(-0.117416\pi\)
							−0.778625	+	0.627489i	\(0.784083\pi\)
\(31\)		−	1.00000i		−	0.179605i
							\(37\)	5.64554	−	3.25945i	0.928120	−	0.535851i	0.0419038	−	0.999122i	\(-0.486658\pi\)
0.886217	+	0.463271i	\(0.153324\pi\)
\(41\)	3.71005	−	2.14200i	0.579412	−	0.334524i	−0.181488	−	0.983393i	\(-0.558091\pi\)
							0.760900	+	0.648870i	\(0.224758\pi\)
\(43\)	2.51627	−	4.35831i	0.383728	−	0.664636i	−0.607864	−	0.794041i	\(-0.707974\pi\)
							0.991592	+	0.129405i	\(0.0413068\pi\)
\(47\)			6.71077i			0.978867i	0.872041	+	0.489433i	\(0.162796\pi\)
							−0.872041	+	0.489433i	\(0.837204\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.r.a.218.10	✓	68
13.2	odd	12		5239.2.a.r.1.8		34
13.4	even	6	inner	403.2.r.a.342.10	yes	68
13.11	odd	12		5239.2.a.q.1.27		34

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.r.a.218.10	✓	68	1.1	even	1	trivial
403.2.r.a.342.10	yes	68	13.4	even	6	inner
5239.2.a.q.1.27		34	13.11	odd	12
5239.2.a.r.1.8		34	13.2	odd	12