Embedded newform 403.2.bf.a.37.16

• Label: 403.2.bf.a.37.16
• Level: $403$
• Weight: $2$
• Character: 403.37
• Analytic conductor: $3.218$
• Analytic rank: $0$
• Dimension: $140$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.bf (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.21797120146\)
Analytic rank:	\(0\)
Dimension:	\(140\)
Relative dimension:	\(35\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			37.16
Character	\(\chi\)	\(=\)	403.37
Dual form			403.2.bf.a.305.16

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.135765 + 0.506680i) q^{2} +(2.44353 + 1.41078i) q^{3} +(1.49376 + 0.862422i) q^{4} +(-0.993746 + 0.266274i) q^{5} +(-1.04656 + 1.04656i) q^{6} +(1.98004 - 1.98004i) q^{7} +(-1.38160 + 1.38160i) q^{8} +(2.48057 + 4.29648i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 6 q^{3} - 12 q^{4} - 2 q^{5} + 12 q^{6} - 12 q^{7} - 10 q^{8} + 62 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(249\)	\(313\)
\(\chi(n)\)	\(e\left(\frac{7}{12}\right)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.135765	+	0.506680i	−0.0960000	+	0.358277i	−0.997169	−	0.0751906i	\(-0.976043\pi\)
							0.901169	+	0.433468i	\(0.142710\pi\)
\(3\)	2.44353	+	1.41078i	1.41078	+	0.814511i	0.995461	−	0.0951667i	\(-0.0303384\pi\)
							0.415314	+	0.909678i	\(0.363672\pi\)
\(5\)	−0.993746	+	0.266274i	−0.444417	+	0.119081i	−0.474086	−	0.880479i	\(-0.657221\pi\)
							0.0296690	+	0.999560i	\(0.490555\pi\)
\(7\)	1.98004	−	1.98004i	0.748385	−	0.748385i	−0.225791	−	0.974176i	\(-0.572497\pi\)
							0.974176	+	0.225791i	\(0.0724967\pi\)
\(11\)	−1.25352	−	1.25352i	−0.377950	−	0.377950i	0.492412	−	0.870362i	\(-0.336115\pi\)
							−0.870362	+	0.492412i	\(0.836115\pi\)
\(13\)	−2.57357	−	2.52522i	−0.713779	−	0.700371i
							\(17\)	3.04080			0.737503			0.368751	−	0.929528i	\(-0.379785\pi\)
0.368751	+	0.929528i	\(0.379785\pi\)
\(19\)	−0.177048	−	0.177048i	−0.0406176	−	0.0406176i	0.686506	−	0.727124i	\(-0.259143\pi\)
							−0.727124	+	0.686506i	\(0.759143\pi\)
\(23\)	−2.63588	−	4.56549i	−0.549620	−	0.951970i	−0.998300	−	0.0582775i	\(-0.981439\pi\)
							0.448680	−	0.893692i	\(-0.351894\pi\)
\(29\)	−7.26207	+	4.19276i	−1.34853	+	0.778575i	−0.988042	−	0.154188i	\(-0.950724\pi\)
							−0.360490	+	0.932763i	\(0.617391\pi\)
\(31\)	−1.21888	−	5.43271i	−0.218918	−	0.975743i
							\(37\)	9.06135	−	2.42798i	1.48968	−	0.399158i	0.580049	−	0.814582i	\(-0.303033\pi\)
0.909628	+	0.415424i	\(0.136367\pi\)
\(41\)	−3.94972	−	3.94972i	−0.616843	−	0.616843i	0.327878	−	0.944720i	\(-0.393667\pi\)
							−0.944720	+	0.327878i	\(0.893667\pi\)
\(43\)	−1.72567			−0.263163			−0.131581	−	0.991305i	\(-0.542005\pi\)
							−0.131581	+	0.991305i	\(0.542005\pi\)
\(47\)	−1.73502	+	1.73502i	−0.253079	+	0.253079i	−0.822232	−	0.569153i	\(-0.807271\pi\)
							0.569153	+	0.822232i	\(0.307271\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bf.a.37.16	yes	140
13.6	odd	12		403.2.ba.a.6.16	✓	140
31.26	odd	6		403.2.ba.a.336.16	yes	140
403.305	even	12	inner	403.2.bf.a.305.16	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.16	✓	140	13.6	odd	12
403.2.ba.a.336.16	yes	140	31.26	odd	6
403.2.bf.a.37.16	yes	140	1.1	even	1	trivial
403.2.bf.a.305.16	yes	140	403.305	even	12	inner