Embedded newform 403.2.ba.a.6.4

• Label: 403.2.ba.a.6.4
• Level: $403$
• Weight: $2$
• Character: 403.6
• 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.ba (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			6.4
Character	\(\chi\)	\(=\)	403.6
Dual form			403.2.ba.a.336.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.616245 - 2.29986i) q^{2} -2.87995i q^{3} +(-3.17753 + 1.83455i) q^{4} +(-0.235608 - 0.879300i) q^{5} +(-6.62347 + 1.77475i) q^{6} +(-1.87834 + 0.503300i) q^{7} +(2.81012 + 2.81012i) q^{8} -5.29412 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 12 q^{4} - 2 q^{5} + 6 q^{6} + 12 q^{7} - 10 q^{8} - 124 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.616245	−	2.29986i	−0.435751	−	1.62624i	−0.739263	−	0.673417i	\(-0.764826\pi\)
							0.303512	−	0.952828i	\(-0.401841\pi\)
\(3\)		−	2.87995i		−	1.66274i	−0.555719	−	0.831370i	\(-0.687557\pi\)
							0.555719	−	0.831370i	\(-0.312443\pi\)
\(5\)	−0.235608	−	0.879300i	−0.105367	−	0.393235i	0.893020	−	0.450018i	\(-0.148582\pi\)
							−0.998387	+	0.0567829i	\(0.981916\pi\)
\(7\)	−1.87834	+	0.503300i	−0.709946	+	0.190229i	−0.595681	−	0.803221i	\(-0.703118\pi\)
							−0.114265	+	0.993450i	\(0.536451\pi\)
\(11\)	2.56477	+	0.687228i	0.773308	+	0.207207i	0.623832	−	0.781558i	\(-0.285575\pi\)
							0.149475	+	0.988765i	\(0.452242\pi\)
\(13\)	1.72646	−	3.16533i	0.478834	−	0.877905i
							\(17\)	−1.73883	+	3.01173i	−0.421727	+	0.730453i	−0.996109	−	0.0881349i	\(-0.971909\pi\)
0.574381	+	0.818588i	\(0.305243\pi\)
\(19\)	−0.883630	−	3.29775i	−0.202719	−	0.756557i	−0.990133	−	0.140132i	\(-0.955247\pi\)
							0.787414	−	0.616424i	\(-0.211419\pi\)
\(23\)	−0.466472	+	0.807953i	−0.0972661	+	0.168470i	−0.910552	−	0.413394i	\(-0.864343\pi\)
							0.813286	+	0.581864i	\(0.197676\pi\)
\(29\)	−5.68854	−	3.28428i	−1.05634	−	0.609876i	−0.131920	−	0.991260i	\(-0.542114\pi\)
							−0.924417	+	0.381384i	\(0.875447\pi\)
\(31\)	−5.54271	−	0.527588i	−0.995500	−	0.0947577i
							\(37\)	4.85930	−	4.85930i	0.798864	−	0.798864i	−0.184053	−	0.982916i	\(-0.558922\pi\)
0.982916	+	0.184053i	\(0.0589217\pi\)
\(41\)	−2.73346	−	0.732428i	−0.426895	−	0.114386i	0.0389735	−	0.999240i	\(-0.487591\pi\)
							−0.465868	+	0.884854i	\(0.654258\pi\)
\(43\)	1.00246	−	1.73631i	0.152874	−	0.264785i	−0.779409	−	0.626515i	\(-0.784481\pi\)
							0.932283	+	0.361731i	\(0.117814\pi\)
\(47\)	7.48134	+	7.48134i	1.09127	+	1.09127i	0.995394	+	0.0958713i	\(0.0305637\pi\)
							0.0958713	+	0.995394i	\(0.469436\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ba.a.6.4	✓	140
13.11	odd	12		403.2.bf.a.37.4	yes	140
31.26	odd	6		403.2.bf.a.305.4	yes	140
403.336	even	12	inner	403.2.ba.a.336.4	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.4	✓	140	1.1	even	1	trivial
403.2.ba.a.336.4	yes	140	403.336	even	12	inner
403.2.bf.a.37.4	yes	140	13.11	odd	12
403.2.bf.a.305.4	yes	140	31.26	odd	6