Embedded newform 403.2.ba.a.6.19

• Label: 403.2.ba.a.6.19
• 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			6.19
Character	\(\chi\)	\(=\)	403.6
Dual form			403.2.ba.a.336.19

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0759990 + 0.283632i) q^{2} -1.83195i q^{3} +(1.65738 - 0.956888i) q^{4} +(-0.350763 - 1.30906i) q^{5} +(0.519599 - 0.139226i) q^{6} +(2.10731 - 0.564653i) q^{7} +(0.812630 + 0.812630i) q^{8} -0.356028 q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 12 q^{4} - 2 q^{5} + 6 q^{6} + 12 q^{7} - 10 q^{8} - 124 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0759990	+	0.283632i	0.0537394	+	0.200558i	0.987576	−	0.157141i	\(-0.0502278\pi\)
							−0.933837	+	0.357700i	\(0.883561\pi\)
\(3\)		−	1.83195i		−	1.05767i	−0.848723	−	0.528837i	\(-0.822628\pi\)
							0.848723	−	0.528837i	\(-0.177372\pi\)
\(5\)	−0.350763	−	1.30906i	−0.156866	−	0.585431i	−0.998938	−	0.0460669i	\(-0.985331\pi\)
							0.842073	−	0.539364i	\(-0.181335\pi\)
\(7\)	2.10731	−	0.564653i	0.796490	−	0.213419i	0.162448	−	0.986717i	\(-0.448061\pi\)
							0.634042	+	0.773298i	\(0.281395\pi\)
\(11\)	0.824279	+	0.220865i	0.248530	+	0.0665933i	0.380933	−	0.924603i	\(-0.375603\pi\)
							−0.132404	+	0.991196i	\(0.542269\pi\)
\(13\)	−2.03397	+	2.97707i	−0.564123	+	0.825691i
							\(17\)	−2.23565	+	3.87226i	−0.542224	+	0.939160i	0.456552	+	0.889697i	\(0.349084\pi\)
−0.998776	+	0.0494630i	\(0.984249\pi\)
\(19\)	−1.28674	−	4.80219i	−0.295199	−	1.10170i	−0.941059	−	0.338242i	\(-0.890168\pi\)
							0.645860	−	0.763455i	\(-0.276499\pi\)
\(23\)	−2.80996	+	4.86699i	−0.585916	+	1.01484i	0.408844	+	0.912604i	\(0.365932\pi\)
							−0.994761	+	0.102232i	\(0.967401\pi\)
\(29\)	0.319806	+	0.184640i	0.0593864	+	0.0342868i	0.529399	−	0.848373i	\(-0.322417\pi\)
							−0.470013	+	0.882660i	\(0.655751\pi\)
\(31\)	−4.50366	+	3.27368i	−0.808882	+	0.587971i
							\(37\)	−5.93867	+	5.93867i	−0.976311	+	0.976311i	−0.999726	−	0.0234145i	\(-0.992546\pi\)
0.0234145	+	0.999726i	\(0.492546\pi\)
\(41\)	−2.04371	−	0.547610i	−0.319174	−	0.0855223i	0.0956753	−	0.995413i	\(-0.469499\pi\)
							−0.414849	+	0.909890i	\(0.636166\pi\)
\(43\)	4.69135	−	8.12565i	0.715424	−	1.23915i	−0.247372	−	0.968921i	\(-0.579567\pi\)
							0.962796	−	0.270230i	\(-0.0870998\pi\)
\(47\)	5.16192	+	5.16192i	0.752943	+	0.752943i	0.975027	−	0.222085i	\(-0.0712861\pi\)
							−0.222085	+	0.975027i	\(0.571286\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.ba.a.6.19	✓	140
13.11	odd	12		403.2.bf.a.37.19	yes	140
31.26	odd	6		403.2.bf.a.305.19	yes	140
403.336	even	12	inner	403.2.ba.a.336.19	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.19	✓	140	1.1	even	1	trivial
403.2.ba.a.336.19	yes	140	403.336	even	12	inner
403.2.bf.a.37.19	yes	140	13.11	odd	12
403.2.bf.a.305.19	yes	140	31.26	odd	6