Embedded newform 403.2.bf.a.37.5

• Label: 403.2.bf.a.37.5
• 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			37.5
Character	\(\chi\)	\(=\)	403.37
Dual form			403.2.bf.a.305.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.610803 + 2.27955i) q^{2} +(-2.48438 - 1.43436i) q^{3} +(-3.09121 - 1.78471i) q^{4} +(1.20512 - 0.322912i) q^{5} +(4.78716 - 4.78716i) q^{6} +(-0.172104 + 0.172104i) q^{7} +(2.61896 - 2.61896i) q^{8} +(2.61477 + 4.52891i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 140 q - 8 q^{2} - 6 q^{3} - 12 q^{4} - 2 q^{5} + 12 q^{6} - 12 q^{7} - 10 q^{8} + 62 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.610803	+	2.27955i	−0.431903	+	1.61188i	0.316468	+	0.948603i	\(0.397503\pi\)
							−0.748371	+	0.663280i	\(0.769164\pi\)
\(3\)	−2.48438	−	1.43436i	−1.43436	−	0.828127i	−0.436909	−	0.899506i	\(-0.643927\pi\)
							−0.997449	+	0.0713786i	\(0.977260\pi\)
\(5\)	1.20512	−	0.322912i	0.538947	−	0.144411i	0.0209305	−	0.999781i	\(-0.493337\pi\)
							0.518017	+	0.855370i	\(0.326670\pi\)
\(7\)	−0.172104	+	0.172104i	−0.0650493	+	0.0650493i	−0.738883	−	0.673834i	\(-0.764646\pi\)
							0.673834	+	0.738883i	\(0.264646\pi\)
\(11\)	4.15332	+	4.15332i	1.25227	+	1.25227i	0.954699	+	0.297573i	\(0.0961771\pi\)
							0.297573	+	0.954699i	\(0.403823\pi\)
\(13\)	0.116542	−	3.60367i	0.0323230	−	0.999477i
							\(17\)	−7.95833			−1.93018			−0.965089	−	0.261921i	\(-0.915644\pi\)
−0.965089	+	0.261921i	\(0.915644\pi\)
\(19\)	−0.291045	−	0.291045i	−0.0667704	−	0.0667704i	0.672933	−	0.739703i	\(-0.265034\pi\)
							−0.739703	+	0.672933i	\(0.765034\pi\)
\(23\)	2.70825	+	4.69083i	0.564710	+	0.978106i	0.997077	+	0.0764084i	\(0.0243453\pi\)
							−0.432367	+	0.901698i	\(0.642321\pi\)
\(29\)	−6.88627	+	3.97579i	−1.27875	+	0.738286i	−0.976618	−	0.214980i	\(-0.931031\pi\)
							−0.302131	+	0.953267i	\(0.597698\pi\)
\(31\)	−1.06092	+	5.46575i	−0.190547	+	0.981678i
							\(37\)	−2.08323	+	0.558200i	−0.342481	+	0.0917675i	−0.425960	−	0.904742i	\(-0.640064\pi\)
0.0834788	+	0.996510i	\(0.473397\pi\)
\(41\)	4.75688	+	4.75688i	0.742899	+	0.742899i	0.973135	−	0.230236i	\(-0.0739497\pi\)
							−0.230236	+	0.973135i	\(0.573950\pi\)
\(43\)	−0.131339			−0.0200291			−0.0100145	−	0.999950i	\(-0.503188\pi\)
							−0.0100145	+	0.999950i	\(0.503188\pi\)
\(47\)	−1.15733	+	1.15733i	−0.168814	+	0.168814i	−0.786458	−	0.617644i	\(-0.788087\pi\)
							0.617644	+	0.786458i	\(0.288087\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.bf.a.37.5	yes	140
13.6	odd	12		403.2.ba.a.6.5	✓	140
31.26	odd	6		403.2.ba.a.336.5	yes	140
403.305	even	12	inner	403.2.bf.a.305.5	yes	140

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.ba.a.6.5	✓	140	13.6	odd	12
403.2.ba.a.336.5	yes	140	31.26	odd	6
403.2.bf.a.37.5	yes	140	1.1	even	1	trivial
403.2.bf.a.305.5	yes	140	403.305	even	12	inner