Embedded newform 37.6.e.a.11.14

• Label: 37.6.e.a.11.14
• Level: $37$
• Weight: $6$
• Character: 37.11
• Analytic conductor: $5.934$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	37.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.93420133308\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			11.14
Character	\(\chi\)	\(=\)	37.11
Dual form			37.6.e.a.27.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(7.81751 - 4.51344i) q^{2} +(10.9075 - 18.8923i) q^{3} +(24.7424 - 42.8550i) q^{4} +(44.4487 + 25.6624i) q^{5} -196.921i q^{6} +(-121.740 + 210.859i) q^{7} -157.833i q^{8} +(-116.446 - 201.690i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 6 q^{2} + 18 q^{3} + 298 q^{4} + 144 q^{5} - 52 q^{7} - 1490 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{5}{6}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	7.81751	−	4.51344i	1.38195	−	0.797872i	0.389564	−	0.921000i	\(-0.372626\pi\)
							0.992391	+	0.123128i	\(0.0392925\pi\)
\(3\)	10.9075	−	18.8923i	0.699714	−	1.21194i	−0.268851	−	0.963182i	\(-0.586644\pi\)
							0.968565	−	0.248759i	\(-0.0800228\pi\)
\(5\)	44.4487	+	25.6624i	0.795122	+	0.459064i	0.841763	−	0.539848i	\(-0.181518\pi\)
							−0.0466407	+	0.998912i	\(0.514852\pi\)
\(7\)	−121.740	+	210.859i	−0.939047	+	1.62648i	−0.171794	+	0.985133i	\(0.554956\pi\)
							−0.767253	+	0.641345i	\(0.778377\pi\)
\(11\)	−232.658			−0.579744			−0.289872	−	0.957065i	\(-0.593613\pi\)
							−0.289872	+	0.957065i	\(0.593613\pi\)
\(13\)	−178.272	−	102.926i	−0.292567	−	0.168914i	0.346532	−	0.938038i	\(-0.387359\pi\)
							−0.639099	+	0.769125i	\(0.720693\pi\)
\(17\)	914.649	−	528.073i	0.767595	−	0.443171i	−0.0644208	−	0.997923i	\(-0.520520\pi\)
							0.832016	+	0.554751i	\(0.187187\pi\)
\(19\)	−1642.03	−	948.028i	−1.04351	−	0.602473i	−0.122687	−	0.992445i	\(-0.539151\pi\)
							−0.920826	+	0.389973i	\(0.872484\pi\)
\(23\)		−	2473.79i		−	0.975086i	−0.873099	−	0.487543i	\(-0.837893\pi\)
							0.873099	−	0.487543i	\(-0.162107\pi\)
\(29\)		−	2938.01i		−	0.648721i	−0.945934	−	0.324360i	\(-0.894851\pi\)
							0.945934	−	0.324360i	\(-0.105149\pi\)
\(31\)			9896.04i			1.84951i	0.380561	+	0.924756i	\(0.375731\pi\)
							−0.380561	+	0.924756i	\(0.624269\pi\)
\(37\)	−7995.85	+	2326.02i	−0.960197	+	0.279324i
							\(41\)	9695.52	−	16793.1i	0.900765	−	1.56017i	0.0742623	−	0.997239i	\(-0.476340\pi\)
0.826503	−	0.562932i	\(-0.190327\pi\)
\(43\)			3753.24i			0.309553i	0.987950	+	0.154777i	\(0.0494657\pi\)
							−0.987950	+	0.154777i	\(0.950534\pi\)
\(47\)	20168.4			1.33176			0.665880	−	0.746059i	\(-0.268056\pi\)
							0.665880	+	0.746059i	\(0.268056\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.e.a.11.14	✓	32
37.27	even	6	inner	37.6.e.a.27.14	yes	32

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.e.a.11.14	✓	32	1.1	even	1	trivial
37.6.e.a.27.14	yes	32	37.27	even	6	inner