Embedded newform 37.6.c.a.10.5

• Label: 37.6.c.a.10.5
• Level: $37$
• Weight: $6$
• Character: 37.10
• Analytic conductor: $5.934$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.93420133308\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(15\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			10.5
Character	\(\chi\)	\(=\)	37.10
Dual form			37.6.c.a.26.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.35757 + 4.08344i) q^{2} +(-14.3898 - 24.9239i) q^{3} +(4.88369 + 8.45880i) q^{4} +(-24.5682 - 42.5534i) q^{5} +135.700 q^{6} +(109.024 + 188.835i) q^{7} -196.939 q^{8} +(-292.634 + 506.856i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 4 q^{2} + 16 q^{3} - 230 q^{4} - 51 q^{5} + 60 q^{6} + 50 q^{7} + 24 q^{8} - 1085 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.35757	+	4.08344i	−0.416764	+	0.721857i	−0.995612	−	0.0935790i	\(-0.970169\pi\)
							0.578848	+	0.815436i	\(0.303503\pi\)
\(3\)	−14.3898	−	24.9239i	−0.923107	−	1.59887i	−0.794578	−	0.607163i	\(-0.792308\pi\)
							−0.128529	−	0.991706i	\(-0.541026\pi\)
\(5\)	−24.5682	−	42.5534i	−0.439490	−	0.761218i	0.558161	−	0.829733i	\(-0.311507\pi\)
							−0.997650	+	0.0685147i	\(0.978174\pi\)
\(7\)	109.024	+	188.835i	0.840962	+	1.45659i	0.889082	+	0.457747i	\(0.151344\pi\)
							−0.0481204	+	0.998842i	\(0.515323\pi\)
\(11\)	122.317			0.304792			0.152396	−	0.988319i	\(-0.451301\pi\)
							0.152396	+	0.988319i	\(0.451301\pi\)
\(13\)	72.0534	+	124.800i	0.118249	+	0.204813i	0.919074	−	0.394086i	\(-0.128939\pi\)
							−0.800825	+	0.598898i	\(0.795605\pi\)
\(17\)	−810.097	+	1403.13i	−0.679852	+	1.17754i	0.295173	+	0.955444i	\(0.404623\pi\)
							−0.975025	+	0.222095i	\(0.928711\pi\)
\(19\)	917.642	+	1589.40i	0.583162	+	1.01007i	0.995102	+	0.0988555i	\(0.0315182\pi\)
							−0.411939	+	0.911211i	\(0.635148\pi\)
\(23\)	−1090.49			−0.429835			−0.214918	−	0.976632i	\(-0.568948\pi\)
							−0.214918	+	0.976632i	\(0.568948\pi\)
\(29\)	376.275			0.0830827			0.0415413	−	0.999137i	\(-0.486773\pi\)
							0.0415413	+	0.999137i	\(0.486773\pi\)
\(31\)	−4695.19			−0.877504			−0.438752	−	0.898608i	\(-0.644579\pi\)
							−0.438752	+	0.898608i	\(0.644579\pi\)
\(37\)	−6162.83	+	5600.32i	−0.740075	+	0.672525i
							\(41\)	4295.62	+	7440.24i	0.399086	+	0.691237i	0.993613	−	0.112838i	\(-0.0359941\pi\)
−0.594527	+	0.804075i	\(0.702661\pi\)
\(43\)	−13440.8			−1.10855			−0.554273	−	0.832335i	\(-0.687004\pi\)
							−0.554273	+	0.832335i	\(0.687004\pi\)
\(47\)	−20120.6			−1.32861			−0.664303	−	0.747463i	\(-0.731272\pi\)
							−0.664303	+	0.747463i	\(0.731272\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.c.a.10.5	✓	30
37.26	even	3	inner	37.6.c.a.26.5	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.c.a.10.5	✓	30	1.1	even	1	trivial
37.6.c.a.26.5	yes	30	37.26	even	3	inner