Embedded newform 37.10.c.a.10.1

• Label: 37.10.c.a.10.1
• Level: $37$
• Weight: $10$
• Character: 37.10
• Analytic conductor: $19.056$
• Analytic rank: $0$
• Dimension: $54$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0563259381\)
Analytic rank:	\(0\)
Dimension:	\(54\)
Relative dimension:	\(27\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			10.1
Character	\(\chi\)	\(=\)	37.10
Dual form			37.10.c.a.26.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-21.2153 + 36.7459i) q^{2} +(86.0505 + 149.044i) q^{3} +(-644.175 - 1115.74i) q^{4} +(1139.39 + 1973.48i) q^{5} -7302.33 q^{6} +(4123.02 + 7141.28i) q^{7} +32941.0 q^{8} +(-4967.86 + 8604.59i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 32 q^{2} + 73 q^{3} - 6486 q^{4} + 2729 q^{5} + 1020 q^{6} + 4231 q^{7} - 7848 q^{8} - 140840 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−21.2153	+	36.7459i	−0.937591	+	1.62396i	−0.167645	+	0.985848i	\(0.553616\pi\)
							−0.769947	+	0.638108i	\(0.779717\pi\)
\(3\)	86.0505	+	149.044i	0.613349	+	1.06235i	0.990672	+	0.136270i	\(0.0435113\pi\)
							−0.377323	+	0.926082i	\(0.623155\pi\)
\(5\)	1139.39	+	1973.48i	0.815280	+	1.41211i	0.909127	+	0.416520i	\(0.136750\pi\)
							−0.0938468	+	0.995587i	\(0.529916\pi\)
\(7\)	4123.02	+	7141.28i	0.649045	+	1.12418i	0.983351	+	0.181714i	\(0.0581645\pi\)
							−0.334307	+	0.942464i	\(0.608502\pi\)
\(11\)	−88433.7			−1.82117			−0.910585	−	0.413321i	\(-0.864369\pi\)
							−0.910585	+	0.413321i	\(0.864369\pi\)
\(13\)	35278.9	+	61104.9i	0.342587	+	0.593378i	0.984912	−	0.173054i	\(-0.0553636\pi\)
							−0.642326	+	0.766432i	\(0.722030\pi\)
\(17\)	42850.1	−	74218.5i	0.124432	−	0.215522i	−0.797079	−	0.603875i	\(-0.793623\pi\)
							0.921511	+	0.388353i	\(0.126956\pi\)
\(19\)	37409.2	+	64794.7i	0.0658548	+	0.114064i	0.897073	−	0.441883i	\(-0.145689\pi\)
							−0.831218	+	0.555947i	\(0.812356\pi\)
\(23\)	2920.35			0.00217600			0.00108800	−	0.999999i	\(-0.499654\pi\)
							0.00108800	+	0.999999i	\(0.499654\pi\)
\(29\)	694304.			0.182288			0.0911441	−	0.995838i	\(-0.470948\pi\)
							0.0911441	+	0.995838i	\(0.470948\pi\)
\(31\)	6.32534e6			1.23014			0.615072	−	0.788471i	\(-0.289127\pi\)
							0.615072	+	0.788471i	\(0.289127\pi\)
\(37\)	1.06276e7	−	4.12493e6i	0.932243	−	0.361834i
							\(41\)	−1.31851e7	−	2.28372e7i	−0.728711	−	1.26216i	−0.957428	−	0.288671i	\(-0.906787\pi\)
0.228718	−	0.973493i	\(-0.426547\pi\)
\(43\)	3.40550e7			1.51905			0.759527	−	0.650476i	\(-0.225431\pi\)
							0.759527	+	0.650476i	\(0.225431\pi\)
\(47\)	−2.65360e7			−0.793223			−0.396611	−	0.917987i	\(-0.629814\pi\)
							−0.396611	+	0.917987i	\(0.629814\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.c.a.10.1	✓	54
37.26	even	3	inner	37.10.c.a.26.1	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.c.a.10.1	✓	54	1.1	even	1	trivial
37.10.c.a.26.1	yes	54	37.26	even	3	inner