Embedded newform 37.10.c.a.10.12

• Label: 37.10.c.a.10.12
• 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			10.12
Character	\(\chi\)	\(=\)	37.10
Dual form			37.10.c.a.26.12

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.64814 + 2.85467i) q^{2} +(22.5023 + 38.9752i) q^{3} +(250.567 + 433.995i) q^{4} +(-142.433 - 246.702i) q^{5} -148.348 q^{6} +(-5192.34 - 8993.40i) q^{7} -3339.58 q^{8} +(8828.79 - 15291.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 32 q^{2} + 73 q^{3} - 6486 q^{4} + 2729 q^{5} + 1020 q^{6} + 4231 q^{7} - 7848 q^{8} - 140840 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.64814	+	2.85467i	−0.0728384	+	0.126160i	−0.900144	−	0.435592i	\(-0.856539\pi\)
							0.827306	+	0.561752i	\(0.189872\pi\)
\(3\)	22.5023	+	38.9752i	0.160392	+	0.277806i	0.935009	−	0.354623i	\(-0.115391\pi\)
							−0.774618	+	0.632430i	\(0.782058\pi\)
\(5\)	−142.433	−	246.702i	−0.101917	−	0.176526i	0.810557	−	0.585659i	\(-0.199164\pi\)
							−0.912474	+	0.409134i	\(0.865831\pi\)
\(7\)	−5192.34	−	8993.40i	−0.817376	−	1.41574i	−0.907609	−	0.419817i	\(-0.862094\pi\)
							0.0902323	−	0.995921i	\(-0.471239\pi\)
\(11\)	−8219.66			−0.169273			−0.0846363	−	0.996412i	\(-0.526973\pi\)
							−0.0846363	+	0.996412i	\(0.526973\pi\)
\(13\)	23278.0	+	40318.6i	0.226048	+	0.391526i	0.956633	−	0.291295i	\(-0.0940862\pi\)
							−0.730586	+	0.682821i	\(0.760753\pi\)
\(17\)	297056.	−	514516.i	0.862618	−	1.49410i	−0.00677571	−	0.999977i	\(-0.502157\pi\)
							0.869393	−	0.494121i	\(-0.164510\pi\)
\(19\)	−69872.1	−	121022.i	−0.123002	−	0.213046i	0.797948	−	0.602726i	\(-0.205919\pi\)
							−0.920950	+	0.389680i	\(0.872586\pi\)
\(23\)	135302.			0.100816			0.0504081	−	0.998729i	\(-0.483948\pi\)
							0.0504081	+	0.998729i	\(0.483948\pi\)
\(29\)	−627001.			−0.164618			−0.0823090	−	0.996607i	\(-0.526229\pi\)
							−0.0823090	+	0.996607i	\(0.526229\pi\)
\(31\)	2.68863e6			0.522882			0.261441	−	0.965219i	\(-0.415802\pi\)
							0.261441	+	0.965219i	\(0.415802\pi\)
\(37\)	5.79733e6	−	9.81594e6i	0.508535	−	0.861042i
							\(41\)	−9.87637e6	−	1.71064e7i	−0.545846	−	0.945433i	−0.998553	−	0.0537734i	\(-0.982875\pi\)
0.452707	−	0.891659i	\(-0.350458\pi\)
\(43\)	−1.02245e7			−0.456074			−0.228037	−	0.973652i	\(-0.573231\pi\)
							−0.228037	+	0.973652i	\(0.573231\pi\)
\(47\)	2.23054e7			0.666761			0.333380	−	0.942792i	\(-0.391811\pi\)
							0.333380	+	0.942792i	\(0.391811\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.c.a.10.12	✓	54
37.26	even	3	inner	37.10.c.a.26.12	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.c.a.10.12	✓	54	1.1	even	1	trivial
37.10.c.a.26.12	yes	54	37.26	even	3	inner