Embedded newform 37.10.c.a.10.23

• Label: 37.10.c.a.10.23
• 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			10.23
Character	\(\chi\)	\(=\)	37.10
Dual form			37.10.c.a.26.23

$q$-expansion

\(f(q)\)	\(=\)	\(q+(15.6613 - 27.1261i) q^{2} +(119.437 + 206.872i) q^{3} +(-234.552 - 406.256i) q^{4} +(-908.086 - 1572.85i) q^{5} +7482.17 q^{6} +(-4896.80 - 8481.51i) q^{7} +1343.62 q^{8} +(-18689.1 + 32370.4i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 32 q^{2} + 73 q^{3} - 6486 q^{4} + 2729 q^{5} + 1020 q^{6} + 4231 q^{7} - 7848 q^{8} - 140840 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	15.6613	−	27.1261i	0.692138	−	1.19882i	−0.278998	−	0.960292i	\(-0.590002\pi\)
							0.971136	−	0.238526i	\(-0.0766643\pi\)
\(3\)	119.437	+	206.872i	0.851323	+	1.47454i	0.880015	+	0.474947i	\(0.157533\pi\)
							−0.0286913	+	0.999588i	\(0.509134\pi\)
\(5\)	−908.086	−	1572.85i	−0.649774	−	1.12544i	−0.983177	−	0.182657i	\(-0.941530\pi\)
							0.333403	−	0.942784i	\(-0.391803\pi\)
\(7\)	−4896.80	−	8481.51i	−0.770853	−	1.33516i	−0.937096	−	0.349071i	\(-0.886497\pi\)
							0.166244	−	0.986085i	\(-0.446836\pi\)
\(11\)	60900.9			1.25417			0.627085	−	0.778951i	\(-0.284248\pi\)
							0.627085	+	0.778951i	\(0.284248\pi\)
\(13\)	−66666.9	−	115470.i	−0.647389	−	1.12131i	−0.983744	−	0.179575i	\(-0.942528\pi\)
							0.336355	−	0.941735i	\(-0.390806\pi\)
\(17\)	130058.	−	225267.i	0.377674	−	0.654150i	−0.613050	−	0.790044i	\(-0.710058\pi\)
							0.990723	+	0.135895i	\(0.0433908\pi\)
\(19\)	11868.3	+	20556.5i	0.0208928	+	0.0361875i	0.876283	−	0.481797i	\(-0.160016\pi\)
							−0.855390	+	0.517985i	\(0.826682\pi\)
\(23\)	−533831.			−0.397767			−0.198883	−	0.980023i	\(-0.563731\pi\)
							−0.198883	+	0.980023i	\(0.563731\pi\)
\(29\)	−3.49933e6			−0.918743			−0.459372	−	0.888244i	\(-0.651925\pi\)
							−0.459372	+	0.888244i	\(0.651925\pi\)
\(31\)	3.89327e6			0.757159			0.378580	−	0.925569i	\(-0.376413\pi\)
							0.378580	+	0.925569i	\(0.376413\pi\)
\(37\)	1.13109e7	+	1.42346e6i	0.992174	+	0.124864i
							\(41\)	1.57435e7	+	2.72686e7i	0.870112	+	1.50708i	0.861880	+	0.507112i	\(0.169287\pi\)
0.00823225	+	0.999966i	\(0.497380\pi\)
\(43\)	−2.67263e7			−1.19215			−0.596075	−	0.802929i	\(-0.703274\pi\)
							−0.596075	+	0.802929i	\(0.703274\pi\)
\(47\)	−1.62723e7			−0.486416			−0.243208	−	0.969974i	\(-0.578200\pi\)
							−0.243208	+	0.969974i	\(0.578200\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.c.a.10.23	✓	54
37.26	even	3	inner	37.10.c.a.26.23	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.c.a.10.23	✓	54	1.1	even	1	trivial
37.10.c.a.26.23	yes	54	37.26	even	3	inner