Embedded newform 37.10.c.a.10.14

• Label: 37.10.c.a.10.14
• 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			10.14
Character	\(\chi\)	\(=\)	37.10
Dual form			37.10.c.a.26.14

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.152284 - 0.263764i) q^{2} +(-87.8341 - 152.133i) q^{3} +(255.954 + 443.325i) q^{4} +(209.008 + 362.012i) q^{5} -53.5031 q^{6} +(-2618.01 - 4534.52i) q^{7} +311.850 q^{8} +(-5588.16 + 9678.98i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 32 q^{2} + 73 q^{3} - 6486 q^{4} + 2729 q^{5} + 1020 q^{6} + 4231 q^{7} - 7848 q^{8} - 140840 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.152284	−	0.263764i	0.00673008	−	0.0116568i	−0.862641	−	0.505817i	\(-0.831191\pi\)
							0.869371	+	0.494160i	\(0.164524\pi\)
\(3\)	−87.8341	−	152.133i	−0.626062	−	1.08437i	−0.988334	−	0.152299i	\(-0.951332\pi\)
							0.362272	−	0.932072i	\(-0.382001\pi\)
\(5\)	209.008	+	362.012i	0.149554	+	0.259034i	0.931063	−	0.364860i	\(-0.118883\pi\)
							−0.781509	+	0.623894i	\(0.785550\pi\)
\(7\)	−2618.01	−	4534.52i	−0.412125	−	0.713822i	0.582997	−	0.812475i	\(-0.301880\pi\)
							−0.995122	+	0.0986525i	\(0.968547\pi\)
\(11\)	60115.3			1.23799			0.618996	−	0.785394i	\(-0.287540\pi\)
							0.618996	+	0.785394i	\(0.287540\pi\)
\(13\)	−67228.3	−	116443.i	−0.652840	−	1.13075i	−0.982431	−	0.186628i	\(-0.940244\pi\)
							0.329590	−	0.944124i	\(-0.393089\pi\)
\(17\)	−256893.	+	444951.i	−0.745988	+	1.29209i	0.203744	+	0.979024i	\(0.434689\pi\)
							−0.949732	+	0.313065i	\(0.898644\pi\)
\(19\)	−343554.	−	595054.i	−0.604790	−	1.04753i	−0.992085	−	0.125571i	\(-0.959924\pi\)
							0.387295	−	0.921956i	\(-0.373410\pi\)
\(23\)	−2.58577e6			−1.92671			−0.963353	−	0.268238i	\(-0.913559\pi\)
							−0.963353	+	0.268238i	\(0.913559\pi\)
\(29\)	2.02055e6			0.530492			0.265246	−	0.964181i	\(-0.414547\pi\)
							0.265246	+	0.964181i	\(0.414547\pi\)
\(31\)	−6.89612e6			−1.34115			−0.670575	−	0.741842i	\(-0.733952\pi\)
							−0.670575	+	0.741842i	\(0.733952\pi\)
\(37\)	6.26648e6	+	9.52328e6i	0.549688	+	0.835370i
							\(41\)	−9.49019e6	−	1.64375e7i	−0.524503	−	0.908465i	−0.999593	−	0.0285283i	\(-0.990918\pi\)
0.475090	−	0.879937i	\(-0.342415\pi\)
\(43\)	−2.46543e7			−1.09973			−0.549864	−	0.835254i	\(-0.685321\pi\)
							−0.549864	+	0.835254i	\(0.685321\pi\)
\(47\)	−1.87506e7			−0.560498			−0.280249	−	0.959927i	\(-0.590417\pi\)
							−0.280249	+	0.959927i	\(0.590417\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.c.a.10.14	✓	54
37.26	even	3	inner	37.10.c.a.26.14	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.c.a.10.14	✓	54	1.1	even	1	trivial
37.10.c.a.26.14	yes	54	37.26	even	3	inner