Embedded newform 37.10.c.a.10.24

• Label: 37.10.c.a.10.24
• 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			10.24
Character	\(\chi\)	\(=\)	37.10
Dual form			37.10.c.a.26.24

$q$-expansion

\(f(q)\)	\(=\)	\(q+(19.2159 - 33.2829i) q^{2} +(97.9371 + 169.632i) q^{3} +(-482.502 - 835.718i) q^{4} +(-39.7366 - 68.8258i) q^{5} +7527.80 q^{6} +(5974.04 + 10347.3i) q^{7} -17409.8 q^{8} +(-9341.85 + 16180.6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 32 q^{2} + 73 q^{3} - 6486 q^{4} + 2729 q^{5} + 1020 q^{6} + 4231 q^{7} - 7848 q^{8} - 140840 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	19.2159	−	33.2829i	0.849231	−	1.47091i	−0.0326645	−	0.999466i	\(-0.510399\pi\)
							0.881896	−	0.471445i	\(-0.156267\pi\)
\(3\)	97.9371	+	169.632i	0.698074	+	1.20910i	0.969133	+	0.246537i	\(0.0792928\pi\)
							−0.271059	+	0.962563i	\(0.587374\pi\)
\(5\)	−39.7366	−	68.8258i	−0.0284332	−	0.0492477i	0.851459	−	0.524422i	\(-0.175718\pi\)
							−0.879892	+	0.475174i	\(0.842385\pi\)
\(7\)	5974.04	+	10347.3i	0.940431	+	1.62887i	0.764651	+	0.644445i	\(0.222911\pi\)
							0.175780	+	0.984429i	\(0.443755\pi\)
\(11\)	−10790.1			−0.222207			−0.111103	−	0.993809i	\(-0.535439\pi\)
							−0.111103	+	0.993809i	\(0.535439\pi\)
\(13\)	58370.0	+	101100.i	0.566819	+	0.981760i	0.996878	+	0.0789587i	\(0.0251595\pi\)
							−0.430059	+	0.902801i	\(0.641507\pi\)
\(17\)	20255.4	−	35083.3i	0.0588193	−	0.101878i	−0.835116	−	0.550073i	\(-0.814600\pi\)
							0.893936	+	0.448195i	\(0.147933\pi\)
\(19\)	22150.8	+	38366.4i	0.0389941	+	0.0675398i	0.884864	−	0.465850i	\(-0.154251\pi\)
							−0.845870	+	0.533390i	\(0.820918\pi\)
\(23\)	1.92237e6			1.43239			0.716197	−	0.697898i	\(-0.245881\pi\)
							0.716197	+	0.697898i	\(0.245881\pi\)
\(29\)	−5.89162e6			−1.54683			−0.773417	−	0.633898i	\(-0.781454\pi\)
							−0.773417	+	0.633898i	\(0.781454\pi\)
\(31\)	−1.85894e6			−0.361524			−0.180762	−	0.983527i	\(-0.557856\pi\)
							−0.180762	+	0.983527i	\(0.557856\pi\)
\(37\)	−1.19395e6	−	1.13374e7i	−0.104732	−	0.994500i
							\(41\)	9.41586e6	+	1.63088e7i	0.520395	+	0.901350i	0.999719	+	0.0237120i	\(0.00754846\pi\)
−0.479324	+	0.877638i	\(0.659118\pi\)
\(43\)	−2.21848e7			−0.989571			−0.494786	−	0.869015i	\(-0.664753\pi\)
							−0.494786	+	0.869015i	\(0.664753\pi\)
\(47\)	−1.10976e7			−0.331734			−0.165867	−	0.986148i	\(-0.553042\pi\)
							−0.165867	+	0.986148i	\(0.553042\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.c.a.10.24	✓	54
37.26	even	3	inner	37.10.c.a.26.24	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.c.a.10.24	✓	54	1.1	even	1	trivial
37.10.c.a.26.24	yes	54	37.26	even	3	inner