Embedded newform 37.10.c.a.10.10

• Label: 37.10.c.a.10.10
• 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			10.10
Character	\(\chi\)	\(=\)	37.10
Dual form			37.10.c.a.26.10

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-9.23482 + 15.9952i) q^{2} +(-68.0812 - 117.920i) q^{3} +(85.4363 + 147.980i) q^{4} +(-1305.65 - 2261.46i) q^{5} +2514.87 q^{6} +(-1262.57 - 2186.84i) q^{7} -12612.4 q^{8} +(571.399 - 989.692i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q + 32 q^{2} + 73 q^{3} - 6486 q^{4} + 2729 q^{5} + 1020 q^{6} + 4231 q^{7} - 7848 q^{8} - 140840 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−9.23482	+	15.9952i	−0.408125	+	0.706893i	−0.994680	−	0.103016i	\(-0.967151\pi\)
							0.586555	+	0.809910i	\(0.300484\pi\)
\(3\)	−68.0812	−	117.920i	−0.485268	−	0.840509i	0.514589	−	0.857437i	\(-0.327945\pi\)
							−0.999857	+	0.0169284i	\(0.994611\pi\)
\(5\)	−1305.65	−	2261.46i	−0.934249	−	1.61817i	−0.775968	−	0.630773i	\(-0.782738\pi\)
							−0.158281	−	0.987394i	\(-0.550595\pi\)
\(7\)	−1262.57	−	2186.84i	−0.198754	−	0.344251i	0.749371	−	0.662150i	\(-0.230356\pi\)
							−0.948125	+	0.317899i	\(0.897023\pi\)
\(11\)	10827.6			0.222980			0.111490	−	0.993766i	\(-0.464438\pi\)
							0.111490	+	0.993766i	\(0.464438\pi\)
\(13\)	1131.95	+	1960.59i	0.0109921	+	0.0190389i	0.871469	−	0.490450i	\(-0.163168\pi\)
							−0.860477	+	0.509489i	\(0.829834\pi\)
\(17\)	16851.6	−	29187.7i	0.0489350	−	0.0847579i	−0.840520	−	0.541780i	\(-0.817751\pi\)
							0.889455	+	0.457022i	\(0.151084\pi\)
\(19\)	240391.	+	416370.i	0.423182	+	0.732974i	0.996249	−	0.0865354i	\(-0.0275796\pi\)
							−0.573066	+	0.819509i	\(0.694246\pi\)
\(23\)	−789542.			−0.588302			−0.294151	−	0.955759i	\(-0.595037\pi\)
							−0.294151	+	0.955759i	\(0.595037\pi\)
\(29\)	−5.28320e6			−1.38710			−0.693548	−	0.720411i	\(-0.743953\pi\)
							−0.693548	+	0.720411i	\(0.743953\pi\)
\(31\)	3.80509e6			0.740010			0.370005	−	0.929030i	\(-0.379356\pi\)
							0.370005	+	0.929030i	\(0.379356\pi\)
\(37\)	−5.40019e6	+	1.00399e7i	−0.473698	+	0.880688i
							\(41\)	1.09901e7	+	1.90353e7i	0.607397	+	1.05204i	0.991668	+	0.128822i	\(0.0411195\pi\)
−0.384271	+	0.923220i	\(0.625547\pi\)
\(43\)	2.77881e7			1.23951			0.619756	−	0.784794i	\(-0.287231\pi\)
							0.619756	+	0.784794i	\(0.287231\pi\)
\(47\)	2.35008e7			0.702492			0.351246	−	0.936283i	\(-0.385758\pi\)
							0.351246	+	0.936283i	\(0.385758\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.10.c.a.10.10	✓	54
37.26	even	3	inner	37.10.c.a.26.10	yes	54

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.10.c.a.10.10	✓	54	1.1	even	1	trivial
37.10.c.a.26.10	yes	54	37.26	even	3	inner