Embedded newform 37.6.c.a.26.3

• Label: 37.6.c.a.26.3
• Level: $37$
• Weight: $6$
• Character: 37.26
• Analytic conductor: $5.934$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.93420133308\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(15\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			26.3
Character	\(\chi\)	\(=\)	37.26
Dual form			37.6.c.a.10.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.72321 - 6.44878i) q^{2} +(5.98171 - 10.3606i) q^{3} +(-11.7245 + 20.3075i) q^{4} +(37.3727 - 64.7314i) q^{5} -89.0845 q^{6} +(68.5781 - 118.781i) q^{7} -63.6740 q^{8} +(49.9383 + 86.4957i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 4 q^{2} + 16 q^{3} - 230 q^{4} - 51 q^{5} + 60 q^{6} + 50 q^{7} + 24 q^{8} - 1085 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.72321	−	6.44878i	−0.658176	−	1.13999i	−0.981087	−	0.193565i	\(-0.937995\pi\)
							0.322911	−	0.946429i	\(-0.395338\pi\)
\(3\)	5.98171	−	10.3606i	0.383727	−	0.664634i	−0.607865	−	0.794041i	\(-0.707974\pi\)
							0.991592	+	0.129406i	\(0.0413071\pi\)
\(5\)	37.3727	−	64.7314i	0.668543	−	1.15795i	−0.309769	−	0.950812i	\(-0.600252\pi\)
							0.978312	−	0.207138i	\(-0.0664149\pi\)
\(7\)	68.5781	−	118.781i	0.528981	−	0.916223i	−0.470447	−	0.882428i	\(-0.655907\pi\)
							0.999429	−	0.0337946i	\(-0.0107592\pi\)
\(11\)	109.599			0.273101			0.136550	−	0.990633i	\(-0.456398\pi\)
							0.136550	+	0.990633i	\(0.456398\pi\)
\(13\)	−500.749	+	867.323i	−0.821791	+	1.42338i	0.0825555	+	0.996586i	\(0.473692\pi\)
							−0.904347	+	0.426798i	\(0.859642\pi\)
\(17\)	−451.495	−	782.012i	−0.378905	−	0.656283i	0.611998	−	0.790859i	\(-0.290366\pi\)
							−0.990903	+	0.134576i	\(0.957033\pi\)
\(19\)	425.074	−	736.250i	0.270135	−	0.467888i	−0.698761	−	0.715355i	\(-0.746265\pi\)
							0.968896	+	0.247467i	\(0.0795983\pi\)
\(23\)	1603.10			0.631890			0.315945	−	0.948778i	\(-0.397679\pi\)
							0.315945	+	0.948778i	\(0.397679\pi\)
\(29\)	6170.70			1.36251			0.681254	−	0.732047i	\(-0.261435\pi\)
							0.681254	+	0.732047i	\(0.261435\pi\)
\(31\)	−8274.52			−1.54646			−0.773230	−	0.634126i	\(-0.781360\pi\)
							−0.773230	+	0.634126i	\(0.781360\pi\)
\(37\)	2858.72	−	7821.23i	0.343295	−	0.939227i
							\(41\)	−3993.80	+	6917.46i	−0.371045	+	0.642668i	−0.989727	−	0.142973i	\(-0.954334\pi\)
0.618682	+	0.785642i	\(0.287667\pi\)
\(43\)	−3863.20			−0.318623			−0.159311	−	0.987228i	\(-0.550927\pi\)
							−0.159311	+	0.987228i	\(0.550927\pi\)
\(47\)	17227.1			1.13754			0.568770	−	0.822497i	\(-0.307420\pi\)
							0.568770	+	0.822497i	\(0.307420\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.c.a.26.3	yes	30
37.10	even	3	inner	37.6.c.a.10.3	✓	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.c.a.10.3	✓	30	37.10	even	3	inner
37.6.c.a.26.3	yes	30	1.1	even	1	trivial