Embedded newform 37.6.c.a.10.8

• Label: 37.6.c.a.10.8
• Level: $37$
• Weight: $6$
• Character: 37.10
• 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			10.8
Character	\(\chi\)	\(=\)	37.10
Dual form			37.6.c.a.26.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.480658 + 0.832525i) q^{2} +(12.5533 + 21.7430i) q^{3} +(15.5379 + 26.9125i) q^{4} +(14.2475 + 24.6775i) q^{5} -24.1355 q^{6} +(-96.5783 - 167.278i) q^{7} -60.6359 q^{8} +(-193.673 + 335.451i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 4 q^{2} + 16 q^{3} - 230 q^{4} - 51 q^{5} + 60 q^{6} + 50 q^{7} + 24 q^{8} - 1085 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.480658	+	0.832525i	−0.0849692	+	0.147171i	−0.905378	−	0.424606i	\(-0.860413\pi\)
							0.820409	+	0.571777i	\(0.193746\pi\)
\(3\)	12.5533	+	21.7430i	0.805297	+	1.39482i	0.916090	+	0.400972i	\(0.131328\pi\)
							−0.110793	+	0.993844i	\(0.535339\pi\)
\(5\)	14.2475	+	24.6775i	0.254868	+	0.441444i	0.964860	−	0.262766i	\(-0.0846348\pi\)
							−0.709992	+	0.704210i	\(0.751301\pi\)
\(7\)	−96.5783	−	167.278i	−0.744962	−	1.29031i	−0.950212	−	0.311603i	\(-0.899134\pi\)
							0.205250	−	0.978710i	\(-0.434199\pi\)
\(11\)	478.653			1.19272			0.596361	−	0.802717i	\(-0.296613\pi\)
							0.596361	+	0.802717i	\(0.296613\pi\)
\(13\)	−111.259	−	192.706i	−0.182590	−	0.316255i	0.760172	−	0.649722i	\(-0.225115\pi\)
							−0.942762	+	0.333467i	\(0.891781\pi\)
\(17\)	447.537	−	775.158i	0.375584	−	0.650530i	−0.614830	−	0.788659i	\(-0.710776\pi\)
							0.990414	+	0.138129i	\(0.0441088\pi\)
\(19\)	1006.01	+	1742.46i	0.639319	+	1.10733i	0.985583	+	0.169195i	\(0.0541169\pi\)
							−0.346264	+	0.938137i	\(0.612550\pi\)
\(23\)	−2891.44			−1.13971			−0.569855	−	0.821745i	\(-0.693001\pi\)
							−0.569855	+	0.821745i	\(0.693001\pi\)
\(29\)	2826.47			0.624094			0.312047	−	0.950067i	\(-0.398985\pi\)
							0.312047	+	0.950067i	\(0.398985\pi\)
\(31\)	4932.55			0.921864			0.460932	−	0.887435i	\(-0.347515\pi\)
							0.460932	+	0.887435i	\(0.347515\pi\)
\(37\)	−5944.60	+	5831.44i	−0.713868	+	0.700280i
							\(41\)	−1594.25	−	2761.31i	−0.148114	−	0.256541i	0.782417	−	0.622755i	\(-0.213987\pi\)
−0.930530	+	0.366215i	\(0.880654\pi\)
\(43\)	19069.5			1.57278			0.786391	−	0.617730i	\(-0.211947\pi\)
							0.786391	+	0.617730i	\(0.211947\pi\)
\(47\)	−25865.4			−1.70794			−0.853972	−	0.520318i	\(-0.825813\pi\)
							−0.853972	+	0.520318i	\(0.825813\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.c.a.10.8	✓	30
37.26	even	3	inner	37.6.c.a.26.8	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.c.a.10.8	✓	30	1.1	even	1	trivial
37.6.c.a.26.8	yes	30	37.26	even	3	inner