Embedded newform 37.8.c.a.10.1

• Label: 37.8.c.a.10.1
• Level: $37$
• Weight: $8$
• Character: 37.10
• Analytic conductor: $11.558$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	37.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5582459429\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(21\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			10.1
Character	\(\chi\)	\(=\)	37.10
Dual form			37.8.c.a.26.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-11.0007 + 19.0538i) q^{2} +(11.3160 + 19.5999i) q^{3} +(-178.031 - 308.359i) q^{4} +(50.3125 + 87.1438i) q^{5} -497.937 q^{6} +(-276.316 - 478.593i) q^{7} +5017.70 q^{8} +(837.396 - 1450.41i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 16 q^{2} - 68 q^{3} - 1126 q^{4} - 337 q^{5} + 252 q^{6} - 158 q^{7} + 6264 q^{8} - 12773 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−11.0007	+	19.0538i	−0.972335	+	1.68413i	−0.283870	+	0.958863i	\(0.591619\pi\)
							−0.688464	+	0.725270i	\(0.741715\pi\)
\(3\)	11.3160	+	19.5999i	0.241974	+	0.419111i	0.961277	−	0.275586i	\(-0.0888718\pi\)
							−0.719302	+	0.694697i	\(0.755538\pi\)
\(5\)	50.3125	+	87.1438i	0.180003	+	0.311775i	0.941881	−	0.335945i	\(-0.109056\pi\)
							−0.761878	+	0.647720i	\(0.775722\pi\)
\(7\)	−276.316	−	478.593i	−0.304483	−	0.527379i	0.672663	−	0.739949i	\(-0.265150\pi\)
							−0.977146	+	0.212569i	\(0.931817\pi\)
\(11\)	5905.03			1.33767			0.668833	−	0.743413i	\(-0.266794\pi\)
							0.668833	+	0.743413i	\(0.266794\pi\)
\(13\)	−4095.32	−	7093.30i	−0.516995	−	0.895461i	−0.999805	−	0.0197360i	\(-0.993717\pi\)
							0.482811	−	0.875725i	\(-0.339616\pi\)
\(17\)	−10678.8	+	18496.3i	−0.527173	+	0.913090i	0.472326	+	0.881424i	\(0.343415\pi\)
							−0.999499	+	0.0316661i	\(0.989919\pi\)
\(19\)	−16344.1	−	28308.8i	−0.546668	−	0.946856i	−0.998500	−	0.0547532i	\(-0.982563\pi\)
							0.451832	−	0.892103i	\(-0.350771\pi\)
\(23\)	45210.0			0.774796			0.387398	−	0.921913i	\(-0.373374\pi\)
							0.387398	+	0.921913i	\(0.373374\pi\)
\(29\)	−51036.3			−0.388585			−0.194293	−	0.980944i	\(-0.562241\pi\)
							−0.194293	+	0.980944i	\(0.562241\pi\)
\(31\)	10053.5			0.0606107			0.0303053	−	0.999541i	\(-0.490352\pi\)
							0.0303053	+	0.999541i	\(0.490352\pi\)
\(37\)	304221.	+	48799.9i	0.987378	+	0.158385i
							\(41\)	138607.	+	240074.i	0.314080	+	0.544003i	0.979242	−	0.202697i	\(-0.0649706\pi\)
−0.665161	+	0.746700i	\(0.731637\pi\)
\(43\)	801271.			1.53688			0.768440	−	0.639922i	\(-0.221033\pi\)
							0.768440	+	0.639922i	\(0.221033\pi\)
\(47\)	−357814.			−0.502707			−0.251353	−	0.967895i	\(-0.580876\pi\)
							−0.251353	+	0.967895i	\(0.580876\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.8.c.a.10.1	✓	42
37.26	even	3	inner	37.8.c.a.26.1	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.8.c.a.10.1	✓	42	1.1	even	1	trivial
37.8.c.a.26.1	yes	42	37.26	even	3	inner