Embedded newform 37.6.h.a.3.3

• Label: 37.6.h.a.3.3
• Level: $37$
• Weight: $6$
• Character: 37.3
• Analytic conductor: $5.934$
• Analytic rank: $0$
• Dimension: $90$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	37.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.93420133308\)
Analytic rank:	\(0\)
Dimension:	\(90\)
Relative dimension:	\(15\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.3
Character	\(\chi\)	\(=\)	37.3
Dual form			37.6.h.a.25.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-5.03691 + 6.00276i) q^{2} +(9.01055 - 7.56075i) q^{3} +(-5.10589 - 28.9569i) q^{4} +(-6.94659 + 19.0856i) q^{5} +92.1710i q^{6} +(-209.145 - 76.1225i) q^{7} +(-17.6194 - 10.1726i) q^{8} +(-18.1714 + 103.055i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 90 q - 3 q^{2} - 9 q^{3} - 39 q^{4} - 153 q^{5} + 201 q^{7} + 792 q^{8} + 87 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−5.03691	+	6.00276i	−0.890409	+	1.06115i	0.107349	+	0.994221i	\(0.465764\pi\)
							−0.997758	+	0.0669268i	\(0.978681\pi\)
\(3\)	9.01055	−	7.56075i	0.578027	−	0.485022i	−0.306271	−	0.951944i	\(-0.599082\pi\)
							0.884299	+	0.466922i	\(0.154637\pi\)
\(5\)	−6.94659	+	19.0856i	−0.124264	+	0.341414i	−0.986189	−	0.165622i	\(-0.947037\pi\)
							0.861925	+	0.507036i	\(0.169259\pi\)
\(7\)	−209.145	−	76.1225i	−1.61325	−	0.587175i	−0.631171	−	0.775644i	\(-0.717425\pi\)
							−0.982079	+	0.188468i	\(0.939648\pi\)
\(11\)	388.545	−	672.979i	0.968187	−	1.67695i	0.267389	−	0.963589i	\(-0.413839\pi\)
							0.700798	−	0.713360i	\(-0.252828\pi\)
\(13\)	−215.921	+	38.0727i	−0.354353	+	0.0624821i	−0.347992	−	0.937498i	\(-0.613136\pi\)
							−0.00636197	+	0.999980i	\(0.502025\pi\)
\(17\)	−2114.20	−	372.790i	−1.77428	−	0.312854i	−0.811747	−	0.584009i	\(-0.801483\pi\)
							−0.962536	+	0.271155i	\(0.912594\pi\)
\(19\)	−622.731	−	742.141i	−0.395746	−	0.471631i	0.530972	−	0.847389i	\(-0.321827\pi\)
							−0.926718	+	0.375758i	\(0.877382\pi\)
\(23\)	−455.950	+	263.243i	−0.179720	+	0.103762i	−0.587161	−	0.809470i	\(-0.699755\pi\)
							0.407441	+	0.913232i	\(0.366421\pi\)
\(29\)	−4468.15	−	2579.69i	−0.986580	−	0.569602i	−0.0823300	−	0.996605i	\(-0.526236\pi\)
							−0.904250	+	0.427003i	\(0.859569\pi\)
\(31\)			449.844i			0.0840732i	0.999116	+	0.0420366i	\(0.0133846\pi\)
							−0.999116	+	0.0420366i	\(0.986615\pi\)
\(37\)	−7216.79	+	4154.75i	−0.866642	+	0.498931i
							\(41\)	1677.76	+	9515.04i	0.155873	+	0.883998i	0.957984	+	0.286823i	\(0.0925991\pi\)
−0.802111	+	0.597175i	\(0.796290\pi\)
\(43\)			5690.30i			0.469315i	0.972078	+	0.234657i	\(0.0753969\pi\)
							−0.972078	+	0.234657i	\(0.924603\pi\)
\(47\)	−7242.28	−	12544.0i	−0.478223	−	0.828307i	0.521465	−	0.853273i	\(-0.325386\pi\)
							−0.999688	+	0.0249655i	\(0.992052\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.6.h.a.3.3	✓	90
37.25	even	18	inner	37.6.h.a.25.3	yes	90

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.6.h.a.3.3	✓	90	1.1	even	1	trivial
37.6.h.a.25.3	yes	90	37.25	even	18	inner