Embedded newform 37.7.i.a.2.17

• Label: 37.7.i.a.2.17
• Level: $37$
• Weight: $7$
• Character: 37.2
• Analytic conductor: $8.512$
• Analytic rank: $0$
• Dimension: $216$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	37.i (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.51200109393\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(18\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			2.17
Character	\(\chi\)	\(=\)	37.2
Dual form			37.7.i.a.19.17

$q$-expansion

\(f(q)\)	\(=\)	\(q+(14.3464 + 1.25514i) q^{2} +(22.3262 - 26.6073i) q^{3} +(141.215 + 24.9000i) q^{4} +(-37.3574 + 80.1132i) q^{5} +(353.696 - 353.696i) q^{6} +(-152.507 + 55.5081i) q^{7} +(1104.40 + 295.923i) q^{8} +(-82.9011 - 470.155i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 216 q - 12 q^{2} - 12 q^{3} - 282 q^{4} + 186 q^{5} - 12 q^{6} - 12 q^{7} + 1782 q^{8} - 972 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	14.3464	+	1.25514i	1.79330	+	0.156893i	0.934625	−	0.355635i	\(-0.115735\pi\)
							0.858671	+	0.512528i	\(0.171291\pi\)
\(3\)	22.3262	−	26.6073i	0.826896	−	0.985456i	−0.173104	−	0.984904i	\(-0.555380\pi\)
							1.00000		0.000552705i	\(-0.000175932\pi\)
\(5\)	−37.3574	+	80.1132i	−0.298859	+	0.640905i	−0.997194	−	0.0748661i	\(-0.976147\pi\)
							0.698334	+	0.715772i	\(0.253925\pi\)
\(7\)	−152.507	+	55.5081i	−0.444628	+	0.161831i	−0.554625	−	0.832101i	\(-0.687138\pi\)
							0.109997	+	0.993932i	\(0.464916\pi\)
\(11\)	−316.323	+	182.629i	−0.237658	+	0.137212i	−0.614100	−	0.789228i	\(-0.710481\pi\)
							0.376442	+	0.926440i	\(0.377148\pi\)
\(13\)	−2407.85	−	3438.76i	−1.09597	−	1.56521i	−0.789939	−	0.613186i	\(-0.789888\pi\)
							−0.306031	−	0.952022i	\(-0.599001\pi\)
\(17\)	4590.22	+	3214.10i	0.934300	+	0.654204i	0.938313	−	0.345788i	\(-0.112388\pi\)
							−0.00401273	+	0.999992i	\(0.501277\pi\)
\(19\)	−7656.33	+	669.842i	−1.11625	+	0.0976589i	−0.630316	−	0.776339i	\(-0.717075\pi\)
							−0.485930	+	0.873998i	\(0.661519\pi\)
\(23\)	3620.56	+	13512.1i	0.297572	+	1.11055i	0.939153	+	0.343498i	\(0.111612\pi\)
							−0.641581	+	0.767055i	\(0.721721\pi\)
\(29\)	6824.72	−	25470.2i	0.279828	−	1.04433i	−0.672709	−	0.739907i	\(-0.734869\pi\)
							0.952537	−	0.304424i	\(-0.0984641\pi\)
\(31\)	−8542.42	−	8542.42i	−0.286745	−	0.286745i	0.549047	−	0.835792i	\(-0.314991\pi\)
							−0.835792	+	0.549047i	\(0.814991\pi\)
\(37\)	−36381.4	+	35243.7i	−0.718248	+	0.695787i
							\(41\)	−49193.6	−	8674.17i	−0.713768	−	0.125857i	−0.195040	−	0.980795i	\(-0.562484\pi\)
−0.518729	+	0.854939i	\(0.673595\pi\)
\(43\)	95245.3	−	95245.3i	1.19795	−	1.19795i	0.223168	−	0.974780i	\(-0.428360\pi\)
							0.974780	−	0.223168i	\(-0.0716399\pi\)
\(47\)	46811.6	−	81080.0i	0.450878	−	0.780944i	−0.547562	−	0.836765i	\(-0.684444\pi\)
							0.998441	+	0.0558205i	\(0.0177775\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.7.i.a.2.17	✓	216
37.19	odd	36	inner	37.7.i.a.19.17	yes	216

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.7.i.a.2.17	✓	216	1.1	even	1	trivial
37.7.i.a.19.17	yes	216	37.19	odd	36	inner