Embedded newform 37.5.i.a.5.8

• Label: 37.5.i.a.5.8
• Level: $37$
• Weight: $5$
• Character: 37.5
• Analytic conductor: $3.825$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	37.i (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.82468863410\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(12\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label			5.8
Character	\(\chi\)	\(=\)	37.5
Dual form			37.5.i.a.15.8

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.479059 - 1.02734i) q^{2} +(-0.611056 + 1.67886i) q^{3} +(9.45866 + 11.2724i) q^{4} +(-16.2436 + 23.1983i) q^{5} +(1.43204 + 1.43204i) q^{6} +(0.371208 + 2.10523i) q^{7} +(33.6307 - 9.01132i) q^{8} +(59.6044 + 50.0140i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 12 q^{3} + 6 q^{4} - 120 q^{5} - 12 q^{6} - 12 q^{7} - 282 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{23}{36}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.479059	−	1.02734i	0.119765	−	0.256836i	−0.837281	−	0.546772i	\(-0.815856\pi\)
							0.957046	+	0.289936i	\(0.0936341\pi\)
\(3\)	−0.611056	+	1.67886i	−0.0678951	+	0.186540i	−0.969000	−	0.247061i	\(-0.920535\pi\)
							0.901105	+	0.433601i	\(0.142757\pi\)
\(5\)	−16.2436	+	23.1983i	−0.649746	+	0.927933i	−0.999962	−	0.00873596i	\(-0.997219\pi\)
							0.350216	+	0.936669i	\(0.386108\pi\)
\(7\)	0.371208	+	2.10523i	0.00757568	+	0.0429638i	0.988361	−	0.152125i	\(-0.0486116\pi\)
							−0.980786	+	0.195089i	\(0.937501\pi\)
\(11\)	70.0911	+	40.4671i	0.579265	+	0.334439i	0.760841	−	0.648938i	\(-0.224787\pi\)
							−0.181576	+	0.983377i	\(0.558120\pi\)
\(13\)	−95.3634	−	8.34322i	−0.564280	−	0.0493681i	−0.198553	−	0.980090i	\(-0.563624\pi\)
							−0.365727	+	0.930722i	\(0.619180\pi\)
\(17\)	−48.3598	−	552.755i	−0.167335	−	1.91265i	−0.362683	−	0.931912i	\(-0.618139\pi\)
							0.195349	−	0.980734i	\(-0.437416\pi\)
\(19\)	36.5236	+	78.3251i	0.101173	+	0.216967i	0.950378	−	0.311096i	\(-0.100696\pi\)
							−0.849205	+	0.528063i	\(0.822918\pi\)
\(23\)	65.8364	−	245.705i	0.124454	−	0.464470i	−0.875365	−	0.483462i	\(-0.839379\pi\)
							0.999820	+	0.0189921i	\(0.00604574\pi\)
\(29\)	−194.135	−	724.523i	−0.230839	−	0.861502i	−0.979981	−	0.199093i	\(-0.936201\pi\)
							0.749142	−	0.662410i	\(-0.230466\pi\)
\(31\)	442.056	−	442.056i	0.459996	−	0.459996i	−0.438658	−	0.898654i	\(-0.644546\pi\)
							0.898654	+	0.438658i	\(0.144546\pi\)
\(37\)	1301.35	+	425.042i	0.950581	+	0.310476i
							\(41\)	974.470	+	1161.33i	0.579697	+	0.690855i	0.973591	−	0.228300i	\(-0.0733168\pi\)
−0.393894	+	0.919156i	\(0.628872\pi\)
\(43\)	1609.68	+	1609.68i	0.870569	+	0.870569i	0.992534	−	0.121965i	\(-0.0389196\pi\)
							−0.121965	+	0.992534i	\(0.538920\pi\)
\(47\)	−1795.11	−	3109.23i	−0.812636	−	1.40753i	−0.911013	−	0.412378i	\(-0.864698\pi\)
							0.0983765	−	0.995149i	\(-0.468635\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.5.i.a.5.8	✓	144
37.15	odd	36	inner	37.5.i.a.15.8	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.5.i.a.5.8	✓	144	1.1	even	1	trivial
37.5.i.a.15.8	yes	144	37.15	odd	36	inner