Embedded newform 37.5.i.a.2.4

• Label: 37.5.i.a.2.4
• Level: $37$
• Weight: $5$
• Character: 37.2
• 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			2.4
Character	\(\chi\)	\(=\)	37.2
Dual form			37.5.i.a.19.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-4.38998 - 0.384074i) q^{2} +(0.247322 - 0.294747i) q^{3} +(3.36752 + 0.593785i) q^{4} +(-2.39998 + 5.14676i) q^{5} +(-1.19894 + 1.19894i) q^{6} +(28.5706 - 10.3989i) q^{7} +(53.5502 + 14.3487i) q^{8} +(14.0398 + 79.6236i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 144 q - 12 q^{2} - 12 q^{3} + 6 q^{4} - 120 q^{5} - 12 q^{6} - 12 q^{7} - 282 q^{8} + 198 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−4.38998	−	0.384074i	−1.09750	−	0.0960185i	−0.476007	−	0.879441i	\(-0.657916\pi\)
							−0.621489	+	0.783423i	\(0.713472\pi\)
\(3\)	0.247322	−	0.294747i	0.0274802	−	0.0327496i	−0.752129	−	0.659016i	\(-0.770973\pi\)
							0.779610	+	0.626266i	\(0.215418\pi\)
\(5\)	−2.39998	+	5.14676i	−0.0959990	+	0.205871i	−0.948443	−	0.316948i	\(-0.897342\pi\)
							0.852444	+	0.522819i	\(0.175120\pi\)
\(7\)	28.5706	−	10.3989i	0.583074	−	0.212222i	−0.0336064	−	0.999435i	\(-0.510699\pi\)
							0.616681	+	0.787213i	\(0.288477\pi\)
\(11\)	172.017	−	99.3142i	1.42163	−	0.820778i	0.425192	−	0.905103i	\(-0.360207\pi\)
							0.996438	+	0.0843249i	\(0.0268734\pi\)
\(13\)	151.563	+	216.455i	0.896825	+	1.28080i	0.959195	+	0.282745i	\(0.0912451\pi\)
							−0.0623705	+	0.998053i	\(0.519866\pi\)
\(17\)	−136.322	−	95.4534i	−0.471701	−	0.330289i	0.313437	−	0.949609i	\(-0.398519\pi\)
							−0.785139	+	0.619320i	\(0.787408\pi\)
\(19\)	−158.545	+	13.8709i	−0.439182	+	0.0384234i	−0.304603	−	0.952479i	\(-0.598524\pi\)
							−0.134579	+	0.990903i	\(0.542968\pi\)
\(23\)	−39.3830	−	146.979i	−0.0744480	−	0.277844i	0.918660	−	0.395050i	\(-0.129273\pi\)
							−0.993108	+	0.117206i	\(0.962606\pi\)
\(29\)	83.4625	−	311.486i	0.0992419	−	0.370376i	−0.898386	−	0.439206i	\(-0.855260\pi\)
							0.997628	+	0.0688298i	\(0.0219266\pi\)
\(31\)	1120.45	+	1120.45i	1.16592	+	1.16592i	0.983157	+	0.182761i	\(0.0585034\pi\)
							0.182761	+	0.983157i	\(0.441497\pi\)
\(37\)	286.305	−	1338.73i	0.209135	−	0.977887i
							\(41\)	730.437	+	128.796i	0.434526	+	0.0766186i	0.386632	−	0.922234i	\(-0.373638\pi\)
0.0478931	+	0.998852i	\(0.484749\pi\)
\(43\)	−1337.22	+	1337.22i	−0.723214	+	0.723214i	−0.969258	−	0.246045i	\(-0.920869\pi\)
							0.246045	+	0.969258i	\(0.420869\pi\)
\(47\)	923.012	−	1598.70i	0.417842	−	0.723723i	−0.577881	−	0.816121i	\(-0.696120\pi\)
							0.995722	+	0.0923986i	\(0.0294534\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.5.i.a.2.4	✓	144
37.19	odd	36	inner	37.5.i.a.19.4	yes	144

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.5.i.a.2.4	✓	144	1.1	even	1	trivial
37.5.i.a.19.4	yes	144	37.19	odd	36	inner