Embedded newform 37.4.h.a.4.4

• Label: 37.4.h.a.4.4
• Level: $37$
• Weight: $4$
• Character: 37.4
• Analytic conductor: $2.183$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	37.h (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.18307067021\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(8\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			4.4
Character	\(\chi\)	\(=\)	37.4
Dual form			37.4.h.a.28.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.09478 - 0.193039i) q^{2} +(0.594427 + 3.37116i) q^{3} +(-6.35626 - 2.31349i) q^{4} +(12.8529 + 15.3175i) q^{5} -3.80543i q^{6} +(-19.5668 + 16.4185i) q^{7} +(14.2140 + 8.20644i) q^{8} +(14.3603 - 5.22672i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 3 q^{2} + 9 q^{4} - 27 q^{5} - 108 q^{7} + 144 q^{8} + 42 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(2\)
\(\chi(n)\)	\(e\left(\frac{1}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.09478	−	0.193039i	−0.387063	−	0.0682496i	−0.0232694	−	0.999729i	\(-0.507408\pi\)
							−0.363794	+	0.931480i	\(0.618519\pi\)
\(3\)	0.594427	+	3.37116i	0.114398	+	0.648781i	0.987047	+	0.160433i	\(0.0512891\pi\)
							−0.872649	+	0.488348i	\(0.837600\pi\)
\(5\)	12.8529	+	15.3175i	1.14960	+	1.37004i	0.917689	+	0.397299i	\(0.130052\pi\)
							0.231908	+	0.972738i	\(0.425503\pi\)
\(7\)	−19.5668	+	16.4185i	−1.05651	+	0.886517i	−0.993763	−	0.111512i	\(-0.964431\pi\)
							−0.0627470	+	0.998029i	\(0.519986\pi\)
\(11\)	−11.6539	+	20.1851i	−0.319435	+	0.553277i	−0.980370	−	0.197166i	\(-0.936826\pi\)
							0.660935	+	0.750443i	\(0.270160\pi\)
\(13\)	14.4253	−	39.6331i	0.307758	−	0.845558i	−0.685335	−	0.728228i	\(-0.740344\pi\)
							0.993093	−	0.117330i	\(-0.0374335\pi\)
\(17\)	−34.2478	−	94.0950i	−0.488606	−	1.34243i	−0.901942	−	0.431856i	\(-0.857859\pi\)
							0.413336	−	0.910579i	\(-0.364363\pi\)
\(19\)	44.0735	−	7.77135i	0.532166	−	0.0938352i	0.0988932	−	0.995098i	\(-0.468470\pi\)
							0.433273	+	0.901263i	\(0.357359\pi\)
\(23\)	51.4464	−	29.7026i	0.466405	−	0.269279i	−0.248328	−	0.968676i	\(-0.579881\pi\)
							0.714734	+	0.699397i	\(0.246548\pi\)
\(29\)	164.567	+	95.0127i	1.05377	+	0.608394i	0.923702	−	0.383111i	\(-0.125147\pi\)
							0.130067	+	0.991505i	\(0.458481\pi\)
\(31\)			126.310i			0.731807i	0.930653	+	0.365903i	\(0.119240\pi\)
							−0.930653	+	0.365903i	\(0.880760\pi\)
\(37\)	198.727	−	105.644i	0.882986	−	0.469399i
							\(41\)	−82.3136	−	29.9597i	−0.313542	−	0.114120i	0.180456	−	0.983583i	\(-0.442243\pi\)
−0.493998	+	0.869463i	\(0.664465\pi\)
\(43\)		−	376.859i		−	1.33652i	−0.743927	−	0.668261i	\(-0.767039\pi\)
							0.743927	−	0.668261i	\(-0.232961\pi\)
\(47\)	86.1076	+	149.143i	0.267236	+	0.462866i	0.968147	−	0.250383i	\(-0.0805564\pi\)
							−0.700911	+	0.713249i	\(0.747223\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	37.4.h.a.4.4	✓	48
37.18	odd	36		1369.4.a.j.1.29		48
37.19	odd	36		1369.4.a.j.1.20		48
37.28	even	18	inner	37.4.h.a.28.4	yes	48

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.h.a.4.4	✓	48	1.1	even	1	trivial
37.4.h.a.28.4	yes	48	37.28	even	18	inner
1369.4.a.j.1.20		48	37.19	odd	36
1369.4.a.j.1.29		48	37.18	odd	36