Embedded newform 38.7.f.a.3.7

• Label: 38.7.f.a.3.7
• Level: $38$
• Weight: $7$
• Character: 38.3
• Analytic conductor: $8.742$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	38.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.74205517755\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(10\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.7
Character	\(\chi\)	\(=\)	38.3
Dual form			38.7.f.a.13.7

$q$-expansion

\(f(q)\)	\(=\)	\(q+(3.63616 - 4.33340i) q^{2} +(-10.8314 - 29.7590i) q^{3} +(-5.55674 - 31.5138i) q^{4} +(-11.6174 + 65.8855i) q^{5} +(-168.342 - 61.2716i) q^{6} +(-205.116 - 355.271i) q^{7} +(-156.767 - 90.5097i) q^{8} +(-209.833 + 176.071i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 60 q + 30 q^{3} - 240 q^{6} + 432 q^{7} + 690 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/38\mathbb{Z}\right)^\times\).

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{13}{18}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	3.63616	−	4.33340i	0.454519	−	0.541675i
							\(3\)	−10.8314	−	29.7590i	−0.401163	−	1.10219i	−0.961711	−	0.274064i	\(-0.911632\pi\)
0.560549	−	0.828122i	\(-0.310590\pi\)
\(5\)	−11.6174	+	65.8855i	−0.0929391	+	0.527084i	0.902420	+	0.430857i	\(0.141789\pi\)
							−0.995359	+	0.0962270i	\(0.969323\pi\)
\(7\)	−205.116	−	355.271i	−0.598004	−	1.03577i	−0.993115	−	0.117141i	\(-0.962627\pi\)
							0.395111	−	0.918633i	\(-0.370706\pi\)
\(11\)	−705.829	+	1222.53i	−0.530300	+	0.918506i	0.469075	+	0.883158i	\(0.344587\pi\)
							−0.999375	+	0.0353482i	\(0.988746\pi\)
\(13\)	−817.579	+	2246.28i	−0.372134	+	1.02243i	0.602400	+	0.798194i	\(0.294211\pi\)
							−0.974535	+	0.224237i	\(0.928011\pi\)
\(17\)	−3580.20	−	3004.14i	−0.728720	−	0.611468i	0.201063	−	0.979578i	\(-0.435561\pi\)
							−0.929782	+	0.368110i	\(0.880005\pi\)
\(19\)	2404.19	−	6423.84i	0.350516	−	0.936557i
							\(23\)	−2726.31	−	15461.7i	−0.224075	−	1.27079i	−0.864447	−	0.502725i	\(-0.832331\pi\)
0.640372	−	0.768065i	\(-0.278780\pi\)
\(29\)	−15718.9	−	18733.1i	−0.644509	−	0.768096i	0.340566	−	0.940221i	\(-0.389381\pi\)
							−0.985075	+	0.172124i	\(0.944937\pi\)
\(31\)	26256.4	−	15159.1i	0.881353	−	0.508850i	0.0102489	−	0.999947i	\(-0.496738\pi\)
							0.871104	+	0.491098i	\(0.163404\pi\)
\(37\)		−	11818.0i		−	0.233312i	−0.993172	−	0.116656i	\(-0.962782\pi\)
							0.993172	−	0.116656i	\(-0.0372176\pi\)
\(41\)	3822.58	+	10502.4i	0.0554632	+	0.152384i	0.964330	−	0.264702i	\(-0.0852736\pi\)
							−0.908867	+	0.417086i	\(0.863051\pi\)
\(43\)	12310.1	−	69814.0i	0.154830	−	0.878087i	−0.804110	−	0.594480i	\(-0.797358\pi\)
							0.958941	−	0.283607i	\(-0.0915310\pi\)
\(47\)	4553.61	−	3820.93i	0.0438594	−	0.0368024i	−0.620594	−	0.784132i	\(-0.713109\pi\)
							0.664454	+	0.747329i	\(0.268664\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.7.f.a.3.7	✓	60
19.13	odd	18	inner	38.7.f.a.13.7	yes	60

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.7.f.a.3.7	✓	60	1.1	even	1	trivial
38.7.f.a.13.7	yes	60	19.13	odd	18	inner