Embedded newform 38.5.f.a.3.4

• Label: 38.5.f.a.3.4
• Level: $38$
• Weight: $5$
• Character: 38.3
• Analytic conductor: $3.928$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	38.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.92805859719\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.4
Character	\(\chi\)	\(=\)	38.3
Dual form			38.5.f.a.13.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.81808 - 2.16670i) q^{2} +(-5.55388 - 15.2592i) q^{3} +(-1.38919 - 7.87846i) q^{4} +(-3.70832 + 21.0309i) q^{5} +(-43.1594 - 15.7087i) q^{6} +(-2.05572 - 3.56062i) q^{7} +(-19.5959 - 11.3137i) q^{8} +(-139.947 + 117.429i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 12 q^{3} + 48 q^{6} + 90 q^{7} - 84 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.81808	−	2.16670i	0.454519	−	0.541675i
							\(3\)	−5.55388	−	15.2592i	−0.617098	−	1.69546i	−0.713980	−	0.700166i	\(-0.753109\pi\)
0.0968828	−	0.995296i	\(-0.469113\pi\)
\(5\)	−3.70832	+	21.0309i	−0.148333	+	0.841236i	0.816298	+	0.577631i	\(0.196023\pi\)
							−0.964631	+	0.263605i	\(0.915088\pi\)
\(7\)	−2.05572	−	3.56062i	−0.0419535	−	0.0726656i	0.844286	−	0.535893i	\(-0.180025\pi\)
							−0.886240	+	0.463227i	\(0.846691\pi\)
\(11\)	95.4230	−	165.277i	0.788620	−	1.36593i	−0.138193	−	0.990405i	\(-0.544129\pi\)
							0.926813	−	0.375524i	\(-0.122537\pi\)
\(13\)	73.9902	−	203.286i	0.437812	−	1.20288i	−0.503101	−	0.864228i	\(-0.667808\pi\)
							0.940913	−	0.338650i	\(-0.109970\pi\)
\(17\)	−195.599	−	164.127i	−0.676813	−	0.567913i	0.238260	−	0.971201i	\(-0.423423\pi\)
							−0.915073	+	0.403288i	\(0.867867\pi\)
\(19\)	119.081	+	340.794i	0.329865	+	0.944028i
							\(23\)	−65.4749	−	371.327i	−0.123771	−	0.701941i	−0.982030	−	0.188724i	\(-0.939565\pi\)
0.858259	−	0.513217i	\(-0.171546\pi\)
\(29\)	744.652	+	887.441i	0.885436	+	1.05522i	0.998102	+	0.0615888i	\(0.0196167\pi\)
							−0.112665	+	0.993633i	\(0.535939\pi\)
\(31\)	628.465	−	362.845i	0.653970	−	0.377570i	−0.136006	−	0.990708i	\(-0.543426\pi\)
							0.789976	+	0.613138i	\(0.210093\pi\)
\(37\)		−	1116.17i		−	0.815319i	−0.913134	−	0.407660i	\(-0.866345\pi\)
							0.913134	−	0.407660i	\(-0.133655\pi\)
\(41\)	871.727	+	2395.05i	0.518577	+	1.42478i	0.872089	+	0.489347i	\(0.162765\pi\)
							−0.353512	+	0.935430i	\(0.615013\pi\)
\(43\)	141.707	−	803.663i	0.0766401	−	0.434647i	−0.922209	−	0.386691i	\(-0.873618\pi\)
							0.998849	−	0.0479564i	\(-0.0152709\pi\)
\(47\)	−617.870	+	518.454i	−0.279706	+	0.234701i	−0.771838	−	0.635820i	\(-0.780662\pi\)
							0.492132	+	0.870521i	\(0.336218\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.5.f.a.3.4	✓	36
19.13	odd	18	inner	38.5.f.a.13.4	yes	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.5.f.a.3.4	✓	36	1.1	even	1	trivial
38.5.f.a.13.4	yes	36	19.13	odd	18	inner