Embedded newform 38.9.f.a.3.3

• Label: 38.9.f.a.3.3
• Level: $38$
• Weight: $9$
• Character: 38.3
• Analytic conductor: $15.480$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	38.f (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(15.4803871823\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(14\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			3.3
Character	\(\chi\)	\(=\)	38.3
Dual form			38.9.f.a.13.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-7.27231 + 8.66680i) q^{2} +(-6.31575 - 17.3524i) q^{3} +(-22.2270 - 126.055i) q^{4} +(-119.553 + 678.019i) q^{5} +(196.320 + 71.4546i) q^{6} +(116.087 + 201.069i) q^{7} +(1254.14 + 724.077i) q^{8} +(4764.80 - 3998.14i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 168 q^{3} - 2688 q^{6} - 6570 q^{7} - 2856 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−7.27231	+	8.66680i	−0.454519	+	0.541675i
							\(3\)	−6.31575	−	17.3524i	−0.0779722	−	0.214227i	0.894582	−	0.446904i	\(-0.147474\pi\)
−0.972554	+	0.232677i	\(0.925251\pi\)
\(5\)	−119.553	+	678.019i	−0.191285	+	1.08483i	0.726326	+	0.687351i	\(0.241226\pi\)
							−0.917611	+	0.397480i	\(0.869885\pi\)
\(7\)	116.087	+	201.069i	0.0483496	+	0.0837439i	0.889187	−	0.457543i	\(-0.151271\pi\)
							−0.840838	+	0.541287i	\(0.817937\pi\)
\(11\)	−6373.68	+	11039.5i	−0.435331	+	0.754015i	−0.997323	−	0.0731277i	\(-0.976702\pi\)
							0.561992	+	0.827143i	\(0.310035\pi\)
\(13\)	−10590.4	+	29097.0i	−0.370801	+	1.01877i	0.604252	+	0.796793i	\(0.293472\pi\)
							−0.975053	+	0.221973i	\(0.928750\pi\)
\(17\)	−93732.7	−	78651.1i	−1.12226	−	0.941692i	−0.123548	−	0.992339i	\(-0.539427\pi\)
							−0.998717	+	0.0506466i	\(0.983872\pi\)
\(19\)	−86304.3	−	97648.0i	−0.662244	−	0.749289i
							\(23\)	−76128.5	−	431746.i	−0.272042	−	1.54283i	−0.748204	−	0.663469i	\(-0.769083\pi\)
0.476162	−	0.879358i	\(-0.342028\pi\)
\(29\)	−292255.	−	348295.i	−0.413209	−	0.492443i	0.518792	−	0.854901i	\(-0.326382\pi\)
							−0.932000	+	0.362458i	\(0.881938\pi\)
\(31\)	−1.33556e6	+	771085.i	−1.44616	+	0.834941i	−0.998250	−	0.0591378i	\(-0.981165\pi\)
							−0.447910	+	0.894079i	\(0.647832\pi\)
\(37\)			536700.i			0.286368i	0.989696	+	0.143184i	\(0.0457341\pi\)
							−0.989696	+	0.143184i	\(0.954266\pi\)
\(41\)	−891557.	−	2.44953e6i	−0.315510	−	0.866858i	−0.991519	−	0.129964i	\(-0.958514\pi\)
							0.676008	−	0.736894i	\(-0.263708\pi\)
\(43\)	−618878.	+	3.50983e6i	−0.181022	+	1.02663i	0.749939	+	0.661507i	\(0.230083\pi\)
							−0.930961	+	0.365119i	\(0.881028\pi\)
\(47\)	−5.46826e6	+	4.58841e6i	−1.12062	+	0.940310i	−0.998636	−	0.0522218i	\(-0.983370\pi\)
							−0.121983	+	0.992532i	\(0.538925\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.9.f.a.3.3	✓	84
19.13	odd	18	inner	38.9.f.a.13.3	yes	84

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.f.a.3.3	✓	84	1.1	even	1	trivial
38.9.f.a.13.3	yes	84	19.13	odd	18	inner