Embedded newform 38.10.e.a.5.1

• Label: 38.10.e.a.5.1
• Level: $38$
• Weight: $10$
• Character: 38.5
• Analytic conductor: $19.571$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.5713617742\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(7\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			5.1
Character	\(\chi\)	\(=\)	38.5
Dual form			38.10.e.a.23.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(15.0351 - 5.47232i) q^{2} +(-40.8678 + 231.773i) q^{3} +(196.107 - 164.554i) q^{4} +(1498.95 + 1257.77i) q^{5} +(653.886 + 3708.37i) q^{6} +(4614.74 + 7992.96i) q^{7} +(2048.00 - 3547.24i) q^{8} +(-33552.6 - 12212.2i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 252 q^{3} + 4032 q^{6} + 14373 q^{7} + 86016 q^{8} + 12852 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{8}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	15.0351	−	5.47232i	0.664463	−	0.241845i
							\(3\)	−40.8678	+	231.773i	−0.291297	+	1.65203i	0.390586	+	0.920567i	\(0.372272\pi\)
−0.681883	+	0.731461i	\(0.738839\pi\)
\(5\)	1498.95	+	1257.77i	1.07256	+	0.899986i	0.995282	−	0.0970231i	\(-0.0309321\pi\)
							0.0772798	+	0.997009i	\(0.475377\pi\)
\(7\)	4614.74	+	7992.96i	0.726450	+	1.25825i	0.958374	+	0.285514i	\(0.0921645\pi\)
							−0.231924	+	0.972734i	\(0.574502\pi\)
\(11\)	28104.0	−	48677.6i	0.578764	−	1.00245i	−0.416858	−	0.908972i	\(-0.636869\pi\)
							0.995621	−	0.0934767i	\(-0.0297981\pi\)
\(13\)	4970.79	+	28190.7i	0.0482703	+	0.273755i	0.999384	−	0.0350854i	\(-0.0111703\pi\)
							−0.951114	+	0.308840i	\(0.900059\pi\)
\(17\)	307210.	−	111815.i	0.892103	−	0.324699i	0.145019	−	0.989429i	\(-0.453676\pi\)
							0.747084	+	0.664730i	\(0.231453\pi\)
\(19\)	−565801.	+	50562.0i	−0.996031	+	0.0890088i
							\(23\)	−1.27457e6	+	1.06950e6i	−0.949708	+	0.796900i	−0.979248	−	0.202664i	\(-0.935040\pi\)
0.0295403	+	0.999564i	\(0.490596\pi\)
\(29\)	4.94722e6	+	1.80064e6i	1.29888	+	0.472755i	0.896633	−	0.442775i	\(-0.146006\pi\)
							0.402251	+	0.915530i	\(0.368228\pi\)
\(31\)	−4.32408e6	−	7.48952e6i	−0.840942	−	1.45655i	−0.889099	−	0.457714i	\(-0.848668\pi\)
							0.0481575	−	0.998840i	\(-0.484665\pi\)
\(37\)	−8.65237e6			−0.758974			−0.379487	−	0.925197i	\(-0.623900\pi\)
							−0.379487	+	0.925197i	\(0.623900\pi\)
\(41\)	5.38288e6	−	3.05278e7i	0.297500	−	1.68721i	−0.359363	−	0.933198i	\(-0.617006\pi\)
							0.656863	−	0.754010i	\(-0.271883\pi\)
\(43\)	1.15676e7	+	9.70638e6i	0.515983	+	0.432961i	0.863229	−	0.504813i	\(-0.168439\pi\)
							−0.347246	+	0.937774i	\(0.612883\pi\)
\(47\)	1.55086e6	+	564467.i	0.0463588	+	0.0168732i	0.365095	−	0.930970i	\(-0.381036\pi\)
							−0.318737	+	0.947843i	\(0.603259\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.10.e.a.5.1	✓	42
19.4	even	9	inner	38.10.e.a.23.1	yes	42

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.10.e.a.5.1	✓	42	1.1	even	1	trivial
38.10.e.a.23.1	yes	42	19.4	even	9	inner