Embedded newform 38.8.e.b.17.5

• Label: 38.8.e.b.17.5
• Level: $38$
• Weight: $8$
• Character: 38.17
• Analytic conductor: $11.871$
• Analytic rank: $0$
• Dimension: $36$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	38.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.8706309684\)
Analytic rank:	\(0\)
Dimension:	\(36\)
Relative dimension:	\(6\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			17.5
Character	\(\chi\)	\(=\)	38.17
Dual form			38.8.e.b.9.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.38919 + 7.87846i) q^{2} +(37.7512 + 31.6770i) q^{3} +(-60.1403 - 21.8893i) q^{4} +(-192.429 + 70.0383i) q^{5} +(-302.010 + 253.416i) q^{6} +(-814.928 - 1411.50i) q^{7} +(256.000 - 443.405i) q^{8} +(41.9510 + 237.916i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 84 q^{3} + 672 q^{6} - 2988 q^{7} + 9216 q^{8} + 1428 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{5}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.38919	+	7.87846i	−0.122788	+	0.696364i
							\(3\)	37.7512	+	31.6770i	0.807247	+	0.677360i	0.949949	−	0.312405i	\(-0.101135\pi\)
−0.142702	+	0.989766i	\(0.545579\pi\)
\(5\)	−192.429	+	70.0383i	−0.688454	+	0.250577i	−0.662473	−	0.749086i	\(-0.730493\pi\)
							−0.0259807	+	0.999662i	\(0.508271\pi\)
\(7\)	−814.928	−	1411.50i	−0.898000	−	1.55538i	−0.830047	−	0.557693i	\(-0.811687\pi\)
							−0.0679524	−	0.997689i	\(-0.521647\pi\)
\(11\)	−1687.55	+	2922.93i	−0.382281	+	0.662131i	−0.991388	−	0.130958i	\(-0.958195\pi\)
							0.609107	+	0.793088i	\(0.291528\pi\)
\(13\)	−6311.72	+	5296.16i	−0.796793	+	0.668589i	−0.947417	−	0.320002i	\(-0.896316\pi\)
							0.150624	+	0.988591i	\(0.451872\pi\)
\(17\)	6352.22	−	36025.2i	0.313584	−	1.77842i	−0.266466	−	0.963844i	\(-0.585856\pi\)
							0.580051	−	0.814580i	\(-0.303033\pi\)
\(19\)	22655.0	+	19509.6i	0.757750	+	0.652545i
							\(23\)	−102048.	−	37142.3i	−1.74886	−	0.636534i	−0.749195	−	0.662349i	\(-0.769560\pi\)
−0.999667	+	0.0258153i	\(0.991782\pi\)
\(29\)	6900.69	+	39135.8i	0.0525412	+	0.297976i	0.999743	−	0.0226616i	\(-0.00721402\pi\)
							−0.947202	+	0.320637i	\(0.896103\pi\)
\(31\)	54036.9	+	93594.6i	0.325780	+	0.564267i	0.981670	−	0.190589i	\(-0.0610398\pi\)
							−0.655890	+	0.754857i	\(0.727706\pi\)
\(37\)	−408636.			−1.32626			−0.663132	−	0.748502i	\(-0.730773\pi\)
							−0.663132	+	0.748502i	\(0.730773\pi\)
\(41\)	−5269.18	−	4421.36i	−0.0119399	−	0.0100187i	0.636798	−	0.771031i	\(-0.280258\pi\)
							−0.648738	+	0.761012i	\(0.724703\pi\)
\(43\)	−95849.5	+	34886.4i	−0.183844	+	0.0669139i	−0.432302	−	0.901729i	\(-0.642299\pi\)
							0.248458	+	0.968643i	\(0.420076\pi\)
\(47\)	124000.	+	703238.i	0.174212	+	0.988007i	0.939049	+	0.343783i	\(0.111708\pi\)
							−0.764837	+	0.644224i	\(0.777180\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.8.e.b.17.5	yes	36
19.9	even	9	inner	38.8.e.b.9.5	✓	36

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.8.e.b.9.5	✓	36	19.9	even	9	inner
38.8.e.b.17.5	yes	36	1.1	even	1	trivial