Embedded newform 38.6.e.b.23.4

• Label: 38.6.e.b.23.4
• Level: $38$
• Weight: $6$
• Character: 38.23
• Analytic conductor: $6.095$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(6.09458515289\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(5\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			23.4
Character	\(\chi\)	\(=\)	38.23
Dual form			38.6.e.b.5.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-3.75877 - 1.36808i) q^{2} +(3.93255 + 22.3026i) q^{3} +(12.2567 + 10.2846i) q^{4} +(61.0402 - 51.2189i) q^{5} +(15.7302 - 89.2104i) q^{6} +(-95.8893 + 166.085i) q^{7} +(-32.0000 - 55.4256i) q^{8} +(-253.595 + 92.3012i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 15 q^{3} + 60 q^{6} - 84 q^{7} - 960 q^{8} - 345 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(21\)
\(\chi(n)\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−3.75877	−	1.36808i	−0.664463	−	0.241845i
							\(3\)	3.93255	+	22.3026i	0.252273	+	1.43071i	0.802976	+	0.596011i	\(0.203248\pi\)
−0.550703	+	0.834701i	\(0.685640\pi\)
\(5\)	61.0402	−	51.2189i	1.09192	−	0.916231i	0.0950656	−	0.995471i	\(-0.469694\pi\)
							0.996856	+	0.0792404i	\(0.0252495\pi\)
\(7\)	−95.8893	+	166.085i	−0.739648	+	1.28111i	0.213006	+	0.977051i	\(0.431675\pi\)
							−0.952654	+	0.304057i	\(0.901659\pi\)
\(11\)	179.141	+	310.281i	0.446388	+	0.773167i	0.998148	−	0.0608359i	\(-0.0193766\pi\)
							−0.551759	+	0.834003i	\(0.686043\pi\)
\(13\)	−77.0619	+	437.039i	−0.126468	+	0.717236i	0.853957	+	0.520344i	\(0.174196\pi\)
							−0.980425	+	0.196893i	\(0.936915\pi\)
\(17\)	−931.877	−	339.176i	−0.782053	−	0.284644i	−0.0800247	−	0.996793i	\(-0.525500\pi\)
							−0.702029	+	0.712149i	\(0.747722\pi\)
\(19\)	−1314.04	+	865.679i	−0.835073	+	0.550140i
							\(23\)	1025.79	+	860.740i	0.404333	+	0.339276i	0.822166	−	0.569248i	\(-0.192766\pi\)
−0.417833	+	0.908524i	\(0.637210\pi\)
\(29\)	4769.55	−	1735.97i	1.05313	−	0.383308i	0.243288	−	0.969954i	\(-0.421774\pi\)
							0.809844	+	0.586646i	\(0.199552\pi\)
\(31\)	3275.66	−	5673.61i	0.612202	−	1.06037i	−0.378666	−	0.925533i	\(-0.623617\pi\)
							0.990868	−	0.134832i	\(-0.0430495\pi\)
\(37\)	13217.2			1.58721			0.793607	−	0.608430i	\(-0.208201\pi\)
							0.793607	+	0.608430i	\(0.208201\pi\)
\(41\)	−1716.25	−	9733.34i	−0.159449	−	0.904278i	−0.954605	−	0.297874i	\(-0.903722\pi\)
							0.795156	−	0.606404i	\(-0.207389\pi\)
\(43\)	8638.67	−	7248.71i	0.712485	−	0.597846i	−0.212810	−	0.977094i	\(-0.568262\pi\)
							0.925295	+	0.379247i	\(0.123817\pi\)
\(47\)	−26465.9	+	9632.81i	−1.74760	+	0.636075i	−0.999618	−	0.0276428i	\(-0.991200\pi\)
							−0.747983	+	0.663718i	\(0.768978\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.6.e.b.23.4	yes	30
19.5	even	9	inner	38.6.e.b.5.4	✓	30
19.9	even	9		722.6.a.r.1.12		15
19.10	odd	18		722.6.a.q.1.4		15

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.e.b.5.4	✓	30	19.5	even	9	inner
38.6.e.b.23.4	yes	30	1.1	even	1	trivial
722.6.a.q.1.4		15	19.10	odd	18
722.6.a.r.1.12		15	19.9	even	9