Embedded newform 138.4.e.c.85.1

• Label: 138.4.e.c.85.1
• Level: $138$
• Weight: $4$
• Character: 138.85
• Analytic conductor: $8.142$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	138.e (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.14226358079\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(3\) over \(\Q(\zeta_{11})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label			85.1
Character	\(\chi\)	\(=\)	138.85
Dual form			138.4.e.c.13.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.30972 - 1.51150i) q^{2} +(2.52376 - 1.62192i) q^{3} +(-0.569259 - 3.95929i) q^{4} +(-4.97931 - 10.9032i) q^{5} +(0.853889 - 5.93893i) q^{6} +(10.5666 - 3.10264i) q^{7} +(-6.73003 - 4.32513i) q^{8} +(3.73874 - 8.18669i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 4 q^{5} + 18 q^{6} + 4 q^{7} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/138\mathbb{Z}\right)^\times\).

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{4}{11}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.30972	−	1.51150i	0.463056	−	0.534396i
							\(3\)	2.52376	−	1.62192i	0.485698	−	0.312139i
\(5\)	−4.97931	−	10.9032i	−0.445363	−	0.975208i								−0.990583	−	0.136915i	\(-0.956281\pi\)
							0.545220	−	0.838293i	\(-0.316446\pi\)
\(7\)	10.5666	−	3.10264i	0.570543	−	0.167527i	0.0162793	−	0.999867i	\(-0.494818\pi\)
							0.554264	+	0.832341i	\(0.313000\pi\)
\(11\)	12.9851	+	14.9856i	0.355924	+	0.410758i	0.905270	−	0.424837i	\(-0.139669\pi\)
							−0.549346	+	0.835595i	\(0.685123\pi\)
\(13\)	−72.7302	−	21.3555i	−1.55167	−	0.455612i	−0.610071	−	0.792347i	\(-0.708859\pi\)
							−0.941599	+	0.336735i	\(0.890677\pi\)
\(17\)	9.18400	−	63.8761i	0.131026	−	0.911308i	−0.813194	−	0.581993i	\(-0.802273\pi\)
							0.944220	−	0.329315i	\(-0.106818\pi\)
\(19\)	7.20524	+	50.1135i	0.0869997	+	0.605096i	0.985949	+	0.167044i	\(0.0534223\pi\)
							−0.898950	+	0.438052i	\(0.855669\pi\)
\(23\)	81.2629	−	74.5878i	0.736717	−	0.676201i
							\(29\)	−0.816574	+	5.67939i	−0.00522876	+	0.0363668i	−0.992269	−	0.124107i	\(-0.960393\pi\)
0.987040	+	0.160474i	\(0.0513023\pi\)
\(31\)	175.793	+	112.975i	1.01849	+	0.654547i	0.939578	−	0.342335i	\(-0.111218\pi\)
							0.0789163	+	0.996881i	\(0.474854\pi\)
\(37\)	80.2765	−	175.781i	0.356686	−	0.781033i	−0.643196	−	0.765701i	\(-0.722392\pi\)
							0.999882	−	0.0153322i	\(-0.00488059\pi\)
\(41\)	61.5008	+	134.668i	0.234264	+	0.512966i	0.989856	−	0.142076i	\(-0.0453779\pi\)
							−0.755592	+	0.655043i	\(0.772651\pi\)
\(43\)	−302.392	+	194.336i	−1.07243	+	0.689207i	−0.952796	−	0.303611i	\(-0.901808\pi\)
							−0.119631	+	0.992818i	\(0.538171\pi\)
\(47\)	604.120			1.87489			0.937446	−	0.348130i	\(-0.113183\pi\)
							0.937446	+	0.348130i	\(0.113183\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.c.85.1	yes	30
23.13	even	11	inner	138.4.e.c.13.1	✓	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.c.13.1	✓	30	23.13	even	11	inner
138.4.e.c.85.1	yes	30	1.1	even	1	trivial