Embedded newform 138.4.e.d.25.1

• Label: 138.4.e.d.25.1
• Level: $138$
• Weight: $4$
• Character: 138.25
• 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			25.1
Character	\(\chi\)	\(=\)	138.25
Dual form			138.4.e.d.127.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68251 - 1.08128i) q^{2} +(0.426945 + 2.96946i) q^{3} +(1.66166 + 3.63853i) q^{4} +(-17.7871 - 5.22276i) q^{5} +(2.49249 - 5.45779i) q^{6} +(3.76192 - 4.34149i) q^{7} +(1.13852 - 7.91857i) q^{8} +(-8.63544 + 2.53559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} + 9 q^{3} - 12 q^{4} - 2 q^{5} - 18 q^{6} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.68251	−	1.08128i	−0.594856	−	0.382291i
							\(3\)	0.426945	+	2.96946i	0.0821655	+	0.571474i
\(5\)	−17.7871	−	5.22276i	−1.59092	−	0.467138i								−0.637922	−	0.770101i	\(-0.720206\pi\)
							−0.953002	+	0.302963i	\(0.902024\pi\)
\(7\)	3.76192	−	4.34149i	0.203125	−	0.234418i	−0.645043	−	0.764146i	\(-0.723161\pi\)
							0.848168	+	0.529728i	\(0.177706\pi\)
\(11\)	56.0818	−	36.0416i	1.53721	−	0.987904i	0.548823	−	0.835939i	\(-0.315076\pi\)
							0.988386	−	0.151965i	\(-0.0485602\pi\)
\(13\)	38.2182	+	44.1062i	0.815371	+	0.940988i	0.999118	−	0.0419851i	\(-0.0133682\pi\)
							−0.183747	+	0.982974i	\(0.558823\pi\)
\(17\)	−32.0227	+	70.1198i	−0.456861	+	1.00039i	0.531331	+	0.847164i	\(0.321692\pi\)
							−0.988192	+	0.153221i	\(0.951035\pi\)
\(19\)	49.2279	+	107.794i	0.594402	+	1.30156i	0.932745	+	0.360537i	\(0.117406\pi\)
							−0.338343	+	0.941023i	\(0.609866\pi\)
\(23\)	108.360	+	20.6180i	0.982375	+	0.186920i
							\(29\)	40.9847	−	89.7440i	0.262437	−	0.574656i	−0.731842	−	0.681475i	\(-0.761339\pi\)
0.994279	+	0.106818i	\(0.0340663\pi\)
\(31\)	40.7265	−	283.259i	0.235958	−	1.64112i	−0.435577	−	0.900151i	\(-0.643456\pi\)
							0.671535	−	0.740973i	\(-0.265635\pi\)
\(37\)	211.074	−	61.9769i	0.937847	−	0.275377i	0.223129	−	0.974789i	\(-0.428373\pi\)
							0.714719	+	0.699412i	\(0.246555\pi\)
\(41\)	−115.521	−	33.9201i	−0.440033	−	0.129205i	0.0542089	−	0.998530i	\(-0.482736\pi\)
							−0.494242	+	0.869324i	\(0.664554\pi\)
\(43\)	−0.462519	−	3.21689i	−0.00164031	−	0.0114086i	0.988985	−	0.148017i	\(-0.0472890\pi\)
							−0.990625	+	0.136608i	\(0.956380\pi\)
\(47\)	235.923			0.732189			0.366095	−	0.930578i	\(-0.380695\pi\)
							0.366095	+	0.930578i	\(0.380695\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.d.25.1	✓	30
23.12	even	11	inner	138.4.e.d.127.1	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.d.25.1	✓	30	1.1	even	1	trivial
138.4.e.d.127.1	yes	30	23.12	even	11	inner