Embedded newform 138.4.e.b.25.1

• Label: 138.4.e.b.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.b.127.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.68251 + 1.08128i) q^{2} +(0.426945 + 2.96946i) q^{3} +(1.66166 + 3.63853i) q^{4} +(-20.3614 - 5.97866i) q^{5} +(-2.49249 + 5.45779i) q^{6} +(5.13322 - 5.92406i) q^{7} +(-1.13852 + 7.91857i) q^{8} +(-8.63544 + 2.53559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 6 q^{2} + 9 q^{3} - 12 q^{4} - 6 q^{5} + 18 q^{6} + 22 q^{7} - 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

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

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
\(5\)	−20.3614	−	5.97866i	−1.82118	−	0.534747i								−0.821796	−	0.569782i	\(-0.807028\pi\)
							−0.999386	+	0.0350350i	\(0.988846\pi\)
\(7\)	5.13322	−	5.92406i	0.277168	−	0.319869i	−0.600049	−	0.799963i	\(-0.704852\pi\)
							0.877217	+	0.480094i	\(0.159398\pi\)
\(11\)	−33.5772	+	21.5787i	−0.920354	+	0.591476i	−0.912760	−	0.408495i	\(-0.866054\pi\)
							−0.00759384	+	0.999971i	\(0.502417\pi\)
\(13\)	−36.4799	−	42.1001i	−0.778286	−	0.898189i	0.218699	−	0.975792i	\(-0.429819\pi\)
							−0.996984	+	0.0776030i	\(0.975273\pi\)
\(17\)	−37.7694	+	82.7034i	−0.538848	+	1.17991i	0.422950	+	0.906153i	\(0.360995\pi\)
							−0.961798	+	0.273760i	\(0.911733\pi\)
\(19\)	−47.3670	−	103.719i	−0.571933	−	1.25236i	−0.945762	−	0.324861i	\(-0.894682\pi\)
							0.373828	−	0.927498i	\(-0.378045\pi\)
\(23\)	107.478	−	24.8105i	0.974375	−	0.224928i
							\(29\)	−62.5068	+	136.871i	−0.400249	+	0.876423i	0.596996	+	0.802244i	\(0.296361\pi\)
−0.997245	+	0.0741786i	\(0.976367\pi\)
\(31\)	0.528140	−	3.67329i	0.00305989	−	0.0212820i	−0.988234	−	0.152948i	\(-0.951123\pi\)
							0.991294	+	0.131666i	\(0.0420325\pi\)
\(37\)	−210.432	+	61.7884i	−0.934995	+	0.274539i	−0.713527	−	0.700628i	\(-0.752903\pi\)
							−0.221469	+	0.975168i	\(0.571085\pi\)
\(41\)	−136.943	−	40.2102i	−0.521633	−	0.153165i	0.0103074	−	0.999947i	\(-0.496719\pi\)
							−0.531941	+	0.846781i	\(0.678537\pi\)
\(43\)	52.5290	+	365.347i	0.186293	+	1.29569i	0.841505	+	0.540250i	\(0.181670\pi\)
							−0.655212	+	0.755445i	\(0.727421\pi\)
\(47\)	−145.843			−0.452626			−0.226313	−	0.974055i	\(-0.572667\pi\)
							−0.226313	+	0.974055i	\(0.572667\pi\)

(See \(a_n\) instead)

Twists

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

    

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