Embedded newform 138.4.e.c.25.3

• Label: 138.4.e.c.25.3
• 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			25.3
Character	\(\chi\)	\(=\)	138.25
Dual form			138.4.e.c.127.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68251 - 1.08128i) q^{2} +(-0.426945 - 2.96946i) q^{3} +(1.66166 + 3.63853i) q^{4} +(19.5175 + 5.73085i) q^{5} +(-2.49249 + 5.45779i) q^{6} +(-20.5565 + 23.7235i) q^{7} +(1.13852 - 7.91857i) q^{8} +(-8.63544 + 2.53559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 4 q^{5} + 18 q^{6} + 4 q^{7} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

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

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
\(5\)	19.5175	+	5.73085i	1.74570	+	0.512582i								0.989844	−	0.142160i	\(-0.0454048\pi\)
							0.755852	+	0.654743i	\(0.227223\pi\)
\(7\)	−20.5565	+	23.7235i	−1.10995	+	1.28095i	−0.153787	+	0.988104i	\(0.549147\pi\)
							−0.956160	+	0.292844i	\(0.905398\pi\)
\(11\)	−48.3718	+	31.0867i	−1.32588	+	0.852090i	−0.995773	−	0.0918537i	\(-0.970721\pi\)
							−0.330106	+	0.943944i	\(0.607084\pi\)
\(13\)	−11.8208	−	13.6420i	−0.252193	−	0.291047i	0.615510	−	0.788129i	\(-0.288950\pi\)
							−0.867703	+	0.497082i	\(0.834405\pi\)
\(17\)	−26.4414	+	57.8985i	−0.377234	+	0.826027i	0.621846	+	0.783140i	\(0.286383\pi\)
							−0.999080	+	0.0428875i	\(0.986344\pi\)
\(19\)	14.4404	+	31.6201i	0.174361	+	0.381797i	0.976556	−	0.215265i	\(-0.0690617\pi\)
							−0.802195	+	0.597063i	\(0.796334\pi\)
\(23\)	−63.6806	+	90.0654i	−0.577318	+	0.816519i
							\(29\)	41.0133	−	89.8066i	0.262620	−	0.575057i	−0.731683	−	0.681645i	\(-0.761265\pi\)
0.994303	+	0.106587i	\(0.0339923\pi\)
\(31\)	9.61944	−	66.9047i	0.0557324	−	0.387627i	−0.942795	−	0.333374i	\(-0.891813\pi\)
							0.998527	−	0.0542536i	\(-0.0172779\pi\)
\(37\)	336.163	−	98.7063i	1.49364	−	0.438573i	0.569942	−	0.821685i	\(-0.306966\pi\)
							0.923702	+	0.383112i	\(0.125148\pi\)
\(41\)	−207.438	−	60.9092i	−0.790154	−	0.232010i	−0.138336	−	0.990385i	\(-0.544175\pi\)
							−0.651818	+	0.758375i	\(0.725993\pi\)
\(43\)	−2.32372	−	16.1618i	−0.00824103	−	0.0573176i	0.985286	−	0.170914i	\(-0.0546720\pi\)
							−0.993527	+	0.113596i	\(0.963763\pi\)
\(47\)	560.969			1.74097			0.870487	−	0.492192i	\(-0.163804\pi\)
							0.870487	+	0.492192i	\(0.163804\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.4.e.c.25.3	✓	30
23.12	even	11	inner	138.4.e.c.127.3	yes	30

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.c.25.3	✓	30	1.1	even	1	trivial
138.4.e.c.127.3	yes	30	23.12	even	11	inner