Embedded newform 138.4.e.c.127.1

• Label: 138.4.e.c.127.1
• Level: $138$
• Weight: $4$
• Character: 138.127
• 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			127.1
Character	\(\chi\)	\(=\)	138.127
Dual form			138.4.e.c.25.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.68251 + 1.08128i) q^{2} +(-0.426945 + 2.96946i) q^{3} +(1.66166 - 3.63853i) q^{4} +(-11.8265 + 3.47259i) q^{5} +(-2.49249 - 5.45779i) q^{6} +(13.0481 + 15.0583i) q^{7} +(1.13852 + 7.91857i) q^{8} +(-8.63544 - 2.53559i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 4 q^{5} + 18 q^{6} + 4 q^{7} + 24 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(47\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{10}{11}\right)\)

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
\(5\)	−11.8265	+	3.47259i	−1.05780	+	0.310598i								−0.763964	−	0.645259i	\(-0.776749\pi\)
							−0.293834	+	0.955856i	\(0.594931\pi\)
\(7\)	13.0481	+	15.0583i	0.704530	+	0.813071i	0.989357	−	0.145508i	\(-0.0464817\pi\)
							−0.284827	+	0.958579i	\(0.591936\pi\)
\(11\)	−32.9545	−	21.1786i	−0.903288	−	0.580508i	0.00447595	−	0.999990i	\(-0.498575\pi\)
							−0.907764	+	0.419482i	\(0.862212\pi\)
\(13\)	−10.0050	+	11.5463i	−0.213452	+	0.246337i	−0.852371	−	0.522937i	\(-0.824836\pi\)
							0.638919	+	0.769274i	\(0.279382\pi\)
\(17\)	−25.9755	−	56.8785i	−0.370588	−	0.811474i	−0.999424	−	0.0339365i	\(-0.989196\pi\)
							0.628836	−	0.777538i	\(-0.283532\pi\)
\(19\)	14.2566	−	31.2177i	0.172142	−	0.376938i	−0.803822	−	0.594870i	\(-0.797204\pi\)
							0.975964	+	0.217931i	\(0.0699309\pi\)
\(23\)	5.55841	−	110.164i	0.0503917	−	0.998730i
							\(29\)	−26.7355	−	58.5426i	−0.171195	−	0.374865i	0.804514	−	0.593933i	\(-0.202426\pi\)
−0.975710	+	0.219068i	\(0.929698\pi\)
\(31\)	−27.1629	−	188.922i	−0.157374	−	1.09456i	−0.903448	−	0.428699i	\(-0.858972\pi\)
							0.746073	−	0.665864i	\(-0.231937\pi\)
\(37\)	−144.999	−	42.5756i	−0.644263	−	0.189173i	−0.0567512	−	0.998388i	\(-0.518074\pi\)
							−0.587512	+	0.809216i	\(0.699892\pi\)
\(41\)	−378.548	+	111.152i	−1.44193	+	0.423390i	−0.906866	−	0.421420i	\(-0.861532\pi\)
							−0.535067	+	0.844809i	\(0.679714\pi\)
\(43\)	−19.5020	+	135.639i	−0.0691633	+	0.481041i	0.925573	+	0.378570i	\(0.123584\pi\)
							−0.994736	+	0.102471i	\(0.967325\pi\)
\(47\)	87.1325			0.270417			0.135208	−	0.990817i	\(-0.456830\pi\)
							0.135208	+	0.990817i	\(0.456830\pi\)

(See \(a_n\) instead)

Twists

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

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.4.e.c.25.1	✓	30	23.2	even	11	inner
138.4.e.c.127.1	yes	30	1.1	even	1	trivial