Embedded newform 114.4.e.a.49.1

• Label: 114.4.e.a.49.1
• Level: $114$
• Weight: $4$
• Character: 114.49
• Analytic conductor: $6.726$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	114.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.72621774065\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\zeta_{6})\)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			49.1
Root			\(0.500000 + 0.866025i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.49
Dual form			114.4.e.a.7.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(-1.50000 + 2.59808i) q^{3} +(-2.00000 - 3.46410i) q^{4} +(-1.00000 + 1.73205i) q^{5} +(3.00000 + 5.19615i) q^{6} -21.0000 q^{7} -8.00000 q^{8} +(-4.50000 - 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} - 2 q^{5} + 6 q^{6} - 42 q^{7} - 16 q^{8} - 9 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	−	1.73205i	0.353553	−	0.612372i
							\(3\)	−1.50000	+	2.59808i	−0.288675	+	0.500000i
\(5\)	−1.00000	+	1.73205i	−0.0894427	+	0.154919i								−0.907276	−	0.420536i	\(-0.861842\pi\)
							0.817833	+	0.575456i	\(0.195175\pi\)
\(7\)	−21.0000			−1.13389			−0.566947	−	0.823754i	\(-0.691875\pi\)
							−0.566947	+	0.823754i	\(0.691875\pi\)
\(11\)	−40.0000			−1.09640			−0.548202	−	0.836346i	\(-0.684688\pi\)
							−0.548202	+	0.836346i	\(0.684688\pi\)
\(13\)	−8.50000	−	14.7224i	−0.181344	−	0.314098i	0.760994	−	0.648759i	\(-0.224712\pi\)
							−0.942339	+	0.334661i	\(0.891378\pi\)
\(17\)	−18.0000	+	31.1769i	−0.256802	+	0.444795i	−0.965384	−	0.260835i	\(-0.916002\pi\)
							0.708581	+	0.705629i	\(0.249336\pi\)
\(19\)	−76.0000	+	32.9090i	−0.917663	+	0.397360i
							\(23\)	−37.0000	−	64.0859i	−0.335436	−	0.580993i	0.648132	−	0.761528i	\(-0.275550\pi\)
−0.983569	+	0.180535i	\(0.942217\pi\)
\(29\)	−50.0000	−	86.6025i	−0.320164	−	0.554541i	0.660357	−	0.750951i	\(-0.270405\pi\)
							−0.980522	+	0.196411i	\(0.937071\pi\)
\(31\)	103.000			0.596753			0.298377	−	0.954448i	\(-0.403555\pi\)
							0.298377	+	0.954448i	\(0.403555\pi\)
\(37\)	187.000			0.830881			0.415441	−	0.909620i	\(-0.363627\pi\)
							0.415441	+	0.909620i	\(0.363627\pi\)
\(41\)	64.0000	−	110.851i	0.243783	−	0.422245i	−0.718005	−	0.696037i	\(-0.754945\pi\)
							0.961789	+	0.273792i	\(0.0882781\pi\)
\(43\)	−60.5000	+	104.789i	−0.214562	+	0.371632i	−0.953137	−	0.302539i	\(-0.902166\pi\)
							0.738575	+	0.674171i	\(0.235499\pi\)
\(47\)	−205.000	−	355.070i	−0.636220	−	1.10196i	−0.986255	−	0.165228i	\(-0.947164\pi\)
							0.350036	−	0.936736i	\(-0.386169\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.4.e.a.49.1	yes	2
3.2	odd	2		342.4.g.b.163.1		2
19.7	even	3	inner	114.4.e.a.7.1	✓	2
19.8	odd	6		2166.4.a.f.1.1		1
19.11	even	3		2166.4.a.c.1.1		1
57.26	odd	6		342.4.g.b.235.1		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.e.a.7.1	✓	2	19.7	even	3	inner
114.4.e.a.49.1	yes	2	1.1	even	1	trivial
342.4.g.b.163.1		2	3.2	odd	2
342.4.g.b.235.1		2	57.26	odd	6
2166.4.a.c.1.1		1	19.11	even	3
2166.4.a.f.1.1		1	19.8	odd	6