Embedded newform 114.4.i.b.61.2

• Label: 114.4.i.b.61.2
• Level: $114$
• Weight: $4$
• Character: 114.61
• Analytic conductor: $6.726$
• Analytic rank: $0$
• Dimension: $12$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	114.i (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.72621774065\)
Analytic rank:	\(0\)
Dimension:	\(12\)
Relative dimension:	\(2\) over \(\Q(\zeta_{9})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial:	\( x^{12} - 252x^{10} + 25236x^{8} - 1271150x^{6} + 34071732x^{4} - 464617557x^{2} + 2560258801 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			61.2
Root			\(4.37438 + 0.642788i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.61
Dual form			114.4.i.b.43.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.87939 + 0.684040i) q^{2} +(-0.520945 - 2.95442i) q^{3} +(3.06418 + 2.57115i) q^{4} +(7.95904 - 6.67843i) q^{5} +(1.04189 - 5.90885i) q^{6} +(4.10394 - 7.10823i) q^{7} +(4.00000 + 6.92820i) q^{8} +(-8.45723 + 3.07818i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 9 q^{5} + 12 q^{7} + 48 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.87939	+	0.684040i	0.664463	+	0.241845i
							\(3\)	−0.520945	−	2.95442i	−0.100256	−	0.568579i
\(5\)	7.95904	−	6.67843i	0.711878	−	0.597337i								−0.213247	−	0.976998i	\(-0.568404\pi\)
							0.925126	+	0.379661i	\(0.123959\pi\)
\(7\)	4.10394	−	7.10823i	0.221592	−	0.383808i	−0.733700	−	0.679474i	\(-0.762208\pi\)
							0.955291	+	0.295666i	\(0.0955414\pi\)
\(11\)	−7.04914	−	12.2095i	−0.193218	−	0.334663i	0.753097	−	0.657909i	\(-0.228559\pi\)
							−0.946315	+	0.323246i	\(0.895226\pi\)
\(13\)	9.12135	−	51.7298i	0.194601	−	1.10363i	−0.718386	−	0.695645i	\(-0.755119\pi\)
							0.912986	−	0.407990i	\(-0.133770\pi\)
\(17\)	86.2773	+	31.4024i	1.23090	+	0.448011i	0.873906	−	0.486095i	\(-0.161579\pi\)
							0.356995	+	0.934106i	\(0.383801\pi\)
\(19\)	82.1079	+	10.8302i	0.991413	+	0.130769i
							\(23\)	−74.2412	−	62.2958i	−0.673059	−	0.564764i	0.240910	−	0.970547i	\(-0.422554\pi\)
−0.913969	+	0.405784i	\(0.866999\pi\)
\(29\)	−123.799	+	45.0592i	−0.792721	+	0.288527i	−0.706467	−	0.707746i	\(-0.749712\pi\)
							−0.0862542	+	0.996273i	\(0.527490\pi\)
\(31\)	−143.166	+	247.971i	−0.829463	+	1.43667i	0.0689976	+	0.997617i	\(0.478020\pi\)
							−0.898460	+	0.439055i	\(0.855313\pi\)
\(37\)	108.337			0.481363			0.240682	−	0.970604i	\(-0.422629\pi\)
							0.240682	+	0.970604i	\(0.422629\pi\)
\(41\)	1.71604	+	9.73212i	0.00653658	+	0.0370708i	0.987902	−	0.155082i	\(-0.0495642\pi\)
							−0.981365	+	0.192153i	\(0.938453\pi\)
\(43\)	−217.922	+	182.858i	−0.772856	+	0.648503i	−0.941439	−	0.337184i	\(-0.890525\pi\)
							0.168583	+	0.985688i	\(0.446081\pi\)
\(47\)	−156.075	+	56.8068i	−0.484381	+	0.176300i	−0.572656	−	0.819796i	\(-0.694087\pi\)
							0.0882746	+	0.996096i	\(0.471865\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.4.i.b.61.2	yes	12
19.5	even	9	inner	114.4.i.b.43.2	✓	12
19.9	even	9		2166.4.a.bb.1.6		6
19.10	odd	18		2166.4.a.bg.1.6		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.i.b.43.2	✓	12	19.5	even	9	inner
114.4.i.b.61.2	yes	12	1.1	even	1	trivial
2166.4.a.bb.1.6		6	19.9	even	9
2166.4.a.bg.1.6		6	19.10	odd	18