Embedded newform 114.4.i.b.55.1

• Label: 114.4.i.b.55.1
• Level: $114$
• Weight: $4$
• Character: 114.55
• 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			55.1
Root			\(5.86912 - 0.342020i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.55
Dual form			114.4.i.b.85.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-0.347296 + 1.96962i) q^{2} +(-2.29813 - 1.92836i) q^{3} +(-3.75877 - 1.36808i) q^{4} +(-4.17612 + 1.51998i) q^{5} +(4.59627 - 3.85673i) q^{6} +(2.30170 + 3.98666i) q^{7} +(4.00000 - 6.92820i) q^{8} +(1.56283 + 8.86327i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 9 q^{5} + 12 q^{7} + 48 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{5}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−0.347296	+	1.96962i	−0.122788	+	0.696364i
							\(3\)	−2.29813	−	1.92836i	−0.442276	−	0.371114i
\(5\)	−4.17612	+	1.51998i	−0.373524	+	0.135952i								−0.521959	−	0.852971i	\(-0.674799\pi\)
							0.148435	+	0.988922i	\(0.452576\pi\)
\(7\)	2.30170	+	3.98666i	0.124280	+	0.215259i	0.921451	−	0.388494i	\(-0.127005\pi\)
							−0.797171	+	0.603753i	\(0.793671\pi\)
\(11\)	28.5180	−	49.3946i	0.781682	−	1.35391i	−0.149279	−	0.988795i	\(-0.547695\pi\)
							0.930961	−	0.365118i	\(-0.118971\pi\)
\(13\)	34.3477	−	28.8211i	0.732794	−	0.614888i	−0.198097	−	0.980182i	\(-0.563476\pi\)
							0.930892	+	0.365295i	\(0.119032\pi\)
\(17\)	14.8505	−	84.2216i	0.211870	−	1.20157i	−0.674387	−	0.738378i	\(-0.735592\pi\)
							0.886257	−	0.463195i	\(-0.153297\pi\)
\(19\)	82.5822	+	6.25877i	0.997140	+	0.0755716i
							\(23\)	10.1682	+	3.70091i	0.0921831	+	0.0335519i	0.387700	−	0.921786i	\(-0.373270\pi\)
−0.295517	+	0.955338i	\(0.595492\pi\)
\(29\)	−16.7053	−	94.7403i	−0.106969	−	0.606649i	−0.990416	−	0.138118i	\(-0.955895\pi\)
							0.883447	−	0.468531i	\(-0.155217\pi\)
\(31\)	−6.77926	−	11.7420i	−0.0392771	−	0.0680300i	0.845719	−	0.533629i	\(-0.179172\pi\)
							−0.884996	+	0.465599i	\(0.845839\pi\)
\(37\)	360.525			1.60189			0.800946	−	0.598737i	\(-0.204331\pi\)
							0.800946	+	0.598737i	\(0.204331\pi\)
\(41\)	−219.061	−	183.814i	−0.834429	−	0.700169i	0.121874	−	0.992546i	\(-0.461109\pi\)
							−0.956303	+	0.292377i	\(0.905554\pi\)
\(43\)	−249.154	+	90.6845i	−0.883618	+	0.321611i	−0.743669	−	0.668548i	\(-0.766916\pi\)
							−0.139949	+	0.990159i	\(0.544694\pi\)
\(47\)	−26.8761	−	152.422i	−0.0834102	−	0.473043i	−0.997688	−	0.0679559i	\(-0.978352\pi\)
							0.914278	−	0.405087i	\(-0.132759\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.4.i.b.55.1	✓	12
19.3	odd	18		2166.4.a.bg.1.4		6
19.9	even	9	inner	114.4.i.b.85.1	yes	12
19.16	even	9		2166.4.a.bb.1.4		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.i.b.55.1	✓	12	1.1	even	1	trivial
114.4.i.b.85.1	yes	12	19.9	even	9	inner
2166.4.a.bb.1.4		6	19.16	even	9
2166.4.a.bg.1.4		6	19.3	odd	18