Embedded newform 114.4.i.d.55.1

• Label: 114.4.i.d.55.1
• Level: $114$
• Weight: $4$
• Character: 114.55
• Analytic conductor: $6.726$
• Analytic rank: $0$
• Dimension: $18$
• 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:	\(18\)
Relative dimension:	\(3\) over \(\Q(\zeta_{9})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)
Defining polynomial:	x^18 + 603*x^16 - 1528*x^15 + 240417*x^14 - 786198*x^13 + 54822410*x^12 - 261322128*x^11 + 9122210535*x^10 - 45012474508*x^9 + 1000170346788*x^8 - 5035551052410*x^7 + 79008124941737*x^6 - 336274219954836*x^5 + 3621639622047612*x^4 - 11991799127581216*x^3 + 107004711561222960*x^2 - 278928948783924960*x + 1161091951621493824
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			55.1
Root			\(-7.03302 + 12.1815i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.55
Dual form			114.4.i.d.85.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.347296 - 1.96962i) q^{2} +(2.29813 + 1.92836i) q^{3} +(-3.75877 - 1.36808i) q^{4} +(-14.2708 + 5.19414i) q^{5} +(4.59627 - 3.85673i) q^{6} +(-0.469536 - 0.813260i) q^{7} +(-4.00000 + 6.92820i) q^{8} +(1.56283 + 8.86327i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 15 q^{5} - 72 q^{8}+O(q^{10}) \)

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
\(5\)	−14.2708	+	5.19414i	−1.27642	+	0.464578i								−0.889246	−	0.457429i	\(-0.848770\pi\)
							−0.387172	+	0.922008i	\(0.626548\pi\)
\(7\)	−0.469536	−	0.813260i	−0.0253525	−	0.0439119i	0.853071	−	0.521795i	\(-0.174737\pi\)
							−0.878423	+	0.477883i	\(0.841404\pi\)
\(11\)	−25.0416	+	43.3733i	−0.686392	+	1.18887i	0.286605	+	0.958049i	\(0.407473\pi\)
							−0.972997	+	0.230817i	\(0.925860\pi\)
\(13\)	−23.4166	+	19.6488i	−0.499584	+	0.419201i	−0.857446	−	0.514574i	\(-0.827950\pi\)
							0.357862	+	0.933774i	\(0.383506\pi\)
\(17\)	−6.41604	+	36.3872i	−0.0915364	+	0.519129i	0.904217	+	0.427072i	\(0.140455\pi\)
							−0.995754	+	0.0920562i	\(0.970656\pi\)
\(19\)	−52.8489	−	63.7652i	−0.638125	−	0.769933i
							\(23\)	22.9568	+	8.35559i	0.208123	+	0.0757504i	0.443978	−	0.896038i	\(-0.353567\pi\)
−0.235855	+	0.971788i	\(0.575789\pi\)
\(29\)	24.8172	+	140.745i	0.158912	+	0.901232i	0.955122	+	0.296212i	\(0.0957237\pi\)
							−0.796211	+	0.605020i	\(0.793165\pi\)
\(31\)	−70.6280	−	122.331i	−0.409199	−	0.708753i	0.585601	−	0.810599i	\(-0.300858\pi\)
							−0.994800	+	0.101846i	\(0.967525\pi\)
\(37\)	126.952			0.564076			0.282038	−	0.959403i	\(-0.408990\pi\)
							0.282038	+	0.959403i	\(0.408990\pi\)
\(41\)	65.7505	+	55.1712i	0.250451	+	0.210154i	0.759367	−	0.650663i	\(-0.225509\pi\)
							−0.508915	+	0.860817i	\(0.669953\pi\)
\(43\)	123.302	−	44.8783i	0.437288	−	0.159160i	−0.113989	−	0.993482i	\(-0.536363\pi\)
							0.551277	+	0.834322i	\(0.314141\pi\)
\(47\)	−23.9842	−	136.021i	−0.0744352	−	0.422143i	−0.999140	−	0.0414586i	\(-0.986800\pi\)
							0.924705	−	0.380684i	\(-0.124312\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.4.i.d.55.1	✓	18
19.3	odd	18		2166.4.a.bj.1.8		9
19.9	even	9	inner	114.4.i.d.85.1	yes	18
19.16	even	9		2166.4.a.bm.1.8		9

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.i.d.55.1	✓	18	1.1	even	1	trivial
114.4.i.d.85.1	yes	18	19.9	even	9	inner
2166.4.a.bj.1.8		9	19.3	odd	18
2166.4.a.bm.1.8		9	19.16	even	9