Embedded newform 114.4.i.a.55.2

• Label: 114.4.i.a.55.2
• 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 - 3*x^11 + 96*x^10 - 97*x^9 + 5532*x^8 - 2793*x^7 + 196003*x^6 + 75186*x^5 + 4841028*x^4 + 2448000*x^3 + 74000304*x^2 + 69242688*x + 675584064
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			55.2
Root			\(3.29438 - 5.70603i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.55
Dual form			114.4.i.a.85.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.347296 - 1.96962i) q^{2} +(-2.29813 - 1.92836i) q^{3} +(-3.75877 - 1.36808i) q^{4} +(-3.67013 + 1.33582i) q^{5} +(-4.59627 + 3.85673i) q^{6} +(-5.36914 - 9.29962i) q^{7} +(-4.00000 + 6.92820i) q^{8} +(1.56283 + 8.86327i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 15 q^{5} + 42 q^{7} - 48 q^{8}+O(q^{10}) \)

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
\(5\)	−3.67013	+	1.33582i	−0.328266	+	0.119479i								−0.500896	−	0.865508i	\(-0.666996\pi\)
							0.172629	+	0.984987i	\(0.444774\pi\)
\(7\)	−5.36914	−	9.29962i	−0.289906	−	0.502132i	0.683881	−	0.729594i	\(-0.260291\pi\)
							−0.973787	+	0.227461i	\(0.926957\pi\)
\(11\)	−11.9734	+	20.7385i	−0.328193	+	0.568446i	−0.982153	−	0.188082i	\(-0.939773\pi\)
							0.653961	+	0.756528i	\(0.273106\pi\)
\(13\)	−60.6978	+	50.9315i	−1.29496	+	1.08660i	−0.303973	+	0.952681i	\(0.598313\pi\)
							−0.990992	+	0.133924i	\(0.957242\pi\)
\(17\)	3.65210	−	20.7121i	0.0521039	−	0.295496i	−0.947610	−	0.319431i	\(-0.896508\pi\)
							0.999714	+	0.0239350i	\(0.00761947\pi\)
\(19\)	55.7463	+	61.2483i	0.673109	+	0.739543i
							\(23\)	−61.6364	−	22.4338i	−0.558786	−	0.203382i	0.0471597	−	0.998887i	\(-0.484983\pi\)
−0.605946	+	0.795506i	\(0.707205\pi\)
\(29\)	−41.8453	−	237.316i	−0.267947	−	1.51961i	−0.760509	−	0.649327i	\(-0.775050\pi\)
							0.492562	−	0.870278i	\(-0.336061\pi\)
\(31\)	−9.65890	−	16.7297i	−0.0559610	−	0.0969273i	0.836688	−	0.547680i	\(-0.184489\pi\)
							−0.892649	+	0.450753i	\(0.851156\pi\)
\(37\)	−272.424			−1.21044			−0.605219	−	0.796059i	\(-0.706915\pi\)
							−0.605219	+	0.796059i	\(0.706915\pi\)
\(41\)	−88.0109	−	73.8499i	−0.335244	−	0.281303i	0.459589	−	0.888132i	\(-0.347997\pi\)
							−0.794832	+	0.606829i	\(0.792441\pi\)
\(43\)	−276.797	+	100.746i	−0.981655	+	0.357293i	−0.782483	−	0.622672i	\(-0.786047\pi\)
							−0.199171	+	0.979965i	\(0.563825\pi\)
\(47\)	−61.9435	−	351.299i	−0.192242	−	1.09026i	−0.916291	−	0.400512i	\(-0.868832\pi\)
							0.724049	−	0.689749i	\(-0.242279\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.4.i.a.55.2	✓	12
19.3	odd	18		2166.4.a.bd.1.3		6
19.9	even	9	inner	114.4.i.a.85.2	yes	12
19.16	even	9		2166.4.a.bf.1.3		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.i.a.55.2	✓	12	1.1	even	1	trivial
114.4.i.a.85.2	yes	12	19.9	even	9	inner
2166.4.a.bd.1.3		6	19.3	odd	18
2166.4.a.bf.1.3		6	19.16	even	9