Embedded newform 114.4.i.a.73.1

• Label: 114.4.i.a.73.1
• Level: $114$
• Weight: $4$
• Character: 114.73
• 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			73.1
Root			\(-2.28952 + 3.96556i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.73
Dual form			114.4.i.a.25.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.53209 + 1.28558i) q^{2} +(2.81908 - 1.02606i) q^{3} +(0.694593 + 3.93923i) q^{4} +(-1.98159 + 11.2381i) q^{5} +(5.63816 + 2.05212i) q^{6} +(4.45295 + 7.71274i) q^{7} +(-4.00000 + 6.92820i) q^{8} +(6.89440 - 5.78509i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 15 q^{5} + 42 q^{7} - 48 q^{8}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{2}{9}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.53209	+	1.28558i	0.541675	+	0.454519i
							\(3\)	2.81908	−	1.02606i	0.542532	−	0.197465i
\(5\)	−1.98159	+	11.2381i	−0.177238	+	1.00517i								0.758290	+	0.651917i	\(0.226035\pi\)
							−0.935528	+	0.353252i	\(0.885076\pi\)
\(7\)	4.45295	+	7.71274i	0.240437	+	0.416449i	0.960839	−	0.277108i	\(-0.0893760\pi\)
							−0.720402	+	0.693557i	\(0.756043\pi\)
\(11\)	−11.2990	+	19.5705i	−0.309708	+	0.536430i	−0.978298	−	0.207201i	\(-0.933565\pi\)
							0.668590	+	0.743631i	\(0.266898\pi\)
\(13\)	10.8015	+	3.93143i	0.230446	+	0.0838756i	0.454662	−	0.890664i	\(-0.349760\pi\)
							−0.224216	+	0.974539i	\(0.571982\pi\)
\(17\)	24.2279	+	20.3297i	0.345655	+	0.290039i	0.799043	−	0.601274i	\(-0.205340\pi\)
							−0.453387	+	0.891314i	\(0.649785\pi\)
\(19\)	75.9180	−	33.0976i	0.916673	−	0.399638i
							\(23\)	−8.61462	−	48.8560i	−0.0780988	−	0.442920i	−0.998634	−	0.0522596i	\(-0.983358\pi\)
0.920535	−	0.390661i	\(-0.127753\pi\)
\(29\)	−18.2700	+	15.3304i	−0.116988	+	0.0981648i	−0.699405	−	0.714726i	\(-0.746552\pi\)
							0.582417	+	0.812890i	\(0.302107\pi\)
\(31\)	−77.5040	−	134.241i	−0.449037	−	0.777754i	0.549287	−	0.835634i	\(-0.314899\pi\)
							−0.998324	+	0.0578797i	\(0.981566\pi\)
\(37\)	−88.9835			−0.395373			−0.197686	−	0.980265i	\(-0.563343\pi\)
							−0.197686	+	0.980265i	\(0.563343\pi\)
\(41\)	1.65163	−	0.601143i	0.00629124	−	0.00228983i	−0.338873	−	0.940832i	\(-0.610046\pi\)
							0.345164	+	0.938542i	\(0.387823\pi\)
\(43\)	87.2480	−	494.808i	0.309423	−	1.75483i	−0.292493	−	0.956268i	\(-0.594485\pi\)
							0.601916	−	0.798559i	\(-0.294404\pi\)
\(47\)	136.913	−	114.884i	0.424912	−	0.356543i	−0.405116	−	0.914265i	\(-0.632769\pi\)
							0.830028	+	0.557722i	\(0.188324\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.4.i.a.73.1	yes	12
19.5	even	9		2166.4.a.bf.1.1		6
19.6	even	9	inner	114.4.i.a.25.1	✓	12
19.14	odd	18		2166.4.a.bd.1.1		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.i.a.25.1	✓	12	19.6	even	9	inner
114.4.i.a.73.1	yes	12	1.1	even	1	trivial
2166.4.a.bd.1.1		6	19.14	odd	18
2166.4.a.bf.1.1		6	19.5	even	9