Embedded newform 114.4.i.b.73.2

• Label: 114.4.i.b.73.2
• 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:	\( x^{12} - 252x^{10} + 25236x^{8} - 1271150x^{6} + 34071732x^{4} - 464617557x^{2} + 2560258801 \)
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			73.2
Root			\(-8.59757 + 0.984808i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.73
Dual form			114.4.i.b.25.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-1.53209 - 1.28558i) q^{2} +(2.81908 - 1.02606i) q^{3} +(0.694593 + 3.93923i) q^{4} +(0.810428 - 4.59616i) q^{5} +(-5.63816 - 2.05212i) q^{6} +(15.9494 + 27.6252i) q^{7} +(4.00000 - 6.92820i) q^{8} +(6.89440 - 5.78509i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 9 q^{5} + 12 q^{7} + 48 q^{8}+O(q^{10}) \)

Character values

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

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

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
\(5\)	0.810428	−	4.59616i	0.0724869	−	0.411093i								−0.926875	−	0.375370i	\(-0.877515\pi\)
							0.999362	−	0.0357230i	\(-0.0113734\pi\)
\(7\)	15.9494	+	27.6252i	0.861188	+	1.49162i	0.870783	+	0.491668i	\(0.163613\pi\)
							−0.00959448	+	0.999954i	\(0.503054\pi\)
\(11\)	4.72098	−	8.17698i	0.129403	−	0.224132i	−0.794043	−	0.607862i	\(-0.792027\pi\)
							0.923445	+	0.383730i	\(0.125361\pi\)
\(13\)	24.0448	+	8.75159i	0.512987	+	0.186712i	0.585526	−	0.810654i	\(-0.300888\pi\)
							−0.0725392	+	0.997366i	\(0.523110\pi\)
\(17\)	70.5281	+	59.1801i	1.00621	+	0.844311i	0.987833	−	0.155520i	\(-0.0497053\pi\)
							0.0183781	+	0.999831i	\(0.494150\pi\)
\(19\)	−14.5473	−	81.5315i	−0.175651	−	0.984453i
							\(23\)	−16.1535	−	91.6111i	−0.146445	−	0.830532i	−0.966195	−	0.257811i	\(-0.916999\pi\)
0.819750	−	0.572721i	\(-0.194112\pi\)
\(29\)	223.189	−	187.277i	1.42914	−	1.19919i	0.482922	−	0.875663i	\(-0.339575\pi\)
							0.946218	−	0.323528i	\(-0.104869\pi\)
\(31\)	33.4084	+	57.8651i	0.193559	+	0.335254i	0.946427	−	0.322917i	\(-0.104664\pi\)
							−0.752868	+	0.658171i	\(0.771330\pi\)
\(37\)	−292.195			−1.29829			−0.649143	−	0.760666i	\(-0.724872\pi\)
							−0.649143	+	0.760666i	\(0.724872\pi\)
\(41\)	−217.333	+	79.1027i	−0.827846	+	0.301311i	−0.720975	−	0.692961i	\(-0.756306\pi\)
							−0.106872	+	0.994273i	\(0.534083\pi\)
\(43\)	−88.0985	+	499.631i	−0.312439	+	1.77193i	0.273793	+	0.961789i	\(0.411722\pi\)
							−0.586232	+	0.810143i	\(0.699389\pi\)
\(47\)	−218.175	+	183.070i	−0.677107	+	0.568160i	−0.915159	−	0.403092i	\(-0.867935\pi\)
							0.238052	+	0.971252i	\(0.423491\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.4.i.b.73.2	yes	12
19.5	even	9		2166.4.a.bb.1.5		6
19.6	even	9	inner	114.4.i.b.25.2	✓	12
19.14	odd	18		2166.4.a.bg.1.5		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.i.b.25.2	✓	12	19.6	even	9	inner
114.4.i.b.73.2	yes	12	1.1	even	1	trivial
2166.4.a.bb.1.5		6	19.5	even	9
2166.4.a.bg.1.5		6	19.14	odd	18