Embedded newform 114.4.i.a.73.2

• Label: 114.4.i.a.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 - 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			73.2
Root			\(2.78952 - 4.83159i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.73
Dual form			114.4.i.a.25.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.53209 + 1.28558i) q^{2} +(2.81908 - 1.02606i) q^{3} +(0.694593 + 3.93923i) q^{4} +(3.09745 - 17.5665i) q^{5} +(5.63816 + 2.05212i) q^{6} +(7.98081 + 13.8232i) q^{7} +(-4.00000 + 6.92820i) q^{8} +(6.89440 - 5.78509i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 15 q^{5} + 42 q^{7} - 48 q^{8}+O(q^{10}) \)

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
\(5\)	3.09745	−	17.5665i	0.277045	−	1.57120i								−0.455347	−	0.890314i	\(-0.650485\pi\)
							0.732391	−	0.680884i	\(-0.238404\pi\)
\(7\)	7.98081	+	13.8232i	0.430923	+	0.746381i	0.996953	−	0.0780036i	\(-0.0248546\pi\)
							−0.566030	+	0.824385i	\(0.691521\pi\)
\(11\)	24.7929	−	42.9426i	0.679578	−	1.17706i	−0.295530	−	0.955333i	\(-0.595496\pi\)
							0.975108	−	0.221730i	\(-0.0711703\pi\)
\(13\)	15.0809	+	5.48898i	0.321744	+	0.117105i	0.497843	−	0.867267i	\(-0.334126\pi\)
							−0.176099	+	0.984373i	\(0.556348\pi\)
\(17\)	−28.4259	−	23.8522i	−0.405547	−	0.340294i	0.417086	−	0.908867i	\(-0.363051\pi\)
							−0.822633	+	0.568573i	\(0.807496\pi\)
\(19\)	31.2930	+	76.6795i	0.377848	+	0.925868i
							\(23\)	31.7101	+	179.837i	0.287479	+	1.63037i	0.696294	+	0.717757i	\(0.254831\pi\)
−0.408814	+	0.912618i	\(0.634058\pi\)
\(29\)	−34.8456	+	29.2389i	−0.223126	+	0.187225i	−0.747498	−	0.664265i	\(-0.768745\pi\)
							0.524371	+	0.851490i	\(0.324300\pi\)
\(31\)	−90.7781	−	157.232i	−0.525943	−	0.910960i	−0.999543	−	0.0302199i	\(-0.990379\pi\)
							0.473600	−	0.880740i	\(-0.342954\pi\)
\(37\)	−294.475			−1.30842			−0.654208	−	0.756315i	\(-0.726998\pi\)
							−0.654208	+	0.756315i	\(0.726998\pi\)
\(41\)	−77.9927	+	28.3870i	−0.297083	+	0.108129i	−0.486262	−	0.873813i	\(-0.661640\pi\)
							0.189178	+	0.981943i	\(0.439418\pi\)
\(43\)	−58.7996	+	333.469i	−0.208532	+	1.18264i	0.683253	+	0.730181i	\(0.260564\pi\)
							−0.891785	+	0.452460i	\(0.850547\pi\)
\(47\)	26.0945	−	21.8959i	0.0809846	−	0.0679542i	−0.601397	−	0.798951i	\(-0.705389\pi\)
							0.682381	+	0.730996i	\(0.260944\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.4.i.a.73.2	yes	12
19.5	even	9		2166.4.a.bf.1.6		6
19.6	even	9	inner	114.4.i.a.25.2	✓	12
19.14	odd	18		2166.4.a.bd.1.6		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.i.a.25.2	✓	12	19.6	even	9	inner
114.4.i.a.73.2	yes	12	1.1	even	1	trivial
2166.4.a.bd.1.6		6	19.14	odd	18
2166.4.a.bf.1.6		6	19.5	even	9