Embedded newform 114.4.e.e.7.2

• Label: 114.4.e.e.7.2
• Level: $114$
• Weight: $4$
• Character: 114.7
• Analytic conductor: $6.726$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	114.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.72621774065\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	6.0.6967728.1
Defining polynomial:	\( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index:	\( 2^{6}\cdot 3^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			7.2
Root			\(-1.13654 + 1.96854i\) of defining polynomial
Character	\(\chi\)	\(=\)	114.7
Dual form			114.4.e.e.49.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(1.00000 + 1.73205i) q^{2} +(1.50000 + 2.59808i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(2.60679 + 4.51510i) q^{5} +(-3.00000 + 5.19615i) q^{6} +31.4905 q^{7} -8.00000 q^{8} +(-4.50000 + 7.79423i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 9 q^{3} - 12 q^{4} - 2 q^{5} - 18 q^{6} - 34 q^{7} - 48 q^{8} - 27 q^{9}+O(q^{10}) \)

Character values

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

\(n\)	\(77\)	\(97\)
\(\chi(n)\)	\(1\)	\(e\left(\frac{1}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	1.00000	+	1.73205i	0.353553	+	0.612372i
							\(3\)	1.50000	+	2.59808i	0.288675	+	0.500000i
\(5\)	2.60679	+	4.51510i	0.233159	+	0.403843i								0.958736	−	0.284298i	\(-0.0917605\pi\)
							−0.725577	+	0.688141i	\(0.758427\pi\)
\(7\)	31.4905			1.70033			0.850163	−	0.526520i	\(-0.176504\pi\)
							0.850163	+	0.526520i	\(0.176504\pi\)
\(11\)	−21.2113			−0.581403			−0.290702	−	0.956814i	\(-0.593889\pi\)
							−0.290702	+	0.956814i	\(0.593889\pi\)
\(13\)	−28.1384	+	48.7372i	−0.600323	+	1.03979i	0.392449	+	0.919774i	\(0.371628\pi\)
							−0.992772	+	0.120016i	\(0.961705\pi\)
\(17\)	−8.63960	−	14.9642i	−0.123259	−	0.213492i	0.797792	−	0.602933i	\(-0.206001\pi\)
							−0.921051	+	0.389441i	\(0.872668\pi\)
\(19\)	−42.3436	+	71.1759i	−0.511279	+	0.859415i
							\(23\)	103.158	−	178.675i	0.935216	−	1.61984i	0.160969	−	0.986960i	\(-0.448538\pi\)
0.774248	−	0.632883i	\(-0.218128\pi\)
\(29\)	−103.257	+	178.846i	−0.661182	+	1.14520i	0.319123	+	0.947713i	\(0.396612\pi\)
							−0.980305	+	0.197488i	\(0.936722\pi\)
\(31\)	129.624			0.751005			0.375503	−	0.926821i	\(-0.377470\pi\)
							0.375503	+	0.926821i	\(0.377470\pi\)
\(37\)	440.212			1.95596			0.977979	−	0.208704i	\(-0.0669245\pi\)
							0.977979	+	0.208704i	\(0.0669245\pi\)
\(41\)	−217.727	−	377.114i	−0.829347	−	1.43647i	−0.898551	−	0.438868i	\(-0.855379\pi\)
							0.0692044	−	0.997602i	\(-0.477954\pi\)
\(43\)	−64.9984	−	112.580i	−0.230515	−	0.399264i	0.727445	−	0.686166i	\(-0.240708\pi\)
							−0.957960	+	0.286902i	\(0.907375\pi\)
\(47\)	−54.2410	+	93.9482i	−0.168338	+	0.291569i	−0.937836	−	0.347080i	\(-0.887173\pi\)
							0.769498	+	0.638649i	\(0.220507\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.4.e.e.7.2	✓	6
3.2	odd	2		342.4.g.g.235.2		6
19.7	even	3		2166.4.a.s.1.2		3
19.11	even	3	inner	114.4.e.e.49.2	yes	6
19.12	odd	6		2166.4.a.w.1.2		3
57.11	odd	6		342.4.g.g.163.2		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.e.e.7.2	✓	6	1.1	even	1	trivial
114.4.e.e.49.2	yes	6	19.11	even	3	inner
342.4.g.g.163.2		6	57.11	odd	6
342.4.g.g.235.2		6	3.2	odd	2
2166.4.a.s.1.2		3	19.7	even	3
2166.4.a.w.1.2		3	19.12	odd	6