Embedded newform 14.6.c.a.11.2

• Label: 14.6.c.a.11.2
• Level: $14$
• Weight: $6$
• Character: 14.11
• Analytic conductor: $2.245$
• Analytic rank: $0$
• Dimension: $4$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 14 = 2 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.24537347738\)
Analytic rank:	\(0\)
Dimension:	\(4\)
Relative dimension:	\(2\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\Q(\sqrt{-3}, \sqrt{79})\)
Defining polynomial:	\( x^{4} + 79x^{2} + 6241 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 2^{2} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			11.2
Root			\(-4.44410 + 7.69740i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.11
Dual form			14.6.c.a.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-2.00000 - 3.46410i) q^{2} +(5.38819 - 9.33263i) q^{3} +(-8.00000 + 13.8564i) q^{4} +(-53.0528 - 91.8901i) q^{5} -43.1056 q^{6} +(124.435 + 36.3731i) q^{7} +64.0000 q^{8} +(63.4347 + 109.872i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} - 14 q^{3} - 32 q^{4} - 70 q^{5} + 112 q^{6} + 256 q^{8} - 244 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

\(n\)	\(3\)
\(\chi(n)\)	\(e\left(\frac{2}{3}\right)\)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−2.00000	−	3.46410i	−0.353553	−	0.612372i
							\(3\)	5.38819	−	9.33263i	0.345653	−	0.598688i	−0.639819	−	0.768525i	\(-0.720991\pi\)
0.985472	+	0.169837i	\(0.0543242\pi\)
\(5\)	−53.0528	−	91.8901i	−0.949037	−	1.64378i	−0.747459	−	0.664308i	\(-0.768726\pi\)
							−0.201578	−	0.979473i	\(-0.564607\pi\)
\(7\)	124.435	+	36.3731i	0.959835	+	0.280566i
							\(11\)	46.7174	−	80.9168i	0.116412	−	0.201631i	−0.801931	−	0.597416i	\(-0.796194\pi\)
0.918343	+	0.395785i	\(0.129527\pi\)
\(13\)	661.131			1.08500			0.542499	−	0.840057i	\(-0.317478\pi\)
							0.542499	+	0.840057i	\(0.317478\pi\)
\(17\)	−227.960	+	394.838i	−0.191309	+	0.331357i	−0.945684	−	0.325086i	\(-0.894607\pi\)
							0.754375	+	0.656444i	\(0.227940\pi\)
\(19\)	−553.140	−	958.066i	−0.351521	−	0.608851i	0.634996	−	0.772516i	\(-0.281002\pi\)
							−0.986516	+	0.163664i	\(0.947669\pi\)
\(23\)	−374.195	−	648.125i	−0.147495	−	0.255470i	0.782806	−	0.622266i	\(-0.213788\pi\)
							−0.930301	+	0.366797i	\(0.880455\pi\)
\(29\)	2804.78			0.619304			0.309652	−	0.950850i	\(-0.399787\pi\)
							0.309652	+	0.950850i	\(0.399787\pi\)
\(31\)	179.649	−	311.162i	0.0335754	−	0.0581543i	−0.848749	−	0.528795i	\(-0.822644\pi\)
							0.882325	+	0.470641i	\(0.155977\pi\)
\(37\)	3406.41	+	5900.08i	0.409066	+	0.708522i	0.994785	−	0.101991i	\(-0.0325214\pi\)
							−0.585720	+	0.810514i	\(0.699188\pi\)
\(41\)	2319.39			0.215484			0.107742	−	0.994179i	\(-0.465638\pi\)
							0.107742	+	0.994179i	\(0.465638\pi\)
\(43\)	−19965.7			−1.64670			−0.823349	−	0.567535i	\(-0.807897\pi\)
							−0.823349	+	0.567535i	\(0.807897\pi\)
\(47\)	7104.77	+	12305.8i	0.469143	+	0.812580i	0.999378	−	0.0352710i	\(-0.0112294\pi\)
							−0.530234	+	0.847851i	\(0.677896\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.6.c.a.11.2	yes	4
3.2	odd	2		126.6.g.j.109.2		4
4.3	odd	2		112.6.i.d.81.1		4
7.2	even	3	inner	14.6.c.a.9.2	✓	4
7.3	odd	6		98.6.a.g.1.2		2
7.4	even	3		98.6.a.h.1.1		2
7.5	odd	6		98.6.c.e.79.1		4
7.6	odd	2		98.6.c.e.67.1		4
21.2	odd	6		126.6.g.j.37.2		4
21.11	odd	6		882.6.a.ba.1.1		2
21.17	even	6		882.6.a.bi.1.2		2
28.3	even	6		784.6.a.bb.1.1		2
28.11	odd	6		784.6.a.s.1.2		2
28.23	odd	6		112.6.i.d.65.1		4

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.c.a.9.2	✓	4	7.2	even	3	inner
14.6.c.a.11.2	yes	4	1.1	even	1	trivial
98.6.a.g.1.2		2	7.3	odd	6
98.6.a.h.1.1		2	7.4	even	3
98.6.c.e.67.1		4	7.6	odd	2
98.6.c.e.79.1		4	7.5	odd	6
112.6.i.d.65.1		4	28.23	odd	6
112.6.i.d.81.1		4	4.3	odd	2
126.6.g.j.37.2		4	21.2	odd	6
126.6.g.j.109.2		4	3.2	odd	2
784.6.a.s.1.2		2	28.11	odd	6
784.6.a.bb.1.1		2	28.3	even	6
882.6.a.ba.1.1		2	21.11	odd	6
882.6.a.bi.1.2		2	21.17	even	6