Embedded newform 14.10.c.b.11.2

• Label: 14.10.c.b.11.2
• Level: $14$
• Weight: $10$
• Character: 14.11
• Analytic conductor: $7.211$
• Analytic rank: $0$
• Dimension: $6$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 14 = 2 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.21050170629\)
Analytic rank:	\(0\)
Dimension:	\(6\)
Relative dimension:	\(3\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial:	\( x^{6} - x^{5} + 4038x^{4} - 137923x^{3} + 16368349x^{2} - 286546260x + 5038160400 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{6}\cdot 7 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			11.2
Root			\(26.1296 - 45.2579i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.11
Dual form			14.10.c.b.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.00000 + 13.8564i) q^{2} +(38.4345 - 66.5705i) q^{3} +(-128.000 + 221.703i) q^{4} +(546.130 + 945.925i) q^{5} +1229.90 q^{6} +(6111.84 + 1731.76i) q^{7} -4096.00 q^{8} +(6887.08 + 11928.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 48 q^{2} + 71 q^{3} - 768 q^{4} - 1085 q^{5} + 2272 q^{6} - 6796 q^{7} - 24576 q^{8} - 9106 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	8.00000	+	13.8564i	0.353553	+	0.612372i
							\(3\)	38.4345	−	66.5705i	0.273953	−	0.474500i	−0.695918	−	0.718121i	\(-0.745002\pi\)
0.969870	+	0.243622i	\(0.0783355\pi\)
\(5\)	546.130	+	945.925i	0.390779	+	0.676849i	0.992552	−	0.121818i	\(-0.0388725\pi\)
							−0.601774	+	0.798667i	\(0.705539\pi\)
\(7\)	6111.84	+	1731.76i	0.962124	+	0.272613i
							\(11\)	−27030.4	+	46818.1i	−0.556655	+	0.964154i	0.441118	+	0.897449i	\(0.354582\pi\)
−0.997773	+	0.0667050i	\(0.978751\pi\)
\(13\)	37802.6			0.367093			0.183547	−	0.983011i	\(-0.441242\pi\)
							0.183547	+	0.983011i	\(0.441242\pi\)
\(17\)	176735.	−	306114.i	0.513218	−	0.888920i	−0.486664	−	0.873589i	\(-0.661786\pi\)
							0.999882	−	0.0153311i	\(-0.00488024\pi\)
\(19\)	−25105.3	−	43483.7i	−0.0441952	−	0.0765483i	0.843082	−	0.537786i	\(-0.180739\pi\)
							−0.887277	+	0.461237i	\(0.847406\pi\)
\(23\)	−1.22926e6	−	2.12914e6i	−0.915944	−	1.58646i	−0.805513	−	0.592579i	\(-0.798110\pi\)
							−0.110432	−	0.993884i	\(-0.535223\pi\)
\(29\)	1.40865e6			0.369840			0.184920	−	0.982754i	\(-0.440797\pi\)
							0.184920	+	0.982754i	\(0.440797\pi\)
\(31\)	−2.89162e6	+	5.00843e6i	−0.562358	+	0.974033i	0.434932	+	0.900463i	\(0.356772\pi\)
							−0.997290	+	0.0735696i	\(0.976561\pi\)
\(37\)	−2.72938e6	−	4.72743e6i	−0.239418	−	0.414684i	0.721129	−	0.692800i	\(-0.243623\pi\)
							−0.960547	+	0.278116i	\(0.910290\pi\)
\(41\)	−1.73066e7			−0.956501			−0.478251	−	0.878223i	\(-0.658729\pi\)
							−0.478251	+	0.878223i	\(0.658729\pi\)
\(43\)	1.43644e6			0.0640735			0.0320367	−	0.999487i	\(-0.489801\pi\)
							0.0320367	+	0.999487i	\(0.489801\pi\)
\(47\)	−1.52194e7	−	2.63607e7i	−0.454942	−	0.787982i	0.543743	−	0.839252i	\(-0.317007\pi\)
							−0.998685	+	0.0512695i	\(0.983673\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.10.c.b.11.2	yes	6
3.2	odd	2		126.10.g.e.109.1		6
4.3	odd	2		112.10.i.a.81.2		6
7.2	even	3	inner	14.10.c.b.9.2	✓	6
7.3	odd	6		98.10.a.h.1.2		3
7.4	even	3		98.10.a.g.1.2		3
7.5	odd	6		98.10.c.l.79.2		6
7.6	odd	2		98.10.c.l.67.2		6
21.2	odd	6		126.10.g.e.37.1		6
28.23	odd	6		112.10.i.a.65.2		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.10.c.b.9.2	✓	6	7.2	even	3	inner
14.10.c.b.11.2	yes	6	1.1	even	1	trivial
98.10.a.g.1.2		3	7.4	even	3
98.10.a.h.1.2		3	7.3	odd	6
98.10.c.l.67.2		6	7.6	odd	2
98.10.c.l.79.2		6	7.5	odd	6
112.10.i.a.65.2		6	28.23	odd	6
112.10.i.a.81.2		6	4.3	odd	2
126.10.g.e.37.1		6	21.2	odd	6
126.10.g.e.109.1		6	3.2	odd	2