Embedded newform 14.10.c.b.11.1

• Label: 14.10.c.b.11.1
• 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			11.1
Root			\(9.63015 - 16.6799i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.11
Dual form			14.10.c.b.9.1

$q$-expansion

\(f(q)\)	\(=\)	\(q+(8.00000 + 13.8564i) q^{2} +(-89.6637 + 155.302i) q^{3} +(-128.000 + 221.703i) q^{4} +(84.1442 + 145.742i) q^{5} -2869.24 q^{6} +(-3162.16 - 5509.48i) q^{7} -4096.00 q^{8} +(-6237.66 - 10803.9i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 48 q^{2} + 71 q^{3} - 768 q^{4} - 1085 q^{5} + 2272 q^{6} - 6796 q^{7} - 24576 q^{8} - 9106 q^{9}+O(q^{10}) \)

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
							\(3\)	−89.6637	+	155.302i	−0.639103	+	1.10696i	0.346527	+	0.938040i	\(0.387361\pi\)
−0.985630	+	0.168919i	\(0.945972\pi\)
\(5\)	84.1442	+	145.742i	0.0602087	+	0.104284i	0.894559	−	0.446951i	\(-0.147490\pi\)
							−0.834350	+	0.551235i	\(0.814157\pi\)
\(7\)	−3162.16	−	5509.48i	−0.497786	−	0.867300i
							\(11\)	−7246.90	+	12552.0i	−0.149240	+	0.258491i	−0.930947	−	0.365155i	\(-0.881016\pi\)
0.781707	+	0.623646i	\(0.214349\pi\)
\(13\)	−109153.			−1.05997			−0.529983	−	0.848008i	\(-0.677802\pi\)
							−0.529983	+	0.848008i	\(0.677802\pi\)
\(17\)	−150208.	+	260168.i	−0.436188	+	0.755499i	−0.997392	−	0.0721783i	\(-0.977005\pi\)
							0.561204	+	0.827677i	\(0.310338\pi\)
\(19\)	245896.	+	425904.i	0.432872	+	0.749757i	0.997119	−	0.0758491i	\(-0.0241667\pi\)
							−0.564247	+	0.825606i	\(0.690833\pi\)
\(23\)	1.13523e6	+	1.96628e6i	0.845880	+	1.46511i	0.884855	+	0.465867i	\(0.154257\pi\)
							−0.0389748	+	0.999240i	\(0.512409\pi\)
\(29\)	−3.80273e6			−0.998399			−0.499199	−	0.866487i	\(-0.666372\pi\)
							−0.499199	+	0.866487i	\(0.666372\pi\)
\(31\)	−3.95368e6	+	6.84798e6i	−0.768908	+	1.33179i	0.169247	+	0.985574i	\(0.445866\pi\)
							−0.938155	+	0.346214i	\(0.887467\pi\)
\(37\)	−7.07756e6	−	1.22587e7i	−0.620835	−	1.07532i	−0.989331	−	0.145688i	\(-0.953461\pi\)
							0.368496	−	0.929629i	\(-0.379873\pi\)
\(41\)	1.11075e7			0.613890			0.306945	−	0.951727i	\(-0.400693\pi\)
							0.306945	+	0.951727i	\(0.400693\pi\)
\(43\)	8.50738e6			0.379479			0.189740	−	0.981834i	\(-0.439236\pi\)
							0.189740	+	0.981834i	\(0.439236\pi\)
\(47\)	2.18728e7	+	3.78848e7i	0.653828	+	1.13246i	0.982186	+	0.187911i	\(0.0601715\pi\)
							−0.328358	+	0.944553i	\(0.606495\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.10.c.b.11.1	yes	6
3.2	odd	2		126.10.g.e.109.2		6
4.3	odd	2		112.10.i.a.81.3		6
7.2	even	3	inner	14.10.c.b.9.1	✓	6
7.3	odd	6		98.10.a.h.1.1		3
7.4	even	3		98.10.a.g.1.3		3
7.5	odd	6		98.10.c.l.79.3		6
7.6	odd	2		98.10.c.l.67.3		6
21.2	odd	6		126.10.g.e.37.2		6
28.23	odd	6		112.10.i.a.65.3		6

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.10.c.b.9.1	✓	6	7.2	even	3	inner
14.10.c.b.11.1	yes	6	1.1	even	1	trivial
98.10.a.g.1.3		3	7.4	even	3
98.10.a.h.1.1		3	7.3	odd	6
98.10.c.l.67.3		6	7.6	odd	2
98.10.c.l.79.3		6	7.5	odd	6
112.10.i.a.65.3		6	28.23	odd	6
112.10.i.a.81.3		6	4.3	odd	2
126.10.g.e.37.2		6	21.2	odd	6
126.10.g.e.109.2		6	3.2	odd	2