Embedded newform 14.14.c.b.11.2

• Label: 14.14.c.b.11.2
• Level: $14$
• Weight: $14$
• Character: 14.11
• Analytic conductor: $15.012$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(15.0123300533\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial:	x^8 + 209077*x^6 - 47718852*x^5 + 40973427094*x^4 - 4988457209802*x^3 + 1142094021456771*x^2 + 65369216338084710*x + 7506311351102577225
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{14}\cdot 3\cdot 7^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			11.2
Root			\(-248.013 + 429.572i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.11
Dual form			14.14.c.b.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(32.0000 + 55.4256i) q^{2} +(-244.702 + 423.837i) q^{3} +(-2048.00 + 3547.24i) q^{4} +(34096.3 + 59056.5i) q^{5} -31321.9 q^{6} +(100530. - 294589. i) q^{7} -262144. q^{8} +(677403. + 1.17330e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 256 q^{2} + 182 q^{3} - 16384 q^{4} + 1792 q^{5} + 23296 q^{6} - 268352 q^{7} - 2097152 q^{8} + 599840 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	32.0000	+	55.4256i	0.353553	+	0.612372i
							\(3\)	−244.702	+	423.837i	−0.193798	+	0.335669i	−0.946506	−	0.322686i	\(-0.895414\pi\)
0.752708	+	0.658355i	\(0.228747\pi\)
\(5\)	34096.3	+	59056.5i	0.975892	+	1.69030i	0.676958	+	0.736022i	\(0.263298\pi\)
							0.298934	+	0.954274i	\(0.403369\pi\)
\(7\)	100530.	−	294589.i	0.322967	−	0.946410i
							\(11\)	−1.01364e6	+	1.75568e6i	−0.172517	+	0.298809i	−0.939299	−	0.343099i	\(-0.888523\pi\)
0.766782	+	0.641908i	\(0.221857\pi\)
\(13\)	−1.22100e7			−0.701591			−0.350796	−	0.936452i	\(-0.614089\pi\)
							−0.350796	+	0.936452i	\(0.614089\pi\)
\(17\)	−4.76426e7	+	8.25194e7i	−0.478715	+	0.829159i	−0.999702	−	0.0244054i	\(-0.992231\pi\)
							0.520987	+	0.853565i	\(0.325564\pi\)
\(19\)	−1.03590e8	−	1.79423e8i	−0.505148	−	0.874942i	−0.999982	−	0.00595495i	\(-0.998104\pi\)
							0.494834	−	0.868988i	\(-0.335229\pi\)
\(23\)	−1.66859e8	−	2.89008e8i	−0.235027	−	0.407079i	0.724253	−	0.689534i	\(-0.242185\pi\)
							−0.959281	+	0.282455i	\(0.908851\pi\)
\(29\)	4.91794e9			1.53531			0.767655	−	0.640864i	\(-0.221424\pi\)
							0.767655	+	0.640864i	\(0.221424\pi\)
\(31\)	2.19527e9	−	3.80232e9i	0.444260	−	0.769481i	−0.553740	−	0.832690i	\(-0.686800\pi\)
							0.998000	+	0.0632083i	\(0.0201332\pi\)
\(37\)	4.86831e9	+	8.43216e9i	0.311937	+	0.540291i	0.978782	−	0.204906i	\(-0.0656889\pi\)
							−0.666845	+	0.745197i	\(0.732356\pi\)
\(41\)	2.57081e10			0.845230			0.422615	−	0.906309i	\(-0.361112\pi\)
							0.422615	+	0.906309i	\(0.361112\pi\)
\(43\)	2.25490e9			0.0543980			0.0271990	−	0.999630i	\(-0.491341\pi\)
							0.0271990	+	0.999630i	\(0.491341\pi\)
\(47\)	6.30189e10	+	1.09152e11i	0.852775	+	1.47705i	0.878694	+	0.477385i	\(0.158415\pi\)
							−0.0259191	+	0.999664i	\(0.508251\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.14.c.b.11.2	yes	8
3.2	odd	2		126.14.g.b.109.1		8
7.2	even	3	inner	14.14.c.b.9.2	✓	8
7.3	odd	6		98.14.a.j.1.2		4
7.4	even	3		98.14.a.h.1.3		4
7.5	odd	6		98.14.c.o.79.3		8
7.6	odd	2		98.14.c.o.67.3		8
21.2	odd	6		126.14.g.b.37.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.14.c.b.9.2	✓	8	7.2	even	3	inner
14.14.c.b.11.2	yes	8	1.1	even	1	trivial
98.14.a.h.1.3		4	7.4	even	3
98.14.a.j.1.2		4	7.3	odd	6
98.14.c.o.67.3		8	7.6	odd	2
98.14.c.o.79.3		8	7.5	odd	6
126.14.g.b.37.1		8	21.2	odd	6
126.14.g.b.109.1		8	3.2	odd	2