Embedded newform 14.14.c.b.9.3

• Label: 14.14.c.b.9.3
• Level: $14$
• Weight: $14$
• Character: 14.9
• 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			9.3
Root			\(169.399 + 293.408i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.9
Dual form			14.14.c.b.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(32.0000 - 55.4256i) q^{2} +(194.343 + 336.611i) q^{3} +(-2048.00 - 3547.24i) q^{4} +(-12951.1 + 22432.0i) q^{5} +24875.9 q^{6} +(286520. - 121636. i) q^{7} -262144. q^{8} +(721623. - 1.24989e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 256 q^{2} + 182 q^{3} - 16384 q^{4} + 1792 q^{5} + 23296 q^{6} - 268352 q^{7} - 2097152 q^{8} + 599840 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	32.0000	−	55.4256i	0.353553	−	0.612372i
							\(3\)	194.343	+	336.611i	0.153915	+	0.266588i	0.932663	−	0.360748i	\(-0.117479\pi\)
−0.778749	+	0.627336i	\(0.784145\pi\)
\(5\)	−12951.1	+	22432.0i	−0.370682	+	0.642040i	−0.989671	−	0.143360i	\(-0.954209\pi\)
							0.618988	+	0.785400i	\(0.287543\pi\)
\(7\)	286520.	−	121636.i	0.920487	−	0.390772i
							\(11\)	112686.	+	195179.i	0.0191787	+	0.0332185i	0.875455	−	0.483299i	\(-0.160562\pi\)
−0.856277	+	0.516517i	\(0.827228\pi\)
\(13\)	2.48426e7			1.42747			0.713733	−	0.700418i	\(-0.247003\pi\)
							0.713733	+	0.700418i	\(0.247003\pi\)
\(17\)	−6.10156e7	−	1.05682e8i	−0.613088	−	1.06190i	−0.990717	−	0.135943i	\(-0.956594\pi\)
							0.377628	−	0.925957i	\(-0.376740\pi\)
\(19\)	1.07218e8	−	1.85707e8i	0.522840	−	0.905586i	−0.476806	−	0.879008i	\(-0.658206\pi\)
							0.999647	−	0.0265777i	\(-0.00846095\pi\)
\(23\)	4.90873e8	−	8.50217e8i	0.691414	−	1.19756i	−0.279960	−	0.960012i	\(-0.590321\pi\)
							0.971375	−	0.237553i	\(-0.0763453\pi\)
\(29\)	−2.47347e9			−0.772183			−0.386092	−	0.922460i	\(-0.626175\pi\)
							−0.386092	+	0.922460i	\(0.626175\pi\)
\(31\)	2.44134e9	+	4.22852e9i	0.494057	+	0.855732i	0.999977	−	0.00684873i	\(-0.00218004\pi\)
							−0.505919	+	0.862581i	\(0.668847\pi\)
\(37\)	−1.35052e10	+	2.33916e10i	−0.865343	+	1.49882i	0.00136270	+	0.999999i	\(0.499566\pi\)
							−0.866706	+	0.498819i	\(0.833767\pi\)
\(41\)	3.27287e10			1.07605			0.538026	−	0.842928i	\(-0.319170\pi\)
							0.538026	+	0.842928i	\(0.319170\pi\)
\(43\)	−5.78165e9			−0.139478			−0.0697392	−	0.997565i	\(-0.522217\pi\)
							−0.0697392	+	0.997565i	\(0.522217\pi\)
\(47\)	−7.58491e9	+	1.31374e10i	−0.102639	+	0.177777i	−0.912771	−	0.408471i	\(-0.866062\pi\)
							0.810132	+	0.586248i	\(0.199395\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.14.c.b.9.3	✓	8
3.2	odd	2		126.14.g.b.37.3		8
7.2	even	3		98.14.a.h.1.2		4
7.3	odd	6		98.14.c.o.67.2		8
7.4	even	3	inner	14.14.c.b.11.3	yes	8
7.5	odd	6		98.14.a.j.1.3		4
7.6	odd	2		98.14.c.o.79.2		8
21.11	odd	6		126.14.g.b.109.3		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.14.c.b.9.3	✓	8	1.1	even	1	trivial
14.14.c.b.11.3	yes	8	7.4	even	3	inner
98.14.a.h.1.2		4	7.2	even	3
98.14.a.j.1.3		4	7.5	odd	6
98.14.c.o.67.2		8	7.3	odd	6
98.14.c.o.79.2		8	7.6	odd	2
126.14.g.b.37.3		8	3.2	odd	2
126.14.g.b.109.3		8	21.11	odd	6