Embedded newform 14.12.c.a.11.3

• Label: 14.12.c.a.11.3
• Level: $14$
• Weight: $12$
• Character: 14.11
• Analytic conductor: $10.757$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 14 = 2 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.7568045278\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial:	x^8 - x^7 + 101803*x^6 + 6576400*x^5 + 8539617914*x^4 + 333205096780*x^3 + 195840371083183*x^2 - 5894448091672309*x + 3315283474489744081
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{12}\cdot 3\cdot 7^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			11.3
Root			\(-100.397 - 173.893i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.11
Dual form			14.12.c.a.9.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.0000 - 27.7128i) q^{2} +(234.294 - 405.809i) q^{3} +(-512.000 + 886.810i) q^{4} +(-3326.37 - 5761.45i) q^{5} -14994.8 q^{6} +(12255.3 - 42745.0i) q^{7} +32768.0 q^{8} +(-21214.1 - 36743.8i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 128 q^{2} + 266 q^{3} - 4096 q^{4} + 7504 q^{5} - 17024 q^{6} - 42224 q^{7} + 262144 q^{8} - 123520 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.0000	−	27.7128i	−0.353553	−	0.612372i
							\(3\)	234.294	−	405.809i	0.556666	−	0.964174i	−0.441106	−	0.897455i	\(-0.645414\pi\)
0.997772	−	0.0667186i	\(-0.0212530\pi\)
\(5\)	−3326.37	−	5761.45i	−0.476032	−	0.824511i	0.523591	−	0.851970i	\(-0.324592\pi\)
							−0.999623	+	0.0274583i	\(0.991259\pi\)
\(7\)	12255.3	−	42745.0i	0.275603	−	0.961271i
							\(11\)	−60907.1	+	105494.i	−0.114027	+	0.197501i	−0.917390	−	0.397988i	\(-0.869708\pi\)
0.803363	+	0.595489i	\(0.203042\pi\)
\(13\)	−1.81677e6			−1.35710			−0.678550	−	0.734554i	\(-0.737391\pi\)
							−0.678550	+	0.734554i	\(0.737391\pi\)
\(17\)	−2.70113e6	+	4.67849e6i	−0.461398	+	0.799165i	−0.999031	−	0.0440143i	\(-0.985985\pi\)
							0.537633	+	0.843179i	\(0.319319\pi\)
\(19\)	5.10442e6	+	8.84111e6i	0.472935	+	0.819148i	0.999520	−	0.0309748i	\(-0.00986115\pi\)
							−0.526585	+	0.850122i	\(0.676528\pi\)
\(23\)	−1.88132e7	−	3.25854e7i	−0.609481	−	1.05565i	−0.991326	−	0.131425i	\(-0.958045\pi\)
							0.381845	−	0.924226i	\(-0.375289\pi\)
\(29\)	−1.87020e7			−0.169316			−0.0846581	−	0.996410i	\(-0.526980\pi\)
							−0.0846581	+	0.996410i	\(0.526980\pi\)
\(31\)	1.22797e8	−	2.12690e8i	0.770367	−	1.33431i	−0.166995	−	0.985958i	\(-0.553406\pi\)
							0.937362	−	0.348357i	\(-0.113260\pi\)
\(37\)	−3.25911e8	−	5.64495e8i	−0.772663	−	1.33829i	−0.936099	−	0.351737i	\(-0.885591\pi\)
							0.163436	−	0.986554i	\(-0.447742\pi\)
\(41\)	−6.38428e8			−0.860598			−0.430299	−	0.902686i	\(-0.641592\pi\)
							−0.430299	+	0.902686i	\(0.641592\pi\)
\(43\)	8.54732e8			0.886652			0.443326	−	0.896360i	\(-0.353798\pi\)
							0.443326	+	0.896360i	\(0.353798\pi\)
\(47\)	4.23660e8	+	7.33801e8i	0.269451	+	0.466702i	0.968720	−	0.248156i	\(-0.0798245\pi\)
							−0.699269	+	0.714858i	\(0.746491\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.12.c.a.11.3	yes	8
3.2	odd	2		126.12.g.e.109.4		8
4.3	odd	2		112.12.i.a.81.2		8
7.2	even	3	inner	14.12.c.a.9.3	✓	8
7.3	odd	6		98.12.a.l.1.3		4
7.4	even	3		98.12.a.j.1.2		4
7.5	odd	6		98.12.c.l.79.2		8
7.6	odd	2		98.12.c.l.67.2		8
21.2	odd	6		126.12.g.e.37.4		8
28.23	odd	6		112.12.i.a.65.2		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.12.c.a.9.3	✓	8	7.2	even	3	inner
14.12.c.a.11.3	yes	8	1.1	even	1	trivial
98.12.a.j.1.2		4	7.4	even	3
98.12.a.l.1.3		4	7.3	odd	6
98.12.c.l.67.2		8	7.6	odd	2
98.12.c.l.79.2		8	7.5	odd	6
112.12.i.a.65.2		8	28.23	odd	6
112.12.i.a.81.2		8	4.3	odd	2
126.12.g.e.37.4		8	21.2	odd	6
126.12.g.e.109.4		8	3.2	odd	2