Embedded newform 14.12.c.b.11.2

• Label: 14.12.c.b.11.2
• 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 + 149344*x^6 + 5578711*x^5 + 20557200983*x^4 + 408905884576*x^3 + 267736482219682*x^2 - 4732823471679688*x + 3039499286293024324
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{12}\cdot 3\cdot 7^{3} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			11.2
Root			\(-62.4133 + 108.103i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.11
Dual form			14.12.c.b.9.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(16.0000 + 27.7128i) q^{2} +(-158.327 + 274.230i) q^{3} +(-512.000 + 886.810i) q^{4} +(-6009.47 - 10408.7i) q^{5} -10132.9 q^{6} +(34168.3 - 28457.9i) q^{7} -32768.0 q^{8} +(38438.9 + 66578.1i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 128 q^{2} - 266 q^{3} - 4096 q^{4} + 3808 q^{5} - 17024 q^{6} + 110328 q^{7} - 262144 q^{8} - 503848 q^{9}+O(q^{10}) \)

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
							\(3\)	−158.327	+	274.230i	−0.376172	+	0.651550i	−0.990502	−	0.137500i	\(-0.956093\pi\)
0.614329	+	0.789050i	\(0.289427\pi\)
\(5\)	−6009.47	−	10408.7i	−0.860005	−	1.48957i	−0.871923	−	0.489644i	\(-0.837127\pi\)
							0.0119176	−	0.999929i	\(-0.496206\pi\)
\(7\)	34168.3	−	28457.9i	0.768394	−	0.639977i
							\(11\)	187066.	−	324007.i	0.350215	−	0.606590i	−0.636072	−	0.771630i	\(-0.719442\pi\)
0.986287	+	0.165040i	\(0.0527753\pi\)
\(13\)	−625709.			−0.467395			−0.233697	−	0.972309i	\(-0.575082\pi\)
							−0.233697	+	0.972309i	\(0.575082\pi\)
\(17\)	3.56150e6	−	6.16869e6i	0.608364	−	1.05372i	−0.383146	−	0.923688i	\(-0.625159\pi\)
							0.991510	−	0.130030i	\(-0.0415072\pi\)
\(19\)	−9.63596e6	−	1.66900e7i	−0.892792	−	1.54636i	−0.836514	−	0.547945i	\(-0.815410\pi\)
							−0.0562775	−	0.998415i	\(-0.517923\pi\)
\(23\)	−1.19610e7	−	2.07171e7i	−0.387494	−	0.671160i	0.604617	−	0.796516i	\(-0.293326\pi\)
							−0.992112	+	0.125356i	\(0.959993\pi\)
\(29\)	7.06535e7			0.639654			0.319827	−	0.947476i	\(-0.396375\pi\)
							0.319827	+	0.947476i	\(0.396375\pi\)
\(31\)	−7.36573e7	+	1.27578e8i	−0.462090	+	0.800363i	−0.999065	−	0.0432348i	\(-0.986234\pi\)
							0.536975	+	0.843598i	\(0.319567\pi\)
\(37\)	−2.68355e8	−	4.64805e8i	−0.636210	−	1.10195i	−0.986257	−	0.165216i	\(-0.947168\pi\)
							0.350048	−	0.936732i	\(-0.386165\pi\)
\(41\)	4.82553e8			0.650480			0.325240	−	0.945632i	\(-0.394555\pi\)
							0.325240	+	0.945632i	\(0.394555\pi\)
\(43\)	5.66805e8			0.587973			0.293986	−	0.955810i	\(-0.405018\pi\)
							0.293986	+	0.955810i	\(0.405018\pi\)
\(47\)	5.35695e8	+	9.27851e8i	0.340706	+	0.590119i	0.984564	−	0.175025i	\(-0.0560007\pi\)
							−0.643858	+	0.765145i	\(0.722667\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.12.c.b.11.2	yes	8
3.2	odd	2		126.12.g.c.109.4		8
4.3	odd	2		112.12.i.b.81.3		8
7.2	even	3	inner	14.12.c.b.9.2	✓	8
7.3	odd	6		98.12.a.h.1.2		4
7.4	even	3		98.12.a.i.1.3		4
7.5	odd	6		98.12.c.n.79.3		8
7.6	odd	2		98.12.c.n.67.3		8
21.2	odd	6		126.12.g.c.37.4		8
28.23	odd	6		112.12.i.b.65.3		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.12.c.b.9.2	✓	8	7.2	even	3	inner
14.12.c.b.11.2	yes	8	1.1	even	1	trivial
98.12.a.h.1.2		4	7.3	odd	6
98.12.a.i.1.3		4	7.4	even	3
98.12.c.n.67.3		8	7.6	odd	2
98.12.c.n.79.3		8	7.5	odd	6
112.12.i.b.65.3		8	28.23	odd	6
112.12.i.b.81.3		8	4.3	odd	2
126.12.g.c.37.4		8	21.2	odd	6
126.12.g.c.109.4		8	3.2	odd	2