Embedded newform 14.12.c.b.11.4

• Label: 14.12.c.b.11.4
• 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			11.4
Root			\(190.275 - 329.566i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.11
Dual form			14.12.c.b.9.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(16.0000 + 27.7128i) q^{2} +(347.050 - 601.109i) q^{3} +(-512.000 + 886.810i) q^{4} +(2646.67 + 4584.17i) q^{5} +22211.2 q^{6} +(22119.1 - 38575.5i) q^{7} -32768.0 q^{8} +(-152314. - 263816. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 128 q^{2} - 266 q^{3} - 4096 q^{4} + 3808 q^{5} - 17024 q^{6} + 110328 q^{7} - 262144 q^{8} - 503848 q^{9}+O(q^{10}) \)

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
							\(3\)	347.050	−	601.109i	0.824566	−	1.42819i	−0.0776844	−	0.996978i	\(-0.524753\pi\)
0.902250	−	0.431212i	\(-0.141914\pi\)
\(5\)	2646.67	+	4584.17i	0.378761	+	0.656033i	0.990882	−	0.134731i	\(-0.0430170\pi\)
							−0.612121	+	0.790764i	\(0.709684\pi\)
\(7\)	22119.1	−	38575.5i	0.497426	−	0.867507i
							\(11\)	317765.	−	550386.i	0.594904	−	1.03040i	−0.398657	−	0.917100i	\(-0.630523\pi\)
0.993560	−	0.113303i	\(-0.0361432\pi\)
\(13\)	1.16481e6			0.870096			0.435048	−	0.900407i	\(-0.356731\pi\)
							0.435048	+	0.900407i	\(0.356731\pi\)
\(17\)	−1.20749e6	+	2.09143e6i	−0.206259	+	0.357252i	−0.950533	−	0.310623i	\(-0.899462\pi\)
							0.744274	+	0.667875i	\(0.232796\pi\)
\(19\)	−2.35558e6	−	4.07999e6i	−0.218250	−	0.378019i	0.736023	−	0.676956i	\(-0.236701\pi\)
							−0.954273	+	0.298937i	\(0.903368\pi\)
\(23\)	2.27961e7	+	3.94840e7i	0.738512	+	1.27914i	0.953165	+	0.302450i	\(0.0978047\pi\)
							−0.214653	+	0.976690i	\(0.568862\pi\)
\(29\)	−2.07428e8			−1.87792			−0.938961	−	0.344024i	\(-0.888210\pi\)
							−0.938961	+	0.344024i	\(0.888210\pi\)
\(31\)	−6.83938e7	+	1.18461e8i	−0.429069	+	0.743169i	−0.996791	−	0.0800511i	\(-0.974492\pi\)
							0.567722	+	0.823221i	\(0.307825\pi\)
\(37\)	4.05104e8	+	7.01661e8i	0.960411	+	1.66348i	0.721470	+	0.692446i	\(0.243467\pi\)
							0.238941	+	0.971034i	\(0.423200\pi\)
\(41\)	−1.17761e9			−1.58742			−0.793709	−	0.608298i	\(-0.791853\pi\)
							−0.793709	+	0.608298i	\(0.791853\pi\)
\(43\)	3.25120e8			0.337262			0.168631	−	0.985679i	\(-0.446065\pi\)
							0.168631	+	0.985679i	\(0.446065\pi\)
\(47\)	9.39647e8	+	1.62752e9i	0.597622	+	1.03511i	0.993171	+	0.116667i	\(0.0372209\pi\)
							−0.395549	+	0.918445i	\(0.629446\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.12.c.b.11.4	yes	8
3.2	odd	2		126.12.g.c.109.2		8
4.3	odd	2		112.12.i.b.81.1		8
7.2	even	3	inner	14.12.c.b.9.4	✓	8
7.3	odd	6		98.12.a.h.1.4		4
7.4	even	3		98.12.a.i.1.1		4
7.5	odd	6		98.12.c.n.79.1		8
7.6	odd	2		98.12.c.n.67.1		8
21.2	odd	6		126.12.g.c.37.2		8
28.23	odd	6		112.12.i.b.65.1		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.12.c.b.9.4	✓	8	7.2	even	3	inner
14.12.c.b.11.4	yes	8	1.1	even	1	trivial
98.12.a.h.1.4		4	7.3	odd	6
98.12.a.i.1.1		4	7.4	even	3
98.12.c.n.67.1		8	7.6	odd	2
98.12.c.n.79.1		8	7.5	odd	6
112.12.i.b.65.1		8	28.23	odd	6
112.12.i.b.81.1		8	4.3	odd	2
126.12.g.c.37.2		8	21.2	odd	6
126.12.g.c.109.2		8	3.2	odd	2