Embedded newform 14.12.c.a.9.2

• Label: 14.12.c.a.9.2
• Level: $14$
• Weight: $12$
• Character: 14.9
• 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			9.2
Root			\(64.1153 - 111.051i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.9
Dual form			14.12.c.a.11.2

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-16.0000 + 27.7128i) q^{2} +(-94.7305 - 164.078i) q^{3} +(-512.000 - 886.810i) q^{4} +(-3211.13 + 5561.84i) q^{5} +6062.75 q^{6} +(-9575.58 - 43423.9i) q^{7} +32768.0 q^{8} +(70625.8 - 122327. i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 128 q^{2} + 266 q^{3} - 4096 q^{4} + 7504 q^{5} - 17024 q^{6} - 42224 q^{7} + 262144 q^{8} - 123520 q^{9}+O(q^{10}) \)

Character values

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

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−16.0000	+	27.7128i	−0.353553	+	0.612372i
							\(3\)	−94.7305	−	164.078i	−0.225073	−	0.389838i	0.731268	−	0.682090i	\(-0.238929\pi\)
−0.956341	+	0.292252i	\(0.905595\pi\)
\(5\)	−3211.13	+	5561.84i	−0.459540	+	0.795946i	−0.998937	−	0.0461054i	\(-0.985319\pi\)
							0.539397	+	0.842052i	\(0.318652\pi\)
\(7\)	−9575.58	−	43423.9i	−0.215341	−	0.976539i
							\(11\)	459958.	+	796671.i	0.861110	+	1.49149i	0.870859	+	0.491533i	\(0.163563\pi\)
−0.00974918	+	0.999952i	\(0.503103\pi\)
\(13\)	1.22771e6			0.917081			0.458541	−	0.888673i	\(-0.348372\pi\)
							0.458541	+	0.888673i	\(0.348372\pi\)
\(17\)	3.31206e6	+	5.73665e6i	0.565755	+	0.979917i	0.996979	+	0.0776717i	\(0.0247486\pi\)
							−0.431224	+	0.902245i	\(0.641918\pi\)
\(19\)	9.33141e6	−	1.61625e7i	0.864574	−	1.49749i	−0.00289490	−	0.999996i	\(-0.500921\pi\)
							0.867469	−	0.497491i	\(-0.165745\pi\)
\(23\)	1.43390e7	−	2.48358e7i	0.464531	−	0.804592i	−0.534649	−	0.845074i	\(-0.679556\pi\)
							0.999180	+	0.0404825i	\(0.0128895\pi\)
\(29\)	−7.73286e6			−0.0700086			−0.0350043	−	0.999387i	\(-0.511144\pi\)
							−0.0350043	+	0.999387i	\(0.511144\pi\)
\(31\)	−5.68702e7	−	9.85021e7i	−0.356776	−	0.617954i	0.630644	−	0.776072i	\(-0.282791\pi\)
							−0.987420	+	0.158118i	\(0.949457\pi\)
\(37\)	9.50304e7	−	1.64597e8i	0.225296	−	0.390224i	−0.731112	−	0.682257i	\(-0.760999\pi\)
							0.956408	+	0.292033i	\(0.0943318\pi\)
\(41\)	−1.74512e8			−0.235241			−0.117621	−	0.993059i	\(-0.537527\pi\)
							−0.117621	+	0.993059i	\(0.537527\pi\)
\(43\)	1.83161e9			1.90001			0.950005	−	0.312235i	\(-0.101078\pi\)
							0.950005	+	0.312235i	\(0.101078\pi\)
\(47\)	−5.75867e8	+	9.97431e8i	−0.366255	+	0.634373i	−0.988977	−	0.148071i	\(-0.952694\pi\)
							0.622721	+	0.782444i	\(0.286027\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.12.c.a.9.2	✓	8
3.2	odd	2		126.12.g.e.37.3		8
4.3	odd	2		112.12.i.a.65.3		8
7.2	even	3		98.12.a.j.1.3		4
7.3	odd	6		98.12.c.l.67.3		8
7.4	even	3	inner	14.12.c.a.11.2	yes	8
7.5	odd	6		98.12.a.l.1.2		4
7.6	odd	2		98.12.c.l.79.3		8
21.11	odd	6		126.12.g.e.109.3		8
28.11	odd	6		112.12.i.a.81.3		8

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.12.c.a.9.2	✓	8	1.1	even	1	trivial
14.12.c.a.11.2	yes	8	7.4	even	3	inner
98.12.a.j.1.3		4	7.2	even	3
98.12.a.l.1.2		4	7.5	odd	6
98.12.c.l.67.3		8	7.3	odd	6
98.12.c.l.79.3		8	7.6	odd	2
112.12.i.a.65.3		8	4.3	odd	2
112.12.i.a.81.3		8	28.11	odd	6
126.12.g.e.37.3		8	3.2	odd	2
126.12.g.e.109.3		8	21.11	odd	6