Embedded newform 14.16.c.a.9.3

• Label: 14.16.c.a.9.3
• Level: $14$
• Weight: $16$
• Character: 14.9
• Analytic conductor: $19.977$
• Analytic rank: $0$
• Dimension: $10$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 14 = 2 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.9770907140\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Relative dimension:	\(5\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial:	x^10 - 2*x^9 + 11041039*x^8 + 10251836788*x^7 + 100086056086567*x^6 + 67517626050179350*x^5 + 266968630608932668831*x^4 + 130403899669863282233290*x^3 + 531639154391012701500584575*x^2 + 238948900825518555440992768500*x + 120052637231856930389048520650625
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{18}\cdot 3^{3}\cdot 5^{2}\cdot 7^{7} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			9.3
Root			\(-253.251 + 438.643i\) of defining polynomial
Character	\(\chi\)	\(=\)	14.9
Dual form			14.16.c.a.11.3

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-64.0000 + 110.851i) q^{2} +(-309.001 - 535.206i) q^{3} +(-8192.00 - 14189.0i) q^{4} +(8985.53 - 15563.4i) q^{5} +79104.3 q^{6} +(-2.17743e6 + 79880.9i) q^{7} +2.09715e6 q^{8} +(6.98349e6 - 1.20958e7i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 640 q^{2} + 1979 q^{3} - 81920 q^{4} + 300953 q^{5} - 506624 q^{6} + 4054316 q^{7} + 20971520 q^{8} - 17367046 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−64.0000	+	110.851i	−0.353553	+	0.612372i
							\(3\)	−309.001	−	535.206i	−0.0815738	−	0.141290i	0.822352	−	0.568979i	\(-0.192661\pi\)
−0.903926	+	0.427689i	\(0.859328\pi\)
\(5\)	8985.53	−	15563.4i	0.0514362	−	0.0890901i	−0.839161	−	0.543883i	\(-0.816953\pi\)
							0.890597	+	0.454793i	\(0.150287\pi\)
\(7\)	−2.17743e6	+	79880.9i	−0.999328	+	0.0366613i
							\(11\)	1.74393e7	+	3.02058e7i	0.269827	+	0.467353i	0.968817	−	0.247778i	\(-0.0797003\pi\)
−0.698990	+	0.715131i	\(0.746367\pi\)
\(13\)	1.86835e8			0.825817			0.412908	−	0.910773i	\(-0.364513\pi\)
							0.412908	+	0.910773i	\(0.364513\pi\)
\(17\)	−4.88772e6	−	8.46579e6i	−0.00288895	−	0.00500381i	0.864577	−	0.502500i	\(-0.167586\pi\)
							−0.867466	+	0.497496i	\(0.834253\pi\)
\(19\)	−1.53089e9	+	2.65158e9i	−0.392909	+	0.680538i	−0.992832	−	0.119520i	\(-0.961865\pi\)
							0.599923	+	0.800058i	\(0.295198\pi\)
\(23\)	−1.45796e10	+	2.52527e10i	−0.892869	+	1.54649i	−0.0564492	+	0.998405i	\(0.517978\pi\)
							−0.836420	+	0.548089i	\(0.815355\pi\)
\(29\)	5.56460e9			0.0599031			0.0299515	−	0.999551i	\(-0.490465\pi\)
							0.0299515	+	0.999551i	\(0.490465\pi\)
\(31\)	1.31711e11	+	2.28130e11i	0.859824	+	1.48926i	0.872097	+	0.489334i	\(0.162760\pi\)
							−0.0122731	+	0.999925i	\(0.503907\pi\)
\(37\)	9.69900e10	−	1.67992e11i	0.167963	−	0.290921i	−0.769740	−	0.638357i	\(-0.779614\pi\)
							0.937704	+	0.347436i	\(0.112948\pi\)
\(41\)	−2.63383e11			−0.211207			−0.105604	−	0.994408i	\(-0.533678\pi\)
							−0.105604	+	0.994408i	\(0.533678\pi\)
\(43\)	6.20794e11			0.348284			0.174142	−	0.984721i	\(-0.444285\pi\)
							0.174142	+	0.984721i	\(0.444285\pi\)
\(47\)	2.11086e12	−	3.65612e12i	0.607752	−	1.05266i	−0.383858	−	0.923392i	\(-0.625405\pi\)
							0.991610	−	0.129265i	\(-0.0412617\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	14.16.c.a.9.3	✓	10
3.2	odd	2		126.16.g.e.37.3		10
7.2	even	3		98.16.a.j.1.3		5
7.3	odd	6		98.16.c.m.67.3		10
7.4	even	3	inner	14.16.c.a.11.3	yes	10
7.5	odd	6		98.16.a.k.1.3		5
7.6	odd	2		98.16.c.m.79.3		10
21.11	odd	6		126.16.g.e.109.3		10

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.16.c.a.9.3	✓	10	1.1	even	1	trivial
14.16.c.a.11.3	yes	10	7.4	even	3	inner
98.16.a.j.1.3		5	7.2	even	3
98.16.a.k.1.3		5	7.5	odd	6
98.16.c.m.67.3		10	7.3	odd	6
98.16.c.m.79.3		10	7.6	odd	2
126.16.g.e.37.3		10	3.2	odd	2
126.16.g.e.109.3		10	21.11	odd	6