Embedded newform 7.14.c.a.2.5

• Label: 7.14.c.a.2.5
• Level: $7$
• Weight: $14$
• Character: 7.2
• Analytic conductor: $7.506$
• Analytic rank: $0$
• Dimension: $16$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(7.50616502663\)
Analytic rank:	\(0\)
Dimension:	\(16\)
Relative dimension:	\(8\) over \(\Q(\zeta_{3})\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial:	x^16 - x^15 + 12266*x^14 - 55087*x^13 + 107030285*x^12 - 640627660*x^11 + 438282746176*x^10 - 5859940491984*x^9 + 1300952456341392*x^8 - 14674000482091008*x^7 + 1878469861868220672*x^6 - 14476511842718976000*x^5 + 1885010114115081494784*x^4 - 9244597941717380241408*x^3 + 656480392494410296000512*x^2 + 3883492623114966780739584*x + 127864623964939499445682176
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 2^{34}\cdot 3^{7}\cdot 7^{11} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2.5
Root			\(-6.80614 + 11.7886i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.2
Dual form			7.14.c.a.4.5

$q$-expansion

\(f(q)\)	\(=\)	\(q+(13.6123 - 23.5772i) q^{2} +(-273.794 - 474.225i) q^{3} +(3725.41 + 6452.60i) q^{4} +(16862.5 - 29206.7i) q^{5} -14907.8 q^{6} +(-174823. - 257538. i) q^{7} +425869. q^{8} +(647235. - 1.12104e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} + 728 q^{3} - 32588 q^{4} + 46928 q^{5} - 30212 q^{6} + 257992 q^{7} - 1093248 q^{8} - 2910808 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

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

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	13.6123	−	23.5772i	0.150396	−	0.260493i	−0.780977	−	0.624560i	\(-0.785278\pi\)
							0.931373	+	0.364066i	\(0.118612\pi\)
\(3\)	−273.794	−	474.225i	−0.216838	−	0.375574i	0.737002	−	0.675891i	\(-0.236241\pi\)
							−0.953840	+	0.300317i	\(0.902908\pi\)
\(5\)	16862.5	−	29206.7i	0.482633	−	0.835945i	−0.517168	−	0.855884i	\(-0.673014\pi\)
							0.999801	+	0.0199389i	\(0.00634717\pi\)
\(7\)	−174823.	−	257538.i	−0.561645	−	0.827378i
							\(11\)	−4.55425e6	−	7.88819e6i	−0.775111	−	1.34253i	−0.934732	−	0.355354i	\(-0.884360\pi\)
0.159620	−	0.987178i	\(-0.448973\pi\)
\(13\)	2.25857e7			1.29778			0.648891	−	0.760881i	\(-0.275233\pi\)
							0.648891	+	0.760881i	\(0.275233\pi\)
\(17\)	2.90226e7	+	5.02687e7i	0.291621	+	0.505102i	0.974193	−	0.225716i	\(-0.0724721\pi\)
							−0.682572	+	0.730818i	\(0.739139\pi\)
\(19\)	−1.30020e8	+	2.25200e8i	−0.634031	+	1.09817i	0.352689	+	0.935741i	\(0.385267\pi\)
							−0.986720	+	0.162433i	\(0.948066\pi\)
\(23\)	4.93676e8	−	8.55072e8i	0.695362	−	1.20440i	−0.274696	−	0.961531i	\(-0.588577\pi\)
							0.970058	−	0.242872i	\(-0.0780895\pi\)
\(29\)	3.04325e8			0.0950060			0.0475030	−	0.998871i	\(-0.484874\pi\)
							0.0475030	+	0.998871i	\(0.484874\pi\)
\(31\)	5.06534e8	+	8.77343e8i	0.102508	+	0.177549i	0.912717	−	0.408592i	\(-0.133980\pi\)
							−0.810209	+	0.586141i	\(0.800647\pi\)
\(37\)	−5.24058e9	+	9.07694e9i	−0.335790	+	0.581605i	−0.983636	−	0.180166i	\(-0.942337\pi\)
							0.647846	+	0.761771i	\(0.275670\pi\)
\(41\)	−8.80757e9			−0.289575			−0.144788	−	0.989463i	\(-0.546250\pi\)
							−0.144788	+	0.989463i	\(0.546250\pi\)
\(43\)	1.24470e10			0.300274			0.150137	−	0.988665i	\(-0.452028\pi\)
							0.150137	+	0.988665i	\(0.452028\pi\)
\(47\)	−1.94424e10	+	3.36753e10i	−0.263096	+	0.455696i	−0.967063	−	0.254537i	\(-0.918077\pi\)
							0.703967	+	0.710233i	\(0.251410\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.14.c.a.2.5	✓	16
3.2	odd	2		63.14.e.c.37.4		16
7.2	even	3		49.14.a.e.1.4		8
7.3	odd	6		49.14.c.g.18.5		16
7.4	even	3	inner	7.14.c.a.4.5	yes	16
7.5	odd	6		49.14.a.f.1.4		8
7.6	odd	2		49.14.c.g.30.5		16
21.11	odd	6		63.14.e.c.46.4		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.14.c.a.2.5	✓	16	1.1	even	1	trivial
7.14.c.a.4.5	yes	16	7.4	even	3	inner
49.14.a.e.1.4		8	7.2	even	3
49.14.a.f.1.4		8	7.5	odd	6
49.14.c.g.18.5		16	7.3	odd	6
49.14.c.g.30.5		16	7.6	odd	2
63.14.e.c.37.4		16	3.2	odd	2
63.14.e.c.46.4		16	21.11	odd	6