Embedded newform 7.14.c.a.2.4

• Label: 7.14.c.a.2.4
• 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			2.4
Root			\(12.1857 - 21.1063i\) of defining polynomial
Character	\(\chi\)	\(=\)	7.2
Dual form			7.14.c.a.4.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(-24.3715 + 42.2127i) q^{2} +(1099.13 + 1903.74i) q^{3} +(2908.06 + 5036.91i) q^{4} +(12992.6 - 22503.8i) q^{5} -107149. q^{6} +(282613. + 130456. i) q^{7} -682798. q^{8} +(-1.61900e6 + 2.80418e6i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} + 728 q^{3} - 32588 q^{4} + 46928 q^{5} - 30212 q^{6} + 257992 q^{7} - 1093248 q^{8} - 2910808 q^{9}+O(q^{10}) \)

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)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	−24.3715	+	42.2127i	−0.269269	+	0.466388i	−0.968673	−	0.248338i	\(-0.920116\pi\)
							0.699404	+	0.714727i	\(0.253449\pi\)
\(3\)	1099.13	+	1903.74i	0.870481	+	1.50772i	0.861500	+	0.507758i	\(0.169526\pi\)
							0.00898176	+	0.999960i	\(0.497141\pi\)
\(5\)	12992.6	−	22503.8i	0.371869	−	0.644096i	−0.617984	−	0.786191i	\(-0.712050\pi\)
							0.989853	+	0.142094i	\(0.0453837\pi\)
\(7\)	282613.	+	130456.i	0.907935	+	0.419110i
							\(11\)	−3.13517e6	−	5.43027e6i	−0.533591	−	0.924207i	−0.999230	−	0.0392324i	\(-0.987509\pi\)
0.465639	−	0.884975i	\(-0.345825\pi\)
\(13\)	6.36730e6			0.365867			0.182934	−	0.983125i	\(-0.441441\pi\)
							0.182934	+	0.983125i	\(0.441441\pi\)
\(17\)	−3.56731e7	−	6.17876e7i	−0.358445	−	0.620845i	0.629256	−	0.777198i	\(-0.283360\pi\)
							−0.987701	+	0.156353i	\(0.950026\pi\)
\(19\)	1.61199e8	−	2.79205e8i	0.786076	−	1.36152i	−0.142279	−	0.989827i	\(-0.545443\pi\)
							0.928354	−	0.371696i	\(-0.121224\pi\)
\(23\)	−3.37943e8	+	5.85335e8i	−0.476006	+	0.824467i	−0.999622	−	0.0274872i	\(-0.991249\pi\)
							0.523616	+	0.851955i	\(0.324583\pi\)
\(29\)	4.62084e9			1.44256			0.721280	−	0.692644i	\(-0.243554\pi\)
							0.721280	+	0.692644i	\(0.243554\pi\)
\(31\)	1.77830e9	+	3.08010e9i	0.359877	+	0.623324i	0.987940	−	0.154838i	\(-0.0494855\pi\)
							−0.628063	+	0.778162i	\(0.716152\pi\)
\(37\)	3.51439e9	−	6.08710e9i	0.225184	−	0.390031i	−0.731190	−	0.682174i	\(-0.761035\pi\)
							0.956375	+	0.292143i	\(0.0943682\pi\)
\(41\)	9.68335e9			0.318369			0.159184	−	0.987249i	\(-0.449114\pi\)
							0.159184	+	0.987249i	\(0.449114\pi\)
\(43\)	−4.40234e10			−1.06203			−0.531017	−	0.847361i	\(-0.678190\pi\)
							−0.531017	+	0.847361i	\(0.678190\pi\)
\(47\)	−1.76007e10	+	3.04854e10i	−0.238174	+	0.412530i	−0.960190	−	0.279346i	\(-0.909882\pi\)
							0.722016	+	0.691876i	\(0.243216\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7.14.c.a.2.4	✓	16
3.2	odd	2		63.14.e.c.37.5		16
7.2	even	3		49.14.a.e.1.5		8
7.3	odd	6		49.14.c.g.18.4		16
7.4	even	3	inner	7.14.c.a.4.4	yes	16
7.5	odd	6		49.14.a.f.1.5		8
7.6	odd	2		49.14.c.g.30.4		16
21.11	odd	6		63.14.e.c.46.5		16

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.14.c.a.2.4	✓	16	1.1	even	1	trivial
7.14.c.a.4.4	yes	16	7.4	even	3	inner
49.14.a.e.1.5		8	7.2	even	3
49.14.a.f.1.5		8	7.5	odd	6
49.14.c.g.18.4		16	7.3	odd	6
49.14.c.g.30.4		16	7.6	odd	2
63.14.e.c.37.5		16	3.2	odd	2
63.14.e.c.46.5		16	21.11	odd	6