Newform orbit 722.2.c.e

• Label: 722.2.c.e
• Level: $722$
• Weight: $2$
• Character orbit: 722.c
• Analytic conductor: $5.765$
• Analytic rank: $1$
• Dimension: $2$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 722 = 2 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	722.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.76519902594\)
Analytic rank:	\(1\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{-3}) \)
Defining polynomial:	\( x^{2} - x + 1 \)
Coefficient ring:	\(\Z[a_1, a_2]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 38)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + ( - \zeta_{6} + 1) q^{2} + (\zeta_{6} - 1) q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} - q^{7} - q^{8} + 2 \zeta_{6} q^{9} +O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{3} - q^{4} + q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/722\mathbb{Z}\right)^\times\).

\(n\)	\(363\)
\(\chi(n)\)	\(-\zeta_{6}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

429.1

0.500000	+	0.866025i
0.500000	−	0.866025i

0.500000

−

0.866025i

−0.500000

+

0.866025i

−0.500000

−

0.866025i

0

0.500000

+

0.866025i

−1.00000

−1.00000

1.00000

+

1.73205i

0

653.1

0.500000

+

0.866025i

−0.500000

−

0.866025i

−0.500000

+

0.866025i

0

0.500000

−

0.866025i

−1.00000

−1.00000

1.00000

−

1.73205i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.c	even	3	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	722.2.c.e		2
19.b	odd	2	1		722.2.c.c		2
19.c	even	3	1		38.2.a.a	✓	1
19.c	even	3	1	inner	722.2.c.e		2
19.d	odd	6	1		722.2.a.e		1
19.d	odd	6	1		722.2.c.c		2
19.e	even	9	6		722.2.e.f		6
19.f	odd	18	6		722.2.e.e		6
57.f	even	6	1		6498.2.a.f		1
57.h	odd	6	1		342.2.a.e		1
76.f	even	6	1		5776.2.a.m		1
76.g	odd	6	1		304.2.a.c		1
95.i	even	6	1		950.2.a.d		1
95.m	odd	12	2		950.2.b.b		2
133.m	odd	6	1		1862.2.a.b		1
152.k	odd	6	1		1216.2.a.m		1
152.p	even	6	1		1216.2.a.e		1
209.h	odd	6	1		4598.2.a.p		1
228.m	even	6	1		2736.2.a.n		1
247.q	even	6	1		6422.2.a.h		1
285.n	odd	6	1		8550.2.a.m		1
380.p	odd	6	1		7600.2.a.n		1

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
38.2.a.a	✓	1	19.c	even	3	1
304.2.a.c		1	76.g	odd	6	1
342.2.a.e		1	57.h	odd	6	1
722.2.a.e		1	19.d	odd	6	1
722.2.c.c		2	19.b	odd	2	1
722.2.c.c		2	19.d	odd	6	1
722.2.c.e		2	1.a	even	1	1	trivial
722.2.c.e		2	19.c	even	3	1	inner
722.2.e.e		6	19.f	odd	18	6
722.2.e.f		6	19.e	even	9	6
950.2.a.d		1	95.i	even	6	1
950.2.b.b		2	95.m	odd	12	2
1216.2.a.e		1	152.p	even	6	1
1216.2.a.m		1	152.k	odd	6	1
1862.2.a.b		1	133.m	odd	6	1
2736.2.a.n		1	228.m	even	6	1
4598.2.a.p		1	209.h	odd	6	1
5776.2.a.m		1	76.f	even	6	1
6422.2.a.h		1	247.q	even	6	1
6498.2.a.f		1	57.f	even	6	1
7600.2.a.n		1	380.p	odd	6	1
8550.2.a.m		1	285.n	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(722, [\chi])\):

\( T_{3}^{2} + T_{3} + 1 \)

\( T_{5} \)

\( T_{7} + 1 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} - T + 1 \)
$3$	\( T^{2} + T + 1 \)
$5$	\( T^{2} \)
$7$	\( (T + 1)^{2} \)
$11$	\( (T + 6)^{2} \)