Newform orbit 722.2.c.n

• Label: 722.2.c.n
• Level: $722$
• Weight: $2$
• Character orbit: 722.c
• Analytic conductor: $5.765$
• Analytic rank: $0$
• Dimension: $8$
• 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:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	8.0.324000000.2
Defining polynomial:	\( x^{8} + 5x^{6} + 20x^{4} + 25x^{2} + 25 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	\( q + (\beta_{4} + 1) q^{2} + ( - \beta_{6} - \beta_{3}) q^{3} + \beta_{4} q^{4} + ( - \beta_{6} + \beta_{3} - \beta_1) q^{5} + (\beta_{7} - \beta_{6} - \beta_{2}) q^{6} + ( - \beta_{5} + \beta_{2} - 1) q^{7} - q^{8} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_1) q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 2 q^{3} - 4 q^{4} + 2 q^{5} - 2 q^{6} - 4 q^{7} - 8 q^{8} - 4 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 20x^{4} + 25x^{2} + 25 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{6} - 15 ) / 20 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{7} - 35\nu ) / 20 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{6} + 4\nu^{4} + 20\nu^{2} + 5 ) / 20 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{7} + 4\nu^{5} + 20\nu^{3} + 5\nu ) / 20 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{6} + 8\nu^{4} + 20\nu^{2} + 25 ) / 20 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{7} + 6\nu^{5} + 20\nu^{3} + 25\nu ) / 10 \)

Character values

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

\(n\)	\(363\)
\(\chi(n)\)	\(\beta_{4}\)

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

−0.587785	+	1.01807i
0.951057	−	1.64728i
0.587785	−	1.01807i
−0.951057	+	1.64728i
−0.587785	−	1.01807i
0.951057	+	1.64728i
0.587785	+	1.01807i
−0.951057	−	1.64728i

0.500000

−

0.866025i

−1.26007

+

2.18251i

−0.500000

−

0.866025i

1.22982

−

2.13012i

1.26007

+

2.18251i

−2.79360

−1.00000

−1.67557

−

2.90217i

−1.22982

−

2.13012i

429.2

0.500000

−

0.866025i

0.221232

−

0.383185i

−0.500000

−

0.866025i

0.445746

−

0.772054i

−0.221232

−

0.383185i

2.52015

−1.00000

1.40211

+

2.42853i

−0.445746

−

0.772054i

429.3

0.500000

−

0.866025i

0.642040

−

1.11205i

−0.500000

−

0.866025i

−1.84786

+

3.20059i

−0.642040

−

1.11205i

−0.442463

−1.00000

0.675571

+

1.17012i

1.84786

+

3.20059i

429.4

0.500000

−

0.866025i

1.39680

−

2.41933i

−0.500000

−

0.866025i

1.17229

−

2.03046i

−1.39680

−

2.41933i

−1.28408

−1.00000

−2.40211

−

4.16058i

−1.17229

−

2.03046i

653.1

0.500000

+

0.866025i

−1.26007

−

2.18251i

−0.500000

+

0.866025i

1.22982

+

2.13012i

1.26007

−

2.18251i

−2.79360

−1.00000

−1.67557

+

2.90217i

−1.22982

+

2.13012i

653.2

0.500000

+

0.866025i

0.221232

+

0.383185i

−0.500000

+

0.866025i

0.445746

+

0.772054i

−0.221232

+

0.383185i

2.52015

−1.00000

1.40211

−

2.42853i

−0.445746

+

0.772054i

653.3

0.500000

+

0.866025i

0.642040

+

1.11205i

−0.500000

+

0.866025i

−1.84786

−

3.20059i

−0.642040

+

1.11205i

−0.442463

−1.00000

0.675571

−

1.17012i

1.84786

−

3.20059i

653.4

0.500000

+

0.866025i

1.39680

+

2.41933i

−0.500000

+

0.866025i

1.17229

+

2.03046i

−1.39680

+

2.41933i

−1.28408

−1.00000

−2.40211

+

4.16058i

−1.17229

+

2.03046i

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.n		8
19.b	odd	2	1		722.2.c.m		8
19.c	even	3	1		722.2.a.m	✓	4
19.c	even	3	1	inner	722.2.c.n		8
19.d	odd	6	1		722.2.a.n	yes	4
19.d	odd	6	1		722.2.c.m		8
19.e	even	9	6		722.2.e.s		24
19.f	odd	18	6		722.2.e.r		24
57.f	even	6	1		6498.2.a.bx		4
57.h	odd	6	1		6498.2.a.ca		4
76.f	even	6	1		5776.2.a.bt		4
76.g	odd	6	1		5776.2.a.bv		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
722.2.a.m	✓	4	19.c	even	3	1
722.2.a.n	yes	4	19.d	odd	6	1
722.2.c.m		8	19.b	odd	2	1
722.2.c.m		8	19.d	odd	6	1
722.2.c.n		8	1.a	even	1	1	trivial
722.2.c.n		8	19.c	even	3	1	inner
722.2.e.r		24	19.f	odd	18	6
722.2.e.s		24	19.e	even	9	6
5776.2.a.bt		4	76.f	even	6	1
5776.2.a.bv		4	76.g	odd	6	1
6498.2.a.bx		4	57.f	even	6	1
6498.2.a.ca		4	57.h	odd	6	1

Hecke kernels

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

\( T_{3}^{8} - 2T_{3}^{7} + 10T_{3}^{6} - 12T_{3}^{5} + 64T_{3}^{4} - 88T_{3}^{3} + 120T_{3}^{2} - 48T_{3} + 16 \)

\( T_{5}^{8} - 2T_{5}^{7} + 15T_{5}^{6} - 42T_{5}^{5} + 204T_{5}^{4} - 428T_{5}^{3} + 815T_{5}^{2} - 608T_{5} + 361 \)

\( T_{7}^{4} + 2T_{7}^{3} - 6T_{7}^{2} - 12T_{7} - 4 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{2} - T + 1)^{4} \)
$3$	T^8 - 2*T^7 + 10*T^6 - 12*T^5 + 64*T^4 - 88*T^3 + 120*T^2 - 48*T + 16
$5$	T^8 - 2*T^7 + 15*T^6 - 42*T^5 + 204*T^4 - 428*T^3 + 815*T^2 - 608*T + 361
$7$	(T^4 + 2*T^3 - 6*T^2 - 12*T - 4)^2
$11$	(T^4 - 2*T^3 - 26*T^2 + 12*T + 76)^2