Newform orbit 735.2.d.c

• Label: 735.2.d.c
• Level: $735$
• Weight: $2$
• Character orbit: 735.d
• Analytic conductor: $5.869$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 735 = 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	735.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.86900454856\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Coefficient field:	8.0.309760000.3
Defining polynomial:	\( x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 2^{4} \)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\)	\(=\)	\( q - \beta_{5} q^{2} - \beta_{4} q^{3} + ( - \beta_{7} + \beta_{6} - 2) q^{4} + \beta_{3} q^{5} + \beta_1 q^{6} + ( - \beta_{7} - \beta_{6} + \beta_{5} + 1) q^{8} - q^{9}+O(q^{10}) \)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} - 8 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1 \) :

\(\beta_{1}\)	\(=\)	\( ( 13\nu^{7} - 47\nu^{6} + 101\nu^{5} - 55\nu^{4} + 143\nu^{3} - 69\nu^{2} - 30\nu + 319 ) / 245 \)
\(\beta_{2}\)	\(=\)	\( ( -10\nu^{7} + 55\nu^{6} - 138\nu^{5} + 129\nu^{4} - 12\nu^{3} - 11\nu^{2} - 188\nu + 75 ) / 49 \)
\(\beta_{3}\)	\(=\)	\( ( -72\nu^{7} + 249\nu^{6} - 386\nu^{5} - 257\nu^{4} + 384\nu^{3} + 9\nu^{2} - 942\nu - 685 ) / 245 \)
\(\beta_{4}\)	\(=\)	\( ( -74\nu^{7} + 309\nu^{6} - 639\nu^{5} + 249\nu^{4} + 19\nu^{3} + 291\nu^{2} - 1401\nu - 229 ) / 245 \)
\(\beta_{5}\)	\(=\)	\( ( 87\nu^{7} - 356\nu^{6} + 740\nu^{5} - 304\nu^{4} + 124\nu^{3} - 360\nu^{2} + 1861\nu + 303 ) / 245 \)
\(\beta_{6}\)	\(=\)	\( ( 34\nu^{7} - 138\nu^{6} + 283\nu^{5} - 76\nu^{4} - 67\nu^{3} + 8\nu^{2} + 649\nu + 186 ) / 49 \)
\(\beta_{7}\)	\(=\)	\( ( 186\nu^{7} - 778\nu^{6} + 1675\nu^{5} - 802\nu^{4} + 37\nu^{3} - 60\nu^{2} + 3193\nu + 369 ) / 245 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/735\mathbb{Z}\right)^\times\).

\(n\)	\(346\)	\(442\)	\(491\)
\(\chi(n)\)	\(1\)	\(-1\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

589.1

1.74861	+	1.74861i
−0.748606	+	0.748606i
1.16407	+	1.16407i
−0.164066	+	0.164066i
−0.164066	−	0.164066i
1.16407	−	1.16407i
−0.748606	−	0.748606i
1.74861	−	1.74861i

−

2.49721i

−

1.00000i

−4.23607

−1.54336

−

1.61803i

−2.49721

0

5.58394i

−1.00000

−4.04057

+

3.85410i

589.2

−

2.49721i

1.00000i

−4.23607

1.54336

+

1.61803i

2.49721

0

5.58394i

−1.00000

4.04057

−

3.85410i

589.3

−

1.32813i

−

1.00000i

0.236068

2.14896

+

0.618034i

−1.32813

0

−

2.96979i

−1.00000

0.820830

−

2.85410i

589.4

−

1.32813i

1.00000i

0.236068

−2.14896

−

0.618034i

1.32813

0

−

2.96979i

−1.00000

−0.820830

+

2.85410i

589.5

1.32813i

−

1.00000i

0.236068

−2.14896

+

0.618034i

1.32813

0

2.96979i

−1.00000

−0.820830

−

2.85410i

589.6

1.32813i

1.00000i

0.236068

2.14896

−

0.618034i

−1.32813

0

2.96979i

−1.00000

0.820830

+

2.85410i

589.7

2.49721i

−

1.00000i

−4.23607

1.54336

−

1.61803i

2.49721

0

−

5.58394i

−1.00000

4.04057

+

3.85410i

589.8

2.49721i

1.00000i

−4.23607

−1.54336

+

1.61803i

−2.49721

0

−

5.58394i

−1.00000

−4.04057

−

3.85410i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
7.b	odd	2	1	inner
35.c	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	735.2.d.c	✓	8
3.b	odd	2	1		2205.2.d.m		8
5.b	even	2	1	inner	735.2.d.c	✓	8
5.c	odd	4	1		3675.2.a.bt		4
5.c	odd	4	1		3675.2.a.bv		4
7.b	odd	2	1	inner	735.2.d.c	✓	8
7.c	even	3	2		735.2.q.h		16
7.d	odd	6	2		735.2.q.h		16
15.d	odd	2	1		2205.2.d.m		8
21.c	even	2	1		2205.2.d.m		8
35.c	odd	2	1	inner	735.2.d.c	✓	8
35.f	even	4	1		3675.2.a.bt		4
35.f	even	4	1		3675.2.a.bv		4
35.i	odd	6	2		735.2.q.h		16
35.j	even	6	2		735.2.q.h		16
105.g	even	2	1		2205.2.d.m		8

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
735.2.d.c	✓	8	1.a	even	1	1	trivial
735.2.d.c	✓	8	5.b	even	2	1	inner
735.2.d.c	✓	8	7.b	odd	2	1	inner
735.2.d.c	✓	8	35.c	odd	2	1	inner
735.2.q.h		16	7.c	even	3	2
735.2.q.h		16	7.d	odd	6	2
735.2.q.h		16	35.i	odd	6	2
735.2.q.h		16	35.j	even	6	2
2205.2.d.m		8	3.b	odd	2	1
2205.2.d.m		8	15.d	odd	2	1
2205.2.d.m		8	21.c	even	2	1
2205.2.d.m		8	105.g	even	2	1
3675.2.a.bt		4	5.c	odd	4	1
3675.2.a.bt		4	35.f	even	4	1
3675.2.a.bv		4	5.c	odd	4	1
3675.2.a.bv		4	35.f	even	4	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(735, [\chi])\):

\( T_{2}^{4} + 8T_{2}^{2} + 11 \)

\( T_{19}^{4} - 68T_{19}^{2} + 176 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( (T^{4} + 8 T^{2} + 11)^{2} \)
$3$	\( (T^{2} + 1)^{4} \)
$5$	T^8 - 8*T^6 + 46*T^4 - 200*T^2 + 625
$7$	\( T^{8} \)
$11$	\( (T^{2} - 20)^{4} \)