Newform orbit 896.2.i.h

• Label: 896.2.i.h
• Level: $896$
• Weight: $2$
• Character orbit: 896.i
• Analytic conductor: $7.155$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 896 = 2^{7} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	896.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.15459602111\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(4\) over \(\Q(\zeta_{3})\)
Coefficient field:	8.0.3010058496.1
Defining polynomial:	\( x^{8} - 4x^{7} + 6x^{6} + 2x^{5} - 17x^{4} + 6x^{3} + 54x^{2} - 108x + 81 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 3 \)
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 + b4 * q^3 + (-b7 - b6 + b5) * q^5 + (-b7 - b4 + b3 + b2 + b1) * q^7 + (-b7 + b5 + 2*b3 + b2 + b1 - 1) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{3} + 2 q^{5} + 2 q^{7} - 4 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 6x^{6} + 2x^{5} - 17x^{4} + 6x^{3} + 54x^{2} - 108x + 81 \) :

\(\beta_{1}\)	\(=\)	\( ( 2\nu^{7} + 10\nu^{6} - 15\nu^{5} - 14\nu^{4} + 191\nu^{3} + 93\nu^{2} - 225\nu + 567 ) / 351 \)
\(\beta_{2}\)	\(=\)	\( ( 8\nu^{7} - 38\nu^{6} + 18\nu^{5} + 61\nu^{4} - 94\nu^{3} - 174\nu^{2} + 504\nu - 189 ) / 351 \)
\(\beta_{3}\)	\(=\)	\( ( 7\nu^{7} - 82\nu^{6} + 123\nu^{5} + 68\nu^{4} - 443\nu^{3} + 150\nu^{2} + 1377\nu - 2052 ) / 351 \)
\(\beta_{4}\)	\(=\)	\( ( 20\nu^{7} - 56\nu^{6} + 6\nu^{5} + 94\nu^{4} - 157\nu^{3} - 162\nu^{2} + 909\nu - 648 ) / 351 \)
\(\beta_{5}\)	\(=\)	\( ( -25\nu^{7} + 70\nu^{6} - 66\nu^{5} - 59\nu^{4} + 284\nu^{3} + 261\nu^{2} - 1107\nu + 1512 ) / 351 \)
\(\beta_{6}\)	\(=\)	\( ( 29\nu^{7} - 89\nu^{6} + 75\nu^{5} + 265\nu^{4} - 331\nu^{3} - 348\nu^{2} + 1710\nu - 1431 ) / 351 \)
\(\beta_{7}\)	\(=\)	\( ( -53\nu^{7} + 125\nu^{6} - 51\nu^{5} - 331\nu^{4} + 457\nu^{3} + 675\nu^{2} - 1818\nu + 1647 ) / 351 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/896\mathbb{Z}\right)^\times\).

\(n\)	\(127\)	\(129\)	\(645\)
\(\chi(n)\)	\(1\)	\(-1 + \beta_{2}\)	\(1\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

513.1

0.745265	−	1.56352i
−1.54667	+	0.779618i
1.65412	−	0.513691i
1.14729	+	1.29759i
0.745265	+	1.56352i
−1.54667	−	0.779618i
1.65412	+	0.513691i
1.14729	−	1.29759i

0

−0.981412

−

1.69985i

0

−1.22668

+

2.12467i

0

1.23615

+

2.33922i

0

−0.426337

+

0.738438i

0

513.2

0

−0.0981673

−

0.170031i

0

1.94851

−

3.37491i

0

2.64484

−

0.0694427i

0

1.48073

−

2.56469i

0

513.3

0

0.382191

+

0.661975i

0

−0.771931

+

1.33702i

0

−1.03631

−

2.43435i

0

1.20786

−

2.09207i

0

513.4

0

1.69739

+

2.93996i

0

1.05010

−

1.81883i

0

−1.84467

+

1.89662i

0

−4.26225

+

7.38243i

0

641.1

0

−0.981412

+

1.69985i

0

−1.22668

−

2.12467i

0

1.23615

−

2.33922i

0

−0.426337

−

0.738438i

0

641.2

0

−0.0981673

+

0.170031i

0

1.94851

+

3.37491i

0

2.64484

+

0.0694427i

0

1.48073

+

2.56469i

0

641.3

0

0.382191

−

0.661975i

0

−0.771931

−

1.33702i

0

−1.03631

+

2.43435i

0

1.20786

+

2.09207i

0

641.4

0

1.69739

−

2.93996i

0

1.05010

+

1.81883i

0

−1.84467

−

1.89662i

0

−4.26225

−

7.38243i

0

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	896.2.i.h	yes	8
4.b	odd	2	1		896.2.i.c	yes	8
7.c	even	3	1	inner	896.2.i.h	yes	8
7.c	even	3	1		6272.2.a.y		4
7.d	odd	6	1		6272.2.a.bs		4
8.b	even	2	1		896.2.i.a	✓	8
8.d	odd	2	1		896.2.i.f	yes	8
28.f	even	6	1		6272.2.a.bf		4
28.g	odd	6	1		896.2.i.c	yes	8
28.g	odd	6	1		6272.2.a.br		4
56.j	odd	6	1		6272.2.a.bb		4
56.k	odd	6	1		896.2.i.f	yes	8
56.k	odd	6	1		6272.2.a.bc		4
56.m	even	6	1		6272.2.a.bo		4
56.p	even	6	1		896.2.i.a	✓	8
56.p	even	6	1		6272.2.a.bv		4

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
896.2.i.a	✓	8	8.b	even	2	1
896.2.i.a	✓	8	56.p	even	6	1
896.2.i.c	yes	8	4.b	odd	2	1
896.2.i.c	yes	8	28.g	odd	6	1
896.2.i.f	yes	8	8.d	odd	2	1
896.2.i.f	yes	8	56.k	odd	6	1
896.2.i.h	yes	8	1.a	even	1	1	trivial
896.2.i.h	yes	8	7.c	even	3	1	inner
6272.2.a.y		4	7.c	even	3	1
6272.2.a.bb		4	56.j	odd	6	1
6272.2.a.bc		4	56.k	odd	6	1
6272.2.a.bf		4	28.f	even	6	1
6272.2.a.bo		4	56.m	even	6	1
6272.2.a.br		4	28.g	odd	6	1
6272.2.a.bs		4	7.d	odd	6	1
6272.2.a.bv		4	56.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(896, [\chi])\):

\( T_{3}^{8} - 2T_{3}^{7} + 10T_{3}^{6} + 4T_{3}^{5} + 43T_{3}^{4} - 20T_{3}^{3} + 22T_{3}^{2} + 4T_{3} + 1 \)

\( T_{5}^{8} - 2T_{5}^{7} + 16T_{5}^{6} + 4T_{5}^{5} + 133T_{5}^{4} + 4T_{5}^{3} + 472T_{5}^{2} + 310T_{5} + 961 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{8} \)
$3$	T^8 - 2*T^7 + 10*T^6 + 4*T^5 + 43*T^4 - 20*T^3 + 22*T^2 + 4*T + 1
$5$	T^8 - 2*T^7 + 16*T^6 + 4*T^5 + 133*T^4 + 4*T^3 + 472*T^2 + 310*T + 961
$7$	T^8 - 2*T^7 + 4*T^6 - 26*T^5 + 58*T^4 - 182*T^3 + 196*T^2 - 686*T + 2401
$11$	T^8 - 4*T^7 + 22*T^6 - 52*T^5 + 223*T^4 - 508*T^3 + 1234*T^2 - 1330*T + 1225