Newform orbit 968.2.i.r

• Label: 968.2.i.r
• Level: $968$
• Weight: $2$
• Character orbit: 968.i
• Analytic conductor: $7.730$
• Analytic rank: $0$
• Dimension: $8$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	968.i (of order \(5\), degree \(4\), not minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.72951891566\)
Analytic rank:	\(0\)
Dimension:	\(8\)
Relative dimension:	\(2\) over \(\Q(\zeta_{5})\)
Coefficient field:	8.0.1305015625.1
Defining polynomial:	\( x^{8} - x^{7} + 5x^{6} - 9x^{5} + 29x^{4} + 36x^{3} + 80x^{2} + 64x + 256 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 88)
Sato-Tate group:	$\mathrm{SU}(2)[C_{5}]$

$q$-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 + b5 * q^3 + (b7 + 2*b6 + b5 - b4 - 2*b3 + 2*b2 - b1 - 2) * q^5 - 2*b7 * q^7 + (b3 - b1) * q^9
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{3} - 3 q^{5} + 2 q^{7} - 3 q^{9}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 5x^{6} - 9x^{5} + 29x^{4} + 36x^{3} + 80x^{2} + 64x + 256 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( -\nu^{7} - 181\nu^{2} ) / 464 \)
\(\beta_{3}\)	\(=\)	\( ( \nu^{6} + 65\nu ) / 116 \)
\(\beta_{4}\)	\(=\)	\( ( \nu^{5} + 36 ) / 29 \)
\(\beta_{5}\)	\(=\)	\( ( -\nu^{7} + 5\nu^{6} - 9\nu^{5} + 29\nu^{4} - 145\nu^{3} + 80\nu^{2} + 64\nu + 256 ) / 464 \)
\(\beta_{6}\)	\(=\)	\( ( -5\nu^{7} + 25\nu^{6} - 45\nu^{5} + 145\nu^{4} - 261\nu^{3} + 400\nu^{2} + 320\nu + 1280 ) / 1856 \)
\(\beta_{7}\)	\(=\)	\( ( -\nu^{7} - 65\nu^{2} ) / 116 \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/968\mathbb{Z}\right)^\times\).

\(n\)	\(485\)	\(727\)	\(849\)
\(\chi(n)\)	\(1\)	\(1\)	\(-\beta_{2}\)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

9.1

−0.791563	+	2.43618i
0.482546	−	1.48512i
−1.26332	−	0.917858i
2.07234	+	1.50564i
−1.26332	+	0.917858i
2.07234	−	1.50564i
−0.791563	−	2.43618i
0.482546	+	1.48512i

0

−2.07234

+

1.50564i

0

−0.173529

−

0.534068i

0

4.14468

+

3.01129i

0

1.10058

−

3.38724i

0

9.2

0

1.26332

−

0.917858i

0

1.10058

+

3.38724i

0

−2.52665

−

1.83572i

0

−0.173529

+

0.534068i

0

81.1

0

−0.482546

+

1.48512i

0

−2.88136

+

2.09343i

0

0.965093

+

2.97025i

0

0.454306

+

0.330072i

0

81.2

0

0.791563

−

2.43618i

0

0.454306

−

0.330072i

0

−1.58313

−

4.87236i

0

−2.88136

−

2.09343i

0

729.1

0

−0.482546

−

1.48512i

0

−2.88136

−

2.09343i

0

0.965093

−

2.97025i

0

0.454306

−

0.330072i

0

729.2

0

0.791563

+

2.43618i

0

0.454306

+

0.330072i

0

−1.58313

+

4.87236i

0

−2.88136

+

2.09343i

0

753.1

0

−2.07234

−

1.50564i

0

−0.173529

+

0.534068i

0

4.14468

−

3.01129i

0

1.10058

+

3.38724i

0

753.2

0

1.26332

+

0.917858i

0

1.10058

−

3.38724i

0

−2.52665

+

1.83572i

0

−0.173529

−

0.534068i

0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	3	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	968.2.i.r		8
11.b	odd	2	1		968.2.i.q		8
11.c	even	5	1		88.2.a.b	✓	2
11.c	even	5	3	inner	968.2.i.r		8
11.d	odd	10	1		968.2.a.j		2
11.d	odd	10	3		968.2.i.q		8
33.f	even	10	1		8712.2.a.bb		2
33.h	odd	10	1		792.2.a.h		2
44.g	even	10	1		1936.2.a.r		2
44.h	odd	10	1		176.2.a.d		2
55.j	even	10	1		2200.2.a.o		2
55.k	odd	20	2		2200.2.b.g		4
77.j	odd	10	1		4312.2.a.n		2
88.k	even	10	1		7744.2.a.cl		2
88.l	odd	10	1		704.2.a.p		2
88.o	even	10	1		704.2.a.m		2
88.p	odd	10	1		7744.2.a.by		2
132.o	even	10	1		1584.2.a.t		2
176.v	odd	20	2		2816.2.c.p		4
176.w	even	20	2		2816.2.c.w		4
220.n	odd	10	1		4400.2.a.bp		2
220.v	even	20	2		4400.2.b.v		4
264.t	odd	10	1		6336.2.a.cu		2
264.w	even	10	1		6336.2.a.cx		2
308.t	even	10	1		8624.2.a.cb		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
88.2.a.b	✓	2	11.c	even	5	1
176.2.a.d		2	44.h	odd	10	1
704.2.a.m		2	88.o	even	10	1
704.2.a.p		2	88.l	odd	10	1
792.2.a.h		2	33.h	odd	10	1
968.2.a.j		2	11.d	odd	10	1
968.2.i.q		8	11.b	odd	2	1
968.2.i.q		8	11.d	odd	10	3
968.2.i.r		8	1.a	even	1	1	trivial
968.2.i.r		8	11.c	even	5	3	inner
1584.2.a.t		2	132.o	even	10	1
1936.2.a.r		2	44.g	even	10	1
2200.2.a.o		2	55.j	even	10	1
2200.2.b.g		4	55.k	odd	20	2
2816.2.c.p		4	176.v	odd	20	2
2816.2.c.w		4	176.w	even	20	2
4312.2.a.n		2	77.j	odd	10	1
4400.2.a.bp		2	220.n	odd	10	1
4400.2.b.v		4	220.v	even	20	2
6336.2.a.cu		2	264.t	odd	10	1
6336.2.a.cx		2	264.w	even	10	1
7744.2.a.by		2	88.p	odd	10	1
7744.2.a.cl		2	88.k	even	10	1
8624.2.a.cb		2	308.t	even	10	1
8712.2.a.bb		2	33.f	even	10	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(968, [\chi])\):

\( T_{3}^{8} + T_{3}^{7} + 5T_{3}^{6} + 9T_{3}^{5} + 29T_{3}^{4} - 36T_{3}^{3} + 80T_{3}^{2} - 64T_{3} + 256 \)

\( T_{7}^{8} - 2T_{7}^{7} + 20T_{7}^{6} - 72T_{7}^{5} + 464T_{7}^{4} + 1152T_{7}^{3} + 5120T_{7}^{2} + 8192T_{7} + 65536 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{8} \)
$3$	T^8 + T^7 + 5*T^6 + 9*T^5 + 29*T^4 - 36*T^3 + 80*T^2 - 64*T + 256
$5$	T^8 + 3*T^7 + 11*T^6 + 39*T^5 + 139*T^4 - 78*T^3 + 44*T^2 - 24*T + 16
$7$	T^8 - 2*T^7 + 20*T^6 - 72*T^5 + 464*T^4 + 1152*T^3 + 5120*T^2 + 8192*T + 65536
$11$	\( T^{8} \)