Newform orbit 8649.2.a.be

• Label: 8649.2.a.be
• Level: $8649$
• Weight: $2$
• Character orbit: 8649.a
• Self dual: yes
• Analytic conductor: $69.063$
• Analytic rank: $1$
• Dimension: $8$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(69.0626127082\)
Analytic rank:	\(1\)
Dimension:	\(8\)
Coefficient field:	8.8.2051578125.1
Defining polynomial:	\( x^{8} - 2x^{7} - 9x^{6} + 19x^{5} + 14x^{4} - 28x^{3} - 11x^{2} + 6x + 1 \)
Coefficient ring:	\(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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 + (b5 - b3 - 1) * q^2 + (-b6 + b1 + 1) * q^4 + (b6 - b5 + b3 + b2 + b1) * q^5 + (-b7 - b5 + b4 - b3 + b1) * q^7 + (b7 + b6 + b5 - 2*b4 + b2) * q^8
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 8 q^{4} - 3 q^{5} - 2 q^{7} + 9 q^{8}+O(q^{10}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 9x^{6} + 19x^{5} + 14x^{4} - 28x^{3} - 11x^{2} + 6x + 1 \) :

\(\beta_{1}\)	\(=\)	\( \nu \)
\(\beta_{2}\)	\(=\)	\( ( \nu^{6} - 7\nu^{4} + 2\nu^{3} + 4\nu^{2} + 5\nu + 1 ) / 3 \)
\(\beta_{3}\)	\(=\)	\( ( -\nu^{7} + \nu^{6} + 7\nu^{5} - 9\nu^{4} + \nu^{3} - \nu^{2} - 11\nu + 4 ) / 3 \)
\(\beta_{4}\)	\(=\)	\( ( -2\nu^{6} + 17\nu^{4} - 4\nu^{3} - 26\nu^{2} - \nu + 1 ) / 3 \)
\(\beta_{5}\)	\(=\)	\( ( \nu^{7} - 10\nu^{5} + 2\nu^{4} + 25\nu^{3} - 4\nu^{2} - 14\nu + 3 ) / 3 \)
\(\beta_{6}\)	\(=\)	\( ( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 26\nu^{4} - 5\nu^{3} - 28\nu^{2} + 10\nu + 6 ) / 3 \)
\(\beta_{7}\)	\(=\)	\( ( \nu^{7} - 4\nu^{6} - 7\nu^{5} + 36\nu^{4} - 4\nu^{3} - 47\nu^{2} - \nu + 2 ) / 3 \)

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  

\(\iota_m(\nu)\)

\( a_{2} \)

\( a_{3} \)

\( a_{4} \)

\( a_{5} \)

\( a_{6} \)

\( a_{7} \)

\( a_{8} \)

\( a_{9} \)

\( a_{10} \)

1.1

2.11562
−0.662608
0.431370
−0.143490
−1.04940
−2.73366
1.76152
2.28064

−2.30753

0

3.32468

2.49846

0

1.60188

−3.05673

0

−5.76526

1.2

−2.07212

0

2.29369

−2.34791

0

−3.67454

−0.608557

0

4.86516

1.3

−1.26660

0

−0.395721

3.80032

0

2.18899

3.03442

0

−4.81349

1.4

−1.23217

0

−0.481752

1.54562

0

−3.80376

3.05795

0

−1.90447

1.5

−0.351432

0

−1.87650

−2.97323

0

−1.08213

1.36233

0

1.04489

1.6

0.689493

0

−1.52460

−3.70752

0

0.763394

−2.43019

0

−2.55631

1.7

1.85021

0

1.42326

−1.20736

0

3.73304

−1.06708

0

−2.23387

1.8

2.69016

0

5.23694

−0.608384

0

−1.72688

8.70786

0

−1.63665

Atkin-Lehner signs

\( p \)	Sign
\(3\)	\(-1\)
\(31\)	\(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	8649.2.a.be		8
3.b	odd	2	1		961.2.a.j		8
31.b	odd	2	1		8649.2.a.bf		8
31.h	odd	30	2		279.2.y.c		16
93.c	even	2	1		961.2.a.i		8
93.g	even	6	2		961.2.c.j		16
93.h	odd	6	2		961.2.c.i		16
93.k	even	10	2		961.2.d.o		16
93.k	even	10	2		961.2.d.p		16
93.l	odd	10	2		961.2.d.n		16
93.l	odd	10	2		961.2.d.q		16
93.o	odd	30	2		961.2.g.j		16
93.o	odd	30	2		961.2.g.l		16
93.o	odd	30	2		961.2.g.m		16
93.o	odd	30	2		961.2.g.n		16
93.p	even	30	2		31.2.g.a	✓	16
93.p	even	30	2		961.2.g.k		16
93.p	even	30	2		961.2.g.s		16
93.p	even	30	2		961.2.g.t		16
372.bc	odd	30	2		496.2.bg.c		16
465.bm	even	30	2		775.2.bl.a		16
465.bv	odd	60	4		775.2.ck.a		32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
31.2.g.a	✓	16	93.p	even	30	2
279.2.y.c		16	31.h	odd	30	2
496.2.bg.c		16	372.bc	odd	30	2
775.2.bl.a		16	465.bm	even	30	2
775.2.ck.a		32	465.bv	odd	60	4
961.2.a.i		8	93.c	even	2	1
961.2.a.j		8	3.b	odd	2	1
961.2.c.i		16	93.h	odd	6	2
961.2.c.j		16	93.g	even	6	2
961.2.d.n		16	93.l	odd	10	2
961.2.d.o		16	93.k	even	10	2
961.2.d.p		16	93.k	even	10	2
961.2.d.q		16	93.l	odd	10	2
961.2.g.j		16	93.o	odd	30	2
961.2.g.k		16	93.p	even	30	2
961.2.g.l		16	93.o	odd	30	2
961.2.g.m		16	93.o	odd	30	2
961.2.g.n		16	93.o	odd	30	2
961.2.g.s		16	93.p	even	30	2
961.2.g.t		16	93.p	even	30	2
8649.2.a.be		8	1.a	even	1	1	trivial
8649.2.a.bf		8	31.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8649))\):

\( T_{2}^{8} + 2T_{2}^{7} - 10T_{2}^{6} - 23T_{2}^{5} + 19T_{2}^{4} + 63T_{2}^{3} + 15T_{2}^{2} - 27T_{2} - 9 \)

\( T_{5}^{8} + 3T_{5}^{7} - 22T_{5}^{6} - 69T_{5}^{5} + 115T_{5}^{4} + 411T_{5}^{3} - 57T_{5}^{2} - 612T_{5} - 279 \)

\( T_{7}^{8} + 2T_{7}^{7} - 25T_{7}^{6} - 38T_{7}^{5} + 184T_{7}^{4} + 153T_{7}^{3} - 435T_{7}^{2} - 162T_{7} + 261 \)

\( T_{11}^{8} + 18T_{11}^{7} + 131T_{11}^{6} + 489T_{11}^{5} + 964T_{11}^{4} + 852T_{11}^{3} - 51T_{11}^{2} - 594T_{11} - 279 \)

\( T_{13}^{8} - 8T_{13}^{7} - 4T_{13}^{6} + 161T_{13}^{5} - 281T_{13}^{4} - 627T_{13}^{3} + 2064T_{13}^{2} - 1296T_{13} - 279 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	T^8 + 2*T^7 - 10*T^6 - 23*T^5 + 19*T^4 + 63*T^3 + 15*T^2 - 27*T - 9
$3$	\( T^{8} \)
$5$	T^8 + 3*T^7 - 22*T^6 - 69*T^5 + 115*T^4 + 411*T^3 - 57*T^2 - 612*T - 279
$7$	T^8 + 2*T^7 - 25*T^6 - 38*T^5 + 184*T^4 + 153*T^3 - 435*T^2 - 162*T + 261
$11$	T^8 + 18*T^7 + 131*T^6 + 489*T^5 + 964*T^4 + 852*T^3 - 51*T^2 - 594*T - 279