Newform orbit 8512.2.a.bf

• Label: 8512.2.a.bf
• Level: $8512$
• Weight: $2$
• Character orbit: 8512.a
• Self dual: yes
• Analytic conductor: $67.969$
• Analytic rank: $0$
• Dimension: $2$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 8512 = 2^{6} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8512.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(67.9686622005\)
Analytic rank:	\(0\)
Dimension:	\(2\)
Coefficient field:	\(\Q(\sqrt{5}) \)
Defining polynomial:	\( x^{2} - x - 1 \)
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)	\(=\)	q + (b + 1) * q^3 + (2*b - 1) * q^5 + q^7 + (3*b - 1) * q^9 + (-3*b + 2) * q^11 + (-2*b + 5) * q^13 + (3*b + 1) * q^15 + (3*b - 4) * q^17 - q^19 + (b + 1) * q^21 + (2*b + 3) * q^23 + (2*b - 1) * q^27 + (-b + 6) * q^29 + (b - 4) * q^31 + (-4*b - 1) * q^33 + (2*b - 1) * q^35 + (-4*b + 5) * q^37 + (b + 3) * q^39 + (-9*b + 4) * q^41 + (8*b - 6) * q^43 + (b + 7) * q^45 + 7 * q^47 + q^49 + (2*b - 1) * q^51 + (3*b - 3) * q^53 + (b - 8) * q^55 + (-b - 1) * q^57 + (6*b + 1) * q^59 + (6*b + 3) * q^61 + (3*b - 1) * q^63 + (8*b - 9) * q^65 + (b + 4) * q^67 + (7*b + 5) * q^69 + (8*b - 9) * q^71 + (3*b - 5) * q^73 + (-3*b + 2) * q^77 + (-4*b + 2) * q^79 + (-6*b + 4) * q^81 + (-b + 13) * q^83 + (-5*b + 10) * q^85 + (4*b + 5) * q^87 + (-2*b + 4) * q^89 + (-2*b + 5) * q^91 + (-2*b - 3) * q^93 + (-2*b + 1) * q^95 + (2*b - 7) * q^97 - 11 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	2 * q + 3 * q^3 + 2 * q^7 + q^9 + q^11 + 8 * q^13 + 5 * q^15 - 5 * q^17 - 2 * q^19 + 3 * q^21 + 8 * q^23 + 11 * q^29 - 7 * q^31 - 6 * q^33 + 6 * q^37 + 7 * q^39 - q^41 - 4 * q^43 + 15 * q^45 + 14 * q^47 + 2 * q^49 - 3 * q^53 - 15 * q^55 - 3 * q^57 + 8 * q^59 + 12 * q^61 + q^63 - 10 * q^65 + 9 * q^67 + 17 * q^69 - 10 * q^71 - 7 * q^73 + q^77 + 2 * q^81 + 25 * q^83 + 15 * q^85 + 14 * q^87 + 6 * q^89 + 8 * q^91 - 8 * q^93 - 12 * q^97 - 22 * q^99

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

−0.618034
1.61803

0

0.381966

0

−2.23607

0

1.00000

0

−2.85410

0

1.2

0

2.61803

0

2.23607

0

1.00000

0

3.85410

0

Atkin-Lehner signs

\( p \)	Sign
\(2\)	\( +1 \)
\(7\)	\( -1 \)
\(19\)	\( +1 \)

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	8512.2.a.bf		2
4.b	odd	2	1		8512.2.a.i		2
8.b	even	2	1		1064.2.a.a	✓	2
8.d	odd	2	1		2128.2.a.n		2
24.h	odd	2	1		9576.2.a.bm		2
56.h	odd	2	1		7448.2.a.bc		2

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
1064.2.a.a	✓	2	8.b	even	2	1
2128.2.a.n		2	8.d	odd	2	1
7448.2.a.bc		2	56.h	odd	2	1
8512.2.a.i		2	4.b	odd	2	1
8512.2.a.bf		2	1.a	even	1	1	trivial
9576.2.a.bm		2	24.h	odd	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8512))\):

\( T_{3}^{2} - 3T_{3} + 1 \)

\( T_{5}^{2} - 5 \)

\( T_{11}^{2} - T_{11} - 11 \)

\( T_{23}^{2} - 8T_{23} + 11 \)

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	\( T^{2} \)
$3$	\( T^{2} - 3T + 1 \)
$5$	\( T^{2} - 5 \)
$7$	\( (T - 1)^{2} \)
$11$	\( T^{2} - T - 11 \)