Properties

Label 3200.2.f.s
Level $3200$
Weight $2$
Character orbit 3200.f
Analytic conductor $25.552$
Analytic rank $0$
Dimension $8$
CM discriminant -8
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \(x^{8} - x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{10}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -\zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{3} + ( 2 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{24} + 2 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{3} + ( 2 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{9} + ( -3 + \zeta_{24} - \zeta_{24}^{3} + 6 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{11} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 3 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{17} + ( 1 + 3 \zeta_{24} - 3 \zeta_{24}^{3} - 2 \zeta_{24}^{4} - 3 \zeta_{24}^{5} ) q^{19} + ( -5 \zeta_{24} + 6 \zeta_{24}^{2} - 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{27} + ( -4 \zeta_{24} + 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 11 \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{33} + ( 3 - 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{5} + 8 \zeta_{24}^{7} ) q^{41} + ( -6 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} ) q^{43} + 7 q^{49} + ( 7 - 9 \zeta_{24} + 9 \zeta_{24}^{3} - 14 \zeta_{24}^{4} + 9 \zeta_{24}^{5} ) q^{51} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 3 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{57} + ( 10 \zeta_{24} - 10 \zeta_{24}^{3} - 10 \zeta_{24}^{5} ) q^{59} + ( -3 \zeta_{24} - 14 \zeta_{24}^{2} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 7 \zeta_{24}^{6} ) q^{67} + ( -6 \zeta_{24} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} - \zeta_{24}^{6} - 12 \zeta_{24}^{7} ) q^{73} + ( 13 - 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{81} + ( -\zeta_{24} - 18 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} + 9 \zeta_{24}^{6} ) q^{83} + ( -9 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{89} -10 \zeta_{24}^{6} q^{97} + ( -10 + 20 \zeta_{24} - 20 \zeta_{24}^{3} + 20 \zeta_{24}^{4} - 20 \zeta_{24}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 16q^{9} + O(q^{10}) \) \( 8q + 16q^{9} + 24q^{41} + 56q^{49} + 104q^{81} - 72q^{89} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−0.258819 0.965926i
−0.258819 + 0.965926i
0.258819 + 0.965926i
0.258819 0.965926i
0.965926 0.258819i
0.965926 + 0.258819i
−0.965926 + 0.258819i
−0.965926 0.258819i
0 −3.14626 0 0 0 0 0 6.89898 0
449.2 0 −3.14626 0 0 0 0 0 6.89898 0
449.3 0 −0.317837 0 0 0 0 0 −2.89898 0
449.4 0 −0.317837 0 0 0 0 0 −2.89898 0
449.5 0 0.317837 0 0 0 0 0 −2.89898 0
449.6 0 0.317837 0 0 0 0 0 −2.89898 0
449.7 0 3.14626 0 0 0 0 0 6.89898 0
449.8 0 3.14626 0 0 0 0 0 6.89898 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
20.d odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.2.f.s 8
4.b odd 2 1 inner 3200.2.f.s 8
5.b even 2 1 inner 3200.2.f.s 8
5.c odd 4 1 3200.2.d.n 4
5.c odd 4 1 3200.2.d.q yes 4
8.b even 2 1 inner 3200.2.f.s 8
8.d odd 2 1 CM 3200.2.f.s 8
20.d odd 2 1 inner 3200.2.f.s 8
20.e even 4 1 3200.2.d.n 4
20.e even 4 1 3200.2.d.q yes 4
40.e odd 2 1 inner 3200.2.f.s 8
40.f even 2 1 inner 3200.2.f.s 8
40.i odd 4 1 3200.2.d.n 4
40.i odd 4 1 3200.2.d.q yes 4
40.k even 4 1 3200.2.d.n 4
40.k even 4 1 3200.2.d.q yes 4
80.i odd 4 1 6400.2.a.cq 4
80.i odd 4 1 6400.2.a.cr 4
80.j even 4 1 6400.2.a.cq 4
80.j even 4 1 6400.2.a.cr 4
80.s even 4 1 6400.2.a.cq 4
80.s even 4 1 6400.2.a.cr 4
80.t odd 4 1 6400.2.a.cq 4
80.t odd 4 1 6400.2.a.cr 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.n 4 5.c odd 4 1
3200.2.d.n 4 20.e even 4 1
3200.2.d.n 4 40.i odd 4 1
3200.2.d.n 4 40.k even 4 1
3200.2.d.q yes 4 5.c odd 4 1
3200.2.d.q yes 4 20.e even 4 1
3200.2.d.q yes 4 40.i odd 4 1
3200.2.d.q yes 4 40.k even 4 1
3200.2.f.s 8 1.a even 1 1 trivial
3200.2.f.s 8 4.b odd 2 1 inner
3200.2.f.s 8 5.b even 2 1 inner
3200.2.f.s 8 8.b even 2 1 inner
3200.2.f.s 8 8.d odd 2 1 CM
3200.2.f.s 8 20.d odd 2 1 inner
3200.2.f.s 8 40.e odd 2 1 inner
3200.2.f.s 8 40.f even 2 1 inner
6400.2.a.cq 4 80.i odd 4 1
6400.2.a.cq 4 80.j even 4 1
6400.2.a.cq 4 80.s even 4 1
6400.2.a.cq 4 80.t odd 4 1
6400.2.a.cr 4 80.i odd 4 1
6400.2.a.cr 4 80.j even 4 1
6400.2.a.cr 4 80.s even 4 1
6400.2.a.cr 4 80.t odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3200, [\chi])\):

\( T_{3}^{4} - 10 T_{3}^{2} + 1 \)
\( T_{7} \)
\( T_{11}^{4} + 58 T_{11}^{2} + 625 \)
\( T_{13} \)
\( T_{31} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 1 - 10 T^{2} + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( T^{8} \)
$11$ \( ( 625 + 58 T^{2} + T^{4} )^{2} \)
$13$ \( T^{8} \)
$17$ \( ( 225 + 66 T^{2} + T^{4} )^{2} \)
$19$ \( ( 225 + 42 T^{2} + T^{4} )^{2} \)
$23$ \( T^{8} \)
$29$ \( T^{8} \)
$31$ \( T^{8} \)
$37$ \( T^{8} \)
$41$ \( ( -87 - 6 T + T^{2} )^{4} \)
$43$ \( ( -72 + T^{2} )^{4} \)
$47$ \( T^{8} \)
$53$ \( T^{8} \)
$59$ \( ( 200 + T^{2} )^{4} \)
$61$ \( T^{8} \)
$67$ \( ( 16641 - 330 T^{2} + T^{4} )^{2} \)
$71$ \( T^{8} \)
$73$ \( ( 46225 + 434 T^{2} + T^{4} )^{2} \)
$79$ \( T^{8} \)
$83$ \( ( 58081 - 490 T^{2} + T^{4} )^{2} \)
$89$ \( ( 57 + 18 T + T^{2} )^{4} \)
$97$ \( ( 100 + T^{2} )^{4} \)
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