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: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} + 2) q^{9} + \beta_{6} q^{11} + ( - \beta_{5} - \beta_{4}) q^{17} + \beta_{7} q^{19} + ( - \beta_{2} + 4 \beta_1) q^{27} + (2 \beta_{5} + 3 \beta_{4}) q^{33} + ( - 2 \beta_{3} + 3) q^{41} + ( - 3 \beta_{2} + 3 \beta_1) q^{43} + 7 q^{49} + ( - 2 \beta_{7} - 3 \beta_{6}) q^{51} + (\beta_{5} + \beta_{4}) q^{57} + (3 \beta_{7} + \beta_{6}) q^{59} + ( - 5 \beta_{2} - 2 \beta_1) q^{67} + (3 \beta_{5} + \beta_{4}) q^{73} + ( - \beta_{3} + 13) q^{81} + ( - 5 \beta_{2} - 4 \beta_1) q^{83} + (\beta_{3} - 9) q^{89} - 2 \beta_{4} q^{97} + (5 \beta_{7} + 5 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{9} + 24 q^{41} + 56 q^{49} + 104 q^{81} - 72 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{24}^{6} + \zeta_{24}^{5} - \zeta_{24}^{3} + 2\zeta_{24}^{2} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -4\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 5\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -4\zeta_{24}^{7} - 2\zeta_{24}^{6} - 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{5} + 6\zeta_{24}^{4} - \zeta_{24}^{3} + \zeta_{24} - 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -3\zeta_{24}^{5} - 2\zeta_{24}^{4} - 3\zeta_{24}^{3} + 3\zeta_{24} + 1 \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( 3\beta_{7} + \beta_{6} - 5\beta_{5} - 2\beta_{4} + 5\beta_{3} + 5\beta_{2} - 5\beta_1 ) / 40 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( 2\beta_{4} + 5\beta_{2} + 5\beta_1 ) / 20 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -3\beta_{7} - \beta_{6} + 5\beta_{2} - 5\beta_1 ) / 20 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( -\beta_{7} + 3\beta_{6} + 10 ) / 20 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -3\beta_{7} - \beta_{6} - 5\beta_{5} - 2\beta_{4} + 5\beta_{3} - 5\beta_{2} + 5\beta_1 ) / 40 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{4} ) / 5 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -3\beta_{7} - \beta_{6} - 5\beta_{5} - 2\beta_{4} - 5\beta_{3} + 5\beta_{2} - 5\beta_1 ) / 40 \) Copy content Toggle raw display

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 \(S_{2}^{\mathrm{new}}(3200, [\chi])\):

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

Hecke characteristic polynomials

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