Properties

Label 3200.2.f.r
Level $3200$
Weight $2$
Character orbit 3200.f
Analytic conductor $25.552$
Analytic rank $0$
Dimension $8$
CM no
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: 8.0.40960000.1
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{2} q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{2} q^{7} + 2 q^{9} + \beta_{3} q^{11} - \beta_{6} q^{13} + \beta_{4} q^{17} + \beta_{3} q^{19} - \beta_{5} q^{21} - 2 \beta_{2} q^{23} + \beta_1 q^{27} - \beta_{5} q^{29} - \beta_{4} q^{33} + \beta_{6} q^{37} - \beta_{7} q^{39} + 3 q^{41} + 4 \beta_1 q^{43} - \beta_{2} q^{47} - q^{49} - 5 \beta_{3} q^{51} - 2 \beta_{6} q^{53} - \beta_{4} q^{57} - 4 \beta_{3} q^{59} - \beta_{5} q^{61} - 2 \beta_{2} q^{63} - 5 \beta_1 q^{67} - 2 \beta_{5} q^{69} + \beta_{7} q^{71} + 3 \beta_{4} q^{73} + \beta_{6} q^{77} - \beta_{7} q^{79} - 11 q^{81} - 3 \beta_1 q^{83} - 5 \beta_{2} q^{87} - q^{89} + 8 \beta_{3} q^{91} - 2 \beta_{4} q^{97} + 2 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{9} + 24 q^{41} - 8 q^{49} - 88 q^{81} - 8 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{4} + 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{7} - 2\nu^{5} - 26\nu^{3} - 10\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{6} - 6\nu^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{6} - 40\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8\nu^{7} + 2\nu^{5} + 58\nu^{3} + 22\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -8\nu^{7} + 2\nu^{5} - 58\nu^{3} + 22\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 20\nu^{7} - 10\nu^{5} + 130\nu^{3} - 50\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 5\beta_{6} + 5\beta_{5} + 5\beta_{2} ) / 40 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{4} + 5\beta_{3} ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} - 5\beta_{6} + 5\beta_{5} + 10\beta_{2} ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -11\beta_{7} - 25\beta_{6} - 25\beta_{5} - 55\beta_{2} ) / 40 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 9\beta_{4} - 20\beta_{3} ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 29\beta_{7} + 65\beta_{6} - 65\beta_{5} - 145\beta_{2} ) / 40 \) 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
−0.437016 0.437016i
0.437016 0.437016i
0.437016 + 0.437016i
−0.437016 + 0.437016i
−1.14412 + 1.14412i
1.14412 + 1.14412i
1.14412 1.14412i
−1.14412 1.14412i
0 −2.23607 0 0 0 2.82843i 0 2.00000 0
449.2 0 −2.23607 0 0 0 2.82843i 0 2.00000 0
449.3 0 −2.23607 0 0 0 2.82843i 0 2.00000 0
449.4 0 −2.23607 0 0 0 2.82843i 0 2.00000 0
449.5 0 2.23607 0 0 0 2.82843i 0 2.00000 0
449.6 0 2.23607 0 0 0 2.82843i 0 2.00000 0
449.7 0 2.23607 0 0 0 2.82843i 0 2.00000 0
449.8 0 2.23607 0 0 0 2.82843i 0 2.00000 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
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 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.r 8
4.b odd 2 1 inner 3200.2.f.r 8
5.b even 2 1 inner 3200.2.f.r 8
5.c odd 4 1 3200.2.d.m 4
5.c odd 4 1 3200.2.d.r yes 4
8.b even 2 1 inner 3200.2.f.r 8
8.d odd 2 1 inner 3200.2.f.r 8
20.d odd 2 1 inner 3200.2.f.r 8
20.e even 4 1 3200.2.d.m 4
20.e even 4 1 3200.2.d.r yes 4
40.e odd 2 1 inner 3200.2.f.r 8
40.f even 2 1 inner 3200.2.f.r 8
40.i odd 4 1 3200.2.d.m 4
40.i odd 4 1 3200.2.d.r yes 4
40.k even 4 1 3200.2.d.m 4
40.k even 4 1 3200.2.d.r yes 4
80.i odd 4 1 6400.2.a.cp 4
80.i odd 4 1 6400.2.a.cs 4
80.j even 4 1 6400.2.a.cp 4
80.j even 4 1 6400.2.a.cs 4
80.s even 4 1 6400.2.a.cp 4
80.s even 4 1 6400.2.a.cs 4
80.t odd 4 1 6400.2.a.cp 4
80.t odd 4 1 6400.2.a.cs 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.m 4 5.c odd 4 1
3200.2.d.m 4 20.e even 4 1
3200.2.d.m 4 40.i odd 4 1
3200.2.d.m 4 40.k even 4 1
3200.2.d.r yes 4 5.c odd 4 1
3200.2.d.r yes 4 20.e even 4 1
3200.2.d.r yes 4 40.i odd 4 1
3200.2.d.r yes 4 40.k even 4 1
3200.2.f.r 8 1.a even 1 1 trivial
3200.2.f.r 8 4.b odd 2 1 inner
3200.2.f.r 8 5.b even 2 1 inner
3200.2.f.r 8 8.b even 2 1 inner
3200.2.f.r 8 8.d odd 2 1 inner
3200.2.f.r 8 20.d odd 2 1 inner
3200.2.f.r 8 40.e odd 2 1 inner
3200.2.f.r 8 40.f even 2 1 inner
6400.2.a.cp 4 80.i odd 4 1
6400.2.a.cp 4 80.j even 4 1
6400.2.a.cp 4 80.s even 4 1
6400.2.a.cp 4 80.t odd 4 1
6400.2.a.cs 4 80.i odd 4 1
6400.2.a.cs 4 80.j even 4 1
6400.2.a.cs 4 80.s even 4 1
6400.2.a.cs 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 \(S_{2}^{\mathrm{new}}(3200, [\chi])\):

\( T_{3}^{2} - 5 \) Copy content Toggle raw display
\( T_{7}^{2} + 8 \) Copy content Toggle raw display
\( T_{11}^{2} + 5 \) Copy content Toggle raw display
\( T_{13}^{2} - 40 \) 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
$3$ \( (T^{2} - 5)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 5)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 40)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 25)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 5)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 40)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} - 40)^{4} \) Copy content Toggle raw display
$41$ \( (T - 3)^{8} \) Copy content Toggle raw display
$43$ \( (T^{2} - 80)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 160)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 80)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 40)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 125)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 200)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 225)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 200)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 45)^{4} \) Copy content Toggle raw display
$89$ \( (T + 1)^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
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