Properties

Label 3200.2.d.r
Level $3200$
Weight $2$
Character orbit 3200.d
Analytic conductor $25.552$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-5})\)
Defining polynomial: \( x^{4} + 4x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_1 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_1 q^{7} - 2 q^{9} - \beta_{3} q^{11} - \beta_{2} q^{13} + 5 q^{17} + \beta_{3} q^{19} + \beta_{2} q^{21} + 2 \beta_1 q^{23} + \beta_{3} q^{27} - \beta_{2} q^{29} + 5 q^{33} - \beta_{2} q^{37} - 5 \beta_1 q^{39} + 3 q^{41} - 4 \beta_{3} q^{43} - \beta_1 q^{47} + q^{49} + 5 \beta_{3} q^{51} - 2 \beta_{2} q^{53} - 5 q^{57} - 4 \beta_{3} q^{59} + \beta_{2} q^{61} + 2 \beta_1 q^{63} - 5 \beta_{3} q^{67} - 2 \beta_{2} q^{69} - 5 \beta_1 q^{71} - 15 q^{73} - \beta_{2} q^{77} - 5 \beta_1 q^{79} - 11 q^{81} + 3 \beta_{3} q^{83} - 5 \beta_1 q^{87} + q^{89} - 8 \beta_{3} q^{91} - 10 q^{97} + 2 \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{9} + 20 q^{17} + 20 q^{33} + 12 q^{41} + 4 q^{49} - 20 q^{57} - 60 q^{73} - 44 q^{81} + 4 q^{89} - 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} + 2\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{3} + 14\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 7\beta_1 ) / 4 \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1601.1
−0.707107 + 1.58114i
0.707107 1.58114i
−0.707107 1.58114i
0.707107 + 1.58114i
0 2.23607i 0 0 0 −2.82843 0 −2.00000 0
1601.2 0 2.23607i 0 0 0 2.82843 0 −2.00000 0
1601.3 0 2.23607i 0 0 0 −2.82843 0 −2.00000 0
1601.4 0 2.23607i 0 0 0 2.82843 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.2.d.r yes 4
4.b odd 2 1 inner 3200.2.d.r yes 4
5.b even 2 1 3200.2.d.m 4
5.c odd 4 2 3200.2.f.r 8
8.b even 2 1 inner 3200.2.d.r yes 4
8.d odd 2 1 inner 3200.2.d.r yes 4
16.e even 4 2 6400.2.a.cs 4
16.f odd 4 2 6400.2.a.cs 4
20.d odd 2 1 3200.2.d.m 4
20.e even 4 2 3200.2.f.r 8
40.e odd 2 1 3200.2.d.m 4
40.f even 2 1 3200.2.d.m 4
40.i odd 4 2 3200.2.f.r 8
40.k even 4 2 3200.2.f.r 8
80.k odd 4 2 6400.2.a.cp 4
80.q even 4 2 6400.2.a.cp 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.m 4 5.b even 2 1
3200.2.d.m 4 20.d odd 2 1
3200.2.d.m 4 40.e odd 2 1
3200.2.d.m 4 40.f even 2 1
3200.2.d.r yes 4 1.a even 1 1 trivial
3200.2.d.r yes 4 4.b odd 2 1 inner
3200.2.d.r yes 4 8.b even 2 1 inner
3200.2.d.r yes 4 8.d odd 2 1 inner
3200.2.f.r 8 5.c odd 4 2
3200.2.f.r 8 20.e even 4 2
3200.2.f.r 8 40.i odd 4 2
3200.2.f.r 8 40.k even 4 2
6400.2.a.cp 4 80.k odd 4 2
6400.2.a.cp 4 80.q even 4 2
6400.2.a.cs 4 16.e even 4 2
6400.2.a.cs 4 16.f odd 4 2

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
\( T_{17} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 40)^{2} \) Copy content Toggle raw display
$17$ \( (T - 5)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 40)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 40)^{2} \) Copy content Toggle raw display
$41$ \( (T - 3)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 160)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 40)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 125)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 200)^{2} \) Copy content Toggle raw display
$73$ \( (T + 15)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 200)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 45)^{2} \) Copy content Toggle raw display
$89$ \( (T - 1)^{4} \) Copy content Toggle raw display
$97$ \( (T + 10)^{4} \) Copy content Toggle raw display
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