Properties

Label 3200.2.d.q
Level $3200$
Weight $2$
Character orbit 3200.d
Analytic conductor $25.552$
Analytic rank $0$
Dimension $4$
CM discriminant -8
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: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} - 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + ( -2 + \beta_{3} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{3} + ( -2 + \beta_{3} ) q^{9} + ( -\beta_{1} - 2 \beta_{2} ) q^{11} + ( 3 - \beta_{3} ) q^{17} + ( -2 \beta_{1} + \beta_{2} ) q^{19} + ( \beta_{1} - 4 \beta_{2} ) q^{27} + ( 11 - 2 \beta_{3} ) q^{33} + ( 3 - 2 \beta_{3} ) q^{41} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{43} -7 q^{49} + ( -\beta_{1} + 8 \beta_{2} ) q^{51} + ( -3 + \beta_{3} ) q^{57} + ( -5 \beta_{1} + 5 \beta_{2} ) q^{59} + ( 5 \beta_{1} + 2 \beta_{2} ) q^{67} + ( -1 - 3 \beta_{3} ) q^{73} + ( 13 - \beta_{3} ) q^{81} + ( -5 \beta_{1} - 4 \beta_{2} ) q^{83} + ( 9 - \beta_{3} ) q^{89} + 10 q^{97} + ( -5 \beta_{1} + 15 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{9} + O(q^{10}) \) \( 4q - 8q^{9} + 12q^{17} + 44q^{33} + 12q^{41} - 28q^{49} - 12q^{57} - 4q^{73} + 52q^{81} + 36q^{89} + 40q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 2 \nu^{2} - 2 \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 2 \nu^{2} - 2 \)\()/2\)
\(\beta_{3}\)\(=\)\( -\nu^{3} + 4 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} - \beta_{2} + \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 2\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{2} + \beta_{1}\)

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
−1.22474 + 0.707107i
1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
0 3.14626i 0 0 0 0 0 −6.89898 0
1601.2 0 0.317837i 0 0 0 0 0 2.89898 0
1601.3 0 0.317837i 0 0 0 0 0 2.89898 0
1601.4 0 3.14626i 0 0 0 0 0 −6.89898 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
4.b odd 2 1 inner
8.b 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.d.q yes 4
4.b odd 2 1 inner 3200.2.d.q yes 4
5.b even 2 1 3200.2.d.n 4
5.c odd 4 2 3200.2.f.s 8
8.b even 2 1 inner 3200.2.d.q yes 4
8.d odd 2 1 CM 3200.2.d.q yes 4
16.e even 4 2 6400.2.a.cr 4
16.f odd 4 2 6400.2.a.cr 4
20.d odd 2 1 3200.2.d.n 4
20.e even 4 2 3200.2.f.s 8
40.e odd 2 1 3200.2.d.n 4
40.f even 2 1 3200.2.d.n 4
40.i odd 4 2 3200.2.f.s 8
40.k even 4 2 3200.2.f.s 8
80.k odd 4 2 6400.2.a.cq 4
80.q even 4 2 6400.2.a.cq 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.n 4 5.b even 2 1
3200.2.d.n 4 20.d odd 2 1
3200.2.d.n 4 40.e odd 2 1
3200.2.d.n 4 40.f even 2 1
3200.2.d.q yes 4 1.a even 1 1 trivial
3200.2.d.q yes 4 4.b odd 2 1 inner
3200.2.d.q yes 4 8.b even 2 1 inner
3200.2.d.q yes 4 8.d odd 2 1 CM
3200.2.f.s 8 5.c odd 4 2
3200.2.f.s 8 20.e even 4 2
3200.2.f.s 8 40.i odd 4 2
3200.2.f.s 8 40.k even 4 2
6400.2.a.cq 4 80.k odd 4 2
6400.2.a.cq 4 80.q even 4 2
6400.2.a.cr 4 16.e even 4 2
6400.2.a.cr 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 \(S_{2}^{\mathrm{new}}(3200, [\chi])\):

\( T_{3}^{4} + 10 T_{3}^{2} + 1 \)
\( T_{7} \)
\( T_{11}^{4} + 58 T_{11}^{2} + 625 \)
\( T_{13} \)
\( T_{17}^{2} - 6 T_{17} - 15 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( 1 + 10 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( 625 + 58 T^{2} + T^{4} \)
$13$ \( T^{4} \)
$17$ \( ( -15 - 6 T + T^{2} )^{2} \)
$19$ \( 225 + 42 T^{2} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( -87 - 6 T + T^{2} )^{2} \)
$43$ \( ( 72 + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( ( 200 + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( 16641 + 330 T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -215 + 2 T + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( 58081 + 490 T^{2} + T^{4} \)
$89$ \( ( 57 - 18 T + T^{2} )^{2} \)
$97$ \( ( -10 + T )^{4} \)
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