Properties

Label 3200.2.d.i
Level $3200$
Weight $2$
Character orbit 3200.d
Analytic conductor $25.552$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $8$

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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: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 640)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - 3 \beta_1 q^{7} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - 3 \beta_1 q^{7} - 7 q^{9} + 3 \beta_{3} q^{21} + \beta_1 q^{23} - 4 \beta_{2} q^{27} + 2 \beta_{3} q^{29} - 12 q^{41} + \beta_{2} q^{43} + 7 \beta_1 q^{47} + 11 q^{49} + 3 \beta_{3} q^{61} + 21 \beta_1 q^{63} + 5 \beta_{2} q^{67} - \beta_{3} q^{69} + 19 q^{81} + 3 \beta_{2} q^{83} + 20 \beta_1 q^{87} + 6 q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 28 q^{9} - 48 q^{41} + 44 q^{49} + 76 q^{81} + 24 q^{89}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 7\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 7\beta_1 ) / 2 \) 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 3.16228i 0 0 0 −4.24264 0 −7.00000 0
1601.2 0 3.16228i 0 0 0 4.24264 0 −7.00000 0
1601.3 0 3.16228i 0 0 0 −4.24264 0 −7.00000 0
1601.4 0 3.16228i 0 0 0 4.24264 0 −7.00000 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
8.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.d.i 4
4.b odd 2 1 inner 3200.2.d.i 4
5.b even 2 1 inner 3200.2.d.i 4
5.c odd 4 2 640.2.f.f 4
8.b even 2 1 inner 3200.2.d.i 4
8.d odd 2 1 inner 3200.2.d.i 4
16.e even 4 2 6400.2.a.cv 4
16.f odd 4 2 6400.2.a.cv 4
20.d odd 2 1 CM 3200.2.d.i 4
20.e even 4 2 640.2.f.f 4
40.e odd 2 1 inner 3200.2.d.i 4
40.f even 2 1 inner 3200.2.d.i 4
40.i odd 4 2 640.2.f.f 4
40.k even 4 2 640.2.f.f 4
80.i odd 4 2 1280.2.c.f 4
80.j even 4 2 1280.2.c.f 4
80.k odd 4 2 6400.2.a.cv 4
80.q even 4 2 6400.2.a.cv 4
80.s even 4 2 1280.2.c.f 4
80.t odd 4 2 1280.2.c.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.f.f 4 5.c odd 4 2
640.2.f.f 4 20.e even 4 2
640.2.f.f 4 40.i odd 4 2
640.2.f.f 4 40.k even 4 2
1280.2.c.f 4 80.i odd 4 2
1280.2.c.f 4 80.j even 4 2
1280.2.c.f 4 80.s even 4 2
1280.2.c.f 4 80.t odd 4 2
3200.2.d.i 4 1.a even 1 1 trivial
3200.2.d.i 4 4.b odd 2 1 inner
3200.2.d.i 4 5.b even 2 1 inner
3200.2.d.i 4 8.b even 2 1 inner
3200.2.d.i 4 8.d odd 2 1 inner
3200.2.d.i 4 20.d odd 2 1 CM
3200.2.d.i 4 40.e odd 2 1 inner
3200.2.d.i 4 40.f even 2 1 inner
6400.2.a.cv 4 16.e even 4 2
6400.2.a.cv 4 16.f odd 4 2
6400.2.a.cv 4 80.k odd 4 2
6400.2.a.cv 4 80.q even 4 2

Hecke kernels

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

\( T_{3}^{2} + 10 \) Copy content Toggle raw display
\( T_{7}^{2} - 18 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 10)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 80)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T + 12)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 10)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 98)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 180)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 250)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 90)^{2} \) Copy content Toggle raw display
$89$ \( (T - 6)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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