Properties

Label 3200.1.bd.a
Level $3200$
Weight $1$
Character orbit 3200.bd
Analytic conductor $1.597$
Analytic rank $0$
Dimension $4$
Projective image $D_{10}$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3200.bd (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.80000000000000000.19

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{10}^{4} q^{5} - \zeta_{10}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{10}^{4} q^{5} - \zeta_{10}^{2} q^{9} + (\zeta_{10}^{3} - \zeta_{10}) q^{13} + ( - \zeta_{10}^{2} + \zeta_{10}) q^{17} - \zeta_{10}^{3} q^{25} + ( - \zeta_{10}^{4} - \zeta_{10}^{3}) q^{29} + ( - \zeta_{10}^{4} + 1) q^{37} + ( - \zeta_{10}^{3} - \zeta_{10}) q^{41} + \zeta_{10} q^{45} + q^{49} + (\zeta_{10}^{2} - 1) q^{53} + (\zeta_{10}^{4} - \zeta_{10}^{2}) q^{61} + ( - \zeta_{10}^{2} + 1) q^{65} + ( - \zeta_{10}^{4} - \zeta_{10}^{2}) q^{73} + \zeta_{10}^{4} q^{81} + (\zeta_{10} - 1) q^{85} + (\zeta_{10} - 1) q^{89} + ( - \zeta_{10}^{4} + \zeta_{10}^{3}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{5} + q^{9} + 2 q^{17} - q^{25} + 5 q^{37} - 2 q^{41} + q^{45} + 4 q^{49} - 5 q^{53} + 5 q^{65} + 2 q^{73} - q^{81} - 3 q^{85} - 3 q^{89} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{10}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−0.309017 0.951057i
0.809017 0.587785i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0 0 0 0.309017 0.951057i 0 0 0 0.809017 0.587785i 0
831.1 0 0 0 −0.809017 0.587785i 0 0 0 −0.309017 + 0.951057i 0
1471.1 0 0 0 −0.809017 + 0.587785i 0 0 0 −0.309017 0.951057i 0
2111.1 0 0 0 0.309017 + 0.951057i 0 0 0 0.809017 + 0.587785i 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 CM by \(\Q(\sqrt{-1}) \)
200.n odd 10 1 inner
200.t even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.1.bd.a 4
4.b odd 2 1 CM 3200.1.bd.a 4
8.b even 2 1 3200.1.bd.b yes 4
8.d odd 2 1 3200.1.bd.b yes 4
25.d even 5 1 3200.1.bd.b yes 4
100.j odd 10 1 3200.1.bd.b yes 4
200.n odd 10 1 inner 3200.1.bd.a 4
200.t even 10 1 inner 3200.1.bd.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.1.bd.a 4 1.a even 1 1 trivial
3200.1.bd.a 4 4.b odd 2 1 CM
3200.1.bd.a 4 200.n odd 10 1 inner
3200.1.bd.a 4 200.t even 10 1 inner
3200.1.bd.b yes 4 8.b even 2 1
3200.1.bd.b yes 4 8.d odd 2 1
3200.1.bd.b yes 4 25.d even 5 1
3200.1.bd.b yes 4 100.j odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{4} + 5T_{13} + 5 \) acting on \(S_{1}^{\mathrm{new}}(3200, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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