Properties

Label 3200.1.m.a
Level $3200$
Weight $1$
Character orbit 3200.m
Analytic conductor $1.597$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
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.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.59700804043\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 640)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.32000.1
Artin image: $C_4\wr C_2$
Artin field: Galois closure of 8.0.131072000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q -i q^{9} +O(q^{10})\) \( q -i q^{9} + ( -1 - i ) q^{13} + ( -1 - i ) q^{17} -2 q^{29} + ( -1 + i ) q^{37} -i q^{49} + ( 1 + i ) q^{53} -2 i q^{61} + ( 1 - i ) q^{73} - q^{81} + ( 1 + i ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + O(q^{10}) \) \( 2 q - 2 q^{13} - 2 q^{17} - 4 q^{29} - 2 q^{37} + 2 q^{53} + 2 q^{73} - 2 q^{81} + 2 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(i\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
193.1
1.00000i
1.00000i
0 0 0 0 0 0 0 1.00000i 0
1857.1 0 0 0 0 0 0 0 1.00000i 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 \(\Q(\sqrt{-1}) \)
40.i odd 4 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.1.m.a 2
4.b odd 2 1 CM 3200.1.m.a 2
5.b even 2 1 640.1.m.b yes 2
5.c odd 4 1 640.1.m.a 2
5.c odd 4 1 3200.1.m.b 2
8.b even 2 1 3200.1.m.b 2
8.d odd 2 1 3200.1.m.b 2
20.d odd 2 1 640.1.m.b yes 2
20.e even 4 1 640.1.m.a 2
20.e even 4 1 3200.1.m.b 2
40.e odd 2 1 640.1.m.a 2
40.f even 2 1 640.1.m.a 2
40.i odd 4 1 640.1.m.b yes 2
40.i odd 4 1 inner 3200.1.m.a 2
40.k even 4 1 640.1.m.b yes 2
40.k even 4 1 inner 3200.1.m.a 2
80.i odd 4 1 1280.1.p.a 2
80.j even 4 1 1280.1.p.b 2
80.k odd 4 1 1280.1.p.a 2
80.k odd 4 1 1280.1.p.b 2
80.q even 4 1 1280.1.p.a 2
80.q even 4 1 1280.1.p.b 2
80.s even 4 1 1280.1.p.a 2
80.t odd 4 1 1280.1.p.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.1.m.a 2 5.c odd 4 1
640.1.m.a 2 20.e even 4 1
640.1.m.a 2 40.e odd 2 1
640.1.m.a 2 40.f even 2 1
640.1.m.b yes 2 5.b even 2 1
640.1.m.b yes 2 20.d odd 2 1
640.1.m.b yes 2 40.i odd 4 1
640.1.m.b yes 2 40.k even 4 1
1280.1.p.a 2 80.i odd 4 1
1280.1.p.a 2 80.k odd 4 1
1280.1.p.a 2 80.q even 4 1
1280.1.p.a 2 80.s even 4 1
1280.1.p.b 2 80.j even 4 1
1280.1.p.b 2 80.k odd 4 1
1280.1.p.b 2 80.q even 4 1
1280.1.p.b 2 80.t odd 4 1
3200.1.m.a 2 1.a even 1 1 trivial
3200.1.m.a 2 4.b odd 2 1 CM
3200.1.m.a 2 40.i odd 4 1 inner
3200.1.m.a 2 40.k even 4 1 inner
3200.1.m.b 2 5.c odd 4 1
3200.1.m.b 2 8.b even 2 1
3200.1.m.b 2 8.d odd 2 1
3200.1.m.b 2 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3200, [\chi])\):

\( T_{3} \)
\( T_{7} \)
\( T_{13}^{2} + 2 T_{13} + 2 \)

Hecke characteristic polynomials

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