Properties

Label 3200.1.g.b
Level $3200$
Weight $1$
Character orbit 3200.g
Analytic conductor $1.597$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -4, -40, 40
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3200.g (of order \(2\), degree \(1\), 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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 640)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(i, \sqrt{10})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.4096000000.2

$q$-expansion

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

\(f(q)\) \(=\) \( q - q^{9} +O(q^{10})\) \( q - q^{9} -2 i q^{13} -2 i q^{37} -2 q^{41} + q^{49} -2 i q^{53} + q^{81} + 2 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{9} - 4 q^{41} + 2 q^{49} + 2 q^{81} + 4 q^{89} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2751.1
1.00000i
1.00000i
0 0 0 0 0 0 0 −1.00000 0
2751.2 0 0 0 0 0 0 0 −1.00000 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 CM by \(\Q(\sqrt{-1}) \)
40.e odd 2 1 CM by \(\Q(\sqrt{-10}) \)
40.f even 2 1 RM by \(\Q(\sqrt{10}) \)
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner

Twists

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

Hecke kernels

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

\( T_{3} \)
\( T_{13}^{2} + 4 \)
\( T_{17} \)

Hecke characteristic polynomials

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