Properties

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

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Newspace parameters

Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3200.e (of order \(2\), degree \(1\), not 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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 128)
Projective image \(D_{2}\)
Projective field Galois closure of \(\Q(\zeta_{8})\)
Artin image $D_4:C_2$
Artin field Galois closure of 8.0.4096000000.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + q^{9} +O(q^{10})\) \( q + q^{9} -2 i q^{17} + 2 q^{41} - q^{49} + 2 i q^{73} + q^{81} + 2 q^{89} -2 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{9} + 4q^{41} - 2q^{49} + 2q^{81} + 4q^{89} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1599.1
1.00000i
1.00000i
0 0 0 0 0 0 0 1.00000 0
1599.2 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 \(\Q(\sqrt{-1}) \)
8.b even 2 1 RM by \(\Q(\sqrt{2}) \)
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
5.b even 2 1 inner
20.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 3200.1.e.a 2
4.b odd 2 1 CM 3200.1.e.a 2
5.b even 2 1 inner 3200.1.e.a 2
5.c odd 4 1 128.1.d.a 1
5.c odd 4 1 3200.1.g.a 1
8.b even 2 1 RM 3200.1.e.a 2
8.d odd 2 1 CM 3200.1.e.a 2
15.e even 4 1 1152.1.b.a 1
20.d odd 2 1 inner 3200.1.e.a 2
20.e even 4 1 128.1.d.a 1
20.e even 4 1 3200.1.g.a 1
40.e odd 2 1 inner 3200.1.e.a 2
40.f even 2 1 inner 3200.1.e.a 2
40.i odd 4 1 128.1.d.a 1
40.i odd 4 1 3200.1.g.a 1
40.k even 4 1 128.1.d.a 1
40.k even 4 1 3200.1.g.a 1
60.l odd 4 1 1152.1.b.a 1
80.i odd 4 1 256.1.c.a 1
80.j even 4 1 256.1.c.a 1
80.s even 4 1 256.1.c.a 1
80.t odd 4 1 256.1.c.a 1
120.q odd 4 1 1152.1.b.a 1
120.w even 4 1 1152.1.b.a 1
160.u even 8 2 1024.1.f.b 2
160.v odd 8 2 1024.1.f.b 2
160.ba even 8 2 1024.1.f.b 2
160.bb odd 8 2 1024.1.f.b 2
240.z odd 4 1 2304.1.g.b 1
240.bb even 4 1 2304.1.g.b 1
240.bd odd 4 1 2304.1.g.b 1
240.bf even 4 1 2304.1.g.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.1.d.a 1 5.c odd 4 1
128.1.d.a 1 20.e even 4 1
128.1.d.a 1 40.i odd 4 1
128.1.d.a 1 40.k even 4 1
256.1.c.a 1 80.i odd 4 1
256.1.c.a 1 80.j even 4 1
256.1.c.a 1 80.s even 4 1
256.1.c.a 1 80.t odd 4 1
1024.1.f.b 2 160.u even 8 2
1024.1.f.b 2 160.v odd 8 2
1024.1.f.b 2 160.ba even 8 2
1024.1.f.b 2 160.bb odd 8 2
1152.1.b.a 1 15.e even 4 1
1152.1.b.a 1 60.l odd 4 1
1152.1.b.a 1 120.q odd 4 1
1152.1.b.a 1 120.w even 4 1
2304.1.g.b 1 240.z odd 4 1
2304.1.g.b 1 240.bb even 4 1
2304.1.g.b 1 240.bd odd 4 1
2304.1.g.b 1 240.bf even 4 1
3200.1.e.a 2 1.a even 1 1 trivial
3200.1.e.a 2 4.b odd 2 1 CM
3200.1.e.a 2 5.b even 2 1 inner
3200.1.e.a 2 8.b even 2 1 RM
3200.1.e.a 2 8.d odd 2 1 CM
3200.1.e.a 2 20.d odd 2 1 inner
3200.1.e.a 2 40.e odd 2 1 inner
3200.1.e.a 2 40.f even 2 1 inner
3200.1.g.a 1 5.c odd 4 1
3200.1.g.a 1 20.e even 4 1
3200.1.g.a 1 40.i odd 4 1
3200.1.g.a 1 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{1}^{\mathrm{new}}(3200, [\chi])\).

Hecke characteristic polynomials

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