Properties

Label 3200.1.g.e
Level $3200$
Weight $1$
Character orbit 3200.g
Analytic conductor $1.597$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -20
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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 640)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.1600.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{3} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{7} + q^{9} +O(q^{10})\) \( q + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{3} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{7} + q^{9} + 2 \zeta_{8}^{2} q^{21} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{23} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{43} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{47} - q^{49} -2 \zeta_{8}^{2} q^{61} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{63} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{67} + 2 \zeta_{8}^{2} q^{69} - q^{81} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{83} -2 q^{89} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{9} - 4 q^{49} - 4 q^{81} - 8 q^{89} + O(q^{100}) \)

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
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i
0 −1.41421 0 0 0 1.41421i 0 1.00000 0
2751.2 0 −1.41421 0 0 0 1.41421i 0 1.00000 0
2751.3 0 1.41421 0 0 0 1.41421i 0 1.00000 0
2751.4 0 1.41421 0 0 0 1.41421i 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
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 3200.1.g.e 4
4.b odd 2 1 inner 3200.1.g.e 4
5.b even 2 1 inner 3200.1.g.e 4
5.c odd 4 2 640.1.e.c 4
8.b even 2 1 inner 3200.1.g.e 4
8.d odd 2 1 inner 3200.1.g.e 4
20.d odd 2 1 CM 3200.1.g.e 4
20.e even 4 2 640.1.e.c 4
40.e odd 2 1 inner 3200.1.g.e 4
40.f even 2 1 inner 3200.1.g.e 4
40.i odd 4 2 640.1.e.c 4
40.k even 4 2 640.1.e.c 4
80.i odd 4 1 1280.1.h.a 2
80.i odd 4 1 1280.1.h.c 2
80.j even 4 1 1280.1.h.a 2
80.j even 4 1 1280.1.h.c 2
80.s even 4 1 1280.1.h.a 2
80.s even 4 1 1280.1.h.c 2
80.t odd 4 1 1280.1.h.a 2
80.t odd 4 1 1280.1.h.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.1.e.c 4 5.c odd 4 2
640.1.e.c 4 20.e even 4 2
640.1.e.c 4 40.i odd 4 2
640.1.e.c 4 40.k even 4 2
1280.1.h.a 2 80.i odd 4 1
1280.1.h.a 2 80.j even 4 1
1280.1.h.a 2 80.s even 4 1
1280.1.h.a 2 80.t odd 4 1
1280.1.h.c 2 80.i odd 4 1
1280.1.h.c 2 80.j even 4 1
1280.1.h.c 2 80.s even 4 1
1280.1.h.c 2 80.t odd 4 1
3200.1.g.e 4 1.a even 1 1 trivial
3200.1.g.e 4 4.b odd 2 1 inner
3200.1.g.e 4 5.b even 2 1 inner
3200.1.g.e 4 8.b even 2 1 inner
3200.1.g.e 4 8.d odd 2 1 inner
3200.1.g.e 4 20.d odd 2 1 CM
3200.1.g.e 4 40.e odd 2 1 inner
3200.1.g.e 4 40.f even 2 1 inner

Hecke kernels

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

\( T_{3}^{2} - 2 \)
\( T_{13} \)
\( T_{17} \)

Hecke characteristic polynomials

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