Properties

Label 1600.4.a.bs
Level $1600$
Weight $4$
Character orbit 1600.a
Self dual yes
Analytic conductor $94.403$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,4,Mod(1,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1600.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(94.4030560092\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 7 q^{3} - 6 q^{7} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 7 q^{3} - 6 q^{7} + 22 q^{9} - 43 q^{11} + 28 q^{13} + 91 q^{17} - 35 q^{19} - 42 q^{21} - 162 q^{23} - 35 q^{27} - 160 q^{29} - 42 q^{31} - 301 q^{33} + 314 q^{37} + 196 q^{39} - 203 q^{41} + 92 q^{43} - 196 q^{47} - 307 q^{49} + 637 q^{51} - 82 q^{53} - 245 q^{57} - 280 q^{59} + 518 q^{61} - 132 q^{63} + 141 q^{67} - 1134 q^{69} - 412 q^{71} - 763 q^{73} + 258 q^{77} - 510 q^{79} - 839 q^{81} + 777 q^{83} - 1120 q^{87} - 945 q^{89} - 168 q^{91} - 294 q^{93} + 1246 q^{97} - 946 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 7.00000 0 0 0 −6.00000 0 22.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.4.a.bs 1
4.b odd 2 1 1600.4.a.i 1
5.b even 2 1 1600.4.a.h 1
8.b even 2 1 400.4.a.c 1
8.d odd 2 1 25.4.a.b yes 1
20.d odd 2 1 1600.4.a.bt 1
24.f even 2 1 225.4.a.c 1
40.e odd 2 1 25.4.a.a 1
40.f even 2 1 400.4.a.s 1
40.i odd 4 2 400.4.c.e 2
40.k even 4 2 25.4.b.b 2
56.e even 2 1 1225.4.a.i 1
120.m even 2 1 225.4.a.e 1
120.q odd 4 2 225.4.b.f 2
280.n even 2 1 1225.4.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 40.e odd 2 1
25.4.a.b yes 1 8.d odd 2 1
25.4.b.b 2 40.k even 4 2
225.4.a.c 1 24.f even 2 1
225.4.a.e 1 120.m even 2 1
225.4.b.f 2 120.q odd 4 2
400.4.a.c 1 8.b even 2 1
400.4.a.s 1 40.f even 2 1
400.4.c.e 2 40.i odd 4 2
1225.4.a.h 1 280.n even 2 1
1225.4.a.i 1 56.e even 2 1
1600.4.a.h 1 5.b even 2 1
1600.4.a.i 1 4.b odd 2 1
1600.4.a.bs 1 1.a even 1 1 trivial
1600.4.a.bt 1 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1600))\):

\( T_{3} - 7 \) Copy content Toggle raw display
\( T_{7} + 6 \) Copy content Toggle raw display
\( T_{11} + 43 \) Copy content Toggle raw display
\( T_{13} - 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 7 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 6 \) Copy content Toggle raw display
$11$ \( T + 43 \) Copy content Toggle raw display
$13$ \( T - 28 \) Copy content Toggle raw display
$17$ \( T - 91 \) Copy content Toggle raw display
$19$ \( T + 35 \) Copy content Toggle raw display
$23$ \( T + 162 \) Copy content Toggle raw display
$29$ \( T + 160 \) Copy content Toggle raw display
$31$ \( T + 42 \) Copy content Toggle raw display
$37$ \( T - 314 \) Copy content Toggle raw display
$41$ \( T + 203 \) Copy content Toggle raw display
$43$ \( T - 92 \) Copy content Toggle raw display
$47$ \( T + 196 \) Copy content Toggle raw display
$53$ \( T + 82 \) Copy content Toggle raw display
$59$ \( T + 280 \) Copy content Toggle raw display
$61$ \( T - 518 \) Copy content Toggle raw display
$67$ \( T - 141 \) Copy content Toggle raw display
$71$ \( T + 412 \) Copy content Toggle raw display
$73$ \( T + 763 \) Copy content Toggle raw display
$79$ \( T + 510 \) Copy content Toggle raw display
$83$ \( T - 777 \) Copy content Toggle raw display
$89$ \( T + 945 \) Copy content Toggle raw display
$97$ \( T - 1246 \) Copy content Toggle raw display
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