Properties

Label 1600.4.a.by
Level $1600$
Weight $4$
Character orbit 1600.a
Self dual yes
Analytic conductor $94.403$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 200)
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)
 
\(f(q)\) \(=\) \( q + 9 q^{3} - 26 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} - 26 q^{7} + 54 q^{9} - 59 q^{11} - 28 q^{13} + 5 q^{17} + 109 q^{19} - 234 q^{21} + 194 q^{23} + 243 q^{27} + 32 q^{29} - 10 q^{31} - 531 q^{33} + 198 q^{37} - 252 q^{39} + 117 q^{41} + 388 q^{43} + 68 q^{47} + 333 q^{49} + 45 q^{51} + 18 q^{53} + 981 q^{57} + 392 q^{59} + 710 q^{61} - 1404 q^{63} - 253 q^{67} + 1746 q^{69} + 612 q^{71} - 549 q^{73} + 1534 q^{77} - 414 q^{79} + 729 q^{81} - 121 q^{83} + 288 q^{87} - 81 q^{89} + 728 q^{91} - 90 q^{93} - 1502 q^{97} - 3186 q^{99}+O(q^{100}) \) 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 9.00000 0 0 0 −26.0000 0 54.0000 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.by 1
4.b odd 2 1 1600.4.a.c 1
5.b even 2 1 1600.4.a.b 1
8.b even 2 1 400.4.a.a 1
8.d odd 2 1 200.4.a.j yes 1
20.d odd 2 1 1600.4.a.bz 1
24.f even 2 1 1800.4.a.bh 1
40.e odd 2 1 200.4.a.b 1
40.f even 2 1 400.4.a.t 1
40.i odd 4 2 400.4.c.b 2
40.k even 4 2 200.4.c.b 2
120.m even 2 1 1800.4.a.c 1
120.q odd 4 2 1800.4.f.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.4.a.b 1 40.e odd 2 1
200.4.a.j yes 1 8.d odd 2 1
200.4.c.b 2 40.k even 4 2
400.4.a.a 1 8.b even 2 1
400.4.a.t 1 40.f even 2 1
400.4.c.b 2 40.i odd 4 2
1600.4.a.b 1 5.b even 2 1
1600.4.a.c 1 4.b odd 2 1
1600.4.a.by 1 1.a even 1 1 trivial
1600.4.a.bz 1 20.d odd 2 1
1800.4.a.c 1 120.m even 2 1
1800.4.a.bh 1 24.f even 2 1
1800.4.f.w 2 120.q odd 4 2

Hecke kernels

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

\( T_{3} - 9 \) Copy content Toggle raw display
\( T_{7} + 26 \) Copy content Toggle raw display
\( T_{11} + 59 \) 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
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 26 \) Copy content Toggle raw display
$11$ \( T + 59 \) Copy content Toggle raw display
$13$ \( T + 28 \) Copy content Toggle raw display
$17$ \( T - 5 \) Copy content Toggle raw display
$19$ \( T - 109 \) Copy content Toggle raw display
$23$ \( T - 194 \) Copy content Toggle raw display
$29$ \( T - 32 \) Copy content Toggle raw display
$31$ \( T + 10 \) Copy content Toggle raw display
$37$ \( T - 198 \) Copy content Toggle raw display
$41$ \( T - 117 \) Copy content Toggle raw display
$43$ \( T - 388 \) Copy content Toggle raw display
$47$ \( T - 68 \) Copy content Toggle raw display
$53$ \( T - 18 \) Copy content Toggle raw display
$59$ \( T - 392 \) Copy content Toggle raw display
$61$ \( T - 710 \) Copy content Toggle raw display
$67$ \( T + 253 \) Copy content Toggle raw display
$71$ \( T - 612 \) Copy content Toggle raw display
$73$ \( T + 549 \) Copy content Toggle raw display
$79$ \( T + 414 \) Copy content Toggle raw display
$83$ \( T + 121 \) Copy content Toggle raw display
$89$ \( T + 81 \) Copy content Toggle raw display
$97$ \( T + 1502 \) Copy content Toggle raw display
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