Properties

Label 900.4.a.f
Level $900$
Weight $4$
Character orbit 900.a
Self dual yes
Analytic conductor $53.102$
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] = [900,4,Mod(1,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, 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("900.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 900.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: \(53.1017190052\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
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)
 
\(f(q)\) \(=\) \( q - 13 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 13 q^{7} - 6 q^{11} + 5 q^{13} + 78 q^{17} + 65 q^{19} - 138 q^{23} - 66 q^{29} + 299 q^{31} - 214 q^{37} - 360 q^{41} + 203 q^{43} - 78 q^{47} - 174 q^{49} - 636 q^{53} - 786 q^{59} + 467 q^{61} - 217 q^{67} + 360 q^{71} - 286 q^{73} + 78 q^{77} + 272 q^{79} - 498 q^{83} - 65 q^{91} - 511 q^{97}+O(q^{100}) \) 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 0 0 0 0 −13.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(5\) \( -1 \)

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 900.4.a.f 1
3.b odd 2 1 300.4.a.a 1
5.b even 2 1 900.4.a.l 1
5.c odd 4 2 900.4.d.e 2
12.b even 2 1 1200.4.a.bi 1
15.d odd 2 1 300.4.a.h yes 1
15.e even 4 2 300.4.d.c 2
60.h even 2 1 1200.4.a.d 1
60.l odd 4 2 1200.4.f.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.4.a.a 1 3.b odd 2 1
300.4.a.h yes 1 15.d odd 2 1
300.4.d.c 2 15.e even 4 2
900.4.a.f 1 1.a even 1 1 trivial
900.4.a.l 1 5.b even 2 1
900.4.d.e 2 5.c odd 4 2
1200.4.a.d 1 60.h even 2 1
1200.4.a.bi 1 12.b even 2 1
1200.4.f.k 2 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(900))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 13 \) Copy content Toggle raw display
$11$ \( T + 6 \) Copy content Toggle raw display
$13$ \( T - 5 \) Copy content Toggle raw display
$17$ \( T - 78 \) Copy content Toggle raw display
$19$ \( T - 65 \) Copy content Toggle raw display
$23$ \( T + 138 \) Copy content Toggle raw display
$29$ \( T + 66 \) Copy content Toggle raw display
$31$ \( T - 299 \) Copy content Toggle raw display
$37$ \( T + 214 \) Copy content Toggle raw display
$41$ \( T + 360 \) Copy content Toggle raw display
$43$ \( T - 203 \) Copy content Toggle raw display
$47$ \( T + 78 \) Copy content Toggle raw display
$53$ \( T + 636 \) Copy content Toggle raw display
$59$ \( T + 786 \) Copy content Toggle raw display
$61$ \( T - 467 \) Copy content Toggle raw display
$67$ \( T + 217 \) Copy content Toggle raw display
$71$ \( T - 360 \) Copy content Toggle raw display
$73$ \( T + 286 \) Copy content Toggle raw display
$79$ \( T - 272 \) Copy content Toggle raw display
$83$ \( T + 498 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 511 \) Copy content Toggle raw display
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