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Properties

Label 2025.4.a.i
Level $2025$
Weight $4$
Character orbit 2025.a
Self dual yes
Analytic conductor $119.479$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2025,4,Mod(1,2025)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2025.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2025.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0,-8,0,0,0,12,0,0,-22,0,64,24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(119.478867762\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 405)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{2} + ( - 2 \beta - 4) q^{4} + 4 \beta q^{7} + ( - 10 \beta + 6) q^{8} + (28 \beta - 11) q^{11} + (4 \beta + 32) q^{13} + ( - 4 \beta + 12) q^{14} + (32 \beta - 4) q^{16} + (56 \beta - 16) q^{17}+ \cdots + ( - 295 \beta + 295) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 8 q^{4} + 12 q^{8} - 22 q^{11} + 64 q^{13} + 24 q^{14} - 8 q^{16} - 32 q^{17} - 10 q^{19} + 190 q^{22} + 84 q^{23} - 40 q^{26} - 48 q^{28} - 170 q^{29} - 258 q^{31} + 104 q^{32} + 368 q^{34}+ \cdots + 590 q^{98}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
−1.73205
1.73205
−2.73205 0 −0.535898 0 0 −6.92820 23.3205 0 0
1.2 0.732051 0 −7.46410 0 0 6.92820 −11.3205 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( +1 \)

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 2025.4.a.i 2
3.b odd 2 1 2025.4.a.m 2
5.b even 2 1 405.4.a.f yes 2
15.d odd 2 1 405.4.a.c ✓ 2
45.h odd 6 2 405.4.e.p 4
45.j even 6 2 405.4.e.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.4.a.c ✓ 2 15.d odd 2 1
405.4.a.f yes 2 5.b even 2 1
405.4.e.o 4 45.j even 6 2
405.4.e.p 4 45.h odd 6 2
2025.4.a.i 2 1.a even 1 1 trivial
2025.4.a.m 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2025))\):

\( T_{2}^{2} + 2T_{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 48 \) Copy content Toggle raw display
\( T_{11}^{2} + 22T_{11} - 2231 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T - 2 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 48 \) Copy content Toggle raw display
$11$ \( T^{2} + 22T - 2231 \) Copy content Toggle raw display
$13$ \( T^{2} - 64T + 976 \) Copy content Toggle raw display
$17$ \( T^{2} + 32T - 9152 \) Copy content Toggle raw display
$19$ \( T^{2} + 10T - 13847 \) Copy content Toggle raw display
$23$ \( T^{2} - 84T - 1308 \) Copy content Toggle raw display
$29$ \( T^{2} + 170T + 5497 \) Copy content Toggle raw display
$31$ \( T^{2} + 258T + 16209 \) Copy content Toggle raw display
$37$ \( T^{2} + 76T - 64268 \) Copy content Toggle raw display
$41$ \( T^{2} + 578T + 76609 \) Copy content Toggle raw display
$43$ \( T^{2} + 380T - 36908 \) Copy content Toggle raw display
$47$ \( T^{2} - 484T + 57796 \) Copy content Toggle raw display
$53$ \( T^{2} - 544T + 43984 \) Copy content Toggle raw display
$59$ \( T^{2} + 706T + 122257 \) Copy content Toggle raw display
$61$ \( T^{2} - 668T + 964 \) Copy content Toggle raw display
$67$ \( T^{2} - 1452 T + 518964 \) Copy content Toggle raw display
$71$ \( T^{2} + 974T + 121921 \) Copy content Toggle raw display
$73$ \( T^{2} + 1184 T + 325072 \) Copy content Toggle raw display
$79$ \( T^{2} - 408T + 9168 \) Copy content Toggle raw display
$83$ \( T^{2} - 444 T - 1030716 \) Copy content Toggle raw display
$89$ \( (T + 513)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 668T - 469244 \) Copy content Toggle raw display
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