Properties

Label 2025.2.a.p
Level $2025$
Weight $2$
Character orbit 2025.a
Self dual yes
Analytic conductor $16.170$
Analytic rank $1$
Dimension $4$
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] = [2025,2,Mod(1,2025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.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: \(16.1697064093\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.11661.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 4x^{2} + 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 225)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{3} - \beta_{2} - 1) q^{7} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 4) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{3} - \beta_{2} - 1) q^{7} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 4) q^{8} + ( - \beta_{3} - \beta_{2}) q^{11} + (2 \beta_{2} + \beta_1 + 1) q^{13} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{14} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 4) q^{16}+ \cdots + ( - 3 \beta_{3} + 3 \beta_{2} + \cdots + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 4 q^{4} - q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 4 q^{4} - q^{7} - 9 q^{8} + q^{11} + 2 q^{13} - 3 q^{14} + 4 q^{16} - 11 q^{17} + 2 q^{19} + 3 q^{22} - 15 q^{23} + 10 q^{26} - 4 q^{28} - q^{29} - 4 q^{31} - 10 q^{32} + 9 q^{34} - q^{37} - 23 q^{38} + 5 q^{41} - 10 q^{43} - 22 q^{44} - 20 q^{47} - 3 q^{49} + 17 q^{52} - 20 q^{53} + 30 q^{56} - 18 q^{58} - 17 q^{59} - 13 q^{61} + 6 q^{62} + 19 q^{64} + 17 q^{67} - 34 q^{68} + 8 q^{71} + 2 q^{73} - 40 q^{74} + 11 q^{76} - 12 q^{77} - 7 q^{79} + 12 q^{82} - 30 q^{83} + 34 q^{86} + 9 q^{88} + 9 q^{89} - 17 q^{91} + 12 q^{92} + 3 q^{94} - 19 q^{97} - 13 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 4x^{2} + 5x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 4\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 5\beta _1 + 2 \) 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
−1.63372
−0.473255
1.47325
2.63372
−2.63372 0 4.93650 0 0 −1.79743 −7.73393 0 0
1.2 −1.47325 0 0.170479 0 0 3.86583 2.69535 0 0
1.3 0.473255 0 −1.77603 0 0 −2.56305 −1.78702 0 0
1.4 1.63372 0 0.669052 0 0 −0.505348 −2.17440 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 2025.2.a.p 4
3.b odd 2 1 2025.2.a.y 4
5.b even 2 1 2025.2.a.z 4
5.c odd 4 2 2025.2.b.o 8
9.c even 3 2 675.2.e.e 8
9.d odd 6 2 225.2.e.c 8
15.d odd 2 1 2025.2.a.q 4
15.e even 4 2 2025.2.b.n 8
45.h odd 6 2 225.2.e.e yes 8
45.j even 6 2 675.2.e.c 8
45.k odd 12 4 675.2.k.c 16
45.l even 12 4 225.2.k.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.e.c 8 9.d odd 6 2
225.2.e.e yes 8 45.h odd 6 2
225.2.k.c 16 45.l even 12 4
675.2.e.c 8 45.j even 6 2
675.2.e.e 8 9.c even 3 2
675.2.k.c 16 45.k odd 12 4
2025.2.a.p 4 1.a even 1 1 trivial
2025.2.a.q 4 15.d odd 2 1
2025.2.a.y 4 3.b odd 2 1
2025.2.a.z 4 5.b even 2 1
2025.2.b.n 8 15.e even 4 2
2025.2.b.o 8 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2025))\):

\( T_{2}^{4} + 2T_{2}^{3} - 4T_{2}^{2} - 5T_{2} + 3 \) Copy content Toggle raw display
\( T_{7}^{4} + T_{7}^{3} - 12T_{7}^{2} - 24T_{7} - 9 \) Copy content Toggle raw display
\( T_{11}^{4} - T_{11}^{3} - 25T_{11}^{2} - 41T_{11} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + T^{3} - 12 T^{2} + \cdots - 9 \) Copy content Toggle raw display
$11$ \( T^{4} - T^{3} - 25 T^{2} + \cdots - 9 \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + \cdots + 107 \) Copy content Toggle raw display
$17$ \( T^{4} + 11 T^{3} + \cdots - 303 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} + \cdots - 25 \) Copy content Toggle raw display
$23$ \( T^{4} + 15 T^{3} + \cdots - 243 \) Copy content Toggle raw display
$29$ \( T^{4} + T^{3} + \cdots - 129 \) Copy content Toggle raw display
$31$ \( T^{4} + 4 T^{3} + \cdots + 243 \) Copy content Toggle raw display
$37$ \( T^{4} + T^{3} + \cdots - 647 \) Copy content Toggle raw display
$41$ \( T^{4} - 5 T^{3} + \cdots - 207 \) Copy content Toggle raw display
$43$ \( T^{4} + 10 T^{3} + \cdots - 673 \) Copy content Toggle raw display
$47$ \( T^{4} + 20 T^{3} + \cdots - 381 \) Copy content Toggle raw display
$53$ \( T^{4} + 20 T^{3} + \cdots - 471 \) Copy content Toggle raw display
$59$ \( T^{4} + 17 T^{3} + \cdots - 2313 \) Copy content Toggle raw display
$61$ \( T^{4} + 13 T^{3} + \cdots - 1 \) Copy content Toggle raw display
$67$ \( T^{4} - 17 T^{3} + \cdots + 243 \) Copy content Toggle raw display
$71$ \( T^{4} - 8 T^{3} + \cdots + 381 \) Copy content Toggle raw display
$73$ \( T^{4} - 2 T^{3} + \cdots + 113 \) Copy content Toggle raw display
$79$ \( T^{4} + 7 T^{3} + \cdots + 207 \) Copy content Toggle raw display
$83$ \( T^{4} + 30 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$89$ \( T^{4} - 9 T^{3} + \cdots + 2025 \) Copy content Toggle raw display
$97$ \( T^{4} + 19 T^{3} + \cdots - 953 \) Copy content Toggle raw display
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