Properties

Label 2025.2.a.z
Level $2025$
Weight $2$
Character orbit 2025.a
Self dual yes
Analytic conductor $16.170$
Analytic rank $0$
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: \(0\)
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} - 3 \beta_1 + 4) q^{16} + (\beta_{3} + \beta_{2} + 3) q^{17} + ( - \beta_{2} - 2 \beta_1 + 1) q^{19} + (\beta_{3} + 2 \beta_1 - 2) q^{22} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{23} + (2 \beta_{3} - \beta_{2} + 3 \beta_1) q^{26} + (2 \beta_{2} - 2 \beta_1 + 3) q^{28} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{29} + (2 \beta_{3} + \beta_1 - 2) q^{31} + (3 \beta_{2} - 2 \beta_1 + 5) q^{32} + ( - \beta_{3} - 5 \beta_1 + 5) q^{34} + (\beta_{3} - 2 \beta_{2} + 4 \beta_1 - 3) q^{37} + (\beta_{3} + \beta_{2} + 6) q^{38} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 2) q^{41} + (2 \beta_{3} - 2 \beta_1 + 3) q^{43} + (2 \beta_{3} - \beta_{2} + \beta_1 - 7) q^{44} + (2 \beta_{3} - 4 \beta_{2} - \beta_1 - 2) q^{46} + ( - 2 \beta_{3} + \beta_{2} + 6) q^{47} + (\beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{49} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 6) q^{52} + ( - 3 \beta_{2} - \beta_1 + 4) q^{53} + ( - 3 \beta_1 + 9) q^{56} + ( - 2 \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 9) q^{58} + (3 \beta_{3} - 3 \beta_{2} + \beta_1 - 7) q^{59} + (\beta_{3} - 3 \beta_{2} + 2 \beta_1 - 6) q^{61} + ( - 3 \beta_{2} - 3) q^{62} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 6) q^{64} + ( - \beta_{3} + \beta_{2} + 3 \beta_1 - 5) q^{67} + ( - 2 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 13) q^{68} + ( - 3 \beta_{2} - \beta_1 + 1) q^{71} + ( - 2 \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{73} + (2 \beta_{3} - 7 \beta_{2} + 4 \beta_1 - 16) q^{74} + ( - \beta_{3} + 2 \beta_{2} - 4 \beta_1 + 6) q^{76} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 2) q^{77} + (\beta_{3} + 2 \beta_{2} - 1) q^{79} + ( - 2 \beta_{3} + 2 \beta_{2} - 3 \beta_1) q^{82} + (3 \beta_1 + 6) q^{83} + ( - 5 \beta_1 + 11) q^{86} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 5) q^{88} + ( - 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 6) q^{89} + ( - 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{91} + (2 \beta_{3} - \beta_{2} + 2 \beta_1 - 5) q^{92} + ( - \beta_{3} + 3 \beta_{2} - 5 \beta_1 + 5) q^{94} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 5) q^{97} + (3 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 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
2.63372
1.47325
−0.473255
−1.63372
−1.63372 0 0.669052 0 0 0.505348 2.17440 0 0
1.2 −0.473255 0 −1.77603 0 0 2.56305 1.78702 0 0
1.3 1.47325 0 0.170479 0 0 −3.86583 −2.69535 0 0
1.4 2.63372 0 4.93650 0 0 1.79743 7.73393 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.z 4
3.b odd 2 1 2025.2.a.q 4
5.b even 2 1 2025.2.a.p 4
5.c odd 4 2 2025.2.b.o 8
9.c even 3 2 675.2.e.c 8
9.d odd 6 2 225.2.e.e yes 8
15.d odd 2 1 2025.2.a.y 4
15.e even 4 2 2025.2.b.n 8
45.h odd 6 2 225.2.e.c 8
45.j even 6 2 675.2.e.e 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 45.h odd 6 2
225.2.e.e yes 8 9.d odd 6 2
225.2.k.c 16 45.l even 12 4
675.2.e.c 8 9.c even 3 2
675.2.e.e 8 45.j even 6 2
675.2.k.c 16 45.k odd 12 4
2025.2.a.p 4 5.b even 2 1
2025.2.a.q 4 3.b odd 2 1
2025.2.a.y 4 15.d odd 2 1
2025.2.a.z 4 1.a even 1 1 trivial
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

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} - 4 T^{2} + 5 T + 3 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - T^{3} - 12 T^{2} + 24 T - 9 \) Copy content Toggle raw display
$11$ \( T^{4} - T^{3} - 25 T^{2} - 41 T - 9 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} - 30 T^{2} - 5 T + 107 \) Copy content Toggle raw display
$17$ \( T^{4} - 11 T^{3} + 20 T^{2} + \cdots - 303 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} - 27 T^{2} + 80 T - 25 \) Copy content Toggle raw display
$23$ \( T^{4} - 15 T^{3} + 57 T^{2} + \cdots - 243 \) Copy content Toggle raw display
$29$ \( T^{4} + T^{3} - 40 T^{2} - 142 T - 129 \) Copy content Toggle raw display
$31$ \( T^{4} + 4 T^{3} - 42 T^{2} - 27 T + 243 \) Copy content Toggle raw display
$37$ \( T^{4} - T^{3} - 99 T^{2} + 503 T - 647 \) Copy content Toggle raw display
$41$ \( T^{4} - 5 T^{3} - 25 T^{2} + 161 T - 207 \) Copy content Toggle raw display
$43$ \( T^{4} - 10 T^{3} - 48 T^{2} + \cdots - 673 \) Copy content Toggle raw display
$47$ \( T^{4} - 20 T^{3} + 107 T^{2} + \cdots - 381 \) Copy content Toggle raw display
$53$ \( T^{4} - 20 T^{3} + 86 T^{2} + \cdots - 471 \) Copy content Toggle raw display
$59$ \( T^{4} + 17 T^{3} + 2 T^{2} + \cdots - 2313 \) Copy content Toggle raw display
$61$ \( T^{4} + 13 T^{3} - 3 T^{2} - 91 T - 1 \) Copy content Toggle raw display
$67$ \( T^{4} + 17 T^{3} + 36 T^{2} + \cdots + 243 \) Copy content Toggle raw display
$71$ \( T^{4} - 8 T^{3} - 40 T^{2} + 263 T + 381 \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{3} - 96 T^{2} + 241 T + 113 \) Copy content Toggle raw display
$79$ \( T^{4} + 7 T^{3} - 33 T^{2} - 69 T + 207 \) Copy content Toggle raw display
$83$ \( T^{4} - 30 T^{3} + 288 T^{2} + \cdots + 729 \) Copy content Toggle raw display
$89$ \( T^{4} - 9 T^{3} - 99 T^{2} + \cdots + 2025 \) Copy content Toggle raw display
$97$ \( T^{4} - 19 T^{3} + 81 T^{2} + \cdots - 953 \) Copy content Toggle raw display
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