Properties

Label 2025.2.b.k
Level $2025$
Weight $2$
Character orbit 2025.b
Analytic conductor $16.170$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2025,2,Mod(649,2025)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2025.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2025 = 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2025.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(16.1697064093\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 81)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - q^{4} - 2 \beta_1 q^{7} + \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - q^{4} - 2 \beta_1 q^{7} + \beta_{2} q^{8} + 2 \beta_{3} q^{11} - \beta_1 q^{13} + 2 \beta_{3} q^{14} - 5 q^{16} + 3 \beta_{2} q^{17} - 2 q^{19} + 6 \beta_1 q^{22} - 2 \beta_{2} q^{23} + \beta_{3} q^{26} + 2 \beta_1 q^{28} + \beta_{3} q^{29} + 8 q^{31} - 3 \beta_{2} q^{32} - 9 q^{34} + 7 \beta_1 q^{37} - 2 \beta_{2} q^{38} + 4 \beta_{3} q^{41} + 2 \beta_1 q^{43} - 2 \beta_{3} q^{44} + 6 q^{46} + 4 \beta_{2} q^{47} + 3 q^{49} + \beta_1 q^{52} + 2 \beta_{3} q^{56} + 3 \beta_1 q^{58} + 8 \beta_{3} q^{59} - 7 q^{61} + 8 \beta_{2} q^{62} - q^{64} + 10 \beta_1 q^{67} - 3 \beta_{2} q^{68} + 6 \beta_{3} q^{71} - 7 \beta_1 q^{73} - 7 \beta_{3} q^{74} + 2 q^{76} - 4 \beta_{2} q^{77} - 2 q^{79} + 12 \beta_1 q^{82} + 8 \beta_{2} q^{83} - 2 \beta_{3} q^{86} + 6 \beta_1 q^{88} - 3 \beta_{3} q^{89} - 2 q^{91} + 2 \beta_{2} q^{92} - 12 q^{94} - 2 \beta_1 q^{97} + 3 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 20 q^{16} - 8 q^{19} + 32 q^{31} - 36 q^{34} + 24 q^{46} + 12 q^{49} - 28 q^{61} - 4 q^{64} + 8 q^{76} - 8 q^{79} - 8 q^{91} - 48 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2025\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(1702\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
−0.866025 + 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
1.73205i 0 −1.00000 0 0 2.00000i 1.73205i 0 0
649.2 1.73205i 0 −1.00000 0 0 2.00000i 1.73205i 0 0
649.3 1.73205i 0 −1.00000 0 0 2.00000i 1.73205i 0 0
649.4 1.73205i 0 −1.00000 0 0 2.00000i 1.73205i 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2025.2.b.k 4
3.b odd 2 1 inner 2025.2.b.k 4
5.b even 2 1 inner 2025.2.b.k 4
5.c odd 4 1 81.2.a.a 2
5.c odd 4 1 2025.2.a.j 2
15.d odd 2 1 inner 2025.2.b.k 4
15.e even 4 1 81.2.a.a 2
15.e even 4 1 2025.2.a.j 2
20.e even 4 1 1296.2.a.o 2
35.f even 4 1 3969.2.a.i 2
40.i odd 4 1 5184.2.a.br 2
40.k even 4 1 5184.2.a.bq 2
45.k odd 12 2 81.2.c.b 4
45.l even 12 2 81.2.c.b 4
55.e even 4 1 9801.2.a.v 2
60.l odd 4 1 1296.2.a.o 2
105.k odd 4 1 3969.2.a.i 2
120.q odd 4 1 5184.2.a.bq 2
120.w even 4 1 5184.2.a.br 2
135.q even 36 6 729.2.e.o 12
135.r odd 36 6 729.2.e.o 12
165.l odd 4 1 9801.2.a.v 2
180.v odd 12 2 1296.2.i.s 4
180.x even 12 2 1296.2.i.s 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.2.a.a 2 5.c odd 4 1
81.2.a.a 2 15.e even 4 1
81.2.c.b 4 45.k odd 12 2
81.2.c.b 4 45.l even 12 2
729.2.e.o 12 135.q even 36 6
729.2.e.o 12 135.r odd 36 6
1296.2.a.o 2 20.e even 4 1
1296.2.a.o 2 60.l odd 4 1
1296.2.i.s 4 180.v odd 12 2
1296.2.i.s 4 180.x even 12 2
2025.2.a.j 2 5.c odd 4 1
2025.2.a.j 2 15.e even 4 1
2025.2.b.k 4 1.a even 1 1 trivial
2025.2.b.k 4 3.b odd 2 1 inner
2025.2.b.k 4 5.b even 2 1 inner
2025.2.b.k 4 15.d odd 2 1 inner
3969.2.a.i 2 35.f even 4 1
3969.2.a.i 2 105.k odd 4 1
5184.2.a.bq 2 40.k even 4 1
5184.2.a.bq 2 120.q odd 4 1
5184.2.a.br 2 40.i odd 4 1
5184.2.a.br 2 120.w even 4 1
9801.2.a.v 2 55.e even 4 1
9801.2.a.v 2 165.l odd 4 1

Hecke kernels

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

\( T_{2}^{2} + 3 \) Copy content Toggle raw display
\( T_{11}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$19$ \( (T + 2)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$61$ \( (T + 7)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$79$ \( (T + 2)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
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