Properties

Label 4000.2.f.a
Level $4000$
Weight $2$
Character orbit 4000.f
Analytic conductor $31.940$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4000,2,Mod(3249,4000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4000.3249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4000.f (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: \(31.9401608085\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1000)
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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - 2) q^{3} + (2 \beta_{3} + 3 \beta_1) q^{7} + (4 \beta_{2} + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - 2) q^{3} + (2 \beta_{3} + 3 \beta_1) q^{7} + (4 \beta_{2} + 5) q^{9} + (3 \beta_{3} + \beta_1) q^{11} + (\beta_{2} - 1) q^{13} + 2 \beta_{3} q^{17} + ( - 4 \beta_{3} - 5 \beta_1) q^{19} + ( - 10 \beta_{3} - 4 \beta_1) q^{21} + ( - 3 \beta_{3} - 3 \beta_1) q^{23} + ( - 4 \beta_{2} - 12) q^{27} + ( - 2 \beta_{3} - 2 \beta_1) q^{29} + ( - 2 \beta_{2} + 2) q^{31} + ( - 8 \beta_{3} - 6 \beta_1) q^{33} + ( - \beta_{2} - 6) q^{37} + 2 \beta_{2} q^{39} + (3 \beta_{2} - 4) q^{41} + 2 \beta_{2} q^{43} + ( - \beta_{3} + 7 \beta_1) q^{47} + ( - 3 \beta_{2} - 6) q^{49} + ( - 4 \beta_{3} - 4 \beta_1) q^{51} + (\beta_{2} - 6) q^{53} + (18 \beta_{3} + 8 \beta_1) q^{57} - 7 \beta_1 q^{59} + ( - 4 \beta_{3} + 2 \beta_1) q^{61} + (22 \beta_{3} + 11 \beta_1) q^{63} + 8 q^{67} + (12 \beta_{3} + 6 \beta_1) q^{69} + (4 \beta_{2} + 10) q^{71} - 8 \beta_1 q^{73} + ( - 8 \beta_{2} - 9) q^{77} + (2 \beta_{2} - 14) q^{79} + (12 \beta_{2} + 17) q^{81} + (6 \beta_{2} + 2) q^{83} + (8 \beta_{3} + 4 \beta_1) q^{87} + (9 \beta_{2} - 3) q^{89} + (\beta_{3} - 4 \beta_1) q^{91} - 4 \beta_{2} q^{93} + ( - 4 \beta_{3} - 12 \beta_1) q^{97} + (19 \beta_{3} + 13 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 12 q^{9} - 6 q^{13} - 40 q^{27} + 12 q^{31} - 22 q^{37} - 4 q^{39} - 22 q^{41} - 4 q^{43} - 18 q^{49} - 26 q^{53} + 32 q^{67} + 32 q^{71} - 20 q^{77} - 60 q^{79} + 44 q^{81} - 4 q^{83} - 30 q^{89} + 8 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times\).

\(n\) \(1377\) \(2501\) \(2751\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3249.1
0.618034i
0.618034i
1.61803i
1.61803i
0 −3.23607 0 0 0 3.85410i 0 7.47214 0
3249.2 0 −3.23607 0 0 0 3.85410i 0 7.47214 0
3249.3 0 1.23607 0 0 0 2.85410i 0 −1.47214 0
3249.4 0 1.23607 0 0 0 2.85410i 0 −1.47214 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
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4000.2.f.a 4
4.b odd 2 1 1000.2.f.b 4
5.b even 2 1 4000.2.f.b 4
5.c odd 4 1 4000.2.d.a 4
5.c odd 4 1 4000.2.d.b 4
8.b even 2 1 4000.2.f.b 4
8.d odd 2 1 1000.2.f.a 4
20.d odd 2 1 1000.2.f.a 4
20.e even 4 1 1000.2.d.a 4
20.e even 4 1 1000.2.d.b yes 4
40.e odd 2 1 1000.2.f.b 4
40.f even 2 1 inner 4000.2.f.a 4
40.i odd 4 1 4000.2.d.a 4
40.i odd 4 1 4000.2.d.b 4
40.k even 4 1 1000.2.d.a 4
40.k even 4 1 1000.2.d.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.2.d.a 4 20.e even 4 1
1000.2.d.a 4 40.k even 4 1
1000.2.d.b yes 4 20.e even 4 1
1000.2.d.b yes 4 40.k even 4 1
1000.2.f.a 4 8.d odd 2 1
1000.2.f.a 4 20.d odd 2 1
1000.2.f.b 4 4.b odd 2 1
1000.2.f.b 4 40.e odd 2 1
4000.2.d.a 4 5.c odd 4 1
4000.2.d.a 4 40.i odd 4 1
4000.2.d.b 4 5.c odd 4 1
4000.2.d.b 4 40.i odd 4 1
4000.2.f.a 4 1.a even 1 1 trivial
4000.2.f.a 4 40.f even 2 1 inner
4000.2.f.b 4 5.b even 2 1
4000.2.f.b 4 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 2T_{3} - 4 \) acting on \(S_{2}^{\mathrm{new}}(4000, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 23T^{2} + 121 \) Copy content Toggle raw display
$11$ \( T^{4} + 15T^{2} + 25 \) Copy content Toggle raw display
$13$ \( (T^{2} + 3 T + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 67T^{2} + 841 \) Copy content Toggle raw display
$23$ \( T^{4} + 27T^{2} + 81 \) Copy content Toggle raw display
$29$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$31$ \( (T^{2} - 6 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 11 T + 29)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 11 T + 19)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 163T^{2} + 1681 \) Copy content Toggle raw display
$53$ \( (T^{2} + 13 T + 41)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 147T^{2} + 2401 \) Copy content Toggle raw display
$61$ \( T^{4} + 60T^{2} + 400 \) Copy content Toggle raw display
$67$ \( (T - 8)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 16 T + 44)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 192T^{2} + 4096 \) Copy content Toggle raw display
$79$ \( (T^{2} + 30 T + 220)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 2 T - 44)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 15 T - 45)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 368 T^{2} + 30976 \) Copy content Toggle raw display
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