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Properties

Label 2600.2.k.a
Level $2600$
Weight $2$
Character orbit 2600.k
Analytic conductor $20.761$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2600,2,Mod(2001,2600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("2600.2001");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2600.k (of order \(2\), degree \(1\), 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: \(20.7611045255\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 104)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{3} + (\beta_{2} - \beta_1) q^{7} + ( - \beta_{3} + 2) q^{9} + ( - \beta_{2} - 2 \beta_1) q^{11} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{13} + (\beta_{3} + 1) q^{17} - \beta_{2} q^{19}+ \cdots + (3 \beta_{2} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 6 q^{9} + 6 q^{13} + 6 q^{17} - 4 q^{23} + 14 q^{27} - 4 q^{29} + 20 q^{39} - 30 q^{43} - 14 q^{49} - 14 q^{51} - 16 q^{53} + 28 q^{61} - 36 q^{69} - 24 q^{77} - 28 q^{79} - 28 q^{81} + 32 q^{87}+ \cdots - 2 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} - 5\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times\).

\(n\) \(1301\) \(1601\) \(1951\) \(1977\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2001.1
− 1.56155i
1.56155i
2.56155i
− 2.56155i
0 −1.56155 0 0 0 − 0.438447i 0 −0.561553 0
2001.2 0 −1.56155 0 0 0 0.438447i 0 −0.561553 0
2001.3 0 2.56155 0 0 0 − 4.56155i 0 3.56155 0
2001.4 0 2.56155 0 0 0 4.56155i 0 3.56155 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2600.2.k.a 4
5.b even 2 1 104.2.f.a ✓ 4
5.c odd 4 1 2600.2.f.a 4
5.c odd 4 1 2600.2.f.b 4
13.b even 2 1 inner 2600.2.k.a 4
15.d odd 2 1 936.2.c.d 4
20.d odd 2 1 208.2.f.b 4
40.e odd 2 1 832.2.f.f 4
40.f even 2 1 832.2.f.i 4
60.h even 2 1 1872.2.c.j 4
65.d even 2 1 104.2.f.a ✓ 4
65.g odd 4 1 1352.2.a.d 2
65.g odd 4 1 1352.2.a.e 2
65.h odd 4 1 2600.2.f.a 4
65.h odd 4 1 2600.2.f.b 4
65.l even 6 2 1352.2.o.e 8
65.n even 6 2 1352.2.o.e 8
65.s odd 12 2 1352.2.i.g 4
65.s odd 12 2 1352.2.i.h 4
195.e odd 2 1 936.2.c.d 4
260.g odd 2 1 208.2.f.b 4
260.u even 4 1 2704.2.a.s 2
260.u even 4 1 2704.2.a.t 2
520.b odd 2 1 832.2.f.f 4
520.p even 2 1 832.2.f.i 4
780.d even 2 1 1872.2.c.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.f.a ✓ 4 5.b even 2 1
104.2.f.a ✓ 4 65.d even 2 1
208.2.f.b 4 20.d odd 2 1
208.2.f.b 4 260.g odd 2 1
832.2.f.f 4 40.e odd 2 1
832.2.f.f 4 520.b odd 2 1
832.2.f.i 4 40.f even 2 1
832.2.f.i 4 520.p even 2 1
936.2.c.d 4 15.d odd 2 1
936.2.c.d 4 195.e odd 2 1
1352.2.a.d 2 65.g odd 4 1
1352.2.a.e 2 65.g odd 4 1
1352.2.i.g 4 65.s odd 12 2
1352.2.i.h 4 65.s odd 12 2
1352.2.o.e 8 65.l even 6 2
1352.2.o.e 8 65.n even 6 2
1872.2.c.j 4 60.h even 2 1
1872.2.c.j 4 780.d even 2 1
2600.2.f.a 4 5.c odd 4 1
2600.2.f.a 4 65.h odd 4 1
2600.2.f.b 4 5.c odd 4 1
2600.2.f.b 4 65.h odd 4 1
2600.2.k.a 4 1.a even 1 1 trivial
2600.2.k.a 4 13.b even 2 1 inner
2704.2.a.s 2 260.u even 4 1
2704.2.a.t 2 260.u even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} - 4 \) acting on \(S_{2}^{\mathrm{new}}(2600, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T - 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 21T^{2} + 4 \) Copy content Toggle raw display
$11$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$13$ \( T^{4} - 6 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} - 3 T - 2)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 2 T - 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T - 16)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$37$ \( T^{4} + 121T^{2} + 2704 \) Copy content Toggle raw display
$41$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 15 T + 52)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 69T^{2} + 676 \) Copy content Toggle raw display
$53$ \( (T^{2} + 8 T - 52)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 14 T + 32)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 84T^{2} + 64 \) Copy content Toggle raw display
$71$ \( T^{4} + 77T^{2} + 1444 \) Copy content Toggle raw display
$73$ \( T^{4} + 324 T^{2} + 20736 \) Copy content Toggle raw display
$79$ \( (T^{2} + 14 T + 32)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 68)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$97$ \( T^{4} + 144T^{2} + 4096 \) Copy content Toggle raw display
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