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Properties

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

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2600,2,Mod(1249,2600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2600.1249"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2600.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-4,0,4,0,0,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.7611045255\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content 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 520)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{3} + 2 \beta_{2} q^{7} + (2 \beta_{3} - 1) q^{9} + (\beta_{3} + 1) q^{11} - \beta_1 q^{13} + (2 \beta_{2} + 4 \beta_1) q^{17} + (\beta_{3} + 1) q^{19} + ( - 2 \beta_{3} + 6) q^{21}+ \cdots + (\beta_{3} + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} + 4 q^{11} + 4 q^{19} + 24 q^{21} - 24 q^{29} + 20 q^{31} + 4 q^{39} + 16 q^{41} - 20 q^{49} + 8 q^{51} + 12 q^{59} - 8 q^{61} + 16 q^{69} + 44 q^{71} + 40 q^{79} + 4 q^{81} + 24 q^{89} + 20 q^{99}+O(q^{100}) \) 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

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) 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

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
−0.866025 − 0.500000i
0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0 − 2.73205i 0 0 0 3.46410i 0 −4.46410 0
1249.2 0 − 0.732051i 0 0 0 3.46410i 0 2.46410 0
1249.3 0 0.732051i 0 0 0 − 3.46410i 0 2.46410 0
1249.4 0 2.73205i 0 0 0 − 3.46410i 0 −4.46410 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
5.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.d.m 4
5.b even 2 1 inner 2600.2.d.m 4
5.c odd 4 1 520.2.a.e ✓ 2
5.c odd 4 1 2600.2.a.v 2
15.e even 4 1 4680.2.a.bd 2
20.e even 4 1 1040.2.a.l 2
20.e even 4 1 5200.2.a.bn 2
40.i odd 4 1 4160.2.a.bk 2
40.k even 4 1 4160.2.a.w 2
60.l odd 4 1 9360.2.a.cr 2
65.h odd 4 1 6760.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.a.e ✓ 2 5.c odd 4 1
1040.2.a.l 2 20.e even 4 1
2600.2.a.v 2 5.c odd 4 1
2600.2.d.m 4 1.a even 1 1 trivial
2600.2.d.m 4 5.b even 2 1 inner
4160.2.a.w 2 40.k even 4 1
4160.2.a.bk 2 40.i odd 4 1
4680.2.a.bd 2 15.e even 4 1
5200.2.a.bn 2 20.e even 4 1
6760.2.a.o 2 65.h odd 4 1
9360.2.a.cr 2 60.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 \(S_{2}^{\mathrm{new}}(2600, [\chi])\):

\( T_{3}^{4} + 8T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} + 12 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 2)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 56T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$29$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 10 T - 2)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 8 T + 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 168T^{2} + 6084 \) Copy content Toggle raw display
$47$ \( T^{4} + 152T^{2} + 2704 \) Copy content Toggle raw display
$53$ \( T^{4} + 56T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} - 6 T - 66)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T - 104)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$71$ \( (T^{2} - 22 T + 118)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$79$ \( (T^{2} - 20 T + 88)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 12 T - 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 392 T^{2} + 35344 \) Copy content Toggle raw display
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