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Properties

Label 2600.2.f.d
Level $2600$
Weight $2$
Character orbit 2600.f
Analytic conductor $20.761$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2600,2,Mod(649,2600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2600.649"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2600.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.7611045255\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) 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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + ( - \beta_{3} - \beta_1 + 1) q^{7} + (\beta_{3} + \beta_1) q^{9} + (2 \beta_{4} + \beta_{2}) q^{11} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3}) q^{13} + (2 \beta_{5} + 2 \beta_{4}) q^{17}+ \cdots + ( - 3 \beta_{5} - 4 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{7} + 2 q^{9} - 2 q^{13} + 12 q^{29} + 36 q^{33} - 16 q^{37} - 8 q^{39} + 12 q^{47} - 18 q^{49} + 24 q^{51} + 28 q^{57} - 36 q^{61} - 20 q^{63} + 20 q^{67} - 8 q^{69} - 24 q^{79} - 26 q^{81} + 28 q^{83}+ \cdots + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 8\nu^{4} - 4\nu^{3} - \nu^{2} + 2\nu + 38 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} + 19\nu^{2} - 38\nu + 14 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5\nu^{5} - 17\nu^{4} + 20\nu^{3} + 5\nu^{2} - 10\nu - 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 18\nu^{5} - 29\nu^{4} + 26\nu^{3} + 41\nu^{2} + 102\nu - 40 ) / 23 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{5} + 4\beta_{4} + 2\beta_{3} + \beta_{2} + 4\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{3} + 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{5} - 8\beta_{4} + 3\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.45161 + 1.45161i
0.403032 − 0.403032i
−0.854638 + 0.854638i
−0.854638 − 0.854638i
0.403032 + 0.403032i
1.45161 − 1.45161i
0 − 2.21432i 0 0 0 2.90321 0 −1.90321 0
649.2 0 − 1.67513i 0 0 0 0.806063 0 0.193937 0
649.3 0 − 0.539189i 0 0 0 −1.70928 0 2.70928 0
649.4 0 0.539189i 0 0 0 −1.70928 0 2.70928 0
649.5 0 1.67513i 0 0 0 0.806063 0 0.193937 0
649.6 0 2.21432i 0 0 0 2.90321 0 −1.90321 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.d 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.f.d 6
5.b even 2 1 2600.2.f.c 6
5.c odd 4 1 520.2.k.a ✓ 6
5.c odd 4 1 2600.2.k.b 6
13.b even 2 1 2600.2.f.c 6
15.e even 4 1 4680.2.g.j 6
20.e even 4 1 1040.2.k.b 6
65.d even 2 1 inner 2600.2.f.d 6
65.f even 4 1 6760.2.a.v 3
65.h odd 4 1 520.2.k.a ✓ 6
65.h odd 4 1 2600.2.k.b 6
65.k even 4 1 6760.2.a.u 3
195.s even 4 1 4680.2.g.j 6
260.p even 4 1 1040.2.k.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.k.a ✓ 6 5.c odd 4 1
520.2.k.a ✓ 6 65.h odd 4 1
1040.2.k.b 6 20.e even 4 1
1040.2.k.b 6 260.p even 4 1
2600.2.f.c 6 5.b even 2 1
2600.2.f.c 6 13.b even 2 1
2600.2.f.d 6 1.a even 1 1 trivial
2600.2.f.d 6 65.d even 2 1 inner
2600.2.k.b 6 5.c odd 4 1
2600.2.k.b 6 65.h odd 4 1
4680.2.g.j 6 15.e even 4 1
4680.2.g.j 6 195.s even 4 1
6760.2.a.u 3 65.k even 4 1
6760.2.a.v 3 65.f even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2600, [\chi])\):

\( T_{3}^{6} + 8T_{3}^{4} + 16T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{3} - 2T_{7}^{2} - 4T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 8 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( (T^{3} - 2 T^{2} - 4 T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{6} + 52 T^{4} + \cdots + 2116 \) Copy content Toggle raw display
$13$ \( T^{6} + 2 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( T^{6} + 48 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( T^{6} + 32 T^{4} + \cdots + 100 \) Copy content Toggle raw display
$23$ \( T^{6} + 60 T^{4} + \cdots + 676 \) Copy content Toggle raw display
$29$ \( (T^{3} - 6 T^{2} + \cdots + 428)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 172 T^{4} + \cdots + 13924 \) Copy content Toggle raw display
$37$ \( (T^{3} + 8 T^{2} + 12 T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 188 T^{4} + \cdots + 40000 \) Copy content Toggle raw display
$43$ \( T^{6} + 180 T^{4} + \cdots + 2916 \) Copy content Toggle raw display
$47$ \( (T^{3} - 6 T^{2} + \cdots + 108)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 188 T^{4} + \cdots + 40000 \) Copy content Toggle raw display
$59$ \( T^{6} + 8 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$61$ \( (T^{3} + 18 T^{2} + \cdots + 52)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 10 T^{2} + 24 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + 336 T^{4} + \cdots + 75076 \) Copy content Toggle raw display
$73$ \( (T^{3} - 100 T - 268)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 12 T^{2} + \cdots + 32)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 14 T^{2} + \cdots + 964)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 64 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$97$ \( (T^{3} - 2 T^{2} + \cdots + 680)^{2} \) Copy content Toggle raw display
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