Properties

Label 520.2.k.a
Level $520$
Weight $2$
Character orbit 520.k
Analytic conductor $4.152$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [520,2,Mod(441,520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(520, 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("520.441");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.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: \(4.15222090511\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{3} q^{5} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{7} + (\beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{3} q^{5} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{7} + (\beta_{2} - \beta_1) q^{9} + ( - 2 \beta_{5} - \beta_{4} - \beta_{3}) q^{11} + (\beta_{4} + 2 \beta_{2} - \beta_1 + 2) q^{13} - \beta_{5} q^{15} + (2 \beta_{2} - 2 \beta_1 + 2) q^{17} + (2 \beta_{5} - \beta_{4} + \beta_{3}) q^{19} + (2 \beta_{5} + 2 \beta_{3}) q^{21} + (2 \beta_{2} - \beta_1 + 4) q^{23} - q^{25} + (2 \beta_1 + 2) q^{27} + (3 \beta_{2} + 3 \beta_1 - 1) q^{29} + ( - 5 \beta_{4} + 3 \beta_{3}) q^{31} + ( - 3 \beta_{5} + 3 \beta_{4} - 7 \beta_{3}) q^{33} + (\beta_{2} - \beta_1 + 1) q^{35} + (\beta_{5} + \beta_{4} - 3 \beta_{3}) q^{37} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \cdots + 1) q^{39}+ \cdots + ( - 4 \beta_{5} + 3 \beta_{4} - 3 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{9} + 8 q^{13} + 8 q^{17} + 20 q^{23} - 6 q^{25} + 12 q^{27} - 12 q^{29} + 4 q^{35} + 8 q^{39} - 36 q^{43} + 18 q^{49} + 24 q^{51} - 4 q^{53} - 8 q^{55} - 36 q^{61} + 2 q^{65} + 8 q^{69} + 20 q^{77} + 24 q^{79} - 26 q^{81} - 60 q^{87} + 8 q^{91} + 4 q^{95}+O(q^{100}) \) 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
\(\beta_{2}\)\(=\) \( ( -5\nu^{5} + 17\nu^{4} - 20\nu^{3} - 5\nu^{2} + 10\nu + 29 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -14\nu^{5} + 20\nu^{4} - 10\nu^{3} - 37\nu^{2} - 64\nu + 26 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - \beta_{4} + 2\beta_{3} - \beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{2} + 5\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{5} + 3\beta_{4} - 9\beta_{3} - 3\beta_{2} + 8\beta _1 - 9 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/520\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(261\) \(391\) \(417\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
441.1
0.403032 + 0.403032i
0.403032 0.403032i
−0.854638 0.854638i
−0.854638 + 0.854638i
1.45161 + 1.45161i
1.45161 1.45161i
0 −1.67513 0 1.00000i 0 0.806063i 0 −0.193937 0
441.2 0 −1.67513 0 1.00000i 0 0.806063i 0 −0.193937 0
441.3 0 −0.539189 0 1.00000i 0 1.70928i 0 −2.70928 0
441.4 0 −0.539189 0 1.00000i 0 1.70928i 0 −2.70928 0
441.5 0 2.21432 0 1.00000i 0 2.90321i 0 1.90321 0
441.6 0 2.21432 0 1.00000i 0 2.90321i 0 1.90321 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 441.6
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 520.2.k.a 6
3.b odd 2 1 4680.2.g.j 6
4.b odd 2 1 1040.2.k.b 6
5.b even 2 1 2600.2.k.b 6
5.c odd 4 1 2600.2.f.c 6
5.c odd 4 1 2600.2.f.d 6
13.b even 2 1 inner 520.2.k.a 6
13.d odd 4 1 6760.2.a.u 3
13.d odd 4 1 6760.2.a.v 3
39.d odd 2 1 4680.2.g.j 6
52.b odd 2 1 1040.2.k.b 6
65.d even 2 1 2600.2.k.b 6
65.h odd 4 1 2600.2.f.c 6
65.h odd 4 1 2600.2.f.d 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.k.a 6 1.a even 1 1 trivial
520.2.k.a 6 13.b even 2 1 inner
1040.2.k.b 6 4.b odd 2 1
1040.2.k.b 6 52.b odd 2 1
2600.2.f.c 6 5.c odd 4 1
2600.2.f.c 6 65.h odd 4 1
2600.2.f.d 6 5.c odd 4 1
2600.2.f.d 6 65.h odd 4 1
2600.2.k.b 6 5.b even 2 1
2600.2.k.b 6 65.d even 2 1
4680.2.g.j 6 3.b odd 2 1
4680.2.g.j 6 39.d odd 2 1
6760.2.a.u 3 13.d odd 4 1
6760.2.a.v 3 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 4T_{3} - 2 \) acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} - 4 T - 2)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 12 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( T^{6} + 52 T^{4} + \cdots + 2116 \) Copy content Toggle raw display
$13$ \( T^{6} - 8 T^{5} + \cdots + 2197 \) Copy content Toggle raw display
$17$ \( (T^{3} - 4 T^{2} - 16 T + 32)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 32 T^{4} + \cdots + 100 \) Copy content Toggle raw display
$23$ \( (T^{3} - 10 T^{2} + \cdots + 26)^{2} \) 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
$37$ \( T^{6} + 40 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$41$ \( T^{6} + 188 T^{4} + \cdots + 40000 \) Copy content Toggle raw display
$43$ \( (T^{3} + 18 T^{2} + \cdots + 54)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 108 T^{4} + \cdots + 11664 \) Copy content Toggle raw display
$53$ \( (T^{3} + 2 T^{2} + \cdots - 200)^{2} \) 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
$67$ \( T^{6} + 52 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$71$ \( T^{6} + 336 T^{4} + \cdots + 75076 \) Copy content Toggle raw display
$73$ \( T^{6} + 200 T^{4} + \cdots + 71824 \) Copy content Toggle raw display
$79$ \( (T^{3} - 12 T^{2} + \cdots - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 372 T^{4} + \cdots + 929296 \) Copy content Toggle raw display
$89$ \( T^{6} + 64 T^{4} + \cdots + 1024 \) Copy content Toggle raw display
$97$ \( T^{6} + 268 T^{4} + \cdots + 462400 \) Copy content Toggle raw display
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