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Properties

Label 520.2.f.a
Level $520$
Weight $2$
Character orbit 520.f
Analytic conductor $4.152$
Analytic rank $0$
Dimension $10$
CM no
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(129,520)]
 
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, 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("520.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 520.f (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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 11x^{8} + 36x^{6} + 42x^{4} + 13x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + \beta_{6} q^{5} - \beta_{2} q^{7} + ( - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + \beta_{6} q^{5} - \beta_{2} q^{7} + ( - \beta_1 - 1) q^{9} + \beta_{4} q^{11} + \beta_{7} q^{13} + (\beta_{7} - \beta_{5} + \beta_{3} + \cdots + 2) q^{15}+ \cdots + ( - \beta_{9} - 2 \beta_{8} + \cdots + 4 \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{5} + 2 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{5} + 2 q^{7} - 8 q^{9} - 2 q^{13} + 13 q^{15} - q^{25} + 8 q^{29} - 8 q^{33} - 3 q^{35} + 22 q^{37} - 12 q^{39} + 4 q^{45} - 2 q^{47} + 12 q^{49} - 30 q^{51} + 12 q^{55} + 12 q^{57} - 16 q^{61} - 24 q^{63} - 9 q^{65} - 16 q^{67} - 12 q^{69} + 4 q^{73} + 21 q^{75} + 28 q^{79} + 22 q^{81} + 4 q^{83} + 25 q^{85} + 2 q^{91} - 12 q^{93} - 10 q^{95} - 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 11x^{8} + 36x^{6} + 42x^{4} + 13x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{8} + 8\nu^{6} + 8\nu^{4} - 18\nu^{2} - 9 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} + 12\nu^{6} + 44\nu^{4} + 50\nu^{2} + 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} - 10\nu^{7} - 26\nu^{5} - 16\nu^{3} + 3\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{9} - 11\nu^{7} - 35\nu^{5} - 33\nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{9} - 5\nu^{8} - 52\nu^{7} - 52\nu^{6} - 152\nu^{5} - 152\nu^{4} - 146\nu^{3} - 146\nu^{2} - 27\nu - 23 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{9} + 5\nu^{8} - 52\nu^{7} + 52\nu^{6} - 152\nu^{5} + 152\nu^{4} - 146\nu^{3} + 146\nu^{2} - 27\nu + 23 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -3\nu^{9} - \nu^{8} - 33\nu^{7} - 11\nu^{6} - 107\nu^{5} - 35\nu^{4} - 117\nu^{3} - 35\nu^{2} - 22\nu - 5 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -3\nu^{9} + \nu^{8} - 33\nu^{7} + 11\nu^{6} - 107\nu^{5} + 35\nu^{4} - 117\nu^{3} + 35\nu^{2} - 22\nu + 5 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( 6\nu^{9} + 65\nu^{7} + 205\nu^{5} + 217\nu^{3} + 41\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} - \beta_{8} - \beta_{7} + \beta_{4} - 2\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} - \beta_{7} - 3\beta_{2} - \beta _1 - 10 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6\beta_{9} + 5\beta_{8} + 5\beta_{7} + 2\beta_{6} + 2\beta_{5} - 2\beta_{4} + 6\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{8} + 4\beta_{7} + \beta_{6} - \beta_{5} + 9\beta_{2} + 2\beta _1 + 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -37\beta_{9} - 29\beta_{8} - 29\beta_{7} - 18\beta_{6} - 18\beta_{5} + 7\beta_{4} - 20\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 55\beta_{8} - 55\beta_{7} - 18\beta_{6} + 18\beta_{5} - 109\beta_{2} - 21\beta _1 - 274 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 230\beta_{9} + 175\beta_{8} + 175\beta_{7} + 128\beta_{6} + 128\beta_{5} - 32\beta_{4} + 84\beta_{3} ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -179\beta_{8} + 179\beta_{7} + 64\beta_{6} - 64\beta_{5} + 337\beta_{2} + 63\beta _1 + 832 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -1437\beta_{9} - 1079\beta_{8} - 1079\beta_{7} - 844\beta_{6} - 844\beta_{5} + 173\beta_{4} - 430\beta_{3} ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
− 1.35685i
− 1.56822i
0.549054i
− 2.50630i
0.341517i
− 0.341517i
2.50630i
− 0.549054i
1.56822i
1.35685i
0 − 3.22659i 0 0.589653 + 2.15692i 0 1.65488 0 −7.41088 0
129.2 0 − 2.14926i 0 −1.82442 − 1.29287i 0 −2.12171 0 −1.61930 0
129.3 0 − 1.57201i 0 −2.14575 + 0.629081i 0 4.19743 0 0.528799 0
129.4 0 − 0.949078i 0 −0.108880 + 2.23342i 0 −3.85660 0 2.09925 0
129.5 0 − 0.773218i 0 1.98940 − 1.02093i 0 1.12600 0 2.40213 0
129.6 0 0.773218i 0 1.98940 + 1.02093i 0 1.12600 0 2.40213 0
129.7 0 0.949078i 0 −0.108880 − 2.23342i 0 −3.85660 0 2.09925 0
129.8 0 1.57201i 0 −2.14575 − 0.629081i 0 4.19743 0 0.528799 0
129.9 0 2.14926i 0 −1.82442 + 1.29287i 0 −2.12171 0 −1.61930 0
129.10 0 3.22659i 0 0.589653 − 2.15692i 0 1.65488 0 −7.41088 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.10
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 520.2.f.a ✓ 10
4.b odd 2 1 1040.2.f.f 10
5.b even 2 1 520.2.f.b yes 10
5.c odd 4 2 2600.2.k.f 20
13.b even 2 1 520.2.f.b yes 10
20.d odd 2 1 1040.2.f.g 10
52.b odd 2 1 1040.2.f.g 10
65.d even 2 1 inner 520.2.f.a ✓ 10
65.h odd 4 2 2600.2.k.f 20
260.g odd 2 1 1040.2.f.f 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
520.2.f.a ✓ 10 1.a even 1 1 trivial
520.2.f.a ✓ 10 65.d even 2 1 inner
520.2.f.b yes 10 5.b even 2 1
520.2.f.b yes 10 13.b even 2 1
1040.2.f.f 10 4.b odd 2 1
1040.2.f.f 10 260.g odd 2 1
1040.2.f.g 10 20.d odd 2 1
1040.2.f.g 10 52.b odd 2 1
2600.2.k.f 20 5.c odd 4 2
2600.2.k.f 20 65.h odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{5} - T_{7}^{4} - 20T_{7}^{3} + 16T_{7}^{2} + 64T_{7} - 64 \) acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 19 T^{8} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{10} + 3 T^{9} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( (T^{5} - T^{4} - 20 T^{3} + \cdots - 64)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} + 64 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$13$ \( T^{10} + 2 T^{9} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( T^{10} + 107 T^{8} + \cdots + 262144 \) Copy content Toggle raw display
$19$ \( T^{10} + 92 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{10} + 44 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{5} - 4 T^{4} - 56 T^{3} + \cdots - 64)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + 108 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$37$ \( (T^{5} - 11 T^{4} + \cdots - 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 208 T^{8} + \cdots + 4194304 \) Copy content Toggle raw display
$43$ \( T^{10} + 211 T^{8} + \cdots + 40246336 \) Copy content Toggle raw display
$47$ \( (T^{5} + T^{4} - 20 T^{3} + \cdots + 64)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 294191104 \) Copy content Toggle raw display
$59$ \( T^{10} + 316 T^{8} + \cdots + 59228416 \) Copy content Toggle raw display
$61$ \( (T^{5} + 8 T^{4} + \cdots + 128)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + 8 T^{4} + \cdots + 10496)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 583319104 \) Copy content Toggle raw display
$73$ \( (T^{5} - 2 T^{4} + \cdots + 6752)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 14 T^{4} + \cdots - 83968)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} - 2 T^{4} + \cdots + 132736)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 524 T^{8} + \cdots + 16777216 \) Copy content Toggle raw display
$97$ \( (T^{5} + 36 T^{4} + \cdots - 4736)^{2} \) Copy content Toggle raw display
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