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Properties

Label 2520.2.f.a
Level $2520$
Weight $2$
Character orbit 2520.f
Analytic conductor $20.122$
Analytic rank $0$
Dimension $8$
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] = [2520,2,Mod(881,2520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 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("2520.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2520.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: \(20.1223013094\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1698758656.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + ( - \beta_{3} + \beta_1 - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + ( - \beta_{3} + \beta_1 - 1) q^{7} + (\beta_{7} + \beta_{3}) q^{11} + ( - \beta_{7} - \beta_{2} + \beta_1) q^{13} + ( - \beta_{4} - 2) q^{17} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{19}+ \cdots + (3 \beta_{7} - 2 \beta_{6} + \cdots - 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} - 8 q^{7} - 16 q^{17} + 8 q^{25} + 8 q^{35} + 8 q^{41} + 8 q^{43} + 24 q^{47} - 4 q^{49} - 8 q^{59} + 16 q^{67} + 4 q^{77} - 48 q^{83} + 16 q^{85} - 24 q^{89} - 28 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 14\nu^{5} + 4\nu^{4} + 37\nu^{3} + 36\nu^{2} - 40\nu + 32 ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 14\nu^{5} + 4\nu^{4} - 37\nu^{3} + 36\nu^{2} + 40\nu + 32 ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 18\nu^{5} + 89\nu^{3} + 104\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 14\nu^{4} - 37\nu^{2} + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 4\nu^{6} + 14\nu^{5} + 60\nu^{4} + 37\nu^{3} + 216\nu^{2} - 8\nu + 160 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 4\nu^{6} + 14\nu^{5} - 60\nu^{4} + 37\nu^{3} - 216\nu^{2} - 8\nu - 160 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} + 46\nu^{5} + 179\nu^{3} + 168\nu ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{2} - \beta _1 - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{7} - 9\beta_{6} - 9\beta_{5} - 4\beta_{3} - 5\beta_{2} + 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 9\beta_{6} - 9\beta_{5} - 18\beta_{4} + 17\beta_{2} + 17\beta _1 + 74 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -52\beta_{7} + 81\beta_{6} + 81\beta_{5} + 68\beta_{3} + 37\beta_{2} - 37\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -89\beta_{6} + 89\beta_{5} + 162\beta_{4} - 201\beta_{2} - 201\beta _1 - 650 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 580\beta_{7} - 761\beta_{6} - 761\beta_{5} - 804\beta_{3} - 325\beta_{2} + 325\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2520\mathbb{Z}\right)^\times\).

\(n\) \(281\) \(631\) \(1081\) \(1261\) \(2017\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
− 2.16053i
2.16053i
1.69230i
− 1.69230i
0.692297i
− 0.692297i
3.16053i
− 3.16053i
0 0 0 −1.00000 0 −2.52773 − 0.781409i 0 0 0
881.2 0 0 0 −1.00000 0 −2.52773 + 0.781409i 0 0 0
881.3 0 0 0 −1.00000 0 −2.19663 − 1.47472i 0 0 0
881.4 0 0 0 −1.00000 0 −2.19663 + 1.47472i 0 0 0
881.5 0 0 0 −1.00000 0 −0.510472 − 2.59604i 0 0 0
881.6 0 0 0 −1.00000 0 −0.510472 + 2.59604i 0 0 0
881.7 0 0 0 −1.00000 0 1.23483 − 2.33991i 0 0 0
881.8 0 0 0 −1.00000 0 1.23483 + 2.33991i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2520.2.f.a ✓ 8
3.b odd 2 1 2520.2.f.c yes 8
4.b odd 2 1 5040.2.f.g 8
7.b odd 2 1 2520.2.f.c yes 8
12.b even 2 1 5040.2.f.j 8
21.c even 2 1 inner 2520.2.f.a ✓ 8
28.d even 2 1 5040.2.f.j 8
84.h odd 2 1 5040.2.f.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2520.2.f.a ✓ 8 1.a even 1 1 trivial
2520.2.f.a ✓ 8 21.c even 2 1 inner
2520.2.f.c yes 8 3.b odd 2 1
2520.2.f.c yes 8 7.b odd 2 1
5040.2.f.g 8 4.b odd 2 1
5040.2.f.g 8 84.h odd 2 1
5040.2.f.j 8 12.b even 2 1
5040.2.f.j 8 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2520, [\chi])\):

\( T_{11}^{4} + 12T_{11}^{2} + 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T + 1)^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 8 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} + 12 T^{2} + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 44 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 4)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} + 104 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( T^{8} + 92 T^{6} + \cdots + 33856 \) Copy content Toggle raw display
$29$ \( T^{8} + 220 T^{6} + \cdots + 1201216 \) Copy content Toggle raw display
$31$ \( T^{8} + 88 T^{6} + \cdots + 16384 \) Copy content Toggle raw display
$37$ \( (T^{4} - 42 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} - 22 T^{2} + \cdots + 8)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{3} + \cdots + 7456)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 12 T^{3} + \cdots - 32)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 180 T^{6} + \cdots + 3717184 \) Copy content Toggle raw display
$59$ \( (T^{4} + 4 T^{3} + \cdots - 752)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 376 T^{6} + \cdots + 9048064 \) Copy content Toggle raw display
$67$ \( (T^{4} - 8 T^{3} + \cdots - 512)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 236 T^{6} + \cdots + 541696 \) Copy content Toggle raw display
$73$ \( T^{8} + 160 T^{6} + \cdots + 246016 \) Copy content Toggle raw display
$79$ \( (T^{4} - 72 T^{2} + \cdots + 512)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 24 T^{3} + \cdots - 5696)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 12 T^{3} + \cdots - 4936)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 400 T^{6} + \cdots + 2408704 \) Copy content Toggle raw display
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