Properties

Label 500.3.d.b
Level $500$
Weight $3$
Character orbit 500.d
Self dual yes
Analytic conductor $13.624$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [500,3,Mod(499,500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(500, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("500.499");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 500.d (of order \(2\), degree \(1\), not 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: yes
Analytic conductor: \(13.6240132180\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \beta_{2} + 1) q^{3} + 4 q^{4} + ( - 2 \beta_{2} + 2) q^{6} + ( - \beta_{3} - \beta_{2} - 3 \beta_1) q^{7} + 8 q^{8} + (\beta_{3} + \beta_{2} + 4 \beta_1 + 8) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + ( - \beta_{2} + 1) q^{3} + 4 q^{4} + ( - 2 \beta_{2} + 2) q^{6} + ( - \beta_{3} - \beta_{2} - 3 \beta_1) q^{7} + 8 q^{8} + (\beta_{3} + \beta_{2} + 4 \beta_1 + 8) q^{9} + ( - 4 \beta_{2} + 4) q^{12} + ( - 2 \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{14} + 16 q^{16} + (2 \beta_{3} + 2 \beta_{2} + 8 \beta_1 + 16) q^{18} + (2 \beta_{3} + 10 \beta_{2} - 4 \beta_1 + 7) q^{21} + (5 \beta_{3} + 5 \beta_{2} + 3 \beta_1 + 12) q^{23} + ( - 8 \beta_{2} + 8) q^{24} + ( - 3 \beta_{3} - 12 \beta_{2} + 7 \beta_1 - 7) q^{27} + ( - 4 \beta_{3} - 4 \beta_{2} - 12 \beta_1) q^{28} + ( - 7 \beta_{3} + 5 \beta_{2} + 2) q^{29} + 32 q^{32} + (4 \beta_{3} + 4 \beta_{2} + 16 \beta_1 + 32) q^{36} + (11 \beta_{3} - \beta_{2} - 10) q^{41} + (4 \beta_{3} + 20 \beta_{2} - 8 \beta_1 + 14) q^{42} + (4 \beta_{3} + 13 \beta_{2} - 17) q^{43} + (10 \beta_{3} + 10 \beta_{2} + 6 \beta_1 + 24) q^{46} + ( - 8 \beta_{3} + 13 \beta_{2} - 5) q^{47} + ( - 16 \beta_{2} + 16) q^{48} + ( - 13 \beta_{3} - \beta_{2} + 63) q^{49} + ( - 6 \beta_{3} - 24 \beta_{2} + 14 \beta_1 - 14) q^{54} + ( - 8 \beta_{3} - 8 \beta_{2} - 24 \beta_1) q^{56} + ( - 14 \beta_{3} + 10 \beta_{2} + 4) q^{58} + (\beta_{3} + \beta_{2} - 24 \beta_1 + 27) q^{61} + (7 \beta_{3} - 4 \beta_{2} - 27 \beta_1 - 145) q^{63} + 64 q^{64} - 116 q^{67} + (2 \beta_{3} - 26 \beta_{2} - 16 \beta_1 - 35) q^{69} + (8 \beta_{3} + 8 \beta_{2} + 32 \beta_1 + 64) q^{72} + ( - 10 \beta_{3} - 2 \beta_{2} + 36 \beta_1 + 102) q^{81} + (22 \beta_{3} - 2 \beta_{2} - 20) q^{82} + ( - \beta_{3} - \beta_{2} + 33 \beta_1 - 36) q^{83} + (8 \beta_{3} + 40 \beta_{2} - 16 \beta_1 + 28) q^{84} + (8 \beta_{3} + 26 \beta_{2} - 34) q^{86} + ( - 19 \beta_{3} - 19 \beta_{2} - 13 \beta_1 - 120) q^{87} + (19 \beta_{3} + 19 \beta_{2} + 24 \beta_1 + 33) q^{89} + (20 \beta_{3} + 20 \beta_{2} + 12 \beta_1 + 48) q^{92} + ( - 16 \beta_{3} + 26 \beta_{2} - 10) q^{94} + ( - 32 \beta_{2} + 32) q^{96} + ( - 26 \beta_{3} - 2 \beta_{2} + 126) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} + 8 q^{6} - 4 q^{7} + 32 q^{8} + 38 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} + 4 q^{3} + 16 q^{4} + 8 q^{6} - 4 q^{7} + 32 q^{8} + 38 q^{9} + 16 q^{12} - 8 q^{14} + 64 q^{16} + 76 q^{18} + 16 q^{21} + 44 q^{23} + 32 q^{24} - 8 q^{27} - 16 q^{28} + 22 q^{29} + 128 q^{32} + 152 q^{36} - 62 q^{41} + 32 q^{42} - 76 q^{43} + 88 q^{46} - 4 q^{47} + 64 q^{48} + 278 q^{49} - 16 q^{54} - 32 q^{56} + 44 q^{58} + 58 q^{61} - 648 q^{63} + 256 q^{64} - 464 q^{67} - 176 q^{69} + 304 q^{72} + 500 q^{81} - 124 q^{82} - 76 q^{83} + 64 q^{84} - 152 q^{86} - 468 q^{87} + 142 q^{89} + 176 q^{92} - 8 q^{94} + 128 q^{96} + 556 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{20} + \zeta_{20}^{-1}\):

\(\beta_{1}\)\(=\) \( -\nu^{3} - \nu^{2} + 5\nu + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 2\nu^{2} - 5\nu - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\nu^{3} + 3\nu^{2} + 5\nu - 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 - 1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 13 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} + 4\beta_{2} + 5\beta _1 - 3 ) / 5 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/500\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(377\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
499.1
1.90211
−1.17557
−1.90211
1.17557
2.00000 −5.48932 4.00000 0 −10.9786 −11.1220 8.00000 21.1327 0
499.2 2.00000 0.607412 4.00000 0 1.21482 13.9958 8.00000 −8.63105 0
499.3 2.00000 3.01719 4.00000 0 6.03437 4.64990 8.00000 0.103412 0
499.4 2.00000 5.86472 4.00000 0 11.7294 −11.5237 8.00000 25.3950 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.3.d.b 4
4.b odd 2 1 500.3.d.a 4
5.b even 2 1 500.3.d.a 4
5.c odd 4 2 500.3.b.a 8
20.d odd 2 1 CM 500.3.d.b 4
20.e even 4 2 500.3.b.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.3.b.a 8 5.c odd 4 2
500.3.b.a 8 20.e even 4 2
500.3.d.a 4 4.b odd 2 1
500.3.d.a 4 5.b even 2 1
500.3.d.b 4 1.a even 1 1 trivial
500.3.d.b 4 20.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4T_{3}^{3} - 29T_{3}^{2} + 116T_{3} - 59 \) acting on \(S_{3}^{\mathrm{new}}(500, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 4 T^{3} - 29 T^{2} + 116 T - 59 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} - 229 T^{2} + \cdots + 8341 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 44 T^{3} - 709 T^{2} + \cdots + 26581 \) Copy content Toggle raw display
$29$ \( T^{4} - 22 T^{3} - 3721 T^{2} + \cdots + 1735441 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 62 T^{3} - 4561 T^{2} + \cdots - 3403679 \) Copy content Toggle raw display
$43$ \( T^{4} + 76 T^{3} - 3469 T^{2} + \cdots - 2942939 \) Copy content Toggle raw display
$47$ \( T^{4} + 4 T^{3} - 11029 T^{2} + \cdots + 24221941 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} - 58 T^{3} - 15241 T^{2} + \cdots + 17958481 \) Copy content Toggle raw display
$67$ \( (T + 116)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 76 T^{3} - 28669 T^{2} + \cdots + 71699461 \) Copy content Toggle raw display
$89$ \( T^{4} - 142 T^{3} + \cdots - 78297119 \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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