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Properties

Label 1500.2.d.a
Level $1500$
Weight $2$
Character orbit 1500.d
Analytic conductor $11.978$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1500,2,Mod(1249,1500)] 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("1500.1249"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1500, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1500 = 2^{2} \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1500.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-4,0,-16,0,0,0,0,0,0,0,30,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.9775603032\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + 2 \beta_1 q^{7} - q^{9} - 4 q^{11} + (2 \beta_{3} + 4 \beta_1) q^{13} + ( - 5 \beta_{3} - 3 \beta_1) q^{17} + (\beta_{2} + 8) q^{19} + 2 \beta_{2} q^{21} + (3 \beta_{3} - \beta_1) q^{23}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} - 16 q^{11} + 30 q^{19} - 4 q^{21} + 4 q^{29} - 14 q^{31} + 4 q^{49} - 14 q^{51} + 24 q^{59} - 22 q^{61} + 14 q^{69} - 2 q^{79} + 4 q^{81} + 20 q^{89} - 40 q^{91} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1500\mathbb{Z}\right)^\times\).

\(n\) \(751\) \(877\) \(1001\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
− 1.61803i
0.618034i
− 0.618034i
1.61803i
0 − 1.00000i 0 0 0 − 3.23607i 0 −1.00000 0
1249.2 0 − 1.00000i 0 0 0 1.23607i 0 −1.00000 0
1249.3 0 1.00000i 0 0 0 − 1.23607i 0 −1.00000 0
1249.4 0 1.00000i 0 0 0 3.23607i 0 −1.00000 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1500.2.d.a 4
3.b odd 2 1 4500.2.d.j 4
4.b odd 2 1 6000.2.f.m 4
5.b even 2 1 inner 1500.2.d.a 4
5.c odd 4 1 1500.2.a.b ✓ 2
5.c odd 4 1 1500.2.a.f yes 2
15.d odd 2 1 4500.2.d.j 4
15.e even 4 1 4500.2.a.g 2
15.e even 4 1 4500.2.a.k 2
20.d odd 2 1 6000.2.f.m 4
20.e even 4 1 6000.2.a.j 2
20.e even 4 1 6000.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1500.2.a.b ✓ 2 5.c odd 4 1
1500.2.a.f yes 2 5.c odd 4 1
1500.2.d.a 4 1.a even 1 1 trivial
1500.2.d.a 4 5.b even 2 1 inner
4500.2.a.g 2 15.e even 4 1
4500.2.a.k 2 15.e even 4 1
4500.2.d.j 4 3.b odd 2 1
4500.2.d.j 4 15.d odd 2 1
6000.2.a.j 2 20.e even 4 1
6000.2.a.t 2 20.e even 4 1
6000.2.f.m 4 4.b odd 2 1
6000.2.f.m 4 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1500, [\chi])\):

\( T_{7}^{4} + 12T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T + 4)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 20)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 47T^{2} + 1 \) Copy content Toggle raw display
$19$ \( (T^{2} - 15 T + 55)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 27T^{2} + 121 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2 T - 44)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 7 T - 19)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 108T^{2} + 1936 \) Copy content Toggle raw display
$41$ \( (T^{2} - 80)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 127T^{2} + 3481 \) Copy content Toggle raw display
$53$ \( T^{4} + 103T^{2} + 121 \) Copy content Toggle raw display
$59$ \( (T - 6)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 11 T + 19)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 188T^{2} + 16 \) Copy content Toggle raw display
$79$ \( (T^{2} + T - 61)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 287T^{2} + 3481 \) Copy content Toggle raw display
$89$ \( (T^{2} - 10 T - 20)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
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