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Properties

Label 2040.1.v.e
Level $2040$
Weight $1$
Character orbit 2040.v
Analytic conductor $1.018$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -120, -255, 136
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2040,1,Mod(509,2040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2040, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1, 1])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2040.509"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2040.v (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.01809262577\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-30}, \sqrt{34})\)
Artin image: $D_4:C_2$
Artin field: Galois closure of 8.0.59927040000.1

$q$-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)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - q^{2} - i q^{3} + q^{4} - i q^{5} + i q^{6} - q^{8} - q^{9} + i q^{10} - 2 i q^{11} - i q^{12} - q^{15} + q^{16} - q^{17} + q^{18} - i q^{20} + 2 i q^{22} + i q^{24} - q^{25} + i q^{27} + 2 i q^{29} + \cdots + 2 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 2 q^{15} + 2 q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{25} + 2 q^{30} - 2 q^{32} - 4 q^{33} + 2 q^{34} - 2 q^{36} + 4 q^{47} - 2 q^{49} + 2 q^{50} - 4 q^{55} - 2 q^{60}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2040\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(511\) \(817\) \(1021\) \(1361\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
509.1
1.00000i
− 1.00000i
−1.00000 − 1.00000i 1.00000 − 1.00000i 1.00000i 0 −1.00000 −1.00000 1.00000i
509.2 −1.00000 1.00000i 1.00000 1.00000i − 1.00000i 0 −1.00000 −1.00000 − 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
120.i odd 2 1 CM by \(\Q(\sqrt{-30}) \)
136.h even 2 1 RM by \(\Q(\sqrt{34}) \)
255.h odd 2 1 CM by \(\Q(\sqrt{-255}) \)
8.b even 2 1 inner
15.d odd 2 1 inner
17.b even 2 1 inner
2040.v odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2040.1.v.e ✓ 2
3.b odd 2 1 2040.1.v.f yes 2
5.b even 2 1 2040.1.v.f yes 2
8.b even 2 1 inner 2040.1.v.e ✓ 2
15.d odd 2 1 inner 2040.1.v.e ✓ 2
17.b even 2 1 inner 2040.1.v.e ✓ 2
24.h odd 2 1 2040.1.v.f yes 2
40.f even 2 1 2040.1.v.f yes 2
51.c odd 2 1 2040.1.v.f yes 2
85.c even 2 1 2040.1.v.f yes 2
120.i odd 2 1 CM 2040.1.v.e ✓ 2
136.h even 2 1 RM 2040.1.v.e ✓ 2
255.h odd 2 1 CM 2040.1.v.e ✓ 2
408.b odd 2 1 2040.1.v.f yes 2
680.h even 2 1 2040.1.v.f yes 2
2040.v odd 2 1 inner 2040.1.v.e ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2040.1.v.e ✓ 2 1.a even 1 1 trivial
2040.1.v.e ✓ 2 8.b even 2 1 inner
2040.1.v.e ✓ 2 15.d odd 2 1 inner
2040.1.v.e ✓ 2 17.b even 2 1 inner
2040.1.v.e ✓ 2 120.i odd 2 1 CM
2040.1.v.e ✓ 2 136.h even 2 1 RM
2040.1.v.e ✓ 2 255.h odd 2 1 CM
2040.1.v.e ✓ 2 2040.v odd 2 1 inner
2040.1.v.f yes 2 3.b odd 2 1
2040.1.v.f yes 2 5.b even 2 1
2040.1.v.f yes 2 24.h odd 2 1
2040.1.v.f yes 2 40.f even 2 1
2040.1.v.f yes 2 51.c odd 2 1
2040.1.v.f yes 2 85.c even 2 1
2040.1.v.f yes 2 408.b odd 2 1
2040.1.v.f yes 2 680.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2040, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} + 4 \) Copy content Toggle raw display
\( T_{47} - 2 \) Copy content Toggle raw display
\( T_{61} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 4 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( (T - 2)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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