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Properties

Label 2940.1.o.a
Level $2940$
Weight $1$
Character orbit 2940.o
Analytic conductor $1.467$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -20
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2940,1,Mod(2939,2940)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2940, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("2940.2939"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2940.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-4,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.46725113714\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.726062400.1

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12}^{3} q^{2} + \zeta_{12}^{5} q^{3} - q^{4} - q^{5} + \zeta_{12}^{2} q^{6} + \zeta_{12}^{3} q^{8} - \zeta_{12}^{4} q^{9} + \zeta_{12}^{3} q^{10} - \zeta_{12}^{5} q^{12} - \zeta_{12}^{5} q^{15} + \cdots + \zeta_{12}^{2} q^{96} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{9} + 4 q^{16} + 4 q^{20} - 2 q^{24} + 4 q^{25} - 2 q^{30} - 2 q^{36} - 4 q^{41} - 2 q^{45} + 4 q^{46} + 4 q^{54} - 4 q^{64} - 2 q^{69} - 4 q^{80} - 2 q^{81} - 4 q^{89}+ \cdots + 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times\).

\(n\) \(1081\) \(1177\) \(1471\) \(1961\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2939.1
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 − 0.500000i
− 1.00000i −0.866025 + 0.500000i −1.00000 −1.00000 0.500000 + 0.866025i 0 1.00000i 0.500000 − 0.866025i 1.00000i
2939.2 − 1.00000i 0.866025 + 0.500000i −1.00000 −1.00000 0.500000 − 0.866025i 0 1.00000i 0.500000 + 0.866025i 1.00000i
2939.3 1.00000i −0.866025 − 0.500000i −1.00000 −1.00000 0.500000 − 0.866025i 0 − 1.00000i 0.500000 + 0.866025i − 1.00000i
2939.4 1.00000i 0.866025 − 0.500000i −1.00000 −1.00000 0.500000 + 0.866025i 0 − 1.00000i 0.500000 − 0.866025i − 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
21.c even 2 1 inner
84.h odd 2 1 inner
105.g even 2 1 inner
420.o odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2940.1.o.a 4
3.b odd 2 1 2940.1.o.b 4
4.b odd 2 1 inner 2940.1.o.a 4
5.b even 2 1 inner 2940.1.o.a 4
7.b odd 2 1 2940.1.o.b 4
7.c even 3 1 420.1.be.b yes 4
7.c even 3 1 2940.1.be.d 4
7.d odd 6 1 420.1.be.a ✓ 4
7.d odd 6 1 2940.1.be.a 4
12.b even 2 1 2940.1.o.b 4
15.d odd 2 1 2940.1.o.b 4
20.d odd 2 1 CM 2940.1.o.a 4
21.c even 2 1 inner 2940.1.o.a 4
21.g even 6 1 420.1.be.b yes 4
21.g even 6 1 2940.1.be.d 4
21.h odd 6 1 420.1.be.a ✓ 4
21.h odd 6 1 2940.1.be.a 4
28.d even 2 1 2940.1.o.b 4
28.f even 6 1 420.1.be.a ✓ 4
28.f even 6 1 2940.1.be.a 4
28.g odd 6 1 420.1.be.b yes 4
28.g odd 6 1 2940.1.be.d 4
35.c odd 2 1 2940.1.o.b 4
35.i odd 6 1 420.1.be.a ✓ 4
35.i odd 6 1 2940.1.be.a 4
35.j even 6 1 420.1.be.b yes 4
35.j even 6 1 2940.1.be.d 4
35.k even 12 1 2100.1.ba.b 2
35.k even 12 1 2100.1.ba.c 2
35.l odd 12 1 2100.1.ba.a 2
35.l odd 12 1 2100.1.ba.d 2
60.h even 2 1 2940.1.o.b 4
84.h odd 2 1 inner 2940.1.o.a 4
84.j odd 6 1 420.1.be.b yes 4
84.j odd 6 1 2940.1.be.d 4
84.n even 6 1 420.1.be.a ✓ 4
84.n even 6 1 2940.1.be.a 4
105.g even 2 1 inner 2940.1.o.a 4
105.o odd 6 1 420.1.be.a ✓ 4
105.o odd 6 1 2940.1.be.a 4
105.p even 6 1 420.1.be.b yes 4
105.p even 6 1 2940.1.be.d 4
105.w odd 12 1 2100.1.ba.a 2
105.w odd 12 1 2100.1.ba.d 2
105.x even 12 1 2100.1.ba.b 2
105.x even 12 1 2100.1.ba.c 2
140.c even 2 1 2940.1.o.b 4
140.p odd 6 1 420.1.be.b yes 4
140.p odd 6 1 2940.1.be.d 4
140.s even 6 1 420.1.be.a ✓ 4
140.s even 6 1 2940.1.be.a 4
140.w even 12 1 2100.1.ba.a 2
140.w even 12 1 2100.1.ba.d 2
140.x odd 12 1 2100.1.ba.b 2
140.x odd 12 1 2100.1.ba.c 2
420.o odd 2 1 inner 2940.1.o.a 4
420.ba even 6 1 420.1.be.a ✓ 4
420.ba even 6 1 2940.1.be.a 4
420.be odd 6 1 420.1.be.b yes 4
420.be odd 6 1 2940.1.be.d 4
420.bp odd 12 1 2100.1.ba.b 2
420.bp odd 12 1 2100.1.ba.c 2
420.br even 12 1 2100.1.ba.a 2
420.br even 12 1 2100.1.ba.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.1.be.a ✓ 4 7.d odd 6 1
420.1.be.a ✓ 4 21.h odd 6 1
420.1.be.a ✓ 4 28.f even 6 1
420.1.be.a ✓ 4 35.i odd 6 1
420.1.be.a ✓ 4 84.n even 6 1
420.1.be.a ✓ 4 105.o odd 6 1
420.1.be.a ✓ 4 140.s even 6 1
420.1.be.a ✓ 4 420.ba even 6 1
420.1.be.b yes 4 7.c even 3 1
420.1.be.b yes 4 21.g even 6 1
420.1.be.b yes 4 28.g odd 6 1
420.1.be.b yes 4 35.j even 6 1
420.1.be.b yes 4 84.j odd 6 1
420.1.be.b yes 4 105.p even 6 1
420.1.be.b yes 4 140.p odd 6 1
420.1.be.b yes 4 420.be odd 6 1
2100.1.ba.a 2 35.l odd 12 1
2100.1.ba.a 2 105.w odd 12 1
2100.1.ba.a 2 140.w even 12 1
2100.1.ba.a 2 420.br even 12 1
2100.1.ba.b 2 35.k even 12 1
2100.1.ba.b 2 105.x even 12 1
2100.1.ba.b 2 140.x odd 12 1
2100.1.ba.b 2 420.bp odd 12 1
2100.1.ba.c 2 35.k even 12 1
2100.1.ba.c 2 105.x even 12 1
2100.1.ba.c 2 140.x odd 12 1
2100.1.ba.c 2 420.bp odd 12 1
2100.1.ba.d 2 35.l odd 12 1
2100.1.ba.d 2 105.w odd 12 1
2100.1.ba.d 2 140.w even 12 1
2100.1.ba.d 2 420.br even 12 1
2940.1.o.a 4 1.a even 1 1 trivial
2940.1.o.a 4 4.b odd 2 1 inner
2940.1.o.a 4 5.b even 2 1 inner
2940.1.o.a 4 20.d odd 2 1 CM
2940.1.o.a 4 21.c even 2 1 inner
2940.1.o.a 4 84.h odd 2 1 inner
2940.1.o.a 4 105.g even 2 1 inner
2940.1.o.a 4 420.o odd 2 1 inner
2940.1.o.b 4 3.b odd 2 1
2940.1.o.b 4 7.b odd 2 1
2940.1.o.b 4 12.b even 2 1
2940.1.o.b 4 15.d odd 2 1
2940.1.o.b 4 28.d even 2 1
2940.1.o.b 4 35.c odd 2 1
2940.1.o.b 4 60.h even 2 1
2940.1.o.b 4 140.c even 2 1
2940.1.be.a 4 7.d odd 6 1
2940.1.be.a 4 21.h odd 6 1
2940.1.be.a 4 28.f even 6 1
2940.1.be.a 4 35.i odd 6 1
2940.1.be.a 4 84.n even 6 1
2940.1.be.a 4 105.o odd 6 1
2940.1.be.a 4 140.s even 6 1
2940.1.be.a 4 420.ba even 6 1
2940.1.be.d 4 7.c even 3 1
2940.1.be.d 4 21.g even 6 1
2940.1.be.d 4 28.g odd 6 1
2940.1.be.d 4 35.j even 6 1
2940.1.be.d 4 84.j odd 6 1
2940.1.be.d 4 105.p even 6 1
2940.1.be.d 4 140.p odd 6 1
2940.1.be.d 4 420.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2940, [\chi])\):

\( T_{17} \) Copy content Toggle raw display
\( T_{41} + 1 \) Copy content Toggle raw display
\( T_{983}^{2} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

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