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Properties

Label 3900.1.cb.a
Level $3900$
Weight $1$
Character orbit 3900.cb
Analytic conductor $1.946$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -3
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3900,1,Mod(1049,3900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3900, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 3, 4])) 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("3900.1049"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3900.cb (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.94635354927\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.50700.1
Artin image: $S_3\times C_{12}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12} q^{3} + 2 \zeta_{12}^{5} q^{7} + \zeta_{12}^{2} q^{9} - \zeta_{12} q^{13} - \zeta_{12}^{2} q^{19} + 2 q^{21} - \zeta_{12}^{3} q^{27} - q^{31} + 2 \zeta_{12} q^{37} + \zeta_{12}^{2} q^{39} + \cdots - \zeta_{12}^{5} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9} - 2 q^{19} + 8 q^{21} - 4 q^{31} + 2 q^{39} + 6 q^{49} + 2 q^{61} + 4 q^{79} - 2 q^{81} + 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1951\) \(3277\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1049.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0 −0.866025 − 0.500000i 0 0 0 −1.73205 + 1.00000i 0 0.500000 + 0.866025i 0
1049.2 0 0.866025 + 0.500000i 0 0 0 1.73205 − 1.00000i 0 0.500000 + 0.866025i 0
3149.1 0 −0.866025 + 0.500000i 0 0 0 −1.73205 − 1.00000i 0 0.500000 − 0.866025i 0
3149.2 0 0.866025 − 0.500000i 0 0 0 1.73205 + 1.00000i 0 0.500000 − 0.866025i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
13.c even 3 1 inner
15.d odd 2 1 inner
39.i odd 6 1 inner
65.n even 6 1 inner
195.x odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3900.1.cb.a 4
3.b odd 2 1 CM 3900.1.cb.a 4
5.b even 2 1 inner 3900.1.cb.a 4
5.c odd 4 1 3900.1.bu.a ✓ 2
5.c odd 4 1 3900.1.bu.c yes 2
13.c even 3 1 inner 3900.1.cb.a 4
15.d odd 2 1 inner 3900.1.cb.a 4
15.e even 4 1 3900.1.bu.a ✓ 2
15.e even 4 1 3900.1.bu.c yes 2
39.i odd 6 1 inner 3900.1.cb.a 4
65.n even 6 1 inner 3900.1.cb.a 4
65.q odd 12 1 3900.1.bu.a ✓ 2
65.q odd 12 1 3900.1.bu.c yes 2
195.x odd 6 1 inner 3900.1.cb.a 4
195.bl even 12 1 3900.1.bu.a ✓ 2
195.bl even 12 1 3900.1.bu.c yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3900.1.bu.a ✓ 2 5.c odd 4 1
3900.1.bu.a ✓ 2 15.e even 4 1
3900.1.bu.a ✓ 2 65.q odd 12 1
3900.1.bu.a ✓ 2 195.bl even 12 1
3900.1.bu.c yes 2 5.c odd 4 1
3900.1.bu.c yes 2 15.e even 4 1
3900.1.bu.c yes 2 65.q odd 12 1
3900.1.bu.c yes 2 195.bl even 12 1
3900.1.cb.a 4 1.a even 1 1 trivial
3900.1.cb.a 4 3.b odd 2 1 CM
3900.1.cb.a 4 5.b even 2 1 inner
3900.1.cb.a 4 13.c even 3 1 inner
3900.1.cb.a 4 15.d odd 2 1 inner
3900.1.cb.a 4 39.i odd 6 1 inner
3900.1.cb.a 4 65.n even 6 1 inner
3900.1.cb.a 4 195.x odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 4T_{7}^{2} + 16 \) acting on \(S_{1}^{\mathrm{new}}(3900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T + 1)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} - 4T^{2} + 16 \) 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
$61$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$79$ \( (T - 1)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
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