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Properties

Label 3900.2.q.e
Level $3900$
Weight $2$
Character orbit 3900.q
Analytic conductor $31.142$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3900,2,Mod(601,3900)] 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, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3900.601"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3900.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1,0,0,0,1,0,-1,0,-2,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(31.1416567883\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) 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 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{3} + \zeta_{6} q^{7} - \zeta_{6} q^{9} + (2 \zeta_{6} - 2) q^{11} + (3 \zeta_{6} - 4) q^{13} - 4 \zeta_{6} q^{17} - 4 \zeta_{6} q^{19} - q^{21} + ( - 6 \zeta_{6} + 6) q^{23} + q^{27} + \cdots + 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + q^{7} - q^{9} - 2 q^{11} - 5 q^{13} - 4 q^{17} - 4 q^{19} - 2 q^{21} + 6 q^{23} + 2 q^{27} - 6 q^{29} - 2 q^{31} - 2 q^{33} + 10 q^{37} - 2 q^{39} + 4 q^{41} + q^{43} + 20 q^{47} + 6 q^{49}+ \cdots + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(301\) \(1301\) \(1951\) \(3277\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
601.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 −0.500000 − 0.866025i 0 0 0 0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0
2401.1 0 −0.500000 + 0.866025i 0 0 0 0.500000 + 0.866025i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3900.2.q.e 2
5.b even 2 1 156.2.i.a ✓ 2
5.c odd 4 2 3900.2.by.c 4
13.c even 3 1 inner 3900.2.q.e 2
15.d odd 2 1 468.2.l.b 2
20.d odd 2 1 624.2.q.d 2
60.h even 2 1 1872.2.t.e 2
65.d even 2 1 2028.2.i.f 2
65.g odd 4 2 2028.2.q.g 4
65.l even 6 1 2028.2.a.a 1
65.l even 6 1 2028.2.i.f 2
65.n even 6 1 156.2.i.a ✓ 2
65.n even 6 1 2028.2.a.b 1
65.q odd 12 2 3900.2.by.c 4
65.s odd 12 2 2028.2.b.c 2
65.s odd 12 2 2028.2.q.g 4
195.x odd 6 1 468.2.l.b 2
195.x odd 6 1 6084.2.a.e 1
195.y odd 6 1 6084.2.a.l 1
195.bh even 12 2 6084.2.b.b 2
260.v odd 6 1 624.2.q.d 2
260.v odd 6 1 8112.2.a.bd 1
260.w odd 6 1 8112.2.a.u 1
780.br even 6 1 1872.2.t.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.i.a ✓ 2 5.b even 2 1
156.2.i.a ✓ 2 65.n even 6 1
468.2.l.b 2 15.d odd 2 1
468.2.l.b 2 195.x odd 6 1
624.2.q.d 2 20.d odd 2 1
624.2.q.d 2 260.v odd 6 1
1872.2.t.e 2 60.h even 2 1
1872.2.t.e 2 780.br even 6 1
2028.2.a.a 1 65.l even 6 1
2028.2.a.b 1 65.n even 6 1
2028.2.b.c 2 65.s odd 12 2
2028.2.i.f 2 65.d even 2 1
2028.2.i.f 2 65.l even 6 1
2028.2.q.g 4 65.g odd 4 2
2028.2.q.g 4 65.s odd 12 2
3900.2.q.e 2 1.a even 1 1 trivial
3900.2.q.e 2 13.c even 3 1 inner
3900.2.by.c 4 5.c odd 4 2
3900.2.by.c 4 65.q odd 12 2
6084.2.a.e 1 195.x odd 6 1
6084.2.a.l 1 195.y odd 6 1
6084.2.b.b 2 195.bh even 12 2
8112.2.a.u 1 260.w odd 6 1
8112.2.a.bd 1 260.v odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3900, [\chi])\):

\( T_{7}^{2} - T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} + 4 \) Copy content Toggle raw display
\( T_{23}^{2} - 6T_{23} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$19$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$31$ \( (T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$43$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$47$ \( (T - 10)^{2} \) Copy content Toggle raw display
$53$ \( (T + 8)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$61$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$67$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$71$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$73$ \( (T - 7)^{2} \) Copy content Toggle raw display
$79$ \( (T - 17)^{2} \) Copy content Toggle raw display
$83$ \( (T + 12)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$97$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
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