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Properties

Label 42.6.e.a
Level $42$
Weight $6$
Character orbit 42.e
Analytic conductor $6.736$
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] = [42,6,Mod(25,42)] 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("42.25"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 42.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.73612043215\)
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: yes
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.

\(f(q)\) \(=\) \( q + (4 \zeta_{6} - 4) q^{2} - 9 \zeta_{6} q^{3} - 16 \zeta_{6} q^{4} + ( - 6 \zeta_{6} + 6) q^{5} + 36 q^{6} + (133 \zeta_{6} - 7) q^{7} + 64 q^{8} + (81 \zeta_{6} - 81) q^{9} + 24 \zeta_{6} q^{10} + 666 \zeta_{6} q^{11} + \cdots - 53946 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 9 q^{3} - 16 q^{4} + 6 q^{5} + 72 q^{6} + 119 q^{7} + 128 q^{8} - 81 q^{9} + 24 q^{10} + 666 q^{11} - 144 q^{12} - 1118 q^{13} - 1036 q^{14} - 108 q^{15} - 256 q^{16} + 1740 q^{17} - 324 q^{18}+ \cdots - 107892 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(31\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
0.500000 − 0.866025i
0.500000 + 0.866025i
−2.00000 − 3.46410i −4.50000 + 7.79423i −8.00000 + 13.8564i 3.00000 + 5.19615i 36.0000 59.5000 − 115.181i 64.0000 −40.5000 − 70.1481i 12.0000 − 20.7846i
37.1 −2.00000 + 3.46410i −4.50000 − 7.79423i −8.00000 − 13.8564i 3.00000 − 5.19615i 36.0000 59.5000 + 115.181i 64.0000 −40.5000 + 70.1481i 12.0000 + 20.7846i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.6.e.a ✓ 2
3.b odd 2 1 126.6.g.c 2
4.b odd 2 1 336.6.q.c 2
7.b odd 2 1 294.6.e.e 2
7.c even 3 1 inner 42.6.e.a ✓ 2
7.c even 3 1 294.6.a.l 1
7.d odd 6 1 294.6.a.j 1
7.d odd 6 1 294.6.e.e 2
21.g even 6 1 882.6.a.e 1
21.h odd 6 1 126.6.g.c 2
21.h odd 6 1 882.6.a.f 1
28.g odd 6 1 336.6.q.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.a ✓ 2 1.a even 1 1 trivial
42.6.e.a ✓ 2 7.c even 3 1 inner
126.6.g.c 2 3.b odd 2 1
126.6.g.c 2 21.h odd 6 1
294.6.a.j 1 7.d odd 6 1
294.6.a.l 1 7.c even 3 1
294.6.e.e 2 7.b odd 2 1
294.6.e.e 2 7.d odd 6 1
336.6.q.c 2 4.b odd 2 1
336.6.q.c 2 28.g odd 6 1
882.6.a.e 1 21.g even 6 1
882.6.a.f 1 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 6T_{5} + 36 \) acting on \(S_{6}^{\mathrm{new}}(42, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$3$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$7$ \( T^{2} - 119T + 16807 \) Copy content Toggle raw display
$11$ \( T^{2} - 666T + 443556 \) Copy content Toggle raw display
$13$ \( (T + 559)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 1740 T + 3027600 \) Copy content Toggle raw display
$19$ \( T^{2} + 1157 T + 1338649 \) Copy content Toggle raw display
$23$ \( T^{2} - 3468 T + 12027024 \) Copy content Toggle raw display
$29$ \( (T - 3372)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 6293 T + 39601849 \) Copy content Toggle raw display
$37$ \( T^{2} + 3131 T + 9803161 \) Copy content Toggle raw display
$41$ \( (T + 4866)^{2} \) Copy content Toggle raw display
$43$ \( (T + 11407)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2310 T + 5336100 \) Copy content Toggle raw display
$53$ \( T^{2} - 28296 T + 800663616 \) Copy content Toggle raw display
$59$ \( T^{2} + 20544 T + 422055936 \) Copy content Toggle raw display
$61$ \( T^{2} - 4630 T + 21436900 \) Copy content Toggle raw display
$67$ \( T^{2} - 18745 T + 351375025 \) Copy content Toggle raw display
$71$ \( (T + 38226)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 4982806921 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 3880417849 \) Copy content Toggle raw display
$83$ \( (T - 79818)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 18120 T + 328334400 \) Copy content Toggle raw display
$97$ \( (T - 124754)^{2} \) Copy content Toggle raw display
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