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Properties

Label 42.8.e.a
Level $42$
Weight $8$
Character orbit 42.e
Analytic conductor $13.120$
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] = [42,8,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, 8); 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, 8, names="a")
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 42.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,8,-27,-64,-165] 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: \(13.1201710703\)
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.

\(f(q)\) \(=\) \( q + ( - 8 \zeta_{6} + 8) q^{2} - 27 \zeta_{6} q^{3} - 64 \zeta_{6} q^{4} + (165 \zeta_{6} - 165) q^{5} - 216 q^{6} + (1029 \zeta_{6} - 686) q^{7} - 512 q^{8} + (729 \zeta_{6} - 729) q^{9} + 1320 \zeta_{6} q^{10} + \cdots - 1498095 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} - 27 q^{3} - 64 q^{4} - 165 q^{5} - 432 q^{6} - 343 q^{7} - 1024 q^{8} - 729 q^{9} + 1320 q^{10} + 2055 q^{11} - 1728 q^{12} + 5440 q^{13} + 10976 q^{14} + 8910 q^{15} - 4096 q^{16} + 15708 q^{17}+ \cdots - 2996190 q^{99}+O(q^{100}) \) 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
4.00000 + 6.92820i −13.5000 + 23.3827i −32.0000 + 55.4256i −82.5000 − 142.894i −216.000 −171.500 − 891.140i −512.000 −364.500 − 631.333i 660.000 − 1143.15i
37.1 4.00000 − 6.92820i −13.5000 − 23.3827i −32.0000 − 55.4256i −82.5000 + 142.894i −216.000 −171.500 + 891.140i −512.000 −364.500 + 631.333i 660.000 + 1143.15i
\(n\): 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 42.8.e.a ✓ 2
3.b odd 2 1 126.8.g.b 2
7.b odd 2 1 294.8.e.p 2
7.c even 3 1 inner 42.8.e.a ✓ 2
7.c even 3 1 294.8.a.i 1
7.d odd 6 1 294.8.a.b 1
7.d odd 6 1 294.8.e.p 2
21.h odd 6 1 126.8.g.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.8.e.a ✓ 2 1.a even 1 1 trivial
42.8.e.a ✓ 2 7.c even 3 1 inner
126.8.g.b 2 3.b odd 2 1
126.8.g.b 2 21.h odd 6 1
294.8.a.b 1 7.d odd 6 1
294.8.a.i 1 7.c even 3 1
294.8.e.p 2 7.b odd 2 1
294.8.e.p 2 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 165T_{5} + 27225 \) acting on \(S_{8}^{\mathrm{new}}(42, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$3$ \( T^{2} + 27T + 729 \) Copy content Toggle raw display
$5$ \( T^{2} + 165T + 27225 \) Copy content Toggle raw display
$7$ \( T^{2} + 343T + 823543 \) Copy content Toggle raw display
$11$ \( T^{2} - 2055 T + 4223025 \) Copy content Toggle raw display
$13$ \( (T - 2720)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 15708 T + 246741264 \) Copy content Toggle raw display
$19$ \( T^{2} + 6752 T + 45589504 \) Copy content Toggle raw display
$23$ \( T^{2} + 30828 T + 950365584 \) Copy content Toggle raw display
$29$ \( (T + 118305)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 21761265289 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 97176839824 \) Copy content Toggle raw display
$41$ \( (T + 491400)^{2} \) Copy content Toggle raw display
$43$ \( (T + 577174)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 730789039044 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 1361615935689 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 27915392241 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 1055291871076 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 10680569116996 \) Copy content Toggle raw display
$71$ \( (T - 3046842)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 10298386992100 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 50808198672169 \) Copy content Toggle raw display
$83$ \( (T - 4365909)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 143909150333796 \) Copy content Toggle raw display
$97$ \( (T - 13343639)^{2} \) Copy content Toggle raw display
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