Properties

Label 42.12.e.a
Level $42$
Weight $12$
Character orbit 42.e
Analytic conductor $32.270$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [42,12,Mod(25,42)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(42, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("42.25");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 42.e (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.2704135835\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 1516x^{4} + 1461x^{3} + 2295252x^{2} - 40905x + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 32 \beta_{2} - 32) q^{2} - 243 \beta_{2} q^{3} + 1024 \beta_{2} q^{4} + ( - 5 \beta_{3} + 350 \beta_{2} + 350) q^{5} - 7776 q^{6} + ( - 19 \beta_{5} - 18 \beta_{4} + \cdots + 7719) q^{7} + 32768 q^{8}+ \cdots + (8798301 \beta_{5} - 8798301 \beta_{4} + \cdots - 3577601763) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 96 q^{2} + 729 q^{3} - 3072 q^{4} + 1045 q^{5} - 46656 q^{6} + 45731 q^{7} + 196608 q^{8} - 177147 q^{9} + 33440 q^{10} + 181565 q^{11} + 746496 q^{12} + 1186364 q^{13} - 703808 q^{14} + 507870 q^{15}+ \cdots - 21442463370 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 1516x^{4} + 1461x^{3} + 2295252x^{2} - 40905x + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -71\nu^{5} + 107636\nu^{4} + 2517211\nu^{3} + 162962892\nu^{2} - 2904255\nu + 168770690484 ) / 331386774 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 765580 \nu^{5} + 765571 \nu^{4} - 1160605636 \nu^{3} - 1139196684 \nu^{2} - 1757178368892 \nu - 368388 ) / 31316050143 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 780125804 \nu^{5} + 779789393 \nu^{4} - 1182160719788 \nu^{3} - 1130021145285 \nu^{2} + \cdots + 31897285120665 ) / 62632100286 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1280769578 \nu^{5} + 1211425091 \nu^{4} - 1940907271766 \nu^{3} - 1958717957451 \nu^{2} + \cdots - 52349083817967 ) / 62632100286 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1280838689 \nu^{5} + 1316197367 \nu^{4} - 1943161791467 \nu^{3} - 1800090796479 \nu^{2} + \cdots + 104821582929939 ) / 62632100286 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{5} - 2\beta_{4} + 5\beta_{3} - 38\beta_{2} - 5\beta _1 + 2 ) / 112 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 6\beta_{5} + 3\beta_{4} + 13\beta_{3} - 14152\beta_{2} - 14155 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -213\beta_{5} + 213\beta_{4} + 1097\beta _1 - 24179 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2181\beta_{5} - 4362\beta_{4} - 10319\beta_{3} + 10730561\beta_{2} + 10319\beta _1 + 4362 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4449234\beta_{5} + 2224617\beta_{4} - 11795981\beta_{3} + 422791574\beta_{2} + 420566957 ) / 112 \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(31\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\)

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.

comment: embeddings in the coefficient field
 
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
0.00891079 0.0154339i
−19.2176 + 33.2859i
19.7087 34.1365i
0.00891079 + 0.0154339i
−19.2176 33.2859i
19.7087 + 34.1365i
−16.0000 27.7128i 121.500 210.444i −512.000 + 886.810i −1098.21 1902.16i −7776.00 39703.5 20023.9i 32768.0 −29524.5 51137.9i −35142.9 + 60869.3i
25.2 −16.0000 27.7128i 121.500 210.444i −512.000 + 886.810i −405.223 701.867i −7776.00 24393.5 + 37179.1i 32768.0 −29524.5 51137.9i −12967.1 + 22459.7i
25.3 −16.0000 27.7128i 121.500 210.444i −512.000 + 886.810i 2025.94 + 3509.03i −7776.00 −41231.5 16652.0i 32768.0 −29524.5 51137.9i 64830.0 112289.i
37.1 −16.0000 + 27.7128i 121.500 + 210.444i −512.000 886.810i −1098.21 + 1902.16i −7776.00 39703.5 + 20023.9i 32768.0 −29524.5 + 51137.9i −35142.9 60869.3i
37.2 −16.0000 + 27.7128i 121.500 + 210.444i −512.000 886.810i −405.223 + 701.867i −7776.00 24393.5 37179.1i 32768.0 −29524.5 + 51137.9i −12967.1 22459.7i
37.3 −16.0000 + 27.7128i 121.500 + 210.444i −512.000 886.810i 2025.94 3509.03i −7776.00 −41231.5 + 16652.0i 32768.0 −29524.5 + 51137.9i 64830.0 + 112289.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.3
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 42.12.e.a 6
3.b odd 2 1 126.12.g.b 6
7.c even 3 1 inner 42.12.e.a 6
21.h odd 6 1 126.12.g.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.12.e.a 6 1.a even 1 1 trivial
42.12.e.a 6 7.c even 3 1 inner
126.12.g.b 6 3.b odd 2 1
126.12.g.b 6 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 1045 T_{5}^{5} + 11495425 T_{5}^{4} + 25296942000 T_{5}^{3} + 100693465807500 T_{5}^{2} + \cdots + 52\!\cdots\!00 \) acting on \(S_{12}^{\mathrm{new}}(42, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 32 T + 1024)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} - 243 T + 59049)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 77\!\cdots\!07 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{3} + \cdots + 39\!\cdots\!80)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 30\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 74\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{3} + \cdots + 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 13\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots + 38\!\cdots\!60)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + \cdots - 41\!\cdots\!70)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 60\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 25\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 79\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots - 30\!\cdots\!52)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 69\!\cdots\!41 \) Copy content Toggle raw display
$83$ \( (T^{3} + \cdots - 51\!\cdots\!80)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots - 35\!\cdots\!04)^{2} \) Copy content Toggle raw display
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