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Properties

Label 54.8.c.a
Level $54$
Weight $8$
Character orbit 54.c
Analytic conductor $16.869$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [54,8,Mod(19,54)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(54, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.19");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 54.c (of order \(3\), degree \(2\), not 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: \(16.8687913761\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.14601465675.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 52x^{4} - 99x^{3} + 709x^{2} - 660x + 1872 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{9} \)
Twist minimal: no (minimal twist has level 18)
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.

\(f(q)\) \(=\) \( q + 8 \beta_1 q^{2} + (64 \beta_1 - 64) q^{4} + ( - 2 \beta_{5} - 3 \beta_{3} + \cdots - 18) q^{5}+ \cdots - 512 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 \beta_1 q^{2} + (64 \beta_1 - 64) q^{4} + ( - 2 \beta_{5} - 3 \beta_{3} + \cdots - 18) q^{5}+ \cdots + ( - 20160 \beta_{5} + 20160 \beta_{4} + \cdots + 1285800) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 24 q^{2} - 192 q^{4} - 54 q^{5} + 210 q^{7} - 3072 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 24 q^{2} - 192 q^{4} - 54 q^{5} + 210 q^{7} - 3072 q^{8} - 864 q^{10} - 6579 q^{11} + 10092 q^{13} - 1680 q^{14} - 12288 q^{16} + 29790 q^{17} - 137490 q^{19} - 3456 q^{20} + 52632 q^{22} + 39654 q^{23} - 7923 q^{25} + 161472 q^{26} - 26880 q^{28} - 239832 q^{29} + 145704 q^{31} + 98304 q^{32} + 119160 q^{34} + 1087776 q^{35} - 720384 q^{37} - 549960 q^{38} + 27648 q^{40} - 1086993 q^{41} - 299967 q^{43} + 842112 q^{44} + 634464 q^{46} - 131634 q^{47} + 482175 q^{49} + 63384 q^{50} + 645888 q^{52} - 1909152 q^{53} - 5494824 q^{55} - 107520 q^{56} + 1918656 q^{58} - 2504853 q^{59} + 7309038 q^{61} + 2331264 q^{62} + 1572864 q^{64} + 1786698 q^{65} + 3433035 q^{67} - 953280 q^{68} + 4351104 q^{70} - 2269368 q^{71} - 15901746 q^{73} - 2881536 q^{74} + 4399680 q^{76} + 6147432 q^{77} + 7076928 q^{79} + 442368 q^{80} - 17391888 q^{82} + 10914444 q^{83} + 17613396 q^{85} + 2399736 q^{86} + 3368448 q^{88} - 23800356 q^{89} - 37170552 q^{91} + 2537856 q^{92} + 1053072 q^{94} + 17342424 q^{95} + 519357 q^{97} + 7714800 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 52x^{4} - 99x^{3} + 709x^{2} - 660x + 1872 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 6\nu^{3} + 4\nu^{2} + 1067\nu + 612 ) / 2292 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{4} + 2\nu^{3} - 35\nu^{2} + 34\nu - 195 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -186\nu^{5} + 83\nu^{4} - 6670\nu^{3} - 2684\nu^{2} - 42491\nu - 48516 ) / 764 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 192\nu^{5} - 671\nu^{4} + 7834\nu^{3} - 12226\nu^{2} + 74243\nu + 12300 ) / 764 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 192\nu^{5} - 289\nu^{4} + 7070\nu^{3} - 9170\nu^{2} + 71569\nu - 81672 ) / 764 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} + 2\beta_{3} - \beta_{2} + 18\beta _1 + 18 ) / 54 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{5} + 5\beta_{4} + 2\beta_{3} - 3\beta_{2} + 18\beta _1 - 864 ) / 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -9\beta_{5} - 5\beta_{4} - 16\beta_{3} + 7\beta_{2} + 432\beta _1 - 648 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 85\beta_{5} - 171\beta_{4} - 98\beta_{3} + 59\beta_{2} + 2574\beta _1 + 16434 ) / 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 821\beta_{5} + 161\beta_{4} + 970\beta_{3} - 455\beta_{2} - 49698\beta _1 + 71316 ) / 54 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/54\mathbb{Z}\right)^\times\).

\(n\) \(29\)
\(\chi(n)\) \(-1 + \beta_{1}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.500000 + 5.21243i
0.500000 − 1.89756i
0.500000 − 4.18090i
0.500000 − 5.21243i
0.500000 + 1.89756i
0.500000 + 4.18090i
4.00000 − 6.92820i 0 −32.0000 − 55.4256i −187.789 − 325.261i 0 −528.867 + 916.024i −512.000 0 −3004.63
19.2 4.00000 − 6.92820i 0 −32.0000 − 55.4256i 1.70819 + 2.95867i 0 407.819 − 706.363i −512.000 0 27.3311
19.3 4.00000 − 6.92820i 0 −32.0000 − 55.4256i 159.081 + 275.536i 0 226.048 − 391.526i −512.000 0 2545.30
37.1 4.00000 + 6.92820i 0 −32.0000 + 55.4256i −187.789 + 325.261i 0 −528.867 − 916.024i −512.000 0 −3004.63
37.2 4.00000 + 6.92820i 0 −32.0000 + 55.4256i 1.70819 − 2.95867i 0 407.819 + 706.363i −512.000 0 27.3311
37.3 4.00000 + 6.92820i 0 −32.0000 + 55.4256i 159.081 − 275.536i 0 226.048 + 391.526i −512.000 0 2545.30
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.3
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 54.8.c.a 6
3.b odd 2 1 18.8.c.a ✓ 6
4.b odd 2 1 432.8.i.a 6
9.c even 3 1 inner 54.8.c.a 6
9.c even 3 1 162.8.a.e 3
9.d odd 6 1 18.8.c.a ✓ 6
9.d odd 6 1 162.8.a.f 3
12.b even 2 1 144.8.i.a 6
36.f odd 6 1 432.8.i.a 6
36.h even 6 1 144.8.i.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.8.c.a ✓ 6 3.b odd 2 1
18.8.c.a ✓ 6 9.d odd 6 1
54.8.c.a 6 1.a even 1 1 trivial
54.8.c.a 6 9.c even 3 1 inner
144.8.i.a 6 12.b even 2 1
144.8.i.a 6 36.h even 6 1
162.8.a.e 3 9.c even 3 1
162.8.a.f 3 9.d odd 6 1
432.8.i.a 6 4.b odd 2 1
432.8.i.a 6 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 54T_{5}^{5} + 122607T_{5}^{4} - 7279794T_{5}^{3} + 14303890521T_{5}^{2} - 48862653840T_{5} + 166659897600 \) acting on \(S_{8}^{\mathrm{new}}(54, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 8 T + 64)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + \cdots + 166659897600 \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$11$ \( T^{6} + \cdots + 35\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 18\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( (T^{3} + \cdots - 2214690411708)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots - 22074972070832)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{6} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{3} + \cdots - 81\!\cdots\!16)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 16\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 14\!\cdots\!69 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( (T^{3} + \cdots - 33\!\cdots\!40)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 94\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 65\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 53\!\cdots\!61 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots - 47\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots + 84\!\cdots\!08)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 87\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 35\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots - 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 88\!\cdots\!61 \) Copy content Toggle raw display
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