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Properties

Label 54.9.d.a
Level $54$
Weight $9$
Character orbit 54.d
Analytic conductor $21.998$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [54,9,Mod(17,54)]
 
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([5]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.17");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 54.d (of order \(6\), 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: \(21.9984449433\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 150208 x^{14} - 1927740 x^{13} + 8702363206 x^{12} + 239206241152 x^{11} + \cdots + 81\!\cdots\!61 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{36} \)
Twist minimal: no (minimal twist has level 18)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - 128 \beta_1 q^{4} + (\beta_{5} + 37 \beta_1 + 74) q^{5} + (\beta_{12} + \beta_{9} + 2 \beta_{5} + \cdots - 230) q^{7} + ( - 128 \beta_{3} + 128 \beta_{2}) q^{8} + ( - 2 \beta_{6} - 2 \beta_{5} + \cdots - 2) q^{10}+ \cdots + (7388 \beta_{15} + 8528 \beta_{14} + \cdots - 19499386) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 1024 q^{4} + 882 q^{5} - 1846 q^{7} - 45756 q^{11} - 3370 q^{13} + 94464 q^{14} - 131072 q^{16} + 362180 q^{19} + 112896 q^{20} - 61824 q^{22} - 1311138 q^{23} + 963394 q^{25} - 472576 q^{28} + 2851290 q^{29}+ \cdots - 89415484 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 150208 x^{14} - 1927740 x^{13} + 8702363206 x^{12} + 239206241152 x^{11} + \cdots + 81\!\cdots\!61 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 15\!\cdots\!75 \nu^{15} + \cdots + 22\!\cdots\!25 ) / 38\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 60\!\cdots\!63 \nu^{15} + \cdots - 74\!\cdots\!80 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 60\!\cdots\!56 \nu^{15} + \cdots + 12\!\cdots\!03 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 34\!\cdots\!99 \nu^{15} + \cdots + 26\!\cdots\!65 ) / 38\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 48\!\cdots\!74 \nu^{15} + \cdots + 49\!\cdots\!82 ) / 38\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 34\!\cdots\!07 \nu^{15} + \cdots - 59\!\cdots\!77 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 57\!\cdots\!19 \nu^{15} + \cdots - 68\!\cdots\!09 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\!\cdots\!37 \nu^{15} + \cdots + 25\!\cdots\!89 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 31\!\cdots\!81 \nu^{15} + \cdots - 18\!\cdots\!91 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 30\!\cdots\!25 \nu^{15} + \cdots - 32\!\cdots\!14 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 34\!\cdots\!89 \nu^{15} + \cdots + 20\!\cdots\!85 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 11\!\cdots\!94 \nu^{15} + \cdots - 33\!\cdots\!71 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 89\!\cdots\!19 \nu^{15} + \cdots + 93\!\cdots\!51 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13\!\cdots\!74 \nu^{15} + \cdots - 93\!\cdots\!69 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 28\!\cdots\!06 \nu^{15} + \cdots - 39\!\cdots\!29 ) / 28\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} + 9\beta _1 + 6 ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 26 \beta_{15} - 26 \beta_{14} + 60 \beta_{13} + 26 \beta_{12} - 60 \beta_{11} + 99 \beta_{9} + \cdots + 506989 ) / 27 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9127 \beta_{15} - 9784 \beta_{14} + 5040 \beta_{13} + 8893 \beta_{12} - 6660 \beta_{11} + \cdots + 40450891 ) / 81 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3563812 \beta_{15} - 3645238 \beta_{14} + 7251660 \beta_{13} + 3367120 \beta_{12} + \cdots + 52500012832 ) / 81 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 563235179 \beta_{15} - 580735304 \beta_{14} + 337886640 \beta_{13} + 514144169 \beta_{12} + \cdots + 2800898068883 ) / 81 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 161469609602 \beta_{15} - 167567515880 \beta_{14} + 293530345920 \beta_{13} + 145318337222 \beta_{12} + \cdots + 20\!\cdots\!13 ) / 81 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 9926360405120 \beta_{15} - 9992185089836 \beta_{14} + 6460224338640 \beta_{13} + \cdots + 53\!\cdots\!54 ) / 27 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 25\!\cdots\!88 \beta_{15} + \cdots + 29\!\cdots\!09 ) / 27 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 15\!\cdots\!25 \beta_{15} + \cdots + 86\!\cdots\!75 ) / 81 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 36\!\cdots\!36 \beta_{15} + \cdots + 38\!\cdots\!52 ) / 81 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 76\!\cdots\!21 \beta_{15} + \cdots + 45\!\cdots\!23 ) / 81 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 17\!\cdots\!60 \beta_{15} + \cdots + 17\!\cdots\!61 ) / 81 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 12\!\cdots\!24 \beta_{15} + \cdots + 79\!\cdots\!10 ) / 27 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 28\!\cdots\!90 \beta_{15} + \cdots + 26\!\cdots\!97 ) / 27 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 18\!\cdots\!47 \beta_{15} + \cdots + 12\!\cdots\!59 ) / 81 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
220.333 + 0.866025i
71.4012 + 0.866025i
−93.2991 + 0.866025i
−197.435 + 0.866025i
164.354 + 0.866025i
−2.97990 + 0.866025i
−13.0653 + 0.866025i
−147.309 + 0.866025i
220.333 − 0.866025i
71.4012 − 0.866025i
−93.2991 − 0.866025i
−197.435 − 0.866025i
164.354 − 0.866025i
−2.97990 − 0.866025i
−13.0653 − 0.866025i
−147.309 − 0.866025i
−9.79796 + 5.65685i 0 64.0000 − 110.851i −935.250 − 539.967i 0 −2056.09 − 3561.25i 1448.15i 0 12218.1
17.2 −9.79796 + 5.65685i 0 64.0000 − 110.851i −265.055 − 153.030i 0 770.107 + 1333.86i 1448.15i 0 3462.67
17.3 −9.79796 + 5.65685i 0 64.0000 − 110.851i 476.096 + 274.874i 0 −1631.36 − 2825.60i 1448.15i 0 −6219.69
17.4 −9.79796 + 5.65685i 0 64.0000 − 110.851i 944.709 + 545.428i 0 1250.70 + 2166.27i 1448.15i 0 −12341.6
17.5 9.79796 − 5.65685i 0 64.0000 − 110.851i −683.343 − 394.528i 0 89.7132 + 155.388i − 1448.15i 0 −8927.15
17.6 9.79796 − 5.65685i 0 64.0000 − 110.851i 69.6596 + 40.2180i 0 370.849 + 642.329i − 1448.15i 0 910.029
17.7 9.79796 − 5.65685i 0 64.0000 − 110.851i 115.044 + 66.4206i 0 −1060.32 − 1836.53i − 1448.15i 0 1502.93
17.8 9.79796 − 5.65685i 0 64.0000 − 110.851i 719.139 + 415.195i 0 1343.41 + 2326.85i − 1448.15i 0 9394.80
35.1 −9.79796 − 5.65685i 0 64.0000 + 110.851i −935.250 + 539.967i 0 −2056.09 + 3561.25i − 1448.15i 0 12218.1
35.2 −9.79796 − 5.65685i 0 64.0000 + 110.851i −265.055 + 153.030i 0 770.107 − 1333.86i − 1448.15i 0 3462.67
35.3 −9.79796 − 5.65685i 0 64.0000 + 110.851i 476.096 − 274.874i 0 −1631.36 + 2825.60i − 1448.15i 0 −6219.69
35.4 −9.79796 − 5.65685i 0 64.0000 + 110.851i 944.709 − 545.428i 0 1250.70 − 2166.27i − 1448.15i 0 −12341.6
35.5 9.79796 + 5.65685i 0 64.0000 + 110.851i −683.343 + 394.528i 0 89.7132 − 155.388i 1448.15i 0 −8927.15
35.6 9.79796 + 5.65685i 0 64.0000 + 110.851i 69.6596 − 40.2180i 0 370.849 − 642.329i 1448.15i 0 910.029
35.7 9.79796 + 5.65685i 0 64.0000 + 110.851i 115.044 − 66.4206i 0 −1060.32 + 1836.53i 1448.15i 0 1502.93
35.8 9.79796 + 5.65685i 0 64.0000 + 110.851i 719.139 − 415.195i 0 1343.41 − 2326.85i 1448.15i 0 9394.80
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 54.9.d.a 16
3.b odd 2 1 18.9.d.a ✓ 16
4.b odd 2 1 432.9.q.c 16
9.c even 3 1 18.9.d.a ✓ 16
9.c even 3 1 162.9.b.c 16
9.d odd 6 1 inner 54.9.d.a 16
9.d odd 6 1 162.9.b.c 16
12.b even 2 1 144.9.q.b 16
36.f odd 6 1 144.9.q.b 16
36.h even 6 1 432.9.q.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.9.d.a ✓ 16 3.b odd 2 1
18.9.d.a ✓ 16 9.c even 3 1
54.9.d.a 16 1.a even 1 1 trivial
54.9.d.a 16 9.d odd 6 1 inner
144.9.q.b 16 12.b even 2 1
144.9.q.b 16 36.f odd 6 1
162.9.b.c 16 9.c even 3 1
162.9.b.c 16 9.d odd 6 1
432.9.q.c 16 4.b odd 2 1
432.9.q.c 16 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(54, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 128 T^{2} + 16384)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 52\!\cdots\!49 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots - 11\!\cdots\!40)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 17\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 63\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 79\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 27\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
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