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Properties

Label 54.5.d.a
Level $54$
Weight $5$
Character orbit 54.d
Analytic conductor $5.582$
Analytic rank $0$
Dimension $8$
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] = [54,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.17");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 5 \)
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: \(5.58197800653\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.221456830464.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 38x^{6} - 100x^{5} + 449x^{4} - 736x^{3} + 1900x^{2} - 1548x + 2307 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{6} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 8 \beta_{2} q^{4} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots - 3) q^{5}+ \cdots + (8 \beta_{3} + 8 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 8 \beta_{2} q^{4} + (\beta_{7} - \beta_{6} - \beta_{5} + \cdots - 3) q^{5}+ \cdots + (57 \beta_{7} + 57 \beta_{6} + \cdots - 1573) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{4} - 18 q^{5} - 26 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 32 q^{4} - 18 q^{5} - 26 q^{7} + 720 q^{11} + 10 q^{13} - 288 q^{14} - 256 q^{16} + 100 q^{19} - 144 q^{20} + 336 q^{22} - 1278 q^{23} + 794 q^{25} - 416 q^{28} + 1854 q^{29} - 1478 q^{31} - 96 q^{34} - 32 q^{37} - 6768 q^{38} + 36 q^{41} - 68 q^{43} + 2112 q^{46} - 2214 q^{47} + 2442 q^{49} + 15552 q^{50} - 80 q^{52} - 3996 q^{55} - 2304 q^{56} - 2400 q^{58} - 9108 q^{59} - 4478 q^{61} - 4096 q^{64} + 22554 q^{65} + 7504 q^{67} + 11088 q^{68} + 6048 q^{70} + 20716 q^{73} - 15264 q^{74} + 400 q^{76} - 34434 q^{77} - 6050 q^{79} + 1152 q^{82} + 3834 q^{83} - 16092 q^{85} + 12528 q^{86} - 2688 q^{88} - 45868 q^{91} - 10224 q^{92} + 672 q^{94} - 20880 q^{95} + 31336 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 38x^{6} - 100x^{5} + 449x^{4} - 736x^{3} + 1900x^{2} - 1548x + 2307 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 19\nu^{7} + 286\nu^{6} - 545\nu^{5} + 8755\nu^{4} - 14939\nu^{3} + 71959\nu^{2} - 65748\nu + 147099 ) / 4230 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -8\nu^{7} + 28\nu^{6} - 290\nu^{5} + 655\nu^{4} - 2912\nu^{3} + 3727\nu^{2} - 7344\nu + 3777 ) / 1410 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 19\nu^{7} - 419\nu^{6} + 1570\nu^{5} - 10985\nu^{4} + 21016\nu^{3} - 78911\nu^{2} + 67497\nu - 146886 ) / 4230 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -589\nu^{7} + 1004\nu^{6} - 14830\nu^{5} + 7070\nu^{4} - 93136\nu^{3} - 69904\nu^{2} - 133917\nu - 289884 ) / 4230 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 589 \nu^{7} + 3119 \nu^{6} - 21175 \nu^{5} + 72635 \nu^{4} - 213691 \nu^{3} + 452501 \nu^{2} + \cdots + 594186 ) / 4230 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -133\nu^{7} + 536\nu^{6} - 4222\nu^{5} + 10202\nu^{4} - 35440\nu^{3} + 41534\nu^{2} - 65553\nu - 4764 ) / 846 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1211 \nu^{7} + 4591 \nu^{6} - 37730 \nu^{5} + 97300 \nu^{4} - 337874 \nu^{3} + 565144 \nu^{2} + \cdots + 725169 ) / 4230 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} - 2\beta_{5} - \beta_{4} + 3\beta_{3} + \beta_{2} + 4\beta _1 + 9 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 2\beta_{5} - \beta_{3} + \beta_{2} + 4\beta _1 - 67 ) / 9 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{7} - 11\beta_{6} + 13\beta_{5} + 8\beta_{4} - 93\beta_{3} + 49\beta_{2} - 89\beta _1 - 234 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -14\beta_{7} - 6\beta_{6} + 31\beta_{5} - 3\beta_{4} - 49\beta_{3} + 43\beta_{2} - 131\beta _1 + 545 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 73\beta_{7} + 118\beta_{6} - 65\beta_{5} - 97\beta_{4} + 1242\beta_{3} - 1238\beta_{2} + 955\beta _1 + 3867 ) / 18 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 178\beta_{7} + 159\beta_{6} - 395\beta_{5} + 48\beta_{4} + 1307\beta_{3} - 1931\beta_{2} + 2407\beta _1 - 3991 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 767 \beta_{7} - 1061 \beta_{6} - 92 \beta_{5} + 1367 \beta_{4} - 13731 \beta_{3} + 17407 \beta_{2} + \cdots - 55887 ) / 18 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\)
\(\chi(n)\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.500000 − 2.20403i
0.500000 + 3.61825i
0.500000 − 3.16175i
0.500000 + 1.74753i
0.500000 + 2.20403i
0.500000 − 3.61825i
0.500000 + 3.16175i
0.500000 − 1.74753i
−2.44949 + 1.41421i 0 4.00000 − 6.92820i −6.41371 − 3.70296i 0 30.1882 + 52.2875i 22.6274i 0 20.9471
17.2 −2.44949 + 1.41421i 0 4.00000 − 6.92820i 1.91371 + 1.10488i 0 −21.9913 − 38.0900i 22.6274i 0 −6.25016
17.3 2.44949 − 1.41421i 0 4.00000 − 6.92820i −37.0033 − 21.3638i 0 −19.8424 − 34.3680i − 22.6274i 0 −120.852
17.4 2.44949 − 1.41421i 0 4.00000 − 6.92820i 32.5033 + 18.7658i 0 −1.35458 − 2.34620i − 22.6274i 0 106.155
35.1 −2.44949 − 1.41421i 0 4.00000 + 6.92820i −6.41371 + 3.70296i 0 30.1882 − 52.2875i − 22.6274i 0 20.9471
35.2 −2.44949 − 1.41421i 0 4.00000 + 6.92820i 1.91371 − 1.10488i 0 −21.9913 + 38.0900i − 22.6274i 0 −6.25016
35.3 2.44949 + 1.41421i 0 4.00000 + 6.92820i −37.0033 + 21.3638i 0 −19.8424 + 34.3680i 22.6274i 0 −120.852
35.4 2.44949 + 1.41421i 0 4.00000 + 6.92820i 32.5033 − 18.7658i 0 −1.35458 + 2.34620i 22.6274i 0 106.155
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.4
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 54.5.d.a 8
3.b odd 2 1 18.5.d.a ✓ 8
4.b odd 2 1 432.5.q.b 8
9.c even 3 1 18.5.d.a ✓ 8
9.c even 3 1 162.5.b.c 8
9.d odd 6 1 inner 54.5.d.a 8
9.d odd 6 1 162.5.b.c 8
12.b even 2 1 144.5.q.b 8
36.f odd 6 1 144.5.q.b 8
36.f odd 6 1 1296.5.e.e 8
36.h even 6 1 432.5.q.b 8
36.h even 6 1 1296.5.e.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.5.d.a ✓ 8 3.b odd 2 1
18.5.d.a ✓ 8 9.c even 3 1
54.5.d.a 8 1.a even 1 1 trivial
54.5.d.a 8 9.d odd 6 1 inner
144.5.q.b 8 12.b even 2 1
144.5.q.b 8 36.f odd 6 1
162.5.b.c 8 9.c even 3 1
162.5.b.c 8 9.d odd 6 1
432.5.q.b 8 4.b odd 2 1
432.5.q.b 8 36.h even 6 1
1296.5.e.e 8 36.f odd 6 1
1296.5.e.e 8 36.h even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(54, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 18 T^{7} + \cdots + 688747536 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 81510250000 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 53\!\cdots\!89 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 54\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{4} - 50 T^{3} + \cdots + 8154234820)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 44\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 52\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{4} + 16 T^{3} + \cdots - 305304165104)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 80\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 62\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 78\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 55\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 13267734129308)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 16\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
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