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Properties

Label 162.5.d.d
Level $162$
Weight $5$
Character orbit 162.d
Analytic conductor $16.746$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,5,Mod(53,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, 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("162.53");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 162.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: \(16.7459340196\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - 2 \beta_{3} q^{2} + ( - 8 \beta_1 + 8) q^{4} + ( - 5 \beta_{6} + 22 \beta_{5}) q^{5} + ( - 13 \beta_{2} + 13 \beta_1) q^{7} + (16 \beta_{5} - 16 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{3} q^{2} + ( - 8 \beta_1 + 8) q^{4} + ( - 5 \beta_{6} + 22 \beta_{5}) q^{5} + ( - 13 \beta_{2} + 13 \beta_1) q^{7} + (16 \beta_{5} - 16 \beta_{3}) q^{8} + (10 \beta_{4} - 78) q^{10} + (10 \beta_{7} - 10 \beta_{6} + 59 \beta_{3}) q^{11} + ( - 14 \beta_{4} + 14 \beta_{2} + \cdots - 143) q^{13}+ \cdots + (1352 \beta_{7} + \cdots + 5338 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{4} + 52 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 32 q^{4} + 52 q^{7} - 624 q^{10} - 572 q^{13} - 256 q^{16} - 248 q^{19} - 864 q^{22} + 1892 q^{25} + 832 q^{28} - 3584 q^{31} + 1608 q^{34} - 18800 q^{37} - 2496 q^{40} - 3020 q^{43} - 4320 q^{46} - 9324 q^{49} + 4576 q^{52} + 22248 q^{55} + 2952 q^{58} + 4144 q^{61} - 4096 q^{64} + 12076 q^{67} - 18096 q^{70} - 3584 q^{73} - 992 q^{76} + 15004 q^{79} + 10752 q^{82} + 24048 q^{85} + 6912 q^{88} + 24440 q^{91} + 12912 q^{94} - 46304 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\zeta_{24}^{6} + 3\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -3\zeta_{24}^{6} + 6\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} + 2\zeta_{24}^{3} + 3\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 3\zeta_{24}^{7} + 2\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} + 5\beta_{3} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_{2} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + 4\beta_{5} - 5\beta_{3} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} - \beta_{6} + 4\beta_{5} + \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{4} + 2\beta_{2} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} - 4\beta_{3} ) / 9 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
−0.965926 − 0.258819i
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.258819 − 0.965926i
−0.965926 + 0.258819i
−2.44949 + 1.41421i 0 4.00000 − 6.92820i 7.97262 + 4.60300i 0 −27.2750 − 47.2417i 22.6274i 0 −26.0385
53.2 −2.44949 + 1.41421i 0 4.00000 − 6.92820i 39.7924 + 22.9742i 0 40.2750 + 69.7583i 22.6274i 0 −129.962
53.3 2.44949 − 1.41421i 0 4.00000 − 6.92820i −39.7924 − 22.9742i 0 40.2750 + 69.7583i − 22.6274i 0 −129.962
53.4 2.44949 − 1.41421i 0 4.00000 − 6.92820i −7.97262 − 4.60300i 0 −27.2750 − 47.2417i − 22.6274i 0 −26.0385
107.1 −2.44949 − 1.41421i 0 4.00000 + 6.92820i 7.97262 − 4.60300i 0 −27.2750 + 47.2417i − 22.6274i 0 −26.0385
107.2 −2.44949 − 1.41421i 0 4.00000 + 6.92820i 39.7924 − 22.9742i 0 40.2750 − 69.7583i − 22.6274i 0 −129.962
107.3 2.44949 + 1.41421i 0 4.00000 + 6.92820i −39.7924 + 22.9742i 0 40.2750 − 69.7583i 22.6274i 0 −129.962
107.4 2.44949 + 1.41421i 0 4.00000 + 6.92820i −7.97262 + 4.60300i 0 −27.2750 + 47.2417i 22.6274i 0 −26.0385
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
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 162.5.d.d 8
3.b odd 2 1 inner 162.5.d.d 8
9.c even 3 1 162.5.b.a ✓ 4
9.c even 3 1 inner 162.5.d.d 8
9.d odd 6 1 162.5.b.a ✓ 4
9.d odd 6 1 inner 162.5.d.d 8
36.f odd 6 1 1296.5.e.b 4
36.h even 6 1 1296.5.e.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.5.b.a ✓ 4 9.c even 3 1
162.5.b.a ✓ 4 9.d odd 6 1
162.5.d.d 8 1.a even 1 1 trivial
162.5.d.d 8 3.b odd 2 1 inner
162.5.d.d 8 9.c even 3 1 inner
162.5.d.d 8 9.d odd 6 1 inner
1296.5.e.b 4 36.f odd 6 1
1296.5.e.b 4 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 2196T_{5}^{6} + 4643487T_{5}^{4} - 392928084T_{5}^{2} + 32015587041 \) acting on \(S_{5}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

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
$5$ \( T^{8} + \cdots + 32015587041 \) Copy content Toggle raw display
$7$ \( (T^{4} - 26 T^{3} + \cdots + 19307236)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 403540761128976 \) Copy content Toggle raw display
$13$ \( (T^{4} + 286 T^{3} + \cdots + 229734649)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 66348 T^{2} + 52229529)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 62 T - 48962)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 15\!\cdots\!01 \) Copy content Toggle raw display
$31$ \( (T^{4} + 1792 T^{3} + \cdots + 524297639056)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 4700 T + 5522473)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( (T^{4} + 1510 T^{3} + \cdots + 76097636164)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{4} + 3072204 T^{2} + 100670405796)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 48\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 5076599303161)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 646328217156196)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 790866794501316)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 896 T - 56958971)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 195836458128964)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 20\!\cdots\!09)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 17\!\cdots\!76)^{2} \) Copy content Toggle raw display
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