Properties

Label 162.8.c.d
Level $162$
Weight $8$
Character orbit 162.c
Analytic conductor $50.606$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,8,Mod(55,162)]
 
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([4]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.55");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 162.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: \(50.6063741284\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (8 \zeta_{6} - 8) q^{2} - 64 \zeta_{6} q^{4} + 114 \zeta_{6} q^{5} + ( - 1576 \zeta_{6} + 1576) q^{7} + 512 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (8 \zeta_{6} - 8) q^{2} - 64 \zeta_{6} q^{4} + 114 \zeta_{6} q^{5} + ( - 1576 \zeta_{6} + 1576) q^{7} + 512 q^{8} - 912 q^{10} + (7332 \zeta_{6} - 7332) q^{11} + 3802 \zeta_{6} q^{13} + 12608 \zeta_{6} q^{14} + (4096 \zeta_{6} - 4096) q^{16} - 6606 q^{17} + 24860 q^{19} + ( - 7296 \zeta_{6} + 7296) q^{20} - 58656 \zeta_{6} q^{22} - 41448 \zeta_{6} q^{23} + ( - 65129 \zeta_{6} + 65129) q^{25} - 30416 q^{26} - 100864 q^{28} + ( - 41610 \zeta_{6} + 41610) q^{29} - 33152 \zeta_{6} q^{31} - 32768 \zeta_{6} q^{32} + ( - 52848 \zeta_{6} + 52848) q^{34} + 179664 q^{35} - 36466 q^{37} + (198880 \zeta_{6} - 198880) q^{38} + 58368 \zeta_{6} q^{40} + 639078 \zeta_{6} q^{41} + ( - 156412 \zeta_{6} + 156412) q^{43} + 469248 q^{44} + 331584 q^{46} + ( - 433776 \zeta_{6} + 433776) q^{47} - 1660233 \zeta_{6} q^{49} + 521032 \zeta_{6} q^{50} + ( - 243328 \zeta_{6} + 243328) q^{52} + 786078 q^{53} - 835848 q^{55} + ( - 806912 \zeta_{6} + 806912) q^{56} + 332880 \zeta_{6} q^{58} - 745140 \zeta_{6} q^{59} + ( - 1660618 \zeta_{6} + 1660618) q^{61} + 265216 q^{62} + 262144 q^{64} + (433428 \zeta_{6} - 433428) q^{65} + 3290836 \zeta_{6} q^{67} + 422784 \zeta_{6} q^{68} + (1437312 \zeta_{6} - 1437312) q^{70} + 5716152 q^{71} + 2659898 q^{73} + ( - 291728 \zeta_{6} + 291728) q^{74} - 1591040 \zeta_{6} q^{76} + 11555232 \zeta_{6} q^{77} + (3807440 \zeta_{6} - 3807440) q^{79} - 466944 q^{80} - 5112624 q^{82} + (2229468 \zeta_{6} - 2229468) q^{83} - 753084 \zeta_{6} q^{85} + 1251296 \zeta_{6} q^{86} + (3753984 \zeta_{6} - 3753984) q^{88} + 5991210 q^{89} + 5991952 q^{91} + (2652672 \zeta_{6} - 2652672) q^{92} + 3470208 \zeta_{6} q^{94} + 2834040 \zeta_{6} q^{95} + ( - 4060126 \zeta_{6} + 4060126) q^{97} + 13281864 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} - 64 q^{4} + 114 q^{5} + 1576 q^{7} + 1024 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} - 64 q^{4} + 114 q^{5} + 1576 q^{7} + 1024 q^{8} - 1824 q^{10} - 7332 q^{11} + 3802 q^{13} + 12608 q^{14} - 4096 q^{16} - 13212 q^{17} + 49720 q^{19} + 7296 q^{20} - 58656 q^{22} - 41448 q^{23} + 65129 q^{25} - 60832 q^{26} - 201728 q^{28} + 41610 q^{29} - 33152 q^{31} - 32768 q^{32} + 52848 q^{34} + 359328 q^{35} - 72932 q^{37} - 198880 q^{38} + 58368 q^{40} + 639078 q^{41} + 156412 q^{43} + 938496 q^{44} + 663168 q^{46} + 433776 q^{47} - 1660233 q^{49} + 521032 q^{50} + 243328 q^{52} + 1572156 q^{53} - 1671696 q^{55} + 806912 q^{56} + 332880 q^{58} - 745140 q^{59} + 1660618 q^{61} + 530432 q^{62} + 524288 q^{64} - 433428 q^{65} + 3290836 q^{67} + 422784 q^{68} - 1437312 q^{70} + 11432304 q^{71} + 5319796 q^{73} + 291728 q^{74} - 1591040 q^{76} + 11555232 q^{77} - 3807440 q^{79} - 933888 q^{80} - 10225248 q^{82} - 2229468 q^{83} - 753084 q^{85} + 1251296 q^{86} - 3753984 q^{88} + 11982420 q^{89} + 11983904 q^{91} - 2652672 q^{92} + 3470208 q^{94} + 2834040 q^{95} + 4060126 q^{97} + 26563728 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(83\)
\(\chi(n)\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 + 0.866025i
0.500000 0.866025i
−4.00000 + 6.92820i 0 −32.0000 55.4256i 57.0000 + 98.7269i 0 788.000 1364.86i 512.000 0 −912.000
109.1 −4.00000 6.92820i 0 −32.0000 + 55.4256i 57.0000 98.7269i 0 788.000 + 1364.86i 512.000 0 −912.000
\(n\): e.g. 2-40 or 990-1000
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 162.8.c.d 2
3.b odd 2 1 162.8.c.i 2
9.c even 3 1 6.8.a.a 1
9.c even 3 1 inner 162.8.c.d 2
9.d odd 6 1 18.8.a.a 1
9.d odd 6 1 162.8.c.i 2
36.f odd 6 1 48.8.a.b 1
36.h even 6 1 144.8.a.h 1
45.h odd 6 1 450.8.a.ba 1
45.j even 6 1 150.8.a.e 1
45.k odd 12 2 150.8.c.k 2
45.l even 12 2 450.8.c.a 2
63.g even 3 1 294.8.e.c 2
63.h even 3 1 294.8.e.c 2
63.k odd 6 1 294.8.e.d 2
63.l odd 6 1 294.8.a.l 1
63.t odd 6 1 294.8.e.d 2
72.j odd 6 1 576.8.a.h 1
72.l even 6 1 576.8.a.i 1
72.n even 6 1 192.8.a.f 1
72.p odd 6 1 192.8.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.8.a.a 1 9.c even 3 1
18.8.a.a 1 9.d odd 6 1
48.8.a.b 1 36.f odd 6 1
144.8.a.h 1 36.h even 6 1
150.8.a.e 1 45.j even 6 1
150.8.c.k 2 45.k odd 12 2
162.8.c.d 2 1.a even 1 1 trivial
162.8.c.d 2 9.c even 3 1 inner
162.8.c.i 2 3.b odd 2 1
162.8.c.i 2 9.d odd 6 1
192.8.a.f 1 72.n even 6 1
192.8.a.n 1 72.p odd 6 1
294.8.a.l 1 63.l odd 6 1
294.8.e.c 2 63.g even 3 1
294.8.e.c 2 63.h even 3 1
294.8.e.d 2 63.k odd 6 1
294.8.e.d 2 63.t odd 6 1
450.8.a.ba 1 45.h odd 6 1
450.8.c.a 2 45.l even 12 2
576.8.a.h 1 72.j odd 6 1
576.8.a.i 1 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 114T_{5} + 12996 \) acting on \(S_{8}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 114T + 12996 \) Copy content Toggle raw display
$7$ \( T^{2} - 1576 T + 2483776 \) Copy content Toggle raw display
$11$ \( T^{2} + 7332 T + 53758224 \) Copy content Toggle raw display
$13$ \( T^{2} - 3802 T + 14455204 \) Copy content Toggle raw display
$17$ \( (T + 6606)^{2} \) Copy content Toggle raw display
$19$ \( (T - 24860)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 1717936704 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 1731392100 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 1099055104 \) Copy content Toggle raw display
$37$ \( (T + 36466)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 408420690084 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 24464713744 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 188161618176 \) Copy content Toggle raw display
$53$ \( (T - 786078)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 555233619600 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 2757652141924 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 10829601578896 \) Copy content Toggle raw display
$71$ \( (T - 5716152)^{2} \) Copy content Toggle raw display
$73$ \( (T - 2659898)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 14496599353600 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 4970527563024 \) Copy content Toggle raw display
$89$ \( (T - 5991210)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 16484623135876 \) Copy content Toggle raw display
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