Properties

Label 162.8.c.f
Level $162$
Weight $8$
Character orbit 162.c
Analytic conductor $50.606$
Analytic rank $0$
Dimension $2$
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] = [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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
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.

\(f(q)\) \(=\) \( q + (8 \zeta_{6} - 8) q^{2} - 64 \zeta_{6} q^{4} + 312 \zeta_{6} q^{5} + (323 \zeta_{6} - 323) q^{7} + 512 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (8 \zeta_{6} - 8) q^{2} - 64 \zeta_{6} q^{4} + 312 \zeta_{6} q^{5} + (323 \zeta_{6} - 323) q^{7} + 512 q^{8} - 2496 q^{10} + ( - 3720 \zeta_{6} + 3720) q^{11} + 14179 \zeta_{6} q^{13} - 2584 \zeta_{6} q^{14} + (4096 \zeta_{6} - 4096) q^{16} - 15912 q^{17} + 22421 q^{19} + ( - 19968 \zeta_{6} + 19968) q^{20} + 29760 \zeta_{6} q^{22} + 57768 \zeta_{6} q^{23} + (19219 \zeta_{6} - 19219) q^{25} - 113432 q^{26} + 20672 q^{28} + ( - 166656 \zeta_{6} + 166656) q^{29} - 94820 \zeta_{6} q^{31} - 32768 \zeta_{6} q^{32} + ( - 127296 \zeta_{6} + 127296) q^{34} - 100776 q^{35} + 453971 q^{37} + (179368 \zeta_{6} - 179368) q^{38} + 159744 \zeta_{6} q^{40} + 627072 \zeta_{6} q^{41} + ( - 42472 \zeta_{6} + 42472) q^{43} - 238080 q^{44} - 462144 q^{46} + (1235256 \zeta_{6} - 1235256) q^{47} + 719214 \zeta_{6} q^{49} - 153752 \zeta_{6} q^{50} + ( - 907456 \zeta_{6} + 907456) q^{52} - 107280 q^{53} + 1160640 q^{55} + (165376 \zeta_{6} - 165376) q^{56} + 1333248 \zeta_{6} q^{58} - 2479224 \zeta_{6} q^{59} + (2874383 \zeta_{6} - 2874383) q^{61} + 758560 q^{62} + 262144 q^{64} + (4423848 \zeta_{6} - 4423848) q^{65} - 1501097 \zeta_{6} q^{67} + 1018368 \zeta_{6} q^{68} + ( - 806208 \zeta_{6} + 806208) q^{70} - 4733136 q^{71} - 85111 q^{73} + (3631768 \zeta_{6} - 3631768) q^{74} - 1434944 \zeta_{6} q^{76} + 1201560 \zeta_{6} q^{77} + ( - 1180819 \zeta_{6} + 1180819) q^{79} - 1277952 q^{80} - 5016576 q^{82} + (1116528 \zeta_{6} - 1116528) q^{83} - 4964544 \zeta_{6} q^{85} + 339776 \zeta_{6} q^{86} + ( - 1904640 \zeta_{6} + 1904640) q^{88} - 9368136 q^{89} - 4579817 q^{91} + ( - 3697152 \zeta_{6} + 3697152) q^{92} - 9882048 \zeta_{6} q^{94} + 6995352 \zeta_{6} q^{95} + ( - 2039995 \zeta_{6} + 2039995) q^{97} - 5753712 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} - 64 q^{4} + 312 q^{5} - 323 q^{7} + 1024 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} - 64 q^{4} + 312 q^{5} - 323 q^{7} + 1024 q^{8} - 4992 q^{10} + 3720 q^{11} + 14179 q^{13} - 2584 q^{14} - 4096 q^{16} - 31824 q^{17} + 44842 q^{19} + 19968 q^{20} + 29760 q^{22} + 57768 q^{23} - 19219 q^{25} - 226864 q^{26} + 41344 q^{28} + 166656 q^{29} - 94820 q^{31} - 32768 q^{32} + 127296 q^{34} - 201552 q^{35} + 907942 q^{37} - 179368 q^{38} + 159744 q^{40} + 627072 q^{41} + 42472 q^{43} - 476160 q^{44} - 924288 q^{46} - 1235256 q^{47} + 719214 q^{49} - 153752 q^{50} + 907456 q^{52} - 214560 q^{53} + 2321280 q^{55} - 165376 q^{56} + 1333248 q^{58} - 2479224 q^{59} - 2874383 q^{61} + 1517120 q^{62} + 524288 q^{64} - 4423848 q^{65} - 1501097 q^{67} + 1018368 q^{68} + 806208 q^{70} - 9466272 q^{71} - 170222 q^{73} - 3631768 q^{74} - 1434944 q^{76} + 1201560 q^{77} + 1180819 q^{79} - 2555904 q^{80} - 10033152 q^{82} - 1116528 q^{83} - 4964544 q^{85} + 339776 q^{86} + 1904640 q^{88} - 18736272 q^{89} - 9159634 q^{91} + 3697152 q^{92} - 9882048 q^{94} + 6995352 q^{95} + 2039995 q^{97} - 11507424 q^{98}+O(q^{100}) \) 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 156.000 + 270.200i 0 −161.500 + 279.726i 512.000 0 −2496.00
109.1 −4.00000 6.92820i 0 −32.0000 + 55.4256i 156.000 270.200i 0 −161.500 279.726i 512.000 0 −2496.00
\(n\): 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.f 2
3.b odd 2 1 162.8.c.g 2
9.c even 3 1 54.8.a.d yes 1
9.c even 3 1 inner 162.8.c.f 2
9.d odd 6 1 54.8.a.c 1
9.d odd 6 1 162.8.c.g 2
36.f odd 6 1 432.8.a.a 1
36.h even 6 1 432.8.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.8.a.c 1 9.d odd 6 1
54.8.a.d yes 1 9.c even 3 1
162.8.c.f 2 1.a even 1 1 trivial
162.8.c.f 2 9.c even 3 1 inner
162.8.c.g 2 3.b odd 2 1
162.8.c.g 2 9.d odd 6 1
432.8.a.a 1 36.f odd 6 1
432.8.a.h 1 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 312T_{5} + 97344 \) 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
$5$ \( T^{2} - 312T + 97344 \) Copy content Toggle raw display
$7$ \( T^{2} + 323T + 104329 \) Copy content Toggle raw display
$11$ \( T^{2} - 3720 T + 13838400 \) Copy content Toggle raw display
$13$ \( T^{2} - 14179 T + 201044041 \) Copy content Toggle raw display
$17$ \( (T + 15912)^{2} \) Copy content Toggle raw display
$19$ \( (T - 22421)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 57768 T + 3337141824 \) Copy content Toggle raw display
$29$ \( T^{2} - 166656 T + 27774222336 \) Copy content Toggle raw display
$31$ \( T^{2} + 94820 T + 8990832400 \) Copy content Toggle raw display
$37$ \( (T - 453971)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 627072 T + 393219293184 \) Copy content Toggle raw display
$43$ \( T^{2} - 42472 T + 1803870784 \) Copy content Toggle raw display
$47$ \( T^{2} + 1235256 T + 1525857385536 \) Copy content Toggle raw display
$53$ \( (T + 107280)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 2479224 T + 6146551642176 \) Copy content Toggle raw display
$61$ \( T^{2} + 2874383 T + 8262077630689 \) Copy content Toggle raw display
$67$ \( T^{2} + 1501097 T + 2253292203409 \) Copy content Toggle raw display
$71$ \( (T + 4733136)^{2} \) Copy content Toggle raw display
$73$ \( (T + 85111)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 1180819 T + 1394333510761 \) Copy content Toggle raw display
$83$ \( T^{2} + 1116528 T + 1246634774784 \) Copy content Toggle raw display
$89$ \( (T + 9368136)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 2039995 T + 4161579600025 \) Copy content Toggle raw display
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