Properties

Label 162.8.c.b
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: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + (8 \zeta_{6} - 8) q^{2} - 64 \zeta_{6} q^{4} - 120 \zeta_{6} q^{5} + (377 \zeta_{6} - 377) q^{7} + 512 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (8 \zeta_{6} - 8) q^{2} - 64 \zeta_{6} q^{4} - 120 \zeta_{6} q^{5} + (377 \zeta_{6} - 377) q^{7} + 512 q^{8} + 960 q^{10} + (600 \zeta_{6} - 600) q^{11} - 5369 \zeta_{6} q^{13} - 3016 \zeta_{6} q^{14} + (4096 \zeta_{6} - 4096) q^{16} + 12168 q^{17} + 16211 q^{19} + (7680 \zeta_{6} - 7680) q^{20} - 4800 \zeta_{6} q^{22} - 106392 \zeta_{6} q^{23} + ( - 63725 \zeta_{6} + 63725) q^{25} + 42952 q^{26} + 24128 q^{28} + (177216 \zeta_{6} - 177216) q^{29} + 268060 \zeta_{6} q^{31} - 32768 \zeta_{6} q^{32} + (97344 \zeta_{6} - 97344) q^{34} + 45240 q^{35} + 114959 q^{37} + (129688 \zeta_{6} - 129688) q^{38} - 61440 \zeta_{6} q^{40} + 112128 \zeta_{6} q^{41} + ( - 115048 \zeta_{6} + 115048) q^{43} + 38400 q^{44} + 851136 q^{46} + (561336 \zeta_{6} - 561336) q^{47} + 681414 \zeta_{6} q^{49} + 509800 \zeta_{6} q^{50} + (343616 \zeta_{6} - 343616) q^{52} - 1787760 q^{53} + 72000 q^{55} + (193024 \zeta_{6} - 193024) q^{56} - 1417728 \zeta_{6} q^{58} + 1786344 \zeta_{6} q^{59} + ( - 1306837 \zeta_{6} + 1306837) q^{61} - 2144480 q^{62} + 262144 q^{64} + (644280 \zeta_{6} - 644280) q^{65} + 2013817 \zeta_{6} q^{67} - 778752 \zeta_{6} q^{68} + (361920 \zeta_{6} - 361920) q^{70} - 4060944 q^{71} - 3850639 q^{73} + (919672 \zeta_{6} - 919672) q^{74} - 1037504 \zeta_{6} q^{76} - 226200 \zeta_{6} q^{77} + (1037231 \zeta_{6} - 1037231) q^{79} + 491520 q^{80} - 897024 q^{82} + (9203568 \zeta_{6} - 9203568) q^{83} - 1460160 \zeta_{6} q^{85} + 920384 \zeta_{6} q^{86} + (307200 \zeta_{6} - 307200) q^{88} - 1289304 q^{89} + 2024113 q^{91} + (6809088 \zeta_{6} - 6809088) q^{92} - 4490688 \zeta_{6} q^{94} - 1945320 \zeta_{6} q^{95} + (8555885 \zeta_{6} - 8555885) q^{97} - 5451312 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} - 64 q^{4} - 120 q^{5} - 377 q^{7} + 1024 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} - 64 q^{4} - 120 q^{5} - 377 q^{7} + 1024 q^{8} + 1920 q^{10} - 600 q^{11} - 5369 q^{13} - 3016 q^{14} - 4096 q^{16} + 24336 q^{17} + 32422 q^{19} - 7680 q^{20} - 4800 q^{22} - 106392 q^{23} + 63725 q^{25} + 85904 q^{26} + 48256 q^{28} - 177216 q^{29} + 268060 q^{31} - 32768 q^{32} - 97344 q^{34} + 90480 q^{35} + 229918 q^{37} - 129688 q^{38} - 61440 q^{40} + 112128 q^{41} + 115048 q^{43} + 76800 q^{44} + 1702272 q^{46} - 561336 q^{47} + 681414 q^{49} + 509800 q^{50} - 343616 q^{52} - 3575520 q^{53} + 144000 q^{55} - 193024 q^{56} - 1417728 q^{58} + 1786344 q^{59} + 1306837 q^{61} - 4288960 q^{62} + 524288 q^{64} - 644280 q^{65} + 2013817 q^{67} - 778752 q^{68} - 361920 q^{70} - 8121888 q^{71} - 7701278 q^{73} - 919672 q^{74} - 1037504 q^{76} - 226200 q^{77} - 1037231 q^{79} + 983040 q^{80} - 1794048 q^{82} - 9203568 q^{83} - 1460160 q^{85} + 920384 q^{86} - 307200 q^{88} - 2578608 q^{89} + 4048226 q^{91} - 6809088 q^{92} - 4490688 q^{94} - 1945320 q^{95} - 8555885 q^{97} - 10902624 q^{98}+O(q^{100}) \) 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 −60.0000 103.923i 0 −188.500 + 326.492i 512.000 0 960.000
109.1 −4.00000 6.92820i 0 −32.0000 + 55.4256i −60.0000 + 103.923i 0 −188.500 326.492i 512.000 0 960.000
\(n\): 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.b 2
3.b odd 2 1 162.8.c.k 2
9.c even 3 1 54.8.a.f yes 1
9.c even 3 1 inner 162.8.c.b 2
9.d odd 6 1 54.8.a.a 1
9.d odd 6 1 162.8.c.k 2
36.f odd 6 1 432.8.a.g 1
36.h even 6 1 432.8.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.8.a.a 1 9.d odd 6 1
54.8.a.f yes 1 9.c even 3 1
162.8.c.b 2 1.a even 1 1 trivial
162.8.c.b 2 9.c even 3 1 inner
162.8.c.k 2 3.b odd 2 1
162.8.c.k 2 9.d odd 6 1
432.8.a.b 1 36.h even 6 1
432.8.a.g 1 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 120T_{5} + 14400 \) 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
$5$ \( T^{2} + 120T + 14400 \) Copy content Toggle raw display
$7$ \( T^{2} + 377T + 142129 \) Copy content Toggle raw display
$11$ \( T^{2} + 600T + 360000 \) Copy content Toggle raw display
$13$ \( T^{2} + 5369 T + 28826161 \) Copy content Toggle raw display
$17$ \( (T - 12168)^{2} \) Copy content Toggle raw display
$19$ \( (T - 16211)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 11319257664 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 31405510656 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 71856163600 \) Copy content Toggle raw display
$37$ \( (T - 114959)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 12572688384 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 13236042304 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 315098104896 \) Copy content Toggle raw display
$53$ \( (T + 1787760)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 3191024886336 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 1707822944569 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 4055458909489 \) Copy content Toggle raw display
$71$ \( (T + 4060944)^{2} \) Copy content Toggle raw display
$73$ \( (T + 3850639)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 1075848147361 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 84705663930624 \) Copy content Toggle raw display
$89$ \( (T + 1289304)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 73203168133225 \) Copy content Toggle raw display
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