Properties

Label 162.8.c.c
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: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + (8 \zeta_{6} - 8) q^{2} - 64 \zeta_{6} q^{4} + 105 \zeta_{6} q^{5} + ( - 937 \zeta_{6} + 937) q^{7} + 512 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (8 \zeta_{6} - 8) q^{2} - 64 \zeta_{6} q^{4} + 105 \zeta_{6} q^{5} + ( - 937 \zeta_{6} + 937) q^{7} + 512 q^{8} - 840 q^{10} + ( - 5943 \zeta_{6} + 5943) q^{11} - 68 \zeta_{6} q^{13} + 7496 \zeta_{6} q^{14} + (4096 \zeta_{6} - 4096) q^{16} + 5400 q^{17} - 48382 q^{19} + ( - 6720 \zeta_{6} + 6720) q^{20} + 47544 \zeta_{6} q^{22} - 642 \zeta_{6} q^{23} + ( - 67100 \zeta_{6} + 67100) q^{25} + 544 q^{26} - 59968 q^{28} + (125934 \zeta_{6} - 125934) q^{29} + 161275 \zeta_{6} q^{31} - 32768 \zeta_{6} q^{32} + (43200 \zeta_{6} - 43200) q^{34} + 98385 q^{35} - 414286 q^{37} + ( - 387056 \zeta_{6} + 387056) q^{38} + 53760 \zeta_{6} q^{40} - 627474 \zeta_{6} q^{41} + (570590 \zeta_{6} - 570590) q^{43} - 380352 q^{44} + 5136 q^{46} + ( - 538698 \zeta_{6} + 538698) q^{47} - 54426 \zeta_{6} q^{49} + 536800 \zeta_{6} q^{50} + (4352 \zeta_{6} - 4352) q^{52} - 356283 q^{53} + 624015 q^{55} + ( - 479744 \zeta_{6} + 479744) q^{56} - 1007472 \zeta_{6} q^{58} - 2910828 \zeta_{6} q^{59} + (2684168 \zeta_{6} - 2684168) q^{61} - 1290200 q^{62} + 262144 q^{64} + ( - 7140 \zeta_{6} + 7140) q^{65} - 2681078 \zeta_{6} q^{67} - 345600 \zeta_{6} q^{68} + (787080 \zeta_{6} - 787080) q^{70} + 3705480 q^{71} - 153151 q^{73} + ( - 3314288 \zeta_{6} + 3314288) q^{74} + 3096448 \zeta_{6} q^{76} - 5568591 \zeta_{6} q^{77} + ( - 7579288 \zeta_{6} + 7579288) q^{79} - 430080 q^{80} + 5019792 q^{82} + ( - 9345999 \zeta_{6} + 9345999) q^{83} + 567000 \zeta_{6} q^{85} - 4564720 \zeta_{6} q^{86} + ( - 3042816 \zeta_{6} + 3042816) q^{88} - 4033602 q^{89} - 63716 q^{91} + (41088 \zeta_{6} - 41088) q^{92} + 4309584 \zeta_{6} q^{94} - 5080110 \zeta_{6} q^{95} + ( - 5754097 \zeta_{6} + 5754097) q^{97} + 435408 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} - 64 q^{4} + 105 q^{5} + 937 q^{7} + 1024 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} - 64 q^{4} + 105 q^{5} + 937 q^{7} + 1024 q^{8} - 1680 q^{10} + 5943 q^{11} - 68 q^{13} + 7496 q^{14} - 4096 q^{16} + 10800 q^{17} - 96764 q^{19} + 6720 q^{20} + 47544 q^{22} - 642 q^{23} + 67100 q^{25} + 1088 q^{26} - 119936 q^{28} - 125934 q^{29} + 161275 q^{31} - 32768 q^{32} - 43200 q^{34} + 196770 q^{35} - 828572 q^{37} + 387056 q^{38} + 53760 q^{40} - 627474 q^{41} - 570590 q^{43} - 760704 q^{44} + 10272 q^{46} + 538698 q^{47} - 54426 q^{49} + 536800 q^{50} - 4352 q^{52} - 712566 q^{53} + 1248030 q^{55} + 479744 q^{56} - 1007472 q^{58} - 2910828 q^{59} - 2684168 q^{61} - 2580400 q^{62} + 524288 q^{64} + 7140 q^{65} - 2681078 q^{67} - 345600 q^{68} - 787080 q^{70} + 7410960 q^{71} - 306302 q^{73} + 3314288 q^{74} + 3096448 q^{76} - 5568591 q^{77} + 7579288 q^{79} - 860160 q^{80} + 10039584 q^{82} + 9345999 q^{83} + 567000 q^{85} - 4564720 q^{86} + 3042816 q^{88} - 8067204 q^{89} - 127432 q^{91} - 41088 q^{92} + 4309584 q^{94} - 5080110 q^{95} + 5754097 q^{97} + 870816 q^{98}+O(q^{100}) \) 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 52.5000 + 90.9327i 0 468.500 811.466i 512.000 0 −840.000
109.1 −4.00000 6.92820i 0 −32.0000 + 55.4256i 52.5000 90.9327i 0 468.500 + 811.466i 512.000 0 −840.000
\(n\): 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.c 2
3.b odd 2 1 162.8.c.j 2
9.c even 3 1 54.8.a.e yes 1
9.c even 3 1 inner 162.8.c.c 2
9.d odd 6 1 54.8.a.b 1
9.d odd 6 1 162.8.c.j 2
36.f odd 6 1 432.8.a.c 1
36.h even 6 1 432.8.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.8.a.b 1 9.d odd 6 1
54.8.a.e yes 1 9.c even 3 1
162.8.c.c 2 1.a even 1 1 trivial
162.8.c.c 2 9.c even 3 1 inner
162.8.c.j 2 3.b odd 2 1
162.8.c.j 2 9.d odd 6 1
432.8.a.c 1 36.f odd 6 1
432.8.a.f 1 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 105T_{5} + 11025 \) 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
$5$ \( T^{2} - 105T + 11025 \) Copy content Toggle raw display
$7$ \( T^{2} - 937T + 877969 \) Copy content Toggle raw display
$11$ \( T^{2} - 5943 T + 35319249 \) Copy content Toggle raw display
$13$ \( T^{2} + 68T + 4624 \) Copy content Toggle raw display
$17$ \( (T - 5400)^{2} \) Copy content Toggle raw display
$19$ \( (T + 48382)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 642T + 412164 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 15859372356 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 26009625625 \) Copy content Toggle raw display
$37$ \( (T + 414286)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 393723620676 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 325572948100 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 290195535204 \) Copy content Toggle raw display
$53$ \( (T + 356283)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 8472919645584 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 7204757852224 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 7188179242084 \) Copy content Toggle raw display
$71$ \( (T - 3705480)^{2} \) Copy content Toggle raw display
$73$ \( (T + 153151)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 57445606586944 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 87347697308001 \) Copy content Toggle raw display
$89$ \( (T + 4033602)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 33109632285409 \) Copy content Toggle raw display
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