Properties

Label 162.8.c.l
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 2)
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.

\(f(q)\) \(=\) \( q + ( - 8 \zeta_{6} + 8) q^{2} - 64 \zeta_{6} q^{4} + 210 \zeta_{6} q^{5} + (1016 \zeta_{6} - 1016) q^{7} - 512 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 8 \zeta_{6} + 8) q^{2} - 64 \zeta_{6} q^{4} + 210 \zeta_{6} q^{5} + (1016 \zeta_{6} - 1016) q^{7} - 512 q^{8} + 1680 q^{10} + (1092 \zeta_{6} - 1092) q^{11} - 1382 \zeta_{6} q^{13} + 8128 \zeta_{6} q^{14} + (4096 \zeta_{6} - 4096) q^{16} + 14706 q^{17} - 39940 q^{19} + ( - 13440 \zeta_{6} + 13440) q^{20} + 8736 \zeta_{6} q^{22} - 68712 \zeta_{6} q^{23} + ( - 34025 \zeta_{6} + 34025) q^{25} - 11056 q^{26} + 65024 q^{28} + ( - 102570 \zeta_{6} + 102570) q^{29} - 227552 \zeta_{6} q^{31} + 32768 \zeta_{6} q^{32} + ( - 117648 \zeta_{6} + 117648) q^{34} - 213360 q^{35} + 160526 q^{37} + (319520 \zeta_{6} - 319520) q^{38} - 107520 \zeta_{6} q^{40} - 10842 \zeta_{6} q^{41} + ( - 630748 \zeta_{6} + 630748) q^{43} + 69888 q^{44} - 549696 q^{46} + (472656 \zeta_{6} - 472656) q^{47} - 208713 \zeta_{6} q^{49} - 272200 \zeta_{6} q^{50} + (88448 \zeta_{6} - 88448) q^{52} - 1494018 q^{53} - 229320 q^{55} + ( - 520192 \zeta_{6} + 520192) q^{56} - 820560 \zeta_{6} q^{58} - 2640660 \zeta_{6} q^{59} + (827702 \zeta_{6} - 827702) q^{61} - 1820416 q^{62} + 262144 q^{64} + ( - 290220 \zeta_{6} + 290220) q^{65} + 126004 \zeta_{6} q^{67} - 941184 \zeta_{6} q^{68} + (1706880 \zeta_{6} - 1706880) q^{70} - 1414728 q^{71} + 980282 q^{73} + ( - 1284208 \zeta_{6} + 1284208) q^{74} + 2556160 \zeta_{6} q^{76} - 1109472 \zeta_{6} q^{77} + ( - 3566800 \zeta_{6} + 3566800) q^{79} - 860160 q^{80} - 86736 q^{82} + (5672892 \zeta_{6} - 5672892) q^{83} + 3088260 \zeta_{6} q^{85} - 5045984 \zeta_{6} q^{86} + ( - 559104 \zeta_{6} + 559104) q^{88} - 11951190 q^{89} + 1404112 q^{91} + (4397568 \zeta_{6} - 4397568) q^{92} + 3781248 \zeta_{6} q^{94} - 8387400 \zeta_{6} q^{95} + (8682146 \zeta_{6} - 8682146) q^{97} - 1669704 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} - 64 q^{4} + 210 q^{5} - 1016 q^{7} - 1024 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} - 64 q^{4} + 210 q^{5} - 1016 q^{7} - 1024 q^{8} + 3360 q^{10} - 1092 q^{11} - 1382 q^{13} + 8128 q^{14} - 4096 q^{16} + 29412 q^{17} - 79880 q^{19} + 13440 q^{20} + 8736 q^{22} - 68712 q^{23} + 34025 q^{25} - 22112 q^{26} + 130048 q^{28} + 102570 q^{29} - 227552 q^{31} + 32768 q^{32} + 117648 q^{34} - 426720 q^{35} + 321052 q^{37} - 319520 q^{38} - 107520 q^{40} - 10842 q^{41} + 630748 q^{43} + 139776 q^{44} - 1099392 q^{46} - 472656 q^{47} - 208713 q^{49} - 272200 q^{50} - 88448 q^{52} - 2988036 q^{53} - 458640 q^{55} + 520192 q^{56} - 820560 q^{58} - 2640660 q^{59} - 827702 q^{61} - 3640832 q^{62} + 524288 q^{64} + 290220 q^{65} + 126004 q^{67} - 941184 q^{68} - 1706880 q^{70} - 2829456 q^{71} + 1960564 q^{73} + 1284208 q^{74} + 2556160 q^{76} - 1109472 q^{77} + 3566800 q^{79} - 1720320 q^{80} - 173472 q^{82} - 5672892 q^{83} + 3088260 q^{85} - 5045984 q^{86} + 559104 q^{88} - 23902380 q^{89} + 2808224 q^{91} - 4397568 q^{92} + 3781248 q^{94} - 8387400 q^{95} - 8682146 q^{97} - 3339408 q^{98}+O(q^{100}) \) 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 105.000 + 181.865i 0 −508.000 + 879.882i −512.000 0 1680.00
109.1 4.00000 + 6.92820i 0 −32.0000 + 55.4256i 105.000 181.865i 0 −508.000 879.882i −512.000 0 1680.00
\(n\): 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.l 2
3.b odd 2 1 162.8.c.a 2
9.c even 3 1 2.8.a.a 1
9.c even 3 1 inner 162.8.c.l 2
9.d odd 6 1 18.8.a.b 1
9.d odd 6 1 162.8.c.a 2
36.f odd 6 1 16.8.a.b 1
36.h even 6 1 144.8.a.i 1
45.h odd 6 1 450.8.a.c 1
45.j even 6 1 50.8.a.g 1
45.k odd 12 2 50.8.b.c 2
45.l even 12 2 450.8.c.g 2
63.g even 3 1 98.8.c.d 2
63.h even 3 1 98.8.c.d 2
63.k odd 6 1 98.8.c.e 2
63.l odd 6 1 98.8.a.a 1
63.t odd 6 1 98.8.c.e 2
72.j odd 6 1 576.8.a.g 1
72.l even 6 1 576.8.a.f 1
72.n even 6 1 64.8.a.c 1
72.p odd 6 1 64.8.a.e 1
99.h odd 6 1 242.8.a.e 1
117.t even 6 1 338.8.a.d 1
117.y odd 12 2 338.8.b.d 2
144.v odd 12 2 256.8.b.f 2
144.x even 12 2 256.8.b.b 2
153.h even 6 1 578.8.a.b 1
180.p odd 6 1 400.8.a.l 1
180.x even 12 2 400.8.c.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.8.a.a 1 9.c even 3 1
16.8.a.b 1 36.f odd 6 1
18.8.a.b 1 9.d odd 6 1
50.8.a.g 1 45.j even 6 1
50.8.b.c 2 45.k odd 12 2
64.8.a.c 1 72.n even 6 1
64.8.a.e 1 72.p odd 6 1
98.8.a.a 1 63.l odd 6 1
98.8.c.d 2 63.g even 3 1
98.8.c.d 2 63.h even 3 1
98.8.c.e 2 63.k odd 6 1
98.8.c.e 2 63.t odd 6 1
144.8.a.i 1 36.h even 6 1
162.8.c.a 2 3.b odd 2 1
162.8.c.a 2 9.d odd 6 1
162.8.c.l 2 1.a even 1 1 trivial
162.8.c.l 2 9.c even 3 1 inner
242.8.a.e 1 99.h odd 6 1
256.8.b.b 2 144.x even 12 2
256.8.b.f 2 144.v odd 12 2
338.8.a.d 1 117.t even 6 1
338.8.b.d 2 117.y odd 12 2
400.8.a.l 1 180.p odd 6 1
400.8.c.j 2 180.x even 12 2
450.8.a.c 1 45.h odd 6 1
450.8.c.g 2 45.l even 12 2
576.8.a.f 1 72.l even 6 1
576.8.a.g 1 72.j odd 6 1
578.8.a.b 1 153.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 210T_{5} + 44100 \) 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
$5$ \( T^{2} - 210T + 44100 \) Copy content Toggle raw display
$7$ \( T^{2} + 1016 T + 1032256 \) Copy content Toggle raw display
$11$ \( T^{2} + 1092 T + 1192464 \) Copy content Toggle raw display
$13$ \( T^{2} + 1382 T + 1909924 \) Copy content Toggle raw display
$17$ \( (T - 14706)^{2} \) Copy content Toggle raw display
$19$ \( (T + 39940)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 4721338944 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 10520604900 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 51779912704 \) Copy content Toggle raw display
$37$ \( (T - 160526)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 10842 T + 117548964 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 397843039504 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 223403694336 \) Copy content Toggle raw display
$53$ \( (T + 1494018)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 6973085235600 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 685090600804 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 15877008016 \) Copy content Toggle raw display
$71$ \( (T + 1414728)^{2} \) Copy content Toggle raw display
$73$ \( (T - 980282)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 12722062240000 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 32181703643664 \) Copy content Toggle raw display
$89$ \( (T + 11951190)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 75379659165316 \) Copy content Toggle raw display
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