Properties

Label 162.6.c.i
Level $162$
Weight $6$
Character orbit 162.c
Analytic conductor $25.982$
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] = [162,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.55");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(25.9821788097\)
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: yes
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.

\(f(q)\) \(=\) \( q + ( - 4 \zeta_{6} + 4) q^{2} - 16 \zeta_{6} q^{4} + 21 \zeta_{6} q^{5} + (74 \zeta_{6} - 74) q^{7} - 64 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 4 \zeta_{6} + 4) q^{2} - 16 \zeta_{6} q^{4} + 21 \zeta_{6} q^{5} + (74 \zeta_{6} - 74) q^{7} - 64 q^{8} + 84 q^{10} + ( - 270 \zeta_{6} + 270) q^{11} + 115 \zeta_{6} q^{13} + 296 \zeta_{6} q^{14} + (256 \zeta_{6} - 256) q^{16} + 861 q^{17} + 1850 q^{19} + ( - 336 \zeta_{6} + 336) q^{20} - 1080 \zeta_{6} q^{22} + 3618 \zeta_{6} q^{23} + ( - 2684 \zeta_{6} + 2684) q^{25} + 460 q^{26} + 1184 q^{28} + ( - 1125 \zeta_{6} + 1125) q^{29} - 5228 \zeta_{6} q^{31} + 1024 \zeta_{6} q^{32} + ( - 3444 \zeta_{6} + 3444) q^{34} - 1554 q^{35} + 9917 q^{37} + ( - 7400 \zeta_{6} + 7400) q^{38} - 1344 \zeta_{6} q^{40} + 10758 \zeta_{6} q^{41} + ( - 19714 \zeta_{6} + 19714) q^{43} - 4320 q^{44} + 14472 q^{46} + ( - 9984 \zeta_{6} + 9984) q^{47} + 11331 \zeta_{6} q^{49} - 10736 \zeta_{6} q^{50} + ( - 1840 \zeta_{6} + 1840) q^{52} - 36726 q^{53} + 5670 q^{55} + ( - 4736 \zeta_{6} + 4736) q^{56} - 4500 \zeta_{6} q^{58} + 26460 \zeta_{6} q^{59} + ( - 53779 \zeta_{6} + 53779) q^{61} - 20912 q^{62} + 4096 q^{64} + (2415 \zeta_{6} - 2415) q^{65} + 12934 \zeta_{6} q^{67} - 13776 \zeta_{6} q^{68} + (6216 \zeta_{6} - 6216) q^{70} + 4254 q^{71} - 17521 q^{73} + ( - 39668 \zeta_{6} + 39668) q^{74} - 29600 \zeta_{6} q^{76} + 19980 \zeta_{6} q^{77} + ( - 36946 \zeta_{6} + 36946) q^{79} - 5376 q^{80} + 43032 q^{82} + (76416 \zeta_{6} - 76416) q^{83} + 18081 \zeta_{6} q^{85} - 78856 \zeta_{6} q^{86} + (17280 \zeta_{6} - 17280) q^{88} + 45357 q^{89} - 8510 q^{91} + ( - 57888 \zeta_{6} + 57888) q^{92} - 39936 \zeta_{6} q^{94} + 38850 \zeta_{6} q^{95} + (127574 \zeta_{6} - 127574) q^{97} + 45324 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{2} - 16 q^{4} + 21 q^{5} - 74 q^{7} - 128 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{2} - 16 q^{4} + 21 q^{5} - 74 q^{7} - 128 q^{8} + 168 q^{10} + 270 q^{11} + 115 q^{13} + 296 q^{14} - 256 q^{16} + 1722 q^{17} + 3700 q^{19} + 336 q^{20} - 1080 q^{22} + 3618 q^{23} + 2684 q^{25} + 920 q^{26} + 2368 q^{28} + 1125 q^{29} - 5228 q^{31} + 1024 q^{32} + 3444 q^{34} - 3108 q^{35} + 19834 q^{37} + 7400 q^{38} - 1344 q^{40} + 10758 q^{41} + 19714 q^{43} - 8640 q^{44} + 28944 q^{46} + 9984 q^{47} + 11331 q^{49} - 10736 q^{50} + 1840 q^{52} - 73452 q^{53} + 11340 q^{55} + 4736 q^{56} - 4500 q^{58} + 26460 q^{59} + 53779 q^{61} - 41824 q^{62} + 8192 q^{64} - 2415 q^{65} + 12934 q^{67} - 13776 q^{68} - 6216 q^{70} + 8508 q^{71} - 35042 q^{73} + 39668 q^{74} - 29600 q^{76} + 19980 q^{77} + 36946 q^{79} - 10752 q^{80} + 86064 q^{82} - 76416 q^{83} + 18081 q^{85} - 78856 q^{86} - 17280 q^{88} + 90714 q^{89} - 17020 q^{91} + 57888 q^{92} - 39936 q^{94} + 38850 q^{95} - 127574 q^{97} + 90648 q^{98}+O(q^{100}) \) 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
2.00000 3.46410i 0 −8.00000 13.8564i 10.5000 + 18.1865i 0 −37.0000 + 64.0859i −64.0000 0 84.0000
109.1 2.00000 + 3.46410i 0 −8.00000 + 13.8564i 10.5000 18.1865i 0 −37.0000 64.0859i −64.0000 0 84.0000
\(n\): 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 162.6.c.i 2
3.b odd 2 1 162.6.c.d 2
9.c even 3 1 162.6.a.a 1
9.c even 3 1 inner 162.6.c.i 2
9.d odd 6 1 162.6.a.b yes 1
9.d odd 6 1 162.6.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.6.a.a 1 9.c even 3 1
162.6.a.b yes 1 9.d odd 6 1
162.6.c.d 2 3.b odd 2 1
162.6.c.d 2 9.d odd 6 1
162.6.c.i 2 1.a even 1 1 trivial
162.6.c.i 2 9.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 21T_{5} + 441 \) acting on \(S_{6}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 21T + 441 \) Copy content Toggle raw display
$7$ \( T^{2} + 74T + 5476 \) Copy content Toggle raw display
$11$ \( T^{2} - 270T + 72900 \) Copy content Toggle raw display
$13$ \( T^{2} - 115T + 13225 \) Copy content Toggle raw display
$17$ \( (T - 861)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1850)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 3618 T + 13089924 \) Copy content Toggle raw display
$29$ \( T^{2} - 1125 T + 1265625 \) Copy content Toggle raw display
$31$ \( T^{2} + 5228 T + 27331984 \) Copy content Toggle raw display
$37$ \( (T - 9917)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 10758 T + 115734564 \) Copy content Toggle raw display
$43$ \( T^{2} - 19714 T + 388641796 \) Copy content Toggle raw display
$47$ \( T^{2} - 9984 T + 99680256 \) Copy content Toggle raw display
$53$ \( (T + 36726)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 26460 T + 700131600 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 2892180841 \) Copy content Toggle raw display
$67$ \( T^{2} - 12934 T + 167288356 \) Copy content Toggle raw display
$71$ \( (T - 4254)^{2} \) Copy content Toggle raw display
$73$ \( (T + 17521)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 1365006916 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 5839405056 \) Copy content Toggle raw display
$89$ \( (T - 45357)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 16275125476 \) Copy content Toggle raw display
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