Properties

Label 162.8.c.e
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: 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.

\(f(q)\) \(=\) \( q + (8 \zeta_{6} - 8) q^{2} - 64 \zeta_{6} q^{4} + 165 \zeta_{6} q^{5} + ( - 508 \zeta_{6} + 508) q^{7} + 512 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (8 \zeta_{6} - 8) q^{2} - 64 \zeta_{6} q^{4} + 165 \zeta_{6} q^{5} + ( - 508 \zeta_{6} + 508) q^{7} + 512 q^{8} - 1320 q^{10} + (3024 \zeta_{6} - 3024) q^{11} - 5039 \zeta_{6} q^{13} + 4064 \zeta_{6} q^{14} + (4096 \zeta_{6} - 4096) q^{16} - 3189 q^{17} + 1508 q^{19} + ( - 10560 \zeta_{6} + 10560) q^{20} - 24192 \zeta_{6} q^{22} + 75600 \zeta_{6} q^{23} + ( - 50900 \zeta_{6} + 50900) q^{25} + 40312 q^{26} - 32512 q^{28} + ( - 82665 \zeta_{6} + 82665) q^{29} + 174892 \zeta_{6} q^{31} - 32768 \zeta_{6} q^{32} + ( - 25512 \zeta_{6} + 25512) q^{34} + 83820 q^{35} - 323569 q^{37} + (12064 \zeta_{6} - 12064) q^{38} + 84480 \zeta_{6} q^{40} + 308118 \zeta_{6} q^{41} + (336680 \zeta_{6} - 336680) q^{43} + 193536 q^{44} - 604800 q^{46} + ( - 383196 \zeta_{6} + 383196) q^{47} + 565479 \zeta_{6} q^{49} + 407200 \zeta_{6} q^{50} + (322496 \zeta_{6} - 322496) q^{52} + 760206 q^{53} - 498960 q^{55} + ( - 260096 \zeta_{6} + 260096) q^{56} + 661320 \zeta_{6} q^{58} + 2225664 \zeta_{6} q^{59} + (2244815 \zeta_{6} - 2244815) q^{61} - 1399136 q^{62} + 262144 q^{64} + ( - 831435 \zeta_{6} + 831435) q^{65} - 1473188 \zeta_{6} q^{67} + 204096 \zeta_{6} q^{68} + (670560 \zeta_{6} - 670560) q^{70} - 5006892 q^{71} - 5898301 q^{73} + ( - 2588552 \zeta_{6} + 2588552) q^{74} - 96512 \zeta_{6} q^{76} + 1536192 \zeta_{6} q^{77} + (7028768 \zeta_{6} - 7028768) q^{79} - 675840 q^{80} - 2464944 q^{82} + ( - 2651196 \zeta_{6} + 2651196) q^{83} - 526185 \zeta_{6} q^{85} - 2693440 \zeta_{6} q^{86} + (1548288 \zeta_{6} - 1548288) q^{88} - 6770901 q^{89} - 2559812 q^{91} + ( - 4838400 \zeta_{6} + 4838400) q^{92} + 3065568 \zeta_{6} q^{94} + 248820 \zeta_{6} q^{95} + (16176386 \zeta_{6} - 16176386) q^{97} - 4523832 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} - 64 q^{4} + 165 q^{5} + 508 q^{7} + 1024 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} - 64 q^{4} + 165 q^{5} + 508 q^{7} + 1024 q^{8} - 2640 q^{10} - 3024 q^{11} - 5039 q^{13} + 4064 q^{14} - 4096 q^{16} - 6378 q^{17} + 3016 q^{19} + 10560 q^{20} - 24192 q^{22} + 75600 q^{23} + 50900 q^{25} + 80624 q^{26} - 65024 q^{28} + 82665 q^{29} + 174892 q^{31} - 32768 q^{32} + 25512 q^{34} + 167640 q^{35} - 647138 q^{37} - 12064 q^{38} + 84480 q^{40} + 308118 q^{41} - 336680 q^{43} + 387072 q^{44} - 1209600 q^{46} + 383196 q^{47} + 565479 q^{49} + 407200 q^{50} - 322496 q^{52} + 1520412 q^{53} - 997920 q^{55} + 260096 q^{56} + 661320 q^{58} + 2225664 q^{59} - 2244815 q^{61} - 2798272 q^{62} + 524288 q^{64} + 831435 q^{65} - 1473188 q^{67} + 204096 q^{68} - 670560 q^{70} - 10013784 q^{71} - 11796602 q^{73} + 2588552 q^{74} - 96512 q^{76} + 1536192 q^{77} - 7028768 q^{79} - 1351680 q^{80} - 4929888 q^{82} + 2651196 q^{83} - 526185 q^{85} - 2693440 q^{86} - 1548288 q^{88} - 13541802 q^{89} - 5119624 q^{91} + 4838400 q^{92} + 3065568 q^{94} + 248820 q^{95} - 16176386 q^{97} - 9047664 q^{98}+O(q^{100}) \) 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 82.5000 + 142.894i 0 254.000 439.941i 512.000 0 −1320.00
109.1 −4.00000 6.92820i 0 −32.0000 + 55.4256i 82.5000 142.894i 0 254.000 + 439.941i 512.000 0 −1320.00
\(n\): 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.e 2
3.b odd 2 1 162.8.c.h 2
9.c even 3 1 162.8.a.b yes 1
9.c even 3 1 inner 162.8.c.e 2
9.d odd 6 1 162.8.a.a 1
9.d odd 6 1 162.8.c.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.8.a.a 1 9.d odd 6 1
162.8.a.b yes 1 9.c even 3 1
162.8.c.e 2 1.a even 1 1 trivial
162.8.c.e 2 9.c even 3 1 inner
162.8.c.h 2 3.b odd 2 1
162.8.c.h 2 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 165T_{5} + 27225 \) 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
$5$ \( T^{2} - 165T + 27225 \) Copy content Toggle raw display
$7$ \( T^{2} - 508T + 258064 \) Copy content Toggle raw display
$11$ \( T^{2} + 3024 T + 9144576 \) Copy content Toggle raw display
$13$ \( T^{2} + 5039 T + 25391521 \) Copy content Toggle raw display
$17$ \( (T + 3189)^{2} \) Copy content Toggle raw display
$19$ \( (T - 1508)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 5715360000 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 6833502225 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 30587211664 \) Copy content Toggle raw display
$37$ \( (T + 323569)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 94936701924 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 113353422400 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 146839174416 \) Copy content Toggle raw display
$53$ \( (T - 760206)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 4953580240896 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 5039194384225 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 2170282883344 \) Copy content Toggle raw display
$71$ \( (T + 5006892)^{2} \) Copy content Toggle raw display
$73$ \( (T + 5898301)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 49403579597824 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 7028840230416 \) Copy content Toggle raw display
$89$ \( (T + 6770901)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 261675464020996 \) Copy content Toggle raw display
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