Properties

Label 162.12.c.b
Level $162$
Weight $12$
Character orbit 162.c
Analytic conductor $124.472$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,12,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, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.55");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 12 \)
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: \(124.471595251\)
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 6)
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.

\(f(q)\) \(=\) \( q + (32 \zeta_{6} - 32) q^{2} - 1024 \zeta_{6} q^{4} - 3630 \zeta_{6} q^{5} + (32936 \zeta_{6} - 32936) q^{7} + 32768 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (32 \zeta_{6} - 32) q^{2} - 1024 \zeta_{6} q^{4} - 3630 \zeta_{6} q^{5} + (32936 \zeta_{6} - 32936) q^{7} + 32768 q^{8} + 116160 q^{10} + ( - 758748 \zeta_{6} + 758748) q^{11} + 2482858 \zeta_{6} q^{13} - 1053952 \zeta_{6} q^{14} + (1048576 \zeta_{6} - 1048576) q^{16} + 8290386 q^{17} - 10867300 q^{19} + (3717120 \zeta_{6} - 3717120) q^{20} + 24279936 \zeta_{6} q^{22} - 20539272 \zeta_{6} q^{23} + ( - 35651225 \zeta_{6} + 35651225) q^{25} - 79451456 q^{26} + 33726464 q^{28} + (28814550 \zeta_{6} - 28814550) q^{29} - 150501392 \zeta_{6} q^{31} - 33554432 \zeta_{6} q^{32} + (265292352 \zeta_{6} - 265292352) q^{34} + 119557680 q^{35} - 319891714 q^{37} + ( - 347753600 \zeta_{6} + 347753600) q^{38} - 118947840 \zeta_{6} q^{40} + 368008998 \zeta_{6} q^{41} + (620469572 \zeta_{6} - 620469572) q^{43} - 776957952 q^{44} + 657256704 q^{46} + (2763110256 \zeta_{6} - 2763110256) q^{47} + 892546647 \zeta_{6} q^{49} + 1140839200 \zeta_{6} q^{50} + ( - 2542446592 \zeta_{6} + 2542446592) q^{52} - 268284258 q^{53} - 2754255240 q^{55} + (1079246848 \zeta_{6} - 1079246848) q^{56} - 922065600 \zeta_{6} q^{58} - 1672894740 \zeta_{6} q^{59} + ( - 7787197498 \zeta_{6} + 7787197498) q^{61} + 4816044544 q^{62} + 1073741824 q^{64} + ( - 9012774540 \zeta_{6} + 9012774540) q^{65} - 18706694156 \zeta_{6} q^{67} - 8489355264 \zeta_{6} q^{68} + (3825845760 \zeta_{6} - 3825845760) q^{70} - 8346990888 q^{71} + 19641746522 q^{73} + ( - 10236534848 \zeta_{6} + 10236534848) q^{74} + 11128115200 \zeta_{6} q^{76} + 24990124128 \zeta_{6} q^{77} + ( - 5873807200 \zeta_{6} + 5873807200) q^{79} + 3806330880 q^{80} - 11776287936 q^{82} + (8492558172 \zeta_{6} - 8492558172) q^{83} - 30094101180 \zeta_{6} q^{85} - 19855026304 \zeta_{6} q^{86} + ( - 24862654464 \zeta_{6} + 24862654464) q^{88} + 75527864010 q^{89} - 81775411088 q^{91} + (21032214528 \zeta_{6} - 21032214528) q^{92} - 88419528192 \zeta_{6} q^{94} + 39448299000 \zeta_{6} q^{95} + ( - 82356782494 \zeta_{6} + 82356782494) q^{97} - 28561492704 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{2} - 1024 q^{4} - 3630 q^{5} - 32936 q^{7} + 65536 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{2} - 1024 q^{4} - 3630 q^{5} - 32936 q^{7} + 65536 q^{8} + 232320 q^{10} + 758748 q^{11} + 2482858 q^{13} - 1053952 q^{14} - 1048576 q^{16} + 16580772 q^{17} - 21734600 q^{19} - 3717120 q^{20} + 24279936 q^{22} - 20539272 q^{23} + 35651225 q^{25} - 158902912 q^{26} + 67452928 q^{28} - 28814550 q^{29} - 150501392 q^{31} - 33554432 q^{32} - 265292352 q^{34} + 239115360 q^{35} - 639783428 q^{37} + 347753600 q^{38} - 118947840 q^{40} + 368008998 q^{41} - 620469572 q^{43} - 1553915904 q^{44} + 1314513408 q^{46} - 2763110256 q^{47} + 892546647 q^{49} + 1140839200 q^{50} + 2542446592 q^{52} - 536568516 q^{53} - 5508510480 q^{55} - 1079246848 q^{56} - 922065600 q^{58} - 1672894740 q^{59} + 7787197498 q^{61} + 9632089088 q^{62} + 2147483648 q^{64} + 9012774540 q^{65} - 18706694156 q^{67} - 8489355264 q^{68} - 3825845760 q^{70} - 16693981776 q^{71} + 39283493044 q^{73} + 10236534848 q^{74} + 11128115200 q^{76} + 24990124128 q^{77} + 5873807200 q^{79} + 7612661760 q^{80} - 23552575872 q^{82} - 8492558172 q^{83} - 30094101180 q^{85} - 19855026304 q^{86} + 24862654464 q^{88} + 151055728020 q^{89} - 163550822176 q^{91} - 21032214528 q^{92} - 88419528192 q^{94} + 39448299000 q^{95} + 82356782494 q^{97} - 57122985408 q^{98}+O(q^{100}) \) 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
−16.0000 + 27.7128i 0 −512.000 886.810i −1815.00 3143.67i 0 −16468.0 + 28523.4i 32768.0 0 116160.
109.1 −16.0000 27.7128i 0 −512.000 + 886.810i −1815.00 + 3143.67i 0 −16468.0 28523.4i 32768.0 0 116160.
\(n\): 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 162.12.c.b 2
3.b odd 2 1 162.12.c.i 2
9.c even 3 1 6.12.a.c 1
9.c even 3 1 inner 162.12.c.b 2
9.d odd 6 1 18.12.a.a 1
9.d odd 6 1 162.12.c.i 2
36.f odd 6 1 48.12.a.d 1
36.h even 6 1 144.12.a.e 1
45.j even 6 1 150.12.a.a 1
45.k odd 12 2 150.12.c.e 2
72.n even 6 1 192.12.a.c 1
72.p odd 6 1 192.12.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.12.a.c 1 9.c even 3 1
18.12.a.a 1 9.d odd 6 1
48.12.a.d 1 36.f odd 6 1
144.12.a.e 1 36.h even 6 1
150.12.a.a 1 45.j even 6 1
150.12.c.e 2 45.k odd 12 2
162.12.c.b 2 1.a even 1 1 trivial
162.12.c.b 2 9.c even 3 1 inner
162.12.c.i 2 3.b odd 2 1
162.12.c.i 2 9.d odd 6 1
192.12.a.c 1 72.n even 6 1
192.12.a.m 1 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3630T_{5} + 13176900 \) acting on \(S_{12}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 32T + 1024 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 3630 T + 13176900 \) Copy content Toggle raw display
$7$ \( T^{2} + 32936 T + 1084780096 \) Copy content Toggle raw display
$11$ \( T^{2} - 758748 T + 575698527504 \) Copy content Toggle raw display
$13$ \( T^{2} - 2482858 T + 6164583848164 \) Copy content Toggle raw display
$17$ \( (T - 8290386)^{2} \) Copy content Toggle raw display
$19$ \( (T + 10867300)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 421861694289984 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 830278291702500 \) Copy content Toggle raw display
$31$ \( T^{2} + 150501392 T + 22\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( (T + 319891714)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 368008998 T + 13\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{2} + 620469572 T + 38\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{2} + 2763110256 T + 76\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T + 268284258)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 1672894740 T + 27\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} - 7787197498 T + 60\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{2} + 18706694156 T + 34\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T + 8346990888)^{2} \) Copy content Toggle raw display
$73$ \( (T - 19641746522)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 5873807200 T + 34\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + 8492558172 T + 72\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T - 75527864010)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 82356782494 T + 67\!\cdots\!36 \) Copy content Toggle raw display
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