Properties

Label 162.12.c.d
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.

\(f(q)\) \(=\) \( q + (32 \zeta_{6} - 32) q^{2} - 1024 \zeta_{6} q^{4} + 5766 \zeta_{6} q^{5} + (72464 \zeta_{6} - 72464) q^{7} + 32768 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (32 \zeta_{6} - 32) q^{2} - 1024 \zeta_{6} q^{4} + 5766 \zeta_{6} q^{5} + (72464 \zeta_{6} - 72464) q^{7} + 32768 q^{8} - 184512 q^{10} + (408948 \zeta_{6} - 408948) q^{11} - 1367558 \zeta_{6} q^{13} - 2318848 \zeta_{6} q^{14} + (1048576 \zeta_{6} - 1048576) q^{16} - 5422914 q^{17} + 15166100 q^{19} + ( - 5904384 \zeta_{6} + 5904384) q^{20} - 13086336 \zeta_{6} q^{22} - 52194072 \zeta_{6} q^{23} + ( - 15581369 \zeta_{6} + 15581369) q^{25} + 43761856 q^{26} + 74203136 q^{28} + ( - 118581150 \zeta_{6} + 118581150) q^{29} + 57652408 \zeta_{6} q^{31} - 33554432 \zeta_{6} q^{32} + ( - 173533248 \zeta_{6} + 173533248) q^{34} - 417827424 q^{35} - 375985186 q^{37} + (485315200 \zeta_{6} - 485315200) q^{38} + 188940288 \zeta_{6} q^{40} + 856316202 \zeta_{6} q^{41} + ( - 1245189172 \zeta_{6} + 1245189172) q^{43} + 418762752 q^{44} + 1670210304 q^{46} + (1306762656 \zeta_{6} - 1306762656) q^{47} - 3273704553 \zeta_{6} q^{49} + 498603808 \zeta_{6} q^{50} + (1400379392 \zeta_{6} - 1400379392) q^{52} - 409556358 q^{53} - 2357994168 q^{55} + (2374500352 \zeta_{6} - 2374500352) q^{56} + 3794596800 \zeta_{6} q^{58} - 2882866260 \zeta_{6} q^{59} + (5731767302 \zeta_{6} - 5731767302) q^{61} - 1844877056 q^{62} + 1073741824 q^{64} + ( - 7885339428 \zeta_{6} + 7885339428) q^{65} - 3893272244 \zeta_{6} q^{67} + 5553063936 \zeta_{6} q^{68} + ( - 13370477568 \zeta_{6} + 13370477568) q^{70} + 9075890088 q^{71} - 15571822822 q^{73} + ( - 12031525952 \zeta_{6} + 12031525952) q^{74} - 15530086400 \zeta_{6} q^{76} - 29634007872 \zeta_{6} q^{77} + ( - 30196762600 \zeta_{6} + 30196762600) q^{79} - 6046089216 q^{80} - 27402118464 q^{82} + ( - 23135252628 \zeta_{6} + 23135252628) q^{83} - 31268522124 \zeta_{6} q^{85} + 39846053504 \zeta_{6} q^{86} + (13400408064 \zeta_{6} - 13400408064) q^{88} + 25614819990 q^{89} + 99098722912 q^{91} + (53446729728 \zeta_{6} - 53446729728) q^{92} - 41816404992 \zeta_{6} q^{94} + 87447732600 \zeta_{6} q^{95} + ( - 61937553406 \zeta_{6} + 61937553406) q^{97} + 104758545696 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{2} - 1024 q^{4} + 5766 q^{5} - 72464 q^{7} + 65536 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{2} - 1024 q^{4} + 5766 q^{5} - 72464 q^{7} + 65536 q^{8} - 369024 q^{10} - 408948 q^{11} - 1367558 q^{13} - 2318848 q^{14} - 1048576 q^{16} - 10845828 q^{17} + 30332200 q^{19} + 5904384 q^{20} - 13086336 q^{22} - 52194072 q^{23} + 15581369 q^{25} + 87523712 q^{26} + 148406272 q^{28} + 118581150 q^{29} + 57652408 q^{31} - 33554432 q^{32} + 173533248 q^{34} - 835654848 q^{35} - 751970372 q^{37} - 485315200 q^{38} + 188940288 q^{40} + 856316202 q^{41} + 1245189172 q^{43} + 837525504 q^{44} + 3340420608 q^{46} - 1306762656 q^{47} - 3273704553 q^{49} + 498603808 q^{50} - 1400379392 q^{52} - 819112716 q^{53} - 4715988336 q^{55} - 2374500352 q^{56} + 3794596800 q^{58} - 2882866260 q^{59} - 5731767302 q^{61} - 3689754112 q^{62} + 2147483648 q^{64} + 7885339428 q^{65} - 3893272244 q^{67} + 5553063936 q^{68} + 13370477568 q^{70} + 18151780176 q^{71} - 31143645644 q^{73} + 12031525952 q^{74} - 15530086400 q^{76} - 29634007872 q^{77} + 30196762600 q^{79} - 12092178432 q^{80} - 54804236928 q^{82} + 23135252628 q^{83} - 31268522124 q^{85} + 39846053504 q^{86} - 13400408064 q^{88} + 51229639980 q^{89} + 198197445824 q^{91} - 53446729728 q^{92} - 41816404992 q^{94} + 87447732600 q^{95} + 61937553406 q^{97} + 209517091392 q^{98}+O(q^{100}) \) 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 2883.00 + 4993.50i 0 −36232.0 + 62755.7i 32768.0 0 −184512.
109.1 −16.0000 27.7128i 0 −512.000 + 886.810i 2883.00 4993.50i 0 −36232.0 62755.7i 32768.0 0 −184512.
\(n\): 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.d 2
3.b odd 2 1 162.12.c.g 2
9.c even 3 1 18.12.a.c 1
9.c even 3 1 inner 162.12.c.d 2
9.d odd 6 1 6.12.a.a 1
9.d odd 6 1 162.12.c.g 2
36.f odd 6 1 144.12.a.b 1
36.h even 6 1 48.12.a.h 1
45.h odd 6 1 150.12.a.g 1
45.l even 12 2 150.12.c.f 2
72.j odd 6 1 192.12.a.l 1
72.l even 6 1 192.12.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.12.a.a 1 9.d odd 6 1
18.12.a.c 1 9.c even 3 1
48.12.a.h 1 36.h even 6 1
144.12.a.b 1 36.f odd 6 1
150.12.a.g 1 45.h odd 6 1
150.12.c.f 2 45.l even 12 2
162.12.c.d 2 1.a even 1 1 trivial
162.12.c.d 2 9.c even 3 1 inner
162.12.c.g 2 3.b odd 2 1
162.12.c.g 2 9.d odd 6 1
192.12.a.b 1 72.l even 6 1
192.12.a.l 1 72.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 5766T_{5} + 33246756 \) 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
$5$ \( T^{2} - 5766 T + 33246756 \) Copy content Toggle raw display
$7$ \( T^{2} + 72464 T + 5251031296 \) Copy content Toggle raw display
$11$ \( T^{2} + 408948 T + 167238466704 \) Copy content Toggle raw display
$13$ \( T^{2} + 1367558 T + 1870214883364 \) Copy content Toggle raw display
$17$ \( (T + 5422914)^{2} \) Copy content Toggle raw display
$19$ \( (T - 15166100)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 52194072 T + 27\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( T^{2} - 118581150 T + 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} - 57652408 T + 33\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( (T + 375985186)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 856316202 T + 73\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{2} - 1245189172 T + 15\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{2} + 1306762656 T + 17\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T + 409556358)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 2882866260 T + 83\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + 5731767302 T + 32\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{2} + 3893272244 T + 15\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T - 9075890088)^{2} \) Copy content Toggle raw display
$73$ \( (T + 15571822822)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 30196762600 T + 91\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} - 23135252628 T + 53\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( (T - 25614819990)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 61937553406 T + 38\!\cdots\!36 \) Copy content Toggle raw display
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