Properties

Label 81.6.c.a
Level $81$
Weight $6$
Character orbit 81.c
Analytic conductor $12.991$
Analytic rank $1$
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] = [81,6,Mod(28,81)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(81, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.28");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 81.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: \(12.9910894049\)
Analytic rank: \(1\)
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 3)
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.

\(f(q)\) \(=\) \( q + (6 \zeta_{6} - 6) q^{2} - 4 \zeta_{6} q^{4} + 6 \zeta_{6} q^{5} + ( - 40 \zeta_{6} + 40) q^{7} - 168 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (6 \zeta_{6} - 6) q^{2} - 4 \zeta_{6} q^{4} + 6 \zeta_{6} q^{5} + ( - 40 \zeta_{6} + 40) q^{7} - 168 q^{8} - 36 q^{10} + (564 \zeta_{6} - 564) q^{11} - 638 \zeta_{6} q^{13} + 240 \zeta_{6} q^{14} + ( - 1136 \zeta_{6} + 1136) q^{16} - 882 q^{17} - 556 q^{19} + ( - 24 \zeta_{6} + 24) q^{20} - 3384 \zeta_{6} q^{22} - 840 \zeta_{6} q^{23} + ( - 3089 \zeta_{6} + 3089) q^{25} + 3828 q^{26} - 160 q^{28} + ( - 4638 \zeta_{6} + 4638) q^{29} - 4400 \zeta_{6} q^{31} + 1440 \zeta_{6} q^{32} + ( - 5292 \zeta_{6} + 5292) q^{34} + 240 q^{35} - 2410 q^{37} + ( - 3336 \zeta_{6} + 3336) q^{38} - 1008 \zeta_{6} q^{40} - 6870 \zeta_{6} q^{41} + (9644 \zeta_{6} - 9644) q^{43} + 2256 q^{44} + 5040 q^{46} + (18672 \zeta_{6} - 18672) q^{47} + 15207 \zeta_{6} q^{49} + 18534 \zeta_{6} q^{50} + (2552 \zeta_{6} - 2552) q^{52} - 33750 q^{53} - 3384 q^{55} + (6720 \zeta_{6} - 6720) q^{56} + 27828 \zeta_{6} q^{58} - 18084 \zeta_{6} q^{59} + (39758 \zeta_{6} - 39758) q^{61} + 26400 q^{62} + 27712 q^{64} + ( - 3828 \zeta_{6} + 3828) q^{65} + 23068 \zeta_{6} q^{67} + 3528 \zeta_{6} q^{68} + (1440 \zeta_{6} - 1440) q^{70} + 4248 q^{71} - 41110 q^{73} + ( - 14460 \zeta_{6} + 14460) q^{74} + 2224 \zeta_{6} q^{76} + 22560 \zeta_{6} q^{77} + (21920 \zeta_{6} - 21920) q^{79} + 6816 q^{80} + 41220 q^{82} + ( - 82452 \zeta_{6} + 82452) q^{83} - 5292 \zeta_{6} q^{85} - 57864 \zeta_{6} q^{86} + ( - 94752 \zeta_{6} + 94752) q^{88} + 94086 q^{89} - 25520 q^{91} + (3360 \zeta_{6} - 3360) q^{92} - 112032 \zeta_{6} q^{94} - 3336 \zeta_{6} q^{95} + (49442 \zeta_{6} - 49442) q^{97} - 91242 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{2} - 4 q^{4} + 6 q^{5} + 40 q^{7} - 336 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{2} - 4 q^{4} + 6 q^{5} + 40 q^{7} - 336 q^{8} - 72 q^{10} - 564 q^{11} - 638 q^{13} + 240 q^{14} + 1136 q^{16} - 1764 q^{17} - 1112 q^{19} + 24 q^{20} - 3384 q^{22} - 840 q^{23} + 3089 q^{25} + 7656 q^{26} - 320 q^{28} + 4638 q^{29} - 4400 q^{31} + 1440 q^{32} + 5292 q^{34} + 480 q^{35} - 4820 q^{37} + 3336 q^{38} - 1008 q^{40} - 6870 q^{41} - 9644 q^{43} + 4512 q^{44} + 10080 q^{46} - 18672 q^{47} + 15207 q^{49} + 18534 q^{50} - 2552 q^{52} - 67500 q^{53} - 6768 q^{55} - 6720 q^{56} + 27828 q^{58} - 18084 q^{59} - 39758 q^{61} + 52800 q^{62} + 55424 q^{64} + 3828 q^{65} + 23068 q^{67} + 3528 q^{68} - 1440 q^{70} + 8496 q^{71} - 82220 q^{73} + 14460 q^{74} + 2224 q^{76} + 22560 q^{77} - 21920 q^{79} + 13632 q^{80} + 82440 q^{82} + 82452 q^{83} - 5292 q^{85} - 57864 q^{86} + 94752 q^{88} + 188172 q^{89} - 51040 q^{91} - 3360 q^{92} - 112032 q^{94} - 3336 q^{95} - 49442 q^{97} - 182484 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
0.500000 0.866025i
0.500000 + 0.866025i
−3.00000 5.19615i 0 −2.00000 + 3.46410i 3.00000 5.19615i 0 20.0000 + 34.6410i −168.000 0 −36.0000
55.1 −3.00000 + 5.19615i 0 −2.00000 3.46410i 3.00000 + 5.19615i 0 20.0000 34.6410i −168.000 0 −36.0000
\(n\): 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 81.6.c.a 2
3.b odd 2 1 81.6.c.c 2
9.c even 3 1 9.6.a.a 1
9.c even 3 1 inner 81.6.c.a 2
9.d odd 6 1 3.6.a.a 1
9.d odd 6 1 81.6.c.c 2
36.f odd 6 1 144.6.a.f 1
36.h even 6 1 48.6.a.a 1
45.h odd 6 1 75.6.a.e 1
45.j even 6 1 225.6.a.a 1
45.k odd 12 2 225.6.b.b 2
45.l even 12 2 75.6.b.b 2
63.i even 6 1 147.6.e.k 2
63.j odd 6 1 147.6.e.h 2
63.l odd 6 1 441.6.a.i 1
63.n odd 6 1 147.6.e.h 2
63.o even 6 1 147.6.a.a 1
63.s even 6 1 147.6.e.k 2
72.j odd 6 1 192.6.a.d 1
72.l even 6 1 192.6.a.l 1
72.n even 6 1 576.6.a.s 1
72.p odd 6 1 576.6.a.t 1
99.g even 6 1 363.6.a.d 1
99.h odd 6 1 1089.6.a.b 1
117.n odd 6 1 507.6.a.b 1
144.u even 12 2 768.6.d.h 2
144.w odd 12 2 768.6.d.k 2
153.i odd 6 1 867.6.a.a 1
171.l even 6 1 1083.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.6.a.a 1 9.d odd 6 1
9.6.a.a 1 9.c even 3 1
48.6.a.a 1 36.h even 6 1
75.6.a.e 1 45.h odd 6 1
75.6.b.b 2 45.l even 12 2
81.6.c.a 2 1.a even 1 1 trivial
81.6.c.a 2 9.c even 3 1 inner
81.6.c.c 2 3.b odd 2 1
81.6.c.c 2 9.d odd 6 1
144.6.a.f 1 36.f odd 6 1
147.6.a.a 1 63.o even 6 1
147.6.e.h 2 63.j odd 6 1
147.6.e.h 2 63.n odd 6 1
147.6.e.k 2 63.i even 6 1
147.6.e.k 2 63.s even 6 1
192.6.a.d 1 72.j odd 6 1
192.6.a.l 1 72.l even 6 1
225.6.a.a 1 45.j even 6 1
225.6.b.b 2 45.k odd 12 2
363.6.a.d 1 99.g even 6 1
441.6.a.i 1 63.l odd 6 1
507.6.a.b 1 117.n odd 6 1
576.6.a.s 1 72.n even 6 1
576.6.a.t 1 72.p odd 6 1
768.6.d.h 2 144.u even 12 2
768.6.d.k 2 144.w odd 12 2
867.6.a.a 1 153.i odd 6 1
1083.6.a.c 1 171.l even 6 1
1089.6.a.b 1 99.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 6T_{2} + 36 \) acting on \(S_{6}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$7$ \( T^{2} - 40T + 1600 \) Copy content Toggle raw display
$11$ \( T^{2} + 564T + 318096 \) Copy content Toggle raw display
$13$ \( T^{2} + 638T + 407044 \) Copy content Toggle raw display
$17$ \( (T + 882)^{2} \) Copy content Toggle raw display
$19$ \( (T + 556)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 840T + 705600 \) Copy content Toggle raw display
$29$ \( T^{2} - 4638 T + 21511044 \) Copy content Toggle raw display
$31$ \( T^{2} + 4400 T + 19360000 \) Copy content Toggle raw display
$37$ \( (T + 2410)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6870 T + 47196900 \) Copy content Toggle raw display
$43$ \( T^{2} + 9644 T + 93006736 \) Copy content Toggle raw display
$47$ \( T^{2} + 18672 T + 348643584 \) Copy content Toggle raw display
$53$ \( (T + 33750)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 18084 T + 327031056 \) Copy content Toggle raw display
$61$ \( T^{2} + 39758 T + 1580698564 \) Copy content Toggle raw display
$67$ \( T^{2} - 23068 T + 532132624 \) Copy content Toggle raw display
$71$ \( (T - 4248)^{2} \) Copy content Toggle raw display
$73$ \( (T + 41110)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 21920 T + 480486400 \) Copy content Toggle raw display
$83$ \( T^{2} - 82452 T + 6798332304 \) Copy content Toggle raw display
$89$ \( (T - 94086)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 49442 T + 2444511364 \) Copy content Toggle raw display
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