Properties

Label 81.2.c.b
Level $81$
Weight $2$
Character orbit 81.c
Analytic conductor $0.647$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 81.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.646788256372\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2}) q^{5} - 2 \beta_1 q^{7} - \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_1 - 1) q^{4} + (\beta_{3} - \beta_{2}) q^{5} - 2 \beta_1 q^{7} - \beta_{3} q^{8} - 3 q^{10} + 2 \beta_{2} q^{11} + ( - \beta_1 + 1) q^{13} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{14} + 5 \beta_1 q^{16} + 3 \beta_{3} q^{17} + 2 q^{19} + \beta_{2} q^{20} + ( - 6 \beta_1 + 6) q^{22} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{23} + 2 \beta_1 q^{25} - \beta_{3} q^{26} + 2 q^{28} - \beta_{2} q^{29} + (8 \beta_1 - 8) q^{31} + (3 \beta_{3} - 3 \beta_{2}) q^{32} - 9 \beta_1 q^{34} - 2 \beta_{3} q^{35} - 7 q^{37} - 2 \beta_{2} q^{38} + (3 \beta_1 - 3) q^{40} + (4 \beta_{3} - 4 \beta_{2}) q^{41} - 2 \beta_1 q^{43} - 2 \beta_{3} q^{44} + 6 q^{46} - 4 \beta_{2} q^{47} + ( - 3 \beta_1 + 3) q^{49} + (2 \beta_{3} - 2 \beta_{2}) q^{50} + \beta_1 q^{52} + 6 q^{55} + 2 \beta_{2} q^{56} + (3 \beta_1 - 3) q^{58} + ( - 8 \beta_{3} + 8 \beta_{2}) q^{59} + 7 \beta_1 q^{61} + 8 \beta_{3} q^{62} + q^{64} - \beta_{2} q^{65} + ( - 10 \beta_1 + 10) q^{67} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{68} + 6 \beta_1 q^{70} - 6 \beta_{3} q^{71} - 7 q^{73} + 7 \beta_{2} q^{74} + (2 \beta_1 - 2) q^{76} + (4 \beta_{3} - 4 \beta_{2}) q^{77} - 2 \beta_1 q^{79} + 5 \beta_{3} q^{80} - 12 q^{82} + 8 \beta_{2} q^{83} + ( - 9 \beta_1 + 9) q^{85} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{86} - 6 \beta_1 q^{88} - 3 \beta_{3} q^{89} - 2 q^{91} - 2 \beta_{2} q^{92} + (12 \beta_1 - 12) q^{94} + (2 \beta_{3} - 2 \beta_{2}) q^{95} - 2 \beta_1 q^{97} - 3 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{4} - 4 q^{7} - 12 q^{10} + 2 q^{13} + 10 q^{16} + 8 q^{19} + 12 q^{22} + 4 q^{25} + 8 q^{28} - 16 q^{31} - 18 q^{34} - 28 q^{37} - 6 q^{40} - 4 q^{43} + 24 q^{46} + 6 q^{49} + 2 q^{52} + 24 q^{55} - 6 q^{58} + 14 q^{61} + 4 q^{64} + 20 q^{67} + 12 q^{70} - 28 q^{73} - 4 q^{76} - 4 q^{79} - 48 q^{82} + 18 q^{85} - 12 q^{88} - 8 q^{91} - 24 q^{94} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{1}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 1.50000i 0 −0.500000 + 0.866025i 0.866025 1.50000i 0 −1.00000 1.73205i −1.73205 0 −3.00000
28.2 0.866025 + 1.50000i 0 −0.500000 + 0.866025i −0.866025 + 1.50000i 0 −1.00000 1.73205i 1.73205 0 −3.00000
55.1 −0.866025 + 1.50000i 0 −0.500000 0.866025i 0.866025 + 1.50000i 0 −1.00000 + 1.73205i −1.73205 0 −3.00000
55.2 0.866025 1.50000i 0 −0.500000 0.866025i −0.866025 1.50000i 0 −1.00000 + 1.73205i 1.73205 0 −3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.2.c.b 4
3.b odd 2 1 inner 81.2.c.b 4
4.b odd 2 1 1296.2.i.s 4
9.c even 3 1 81.2.a.a 2
9.c even 3 1 inner 81.2.c.b 4
9.d odd 6 1 81.2.a.a 2
9.d odd 6 1 inner 81.2.c.b 4
12.b even 2 1 1296.2.i.s 4
27.e even 9 6 729.2.e.o 12
27.f odd 18 6 729.2.e.o 12
36.f odd 6 1 1296.2.a.o 2
36.f odd 6 1 1296.2.i.s 4
36.h even 6 1 1296.2.a.o 2
36.h even 6 1 1296.2.i.s 4
45.h odd 6 1 2025.2.a.j 2
45.j even 6 1 2025.2.a.j 2
45.k odd 12 2 2025.2.b.k 4
45.l even 12 2 2025.2.b.k 4
63.l odd 6 1 3969.2.a.i 2
63.o even 6 1 3969.2.a.i 2
72.j odd 6 1 5184.2.a.br 2
72.l even 6 1 5184.2.a.bq 2
72.n even 6 1 5184.2.a.br 2
72.p odd 6 1 5184.2.a.bq 2
99.g even 6 1 9801.2.a.v 2
99.h odd 6 1 9801.2.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.2.a.a 2 9.c even 3 1
81.2.a.a 2 9.d odd 6 1
81.2.c.b 4 1.a even 1 1 trivial
81.2.c.b 4 3.b odd 2 1 inner
81.2.c.b 4 9.c even 3 1 inner
81.2.c.b 4 9.d odd 6 1 inner
729.2.e.o 12 27.e even 9 6
729.2.e.o 12 27.f odd 18 6
1296.2.a.o 2 36.f odd 6 1
1296.2.a.o 2 36.h even 6 1
1296.2.i.s 4 4.b odd 2 1
1296.2.i.s 4 12.b even 2 1
1296.2.i.s 4 36.f odd 6 1
1296.2.i.s 4 36.h even 6 1
2025.2.a.j 2 45.h odd 6 1
2025.2.a.j 2 45.j even 6 1
2025.2.b.k 4 45.k odd 12 2
2025.2.b.k 4 45.l even 12 2
3969.2.a.i 2 63.l odd 6 1
3969.2.a.i 2 63.o even 6 1
5184.2.a.bq 2 72.l even 6 1
5184.2.a.bq 2 72.p odd 6 1
5184.2.a.br 2 72.j odd 6 1
5184.2.a.br 2 72.n even 6 1
9801.2.a.v 2 99.g even 6 1
9801.2.a.v 2 99.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 3T_{2}^{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$13$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$19$ \( (T - 2)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$29$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$37$ \( (T + 7)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$43$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 192 T^{2} + 36864 \) Copy content Toggle raw display
$61$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$73$ \( (T + 7)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 192 T^{2} + 36864 \) Copy content Toggle raw display
$89$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
show more
show less