Properties

Label 81.3.b.a
Level $81$
Weight $3$
Character orbit 81.b
Analytic conductor $2.207$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 81.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.20709014132\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{2} + q^{4} + ( 2 - 4 \zeta_{6} ) q^{5} + 2 q^{7} + ( 5 - 10 \zeta_{6} ) q^{8} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{2} + q^{4} + ( 2 - 4 \zeta_{6} ) q^{5} + 2 q^{7} + ( 5 - 10 \zeta_{6} ) q^{8} -6 q^{10} + ( 1 - 2 \zeta_{6} ) q^{11} -4 q^{13} + ( 2 - 4 \zeta_{6} ) q^{14} -11 q^{16} + ( -9 + 18 \zeta_{6} ) q^{17} + 11 q^{19} + ( 2 - 4 \zeta_{6} ) q^{20} -3 q^{22} + ( -16 + 32 \zeta_{6} ) q^{23} + 13 q^{25} + ( -4 + 8 \zeta_{6} ) q^{26} + 2 q^{28} + ( -26 + 52 \zeta_{6} ) q^{29} + 32 q^{31} + ( 9 - 18 \zeta_{6} ) q^{32} + 27 q^{34} + ( 4 - 8 \zeta_{6} ) q^{35} -34 q^{37} + ( 11 - 22 \zeta_{6} ) q^{38} -30 q^{40} + ( -7 + 14 \zeta_{6} ) q^{41} -61 q^{43} + ( 1 - 2 \zeta_{6} ) q^{44} + 48 q^{46} + ( 28 - 56 \zeta_{6} ) q^{47} -45 q^{49} + ( 13 - 26 \zeta_{6} ) q^{50} -4 q^{52} -6 q^{55} + ( 10 - 20 \zeta_{6} ) q^{56} + 78 q^{58} + ( 29 - 58 \zeta_{6} ) q^{59} + 56 q^{61} + ( 32 - 64 \zeta_{6} ) q^{62} -71 q^{64} + ( -8 + 16 \zeta_{6} ) q^{65} -31 q^{67} + ( -9 + 18 \zeta_{6} ) q^{68} -12 q^{70} + ( 18 - 36 \zeta_{6} ) q^{71} + 65 q^{73} + ( -34 + 68 \zeta_{6} ) q^{74} + 11 q^{76} + ( 2 - 4 \zeta_{6} ) q^{77} + 38 q^{79} + ( -22 + 44 \zeta_{6} ) q^{80} + 21 q^{82} + ( 28 - 56 \zeta_{6} ) q^{83} + 54 q^{85} + ( -61 + 122 \zeta_{6} ) q^{86} -15 q^{88} + ( -72 + 144 \zeta_{6} ) q^{89} -8 q^{91} + ( -16 + 32 \zeta_{6} ) q^{92} -84 q^{94} + ( 22 - 44 \zeta_{6} ) q^{95} -115 q^{97} + ( -45 + 90 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 4 q^{7} + O(q^{10}) \) \( 2 q + 2 q^{4} + 4 q^{7} - 12 q^{10} - 8 q^{13} - 22 q^{16} + 22 q^{19} - 6 q^{22} + 26 q^{25} + 4 q^{28} + 64 q^{31} + 54 q^{34} - 68 q^{37} - 60 q^{40} - 122 q^{43} + 96 q^{46} - 90 q^{49} - 8 q^{52} - 12 q^{55} + 156 q^{58} + 112 q^{61} - 142 q^{64} - 62 q^{67} - 24 q^{70} + 130 q^{73} + 22 q^{76} + 76 q^{79} + 42 q^{82} + 108 q^{85} - 30 q^{88} - 16 q^{91} - 168 q^{94} - 230 q^{97} + O(q^{100}) \)

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1
0.500000 + 0.866025i
0.500000 0.866025i
1.73205i 0 1.00000 3.46410i 0 2.00000 8.66025i 0 −6.00000
80.2 1.73205i 0 1.00000 3.46410i 0 2.00000 8.66025i 0 −6.00000
\(n\): 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.3.b.a 2
3.b odd 2 1 inner 81.3.b.a 2
4.b odd 2 1 1296.3.e.a 2
9.c even 3 1 9.3.d.a 2
9.c even 3 1 27.3.d.a 2
9.d odd 6 1 9.3.d.a 2
9.d odd 6 1 27.3.d.a 2
12.b even 2 1 1296.3.e.a 2
36.f odd 6 1 144.3.q.a 2
36.f odd 6 1 432.3.q.a 2
36.h even 6 1 144.3.q.a 2
36.h even 6 1 432.3.q.a 2
45.h odd 6 1 225.3.j.a 2
45.h odd 6 1 675.3.j.a 2
45.j even 6 1 225.3.j.a 2
45.j even 6 1 675.3.j.a 2
45.k odd 12 2 225.3.i.a 4
45.k odd 12 2 675.3.i.a 4
45.l even 12 2 225.3.i.a 4
45.l even 12 2 675.3.i.a 4
63.g even 3 1 441.3.n.b 2
63.h even 3 1 441.3.j.a 2
63.i even 6 1 441.3.j.b 2
63.j odd 6 1 441.3.j.a 2
63.k odd 6 1 441.3.n.a 2
63.l odd 6 1 441.3.r.a 2
63.n odd 6 1 441.3.n.b 2
63.o even 6 1 441.3.r.a 2
63.s even 6 1 441.3.n.a 2
63.t odd 6 1 441.3.j.b 2
72.j odd 6 1 576.3.q.b 2
72.j odd 6 1 1728.3.q.a 2
72.l even 6 1 576.3.q.a 2
72.l even 6 1 1728.3.q.b 2
72.n even 6 1 576.3.q.b 2
72.n even 6 1 1728.3.q.a 2
72.p odd 6 1 576.3.q.a 2
72.p odd 6 1 1728.3.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.3.d.a 2 9.c even 3 1
9.3.d.a 2 9.d odd 6 1
27.3.d.a 2 9.c even 3 1
27.3.d.a 2 9.d odd 6 1
81.3.b.a 2 1.a even 1 1 trivial
81.3.b.a 2 3.b odd 2 1 inner
144.3.q.a 2 36.f odd 6 1
144.3.q.a 2 36.h even 6 1
225.3.i.a 4 45.k odd 12 2
225.3.i.a 4 45.l even 12 2
225.3.j.a 2 45.h odd 6 1
225.3.j.a 2 45.j even 6 1
432.3.q.a 2 36.f odd 6 1
432.3.q.a 2 36.h even 6 1
441.3.j.a 2 63.h even 3 1
441.3.j.a 2 63.j odd 6 1
441.3.j.b 2 63.i even 6 1
441.3.j.b 2 63.t odd 6 1
441.3.n.a 2 63.k odd 6 1
441.3.n.a 2 63.s even 6 1
441.3.n.b 2 63.g even 3 1
441.3.n.b 2 63.n odd 6 1
441.3.r.a 2 63.l odd 6 1
441.3.r.a 2 63.o even 6 1
576.3.q.a 2 72.l even 6 1
576.3.q.a 2 72.p odd 6 1
576.3.q.b 2 72.j odd 6 1
576.3.q.b 2 72.n even 6 1
675.3.i.a 4 45.k odd 12 2
675.3.i.a 4 45.l even 12 2
675.3.j.a 2 45.h odd 6 1
675.3.j.a 2 45.j even 6 1
1296.3.e.a 2 4.b odd 2 1
1296.3.e.a 2 12.b even 2 1
1728.3.q.a 2 72.j odd 6 1
1728.3.q.a 2 72.n even 6 1
1728.3.q.b 2 72.l even 6 1
1728.3.q.b 2 72.p odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 12 + T^{2} \)
$7$ \( ( -2 + T )^{2} \)
$11$ \( 3 + T^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( 243 + T^{2} \)
$19$ \( ( -11 + T )^{2} \)
$23$ \( 768 + T^{2} \)
$29$ \( 2028 + T^{2} \)
$31$ \( ( -32 + T )^{2} \)
$37$ \( ( 34 + T )^{2} \)
$41$ \( 147 + T^{2} \)
$43$ \( ( 61 + T )^{2} \)
$47$ \( 2352 + T^{2} \)
$53$ \( T^{2} \)
$59$ \( 2523 + T^{2} \)
$61$ \( ( -56 + T )^{2} \)
$67$ \( ( 31 + T )^{2} \)
$71$ \( 972 + T^{2} \)
$73$ \( ( -65 + T )^{2} \)
$79$ \( ( -38 + T )^{2} \)
$83$ \( 2352 + T^{2} \)
$89$ \( 15552 + T^{2} \)
$97$ \( ( 115 + T )^{2} \)
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