Properties

Label 81.4.c.b
Level $81$
Weight $4$
Character orbit 81.c
Analytic conductor $4.779$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(4.77915471046\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + 8 \zeta_{6} q^{4} + ( -20 + 20 \zeta_{6} ) q^{7} +O(q^{10})\) \( q + 8 \zeta_{6} q^{4} + ( -20 + 20 \zeta_{6} ) q^{7} + 70 \zeta_{6} q^{13} + ( -64 + 64 \zeta_{6} ) q^{16} + 56 q^{19} + ( 125 - 125 \zeta_{6} ) q^{25} -160 q^{28} -308 \zeta_{6} q^{31} + 110 q^{37} + ( 520 - 520 \zeta_{6} ) q^{43} -57 \zeta_{6} q^{49} + ( -560 + 560 \zeta_{6} ) q^{52} + ( -182 + 182 \zeta_{6} ) q^{61} -512 q^{64} + 880 \zeta_{6} q^{67} + 1190 q^{73} + 448 \zeta_{6} q^{76} + ( -884 + 884 \zeta_{6} ) q^{79} -1400 q^{91} + ( 1330 - 1330 \zeta_{6} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 8q^{4} - 20q^{7} + O(q^{10}) \) \( 2q + 8q^{4} - 20q^{7} + 70q^{13} - 64q^{16} + 112q^{19} + 125q^{25} - 320q^{28} - 308q^{31} + 220q^{37} + 520q^{43} - 57q^{49} - 560q^{52} - 182q^{61} - 1024q^{64} + 880q^{67} + 2380q^{73} + 448q^{76} - 884q^{79} - 2800q^{91} + 1330q^{97} + O(q^{100}) \)

Character values

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

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

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.

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
0 0 4.00000 6.92820i 0 0 −10.0000 17.3205i 0 0 0
55.1 0 0 4.00000 + 6.92820i 0 0 −10.0000 + 17.3205i 0 0 0
\(n\): 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 CM by \(\Q(\sqrt{-3}) \)
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 81.4.c.b 2
3.b odd 2 1 CM 81.4.c.b 2
9.c even 3 1 9.4.a.a 1
9.c even 3 1 inner 81.4.c.b 2
9.d odd 6 1 9.4.a.a 1
9.d odd 6 1 inner 81.4.c.b 2
36.f odd 6 1 144.4.a.d 1
36.h even 6 1 144.4.a.d 1
45.h odd 6 1 225.4.a.d 1
45.j even 6 1 225.4.a.d 1
45.k odd 12 2 225.4.b.g 2
45.l even 12 2 225.4.b.g 2
63.g even 3 1 441.4.e.i 2
63.h even 3 1 441.4.e.i 2
63.i even 6 1 441.4.e.j 2
63.j odd 6 1 441.4.e.i 2
63.k odd 6 1 441.4.e.j 2
63.l odd 6 1 441.4.a.f 1
63.n odd 6 1 441.4.e.i 2
63.o even 6 1 441.4.a.f 1
63.s even 6 1 441.4.e.j 2
63.t odd 6 1 441.4.e.j 2
72.j odd 6 1 576.4.a.m 1
72.l even 6 1 576.4.a.l 1
72.n even 6 1 576.4.a.m 1
72.p odd 6 1 576.4.a.l 1
99.g even 6 1 1089.4.a.g 1
99.h odd 6 1 1089.4.a.g 1
117.n odd 6 1 1521.4.a.g 1
117.t even 6 1 1521.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.a.a 1 9.c even 3 1
9.4.a.a 1 9.d odd 6 1
81.4.c.b 2 1.a even 1 1 trivial
81.4.c.b 2 3.b odd 2 1 CM
81.4.c.b 2 9.c even 3 1 inner
81.4.c.b 2 9.d odd 6 1 inner
144.4.a.d 1 36.f odd 6 1
144.4.a.d 1 36.h even 6 1
225.4.a.d 1 45.h odd 6 1
225.4.a.d 1 45.j even 6 1
225.4.b.g 2 45.k odd 12 2
225.4.b.g 2 45.l even 12 2
441.4.a.f 1 63.l odd 6 1
441.4.a.f 1 63.o even 6 1
441.4.e.i 2 63.g even 3 1
441.4.e.i 2 63.h even 3 1
441.4.e.i 2 63.j odd 6 1
441.4.e.i 2 63.n odd 6 1
441.4.e.j 2 63.i even 6 1
441.4.e.j 2 63.k odd 6 1
441.4.e.j 2 63.s even 6 1
441.4.e.j 2 63.t odd 6 1
576.4.a.l 1 72.l even 6 1
576.4.a.l 1 72.p odd 6 1
576.4.a.m 1 72.j odd 6 1
576.4.a.m 1 72.n even 6 1
1089.4.a.g 1 99.g even 6 1
1089.4.a.g 1 99.h odd 6 1
1521.4.a.g 1 117.n odd 6 1
1521.4.a.g 1 117.t even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 400 + 20 T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( 4900 - 70 T + T^{2} \)
$17$ \( T^{2} \)
$19$ \( ( -56 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 94864 + 308 T + T^{2} \)
$37$ \( ( -110 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 270400 - 520 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 33124 + 182 T + T^{2} \)
$67$ \( 774400 - 880 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -1190 + T )^{2} \)
$79$ \( 781456 + 884 T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 1768900 - 1330 T + T^{2} \)
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