Properties

Label 81.4.a.d
Level $81$
Weight $4$
Character orbit 81.a
Self dual yes
Analytic conductor $4.779$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 81.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.77915471046\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 + \beta ) q^{2} + ( 1 + 3 \beta ) q^{4} + ( 8 - \beta ) q^{5} + ( 2 + 3 \beta ) q^{7} + ( 17 - \beta ) q^{8} +O(q^{10})\) \( q + ( 1 + \beta ) q^{2} + ( 1 + 3 \beta ) q^{4} + ( 8 - \beta ) q^{5} + ( 2 + 3 \beta ) q^{7} + ( 17 - \beta ) q^{8} + 6 \beta q^{10} + ( 37 - 8 \beta ) q^{11} + ( 2 - 15 \beta ) q^{13} + ( 26 + 8 \beta ) q^{14} + ( 1 - 9 \beta ) q^{16} + ( 45 + 9 \beta ) q^{17} + ( -25 - 27 \beta ) q^{19} + ( -16 + 20 \beta ) q^{20} + ( -27 + 21 \beta ) q^{22} + ( 26 - 19 \beta ) q^{23} + ( -53 - 15 \beta ) q^{25} + ( -118 - 28 \beta ) q^{26} + ( 74 + 18 \beta ) q^{28} + ( -26 + \beta ) q^{29} + ( 20 + 3 \beta ) q^{31} + ( -207 - 9 \beta ) q^{32} + ( 117 + 63 \beta ) q^{34} + ( -8 + 19 \beta ) q^{35} + ( -52 + 54 \beta ) q^{37} + ( -241 - 79 \beta ) q^{38} + ( 144 - 24 \beta ) q^{40} + ( 17 + 98 \beta ) q^{41} + ( 47 - 6 \beta ) q^{43} + ( -155 + 79 \beta ) q^{44} + ( -126 - 12 \beta ) q^{46} + ( 154 + 91 \beta ) q^{47} + ( -267 + 21 \beta ) q^{49} + ( -173 - 83 \beta ) q^{50} + ( -358 - 54 \beta ) q^{52} + ( 108 - 162 \beta ) q^{53} + ( 360 - 93 \beta ) q^{55} + ( 10 + 46 \beta ) q^{56} + ( -18 - 24 \beta ) q^{58} + ( 467 - 136 \beta ) q^{59} + ( 272 - 105 \beta ) q^{61} + ( 44 + 26 \beta ) q^{62} + ( -287 - 153 \beta ) q^{64} + ( 136 - 107 \beta ) q^{65} + ( 461 + 66 \beta ) q^{67} + ( 261 + 171 \beta ) q^{68} + ( 144 + 30 \beta ) q^{70} + ( 612 + 144 \beta ) q^{71} + ( -349 + 243 \beta ) q^{73} + ( 380 + 56 \beta ) q^{74} + ( -673 - 183 \beta ) q^{76} + ( -118 + 71 \beta ) q^{77} + ( -556 + 309 \beta ) q^{79} + ( 80 - 64 \beta ) q^{80} + ( 801 + 213 \beta ) q^{82} + ( 460 - 107 \beta ) q^{83} + ( 288 + 18 \beta ) q^{85} + ( -1 + 35 \beta ) q^{86} + ( 693 - 165 \beta ) q^{88} + ( -234 + 72 \beta ) q^{89} + ( -356 - 69 \beta ) q^{91} + ( -430 + 2 \beta ) q^{92} + ( 882 + 336 \beta ) q^{94} + ( 16 - 164 \beta ) q^{95} + ( 317 + 102 \beta ) q^{97} + ( -99 - 225 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 3q^{2} + 5q^{4} + 15q^{5} + 7q^{7} + 33q^{8} + O(q^{10}) \) \( 2q + 3q^{2} + 5q^{4} + 15q^{5} + 7q^{7} + 33q^{8} + 6q^{10} + 66q^{11} - 11q^{13} + 60q^{14} - 7q^{16} + 99q^{17} - 77q^{19} - 12q^{20} - 33q^{22} + 33q^{23} - 121q^{25} - 264q^{26} + 166q^{28} - 51q^{29} + 43q^{31} - 423q^{32} + 297q^{34} + 3q^{35} - 50q^{37} - 561q^{38} + 264q^{40} + 132q^{41} + 88q^{43} - 231q^{44} - 264q^{46} + 399q^{47} - 513q^{49} - 429q^{50} - 770q^{52} + 54q^{53} + 627q^{55} + 66q^{56} - 60q^{58} + 798q^{59} + 439q^{61} + 114q^{62} - 727q^{64} + 165q^{65} + 988q^{67} + 693q^{68} + 318q^{70} + 1368q^{71} - 455q^{73} + 816q^{74} - 1529q^{76} - 165q^{77} - 803q^{79} + 96q^{80} + 1815q^{82} + 813q^{83} + 594q^{85} + 33q^{86} + 1221q^{88} - 396q^{89} - 781q^{91} - 858q^{92} + 2100q^{94} - 132q^{95} + 736q^{97} - 423q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.37228
3.37228
−1.37228 0 −6.11684 10.3723 0 −5.11684 19.3723 0 −14.2337
1.2 4.37228 0 11.1168 4.62772 0 12.1168 13.6277 0 20.2337
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.4.a.d 2
3.b odd 2 1 81.4.a.a 2
4.b odd 2 1 1296.4.a.u 2
5.b even 2 1 2025.4.a.g 2
9.c even 3 2 9.4.c.a 4
9.d odd 6 2 27.4.c.a 4
12.b even 2 1 1296.4.a.i 2
15.d odd 2 1 2025.4.a.n 2
36.f odd 6 2 144.4.i.c 4
36.h even 6 2 432.4.i.c 4
45.j even 6 2 225.4.e.b 4
45.k odd 12 4 225.4.k.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a 4 9.c even 3 2
27.4.c.a 4 9.d odd 6 2
81.4.a.a 2 3.b odd 2 1
81.4.a.d 2 1.a even 1 1 trivial
144.4.i.c 4 36.f odd 6 2
225.4.e.b 4 45.j even 6 2
225.4.k.b 8 45.k odd 12 4
432.4.i.c 4 36.h even 6 2
1296.4.a.i 2 12.b even 2 1
1296.4.a.u 2 4.b odd 2 1
2025.4.a.g 2 5.b even 2 1
2025.4.a.n 2 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3 T_{2} - 6 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(81))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -6 - 3 T + T^{2} \)
$3$ \( T^{2} \)
$5$ \( 48 - 15 T + T^{2} \)
$7$ \( -62 - 7 T + T^{2} \)
$11$ \( 561 - 66 T + T^{2} \)
$13$ \( -1826 + 11 T + T^{2} \)
$17$ \( 1782 - 99 T + T^{2} \)
$19$ \( -4532 + 77 T + T^{2} \)
$23$ \( -2706 - 33 T + T^{2} \)
$29$ \( 642 + 51 T + T^{2} \)
$31$ \( 388 - 43 T + T^{2} \)
$37$ \( -23432 + 50 T + T^{2} \)
$41$ \( -74877 - 132 T + T^{2} \)
$43$ \( 1639 - 88 T + T^{2} \)
$47$ \( -28518 - 399 T + T^{2} \)
$53$ \( -215784 - 54 T + T^{2} \)
$59$ \( 6609 - 798 T + T^{2} \)
$61$ \( -42776 - 439 T + T^{2} \)
$67$ \( 208099 - 988 T + T^{2} \)
$71$ \( 296784 - 1368 T + T^{2} \)
$73$ \( -435398 + 455 T + T^{2} \)
$79$ \( -626516 + 803 T + T^{2} \)
$83$ \( 70788 - 813 T + T^{2} \)
$89$ \( -3564 + 396 T + T^{2} \)
$97$ \( 49591 - 736 T + T^{2} \)
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