Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,4,Mod(1,81)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(81, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 81.a (trivial)

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: yes
Analytic conductor: \(4.77915471046\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
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

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.37228
−2.37228
−4.37228 0 11.1168 −4.62772 0 12.1168 −13.6277 0 20.2337
1.2 1.37228 0 −6.11684 −10.3723 0 −5.11684 −19.3723 0 −14.2337
\(n\): 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.a 2
3.b odd 2 1 81.4.a.d 2
4.b odd 2 1 1296.4.a.i 2
5.b even 2 1 2025.4.a.n 2
9.c even 3 2 27.4.c.a 4
9.d odd 6 2 9.4.c.a 4
12.b even 2 1 1296.4.a.u 2
15.d odd 2 1 2025.4.a.g 2
36.f odd 6 2 432.4.i.c 4
36.h even 6 2 144.4.i.c 4
45.h odd 6 2 225.4.e.b 4
45.l even 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.d odd 6 2
27.4.c.a 4 9.c even 3 2
81.4.a.a 2 1.a even 1 1 trivial
81.4.a.d 2 3.b odd 2 1
144.4.i.c 4 36.h even 6 2
225.4.e.b 4 45.h odd 6 2
225.4.k.b 8 45.l even 12 4
432.4.i.c 4 36.f odd 6 2
1296.4.a.i 2 4.b odd 2 1
1296.4.a.u 2 12.b even 2 1
2025.4.a.g 2 15.d odd 2 1
2025.4.a.n 2 5.b even 2 1

Hecke kernels

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

Hecke characteristic polynomials

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