Properties

Label 81.9.d.e
Level $81$
Weight $9$
Character orbit 81.d
Analytic conductor $32.998$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,9,Mod(26,81)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(81, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.26");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 81.d (of order \(6\), degree \(2\), not minimal)

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: no
Analytic conductor: \(32.9976674150\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (608 \beta_1 + 608) q^{4} + (28 \beta_{3} - 28 \beta_{2}) q^{5} + 1967 \beta_1 q^{7} + 352 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (608 \beta_1 + 608) q^{4} + (28 \beta_{3} - 28 \beta_{2}) q^{5} + 1967 \beta_1 q^{7} + 352 \beta_{2} q^{8} + 24192 q^{10} - 428 \beta_{3} q^{11} + (45505 \beta_1 + 45505) q^{13} + ( - 1967 \beta_{3} + 1967 \beta_{2}) q^{14} + 148480 \beta_1 q^{16} + 2028 \beta_{2} q^{17} + 152399 q^{19} + 17024 \beta_{3} q^{20} + ( - 369792 \beta_1 - 369792) q^{22} + (4468 \beta_{3} - 4468 \beta_{2}) q^{23} - 286751 \beta_1 q^{25} + 45505 \beta_{2} q^{26} - 1195936 q^{28} - 20024 \beta_{3} q^{29} + (164350 \beta_1 + 164350) q^{31} + ( - 58368 \beta_{3} + 58368 \beta_{2}) q^{32} + 1752192 \beta_1 q^{34} + 55076 \beta_{2} q^{35} - 663937 q^{37} + 152399 \beta_{3} q^{38} + (8515584 \beta_1 + 8515584) q^{40} + (31912 \beta_{3} - 31912 \beta_{2}) q^{41} + 575330 \beta_1 q^{43} - 260224 \beta_{2} q^{44} + 3860352 q^{46} - 314156 \beta_{3} q^{47} + (1895712 \beta_1 + 1895712) q^{49} + (286751 \beta_{3} - 286751 \beta_{2}) q^{50} + 27667040 \beta_1 q^{52} - 353016 \beta_{2} q^{53} - 10354176 q^{55} - 692384 \beta_{3} q^{56} + ( - 17300736 \beta_1 - 17300736) q^{58} + (171460 \beta_{3} - 171460 \beta_{2}) q^{59} - 19212961 \beta_1 q^{61} + 164350 \beta_{2} q^{62} - 12419072 q^{64} + 1274140 \beta_{3} q^{65} + (598033 \beta_1 + 598033) q^{67} + ( - 1233024 \beta_{3} + 1233024 \beta_{2}) q^{68} + 47585664 \beta_1 q^{70} - 995856 \beta_{2} q^{71} + 12850175 q^{73} - 663937 \beta_{3} q^{74} + (92658592 \beta_1 + 92658592) q^{76} + (841876 \beta_{3} - 841876 \beta_{2}) q^{77} - 23584657 \beta_1 q^{79} + 4157440 \beta_{2} q^{80} + 27571968 q^{82} + 1138024 \beta_{3} q^{83} + (49061376 \beta_1 + 49061376) q^{85} + ( - 575330 \beta_{3} + 575330 \beta_{2}) q^{86} - 130166784 \beta_1 q^{88} + 962268 \beta_{2} q^{89} - 89508335 q^{91} + 2716544 \beta_{3} q^{92} + ( - 271430784 \beta_1 - 271430784) q^{94} + (4267172 \beta_{3} - 4267172 \beta_{2}) q^{95} + 136489631 \beta_1 q^{97} + 1895712 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 1216 q^{4} - 3934 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 1216 q^{4} - 3934 q^{7} + 96768 q^{10} + 91010 q^{13} - 296960 q^{16} + 609596 q^{19} - 739584 q^{22} + 573502 q^{25} - 4783744 q^{28} + 328700 q^{31} - 3504384 q^{34} - 2655748 q^{37} + 17031168 q^{40} - 1150660 q^{43} + 15441408 q^{46} + 3791424 q^{49} - 55334080 q^{52} - 41416704 q^{55} - 34601472 q^{58} + 38425922 q^{61} - 49676288 q^{64} + 1196066 q^{67} - 95171328 q^{70} + 51400700 q^{73} + 185317184 q^{76} + 47169314 q^{79} + 110287872 q^{82} + 98122752 q^{85} + 260333568 q^{88} - 358033340 q^{91} - 542861568 q^{94} - 272979262 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 6\nu^{3} + 24\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -6\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{3} + \beta_{2} ) / 18 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
−25.4558 14.6969i 0 304.000 + 526.543i −712.764 + 411.514i 0 −983.500 + 1703.47i 10346.6i 0 24192.0
26.2 25.4558 + 14.6969i 0 304.000 + 526.543i 712.764 411.514i 0 −983.500 + 1703.47i 10346.6i 0 24192.0
53.1 −25.4558 + 14.6969i 0 304.000 526.543i −712.764 411.514i 0 −983.500 1703.47i 10346.6i 0 24192.0
53.2 25.4558 14.6969i 0 304.000 526.543i 712.764 + 411.514i 0 −983.500 1703.47i 10346.6i 0 24192.0
\(n\): 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 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 81.9.d.e 4
3.b odd 2 1 inner 81.9.d.e 4
9.c even 3 1 27.9.b.b 2
9.c even 3 1 inner 81.9.d.e 4
9.d odd 6 1 27.9.b.b 2
9.d odd 6 1 inner 81.9.d.e 4
36.f odd 6 1 432.9.e.d 2
36.h even 6 1 432.9.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.9.b.b 2 9.c even 3 1
27.9.b.b 2 9.d odd 6 1
81.9.d.e 4 1.a even 1 1 trivial
81.9.d.e 4 3.b odd 2 1 inner
81.9.d.e 4 9.c even 3 1 inner
81.9.d.e 4 9.d odd 6 1 inner
432.9.e.d 2 36.f odd 6 1
432.9.e.d 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 864T_{2}^{2} + 746496 \) acting on \(S_{9}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 864 T^{2} + 746496 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 458838245376 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1967 T + 3869089)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{2} - 45505 T + 2070705025)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 3553445376)^{2} \) Copy content Toggle raw display
$19$ \( (T - 152399)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 29\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( (T^{2} - 164350 T + 27010922500)^{2} \) Copy content Toggle raw display
$37$ \( (T + 663937)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 77\!\cdots\!56 \) Copy content Toggle raw display
$43$ \( (T^{2} + 575330 T + 331004608900)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 72\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{2} + 107671935965184)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 369137870387521)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 598033 T + 357643469089)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 856854005243904)^{2} \) Copy content Toggle raw display
$73$ \( (T - 12850175)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 556236045807649)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{2} + 800029184103936)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 18\!\cdots\!61)^{2} \) Copy content Toggle raw display
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