Properties

Label 81.9.d.c
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: \(\Q(\sqrt{-3}, \sqrt{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 10x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{3} \)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (14 \beta_1 + 14) q^{4} + ( - 26 \beta_{3} + 26 \beta_{2}) q^{5} - 679 \beta_1 q^{7} - 242 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (14 \beta_1 + 14) q^{4} + ( - 26 \beta_{3} + 26 \beta_{2}) q^{5} - 679 \beta_1 q^{7} - 242 \beta_{2} q^{8} - 7020 q^{10} + 814 \beta_{3} q^{11} + (30817 \beta_1 + 30817) q^{13} + (679 \beta_{3} - 679 \beta_{2}) q^{14} - 68924 \beta_1 q^{16} + 7806 \beta_{2} q^{17} - 138391 q^{19} - 364 \beta_{3} q^{20} + (219780 \beta_1 + 219780) q^{22} + ( - 18482 \beta_{3} + 18482 \beta_{2}) q^{23} + 208105 \beta_1 q^{25} + 30817 \beta_{2} q^{26} + 9506 q^{28} + 80740 \beta_{3} q^{29} + ( - 352214 \beta_1 - 352214) q^{31} + (6972 \beta_{3} - 6972 \beta_{2}) q^{32} + 2107620 \beta_1 q^{34} + 17654 \beta_{2} q^{35} + 1189991 q^{37} - 138391 \beta_{3} q^{38} + (1698840 \beta_1 + 1698840) q^{40} + (66580 \beta_{3} - 66580 \beta_{2}) q^{41} + 6246086 \beta_1 q^{43} + 11396 \beta_{2} q^{44} - 4990140 q^{46} - 146 \beta_{3} q^{47} + (5303760 \beta_1 + 5303760) q^{49} + ( - 208105 \beta_{3} + 208105 \beta_{2}) q^{50} + 431438 \beta_1 q^{52} - 765468 \beta_{2} q^{53} - 5714280 q^{55} - 164318 \beta_{3} q^{56} + (21799800 \beta_1 + 21799800) q^{58} + (641206 \beta_{3} - 641206 \beta_{2}) q^{59} + 16580399 \beta_1 q^{61} - 352214 \beta_{2} q^{62} - 15762104 q^{64} - 801242 \beta_{3} q^{65} + ( - 7667153 \beta_1 - 7667153) q^{67} + ( - 109284 \beta_{3} + 109284 \beta_{2}) q^{68} + 4766580 \beta_1 q^{70} + 1413192 \beta_{2} q^{71} + 24949631 q^{73} + 1189991 \beta_{3} q^{74} + ( - 1937474 \beta_1 - 1937474) q^{76} + (552706 \beta_{3} - 552706 \beta_{2}) q^{77} + 41685089 \beta_1 q^{79} + 1792024 \beta_{2} q^{80} + 17976600 q^{82} + 2722060 \beta_{3} q^{83} + ( - 54798120 \beta_1 - 54798120) q^{85} + ( - 6246086 \beta_{3} + 6246086 \beta_{2}) q^{86} - 53186760 \beta_1 q^{88} + 45078 \beta_{2} q^{89} + 20924743 q^{91} - 258748 \beta_{3} q^{92} + ( - 39420 \beta_1 - 39420) q^{94} + (3598166 \beta_{3} - 3598166 \beta_{2}) q^{95} - 105926089 \beta_1 q^{97} + 5303760 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 28 q^{4} + 1358 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 28 q^{4} + 1358 q^{7} - 28080 q^{10} + 61634 q^{13} + 137848 q^{16} - 553564 q^{19} + 439560 q^{22} - 416210 q^{25} + 38024 q^{28} - 704428 q^{31} - 4215240 q^{34} + 4759964 q^{37} + 3397680 q^{40} - 12492172 q^{43} - 19960560 q^{46} + 10607520 q^{49} - 862876 q^{52} - 22857120 q^{55} + 43599600 q^{58} - 33160798 q^{61} - 63048416 q^{64} - 15334306 q^{67} - 9533160 q^{70} + 99798524 q^{73} - 3874948 q^{76} - 83370178 q^{79} + 71906400 q^{82} - 109596240 q^{85} + 106373520 q^{88} + 83698972 q^{91} - 78840 q^{94} + 211852178 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 10x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{3} + 60\nu ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{3} + 30\nu ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 9 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 10\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -20\beta_{3} + 10\beta_{2} ) / 9 \) 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
−1.58114 2.73861i
1.58114 + 2.73861i
−1.58114 + 2.73861i
1.58114 2.73861i
−14.2302 8.21584i 0 7.00000 + 12.1244i 369.986 213.612i 0 339.500 588.031i 3976.47i 0 −7020.00
26.2 14.2302 + 8.21584i 0 7.00000 + 12.1244i −369.986 + 213.612i 0 339.500 588.031i 3976.47i 0 −7020.00
53.1 −14.2302 + 8.21584i 0 7.00000 12.1244i 369.986 + 213.612i 0 339.500 + 588.031i 3976.47i 0 −7020.00
53.2 14.2302 8.21584i 0 7.00000 12.1244i −369.986 213.612i 0 339.500 + 588.031i 3976.47i 0 −7020.00
\(n\): 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.c 4
3.b odd 2 1 inner 81.9.d.c 4
9.c even 3 1 27.9.b.c 2
9.c even 3 1 inner 81.9.d.c 4
9.d odd 6 1 27.9.b.c 2
9.d odd 6 1 inner 81.9.d.c 4
36.f odd 6 1 432.9.e.f 2
36.h even 6 1 432.9.e.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.9.b.c 2 9.c even 3 1
27.9.b.c 2 9.d odd 6 1
81.9.d.c 4 1.a even 1 1 trivial
81.9.d.c 4 3.b odd 2 1 inner
81.9.d.c 4 9.c even 3 1 inner
81.9.d.c 4 9.d odd 6 1 inner
432.9.e.f 2 36.f odd 6 1
432.9.e.f 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 270T_{2}^{2} + 72900 \) acting on \(S_{9}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 270 T^{2} + 72900 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots + 33313550400 \) Copy content Toggle raw display
$7$ \( (T^{2} - 679 T + 461041)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{2} - 30817 T + 949687489)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 16452081720)^{2} \) Copy content Toggle raw display
$19$ \( (T + 138391)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{2} + 352214 T + 124054701796)^{2} \) Copy content Toggle raw display
$37$ \( (T - 1189991)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{2} + \cdots + 39013590319396)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 33123708302400 \) Copy content Toggle raw display
$53$ \( (T^{2} + 158204139936480)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots + 274909630999201)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + \cdots + 58785235125409)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 539220139793280)^{2} \) Copy content Toggle raw display
$73$ \( (T - 24949631)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 17\!\cdots\!21)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{2} + 548647042680)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + \cdots + 11\!\cdots\!21)^{2} \) Copy content Toggle raw display
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