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Properties

Label 81.9.d.f
Level $81$
Weight $9$
Character orbit 81.d
Analytic conductor $32.998$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 2 x^{10} - 84 x^{9} - 141 x^{8} + 1130 x^{7} + 1550 x^{6} + 8300 x^{5} + 44525 x^{4} + \cdots + 2500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{42} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + (\beta_{7} - 131 \beta_1) q^{4} + (\beta_{6} + 20 \beta_{2}) q^{5} + (\beta_{11} + 2 \beta_{7} + \beta_{5} + \cdots + 283) q^{7} + (\beta_{10} + 7 \beta_{9} + \cdots + 161 \beta_{2}) q^{8}+ \cdots + (28240 \beta_{10} + \cdots + 1372456 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 786 q^{4} + 1698 q^{7} - 92556 q^{10} - 41844 q^{13} - 170466 q^{16} - 72768 q^{19} - 647298 q^{22} + 1646688 q^{25} - 1677972 q^{28} - 2058474 q^{31} + 1320300 q^{34} - 18791760 q^{37} - 29560086 q^{40}+ \cdots - 53735106 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 2 x^{10} - 84 x^{9} - 141 x^{8} + 1130 x^{7} + 1550 x^{6} + 8300 x^{5} + 44525 x^{4} + \cdots + 2500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 36359663 \nu^{11} + 86582291 \nu^{10} - 109850966 \nu^{9} + 3107263582 \nu^{8} + \cdots - 254024381500 ) / 118206320250 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 9565171136 \nu^{11} + 69328993967 \nu^{10} - 153334291592 \nu^{9} + 982947594424 \nu^{8} + \cdots - 4887640255000 ) / 25650771494250 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 545995374 \nu^{11} - 1664112534 \nu^{10} + 5414636868 \nu^{9} - 66349445169 \nu^{8} + \cdots - 112560389309850 ) / 855025716475 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 124544231765 \nu^{11} - 288279071309 \nu^{10} + 320627145188 \nu^{9} + \cdots + 467513871568750 ) / 25650771494250 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5109813018 \nu^{11} - 15779438388 \nu^{10} + 47454682626 \nu^{9} - 640213387758 \nu^{8} + \cdots - 37\!\cdots\!00 ) / 855025716475 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20005351192 \nu^{11} - 147015937099 \nu^{10} + 325165769224 \nu^{9} - 2067073015928 \nu^{8} + \cdots + 10328107235000 ) / 1710051432950 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 171819166089 \nu^{11} + 406679127573 \nu^{10} - 508959146298 \nu^{9} + \cdots - 12\!\cdots\!00 ) / 4275128582375 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1580212999724 \nu^{11} + 11535129466853 \nu^{10} - 25512624126728 \nu^{9} + \cdots - 811743383545000 ) / 25650771494250 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 61359099583 \nu^{11} - 163761564148 \nu^{10} + 220848638212 \nu^{9} - 5276121240110 \nu^{8} + \cdots + 217040473279250 ) / 342010286590 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 22087186843973 \nu^{11} - 51615953404919 \nu^{10} + 59261341856192 \nu^{9} + \cdots + 82\!\cdots\!50 ) / 25650771494250 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5754602301876 \nu^{11} - 13680306763182 \nu^{10} + 17291586766932 \nu^{9} + \cdots + 40\!\cdots\!00 ) / 4275128582375 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{11} + \beta_{10} - 9 \beta_{9} + 18 \beta_{7} + 9 \beta_{6} + \beta_{5} + 68 \beta_{4} + \cdots + 972 ) / 2916 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 10\beta_{8} + 27\beta_{6} - 805\beta_{2} ) / 2187 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 35 \beta_{10} + 171 \beta_{9} + 35 \beta_{8} + 23 \beta_{5} + 320 \beta_{4} - 306 \beta_{3} + \cdots + 60264 ) / 2916 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 44\beta_{11} + 333\beta_{7} + 44\beta_{5} - 333\beta_{3} + 149202\beta _1 + 149202 ) / 729 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1683\beta_{11} + 2615\beta_{8} + 18306\beta_{7} + 11151\beta_{6} - 78680\beta_{2} + 4968864\beta_1 ) / 8748 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -4720\beta_{10} + 17793\beta_{9} + 4720\beta_{8} + 222415\beta_{4} - 222415\beta_{2} ) / 2187 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 39861 \beta_{11} - 62305 \beta_{10} + 253017 \beta_{9} + 400302 \beta_{7} - 253017 \beta_{6} + \cdots + 122087088 ) / 8748 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 7672\beta_{11} + 71379\beta_{7} + 24247026\beta_1 ) / 243 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 487645 \beta_{10} + 1945413 \beta_{9} + 487645 \beta_{8} - 311329 \beta_{5} + 19287640 \beta_{4} + \cdots - 965736432 ) / 2916 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -2514580\beta_{10} + 9869877\beta_{9} - 9869877\beta_{6} + 105032935\beta_{4} ) / 2187 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7266743 \beta_{11} - 11390715 \beta_{8} + 70048026 \beta_{7} - 45149571 \beta_{6} + \cdots + 22643762544 \beta_1 ) / 2916 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{1}\)

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
0.752622 + 2.80882i
−0.486700 + 0.130411i
−1.24906 − 4.66155i
4.66155 − 1.24906i
0.130411 + 0.486700i
−2.80882 + 0.752622i
0.752622 − 2.80882i
−0.486700 − 0.130411i
−1.24906 + 4.66155i
4.66155 + 1.24906i
0.130411 − 0.486700i
−2.80882 − 0.752622i
−25.1513 − 14.5211i 0 293.725 + 508.746i 988.609 − 570.773i 0 309.155 − 535.472i − 9626.03i 0 −33153.0
26.2 −13.8243 − 7.98144i 0 −0.593312 − 1.02765i −199.822 + 115.367i 0 −1921.46 + 3328.07i 4105.44i 0 3683.19
26.3 −6.85950 − 3.96034i 0 −96.6315 − 167.371i −692.198 + 399.641i 0 2036.81 − 3527.85i 3558.46i 0 6330.85
26.4 6.85950 + 3.96034i 0 −96.6315 − 167.371i 692.198 − 399.641i 0 2036.81 − 3527.85i − 3558.46i 0 6330.85
26.5 13.8243 + 7.98144i 0 −0.593312 − 1.02765i 199.822 − 115.367i 0 −1921.46 + 3328.07i − 4105.44i 0 3683.19
26.6 25.1513 + 14.5211i 0 293.725 + 508.746i −988.609 + 570.773i 0 309.155 − 535.472i 9626.03i 0 −33153.0
53.1 −25.1513 + 14.5211i 0 293.725 − 508.746i 988.609 + 570.773i 0 309.155 + 535.472i 9626.03i 0 −33153.0
53.2 −13.8243 + 7.98144i 0 −0.593312 + 1.02765i −199.822 − 115.367i 0 −1921.46 − 3328.07i − 4105.44i 0 3683.19
53.3 −6.85950 + 3.96034i 0 −96.6315 + 167.371i −692.198 − 399.641i 0 2036.81 + 3527.85i − 3558.46i 0 6330.85
53.4 6.85950 − 3.96034i 0 −96.6315 + 167.371i 692.198 + 399.641i 0 2036.81 + 3527.85i 3558.46i 0 6330.85
53.5 13.8243 − 7.98144i 0 −0.593312 + 1.02765i 199.822 + 115.367i 0 −1921.46 − 3328.07i 4105.44i 0 3683.19
53.6 25.1513 − 14.5211i 0 293.725 − 508.746i −988.609 − 570.773i 0 309.155 + 535.472i − 9626.03i 0 −33153.0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.6
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.f 12
3.b odd 2 1 inner 81.9.d.f 12
9.c even 3 1 27.9.b.d ✓ 6
9.c even 3 1 inner 81.9.d.f 12
9.d odd 6 1 27.9.b.d ✓ 6
9.d odd 6 1 inner 81.9.d.f 12
36.f odd 6 1 432.9.e.k 6
36.h even 6 1 432.9.e.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.9.b.d ✓ 6 9.c even 3 1
27.9.b.d ✓ 6 9.d odd 6 1
81.9.d.f 12 1.a even 1 1 trivial
81.9.d.f 12 3.b odd 2 1 inner
81.9.d.f 12 9.c even 3 1 inner
81.9.d.f 12 9.d odd 6 1 inner
432.9.e.k 6 36.f odd 6 1
432.9.e.k 6 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 1161 T_{2}^{10} + 1064097 T_{2}^{8} - 302552496 T_{2}^{6} + 64901621952 T_{2}^{4} + \cdots + 181807037485056 \) acting on \(S_{9}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 181807037485056 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 19\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 93\!\cdots\!25)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 48\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 16\!\cdots\!76)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots + 54072465923584)^{4} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 69\!\cdots\!01)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + \cdots + 22\!\cdots\!00)^{4} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 37\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 28\!\cdots\!89)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 95\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 66\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + \cdots + 43\!\cdots\!75)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 21\!\cdots\!01 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 48\!\cdots\!25)^{2} \) Copy content Toggle raw display
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