Properties

Label 81.9.b.a
Level $81$
Weight $9$
Character orbit 81.b
Analytic conductor $32.998$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,9,Mod(80,81)]
 
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([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.80");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 81.b (of order \(2\), degree \(1\), 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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 427 x^{12} - 1362 x^{11} + 135762 x^{10} - 371244 x^{9} + 18261508 x^{8} + \cdots + 872385888256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{63} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} - 110) q^{4} + \beta_{6} q^{5} + (\beta_{4} + \beta_{2} - 132) q^{7} + (\beta_{3} + 102 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} - 110) q^{4} + \beta_{6} q^{5} + (\beta_{4} + \beta_{2} - 132) q^{7} + (\beta_{3} + 102 \beta_1) q^{8} + ( - \beta_{5} - \beta_{4} - \beta_{2} - 37) q^{10} + (\beta_{10} - \beta_{6} - 24 \beta_1) q^{11} + (\beta_{7} - 2 \beta_{4} + 8 \beta_{2} - 244) q^{13} + (\beta_{11} + \beta_{10} + \cdots + 303 \beta_1) q^{14}+ \cdots + ( - 259 \beta_{13} + \cdots - 1304551 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 1534 q^{4} - 1844 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 1534 q^{4} - 1844 q^{7} - 516 q^{10} - 3368 q^{13} + 130562 q^{16} - 269630 q^{19} - 122622 q^{22} - 130354 q^{25} + 1075708 q^{28} + 328264 q^{31} - 1309986 q^{34} - 1671668 q^{37} - 1226652 q^{40} - 1583630 q^{43} + 1189536 q^{46} + 9653274 q^{49} + 11105440 q^{52} + 8107476 q^{55} - 28423644 q^{58} + 10511200 q^{61} - 26813830 q^{64} + 16577710 q^{67} - 55627512 q^{70} - 36721682 q^{73} + 85645918 q^{76} + 65543644 q^{79} + 236099418 q^{82} - 194972292 q^{85} - 49911654 q^{88} - 201514504 q^{91} + 73396488 q^{94} - 254098322 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 427 x^{12} - 1362 x^{11} + 135762 x^{10} - 371244 x^{9} + 18261508 x^{8} + \cdots + 872385888256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 44\!\cdots\!83 \nu^{13} + \cdots - 86\!\cdots\!64 ) / 10\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 60\!\cdots\!51 \nu^{13} + \cdots + 33\!\cdots\!64 ) / 90\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 51\!\cdots\!63 \nu^{13} + \cdots + 27\!\cdots\!20 ) / 54\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 34\!\cdots\!55 \nu^{13} + \cdots + 21\!\cdots\!40 ) / 10\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 25\!\cdots\!57 \nu^{13} + \cdots - 46\!\cdots\!80 ) / 33\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31\!\cdots\!15 \nu^{13} + \cdots + 16\!\cdots\!00 ) / 32\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12\!\cdots\!71 \nu^{13} + \cdots + 96\!\cdots\!52 ) / 42\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 30\!\cdots\!53 \nu^{13} + \cdots - 43\!\cdots\!44 ) / 37\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 20\!\cdots\!87 \nu^{13} + \cdots + 15\!\cdots\!88 ) / 15\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 30\!\cdots\!29 \nu^{13} + \cdots + 16\!\cdots\!48 ) / 10\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 97\!\cdots\!71 \nu^{13} + \cdots - 59\!\cdots\!84 ) / 32\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 14\!\cdots\!01 \nu^{13} + \cdots - 90\!\cdots\!16 ) / 36\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 11\!\cdots\!09 \nu^{13} + \cdots + 49\!\cdots\!20 ) / 82\!\cdots\!92 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} - 4\beta_{7} + 2\beta_{5} + 16\beta_{4} + 146\beta_{2} - 19683\beta _1 + 2752 ) / 39366 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 9 \beta_{13} + 91 \beta_{12} + 36 \beta_{11} + 162 \beta_{10} + 1515 \beta_{6} + 524 \beta_{3} + \cdots - 2401326 ) / 39366 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -136\beta_{9} - 253\beta_{8} + 886\beta_{7} - 668\beta_{5} - 8098\beta_{4} - 62048\beta_{2} + 3970307 ) / 19683 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2178 \beta_{13} - 16120 \beta_{12} - 13410 \beta_{11} - 33048 \beta_{10} + 2187 \beta_{8} + \cdots - 491915349 ) / 39366 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8748 \beta_{13} + 30618 \beta_{12} + 39366 \beta_{11} + 26244 \beta_{10} + 23392 \beta_{9} + \cdots - 2297763893 ) / 39366 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -32\beta_{9} - 1277\beta_{8} + 248\beta_{7} - 256\beta_{5} - 16256\beta_{4} - 761968\beta_{2} + 162690127 ) / 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 3018060 \beta_{13} - 15406686 \beta_{12} - 21795642 \beta_{11} - 19954188 \beta_{10} + \cdots - 892619708321 ) / 39366 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 111112164 \beta_{13} + 858750346 \beta_{12} + 1113772230 \beta_{11} + 1728042660 \beta_{10} + \cdots - 31201524494667 ) / 39366 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1100544544 \beta_{9} - 6470931979 \beta_{8} + 11120417176 \beta_{7} - 19598390336 \beta_{5} + \cdots + 309198764357057 ) / 19683 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 25975369404 \beta_{13} - 226323473842 \beta_{12} - 313203600198 \beta_{11} - 444116029140 \beta_{10} + \cdots - 86\!\cdots\!35 ) / 39366 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 156166835868 \beta_{13} + 2097126512694 \beta_{12} + 3081730297698 \beta_{11} + \cdots - 10\!\cdots\!45 ) / 39366 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 8907302432 \beta_{9} - 128211608939 \beta_{8} + 89368085768 \beta_{7} - 138405126304 \beta_{5} + \cdots + 10\!\cdots\!01 ) / 81 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 28004378162940 \beta_{13} - 697436660514150 \beta_{12} + \cdots - 32\!\cdots\!21 ) / 39366 \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
80.1
−8.68602 15.0446i
7.38374 12.7890i
5.69757 9.86849i
−5.49482 9.51731i
4.05115 7.01679i
−2.00397 3.47098i
−0.447645 0.775344i
−0.447645 + 0.775344i
−2.00397 + 3.47098i
4.05115 + 7.01679i
−5.49482 + 9.51731i
5.69757 + 9.86849i
7.38374 + 12.7890i
−8.68602 + 15.0446i
30.0893i 0 −649.363 218.389i 0 −2809.34 11836.0i 0 6571.17
80.2 25.5780i 0 −398.235 658.586i 0 3024.83 3638.08i 0 −16845.3
80.3 19.7370i 0 −133.548 1035.25i 0 43.8263 2416.83i 0 20432.8
80.4 19.0346i 0 −106.317 54.2580i 0 1842.02 2849.17i 0 −1032.78
80.5 14.0336i 0 59.0584 780.826i 0 −4337.21 4421.40i 0 −10957.8
80.6 6.94197i 0 207.809 382.663i 0 −935.032 3219.75i 0 2656.43
80.7 1.55069i 0 253.595 698.073i 0 2248.91 790.223i 0 −1082.49
80.8 1.55069i 0 253.595 698.073i 0 2248.91 790.223i 0 −1082.49
80.9 6.94197i 0 207.809 382.663i 0 −935.032 3219.75i 0 2656.43
80.10 14.0336i 0 59.0584 780.826i 0 −4337.21 4421.40i 0 −10957.8
80.11 19.0346i 0 −106.317 54.2580i 0 1842.02 2849.17i 0 −1032.78
80.12 19.7370i 0 −133.548 1035.25i 0 43.8263 2416.83i 0 20432.8
80.13 25.5780i 0 −398.235 658.586i 0 3024.83 3638.08i 0 −16845.3
80.14 30.0893i 0 −649.363 218.389i 0 −2809.34 11836.0i 0 6571.17
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 80.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.9.b.a 14
3.b odd 2 1 inner 81.9.b.a 14
9.c even 3 1 9.9.d.a 14
9.c even 3 1 27.9.d.a 14
9.d odd 6 1 9.9.d.a 14
9.d odd 6 1 27.9.d.a 14
36.f odd 6 1 144.9.q.a 14
36.f odd 6 1 432.9.q.a 14
36.h even 6 1 144.9.q.a 14
36.h even 6 1 432.9.q.a 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.9.d.a 14 9.c even 3 1
9.9.d.a 14 9.d odd 6 1
27.9.d.a 14 9.c even 3 1
27.9.d.a 14 9.d odd 6 1
81.9.b.a 14 1.a even 1 1 trivial
81.9.b.a 14 3.b odd 2 1 inner
144.9.q.a 14 36.f odd 6 1
144.9.q.a 14 36.h even 6 1
432.9.q.a 14 36.f odd 6 1
432.9.q.a 14 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 2559 T_{2}^{12} + 2488320 T_{2}^{10} + 1160611308 T_{2}^{8} + 267593880960 T_{2}^{6} + \cdots + 19\!\cdots\!72 \) acting on \(S_{9}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 19\!\cdots\!72 \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{7} + \cdots + 62\!\cdots\!56)^{2} \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 15\!\cdots\!43 \) Copy content Toggle raw display
$13$ \( (T^{7} + \cdots + 20\!\cdots\!36)^{2} \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 19\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots + 18\!\cdots\!88)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 83\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 26\!\cdots\!88 \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots - 15\!\cdots\!28)^{2} \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 20\!\cdots\!23 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 34\!\cdots\!13)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 35\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 10\!\cdots\!67 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 51\!\cdots\!48)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} + \cdots + 69\!\cdots\!93)^{2} \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots - 22\!\cdots\!72)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 55\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 78\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 13\!\cdots\!25)^{2} \) Copy content Toggle raw display
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