Properties

Label 81.9.b.b
Level $81$
Weight $9$
Character orbit 81.b
Analytic conductor $32.998$
Analytic rank $0$
Dimension $16$
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: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1024 x^{14} + 419701 x^{12} + 88203292 x^{10} + 10121979748 x^{8} + 629108384896 x^{6} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{84} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} + (\beta_1 - 128) q^{4} + ( - \beta_{9} - 2 \beta_{8}) q^{5} + (\beta_{3} + 231) q^{7} + ( - \beta_{11} - 101 \beta_{8}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} + (\beta_1 - 128) q^{4} + ( - \beta_{9} - 2 \beta_{8}) q^{5} + (\beta_{3} + 231) q^{7} + ( - \beta_{11} - 101 \beta_{8}) q^{8} + ( - \beta_{4} - 3 \beta_1 + 672) q^{10} + ( - \beta_{12} + \beta_{11} + \cdots + 112 \beta_{8}) q^{11}+ \cdots + (236 \beta_{15} + \cdots + 1566103 \beta_{8}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2048 q^{4} + 3692 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 2048 q^{4} + 3692 q^{7} + 10752 q^{10} + 63860 q^{13} + 95116 q^{16} + 185108 q^{19} - 691980 q^{22} - 541712 q^{25} - 997132 q^{28} + 571136 q^{31} + 1027656 q^{34} + 4354268 q^{37} - 2973768 q^{40} + 5453084 q^{43} - 4234044 q^{46} + 14602560 q^{49} - 25384960 q^{52} - 14452572 q^{55} + 1213068 q^{58} + 31037996 q^{61} + 24253000 q^{64} - 62356828 q^{67} + 11075124 q^{70} - 21273664 q^{73} + 69593876 q^{76} - 151045036 q^{79} - 40201140 q^{82} + 160995564 q^{85} + 366976068 q^{88} + 373680796 q^{91} - 544741584 q^{94} - 133878688 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 1024 x^{14} + 419701 x^{12} + 88203292 x^{10} + 10121979748 x^{8} + 629108384896 x^{6} + \cdots + 16\!\cdots\!24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu^{2} + 384 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 16592055498945 \nu^{14} + \cdots + 31\!\cdots\!08 ) / 24\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2497774802274 \nu^{14} + \cdots + 11\!\cdots\!08 ) / 30\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 25410686450643 \nu^{14} + \cdots - 98\!\cdots\!16 ) / 98\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 143265531846591 \nu^{14} + \cdots - 85\!\cdots\!20 ) / 24\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 23\!\cdots\!33 \nu^{14} + \cdots - 14\!\cdots\!52 ) / 49\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 594985394614059 \nu^{14} + \cdots + 28\!\cdots\!08 ) / 12\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 55090601 \nu^{15} - 55395890770 \nu^{13} - 21996486413661 \nu^{11} + \cdots - 34\!\cdots\!12 \nu ) / 83\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 69\!\cdots\!75 \nu^{15} + \cdots - 63\!\cdots\!48 \nu ) / 60\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 11\!\cdots\!79 \nu^{15} + \cdots + 35\!\cdots\!56 \nu ) / 50\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 30719884451 \nu^{15} + 30582399292090 \nu^{13} + \cdots + 18\!\cdots\!84 \nu ) / 83\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11\!\cdots\!63 \nu^{15} + \cdots + 94\!\cdots\!84 \nu ) / 30\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 636078186037275 \nu^{15} + \cdots + 30\!\cdots\!84 \nu ) / 33\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 74\!\cdots\!17 \nu^{15} + \cdots - 39\!\cdots\!36 \nu ) / 30\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 26\!\cdots\!27 \nu^{15} + \cdots - 12\!\cdots\!72 \nu ) / 30\!\cdots\!20 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{15} - 26 \beta_{14} - 27 \beta_{13} - 33 \beta_{12} + 62 \beta_{11} + 4 \beta_{10} + \cdots + 10850 \beta_{8} ) / 177147 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta _1 - 384 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 101 \beta_{15} + 3922 \beta_{14} + 6453 \beta_{13} + 7959 \beta_{12} - 31039 \beta_{11} + \cdots - 3536290 \beta_{8} ) / 177147 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{5} + 8\beta_{3} + \beta_{2} - 852\beta _1 + 235323 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 69514 \beta_{15} - 726014 \beta_{14} - 1602774 \beta_{13} - 1954284 \beta_{12} + \cdots + 1324336535 \beta_{8} ) / 177147 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 29 \beta_{7} - 86 \beta_{6} - 1402 \beta_{5} - 206 \beta_{4} - 15202 \beta_{3} - 1528 \beta_{2} + \cdots - 165481192 ) / 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 27108917 \beta_{15} + 145439524 \beta_{14} + 412743735 \beta_{13} + \cdots - 477292582570 \beta_{8} ) / 177147 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 44142 \beta_{7} + 151412 \beta_{6} + 1491915 \beta_{5} + 256900 \beta_{4} + 19087316 \beta_{3} + \cdots + 124022951365 ) / 81 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 8996587270 \beta_{15} - 30640114838 \beta_{14} - 109372233150 \beta_{13} + \cdots + 162516241907297 \beta_{8} ) / 177147 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 15281869 \beta_{7} - 62299606 \beta_{6} - 481409782 \beta_{5} - 66055758 \beta_{4} + \cdots - 32351942692828 ) / 81 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2819717649515 \beta_{15} + 6750809798956 \beta_{14} + 29650652884593 \beta_{13} + \cdots - 53\!\cdots\!50 \beta_{8} ) / 177147 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 4527415742 \beta_{7} + 22266002740 \beta_{6} + 148886537917 \beta_{5} + 11886560580 \beta_{4} + \cdots + 87\!\cdots\!59 ) / 81 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 862090036459174 \beta_{15} + \cdots + 16\!\cdots\!71 \beta_{8} ) / 177147 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 3713925118737 \beta_{7} - 22225534276430 \beta_{6} - 135217317246954 \beta_{5} + \cdots - 72\!\cdots\!20 ) / 243 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 26\!\cdots\!65 \beta_{15} + \cdots - 52\!\cdots\!02 \beta_{8} ) / 177147 \) 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
17.2054i
14.8268i
13.9244i
12.9444i
8.62663i
5.78233i
5.26228i
3.33872i
3.33872i
5.26228i
5.78233i
8.62663i
12.9444i
13.9244i
14.8268i
17.2054i
29.8006i 0 −632.074 177.778i 0 3390.81 11207.2i 0 5297.89
80.2 25.6809i 0 −403.506 204.369i 0 −3702.42 3788.08i 0 −5248.38
80.3 24.1178i 0 −325.666 436.510i 0 550.753 1680.18i 0 −10527.6
80.4 22.4204i 0 −246.676 1080.12i 0 −1006.64 209.045i 0 24216.8
80.5 14.9418i 0 32.7440 1089.48i 0 1308.49 4314.34i 0 −16278.7
80.6 10.0153i 0 155.694 594.387i 0 4409.51 4123.23i 0 5952.96
80.7 9.11454i 0 172.925 181.837i 0 −873.072 3909.46i 0 −1657.36
80.8 5.78284i 0 222.559 626.068i 0 −2231.44 2767.43i 0 3620.45
80.9 5.78284i 0 222.559 626.068i 0 −2231.44 2767.43i 0 3620.45
80.10 9.11454i 0 172.925 181.837i 0 −873.072 3909.46i 0 −1657.36
80.11 10.0153i 0 155.694 594.387i 0 4409.51 4123.23i 0 5952.96
80.12 14.9418i 0 32.7440 1089.48i 0 1308.49 4314.34i 0 −16278.7
80.13 22.4204i 0 −246.676 1080.12i 0 −1006.64 209.045i 0 24216.8
80.14 24.1178i 0 −325.666 436.510i 0 550.753 1680.18i 0 −10527.6
80.15 25.6809i 0 −403.506 204.369i 0 −3702.42 3788.08i 0 −5248.38
80.16 29.8006i 0 −632.074 177.778i 0 3390.81 11207.2i 0 5297.89
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 80.16
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.b 16
3.b odd 2 1 inner 81.9.b.b 16
9.c even 3 2 81.9.d.g 32
9.d odd 6 2 81.9.d.g 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.9.b.b 16 1.a even 1 1 trivial
81.9.b.b 16 3.b odd 2 1 inner
81.9.d.g 32 9.c even 3 2
81.9.d.g 32 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 3072 T_{2}^{14} + 3777309 T_{2}^{12} + 2381488884 T_{2}^{10} + 819880359588 T_{2}^{8} + \cdots + 10\!\cdots\!64 \) acting on \(S_{9}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 15\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 78\!\cdots\!56)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 11\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 40\!\cdots\!93)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 10\!\cdots\!69 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 24\!\cdots\!92)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 11\!\cdots\!09 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots - 14\!\cdots\!44)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 11\!\cdots\!63)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 24\!\cdots\!52)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 35\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 46\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 11\!\cdots\!77)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 33\!\cdots\!68)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 97\!\cdots\!99)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots - 12\!\cdots\!84)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 62\!\cdots\!81 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 17\!\cdots\!84)^{2} \) Copy content Toggle raw display
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