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Properties

Label 81.5.d.e
Level $81$
Weight $5$
Character orbit 81.d
Analytic conductor $8.373$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [81,5,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, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("81.26");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 5 \)
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: \(8.37296700979\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 28x^{14} + 519x^{12} - 5404x^{10} + 40705x^{8} - 194544x^{6} + 672624x^{4} - 1306368x^{2} + 1679616 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{40} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + ( - \beta_{9} - \beta_{4} - 8 \beta_1 + 8) q^{4} + (\beta_{6} + \beta_{3}) q^{5} + ( - \beta_{8} - \beta_{4} - 7 \beta_1) q^{7} + (\beta_{13} - \beta_{11} + \cdots - 6 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + ( - \beta_{9} - \beta_{4} - 8 \beta_1 + 8) q^{4} + (\beta_{6} + \beta_{3}) q^{5} + ( - \beta_{8} - \beta_{4} - 7 \beta_1) q^{7} + (\beta_{13} - \beta_{11} + \cdots - 6 \beta_{3}) q^{8}+ \cdots + ( - 28 \beta_{15} + \cdots - 65 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 64 q^{4} - 52 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 64 q^{4} - 52 q^{7} + 384 q^{10} + 260 q^{13} - 140 q^{16} - 616 q^{19} - 1140 q^{22} + 1672 q^{25} - 4504 q^{28} - 2944 q^{31} - 3528 q^{34} + 9176 q^{37} + 11208 q^{40} + 3836 q^{43} + 6792 q^{46} - 3432 q^{49} - 6400 q^{52} - 20424 q^{55} - 11148 q^{58} + 3476 q^{61} + 16 q^{64} + 13508 q^{67} + 19596 q^{70} + 33152 q^{73} - 27412 q^{76} - 40876 q^{79} - 65640 q^{82} - 684 q^{85} + 32892 q^{88} - 26168 q^{91} + 55632 q^{94} + 27488 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 28x^{14} + 519x^{12} - 5404x^{10} + 40705x^{8} - 194544x^{6} + 672624x^{4} - 1306368x^{2} + 1679616 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 236299 \nu^{14} + 6070207 \nu^{12} - 109750641 \nu^{10} + 1048998769 \nu^{8} + \cdots + 236767256064 ) / 136431603504 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 786761 \nu^{14} - 287085772 \nu^{12} + 6904749201 \nu^{10} - 124768312060 \nu^{8} + \cdots - 39680542994688 ) / 136431603504 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 5017367 \nu^{15} + 156436684 \nu^{13} - 4649545887 \nu^{11} + 96817310380 \nu^{9} + \cdots + 32548195858176 \nu ) / 3274358484096 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 8909729 \nu^{14} + 78362972 \nu^{12} - 369172839 \nu^{10} - 23270321956 \nu^{8} + \cdots - 10894558392576 ) / 181908804672 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10620935 \nu^{14} - 249264980 \nu^{12} + 4433017953 \nu^{10} - 37882510484 \nu^{8} + \cdots + 6392135909664 ) / 90954402336 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7113223 \nu^{15} - 93953584 \nu^{13} + 1100287041 \nu^{11} + 5475988616 \nu^{9} + \cdots + 4563827774208 \nu ) / 363817609344 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 73440775 \nu^{15} + 1796118580 \nu^{13} - 30653070945 \nu^{11} + 261946893460 \nu^{9} + \cdots - 274023565056 \nu ) / 3274358484096 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 41211559 \nu^{14} - 1128270592 \nu^{12} + 20620612257 \nu^{10} - 212000064808 \nu^{8} + \cdots - 50459129314560 ) / 272863207008 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 34935745 \nu^{14} - 864442156 \nu^{12} + 14581652631 \nu^{10} - 124608024268 \nu^{8} + \cdots - 2098699974144 ) / 181908804672 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1221311 \nu^{15} - 36807023 \nu^{13} + 665478849 \nu^{11} - 6926750369 \nu^{9} + \cdots - 608390565840 \nu ) / 30318134112 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 30494875 \nu^{15} - 710922892 \nu^{13} + 12728100525 \nu^{11} - 108768429700 \nu^{9} + \cdots + 19779671669400 \nu ) / 409294810512 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 72028483 \nu^{15} + 3274107514 \nu^{13} - 68698565565 \nu^{11} + \cdots + 259674833809152 \nu ) / 818589621024 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 84808181 \nu^{15} + 2301006503 \nu^{13} - 41853764631 \nu^{11} + \cdots + 101095105397760 \nu ) / 818589621024 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 29273755 \nu^{14} - 707392948 \nu^{12} + 12218423469 \nu^{10} - 104412966532 \nu^{8} + \cdots - 2078979696 ) / 45477201168 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 358104835 \nu^{15} - 8792459812 \nu^{13} + 149467552773 \nu^{11} - 1277280217444 \nu^{9} + \cdots + 60224154975072 \nu ) / 818589621024 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{15} - \beta_{11} - 19\beta_{10} - 15\beta_{7} - 19\beta_{6} ) / 243 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{14} - 2\beta_{9} - 5\beta_{8} - 5\beta_{5} - 2\beta_{4} + \beta_{2} - 570\beta _1 + 564 ) / 81 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{15} + 26\beta_{13} - 7\beta_{12} - 26\beta_{11} - 188\beta_{10} - 408\beta_{7} - 408\beta_{3} ) / 243 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -71\beta_{8} - 38\beta_{4} + 22\beta_{2} - 5190\beta_1 ) / 81 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 301\beta_{13} + 10\beta_{12} + 2173\beta_{6} - 6801\beta_{3} ) / 243 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -117\beta_{14} + 214\beta_{9} + 293\beta_{5} - 18128 ) / 27 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 767\beta_{15} + 3194\beta_{11} + 26936\beta_{10} + 97428\beta_{7} + 26936\beta_{6} ) / 243 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 5006 \beta_{14} + 9922 \beta_{9} + 10837 \beta_{8} + 10837 \beta_{5} + 9922 \beta_{4} + \cdots - 635583 ) / 81 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 14362 \beta_{15} - 34579 \beta_{13} - 14362 \beta_{12} + 34579 \beta_{11} + 343243 \beta_{10} + \cdots + 1317543 \beta_{3} ) / 243 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 135589\beta_{8} + 143398\beta_{4} - 68033\beta_{2} + 7995018\beta_1 ) / 81 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -391514\beta_{13} - 218033\beta_{12} - 4419500\beta_{6} + 17399424\beta_{3} ) / 243 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 301210\beta_{14} - 661234\beta_{9} - 572901\beta_{5} + 32647651 ) / 27 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( -3059170\beta_{15} - 4628413\beta_{11} - 57121885\beta_{10} - 227428305\beta_{7} - 57121885\beta_{6} ) / 243 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 11864047 \beta_{14} - 26712530 \beta_{9} - 21977111 \beta_{8} - 21977111 \beta_{5} - 26712530 \beta_{4} + \cdots + 1247868744 ) / 81 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 41360879 \beta_{15} + 56560562 \beta_{13} + 41360879 \beta_{12} - 56560562 \beta_{11} + \cdots - 2958961836 \beta_{3} ) / 243 \) Copy content Toggle raw display

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

Character values

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

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

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
1.90899 + 1.10215i
−3.11681 − 1.79949i
−1.44378 − 0.833568i
2.35727 + 1.36097i
−2.35727 − 1.36097i
1.44378 + 0.833568i
3.11681 + 1.79949i
−1.90899 − 1.10215i
1.90899 − 1.10215i
−3.11681 + 1.79949i
−1.44378 + 0.833568i
2.35727 − 1.36097i
−2.35727 + 1.36097i
1.44378 − 0.833568i
3.11681 − 1.79949i
−1.90899 + 1.10215i
−6.20424 − 3.58202i 0 17.6617 + 30.5910i −34.4078 + 19.8654i 0 −7.32022 + 12.6790i − 138.434i 0 284.632
26.2 −4.62202 − 2.66853i 0 6.24207 + 10.8116i 34.4520 − 19.8909i 0 −32.8605 + 56.9160i 18.7643i 0 −212.317
26.3 −3.27716 − 1.89207i 0 −0.840146 − 1.45517i −1.30583 + 0.753920i 0 39.3509 − 68.1577i 66.9047i 0 5.70588
26.4 −1.18514 − 0.684241i 0 −7.06363 − 12.2346i −11.3779 + 6.56903i 0 −12.1702 + 21.0793i 41.2286i 0 17.9792
26.5 1.18514 + 0.684241i 0 −7.06363 − 12.2346i 11.3779 − 6.56903i 0 −12.1702 + 21.0793i − 41.2286i 0 17.9792
26.6 3.27716 + 1.89207i 0 −0.840146 − 1.45517i 1.30583 − 0.753920i 0 39.3509 − 68.1577i − 66.9047i 0 5.70588
26.7 4.62202 + 2.66853i 0 6.24207 + 10.8116i −34.4520 + 19.8909i 0 −32.8605 + 56.9160i − 18.7643i 0 −212.317
26.8 6.20424 + 3.58202i 0 17.6617 + 30.5910i 34.4078 − 19.8654i 0 −7.32022 + 12.6790i 138.434i 0 284.632
53.1 −6.20424 + 3.58202i 0 17.6617 − 30.5910i −34.4078 − 19.8654i 0 −7.32022 − 12.6790i 138.434i 0 284.632
53.2 −4.62202 + 2.66853i 0 6.24207 − 10.8116i 34.4520 + 19.8909i 0 −32.8605 − 56.9160i − 18.7643i 0 −212.317
53.3 −3.27716 + 1.89207i 0 −0.840146 + 1.45517i −1.30583 − 0.753920i 0 39.3509 + 68.1577i − 66.9047i 0 5.70588
53.4 −1.18514 + 0.684241i 0 −7.06363 + 12.2346i −11.3779 − 6.56903i 0 −12.1702 − 21.0793i − 41.2286i 0 17.9792
53.5 1.18514 − 0.684241i 0 −7.06363 + 12.2346i 11.3779 + 6.56903i 0 −12.1702 − 21.0793i 41.2286i 0 17.9792
53.6 3.27716 − 1.89207i 0 −0.840146 + 1.45517i 1.30583 + 0.753920i 0 39.3509 + 68.1577i 66.9047i 0 5.70588
53.7 4.62202 − 2.66853i 0 6.24207 − 10.8116i −34.4520 − 19.8909i 0 −32.8605 − 56.9160i 18.7643i 0 −212.317
53.8 6.20424 − 3.58202i 0 17.6617 − 30.5910i 34.4078 + 19.8654i 0 −7.32022 − 12.6790i − 138.434i 0 284.632
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 26.8
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 81.5.d.e 16
3.b odd 2 1 inner 81.5.d.e 16
9.c even 3 1 81.5.b.b ✓ 8
9.c even 3 1 inner 81.5.d.e 16
9.d odd 6 1 81.5.b.b ✓ 8
9.d odd 6 1 inner 81.5.d.e 16
36.f odd 6 1 1296.5.e.d 8
36.h even 6 1 1296.5.e.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
81.5.b.b ✓ 8 9.c even 3 1
81.5.b.b ✓ 8 9.d odd 6 1
81.5.d.e 16 1.a even 1 1 trivial
81.5.d.e 16 3.b odd 2 1 inner
81.5.d.e 16 9.c even 3 1 inner
81.5.d.e 16 9.d odd 6 1 inner
1296.5.e.d 8 36.f odd 6 1
1296.5.e.d 8 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 96 T_{2}^{14} + 6435 T_{2}^{12} - 215352 T_{2}^{10} + 5216805 T_{2}^{8} - 64256004 T_{2}^{6} + \cdots + 1536953616 \) acting on \(S_{5}^{\mathrm{new}}(81, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 1536953616 \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 96\!\cdots\!41 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 3397327257856)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 44\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 98\!\cdots\!69)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 16\!\cdots\!89)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 154 T^{3} + \cdots + 22562369872)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 10\!\cdots\!81 \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 29\!\cdots\!36)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 3469511755343)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 16\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 16\!\cdots\!84)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 18\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 29\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 33\!\cdots\!29)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 37\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 25\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 214837668024959)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 24\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 83\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 57\!\cdots\!61)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 23\!\cdots\!36)^{2} \) Copy content Toggle raw display
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