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Properties

Label 3.27.b.a
Level $3$
Weight $27$
Character orbit 3.b
Analytic conductor $12.849$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3,27,Mod(2,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 27, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.2");
 
S:= CuspForms(chi, 27);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 27 \)
Character orbit: \([\chi]\) \(=\) 3.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: \(12.8487876219\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 2728193x^{6} + 2071419806976x^{4} + 503906274711956480x^{2} + 26754716118873014272000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{33}\cdot 3^{38}\cdot 5^{3} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} - 5 \beta_1 - 157635) q^{3} + (\beta_{3} - 16 \beta_{2} + \cdots - 31106084) q^{4}+ \cdots + (9 \beta_{7} + 15 \beta_{6} + \cdots + 441904690665) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} - 5 \beta_1 - 157635) q^{3} + (\beta_{3} - 16 \beta_{2} + \cdots - 31106084) q^{4}+ \cdots + (187451750304954 \beta_{7} + \cdots + 15\!\cdots\!40) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 1261080 q^{3} - 248848672 q^{4} + 4268261088 q^{6} - 83723264240 q^{7} + 3535237525320 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 1261080 q^{3} - 248848672 q^{4} + 4268261088 q^{6} - 83723264240 q^{7} + 3535237525320 q^{9} - 19160376580800 q^{10} + 361652147111520 q^{12} - 397915354631600 q^{13} - 719893807027200 q^{15} + 14\!\cdots\!24 q^{16}+ \cdots + 12\!\cdots\!20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 2728193x^{6} + 2071419806976x^{4} + 503906274711956480x^{2} + 26754716118873014272000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 12\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 471665 \nu^{7} + 20280832 \nu^{6} - 1161523912305 \nu^{5} + 96171313767936 \nu^{4} + \cdots + 16\!\cdots\!80 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 471665 \nu^{7} + 20280832 \nu^{6} - 1161523912305 \nu^{5} + 96171313767936 \nu^{4} + \cdots + 81\!\cdots\!00 ) / 65\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 33155731 \nu^{7} - 831514112 \nu^{6} + 161318375210643 \nu^{5} + \cdots - 68\!\cdots\!80 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 285348169 \nu^{7} - 1501425874432 \nu^{6} + \cdots - 14\!\cdots\!00 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1134243497 \nu^{7} + 40135766528 \nu^{6} + \cdots + 44\!\cdots\!00 ) / 35\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 798776683 \nu^{7} + 4240177508864 \nu^{6} + \cdots + 44\!\cdots\!80 ) / 21\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 16\beta_{2} + 7\beta _1 - 98214948 ) / 144 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + \beta_{6} + 42\beta_{5} + 372\beta_{4} - 185\beta_{3} - 14488\beta_{2} - 174185229\beta_1 ) / 1728 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 208 \beta_{7} + 21168 \beta_{6} + 832 \beta_{5} - 20960 \beta_{4} - 15509145 \beta_{3} + \cdots + 10\!\cdots\!28 ) / 1296 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 60258747 \beta_{7} - 24668905 \beta_{6} - 843622458 \beta_{5} - 5414815892 \beta_{4} + \cdots + 23\!\cdots\!17 \beta_1 ) / 15552 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 328777328 \beta_{7} - 16389815568 \beta_{6} - 1315109312 \beta_{5} + 16061038240 \beta_{4} + \cdots - 52\!\cdots\!44 ) / 432 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 36368554359273 \beta_{7} + 24692188805027 \beta_{6} + 509159761029822 \beta_{5} + \cdots - 12\!\cdots\!55 \beta_1 ) / 5184 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1
− 1286.17i
− 818.442i
− 575.513i
− 269.997i
269.997i
575.513i
818.442i
1286.17i
− 15434.0i −1.57937e6 + 217880.i −1.71101e8 − 7.56364e8i 3.36276e9 + 2.43760e10i −6.43856e10 1.60501e12i 2.44692e12 − 6.88223e11i −1.16738e13
2.2 − 9821.30i 1.39737e6 − 767608.i −2.93491e7 − 1.01800e9i −7.53891e9 − 1.37240e10i 1.32967e11 − 3.70850e11i 1.36342e12 − 2.14527e12i −9.99808e12
2.3 − 6906.16i 618315. + 1.46954e6i 1.94139e7 1.04255e9i 1.01489e10 − 4.27018e9i −1.82788e11 − 5.97540e11i −1.77724e12 + 1.81728e12i 7.19999e12
2.4 − 3239.96i −1.06686e6 − 1.18477e6i 5.66115e7 1.50979e9i −3.83861e9 + 3.45659e9i 7.23448e10 − 4.00850e11i −2.65486e11 + 2.52796e12i 4.89166e12
2.5 3239.96i −1.06686e6 + 1.18477e6i 5.66115e7 − 1.50979e9i −3.83861e9 − 3.45659e9i 7.23448e10 4.00850e11i −2.65486e11 − 2.52796e12i 4.89166e12
2.6 6906.16i 618315. − 1.46954e6i 1.94139e7 − 1.04255e9i 1.01489e10 + 4.27018e9i −1.82788e11 5.97540e11i −1.77724e12 − 1.81728e12i 7.19999e12
2.7 9821.30i 1.39737e6 + 767608.i −2.93491e7 1.01800e9i −7.53891e9 + 1.37240e10i 1.32967e11 3.70850e11i 1.36342e12 + 2.14527e12i −9.99808e12
2.8 15434.0i −1.57937e6 − 217880.i −1.71101e8 7.56364e8i 3.36276e9 − 2.43760e10i −6.43856e10 − 1.60501e12i 2.44692e12 + 6.88223e11i −1.16738e13
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.8
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 3.27.b.a ✓ 8
3.b odd 2 1 inner 3.27.b.a ✓ 8
4.b odd 2 1 48.27.e.d 8
12.b even 2 1 48.27.e.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.27.b.a ✓ 8 1.a even 1 1 trivial
3.27.b.a ✓ 8 3.b odd 2 1 inner
48.27.e.d 8 4.b odd 2 1
48.27.e.d 8 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{27}^{\mathrm{new}}(3, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 41\!\cdots\!81 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{4} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 66\!\cdots\!36)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 97\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 99\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 62\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 57\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 18\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 84\!\cdots\!00)^{2} \) Copy content Toggle raw display
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