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Properties

Label 5.28.b.a
Level $5$
Weight $28$
Character orbit 5.b
Analytic conductor $23.093$
Analytic rank $0$
Dimension $12$
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] = [5,28,Mod(4,5)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 28, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5.4");
 
S:= CuspForms(chi, 28);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 5.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: \(23.0927787419\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 300081849 x^{10} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{50}\cdot 3^{18}\cdot 5^{31}\cdot 7^{4}\cdot 11 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} - 65836838) q^{4} + (\beta_{4} + 133 \beta_{3} + \cdots - 136211925) q^{5}+ \cdots + (5 \beta_{7} + \beta_{6} + \cdots - 1905946534857) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{2} - 65836838) q^{4} + (\beta_{4} + 133 \beta_{3} + \cdots - 136211925) q^{5}+ \cdots + (2291432469039 \beta_{11} + \cdots + 11\!\cdots\!96) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 790042056 q^{4} - 1634543100 q^{5} + 493654824 q^{6} - 22871358418284 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 790042056 q^{4} - 1634543100 q^{5} + 493654824 q^{6} - 22871358418284 q^{9} - 7984385278200 q^{10} - 39081130109136 q^{11} + 89\!\cdots\!88 q^{14}+ \cdots + 13\!\cdots\!52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 300081849 x^{10} + \cdots + 15\!\cdots\!96 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} + 200054566 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!69 \nu^{11} + \cdots + 12\!\cdots\!56 \nu ) / 18\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 21\!\cdots\!13 \nu^{11} + \cdots - 93\!\cdots\!32 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 83\!\cdots\!27 \nu^{11} + \cdots - 16\!\cdots\!28 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 71\!\cdots\!67 \nu^{11} + \cdots + 29\!\cdots\!12 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21\!\cdots\!13 \nu^{11} + \cdots - 17\!\cdots\!68 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 30\!\cdots\!79 \nu^{11} + \cdots + 74\!\cdots\!56 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 32\!\cdots\!43 \nu^{11} + \cdots + 93\!\cdots\!32 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 25\!\cdots\!61 \nu^{11} + \cdots - 29\!\cdots\!04 ) / 58\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 25\!\cdots\!81 \nu^{11} + \cdots + 30\!\cdots\!16 ) / 58\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 200054566 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} - \beta_{8} - \beta_{7} + 3\beta_{5} + 142\beta_{4} + 12157\beta_{3} - 10\beta_{2} - 303361963\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 67 \beta_{11} - 67 \beta_{10} + 4666 \beta_{7} + 408 \beta_{6} - 77118 \beta_{5} + 771797 \beta_{4} + \cdots + 30\!\cdots\!28 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 690708 \beta_{11} + 690708 \beta_{10} - 115124921 \beta_{9} + 138913073 \beta_{8} + \cdots + 26\!\cdots\!35 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 11639131563 \beta_{11} + 11639131563 \beta_{10} - 951491971434 \beta_{7} - 54043191000 \beta_{6} + \cdots - 26\!\cdots\!12 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 133412968530036 \beta_{11} - 133412968530036 \beta_{10} + \cdots - 23\!\cdots\!11 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 15\!\cdots\!39 \beta_{11} + \cdots + 23\!\cdots\!48 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 18\!\cdots\!40 \beta_{11} + \cdots + 21\!\cdots\!27 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 18\!\cdots\!55 \beta_{11} + \cdots - 21\!\cdots\!92 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 22\!\cdots\!60 \beta_{11} + \cdots - 20\!\cdots\!11 \beta_1 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1
− 10141.2i
− 8962.26i
− 8317.42i
− 5113.95i
− 3807.62i
− 2662.06i
2662.06i
3807.62i
5113.95i
8317.42i
8962.26i
10141.2i
− 20282.3i − 2.08215e6i −2.77155e8 1.59030e9 + 2.21845e9i −4.22309e10 1.79255e11i 2.89910e12i 3.29024e12 4.49953e13 − 3.22550e13i
4.2 − 17924.5i 395547.i −1.87071e8 −6.43856e8 − 2.65255e9i 7.08998e9 − 4.81533e11i 9.47366e11i 7.46914e12 −4.75457e13 + 1.15408e13i
4.3 − 16634.8i 4.35412e6i −1.42500e8 −2.54595e9 + 9.84226e8i 7.24302e10 4.16636e11i 1.37781e11i −1.13328e13 1.63725e13 + 4.23515e13i
4.4 − 10227.9i − 4.81820e6i 2.96077e7 −2.56072e8 − 2.71754e9i −4.92801e10 3.32075e11i − 1.67559e12i −1.55894e13 −2.77947e13 + 2.61908e12i
4.5 − 7615.23i 2.77813e6i 7.62260e7 2.72289e9 − 1.90980e8i 2.11561e10 2.51935e9i − 1.60258e12i −9.24181e10 −1.45436e12 − 2.07354e13i
4.6 − 5324.11i − 1.67512e6i 1.05872e8 −1.68458e9 + 2.14774e9i −8.91852e9 − 1.62340e11i − 1.27826e12i 4.81957e12 1.14348e13 + 8.96887e12i
4.7 5324.11i 1.67512e6i 1.05872e8 −1.68458e9 − 2.14774e9i −8.91852e9 1.62340e11i 1.27826e12i 4.81957e12 1.14348e13 − 8.96887e12i
4.8 7615.23i − 2.77813e6i 7.62260e7 2.72289e9 + 1.90980e8i 2.11561e10 − 2.51935e9i 1.60258e12i −9.24181e10 −1.45436e12 + 2.07354e13i
4.9 10227.9i 4.81820e6i 2.96077e7 −2.56072e8 + 2.71754e9i −4.92801e10 − 3.32075e11i 1.67559e12i −1.55894e13 −2.77947e13 − 2.61908e12i
4.10 16634.8i − 4.35412e6i −1.42500e8 −2.54595e9 − 9.84226e8i 7.24302e10 − 4.16636e11i − 1.37781e11i −1.13328e13 1.63725e13 − 4.23515e13i
4.11 17924.5i − 395547.i −1.87071e8 −6.43856e8 + 2.65255e9i 7.08998e9 4.81533e11i − 9.47366e11i 7.46914e12 −4.75457e13 − 1.15408e13i
4.12 20282.3i 2.08215e6i −2.77155e8 1.59030e9 − 2.21845e9i −4.22309e10 − 1.79255e11i − 2.89910e12i 3.29024e12 4.49953e13 + 3.22550e13i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.28.b.a ✓ 12
3.b odd 2 1 45.28.b.b 12
5.b even 2 1 inner 5.28.b.a ✓ 12
5.c odd 4 2 25.28.a.f 12
15.d odd 2 1 45.28.b.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.28.b.a ✓ 12 1.a even 1 1 trivial
5.28.b.a ✓ 12 5.b even 2 1 inner
25.28.a.f 12 5.c odd 4 2
45.28.b.b 12 3.b odd 2 1
45.28.b.b 12 15.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{28}^{\mathrm{new}}(5, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 62\!\cdots\!16 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 64\!\cdots\!56 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 10\!\cdots\!96)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 78\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 66\!\cdots\!76)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 21\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 14\!\cdots\!04)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 42\!\cdots\!56)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 30\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 22\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 97\!\cdots\!96 \) Copy content Toggle raw display
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