Properties

Label 5.28.a.b
Level $5$
Weight $28$
Character orbit 5.a
Self dual yes
Analytic conductor $23.093$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 28 \)
Character orbit: \([\chi]\) \(=\) 5.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.0927787419\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - 2x^{4} - 105406182x^{3} - 8285617904x^{2} + 1593173725628800x - 1939393055148057600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{5}\cdot 5^{4}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 3983) q^{2} + (\beta_{2} - 24 \beta_1 + 974132) q^{3} + (\beta_{3} + 16 \beta_{2} - 7732 \beta_1 + 50296508) q^{4} - 1220703125 q^{5} + (84 \beta_{4} + 132 \beta_{3} + 6524 \beta_{2} + \cdots + 7876627738) q^{6}+ \cdots + (3542 \beta_{4} - 27834 \beta_{3} + 2225862 \beta_{2} + \cdots + 1851249619275) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 3983) q^{2} + (\beta_{2} - 24 \beta_1 + 974132) q^{3} + (\beta_{3} + 16 \beta_{2} - 7732 \beta_1 + 50296508) q^{4} - 1220703125 q^{5} + (84 \beta_{4} + 132 \beta_{3} + 6524 \beta_{2} + \cdots + 7876627738) q^{6}+ \cdots + ( - 81\!\cdots\!87 \beta_{4} + \cdots + 53\!\cdots\!51) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 19916 q^{2} + 4870682 q^{3} + 251490240 q^{4} - 6103515625 q^{5} + 39384982360 q^{6} + 155646348206 q^{7} + 4844427693600 q^{8} + 9256436775085 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 19916 q^{2} + 4870682 q^{3} + 251490240 q^{4} - 6103515625 q^{5} + 39384982360 q^{6} + 155646348206 q^{7} + 4844427693600 q^{8} + 9256436775085 q^{9} - 24311523437500 q^{10} - 34307841041440 q^{11} + 10\!\cdots\!96 q^{12}+ \cdots + 26\!\cdots\!20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 105406182x^{3} - 8285617904x^{2} + 1593173725628800x - 1939393055148057600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -119\nu^{4} - 488562\nu^{3} + 12666282090\nu^{2} + 43203932098384\nu - 154378926928912992 ) / 21119977584 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 119\nu^{4} + 488562\nu^{3} - 7386287694\nu^{2} - 43826971437112\nu - 68238456632840496 ) / 1319998599 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 85711 \nu^{4} - 3065694 \nu^{3} - 8248407182202 \nu^{2} + \cdots + 76\!\cdots\!60 ) / 147839843088 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 16\beta_{2} + 236\beta _1 + 168649948 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -833\beta_{4} + 1232\beta_{3} - 65999\beta_{2} + 75196676\beta _1 + 10210729427 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3419934\beta_{4} + 48161619\beta_{3} + 767520210\beta_{2} + 66893403068\beta _1 + 6339333475629194 ) / 2 \) Copy content Toggle raw display

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
9540.66
3161.03
1409.38
−5020.22
−9088.85
−15097.3 540259. 9.37113e7 −1.22070e9 −8.15645e9 2.87369e11 6.11538e11 −7.33372e12 1.84293e13
1.2 −2338.06 4.67850e6 −1.28751e8 −1.22070e9 −1.09386e10 −1.25721e10 6.14836e11 1.42627e13 2.85407e12
1.3 1165.24 −2.41575e6 −1.32860e8 −1.22070e9 −2.81493e9 −4.04373e11 −3.11211e11 −1.78977e12 −1.42242e12
1.4 14024.4 −1.90136e6 6.24671e7 −1.22070e9 −2.66655e10 3.73002e11 −1.00626e12 −4.01043e12 −1.71197e13
1.5 22161.7 3.96903e6 3.56923e8 −1.22070e9 8.79605e10 −8.77786e10 4.93553e12 8.12762e12 −2.70528e13
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5.28.a.b 5
3.b odd 2 1 45.28.a.d 5
5.b even 2 1 25.28.a.c 5
5.c odd 4 2 25.28.b.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.28.a.b 5 1.a even 1 1 trivial
25.28.a.c 5 5.b even 2 1
25.28.b.c 10 5.c odd 4 2
45.28.a.d 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 19916T_{2}^{4} - 262965912T_{2}^{3} + 4473573947392T_{2}^{2} + 6144845117710336T_{2} - 12783803780642635776 \) acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(5))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 19916 T^{4} + \cdots - 12\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( T^{5} - 4870682 T^{4} + \cdots - 46\!\cdots\!32 \) Copy content Toggle raw display
$5$ \( (T + 1220703125)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 155646348206 T^{4} + \cdots + 47\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{5} + 34307841041440 T^{4} + \cdots + 40\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( T^{5} + \cdots - 71\!\cdots\!32 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots - 63\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{5} + \cdots - 26\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots + 33\!\cdots\!68 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots + 56\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots + 10\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots - 14\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots - 49\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots + 18\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots - 27\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots + 27\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots - 71\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots - 66\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots + 20\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots - 20\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots - 29\!\cdots\!76 \) Copy content Toggle raw display
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