Properties

Label 5.28.a.a
Level $5$
Weight $28$
Character orbit 5.a
Self dual yes
Analytic conductor $23.093$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 19275662x^{2} - 30468026939x + 4134032404260 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4}\cdot 5^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 2887) q^{2} + (\beta_{2} - 195 \beta_1 - 618547) q^{3} + (11 \beta_{3} + 72 \beta_{2} - 389 \beta_1 + 89657800) q^{4} + 1220703125 q^{5} + ( - 2466 \beta_{3} - 15632 \beta_{2} + \cdots - 40237502620) q^{6}+ \cdots + (772992 \beta_{3} - 1222116 \beta_{2} + \cdots + 6447848475921) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 2887) q^{2} + (\beta_{2} - 195 \beta_1 - 618547) q^{3} + (11 \beta_{3} + 72 \beta_{2} - 389 \beta_1 + 89657800) q^{4} + 1220703125 q^{5} + ( - 2466 \beta_{3} - 15632 \beta_{2} + \cdots - 40237502620) q^{6}+ \cdots + ( - 34\!\cdots\!76 \beta_{3} + \cdots - 86\!\cdots\!38) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 11550 q^{2} - 2473800 q^{3} + 358631812 q^{4} + 4882812500 q^{5} - 160951168752 q^{6} - 215015185000 q^{7} + 183282059400 q^{8} + 25791757037748 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 11550 q^{2} - 2473800 q^{3} + 358631812 q^{4} + 4882812500 q^{5} - 160951168752 q^{6} - 215015185000 q^{7} + 183282059400 q^{8} + 25791757037748 q^{9} - 14099121093750 q^{10} - 107427307660512 q^{11} + 13\!\cdots\!00 q^{12}+ \cdots - 34\!\cdots\!44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 19275662x^{2} - 30468026939x + 4134032404260 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -14\nu^{3} + 380152\nu^{2} - 82619548\nu - 3344017320395 ) / 181745885 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 64542\nu^{3} - 194733156\nu^{2} - 664410556656\nu + 401954610728990 ) / 181745885 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -99966\nu^{3} + 221933208\nu^{2} + 1396595469348\nu + 145370315571619 ) / 36349177 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 8\beta_{2} + 1179\beta _1 + 594 ) / 8640 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 671\beta_{3} + 6376\beta_{2} + 5438133\beta _1 + 83273582430 ) / 8640 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6159373\beta_{3} + 62960456\beta_{2} + 14272294383\beta _1 + 98723577989466 ) / 4320 \) 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
125.697
−2324.55
−2819.41
5018.26
−21310.7 4.70996e6 3.19926e8 1.22070e9 −1.00372e11 −2.75207e11 −3.95757e12 1.45581e13 −2.60140e13
1.2 −7959.76 829899. −7.08599e7 1.22070e9 −6.60580e9 −2.73788e10 1.63237e12 −6.93687e12 −9.71651e12
1.3 −1651.55 −4.81686e6 −1.31490e8 1.22070e9 7.95528e9 4.20350e11 4.38829e11 1.55766e13 −2.01605e12
1.4 19372.0 −3.19680e6 2.41055e8 1.22070e9 −6.19283e10 −3.32780e11 2.06966e12 2.59392e12 2.36474e13
\(n\): e.g. 2-40 or 990-1000
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.a 4
3.b odd 2 1 45.28.a.b 4
5.b even 2 1 25.28.a.b 4
5.c odd 4 2 25.28.b.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.28.a.a 4 1.a even 1 1 trivial
25.28.a.b 4 5.b even 2 1
25.28.b.b 8 5.c odd 4 2
45.28.a.b 4 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 11550T_{2}^{3} - 381050112T_{2}^{2} - 3942345932800T_{2} - 5427025739710464 \) acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(5))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 11550 T^{3} + \cdots - 54\!\cdots\!64 \) Copy content Toggle raw display
$3$ \( T^{4} + 2473800 T^{3} + \cdots + 60\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( (T - 1220703125)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 215015185000 T^{3} + \cdots - 10\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{4} + 107427307660512 T^{3} + \cdots + 23\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots - 38\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots - 19\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots - 60\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 49\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 33\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 86\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 27\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 82\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 94\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 20\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 13\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 13\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 11\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 56\!\cdots\!24 \) Copy content Toggle raw display
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