Properties

Label 5.26.a.b
Level $5$
Weight $26$
Character orbit 5.a
Self dual yes
Analytic conductor $19.800$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 5.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.7998389976\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - x^{4} - 31823342x^{3} + 3040467992x^{2} + 155755658754016x - 41401144140044416 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{12}\cdot 3^{3}\cdot 5^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 920) q^{2} + ( - \beta_{2} - 25 \beta_1 + 125251) q^{3} + (\beta_{3} + 7 \beta_{2} + 1558 \beta_1 + 18209427) q^{4} + 244140625 q^{5} + (\beta_{4} + 92 \beta_{3} - 1007 \beta_{2} + 47 \beta_1 + 1139315098) q^{6} + ( - 18 \beta_{4} + 612 \beta_{3} - 18297 \beta_{2} + \cdots + 11095633501) q^{7}+ \cdots + ( - 368 \beta_{4} - 11296 \beta_{3} - 466004 \beta_{2} + \cdots + 55614233065) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 920) q^{2} + ( - \beta_{2} - 25 \beta_1 + 125251) q^{3} + (\beta_{3} + 7 \beta_{2} + 1558 \beta_1 + 18209427) q^{4} + 244140625 q^{5} + (\beta_{4} + 92 \beta_{3} - 1007 \beta_{2} + 47 \beta_1 + 1139315098) q^{6} + ( - 18 \beta_{4} + 612 \beta_{3} - 18297 \beta_{2} + \cdots + 11095633501) q^{7}+ \cdots + (30\!\cdots\!78 \beta_{4} + \cdots - 45\!\cdots\!06) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4602 q^{2} + 626204 q^{3} + 91050260 q^{4} + 1220703125 q^{5} + 5696574760 q^{6} + 55481235808 q^{7} - 325515457080 q^{8} + 277996846465 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 4602 q^{2} + 626204 q^{3} + 91050260 q^{4} + 1220703125 q^{5} + 5696574760 q^{6} + 55481235808 q^{7} - 325515457080 q^{8} + 277996846465 q^{9} - 1123535156250 q^{10} - 4144958451540 q^{11} - 26370992065712 q^{12} + 111211249076614 q^{13} - 445566653510880 q^{14} + 152881835937500 q^{15} + 30\!\cdots\!80 q^{16}+ \cdots - 22\!\cdots\!20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 31823342x^{3} + 3040467992x^{2} + 155755658754016x - 41401144140044416 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{4} + 5339\nu^{3} + 204537534\nu^{2} - 118916988560\nu - 630583703059808 ) / 446010624 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 49\nu^{4} - 37373\nu^{3} + 352279758\nu^{2} + 1083968911856\nu - 18295641739665760 ) / 446010624 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -7333\nu^{4} - 22725151\nu^{3} + 211067727450\nu^{2} + 499281649962832\nu - 671092626032517152 ) / 446010624 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 7\beta_{2} - 282\beta _1 + 50917459 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -63\beta_{4} - 113\beta_{3} + 65206\beta_{2} + 44092373\beta _1 - 7238397117 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -48051\beta_{4} + 29133461\beta_{3} - 592132\beta_{2} - 8586340323\beta _1 + 1121935840994173 ) / 4 \) 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
5041.97
2350.27
268.342
−2546.26
−5113.32
−11003.9 −418448. 8.75320e7 2.44141e8 4.60457e9 7.06271e10 −5.93966e11 −6.72190e11 −2.68651e12
1.2 −5620.55 −161492. −1.96387e6 2.44141e8 9.07675e8 −4.86926e10 1.99632e11 −8.21209e11 −1.37220e12
1.3 −1456.68 1.56404e6 −3.14325e7 2.44141e8 −2.27831e9 3.49059e10 9.46654e10 1.59893e12 −3.55636e11
1.4 4172.52 −1.12842e6 −1.61445e7 2.44141e8 −4.70836e9 −2.36507e10 −2.07370e11 4.26045e11 1.01868e12
1.5 9306.64 770525. 5.30591e7 2.44141e8 7.17100e9 2.22916e10 1.81523e11 −2.53579e11 2.27213e12
\(n\): 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 5.26.a.b 5
3.b odd 2 1 45.26.a.f 5
5.b even 2 1 25.26.a.c 5
5.c odd 4 2 25.26.b.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.26.a.b 5 1.a even 1 1 trivial
25.26.a.c 5 5.b even 2 1
25.26.b.c 10 5.c odd 4 2
45.26.a.f 5 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} + 4602T_{2}^{4} - 118822008T_{2}^{3} - 367856402816T_{2}^{2} + 2127699725500416T_{2} + 3498510988641042432 \) acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(5))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} + 4602 T^{4} + \cdots + 34\!\cdots\!32 \) Copy content Toggle raw display
$3$ \( T^{5} - 626204 T^{4} + \cdots + 91\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( (T - 244140625)^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 55481235808 T^{4} + \cdots - 63\!\cdots\!68 \) Copy content Toggle raw display
$11$ \( T^{5} + 4144958451540 T^{4} + \cdots + 87\!\cdots\!68 \) Copy content Toggle raw display
$13$ \( T^{5} - 111211249076614 T^{4} + \cdots - 56\!\cdots\!24 \) Copy content Toggle raw display
$17$ \( T^{5} + \cdots + 11\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{5} + \cdots - 68\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{5} + \cdots + 70\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{5} + \cdots - 52\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{5} + \cdots + 12\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( T^{5} + \cdots - 97\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( T^{5} + \cdots - 29\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( T^{5} + \cdots + 65\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{5} + \cdots + 44\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{5} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{5} + \cdots + 84\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{5} + \cdots + 53\!\cdots\!68 \) Copy content Toggle raw display
$67$ \( T^{5} + \cdots - 16\!\cdots\!68 \) Copy content Toggle raw display
$71$ \( T^{5} + \cdots + 32\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{5} + \cdots - 19\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{5} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{5} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( T^{5} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{5} + \cdots - 47\!\cdots\!68 \) Copy content Toggle raw display
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