Properties

Label 5.20.a.b
Level $5$
Weight $20$
Character orbit 5.a
Self dual yes
Analytic conductor $11.441$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 5.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.4408348278\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 2x^{3} - 323561x^{2} - 26738538x + 10870990650 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 5^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 105) q^{2} + ( - \beta_{2} + 14 \beta_1 + 770) q^{3} + ( - \beta_{3} - 26 \beta_{2} - 18 \beta_1 + 521248) q^{4} - 1953125 q^{5} + (96 \beta_{3} + 256 \beta_{2} - 24482 \beta_1 - 14517938) q^{6} + ( - 720 \beta_{3} - 81 \beta_{2} - 49970 \beta_1 + 53505350) q^{7} + (2660 \beta_{3} - 968 \beta_{2} - 698512 \beta_1 + 20434440) q^{8} + ( - 5472 \beta_{3} + 10108 \beta_{2} + 508024 \beta_1 + 38516137) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 105) q^{2} + ( - \beta_{2} + 14 \beta_1 + 770) q^{3} + ( - \beta_{3} - 26 \beta_{2} - 18 \beta_1 + 521248) q^{4} - 1953125 q^{5} + (96 \beta_{3} + 256 \beta_{2} - 24482 \beta_1 - 14517938) q^{6} + ( - 720 \beta_{3} - 81 \beta_{2} - 49970 \beta_1 + 53505350) q^{7} + (2660 \beta_{3} - 968 \beta_{2} - 698512 \beta_1 + 20434440) q^{8} + ( - 5472 \beta_{3} + 10108 \beta_{2} + 508024 \beta_1 + 38516137) q^{9} + (1953125 \beta_1 + 205078125) q^{10} + (40240 \beta_{3} - 77230 \beta_{2} + 1619940 \beta_1 + 2896428192) q^{11} + ( - 97890 \beta_{3} - 306164 \beta_{2} + \cdots + 26407339840) q^{12}+ \cdots + ( - 26293460644944 \beta_{3} + \cdots - 63\!\cdots\!96) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 420 q^{2} + 3080 q^{3} + 2084992 q^{4} - 7812500 q^{5} - 58071752 q^{6} + 214021400 q^{7} + 81737760 q^{8} + 154064548 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 420 q^{2} + 3080 q^{3} + 2084992 q^{4} - 7812500 q^{5} - 58071752 q^{6} + 214021400 q^{7} + 81737760 q^{8} + 154064548 q^{9} + 820312500 q^{10} + 11585712768 q^{11} + 105629359360 q^{12} + 14812333160 q^{13} + 185174155944 q^{14} - 6015625000 q^{15} + 1785780098944 q^{16} + 849033742440 q^{17} - 2113778999620 q^{18} + 1978167708560 q^{19} - 4072250000000 q^{20} - 1487020185552 q^{21} - 7953348762240 q^{22} - 26569906952760 q^{23} - 39774243472320 q^{24} + 15258789062500 q^{25} - 48695658207912 q^{26} - 7557605929360 q^{27} + 236612033519360 q^{28} + 116267174339640 q^{29} + 113421390625000 q^{30} + 251049672388688 q^{31} - 142495342974720 q^{32} + 359905680636160 q^{33} + 411849015040344 q^{34} - 418010546875000 q^{35} - 168308645735296 q^{36} + 53471657716520 q^{37} - 52\!\cdots\!60 q^{38}+ \cdots - 25\!\cdots\!84 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 323561x^{2} - 26738538x + 10870990650 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -50\nu^{3} + 12712\nu^{2} + 10576678\nu - 1034892417 ) / 1360737 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -100\nu^{3} + 25424\nu^{2} + 57439676\nu - 2087927994 ) / 453579 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -340\nu^{3} - 224584\nu^{2} + 120441404\nu + 43256659674 ) / 194391 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 6\beta _1 + 40 ) / 80 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -25\beta_{3} + 78\beta_{2} + 722\beta _1 + 6471260 ) / 40 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -12712\beta_{3} + 251195\beta_{2} - 3079258\beta _1 + 1643139760 ) / 80 \) 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
−462.664
150.720
581.867
−267.923
−1387.10 48079.9 1.39976e6 −1.95312e6 −6.66917e7 9.13351e7 −1.21437e9 1.14942e9 2.70918e9
1.2 −602.379 −7268.55 −161427. −1.95312e6 4.37842e6 −1.76809e8 4.13061e8 −1.10943e9 1.17652e9
1.3 208.721 −48249.1 −480724. −1.95312e6 −1.00706e7 1.75500e8 −2.09767e8 1.16572e9 −4.07657e8
1.4 1360.76 10517.7 1.32738e6 −1.95312e6 1.43121e7 1.23996e8 1.09282e9 −1.05164e9 −2.65773e9
\(n\): 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 5.20.a.b 4
3.b odd 2 1 45.20.a.f 4
4.b odd 2 1 80.20.a.g 4
5.b even 2 1 25.20.a.c 4
5.c odd 4 2 25.20.b.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.20.a.b 4 1.a even 1 1 trivial
25.20.a.c 4 5.b even 2 1
25.20.b.c 8 5.c odd 4 2
45.20.a.f 4 3.b odd 2 1
80.20.a.g 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 420T_{2}^{3} - 2002872T_{2}^{2} - 746347520T_{2} + 237314973696 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(5))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 420 T^{3} + \cdots + 237314973696 \) Copy content Toggle raw display
$3$ \( T^{4} - 3080 T^{3} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$5$ \( (T + 1953125)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 214021400 T^{3} + \cdots - 35\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{4} - 11585712768 T^{3} + \cdots - 53\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{4} - 14812333160 T^{3} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{4} - 849033742440 T^{3} + \cdots - 65\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{4} - 1978167708560 T^{3} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} + 26569906952760 T^{3} + \cdots - 42\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{4} - 116267174339640 T^{3} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} - 251049672388688 T^{3} + \cdots - 30\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{4} - 53471657716520 T^{3} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 54\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 43\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 52\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 59\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 53\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 98\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 18\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 22\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 27\!\cdots\!96 \) Copy content Toggle raw display
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