Properties

Label 5.22.a.b
Level $5$
Weight $22$
Character orbit 5.a
Self dual yes
Analytic conductor $13.974$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 5.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.9738672144\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - x^{3} - 1929606x^{2} - 743130000x + 239341586400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6}\cdot 3\cdot 5^{2}\cdot 7 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 728) q^{2} + (\beta_{3} + 14 \beta_1 + 20803) q^{3} + ( - 5 \beta_{3} + \beta_{2} - 297 \beta_1 + 2291466) q^{4} + 9765625 q^{5} + (2450 \beta_{3} + 6 \beta_{2} - 10022 \beta_1 - 39626164) q^{6} + ( - 3741 \beta_{3} - 240 \beta_{2} - 151718 \beta_1 + 128229189) q^{7} + ( - 4742 \beta_{3} + 430 \beta_{2} - 1594534 \beta_1 + 1292638252) q^{8} + ( - 30700 \beta_{3} + 2112 \beta_{2} - 5979944 \beta_1 + 5436213161) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 728) q^{2} + (\beta_{3} + 14 \beta_1 + 20803) q^{3} + ( - 5 \beta_{3} + \beta_{2} - 297 \beta_1 + 2291466) q^{4} + 9765625 q^{5} + (2450 \beta_{3} + 6 \beta_{2} - 10022 \beta_1 - 39626164) q^{6} + ( - 3741 \beta_{3} - 240 \beta_{2} - 151718 \beta_1 + 128229189) q^{7} + ( - 4742 \beta_{3} + 430 \beta_{2} - 1594534 \beta_1 + 1292638252) q^{8} + ( - 30700 \beta_{3} + 2112 \beta_{2} - 5979944 \beta_1 + 5436213161) q^{9} + ( - 9765625 \beta_1 + 7109375000) q^{10} + ( - 498310 \beta_{3} - 34960 \beta_{2} + \cdots + 8426594482) q^{11}+ \cdots + ( - 10\!\cdots\!30 \beta_{3} + \cdots - 37\!\cdots\!58) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2910 q^{2} + 83240 q^{3} + 9165268 q^{4} + 39062500 q^{5} - 158524712 q^{6} + 512613800 q^{7} + 5167363080 q^{8} + 21732888532 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2910 q^{2} + 83240 q^{3} + 9165268 q^{4} + 39062500 q^{5} - 158524712 q^{6} + 512613800 q^{7} + 5167363080 q^{8} + 21732888532 q^{9} + 28417968750 q^{10} + 33727076448 q^{11} - 142435377680 q^{12} + 863532165080 q^{13} + 2725405637616 q^{14} + 812890625000 q^{15} + 9168135122704 q^{16} + 17694691101480 q^{17} + 108219081471590 q^{18} + 65217596849840 q^{19} + 89504570312500 q^{20} - 248634744508992 q^{21} - 133302721028280 q^{22} + 306130984922520 q^{23} - 509427036802080 q^{24} + 381469726562500 q^{25} - 19\!\cdots\!12 q^{26}+ \cdots - 15\!\cdots\!16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 1929606x^{2} - 743130000x + 239341586400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{3} - 3089\nu^{2} - 6622602\nu + 187959204 ) / 324 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 877\nu^{2} - 1174314\nu + 287631324 ) / 324 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -5\beta_{3} + \beta_{2} + 1159\beta _1 + 3858634 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3089\beta_{3} + 877\beta_{2} + 3365071\beta _1 + 2233496722 ) / 4 \) 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
1521.87
210.082
−844.370
−886.582
−2315.74 45067.6 3.26550e6 9.76562e6 −1.04365e8 −6.93632e8 −2.70560e9 −8.42927e9 −2.26147e10
1.2 307.836 62164.3 −2.00239e6 9.76562e6 1.91364e7 8.89757e8 −1.26199e9 −6.59596e9 3.00621e9
1.3 2416.74 157402. 3.74348e6 9.76562e6 3.80399e8 −6.35419e8 3.97874e9 1.43149e10 2.36010e10
1.4 2501.16 −181393. 4.15867e6 9.76562e6 −4.53695e8 9.51907e8 5.15621e9 2.24432e10 2.44254e10
\(n\): 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 5.22.a.b 4
3.b odd 2 1 45.22.a.f 4
4.b odd 2 1 80.22.a.g 4
5.b even 2 1 25.22.a.c 4
5.c odd 4 2 25.22.b.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.22.a.b 4 1.a even 1 1 trivial
25.22.a.c 4 5.b even 2 1
25.22.b.c 8 5.c odd 4 2
45.22.a.f 4 3.b odd 2 1
80.22.a.g 4 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 2910T_{2}^{3} - 4542888T_{2}^{2} + 15642931840T_{2} - 4309053579264 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(5))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2910 T^{3} + \cdots - 4309053579264 \) Copy content Toggle raw display
$3$ \( T^{4} - 83240 T^{3} + \cdots - 79\!\cdots\!04 \) Copy content Toggle raw display
$5$ \( (T - 9765625)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 512613800 T^{3} + \cdots + 37\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{4} - 33727076448 T^{3} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{4} - 863532165080 T^{3} + \cdots - 19\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{4} - 17694691101480 T^{3} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{4} - 65217596849840 T^{3} + \cdots - 27\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} - 306130984922520 T^{3} + \cdots - 10\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots - 37\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 70\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 44\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 72\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 58\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 28\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 21\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 46\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
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