Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [25,22,Mod(1,25)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("25.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(25, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(69.8693360718\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 3780655 x^{8} + 4653816871660 x^{6} + \cdots - 14\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{10}\cdot 5^{24}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{4} + 6 \beta_1) q^{3} + (\beta_{2} + 927372) q^{4} + (\beta_{3} + 17 \beta_{2} + 19018332) q^{6} + (\beta_{7} - 1072 \beta_{4} - 39884 \beta_1) q^{7} + (\beta_{8} + 2 \beta_{7} + \cdots + 1079295 \beta_1) q^{8}+ \cdots + (22480878633 \beta_{6} + \cdots + 22\!\cdots\!56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 9273720 q^{4} + 190183320 q^{6} + 46796905530 q^{9} + 150626450520 q^{11} - 1196972791560 q^{14} + 13236859984160 q^{16} + 111339219544600 q^{19} + 153512457036120 q^{21} + 925398618703200 q^{24}+ \cdots + 22\!\cdots\!60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3780655 x^{8} + 4653816871660 x^{6} + \cdots - 14\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} - 3024524 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10621 \nu^{8} - 22795614323 \nu^{6} + \cdots - 18\!\cdots\!20 ) / 56\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2433333971 \nu^{9} + \cdots + 17\!\cdots\!04 \nu ) / 10\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 18438859 \nu^{8} + 48574256094533 \nu^{6} + \cdots + 13\!\cdots\!44 ) / 45\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 37049771 \nu^{8} + 121243334678629 \nu^{6} + \cdots - 25\!\cdots\!40 ) / 45\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 25499881709789 \nu^{9} + \cdots - 42\!\cdots\!08 \nu ) / 10\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19669613515273 \nu^{9} + \cdots - 96\!\cdots\!96 \nu ) / 53\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 593309725264171 \nu^{9} + \cdots + 92\!\cdots\!44 \nu ) / 35\!\cdots\!04 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 3024524 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} + 2\beta_{7} - 4792\beta_{4} + 5273599\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -11\beta_{6} + 51\beta_{5} + 6271\beta_{3} + 1977650\beta_{2} + 3988574788564 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 29944\beta_{9} + 2473671\beta_{8} + 4044766\beta_{7} - 24298739680\beta_{4} + 8430271666553\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 24560565 \beta_{6} + 133893261 \beta_{5} + 18346620513 \beta_{3} + 3738591680370 \beta_{2} + 63\!\cdots\!24 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 83036493192 \beta_{9} + 5110287943581 \beta_{8} + 7933367385066 \beta_{7} + \cdots + 14\!\cdots\!83 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 51129798555159 \beta_{6} + 280028144083167 \beta_{5} + \cdots + 11\!\cdots\!04 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 18\!\cdots\!60 \beta_{9} + \cdots + 27\!\cdots\!25 \beta_1 ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1378.95
−934.563
−867.225
−441.894
−241.650
241.650
441.894
867.225
934.563
1378.95
−2757.90 −90721.2 5.50886e6 0 2.50200e8 −2.49542e8 −9.40915e9 −2.23002e9 0
1.2 −1869.13 163448. 1.39648e6 0 −3.05505e8 1.05403e9 1.30964e9 1.62549e10 0
1.3 −1734.45 21936.5 911162. 0 −3.80478e7 −8.17780e8 2.05704e9 −9.97914e9 0
1.4 −883.789 −198986. −1.31607e6 0 1.75862e8 4.00513e8 3.01657e9 2.91352e10 0
1.5 −483.300 −26035.4 −1.86357e6 0 1.25829e7 7.88368e8 1.91422e9 −9.78251e9 0
1.6 483.300 26035.4 −1.86357e6 0 1.25829e7 −7.88368e8 −1.91422e9 −9.78251e9 0
1.7 883.789 198986. −1.31607e6 0 1.75862e8 −4.00513e8 −3.01657e9 2.91352e10 0
1.8 1734.45 −21936.5 911162. 0 −3.80478e7 8.17780e8 −2.05704e9 −9.97914e9 0
1.9 1869.13 −163448. 1.39648e6 0 −3.05505e8 −1.05403e9 −1.30964e9 1.62549e10 0
1.10 2757.90 90721.2 5.50886e6 0 2.50200e8 2.49542e8 9.40915e9 −2.23002e9 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.22.a.f 10
5.b even 2 1 inner 25.22.a.f 10
5.c odd 4 2 5.22.b.a ✓ 10
15.e even 4 2 45.22.b.b 10
20.e even 4 2 80.22.c.a 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.22.b.a ✓ 10 5.c odd 4 2
25.22.a.f 10 1.a even 1 1 trivial
25.22.a.f 10 5.b even 2 1 inner
45.22.b.b 10 15.e even 4 2
80.22.c.a 10 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 15122620 T_{2}^{8} + 74461069946560 T_{2}^{6} + \cdots - 14\!\cdots\!24 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(25))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots - 14\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots - 28\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots - 46\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots - 21\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 43\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 31\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots - 44\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 83\!\cdots\!32)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots - 23\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 12\!\cdots\!68)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots - 38\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 50\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 55\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 70\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 10\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots - 69\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 63\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 20\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 15\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 37\!\cdots\!24 \) Copy content Toggle raw display
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