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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [25,22,Mod(24,25)] 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("25.24"); 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([1])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(69.8693360718\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3859213x^{6} + 4200576228036x^{4} + 1475912119233916800x^{2} + 57284394980468664960000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{2}\cdot 5^{14}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{4}) q^{2} + ( - \beta_{6} + 11 \beta_{5} - 14 \beta_{4}) q^{3} + (\beta_{3} + \beta_1 - 2291317) q^{4} + (162 \beta_{3} - 80 \beta_{2} + \cdots - 39631178) q^{6} + ( - 24 \beta_{7} + \cdots - 151646 \beta_{4}) q^{7}+ \cdots + (710801478320778 \beta_{3} + \cdots + 37\!\cdots\!04) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 18330536 q^{4} - 317049424 q^{6} - 43465777064 q^{9} + 67454152896 q^{11} - 5450811275232 q^{14} + 18336270245408 q^{16} - 130435193699680 q^{19} - 497269489017984 q^{21} + 10\!\cdots\!60 q^{24} - 38\!\cdots\!24 q^{26}+ \cdots + 30\!\cdots\!32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3859213x^{6} + 4200576228036x^{4} + 1475912119233916800x^{2} + 57284394980468664960000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 620312239 \nu^{6} + \cdots + 10\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 124060223 \nu^{6} + 392250951980099 \nu^{4} + \cdots + 10\!\cdots\!00 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 124046723 \nu^{6} - 382166597604599 \nu^{4} + \cdots - 76\!\cdots\!00 ) / 29\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3601121429 \nu^{7} + \cdots - 61\!\cdots\!00 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 124176601 \nu^{7} + 479184062610613 \nu^{5} + \cdots + 12\!\cdots\!00 \nu ) / 17\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 27524565526849 \nu^{7} + \cdots - 31\!\cdots\!00 \nu ) / 55\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7669365595877 \nu^{7} + \cdots + 12\!\cdots\!00 \nu ) / 61\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -261\beta_{5} - 625\beta_{4} ) / 1250 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -103\beta_{3} - 728\beta_{2} + 625\beta _1 - 2412008125 ) / 2500 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 109625\beta_{7} - 3422750\beta_{6} - 476336659\beta_{5} + 4206009875\beta_{4} ) / 5000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 943201087\beta_{3} + 2149671962\beta_{2} - 1206551875\beta _1 + 4057732827060625 ) / 2500 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 304665618125 \beta_{7} + 10888655093750 \beta_{6} + \cdots - 85\!\cdots\!75 \beta_{4} ) / 5000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 27\!\cdots\!23 \beta_{3} + \cdots - 82\!\cdots\!25 ) / 2500 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 74\!\cdots\!25 \beta_{7} + \cdots + 19\!\cdots\!75 \beta_{4} ) / 5000 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
− 886.582i
− 844.370i
− 1521.87i
210.082i
− 210.082i
1521.87i
844.370i
886.582i
− 2501.16i − 181393.i −4.15867e6 0 −4.53695e8 − 9.51907e8i 5.15621e9i −2.24432e10 0
24.2 − 2416.74i 157402.i −3.74348e6 0 3.80399e8 6.35419e8i 3.97874e9i −1.43149e10 0
24.3 − 2315.74i − 45067.6i −3.26550e6 0 −1.04365e8 − 6.93632e8i 2.70560e9i 8.42927e9 0
24.4 − 307.836i 62164.3i 2.00239e6 0 1.91364e7 − 8.89757e8i − 1.26199e9i 6.59596e9 0
24.5 307.836i − 62164.3i 2.00239e6 0 1.91364e7 8.89757e8i 1.26199e9i 6.59596e9 0
24.6 2315.74i 45067.6i −3.26550e6 0 −1.04365e8 6.93632e8i − 2.70560e9i 8.42927e9 0
24.7 2416.74i − 157402.i −3.74348e6 0 3.80399e8 − 6.35419e8i − 3.97874e9i −1.43149e10 0
24.8 2501.16i 181393.i −4.15867e6 0 −4.53695e8 9.51907e8i − 5.15621e9i −2.24432e10 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 24.8
Significant digits:
Format:

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.b.c 8
5.b even 2 1 inner 25.22.b.c 8
5.c odd 4 1 5.22.a.b ✓ 4
5.c odd 4 1 25.22.a.c 4
15.e even 4 1 45.22.a.f 4
20.e even 4 1 80.22.a.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.22.a.b ✓ 4 5.c odd 4 1
25.22.a.c 4 5.c odd 4 1
25.22.b.c 8 1.a even 1 1 trivial
25.22.b.c 8 5.b even 2 1 inner
45.22.a.f 4 15.e even 4 1
80.22.a.g 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 17553876 T_{2}^{6} + 103061587530816 T_{2}^{4} + \cdots + 18\!\cdots\!96 \) acting on \(S_{22}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 63\!\cdots\!16 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 17\!\cdots\!36)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 37\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 37\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 44\!\cdots\!96)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 52\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 34\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 37\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 47\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 28\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 57\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 98\!\cdots\!76 \) Copy content Toggle raw display
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