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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 = [14] 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: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 22352378 x^{12} + 189802834808311 x^{10} + \cdots + 98\!\cdots\!84 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{6}\cdot 5^{50}\cdot 7^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + (\beta_{9} - \beta_{7}) q^{3} + (\beta_1 - 1107130) q^{4} + ( - \beta_{3} + 9 \beta_1 - 4348991) q^{6} + (\beta_{11} - 2082 \beta_{9} + \cdots - 1906 \beta_{7}) q^{7} + (\beta_{11} + \beta_{10} + \cdots + 1083449 \beta_{7}) q^{8}+ \cdots + (3625952931 \beta_{6} + \cdots - 37\!\cdots\!60) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 15499818 q^{4} - 60885862 q^{6} - 3712208372 q^{9} - 34053861182 q^{11} - 26807291796 q^{14} + 16303919085474 q^{16} + 147855581981790 q^{19} + 322845385573788 q^{21} + 779473547408370 q^{24} + 30\!\cdots\!48 q^{26}+ \cdots - 52\!\cdots\!64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 22352378 x^{12} + 189802834808311 x^{10} + \cdots + 98\!\cdots\!84 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 58\!\cdots\!59 \nu^{12} + \cdots + 20\!\cdots\!04 ) / 59\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 13\!\cdots\!41 \nu^{12} + \cdots - 30\!\cdots\!96 ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 10\!\cdots\!43 \nu^{12} + \cdots - 26\!\cdots\!08 ) / 71\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23\!\cdots\!73 \nu^{12} + \cdots + 50\!\cdots\!88 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 34\!\cdots\!41 \nu^{12} + \cdots + 88\!\cdots\!96 ) / 17\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 44\!\cdots\!77 \nu^{12} + \cdots + 10\!\cdots\!12 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 65\!\cdots\!01 \nu^{13} + \cdots - 76\!\cdots\!44 \nu ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 23\!\cdots\!93 \nu^{13} + \cdots + 38\!\cdots\!92 \nu ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 41\!\cdots\!71 \nu^{13} + \cdots - 10\!\cdots\!76 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 41\!\cdots\!91 \nu^{13} + \cdots + 10\!\cdots\!96 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 36\!\cdots\!43 \nu^{13} + \cdots - 91\!\cdots\!08 \nu ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 20\!\cdots\!59 \nu^{13} + \cdots + 60\!\cdots\!04 \nu ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 34\!\cdots\!31 \nu^{13} + \cdots + 81\!\cdots\!36 \nu ) / 62\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -21\beta_{8} - 15562\beta_{7} ) / 15625 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 42\beta_{2} + 15793\beta _1 - 49893703143 ) / 15625 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 1575 \beta_{12} + 14050 \beta_{11} + 17200 \beta_{10} - 83393825 \beta_{9} + \cdots + 82919421954 \beta_{7} ) / 15625 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1008125 \beta_{6} + 6253750 \beta_{5} + 45704825 \beta_{4} + 236639550 \beta_{3} + \cdots + 26\!\cdots\!79 ) / 15625 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 349890000 \beta_{13} + 6112773450 \beta_{12} - 24464181550 \beta_{11} + \cdots - 10\!\cdots\!52 \beta_{7} ) / 3125 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 11928633088750 \beta_{6} - 69060925002500 \beta_{5} - 634394774680150 \beta_{4} + \cdots - 17\!\cdots\!59 ) / 15625 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 23\!\cdots\!00 \beta_{13} + \cdots + 36\!\cdots\!46 \beta_{7} ) / 15625 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 11\!\cdots\!75 \beta_{6} + \cdots + 11\!\cdots\!43 ) / 15625 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 24\!\cdots\!00 \beta_{13} + \cdots - 26\!\cdots\!28 \beta_{7} ) / 15625 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 19\!\cdots\!00 \beta_{6} + \cdots - 17\!\cdots\!63 ) / 3125 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 22\!\cdots\!00 \beta_{13} + \cdots + 19\!\cdots\!18 \beta_{7} ) / 15625 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 82\!\cdots\!25 \beta_{6} + \cdots + 66\!\cdots\!67 ) / 15625 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 20\!\cdots\!00 \beta_{13} + \cdots - 15\!\cdots\!16 \beta_{7} ) / 15625 \) 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
− 2832.50i
− 2328.48i
− 2028.11i
− 1647.62i
− 1134.06i
− 640.447i
− 619.148i
619.148i
640.447i
1134.06i
1647.62i
2028.11i
2328.48i
2832.50i
− 2727.50i − 50838.6i −5.34213e6 0 −1.38663e8 3.19642e8i 8.85070e9i 7.87579e9 0
24.2 − 2433.48i − 81248.2i −3.82468e6 0 −1.97716e8 9.23572e8i 4.20390e9i 3.85908e9 0
24.3 − 2133.11i 148627.i −2.45299e6 0 3.17038e8 − 1.25395e9i 7.59041e8i −1.16297e10 0
24.4 − 1542.62i 110812.i −282527. 0 1.70941e8 − 3.07836e8i − 2.79928e9i −1.81900e9 0
24.5 − 1029.06i − 120096.i 1.03819e6 0 −1.23586e8 2.19729e8i − 3.22645e9i −3.96271e9 0
24.6 − 745.447i − 122603.i 1.54146e6 0 −9.13943e7 − 1.12071e9i − 2.71239e9i −4.57120e9 0
24.7 − 724.148i 45482.7i 1.57276e6 0 3.29362e7 8.64838e8i − 2.65756e9i 8.39168e9 0
24.8 724.148i − 45482.7i 1.57276e6 0 3.29362e7 − 8.64838e8i 2.65756e9i 8.39168e9 0
24.9 745.447i 122603.i 1.54146e6 0 −9.13943e7 1.12071e9i 2.71239e9i −4.57120e9 0
24.10 1029.06i 120096.i 1.03819e6 0 −1.23586e8 − 2.19729e8i 3.22645e9i −3.96271e9 0
24.11 1542.62i − 110812.i −282527. 0 1.70941e8 3.07836e8i 2.79928e9i −1.81900e9 0
24.12 2133.11i − 148627.i −2.45299e6 0 3.17038e8 1.25395e9i − 7.59041e8i −1.16297e10 0
24.13 2433.48i 81248.2i −3.82468e6 0 −1.97716e8 − 9.23572e8i − 4.20390e9i 3.85908e9 0
24.14 2727.50i 50838.6i −5.34213e6 0 −1.38663e8 − 3.19642e8i − 8.85070e9i 7.87579e9 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 24.14
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.d 14
5.b even 2 1 inner 25.22.b.d 14
5.c odd 4 1 25.22.a.d ✓ 7
5.c odd 4 1 25.22.a.e yes 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.22.a.d ✓ 7 5.c odd 4 1
25.22.a.e yes 7 5.c odd 4 1
25.22.b.d 14 1.a even 1 1 trivial
25.22.b.d 14 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + 22429973 T_{2}^{12} + 192310433302516 T_{2}^{10} + \cdots + 14\!\cdots\!04 \) acting on \(S_{22}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 20\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 58\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( (T^{7} + \cdots - 22\!\cdots\!83)^{2} \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 75\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 74\!\cdots\!29 \) Copy content Toggle raw display
$19$ \( (T^{7} + \cdots - 88\!\cdots\!25)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 53\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots + 47\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} + \cdots - 13\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots - 69\!\cdots\!73)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{7} + \cdots - 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 64\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 20\!\cdots\!29 \) Copy content Toggle raw display
$71$ \( (T^{7} + \cdots - 75\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 25\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots + 65\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 28\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots - 60\!\cdots\!25)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 24\!\cdots\!04 \) Copy content Toggle raw display
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