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Properties

Label 25.14.b.c
Level $25$
Weight $14$
Character orbit 25.b
Analytic conductor $26.808$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [25,14,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, 14); 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, 14, names="a")
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 14 \)
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: \(26.8077322380\)
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: \( x^{8} + 35445x^{6} + 335091828x^{4} + 383886844480x^{2} + 107862016720896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{14} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{5} - \beta_{4}) q^{3} + ( - \beta_1 - 933) q^{4} + (\beta_{3} + \beta_{2} + 6 \beta_1 + 10082) q^{6} + (8 \beta_{7} + 44 \beta_{6} + \cdots + 1018 \beta_{4}) q^{7} + ( - 9 \beta_{7} - 54 \beta_{6} + \cdots - 576 \beta_{4}) q^{8}+ \cdots + ( - 376257200 \beta_{3} + \cdots - 356677243450) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 7466 q^{4} + 80666 q^{6} - 95744 q^{9} + 13350856 q^{11} - 74206932 q^{14} - 16881502 q^{16} - 565297720 q^{19} - 877774944 q^{21} - 2830688910 q^{24} - 6534908744 q^{26} - 15261297680 q^{29} - 25432078064 q^{31}+ \cdots - 2851279676608 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 35445x^{6} + 335091828x^{4} + 383886844480x^{2} + 107862016720896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} - 180326\nu^{4} - 4650496987\nu^{2} - 9758372148351 ) / 847608147 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 97135\nu^{6} + 2543458955\nu^{4} + 15553446352540\nu^{2} + 9175915578222720 ) / 515345753376 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 267655\nu^{6} + 9771748115\nu^{4} + 91819644032620\nu^{2} + 54219249646914816 ) / 515345753376 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -17981\nu^{7} - 639067489\nu^{5} - 5869219055396\nu^{3} - 2144213304061632\nu ) / 1467162236400768 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 155084933 \nu^{7} + 5566311488329 \nu^{5} + \cdots + 64\!\cdots\!24 \nu ) / 44\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3650143\nu^{7} - 129730700267\nu^{5} - 1191451468245388\nu^{3} - 740934099974671296\nu ) / 7824865260804096 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3796927853 \nu^{7} + 129752728985041 \nu^{5} + \cdots + 69\!\cdots\!72 \nu ) / 11\!\cdots\!68 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -16\beta_{6} + 609\beta_{4} ) / 625 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -32\beta_{3} + 96\beta_{2} - 625\beta _1 - 5538133 ) / 625 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 375\beta_{7} + 396298\beta_{6} + 143250\beta_{5} - 9977952\beta_{4} ) / 625 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 136714\beta_{3} - 404442\beta_{2} + 2214875\beta _1 + 18317116991 ) / 125 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -20131875\beta_{7} - 7850793946\beta_{6} - 2565791250\beta_{5} + 170089809804\beta_{4} ) / 625 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -12775229882\beta_{3} + 40895335146\beta_{2} - 189903889375\beta _1 - 1570413022395283 ) / 625 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 593107705875\beta_{7} + 151578600376858\beta_{6} + 44432909279250\beta_{5} - 2911900558568892\beta_{4} ) / 625 \) 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
− 127.370i
− 133.966i
− 28.7600i
− 21.1633i
21.1633i
28.7600i
133.966i
127.370i
− 143.370i 618.204i −12362.8 0 88631.7 − 474578.i 597973.i 1.21215e6 0
24.2 − 117.966i 235.672i −5724.04 0 27801.3 227749.i − 291136.i 1.53878e6 0
24.3 − 44.7600i − 1474.96i 6188.54 0 −66019.0 143529.i − 643673.i −581170. 0
24.4 − 5.16335i − 1952.42i 8165.34 0 −10081.0 − 455997.i − 84458.7i −2.21763e6 0
24.5 5.16335i 1952.42i 8165.34 0 −10081.0 455997.i 84458.7i −2.21763e6 0
24.6 44.7600i 1474.96i 6188.54 0 −66019.0 − 143529.i 643673.i −581170. 0
24.7 117.966i − 235.672i −5724.04 0 27801.3 − 227749.i 291136.i 1.53878e6 0
24.8 143.370i − 618.204i −12362.8 0 88631.7 474578.i − 597973.i 1.21215e6 0
\(n\): e.g. 2-40 or 990-1000
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 25.14.b.c 8
5.b even 2 1 inner 25.14.b.c 8
5.c odd 4 1 25.14.a.c ✓ 4
5.c odd 4 1 25.14.a.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.14.a.c ✓ 4 5.c odd 4 1
25.14.a.d yes 4 5.c odd 4 1
25.14.b.c 8 1.a even 1 1 trivial
25.14.b.c 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 36501T_{2}^{6} + 356075316T_{2}^{4} + 582539676736T_{2}^{2} + 15278217117696 \) acting on \(S_{14}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 15278217117696 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 17\!\cdots\!61 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 50\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 60\!\cdots\!01)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 96\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 75\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 91\!\cdots\!25)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 52\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 30\!\cdots\!04)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 76\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 25\!\cdots\!81)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 53\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 22\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 47\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 90\!\cdots\!81 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 91\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 21\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 18\!\cdots\!21 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 17\!\cdots\!25)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
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