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Properties

Label 25.26.b.c
Level $25$
Weight $26$
Character orbit 25.b
Analytic conductor $98.999$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,26,Mod(24,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.24");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(98.9991949881\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 63646685 x^{8} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{29}\cdot 3^{6}\cdot 5^{24} \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + ( - \beta_{7} - 24 \beta_{6} + 25 \beta_{5}) q^{3} + (\beta_1 - 18210052) q^{4} + ( - \beta_{4} - 7 \beta_{3} + \cdots + 1139314952) q^{6}+ \cdots + ( - 368 \beta_{4} + 1402 \beta_{3} + \cdots - 55599369293) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + ( - \beta_{7} - 24 \beta_{6} + 25 \beta_{5}) q^{3} + (\beta_1 - 18210052) q^{4} + ( - \beta_{4} - 7 \beta_{3} + \cdots + 1139314952) q^{6}+ \cdots + (30\!\cdots\!78 \beta_{4} + \cdots + 45\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 182100520 q^{4} + 11393149520 q^{6} - 555993692930 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 182100520 q^{4} + 11393149520 q^{6} - 555993692930 q^{9} - 8289916903080 q^{11} + 891133307021760 q^{14} + 60\!\cdots\!60 q^{16}+ \cdots + 45\!\cdots\!40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 63646685 x^{8} + \cdots + 17\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 13979561 \nu^{8} + 698107475997269 \nu^{6} + \cdots + 55\!\cdots\!36 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 607807 \nu^{8} + 30352498956403 \nu^{6} + \cdots + 76\!\cdots\!32 ) / 13\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 346438941304801 \nu^{8} + \cdots - 12\!\cdots\!24 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!31 \nu^{8} + \cdots + 14\!\cdots\!56 ) / 78\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 755585045058991 \nu^{9} + \cdots - 16\!\cdots\!84 \nu ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 32851523698217 \nu^{9} + \cdots - 46\!\cdots\!08 \nu ) / 20\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14\!\cdots\!27 \nu^{9} + \cdots - 16\!\cdots\!52 \nu ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!63 \nu^{9} + \cdots - 12\!\cdots\!12 \nu ) / 96\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 24\!\cdots\!17 \nu^{9} + \cdots - 52\!\cdots\!92 \nu ) / 14\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 92\beta_{6} - 625\beta_{5} ) / 1250 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -184\beta_{2} + 625\beta _1 - 31823342500 ) / 2500 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11275\beta_{9} - 16825\beta_{8} - 81572225\beta_{7} - 9583498524\beta_{6} + 55115424775\beta_{5} ) / 5000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 30031875\beta_{4} - 499423275\beta_{3} + 5144877458\beta_{2} - 17708989850\beta _1 + 701207761251632500 ) / 2500 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 74774604475 \beta_{9} + 101065615675 \beta_{8} + 514202487940775 \beta_{7} + \cdots - 27\!\cdots\!25 \beta_{5} ) / 1000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 837249275964375 \beta_{4} + \cdots - 17\!\cdots\!00 ) / 2500 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10\!\cdots\!75 \beta_{9} + \cdots + 34\!\cdots\!25 \beta_{5} ) / 5000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 19\!\cdots\!75 \beta_{4} + \cdots + 44\!\cdots\!00 ) / 2500 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 27\!\cdots\!75 \beta_{9} + \cdots - 89\!\cdots\!25 \beta_{5} ) / 5000 \) 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.

comment: embeddings in the coefficient field
 
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
− 5041.97i
− 5113.32i
− 2350.27i
− 2546.26i
− 268.342i
268.342i
2546.26i
2350.27i
5113.32i
5041.97i
− 11003.9i 418448.i −8.75320e7 0 4.60457e9 7.06271e10i 5.93966e11i 6.72190e11 0
24.2 − 9306.64i 770525.i −5.30591e7 0 7.17100e9 − 2.22916e10i 1.81523e11i 2.53579e11 0
24.3 − 5620.55i 161492.i 1.96387e6 0 9.07675e8 − 4.86926e10i − 1.99632e11i 8.21209e11 0
24.4 − 4172.52i − 1.12842e6i 1.61445e7 0 −4.70836e9 2.36507e10i − 2.07370e11i −4.26045e11 0
24.5 − 1456.68i − 1.56404e6i 3.14325e7 0 −2.27831e9 3.49059e10i − 9.46654e10i −1.59893e12 0
24.6 1456.68i 1.56404e6i 3.14325e7 0 −2.27831e9 − 3.49059e10i 9.46654e10i −1.59893e12 0
24.7 4172.52i 1.12842e6i 1.61445e7 0 −4.70836e9 − 2.36507e10i 2.07370e11i −4.26045e11 0
24.8 5620.55i − 161492.i 1.96387e6 0 9.07675e8 4.86926e10i 1.99632e11i 8.21209e11 0
24.9 9306.64i − 770525.i −5.30591e7 0 7.17100e9 2.22916e10i − 1.81523e11i 2.53579e11 0
24.10 11003.9i − 418448.i −8.75320e7 0 4.60457e9 − 7.06271e10i − 5.93966e11i 6.72190e11 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.10
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 25.26.b.c 10
5.b even 2 1 inner 25.26.b.c 10
5.c odd 4 1 5.26.a.b ✓ 5
5.c odd 4 1 25.26.a.c 5
15.e even 4 1 45.26.a.f 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.26.a.b ✓ 5 5.c odd 4 1
25.26.a.c 5 5.c odd 4 1
25.26.b.c 10 1.a even 1 1 trivial
25.26.b.c 10 5.b even 2 1 inner
45.26.a.f 5 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 258822420 T_{2}^{8} + \cdots + 12\!\cdots\!24 \) acting on \(S_{26}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{10} + \cdots + 84\!\cdots\!76 \) Copy content Toggle raw display
$5$ \( T^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 40\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots + 87\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots + 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 50\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{5} + \cdots + 52\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 12\!\cdots\!68)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 94\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 29\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 20\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 24\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 84\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 53\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 32\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 39\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 45\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
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