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Properties

Label 25.26.b.b
Level $25$
Weight $26$
Character orbit 25.b
Analytic conductor $98.999$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3539713x^{6} + 4388685173736x^{4} + 2234496951580358425x^{2} + 394435020401386431000625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{4}\cdot 5^{16} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + (\beta_{6} + 158 \beta_{5} + 17 \beta_{4}) q^{3} + ( - \beta_{2} + \beta_1 - 23103472) q^{4} + ( - 64 \beta_{3} - 324 \beta_{2} + \cdots - 991293708) q^{6}+ \cdots + (11058 \beta_{3} + 8208 \beta_{2} + \cdots - 521891451873) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + (\beta_{6} + 158 \beta_{5} + 17 \beta_{4}) q^{3} + ( - \beta_{2} + \beta_1 - 23103472) q^{4} + ( - 64 \beta_{3} - 324 \beta_{2} + \cdots - 991293708) q^{6}+ \cdots + ( - 11\!\cdots\!03 \beta_{3} + \cdots + 48\!\cdots\!04) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 184827776 q^{4} - 7930349664 q^{6} - 4175131614984 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 184827776 q^{4} - 7930349664 q^{6} - 4175131614984 q^{9} - 47282907581184 q^{11} - 219961002072672 q^{14} - 56\!\cdots\!52 q^{16}+ \cdots + 38\!\cdots\!32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3539713x^{6} + 4388685173736x^{4} + 2234496951580358425x^{2} + 394435020401386431000625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 200\nu^{6} - 20659352400\nu^{4} - 37064927716072800\nu^{2} - 13197782186900826497500 ) / 224511505094961 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3784 \nu^{6} - 390874947408 \nu^{4} + \cdots - 56\!\cdots\!00 ) / 56\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10450736 \nu^{6} + 29171601275568 \nu^{4} + \cdots + 63\!\cdots\!00 ) / 51\!\cdots\!03 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 60038068 \nu^{7} + 212876410148784 \nu^{5} + \cdots + 27\!\cdots\!00 \nu ) / 26\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15009517 \nu^{7} + 53219102537196 \nu^{5} + \cdots + 16\!\cdots\!25 \nu ) / 80\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14490660394919 \nu^{7} + \cdots + 46\!\cdots\!75 \nu ) / 77\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 53\!\cdots\!13 \nu^{7} + \cdots - 34\!\cdots\!25 \nu ) / 11\!\cdots\!25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -76\beta_{5} + 625\beta_{4} ) / 5000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -625\beta_{2} + 473\beta _1 - 35397130000 ) / 40000 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5225\beta_{7} - 10292075\beta_{6} - 1888830818\beta_{5} - 5300517675\beta_{4} ) / 40000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 93150\beta_{3} + 552949375\beta_{2} - 630099491\beta _1 + 18760988874485000 ) / 20000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 21121671125 \beta_{7} + 36922188410375 \beta_{6} + \cdots + 12\!\cdots\!75 \beta_{4} ) / 80000 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2405523345075\beta_{3} - 199017392235000\beta_{2} + 298297593901011\beta _1 - 5563135168418102585000 ) / 5000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 44\!\cdots\!75 \beta_{7} + \cdots - 19\!\cdots\!75 \beta_{4} ) / 10000 \) 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
− 1189.32i
− 1046.70i
− 785.715i
− 642.094i
642.094i
785.715i
1046.70i
1189.32i
− 9666.59i − 1.07197e6i −5.98886e7 0 −1.03623e10 − 1.87952e10i 2.54562e11i −3.01839e11 0
24.2 − 8221.63i − 987550.i −3.40407e7 0 −8.11927e9 − 1.93681e10i 3.99805e9i −1.27967e11 0
24.3 − 6133.72i 1.60154e6i −4.06809e6 0 9.82339e9 − 2.17553e9i − 1.80861e11i −1.71764e12 0
24.4 − 5288.75i 887362.i 5.58351e6 0 4.69304e9 4.61897e10i − 2.06991e11i 5.98771e10 0
24.5 5288.75i − 887362.i 5.58351e6 0 4.69304e9 − 4.61897e10i 2.06991e11i 5.98771e10 0
24.6 6133.72i − 1.60154e6i −4.06809e6 0 9.82339e9 2.17553e9i 1.80861e11i −1.71764e12 0
24.7 8221.63i 987550.i −3.40407e7 0 −8.11927e9 1.93681e10i − 3.99805e9i −1.27967e11 0
24.8 9666.59i 1.07197e6i −5.98886e7 0 −1.03623e10 1.87952e10i − 2.54562e11i −3.01839e11 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 25.26.b.b 8
5.b even 2 1 inner 25.26.b.b 8
5.c odd 4 1 5.26.a.a ✓ 4
5.c odd 4 1 25.26.a.b 4
15.e even 4 1 45.26.a.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.26.a.a ✓ 4 5.c odd 4 1
25.26.a.b 4 5.c odd 4 1
25.26.b.b 8 1.a even 1 1 trivial
25.26.b.b 8 5.b even 2 1 inner
45.26.a.c 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 226631616 T_{2}^{6} + \cdots + 66\!\cdots\!56 \) acting on \(S_{26}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 66\!\cdots\!56 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 22\!\cdots\!36 \) 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
$11$ \( (T^{4} + \cdots - 12\!\cdots\!84)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 76\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 56\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 31\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 71\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 88\!\cdots\!84)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 14\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 80\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 89\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 71\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
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