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Properties

Label 25.26.a.f
Level $25$
Weight $26$
Character orbit 25.a
Self dual yes
Analytic conductor $98.999$
Analytic rank $0$
Dimension $12$
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(1,25)]
 
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([0]))
 
N = Newforms(chi, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.1");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

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: yes
Analytic conductor: \(98.9991949881\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 71168091 x^{10} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{46}\cdot 3^{20}\cdot 5^{36} \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{7} + 19 \beta_1) q^{3} + (\beta_{2} + 13890962) q^{4} + ( - \beta_{3} + 76 \beta_{2} + 882628682) q^{6} + ( - \beta_{8} - 218 \beta_{7} + 1564250 \beta_1) q^{7} + (\beta_{9} + 78575 \beta_{7} + 13730072 \beta_1) q^{8} + ( - \beta_{4} - 12 \beta_{3} + \cdots + 329038836753) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{7} + 19 \beta_1) q^{3} + (\beta_{2} + 13890962) q^{4} + ( - \beta_{3} + 76 \beta_{2} + 882628682) q^{6} + ( - \beta_{8} - 218 \beta_{7} + 1564250 \beta_1) q^{7} + (\beta_{9} + 78575 \beta_{7} + 13730072 \beta_1) q^{8} + ( - \beta_{4} - 12 \beta_{3} + \cdots + 329038836753) q^{9}+ \cdots + ( - 385515624231 \beta_{6} + \cdots + 83\!\cdots\!56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 166691544 q^{4} + 10591544184 q^{6} + 3948466041036 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 166691544 q^{4} + 10591544184 q^{6} + 3948466041036 q^{9} - 1090673824176 q^{11} + 890646861445848 q^{14} + 22\!\cdots\!32 q^{16}+ \cdots + 10\!\cdots\!72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 71168091 x^{10} + \cdots + 10\!\cdots\!36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} - 47445394 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 7813429837 \nu^{10} + \cdots + 10\!\cdots\!52 ) / 14\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 10342832454163 \nu^{10} + \cdots + 30\!\cdots\!12 ) / 15\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6263041283885 \nu^{10} + \cdots - 50\!\cdots\!68 ) / 45\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 765300186695791 \nu^{10} + \cdots - 38\!\cdots\!56 ) / 45\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\!\cdots\!09 \nu^{11} + \cdots - 26\!\cdots\!04 \nu ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20\!\cdots\!03 \nu^{11} + \cdots + 43\!\cdots\!28 \nu ) / 36\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 38\!\cdots\!05 \nu^{11} + \cdots - 10\!\cdots\!68 \nu ) / 10\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 68\!\cdots\!33 \nu^{11} + \cdots + 49\!\cdots\!28 \nu ) / 40\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16\!\cdots\!93 \nu^{11} + \cdots - 10\!\cdots\!48 \nu ) / 23\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 47445394 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} + 78575\beta_{7} + 80838936\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -23\beta_{6} + 49\beta_{5} - 49\beta_{4} - 64206\beta_{3} + 57590786\beta_{2} + 1916977661425068 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 14826 \beta_{11} + 296842 \beta_{10} + 32549349 \beta_{9} - 17687988 \beta_{8} + \cdots + 19\!\cdots\!32 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 822527133 \beta_{6} + 967635291 \beta_{5} - 4636522299 \beta_{4} - 2850625083546 \beta_{3} + \cdots + 45\!\cdots\!48 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1112844851118 \beta_{11} + 12152288046990 \beta_{10} + 924312190491099 \beta_{9} + \cdots + 48\!\cdots\!88 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 23\!\cdots\!11 \beta_{6} - 628529706303219 \beta_{5} + \cdots + 11\!\cdots\!28 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 50\!\cdots\!10 \beta_{11} + \cdots + 12\!\cdots\!76 \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 63\!\cdots\!05 \beta_{6} + \cdots + 30\!\cdots\!08 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 18\!\cdots\!30 \beta_{11} + \cdots + 33\!\cdots\!64 \beta_1 ) / 8 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−5297.67
−4680.77
−3156.95
−2840.59
−1460.97
−1011.64
1011.64
1460.97
2840.59
3156.95
4680.77
5297.67
−10595.3 −1.42235e6 7.87068e7 0 1.50702e10 −3.45094e10 −4.78405e11 1.17578e12 0
1.2 −9361.55 531494. 5.40841e7 0 −4.97561e9 1.37587e10 −1.92189e11 −5.64803e11 0
1.3 −6313.90 71116.0 6.31093e6 0 −4.49019e8 3.34630e10 1.72013e11 −8.42231e11 0
1.4 −5681.18 1.63997e6 −1.27857e6 0 −9.31697e9 −6.01315e10 1.97893e11 1.84221e12 0
1.5 −2921.94 −1.33312e6 −2.50167e7 0 3.89530e9 −4.60230e10 1.71141e11 9.29927e11 0
1.6 −2023.27 −529751. −2.94608e7 0 1.07183e9 2.78390e10 1.27497e11 −5.66653e11 0
1.7 2023.27 529751. −2.94608e7 0 1.07183e9 −2.78390e10 −1.27497e11 −5.66653e11 0
1.8 2921.94 1.33312e6 −2.50167e7 0 3.89530e9 4.60230e10 −1.71141e11 9.29927e11 0
1.9 5681.18 −1.63997e6 −1.27857e6 0 −9.31697e9 6.01315e10 −1.97893e11 1.84221e12 0
1.10 6313.90 −71116.0 6.31093e6 0 −4.49019e8 −3.34630e10 −1.72013e11 −8.42231e11 0
1.11 9361.55 −531494. 5.40841e7 0 −4.97561e9 −1.37587e10 1.92189e11 −5.64803e11 0
1.12 10595.3 1.42235e6 7.87068e7 0 1.50702e10 3.45094e10 4.78405e11 1.17578e12 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)

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.a.f 12
5.b even 2 1 inner 25.26.a.f 12
5.c odd 4 2 5.26.b.a ✓ 12
15.e even 4 2 45.26.b.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.26.b.a ✓ 12 5.c odd 4 2
25.26.a.f 12 1.a even 1 1 trivial
25.26.a.f 12 5.b even 2 1 inner
45.26.b.b 12 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 284672364 T_{2}^{10} + \cdots + 44\!\cdots\!56 \) acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(25))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 44\!\cdots\!56 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 68\!\cdots\!36)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 63\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 15\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 29\!\cdots\!36)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 13\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 43\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 66\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 75\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 14\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
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