Properties

Label 25.38.a.a
Level $25$
Weight $38$
Character orbit 25.a
Self dual yes
Analytic conductor $216.785$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,38,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, 38, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.1");
 
S:= CuspForms(chi, 38);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 38 \)
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: \(216.785095312\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 15934380 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: no (minimal twist has level 1)
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 \(\beta = 48\sqrt{63737521}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 97200) q^{2} + (72 \beta - 6995700) q^{3} + ( - 194400 \beta + 18860134912) q^{4} + (13994100 \beta - 11253271923648) q^{6} + ( - 9650004336 \beta + 17\!\cdots\!00) q^{7}+ \cdots + ( - 1007380800 \beta - 44\!\cdots\!07) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 97200) q^{2} + (72 \beta - 6995700) q^{3} + ( - 194400 \beta + 18860134912) q^{4} + (13994100 \beta - 11253271923648) q^{6} + ( - 9650004336 \beta + 17\!\cdots\!00) q^{7}+ \cdots + (14\!\cdots\!00 \beta + 60\!\cdots\!96) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 194400 q^{2} - 13991400 q^{3} + 37720269824 q^{4} - 22506543847296 q^{6} + 34\!\cdots\!00 q^{7}+ \cdots - 89\!\cdots\!14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 194400 q^{2} - 13991400 q^{3} + 37720269824 q^{4} - 22506543847296 q^{6} + 34\!\cdots\!00 q^{7}+ \cdots + 12\!\cdots\!92 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
3992.29
−3991.29
−286012. 2.05955e7 −5.56362e10 0 −5.89057e12 −1.97377e15 5.52218e16 −4.49860e17 0
1.2 480412. −3.45869e7 9.33565e10 0 −1.66160e13 5.42222e15 −2.11777e16 −4.49088e17 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.38.a.a 2
5.b even 2 1 1.38.a.a 2
5.c odd 4 2 25.38.b.a 4
15.d odd 2 1 9.38.a.a 2
20.d odd 2 1 16.38.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.38.a.a 2 5.b even 2 1
9.38.a.a 2 15.d odd 2 1
16.38.a.b 2 20.d odd 2 1
25.38.a.a 2 1.a even 1 1 trivial
25.38.b.a 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 194400T_{2} - 137403408384 \) acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(25))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 194400 T - 137403408384 \) Copy content Toggle raw display
$3$ \( T^{2} + \cdots - 712337053132656 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 10\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 66\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 21\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 37\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 61\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 80\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 34\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 10\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 23\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 44\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 12\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 44\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 96\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 46\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 35\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 75\!\cdots\!04 \) Copy content Toggle raw display
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