Properties

Label 10.26.a.b
Level $10$
Weight $26$
Character orbit 10.a
Self dual yes
Analytic conductor $39.600$
Analytic rank $0$
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] = [10,26,Mod(1,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, 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("10.1");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 10.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: \(39.5996779952\)
Analytic rank: \(0\)
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: \( x^{2} - x - 6900880 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3}\cdot 5 \)
Twist minimal: yes
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 \(\beta = 270\sqrt{27603521}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4096 q^{2} + ( - \beta - 219294) q^{3} + 16777216 q^{4} + 244140625 q^{5} + (4096 \beta + 898228224) q^{6} + ( - 2667 \beta - 17004298318) q^{7} - 68719476736 q^{8} + (438588 \beta + 1213097929893) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4096 q^{2} + ( - \beta - 219294) q^{3} + 16777216 q^{4} + 244140625 q^{5} + (4096 \beta + 898228224) q^{6} + ( - 2667 \beta - 17004298318) q^{7} - 68719476736 q^{8} + (438588 \beta + 1213097929893) q^{9} - 1000000000000 q^{10} + ( - 3895722 \beta + 9672341395992) q^{11} + ( - 16777216 \beta - 3679142805504) q^{12} + ( - 47271012 \beta - 63898024016794) q^{13} + (10924032 \beta + 69649605910528) q^{14} + ( - 244140625 \beta - 53538574218750) q^{15} + 281474976710656 q^{16} + ( - 2203187748 \beta - 14\!\cdots\!38) q^{17}+ \cdots + ( - 48\!\cdots\!50 \beta + 82\!\cdots\!56) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8192 q^{2} - 438588 q^{3} + 33554432 q^{4} + 488281250 q^{5} + 1796456448 q^{6} - 34008596636 q^{7} - 137438953472 q^{8} + 2426195859786 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8192 q^{2} - 438588 q^{3} + 33554432 q^{4} + 488281250 q^{5} + 1796456448 q^{6} - 34008596636 q^{7} - 137438953472 q^{8} + 2426195859786 q^{9} - 2000000000000 q^{10} + 19344682791984 q^{11} - 7358285611008 q^{12} - 127796048033588 q^{13} + 139299211821056 q^{14} - 107077148437500 q^{15} + 562949953421312 q^{16} - 29\!\cdots\!76 q^{17}+ \cdots + 16\!\cdots\!12 q^{99}+O(q^{100}) \) 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
2627.45
−2626.45
−4096.00 −1.63785e6 1.67772e7 2.44141e8 6.70863e9 −2.07876e10 −6.87195e10 1.83526e12 −1.00000e12
1.2 −4096.00 1.19926e6 1.67772e7 2.44141e8 −4.91217e9 −1.32210e10 −6.87195e10 5.90937e11 −1.00000e12
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( -1 \)

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 10.26.a.b 2
5.b even 2 1 50.26.a.f 2
5.c odd 4 2 50.26.b.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.26.a.b 2 1.a even 1 1 trivial
50.26.a.f 2 5.b even 2 1
50.26.b.f 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 438588T_{3} - 1964206822464 \) acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(10))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4096)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + \cdots - 1964206822464 \) Copy content Toggle raw display
$5$ \( (T - 244140625)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots + 63\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 41\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 75\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 37\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 34\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 45\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 54\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 86\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 55\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 41\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 17\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 22\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 10\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 13\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 70\!\cdots\!76 \) Copy content Toggle raw display
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