Properties

Label 18.26.a.e
Level $18$
Weight $26$
Character orbit 18.a
Self dual yes
Analytic conductor $71.279$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [18,26,Mod(1,18)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(18, 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("18.1");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 18.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: \(71.2794203914\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{106705}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 26676 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{5}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 2)
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 \(\beta = 388800\sqrt{106705}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4096 q^{2} + 16777216 q^{4} + ( - 4 \beta - 370976550) q^{5} + (238 \beta - 188268472) q^{7} - 68719476736 q^{8} + (16384 \beta + 1519519948800) q^{10} + ( - 12397 \beta - 4161517305132) q^{11} + (334628 \beta - 53233526576146) q^{13}+ \cdots + (367066286784512 \beta + 17\!\cdots\!08) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8192 q^{2} + 33554432 q^{4} - 741953100 q^{5} - 376536944 q^{7} - 137438953472 q^{8} + 3039039897600 q^{10} - 8323034610264 q^{11} - 106467053152292 q^{13} + 1542295322624 q^{14} + 562949953421312 q^{16}+ \cdots + 35\!\cdots\!16 q^{98}+O(q^{100}) \) 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
163.829
−162.829
−4096.00 0 1.67772e7 −8.78994e8 0 3.00388e10 −6.87195e10 0 3.60036e12
1.2 −4096.00 0 1.67772e7 1.37041e8 0 −3.04153e10 −6.87195e10 0 −5.61320e11
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)

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 18.26.a.e 2
3.b odd 2 1 2.26.a.b 2
12.b even 2 1 16.26.a.c 2
15.d odd 2 1 50.26.a.c 2
15.e even 4 2 50.26.b.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.26.a.b 2 3.b odd 2 1
16.26.a.c 2 12.b even 2 1
18.26.a.e 2 1.a even 1 1 trivial
50.26.a.c 2 15.d odd 2 1
50.26.b.e 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 741953100T_{5} - 120458131753297500 \) acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(18))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4096)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 91\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots - 48\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 26\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots + 11\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 42\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 45\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 12\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 25\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 65\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 24\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots - 81\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 46\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 97\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 19\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 36\!\cdots\!24 \) Copy content Toggle raw display
show more
show less