Properties

Label 3.26.a.b
Level $3$
Weight $26$
Character orbit 3.a
Self dual yes
Analytic conductor $11.880$
Analytic rank $0$
Dimension $3$
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] = [3,26,Mod(1,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, 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("3.1");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 3.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: \(11.8799033986\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 783420x + 148321440 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{6}\cdot 3^{5}\cdot 5^{2} \)
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 a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1226) q^{2} + 531441 q^{3} + (\beta_{2} - 1499 \beta_1 + 30249196) q^{4} + ( - 12 \beta_{2} + 43676 \beta_1 - 54384250) q^{5} + (531441 \beta_1 - 651546666) q^{6} + (676 \beta_{2} + 3973036 \beta_1 - 3207524248) q^{7} + ( - 3678 \beta_{2} + 27346058 \beta_1 - 89337610216) q^{8} + 282429536481 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1226) q^{2} + 531441 q^{3} + (\beta_{2} - 1499 \beta_1 + 30249196) q^{4} + ( - 12 \beta_{2} + 43676 \beta_1 - 54384250) q^{5} + (531441 \beta_1 - 651546666) q^{6} + (676 \beta_{2} + 3973036 \beta_1 - 3207524248) q^{7} + ( - 3678 \beta_{2} + 27346058 \beta_1 - 89337610216) q^{8} + 282429536481 q^{9} + (69824 \beta_{2} - 429212602 \beta_1 + 2787729624900) q^{10} + ( - 290328 \beta_{2} - 398988488 \beta_1 - 1982332710260) q^{11} + (531441 \beta_{2} - 796630059 \beta_1 + 16075662971436) q^{12} + ( - 730664 \beta_{2} + 6520110088 \beta_1 + 82712591469230) q^{13} + (2500032 \beta_{2} + 16151474216 \beta_1 + 251453880421616) q^{14} + ( - 6377292 \beta_{2} + 23211217116 \beta_1 - 28902020204250) q^{15} + (1805988 \beta_{2} - 157735312908 \beta_1 + 798212617450672) q^{16} + (14873352 \beta_{2} - 120931939176 \beta_1 + 22\!\cdots\!58) q^{17}+ \cdots + ( - 81\!\cdots\!68 \beta_{2} + \cdots - 55\!\cdots\!60) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3678 q^{2} + 1594323 q^{3} + 90747588 q^{4} - 163152750 q^{5} - 1954639998 q^{6} - 9622572744 q^{7} - 268012830648 q^{8} + 847288609443 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3678 q^{2} + 1594323 q^{3} + 90747588 q^{4} - 163152750 q^{5} - 1954639998 q^{6} - 9622572744 q^{7} - 268012830648 q^{8} + 847288609443 q^{9} + 8363188874700 q^{10} - 5946998130780 q^{11} + 48226988914308 q^{12} + 248137774407690 q^{13} + 754361641264848 q^{14} - 86706060612750 q^{15} + 23\!\cdots\!16 q^{16}+ \cdots - 16\!\cdots\!80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 783420x + 148321440 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} + 455\nu - 522432 ) / 42 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2227\nu^{2} + 1708315\nu + 1162548864 ) / 42 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2227\beta _1 + 21600 ) / 64800 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -91\beta_{2} + 341663\beta _1 + 6768753120 ) / 12960 \) 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
199.394
−967.357
768.963
−10558.1 531441. 7.79199e7 −8.66156e8 −5.61103e9 −1.75156e10 −4.68416e11 2.82430e11 9.14500e12
1.2 −1864.10 531441. −3.00796e7 6.53169e8 −9.90659e8 −4.71716e10 1.18620e11 2.82430e11 −1.21757e12
1.3 8744.24 531441. 4.29073e7 4.98342e7 4.64705e9 5.50645e10 8.17835e10 2.82430e11 4.35762e11
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 3.26.a.b 3
3.b odd 2 1 9.26.a.c 3
4.b odd 2 1 48.26.a.i 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.26.a.b 3 1.a even 1 1 trivial
9.26.a.c 3 3.b odd 2 1
48.26.a.i 3 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} + 3678T_{2}^{2} - 88941600T_{2} - 172099067904 \) acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + 3678 T^{2} + \cdots - 172099067904 \) Copy content Toggle raw display
$3$ \( (T - 531441)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 163152750 T^{2} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{3} + 9622572744 T^{2} + \cdots - 45\!\cdots\!80 \) Copy content Toggle raw display
$11$ \( T^{3} + 5946998130780 T^{2} + \cdots - 18\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( T^{3} - 248137774407690 T^{2} + \cdots + 43\!\cdots\!48 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots - 75\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots + 17\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( T^{3} + 506350856671782 T^{2} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 78\!\cdots\!60 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 20\!\cdots\!80 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots - 12\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 14\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 28\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots + 16\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 19\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 19\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 90\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots + 45\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 44\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots - 69\!\cdots\!52 \) Copy content Toggle raw display
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