Properties

Label 1296.3.g.g
Level $1296$
Weight $3$
Character orbit 1296.g
Analytic conductor $35.313$
Analytic rank $0$
Dimension $4$
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] = [1296,3,Mod(1135,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1296.1135");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1296.g (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(35.3134422611\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - x^{2} - 2x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + \beta_{3} q^{7} + ( - \beta_{3} - \beta_{2}) q^{11} + (3 \beta_1 - 2) q^{13} + (4 \beta_1 + 9) q^{17} + (\beta_{3} - \beta_{2}) q^{19} - 3 \beta_{3} q^{23} - 4 q^{25} + (\beta_1 - 18) q^{29} + ( - 4 \beta_{3} - 4 \beta_{2}) q^{31} + ( - \beta_{3} + 5 \beta_{2}) q^{35} + (3 \beta_1 - 22) q^{37} + 6 \beta_1 q^{41} + (\beta_{3} - 5 \beta_{2}) q^{43} + (2 \beta_{3} - 10 \beta_{2}) q^{47} + ( - 6 \beta_1 - 17) q^{49} + ( - 12 \beta_1 + 18) q^{53} + ( - 3 \beta_{3} - 6 \beta_{2}) q^{55} + (10 \beta_{3} + 4 \beta_{2}) q^{59} + ( - 9 \beta_1 - 46) q^{61} + (2 \beta_1 - 63) q^{65} + (9 \beta_{3} - 5 \beta_{2}) q^{67} + ( - \beta_{3} + 14 \beta_{2}) q^{71} + ( - 18 \beta_1 - 23) q^{73} + ( - 6 \beta_1 + 54) q^{77} + ( - 11 \beta_{3} - 10 \beta_{2}) q^{79} + ( - 2 \beta_{3} - 20 \beta_{2}) q^{83} + ( - 9 \beta_1 - 84) q^{85} + ( - 4 \beta_1 + 117) q^{89} + (\beta_{3} - 15 \beta_{2}) q^{91} + ( - 5 \beta_{3} + 4 \beta_{2}) q^{95} + (30 \beta_1 - 20) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{13} + 36 q^{17} - 16 q^{25} - 72 q^{29} - 88 q^{37} - 68 q^{49} + 72 q^{53} - 184 q^{61} - 252 q^{65} - 92 q^{73} + 216 q^{77} - 336 q^{85} + 468 q^{89} - 80 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - x^{2} - 2x + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -\nu^{3} + \nu^{2} + 3\nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{3} + 4\nu^{2} - 4\nu - 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -7\nu^{3} - \nu^{2} + \nu + 18 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} - \beta_{2} + 3\beta _1 + 3 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + 5\beta_{2} + 6\beta _1 + 18 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{3} - \beta_{2} + 60 ) / 24 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1135.1
1.39564 + 0.228425i
1.39564 0.228425i
−0.895644 + 1.09445i
−0.895644 1.09445i
0 0 0 −4.58258 0 9.66930i 0 0 0
1135.2 0 0 0 −4.58258 0 9.66930i 0 0 0
1135.3 0 0 0 4.58258 0 6.20520i 0 0 0
1135.4 0 0 0 4.58258 0 6.20520i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.3.g.g yes 4
3.b odd 2 1 1296.3.g.f 4
4.b odd 2 1 inner 1296.3.g.g yes 4
9.c even 3 1 1296.3.o.u 4
9.c even 3 1 1296.3.o.bb 4
9.d odd 6 1 1296.3.o.w 4
9.d odd 6 1 1296.3.o.z 4
12.b even 2 1 1296.3.g.f 4
36.f odd 6 1 1296.3.o.u 4
36.f odd 6 1 1296.3.o.bb 4
36.h even 6 1 1296.3.o.w 4
36.h even 6 1 1296.3.o.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1296.3.g.f 4 3.b odd 2 1
1296.3.g.f 4 12.b even 2 1
1296.3.g.g yes 4 1.a even 1 1 trivial
1296.3.g.g yes 4 4.b odd 2 1 inner
1296.3.o.u 4 9.c even 3 1
1296.3.o.u 4 36.f odd 6 1
1296.3.o.w 4 9.d odd 6 1
1296.3.o.w 4 36.h even 6 1
1296.3.o.z 4 9.d odd 6 1
1296.3.o.z 4 36.h even 6 1
1296.3.o.bb 4 9.c even 3 1
1296.3.o.bb 4 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5}^{2} - 21 \) Copy content Toggle raw display
\( T_{17}^{2} - 18T_{17} - 255 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 21)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 132T^{2} + 3600 \) Copy content Toggle raw display
$11$ \( T^{4} + 180T^{2} + 1296 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T - 185)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 18 T - 255)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 276T^{2} + 144 \) Copy content Toggle raw display
$23$ \( T^{4} + 1188 T^{2} + 291600 \) Copy content Toggle raw display
$29$ \( (T^{2} + 36 T + 303)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2880 T^{2} + 331776 \) Copy content Toggle raw display
$37$ \( (T^{2} + 44 T + 295)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 756)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 2772 T^{2} + \cdots + 1587600 \) Copy content Toggle raw display
$47$ \( T^{4} + 11088 T^{2} + \cdots + 25401600 \) Copy content Toggle raw display
$53$ \( (T^{2} - 36 T - 2700)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 12816 T^{2} + \cdots + 38340864 \) Copy content Toggle raw display
$61$ \( (T^{2} + 92 T + 415)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 15252 T^{2} + \cdots + 6656400 \) Copy content Toggle raw display
$71$ \( T^{4} + 19620 T^{2} + \cdots + 93779856 \) Copy content Toggle raw display
$73$ \( (T^{2} + 46 T - 6275)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 20292 T^{2} + \cdots + 26010000 \) Copy content Toggle raw display
$83$ \( T^{4} + 37008 T^{2} + \cdots + 324000000 \) Copy content Toggle raw display
$89$ \( (T^{2} - 234 T + 13353)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 40 T - 18500)^{2} \) Copy content Toggle raw display
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