Properties

Label 1296.3.e.f
Level $1296$
Weight $3$
Character orbit 1296.e
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(161,1296)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1296.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1296.e (of order \(2\), degree \(1\), not 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: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 324)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + ( - \beta_{2} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + ( - \beta_{2} + 1) q^{7} + \beta_{3} q^{11} + (2 \beta_{2} - 1) q^{13} + (2 \beta_{3} - 5 \beta_1) q^{17} + ( - 5 \beta_{2} + 7) q^{19} + ( - \beta_{3} - 6 \beta_1) q^{23} + (3 \beta_{2} + 7) q^{25} + ( - 3 \beta_{3} - 5 \beta_1) q^{29} + ( - 6 \beta_{2} - 8) q^{31} + ( - \beta_{3} + 6 \beta_1) q^{35} + (\beta_{2} + 14) q^{37} + ( - \beta_{3} + 12 \beta_1) q^{41} + ( - 9 \beta_{2} + 13) q^{43} + (2 \beta_{3} - 10 \beta_1) q^{47} + ( - 2 \beta_{2} - 21) q^{49} + (3 \beta_{3} + 4 \beta_1) q^{53} + ( - 3 \beta_{2} - 9) q^{55} + ( - 2 \beta_{3} + 14 \beta_1) q^{59} + ( - 9 \beta_{2} - 16) q^{61} + (2 \beta_{3} - 11 \beta_1) q^{65} + ( - 3 \beta_{2} - 5) q^{67} + (7 \beta_{3} + 2 \beta_1) q^{71} + ( - 9 \beta_{2} - 64) q^{73} + ( - 4 \beta_{3} - 2 \beta_1) q^{77} + (7 \beta_{2} - 77) q^{79} + ( - 10 \beta_{3} - 4 \beta_1) q^{83} + ( - 21 \beta_{2} + 72) q^{85} + ( - \beta_{3} + 19 \beta_1) q^{89} + (3 \beta_{2} - 55) q^{91} + ( - 5 \beta_{3} + 32 \beta_1) q^{95} + ( - 22 \beta_{2} - 40) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{7} - 4 q^{13} + 28 q^{19} + 28 q^{25} - 32 q^{31} + 56 q^{37} + 52 q^{43} - 84 q^{49} - 36 q^{55} - 64 q^{61} - 20 q^{67} - 256 q^{73} - 308 q^{79} + 288 q^{85} - 220 q^{91} - 160 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 9\nu^{3} + 33\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} - 11\beta_1 ) / 9 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
1.93185i
0.517638i
0.517638i
1.93185i
0 0 0 5.79555i 0 6.19615 0 0 0
161.2 0 0 0 1.55291i 0 −4.19615 0 0 0
161.3 0 0 0 1.55291i 0 −4.19615 0 0 0
161.4 0 0 0 5.79555i 0 6.19615 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
3.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.e.f 4
3.b odd 2 1 inner 1296.3.e.f 4
4.b odd 2 1 324.3.c.a 4
9.c even 3 2 1296.3.q.n 8
9.d odd 6 2 1296.3.q.n 8
12.b even 2 1 324.3.c.a 4
36.f odd 6 2 324.3.g.d 8
36.h even 6 2 324.3.g.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.3.c.a 4 4.b odd 2 1
324.3.c.a 4 12.b even 2 1
324.3.g.d 8 36.f odd 6 2
324.3.g.d 8 36.h even 6 2
1296.3.e.f 4 1.a even 1 1 trivial
1296.3.e.f 4 3.b odd 2 1 inner
1296.3.q.n 8 9.c even 3 2
1296.3.q.n 8 9.d odd 6 2

Hecke kernels

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

\( T_{5}^{4} + 36T_{5}^{2} + 81 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 26 \) 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
$5$ \( T^{4} + 36T^{2} + 81 \) Copy content Toggle raw display
$7$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 252T^{2} + 324 \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T - 107)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 1548 T^{2} + 558009 \) Copy content Toggle raw display
$19$ \( (T^{2} - 14 T - 626)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1764 T^{2} + 715716 \) Copy content Toggle raw display
$29$ \( T^{4} + 3708 T^{2} + \cdots + 1996569 \) Copy content Toggle raw display
$31$ \( (T^{2} + 16 T - 908)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 28 T + 169)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 5004 T^{2} + 39204 \) Copy content Toggle raw display
$43$ \( (T^{2} - 26 T - 2018)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 3888 T^{2} + 944784 \) Copy content Toggle raw display
$53$ \( T^{4} + 3276 T^{2} + \cdots + 1127844 \) Copy content Toggle raw display
$59$ \( T^{4} + 7056 T^{2} + 685584 \) Copy content Toggle raw display
$61$ \( (T^{2} + 32 T - 1931)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 10 T - 218)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 12996 T^{2} + \cdots + 171396 \) Copy content Toggle raw display
$73$ \( (T^{2} + 128 T + 1909)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 154 T + 4606)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 27216 T^{2} + \cdots + 3779136 \) Copy content Toggle raw display
$89$ \( T^{4} + 12564 T^{2} + \cdots + 2313441 \) Copy content Toggle raw display
$97$ \( (T^{2} + 80 T - 11468)^{2} \) Copy content Toggle raw display
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