Properties

Label 2736.3.o.k
Level $2736$
Weight $3$
Character orbit 2736.o
Self dual yes
Analytic conductor $74.551$
Analytic rank $0$
Dimension $4$
CM discriminant -19
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,3,Mod(721,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2736.721");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.o (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: yes
Analytic conductor: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 11x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 684)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + (\beta_{3} + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + (\beta_{3} + 3) q^{7} + \beta_{2} q^{11} + (\beta_{2} + 2 \beta_1) q^{17} - 19 q^{19} + ( - 3 \beta_{2} - \beta_1) q^{23} + (3 \beta_{3} + 42) q^{25} + (7 \beta_{2} + 8 \beta_1) q^{35} + ( - \beta_{3} - 43) q^{43} + ( - 4 \beta_{2} + 7 \beta_1) q^{47} + (5 \beta_{3} + 88) q^{49} + (7 \beta_{3} + 7) q^{55} + ( - 5 \beta_{3} + 49) q^{61} + ( - 11 \beta_{3} + 7) q^{73} + ( - 3 \beta_{2} + 14 \beta_1) q^{77} + ( - 12 \beta_{2} - 4 \beta_1) q^{83} + (13 \beta_{3} + 141) q^{85} - 19 \beta_1 q^{95}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 10 q^{7} - 76 q^{19} + 162 q^{25} - 170 q^{43} + 342 q^{49} + 14 q^{55} + 206 q^{61} + 50 q^{73} + 538 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 11x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 3\nu^{2} - 17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 17 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 2\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1
−3.04547
1.31342
−1.31342
3.04547
0 0 0 −9.97368 0 13.8248 0 0 0
721.2 0 0 0 −5.61478 0 −8.82475 0 0 0
721.3 0 0 0 5.61478 0 −8.82475 0 0 0
721.4 0 0 0 9.97368 0 13.8248 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
3.b odd 2 1 inner
57.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.o.k 4
3.b odd 2 1 inner 2736.3.o.k 4
4.b odd 2 1 684.3.h.d 4
12.b even 2 1 684.3.h.d 4
19.b odd 2 1 CM 2736.3.o.k 4
57.d even 2 1 inner 2736.3.o.k 4
76.d even 2 1 684.3.h.d 4
228.b odd 2 1 684.3.h.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.h.d 4 4.b odd 2 1
684.3.h.d 4 12.b even 2 1
684.3.h.d 4 76.d even 2 1
684.3.h.d 4 228.b odd 2 1
2736.3.o.k 4 1.a even 1 1 trivial
2736.3.o.k 4 3.b odd 2 1 inner
2736.3.o.k 4 19.b odd 2 1 CM
2736.3.o.k 4 57.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2736, [\chi])\):

\( T_{5}^{4} - 131T_{5}^{2} + 3136 \) Copy content Toggle raw display
\( T_{7}^{2} - 5T_{7} - 122 \) 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
$5$ \( T^{4} - 131T^{2} + 3136 \) Copy content Toggle raw display
$7$ \( (T^{2} - 5 T - 122)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 251 T^{2} + 12544 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 803T^{2} + 4096 \) Copy content Toggle raw display
$19$ \( (T + 19)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 1216)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 85 T + 1678)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 10043 T^{2} + \cdots + 11669056 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 103 T - 554)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 25 T - 15362)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 19456)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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