Properties

Label 2736.3.o.h
Level $2736$
Weight $3$
Character orbit 2736.o
Analytic conductor $74.551$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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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: no
Analytic conductor: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 38)
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 \(\beta = 6\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} - 5 q^{7} + 5 q^{11} - 2 \beta q^{13} + 25 q^{17} - 19 q^{19} - 10 q^{23} - 24 q^{25} - 5 \beta q^{29} + 5 \beta q^{31} - 5 q^{35} - 3 \beta q^{37} + 5 \beta q^{41} - 5 q^{43} + 5 q^{47} - 24 q^{49} + 3 \beta q^{53} + 5 q^{55} + 10 \beta q^{59} + 95 q^{61} - 2 \beta q^{65} - 13 \beta q^{67} - 25 q^{73} - 25 q^{77} - 5 \beta q^{79} - 130 q^{83} + 25 q^{85} + 15 \beta q^{89} + 10 \beta q^{91} - 19 q^{95} + 2 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 10 q^{7} + 10 q^{11} + 50 q^{17} - 38 q^{19} - 20 q^{23} - 48 q^{25} - 10 q^{35} - 10 q^{43} + 10 q^{47} - 48 q^{49} + 10 q^{55} + 190 q^{61} - 50 q^{73} - 50 q^{77} - 260 q^{83} + 50 q^{85} - 38 q^{95}+O(q^{100}) \) 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
1.41421i
1.41421i
0 0 0 1.00000 0 −5.00000 0 0 0
721.2 0 0 0 1.00000 0 −5.00000 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2736.3.o.h 2
3.b odd 2 1 304.3.e.c 2
4.b odd 2 1 342.3.d.a 2
12.b even 2 1 38.3.b.a 2
19.b odd 2 1 inner 2736.3.o.h 2
24.f even 2 1 1216.3.e.j 2
24.h odd 2 1 1216.3.e.i 2
57.d even 2 1 304.3.e.c 2
60.h even 2 1 950.3.c.a 2
60.l odd 4 2 950.3.d.a 4
76.d even 2 1 342.3.d.a 2
228.b odd 2 1 38.3.b.a 2
456.l odd 2 1 1216.3.e.j 2
456.p even 2 1 1216.3.e.i 2
1140.p odd 2 1 950.3.c.a 2
1140.w even 4 2 950.3.d.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.b.a 2 12.b even 2 1
38.3.b.a 2 228.b odd 2 1
304.3.e.c 2 3.b odd 2 1
304.3.e.c 2 57.d even 2 1
342.3.d.a 2 4.b odd 2 1
342.3.d.a 2 76.d even 2 1
950.3.c.a 2 60.h even 2 1
950.3.c.a 2 1140.p odd 2 1
950.3.d.a 4 60.l odd 4 2
950.3.d.a 4 1140.w even 4 2
1216.3.e.i 2 24.h odd 2 1
1216.3.e.i 2 456.p even 2 1
1216.3.e.j 2 24.f even 2 1
1216.3.e.j 2 456.l odd 2 1
2736.3.o.h 2 1.a even 1 1 trivial
2736.3.o.h 2 19.b odd 2 1 inner

Hecke kernels

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

\( T_{5} - 1 \) Copy content Toggle raw display
\( T_{7} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 5)^{2} \) Copy content Toggle raw display
$11$ \( (T - 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 288 \) Copy content Toggle raw display
$17$ \( (T - 25)^{2} \) Copy content Toggle raw display
$19$ \( (T + 19)^{2} \) Copy content Toggle raw display
$23$ \( (T + 10)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 1800 \) Copy content Toggle raw display
$31$ \( T^{2} + 1800 \) Copy content Toggle raw display
$37$ \( T^{2} + 648 \) Copy content Toggle raw display
$41$ \( T^{2} + 1800 \) Copy content Toggle raw display
$43$ \( (T + 5)^{2} \) Copy content Toggle raw display
$47$ \( (T - 5)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 648 \) Copy content Toggle raw display
$59$ \( T^{2} + 7200 \) Copy content Toggle raw display
$61$ \( (T - 95)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 12168 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 25)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 1800 \) Copy content Toggle raw display
$83$ \( (T + 130)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 16200 \) Copy content Toggle raw display
$97$ \( T^{2} + 288 \) Copy content Toggle raw display
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