Properties

Label 2704.2.a.o
Level $2704$
Weight $2$
Character orbit 2704.a
Self dual yes
Analytic conductor $21.592$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2704,2,Mod(1,2704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2704, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2704.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2704 = 2^{4} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2704.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: \(21.5915487066\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{3} - \beta q^{5} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{3} - \beta q^{5} + q^{9} + 2 \beta q^{15} + 3 q^{17} + 2 \beta q^{19} - 6 q^{23} - 2 q^{25} + 4 q^{27} + 3 q^{29} - 2 \beta q^{31} + 5 \beta q^{37} + 3 \beta q^{41} + 8 q^{43} - \beta q^{45} - 2 \beta q^{47} - 7 q^{49} - 6 q^{51} - 3 q^{53} - 4 \beta q^{57} + 4 \beta q^{59} + q^{61} + 2 \beta q^{67} + 12 q^{69} - 2 \beta q^{71} - \beta q^{73} + 4 q^{75} - 4 q^{79} - 11 q^{81} - 8 \beta q^{83} - 3 \beta q^{85} - 6 q^{87} - 4 \beta q^{89} + 4 \beta q^{93} - 6 q^{95} + 4 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} + 2 q^{9} + 6 q^{17} - 12 q^{23} - 4 q^{25} + 8 q^{27} + 6 q^{29} + 16 q^{43} - 14 q^{49} - 12 q^{51} - 6 q^{53} + 2 q^{61} + 24 q^{69} + 8 q^{75} - 8 q^{79} - 22 q^{81} - 12 q^{87} - 12 q^{95}+O(q^{100}) \) 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
1.73205
−1.73205
0 −2.00000 0 −1.73205 0 0 0 1.00000 0
1.2 0 −2.00000 0 1.73205 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(13\) \( -1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2704.2.a.o 2
4.b odd 2 1 169.2.a.a 2
12.b even 2 1 1521.2.a.k 2
13.b even 2 1 inner 2704.2.a.o 2
13.d odd 4 2 2704.2.f.b 2
13.f odd 12 2 208.2.w.b 2
20.d odd 2 1 4225.2.a.v 2
28.d even 2 1 8281.2.a.q 2
39.k even 12 2 1872.2.by.d 2
52.b odd 2 1 169.2.a.a 2
52.f even 4 2 169.2.b.a 2
52.i odd 6 2 169.2.c.a 4
52.j odd 6 2 169.2.c.a 4
52.l even 12 2 13.2.e.a 2
52.l even 12 2 169.2.e.a 2
104.u even 12 2 832.2.w.d 2
104.x odd 12 2 832.2.w.a 2
156.h even 2 1 1521.2.a.k 2
156.l odd 4 2 1521.2.b.a 2
156.v odd 12 2 117.2.q.c 2
260.g odd 2 1 4225.2.a.v 2
260.bc even 12 2 325.2.n.a 2
260.be odd 12 2 325.2.m.a 4
260.bl odd 12 2 325.2.m.a 4
364.h even 2 1 8281.2.a.q 2
364.bt even 12 2 637.2.u.c 2
364.bv odd 12 2 637.2.q.a 2
364.bz odd 12 2 637.2.k.c 2
364.ca even 12 2 637.2.k.a 2
364.cg odd 12 2 637.2.u.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 52.l even 12 2
117.2.q.c 2 156.v odd 12 2
169.2.a.a 2 4.b odd 2 1
169.2.a.a 2 52.b odd 2 1
169.2.b.a 2 52.f even 4 2
169.2.c.a 4 52.i odd 6 2
169.2.c.a 4 52.j odd 6 2
169.2.e.a 2 52.l even 12 2
208.2.w.b 2 13.f odd 12 2
325.2.m.a 4 260.be odd 12 2
325.2.m.a 4 260.bl odd 12 2
325.2.n.a 2 260.bc even 12 2
637.2.k.a 2 364.ca even 12 2
637.2.k.c 2 364.bz odd 12 2
637.2.q.a 2 364.bv odd 12 2
637.2.u.b 2 364.cg odd 12 2
637.2.u.c 2 364.bt even 12 2
832.2.w.a 2 104.x odd 12 2
832.2.w.d 2 104.u even 12 2
1521.2.a.k 2 12.b even 2 1
1521.2.a.k 2 156.h even 2 1
1521.2.b.a 2 156.l odd 4 2
1872.2.by.d 2 39.k even 12 2
2704.2.a.o 2 1.a even 1 1 trivial
2704.2.a.o 2 13.b even 2 1 inner
2704.2.f.b 2 13.d odd 4 2
4225.2.a.v 2 20.d odd 2 1
4225.2.a.v 2 260.g odd 2 1
8281.2.a.q 2 28.d even 2 1
8281.2.a.q 2 364.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2704))\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{5}^{2} - 3 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 3 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 12 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( (T - 3)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 12 \) Copy content Toggle raw display
$37$ \( T^{2} - 75 \) Copy content Toggle raw display
$41$ \( T^{2} - 27 \) Copy content Toggle raw display
$43$ \( (T - 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 12 \) Copy content Toggle raw display
$53$ \( (T + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 48 \) Copy content Toggle raw display
$61$ \( (T - 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 12 \) Copy content Toggle raw display
$71$ \( T^{2} - 12 \) Copy content Toggle raw display
$73$ \( T^{2} - 3 \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 192 \) Copy content Toggle raw display
$89$ \( T^{2} - 48 \) Copy content Toggle raw display
$97$ \( T^{2} - 48 \) Copy content Toggle raw display
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