Properties

Label 2704.2.f.b
Level $2704$
Weight $2$
Character orbit 2704.f
Analytic conductor $21.592$
Analytic rank $0$
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(337,2704)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2704.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2704 = 2^{4} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2704.f (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: \(21.5915487066\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 13)
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 \(\beta = \sqrt{-3}\). 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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1185\) \(2367\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −2.00000 0 1.73205i 0 0 0 1.00000 0
337.2 0 −2.00000 0 1.73205i 0 0 0 1.00000 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
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.f.b 2
4.b odd 2 1 169.2.b.a 2
12.b even 2 1 1521.2.b.a 2
13.b even 2 1 inner 2704.2.f.b 2
13.c even 3 1 208.2.w.b 2
13.d odd 4 2 2704.2.a.o 2
13.e even 6 1 208.2.w.b 2
39.h odd 6 1 1872.2.by.d 2
39.i odd 6 1 1872.2.by.d 2
52.b odd 2 1 169.2.b.a 2
52.f even 4 2 169.2.a.a 2
52.i odd 6 1 13.2.e.a 2
52.i odd 6 1 169.2.e.a 2
52.j odd 6 1 13.2.e.a 2
52.j odd 6 1 169.2.e.a 2
52.l even 12 4 169.2.c.a 4
104.n odd 6 1 832.2.w.d 2
104.p odd 6 1 832.2.w.d 2
104.r even 6 1 832.2.w.a 2
104.s even 6 1 832.2.w.a 2
156.h even 2 1 1521.2.b.a 2
156.l odd 4 2 1521.2.a.k 2
156.p even 6 1 117.2.q.c 2
156.r even 6 1 117.2.q.c 2
260.u even 4 2 4225.2.a.v 2
260.v odd 6 1 325.2.n.a 2
260.w odd 6 1 325.2.n.a 2
260.bg even 12 2 325.2.m.a 4
260.bj even 12 2 325.2.m.a 4
364.p odd 4 2 8281.2.a.q 2
364.q odd 6 1 637.2.u.c 2
364.s odd 6 1 637.2.u.c 2
364.v even 6 1 637.2.q.a 2
364.w even 6 1 637.2.k.c 2
364.ba even 6 1 637.2.k.c 2
364.bc even 6 1 637.2.q.a 2
364.bi odd 6 1 637.2.k.a 2
364.bk odd 6 1 637.2.k.a 2
364.bp even 6 1 637.2.u.b 2
364.br even 6 1 637.2.u.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 52.i odd 6 1
13.2.e.a 2 52.j odd 6 1
117.2.q.c 2 156.p even 6 1
117.2.q.c 2 156.r even 6 1
169.2.a.a 2 52.f even 4 2
169.2.b.a 2 4.b odd 2 1
169.2.b.a 2 52.b odd 2 1
169.2.c.a 4 52.l even 12 4
169.2.e.a 2 52.i odd 6 1
169.2.e.a 2 52.j odd 6 1
208.2.w.b 2 13.c even 3 1
208.2.w.b 2 13.e even 6 1
325.2.m.a 4 260.bg even 12 2
325.2.m.a 4 260.bj even 12 2
325.2.n.a 2 260.v odd 6 1
325.2.n.a 2 260.w odd 6 1
637.2.k.a 2 364.bi odd 6 1
637.2.k.a 2 364.bk odd 6 1
637.2.k.c 2 364.w even 6 1
637.2.k.c 2 364.ba even 6 1
637.2.q.a 2 364.v even 6 1
637.2.q.a 2 364.bc even 6 1
637.2.u.b 2 364.bp even 6 1
637.2.u.b 2 364.br even 6 1
637.2.u.c 2 364.q odd 6 1
637.2.u.c 2 364.s odd 6 1
832.2.w.a 2 104.r even 6 1
832.2.w.a 2 104.s even 6 1
832.2.w.d 2 104.n odd 6 1
832.2.w.d 2 104.p odd 6 1
1521.2.a.k 2 156.l odd 4 2
1521.2.b.a 2 12.b even 2 1
1521.2.b.a 2 156.h even 2 1
1872.2.by.d 2 39.h odd 6 1
1872.2.by.d 2 39.i odd 6 1
2704.2.a.o 2 13.d odd 4 2
2704.2.f.b 2 1.a even 1 1 trivial
2704.2.f.b 2 13.b even 2 1 inner
4225.2.a.v 2 260.u even 4 2
8281.2.a.q 2 364.p odd 4 2

Hecke kernels

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

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{5}^{2} + 3 \) Copy content Toggle raw display
\( T_{11} \) 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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