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Properties

Label 2704.2.f.k
Level $2704$
Weight $2$
Character orbit 2704.f
Analytic conductor $21.592$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2704,2,Mod(337,2704)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
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

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-2,0,0,0,0,0,6,0,0,0,0,0,0,0,2,0,0,0,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(21.5915487066\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{2} + \beta_1) q^{5} + \beta_1 q^{7} + (\beta_{3} + 1) q^{9} + 2 \beta_1 q^{11} + (2 \beta_{2} + \beta_1) q^{15} + ( - 3 \beta_{3} + 2) q^{17} + 2 \beta_1 q^{19} + (2 \beta_{2} - \beta_1) q^{21}+ \cdots + ( - 4 \beta_{2} + 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} + 6 q^{9} + 2 q^{17} - 32 q^{23} - 6 q^{25} - 14 q^{27} - 8 q^{29} - 14 q^{35} + 30 q^{43} + 10 q^{49} + 50 q^{51} - 4 q^{53} - 28 q^{55} + 28 q^{61} + 16 q^{69} - 48 q^{75} - 36 q^{77} - 32 q^{79}+ \cdots - 28 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} - 9\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{2} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{2} + 10\beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
1.56155i
− 1.56155i
2.56155i
− 2.56155i
0 −2.56155 0 − 0.561553i 0 − 2.56155i 0 3.56155 0
337.2 0 −2.56155 0 0.561553i 0 2.56155i 0 3.56155 0
337.3 0 1.56155 0 − 3.56155i 0 − 1.56155i 0 −0.561553 0
337.4 0 1.56155 0 3.56155i 0 1.56155i 0 −0.561553 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.k 4
4.b odd 2 1 1352.2.f.c 4
13.b even 2 1 inner 2704.2.f.k 4
13.d odd 4 1 208.2.a.e 2
13.d odd 4 1 2704.2.a.p 2
39.f even 4 1 1872.2.a.u 2
52.b odd 2 1 1352.2.f.c 4
52.f even 4 1 104.2.a.b ✓ 2
52.f even 4 1 1352.2.a.g 2
52.i odd 6 2 1352.2.o.d 8
52.j odd 6 2 1352.2.o.d 8
52.l even 12 2 1352.2.i.d 4
52.l even 12 2 1352.2.i.f 4
65.g odd 4 1 5200.2.a.bw 2
104.j odd 4 1 832.2.a.n 2
104.m even 4 1 832.2.a.k 2
156.l odd 4 1 936.2.a.j 2
208.l even 4 1 3328.2.b.y 4
208.m odd 4 1 3328.2.b.w 4
208.r odd 4 1 3328.2.b.w 4
208.s even 4 1 3328.2.b.y 4
260.l odd 4 1 2600.2.d.k 4
260.s odd 4 1 2600.2.d.k 4
260.u even 4 1 2600.2.a.p 2
312.w odd 4 1 7488.2.a.cu 2
312.y even 4 1 7488.2.a.cv 2
364.p odd 4 1 5096.2.a.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.a.b ✓ 2 52.f even 4 1
208.2.a.e 2 13.d odd 4 1
832.2.a.k 2 104.m even 4 1
832.2.a.n 2 104.j odd 4 1
936.2.a.j 2 156.l odd 4 1
1352.2.a.g 2 52.f even 4 1
1352.2.f.c 4 4.b odd 2 1
1352.2.f.c 4 52.b odd 2 1
1352.2.i.d 4 52.l even 12 2
1352.2.i.f 4 52.l even 12 2
1352.2.o.d 8 52.i odd 6 2
1352.2.o.d 8 52.j odd 6 2
1872.2.a.u 2 39.f even 4 1
2600.2.a.p 2 260.u even 4 1
2600.2.d.k 4 260.l odd 4 1
2600.2.d.k 4 260.s odd 4 1
2704.2.a.p 2 13.d odd 4 1
2704.2.f.k 4 1.a even 1 1 trivial
2704.2.f.k 4 13.b even 2 1 inner
3328.2.b.w 4 208.m odd 4 1
3328.2.b.w 4 208.r odd 4 1
3328.2.b.y 4 208.l even 4 1
3328.2.b.y 4 208.s even 4 1
5096.2.a.m 2 364.p odd 4 1
5200.2.a.bw 2 65.g odd 4 1
7488.2.a.cu 2 312.w odd 4 1
7488.2.a.cv 2 312.y even 4 1

Hecke kernels

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

\( T_{3}^{2} + T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{4} + 13T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{4} + 36T_{11}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + T - 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 13T^{2} + 4 \) Copy content Toggle raw display
$7$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - T - 38)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$23$ \( (T + 8)^{4} \) Copy content Toggle raw display
$29$ \( (T + 2)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 101T^{2} + 676 \) Copy content Toggle raw display
$41$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$43$ \( (T^{2} - 15 T + 52)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 161T^{2} + 16 \) Copy content Toggle raw display
$53$ \( (T^{2} + 2 T - 16)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$61$ \( (T^{2} - 14 T + 32)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$71$ \( T^{4} + 81T^{2} + 1296 \) Copy content Toggle raw display
$73$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 208T^{2} + 1024 \) Copy content Toggle raw display
$89$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 68)^{2} \) Copy content Toggle raw display
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