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Properties

Label 2704.2.f.g
Level $2704$
Weight $2$
Character orbit 2704.f
Analytic conductor $21.592$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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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 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")
 
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);
 
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 = [2,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,0,-6,0,0,0,0,0,-8] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{5} - 4 i q^{7} - 3 q^{9} - 4 i q^{11} - 3 q^{17} - 4 q^{23} + 4 q^{25} - q^{29} + 4 i q^{31} + 4 q^{35} + 3 i q^{37} + 9 i q^{41} - 8 q^{43} - 3 i q^{45} + 8 i q^{47} - 9 q^{49} - 9 q^{53} + 4 q^{55} + \cdots + 12 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{9} - 6 q^{17} - 8 q^{23} + 8 q^{25} - 2 q^{29} + 8 q^{35} - 16 q^{43} - 18 q^{49} - 18 q^{53} + 8 q^{55} + 14 q^{61} - 32 q^{77} + 8 q^{79} + 18 q^{81}+O(q^{100}) \) 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.

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
− 1.00000i
1.00000i
0 0 0 − 1.00000i 0 4.00000i 0 −3.00000 0
337.2 0 0 0 1.00000i 0 − 4.00000i 0 −3.00000 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.g 2
4.b odd 2 1 338.2.b.b 2
12.b even 2 1 3042.2.b.e 2
13.b even 2 1 inner 2704.2.f.g 2
13.d odd 4 1 2704.2.a.h 1
13.d odd 4 1 2704.2.a.i 1
13.f odd 12 2 208.2.i.b 2
39.k even 12 2 1872.2.t.k 2
52.b odd 2 1 338.2.b.b 2
52.f even 4 1 338.2.a.c 1
52.f even 4 1 338.2.a.e 1
52.i odd 6 2 338.2.e.b 4
52.j odd 6 2 338.2.e.b 4
52.l even 12 2 26.2.c.a ✓ 2
52.l even 12 2 338.2.c.e 2
104.u even 12 2 832.2.i.e 2
104.x odd 12 2 832.2.i.f 2
156.h even 2 1 3042.2.b.e 2
156.l odd 4 1 3042.2.a.e 1
156.l odd 4 1 3042.2.a.k 1
156.v odd 12 2 234.2.h.c 2
260.u even 4 1 8450.2.a.f 1
260.u even 4 1 8450.2.a.s 1
260.bc even 12 2 650.2.e.c 2
260.be odd 12 2 650.2.o.c 4
260.bl odd 12 2 650.2.o.c 4
364.bt even 12 2 1274.2.h.b 2
364.bv odd 12 2 1274.2.g.a 2
364.bz odd 12 2 1274.2.e.m 2
364.ca even 12 2 1274.2.e.n 2
364.cg odd 12 2 1274.2.h.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.c.a ✓ 2 52.l even 12 2
208.2.i.b 2 13.f odd 12 2
234.2.h.c 2 156.v odd 12 2
338.2.a.c 1 52.f even 4 1
338.2.a.e 1 52.f even 4 1
338.2.b.b 2 4.b odd 2 1
338.2.b.b 2 52.b odd 2 1
338.2.c.e 2 52.l even 12 2
338.2.e.b 4 52.i odd 6 2
338.2.e.b 4 52.j odd 6 2
650.2.e.c 2 260.bc even 12 2
650.2.o.c 4 260.be odd 12 2
650.2.o.c 4 260.bl odd 12 2
832.2.i.e 2 104.u even 12 2
832.2.i.f 2 104.x odd 12 2
1274.2.e.m 2 364.bz odd 12 2
1274.2.e.n 2 364.ca even 12 2
1274.2.g.a 2 364.bv odd 12 2
1274.2.h.a 2 364.cg odd 12 2
1274.2.h.b 2 364.bt even 12 2
1872.2.t.k 2 39.k even 12 2
2704.2.a.h 1 13.d odd 4 1
2704.2.a.i 1 13.d odd 4 1
2704.2.f.g 2 1.a even 1 1 trivial
2704.2.f.g 2 13.b even 2 1 inner
3042.2.a.e 1 156.l odd 4 1
3042.2.a.k 1 156.l odd 4 1
3042.2.b.e 2 12.b even 2 1
3042.2.b.e 2 156.h even 2 1
8450.2.a.f 1 260.u even 4 1
8450.2.a.s 1 260.u even 4 1

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} + 16 \) 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
$5$ \( T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( (T + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 9 \) Copy content Toggle raw display
$41$ \( T^{2} + 81 \) Copy content Toggle raw display
$43$ \( (T + 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( (T + 9)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 16 \) Copy content Toggle raw display
$61$ \( (T - 7)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 64 \) Copy content Toggle raw display
$73$ \( T^{2} + 121 \) Copy content Toggle raw display
$79$ \( (T - 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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