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Properties

Label 1600.2.f.a
Level $1600$
Weight $2$
Character orbit 1600.f
Analytic conductor $12.776$
Analytic rank $1$
Dimension $2$
CM discriminant -8
Inner twists $4$

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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] = [1600,2,Mod(1249,1600)] 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("1600.1249"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0, 0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7760643234\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 64)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{3} + q^{9} + 3 \beta q^{11} - 3 \beta q^{17} + \beta q^{19} + 4 q^{27} - 6 \beta q^{33} - 6 q^{41} - 10 q^{43} + 7 q^{49} + 6 \beta q^{51} - 2 \beta q^{57} - 3 \beta q^{59} - 14 q^{67} - \beta q^{73} + \cdots + 3 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 2 q^{9} + 8 q^{27} - 12 q^{41} - 20 q^{43} + 14 q^{49} - 28 q^{67} - 22 q^{81} - 36 q^{83} - 36 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
− 1.00000i
1.00000i
0 −2.00000 0 0 0 0 0 1.00000 0
1249.2 0 −2.00000 0 0 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
20.d odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.f.a 2
4.b odd 2 1 1600.2.f.b 2
5.b even 2 1 1600.2.f.b 2
5.c odd 4 1 64.2.b.a ✓ 2
5.c odd 4 1 1600.2.d.a 2
8.b even 2 1 1600.2.f.b 2
8.d odd 2 1 CM 1600.2.f.a 2
15.e even 4 1 576.2.d.a 2
20.d odd 2 1 inner 1600.2.f.a 2
20.e even 4 1 64.2.b.a ✓ 2
20.e even 4 1 1600.2.d.a 2
35.f even 4 1 3136.2.b.b 2
40.e odd 2 1 1600.2.f.b 2
40.f even 2 1 inner 1600.2.f.a 2
40.i odd 4 1 64.2.b.a ✓ 2
40.i odd 4 1 1600.2.d.a 2
40.k even 4 1 64.2.b.a ✓ 2
40.k even 4 1 1600.2.d.a 2
60.l odd 4 1 576.2.d.a 2
80.i odd 4 1 256.2.a.d 1
80.i odd 4 1 6400.2.a.x 1
80.j even 4 1 256.2.a.d 1
80.j even 4 1 6400.2.a.x 1
80.s even 4 1 256.2.a.a 1
80.s even 4 1 6400.2.a.a 1
80.t odd 4 1 256.2.a.a 1
80.t odd 4 1 6400.2.a.a 1
120.q odd 4 1 576.2.d.a 2
120.w even 4 1 576.2.d.a 2
140.j odd 4 1 3136.2.b.b 2
160.u even 8 2 1024.2.e.l 4
160.v odd 8 2 1024.2.e.l 4
160.ba even 8 2 1024.2.e.l 4
160.bb odd 8 2 1024.2.e.l 4
240.z odd 4 1 2304.2.a.i 1
240.bb even 4 1 2304.2.a.h 1
240.bd odd 4 1 2304.2.a.h 1
240.bf even 4 1 2304.2.a.i 1
280.s even 4 1 3136.2.b.b 2
280.y odd 4 1 3136.2.b.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.2.b.a ✓ 2 5.c odd 4 1
64.2.b.a ✓ 2 20.e even 4 1
64.2.b.a ✓ 2 40.i odd 4 1
64.2.b.a ✓ 2 40.k even 4 1
256.2.a.a 1 80.s even 4 1
256.2.a.a 1 80.t odd 4 1
256.2.a.d 1 80.i odd 4 1
256.2.a.d 1 80.j even 4 1
576.2.d.a 2 15.e even 4 1
576.2.d.a 2 60.l odd 4 1
576.2.d.a 2 120.q odd 4 1
576.2.d.a 2 120.w even 4 1
1024.2.e.l 4 160.u even 8 2
1024.2.e.l 4 160.v odd 8 2
1024.2.e.l 4 160.ba even 8 2
1024.2.e.l 4 160.bb odd 8 2
1600.2.d.a 2 5.c odd 4 1
1600.2.d.a 2 20.e even 4 1
1600.2.d.a 2 40.i odd 4 1
1600.2.d.a 2 40.k even 4 1
1600.2.f.a 2 1.a even 1 1 trivial
1600.2.f.a 2 8.d odd 2 1 CM
1600.2.f.a 2 20.d odd 2 1 inner
1600.2.f.a 2 40.f even 2 1 inner
1600.2.f.b 2 4.b odd 2 1
1600.2.f.b 2 5.b even 2 1
1600.2.f.b 2 8.b even 2 1
1600.2.f.b 2 40.e odd 2 1
2304.2.a.h 1 240.bb even 4 1
2304.2.a.h 1 240.bd odd 4 1
2304.2.a.i 1 240.z odd 4 1
2304.2.a.i 1 240.bf even 4 1
3136.2.b.b 2 35.f even 4 1
3136.2.b.b 2 140.j odd 4 1
3136.2.b.b 2 280.s even 4 1
3136.2.b.b 2 280.y odd 4 1
6400.2.a.a 1 80.s even 4 1
6400.2.a.a 1 80.t odd 4 1
6400.2.a.x 1 80.i odd 4 1
6400.2.a.x 1 80.j even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{31} \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 36 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 36 \) Copy content Toggle raw display
$19$ \( T^{2} + 4 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( (T + 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 36 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( (T + 14)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T + 18)^{2} \) Copy content Toggle raw display
$89$ \( (T + 18)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 100 \) Copy content Toggle raw display
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