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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3200,2,Mod(449,3200)] 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("3200.449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3200 = 2^{7} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3200.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,-6,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,-24,0,0,0,20,0,0,0,0,0,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(49)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.5521286468\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 128)
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.

\(f(q)\) \(=\) \( q - 3 q^{9} + 4 q^{13} + \beta q^{17} + 2 \beta q^{29} - 12 q^{37} + 10 q^{41} + 7 q^{49} - 4 q^{53} + 6 \beta q^{61} + 3 \beta q^{73} + 9 q^{81} + 10 q^{89} + 9 \beta q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{9} + 8 q^{13} - 24 q^{37} + 20 q^{41} + 14 q^{49} - 8 q^{53} + 18 q^{81} + 20 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1151\) \(2177\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
− 1.00000i
1.00000i
0 0 0 0 0 0 0 −3.00000 0
449.2 0 0 0 0 0 0 0 −3.00000 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
40.e 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 3200.2.f.d 2
4.b odd 2 1 CM 3200.2.f.d 2
5.b even 2 1 3200.2.f.c 2
5.c odd 4 1 128.2.b.b ✓ 2
5.c odd 4 1 3200.2.d.e 2
8.b even 2 1 3200.2.f.c 2
8.d odd 2 1 3200.2.f.c 2
15.e even 4 1 1152.2.d.d 2
20.d odd 2 1 3200.2.f.c 2
20.e even 4 1 128.2.b.b ✓ 2
20.e even 4 1 3200.2.d.e 2
40.e odd 2 1 inner 3200.2.f.d 2
40.f even 2 1 inner 3200.2.f.d 2
40.i odd 4 1 128.2.b.b ✓ 2
40.i odd 4 1 3200.2.d.e 2
40.k even 4 1 128.2.b.b ✓ 2
40.k even 4 1 3200.2.d.e 2
60.l odd 4 1 1152.2.d.d 2
80.i odd 4 1 256.2.a.b 1
80.i odd 4 1 6400.2.a.l 1
80.j even 4 1 256.2.a.c 1
80.j even 4 1 6400.2.a.m 1
80.s even 4 1 256.2.a.b 1
80.s even 4 1 6400.2.a.l 1
80.t odd 4 1 256.2.a.c 1
80.t odd 4 1 6400.2.a.m 1
120.q odd 4 1 1152.2.d.d 2
120.w even 4 1 1152.2.d.d 2
160.u even 8 2 1024.2.e.k 4
160.v odd 8 2 1024.2.e.k 4
160.ba even 8 2 1024.2.e.k 4
160.bb odd 8 2 1024.2.e.k 4
240.z odd 4 1 2304.2.a.p 1
240.bb even 4 1 2304.2.a.p 1
240.bd odd 4 1 2304.2.a.a 1
240.bf even 4 1 2304.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
128.2.b.b ✓ 2 5.c odd 4 1
128.2.b.b ✓ 2 20.e even 4 1
128.2.b.b ✓ 2 40.i odd 4 1
128.2.b.b ✓ 2 40.k even 4 1
256.2.a.b 1 80.i odd 4 1
256.2.a.b 1 80.s even 4 1
256.2.a.c 1 80.j even 4 1
256.2.a.c 1 80.t odd 4 1
1024.2.e.k 4 160.u even 8 2
1024.2.e.k 4 160.v odd 8 2
1024.2.e.k 4 160.ba even 8 2
1024.2.e.k 4 160.bb odd 8 2
1152.2.d.d 2 15.e even 4 1
1152.2.d.d 2 60.l odd 4 1
1152.2.d.d 2 120.q odd 4 1
1152.2.d.d 2 120.w even 4 1
2304.2.a.a 1 240.bd odd 4 1
2304.2.a.a 1 240.bf even 4 1
2304.2.a.p 1 240.z odd 4 1
2304.2.a.p 1 240.bb even 4 1
3200.2.d.e 2 5.c odd 4 1
3200.2.d.e 2 20.e even 4 1
3200.2.d.e 2 40.i odd 4 1
3200.2.d.e 2 40.k even 4 1
3200.2.f.c 2 5.b even 2 1
3200.2.f.c 2 8.b even 2 1
3200.2.f.c 2 8.d odd 2 1
3200.2.f.c 2 20.d odd 2 1
3200.2.f.d 2 1.a even 1 1 trivial
3200.2.f.d 2 4.b odd 2 1 CM
3200.2.f.d 2 40.e odd 2 1 inner
3200.2.f.d 2 40.f even 2 1 inner
6400.2.a.l 1 80.i odd 4 1
6400.2.a.l 1 80.s even 4 1
6400.2.a.m 1 80.j even 4 1
6400.2.a.m 1 80.t odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3200, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} - 4 \) 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
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 16 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 12)^{2} \) Copy content Toggle raw display
$41$ \( (T - 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T + 4)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 144 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 36 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 324 \) Copy content Toggle raw display
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