Properties

Label 1600.1.bh.b
Level $1600$
Weight $1$
Character orbit 1600.bh
Analytic conductor $0.799$
Analytic rank $0$
Dimension $8$
Projective image $A_{5}$
CM/RM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,1,Mod(191,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.191");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1600.bh (of order \(10\), degree \(4\), 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: \(0.798504020213\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 800)
Projective image: \(A_{5}\)
Projective field: Galois closure of 5.1.25000000.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{20}^{9} q^{3} - \zeta_{20}^{8} q^{5} + ( - \zeta_{20}^{7} - \zeta_{20}^{3}) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{20}^{9} q^{3} - \zeta_{20}^{8} q^{5} + ( - \zeta_{20}^{7} - \zeta_{20}^{3}) q^{7} + (\zeta_{20}^{6} - 1) q^{13} - \zeta_{20}^{7} q^{15} + ( - \zeta_{20}^{7} + \zeta_{20}^{5}) q^{19} + ( - \zeta_{20}^{6} - \zeta_{20}^{2}) q^{21} - \zeta_{20}^{7} q^{23} - \zeta_{20}^{6} q^{25} + \zeta_{20}^{7} q^{27} + (\zeta_{20}^{8} + 1) q^{29} + \zeta_{20} q^{31} + ( - \zeta_{20}^{5} - \zeta_{20}) q^{35} + \zeta_{20}^{8} q^{37} + (\zeta_{20}^{9} + \zeta_{20}^{5}) q^{39} + ( - \zeta_{20}^{9} - \zeta_{20}) q^{43} + (\zeta_{20}^{7} - \zeta_{20}) q^{47} + (\zeta_{20}^{6} - \zeta_{20}^{4} - 1) q^{49} + \zeta_{20}^{4} q^{53} + ( - \zeta_{20}^{6} + \zeta_{20}^{4}) q^{57} + (\zeta_{20}^{9} - \zeta_{20}^{7}) q^{59} + \zeta_{20}^{2} q^{61} + (\zeta_{20}^{8} + \zeta_{20}^{4}) q^{65} - \zeta_{20}^{6} q^{69} + ( - \zeta_{20}^{7} + \zeta_{20}) q^{71} - \zeta_{20}^{2} q^{73} - \zeta_{20}^{5} q^{75} + (\zeta_{20}^{5} - \zeta_{20}^{3}) q^{79} + \zeta_{20}^{6} q^{81} + \zeta_{20} q^{83} + ( - \zeta_{20}^{9} + \zeta_{20}^{7}) q^{87} + ( - \zeta_{20}^{9} + \zeta_{20}^{7} + \zeta_{20}^{3}) q^{91} + q^{93} + ( - \zeta_{20}^{5} + \zeta_{20}^{3}) q^{95} + (\zeta_{20}^{6} + \zeta_{20}^{2}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{5} - 6 q^{13} - 4 q^{21} - 2 q^{25} + 6 q^{29} - 2 q^{37} - 4 q^{49} - 2 q^{53} - 4 q^{57} + 2 q^{61} - 4 q^{65} - 2 q^{69} - 2 q^{73} + 2 q^{81} + 8 q^{93} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(\zeta_{20}^{4}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−0.951057 0.309017i
0.951057 + 0.309017i
−0.951057 + 0.309017i
0.951057 0.309017i
−0.587785 + 0.809017i
0.587785 0.809017i
−0.587785 0.809017i
0.587785 + 0.809017i
0 −0.951057 + 0.309017i 0 0.809017 0.587785i 0 1.61803i 0 0 0
191.2 0 0.951057 0.309017i 0 0.809017 0.587785i 0 1.61803i 0 0 0
511.1 0 −0.951057 0.309017i 0 0.809017 + 0.587785i 0 1.61803i 0 0 0
511.2 0 0.951057 + 0.309017i 0 0.809017 + 0.587785i 0 1.61803i 0 0 0
831.1 0 −0.587785 0.809017i 0 −0.309017 + 0.951057i 0 0.618034i 0 0 0
831.2 0 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 0 0.618034i 0 0 0
1471.1 0 −0.587785 + 0.809017i 0 −0.309017 0.951057i 0 0.618034i 0 0 0
1471.2 0 0.587785 0.809017i 0 −0.309017 0.951057i 0 0.618034i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
25.d even 5 1 inner
100.j odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.1.bh.b 8
4.b odd 2 1 inner 1600.1.bh.b 8
8.b even 2 1 800.1.bh.a 8
8.d odd 2 1 800.1.bh.a 8
25.d even 5 1 inner 1600.1.bh.b 8
40.e odd 2 1 4000.1.bh.a 8
40.f even 2 1 4000.1.bh.a 8
40.i odd 4 1 4000.1.bf.a 8
40.i odd 4 1 4000.1.bf.b 8
40.k even 4 1 4000.1.bf.a 8
40.k even 4 1 4000.1.bf.b 8
100.j odd 10 1 inner 1600.1.bh.b 8
200.n odd 10 1 800.1.bh.a 8
200.o even 10 1 4000.1.bh.a 8
200.s odd 10 1 4000.1.bh.a 8
200.t even 10 1 800.1.bh.a 8
200.v even 20 1 4000.1.bf.a 8
200.v even 20 1 4000.1.bf.b 8
200.x odd 20 1 4000.1.bf.a 8
200.x odd 20 1 4000.1.bf.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.1.bh.a 8 8.b even 2 1
800.1.bh.a 8 8.d odd 2 1
800.1.bh.a 8 200.n odd 10 1
800.1.bh.a 8 200.t even 10 1
1600.1.bh.b 8 1.a even 1 1 trivial
1600.1.bh.b 8 4.b odd 2 1 inner
1600.1.bh.b 8 25.d even 5 1 inner
1600.1.bh.b 8 100.j odd 10 1 inner
4000.1.bf.a 8 40.i odd 4 1
4000.1.bf.a 8 40.k even 4 1
4000.1.bf.a 8 200.v even 20 1
4000.1.bf.a 8 200.x odd 20 1
4000.1.bf.b 8 40.i odd 4 1
4000.1.bf.b 8 40.k even 4 1
4000.1.bf.b 8 200.v even 20 1
4000.1.bf.b 8 200.x odd 20 1
4000.1.bh.a 8 40.e odd 2 1
4000.1.bh.a 8 40.f even 2 1
4000.1.bh.a 8 200.o even 10 1
4000.1.bh.a 8 200.s odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - T_{3}^{6} + T_{3}^{4} - T_{3}^{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1600, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} + 3 T^{3} + 4 T^{2} + 2 T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + T^{6} + 6 T^{4} - 4 T^{2} + 1 \) Copy content Toggle raw display
$23$ \( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$37$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1 \) Copy content Toggle raw display
$53$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1 \) Copy content Toggle raw display
$61$ \( (T^{4} - T^{3} + T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} - 4 T^{6} + 6 T^{4} + T^{2} + 1 \) Copy content Toggle raw display
$73$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + T^{6} + 6 T^{4} - 4 T^{2} + 1 \) Copy content Toggle raw display
$83$ \( T^{8} - T^{6} + T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1)^{2} \) Copy content Toggle raw display
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