↑

Properties

Label 1600.2.o.i
Level $1600$
Weight $2$
Character orbit 1600.o
Analytic conductor $12.776$
Analytic rank $0$
Dimension $8$
Inner twists $8$

Related objects

Downloads

Learn more

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(543,1600)] 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("1600.543"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.o (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(39)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + 2 \beta_{3} q^{9} - \beta_{6} q^{11} - 4 \beta_1 q^{13} + \beta_{2} q^{17} - 3 \beta_{4} q^{19} - 4 \beta_{2} q^{23} - 5 \beta_1 q^{27} + 4 \beta_{6} q^{29} + 4 \beta_{3} q^{31} - \beta_{7} q^{33}+ \cdots - 2 \beta_{4} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 32 q^{39} + 24 q^{41} + 64 q^{79} - 8 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} - \zeta_{24}^{3} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(\beta_{3}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
543.1
0.965926 − 0.258819i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.258819 − 0.965926i
0.965926 + 0.258819i
−0.258819 − 0.965926i
−0.965926 − 0.258819i
0.258819 + 0.965926i
0 −0.707107 − 0.707107i 0 0 0 0 0 − 2.00000i 0
543.2 0 −0.707107 − 0.707107i 0 0 0 0 0 − 2.00000i 0
543.3 0 0.707107 + 0.707107i 0 0 0 0 0 − 2.00000i 0
543.4 0 0.707107 + 0.707107i 0 0 0 0 0 − 2.00000i 0
607.1 0 −0.707107 + 0.707107i 0 0 0 0 0 2.00000i 0
607.2 0 −0.707107 + 0.707107i 0 0 0 0 0 2.00000i 0
607.3 0 0.707107 − 0.707107i 0 0 0 0 0 2.00000i 0
607.4 0 0.707107 − 0.707107i 0 0 0 0 0 2.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 543.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
20.e even 4 2 inner
40.f even 2 1 inner
40.k even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.o.i yes 8
4.b odd 2 1 1600.2.o.h ✓ 8
5.b even 2 1 inner 1600.2.o.i yes 8
5.c odd 4 2 1600.2.o.h ✓ 8
8.b even 2 1 inner 1600.2.o.i yes 8
8.d odd 2 1 1600.2.o.h ✓ 8
20.d odd 2 1 1600.2.o.h ✓ 8
20.e even 4 2 inner 1600.2.o.i yes 8
40.e odd 2 1 1600.2.o.h ✓ 8
40.f even 2 1 inner 1600.2.o.i yes 8
40.i odd 4 2 1600.2.o.h ✓ 8
40.k even 4 2 inner 1600.2.o.i yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.2.o.h ✓ 8 4.b odd 2 1
1600.2.o.h ✓ 8 5.c odd 4 2
1600.2.o.h ✓ 8 8.d odd 2 1
1600.2.o.h ✓ 8 20.d odd 2 1
1600.2.o.h ✓ 8 40.e odd 2 1
1600.2.o.h ✓ 8 40.i odd 4 2
1600.2.o.i yes 8 1.a even 1 1 trivial
1600.2.o.i yes 8 5.b even 2 1 inner
1600.2.o.i yes 8 8.b even 2 1 inner
1600.2.o.i yes 8 20.e even 4 2 inner
1600.2.o.i yes 8 40.f even 2 1 inner
1600.2.o.i yes 8 40.k even 4 2 inner

Hecke kernels

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

\( T_{3}^{4} + 1 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{79} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 27)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4096)^{2} \) Copy content Toggle raw display
$41$ \( (T - 3)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 4096)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 20736)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 108)^{4} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 2401)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 144)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 5625)^{2} \) Copy content Toggle raw display
$79$ \( (T - 8)^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} + 50625)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
show more
show less