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Properties

Label 1600.2.o.l
Level $1600$
Weight $2$
Character orbit 1600.o
Analytic conductor $12.776$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(543,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.o (of order \(4\), degree \(2\), 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: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + ( - \beta_{4} + \beta_1) q^{7} + ( - \beta_{5} + \beta_{4} + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + ( - \beta_{4} + \beta_1) q^{7} + ( - \beta_{5} + \beta_{4} + 2 \beta_1) q^{9} + (\beta_{7} - \beta_{6}) q^{11} + (\beta_{7} + \beta_{3}) q^{13} + (3 \beta_1 - 3) q^{17} + (\beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{19}+ \cdots + (3 \beta_{7} + 3 \beta_{6} + \cdots - 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{7} - 24 q^{17} - 12 q^{23} + 16 q^{33} - 24 q^{39} - 24 q^{41} + 12 q^{47} - 24 q^{57} + 76 q^{63} + 32 q^{73} + 96 q^{79} - 80 q^{81} + 56 q^{87} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{7} + 5\nu^{5} - 5\nu^{3} + 16\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{7} + 2\nu^{6} - 15\nu^{5} - 10\nu^{4} - 25\nu^{3} - 30\nu^{2} - 20\nu - 16 ) / 40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} - 2\nu^{6} - 15\nu^{5} + 10\nu^{4} - 25\nu^{3} + 30\nu^{2} - 20\nu + 16 ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 5\nu^{6} - 5\nu^{5} - 15\nu^{4} - 15\nu^{3} - 5\nu^{2} - 42\nu - 40 ) / 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 3\nu^{5} - 6\nu^{4} + 5\nu^{3} - 2\nu^{2} + 20\nu - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} + 7\nu^{6} - 5\nu^{5} + 5\nu^{4} - 15\nu^{3} + 15\nu^{2} - 30\nu + 44 ) / 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5\nu^{7} + 7\nu^{6} + 5\nu^{5} + 5\nu^{4} + 15\nu^{3} + 15\nu^{2} + 30\nu + 44 ) / 20 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} - 2\beta_{2} - \beta _1 - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{3} - \beta_{2} - 10\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{7} - 2\beta_{6} - 3\beta_{5} - 3\beta_{4} + \beta_{3} - \beta_{2} + 3\beta _1 - 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{7} + 5\beta_{6} + \beta_{5} - \beta_{4} - 6\beta_{3} - 6\beta_{2} + 11\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5\beta_{7} + 5\beta_{6} - 5\beta_{3} + 5\beta_{2} - 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10\beta_{7} - 10\beta_{6} - 7\beta_{5} + 7\beta_{4} + 3\beta_{3} + 3\beta_{2} + 13\beta_1 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
543.1
1.09445 − 0.895644i
0.228425 + 1.39564i
−0.228425 + 1.39564i
−1.09445 − 0.895644i
1.09445 + 0.895644i
0.228425 − 1.39564i
−0.228425 − 1.39564i
−1.09445 + 0.895644i
0 −2.18890 − 2.18890i 0 0 0 −1.79129 − 1.79129i 0 6.58258i 0
543.2 0 −0.456850 − 0.456850i 0 0 0 2.79129 + 2.79129i 0 − 2.58258i 0
543.3 0 0.456850 + 0.456850i 0 0 0 2.79129 + 2.79129i 0 − 2.58258i 0
543.4 0 2.18890 + 2.18890i 0 0 0 −1.79129 − 1.79129i 0 6.58258i 0
607.1 0 −2.18890 + 2.18890i 0 0 0 −1.79129 + 1.79129i 0 − 6.58258i 0
607.2 0 −0.456850 + 0.456850i 0 0 0 2.79129 − 2.79129i 0 2.58258i 0
607.3 0 0.456850 − 0.456850i 0 0 0 2.79129 − 2.79129i 0 2.58258i 0
607.4 0 2.18890 − 2.18890i 0 0 0 −1.79129 + 1.79129i 0 − 6.58258i 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
8.b even 2 1 inner
20.e even 4 1 inner
40.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.o.l 8
4.b odd 2 1 1600.2.o.e 8
5.b even 2 1 320.2.o.e ✓ 8
5.c odd 4 1 320.2.o.f yes 8
5.c odd 4 1 1600.2.o.e 8
8.b even 2 1 inner 1600.2.o.l 8
8.d odd 2 1 1600.2.o.e 8
20.d odd 2 1 320.2.o.f yes 8
20.e even 4 1 320.2.o.e ✓ 8
20.e even 4 1 inner 1600.2.o.l 8
40.e odd 2 1 320.2.o.f yes 8
40.f even 2 1 320.2.o.e ✓ 8
40.i odd 4 1 320.2.o.f yes 8
40.i odd 4 1 1600.2.o.e 8
40.k even 4 1 320.2.o.e ✓ 8
40.k even 4 1 inner 1600.2.o.l 8
80.i odd 4 1 1280.2.n.n 8
80.j even 4 1 1280.2.n.p 8
80.k odd 4 2 1280.2.n.n 8
80.q even 4 2 1280.2.n.p 8
80.s even 4 1 1280.2.n.p 8
80.t odd 4 1 1280.2.n.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.o.e ✓ 8 5.b even 2 1
320.2.o.e ✓ 8 20.e even 4 1
320.2.o.e ✓ 8 40.f even 2 1
320.2.o.e ✓ 8 40.k even 4 1
320.2.o.f yes 8 5.c odd 4 1
320.2.o.f yes 8 20.d odd 2 1
320.2.o.f yes 8 40.e odd 2 1
320.2.o.f yes 8 40.i odd 4 1
1280.2.n.n 8 80.i odd 4 1
1280.2.n.n 8 80.k odd 4 2
1280.2.n.n 8 80.t odd 4 1
1280.2.n.p 8 80.j even 4 1
1280.2.n.p 8 80.q even 4 2
1280.2.n.p 8 80.s even 4 1
1600.2.o.e 8 4.b odd 2 1
1600.2.o.e 8 5.c odd 4 1
1600.2.o.e 8 8.d odd 2 1
1600.2.o.e 8 40.i odd 4 1
1600.2.o.l 8 1.a even 1 1 trivial
1600.2.o.l 8 8.b even 2 1 inner
1600.2.o.l 8 20.e even 4 1 inner
1600.2.o.l 8 40.k even 4 1 inner

Hecke kernels

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

\( T_{3}^{8} + 92T_{3}^{4} + 16 \) Copy content Toggle raw display
\( T_{7}^{4} - 2T_{7}^{3} + 2T_{7}^{2} + 20T_{7} + 100 \) Copy content Toggle raw display
\( T_{79} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 92T^{4} + 16 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 20 T^{2} + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 36)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 6 T + 18)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} + 18 T^{2} + \cdots + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 60 T^{2} + 144)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2916)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 12)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} + 18972 T^{4} + 1296 \) Copy content Toggle raw display
$47$ \( (T^{4} - 6 T^{3} + 18 T^{2} + \cdots + 36)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 196)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 132 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 18972 T^{4} + 1296 \) Copy content Toggle raw display
$71$ \( (T^{4} + 204 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 16 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$79$ \( (T - 12)^{8} \) Copy content Toggle raw display
$83$ \( T^{8} + 2556 T^{4} + 810000 \) Copy content Toggle raw display
$89$ \( (T^{4} + 240 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 16 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
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