Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(1343,1600)]
 
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, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.1343");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.n (of order \(4\), degree \(2\), 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: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.3317760000.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 7x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 100)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + 2 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + 2 \beta_{3} q^{9} + \beta_{7} q^{11} + 2 \beta_{4} q^{13} - \beta_1 q^{17} - \beta_{5} q^{19} + 2 \beta_{2} q^{23} + \beta_{6} q^{27} + 6 \beta_{3} q^{29} + 2 \beta_{7} q^{31} + 5 \beta_{4} q^{33} - 4 \beta_1 q^{37} - 2 \beta_{5} q^{39} - 3 q^{41} + 2 \beta_{6} q^{47} + 7 \beta_{3} q^{49} - \beta_{7} q^{51} + 2 \beta_{4} q^{53} - 5 \beta_1 q^{57} + 2 \beta_{5} q^{59} + 8 q^{61} - 3 \beta_{6} q^{67} + 10 \beta_{3} q^{69} - 2 \beta_{7} q^{71} + 3 \beta_{4} q^{73} + 4 \beta_{5} q^{79} + 11 q^{81} - \beta_{2} q^{83} - 6 \beta_{6} q^{87} + 9 \beta_{3} q^{89} + 10 \beta_{4} q^{93} + 4 \beta_1 q^{97} - 2 \beta_{5} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{41} + 64 q^{61} + 88 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 7x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + \nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 3\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 5\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 11\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 13\nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\nu^{4} - 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{6} + 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{4} + 5\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3\beta_{5} + 11\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -\beta_{6} + 13\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1343.1
−1.40294 + 0.178197i
−0.178197 + 1.40294i
1.40294 0.178197i
0.178197 1.40294i
−0.178197 1.40294i
−1.40294 0.178197i
0.178197 + 1.40294i
1.40294 + 0.178197i
0 −1.58114 + 1.58114i 0 0 0 0 0 2.00000i 0
1343.2 0 −1.58114 + 1.58114i 0 0 0 0 0 2.00000i 0
1343.3 0 1.58114 1.58114i 0 0 0 0 0 2.00000i 0
1343.4 0 1.58114 1.58114i 0 0 0 0 0 2.00000i 0
1407.1 0 −1.58114 1.58114i 0 0 0 0 0 2.00000i 0
1407.2 0 −1.58114 1.58114i 0 0 0 0 0 2.00000i 0
1407.3 0 1.58114 + 1.58114i 0 0 0 0 0 2.00000i 0
1407.4 0 1.58114 + 1.58114i 0 0 0 0 0 2.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1343.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e 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.n.v 8
4.b odd 2 1 inner 1600.2.n.v 8
5.b even 2 1 inner 1600.2.n.v 8
5.c odd 4 2 inner 1600.2.n.v 8
8.b even 2 1 100.2.e.d 8
8.d odd 2 1 100.2.e.d 8
20.d odd 2 1 inner 1600.2.n.v 8
20.e even 4 2 inner 1600.2.n.v 8
24.f even 2 1 900.2.k.j 8
24.h odd 2 1 900.2.k.j 8
40.e odd 2 1 100.2.e.d 8
40.f even 2 1 100.2.e.d 8
40.i odd 4 2 100.2.e.d 8
40.k even 4 2 100.2.e.d 8
120.i odd 2 1 900.2.k.j 8
120.m even 2 1 900.2.k.j 8
120.q odd 4 2 900.2.k.j 8
120.w even 4 2 900.2.k.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.e.d 8 8.b even 2 1
100.2.e.d 8 8.d odd 2 1
100.2.e.d 8 40.e odd 2 1
100.2.e.d 8 40.f even 2 1
100.2.e.d 8 40.i odd 4 2
100.2.e.d 8 40.k even 4 2
900.2.k.j 8 24.f even 2 1
900.2.k.j 8 24.h odd 2 1
900.2.k.j 8 120.i odd 2 1
900.2.k.j 8 120.m even 2 1
900.2.k.j 8 120.q odd 4 2
900.2.k.j 8 120.w even 4 2
1600.2.n.v 8 1.a even 1 1 trivial
1600.2.n.v 8 4.b odd 2 1 inner
1600.2.n.v 8 5.b even 2 1 inner
1600.2.n.v 8 5.c odd 4 2 inner
1600.2.n.v 8 20.d odd 2 1 inner
1600.2.n.v 8 20.e 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 \(S_{2}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3}^{4} + 25 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{11}^{2} + 15 \) Copy content Toggle raw display
\( T_{13}^{4} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 25)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + 15)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 15)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 400)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
$41$ \( (T + 3)^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} + 400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 144)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 60)^{4} \) Copy content Toggle raw display
$61$ \( (T - 8)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 2025)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 729)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 240)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 25)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 81)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 2304)^{2} \) Copy content Toggle raw display
show more
show less