Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(801,1600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.801");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.d (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
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: \( 2^{12} \)
Twist minimal: no (minimal twist has level 320)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} - \beta_{4} q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} - \beta_{4} q^{7} - 3 q^{9} - \beta_1 q^{11} - 2 \beta_{6} q^{13} - \beta_{7} q^{17} + 3 \beta_1 q^{19} - \beta_{2} q^{21} - 5 \beta_{4} q^{23} - 2 \beta_{2} q^{29} - \beta_{3} q^{31} - \beta_{7} q^{33} - \beta_{6} q^{37} - 2 \beta_{3} q^{39} - 4 q^{41} - \beta_{5} q^{43} - 3 \beta_{4} q^{47} - 5 q^{49} + 6 \beta_1 q^{51} + 3 \beta_{7} q^{57} - \beta_1 q^{59} - \beta_{2} q^{61} + 3 \beta_{4} q^{63} - \beta_{5} q^{67} - 5 \beta_{2} q^{69} + \beta_{3} q^{71} + \beta_{7} q^{73} - \beta_{6} q^{77} - \beta_{3} q^{79} - 9 q^{81} + 5 \beta_{5} q^{83} + 12 \beta_{4} q^{87} + 2 q^{89} - 4 \beta_1 q^{91} + 6 \beta_{6} q^{93} - 3 \beta_{7} q^{97} + 3 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{9} - 32 q^{41} - 40 q^{49} - 72 q^{81} + 16 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\zeta_{24}^{4} - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -4\zeta_{24}^{6} + 8\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{5} - \zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -4\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} - \beta_{6} + 2\beta_{5} - 2\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{3} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{6} - 2\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{5} + 2\beta_{4} ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + \beta_{6} + 2\beta_{5} - 2\beta_{4} ) / 8 \) Copy content Toggle raw display

Character values

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

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
801.1
0.258819 + 0.965926i
−0.965926 + 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 0.258819i
0.258819 0.965926i
−0.258819 0.965926i
0.965926 0.258819i
0 2.44949i 0 0 0 −1.41421 0 −3.00000 0
801.2 0 2.44949i 0 0 0 −1.41421 0 −3.00000 0
801.3 0 2.44949i 0 0 0 1.41421 0 −3.00000 0
801.4 0 2.44949i 0 0 0 1.41421 0 −3.00000 0
801.5 0 2.44949i 0 0 0 −1.41421 0 −3.00000 0
801.6 0 2.44949i 0 0 0 −1.41421 0 −3.00000 0
801.7 0 2.44949i 0 0 0 1.41421 0 −3.00000 0
801.8 0 2.44949i 0 0 0 1.41421 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 801.8
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
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
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 1600.2.d.i 8
4.b odd 2 1 inner 1600.2.d.i 8
5.b even 2 1 inner 1600.2.d.i 8
5.c odd 4 2 320.2.f.b 8
8.b even 2 1 inner 1600.2.d.i 8
8.d odd 2 1 inner 1600.2.d.i 8
15.e even 4 2 2880.2.d.g 8
16.e even 4 1 6400.2.a.ct 4
16.e even 4 1 6400.2.a.cu 4
16.f odd 4 1 6400.2.a.ct 4
16.f odd 4 1 6400.2.a.cu 4
20.d odd 2 1 inner 1600.2.d.i 8
20.e even 4 2 320.2.f.b 8
40.e odd 2 1 inner 1600.2.d.i 8
40.f even 2 1 inner 1600.2.d.i 8
40.i odd 4 2 320.2.f.b 8
40.k even 4 2 320.2.f.b 8
60.l odd 4 2 2880.2.d.g 8
80.i odd 4 1 1280.2.c.g 4
80.i odd 4 1 1280.2.c.h 4
80.j even 4 1 1280.2.c.g 4
80.j even 4 1 1280.2.c.h 4
80.k odd 4 1 6400.2.a.ct 4
80.k odd 4 1 6400.2.a.cu 4
80.q even 4 1 6400.2.a.ct 4
80.q even 4 1 6400.2.a.cu 4
80.s even 4 1 1280.2.c.g 4
80.s even 4 1 1280.2.c.h 4
80.t odd 4 1 1280.2.c.g 4
80.t odd 4 1 1280.2.c.h 4
120.q odd 4 2 2880.2.d.g 8
120.w even 4 2 2880.2.d.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.f.b 8 5.c odd 4 2
320.2.f.b 8 20.e even 4 2
320.2.f.b 8 40.i odd 4 2
320.2.f.b 8 40.k even 4 2
1280.2.c.g 4 80.i odd 4 1
1280.2.c.g 4 80.j even 4 1
1280.2.c.g 4 80.s even 4 1
1280.2.c.g 4 80.t odd 4 1
1280.2.c.h 4 80.i odd 4 1
1280.2.c.h 4 80.j even 4 1
1280.2.c.h 4 80.s even 4 1
1280.2.c.h 4 80.t odd 4 1
1600.2.d.i 8 1.a even 1 1 trivial
1600.2.d.i 8 4.b odd 2 1 inner
1600.2.d.i 8 5.b even 2 1 inner
1600.2.d.i 8 8.b even 2 1 inner
1600.2.d.i 8 8.d odd 2 1 inner
1600.2.d.i 8 20.d odd 2 1 inner
1600.2.d.i 8 40.e odd 2 1 inner
1600.2.d.i 8 40.f even 2 1 inner
2880.2.d.g 8 15.e even 4 2
2880.2.d.g 8 60.l odd 4 2
2880.2.d.g 8 120.q odd 4 2
2880.2.d.g 8 120.w even 4 2
6400.2.a.ct 4 16.e even 4 1
6400.2.a.ct 4 16.f odd 4 1
6400.2.a.ct 4 80.k odd 4 1
6400.2.a.ct 4 80.q even 4 1
6400.2.a.cu 4 16.e even 4 1
6400.2.a.cu 4 16.f odd 4 1
6400.2.a.cu 4 80.k odd 4 1
6400.2.a.cu 4 80.q even 4 1

Hecke kernels

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

\( T_{3}^{2} + 6 \) Copy content Toggle raw display
\( T_{7}^{2} - 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 24)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 50)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 48)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$41$ \( (T + 4)^{8} \) Copy content Toggle raw display
$43$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 18)^{4} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 24)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 150)^{4} \) Copy content Toggle raw display
$89$ \( (T - 2)^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} - 216)^{4} \) Copy content Toggle raw display
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