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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1600,2,Mod(1249,1600)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.f (of order \(2\), degree \(1\), 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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + (\beta_{3} - \beta_{2}) q^{7} + ( - 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + (\beta_{3} - \beta_{2}) q^{7} + ( - 2 \beta_1 + 1) q^{9} + (2 \beta_{3} + \beta_{2}) q^{11} + 2 \beta_1 q^{13} + ( - 2 \beta_{3} - \beta_{2}) q^{17} - \beta_{2} q^{19} + (4 \beta_{3} - \beta_{2}) q^{21} + (3 \beta_{3} + 3 \beta_{2}) q^{23} + 4 q^{27} + ( - 2 \beta_1 - 6) q^{31} + (2 \beta_{3} - 2 \beta_{2}) q^{33} + 6 q^{37} + (2 \beta_1 - 6) q^{39} + ( - 2 \beta_1 + 6) q^{41} + (3 \beta_1 + 5) q^{43} + (3 \beta_{3} + 3 \beta_{2}) q^{47} + (6 \beta_1 - 5) q^{49} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{51} + 6 \beta_1 q^{53} + 2 \beta_{3} q^{57} + 3 \beta_{2} q^{59} + (4 \beta_{3} + 5 \beta_{2}) q^{61} + (7 \beta_{3} - \beta_{2}) q^{63} + ( - 3 \beta_1 - 5) q^{67} - 3 \beta_{2} q^{69} + ( - 6 \beta_1 + 6) q^{71} + ( - 6 \beta_{3} - 5 \beta_{2}) q^{73} + (6 \beta_1 - 6) q^{77} + 12 q^{79} + (2 \beta_1 + 1) q^{81} + (\beta_1 + 3) q^{83} + ( - 4 \beta_1 + 6) q^{89} - 6 \beta_{3} q^{91} + 4 \beta_1 q^{93} + (6 \beta_{3} + \beta_{2}) q^{97} + (2 \beta_{3} - 5 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{9} + 16 q^{27} - 24 q^{31} + 24 q^{37} - 24 q^{39} + 24 q^{41} + 20 q^{43} - 20 q^{49} - 20 q^{67} + 24 q^{71} - 24 q^{77} + 48 q^{79} + 4 q^{81} + 12 q^{83} + 24 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

\(\zeta_{12}\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( 2\beta_{3} + \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{2} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
0.866025 + 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
−0.866025 − 0.500000i
0 −0.732051 0 0 0 − 1.26795i 0 −2.46410 0
1249.2 0 −0.732051 0 0 0 1.26795i 0 −2.46410 0
1249.3 0 2.73205 0 0 0 − 4.73205i 0 4.46410 0
1249.4 0 2.73205 0 0 0 4.73205i 0 4.46410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.f.h 4
4.b odd 2 1 1600.2.f.e 4
5.b even 2 1 1600.2.f.d 4
5.c odd 4 1 320.2.d.b yes 4
5.c odd 4 1 1600.2.d.b 4
8.b even 2 1 1600.2.f.d 4
8.d odd 2 1 1600.2.f.i 4
15.e even 4 1 2880.2.k.l 4
20.d odd 2 1 1600.2.f.i 4
20.e even 4 1 320.2.d.a ✓ 4
20.e even 4 1 1600.2.d.h 4
40.e odd 2 1 1600.2.f.e 4
40.f even 2 1 inner 1600.2.f.h 4
40.i odd 4 1 320.2.d.b yes 4
40.i odd 4 1 1600.2.d.b 4
40.k even 4 1 320.2.d.a ✓ 4
40.k even 4 1 1600.2.d.h 4
60.l odd 4 1 2880.2.k.e 4
80.i odd 4 1 1280.2.a.c 2
80.i odd 4 1 6400.2.a.bf 2
80.j even 4 1 1280.2.a.b 2
80.j even 4 1 6400.2.a.y 2
80.s even 4 1 1280.2.a.p 2
80.s even 4 1 6400.2.a.cd 2
80.t odd 4 1 1280.2.a.m 2
80.t odd 4 1 6400.2.a.ck 2
120.q odd 4 1 2880.2.k.e 4
120.w even 4 1 2880.2.k.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.d.a ✓ 4 20.e even 4 1
320.2.d.a ✓ 4 40.k even 4 1
320.2.d.b yes 4 5.c odd 4 1
320.2.d.b yes 4 40.i odd 4 1
1280.2.a.b 2 80.j even 4 1
1280.2.a.c 2 80.i odd 4 1
1280.2.a.m 2 80.t odd 4 1
1280.2.a.p 2 80.s even 4 1
1600.2.d.b 4 5.c odd 4 1
1600.2.d.b 4 40.i odd 4 1
1600.2.d.h 4 20.e even 4 1
1600.2.d.h 4 40.k even 4 1
1600.2.f.d 4 5.b even 2 1
1600.2.f.d 4 8.b even 2 1
1600.2.f.e 4 4.b odd 2 1
1600.2.f.e 4 40.e odd 2 1
1600.2.f.h 4 1.a even 1 1 trivial
1600.2.f.h 4 40.f even 2 1 inner
1600.2.f.i 4 8.d odd 2 1
1600.2.f.i 4 20.d odd 2 1
2880.2.k.e 4 60.l odd 4 1
2880.2.k.e 4 120.q odd 4 1
2880.2.k.l 4 15.e even 4 1
2880.2.k.l 4 120.w even 4 1
6400.2.a.y 2 80.j even 4 1
6400.2.a.bf 2 80.i odd 4 1
6400.2.a.cd 2 80.s even 4 1
6400.2.a.ck 2 80.t odd 4 1

Hecke kernels

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

\( T_{3}^{2} - 2T_{3} - 2 \) Copy content Toggle raw display
\( T_{31}^{2} + 12T_{31} + 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2 T - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 72T^{2} + 324 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$37$ \( (T - 6)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12 T + 24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 10 T - 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 72T^{2} + 324 \) Copy content Toggle raw display
$53$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$67$ \( (T^{2} + 10 T - 2)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 12 T - 72)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 248T^{2} + 8464 \) Copy content Toggle raw display
$79$ \( (T - 12)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 12 T - 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 248T^{2} + 8464 \) Copy content Toggle raw display
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