Properties

Label 1600.3.e.c
Level $1600$
Weight $3$
Character orbit 1600.e
Analytic conductor $43.597$
Analytic rank $0$
Dimension $4$
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] = [1600,3,Mod(799,1600)]
 
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([1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.799");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1600.e (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: \(43.5968422976\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} - 2 q^{7} - 2 q^{9} + \beta_{2} q^{11} + 2 \beta_{2} q^{13} + 11 \beta_1 q^{17} - 7 \beta_{2} q^{19} + 2 \beta_{3} q^{21} + 12 q^{23} - 7 \beta_{3} q^{27} - 14 \beta_{3} q^{29} - 24 \beta_1 q^{31} - 11 \beta_1 q^{33} + 14 \beta_{2} q^{37} - 22 \beta_1 q^{39} - 29 q^{41} - 12 \beta_{3} q^{43} - 58 q^{47} - 45 q^{49} + 11 \beta_{2} q^{51} + 20 \beta_{2} q^{53} + 77 \beta_1 q^{57} - 20 \beta_{2} q^{59} + 18 \beta_{3} q^{61} + 4 q^{63} - 13 \beta_{3} q^{67} - 12 \beta_{3} q^{69} - 34 \beta_1 q^{71} - 35 \beta_1 q^{73} - 2 \beta_{2} q^{77} - 110 \beta_1 q^{79} - 95 q^{81} - 7 \beta_{3} q^{83} - 154 q^{87} - 65 q^{89} - 4 \beta_{2} q^{91} - 24 \beta_{2} q^{93} + 134 \beta_1 q^{97} - 2 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{7} - 8 q^{9} + 48 q^{23} - 116 q^{41} - 232 q^{47} - 180 q^{49} + 16 q^{63} - 380 q^{81} - 616 q^{87} - 260 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 2\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 8\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{2} + 4\beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
−1.65831 0.500000i
1.65831 + 0.500000i
−1.65831 + 0.500000i
1.65831 0.500000i
0 3.31662i 0 0 0 −2.00000 0 −2.00000 0
799.2 0 3.31662i 0 0 0 −2.00000 0 −2.00000 0
799.3 0 3.31662i 0 0 0 −2.00000 0 −2.00000 0
799.4 0 3.31662i 0 0 0 −2.00000 0 −2.00000 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
8.b even 2 1 inner
20.d odd 2 1 inner
40.e odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.3.e.c 4
4.b odd 2 1 1600.3.e.e 4
5.b even 2 1 1600.3.e.e 4
5.c odd 4 1 1600.3.g.a 4
5.c odd 4 1 1600.3.g.b yes 4
8.b even 2 1 inner 1600.3.e.c 4
8.d odd 2 1 1600.3.e.e 4
20.d odd 2 1 inner 1600.3.e.c 4
20.e even 4 1 1600.3.g.a 4
20.e even 4 1 1600.3.g.b yes 4
40.e odd 2 1 inner 1600.3.e.c 4
40.f even 2 1 1600.3.e.e 4
40.i odd 4 1 1600.3.g.a 4
40.i odd 4 1 1600.3.g.b yes 4
40.k even 4 1 1600.3.g.a 4
40.k even 4 1 1600.3.g.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.3.e.c 4 1.a even 1 1 trivial
1600.3.e.c 4 8.b even 2 1 inner
1600.3.e.c 4 20.d odd 2 1 inner
1600.3.e.c 4 40.e odd 2 1 inner
1600.3.e.e 4 4.b odd 2 1
1600.3.e.e 4 5.b even 2 1
1600.3.e.e 4 8.d odd 2 1
1600.3.e.e 4 40.f even 2 1
1600.3.g.a 4 5.c odd 4 1
1600.3.g.a 4 20.e even 4 1
1600.3.g.a 4 40.i odd 4 1
1600.3.g.a 4 40.k even 4 1
1600.3.g.b yes 4 5.c odd 4 1
1600.3.g.b yes 4 20.e even 4 1
1600.3.g.b yes 4 40.i odd 4 1
1600.3.g.b yes 4 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1600, [\chi])\):

\( T_{3}^{2} + 11 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 11)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T + 2)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 11)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 44)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 121)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 539)^{2} \) Copy content Toggle raw display
$23$ \( (T - 12)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2156)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 576)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 2156)^{2} \) Copy content Toggle raw display
$41$ \( (T + 29)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1584)^{2} \) Copy content Toggle raw display
$47$ \( (T + 58)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 4400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 4400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 3564)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 1859)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 1156)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 1225)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 12100)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 539)^{2} \) Copy content Toggle raw display
$89$ \( (T + 65)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 17956)^{2} \) Copy content Toggle raw display
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