Properties

Label 1600.2.j.b
Level $1600$
Weight $2$
Character orbit 1600.j
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(143,1600)]
 
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, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1600.143");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.j (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: \(\Q(\zeta_{24})\)
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: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 400)
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.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1) q^{3} + (\beta_{7} + \beta_{4} + 3 \beta_1) q^{7} + (\beta_{6} - \beta_{3} + \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_1) q^{3} + (\beta_{7} + \beta_{4} + 3 \beta_1) q^{7} + (\beta_{6} - \beta_{3} + \beta_{2} + 1) q^{9} + ( - 3 \beta_{3} + 3 \beta_{2} - 3) q^{11} + (4 \beta_{7} + 2 \beta_{5} + 2 \beta_1) q^{13} - \beta_1 q^{17} + (2 \beta_{3} - 3 \beta_{2} + 2) q^{19} + ( - 2 \beta_{6} + \beta_{3} - 1) q^{21} + (3 \beta_{7} + 4 \beta_{5} - 3 \beta_{4} + 3 \beta_1) q^{23} + ( - \beta_{5} + 3 \beta_{4} - 2 \beta_1) q^{27} + ( - 2 \beta_{6} + 4 \beta_{3} - 4) q^{29} + (3 \beta_{6} - \beta_{3} - 3 \beta_{2} + 3) q^{31} + (3 \beta_{7} - 3 \beta_{5} - 3 \beta_{4} + 3 \beta_1) q^{33} + (2 \beta_{7} - 5 \beta_{5} - 5 \beta_1) q^{37} + (2 \beta_{6} + 6 \beta_{3} - 2 \beta_{2} + 2) q^{39} + ( - 2 \beta_{6} - 3 \beta_{3} + 2 \beta_{2} - 2) q^{41} + (4 \beta_{7} - \beta_{5} - \beta_1) q^{43} + 2 \beta_{5} q^{47} + ( - 6 \beta_{6} + 5 \beta_{3} + 6 \beta_{2} - 6) q^{49} + \beta_{6} q^{51} + (7 \beta_{5} - 7 \beta_1) q^{53} + ( - 2 \beta_{7} + 3 \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{57} + ( - 2 \beta_{6} + 7 \beta_{3} - 7) q^{59} + (5 \beta_{3} - 6 \beta_{2} + 5) q^{61} + (4 \beta_{7} + 4 \beta_{4} + 6 \beta_1) q^{63} + (3 \beta_{7} + 3 \beta_{5} + 3 \beta_1) q^{67} + (5 \beta_{3} - 2 \beta_{2} + 5) q^{69} + ( - 4 \beta_{6} + 4 \beta_{3} - 4 \beta_{2} + 6) q^{71} + (2 \beta_{7} + 2 \beta_{4} + 9 \beta_1) q^{73} + ( - 3 \beta_{5} + 6 \beta_{4} - 3 \beta_1) q^{77} + (3 \beta_{6} - 3 \beta_{3} + 3 \beta_{2} - 3) q^{79} + (5 \beta_{6} - 5 \beta_{3} + 5 \beta_{2} - 2) q^{81} + (9 \beta_{5} - 9 \beta_{4}) q^{83} + ( - 2 \beta_{7} - 2 \beta_{4}) q^{87} + (\beta_{6} - \beta_{3} + \beta_{2} - 12) q^{89} + 12 \beta_{2} q^{91} + ( - 2 \beta_{7} + 3 \beta_{5} + 3 \beta_1) q^{93} + (6 \beta_{7} + 6 \beta_{4} + 6 \beta_1) q^{97} + (3 \beta_{3} + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} - 12 q^{11} + 4 q^{19} - 24 q^{29} - 4 q^{51} - 48 q^{59} + 16 q^{61} + 32 q^{69} + 48 q^{71} - 24 q^{79} - 16 q^{81} - 96 q^{89} + 48 q^{91} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
0.258819 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.258819 0.965926i
−0.965926 0.258819i
0.965926 + 0.258819i
0.258819 + 0.965926i
0 1.93185i 0 0 0 −0.896575 + 0.896575i 0 −0.732051 0
143.2 0 0.517638i 0 0 0 3.34607 3.34607i 0 2.73205 0
143.3 0 0.517638i 0 0 0 −3.34607 + 3.34607i 0 2.73205 0
143.4 0 1.93185i 0 0 0 0.896575 0.896575i 0 −0.732051 0
1007.1 0 1.93185i 0 0 0 0.896575 + 0.896575i 0 −0.732051 0
1007.2 0 0.517638i 0 0 0 −3.34607 3.34607i 0 2.73205 0
1007.3 0 0.517638i 0 0 0 3.34607 + 3.34607i 0 2.73205 0
1007.4 0 1.93185i 0 0 0 −0.896575 0.896575i 0 −0.732051 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 143.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
80.j even 4 1 inner
80.s 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.j.b 8
4.b odd 2 1 400.2.j.b 8
5.b even 2 1 inner 1600.2.j.b 8
5.c odd 4 2 1600.2.s.b 8
16.e even 4 1 400.2.s.b yes 8
16.f odd 4 1 1600.2.s.b 8
20.d odd 2 1 400.2.j.b 8
20.e even 4 2 400.2.s.b yes 8
80.i odd 4 1 400.2.j.b 8
80.j even 4 1 inner 1600.2.j.b 8
80.k odd 4 1 1600.2.s.b 8
80.q even 4 1 400.2.s.b yes 8
80.s even 4 1 inner 1600.2.j.b 8
80.t odd 4 1 400.2.j.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.j.b 8 4.b odd 2 1
400.2.j.b 8 20.d odd 2 1
400.2.j.b 8 80.i odd 4 1
400.2.j.b 8 80.t odd 4 1
400.2.s.b yes 8 16.e even 4 1
400.2.s.b yes 8 20.e even 4 2
400.2.s.b yes 8 80.q even 4 1
1600.2.j.b 8 1.a even 1 1 trivial
1600.2.j.b 8 5.b even 2 1 inner
1600.2.j.b 8 80.j even 4 1 inner
1600.2.j.b 8 80.s even 4 1 inner
1600.2.s.b 8 5.c odd 4 2
1600.2.s.b 8 16.f odd 4 1
1600.2.s.b 8 80.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 4T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1600, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 504T^{4} + 1296 \) Copy content Toggle raw display
$11$ \( (T^{4} + 6 T^{3} + 18 T^{2} - 54 T + 81)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 24)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 2 T^{3} + 2 T^{2} + 26 T + 169)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 1784 T^{4} + 456976 \) Copy content Toggle raw display
$29$ \( (T^{4} + 12 T^{3} + 72 T^{2} + 144 T + 144)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 56 T^{2} + 676)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 156 T^{2} + 4356)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 42 T^{2} + 9)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 84 T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 98)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 24 T^{3} + 288 T^{2} + 1584 T + 4356)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 8 T^{3} + 32 T^{2} + 368 T + 2116)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 36 T^{2} + 81)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 12 T - 12)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + 25074 T^{4} + \cdots + 22667121 \) Copy content Toggle raw display
$79$ \( (T^{2} + 6 T - 18)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 324 T^{2} + 6561)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 24 T + 141)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} + 72576 T^{4} + \cdots + 26873856 \) Copy content Toggle raw display
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