Properties

Label 1000.2.c.a
Level $1000$
Weight $2$
Character orbit 1000.c
Analytic conductor $7.985$
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] = [1000,2,Mod(249,1000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1000.249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.c (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: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{3} + 2 \beta_1) q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{3} + 2 \beta_1) q^{7} + (\beta_{2} + 2) q^{9} + (2 \beta_{2} - 3) q^{11} + ( - 4 \beta_{3} - \beta_1) q^{13} + ( - \beta_{3} - 2 \beta_1) q^{17} + (5 \beta_{2} + 4) q^{19} + (3 \beta_{2} - 2) q^{21} + (2 \beta_{3} - 2 \beta_1) q^{23} + (\beta_{3} + 4 \beta_1) q^{27} + ( - 2 \beta_{2} + 7) q^{29} + (3 \beta_{2} - 1) q^{31} + (2 \beta_{3} - 5 \beta_1) q^{33} + (2 \beta_{3} + 2 \beta_1) q^{37} + (3 \beta_{2} + 1) q^{39} - 3 q^{41} + (7 \beta_{3} + 6 \beta_1) q^{43} + ( - 5 \beta_{3} - 7 \beta_1) q^{47} + (8 \beta_{2} + 2) q^{49} + ( - \beta_{2} + 2) q^{51} + ( - 5 \beta_{3} + 3 \beta_1) q^{53} + (5 \beta_{3} - \beta_1) q^{57} + (\beta_{2} + 5) q^{59} + (9 \beta_{2} + 2) q^{61} + \beta_1 q^{63} + ( - \beta_{3} + \beta_1) q^{67} + ( - 4 \beta_{2} + 2) q^{69} + ( - 2 \beta_{2} - 3) q^{71} + ( - 11 \beta_{3} - 3 \beta_1) q^{73} + (7 \beta_{3} - 12 \beta_1) q^{77} + (2 \beta_{2} + 5) q^{79} + (6 \beta_{2} + 2) q^{81} + (6 \beta_{3} + 8 \beta_1) q^{83} + ( - 2 \beta_{3} + 9 \beta_1) q^{87} + (4 \beta_{2} + 6) q^{89} + (5 \beta_{2} - 2) q^{91} + (3 \beta_{3} - 4 \beta_1) q^{93} + ( - 3 \beta_{3} - 9 \beta_1) q^{97} + ( - \beta_{2} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{9} - 16 q^{11} + 6 q^{19} - 14 q^{21} + 32 q^{29} - 10 q^{31} - 2 q^{39} - 12 q^{41} - 8 q^{49} + 10 q^{51} + 18 q^{59} - 10 q^{61} + 16 q^{69} - 8 q^{71} + 16 q^{79} - 4 q^{81} + 16 q^{89} - 18 q^{91} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 2\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

\(n\) \(377\) \(501\) \(751\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
249.1
1.61803i
0.618034i
0.618034i
1.61803i
0 1.61803i 0 0 0 4.23607i 0 0.381966 0
249.2 0 0.618034i 0 0 0 0.236068i 0 2.61803 0
249.3 0 0.618034i 0 0 0 0.236068i 0 2.61803 0
249.4 0 1.61803i 0 0 0 4.23607i 0 0.381966 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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.2.c.a 4
4.b odd 2 1 2000.2.c.h 4
5.b even 2 1 inner 1000.2.c.a 4
5.c odd 4 1 1000.2.a.b 2
5.c odd 4 1 1000.2.a.c yes 2
15.e even 4 1 9000.2.a.c 2
15.e even 4 1 9000.2.a.o 2
20.d odd 2 1 2000.2.c.h 4
20.e even 4 1 2000.2.a.f 2
20.e even 4 1 2000.2.a.g 2
40.i odd 4 1 8000.2.a.h 2
40.i odd 4 1 8000.2.a.r 2
40.k even 4 1 8000.2.a.g 2
40.k even 4 1 8000.2.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1000.2.a.b 2 5.c odd 4 1
1000.2.a.c yes 2 5.c odd 4 1
1000.2.c.a 4 1.a even 1 1 trivial
1000.2.c.a 4 5.b even 2 1 inner
2000.2.a.f 2 20.e even 4 1
2000.2.a.g 2 20.e even 4 1
2000.2.c.h 4 4.b odd 2 1
2000.2.c.h 4 20.d odd 2 1
8000.2.a.g 2 40.k even 4 1
8000.2.a.h 2 40.i odd 4 1
8000.2.a.q 2 40.k even 4 1
8000.2.a.r 2 40.i odd 4 1
9000.2.a.c 2 15.e even 4 1
9000.2.a.o 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\):

\( T_{3}^{4} + 3T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 18T_{7}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 18T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} + 8 T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 27T^{2} + 121 \) Copy content Toggle raw display
$17$ \( (T^{2} + 5)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 3 T - 29)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 28T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T^{2} - 16 T + 59)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 5 T - 5)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T + 3)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 122T^{2} + 841 \) Copy content Toggle raw display
$47$ \( T^{4} + 127T^{2} + 3481 \) Copy content Toggle raw display
$53$ \( T^{4} + 107T^{2} + 961 \) Copy content Toggle raw display
$59$ \( (T^{2} - 9 T + 19)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 5 T - 95)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 7T^{2} + 1 \) Copy content Toggle raw display
$71$ \( (T^{2} + 4 T - 1)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 203T^{2} + 6241 \) Copy content Toggle raw display
$79$ \( (T^{2} - 8 T + 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 168T^{2} + 5776 \) Copy content Toggle raw display
$89$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 207T^{2} + 9801 \) Copy content Toggle raw display
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