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Properties

Label 4000.2.c.d
Level $4000$
Weight $2$
Character orbit 4000.c
Analytic conductor $31.940$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4000,2,Mod(1249,4000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, 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("4000.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4000.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: \(31.9401608085\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - \beta_{7} q^{7} - \beta_{5} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - \beta_{7} q^{7} - \beta_{5} q^{9} + \beta_{4} q^{11} + \beta_{3} q^{13} - \beta_1 q^{17} + ( - \beta_{6} + \beta_{4}) q^{19} + (2 \beta_{5} + 1) q^{21} - \beta_{7} q^{23} + ( - \beta_{7} + 3 \beta_{2}) q^{27} + (\beta_{5} - 2) q^{29} + (2 \beta_{6} + \beta_{4}) q^{31} + (2 \beta_{3} - \beta_1) q^{33} + (3 \beta_{3} - 3 \beta_1) q^{37} + ( - \beta_{6} - \beta_{4}) q^{39} + ( - 3 \beta_{5} - 4) q^{41} + (4 \beta_{7} - 3 \beta_{2}) q^{43} + (2 \beta_{7} - 5 \beta_{2}) q^{47} + (\beta_{5} + 5) q^{49} + \beta_{6} q^{51} + 2 \beta_1 q^{53} + (\beta_{3} - 3 \beta_1) q^{57} + \beta_{4} q^{59} + (\beta_{5} + 3) q^{61} + ( - \beta_{7} + \beta_{2}) q^{63} + (2 \beta_{7} + 2 \beta_{2}) q^{67} + (2 \beta_{5} + 1) q^{69} + (\beta_{6} + 3 \beta_{4}) q^{71} + (\beta_{3} + 4 \beta_1) q^{73} + (\beta_{3} - 3 \beta_1) q^{77} + (\beta_{6} + 4 \beta_{4}) q^{79} + ( - 4 \beta_{5} - 8) q^{81} + ( - \beta_{7} + 2 \beta_{2}) q^{83} + (\beta_{7} - 2 \beta_{2}) q^{87} + ( - 5 \beta_{5} - 5) q^{89} + \beta_{6} q^{91} + (4 \beta_{3} + 3 \beta_1) q^{93} + (5 \beta_{3} - 4 \beta_1) q^{97} + ( - \beta_{6} + \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} - 20 q^{29} - 20 q^{41} + 36 q^{49} + 20 q^{61} - 48 q^{81} - 20 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{20}^{5} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{20}^{6} + \zeta_{20}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{20}^{7} + 2\zeta_{20}^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{20}^{7} + 2\zeta_{20}^{3} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{20}^{6} + \zeta_{20}^{4} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{20}^{7} + 2\zeta_{20}^{5} - 2\zeta_{20}^{3} + 4\zeta_{20} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \zeta_{20}^{6} - \zeta_{20}^{4} + 2\zeta_{20}^{2} - 1 \) Copy content Toggle raw display

Display \(\zeta_{20}^j\) in terms of \(\beta_i\)

\(\zeta_{20}\)\(=\) \( ( \beta_{6} + \beta_{3} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{20}^{2}\)\(=\) \( ( \beta_{7} + \beta_{5} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{3}\)\(=\) \( ( \beta_{4} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{20}^{4}\)\(=\) \( ( \beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{5}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{6}\)\(=\) \( ( -\beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{7}\)\(=\) \( ( -\beta_{4} + \beta_{3} ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{20}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times\).

\(n\) \(1377\) \(2501\) \(2751\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
−0.951057 + 0.309017i
0.951057 − 0.309017i
0.587785 + 0.809017i
−0.587785 − 0.809017i
0.587785 − 0.809017i
−0.587785 + 0.809017i
−0.951057 − 0.309017i
0.951057 + 0.309017i
0 − 1.90211i 0 0 0 1.17557i 0 −0.618034 0
1249.2 0 − 1.90211i 0 0 0 1.17557i 0 −0.618034 0
1249.3 0 − 1.17557i 0 0 0 − 1.90211i 0 1.61803 0
1249.4 0 − 1.17557i 0 0 0 − 1.90211i 0 1.61803 0
1249.5 0 1.17557i 0 0 0 1.90211i 0 1.61803 0
1249.6 0 1.17557i 0 0 0 1.90211i 0 1.61803 0
1249.7 0 1.90211i 0 0 0 − 1.17557i 0 −0.618034 0
1249.8 0 1.90211i 0 0 0 − 1.17557i 0 −0.618034 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.8
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
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4000.2.c.d 8
4.b odd 2 1 inner 4000.2.c.d 8
5.b even 2 1 inner 4000.2.c.d 8
5.c odd 4 1 4000.2.a.e ✓ 4
5.c odd 4 1 4000.2.a.f yes 4
20.d odd 2 1 inner 4000.2.c.d 8
20.e even 4 1 4000.2.a.e ✓ 4
20.e even 4 1 4000.2.a.f yes 4
40.i odd 4 1 8000.2.a.bg 4
40.i odd 4 1 8000.2.a.bh 4
40.k even 4 1 8000.2.a.bg 4
40.k even 4 1 8000.2.a.bh 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4000.2.a.e ✓ 4 5.c odd 4 1
4000.2.a.e ✓ 4 20.e even 4 1
4000.2.a.f yes 4 5.c odd 4 1
4000.2.a.f yes 4 20.e even 4 1
4000.2.c.d 8 1.a even 1 1 trivial
4000.2.c.d 8 4.b odd 2 1 inner
4000.2.c.d 8 5.b even 2 1 inner
4000.2.c.d 8 20.d odd 2 1 inner
8000.2.a.bg 4 40.i odd 4 1
8000.2.a.bg 4 40.k even 4 1
8000.2.a.bh 4 40.i odd 4 1
8000.2.a.bh 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 \(S_{2}^{\mathrm{new}}(4000, [\chi])\):

\( T_{3}^{4} + 5T_{3}^{2} + 5 \) Copy content Toggle raw display
\( T_{7}^{4} + 5T_{7}^{2} + 5 \) Copy content Toggle raw display
\( T_{11}^{4} - 20T_{11}^{2} + 80 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 5 T^{2} + 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 5 T^{2} + 5)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 20 T^{2} + 80)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 12 T^{2} + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 40 T^{2} + 80)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 5 T^{2} + 5)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 5 T + 5)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 100 T^{2} + 80)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 108 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 5 T - 5)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 125 T^{2} + 125)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 145 T^{2} + 4805)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 20 T^{2} + 80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 5 T + 5)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 40 T^{2} + 80)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 200 T^{2} + 9680)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 172 T^{2} + 5776)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 340 T^{2} + 28880)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 25 T^{2} + 125)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 5 T - 25)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 268 T^{2} + 13456)^{2} \) Copy content Toggle raw display
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