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Properties

Label 6000.2.f.p
Level $6000$
Weight $2$
Character orbit 6000.f
Analytic conductor $47.910$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6000,2,Mod(1249,6000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6000, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6000.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6000 = 2^{4} \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6000.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: \(47.9102412128\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.566105760000.21
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 45x^{6} + 728x^{4} + 4965x^{2} + 11881 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3000)
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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + ( - \beta_{7} + \beta_1) q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + ( - \beta_{7} + \beta_1) q^{7} - q^{9} + (\beta_{6} + \beta_{2} - 1) q^{11} + (\beta_{7} + \beta_{5}) q^{13} - \beta_{4} q^{17} + ( - \beta_{6} - \beta_{3} + 2 \beta_{2} - 1) q^{19} + (\beta_{6} + 1) q^{21} + (2 \beta_{7} + \beta_{4} - \beta_1) q^{23} - \beta_{7} q^{27} + ( - \beta_{3} - \beta_{2} - 3) q^{29} + ( - \beta_{6} - \beta_{3} + 3 \beta_{2} - 3) q^{31} + (\beta_{4} - \beta_1) q^{33} + (4 \beta_{7} + 2 \beta_{5} + \cdots - \beta_1) q^{37}+ \cdots + ( - \beta_{6} - \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{9} - 6 q^{11} + 6 q^{21} - 30 q^{29} - 12 q^{31} - 6 q^{39} + 26 q^{41} - 38 q^{49} - 4 q^{51} + 16 q^{59} + 26 q^{61} - 10 q^{69} + 18 q^{71} + 42 q^{79} + 8 q^{81} - 24 q^{89} - 34 q^{91} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 45x^{6} + 728x^{4} + 4965x^{2} + 11881 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 24\nu^{4} + 144\nu^{2} + 181 ) / 80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{2} - 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9\nu^{7} + 296\nu^{5} + 3936\nu^{3} + 20269\nu ) / 8720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -19\nu^{7} + 344\nu^{5} + 14944\nu^{3} + 87041\nu ) / 8720 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 32\nu^{4} + 320\nu^{2} + 981 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -9\nu^{7} - 296\nu^{5} - 3064\nu^{3} - 9805\nu ) / 872 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 10\beta_{4} - 12\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 22\beta_{3} - 10\beta_{2} + 142 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -24\beta_{7} + 9\beta_{5} - 221\beta_{4} + 154\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -24\beta_{6} - 384\beta_{3} + 320\beta_{2} - 2005 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 352\beta_{7} - 296\beta_{5} + 3864\beta_{4} - 2069\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6000\mathbb{Z}\right)^\times\).

\(n\) \(751\) \(4001\) \(4501\) \(5377\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
− 3.97706i
− 3.11349i
2.35902i
3.73152i
− 3.73152i
− 2.35902i
3.11349i
3.97706i
0 − 1.00000i 0 0 0 − 2.97706i 0 −1.00000 0
1249.2 0 − 1.00000i 0 0 0 − 2.11349i 0 −1.00000 0
1249.3 0 − 1.00000i 0 0 0 3.35902i 0 −1.00000 0
1249.4 0 − 1.00000i 0 0 0 4.73152i 0 −1.00000 0
1249.5 0 1.00000i 0 0 0 − 4.73152i 0 −1.00000 0
1249.6 0 1.00000i 0 0 0 − 3.35902i 0 −1.00000 0
1249.7 0 1.00000i 0 0 0 2.11349i 0 −1.00000 0
1249.8 0 1.00000i 0 0 0 2.97706i 0 −1.00000 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
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 6000.2.f.p 8
4.b odd 2 1 3000.2.f.h 8
5.b even 2 1 inner 6000.2.f.p 8
5.c odd 4 1 6000.2.a.bc 4
5.c odd 4 1 6000.2.a.bl 4
20.d odd 2 1 3000.2.f.h 8
20.e even 4 1 3000.2.a.j ✓ 4
20.e even 4 1 3000.2.a.o yes 4
60.l odd 4 1 9000.2.a.t 4
60.l odd 4 1 9000.2.a.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3000.2.a.j ✓ 4 20.e even 4 1
3000.2.a.o yes 4 20.e even 4 1
3000.2.f.h 8 4.b odd 2 1
3000.2.f.h 8 20.d odd 2 1
6000.2.a.bc 4 5.c odd 4 1
6000.2.a.bl 4 5.c odd 4 1
6000.2.f.p 8 1.a even 1 1 trivial
6000.2.f.p 8 5.b even 2 1 inner
9000.2.a.t 4 60.l odd 4 1
9000.2.a.y 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6000, [\chi])\):

\( T_{7}^{8} + 47T_{7}^{6} + 741T_{7}^{4} + 4700T_{7}^{2} + 10000 \) Copy content Toggle raw display
\( T_{11}^{4} + 3T_{11}^{3} - 24T_{11}^{2} - 45T_{11} + 45 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 47 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$11$ \( (T^{4} + 3 T^{3} - 24 T^{2} + \cdots + 45)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 65 T^{6} + \cdots + 11881 \) Copy content Toggle raw display
$17$ \( (T^{4} + 3 T^{2} + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 37 T^{2} + \cdots + 76)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 61 T^{6} + \cdots + 961 \) Copy content Toggle raw display
$29$ \( (T^{4} + 15 T^{3} + \cdots - 549)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 6 T^{3} + \cdots - 305)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 249 T^{6} + \cdots + 4669921 \) Copy content Toggle raw display
$41$ \( (T^{4} - 13 T^{3} + \cdots - 500)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 47 T^{2} + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 193 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$53$ \( T^{8} + 123 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$59$ \( (T^{4} - 8 T^{3} + \cdots - 4805)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 13 T^{3} + \cdots - 4196)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 539 T^{6} + \cdots + 149426176 \) Copy content Toggle raw display
$71$ \( (T^{4} - 9 T^{3} - T^{2} + \cdots - 80)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 207 T^{6} + \cdots + 2310400 \) Copy content Toggle raw display
$79$ \( (T^{4} - 21 T^{3} + \cdots - 36)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 435 T^{6} + \cdots + 57214096 \) Copy content Toggle raw display
$89$ \( (T^{4} + 12 T^{3} + \cdots - 36)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 151 T^{6} + \cdots + 5776 \) Copy content Toggle raw display
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