Properties

Label 8100.2.d.l
Level $8100$
Weight $2$
Character orbit 8100.d
Analytic conductor $64.679$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8100,2,Mod(649,8100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8100, 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("8100.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8100.d (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: \(64.6788256372\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1620)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{7} + \beta_1 q^{11} - 2 \beta_{3} q^{13} + (\beta_{3} - \beta_{2}) q^{17} + ( - 2 \beta_1 + 1) q^{19} + (2 \beta_{3} + \beta_{2}) q^{23} + (\beta_1 - 6) q^{29} + ( - 4 \beta_1 - 1) q^{31} + ( - 3 \beta_{3} - \beta_{2}) q^{37} + ( - 3 \beta_1 + 6) q^{41} + (\beta_{3} + 3 \beta_{2}) q^{43} + ( - \beta_{3} - 2 \beta_{2}) q^{47} + (2 \beta_1 + 3) q^{49} + (\beta_{3} + 5 \beta_{2}) q^{53} + (\beta_1 - 6) q^{59} - 4 q^{61} + (6 \beta_{3} + 5 \beta_{2}) q^{67} + ( - 3 \beta_1 + 6) q^{71} + (3 \beta_{3} + 4 \beta_{2}) q^{73} + (\beta_{3} - \beta_{2}) q^{77} + ( - 6 \beta_1 + 4) q^{79} + ( - 7 \beta_{3} - 2 \beta_{2}) q^{83} - 3 \beta_1 q^{89} + (4 \beta_1 - 8) q^{91} + (\beta_{3} + \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{19} - 24 q^{29} - 4 q^{31} + 24 q^{41} + 12 q^{49} - 24 q^{59} - 16 q^{61} + 24 q^{71} + 16 q^{79} - 32 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( 2\beta_{3} + \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times\).

\(n\) \(4051\) \(6401\) \(7777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0 0 0 0 2.73205i 0 0 0
649.2 0 0 0 0 0 0.732051i 0 0 0
649.3 0 0 0 0 0 0.732051i 0 0 0
649.4 0 0 0 0 0 2.73205i 0 0 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 8100.2.d.l 4
3.b odd 2 1 8100.2.d.m 4
5.b even 2 1 inner 8100.2.d.l 4
5.c odd 4 1 1620.2.a.h yes 2
5.c odd 4 1 8100.2.a.s 2
15.d odd 2 1 8100.2.d.m 4
15.e even 4 1 1620.2.a.g 2
15.e even 4 1 8100.2.a.t 2
20.e even 4 1 6480.2.a.bp 2
45.k odd 12 2 1620.2.i.m 4
45.l even 12 2 1620.2.i.n 4
60.l odd 4 1 6480.2.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1620.2.a.g 2 15.e even 4 1
1620.2.a.h yes 2 5.c odd 4 1
1620.2.i.m 4 45.k odd 12 2
1620.2.i.n 4 45.l even 12 2
6480.2.a.bh 2 60.l odd 4 1
6480.2.a.bp 2 20.e even 4 1
8100.2.a.s 2 5.c odd 4 1
8100.2.a.t 2 15.e even 4 1
8100.2.d.l 4 1.a even 1 1 trivial
8100.2.d.l 4 5.b even 2 1 inner
8100.2.d.m 4 3.b odd 2 1
8100.2.d.m 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(8100, [\chi])\):

\( T_{7}^{4} + 8T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 3 \) Copy content Toggle raw display
\( T_{29}^{2} + 12T_{29} + 33 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 32T^{2} + 64 \) Copy content Toggle raw display
$17$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 11)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 12 T + 33)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T - 47)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 56T^{2} + 676 \) Copy content Toggle raw display
$41$ \( (T^{2} - 12 T + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$47$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$53$ \( T^{4} + 168T^{2} + 6084 \) Copy content Toggle raw display
$59$ \( (T^{2} + 12 T + 33)^{2} \) Copy content Toggle raw display
$61$ \( (T + 4)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 248T^{2} + 8464 \) Copy content Toggle raw display
$71$ \( (T^{2} - 12 T + 9)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 104T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T^{2} - 8 T - 92)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 312 T^{2} + 19044 \) Copy content Toggle raw display
$89$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
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