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Properties

Label 2100.2.d.l
Level $2100$
Weight $2$
Character orbit 2100.d
Analytic conductor $16.769$
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] = [2100,2,Mod(1301,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("2100.1301");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.d (of order \(2\), degree \(1\), 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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.14786560000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 6x^{6} + 22x^{4} - 54x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{3} + \beta_1) q^{3} + ( - \beta_{5} + \beta_{3}) q^{7} + (\beta_{5} - \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{3} + \beta_1) q^{3} + ( - \beta_{5} + \beta_{3}) q^{7} + (\beta_{5} - \beta_{2} + 2) q^{9} + (\beta_{5} - \beta_{4} - \beta_{2} + 1) q^{11} + (2 \beta_{6} + \beta_{3}) q^{13} + \beta_{7} q^{17} + (\beta_{6} + 2 \beta_{3}) q^{19} + ( - \beta_{7} + \beta_{4} + \beta_{3} + \cdots - 1) q^{21}+ \cdots + (2 \beta_{5} - 3 \beta_{4} - 2 \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{9} - 4 q^{21} + 12 q^{39} + 40 q^{43} + 24 q^{49} - 8 q^{51} - 20 q^{63} - 24 q^{79} - 16 q^{81} - 32 q^{91} + 20 q^{93} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 6x^{6} + 22x^{4} - 54x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 3\nu^{5} - 5\nu^{3} + 9\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 3\nu^{4} - 5\nu^{2} + 27 ) / 18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 6\nu^{4} - 13\nu^{2} + 27 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 3\nu^{5} + 13\nu^{3} - 15\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 21\nu^{5} - 29\nu^{3} + 45\nu ) / 54 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 2\beta_{6} - \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 2\beta_{4} + 2\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{7} + 3\beta_{6} + 6\beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -3\beta_{5} + 12\beta_{4} - \beta_{2} - 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -4\beta_{7} + \beta_{6} + 31\beta_{3} - \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1301.1
−1.67601 + 0.437016i
−1.67601 − 0.437016i
−1.30038 + 1.14412i
−1.30038 − 1.14412i
1.30038 + 1.14412i
1.30038 − 1.14412i
1.67601 + 0.437016i
1.67601 − 0.437016i
0 −1.67601 − 0.437016i 0 0 0 −2.23607 + 1.41421i 0 2.61803 + 1.46489i 0
1301.2 0 −1.67601 + 0.437016i 0 0 0 −2.23607 − 1.41421i 0 2.61803 − 1.46489i 0
1301.3 0 −1.30038 − 1.14412i 0 0 0 2.23607 − 1.41421i 0 0.381966 + 2.97558i 0
1301.4 0 −1.30038 + 1.14412i 0 0 0 2.23607 + 1.41421i 0 0.381966 − 2.97558i 0
1301.5 0 1.30038 − 1.14412i 0 0 0 2.23607 − 1.41421i 0 0.381966 − 2.97558i 0
1301.6 0 1.30038 + 1.14412i 0 0 0 2.23607 + 1.41421i 0 0.381966 + 2.97558i 0
1301.7 0 1.67601 − 0.437016i 0 0 0 −2.23607 + 1.41421i 0 2.61803 − 1.46489i 0
1301.8 0 1.67601 + 0.437016i 0 0 0 −2.23607 − 1.41421i 0 2.61803 + 1.46489i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1301.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.d.l yes 8
3.b odd 2 1 inner 2100.2.d.l yes 8
5.b even 2 1 2100.2.d.k ✓ 8
5.c odd 4 2 2100.2.f.i 16
7.b odd 2 1 inner 2100.2.d.l yes 8
15.d odd 2 1 2100.2.d.k ✓ 8
15.e even 4 2 2100.2.f.i 16
21.c even 2 1 inner 2100.2.d.l yes 8
35.c odd 2 1 2100.2.d.k ✓ 8
35.f even 4 2 2100.2.f.i 16
105.g even 2 1 2100.2.d.k ✓ 8
105.k odd 4 2 2100.2.f.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.d.k ✓ 8 5.b even 2 1
2100.2.d.k ✓ 8 15.d odd 2 1
2100.2.d.k ✓ 8 35.c odd 2 1
2100.2.d.k ✓ 8 105.g even 2 1
2100.2.d.l yes 8 1.a even 1 1 trivial
2100.2.d.l yes 8 3.b odd 2 1 inner
2100.2.d.l yes 8 7.b odd 2 1 inner
2100.2.d.l yes 8 21.c even 2 1 inner
2100.2.f.i 16 5.c odd 4 2
2100.2.f.i 16 15.e even 4 2
2100.2.f.i 16 35.f even 4 2
2100.2.f.i 16 105.k odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2100, [\chi])\):

\( T_{11}^{4} + 16T_{11}^{2} + 19 \) Copy content Toggle raw display
\( T_{13}^{4} + 36T_{13}^{2} + 4 \) Copy content Toggle raw display
\( T_{17}^{4} - 22T_{17}^{2} + 76 \) Copy content Toggle raw display
\( T_{37}^{2} - 5 \) Copy content Toggle raw display
\( T_{41}^{4} - 90T_{41}^{2} + 1900 \) Copy content Toggle raw display
\( T_{43}^{2} - 10T_{43} + 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 6 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 6 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 16 T^{2} + 19)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 36 T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 22 T^{2} + 76)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 30 T^{2} + 100)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 80 T^{2} + 475)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 104 T^{2} + 2299)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 30 T^{2} + 100)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 5)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 90 T^{2} + 1900)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 10 T + 5)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 222 T^{2} + 9196)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 180 T^{2} + 7600)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 160 T^{2} + 1900)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 150 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 45)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 120 T^{2} + 475)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 6 T - 11)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 90 T^{2} + 1900)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 240 T^{2} + 1900)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 94 T^{2} + 4)^{2} \) Copy content Toggle raw display
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