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Properties

Label 2100.2.f.d
Level $2100$
Weight $2$
Character orbit 2100.f
Analytic conductor $16.769$
Analytic rank $0$
Dimension $4$
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(1049,2100)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.1049");
 
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.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: \(16.7685844245\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \zeta_{12}^{3} - \zeta_{12}) q^{3} + ( - \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} - 3 \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{12}^{3} - \zeta_{12}) q^{3} + ( - \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} - 3 \zeta_{12}^{2} q^{9} + ( - 6 \zeta_{12}^{2} + 3) q^{11} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{13} + 3 \zeta_{12}^{3} q^{17} + ( - 4 \zeta_{12}^{2} + 2) q^{19} + ( - \zeta_{12}^{2} + 5) q^{21} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} + (6 \zeta_{12}^{2} - 3) q^{29} + (12 \zeta_{12}^{2} - 6) q^{31} + 9 \zeta_{12} q^{33} + 8 \zeta_{12}^{3} q^{37} + ( - 3 \zeta_{12}^{2} + 3) q^{39} - 6 q^{41} + 10 \zeta_{12}^{3} q^{43} + 3 \zeta_{12}^{3} q^{47} + (8 \zeta_{12}^{2} - 5) q^{49} + ( - 3 \zeta_{12}^{2} - 3) q^{51} + (6 \zeta_{12}^{3} - 12 \zeta_{12}) q^{53} + 6 \zeta_{12} q^{57} + 6 q^{59} + ( - 8 \zeta_{12}^{2} + 4) q^{61} + (9 \zeta_{12}^{3} - 3 \zeta_{12}) q^{63} - 2 \zeta_{12}^{3} q^{67} + ( - 12 \zeta_{12}^{2} + 6) q^{71} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{73} + (15 \zeta_{12}^{3} - 12 \zeta_{12}) q^{77} + 13 q^{79} + (9 \zeta_{12}^{2} - 9) q^{81} + 12 \zeta_{12}^{3} q^{83} - 9 \zeta_{12} q^{87} + (4 \zeta_{12}^{2} + 1) q^{91} - 18 \zeta_{12} q^{93} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{97} + (9 \zeta_{12}^{2} - 18) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{9} + 18 q^{21} + 6 q^{39} - 24 q^{41} - 4 q^{49} - 18 q^{51} + 24 q^{59} + 52 q^{79} - 18 q^{81} + 12 q^{91} - 54 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1049.1
0.866025 − 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
−0.866025 + 0.500000i
0 −0.866025 − 1.50000i 0 0 0 −1.73205 + 2.00000i 0 −1.50000 + 2.59808i 0
1049.2 0 −0.866025 + 1.50000i 0 0 0 −1.73205 − 2.00000i 0 −1.50000 − 2.59808i 0
1049.3 0 0.866025 − 1.50000i 0 0 0 1.73205 + 2.00000i 0 −1.50000 − 2.59808i 0
1049.4 0 0.866025 + 1.50000i 0 0 0 1.73205 − 2.00000i 0 −1.50000 + 2.59808i 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
21.c even 2 1 inner
105.g 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.f.d 4
3.b odd 2 1 2100.2.f.c 4
5.b even 2 1 inner 2100.2.f.d 4
5.c odd 4 1 420.2.d.b yes 2
5.c odd 4 1 2100.2.d.a 2
7.b odd 2 1 2100.2.f.c 4
15.d odd 2 1 2100.2.f.c 4
15.e even 4 1 420.2.d.a ✓ 2
15.e even 4 1 2100.2.d.e 2
20.e even 4 1 1680.2.f.a 2
21.c even 2 1 inner 2100.2.f.d 4
35.c odd 2 1 2100.2.f.c 4
35.f even 4 1 420.2.d.a ✓ 2
35.f even 4 1 2100.2.d.e 2
60.l odd 4 1 1680.2.f.d 2
105.g even 2 1 inner 2100.2.f.d 4
105.k odd 4 1 420.2.d.b yes 2
105.k odd 4 1 2100.2.d.a 2
140.j odd 4 1 1680.2.f.d 2
420.w even 4 1 1680.2.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.d.a ✓ 2 15.e even 4 1
420.2.d.a ✓ 2 35.f even 4 1
420.2.d.b yes 2 5.c odd 4 1
420.2.d.b yes 2 105.k odd 4 1
1680.2.f.a 2 20.e even 4 1
1680.2.f.a 2 420.w even 4 1
1680.2.f.d 2 60.l odd 4 1
1680.2.f.d 2 140.j odd 4 1
2100.2.d.a 2 5.c odd 4 1
2100.2.d.a 2 105.k odd 4 1
2100.2.d.e 2 15.e even 4 1
2100.2.d.e 2 35.f even 4 1
2100.2.f.c 4 3.b odd 2 1
2100.2.f.c 4 7.b odd 2 1
2100.2.f.c 4 15.d odd 2 1
2100.2.f.c 4 35.c odd 2 1
2100.2.f.d 4 1.a even 1 1 trivial
2100.2.f.d 4 5.b even 2 1 inner
2100.2.f.d 4 21.c even 2 1 inner
2100.2.f.d 4 105.g even 2 1 inner

Hecke kernels

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

\( T_{11}^{2} + 27 \) Copy content Toggle raw display
\( T_{13}^{2} - 3 \) Copy content Toggle raw display
\( T_{23} \) Copy content Toggle raw display
\( T_{41} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$41$ \( (T + 6)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$59$ \( (T - 6)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 48)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$79$ \( (T - 13)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
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