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Properties

Label 2100.1.ba.a
Level $2100$
Weight $1$
Character orbit 2100.ba
Analytic conductor $1.048$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -20
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2100,1,Mod(551,2100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2100, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 0, 5])) N = Newforms(chi, 1, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2100.551"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2100.ba (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.04803652653\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.726062400.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} - q^{3} - \zeta_{6} q^{4} - \zeta_{6}^{2} q^{6} - \zeta_{6} q^{7} + q^{8} + q^{9} + \zeta_{6} q^{12} + q^{14} + \zeta_{6}^{2} q^{16} + \zeta_{6}^{2} q^{18} + \zeta_{6} q^{21} + \cdots - \zeta_{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - 2 q^{3} - q^{4} + q^{6} - q^{7} + 2 q^{8} + 2 q^{9} + q^{12} + 2 q^{14} - q^{16} - q^{18} + q^{21} - q^{23} - 2 q^{24} - 2 q^{27} - q^{28} - q^{32} - q^{36} - 2 q^{41} - 2 q^{42}+ \cdots - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-\zeta_{6}^{2}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
551.1
0.500000 + 0.866025i
0.500000 − 0.866025i
−0.500000 + 0.866025i −1.00000 −0.500000 − 0.866025i 0 0.500000 − 0.866025i −0.500000 − 0.866025i 1.00000 1.00000 0
1151.1 −0.500000 − 0.866025i −1.00000 −0.500000 + 0.866025i 0 0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 1.00000 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
84.j odd 6 1 inner
105.p even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.1.ba.a 2
3.b odd 2 1 2100.1.ba.c 2
4.b odd 2 1 2100.1.ba.d 2
5.b even 2 1 2100.1.ba.d 2
5.c odd 4 2 420.1.be.b yes 4
7.d odd 6 1 2100.1.ba.b 2
12.b even 2 1 2100.1.ba.b 2
15.d odd 2 1 2100.1.ba.b 2
15.e even 4 2 420.1.be.a ✓ 4
20.d odd 2 1 CM 2100.1.ba.a 2
20.e even 4 2 420.1.be.b yes 4
21.g even 6 1 2100.1.ba.d 2
28.f even 6 1 2100.1.ba.c 2
35.f even 4 2 2940.1.be.a 4
35.i odd 6 1 2100.1.ba.c 2
35.k even 12 2 420.1.be.a ✓ 4
35.k even 12 2 2940.1.o.b 4
35.l odd 12 2 2940.1.o.a 4
35.l odd 12 2 2940.1.be.d 4
60.h even 2 1 2100.1.ba.c 2
60.l odd 4 2 420.1.be.a ✓ 4
84.j odd 6 1 inner 2100.1.ba.a 2
105.k odd 4 2 2940.1.be.d 4
105.p even 6 1 inner 2100.1.ba.a 2
105.w odd 12 2 420.1.be.b yes 4
105.w odd 12 2 2940.1.o.a 4
105.x even 12 2 2940.1.o.b 4
105.x even 12 2 2940.1.be.a 4
140.j odd 4 2 2940.1.be.a 4
140.s even 6 1 2100.1.ba.b 2
140.w even 12 2 2940.1.o.a 4
140.w even 12 2 2940.1.be.d 4
140.x odd 12 2 420.1.be.a ✓ 4
140.x odd 12 2 2940.1.o.b 4
420.w even 4 2 2940.1.be.d 4
420.be odd 6 1 2100.1.ba.d 2
420.bp odd 12 2 2940.1.o.b 4
420.bp odd 12 2 2940.1.be.a 4
420.br even 12 2 420.1.be.b yes 4
420.br even 12 2 2940.1.o.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.1.be.a ✓ 4 15.e even 4 2
420.1.be.a ✓ 4 35.k even 12 2
420.1.be.a ✓ 4 60.l odd 4 2
420.1.be.a ✓ 4 140.x odd 12 2
420.1.be.b yes 4 5.c odd 4 2
420.1.be.b yes 4 20.e even 4 2
420.1.be.b yes 4 105.w odd 12 2
420.1.be.b yes 4 420.br even 12 2
2100.1.ba.a 2 1.a even 1 1 trivial
2100.1.ba.a 2 20.d odd 2 1 CM
2100.1.ba.a 2 84.j odd 6 1 inner
2100.1.ba.a 2 105.p even 6 1 inner
2100.1.ba.b 2 7.d odd 6 1
2100.1.ba.b 2 12.b even 2 1
2100.1.ba.b 2 15.d odd 2 1
2100.1.ba.b 2 140.s even 6 1
2100.1.ba.c 2 3.b odd 2 1
2100.1.ba.c 2 28.f even 6 1
2100.1.ba.c 2 35.i odd 6 1
2100.1.ba.c 2 60.h even 2 1
2100.1.ba.d 2 4.b odd 2 1
2100.1.ba.d 2 5.b even 2 1
2100.1.ba.d 2 21.g even 6 1
2100.1.ba.d 2 420.be odd 6 1
2940.1.o.a 4 35.l odd 12 2
2940.1.o.a 4 105.w odd 12 2
2940.1.o.a 4 140.w even 12 2
2940.1.o.a 4 420.br even 12 2
2940.1.o.b 4 35.k even 12 2
2940.1.o.b 4 105.x even 12 2
2940.1.o.b 4 140.x odd 12 2
2940.1.o.b 4 420.bp odd 12 2
2940.1.be.a 4 35.f even 4 2
2940.1.be.a 4 105.x even 12 2
2940.1.be.a 4 140.j odd 4 2
2940.1.be.a 4 420.bp odd 12 2
2940.1.be.d 4 35.l odd 12 2
2940.1.be.d 4 105.k odd 4 2
2940.1.be.d 4 140.w even 12 2
2940.1.be.d 4 420.w even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2100, [\chi])\):

\( T_{23}^{2} + T_{23} + 1 \) Copy content Toggle raw display
\( T_{41} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} + 3 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( (T + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 3 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$67$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 3 \) Copy content Toggle raw display
$89$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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