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Properties

Label 2100.1.bh.b
Level $2100$
Weight $1$
Character orbit 2100.bh
Analytic conductor $1.048$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -3
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,1,Mod(149,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.149");
 
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.bh (of order \(6\), degree \(2\), 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: \(1.04803652653\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.14700.1

$q$-expansion

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

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

\(f(q)\) \(=\) \( q - \zeta_{12} q^{3} - \zeta_{12}^{3} q^{7} + \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12} q^{3} - \zeta_{12}^{3} q^{7} + \zeta_{12}^{2} q^{9} + \zeta_{12}^{3} q^{13} + \zeta_{12}^{2} q^{19} + \zeta_{12}^{4} q^{21} - \zeta_{12}^{3} q^{27} - \zeta_{12}^{4} q^{31} - \zeta_{12}^{5} q^{37} - 2 \zeta_{12}^{4} q^{39} - \zeta_{12}^{3} q^{43} - q^{49} - 2 \zeta_{12}^{3} q^{57} + \zeta_{12}^{2} q^{61} - \zeta_{12}^{5} q^{63} + \zeta_{12} q^{67} + \zeta_{12} q^{73} - \zeta_{12}^{2} q^{79} + \zeta_{12}^{4} q^{81} + 2 q^{91} + \zeta_{12}^{5} q^{93} + \zeta_{12}^{3} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{9} + 4 q^{19} - 2 q^{21} + 2 q^{31} + 4 q^{39} - 4 q^{49} + 2 q^{61} - 2 q^{79} - 2 q^{81} + 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-\zeta_{12}^{2}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
0 −0.866025 + 0.500000i 0 0 0 1.00000i 0 0.500000 − 0.866025i 0
149.2 0 0.866025 − 0.500000i 0 0 0 − 1.00000i 0 0.500000 − 0.866025i 0
1649.1 0 −0.866025 − 0.500000i 0 0 0 − 1.00000i 0 0.500000 + 0.866025i 0
1649.2 0 0.866025 + 0.500000i 0 0 0 1.00000i 0 0.500000 + 0.866025i 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
5.b even 2 1 inner
7.c even 3 1 inner
15.d odd 2 1 inner
21.h odd 6 1 inner
35.j even 6 1 inner
105.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.1.bh.b 4
3.b odd 2 1 CM 2100.1.bh.b 4
5.b even 2 1 inner 2100.1.bh.b 4
5.c odd 4 1 2100.1.bn.a ✓ 2
5.c odd 4 1 2100.1.bn.b yes 2
7.c even 3 1 inner 2100.1.bh.b 4
15.d odd 2 1 inner 2100.1.bh.b 4
15.e even 4 1 2100.1.bn.a ✓ 2
15.e even 4 1 2100.1.bn.b yes 2
21.h odd 6 1 inner 2100.1.bh.b 4
35.j even 6 1 inner 2100.1.bh.b 4
35.l odd 12 1 2100.1.bn.a ✓ 2
35.l odd 12 1 2100.1.bn.b yes 2
105.o odd 6 1 inner 2100.1.bh.b 4
105.x even 12 1 2100.1.bn.a ✓ 2
105.x even 12 1 2100.1.bn.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.1.bh.b 4 1.a even 1 1 trivial
2100.1.bh.b 4 3.b odd 2 1 CM
2100.1.bh.b 4 5.b even 2 1 inner
2100.1.bh.b 4 7.c even 3 1 inner
2100.1.bh.b 4 15.d odd 2 1 inner
2100.1.bh.b 4 21.h odd 6 1 inner
2100.1.bh.b 4 35.j even 6 1 inner
2100.1.bh.b 4 105.o odd 6 1 inner
2100.1.bn.a ✓ 2 5.c odd 4 1
2100.1.bn.a ✓ 2 15.e even 4 1
2100.1.bn.a ✓ 2 35.l odd 12 1
2100.1.bn.a ✓ 2 105.x even 12 1
2100.1.bn.b yes 2 5.c odd 4 1
2100.1.bn.b yes 2 15.e even 4 1
2100.1.bn.b yes 2 35.l odd 12 1
2100.1.bn.b yes 2 105.x even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(2100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$79$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
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