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Properties

Label 2100.2.bc.d
Level $2100$
Weight $2$
Character orbit 2100.bc
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(949,2100)]
 
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, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.949");
 
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.bc (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: \(16.7685844245\)
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: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}) q^{3} + (2 \zeta_{12}^{3} + \zeta_{12}) q^{7} + ( - \zeta_{12}^{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12}) q^{3} + (2 \zeta_{12}^{3} + \zeta_{12}) q^{7} + ( - \zeta_{12}^{2} + 1) q^{9} + 4 \zeta_{12}^{2} q^{11} + 7 \zeta_{12}^{3} q^{13} + ( - 6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{17} + ( - 3 \zeta_{12}^{2} + 3) q^{19} + ( - 2 \zeta_{12}^{2} - 1) q^{21} + 2 \zeta_{12} q^{23} + \zeta_{12}^{3} q^{27} + 2 q^{29} - 7 \zeta_{12}^{2} q^{31} - 4 \zeta_{12} q^{33} - 7 \zeta_{12} q^{37} - 7 \zeta_{12}^{2} q^{39} - 8 q^{41} + 5 \zeta_{12}^{3} q^{43} + 10 \zeta_{12} q^{47} + (5 \zeta_{12}^{2} - 8) q^{49} + (6 \zeta_{12}^{2} - 6) q^{51} + (8 \zeta_{12}^{3} - 8 \zeta_{12}) q^{53} + 3 \zeta_{12}^{3} q^{57} + 10 \zeta_{12}^{2} q^{59} + ( - 6 \zeta_{12}^{2} + 6) q^{61} + ( - \zeta_{12}^{3} + 3 \zeta_{12}) q^{63} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{67} - 2 q^{69} + ( - 15 \zeta_{12}^{3} + 15 \zeta_{12}) q^{73} + (12 \zeta_{12}^{3} - 8 \zeta_{12}) q^{77} + ( - \zeta_{12}^{2} + 1) q^{79} - \zeta_{12}^{2} q^{81} + 8 \zeta_{12}^{3} q^{83} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{87} + ( - 2 \zeta_{12}^{2} + 2) q^{89} + (7 \zeta_{12}^{2} - 21) q^{91} + 7 \zeta_{12} q^{93} + 10 \zeta_{12}^{3} q^{97} + 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{9} + 8 q^{11} + 6 q^{19} - 8 q^{21} + 8 q^{29} - 14 q^{31} - 14 q^{39} - 32 q^{41} - 22 q^{49} - 12 q^{51} + 20 q^{59} + 12 q^{61} - 8 q^{69} + 2 q^{79} - 2 q^{81} + 4 q^{89} - 70 q^{91} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
949.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
0 −0.866025 − 0.500000i 0 0 0 0.866025 − 2.50000i 0 0.500000 + 0.866025i 0
949.2 0 0.866025 + 0.500000i 0 0 0 −0.866025 + 2.50000i 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 0 0 0.866025 + 2.50000i 0 0.500000 − 0.866025i 0
1549.2 0 0.866025 − 0.500000i 0 0 0 −0.866025 − 2.50000i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.bc.d 4
5.b even 2 1 inner 2100.2.bc.d 4
5.c odd 4 1 420.2.q.b ✓ 2
5.c odd 4 1 2100.2.q.d 2
7.c even 3 1 inner 2100.2.bc.d 4
15.e even 4 1 1260.2.s.a 2
20.e even 4 1 1680.2.bg.j 2
35.f even 4 1 2940.2.q.c 2
35.j even 6 1 inner 2100.2.bc.d 4
35.k even 12 1 2940.2.a.k 1
35.k even 12 1 2940.2.q.c 2
35.l odd 12 1 420.2.q.b ✓ 2
35.l odd 12 1 2100.2.q.d 2
35.l odd 12 1 2940.2.a.b 1
105.w odd 12 1 8820.2.a.l 1
105.x even 12 1 1260.2.s.a 2
105.x even 12 1 8820.2.a.bb 1
140.w even 12 1 1680.2.bg.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.b ✓ 2 5.c odd 4 1
420.2.q.b ✓ 2 35.l odd 12 1
1260.2.s.a 2 15.e even 4 1
1260.2.s.a 2 105.x even 12 1
1680.2.bg.j 2 20.e even 4 1
1680.2.bg.j 2 140.w even 12 1
2100.2.q.d 2 5.c odd 4 1
2100.2.q.d 2 35.l odd 12 1
2100.2.bc.d 4 1.a even 1 1 trivial
2100.2.bc.d 4 5.b even 2 1 inner
2100.2.bc.d 4 7.c even 3 1 inner
2100.2.bc.d 4 35.j even 6 1 inner
2940.2.a.b 1 35.l odd 12 1
2940.2.a.k 1 35.k even 12 1
2940.2.q.c 2 35.f even 4 1
2940.2.q.c 2 35.k even 12 1
8820.2.a.l 1 105.w odd 12 1
8820.2.a.bb 1 105.x even 12 1

Hecke kernels

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

\( T_{11}^{2} - 4T_{11} + 16 \) Copy content Toggle raw display
\( T_{13}^{2} + 49 \) 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
$7$ \( T^{4} + 11T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T - 2)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 49T^{2} + 2401 \) Copy content Toggle raw display
$41$ \( (T + 8)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$53$ \( T^{4} - 64T^{2} + 4096 \) Copy content Toggle raw display
$59$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 225 T^{2} + 50625 \) Copy content Toggle raw display
$79$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
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