Properties

Label 2100.2.bc.g
Level $2100$
Weight $2$
Character orbit 2100.bc
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(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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: yes
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{6} - \beta_{5} + \beta_1) q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{6} - \beta_{5} + \beta_1) q^{7} + \beta_{2} q^{9} + ( - 2 \beta_{7} + \beta_{4}) q^{11} + 5 \beta_{3} q^{13} + ( - \beta_{6} - \beta_{5}) q^{17} + (\beta_{7} + \beta_{4} - \beta_{2}) q^{19} + (\beta_{7} - \beta_{4} - \beta_{2}) q^{21} + ( - 6 \beta_{3} + 6 \beta_1) q^{23} - \beta_{3} q^{27} + ( - 2 \beta_{7} + 4 \beta_{4}) q^{29} + ( - 4 \beta_{2} + 4) q^{31} + ( - 2 \beta_{6} + \beta_{5}) q^{33} + (5 \beta_{3} - 5 \beta_1) q^{37} + ( - 5 \beta_{2} + 5) q^{39} + ( - \beta_{7} + 2 \beta_{4} - 6) q^{41} + ( - 2 \beta_{6} + 4 \beta_{5} - 4 \beta_{3}) q^{43} + (2 \beta_{6} - \beta_{5} + 6 \beta_{3} - 6 \beta_1) q^{47} + ( - 2 \beta_{7} + 2 \beta_{4} - 5 \beta_{2}) q^{49} + (\beta_{7} + \beta_{4}) q^{51} + (\beta_{6} + \beta_{5}) q^{53} + (\beta_{6} - 2 \beta_{5} + \beta_{3}) q^{57} + (2 \beta_{7} - \beta_{4} - 6 \beta_{2} + 6) q^{59} + \beta_{2} q^{61} + (\beta_{6} + \beta_{3}) q^{63} + ( - 3 \beta_{6} - 3 \beta_{5} - \beta_1) q^{67} - 6 q^{69} + ( - 2 \beta_{7} + 4 \beta_{4}) q^{71} + ( - 2 \beta_{6} - 2 \beta_{5} - 5 \beta_1) q^{73} + (2 \beta_{6} - \beta_{5} - 6 \beta_{3} - 6 \beta_1) q^{77} + ( - \beta_{7} - \beta_{4} + 5 \beta_{2}) q^{79} + (\beta_{2} - 1) q^{81} + ( - \beta_{6} + 2 \beta_{5} + 6 \beta_{3}) q^{83} + ( - 2 \beta_{6} - 2 \beta_{5}) q^{87} + (\beta_{7} + \beta_{4} + 12 \beta_{2}) q^{89} + (5 \beta_{4} + 5 \beta_{2} - 5) q^{91} + (4 \beta_{3} - 4 \beta_1) q^{93} + (4 \beta_{6} - 8 \beta_{5} + \beta_{3}) q^{97} + ( - \beta_{7} + 2 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} - 4 q^{19} - 4 q^{21} + 16 q^{31} + 20 q^{39} - 48 q^{41} - 20 q^{49} + 24 q^{59} + 4 q^{61} - 48 q^{69} + 20 q^{79} - 4 q^{81} + 48 q^{89} - 20 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} + 2\beta_{5} + 2\beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} ) / 6 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} ) / 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-\beta_{2}\)

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
−0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
0.258819 + 0.965926i
0 −0.866025 0.500000i 0 0 0 −0.358719 + 2.62132i 0 0.500000 + 0.866025i 0
949.2 0 −0.866025 0.500000i 0 0 0 2.09077 1.62132i 0 0.500000 + 0.866025i 0
949.3 0 0.866025 + 0.500000i 0 0 0 −2.09077 + 1.62132i 0 0.500000 + 0.866025i 0
949.4 0 0.866025 + 0.500000i 0 0 0 0.358719 2.62132i 0 0.500000 + 0.866025i 0
1549.1 0 −0.866025 + 0.500000i 0 0 0 −0.358719 2.62132i 0 0.500000 0.866025i 0
1549.2 0 −0.866025 + 0.500000i 0 0 0 2.09077 + 1.62132i 0 0.500000 0.866025i 0
1549.3 0 0.866025 0.500000i 0 0 0 −2.09077 1.62132i 0 0.500000 0.866025i 0
1549.4 0 0.866025 0.500000i 0 0 0 0.358719 + 2.62132i 0 0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 949.4
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.g 8
5.b even 2 1 inner 2100.2.bc.g 8
5.c odd 4 1 2100.2.q.f 4
5.c odd 4 1 2100.2.q.j yes 4
7.c even 3 1 inner 2100.2.bc.g 8
35.j even 6 1 inner 2100.2.bc.g 8
35.l odd 12 1 2100.2.q.f 4
35.l odd 12 1 2100.2.q.j yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.q.f 4 5.c odd 4 1
2100.2.q.f 4 35.l odd 12 1
2100.2.q.j yes 4 5.c odd 4 1
2100.2.q.j yes 4 35.l odd 12 1
2100.2.bc.g 8 1.a even 1 1 trivial
2100.2.bc.g 8 5.b even 2 1 inner
2100.2.bc.g 8 7.c even 3 1 inner
2100.2.bc.g 8 35.j even 6 1 inner

Hecke kernels

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

\( T_{11}^{4} + 18T_{11}^{2} + 324 \) Copy content Toggle raw display
\( T_{13}^{2} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 10 T^{6} + 51 T^{4} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{4} + 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 25)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 2 T^{3} + 21 T^{2} - 34 T + 289)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 36 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 72)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T + 16)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 25 T^{2} + 625)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12 T + 18)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 176 T^{2} + 3136)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 108 T^{6} + 11340 T^{4} + \cdots + 104976 \) Copy content Toggle raw display
$53$ \( (T^{4} - 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 12 T^{3} + 126 T^{2} - 216 T + 324)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 326 T^{6} + \cdots + 671898241 \) Copy content Toggle raw display
$71$ \( (T^{2} - 72)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} - 194 T^{6} + 35427 T^{4} + \cdots + 4879681 \) Copy content Toggle raw display
$79$ \( (T^{4} - 10 T^{3} + 93 T^{2} - 70 T + 49)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 108 T^{2} + 324)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 24 T^{3} + 450 T^{2} + \cdots + 15876)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 578 T^{2} + 82369)^{2} \) Copy content Toggle raw display
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