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Properties

Label 2100.2.q.k
Level $2100$
Weight $2$
Character orbit 2100.q
Analytic conductor $16.769$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2100,2,Mod(1201,2100)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2100.1201"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2100, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,2,0,0,0,2,0,-2,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content 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 \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} + ( - \beta_{3} - \beta_1) q^{7} + \beta_1 q^{9} + (\beta_{3} + \beta_{2}) q^{11} + ( - 2 \beta_{3} + \beta_{2} + 1) q^{13} + (\beta_{3} + \beta_{2}) q^{17} - 7 \beta_1 q^{19} + ( - \beta_{2} + 1) q^{21}+ \cdots + ( - 2 \beta_{3} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 2 q^{7} - 2 q^{9} + 4 q^{13} + 14 q^{19} + 4 q^{21} - 4 q^{27} - 24 q^{29} + 2 q^{31} - 2 q^{37} + 2 q^{39} + 4 q^{43} - 12 q^{47} + 10 q^{49} + 28 q^{57} + 12 q^{59} + 8 q^{61} + 2 q^{63}+ \cdots - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1201.1
0.707107 − 1.22474i
−0.707107 + 1.22474i
0.707107 + 1.22474i
−0.707107 − 1.22474i
0 0.500000 − 0.866025i 0 0 0 −1.62132 + 2.09077i 0 −0.500000 − 0.866025i 0
1201.2 0 0.500000 − 0.866025i 0 0 0 2.62132 − 0.358719i 0 −0.500000 − 0.866025i 0
1801.1 0 0.500000 + 0.866025i 0 0 0 −1.62132 − 2.09077i 0 −0.500000 + 0.866025i 0
1801.2 0 0.500000 + 0.866025i 0 0 0 2.62132 + 0.358719i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.q.k 4
5.b even 2 1 420.2.q.d ✓ 4
5.c odd 4 2 2100.2.bc.f 8
7.c even 3 1 inner 2100.2.q.k 4
15.d odd 2 1 1260.2.s.e 4
20.d odd 2 1 1680.2.bg.t 4
35.c odd 2 1 2940.2.q.q 4
35.i odd 6 1 2940.2.a.p 2
35.i odd 6 1 2940.2.q.q 4
35.j even 6 1 420.2.q.d ✓ 4
35.j even 6 1 2940.2.a.r 2
35.l odd 12 2 2100.2.bc.f 8
105.o odd 6 1 1260.2.s.e 4
105.o odd 6 1 8820.2.a.bk 2
105.p even 6 1 8820.2.a.bf 2
140.p odd 6 1 1680.2.bg.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.d ✓ 4 5.b even 2 1
420.2.q.d ✓ 4 35.j even 6 1
1260.2.s.e 4 15.d odd 2 1
1260.2.s.e 4 105.o odd 6 1
1680.2.bg.t 4 20.d odd 2 1
1680.2.bg.t 4 140.p odd 6 1
2100.2.q.k 4 1.a even 1 1 trivial
2100.2.q.k 4 7.c even 3 1 inner
2100.2.bc.f 8 5.c odd 4 2
2100.2.bc.f 8 35.l odd 12 2
2940.2.a.p 2 35.i odd 6 1
2940.2.a.r 2 35.j even 6 1
2940.2.q.q 4 35.c odd 2 1
2940.2.q.q 4 35.i odd 6 1
8820.2.a.bf 2 105.p even 6 1
8820.2.a.bk 2 105.o odd 6 1

Hecke kernels

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

\( T_{11}^{4} + 18T_{11}^{2} + 324 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 17 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 2 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T - 17)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$19$ \( (T^{2} - 7 T + 49)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$29$ \( (T^{2} + 12 T + 18)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 2 T^{3} + \cdots + 5041 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$41$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T - 17)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 72T^{2} + 5184 \) Copy content Toggle raw display
$59$ \( T^{4} - 12 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} + \cdots + 3136 \) Copy content Toggle raw display
$67$ \( T^{4} + 2 T^{3} + \cdots + 289 \) Copy content Toggle raw display
$71$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 10 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$79$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 12 T + 18)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 12 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$97$ \( (T^{2} + 16 T - 8)^{2} \) Copy content Toggle raw display
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