↑

Properties

Label 4212.2.i.n
Level $4212$
Weight $2$
Character orbit 4212.i
Analytic conductor $33.633$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4212,2,Mod(1405,4212)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4212, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4212.1405"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 4212 = 2^{2} \cdot 3^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4212.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-1,0,0,0,0,0,-7,0,-2,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(33.6329893314\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1404)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + (2 \beta_{3} - \beta_{2} + 2 \beta_1 - 2) q^{7} + (\beta_{3} + 3 \beta_{2} + \beta_1 - 1) q^{11} + ( - \beta_{2} - 1) q^{13} + ( - \beta_{3} + 1) q^{17} + (\beta_{3} - 2) q^{19} + ( - 3 \beta_{2} + 3 \beta_1 - 3) q^{23}+ \cdots + ( - 4 \beta_{3} - 4 \beta_{2} + \cdots + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{5} - 7 q^{11} - 2 q^{13} + 2 q^{17} - 6 q^{19} - 3 q^{23} + 3 q^{25} - 6 q^{29} - 4 q^{31} + 26 q^{35} - 12 q^{37} - 4 q^{41} + 15 q^{43} - 7 q^{47} - 12 q^{49} + 2 q^{53} + 20 q^{55} - 14 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 7 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} - 7 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4212\mathbb{Z}\right)^\times\).

\(n\) \(2107\) \(3485\) \(3889\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1405.1
1.15139 − 1.99426i
−0.651388 + 1.12824i
1.15139 + 1.99426i
−0.651388 − 1.12824i
0 0 0 −1.15139 + 1.99426i 0 −1.80278 − 3.12250i 0 0 0
1405.2 0 0 0 0.651388 − 1.12824i 0 1.80278 + 3.12250i 0 0 0
2809.1 0 0 0 −1.15139 − 1.99426i 0 −1.80278 + 3.12250i 0 0 0
2809.2 0 0 0 0.651388 + 1.12824i 0 1.80278 − 3.12250i 0 0 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4212.2.i.n 4
3.b odd 2 1 4212.2.i.s 4
9.c even 3 1 1404.2.a.i yes 2
9.c even 3 1 inner 4212.2.i.n 4
9.d odd 6 1 1404.2.a.f ✓ 2
9.d odd 6 1 4212.2.i.s 4
36.f odd 6 1 5616.2.a.bt 2
36.h even 6 1 5616.2.a.bp 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1404.2.a.f ✓ 2 9.d odd 6 1
1404.2.a.i yes 2 9.c even 3 1
4212.2.i.n 4 1.a even 1 1 trivial
4212.2.i.n 4 9.c even 3 1 inner
4212.2.i.s 4 3.b odd 2 1
4212.2.i.s 4 9.d odd 6 1
5616.2.a.bp 2 36.h even 6 1
5616.2.a.bt 2 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4212, [\chi])\):

\( T_{5}^{4} + T_{5}^{3} + 4T_{5}^{2} - 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{7}^{4} + 13T_{7}^{2} + 169 \) Copy content Toggle raw display
\( T_{17}^{2} - T_{17} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} + 4 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$11$ \( T^{4} + 7 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - T - 3)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 3 T - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 3 T^{3} + \cdots + 729 \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 6 T - 43)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( T^{4} - 15 T^{3} + \cdots + 2809 \) Copy content Toggle raw display
$47$ \( T^{4} + 7 T^{3} + \cdots + 4761 \) Copy content Toggle raw display
$53$ \( (T^{2} - T - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 14 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$61$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$67$ \( T^{4} + 13 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$71$ \( (T^{2} - 26 T + 156)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 11 T + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 5 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$83$ \( T^{4} + 117 T^{2} + 13689 \) Copy content Toggle raw display
$89$ \( (T^{2} + 2 T - 12)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 12 T^{3} + \cdots + 256 \) Copy content Toggle raw display
show more
show less