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Properties

Label 444.2.e.b
Level $444$
Weight $2$
Character orbit 444.e
Analytic conductor $3.545$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [444,2,Mod(73,444)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) 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("444.73"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.32448.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 10x^{2} + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_1 q^{5} + (\beta_{2} - 1) q^{7} + q^{9} + 2 \beta_1 q^{13} + \beta_1 q^{15} + ( - \beta_{3} + \beta_1) q^{17} + \beta_{3} q^{19} + (\beta_{2} - 1) q^{21} - \beta_1 q^{23} + \beta_{2} q^{25}+ \cdots - 2 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{7} + 4 q^{9} - 4 q^{21} + 4 q^{27} + 8 q^{37} + 16 q^{41} - 8 q^{47} + 28 q^{49} - 16 q^{53} - 4 q^{63} - 40 q^{65} - 12 q^{67} - 8 q^{71} + 12 q^{73} + 4 q^{81} - 8 q^{83} - 12 q^{85}+ \cdots - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 10x^{2} + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 8\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 8\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/444\mathbb{Z}\right)^\times\).

\(n\) \(149\) \(223\) \(409\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
− 2.93352i
− 1.18087i
1.18087i
2.93352i
0 1.00000 0 − 2.93352i 0 −4.60555 0 1.00000 0
73.2 0 1.00000 0 − 1.18087i 0 2.60555 0 1.00000 0
73.3 0 1.00000 0 1.18087i 0 2.60555 0 1.00000 0
73.4 0 1.00000 0 2.93352i 0 −4.60555 0 1.00000 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
37.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 444.2.e.b ✓ 4
3.b odd 2 1 1332.2.e.d 4
4.b odd 2 1 1776.2.h.d 4
12.b even 2 1 5328.2.h.j 4
37.b even 2 1 inner 444.2.e.b ✓ 4
111.d odd 2 1 1332.2.e.d 4
148.b odd 2 1 1776.2.h.d 4
444.g even 2 1 5328.2.h.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
444.2.e.b ✓ 4 1.a even 1 1 trivial
444.2.e.b ✓ 4 37.b even 2 1 inner
1332.2.e.d 4 3.b odd 2 1
1332.2.e.d 4 111.d odd 2 1
1776.2.h.d 4 4.b odd 2 1
1776.2.h.d 4 148.b odd 2 1
5328.2.h.j 4 12.b even 2 1
5328.2.h.j 4 444.g even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 10T_{5}^{2} + 12 \) acting on \(S_{2}^{\mathrm{new}}(444, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 10T^{2} + 12 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T - 12)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 40T^{2} + 192 \) Copy content Toggle raw display
$17$ \( T^{4} + 66T^{2} + 972 \) Copy content Toggle raw display
$19$ \( T^{4} + 64T^{2} + 192 \) Copy content Toggle raw display
$23$ \( T^{4} + 10T^{2} + 12 \) Copy content Toggle raw display
$29$ \( T^{4} + 66T^{2} + 972 \) Copy content Toggle raw display
$31$ \( T^{4} + 88T^{2} + 1728 \) Copy content Toggle raw display
$37$ \( T^{4} - 8 T^{3} + \cdots + 1369 \) Copy content Toggle raw display
$41$ \( (T^{2} - 8 T - 36)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 88T^{2} + 1728 \) Copy content Toggle raw display
$47$ \( (T^{2} + 4 T - 48)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 8 T - 36)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 354 T^{2} + 31212 \) Copy content Toggle raw display
$61$ \( T^{4} + 160T^{2} + 3072 \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T - 4)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4 T - 48)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 6 T - 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 64T^{2} + 192 \) Copy content Toggle raw display
$83$ \( (T^{2} + 4 T - 48)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 90T^{2} + 972 \) Copy content Toggle raw display
$97$ \( T^{4} + 40T^{2} + 192 \) Copy content Toggle raw display
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