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Properties

Label 544.2.bb.a
Level $544$
Weight $2$
Character orbit 544.bb
Analytic conductor $4.344$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [544,2,Mod(161,544)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(544, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 0, 5])) 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("544.161"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 544 = 2^{5} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 544.bb (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-8,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.34386186996\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{8}]$

$q$-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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{8}^{3} - \zeta_{8}^{2} + \cdots - 2) q^{5} - 3 \zeta_{8} q^{9} + ( - 5 \zeta_{8}^{3} - 5 \zeta_{8}) q^{13} + ( - \zeta_{8}^{2} + 4) q^{17} + (7 \zeta_{8}^{2} - 5 \zeta_{8} + 7) q^{25} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + \cdots + 5) q^{29} + \cdots + (9 \zeta_{8}^{3} + 9 \zeta_{8}^{2} + \cdots + 4) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{5} + 16 q^{17} + 28 q^{25} + 20 q^{29} - 24 q^{37} - 20 q^{41} - 12 q^{45} + 20 q^{53} - 24 q^{61} + 20 q^{65} - 12 q^{73} - 36 q^{85} + 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/544\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(511\) \(513\)
\(\chi(n)\) \(1\) \(1\) \(\zeta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−0.707107 − 0.707107i
0.707107 − 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
0 0 0 −4.12132 − 1.70711i 0 0 0 2.12132 + 2.12132i 0
257.1 0 0 0 0.121320 + 0.292893i 0 0 0 −2.12132 + 2.12132i 0
321.1 0 0 0 −4.12132 + 1.70711i 0 0 0 2.12132 − 2.12132i 0
417.1 0 0 0 0.121320 − 0.292893i 0 0 0 −2.12132 − 2.12132i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
17.d even 8 1 inner
68.g odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 544.2.bb.a ✓ 4
4.b odd 2 1 CM 544.2.bb.a ✓ 4
17.d even 8 1 inner 544.2.bb.a ✓ 4
17.e odd 16 2 9248.2.a.bc 4
68.g odd 8 1 inner 544.2.bb.a ✓ 4
68.i even 16 2 9248.2.a.bc 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
544.2.bb.a ✓ 4 1.a even 1 1 trivial
544.2.bb.a ✓ 4 4.b odd 2 1 CM
544.2.bb.a ✓ 4 17.d even 8 1 inner
544.2.bb.a ✓ 4 68.g odd 8 1 inner
9248.2.a.bc 4 17.e odd 16 2
9248.2.a.bc 4 68.i even 16 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(544, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{4} + 8T_{5}^{3} + 18T_{5}^{2} - 4T_{5} + 2 \) 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
$5$ \( T^{4} + 8 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 50)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 8 T + 17)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 20 T^{3} + \cdots + 3362 \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 24 T^{3} + \cdots + 1058 \) Copy content Toggle raw display
$41$ \( T^{4} + 20 T^{3} + \cdots + 1922 \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 10 T + 50)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( T^{4} + 24 T^{3} + \cdots + 4802 \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 12 T^{3} + \cdots + 98 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 16 T^{3} + \cdots + 37538 \) Copy content Toggle raw display
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