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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [544,2,Mod(273,544)] 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("544.273"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(544, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 544 = 2^{5} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 544.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.34386186996\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1649659456.5
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2x^{5} + 4x^{4} - 4x^{3} - 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 136)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} - \beta_{7} q^{5} + ( - \beta_{5} - 1) q^{7} + ( - \beta_{3} - 1) q^{9} + ( - \beta_{7} - \beta_1) q^{11} + ( - \beta_{2} - \beta_1) q^{13} + ( - \beta_{6} - \beta_{5} + 2 \beta_{3} + 2) q^{15}+ \cdots + (2 \beta_{7} - \beta_{4} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7} - 8 q^{9} + 12 q^{15} - 8 q^{17} - 12 q^{23} - 8 q^{25} + 12 q^{31} + 8 q^{33} - 20 q^{39} + 28 q^{47} - 8 q^{49} - 44 q^{55} + 8 q^{57} + 20 q^{63} + 24 q^{65} + 8 q^{73} + 44 q^{79} - 40 q^{81}+ \cdots + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 2x^{5} + 4x^{4} - 4x^{3} - 8x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - \nu^{6} - 2\nu^{5} + 4\nu^{2} ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - \nu^{6} - 4\nu^{4} + 4\nu^{3} - 4\nu^{2} + 16\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - \nu^{5} - 2\nu^{3} ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + \nu^{6} - 4\nu^{4} + 4\nu^{3} + 4\nu^{2} - 16 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 2\nu^{5} + 4\nu^{4} + 4\nu^{2} + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + \nu^{6} + 2\nu^{5} - 8\nu^{3} + 4\nu^{2} + 16\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + \nu^{5} + 2\nu^{3} - 4\nu^{2} - 8 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{4} - 2\beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - \beta_{6} + 3\beta_{5} + 2\beta_{4} - 2\beta_{3} + \beta_{2} - \beta _1 + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{7} + \beta_{6} + 5\beta_{5} - 2\beta_{4} - 2\beta_{3} - \beta_{2} + \beta _1 - 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} + 3\beta_{6} - \beta_{5} + 2\beta_{4} - 6\beta_{3} - 3\beta_{2} - 5\beta _1 + 4 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3\beta_{7} + \beta_{6} + 5\beta_{5} + 6\beta_{4} + 6\beta_{3} - \beta_{2} - 7\beta _1 + 12 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 9\beta_{7} + 3\beta_{6} - \beta_{5} + 2\beta_{4} + 2\beta_{3} - 3\beta_{2} + 11\beta _1 + 20 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(69\) \(511\) \(513\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
273.1
0.814732 + 1.15595i
−0.578647 − 1.29041i
1.40014 + 0.199044i
−1.13622 + 0.842022i
−1.13622 − 0.842022i
1.40014 − 0.199044i
−0.578647 + 1.29041i
0.814732 − 1.15595i
0 − 2.64089i 0 1.77580i 0 0.423267 0 −3.97431 0
273.2 0 − 2.34593i 0 3.12087i 0 −2.86993 0 −2.50338 0
273.3 0 − 1.72010i 0 − 3.30391i 0 −3.45790 0 0.0412530 0
273.4 0 − 0.750707i 0 − 0.436910i 0 1.90455 0 2.43644 0
273.5 0 0.750707i 0 0.436910i 0 1.90455 0 2.43644 0
273.6 0 1.72010i 0 3.30391i 0 −3.45790 0 0.0412530 0
273.7 0 2.34593i 0 − 3.12087i 0 −2.86993 0 −2.50338 0
273.8 0 2.64089i 0 − 1.77580i 0 0.423267 0 −3.97431 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 273.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 544.2.c.a 8
3.b odd 2 1 4896.2.f.c 8
4.b odd 2 1 136.2.c.a ✓ 8
8.b even 2 1 inner 544.2.c.a 8
8.d odd 2 1 136.2.c.a ✓ 8
12.b even 2 1 1224.2.f.d 8
16.e even 4 2 4352.2.a.be 8
16.f odd 4 2 4352.2.a.bc 8
24.f even 2 1 1224.2.f.d 8
24.h odd 2 1 4896.2.f.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.c.a ✓ 8 4.b odd 2 1
136.2.c.a ✓ 8 8.d odd 2 1
544.2.c.a 8 1.a even 1 1 trivial
544.2.c.a 8 8.b even 2 1 inner
1224.2.f.d 8 12.b even 2 1
1224.2.f.d 8 24.f even 2 1
4352.2.a.bc 8 16.f odd 4 2
4352.2.a.be 8 16.e even 4 2
4896.2.f.c 8 3.b odd 2 1
4896.2.f.c 8 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 16T_{3}^{6} + 84T_{3}^{4} + 156T_{3}^{2} + 64 \) acting on \(S_{2}^{\mathrm{new}}(544, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 16 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{8} + 24 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$7$ \( (T^{4} + 4 T^{3} - 4 T^{2} + \cdots + 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 44 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( T^{8} + 60 T^{6} + \cdots + 9216 \) Copy content Toggle raw display
$17$ \( (T + 1)^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 112 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} - 12 T^{2} + \cdots + 48)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 152 T^{6} + \cdots + 1032256 \) Copy content Toggle raw display
$31$ \( (T^{4} - 6 T^{3} - 40 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 64 T^{6} + \cdots + 5184 \) Copy content Toggle raw display
$41$ \( (T^{4} - 64 T^{2} + \cdots - 48)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 156 T^{6} + \cdots + 589824 \) Copy content Toggle raw display
$47$ \( (T^{4} - 14 T^{3} + \cdots - 192)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 176 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$59$ \( T^{8} + 140 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$61$ \( T^{8} + 504 T^{6} + \cdots + 25644096 \) Copy content Toggle raw display
$67$ \( T^{8} + 240 T^{6} + \cdots + 2304 \) Copy content Toggle raw display
$71$ \( (T^{4} - 208 T^{2} + \cdots + 4608)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 4 T^{3} + \cdots + 30704)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 22 T^{3} + \cdots + 216)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 444 T^{6} + \cdots + 34668544 \) Copy content Toggle raw display
$89$ \( (T^{4} - 4 T^{3} - 88 T^{2} + \cdots - 72)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{3} + \cdots - 1168)^{2} \) Copy content Toggle raw display
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