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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [272,4,Mod(33,272)] 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("272.33"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(272, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 272 = 2^{4} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 272.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.0485195216\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-37 +3 \sqrt{33}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 74x^{2} + 1072 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 17)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{2} q^{5} + (\beta_{2} - \beta_1) q^{7} + (3 \beta_{3} - 10) q^{9} + 7 \beta_1 q^{11} + ( - 7 \beta_{3} + 35) q^{13} + ( - 14 \beta_{3} - 6) q^{15} + (7 \beta_{3} - \beta_{2} + 8 \beta_1 + 28) q^{17}+ \cdots + (42 \beta_{2} - 133 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 40 q^{9} + 140 q^{13} - 24 q^{15} + 112 q^{17} + 112 q^{19} + 124 q^{21} - 460 q^{25} - 1036 q^{33} - 936 q^{35} + 520 q^{43} - 952 q^{47} + 312 q^{49} - 1160 q^{51} - 576 q^{53} - 168 q^{55} + 1176 q^{59}+ \cdots + 1060 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 74x^{2} + 1072 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 46\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{2} + 37 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{3} - 37 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 6\beta_{2} - 46\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/272\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(239\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
− 7.36435i
− 4.44593i
4.44593i
7.36435i
0 − 7.36435i 0 10.1060i 0 17.4703i 0 −27.2337 0
33.2 0 − 4.44593i 0 − 19.4389i 0 − 14.9929i 0 7.23369 0
33.3 0 4.44593i 0 19.4389i 0 14.9929i 0 7.23369 0
33.4 0 7.36435i 0 − 10.1060i 0 − 17.4703i 0 −27.2337 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 272.4.b.d 4
4.b odd 2 1 17.4.b.a ✓ 4
12.b even 2 1 153.4.d.b 4
17.b even 2 1 inner 272.4.b.d 4
20.d odd 2 1 425.4.d.c 4
20.e even 4 2 425.4.c.c 8
68.d odd 2 1 17.4.b.a ✓ 4
68.f odd 4 2 289.4.a.e 4
204.h even 2 1 153.4.d.b 4
340.d odd 2 1 425.4.d.c 4
340.r even 4 2 425.4.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.b.a ✓ 4 4.b odd 2 1
17.4.b.a ✓ 4 68.d odd 2 1
153.4.d.b 4 12.b even 2 1
153.4.d.b 4 204.h even 2 1
272.4.b.d 4 1.a even 1 1 trivial
272.4.b.d 4 17.b even 2 1 inner
289.4.a.e 4 68.f odd 4 2
425.4.c.c 8 20.e even 4 2
425.4.c.c 8 340.r even 4 2
425.4.d.c 4 20.d odd 2 1
425.4.d.c 4 340.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 74T_{3}^{2} + 1072 \) acting on \(S_{4}^{\mathrm{new}}(272, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 74T^{2} + 1072 \) Copy content Toggle raw display
$5$ \( T^{4} + 480 T^{2} + 38592 \) Copy content Toggle raw display
$7$ \( T^{4} + 530 T^{2} + 68608 \) Copy content Toggle raw display
$11$ \( T^{4} + 3626 T^{2} + 2573872 \) Copy content Toggle raw display
$13$ \( (T^{2} - 70 T - 392)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 112 T^{3} + \cdots + 24137569 \) Copy content Toggle raw display
$19$ \( (T - 28)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 28322 T^{2} + 10295488 \) Copy content Toggle raw display
$29$ \( T^{4} + 23520 T^{2} + 92659392 \) Copy content Toggle raw display
$31$ \( T^{4} + 4274 T^{2} + 4390912 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 1245754048 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 2799480832 \) Copy content Toggle raw display
$43$ \( (T^{2} - 260 T - 30752)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 476 T + 50176)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 288 T - 49092)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 588 T - 75264)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 7146728128 \) Copy content Toggle raw display
$67$ \( (T^{2} - 120 T - 30192)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 52626 T^{2} + 92659392 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 647466319872 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 70925616832 \) Copy content Toggle raw display
$83$ \( (T^{2} - 1428 T + 451584)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 994 T + 232456)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 16878665728 \) Copy content Toggle raw display
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