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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [176,4,Mod(1,176)] 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("176.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(176, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 176 = 2^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 176.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,2,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.3843361610\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 88)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + (4 \beta - 3) q^{5} + (\beta + 28) q^{7} + (2 \beta - 6) q^{9} - 11 q^{11} + ( - 11 \beta - 22) q^{13} + (\beta + 77) q^{15} + ( - 15 \beta - 4) q^{17} + (9 \beta + 94) q^{19} + (29 \beta + 48) q^{21}+ \cdots + ( - 22 \beta + 66) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 6 q^{5} + 56 q^{7} - 12 q^{9} - 22 q^{11} - 44 q^{13} + 154 q^{15} - 8 q^{17} + 188 q^{19} + 96 q^{21} + 66 q^{23} + 408 q^{25} + 14 q^{27} - 56 q^{29} - 206 q^{31} - 22 q^{33} - 8 q^{35}+ \cdots + 132 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.618034
1.61803
0 −3.47214 0 −20.8885 0 23.5279 0 −14.9443 0
1.2 0 5.47214 0 14.8885 0 32.4721 0 2.94427 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(11\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 176.4.a.h 2
3.b odd 2 1 1584.4.a.bg 2
4.b odd 2 1 88.4.a.c ✓ 2
8.b even 2 1 704.4.a.o 2
8.d odd 2 1 704.4.a.q 2
11.b odd 2 1 1936.4.a.t 2
12.b even 2 1 792.4.a.g 2
20.d odd 2 1 2200.4.a.k 2
44.c even 2 1 968.4.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.4.a.c ✓ 2 4.b odd 2 1
176.4.a.h 2 1.a even 1 1 trivial
704.4.a.o 2 8.b even 2 1
704.4.a.q 2 8.d odd 2 1
792.4.a.g 2 12.b even 2 1
968.4.a.f 2 44.c even 2 1
1584.4.a.bg 2 3.b odd 2 1
1936.4.a.t 2 11.b odd 2 1
2200.4.a.k 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 2T_{3} - 19 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(176))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 19 \) Copy content Toggle raw display
$5$ \( T^{2} + 6T - 311 \) Copy content Toggle raw display
$7$ \( T^{2} - 56T + 764 \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 44T - 1936 \) Copy content Toggle raw display
$17$ \( T^{2} + 8T - 4484 \) Copy content Toggle raw display
$19$ \( T^{2} - 188T + 7216 \) Copy content Toggle raw display
$23$ \( T^{2} - 66T - 15731 \) Copy content Toggle raw display
$29$ \( T^{2} + 56T - 41536 \) Copy content Toggle raw display
$31$ \( T^{2} + 206T + 9629 \) Copy content Toggle raw display
$37$ \( T^{2} - 142T - 49039 \) Copy content Toggle raw display
$41$ \( T^{2} - 252T + 12496 \) Copy content Toggle raw display
$43$ \( T^{2} - 372T + 18916 \) Copy content Toggle raw display
$47$ \( T^{2} - 200T + 3520 \) Copy content Toggle raw display
$53$ \( T^{2} + 1060 T + 276980 \) Copy content Toggle raw display
$59$ \( T^{2} - 782T + 108701 \) Copy content Toggle raw display
$61$ \( T^{2} + 240T - 83600 \) Copy content Toggle raw display
$67$ \( T^{2} + 282T - 64619 \) Copy content Toggle raw display
$71$ \( T^{2} + 754T + 137629 \) Copy content Toggle raw display
$73$ \( T^{2} - 68T - 517264 \) Copy content Toggle raw display
$79$ \( T^{2} + 52T - 154204 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T - 551116 \) Copy content Toggle raw display
$89$ \( T^{2} - 278T - 672599 \) Copy content Toggle raw display
$97$ \( T^{2} - 306T - 192911 \) Copy content Toggle raw display
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