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Properties

Label 352.2.s.b
Level $352$
Weight $2$
Character orbit 352.s
Analytic conductor $2.811$
Analytic rank $0$
Dimension $32$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [352,2,Mod(79,352)] 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("352.79"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(352, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 5, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 352.s (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.81073415115\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 88)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 2 q^{3} - 10 q^{9} + 18 q^{11} - 10 q^{17} + 6 q^{25} + 32 q^{27} + 32 q^{33} + 10 q^{35} - 10 q^{41} - 18 q^{49} - 60 q^{51} - 80 q^{57} - 28 q^{59} + 28 q^{67} - 10 q^{73} - 4 q^{75} + 28 q^{81}+ \cdots - 122 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1 0 −1.70387 + 1.23794i 0 −1.49538 + 0.485879i 0 −1.63043 − 1.18458i 0 0.443645 − 1.36540i 0
79.2 0 −1.70387 + 1.23794i 0 1.49538 − 0.485879i 0 1.63043 + 1.18458i 0 0.443645 − 1.36540i 0
79.3 0 −0.0248408 + 0.0180479i 0 −1.78547 + 0.580134i 0 −0.623146 − 0.452742i 0 −0.926760 + 2.85227i 0
79.4 0 −0.0248408 + 0.0180479i 0 1.78547 − 0.580134i 0 0.623146 + 0.452742i 0 −0.926760 + 2.85227i 0
79.5 0 0.903665 − 0.656551i 0 −3.74056 + 1.21538i 0 −2.25832 − 1.64076i 0 −0.541500 + 1.66657i 0
79.6 0 0.903665 − 0.656551i 0 3.74056 − 1.21538i 0 2.25832 + 1.64076i 0 −0.541500 + 1.66657i 0
79.7 0 1.63407 − 1.18722i 0 −1.62415 + 0.527718i 0 3.70164 + 2.68940i 0 0.333632 − 1.02681i 0
79.8 0 1.63407 − 1.18722i 0 1.62415 − 0.527718i 0 −3.70164 − 2.68940i 0 0.333632 − 1.02681i 0
239.1 0 −0.852681 − 2.62428i 0 −1.35292 + 1.86213i 0 −1.08979 + 3.35402i 0 −3.73274 + 2.71199i 0
239.2 0 −0.852681 − 2.62428i 0 1.35292 − 1.86213i 0 1.08979 − 3.35402i 0 −3.73274 + 2.71199i 0
239.3 0 −0.385688 − 1.18702i 0 −2.03454 + 2.80031i 0 0.442181 − 1.36089i 0 1.16678 − 0.847714i 0
239.4 0 −0.385688 − 1.18702i 0 2.03454 − 2.80031i 0 −0.442181 + 1.36089i 0 1.16678 − 0.847714i 0
239.5 0 0.303809 + 0.935028i 0 −0.398383 + 0.548327i 0 −1.40393 + 4.32085i 0 1.64507 − 1.19522i 0
239.6 0 0.303809 + 0.935028i 0 0.398383 − 0.548327i 0 1.40393 − 4.32085i 0 1.64507 − 1.19522i 0
239.7 0 0.625543 + 1.92522i 0 −1.75152 + 2.41076i 0 −0.216882 + 0.667493i 0 −0.888128 + 0.645263i 0
239.8 0 0.625543 + 1.92522i 0 1.75152 − 2.41076i 0 0.216882 − 0.667493i 0 −0.888128 + 0.645263i 0
271.1 0 −0.852681 + 2.62428i 0 −1.35292 − 1.86213i 0 −1.08979 − 3.35402i 0 −3.73274 − 2.71199i 0
271.2 0 −0.852681 + 2.62428i 0 1.35292 + 1.86213i 0 1.08979 + 3.35402i 0 −3.73274 − 2.71199i 0
271.3 0 −0.385688 + 1.18702i 0 −2.03454 − 2.80031i 0 0.442181 + 1.36089i 0 1.16678 + 0.847714i 0
271.4 0 −0.385688 + 1.18702i 0 2.03454 + 2.80031i 0 −0.442181 − 1.36089i 0 1.16678 + 0.847714i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
11.d odd 10 1 inner
88.k even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 352.2.s.b 32
4.b odd 2 1 88.2.k.b ✓ 32
8.b even 2 1 88.2.k.b ✓ 32
8.d odd 2 1 inner 352.2.s.b 32
11.c even 5 1 3872.2.g.d 32
11.d odd 10 1 inner 352.2.s.b 32
11.d odd 10 1 3872.2.g.d 32
12.b even 2 1 792.2.bp.b 32
24.h odd 2 1 792.2.bp.b 32
44.c even 2 1 968.2.k.h 32
44.g even 10 1 88.2.k.b ✓ 32
44.g even 10 1 968.2.g.e 32
44.g even 10 1 968.2.k.e 32
44.g even 10 1 968.2.k.i 32
44.h odd 10 1 968.2.g.e 32
44.h odd 10 1 968.2.k.e 32
44.h odd 10 1 968.2.k.h 32
44.h odd 10 1 968.2.k.i 32
88.b odd 2 1 968.2.k.h 32
88.k even 10 1 inner 352.2.s.b 32
88.k even 10 1 3872.2.g.d 32
88.l odd 10 1 3872.2.g.d 32
88.o even 10 1 968.2.g.e 32
88.o even 10 1 968.2.k.e 32
88.o even 10 1 968.2.k.h 32
88.o even 10 1 968.2.k.i 32
88.p odd 10 1 88.2.k.b ✓ 32
88.p odd 10 1 968.2.g.e 32
88.p odd 10 1 968.2.k.e 32
88.p odd 10 1 968.2.k.i 32
132.n odd 10 1 792.2.bp.b 32
264.u even 10 1 792.2.bp.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.k.b ✓ 32 4.b odd 2 1
88.2.k.b ✓ 32 8.b even 2 1
88.2.k.b ✓ 32 44.g even 10 1
88.2.k.b ✓ 32 88.p odd 10 1
352.2.s.b 32 1.a even 1 1 trivial
352.2.s.b 32 8.d odd 2 1 inner
352.2.s.b 32 11.d odd 10 1 inner
352.2.s.b 32 88.k even 10 1 inner
792.2.bp.b 32 12.b even 2 1
792.2.bp.b 32 24.h odd 2 1
792.2.bp.b 32 132.n odd 10 1
792.2.bp.b 32 264.u even 10 1
968.2.g.e 32 44.g even 10 1
968.2.g.e 32 44.h odd 10 1
968.2.g.e 32 88.o even 10 1
968.2.g.e 32 88.p odd 10 1
968.2.k.e 32 44.g even 10 1
968.2.k.e 32 44.h odd 10 1
968.2.k.e 32 88.o even 10 1
968.2.k.e 32 88.p odd 10 1
968.2.k.h 32 44.c even 2 1
968.2.k.h 32 44.h odd 10 1
968.2.k.h 32 88.b odd 2 1
968.2.k.h 32 88.o even 10 1
968.2.k.i 32 44.g even 10 1
968.2.k.i 32 44.h odd 10 1
968.2.k.i 32 88.o even 10 1
968.2.k.i 32 88.p odd 10 1
3872.2.g.d 32 11.c even 5 1
3872.2.g.d 32 11.d odd 10 1
3872.2.g.d 32 88.k even 10 1
3872.2.g.d 32 88.l odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - T_{3}^{15} + 9 T_{3}^{14} - 16 T_{3}^{13} + 51 T_{3}^{12} - 58 T_{3}^{11} + 181 T_{3}^{10} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(352, [\chi])\). Copy content Toggle raw display