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Properties

Label 352.2.w.a
Level $352$
Weight $2$
Character orbit 352.w
Analytic conductor $2.811$
Analytic rank $0$
Dimension $40$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [352,2,Mod(49,352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(352, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("352.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 352.w (of order \(10\), degree \(4\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.81073415115\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(10\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 10 q^{7} + 18 q^{15} - 6 q^{17} + 8 q^{23} - 4 q^{25} + 6 q^{31} - 10 q^{33} + 34 q^{39} - 14 q^{41} + 6 q^{47} - 4 q^{49} + 2 q^{55} - 26 q^{57} - 60 q^{63} - 36 q^{65} - 22 q^{71} - 6 q^{73} - 74 q^{79}+ \cdots + 10 q^{97}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1 0 −1.67918 + 2.31119i 0 −2.48027 − 0.805887i 0 0.158860 − 0.115419i 0 −1.59492 − 4.90866i 0
49.2 0 −1.50170 + 2.06692i 0 2.56349 + 0.832928i 0 3.19487 − 2.32121i 0 −1.08998 − 3.35462i 0
49.3 0 −1.10501 + 1.52092i 0 −0.468972 − 0.152378i 0 0.141392 − 0.102727i 0 −0.165095 − 0.508111i 0
49.4 0 −0.317655 + 0.437215i 0 −2.75892 − 0.896426i 0 2.10709 − 1.53089i 0 0.836799 + 2.57540i 0
49.5 0 −0.188809 + 0.259874i 0 −1.22432 − 0.397805i 0 −2.67516 + 1.94362i 0 0.895166 + 2.75504i 0
49.6 0 0.188809 − 0.259874i 0 1.22432 + 0.397805i 0 −2.67516 + 1.94362i 0 0.895166 + 2.75504i 0
49.7 0 0.317655 − 0.437215i 0 2.75892 + 0.896426i 0 2.10709 − 1.53089i 0 0.836799 + 2.57540i 0
49.8 0 1.10501 − 1.52092i 0 0.468972 + 0.152378i 0 0.141392 − 0.102727i 0 −0.165095 − 0.508111i 0
49.9 0 1.50170 − 2.06692i 0 −2.56349 − 0.832928i 0 3.19487 − 2.32121i 0 −1.08998 − 3.35462i 0
49.10 0 1.67918 − 2.31119i 0 2.48027 + 0.805887i 0 0.158860 − 0.115419i 0 −1.59492 − 4.90866i 0
81.1 0 −2.62285 − 0.852217i 0 −1.70795 − 2.35079i 0 −0.730368 − 2.24784i 0 3.72604 + 2.70713i 0
81.2 0 −2.32812 − 0.756451i 0 0.117836 + 0.162187i 0 0.725001 + 2.23132i 0 2.42086 + 1.75886i 0
81.3 0 −1.22337 − 0.397498i 0 −0.929834 − 1.27981i 0 0.577320 + 1.77681i 0 −1.08841 − 0.790780i 0
81.4 0 −0.826127 − 0.268425i 0 2.15963 + 2.97247i 0 0.369362 + 1.13678i 0 −1.81662 − 1.31985i 0
81.5 0 −0.582243 − 0.189182i 0 0.858340 + 1.18140i 0 −1.36837 − 4.21140i 0 −2.12383 − 1.54306i 0
81.6 0 0.582243 + 0.189182i 0 −0.858340 − 1.18140i 0 −1.36837 − 4.21140i 0 −2.12383 − 1.54306i 0
81.7 0 0.826127 + 0.268425i 0 −2.15963 − 2.97247i 0 0.369362 + 1.13678i 0 −1.81662 − 1.31985i 0
81.8 0 1.22337 + 0.397498i 0 0.929834 + 1.27981i 0 0.577320 + 1.77681i 0 −1.08841 − 0.790780i 0
81.9 0 2.32812 + 0.756451i 0 −0.117836 − 0.162187i 0 0.725001 + 2.23132i 0 2.42086 + 1.75886i 0
81.10 0 2.62285 + 0.852217i 0 1.70795 + 2.35079i 0 −0.730368 − 2.24784i 0 3.72604 + 2.70713i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
11.c even 5 1 inner
88.o 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.w.a 40
4.b odd 2 1 88.2.o.a ✓ 40
8.b even 2 1 inner 352.2.w.a 40
8.d odd 2 1 88.2.o.a ✓ 40
11.c even 5 1 inner 352.2.w.a 40
11.c even 5 1 3872.2.c.h 20
11.d odd 10 1 3872.2.c.i 20
12.b even 2 1 792.2.br.b 40
24.f even 2 1 792.2.br.b 40
44.c even 2 1 968.2.o.i 40
44.g even 10 1 968.2.c.i 20
44.g even 10 2 968.2.o.d 40
44.g even 10 1 968.2.o.i 40
44.h odd 10 1 88.2.o.a ✓ 40
44.h odd 10 1 968.2.c.h 20
44.h odd 10 2 968.2.o.j 40
88.g even 2 1 968.2.o.i 40
88.k even 10 1 968.2.c.i 20
88.k even 10 2 968.2.o.d 40
88.k even 10 1 968.2.o.i 40
88.l odd 10 1 88.2.o.a ✓ 40
88.l odd 10 1 968.2.c.h 20
88.l odd 10 2 968.2.o.j 40
88.o even 10 1 inner 352.2.w.a 40
88.o even 10 1 3872.2.c.h 20
88.p odd 10 1 3872.2.c.i 20
132.o even 10 1 792.2.br.b 40
264.w even 10 1 792.2.br.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.o.a ✓ 40 4.b odd 2 1
88.2.o.a ✓ 40 8.d odd 2 1
88.2.o.a ✓ 40 44.h odd 10 1
88.2.o.a ✓ 40 88.l odd 10 1
352.2.w.a 40 1.a even 1 1 trivial
352.2.w.a 40 8.b even 2 1 inner
352.2.w.a 40 11.c even 5 1 inner
352.2.w.a 40 88.o even 10 1 inner
792.2.br.b 40 12.b even 2 1
792.2.br.b 40 24.f even 2 1
792.2.br.b 40 132.o even 10 1
792.2.br.b 40 264.w even 10 1
968.2.c.h 20 44.h odd 10 1
968.2.c.h 20 88.l odd 10 1
968.2.c.i 20 44.g even 10 1
968.2.c.i 20 88.k even 10 1
968.2.o.d 40 44.g even 10 2
968.2.o.d 40 88.k even 10 2
968.2.o.i 40 44.c even 2 1
968.2.o.i 40 44.g even 10 1
968.2.o.i 40 88.g even 2 1
968.2.o.i 40 88.k even 10 1
968.2.o.j 40 44.h odd 10 2
968.2.o.j 40 88.l odd 10 2
3872.2.c.h 20 11.c even 5 1
3872.2.c.h 20 88.o even 10 1
3872.2.c.i 20 11.d odd 10 1
3872.2.c.i 20 88.p odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(352, [\chi])\).