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Properties

Label 775.2.bs.c
Level $775$
Weight $2$
Character orbit 775.bs
Analytic conductor $6.188$
Analytic rank $0$
Dimension $160$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [775,2,Mod(182,775)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(775, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([5, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("775.182");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 775 = 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 775.bs (of order \(20\), degree \(8\), 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: \(6.18840615665\)
Analytic rank: \(0\)
Dimension: \(160\)
Relative dimension: \(20\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 160 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 160 q + 20 q^{11} + 48 q^{16} + 60 q^{21} + 16 q^{31} - 40 q^{36} + 12 q^{41} + 80 q^{46} - 16 q^{51} + 24 q^{56} + 72 q^{66} - 104 q^{71} + 40 q^{76} - 104 q^{81} - 120 q^{86} - 100 q^{91} - 240 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
182.1 −1.25767 + 2.46833i 0.699905 − 0.356619i −3.33531 − 4.59067i 0 2.17610i −0.450463 − 0.0713463i 10.0537 − 1.59235i −1.40067 + 1.92785i 0
182.2 −1.02621 + 2.01404i 2.36184 − 1.20342i −1.82770 − 2.51562i 0 5.99180i −1.29603 − 0.205271i 2.47700 − 0.392318i 2.36671 − 3.25750i 0
182.3 −0.877741 + 1.72266i −2.50905 + 1.27843i −1.02157 − 1.40607i 0 − 5.44439i 1.90036 + 0.300987i −0.500308 + 0.0792410i 2.89762 − 3.98823i 0
182.4 −0.809045 + 1.58784i −1.61971 + 0.825285i −0.691112 − 0.951235i 0 − 3.23954i −1.04427 − 0.165395i −1.45072 + 0.229771i 0.179018 − 0.246398i 0
182.5 −0.790268 + 1.55099i 0.972682 − 0.495606i −0.605472 − 0.833360i 0 1.90028i 3.12747 + 0.495343i −1.66755 + 0.264114i −1.06287 + 1.46292i 0
182.6 −0.648920 + 1.27358i 0.672494 − 0.342653i −0.0253321 − 0.0348667i 0 1.07883i 2.21132 + 0.350238i −2.76270 + 0.437569i −1.42852 + 1.96619i 0
182.7 −0.591301 + 1.16049i −0.341886 + 0.174200i 0.178462 + 0.245632i 0 − 0.499761i −3.08971 − 0.489362i −2.96341 + 0.469359i −1.67682 + 2.30794i 0
182.8 −0.212930 + 0.417899i 0.862687 − 0.439561i 1.04627 + 1.44007i 0 0.454111i −4.59691 − 0.728079i −1.75107 + 0.277343i −1.21234 + 1.66864i 0
182.9 −0.117934 + 0.231459i 2.14921 − 1.09508i 1.13591 + 1.56344i 0 0.626601i 2.67722 + 0.424030i −1.00898 + 0.159807i 1.65657 − 2.28007i 0
182.10 −0.0436826 + 0.0857319i 2.88138 − 1.46814i 1.17013 + 1.61054i 0 0.311158i −0.0128774 − 0.00203958i −0.379258 + 0.0600686i 4.38358 − 6.03348i 0
182.11 0.0436826 − 0.0857319i −2.88138 + 1.46814i 1.17013 + 1.61054i 0 0.311158i 0.0128774 + 0.00203958i 0.379258 − 0.0600686i 4.38358 − 6.03348i 0
182.12 0.117934 − 0.231459i −2.14921 + 1.09508i 1.13591 + 1.56344i 0 0.626601i −2.67722 − 0.424030i 1.00898 − 0.159807i 1.65657 − 2.28007i 0
182.13 0.212930 − 0.417899i −0.862687 + 0.439561i 1.04627 + 1.44007i 0 0.454111i 4.59691 + 0.728079i 1.75107 − 0.277343i −1.21234 + 1.66864i 0
182.14 0.591301 − 1.16049i 0.341886 − 0.174200i 0.178462 + 0.245632i 0 − 0.499761i 3.08971 + 0.489362i 2.96341 − 0.469359i −1.67682 + 2.30794i 0
182.15 0.648920 − 1.27358i −0.672494 + 0.342653i −0.0253321 − 0.0348667i 0 1.07883i −2.21132 − 0.350238i 2.76270 − 0.437569i −1.42852 + 1.96619i 0
182.16 0.790268 − 1.55099i −0.972682 + 0.495606i −0.605472 − 0.833360i 0 1.90028i −3.12747 − 0.495343i 1.66755 − 0.264114i −1.06287 + 1.46292i 0
182.17 0.809045 − 1.58784i 1.61971 − 0.825285i −0.691112 − 0.951235i 0 − 3.23954i 1.04427 + 0.165395i 1.45072 − 0.229771i 0.179018 − 0.246398i 0
182.18 0.877741 − 1.72266i 2.50905 − 1.27843i −1.02157 − 1.40607i 0 − 5.44439i −1.90036 − 0.300987i 0.500308 − 0.0792410i 2.89762 − 3.98823i 0
182.19 1.02621 − 2.01404i −2.36184 + 1.20342i −1.82770 − 2.51562i 0 5.99180i 1.29603 + 0.205271i −2.47700 + 0.392318i 2.36671 − 3.25750i 0
182.20 1.25767 − 2.46833i −0.699905 + 0.356619i −3.33531 − 4.59067i 0 2.17610i 0.450463 + 0.0713463i −10.0537 + 1.59235i −1.40067 + 1.92785i 0
See next 80 embeddings (of 160 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 182.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
31.f odd 10 1 inner
155.m odd 10 1 inner
155.r even 20 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 775.2.bs.c ✓ 160
5.b even 2 1 inner 775.2.bs.c ✓ 160
5.c odd 4 2 inner 775.2.bs.c ✓ 160
31.f odd 10 1 inner 775.2.bs.c ✓ 160
155.m odd 10 1 inner 775.2.bs.c ✓ 160
155.r even 20 2 inner 775.2.bs.c ✓ 160
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
775.2.bs.c ✓ 160 1.a even 1 1 trivial
775.2.bs.c ✓ 160 5.b even 2 1 inner
775.2.bs.c ✓ 160 5.c odd 4 2 inner
775.2.bs.c ✓ 160 31.f odd 10 1 inner
775.2.bs.c ✓ 160 155.m odd 10 1 inner
775.2.bs.c ✓ 160 155.r even 20 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{160} - 176 T_{2}^{156} + 17616 T_{2}^{152} - 1369159 T_{2}^{148} + 90979600 T_{2}^{144} + \cdots + 16983563041 \) acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\). Copy content Toggle raw display