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Properties

Label 800.2.y.c
Level $800$
Weight $2$
Character orbit 800.y
Analytic conductor $6.388$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,2,Mod(101,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.y (of order \(8\), degree \(4\), 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: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 16 q^{14} - 40 q^{16} + 40 q^{18} - 8 q^{22} + 16 q^{23} + 32 q^{24} + 48 q^{27} - 40 q^{28} - 40 q^{32} + 40 q^{34} - 24 q^{36} - 40 q^{38} - 48 q^{39} + 40 q^{42} + 16 q^{43} - 80 q^{44} - 64 q^{46} + 16 q^{51} + 16 q^{52} + 32 q^{53} - 56 q^{54} - 72 q^{56} - 72 q^{58} + 32 q^{59} - 32 q^{61} - 24 q^{62} - 80 q^{63} + 48 q^{64} + 16 q^{66} - 80 q^{67} - 48 q^{68} - 32 q^{69} + 32 q^{71} - 72 q^{72} + 8 q^{74} + 32 q^{77} - 72 q^{78} + 80 q^{83} + 56 q^{84} + 72 q^{86} - 80 q^{88} - 96 q^{91} + 8 q^{92} - 40 q^{94} + 104 q^{96} - 80 q^{98} - 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −1.39623 + 0.224797i 0.528683 + 1.27635i 1.89893 − 0.627737i 0 −1.02508 − 1.66324i 0.511565 + 0.511565i −2.51024 + 1.30334i 0.771749 − 0.771749i 0
101.2 −1.33518 − 0.466153i 0.811208 + 1.95843i 1.56540 + 1.24479i 0 −0.170181 − 2.99300i 0.304493 + 0.304493i −1.50983 − 2.39174i −1.05607 + 1.05607i 0
101.3 −1.28237 + 0.596257i −1.24322 − 3.00140i 1.28895 − 1.52925i 0 3.38387 + 3.10763i 1.78484 + 1.78484i −0.741093 + 2.72961i −5.34146 + 5.34146i 0
101.4 −1.13608 − 0.842214i −0.383809 − 0.926597i 0.581351 + 1.91364i 0 −0.344356 + 1.37594i 0.119270 + 0.119270i 0.951237 − 2.66367i 1.41005 − 1.41005i 0
101.5 −0.975074 + 1.02432i −0.249341 − 0.601962i −0.0984624 − 1.99757i 0 0.859728 + 0.331553i −0.169889 − 0.169889i 2.14216 + 1.84693i 1.82113 − 1.82113i 0
101.6 −0.669185 + 1.24587i 0.850447 + 2.05316i −1.10438 − 1.66743i 0 −3.12708 − 0.314397i −2.30853 − 2.30853i 2.81644 − 0.260097i −1.37089 + 1.37089i 0
101.7 −0.613883 − 1.27403i −0.755932 − 1.82498i −1.24629 + 1.56421i 0 −1.86102 + 2.08341i 1.73757 + 1.73757i 2.75793 + 0.627573i −0.637806 + 0.637806i 0
101.8 −0.419847 − 1.35046i 1.23041 + 2.97048i −1.64746 + 1.13397i 0 3.49491 − 2.90876i 2.69077 + 2.69077i 2.22305 + 1.74872i −5.18850 + 5.18850i 0
101.9 −0.401490 − 1.35603i 0.491276 + 1.18605i −1.67761 + 1.08886i 0 1.41107 − 1.14237i −3.65272 − 3.65272i 2.15007 + 1.83772i 0.955967 − 0.955967i 0
101.10 0.0405868 + 1.41363i 0.850240 + 2.05266i −1.99671 + 0.114750i 0 −2.86720 + 1.28524i 2.55327 + 2.55327i −0.243254 − 2.81795i −1.36919 + 1.36919i 0
101.11 0.783304 − 1.17747i 0.369438 + 0.891903i −0.772870 − 1.84463i 0 1.33957 + 0.263629i −2.01477 − 2.01477i −2.77739 − 0.534877i 1.46231 − 1.46231i 0
101.12 1.01850 − 0.981151i −1.03105 − 2.48917i 0.0746864 − 1.99861i 0 −3.49238 − 1.52361i −1.75057 − 1.75057i −1.88486 − 2.10886i −3.01160 + 3.01160i 0
101.13 1.07656 + 0.917072i −0.00330482 − 0.00797855i 0.317958 + 1.97456i 0 0.00375907 − 0.0116201i 3.03324 + 3.03324i −1.46852 + 2.41732i 2.12127 − 2.12127i 0
101.14 1.10796 + 0.878876i −1.14037 − 2.75309i 0.455155 + 1.94752i 0 1.15614 − 4.05255i 1.26936 + 1.26936i −1.20733 + 2.55780i −4.15773 + 4.15773i 0
101.15 1.37535 − 0.329260i 0.258596 + 0.624307i 1.78318 − 0.905695i 0 0.561220 + 0.773496i 1.12270 + 1.12270i 2.15428 − 1.83278i 1.79843 − 1.79843i 0
101.16 1.41287 + 0.0617194i −0.583280 − 1.40816i 1.99238 + 0.174403i 0 −0.737186 − 2.02555i −2.40218 − 2.40218i 2.80420 + 0.369376i 0.478612 − 0.478612i 0
301.1 −1.39623 − 0.224797i 0.528683 − 1.27635i 1.89893 + 0.627737i 0 −1.02508 + 1.66324i 0.511565 − 0.511565i −2.51024 − 1.30334i 0.771749 + 0.771749i 0
301.2 −1.33518 + 0.466153i 0.811208 − 1.95843i 1.56540 − 1.24479i 0 −0.170181 + 2.99300i 0.304493 − 0.304493i −1.50983 + 2.39174i −1.05607 − 1.05607i 0
301.3 −1.28237 − 0.596257i −1.24322 + 3.00140i 1.28895 + 1.52925i 0 3.38387 − 3.10763i 1.78484 − 1.78484i −0.741093 − 2.72961i −5.34146 − 5.34146i 0
301.4 −1.13608 + 0.842214i −0.383809 + 0.926597i 0.581351 − 1.91364i 0 −0.344356 − 1.37594i 0.119270 − 0.119270i 0.951237 + 2.66367i 1.41005 + 1.41005i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.y.c 64
5.b even 2 1 160.2.x.a ✓ 64
5.c odd 4 1 800.2.ba.e 64
5.c odd 4 1 800.2.ba.g 64
20.d odd 2 1 640.2.x.a 64
32.g even 8 1 inner 800.2.y.c 64
160.v odd 8 1 800.2.ba.g 64
160.y odd 8 1 640.2.x.a 64
160.z even 8 1 160.2.x.a ✓ 64
160.bb odd 8 1 800.2.ba.e 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.x.a ✓ 64 5.b even 2 1
160.2.x.a ✓ 64 160.z even 8 1
640.2.x.a 64 20.d odd 2 1
640.2.x.a 64 160.y odd 8 1
800.2.y.c 64 1.a even 1 1 trivial
800.2.y.c 64 32.g even 8 1 inner
800.2.ba.e 64 5.c odd 4 1
800.2.ba.e 64 160.bb odd 8 1
800.2.ba.g 64 5.c odd 4 1
800.2.ba.g 64 160.v odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} - 16 T_{3}^{61} + 112 T_{3}^{59} + 96 T_{3}^{58} - 336 T_{3}^{57} + 14880 T_{3}^{56} + \cdots + 1597696 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display