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Properties

Label 800.2.bb.a
Level $800$
Weight $2$
Character orbit 800.bb
Analytic conductor $6.388$
Analytic rank $0$
Dimension $64$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [800,2,Mod(107,800)]
 
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([4, 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("800.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.bb (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: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 8 q^{4} - 8 q^{14} + 40 q^{16} + 16 q^{19} - 32 q^{24} + 32 q^{29} + 8 q^{34} - 24 q^{36} - 48 q^{39} + 16 q^{44} - 64 q^{46} + 64 q^{49} - 16 q^{51} - 56 q^{54} - 88 q^{56} + 48 q^{59} + 32 q^{61} - 64 q^{64} - 80 q^{66} - 32 q^{71} + 152 q^{74} - 64 q^{76} - 184 q^{84} - 136 q^{86} + 96 q^{91} + 64 q^{94} - 168 q^{96} - 80 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −1.37688 + 0.322819i 2.90764 + 1.20438i 1.79158 − 0.888965i 0 −4.39226 − 0.719644i −3.66122 −2.17980 + 1.80235i 4.88250 + 4.88250i 0
107.2 −1.37681 − 0.323096i 0.961950 + 0.398453i 1.79122 + 0.889684i 0 −1.19569 − 0.859396i 4.93988 −2.17872 − 1.80366i −1.35474 − 1.35474i 0
107.3 −1.33577 + 0.464441i −1.95643 − 0.810381i 1.56859 − 1.24078i 0 2.98973 + 0.173839i 1.26392 −1.51901 + 2.38592i 1.04960 + 1.04960i 0
107.4 −1.18499 − 0.771881i −2.16069 − 0.894988i 0.808400 + 1.82934i 0 1.86957 + 2.72835i −2.35189 0.454089 − 2.79174i 1.74627 + 1.74627i 0
107.5 −1.07355 − 0.920591i 1.32907 + 0.550517i 0.305023 + 1.97660i 0 −0.920018 − 1.81453i −1.99132 1.49219 − 2.40279i −0.657973 − 0.657973i 0
107.6 −0.834038 + 1.14209i 0.720668 + 0.298510i −0.608760 − 1.90510i 0 −0.941991 + 0.574102i −0.532816 2.68354 + 0.893665i −1.69107 − 1.69107i 0
107.7 −0.120950 − 1.40903i 0.0931833 + 0.0385978i −1.97074 + 0.340845i 0 0.0431150 − 0.135967i −2.95538 0.718624 + 2.73561i −2.11413 − 2.11413i 0
107.8 −0.104415 + 1.41035i −2.67270 − 1.10707i −1.97820 − 0.294524i 0 1.84043 − 3.65385i −2.46590 0.621936 − 2.75920i 3.79639 + 3.79639i 0
107.9 0.104415 − 1.41035i 2.67270 + 1.10707i −1.97820 − 0.294524i 0 1.84043 − 3.65385i 2.46590 −0.621936 + 2.75920i 3.79639 + 3.79639i 0
107.10 0.120950 + 1.40903i −0.0931833 − 0.0385978i −1.97074 + 0.340845i 0 0.0431150 − 0.135967i 2.95538 −0.718624 − 2.73561i −2.11413 − 2.11413i 0
107.11 0.834038 − 1.14209i −0.720668 − 0.298510i −0.608760 − 1.90510i 0 −0.941991 + 0.574102i 0.532816 −2.68354 − 0.893665i −1.69107 − 1.69107i 0
107.12 1.07355 + 0.920591i −1.32907 − 0.550517i 0.305023 + 1.97660i 0 −0.920018 − 1.81453i 1.99132 −1.49219 + 2.40279i −0.657973 − 0.657973i 0
107.13 1.18499 + 0.771881i 2.16069 + 0.894988i 0.808400 + 1.82934i 0 1.86957 + 2.72835i 2.35189 −0.454089 + 2.79174i 1.74627 + 1.74627i 0
107.14 1.33577 − 0.464441i 1.95643 + 0.810381i 1.56859 − 1.24078i 0 2.98973 + 0.173839i −1.26392 1.51901 − 2.38592i 1.04960 + 1.04960i 0
107.15 1.37681 + 0.323096i −0.961950 − 0.398453i 1.79122 + 0.889684i 0 −1.19569 − 0.859396i −4.93988 2.17872 + 1.80366i −1.35474 − 1.35474i 0
107.16 1.37688 − 0.322819i −2.90764 − 1.20438i 1.79158 − 0.888965i 0 −4.39226 − 0.719644i 3.66122 2.17980 − 1.80235i 4.88250 + 4.88250i 0
243.1 −1.39837 + 0.211115i 0.851236 + 2.05507i 1.91086 − 0.590432i 0 −1.62419 − 2.69403i 1.56191 −2.54744 + 1.22905i −1.37737 + 1.37737i 0
243.2 −1.37378 − 0.335731i −1.26810 − 3.06146i 1.77457 + 0.922445i 0 0.714265 + 4.63152i 2.27663 −2.12818 − 1.86302i −5.64312 + 5.64312i 0
243.3 −1.28234 + 0.596331i −0.379767 − 0.916838i 1.28878 − 1.52940i 0 1.03373 + 0.949229i −2.31537 −0.740621 + 2.72974i 1.42495 − 1.42495i 0
243.4 −1.27313 − 0.615752i 0.133851 + 0.323146i 1.24170 + 1.56786i 0 0.0285681 − 0.493824i 1.30191 −0.615425 − 2.76066i 2.03481 − 2.03481i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
160.u even 8 1 inner
160.ba 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.bb.a yes 64
5.b even 2 1 inner 800.2.bb.a yes 64
5.c odd 4 2 800.2.v.a ✓ 64
32.h odd 8 1 800.2.v.a ✓ 64
160.u even 8 1 inner 800.2.bb.a yes 64
160.y odd 8 1 800.2.v.a ✓ 64
160.ba even 8 1 inner 800.2.bb.a yes 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.v.a ✓ 64 5.c odd 4 2
800.2.v.a ✓ 64 32.h odd 8 1
800.2.v.a ✓ 64 160.y odd 8 1
800.2.bb.a yes 64 1.a even 1 1 trivial
800.2.bb.a yes 64 5.b even 2 1 inner
800.2.bb.a yes 64 160.u even 8 1 inner
800.2.bb.a yes 64 160.ba even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} - 32 T_{3}^{58} + 16800 T_{3}^{56} - 9920 T_{3}^{54} + 512 T_{3}^{52} + 973120 T_{3}^{50} + \cdots + 71639296 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display