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Properties

Label 800.2.bq.b
Level $800$
Weight $2$
Character orbit 800.bq
Analytic conductor $6.388$
Analytic rank $0$
Dimension $56$
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(63,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 0, 19]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.63");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.bq (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: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(7\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 2 q^{5} + 4 q^{7} - 6 q^{13} + 42 q^{15} - 2 q^{17} + 18 q^{19} + 16 q^{21} + 8 q^{23} - 34 q^{25} - 42 q^{27} - 20 q^{31} - 36 q^{33} - 22 q^{35} + 20 q^{37} + 36 q^{39} + 16 q^{41} + 32 q^{43} - 4 q^{45} + 8 q^{47} + 48 q^{53} - 8 q^{55} - 8 q^{57} + 4 q^{59} + 36 q^{61} - 2 q^{63} + 2 q^{65} - 6 q^{67} - 60 q^{69} + 10 q^{71} - 2 q^{73} - 26 q^{75} - 72 q^{77} - 16 q^{79} + 54 q^{81} + 28 q^{83} + 62 q^{85} - 24 q^{87} + 30 q^{89} + 30 q^{91} + 20 q^{93} - 8 q^{95} - 46 q^{97} - 132 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1 0 −2.26230 − 1.15270i 0 −0.350884 + 2.20837i 0 −2.25510 − 2.25510i 0 2.02593 + 2.78846i 0
63.2 0 −1.89091 − 0.963465i 0 −1.89826 − 1.18178i 0 2.21349 + 2.21349i 0 0.883909 + 1.21660i 0
63.3 0 −1.46116 − 0.744497i 0 1.78517 − 1.34654i 0 0.112121 + 0.112121i 0 −0.182649 − 0.251395i 0
63.4 0 0.483226 + 0.246216i 0 0.793395 + 2.09058i 0 1.92115 + 1.92115i 0 −1.59047 − 2.18909i 0
63.5 0 0.988556 + 0.503694i 0 −2.10284 + 0.760299i 0 −1.41729 − 1.41729i 0 −1.03982 − 1.43119i 0
63.6 0 1.96776 + 1.00262i 0 −0.797421 − 2.08905i 0 3.41975 + 3.41975i 0 1.10348 + 1.51880i 0
63.7 0 2.17482 + 1.10813i 0 0.977747 − 2.01097i 0 −2.71003 − 2.71003i 0 1.73855 + 2.39291i 0
127.1 0 −2.26230 + 1.15270i 0 −0.350884 − 2.20837i 0 −2.25510 + 2.25510i 0 2.02593 − 2.78846i 0
127.2 0 −1.89091 + 0.963465i 0 −1.89826 + 1.18178i 0 2.21349 − 2.21349i 0 0.883909 − 1.21660i 0
127.3 0 −1.46116 + 0.744497i 0 1.78517 + 1.34654i 0 0.112121 − 0.112121i 0 −0.182649 + 0.251395i 0
127.4 0 0.483226 − 0.246216i 0 0.793395 − 2.09058i 0 1.92115 − 1.92115i 0 −1.59047 + 2.18909i 0
127.5 0 0.988556 − 0.503694i 0 −2.10284 − 0.760299i 0 −1.41729 + 1.41729i 0 −1.03982 + 1.43119i 0
127.6 0 1.96776 − 1.00262i 0 −0.797421 + 2.08905i 0 3.41975 − 3.41975i 0 1.10348 − 1.51880i 0
127.7 0 2.17482 − 1.10813i 0 0.977747 + 2.01097i 0 −2.71003 + 2.71003i 0 1.73855 − 2.39291i 0
223.1 0 −1.18617 − 2.32800i 0 0.966683 + 2.01631i 0 0.0730276 + 0.0730276i 0 −2.24921 + 3.09578i 0
223.2 0 −0.802897 − 1.57577i 0 1.58800 − 1.57424i 0 1.63701 + 1.63701i 0 −0.0750643 + 0.103317i 0
223.3 0 −0.640680 − 1.25741i 0 −2.21664 + 0.294142i 0 1.10768 + 1.10768i 0 0.592758 − 0.815862i 0
223.4 0 0.0719261 + 0.141163i 0 −0.475770 + 2.18487i 0 −2.98667 − 2.98667i 0 1.74860 − 2.40674i 0
223.5 0 0.272462 + 0.534736i 0 1.53374 − 1.62716i 0 −2.42260 − 2.42260i 0 1.55165 − 2.13566i 0
223.6 0 0.821892 + 1.61305i 0 −1.22855 − 1.86833i 0 1.10068 + 1.10068i 0 −0.163081 + 0.224462i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
100.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.bq.b yes 56
4.b odd 2 1 800.2.bq.a ✓ 56
25.f odd 20 1 800.2.bq.a ✓ 56
100.l even 20 1 inner 800.2.bq.b yes 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.bq.a ✓ 56 4.b odd 2 1
800.2.bq.a ✓ 56 25.f odd 20 1
800.2.bq.b yes 56 1.a even 1 1 trivial
800.2.bq.b yes 56 100.l even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{56} + 14 T_{3}^{53} - 99 T_{3}^{52} - 164 T_{3}^{51} - 82 T_{3}^{50} - 770 T_{3}^{49} + \cdots + 87684496 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display