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Properties

Label 800.2.bq.c
Level $800$
Weight $2$
Character orbit 800.bq
Analytic conductor $6.388$
Analytic rank $0$
Dimension $64$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [800,2,Mod(63,800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://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);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.bq (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64,0,0,0,2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 2 q^{5} - 4 q^{7} + 4 q^{13} - 22 q^{15} + 8 q^{17} + 18 q^{19} - 16 q^{21} - 8 q^{23} + 40 q^{25} - 18 q^{27} + 20 q^{31} + 44 q^{33} - 38 q^{35} - 10 q^{37} + 36 q^{39} - 16 q^{41} - 32 q^{43}+ \cdots - 132 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 −2.49551 − 1.27153i 0 −1.46592 + 1.68851i 0 1.17365 + 1.17365i 0 2.84744 + 3.91917i 0
63.2 0 −2.31654 − 1.18033i 0 2.18236 − 0.487149i 0 −2.41634 − 2.41634i 0 2.20979 + 3.04152i 0
63.3 0 −0.696591 − 0.354931i 0 −2.17155 − 0.533278i 0 −1.62475 − 1.62475i 0 −1.40409 − 1.93257i 0
63.4 0 −0.685472 − 0.349265i 0 1.93228 + 1.12530i 0 3.37195 + 3.37195i 0 −1.41547 − 1.94823i 0
63.5 0 0.306774 + 0.156309i 0 0.187686 − 2.22818i 0 1.04191 + 1.04191i 0 −1.69368 − 2.33115i 0
63.6 0 0.675433 + 0.344150i 0 −0.294150 + 2.21664i 0 −2.26261 − 2.26261i 0 −1.42559 − 1.96215i 0
63.7 0 2.21160 + 1.12686i 0 2.10230 + 0.761808i 0 −0.390880 − 0.390880i 0 1.85798 + 2.55729i 0
63.8 0 3.00031 + 1.52873i 0 −1.83096 − 1.28358i 0 −0.177009 − 0.177009i 0 4.90147 + 6.74629i 0
127.1 0 −2.49551 + 1.27153i 0 −1.46592 − 1.68851i 0 1.17365 − 1.17365i 0 2.84744 − 3.91917i 0
127.2 0 −2.31654 + 1.18033i 0 2.18236 + 0.487149i 0 −2.41634 + 2.41634i 0 2.20979 − 3.04152i 0
127.3 0 −0.696591 + 0.354931i 0 −2.17155 + 0.533278i 0 −1.62475 + 1.62475i 0 −1.40409 + 1.93257i 0
127.4 0 −0.685472 + 0.349265i 0 1.93228 − 1.12530i 0 3.37195 − 3.37195i 0 −1.41547 + 1.94823i 0
127.5 0 0.306774 − 0.156309i 0 0.187686 + 2.22818i 0 1.04191 − 1.04191i 0 −1.69368 + 2.33115i 0
127.6 0 0.675433 − 0.344150i 0 −0.294150 − 2.21664i 0 −2.26261 + 2.26261i 0 −1.42559 + 1.96215i 0
127.7 0 2.21160 − 1.12686i 0 2.10230 − 0.761808i 0 −0.390880 + 0.390880i 0 1.85798 − 2.55729i 0
127.8 0 3.00031 − 1.52873i 0 −1.83096 + 1.28358i 0 −0.177009 + 0.177009i 0 4.90147 − 6.74629i 0
223.1 0 −1.39978 − 2.74723i 0 −1.70274 − 1.44937i 0 2.85341 + 2.85341i 0 −3.82451 + 5.26398i 0
223.2 0 −1.07219 − 2.10429i 0 1.72608 − 1.42149i 0 −3.37202 − 3.37202i 0 −1.51510 + 2.08536i 0
223.3 0 −0.751816 − 1.47552i 0 −0.850652 + 2.06794i 0 −0.0884893 − 0.0884893i 0 0.151419 − 0.208410i 0
223.4 0 0.0251148 + 0.0492906i 0 1.96222 + 1.07225i 0 0.693303 + 0.693303i 0 1.76156 − 2.42458i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.8
Significant digits:
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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.c ✓ 64
4.b odd 2 1 800.2.bq.d yes 64
25.f odd 20 1 800.2.bq.d yes 64
100.l even 20 1 inner 800.2.bq.c ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.bq.c ✓ 64 1.a even 1 1 trivial
800.2.bq.c ✓ 64 100.l even 20 1 inner
800.2.bq.d yes 64 4.b odd 2 1
800.2.bq.d yes 64 25.f odd 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{64} + 6 T_{3}^{61} - 111 T_{3}^{60} + 148 T_{3}^{59} - 42 T_{3}^{58} + 358 T_{3}^{57} + \cdots + 110166016 \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display