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Properties

Label 888.2.o.a
Level $888$
Weight $2$
Character orbit 888.o
Analytic conductor $7.091$
Analytic rank $0$
Dimension $76$
Inner twists $4$

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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] = [888,2,Mod(517,888)] 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("888.517"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.o (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 76 q + 4 q^{4} + 8 q^{7} - 76 q^{9} + 4 q^{16} + 84 q^{25} + 12 q^{28} + 8 q^{30} - 12 q^{34} - 4 q^{36} + 8 q^{38} - 56 q^{40} - 8 q^{41} - 24 q^{44} - 44 q^{46} - 8 q^{48} + 60 q^{49} - 40 q^{58} + 32 q^{62}+ \cdots - 56 q^{86}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
517.1 −1.40736 − 0.139090i 1.00000i 1.96131 + 0.391499i −2.38070 0.139090 − 1.40736i 1.41343 −2.70581 − 0.823778i −1.00000 3.35049 + 0.331132i
517.2 −1.40736 + 0.139090i − 1.00000i 1.96131 − 0.391499i −2.38070 0.139090 + 1.40736i 1.41343 −2.70581 + 0.823778i −1.00000 3.35049 − 0.331132i
517.3 −1.39994 − 0.200399i 1.00000i 1.91968 + 0.561094i 2.44038 0.200399 − 1.39994i 4.61327 −2.57500 − 1.17020i −1.00000 −3.41639 − 0.489048i
517.4 −1.39994 + 0.200399i − 1.00000i 1.91968 − 0.561094i 2.44038 0.200399 + 1.39994i 4.61327 −2.57500 + 1.17020i −1.00000 −3.41639 + 0.489048i
517.5 −1.39216 − 0.248794i − 1.00000i 1.87620 + 0.692722i 1.62102 −0.248794 + 1.39216i −0.682716 −2.43962 − 1.43117i −1.00000 −2.25671 − 0.403300i
517.6 −1.39216 + 0.248794i 1.00000i 1.87620 − 0.692722i 1.62102 −0.248794 − 1.39216i −0.682716 −2.43962 + 1.43117i −1.00000 −2.25671 + 0.403300i
517.7 −1.38047 − 0.307088i − 1.00000i 1.81139 + 0.847851i 3.72444 −0.307088 + 1.38047i −1.69680 −2.24021 − 1.72669i −1.00000 −5.14148 − 1.14373i
517.8 −1.38047 + 0.307088i 1.00000i 1.81139 − 0.847851i 3.72444 −0.307088 − 1.38047i −1.69680 −2.24021 + 1.72669i −1.00000 −5.14148 + 1.14373i
517.9 −1.34547 − 0.435559i 1.00000i 1.62058 + 1.17206i 0.598100 0.435559 − 1.34547i −4.50845 −1.66994 − 2.28283i −1.00000 −0.804725 − 0.260508i
517.10 −1.34547 + 0.435559i − 1.00000i 1.62058 − 1.17206i 0.598100 0.435559 + 1.34547i −4.50845 −1.66994 + 2.28283i −1.00000 −0.804725 + 0.260508i
517.11 −1.28459 − 0.591458i − 1.00000i 1.30036 + 1.51956i −4.07371 −0.591458 + 1.28459i −2.80969 −0.771669 − 2.72113i −1.00000 5.23305 + 2.40942i
517.12 −1.28459 + 0.591458i 1.00000i 1.30036 − 1.51956i −4.07371 −0.591458 − 1.28459i −2.80969 −0.771669 + 2.72113i −1.00000 5.23305 − 2.40942i
517.13 −1.21738 − 0.719718i − 1.00000i 0.964012 + 1.75234i −1.66630 −0.719718 + 1.21738i 4.35248 0.0876211 − 2.82707i −1.00000 2.02852 + 1.19927i
517.14 −1.21738 + 0.719718i 1.00000i 0.964012 − 1.75234i −1.66630 −0.719718 − 1.21738i 4.35248 0.0876211 + 2.82707i −1.00000 2.02852 − 1.19927i
517.15 −1.15841 − 0.811225i 1.00000i 0.683827 + 1.87946i −0.690214 0.811225 − 1.15841i 3.18564 0.732517 − 2.73193i −1.00000 0.799551 + 0.559919i
517.16 −1.15841 + 0.811225i − 1.00000i 0.683827 − 1.87946i −0.690214 0.811225 + 1.15841i 3.18564 0.732517 + 2.73193i −1.00000 0.799551 − 0.559919i
517.17 −1.07422 − 0.919807i 1.00000i 0.307910 + 1.97616i 1.74292 0.919807 − 1.07422i −1.43072 1.48692 − 2.40605i −1.00000 −1.87229 − 1.60315i
517.18 −1.07422 + 0.919807i − 1.00000i 0.307910 − 1.97616i 1.74292 0.919807 + 1.07422i −1.43072 1.48692 + 2.40605i −1.00000 −1.87229 + 1.60315i
517.19 −1.02048 − 0.979091i 1.00000i 0.0827621 + 1.99829i −3.42335 0.979091 − 1.02048i 1.31476 1.87205 − 2.12024i −1.00000 3.49346 + 3.35177i
517.20 −1.02048 + 0.979091i − 1.00000i 0.0827621 − 1.99829i −3.42335 0.979091 + 1.02048i 1.31476 1.87205 + 2.12024i −1.00000 3.49346 − 3.35177i
See all 76 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 517.76
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
37.b even 2 1 inner
296.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 888.2.o.a ✓ 76
4.b odd 2 1 3552.2.o.a 76
8.b even 2 1 inner 888.2.o.a ✓ 76
8.d odd 2 1 3552.2.o.a 76
37.b even 2 1 inner 888.2.o.a ✓ 76
148.b odd 2 1 3552.2.o.a 76
296.e even 2 1 inner 888.2.o.a ✓ 76
296.h odd 2 1 3552.2.o.a 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
888.2.o.a ✓ 76 1.a even 1 1 trivial
888.2.o.a ✓ 76 8.b even 2 1 inner
888.2.o.a ✓ 76 37.b even 2 1 inner
888.2.o.a ✓ 76 296.e even 2 1 inner
3552.2.o.a 76 4.b odd 2 1
3552.2.o.a 76 8.d odd 2 1
3552.2.o.a 76 148.b odd 2 1
3552.2.o.a 76 296.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(888, [\chi])\).