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Properties

Label 888.2.r.e
Level $888$
Weight $2$
Character orbit 888.r
Analytic conductor $7.091$
Analytic rank $0$
Dimension $72$
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(43,888)] 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("888.43"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(888, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 0, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 888 = 2^{3} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 888.r (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,2,0,0,0,-2,0,2,-72,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.09071569949\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 2 q^{2} - 2 q^{6} + 2 q^{8} - 72 q^{9} + 8 q^{10} - 8 q^{12} + 20 q^{14} + 12 q^{16} + 12 q^{17} - 2 q^{18} + 20 q^{19} - 22 q^{20} - 16 q^{22} + 2 q^{24} - 32 q^{26} - 8 q^{32} - 16 q^{33} + 8 q^{34}+ \cdots - 110 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.41405 + 0.0214268i − 1.00000i 1.99908 − 0.0605971i −2.93098 + 2.93098i 0.0214268 + 1.41405i 1.80046i −2.82551 + 0.128521i −1.00000 4.08175 − 4.20735i
43.2 −1.41194 − 0.0802186i − 1.00000i 1.98713 + 0.226527i 0.922211 − 0.922211i −0.0802186 + 1.41194i 1.23837i −2.78753 − 0.479246i −1.00000 −1.37608 + 1.22813i
43.3 −1.40508 + 0.160452i − 1.00000i 1.94851 − 0.450896i 2.38279 − 2.38279i 0.160452 + 1.40508i − 3.14379i −2.66547 + 0.946188i −1.00000 −2.96569 + 3.73034i
43.4 −1.27341 + 0.615171i − 1.00000i 1.24313 − 1.56673i 0.125657 − 0.125657i 0.615171 + 1.27341i 2.31726i −0.619204 + 2.75982i −1.00000 −0.0827118 + 0.237313i
43.5 −1.27302 − 0.615974i − 1.00000i 1.24115 + 1.56829i −0.116713 + 0.116713i −0.615974 + 1.27302i 4.08372i −0.613982 − 2.76098i −1.00000 0.220470 − 0.0766855i
43.6 −1.25521 + 0.651493i − 1.00000i 1.15111 − 1.63552i 1.15383 − 1.15383i 0.651493 + 1.25521i 1.69453i −0.379359 + 2.80287i −1.00000 −0.696586 + 2.20000i
43.7 −1.25296 − 0.655808i − 1.00000i 1.13983 + 1.64341i −1.22832 + 1.22832i −0.655808 + 1.25296i − 2.71379i −0.350405 − 2.80664i −1.00000 2.34458 − 0.733495i
43.8 −1.12946 + 0.851073i − 1.00000i 0.551349 − 1.92250i −1.41934 + 1.41934i 0.851073 + 1.12946i − 1.22240i 1.01347 + 2.64062i −1.00000 0.395121 − 2.81104i
43.9 −1.06809 − 0.926925i − 1.00000i 0.281621 + 1.98007i −2.48900 + 2.48900i −0.926925 + 1.06809i − 1.23837i 1.53458 − 2.37593i −1.00000 4.96558 − 0.351353i
43.10 −0.851073 + 1.12946i − 1.00000i −0.551349 − 1.92250i 1.41934 − 1.41934i 1.12946 + 0.851073i 1.22240i 2.64062 + 1.01347i −1.00000 0.395121 + 2.81104i
43.11 −0.765018 − 1.18943i − 1.00000i −0.829494 + 1.81987i −0.186808 + 0.186808i −1.18943 + 0.765018i − 1.09998i 2.79919 − 0.405610i −1.00000 0.365108 + 0.0792839i
43.12 −0.651493 + 1.25521i − 1.00000i −1.15111 − 1.63552i −1.15383 + 1.15383i 1.25521 + 0.651493i − 1.69453i 2.80287 − 0.379359i −1.00000 −0.696586 − 2.20000i
43.13 −0.615171 + 1.27341i − 1.00000i −1.24313 − 1.56673i −0.125657 + 0.125657i 1.27341 + 0.615171i − 2.31726i 2.75982 − 0.619204i −1.00000 −0.0827118 − 0.237313i
43.14 −0.476177 − 1.33164i − 1.00000i −1.54651 + 1.26819i 0.0505532 − 0.0505532i −1.33164 + 0.476177i 3.49253i 2.42518 + 1.45551i −1.00000 −0.0913908 − 0.0432462i
43.15 −0.398861 − 1.35680i − 1.00000i −1.68182 + 1.08235i 2.91128 − 2.91128i −1.35680 + 0.398861i 4.26853i 2.13935 + 1.85019i −1.00000 −5.11123 − 2.78884i
43.16 −0.160452 + 1.40508i − 1.00000i −1.94851 − 0.450896i −2.38279 + 2.38279i 1.40508 + 0.160452i 3.14379i 0.946188 − 2.66547i −1.00000 −2.96569 − 3.73034i
43.17 −0.0909954 − 1.41128i − 1.00000i −1.98344 + 0.256841i 1.25714 − 1.25714i −1.41128 + 0.0909954i − 5.01253i 0.542959 + 2.77582i −1.00000 −1.88857 − 1.65978i
43.18 −0.0214268 + 1.41405i − 1.00000i −1.99908 − 0.0605971i 2.93098 − 2.93098i 1.41405 + 0.0214268i − 1.80046i 0.128521 − 2.82551i −1.00000 4.08175 + 4.20735i
43.19 0.0571571 − 1.41306i − 1.00000i −1.99347 − 0.161533i −0.322645 + 0.322645i −1.41306 − 0.0571571i 0.857064i −0.342196 + 2.80765i −1.00000 0.437474 + 0.474357i
43.20 0.0802186 + 1.41194i − 1.00000i −1.98713 + 0.226527i −0.922211 + 0.922211i 1.41194 − 0.0802186i − 1.23837i −0.479246 − 2.78753i −1.00000 −1.37608 − 1.22813i
See all 72 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 43.36
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
37.d odd 4 1 inner
296.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 888.2.r.e ✓ 72
8.d odd 2 1 inner 888.2.r.e ✓ 72
37.d odd 4 1 inner 888.2.r.e ✓ 72
296.j even 4 1 inner 888.2.r.e ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
888.2.r.e ✓ 72 1.a even 1 1 trivial
888.2.r.e ✓ 72 8.d odd 2 1 inner
888.2.r.e ✓ 72 37.d odd 4 1 inner
888.2.r.e ✓ 72 296.j even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(888, [\chi])\):

\( T_{5}^{72} + 1276 T_{5}^{68} + 677168 T_{5}^{64} + 194258144 T_{5}^{60} + 32835461504 T_{5}^{56} + \cdots + 1048576 \) Copy content Toggle raw display
\( T_{11}^{36} + 244 T_{11}^{34} + 26542 T_{11}^{32} + 1701956 T_{11}^{30} + 71699385 T_{11}^{28} + \cdots + 18806446489600 \) Copy content Toggle raw display