Properties

Label 275.3.bk.a
Level $275$
Weight $3$
Character orbit 275.bk
Analytic conductor $7.493$
Analytic rank $0$
Dimension $64$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,3,Mod(82,275)] 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("275.82"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([5, 8])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 275.bk (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.49320726991\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 48 q^{6} - 20 q^{11} - 16 q^{16} - 160 q^{21} - 216 q^{26} - 80 q^{31} + 192 q^{36} + 820 q^{41} + 180 q^{46} - 80 q^{51} + 1040 q^{56} + 824 q^{61} - 220 q^{66} - 1304 q^{71} - 3648 q^{76} - 1652 q^{81}+ \cdots + 2252 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1 −1.71217 + 3.36033i −2.07939 + 0.329343i −6.00911 − 8.27084i 0 2.45357 − 7.55131i 6.84618 + 1.08433i 23.1815 − 3.67159i −4.34412 + 1.41149i 0
82.2 −1.24641 + 2.44622i 3.36674 − 0.533239i −2.07931 − 2.86192i 0 −2.89192 + 8.90041i −13.0610 − 2.06866i −1.25407 + 0.198625i 2.49108 − 0.809399i 0
82.3 −0.680679 + 1.33591i −4.31037 + 0.682695i 1.02982 + 1.41742i 0 2.02196 − 6.22295i −0.990655 − 0.156904i −8.51798 + 1.34911i 9.55368 − 3.10418i 0
82.4 −0.616707 + 1.21036i 1.92862 − 0.305464i 1.26651 + 1.74320i 0 −0.819676 + 2.52270i 8.20578 + 1.29967i −8.25771 + 1.30789i −4.93322 + 1.60290i 0
82.5 0.616707 − 1.21036i −1.92862 + 0.305464i 1.26651 + 1.74320i 0 −0.819676 + 2.52270i −8.20578 − 1.29967i 8.25771 − 1.30789i −4.93322 + 1.60290i 0
82.6 0.680679 − 1.33591i 4.31037 − 0.682695i 1.02982 + 1.41742i 0 2.02196 − 6.22295i 0.990655 + 0.156904i 8.51798 − 1.34911i 9.55368 − 3.10418i 0
82.7 1.24641 − 2.44622i −3.36674 + 0.533239i −2.07931 − 2.86192i 0 −2.89192 + 8.90041i 13.0610 + 2.06866i 1.25407 − 0.198625i 2.49108 − 0.809399i 0
82.8 1.71217 − 3.36033i 2.07939 − 0.329343i −6.00911 − 8.27084i 0 2.45357 − 7.55131i −6.84618 − 1.08433i −23.1815 + 3.67159i −4.34412 + 1.41149i 0
93.1 −3.36033 − 1.71217i 0.329343 + 2.07939i 6.00911 + 8.27084i 0 2.45357 − 7.55131i −1.08433 + 6.84618i −3.67159 − 23.1815i 4.34412 − 1.41149i 0
93.2 −2.44622 − 1.24641i −0.533239 − 3.36674i 2.07931 + 2.86192i 0 −2.89192 + 8.90041i 2.06866 − 13.0610i 0.198625 + 1.25407i −2.49108 + 0.809399i 0
93.3 −1.33591 − 0.680679i 0.682695 + 4.31037i −1.02982 − 1.41742i 0 2.02196 − 6.22295i 0.156904 − 0.990655i 1.34911 + 8.51798i −9.55368 + 3.10418i 0
93.4 −1.21036 − 0.616707i −0.305464 − 1.92862i −1.26651 − 1.74320i 0 −0.819676 + 2.52270i −1.29967 + 8.20578i 1.30789 + 8.25771i 4.93322 − 1.60290i 0
93.5 1.21036 + 0.616707i 0.305464 + 1.92862i −1.26651 − 1.74320i 0 −0.819676 + 2.52270i 1.29967 − 8.20578i −1.30789 − 8.25771i 4.93322 − 1.60290i 0
93.6 1.33591 + 0.680679i −0.682695 − 4.31037i −1.02982 − 1.41742i 0 2.02196 − 6.22295i −0.156904 + 0.990655i −1.34911 − 8.51798i −9.55368 + 3.10418i 0
93.7 2.44622 + 1.24641i 0.533239 + 3.36674i 2.07931 + 2.86192i 0 −2.89192 + 8.90041i −2.06866 + 13.0610i −0.198625 − 1.25407i −2.49108 + 0.809399i 0
93.8 3.36033 + 1.71217i −0.329343 − 2.07939i 6.00911 + 8.27084i 0 2.45357 − 7.55131i 1.08433 − 6.84618i 3.67159 + 23.1815i 4.34412 − 1.41149i 0
157.1 −3.36545 − 0.533034i −3.29832 − 1.68058i 7.23787 + 2.35173i 0 10.2045 + 7.41400i 3.14032 − 1.60007i −10.9611 − 5.58494i 2.76449 + 3.80499i 0
157.2 −2.40167 − 0.380388i 1.53559 + 0.782422i 1.81912 + 0.591068i 0 −3.39036 − 2.46324i −8.81808 + 4.49303i 4.52223 + 2.30419i −3.54422 − 4.87819i 0
157.3 −1.09452 − 0.173355i −1.26289 − 0.643477i −2.63631 − 0.856588i 0 1.27071 + 0.923226i 6.44198 − 3.28236i 6.68651 + 3.40695i −4.10923 − 5.65587i 0
157.4 −0.620587 − 0.0982913i 4.99343 + 2.54428i −3.42876 − 1.11407i 0 −2.84878 − 2.06976i 9.44698 − 4.81348i 4.25770 + 2.16941i 13.1710 + 18.1283i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 82.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
11.c even 5 1 inner
55.j even 10 1 inner
55.k odd 20 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.3.bk.a ✓ 64
5.b even 2 1 inner 275.3.bk.a ✓ 64
5.c odd 4 2 inner 275.3.bk.a ✓ 64
11.c even 5 1 inner 275.3.bk.a ✓ 64
55.j even 10 1 inner 275.3.bk.a ✓ 64
55.k odd 20 2 inner 275.3.bk.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.3.bk.a ✓ 64 1.a even 1 1 trivial
275.3.bk.a ✓ 64 5.b even 2 1 inner
275.3.bk.a ✓ 64 5.c odd 4 2 inner
275.3.bk.a ✓ 64 11.c even 5 1 inner
275.3.bk.a ✓ 64 55.j even 10 1 inner
275.3.bk.a ✓ 64 55.k odd 20 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{64} - 112 T_{2}^{60} + 36030 T_{2}^{56} - 7618144 T_{2}^{52} + 963451671 T_{2}^{48} + \cdots + 47\!\cdots\!01 \) acting on \(S_{3}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display