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Properties

Label 528.2.bv.a
Level $528$
Weight $2$
Character orbit 528.bv
Analytic conductor $4.216$
Analytic rank $0$
Dimension $384$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [528,2,Mod(19,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 15, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("528.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 528.bv (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: \(4.21610122672\)
Analytic rank: \(0\)
Dimension: \(384\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 384 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 384 q + 8 q^{4} - 8 q^{11} + 16 q^{12} - 4 q^{14} - 8 q^{16} - 20 q^{18} + 16 q^{20} + 60 q^{22} - 32 q^{23} - 60 q^{26} + 40 q^{30} - 40 q^{34} + 48 q^{37} - 56 q^{38} + 32 q^{44} + 96 q^{49} + 40 q^{52} - 48 q^{53} - 32 q^{55} + 56 q^{56} - 8 q^{58} - 32 q^{59} - 24 q^{60} - 16 q^{64} - 24 q^{66} + 16 q^{67} - 40 q^{70} - 64 q^{71} - 20 q^{72} - 100 q^{74} - 48 q^{75} - 16 q^{77} - 72 q^{78} - 92 q^{80} + 96 q^{81} - 20 q^{82} + 200 q^{83} - 120 q^{84} - 112 q^{86} - 120 q^{88} - 40 q^{90} - 104 q^{91} - 88 q^{92} - 260 q^{94} - 100 q^{96} + 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.40692 + 0.143465i 0.891007 + 0.453990i 1.95884 − 0.403687i −2.64546 + 0.419000i −1.31870 − 0.510899i −0.432379 − 0.140488i −2.69801 + 0.848980i 0.587785 + 0.809017i 3.66184 − 0.969030i
19.2 −1.39300 − 0.244049i −0.891007 − 0.453990i 1.88088 + 0.679919i −3.79815 + 0.601568i 1.13037 + 0.849857i −0.636765 − 0.206898i −2.45413 − 1.40615i 0.587785 + 0.809017i 5.43762 + 0.0889525i
19.3 −1.37350 + 0.336891i 0.891007 + 0.453990i 1.77301 − 0.925440i 2.58767 − 0.409847i −1.37674 − 0.323384i −0.130795 − 0.0424978i −2.12346 + 1.86840i 0.587785 + 0.809017i −3.41610 + 1.43469i
19.4 −1.36829 + 0.357471i −0.891007 − 0.453990i 1.74443 − 0.978247i −0.754949 + 0.119572i 1.38144 + 0.302681i −1.07145 − 0.348136i −2.03719 + 1.96211i 0.587785 + 0.809017i 0.990245 − 0.433482i
19.5 −1.36828 − 0.357507i −0.891007 − 0.453990i 1.74438 + 0.978339i 4.22012 − 0.668402i 1.05684 + 0.939727i −2.35447 − 0.765014i −2.03703 − 1.96227i 0.587785 + 0.809017i −6.01327 − 0.594163i
19.6 −1.33556 + 0.465049i −0.891007 − 0.453990i 1.56746 − 1.24221i 2.64905 − 0.419568i 1.40112 + 0.191971i 4.12108 + 1.33902i −1.51575 + 2.38799i 0.587785 + 0.809017i −3.34285 + 1.79230i
19.7 −1.30063 + 0.555307i 0.891007 + 0.453990i 1.38327 − 1.44450i 1.43499 − 0.227280i −1.41097 − 0.0956911i −4.59288 − 1.49232i −0.996981 + 2.64689i 0.587785 + 0.809017i −1.74018 + 1.09247i
19.8 −1.27848 − 0.604558i −0.891007 − 0.453990i 1.26902 + 1.54583i 0.485072 − 0.0768279i 0.864670 + 1.11908i 4.08037 + 1.32579i −0.687874 − 2.74351i 0.587785 + 0.809017i −0.666602 − 0.195031i
19.9 −1.27586 − 0.610077i 0.891007 + 0.453990i 1.25561 + 1.55674i −1.75312 + 0.277667i −0.859826 − 1.12281i −1.26259 − 0.410241i −0.652250 − 2.75219i 0.587785 + 0.809017i 2.40612 + 0.715275i
19.10 −1.24120 + 0.677808i 0.891007 + 0.453990i 1.08115 − 1.68259i −2.00350 + 0.317323i −1.41364 + 0.0404387i 3.88637 + 1.26276i −0.201453 + 2.82124i 0.587785 + 0.809017i 2.27166 − 1.75185i
19.11 −1.22068 − 0.714096i 0.891007 + 0.453990i 0.980135 + 1.74337i 0.525648 − 0.0832545i −0.763444 − 1.19044i −1.43759 − 0.467102i 0.0484985 − 2.82801i 0.587785 + 0.809017i −0.701102 − 0.273736i
19.12 −1.11063 + 0.875502i −0.891007 − 0.453990i 0.466992 − 1.94472i −0.283198 + 0.0448542i 1.38705 − 0.275863i −3.75723 − 1.22080i 1.18395 + 2.56871i 0.587785 + 0.809017i 0.275258 − 0.297757i
19.13 −0.903428 + 1.08803i −0.891007 − 0.453990i −0.367637 − 1.96592i −2.02216 + 0.320278i 1.29892 − 0.559298i 2.64421 + 0.859156i 2.47112 + 1.37607i 0.587785 + 0.809017i 1.47840 − 2.48952i
19.14 −0.902926 − 1.08845i 0.891007 + 0.453990i −0.369448 + 1.96558i 2.70476 − 0.428391i −0.310367 − 1.37974i 3.51756 + 1.14293i 2.47302 − 1.37265i 0.587785 + 0.809017i −2.90848 − 2.55719i
19.15 −0.889047 − 1.09982i −0.891007 − 0.453990i −0.419193 + 1.95558i 1.53559 − 0.243214i 0.292840 + 1.38356i −1.39105 − 0.451980i 2.52346 − 1.27756i 0.587785 + 0.809017i −1.63270 − 1.47264i
19.16 −0.848087 + 1.13170i 0.891007 + 0.453990i −0.561496 − 1.91956i 2.73337 − 0.432923i −1.26943 + 0.623330i 2.88019 + 0.935832i 2.64857 + 0.992512i 0.587785 + 0.809017i −1.82820 + 3.46051i
19.17 −0.798575 + 1.16717i −0.891007 − 0.453990i −0.724557 − 1.86414i 3.92447 − 0.621574i 1.24142 − 0.677408i −1.38050 − 0.448553i 2.75438 + 0.642975i 0.587785 + 0.809017i −2.40850 + 5.07688i
19.18 −0.748418 − 1.19995i −0.891007 − 0.453990i −0.879742 + 1.79612i −3.69216 + 0.584781i 0.122081 + 1.40893i −3.99843 − 1.29917i 2.81366 − 0.288605i 0.587785 + 0.809017i 3.46499 + 3.99274i
19.19 −0.656442 + 1.25263i 0.891007 + 0.453990i −1.13817 − 1.64456i −3.76947 + 0.597025i −1.15358 + 0.818083i −1.77949 − 0.578191i 2.80717 − 0.346143i 0.587785 + 0.809017i 1.72659 − 5.11366i
19.20 −0.571883 − 1.29343i 0.891007 + 0.453990i −1.34590 + 1.47938i −0.851658 + 0.134889i 0.0776520 − 1.41208i −1.94442 − 0.631780i 2.68316 + 0.894794i 0.587785 + 0.809017i 0.661517 + 1.02442i
See next 80 embeddings (of 384 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner
16.f odd 4 1 inner
176.x even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.2.bv.a ✓ 384
11.d odd 10 1 inner 528.2.bv.a ✓ 384
16.f odd 4 1 inner 528.2.bv.a ✓ 384
176.x even 20 1 inner 528.2.bv.a ✓ 384
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
528.2.bv.a ✓ 384 1.a even 1 1 trivial
528.2.bv.a ✓ 384 11.d odd 10 1 inner
528.2.bv.a ✓ 384 16.f odd 4 1 inner
528.2.bv.a ✓ 384 176.x even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(528, [\chi])\).