↑

Properties

Label 528.2.q.a
Level $528$
Weight $2$
Character orbit 528.q
Analytic conductor $4.216$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [528,2,Mod(43,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("528.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 528.q (of order \(4\), degree \(2\), 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: \(96\)
Relative dimension: \(48\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 96 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q - 8 q^{4} + 8 q^{11} - 16 q^{12} + 24 q^{14} + 8 q^{16} - 16 q^{20} + 32 q^{23} - 40 q^{26} + 40 q^{34} + 32 q^{37} + 56 q^{38} - 32 q^{44} - 96 q^{49} - 32 q^{53} + 32 q^{55} - 56 q^{56} + 8 q^{58} + 32 q^{59} + 24 q^{60} - 104 q^{64} + 24 q^{66} - 16 q^{67} + 40 q^{70} + 64 q^{71} - 32 q^{75} + 16 q^{77} - 48 q^{78} - 128 q^{80} - 96 q^{81} - 80 q^{82} - 48 q^{86} + 80 q^{88} - 16 q^{91} - 72 q^{92} + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −1.41392 − 0.0288740i 0.707107 + 0.707107i 1.99833 + 0.0816511i 1.18924 + 1.18924i −0.979375 − 1.02021i − 0.0841373i −2.82312 − 0.173148i 1.00000i −1.64715 − 1.71582i
43.2 −1.39448 + 0.235445i −0.707107 − 0.707107i 1.88913 − 0.656646i −1.83785 − 1.83785i 1.15253 + 0.819559i 1.24015i −2.47975 + 1.36046i 1.00000i 2.99555 + 2.13012i
43.3 −1.38494 + 0.286230i −0.707107 − 0.707107i 1.83614 − 0.792827i 0.771368 + 0.771368i 1.18170 + 0.776908i − 5.09428i −2.31603 + 1.62358i 1.00000i −1.28909 − 0.847513i
43.4 −1.36170 − 0.381807i −0.707107 − 0.707107i 1.70845 + 1.03981i −1.23412 − 1.23412i 0.692888 + 1.23284i 2.75554i −1.92938 − 2.06821i 1.00000i 1.20930 + 2.15169i
43.5 −1.30694 − 0.540296i −0.707107 − 0.707107i 1.41616 + 1.41226i 0.538424 + 0.538424i 0.542096 + 1.30619i − 2.73758i −1.08779 − 2.61088i 1.00000i −0.412777 − 0.994594i
43.6 −1.29387 − 0.570886i 0.707107 + 0.707107i 1.34818 + 1.47730i −1.28258 − 1.28258i −0.511224 − 1.31858i 0.0390261i −0.900993 − 2.68108i 1.00000i 0.927283 + 2.39170i
43.7 −1.26990 + 0.622385i 0.707107 + 0.707107i 1.22527 − 1.58073i 0.355264 + 0.355264i −1.33805 − 0.457859i − 2.95587i −0.572146 + 2.76995i 1.00000i −0.672259 − 0.230037i
43.8 −1.22712 + 0.702982i 0.707107 + 0.707107i 1.01163 − 1.72528i −3.12937 − 3.12937i −1.36479 − 0.370619i 0.642297i −0.0285474 + 2.82828i 1.00000i 6.04000 + 1.64021i
43.9 −1.21973 + 0.715719i −0.707107 − 0.707107i 0.975493 − 1.74597i 1.71794 + 1.71794i 1.36857 + 0.356391i 2.63943i 0.0597843 + 2.82780i 1.00000i −3.32499 − 0.865864i
43.10 −1.09485 + 0.895151i 0.707107 + 0.707107i 0.397410 − 1.96012i 2.11423 + 2.11423i −1.40715 − 0.141211i 4.78222i 1.31950 + 2.50179i 1.00000i −4.20732 − 0.422217i
43.11 −1.06025 − 0.935876i −0.707107 − 0.707107i 0.248271 + 1.98453i 2.67631 + 2.67631i 0.0879473 + 1.41148i 0.550981i 1.59405 − 2.33645i 1.00000i −0.332869 − 5.34226i
43.12 −1.05997 + 0.936192i −0.707107 − 0.707107i 0.247089 − 1.98468i −0.831486 − 0.831486i 1.41150 + 0.0875268i 1.18501i 1.59613 + 2.33503i 1.00000i 1.65978 + 0.102923i
43.13 −1.04428 − 0.953669i 0.707107 + 0.707107i 0.181030 + 1.99179i 2.39017 + 2.39017i −0.0640695 − 1.41276i 3.58593i 1.71046 − 2.25262i 1.00000i −0.216569 − 4.77543i
43.14 −0.960149 − 1.03832i 0.707107 + 0.707107i −0.156229 + 1.99389i 0.671872 + 0.671872i 0.0552774 − 1.41313i − 2.65911i 2.22030 − 1.75221i 1.00000i 0.0525230 − 1.34272i
43.15 −0.682302 + 1.23873i −0.707107 − 0.707107i −1.06893 − 1.69038i −2.31495 − 2.31495i 1.35838 − 0.393457i − 3.64060i 2.82327 − 0.170764i 1.00000i 4.44710 − 1.28811i
43.16 −0.617329 − 1.27236i 0.707107 + 0.707107i −1.23781 + 1.57093i −1.55533 − 1.55533i 0.463178 − 1.33621i − 3.21056i 2.76293 + 0.605160i 1.00000i −1.01879 + 2.93908i
43.17 −0.485241 − 1.32836i −0.707107 − 0.707107i −1.52908 + 1.28915i 0.674873 + 0.674873i −0.596175 + 1.28241i 3.13957i 2.45443 + 1.40562i 1.00000i 0.568998 − 1.22395i
43.18 −0.474411 + 1.33227i −0.707107 − 0.707107i −1.54987 − 1.26408i 2.32179 + 2.32179i 1.27751 − 0.606595i − 2.41984i 2.41937 − 1.46514i 1.00000i −4.19473 + 1.99176i
43.19 −0.434706 + 1.34575i 0.707107 + 0.707107i −1.62206 − 1.17001i −0.490377 − 0.490377i −1.25897 + 0.644202i 1.20432i 2.27965 − 1.67427i 1.00000i 0.873092 − 0.446753i
43.20 −0.420924 − 1.35012i 0.707107 + 0.707107i −1.64565 + 1.13660i −2.23973 − 2.23973i 0.657041 − 1.25232i 2.85710i 2.22723 + 1.74340i 1.00000i −2.08115 + 3.96666i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
16.f odd 4 1 inner
176.i even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 528.2.q.a ✓ 96
4.b odd 2 1 2112.2.q.a 96
11.b odd 2 1 inner 528.2.q.a ✓ 96
16.e even 4 1 2112.2.q.a 96
16.f odd 4 1 inner 528.2.q.a ✓ 96
44.c even 2 1 2112.2.q.a 96
176.i even 4 1 inner 528.2.q.a ✓ 96
176.l odd 4 1 2112.2.q.a 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
528.2.q.a ✓ 96 1.a even 1 1 trivial
528.2.q.a ✓ 96 11.b odd 2 1 inner
528.2.q.a ✓ 96 16.f odd 4 1 inner
528.2.q.a ✓ 96 176.i even 4 1 inner
2112.2.q.a 96 4.b odd 2 1
2112.2.q.a 96 16.e even 4 1
2112.2.q.a 96 44.c even 2 1
2112.2.q.a 96 176.l odd 4 1

Hecke kernels

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