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Properties

Label 720.2.bc.a
Level $720$
Weight $2$
Character orbit 720.bc
Analytic conductor $5.749$
Analytic rank $0$
Dimension $96$
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] = [720,2,Mod(197,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.197");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.bc (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: \(5.74922894553\)
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
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 8 q^{4} + 8 q^{16} + 16 q^{19} + 16 q^{22} + 32 q^{28} - 8 q^{34} + 40 q^{40} + 64 q^{43} + 8 q^{46} - 24 q^{52} - 80 q^{58} + 32 q^{61} + 8 q^{64} - 8 q^{70} + 8 q^{76} + 32 q^{88} + 40 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1 −1.41326 + 0.0520118i 0 1.99459 − 0.147012i 2.14437 − 0.633793i 0 0.431588 − 0.431588i −2.81122 + 0.311508i 0 −2.99758 + 1.00725i
197.2 −1.41043 − 0.103327i 0 1.97865 + 0.291473i −2.21174 − 0.328918i 0 −2.33595 + 2.33595i −2.76063 − 0.615551i 0 3.08553 + 0.692451i
197.3 −1.40624 − 0.149965i 0 1.95502 + 0.421775i −1.29201 − 1.82503i 0 −2.02860 + 2.02860i −2.68598 − 0.886302i 0 1.54318 + 2.76018i
197.4 −1.39508 + 0.231832i 0 1.89251 − 0.646849i 0.244063 + 2.22271i 0 0.718424 − 0.718424i −2.49024 + 1.34115i 0 −0.855782 − 3.04428i
197.5 −1.39415 + 0.237394i 0 1.88729 − 0.661925i 0.257710 − 2.22117i 0 3.41086 − 3.41086i −2.47402 + 1.37085i 0 0.168007 + 3.15781i
197.6 −1.34579 − 0.434580i 0 1.62228 + 1.16970i −1.74165 + 1.40237i 0 2.18248 − 2.18248i −1.67491 − 2.27918i 0 2.95334 − 1.13040i
197.7 −1.33026 − 0.480016i 0 1.53917 + 1.27709i 2.06437 + 0.859295i 0 −1.97431 + 1.97431i −1.43447 − 2.43768i 0 −2.33366 − 2.13401i
197.8 −1.20197 + 0.745171i 0 0.889441 − 1.79134i −0.470363 + 2.18604i 0 −0.177389 + 0.177389i 0.265776 + 2.81591i 0 −1.06361 − 2.97804i
197.9 −1.17483 + 0.787263i 0 0.760434 − 1.84979i −2.23567 − 0.0422885i 0 1.80532 − 1.80532i 0.562896 + 2.77185i 0 2.65981 − 1.71038i
197.10 −1.16790 − 0.797498i 0 0.727993 + 1.86280i −2.02484 + 0.948701i 0 1.06871 − 1.06871i 0.635356 − 2.75614i 0 3.12140 + 0.506814i
197.11 −1.13002 + 0.850328i 0 0.553884 − 1.92177i 0.885098 − 2.05344i 0 −2.95475 + 2.95475i 1.00824 + 2.64262i 0 0.745918 + 3.07305i
197.12 −1.03923 − 0.959172i 0 0.159980 + 1.99359i 0.117317 − 2.23299i 0 −0.303980 + 0.303980i 1.74594 − 2.22524i 0 −2.26374 + 2.20805i
197.13 −1.02160 − 0.977925i 0 0.0873238 + 1.99809i 1.73447 − 1.41125i 0 −2.32146 + 2.32146i 1.86478 − 2.12664i 0 −3.15202 − 0.254450i
197.14 −0.981505 + 1.01816i 0 −0.0732970 − 1.99866i 1.78433 + 1.34765i 0 −3.52140 + 3.52140i 2.10689 + 1.88706i 0 −3.12345 + 0.494009i
197.15 −0.910521 − 1.08211i 0 −0.341905 + 1.97056i 2.04823 + 0.897076i 0 3.15125 − 3.15125i 2.44366 − 1.42426i 0 −0.894226 − 3.03321i
197.16 −0.898132 + 1.09241i 0 −0.386719 − 1.96226i 1.29671 − 1.82168i 0 2.16011 − 2.16011i 2.49091 + 1.33991i 0 0.825405 + 3.05266i
197.17 −0.663127 + 1.24910i 0 −1.12053 − 1.65663i −1.95867 − 1.07871i 0 1.03362 − 1.03362i 2.81235 − 0.301100i 0 2.64626 − 1.73127i
197.18 −0.572467 − 1.29317i 0 −1.34456 + 1.48059i −0.0482170 − 2.23555i 0 2.70126 − 2.70126i 2.68437 + 0.891154i 0 −2.86333 + 1.34213i
197.19 −0.487044 − 1.32770i 0 −1.52558 + 1.29330i 1.08017 + 1.95786i 0 0.362166 − 0.362166i 2.46013 + 1.39562i 0 2.07336 − 2.38771i
197.20 −0.427613 + 1.34802i 0 −1.63429 − 1.15286i −1.09255 − 1.95099i 0 −1.70979 + 1.70979i 2.25292 − 1.71008i 0 3.09715 − 0.638502i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
80.i odd 4 1 inner
240.bb even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.bc.a ✓ 96
3.b odd 2 1 inner 720.2.bc.a ✓ 96
4.b odd 2 1 2880.2.bc.a 96
5.c odd 4 1 720.2.bg.a yes 96
12.b even 2 1 2880.2.bc.a 96
15.e even 4 1 720.2.bg.a yes 96
16.e even 4 1 720.2.bg.a yes 96
16.f odd 4 1 2880.2.bg.a 96
20.e even 4 1 2880.2.bg.a 96
48.i odd 4 1 720.2.bg.a yes 96
48.k even 4 1 2880.2.bg.a 96
60.l odd 4 1 2880.2.bg.a 96
80.i odd 4 1 inner 720.2.bc.a ✓ 96
80.s even 4 1 2880.2.bc.a 96
240.z odd 4 1 2880.2.bc.a 96
240.bb even 4 1 inner 720.2.bc.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.bc.a ✓ 96 1.a even 1 1 trivial
720.2.bc.a ✓ 96 3.b odd 2 1 inner
720.2.bc.a ✓ 96 80.i odd 4 1 inner
720.2.bc.a ✓ 96 240.bb even 4 1 inner
720.2.bg.a yes 96 5.c odd 4 1
720.2.bg.a yes 96 15.e even 4 1
720.2.bg.a yes 96 16.e even 4 1
720.2.bg.a yes 96 48.i odd 4 1
2880.2.bc.a 96 4.b odd 2 1
2880.2.bc.a 96 12.b even 2 1
2880.2.bc.a 96 80.s even 4 1
2880.2.bc.a 96 240.z odd 4 1
2880.2.bg.a 96 16.f odd 4 1
2880.2.bg.a 96 20.e even 4 1
2880.2.bg.a 96 48.k even 4 1
2880.2.bg.a 96 60.l odd 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(720, [\chi])\).