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Properties

Label 555.2.bb.a
Level $555$
Weight $2$
Character orbit 555.bb
Analytic conductor $4.432$
Analytic rank $0$
Dimension $72$
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] = [555,2,Mod(454,555)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(555, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("555.454");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.bb (of order \(6\), 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: \(4.43169731218\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 32 q^{4} - 2 q^{5} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 32 q^{4} - 2 q^{5} + 36 q^{9} - 8 q^{11} - 16 q^{14} - 24 q^{16} + 4 q^{19} + 4 q^{20} + 12 q^{21} + 10 q^{25} + 24 q^{26} + 16 q^{29} + 12 q^{30} + 24 q^{31} - 44 q^{34} - 2 q^{35} + 64 q^{36} - 8 q^{39} + 46 q^{40} - 48 q^{41} - 8 q^{44} - 4 q^{45} - 64 q^{46} + 32 q^{49} - 46 q^{50} - 8 q^{51} - 12 q^{55} - 72 q^{56} + 32 q^{59} - 68 q^{60} - 8 q^{61} - 136 q^{64} - 42 q^{65} - 32 q^{66} - 4 q^{69} + 4 q^{70} + 12 q^{71} - 96 q^{74} - 8 q^{75} + 8 q^{76} - 8 q^{79} - 44 q^{80} - 36 q^{81} + 136 q^{84} + 4 q^{85} + 8 q^{86} + 8 q^{91} - 28 q^{94} + 18 q^{95} - 20 q^{96} - 4 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
454.1 −2.37811 − 1.37300i 0.866025 − 0.500000i 2.77028 + 4.79826i −2.00871 + 0.982395i −2.74601 4.01945 − 2.32063i − 9.72238i 0.500000 − 0.866025i 6.12576 + 0.421716i
454.2 −2.28225 − 1.31765i −0.866025 + 0.500000i 2.47243 + 4.28237i 2.17287 − 0.527852i 2.63531 −2.57365 + 1.48590i − 7.76061i 0.500000 − 0.866025i −5.65455 − 1.65841i
454.3 −2.16333 − 1.24900i −0.866025 + 0.500000i 2.11999 + 3.67194i −2.13789 − 0.655312i 2.49800 −0.174209 + 0.100580i − 5.59548i 0.500000 − 0.866025i 3.80647 + 4.08787i
454.4 −1.98150 − 1.14402i 0.866025 − 0.500000i 1.61755 + 2.80168i 2.23334 + 0.110439i −2.28803 1.71703 − 0.991326i − 2.82596i 0.500000 − 0.866025i −4.29901 − 2.77381i
454.5 −1.96173 − 1.13261i −0.866025 + 0.500000i 1.56559 + 2.71168i 0.414545 + 2.19731i 2.26521 1.72326 − 0.994927i − 2.56237i 0.500000 − 0.866025i 1.67546 − 4.78004i
454.6 −1.73475 − 1.00156i 0.866025 − 0.500000i 1.00624 + 1.74286i −1.32535 − 1.80096i −2.00312 −3.12135 + 1.80211i − 0.0249930i 0.500000 − 0.866025i 0.495375 + 4.45163i
454.7 −1.63447 − 0.943665i 0.866025 − 0.500000i 0.781006 + 1.35274i −1.89133 + 1.19284i −1.88733 −1.09596 + 0.632750i 0.826629i 0.500000 − 0.866025i 4.21697 − 0.164890i
454.8 −1.45470 − 0.839870i 0.866025 − 0.500000i 0.410763 + 0.711463i −0.906648 − 2.04401i −1.67974 3.05983 − 1.76660i 1.97953i 0.500000 − 0.866025i −0.397806 + 3.73489i
454.9 −1.34809 − 0.778318i −0.866025 + 0.500000i 0.211559 + 0.366431i 0.197466 − 2.22733i 1.55664 0.151651 − 0.0875559i 2.45463i 0.500000 − 0.866025i −1.99977 + 2.84895i
454.10 −1.22378 − 0.706551i −0.866025 + 0.500000i −0.00157058 − 0.00272032i 2.18051 + 0.495360i 1.41310 −1.52412 + 0.879952i 2.83064i 0.500000 − 0.866025i −2.31847 − 2.14685i
454.11 −1.07810 − 0.622440i 0.866025 − 0.500000i −0.225137 − 0.389949i 1.69944 − 1.45324i −1.24488 −2.38473 + 1.37683i 3.05030i 0.500000 − 0.866025i −2.73671 + 0.508942i
454.12 −1.04703 − 0.604502i −0.866025 + 0.500000i −0.269155 − 0.466190i −2.14730 − 0.623789i 1.20900 0.944369 − 0.545232i 3.06883i 0.500000 − 0.866025i 1.87120 + 1.95117i
454.13 −0.950972 − 0.549044i 0.866025 − 0.500000i −0.397102 − 0.687801i 0.837757 + 2.07320i −1.09809 −1.35262 + 0.780936i 3.06828i 0.500000 − 0.866025i 0.341594 − 2.43152i
454.14 −0.931961 − 0.538068i −0.866025 + 0.500000i −0.420965 − 0.729134i −0.374645 + 2.20446i 1.07614 −3.54514 + 2.04679i 3.05830i 0.500000 − 0.866025i 1.53530 − 1.85289i
454.15 −0.625647 − 0.361218i −0.866025 + 0.500000i −0.739044 − 1.28006i 2.23591 − 0.0267130i 0.722435 4.45380 − 2.57140i 2.51269i 0.500000 − 0.866025i −1.40854 − 0.790937i
454.16 −0.331339 − 0.191298i 0.866025 − 0.500000i −0.926810 − 1.60528i 1.22222 − 1.87248i −0.382597 3.21815 − 1.85800i 1.47438i 0.500000 − 0.866025i −0.763172 + 0.386614i
454.17 −0.153667 − 0.0887196i 0.866025 − 0.500000i −0.984258 − 1.70478i 2.22191 + 0.251234i −0.177439 −1.07771 + 0.622215i 0.704171i 0.500000 − 0.866025i −0.319145 − 0.235733i
454.18 −0.113794 − 0.0656990i −0.866025 + 0.500000i −0.991367 − 1.71710i 0.217380 − 2.22548i 0.131398 −1.67003 + 0.964193i 0.523323i 0.500000 − 0.866025i −0.170948 + 0.238964i
454.19 0.113794 + 0.0656990i 0.866025 − 0.500000i −0.991367 − 1.71710i −2.03601 − 0.924481i 0.131398 1.67003 − 0.964193i − 0.523323i 0.500000 − 0.866025i −0.170948 − 0.238964i
454.20 0.153667 + 0.0887196i −0.866025 + 0.500000i −0.984258 − 1.70478i −0.893380 + 2.04985i −0.177439 1.07771 − 0.622215i − 0.704171i 0.500000 − 0.866025i −0.319145 + 0.235733i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 454.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.c even 3 1 inner
185.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 555.2.bb.a ✓ 72
5.b even 2 1 inner 555.2.bb.a ✓ 72
37.c even 3 1 inner 555.2.bb.a ✓ 72
185.n even 6 1 inner 555.2.bb.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.bb.a ✓ 72 1.a even 1 1 trivial
555.2.bb.a ✓ 72 5.b even 2 1 inner
555.2.bb.a ✓ 72 37.c even 3 1 inner
555.2.bb.a ✓ 72 185.n even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(555, [\chi])\).