Properties

Label 555.2.bp.a
Level $555$
Weight $2$
Character orbit 555.bp
Analytic conductor $4.432$
Analytic rank $0$
Dimension $240$
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(4,555)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(555, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 9, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("555.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.bp (of order \(18\), degree \(6\), 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: \(240\)
Relative dimension: \(40\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 12 q^{5} + 36 q^{14} + 12 q^{19} - 30 q^{20} + 12 q^{21} - 48 q^{25} - 36 q^{26} + 24 q^{30} + 48 q^{34} + 12 q^{35} - 264 q^{36} + 36 q^{39} + 66 q^{40} - 144 q^{41} + 192 q^{44} + 24 q^{46} - 12 q^{49} + 42 q^{50} - 6 q^{55} - 36 q^{59} + 108 q^{61} - 276 q^{64} - 66 q^{65} + 12 q^{69} - 42 q^{70} - 24 q^{71} - 108 q^{74} + 96 q^{76} - 96 q^{79} + 24 q^{84} - 12 q^{85} + 72 q^{86} - 300 q^{89} + 6 q^{90} + 72 q^{91} - 48 q^{94} - 42 q^{95} + 120 q^{96} + 12 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −0.483654 + 2.74294i −0.984808 + 0.173648i −5.41042 1.96923i 2.17371 0.524382i 2.78526i 2.76078 + 3.29017i 5.23300 9.06382i 0.939693 0.342020i 0.387024 + 6.21598i
4.2 −0.482121 + 2.73425i 0.984808 0.173648i −5.36427 1.95244i −1.84866 1.25795i 2.77643i −0.622659 0.742056i 5.14824 8.91702i 0.939693 0.342020i 4.33083 4.44822i
4.3 −0.444573 + 2.52130i −0.984808 + 0.173648i −4.27990 1.55776i −1.33654 + 1.79267i 2.56019i −0.953334 1.13614i 3.27010 5.66399i 0.939693 0.342020i −3.92566 4.16679i
4.4 −0.431485 + 2.44707i 0.984808 0.173648i −3.92261 1.42771i 0.791716 + 2.09122i 2.48482i −2.01383 2.39999i 2.70145 4.67904i 0.939693 0.342020i −5.45898 + 1.03506i
4.5 −0.396444 + 2.24835i 0.984808 0.173648i −3.01852 1.09865i 1.48813 + 1.66898i 2.28303i 2.52076 + 3.00412i 1.38379 2.39679i 0.939693 0.342020i −4.34240 + 2.68417i
4.6 −0.392039 + 2.22336i −0.984808 + 0.173648i −2.91026 1.05925i 2.23508 + 0.0665805i 2.25766i −3.20631 3.82113i 1.23837 2.14492i 0.939693 0.342020i −1.02427 + 4.94328i
4.7 −0.333537 + 1.89158i −0.984808 + 0.173648i −1.58745 0.577784i −2.21455 0.309466i 1.92076i 2.01071 + 2.39627i −0.298365 + 0.516783i 0.939693 0.342020i 1.32401 4.08578i
4.8 −0.328062 + 1.86053i −0.984808 + 0.173648i −1.47457 0.536701i 0.674706 + 2.13185i 1.88923i 0.456273 + 0.543765i −0.406933 + 0.704829i 0.939693 0.342020i −4.18772 + 0.555934i
4.9 −0.326358 + 1.85087i 0.984808 0.173648i −1.43983 0.524054i −2.21686 0.292450i 1.87942i −2.07200 2.46931i −0.439568 + 0.761354i 0.939693 0.342020i 1.26478 4.00768i
4.10 −0.315810 + 1.79105i 0.984808 0.173648i −1.22873 0.447221i 1.61367 1.54792i 1.81868i −1.35469 1.61445i −0.629637 + 1.09056i 0.939693 0.342020i 2.26279 + 3.37901i
4.11 −0.277733 + 1.57510i 0.984808 0.173648i −0.524428 0.190876i −1.34715 1.78471i 1.59940i 1.57453 + 1.87646i −1.15310 + 1.99723i 0.939693 0.342020i 3.18525 1.62623i
4.12 −0.208975 + 1.18516i 0.984808 0.173648i 0.518457 + 0.188703i 2.23588 0.0286419i 1.20344i 1.21615 + 1.44935i −1.53543 + 2.65944i 0.939693 0.342020i −0.433299 + 2.65586i
4.13 −0.204840 + 1.16171i −0.984808 + 0.173648i 0.571779 + 0.208111i 1.68974 1.46451i 1.17963i 2.65369 + 3.16254i −1.53852 + 2.66479i 0.939693 0.342020i 1.35520 + 2.26297i
4.14 −0.190715 + 1.08160i −0.984808 + 0.173648i 0.745904 + 0.271487i 0.174198 2.22927i 1.09828i −2.17481 2.59184i −1.53418 + 2.65727i 0.939693 0.342020i 2.37795 + 0.613567i
4.15 −0.185254 + 1.05063i −0.984808 + 0.173648i 0.809888 + 0.294775i −0.663270 + 2.13543i 1.06683i −0.415188 0.494802i −1.52657 + 2.64409i 0.939693 0.342020i −2.12067 1.09245i
4.16 −0.169137 + 0.959222i 0.984808 0.173648i 0.987886 + 0.359561i −1.41362 + 1.73254i 0.974019i 0.930024 + 1.10836i −1.48601 + 2.57384i 0.939693 0.342020i −1.42279 1.64901i
4.17 −0.101920 + 0.578017i −0.984808 + 0.173648i 1.55567 + 0.566218i −2.03763 0.920904i 0.586933i −1.49229 1.77844i −1.07277 + 1.85809i 0.939693 0.342020i 0.739973 1.08393i
4.18 −0.0741078 + 0.420286i 0.984808 0.173648i 1.70824 + 0.621747i 1.35836 + 1.77619i 0.426770i −2.53805 3.02473i −0.814676 + 1.41106i 0.939693 0.342020i −0.847173 + 0.439272i
4.19 −0.0600890 + 0.340781i −0.984808 + 0.173648i 1.76686 + 0.643086i 1.86567 + 1.23259i 0.346039i −1.02117 1.21698i −0.671359 + 1.16283i 0.939693 0.342020i −0.532151 + 0.561720i
4.20 −0.0425104 + 0.241089i 0.984808 0.173648i 1.82307 + 0.663543i 0.433786 2.19359i 0.244808i −2.03953 2.43062i −0.482280 + 0.835333i 0.939693 0.342020i 0.510409 + 0.197831i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.40
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.h even 18 1 inner
185.v even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 555.2.bp.a 240
5.b even 2 1 inner 555.2.bp.a 240
37.h even 18 1 inner 555.2.bp.a 240
185.v even 18 1 inner 555.2.bp.a 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
555.2.bp.a 240 1.a even 1 1 trivial
555.2.bp.a 240 5.b even 2 1 inner
555.2.bp.a 240 37.h even 18 1 inner
555.2.bp.a 240 185.v even 18 1 inner

Hecke kernels

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