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Properties

Label 675.2.ba.b
Level $675$
Weight $2$
Character orbit 675.ba
Analytic conductor $5.390$
Analytic rank $0$
Dimension $192$
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] = [675,2,Mod(32,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([10, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.32");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.ba (of order \(36\), degree \(12\), 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(16\) over \(\Q(\zeta_{36})\)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q + 12 q^{2} + 12 q^{3} - 36 q^{6} + 12 q^{7} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 192 q + 12 q^{2} + 12 q^{3} - 36 q^{6} + 12 q^{7} + 18 q^{8} - 36 q^{11} + 12 q^{12} + 12 q^{13} - 24 q^{16} + 18 q^{17} + 54 q^{18} - 24 q^{21} + 12 q^{22} + 36 q^{23} + 36 q^{27} + 24 q^{28} - 24 q^{31} + 48 q^{32} + 6 q^{33} + 12 q^{36} + 6 q^{37} - 12 q^{38} + 24 q^{41} + 24 q^{42} + 12 q^{43} - 12 q^{46} + 6 q^{47} - 12 q^{48} + 144 q^{51} - 12 q^{52} + 180 q^{56} + 12 q^{57} + 12 q^{58} - 60 q^{61} + 18 q^{62} + 54 q^{63} + 72 q^{66} - 24 q^{67} + 60 q^{68} - 36 q^{71} - 180 q^{72} + 6 q^{73} - 72 q^{76} - 132 q^{77} - 78 q^{78} + 12 q^{81} + 24 q^{82} - 48 q^{83} + 12 q^{86} - 144 q^{87} + 48 q^{88} - 12 q^{91} - 258 q^{92} - 180 q^{93} - 12 q^{96} - 24 q^{97} - 324 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −0.237511 + 2.71476i −0.914466 + 1.47097i −5.34392 − 0.942278i 0 −3.77614 − 2.83193i −0.907744 + 1.29639i 2.41667 − 9.01913i −1.32750 − 2.69030i 0
32.2 −0.214747 + 2.45457i 1.63667 − 0.566833i −4.00918 − 0.706926i 0 1.03986 + 4.13905i 0.719449 − 1.02748i 1.32073 − 4.92901i 2.35740 − 1.85544i 0
32.3 −0.191628 + 2.19031i −0.699511 − 1.58451i −2.79113 − 0.492152i 0 3.60463 − 1.22851i 0.000371431 0 0.000530458i 0.474704 − 1.77162i −2.02137 + 2.21677i 0
32.4 −0.153697 + 1.75676i −1.70488 + 0.305576i −1.09297 − 0.192720i 0 −0.274789 − 3.04204i 1.93217 − 2.75943i −0.406292 + 1.51630i 2.81325 − 1.04194i 0
32.5 −0.119567 + 1.36666i 1.50279 − 0.861169i 0.116159 + 0.0204819i 0 0.997239 + 2.15677i −1.87969 + 2.68447i −0.752017 + 2.80657i 1.51678 − 2.58832i 0
32.6 −0.0904240 + 1.03355i 0.538569 + 1.64619i 0.909563 + 0.160381i 0 −1.75012 + 0.407783i −1.66856 + 2.38295i −0.785057 + 2.92987i −2.41989 + 1.77317i 0
32.7 −0.0559313 + 0.639298i −0.432570 + 1.67717i 1.56404 + 0.275783i 0 −1.04801 − 0.370347i 2.47086 − 3.52875i −0.595975 + 2.22421i −2.62577 − 1.45098i 0
32.8 −0.0504771 + 0.576956i 0.0483220 − 1.73138i 1.63929 + 0.289050i 0 0.996489 + 0.115275i 1.94671 − 2.78018i −0.549311 + 2.05006i −2.99533 − 0.167327i 0
32.9 −0.00133058 + 0.0152086i 1.71244 + 0.259886i 1.96939 + 0.347256i 0 −0.00623106 + 0.0256981i 0.148030 − 0.211408i −0.0158044 + 0.0589827i 2.86492 + 0.890079i 0
32.10 0.0140240 − 0.160295i −0.850222 − 1.50901i 1.94412 + 0.342800i 0 −0.253811 + 0.115124i −1.66092 + 2.37204i 0.165506 − 0.617675i −1.55424 + 2.56599i 0
32.11 0.0909691 − 1.03978i −1.49497 + 0.874685i 0.896746 + 0.158121i 0 0.773485 + 1.63401i −0.397456 + 0.567627i 0.786273 − 2.93441i 1.46985 − 2.61525i 0
32.12 0.118646 − 1.35613i 1.53542 + 0.801546i 0.144600 + 0.0254969i 0 1.26917 − 1.98714i 0.346787 − 0.495263i 0.756400 − 2.82292i 1.71505 + 2.46142i 0
32.13 0.162710 − 1.85978i −0.207283 + 1.71960i −1.46269 − 0.257912i 0 3.16436 + 0.665296i −1.79546 + 2.56418i 0.248717 − 0.928224i −2.91407 − 0.712888i 0
32.14 0.174019 − 1.98905i 1.19471 − 1.25406i −1.95641 − 0.344968i 0 −2.28648 − 2.59457i 0.808730 − 1.15499i 0.00692938 − 0.0258608i −0.145327 − 2.99648i 0
32.15 0.182563 − 2.08671i −1.65611 − 0.507262i −2.35140 − 0.414615i 0 −1.36085 + 3.36320i 2.36759 − 3.38127i −0.210175 + 0.784385i 2.48537 + 1.68016i 0
32.16 0.213925 − 2.44518i −0.617760 − 1.61814i −3.96351 − 0.698873i 0 −4.08879 + 1.16437i −2.71258 + 3.87397i −1.28621 + 4.80021i −2.23674 + 1.99924i 0
68.1 −1.43892 + 2.05499i −1.64201 − 0.551185i −1.46845 − 4.03452i 0 3.49539 − 2.58120i 2.44940 − 1.14217i 5.55747 + 1.48912i 2.39239 + 1.81010i 0
68.2 −1.36466 + 1.94894i 1.63096 + 0.583078i −1.25203 − 3.43992i 0 −3.36209 + 2.38294i 2.19758 − 1.02475i 3.81650 + 1.02263i 2.32004 + 1.90195i 0
68.3 −1.15565 + 1.65044i 0.352907 + 1.69572i −0.704386 − 1.93528i 0 −3.20652 − 1.37721i −1.32708 + 0.618828i 0.115769 + 0.0310202i −2.75091 + 1.19686i 0
68.4 −0.866263 + 1.23715i 0.938839 − 1.45553i −0.0960923 − 0.264011i 0 0.987435 + 2.42236i −1.45360 + 0.677826i −2.50778 − 0.671958i −1.23716 − 2.73303i 0
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
27.f odd 18 1 inner
135.q even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.ba.b 192
5.b even 2 1 135.2.q.a ✓ 192
5.c odd 4 1 135.2.q.a ✓ 192
5.c odd 4 1 inner 675.2.ba.b 192
15.d odd 2 1 405.2.r.a 192
15.e even 4 1 405.2.r.a 192
27.f odd 18 1 inner 675.2.ba.b 192
135.n odd 18 1 135.2.q.a ✓ 192
135.p even 18 1 405.2.r.a 192
135.q even 36 1 135.2.q.a ✓ 192
135.q even 36 1 inner 675.2.ba.b 192
135.r odd 36 1 405.2.r.a 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.q.a ✓ 192 5.b even 2 1
135.2.q.a ✓ 192 5.c odd 4 1
135.2.q.a ✓ 192 135.n odd 18 1
135.2.q.a ✓ 192 135.q even 36 1
405.2.r.a 192 15.d odd 2 1
405.2.r.a 192 15.e even 4 1
405.2.r.a 192 135.p even 18 1
405.2.r.a 192 135.r odd 36 1
675.2.ba.b 192 1.a even 1 1 trivial
675.2.ba.b 192 5.c odd 4 1 inner
675.2.ba.b 192 27.f odd 18 1 inner
675.2.ba.b 192 135.q even 36 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{192} - 12 T_{2}^{191} + 72 T_{2}^{190} - 294 T_{2}^{189} + 942 T_{2}^{188} - 2544 T_{2}^{187} + \cdots + 130321 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display