Properties

Label 945.2.cz.a
Level $945$
Weight $2$
Character orbit 945.cz
Analytic conductor $7.546$
Analytic rank $0$
Dimension $288$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [945,2,Mod(41,945)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(945, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([17, 0, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("945.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 945 = 3^{3} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 945.cz (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: \(7.54586299101\)
Analytic rank: \(0\)
Dimension: \(288\)
Relative dimension: \(48\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 288 q - 3 q^{3} - 6 q^{6} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 288 q - 3 q^{3} - 6 q^{6} - 3 q^{9} + 3 q^{11} + 42 q^{12} - 3 q^{13} - 21 q^{14} + 12 q^{21} - 36 q^{23} + 24 q^{27} + 18 q^{29} + 24 q^{31} + 30 q^{33} - 60 q^{34} + 42 q^{36} - 12 q^{39} - 108 q^{41} - 78 q^{42} + 6 q^{45} + 9 q^{47} + 54 q^{49} - 18 q^{51} - 102 q^{52} - 30 q^{54} - 36 q^{56} - 12 q^{57} + 42 q^{61} - 18 q^{63} + 144 q^{64} - 6 q^{65} - 54 q^{66} - 36 q^{68} + 180 q^{69} + 39 q^{70} - 36 q^{71} - 102 q^{72} + 6 q^{74} + 108 q^{76} + 27 q^{77} - 48 q^{78} - 18 q^{79} + 288 q^{80} - 3 q^{81} - 72 q^{83} - 9 q^{84} - 18 q^{85} + 6 q^{86} - 60 q^{90} - 9 q^{91} + 192 q^{92} - 24 q^{93} + 48 q^{94} - 120 q^{96} + 45 q^{97} + 36 q^{98} + 162 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −2.77060 + 0.488532i 1.05154 1.37632i 5.55819 2.02302i 0.766044 + 0.642788i −2.24103 + 4.32695i −0.657315 + 2.56280i −9.53839 + 5.50699i −0.788524 2.89452i −2.43643 1.40667i
41.2 −2.62818 + 0.463420i 1.64365 + 0.546268i 4.81321 1.75186i 0.766044 + 0.642788i −4.57297 0.673993i 2.06919 1.64877i −7.21577 + 4.16603i 2.40318 + 1.79575i −2.31119 1.33436i
41.3 −2.50444 + 0.441600i −0.416120 + 1.68132i 4.19781 1.52788i 0.766044 + 0.642788i 0.299675 4.39452i −0.291249 + 2.62967i −5.43371 + 3.13716i −2.65369 1.39926i −2.20237 1.27154i
41.4 −2.45634 + 0.433119i −1.54557 0.781793i 3.96663 1.44374i 0.766044 + 0.642788i 4.13507 + 1.25093i 2.46317 0.965811i −4.79795 + 2.77010i 1.77760 + 2.41664i −2.16007 1.24712i
41.5 −2.35105 + 0.414554i −1.68744 + 0.390583i 3.47620 1.26523i 0.766044 + 0.642788i 3.80533 1.61781i −2.21886 + 1.44107i −3.51326 + 2.02838i 2.69489 1.31817i −2.06748 1.19366i
41.6 −2.21258 + 0.390137i 1.63183 0.580635i 2.86391 1.04238i 0.766044 + 0.642788i −3.38402 + 1.92134i −2.25277 1.38745i −2.03854 + 1.17695i 2.32572 1.89499i −1.94571 1.12336i
41.7 −2.06315 + 0.363789i −0.333455 1.69965i 2.24487 0.817065i 0.766044 + 0.642788i 1.30628 + 3.38533i 1.06350 + 2.42260i −0.705652 + 0.407408i −2.77762 + 1.13351i −1.81431 1.04749i
41.8 −2.00826 + 0.354110i 0.172950 1.72339i 2.02831 0.738245i 0.766044 + 0.642788i 0.262942 + 3.52226i −2.49574 0.878224i −0.279887 + 0.161593i −2.94018 0.596123i −1.76603 1.01962i
41.9 −1.94739 + 0.343377i −1.70090 + 0.327000i 1.79502 0.653332i 0.766044 + 0.642788i 3.20003 1.22085i 2.06903 1.64898i 0.153755 0.0887708i 2.78614 1.11239i −1.71250 0.988713i
41.10 −1.88313 + 0.332047i 1.38750 + 1.03675i 1.55654 0.566534i 0.766044 + 0.642788i −2.95709 1.49162i −2.51623 + 0.817667i 0.568941 0.328478i 0.850310 + 2.87697i −1.65600 0.956090i
41.11 −1.86641 + 0.329098i −0.963447 + 1.43936i 1.49579 0.544424i 0.766044 + 0.642788i 1.32449 3.00351i −0.286823 2.63016i 0.669993 0.386821i −1.14354 2.77350i −1.64129 0.947601i
41.12 −1.71438 + 0.302292i −0.0407670 1.73157i 0.968344 0.352448i 0.766044 + 0.642788i 0.593330 + 2.95625i 1.44268 2.21781i 1.46164 0.843876i −2.99668 + 0.141182i −1.50760 0.870415i
41.13 −1.62778 + 0.287022i 1.72641 0.139631i 0.687901 0.250375i 0.766044 + 0.642788i −2.77014 + 0.722807i 2.62263 + 0.348987i 1.81500 1.04789i 2.96101 0.482122i −1.43145 0.826446i
41.14 −1.48500 + 0.261846i 1.03981 + 1.38520i 0.257282 0.0936430i 0.766044 + 0.642788i −1.90684 1.78476i 1.48945 + 2.18667i 2.25423 1.30148i −0.837570 + 2.88071i −1.30589 0.753955i
41.15 −1.18606 + 0.209134i −0.574634 + 1.63395i −0.516395 + 0.187952i 0.766044 + 0.642788i 0.339834 2.05813i −2.16990 + 1.51378i 2.65916 1.53527i −2.33959 1.87785i −1.04300 0.602176i
41.16 −1.11418 + 0.196460i −1.59502 0.675213i −0.676583 + 0.246256i 0.766044 + 0.642788i 1.90979 + 0.438951i −1.89536 1.84597i 2.66504 1.53866i 2.08818 + 2.15396i −0.979794 0.565684i
41.17 −1.04094 + 0.183545i 1.44985 0.947593i −0.829527 + 0.301923i 0.766044 + 0.642788i −1.33527 + 1.25250i −1.02871 + 2.43757i 2.63883 1.52353i 1.20413 2.74774i −0.915383 0.528497i
41.18 −0.848391 + 0.149594i −0.355775 + 1.69512i −1.18200 + 0.430211i 0.766044 + 0.642788i 0.0482561 1.49135i 2.57136 + 0.622983i 2.43056 1.40329i −2.74685 1.20616i −0.746063 0.430740i
41.19 −0.745797 + 0.131504i 1.19311 1.25558i −1.34047 + 0.487889i 0.766044 + 0.642788i −0.724707 + 1.09331i 1.50606 2.17527i 2.24724 1.29745i −0.152958 2.99610i −0.655843 0.378651i
41.20 −0.742439 + 0.130912i −0.816350 1.52760i −1.34531 + 0.489652i 0.766044 + 0.642788i 0.806072 + 1.02728i −2.43680 + 1.03055i 2.24049 1.29355i −1.66715 + 2.49412i −0.652890 0.376946i
See next 80 embeddings (of 288 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
189.be even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 945.2.cz.a 288
7.b odd 2 1 945.2.cz.b yes 288
27.f odd 18 1 945.2.cz.b yes 288
189.be even 18 1 inner 945.2.cz.a 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.cz.a 288 1.a even 1 1 trivial
945.2.cz.a 288 189.be even 18 1 inner
945.2.cz.b yes 288 7.b odd 2 1
945.2.cz.b yes 288 27.f odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{288} + 3 T_{13}^{287} + 81 T_{13}^{286} - 228 T_{13}^{285} + 2016 T_{13}^{284} - 14346 T_{13}^{283} - 183213 T_{13}^{282} - 734274 T_{13}^{281} - 21134637 T_{13}^{280} + 82135332 T_{13}^{279} - 675305091 T_{13}^{278} + \cdots + 59\!\cdots\!24 \) acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\). Copy content Toggle raw display