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Properties

Label 945.2.cz.b
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
\(\operatorname{Tr}(f)(q) = \) \( 288 q + 3 q^{3} + 6 q^{6} - 3 q^{9} + 3 q^{11} - 42 q^{12} + 3 q^{13} - 3 q^{14} - 36 q^{21} - 36 q^{23} - 24 q^{27} + 18 q^{29} - 24 q^{31} - 30 q^{33} + 60 q^{34} + 42 q^{36} - 12 q^{39} + 108 q^{41} + 18 q^{42} - 6 q^{45} - 9 q^{47} - 36 q^{49} - 18 q^{51} + 102 q^{52} + 30 q^{54} - 90 q^{56} - 12 q^{57} - 42 q^{61} + 6 q^{63} + 144 q^{64} - 6 q^{65} + 54 q^{66} + 36 q^{68} - 180 q^{69} - 3 q^{70} - 36 q^{71} - 102 q^{72} + 6 q^{74} - 108 q^{76} + 27 q^{77} - 48 q^{78} - 18 q^{79} - 288 q^{80} - 3 q^{81} + 72 q^{83} + 159 q^{84} - 18 q^{85} + 6 q^{86} + 60 q^{90} - 9 q^{91} + 192 q^{92} - 24 q^{93} - 48 q^{94} + 120 q^{96} - 45 q^{97} + 36 q^{98} + 162 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 −2.77060 + 0.488532i −1.05154 + 1.37632i 5.55819 − 2.02302i −0.766044 − 0.642788i 2.24103 − 4.32695i 1.14380 − 2.38573i −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 0.525287 + 2.59308i −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 1.46721 − 2.20166i −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 1.26609 + 2.32315i −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 −0.773440 − 2.53018i −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 −2.61756 − 0.385204i −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 2.37190 − 1.17221i −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 −2.47636 − 0.931473i −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 0.525022 + 2.59314i 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 −1.40196 − 2.24377i 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 −1.91035 + 1.83045i 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 −0.320421 + 2.62628i 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 2.23338 + 1.41846i 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 2.54655 − 0.717692i 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 −0.689206 − 2.55441i 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 −2.63850 + 0.195786i 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 0.778803 − 2.52853i 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 2.37022 + 1.17561i 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 −0.244525 + 2.63443i 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 −1.20427 − 2.35579i 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.b yes 288
7.b odd 2 1 945.2.cz.a ✓ 288
27.f odd 18 1 945.2.cz.a ✓ 288
189.be even 18 1 inner 945.2.cz.b yes 288
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
945.2.cz.a ✓ 288 7.b odd 2 1
945.2.cz.a ✓ 288 27.f odd 18 1
945.2.cz.b yes 288 1.a even 1 1 trivial
945.2.cz.b yes 288 189.be even 18 1 inner

Hecke kernels

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