Properties

Label 912.2.cn.a
Level $912$
Weight $2$
Character orbit 912.cn
Analytic conductor $7.282$
Analytic rank $0$
Dimension $960$
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] = [912,2,Mod(67,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([18, 27, 0, 34]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.67");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.cn (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: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(960\)
Relative dimension: \(80\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 960 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 960 q + 48 q^{10} + 12 q^{16} - 60 q^{32} - 120 q^{34} + 12 q^{36} + 108 q^{38} + 60 q^{40} + 180 q^{46} - 480 q^{49} - 108 q^{50} - 24 q^{51} + 48 q^{52} + 12 q^{54} + 72 q^{68} - 144 q^{69} - 72 q^{70} - 48 q^{76} + 72 q^{78} - 120 q^{80} + 120 q^{82} + 96 q^{85} + 84 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 −1.41411 0.0169281i −0.996195 0.0871557i 1.99943 + 0.0478763i 2.68717 1.25305i 1.40726 + 0.140112i 0.522822 0.905554i −2.82660 0.101549i 0.984808 + 0.173648i −3.82117 + 1.72646i
67.2 −1.41324 0.0523593i 0.996195 + 0.0871557i 1.99452 + 0.147993i 0.825343 0.384864i −1.40330 0.175332i −1.56239 + 2.70614i −2.81099 0.313582i 0.984808 + 0.173648i −1.18656 + 0.500692i
67.3 −1.39604 + 0.225998i −0.996195 0.0871557i 1.89785 0.631005i −3.02036 + 1.40842i 1.41042 0.103465i 0.0696086 0.120566i −2.50687 + 1.30982i 0.984808 + 0.173648i 3.89824 2.64880i
67.4 −1.39537 0.230080i 0.996195 + 0.0871557i 1.89413 + 0.642094i 1.80595 0.842128i −1.37001 0.350819i 0.971129 1.68204i −2.49528 1.33176i 0.984808 + 0.173648i −2.71373 + 0.759570i
67.5 −1.38343 + 0.293462i 0.996195 + 0.0871557i 1.82776 0.811970i −1.56884 + 0.731561i −1.40374 + 0.171772i 2.36990 4.10479i −2.29030 + 1.65968i 0.984808 + 0.173648i 1.95569 1.47246i
67.6 −1.38263 0.297198i −0.996195 0.0871557i 1.82335 + 0.821833i −1.83893 + 0.857507i 1.35147 + 0.416572i −2.27979 + 3.94871i −2.27677 1.67819i 0.984808 + 0.173648i 2.79741 0.639090i
67.7 −1.36953 + 0.352702i −0.996195 0.0871557i 1.75120 0.966068i 0.690327 0.321905i 1.39505 0.231997i 1.45929 2.52757i −2.05758 + 1.94071i 0.984808 + 0.173648i −0.831884 + 0.684336i
67.8 −1.36898 + 0.354799i 0.996195 + 0.0871557i 1.74824 0.971427i −0.640238 + 0.298548i −1.39470 + 0.234134i −1.64256 + 2.84500i −2.04865 + 1.95014i 0.984808 + 0.173648i 0.770552 0.635863i
67.9 −1.36646 + 0.364412i −0.996195 0.0871557i 1.73441 0.995906i 1.59591 0.744183i 1.39302 0.243931i −0.878950 + 1.52239i −2.00707 + 1.99290i 0.984808 + 0.173648i −1.90955 + 1.59846i
67.10 −1.32152 0.503585i 0.996195 + 0.0871557i 1.49280 + 1.33099i −3.76611 + 1.75616i −1.27260 0.616846i 0.782781 1.35582i −1.30250 2.51068i 0.984808 + 0.173648i 5.86135 0.424243i
67.11 −1.31402 0.522838i −0.996195 0.0871557i 1.45328 + 1.37404i −1.75107 + 0.816536i 1.26345 + 0.635372i −0.825112 + 1.42914i −1.19124 2.56534i 0.984808 + 0.173648i 2.72785 0.157418i
67.12 −1.27607 + 0.609618i 0.996195 + 0.0871557i 1.25673 1.55584i 3.88530 1.81174i −1.32435 + 0.496082i 0.817991 1.41680i −0.655215 + 2.75149i 0.984808 + 0.173648i −3.85346 + 4.68047i
67.13 −1.22595 0.705018i 0.996195 + 0.0871557i 1.00590 + 1.72863i −0.977951 + 0.456026i −1.15984 0.809183i −0.576210 + 0.998025i −0.0144655 2.82839i 0.984808 + 0.173648i 1.52042 + 0.130408i
67.14 −1.18996 + 0.764195i 0.996195 + 0.0871557i 0.832013 1.81872i −1.97149 + 0.919319i −1.25204 + 0.657575i 0.262357 0.454416i 0.399795 + 2.80003i 0.984808 + 0.173648i 1.64345 2.60055i
67.15 −1.16246 + 0.805415i −0.996195 0.0871557i 0.702613 1.87252i −2.99568 + 1.39691i 1.22823 0.701035i 0.753682 1.30542i 0.691398 + 2.74262i 0.984808 + 0.173648i 2.35726 4.03661i
67.16 −1.12072 0.862545i 0.996195 + 0.0871557i 0.512032 + 1.93335i 2.97317 1.38641i −1.04128 0.956940i −0.314742 + 0.545150i 1.09375 2.60839i 0.984808 + 0.173648i −4.52794 1.01071i
67.17 −1.10948 0.876960i −0.996195 0.0871557i 0.461884 + 1.94594i 1.24808 0.581989i 1.02882 + 0.970320i 1.65003 2.85793i 1.19406 2.56403i 0.984808 + 0.173648i −1.89510 0.448811i
67.18 −1.09515 + 0.894789i −0.996195 0.0871557i 0.398706 1.95986i 1.67888 0.782875i 1.16897 0.795935i −1.05656 + 1.83001i 1.31701 + 2.50309i 0.984808 + 0.173648i −1.13812 + 2.35961i
67.19 −1.06269 0.933102i −0.996195 0.0871557i 0.258640 + 1.98321i −0.264715 + 0.123438i 0.977326 + 1.02217i −0.251352 + 0.435354i 1.57568 2.34888i 0.984808 + 0.173648i 0.396492 + 0.115828i
67.20 −0.992328 + 1.00761i −0.996195 0.0871557i −0.0305710 1.99977i −0.810516 + 0.377950i 1.07637 0.917292i −2.00575 + 3.47406i 2.04533 + 1.95362i 0.984808 + 0.173648i 0.423470 1.19174i
See next 80 embeddings (of 960 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.80
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner
19.f odd 18 1 inner
304.bg even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.cn.a 960
16.f odd 4 1 inner 912.2.cn.a 960
19.f odd 18 1 inner 912.2.cn.a 960
304.bg even 36 1 inner 912.2.cn.a 960
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.cn.a 960 1.a even 1 1 trivial
912.2.cn.a 960 16.f odd 4 1 inner
912.2.cn.a 960 19.f odd 18 1 inner
912.2.cn.a 960 304.bg even 36 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(912, [\chi])\).