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Properties

Label 297.2.o.b
Level $297$
Weight $2$
Character orbit 297.o
Analytic conductor $2.372$
Analytic rank $0$
Dimension $192$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [297,2,Mod(32,297)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(297, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([5, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("297.32");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 297 = 3^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 297.o (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: \(2.37155694003\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(32\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 12 q^{3} - 12 q^{4} - 18 q^{5} - 24 q^{9} + 3 q^{11} - 30 q^{12} - 12 q^{14} + 12 q^{15} + 36 q^{20} - 15 q^{22} + 24 q^{23} - 6 q^{25} - 90 q^{27} - 6 q^{31} - 18 q^{34} - 126 q^{36} - 6 q^{37}+ \cdots - 63 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1 −2.09243 + 1.75576i −1.59250 + 0.681145i 0.948290 − 5.37802i −0.514806 − 1.41442i 2.13626 − 4.22129i 1.28107 − 0.225888i 4.72680 + 8.18706i 2.07208 − 2.16944i 3.56058 + 2.05570i
32.2 −1.91980 + 1.61091i −0.858067 − 1.50457i 0.743329 − 4.21563i 0.397283 + 1.09153i 4.07103 + 1.50621i −5.05912 + 0.892060i 2.85781 + 4.94987i −1.52744 + 2.58204i −2.52105 − 1.45553i
32.3 −1.91450 + 1.60646i 0.506021 + 1.65649i 0.737314 − 4.18151i 1.08239 + 2.97385i −3.62985 − 2.35844i 0.365561 − 0.0644583i 2.80663 + 4.86123i −2.48789 + 1.67643i −6.84961 − 3.95463i
32.4 −1.87415 + 1.57260i 0.781035 − 1.54596i 0.692074 − 3.92495i 0.806177 + 2.21495i 0.967392 + 4.12561i 3.52553 − 0.621646i 2.42878 + 4.20678i −1.77997 − 2.41489i −4.99413 − 2.88336i
32.5 −1.76104 + 1.47768i 1.05002 + 1.37748i 0.570400 − 3.23490i −1.18284 − 3.24983i −3.88461 − 0.874191i 0.136995 − 0.0241558i 1.47679 + 2.55788i −0.794905 + 2.89277i 6.88525 + 3.97520i
32.6 −1.47684 + 1.23921i 1.68551 − 0.398808i 0.298101 − 1.69061i −0.265754 − 0.730152i −1.99502 + 2.67768i 0.802991 − 0.141589i −0.273090 − 0.473006i 2.68191 − 1.34439i 1.29729 + 0.748990i
32.7 −1.34667 + 1.12999i 0.0420361 − 1.73154i 0.189349 − 1.07385i −1.03215 − 2.83581i 1.90002 + 2.37932i 0.573955 − 0.101204i −0.799502 − 1.38478i −2.99647 − 0.145575i 4.59442 + 2.65259i
32.8 −1.31148 + 1.10046i −0.659983 + 1.60138i 0.161662 − 0.916831i 0.164690 + 0.452483i −0.896703 − 2.82646i −1.83599 + 0.323735i −0.915090 − 1.58498i −2.12885 − 2.11377i −0.713926 − 0.412186i
32.9 −1.23479 + 1.03611i −1.70861 − 0.283968i 0.103880 − 0.589132i 1.27393 + 3.50009i 2.40400 − 1.41967i 3.32295 − 0.585925i −1.12976 − 1.95681i 2.83872 + 0.970383i −5.19951 − 3.00194i
32.10 −1.19486 + 1.00261i −1.71312 + 0.255401i 0.0751763 − 0.426346i −0.0532311 − 0.146251i 1.79087 − 2.02276i −2.48842 + 0.438776i −1.22215 − 2.11683i 2.86954 − 0.875063i 0.210237 + 0.121380i
32.11 −0.710212 + 0.595938i −1.04133 + 1.38406i −0.198038 + 1.12313i −1.33080 − 3.65633i −0.0852474 − 1.60355i 4.41264 − 0.778068i −1.45578 − 2.52149i −0.831248 − 2.88254i 3.12410 + 1.80370i
32.12 −0.685774 + 0.575433i 1.61836 + 0.617178i −0.208133 + 1.18038i 0.431823 + 1.18642i −1.46498 + 0.508014i 4.41551 − 0.778573i −1.43171 − 2.47980i 2.23818 + 1.99763i −0.978841 − 0.565134i
32.13 −0.675289 + 0.566635i −0.411537 − 1.68245i −0.212356 + 1.20433i 0.912254 + 2.50640i 1.23124 + 0.902949i −1.93778 + 0.341683i −1.42054 − 2.46045i −2.66128 + 1.38478i −2.03625 − 1.17563i
32.14 −0.575682 + 0.483055i 0.756929 + 1.55790i −0.249228 + 1.41344i −0.213646 − 0.586988i −1.18830 − 0.531218i −2.12623 + 0.374911i −1.29079 − 2.23572i −1.85412 + 2.35844i 0.406540 + 0.234716i
32.15 −0.558704 + 0.468808i 1.05330 − 1.37498i −0.254928 + 1.44577i −0.813006 − 2.23371i 0.0561209 + 1.26200i −3.18953 + 0.562400i −1.26469 − 2.19051i −0.781135 − 2.89652i 1.50141 + 0.866841i
32.16 −0.230822 + 0.193682i −1.44776 − 0.950785i −0.331531 + 1.88020i −0.569924 − 1.56585i 0.518325 − 0.0609437i 2.33309 − 0.411387i −0.588954 − 1.02010i 1.19202 + 2.75302i 0.434829 + 0.251049i
32.17 0.230822 − 0.193682i −1.44776 − 0.950785i −0.331531 + 1.88020i −0.569924 − 1.56585i −0.518325 + 0.0609437i −2.33309 + 0.411387i 0.588954 + 1.02010i 1.19202 + 2.75302i −0.434829 − 0.251049i
32.18 0.558704 − 0.468808i 1.05330 − 1.37498i −0.254928 + 1.44577i −0.813006 − 2.23371i −0.0561209 − 1.26200i 3.18953 − 0.562400i 1.26469 + 2.19051i −0.781135 − 2.89652i −1.50141 − 0.866841i
32.19 0.575682 − 0.483055i 0.756929 + 1.55790i −0.249228 + 1.41344i −0.213646 − 0.586988i 1.18830 + 0.531218i 2.12623 − 0.374911i 1.29079 + 2.23572i −1.85412 + 2.35844i −0.406540 − 0.234716i
32.20 0.675289 − 0.566635i −0.411537 − 1.68245i −0.212356 + 1.20433i 0.912254 + 2.50640i −1.23124 − 0.902949i 1.93778 − 0.341683i 1.42054 + 2.46045i −2.66128 + 1.38478i 2.03625 + 1.17563i
See next 80 embeddings (of 192 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
27.f odd 18 1 inner
297.o even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 297.2.o.b ✓ 192
3.b odd 2 1 891.2.o.b 192
11.b odd 2 1 inner 297.2.o.b ✓ 192
27.e even 9 1 891.2.o.b 192
27.f odd 18 1 inner 297.2.o.b ✓ 192
33.d even 2 1 891.2.o.b 192
297.o even 18 1 inner 297.2.o.b ✓ 192
297.q odd 18 1 891.2.o.b 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.o.b ✓ 192 1.a even 1 1 trivial
297.2.o.b ✓ 192 11.b odd 2 1 inner
297.2.o.b ✓ 192 27.f odd 18 1 inner
297.2.o.b ✓ 192 297.o even 18 1 inner
891.2.o.b 192 3.b odd 2 1
891.2.o.b 192 27.e even 9 1
891.2.o.b 192 33.d even 2 1
891.2.o.b 192 297.q odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{192} + 6 T_{2}^{190} + 9 T_{2}^{188} + 1389 T_{2}^{186} + 7974 T_{2}^{184} + \cdots + 48\!\cdots\!01 \) acting on \(S_{2}^{\mathrm{new}}(297, [\chi])\). Copy content Toggle raw display