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Properties

Label 900.2.bs.a
Level $900$
Weight $2$
Character orbit 900.bs
Analytic conductor $7.187$
Analytic rank $0$
Dimension $2816$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [900,2,Mod(67,900)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("900.67"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(900, base_ring=CyclotomicField(60)) chi = DirichletCharacter(H, H._module([30, 20, 39])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 900.bs (of order \(60\), degree \(16\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.18653618192\)
Analytic rank: \(0\)
Dimension: \(2816\)
Relative dimension: \(176\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2816 q - 8 q^{2} - 10 q^{4} - 16 q^{5} - 12 q^{6} - 32 q^{8} - 40 q^{9} - 32 q^{10} - 22 q^{12} - 16 q^{13} - 10 q^{14} - 6 q^{16} - 64 q^{17} + 16 q^{18} + 8 q^{20} - 24 q^{21} - 16 q^{25} - 64 q^{26}+ \cdots - 244 q^{98}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1 −1.41285 − 0.0621000i 1.70498 − 0.305019i 1.99229 + 0.175476i 1.99392 − 1.01207i −2.42782 + 0.325067i 1.73368 + 0.464537i −2.80390 − 0.371642i 2.81393 − 1.04010i −2.87996 + 1.30608i
67.2 −1.41239 + 0.0718146i 1.59967 + 0.664132i 1.98969 − 0.202860i −1.20871 − 1.88123i −2.30704 − 0.823133i −3.32278 − 0.890336i −2.79564 + 0.429406i 2.11786 + 2.12478i 1.84227 + 2.57022i
67.3 −1.41216 + 0.0762136i −1.59967 − 0.664132i 1.98838 − 0.215251i −1.20871 − 1.88123i 2.30960 + 0.815943i 3.32278 + 0.890336i −2.79151 + 0.455511i 2.11786 + 2.12478i 1.85027 + 2.56447i
67.4 −1.40754 − 0.137222i 1.08120 + 1.35315i 1.96234 + 0.386291i −1.57853 + 1.58374i −1.33615 − 2.05298i 1.23005 + 0.329590i −2.70907 − 0.812996i −0.662028 + 2.92604i 2.43917 − 2.01257i
67.5 −1.40725 − 0.140201i −0.474364 + 1.66583i 1.96069 + 0.394594i 1.42970 + 1.71929i 0.901097 − 2.27772i −4.19553 − 1.12419i −2.70385 − 0.830181i −2.54996 − 1.58042i −1.77089 − 2.61991i
67.6 −1.40660 − 0.146596i −0.421771 − 1.67991i 1.95702 + 0.412402i 2.16013 + 0.577786i 0.346992 + 2.42479i −4.94716 − 1.32559i −2.69228 − 0.866975i −2.64422 + 1.41708i −2.95373 − 1.12938i
67.7 −1.39862 + 0.209443i −1.70498 + 0.305019i 1.91227 − 0.585861i 1.99392 − 1.01207i 2.32074 − 0.783702i −1.73368 − 0.464537i −2.55183 + 1.21991i 2.81393 − 1.04010i −2.57676 + 1.83311i
67.8 −1.39516 − 0.231340i −0.881359 + 1.49104i 1.89296 + 0.645515i −0.772730 + 2.09831i 1.57458 − 1.87635i 2.07222 + 0.555251i −2.49166 − 1.33852i −1.44641 − 2.62829i 1.56351 − 2.74872i
67.9 −1.38549 + 0.283598i −1.08120 − 1.35315i 1.83914 − 0.785843i −1.57853 + 1.58374i 1.88173 + 1.56814i −1.23005 − 0.329590i −2.32525 + 1.61035i −0.662028 + 2.92604i 1.73789 − 2.64192i
67.10 −1.38488 + 0.286530i 0.474364 − 1.66583i 1.83580 − 0.793621i 1.42970 + 1.71929i −0.179629 + 2.44289i 4.19553 + 1.12419i −2.31497 + 1.62508i −2.54996 − 1.58042i −2.47259 − 1.97137i
67.11 −1.38357 + 0.292822i 0.421771 + 1.67991i 1.82851 − 0.810278i 2.16013 + 0.577786i −1.07546 − 2.20077i 4.94716 + 1.32559i −2.29260 + 1.65650i −2.64422 + 1.41708i −3.15787 − 0.166871i
67.12 −1.38261 − 0.297324i −1.49849 − 0.868640i 1.82320 + 0.822163i 1.02261 + 1.98853i 1.81355 + 1.64652i 1.76051 + 0.471728i −2.27631 − 1.67881i 1.49093 + 2.60329i −0.822634 − 3.05340i
67.13 −1.37555 − 0.328415i 1.03424 − 1.38937i 1.78429 + 0.903503i −2.23342 + 0.108687i −1.87894 + 1.57149i 4.18156 + 1.12044i −2.15766 − 1.82880i −0.860685 − 2.87389i 3.10789 + 0.583985i
67.14 −1.37445 − 0.333012i 0.396300 + 1.68610i 1.77821 + 0.915415i −0.302252 − 2.21555i 0.0167997 − 2.44943i −1.39229 − 0.373064i −2.13920 − 1.85035i −2.68589 + 1.33641i −0.322375 + 3.14580i
67.15 −1.36739 − 0.360908i −0.388549 − 1.68791i 1.73949 + 0.987002i −2.22817 − 0.187726i −0.0778834 + 2.44825i −0.453284 − 0.121457i −2.02234 − 1.97741i −2.69806 + 1.31167i 2.97902 + 1.06086i
67.16 −1.36334 + 0.375907i 0.881359 − 1.49104i 1.71739 − 1.02498i −0.772730 + 2.09831i −0.641098 + 2.36411i −2.07222 − 0.555251i −1.95609 + 2.04297i −1.44641 − 2.62829i 0.264724 − 3.15118i
67.17 −1.35173 − 0.415714i −0.643162 − 1.60821i 1.65436 + 1.12387i 1.35811 − 1.77638i 0.200828 + 2.44124i 1.54287 + 0.413410i −1.76905 − 2.20691i −2.17268 + 2.06868i −2.57427 + 1.83661i
67.18 −1.35051 − 0.419679i −1.65914 + 0.497255i 1.64774 + 1.13356i −1.86014 − 1.24091i 2.44936 + 0.0247591i −1.13575 − 0.304324i −1.74955 − 2.22240i 2.50547 − 1.65003i 1.99135 + 2.45652i
67.19 −1.34395 + 0.440217i 1.49849 + 0.868640i 1.61242 − 1.18326i 1.02261 + 1.98853i −2.39629 − 0.507751i −1.76051 − 0.471728i −1.64612 + 2.30006i 1.49093 + 2.60329i −2.24973 − 2.22232i
67.20 −1.34364 − 0.441157i 1.69434 − 0.359467i 1.61076 + 1.18552i −1.88764 + 1.19868i −2.43517 − 0.264474i −3.81595 − 1.02248i −1.64129 − 2.30351i 2.74157 − 1.21812i 3.06512 − 0.777853i
See next 80 embeddings (of 2816 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.176
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
9.c even 3 1 inner
25.f odd 20 1 inner
36.f odd 6 1 inner
100.l even 20 1 inner
225.x odd 60 1 inner
900.bs even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.2.bs.a ✓ 2816
4.b odd 2 1 inner 900.2.bs.a ✓ 2816
9.c even 3 1 inner 900.2.bs.a ✓ 2816
25.f odd 20 1 inner 900.2.bs.a ✓ 2816
36.f odd 6 1 inner 900.2.bs.a ✓ 2816
100.l even 20 1 inner 900.2.bs.a ✓ 2816
225.x odd 60 1 inner 900.2.bs.a ✓ 2816
900.bs even 60 1 inner 900.2.bs.a ✓ 2816
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.2.bs.a ✓ 2816 1.a even 1 1 trivial
900.2.bs.a ✓ 2816 4.b odd 2 1 inner
900.2.bs.a ✓ 2816 9.c even 3 1 inner
900.2.bs.a ✓ 2816 25.f odd 20 1 inner
900.2.bs.a ✓ 2816 36.f odd 6 1 inner
900.2.bs.a ✓ 2816 100.l even 20 1 inner
900.2.bs.a ✓ 2816 225.x odd 60 1 inner
900.2.bs.a ✓ 2816 900.bs even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(900, [\chi])\).