↑

Properties

Label 936.2.em.a
Level $936$
Weight $2$
Character orbit 936.em
Analytic conductor $7.474$
Analytic rank $0$
Dimension $656$
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [936,2,Mod(5,936)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(936, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 6, 10, 9])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("936.5"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.em (of order \(12\), degree \(4\), 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: \(7.47399762919\)
Analytic rank: \(0\)
Dimension: \(656\)
Relative dimension: \(164\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 656 q - 6 q^{2} + 6 q^{6} - 4 q^{7} - 16 q^{9} - 12 q^{14} + 4 q^{15} - 4 q^{16} - 2 q^{18} - 6 q^{20} - 4 q^{22} - 32 q^{24} - 24 q^{28} - 4 q^{31} - 6 q^{32} + 4 q^{33} - 6 q^{34} - 20 q^{39} - 4 q^{40}+ \cdots - 4 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −1.41376 − 0.0358025i −0.700335 − 1.58415i 1.99744 + 0.101232i 0.0180874 + 0.0675031i 0.933390 + 2.26468i −0.695957 + 2.59735i −2.82027 − 0.214631i −2.01906 + 2.21887i −0.0231545 − 0.0960807i
5.2 −1.41321 + 0.0531357i 1.72013 + 0.202834i 1.99435 − 0.150184i 0.575126 + 2.14640i −2.44170 − 0.195248i 0.751447 − 2.80444i −2.81047 + 0.318214i 2.91772 + 0.697804i −0.926827 − 3.00276i
5.3 −1.41302 − 0.0579876i 1.40761 + 1.00927i 1.99327 + 0.163876i 0.881637 + 3.29032i −1.93046 − 1.50775i −0.863534 + 3.22275i −2.80704 − 0.347146i 0.962732 + 2.84133i −1.05498 − 4.70042i
5.4 −1.41208 − 0.0775900i −0.226318 − 1.71720i 1.98796 + 0.219127i 0.825410 + 3.08047i 0.186343 + 2.44239i 1.08514 − 4.04979i −2.79016 − 0.463671i −2.89756 + 0.777268i −0.926534 − 4.41393i
5.5 −1.41099 − 0.0955053i 1.62333 − 0.604000i 1.98176 + 0.269513i −0.824037 − 3.07535i −2.34817 + 0.697198i −0.387824 + 1.44738i −2.77049 − 0.569547i 2.27037 − 1.96098i 0.868992 + 4.41797i
5.6 −1.40893 + 0.122095i −1.68850 − 0.385951i 1.97019 − 0.344048i 0.241254 + 0.900372i 2.42611 + 0.337621i −0.282677 + 1.05496i −2.73385 + 0.725290i 2.70208 + 1.30336i −0.449842 − 1.23911i
5.7 −1.40144 − 0.189655i 0.528267 − 1.64953i 1.92806 + 0.531579i −0.431121 − 1.60897i −1.05317 + 2.21152i 0.704291 − 2.62845i −2.60125 − 1.11064i −2.44187 − 1.74278i 0.299042 + 2.33663i
5.8 −1.39890 − 0.207571i 0.710910 + 1.57943i 1.91383 + 0.580741i −0.727736 − 2.71595i −0.666646 − 2.35703i −0.267883 + 0.999754i −2.55671 − 1.20965i −1.98921 + 2.24567i 0.454277 + 3.95039i
5.9 −1.39487 − 0.233077i −0.935379 + 1.45776i 1.89135 + 0.650225i −0.182733 − 0.681970i 1.64451 − 1.81538i 0.331919 − 1.23874i −2.48664 − 1.34781i −1.25013 − 2.72712i 0.0959388 + 0.993853i
5.10 −1.38966 + 0.262383i −1.41602 − 0.997447i 1.86231 − 0.729247i −0.647958 − 2.41821i 2.22949 + 1.01457i 0.308834 − 1.15258i −2.39664 + 1.50204i 1.01020 + 2.82480i 1.53494 + 3.19048i
5.11 −1.38863 − 0.267791i 0.785236 + 1.54383i 1.85658 + 0.743724i −0.228862 − 0.854125i −0.676979 − 2.35408i 1.00594 − 3.75423i −2.37893 − 1.52993i −1.76681 + 2.42454i 0.0890775 + 1.24735i
5.12 −1.38429 − 0.289375i −1.45142 + 0.945190i 1.83252 + 0.801158i 0.0701899 + 0.261952i 2.28270 − 0.888415i −1.31796 + 4.91869i −2.30491 − 1.63932i 1.21323 − 2.74373i −0.0213609 − 0.382929i
5.13 −1.38425 + 0.289595i −1.55312 + 0.766689i 1.83227 − 0.801742i −0.412571 − 1.53974i 1.92787 − 1.51106i 1.10981 − 4.14187i −2.30413 + 1.64042i 1.82438 − 2.38152i 1.01700 + 2.01189i
5.14 −1.36456 − 0.371460i 1.71262 − 0.258744i 1.72403 + 1.01376i 0.276289 + 1.03112i −2.43308 − 0.283098i −0.714095 + 2.66504i −1.97597 − 2.02374i 2.86610 − 0.886258i 0.00600982 − 1.50966i
5.15 −1.35327 + 0.410691i 1.17824 − 1.26955i 1.66267 − 1.11155i 0.593075 + 2.21339i −1.07307 + 2.20193i −0.832292 + 3.10616i −1.79353 + 2.18707i −0.223521 − 2.99166i −1.71161 − 2.75173i
5.16 −1.34560 + 0.435144i −0.114709 + 1.72825i 1.62130 − 1.17106i 0.0471805 + 0.176080i −0.597684 − 2.37545i −0.351300 + 1.31107i −1.67204 + 2.28129i −2.97368 − 0.396493i −0.140107 − 0.216404i
5.17 −1.32146 + 0.503724i −0.935678 − 1.45757i 1.49252 − 1.33130i 1.08622 + 4.05383i 1.97068 + 1.45480i −0.190925 + 0.712543i −1.30170 + 2.51109i −1.24901 + 2.72763i −3.47741 − 4.80982i
5.18 −1.32034 − 0.506666i −0.754784 + 1.55894i 1.48658 + 1.33794i 1.03054 + 3.84603i 1.78643 − 1.67590i −0.0344252 + 0.128476i −1.28490 − 2.51973i −1.86060 − 2.35333i 0.587992 − 5.60019i
5.19 −1.31716 + 0.514870i 1.47405 + 0.909496i 1.46982 − 1.35633i −0.633992 − 2.36609i −2.40983 − 0.439008i −0.466230 + 1.73999i −1.23765 + 2.54327i 1.34563 + 2.68128i 2.05330 + 2.79009i
5.20 −1.31161 + 0.528849i −1.43144 + 0.975187i 1.44064 − 1.38729i 0.781042 + 2.91489i 1.36176 − 2.03608i 0.165226 − 0.616633i −1.15589 + 2.58146i 1.09802 − 2.79184i −2.56596 − 3.41014i
See next 80 embeddings (of 656 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.164
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
9.d odd 6 1 inner
13.d odd 4 1 inner
72.j odd 6 1 inner
104.j odd 4 1 inner
117.z even 12 1 inner
936.em even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.em.a ✓ 656
8.b even 2 1 inner 936.2.em.a ✓ 656
9.d odd 6 1 inner 936.2.em.a ✓ 656
13.d odd 4 1 inner 936.2.em.a ✓ 656
72.j odd 6 1 inner 936.2.em.a ✓ 656
104.j odd 4 1 inner 936.2.em.a ✓ 656
117.z even 12 1 inner 936.2.em.a ✓ 656
936.em even 12 1 inner 936.2.em.a ✓ 656
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
936.2.em.a ✓ 656 1.a even 1 1 trivial
936.2.em.a ✓ 656 8.b even 2 1 inner
936.2.em.a ✓ 656 9.d odd 6 1 inner
936.2.em.a ✓ 656 13.d odd 4 1 inner
936.2.em.a ✓ 656 72.j odd 6 1 inner
936.2.em.a ✓ 656 104.j odd 4 1 inner
936.2.em.a ✓ 656 117.z even 12 1 inner
936.2.em.a ✓ 656 936.em even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(936, [\chi])\).