Properties

Label 296.2.s.a
Level $296$
Weight $2$
Character orbit 296.s
Analytic conductor $2.364$
Analytic rank $0$
Dimension $72$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [296,2,Mod(269,296)] 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("296.269"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(296, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 296 = 2^{3} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 296.s (of order \(6\), degree \(2\), 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: \(2.36357189983\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 2 q^{4} - 8 q^{6} - 6 q^{7} + 30 q^{9} - 8 q^{10} - 2 q^{12} - 8 q^{14} - 14 q^{15} - 6 q^{16} + 18 q^{18} - 16 q^{20} + 8 q^{22} + 20 q^{24} + 24 q^{25} + 8 q^{26} + 4 q^{28} + 4 q^{30} - 8 q^{31}+ \cdots - 30 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1 −1.39361 + 0.240527i 0.555075 − 0.320472i 1.88429 − 0.670401i 0.236166 − 0.136351i −0.696475 + 0.580124i 1.38239 + 2.39437i −2.46472 + 1.38750i −1.29459 + 2.24230i −0.296327 + 0.246824i
269.2 −1.37353 − 0.336789i −0.975356 + 0.563122i 1.77315 + 0.925178i −0.721676 + 0.416660i 1.52933 − 0.444973i 0.341595 + 0.591659i −2.12387 − 1.86793i −0.865787 + 1.49959i 1.13157 − 0.329240i
269.3 −1.34953 − 0.422796i 1.20920 − 0.698134i 1.64249 + 1.14116i 1.89921 − 1.09651i −1.92703 + 0.430909i −2.24551 − 3.88934i −1.73411 − 2.23447i −0.525218 + 0.909704i −3.02665 + 0.676799i
269.4 −1.32618 + 0.491157i −1.92368 + 1.11064i 1.51753 − 1.30273i −2.53462 + 1.46336i 2.00566 − 2.41774i −0.853086 − 1.47759i −1.37268 + 2.47300i 0.967030 − 1.67494i 2.64263 − 3.18558i
269.5 −1.31305 − 0.525256i −2.81249 + 1.62379i 1.44821 + 1.37938i 1.27213 − 0.734465i 4.54585 − 0.654844i −0.481147 − 0.833370i −1.17705 − 2.57188i 3.77339 − 6.53570i −2.05616 + 0.296196i
269.6 −1.29250 + 0.573980i 2.62536 − 1.51575i 1.34109 − 1.48373i −2.41637 + 1.39509i −2.52326 + 3.46601i −1.97190 − 3.41544i −0.881723 + 2.68748i 3.09502 − 5.36073i 2.32240 − 3.19010i
269.7 −1.25014 − 0.661170i 1.47475 − 0.851446i 1.12571 + 1.65311i −3.51786 + 2.03104i −2.40659 + 0.0893698i 0.408571 + 0.707665i −0.314309 − 2.81091i −0.0500808 + 0.0867425i 5.74068 − 0.213182i
269.8 −1.13598 + 0.842344i 2.01339 − 1.16243i 0.580912 − 1.91378i 3.12944 − 1.80678i −1.30801 + 3.01646i 0.259310 + 0.449138i 0.952154 + 2.66334i 1.20248 − 2.08276i −2.03305 + 4.68854i
269.9 −0.981975 − 1.01771i −0.114166 + 0.0659138i −0.0714485 + 1.99872i 3.62817 − 2.09472i 0.179189 + 0.0514617i 1.60196 + 2.77467i 2.10427 − 1.88998i −1.49131 + 2.58303i −5.69458 − 1.63544i
269.10 −0.958354 + 1.03998i −2.28803 + 1.32100i −0.163114 − 1.99334i 1.31467 − 0.759022i 0.818937 − 3.64549i 2.53121 + 4.38418i 2.22935 + 1.74069i 1.99006 − 3.44689i −0.470547 + 2.09464i
269.11 −0.758405 + 1.19366i 0.434413 − 0.250808i −0.849644 − 1.81055i −1.29153 + 0.745668i −0.0300811 + 0.708755i −0.955971 − 1.65579i 2.80556 + 0.358948i −1.37419 + 2.38017i 0.0894329 − 2.10717i
269.12 −0.739652 − 1.20537i 2.07843 − 1.19998i −0.905829 + 1.78311i 0.0352460 − 0.0203493i −2.98373 − 1.61770i 0.694063 + 1.20215i 2.81930 − 0.227022i 1.37991 − 2.39007i −0.0505981 − 0.0274330i
269.13 −0.654537 + 1.25363i −0.434413 + 0.250808i −1.14316 − 1.64109i 1.29153 − 0.745668i −0.0300811 − 0.708755i −0.955971 − 1.65579i 2.80556 − 0.358948i −1.37419 + 2.38017i 0.0894329 + 2.10717i
269.14 −0.601535 − 1.27990i −1.13850 + 0.657311i −1.27631 + 1.53981i 0.645687 − 0.372788i 1.52614 + 1.06177i −1.56385 − 2.70867i 2.73856 + 0.707306i −0.635883 + 1.10138i −0.865536 − 0.602173i
269.15 −0.421471 + 1.34995i 2.28803 − 1.32100i −1.64472 − 1.13793i −1.31467 + 0.759022i 0.818937 + 3.64549i 2.53121 + 4.38418i 2.22935 − 1.74069i 1.99006 − 3.44689i −0.470547 − 2.09464i
269.16 −0.294508 − 1.38321i 0.158616 − 0.0915768i −1.82653 + 0.814732i −1.79163 + 1.03440i −0.173383 − 0.192428i 0.499734 + 0.865565i 1.66487 + 2.28653i −1.48323 + 2.56903i 1.95843 + 2.17355i
269.17 −0.161500 + 1.40496i −2.01339 + 1.16243i −1.94784 − 0.453804i −3.12944 + 1.80678i −1.30801 − 3.01646i 0.259310 + 0.449138i 0.952154 − 2.66334i 1.20248 − 2.08276i −2.03305 − 4.68854i
269.18 0.0178034 − 1.41410i −2.15787 + 1.24585i −1.99937 − 0.0503517i 1.75942 − 1.01580i 1.72334 + 3.07363i 0.675600 + 1.17017i −0.106798 + 2.82641i 1.60427 − 2.77867i −1.40512 − 2.50608i
269.19 0.0384450 − 1.41369i 2.54408 − 1.46882i −1.99704 − 0.108699i 1.20962 − 0.698375i −1.97866 − 3.65301i −1.27124 − 2.20186i −0.230443 + 2.81902i 2.81489 − 4.87553i −0.940782 − 1.73688i
269.20 0.149167 + 1.40632i −2.62536 + 1.51575i −1.95550 + 0.419553i 2.41637 − 1.39509i −2.52326 − 3.46601i −1.97190 − 3.41544i −0.881723 − 2.68748i 3.09502 − 5.36073i 2.32240 + 3.19010i
See all 72 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 269.36
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
37.c even 3 1 inner
296.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 296.2.s.a ✓ 72
4.b odd 2 1 1184.2.ba.a 72
8.b even 2 1 inner 296.2.s.a ✓ 72
8.d odd 2 1 1184.2.ba.a 72
37.c even 3 1 inner 296.2.s.a ✓ 72
148.i odd 6 1 1184.2.ba.a 72
296.p odd 6 1 1184.2.ba.a 72
296.s even 6 1 inner 296.2.s.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
296.2.s.a ✓ 72 1.a even 1 1 trivial
296.2.s.a ✓ 72 8.b even 2 1 inner
296.2.s.a ✓ 72 37.c even 3 1 inner
296.2.s.a ✓ 72 296.s even 6 1 inner
1184.2.ba.a 72 4.b odd 2 1
1184.2.ba.a 72 8.d odd 2 1
1184.2.ba.a 72 148.i odd 6 1
1184.2.ba.a 72 296.p odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(296, [\chi])\).