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Properties

Label 833.2.t.a
Level $833$
Weight $2$
Character orbit 833.t
Analytic conductor $6.652$
Analytic rank $0$
Dimension $72$
Inner twists $2$

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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] = [833,2,Mod(48,833)] 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("833.48"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(833, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([8, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 833 = 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 833.t (of order \(16\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,0,-8,0,-8,-8,0,0,24,0,8,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(12)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.65153848837\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(9\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 8 q^{3} - 8 q^{5} - 8 q^{6} + 24 q^{9} + 8 q^{11} + 32 q^{13} - 40 q^{18} - 24 q^{19} + 24 q^{22} + 8 q^{23} + 24 q^{24} + 8 q^{25} - 8 q^{27} - 80 q^{31} - 40 q^{32} + 80 q^{33} + 120 q^{36} + 48 q^{37}+ \cdots + 40 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
48.1 −2.56233 + 1.06135i −1.71514 − 0.341162i 4.02484 − 4.02484i −0.155648 − 0.232944i 4.75684 − 0.946194i 0 −3.91849 + 9.46008i 0.0536674 + 0.0222297i 0.646056 + 0.431681i
48.2 −2.18495 + 0.905036i 3.23570 + 0.643620i 2.54070 − 2.54070i −1.04445 − 1.56313i −7.65233 + 1.52214i 0 −1.44181 + 3.48083i 7.28385 + 3.01707i 3.69676 + 2.47009i
48.3 −1.52417 + 0.631334i 0.480778 + 0.0956326i 0.510311 − 0.510311i 1.43429 + 2.14657i −0.793165 + 0.157770i 0 0.807041 − 1.94837i −2.54964 − 1.05609i −3.54131 − 2.36623i
48.4 −0.985259 + 0.408108i −2.83391 − 0.563700i −0.610030 + 0.610030i 0.380055 + 0.568792i 3.02219 − 0.601151i 0 1.16830 − 2.82051i 4.94167 + 2.04690i −0.606580 − 0.405304i
48.5 −0.355236 + 0.147143i 1.41465 + 0.281392i −1.30967 + 1.30967i −0.839778 − 1.25682i −0.543939 + 0.108196i 0 0.566820 − 1.36842i −0.849583 − 0.351909i 0.483251 + 0.322898i
48.6 0.796325 − 0.329849i −1.48546 − 0.295477i −0.888880 + 0.888880i −1.62687 − 2.43478i −1.28037 + 0.254682i 0 −1.07434 + 2.59368i −0.652344 − 0.270210i −2.09862 − 1.40226i
48.7 1.36887 − 0.567005i 2.16558 + 0.430761i 0.138099 − 0.138099i 1.47955 + 2.21431i 3.20864 − 0.638239i 0 −1.02327 + 2.47040i 1.73255 + 0.717644i 3.28084 + 2.19219i
48.8 2.18677 − 0.905789i −3.01102 − 0.598929i 2.54729 − 2.54729i 1.18717 + 1.77673i −7.12690 + 1.41763i 0 1.45144 − 3.50409i 5.93588 + 2.45872i 4.20540 + 2.80996i
48.9 2.33610 − 0.967645i 0.914736 + 0.181952i 3.10682 − 3.10682i −0.998100 − 1.49376i 2.31298 − 0.460081i 0 2.31626 − 5.59195i −1.96800 − 0.815173i −3.77709 − 2.52377i
97.1 −1.02069 − 2.46415i 0.692606 + 1.03656i −3.61604 + 3.61604i 0.102295 − 0.514273i 1.84730 − 2.76469i 0 7.67299 + 3.17826i 0.553301 − 1.33579i −1.37166 + 0.272840i
97.2 −0.684321 − 1.65210i −0.302826 − 0.453212i −0.846919 + 0.846919i −0.523006 + 2.62933i −0.541520 + 0.810442i 0 −1.32544 − 0.549014i 1.03435 − 2.49715i 4.70181 − 0.935249i
97.3 −0.601251 − 1.45155i −1.18492 − 1.77336i −0.331273 + 0.331273i 0.240255 − 1.20785i −1.86168 + 2.78620i 0 −2.22306 − 0.920820i −0.592708 + 1.43092i −1.89770 + 0.377476i
97.4 −0.252613 − 0.609862i 0.947770 + 1.41844i 1.10610 − 1.10610i 0.849136 − 4.26890i 0.625632 − 0.936325i 0 −2.17370 − 0.900378i 0.0343520 − 0.0829330i −2.81794 + 0.560523i
97.5 0.0230395 + 0.0556222i 0.638303 + 0.955287i 1.41165 − 1.41165i −0.456570 + 2.29533i −0.0384290 + 0.0575131i 0 0.222287 + 0.0920744i 0.642907 − 1.55211i −0.138191 + 0.0274878i
97.6 0.422335 + 1.01961i −1.37804 − 2.06239i 0.552980 − 0.552980i 0.370260 − 1.86142i 1.52083 − 2.27608i 0 2.83658 + 1.17495i −1.20639 + 2.91248i 2.05430 − 0.408625i
97.7 0.672106 + 1.62261i 1.35675 + 2.03053i −0.766915 + 0.766915i 0.0982159 − 0.493765i −2.38286 + 3.56621i 0 1.48536 + 0.615258i −1.13420 + 2.73821i 0.867198 − 0.172496i
97.8 0.768113 + 1.85439i −0.823215 − 1.23203i −1.43455 + 1.43455i −0.743583 + 3.73825i 1.65234 − 2.47290i 0 −0.0533216 − 0.0220865i 0.307840 − 0.743193i −7.50332 + 1.49250i
97.9 1.05596 + 2.54931i −0.346973 − 0.519282i −3.96973 + 3.96973i 0.635722 − 3.19599i 0.957422 − 1.43288i 0 −9.21334 − 3.81629i 0.998787 − 2.41129i 8.81888 − 1.75418i
146.1 −1.02069 + 2.46415i 0.692606 − 1.03656i −3.61604 − 3.61604i 0.102295 + 0.514273i 1.84730 + 2.76469i 0 7.67299 − 3.17826i 0.553301 + 1.33579i −1.37166 − 0.272840i
146.2 −0.684321 + 1.65210i −0.302826 + 0.453212i −0.846919 − 0.846919i −0.523006 − 2.62933i −0.541520 − 0.810442i 0 −1.32544 + 0.549014i 1.03435 + 2.49715i 4.70181 + 0.935249i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 48.9
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
119.p even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 833.2.t.a ✓ 72
7.b odd 2 1 833.2.t.d yes 72
7.c even 3 2 833.2.bc.d 144
7.d odd 6 2 833.2.bc.a 144
17.e odd 16 1 833.2.t.d yes 72
119.p even 16 1 inner 833.2.t.a ✓ 72
119.s even 48 2 833.2.bc.d 144
119.t odd 48 2 833.2.bc.a 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
833.2.t.a ✓ 72 1.a even 1 1 trivial
833.2.t.a ✓ 72 119.p even 16 1 inner
833.2.t.d yes 72 7.b odd 2 1
833.2.t.d yes 72 17.e odd 16 1
833.2.bc.a 144 7.d odd 6 2
833.2.bc.a 144 119.t odd 48 2
833.2.bc.d 144 7.c even 3 2
833.2.bc.d 144 119.s even 48 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(833, [\chi])\):

\( T_{2}^{72} + 8 T_{2}^{67} - 72 T_{2}^{65} + 5079 T_{2}^{64} - 576 T_{2}^{63} - 432 T_{2}^{62} + \cdots + 16129 \) Copy content Toggle raw display
\( T_{3}^{72} + 8 T_{3}^{71} + 20 T_{3}^{70} - 8 T_{3}^{69} - 160 T_{3}^{68} - 624 T_{3}^{67} + \cdots + 358982369792 \) Copy content Toggle raw display