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Properties

Label 833.2.t.d
Level $833$
Weight $2$
Character orbit 833.t
Analytic conductor $6.652$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [833,2,Mod(48,833)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 833 = 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 833.t (of order \(16\), degree \(8\), 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: \(6.65153848837\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 8 q^{3} + 8 q^{5} + 8 q^{6} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 8 q^{3} + 8 q^{5} + 8 q^{6} + 24 q^{9} + 8 q^{11} - 32 q^{13} - 40 q^{18} + 24 q^{19} + 24 q^{22} + 8 q^{23} - 24 q^{24} + 8 q^{25} + 8 q^{27} + 80 q^{31} - 40 q^{32} - 80 q^{33} + 120 q^{36} + 48 q^{37} - 56 q^{38} - 16 q^{39} - 32 q^{40} + 24 q^{41} + 24 q^{43} + 32 q^{44} - 48 q^{45} - 40 q^{47} - 40 q^{51} - 24 q^{53} + 200 q^{54} - 48 q^{57} + 32 q^{58} + 8 q^{59} - 80 q^{60} + 128 q^{62} - 24 q^{65} - 8 q^{66} - 40 q^{68} - 48 q^{73} - 32 q^{74} + 128 q^{75} + 72 q^{76} - 48 q^{78} - 32 q^{79} + 256 q^{80} + 16 q^{81} - 152 q^{82} - 32 q^{85} - 80 q^{86} - 64 q^{87} + 56 q^{88} - 64 q^{89} - 376 q^{90} - 216 q^{92} - 32 q^{93} - 128 q^{94} + 32 q^{95} + 32 q^{96} - 8 q^{97} + 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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:
Format:

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