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Properties

Label 833.2.bc.b
Level $833$
Weight $2$
Character orbit 833.bc
Analytic conductor $6.652$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [833,2,Mod(31,833)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(833, base_ring=CyclotomicField(48))
 
chi = DirichletCharacter(H, H._module([8, 27]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("833.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 833 = 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 833.bc (of order \(48\), degree \(16\), not 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: \(144\)
Relative dimension: \(9\) over \(\Q(\zeta_{48})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{48}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + 24 q^{9} + 8 q^{11} - 24 q^{12} + 64 q^{13} + 32 q^{17} - 40 q^{18} - 40 q^{19} - 80 q^{20} - 48 q^{22} + 8 q^{23} + 16 q^{24} + 8 q^{25} + 48 q^{27} + 8 q^{31} - 40 q^{32} + 48 q^{34} - 240 q^{36} + 48 q^{37} - 56 q^{38} + 16 q^{39} - 48 q^{40} - 112 q^{41} - 48 q^{43} - 96 q^{44} + 40 q^{45} + 64 q^{46} + 24 q^{47} + 448 q^{48} - 104 q^{51} + 40 q^{53} + 48 q^{54} + 96 q^{57} - 32 q^{58} + 8 q^{59} + 48 q^{60} - 128 q^{61} - 96 q^{62} - 56 q^{65} + 120 q^{66} + 32 q^{68} - 120 q^{73} - 48 q^{74} + 32 q^{75} - 16 q^{76} + 96 q^{78} - 32 q^{79} - 32 q^{80} - 80 q^{81} + 64 q^{82} + 64 q^{83} - 64 q^{85} + 80 q^{86} - 64 q^{87} - 136 q^{88} - 16 q^{89} - 64 q^{90} - 16 q^{92} + 96 q^{93} + 32 q^{94} - 32 q^{95} + 40 q^{96} - 80 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −1.91484 − 1.46931i 0.364424 − 0.415546i 0.990110 + 3.69514i −1.59065 + 3.22553i −1.30838 + 0.260253i 0 1.68611 − 4.07063i 0.351705 + 2.67146i 7.78515 − 3.83921i
31.2 −1.68105 − 1.28992i −1.52675 + 1.74093i 0.644413 + 2.40498i 0.749763 − 1.52037i 4.81221 − 0.957207i 0 0.397183 − 0.958885i −0.308279 − 2.34161i −3.22154 + 1.58869i
31.3 −0.872917 − 0.669813i 2.15498 − 2.45729i −0.204303 − 0.762470i −0.160571 + 0.325606i −3.52704 + 0.701572i 0 −1.17450 + 2.83548i −1.00273 − 7.61648i 0.358260 − 0.176674i
31.4 −0.543824 − 0.417291i −0.0827370 + 0.0943435i −0.396025 − 1.47799i 0.455770 − 0.924209i 0.0843631 − 0.0167809i 0 −0.926022 + 2.23562i 0.389523 + 2.95872i −0.633523 + 0.312419i
31.5 −0.305780 − 0.234633i −1.84733 + 2.10648i −0.479190 − 1.78836i −1.55079 + 3.14470i 1.05913 − 0.210673i 0 −0.568075 + 1.37145i −0.633039 − 4.80841i 1.21205 − 0.597717i
31.6 0.546565 + 0.419394i 0.584020 − 0.665947i −0.394796 − 1.47340i −1.18809 + 2.40922i 0.598499 − 0.119049i 0 0.929438 − 2.24386i 0.289172 + 2.19648i −1.65978 + 0.818514i
31.7 0.860961 + 0.660639i 1.05295 − 1.20066i −0.212828 − 0.794283i 1.83214 − 3.71521i 1.69976 − 0.338102i 0 1.17209 − 2.82967i 0.0586975 + 0.445852i 4.03182 − 1.98827i
31.8 1.27476 + 0.978157i −1.85836 + 2.11905i 0.150581 + 0.561975i 0.992455 − 2.01250i −4.44172 + 0.883512i 0 0.872044 − 2.10530i −0.645302 − 4.90156i 3.23368 − 1.59467i
31.9 1.84278 + 1.41401i −0.194915 + 0.222258i 0.878756 + 3.27956i −0.539679 + 1.09436i −0.673459 + 0.133959i 0 −1.24022 + 2.99415i 0.380172 + 2.88769i −2.54195 + 1.25355i
80.1 −1.44161 + 1.87874i −1.57299 + 0.103099i −0.933791 − 3.48496i 0.355827 − 0.120787i 2.07394 − 3.10387i 0 3.51781 + 1.45712i −0.510656 + 0.0672292i −0.286036 + 0.842634i
80.2 −1.37317 + 1.78956i 2.68168 − 0.175766i −0.799265 − 2.98290i 3.74174 − 1.27015i −3.36787 + 5.04037i 0 2.26763 + 0.939281i 4.18617 − 0.551120i −2.86506 + 8.44020i
80.3 −1.04811 + 1.36592i 1.96798 − 0.128988i −0.249574 − 0.931424i −3.73402 + 1.26753i −1.88647 + 2.82330i 0 −1.64747 − 0.682403i 0.881965 − 0.116113i 2.18232 − 6.42889i
80.4 −0.261624 + 0.340955i 0.481139 − 0.0315355i 0.469835 + 1.75345i 0.618139 − 0.209830i −0.115125 + 0.172297i 0 −1.51487 − 0.627480i −2.74383 + 0.361233i −0.0901776 + 0.265655i
80.5 −0.112383 + 0.146460i −3.23999 + 0.212360i 0.508817 + 1.89893i 3.33652 − 1.13259i 0.333017 − 0.498395i 0 −0.676412 − 0.280179i 7.47813 − 0.984514i −0.209087 + 0.615949i
80.6 0.433555 − 0.565020i −1.62941 + 0.106797i 0.386361 + 1.44192i −2.99371 + 1.01623i −0.646095 + 0.966950i 0 2.29818 + 0.951937i −0.330774 + 0.0435472i −0.723749 + 2.13210i
80.7 0.559738 − 0.729465i 3.05045 − 0.199937i 0.298826 + 1.11523i 0.905309 − 0.307311i 1.56160 − 2.33711i 0 2.67975 + 1.10999i 6.29091 − 0.828214i 0.282564 − 0.832405i
80.8 1.10794 − 1.44390i 0.615539 − 0.0403446i −0.339669 − 1.26766i 1.15379 − 0.391659i 0.623729 − 0.933476i 0 1.15620 + 0.478915i −2.59707 + 0.341911i 0.712817 − 2.09989i
80.9 1.52690 − 1.98990i −2.05639 + 0.134783i −1.11062 − 4.14490i 0.578222 − 0.196280i −2.87171 + 4.29781i 0 −5.30917 − 2.19913i 1.23625 − 0.162756i 0.492311 − 1.45030i
129.1 −2.48675 + 0.327387i 0.911471 + 1.84828i 4.14490 − 1.11062i −0.459094 + 0.402615i −2.87171 − 4.29781i 0 −5.30917 + 2.19913i −0.759077 + 0.989249i 1.00984 − 1.15150i
129.2 −1.80442 + 0.237557i −0.272830 − 0.553245i 1.26766 − 0.339669i −0.916082 + 0.803382i 0.623729 + 0.933476i 0 1.15620 − 0.478915i 1.59464 − 2.07818i 1.46215 − 1.66726i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
119.p even 16 1 inner
119.s even 48 1 inner

Twists

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

Hecke kernels

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

\( T_{2}^{144} + 8 T_{2}^{139} - 72 T_{2}^{137} - 4695 T_{2}^{136} + 1088 T_{2}^{135} + 16 T_{2}^{134} + \cdots + 407555836801 \) Copy content Toggle raw display
\( T_{3}^{144} - 12 T_{3}^{142} - 48 T_{3}^{141} + 128 T_{3}^{140} + 504 T_{3}^{139} + \cdots + 23207419838464 \) Copy content Toggle raw display