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Properties

Label 833.2.t.b
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 = [72,0,0,0,0,0,0,0,-24,0,-8,-24] 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 24 q^{9} - 8 q^{11} - 24 q^{12} - 32 q^{13} + 32 q^{17} + 40 q^{18} - 40 q^{19} + 40 q^{20} - 24 q^{22} - 8 q^{23} + 16 q^{24} - 8 q^{25} - 24 q^{27} + 8 q^{31} + 40 q^{32} - 24 q^{34} - 120 q^{36}+ \cdots - 40 q^{99}+O(q^{100}) \) 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 −2.14596 + 0.888885i −0.289938 − 0.0576723i 2.40081 − 2.40081i 0.677905 + 1.01456i 0.673459 − 0.133959i 0 −1.24022 + 2.99415i −2.69090 − 1.11461i −2.35658 − 1.57462i
48.2 −1.48449 + 0.614895i −2.76433 − 0.549859i 0.411394 − 0.411394i −1.24665 − 1.86574i 4.44172 − 0.883512i 0 0.872044 − 2.10530i 4.56752 + 1.89193i 2.99787 + 2.00311i
48.3 −1.00261 + 0.415295i 1.56628 + 0.311553i −0.581456 + 0.581456i −2.30140 − 3.44429i −1.69976 + 0.338102i 0 1.17209 − 2.82967i −0.415468 − 0.172092i 3.73780 + 2.49752i
48.4 −0.636489 + 0.263642i 0.868736 + 0.172802i −1.07860 + 1.07860i 1.49239 + 2.23353i −0.598499 + 0.119049i 0 0.929438 − 2.24386i −2.04680 − 0.847811i −1.53875 − 1.02816i
48.5 0.356088 − 0.147496i −2.74793 − 0.546598i −1.30917 + 1.30917i 1.94799 + 2.91537i −1.05913 + 0.210673i 0 −0.568075 + 1.37145i 4.48072 + 1.85598i 1.12366 + 0.750808i
48.6 0.633297 − 0.262320i −0.123072 − 0.0244806i −1.08196 + 1.08196i −0.572504 − 0.856813i −0.0843631 + 0.0167809i 0 −0.926022 + 2.23562i −2.75709 − 1.14202i −0.587324 − 0.392437i
48.7 1.01653 − 0.421062i 3.20556 + 0.637626i −0.558167 + 0.558167i 0.201698 + 0.301862i 3.52704 − 0.701572i 0 −1.17450 + 2.83548i 7.09743 + 2.93985i 0.332135 + 0.221925i
48.8 1.95763 − 0.810875i −2.27107 − 0.451743i 1.76057 − 1.76057i −0.941797 − 1.40950i −4.81221 + 0.957207i 0 0.397183 − 0.958885i 2.18203 + 0.903827i −2.98661 − 1.99559i
48.9 2.22988 − 0.923647i 0.542085 + 0.107827i 2.70503 − 2.70503i 1.99806 + 2.99031i 1.30838 − 0.260253i 0 1.68611 − 4.07063i −2.48941 − 1.03115i 7.21743 + 4.82253i
97.1 −0.906232 − 2.18784i −0.875783 − 1.31070i −2.55116 + 2.55116i 0.0733090 − 0.368549i −2.07394 + 3.10387i 0 3.51781 + 1.45712i 0.197106 − 0.475856i −0.872761 + 0.173603i
97.2 −0.863214 − 2.08398i 1.49306 + 2.23452i −2.18363 + 2.18363i 0.770889 − 3.87552i 3.36787 − 5.04037i 0 2.26763 + 0.939281i −1.61580 + 3.90089i −8.74195 + 1.73888i
97.3 −0.658869 − 1.59065i 1.09570 + 1.63982i −0.681850 + 0.681850i −0.769298 + 3.86752i 1.88647 − 2.82330i 0 −1.64747 − 0.682403i −0.340426 + 0.821860i 6.65874 − 1.32451i
97.4 −0.164464 − 0.397051i 0.267880 + 0.400911i 1.28361 − 1.28361i 0.127352 − 0.640239i 0.115125 − 0.172297i 0 −1.51487 − 0.627480i 1.05908 − 2.55685i −0.275152 + 0.0547312i
97.5 −0.0706467 − 0.170556i −1.80391 − 2.69974i 1.39012 − 1.39012i 0.687402 − 3.45580i −0.333017 + 0.498395i 0 −0.676412 − 0.280179i −2.88645 + 6.96851i −0.637971 + 0.126900i
97.6 0.272544 + 0.657980i −0.907192 − 1.35771i 1.05556 − 1.05556i −0.616776 + 3.10074i 0.646095 − 0.966950i 0 2.29818 + 0.951937i 0.127674 − 0.308232i −2.20832 + 0.439263i
97.7 0.351866 + 0.849480i 1.69837 + 2.54179i 0.816407 − 0.816407i 0.186515 − 0.937676i −1.56160 + 2.33711i 0 2.67975 + 1.10999i −2.42820 + 5.86219i 0.862166 − 0.171495i
97.8 0.696482 + 1.68146i 0.342709 + 0.512900i −0.927993 + 0.927993i 0.237708 − 1.19504i −0.623729 + 0.933476i 0 1.15620 + 0.478915i 1.00243 − 2.42009i 2.17497 − 0.432628i
97.9 0.959851 + 2.31728i −1.14492 − 1.71350i −3.03428 + 3.03428i 0.119128 − 0.598894i 2.87171 − 4.29781i 0 −5.30917 − 2.19913i −0.477176 + 1.15200i 1.50215 − 0.298797i
146.1 −0.906232 + 2.18784i −0.875783 + 1.31070i −2.55116 − 2.55116i 0.0733090 + 0.368549i −2.07394 − 3.10387i 0 3.51781 − 1.45712i 0.197106 + 0.475856i −0.872761 − 0.173603i
146.2 −0.863214 + 2.08398i 1.49306 − 2.23452i −2.18363 − 2.18363i 0.770889 + 3.87552i 3.36787 + 5.04037i 0 2.26763 − 0.939281i −1.61580 − 3.90089i −8.74195 − 1.73888i
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.b ✓ 72
7.b odd 2 1 833.2.t.c yes 72
7.c even 3 2 833.2.bc.c 144
7.d odd 6 2 833.2.bc.b 144
17.e odd 16 1 833.2.t.c yes 72
119.p even 16 1 inner 833.2.t.b ✓ 72
119.s even 48 2 833.2.bc.c 144
119.t odd 48 2 833.2.bc.b 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
833.2.t.b ✓ 72 1.a even 1 1 trivial
833.2.t.b ✓ 72 119.p even 16 1 inner
833.2.t.c yes 72 7.b odd 2 1
833.2.t.c yes 72 17.e odd 16 1
833.2.bc.b 144 7.d odd 6 2
833.2.bc.b 144 119.t odd 48 2
833.2.bc.c 144 7.c even 3 2
833.2.bc.c 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 \(S_{2}^{\mathrm{new}}(833, [\chi])\):

\( T_{2}^{72} - 8 T_{2}^{67} + 72 T_{2}^{65} + 4695 T_{2}^{64} + 544 T_{2}^{63} + 48 T_{2}^{62} + \cdots + 638401 \) Copy content Toggle raw display
\( T_{3}^{72} + 12 T_{3}^{70} + 24 T_{3}^{69} + 16 T_{3}^{68} + 216 T_{3}^{67} - 632 T_{3}^{66} + \cdots + 4817408 \) Copy content Toggle raw display