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Properties

Label 83.11.b.b
Level $83$
Weight $11$
Character orbit 83.b
Analytic conductor $52.735$
Analytic rank $0$
Dimension $66$
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] = [83,11,Mod(82,83)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(83, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("83.82");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 83 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 83.b (of order \(2\), degree \(1\), 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: \(52.7346519719\)
Analytic rank: \(0\)
Dimension: \(66\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66 q + 42 q^{3} - 36670 q^{4} - 22288 q^{7} + 940700 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q + 42 q^{3} - 36670 q^{4} - 22288 q^{7} + 940700 q^{9} + 233418 q^{10} + 222202 q^{11} - 83282 q^{12} + 13541162 q^{16} - 4357164 q^{17} + 15140400 q^{21} - 40124428 q^{23} - 185868936 q^{25} - 37464430 q^{26} + 58800732 q^{27} + 43092988 q^{28} + 22238750 q^{29} + 70529600 q^{30} - 9801784 q^{31} + 189491640 q^{33} - 244288080 q^{36} - 72864542 q^{37} - 329159554 q^{38} - 174616954 q^{40} - 313076740 q^{41} - 223619034 q^{44} + 1020340858 q^{48} + 2398563470 q^{49} - 1155712104 q^{51} + 1248479518 q^{59} - 1939925070 q^{61} - 678082712 q^{63} - 15650089674 q^{64} + 1053278308 q^{65} + 2522554000 q^{68} - 2466463576 q^{69} + 5989525120 q^{70} - 22765636350 q^{75} - 6910771176 q^{77} - 22308083652 q^{78} + 13841526454 q^{81} + 13367555910 q^{83} - 4161985954 q^{84} - 39464784186 q^{86} - 233448374 q^{87} + 39409344938 q^{90} - 29805283946 q^{92} - 16377731956 q^{93} - 50984719324 q^{94} + 60940707240 q^{95} - 24272968988 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1 − 63.7531i 256.799 −3040.46 4916.24i − 16371.7i 1838.31 128556.i 6896.65 313426.
82.2 − 62.4203i 94.9562 −2872.29 − 4289.81i − 5927.19i −17998.1 115371.i −50032.3 −267772.
82.3 − 60.2687i −386.833 −2608.31 1682.86i 23313.9i 3656.05 95484.5i 90590.6 101424.
82.4 − 57.9725i −86.6144 −2336.82 − 2442.63i 5021.26i 16426.0 76107.3i −51546.9 −141605.
82.5 − 54.3951i −136.287 −1934.83 4240.21i 7413.34i −31306.6 49544.8i −40474.9 230647.
82.6 − 53.8380i 378.536 −1874.53 − 3223.05i − 20379.6i 18470.2 45790.8i 84240.2 −173523.
82.7 − 53.1204i 34.6398 −1797.78 1896.77i − 1840.08i 28546.4 41103.5i −57849.1 100757.
82.8 − 53.0309i −315.093 −1788.28 − 1983.19i 16709.7i −6735.07 40530.4i 40234.5 −105170.
82.9 − 52.1106i 398.377 −1691.51 − 677.243i − 20759.7i −15666.7 34784.5i 99655.4 −35291.5
82.10 − 50.0296i −93.5597 −1478.96 − 1140.94i 4680.75i −19597.7 22761.5i −50295.6 −57080.6
82.11 − 48.7353i 196.042 −1351.13 2324.45i − 9554.17i 539.790 15942.6i −20616.4 113283.
82.12 − 46.7232i −196.633 −1159.06 5054.02i 9187.32i 11043.0 6310.35i −20384.5 236140.
82.13 − 43.9947i −316.745 −911.534 − 5921.42i 13935.1i 18284.6 − 4947.93i 41278.5 −260511.
82.14 − 38.4813i 81.8765 −456.813 − 1982.05i − 3150.72i −3904.64 − 21826.1i −52345.2 −76271.9
82.15 − 38.3907i −407.665 −449.843 − 2135.94i 15650.5i −24664.9 − 22042.3i 107142. −82000.1
82.16 − 38.1158i 103.275 −428.813 − 5945.57i − 3936.39i −19362.1 − 22686.0i −48383.4 −226620.
82.17 − 36.2877i −435.615 −292.796 2665.94i 15807.4i 17407.6 − 26533.7i 130711. 96740.7
82.18 − 34.9125i 471.751 −194.881 5546.40i − 16470.0i 28094.2 − 28946.6i 163500. 193639.
82.19 − 33.6700i 340.004 −109.667 3372.69i − 11447.9i −31577.9 − 30785.6i 56554.0 113558.
82.20 − 32.4174i −151.410 −26.8860 4541.97i 4908.30i −7429.63 − 32323.8i −36124.2 147239.
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.66
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
83.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 83.11.b.b ✓ 66
83.b odd 2 1 inner 83.11.b.b ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.11.b.b ✓ 66 1.a even 1 1 trivial
83.11.b.b ✓ 66 83.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{66} + 52127 T_{2}^{64} + 1292461206 T_{2}^{62} + 20289440757420 T_{2}^{60} + \cdots + 62\!\cdots\!00 \) acting on \(S_{11}^{\mathrm{new}}(83, [\chi])\). Copy content Toggle raw display