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Properties

Label 83.7.b.c
Level $83$
Weight $7$
Character orbit 83.b
Analytic conductor $19.094$
Analytic rank $0$
Dimension $38$
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] = [83,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("83.82");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 83 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(19.0944889404\)
Analytic rank: \(0\)
Dimension: \(38\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 18 q^{3} - 1342 q^{4} + 262 q^{7} + 9068 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 38 q + 18 q^{3} - 1342 q^{4} + 262 q^{7} + 9068 q^{9} + 2058 q^{10} - 998 q^{11} + 2542 q^{12} + 4394 q^{16} + 12194 q^{17} + 9870 q^{21} + 2376 q^{23} - 179566 q^{25} + 69298 q^{26} - 65442 q^{27} + 13948 q^{28} - 35418 q^{29} + 15200 q^{30} - 81746 q^{31} - 282666 q^{33} - 341040 q^{36} - 27650 q^{37} + 60478 q^{38} - 73434 q^{40} - 164988 q^{41} + 165158 q^{44} - 388934 q^{48} + 391272 q^{49} + 545358 q^{51} + 668362 q^{59} - 850010 q^{61} + 1793212 q^{63} - 54346 q^{64} + 1043648 q^{65} - 82256 q^{68} - 147292 q^{69} - 3597600 q^{70} - 698790 q^{75} + 4155622 q^{77} + 953724 q^{78} + 2779834 q^{81} + 2225038 q^{83} + 4158878 q^{84} + 18150 q^{86} + 1384174 q^{87} + 1165418 q^{90} - 3404138 q^{92} + 5904566 q^{93} - 5928444 q^{94} + 1075920 q^{95} - 10192352 q^{99}+O(q^{100}) \) 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 − 15.4022i 13.4607 −173.228 − 27.7156i − 207.324i 38.0665 1682.36i −547.811 −426.882
82.2 − 14.5796i −31.2601 −148.566 84.8726i 455.761i −458.266 1232.94i 248.192 1237.41
82.3 − 13.6207i −46.3702 −121.523 − 60.2937i 631.594i 519.118 783.509i 1421.19 −821.242
82.4 − 13.1617i 38.1108 −109.229 63.5646i − 501.601i 434.882 595.292i 723.433 836.616
82.5 − 12.7586i −11.8648 −98.7826 − 221.852i 151.379i 69.8282 443.778i −588.226 −2830.53
82.6 − 12.2215i −16.8683 −85.3640 185.624i 206.155i 285.720 261.099i −444.462 2268.59
82.7 − 12.1844i 39.2112 −84.4597 230.780i − 477.766i −594.254 249.289i 808.522 2811.92
82.8 − 11.7113i 28.2216 −73.1551 − 114.627i − 330.513i −387.001 107.218i 67.4606 −1342.44
82.9 − 9.35568i −5.01535 −23.5288 55.2230i 46.9220i 270.229 − 378.636i −703.846 516.649
82.10 − 9.28916i 52.1430 −22.2886 − 179.470i − 484.365i 290.695 − 387.464i 1989.90 −1667.13
82.11 − 8.77656i −37.2311 −13.0281 − 87.6908i 326.761i −497.308 − 447.358i 657.152 −769.624
82.12 − 8.66609i −3.35057 −11.1010 101.654i 29.0363i −313.942 − 458.427i −717.774 880.942
82.13 − 6.38460i −40.8423 23.2369 − 89.6703i 260.762i 138.112 − 556.973i 939.094 −572.509
82.14 − 5.39000i −49.7612 34.9479 230.475i 268.213i 47.4802 − 533.329i 1747.18 1242.26
82.15 − 5.07231i 20.3703 38.2717 − 84.1308i − 103.325i −259.782 − 518.753i −314.050 −426.737
82.16 − 5.05951i 31.5953 38.4014 49.5617i − 159.856i 121.438 − 518.101i 269.261 250.758
82.17 − 4.77610i 0.547506 41.1888 − 179.066i − 2.61494i 529.517 − 502.393i −728.700 −855.238
82.18 − 3.19655i 33.6517 53.7821 173.608i − 107.569i 321.578 − 376.496i 403.440 554.947
82.19 − 0.757570i −5.74837 63.4261 186.432i 4.35479i −425.111 − 96.5342i −695.956 141.235
82.20 0.757570i −5.74837 63.4261 − 186.432i − 4.35479i −425.111 96.5342i −695.956 141.235
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.38
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 83.7.b.c ✓ 38
83.b odd 2 1 inner 83.7.b.c ✓ 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.7.b.c ✓ 38 1.a even 1 1 trivial
83.7.b.c ✓ 38 83.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(83, [\chi])\):

\( T_{2}^{38} + 1887 T_{2}^{36} + 1637046 T_{2}^{34} + 866171980 T_{2}^{32} + 312672844455 T_{2}^{30} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
\( T_{3}^{19} - 9 T_{3}^{18} - 9152 T_{3}^{17} + 89144 T_{3}^{16} + 33945075 T_{3}^{15} + \cdots - 74\!\cdots\!00 \) Copy content Toggle raw display