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Properties

Label 83.15.b.b
Level $83$
Weight $15$
Character orbit 83.b
Analytic conductor $103.193$
Analytic rank $0$
Dimension $94$
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] = [83,15,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, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("83.82");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 83 \)
Weight: \( k \) \(=\) \( 15 \)
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: \(103.193043566\)
Analytic rank: \(0\)
Dimension: \(94\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 94 q - 2238 q^{3} - 848638 q^{4} - 386388 q^{7} + 128434688 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 94 q - 2238 q^{3} - 848638 q^{4} - 386388 q^{7} + 128434688 q^{9} + 27080778 q^{10} - 4126190 q^{11} + 137981038 q^{12} + 6547031210 q^{16} + 476615096 q^{17} - 2060776290 q^{21} - 4275587850 q^{23} - 141017170856 q^{25} + 41418028210 q^{26} + 32923822938 q^{27} - 16733241092 q^{28} - 4611391938 q^{29} + 3945099200 q^{30} - 5373267756 q^{31} - 278000003802 q^{33} - 1083507277488 q^{36} + 69397610938 q^{37} + 530718022846 q^{38} - 267283870874 q^{40} + 192951253782 q^{41} - 459195790042 q^{44} - 3660991704902 q^{48} + 2671366947934 q^{49} + 6707783816286 q^{51} + 3125896638094 q^{59} + 4695509611274 q^{61} + 8744666360986 q^{63} - 78695375154122 q^{64} - 2806655221732 q^{65} - 9074879365520 q^{68} + 14075787112208 q^{69} - 46362798885760 q^{70} - 9403611017910 q^{75} - 74894712297290 q^{77} - 43317446184612 q^{78} + 156566775379330 q^{81} + 50744578884442 q^{83} + 9960853502366 q^{84} + 97975064390694 q^{86} + 4328716498978 q^{87} + 199972790786858 q^{90} - 216389450837354 q^{92} - 435468107782558 q^{93} - 366723106357244 q^{94} + 79196072103960 q^{95} + 297860699106964 q^{99}+O(q^{100}) \) 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 − 250.287i −1758.00 −46259.5 − 29937.2i 440004.i −904604. 7.47743e6i −1.69241e6 −7.49289e6
82.2 − 249.297i −1527.31 −45765.0 − 139359.i 380754.i 1.09714e6 7.32460e6i −2.45029e6 −3.47417e7
82.3 − 240.725i 923.676 −41564.8 73393.4i − 222352.i −458079. 6.06165e6i −3.92979e6 1.76677e7
82.4 − 239.505i 2086.40 −40978.4 − 27655.9i − 499703.i 541195. 5.89048e6i −429894. −6.62371e6
82.5 − 234.559i −3064.15 −38633.9 96290.2i 718724.i 1.25505e6 5.21893e6i 4.60604e6 2.25857e7
82.6 − 233.781i 2517.60 −38269.5 − 37402.4i − 588567.i −1.03897e6 5.11642e6i 1.55534e6 −8.74396e6
82.7 − 232.224i −4125.13 −37543.8 8065.35i 957951.i −914440. 4.91379e6i 1.22337e7 1.87296e6
82.8 − 230.140i 1875.44 −36580.6 125959.i − 431616.i 1.12149e6 4.64806e6i −1.26568e6 2.89884e7
82.9 − 224.688i −1078.20 −34100.5 144168.i 242259.i −957188. 3.98068e6i −3.62045e6 3.23927e7
82.10 − 222.670i 4339.25 −33198.1 70326.9i − 966222.i −268125. 3.74401e6i 1.40461e7 1.56597e7
82.11 − 213.191i 3113.92 −29066.6 − 141095.i − 663861.i 20724.3 2.70382e6i 4.91354e6 −3.00802e7
82.12 − 206.742i −2985.40 −26358.2 − 2773.16i 617208.i 317129. 2.06209e6i 4.12966e6 −573328.
82.13 − 203.849i −536.710 −25170.6 17012.2i 109408.i 96429.2 1.79113e6i −4.49491e6 3.46793e6
82.14 − 197.208i −604.680 −22507.0 − 103620.i 119248.i −1.20885e6 1.20750e6i −4.41733e6 −2.04347e7
82.15 − 196.644i −910.581 −22285.0 − 15211.9i 179061.i 1.02145e6 1.16041e6i −3.95381e6 −2.99134e6
82.16 − 189.442i −3681.56 −19504.2 − 117215.i 697441.i −55456.8 591093.i 8.77092e6 −2.22054e7
82.17 − 187.318i 1054.97 −18704.0 − 65758.7i − 197615.i 889020. 434571.i −3.67001e6 −1.23178e7
82.18 − 172.117i −2275.84 −13240.4 70183.2i 391712.i −589257. − 541074.i 396487. 1.20797e7
82.19 − 170.296i 1619.44 −12616.6 65695.8i − 275784.i −1.27726e6 − 641568.i −2.16037e6 1.11877e7
82.20 − 168.570i 3545.45 −12031.9 − 31559.7i − 597658.i 1.33420e6 − 733637.i 7.78728e6 −5.32002e6
See all 94 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.94
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 83.15.b.b ✓ 94
83.b odd 2 1 inner 83.15.b.b ✓ 94
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.15.b.b ✓ 94 1.a even 1 1 trivial
83.15.b.b ✓ 94 83.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{94} + 1194367 T_{2}^{92} + 688574977366 T_{2}^{90} + \cdots + 24\!\cdots\!00 \) acting on \(S_{15}^{\mathrm{new}}(83, [\chi])\). Copy content Toggle raw display