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Properties

Label 799.2.g.d
Level $799$
Weight $2$
Character orbit 799.g
Analytic conductor $6.380$
Analytic rank $0$
Dimension $152$
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] = [799,2,Mod(189,799)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(799, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([7, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("799.189");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 799 = 17 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 799.g (of order \(8\), degree \(4\), 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: \(6.38004712150\)
Analytic rank: \(0\)
Dimension: \(152\)
Relative dimension: \(38\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 152 q - 4 q^{2} + 4 q^{6} + 4 q^{7} + 4 q^{8} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 152 q - 4 q^{2} + 4 q^{6} + 4 q^{7} + 4 q^{8} - 8 q^{9} - 8 q^{11} + 8 q^{12} - 12 q^{14} + 20 q^{15} - 160 q^{16} - 12 q^{17} + 96 q^{18} + 16 q^{19} + 96 q^{20} - 4 q^{22} - 16 q^{23} - 76 q^{24} - 12 q^{25} - 8 q^{26} + 8 q^{28} - 4 q^{29} - 4 q^{31} - 16 q^{32} - 72 q^{33} - 20 q^{34} + 48 q^{35} - 16 q^{36} + 16 q^{37} + 64 q^{40} - 68 q^{41} - 144 q^{42} + 4 q^{43} + 12 q^{44} - 4 q^{46} - 12 q^{48} - 4 q^{49} + 96 q^{50} - 52 q^{51} + 168 q^{52} - 16 q^{53} + 168 q^{54} + 108 q^{56} - 104 q^{58} - 84 q^{59} - 4 q^{60} + 4 q^{61} - 28 q^{62} + 60 q^{63} + 60 q^{65} - 44 q^{66} - 160 q^{67} - 96 q^{68} + 160 q^{69} + 36 q^{70} + 40 q^{71} + 4 q^{73} + 76 q^{74} - 116 q^{75} - 148 q^{76} - 4 q^{77} + 68 q^{78} - 76 q^{80} + 64 q^{82} - 124 q^{83} + 208 q^{84} - 68 q^{85} + 80 q^{86} - 72 q^{87} + 188 q^{88} + 236 q^{90} - 32 q^{91} - 196 q^{92} - 152 q^{93} + 4 q^{94} - 48 q^{95} - 56 q^{96} - 20 q^{97} + 244 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
189.1 −1.88952 − 1.88952i −2.62295 + 1.08646i 5.14055i 0.619807 + 1.49635i 7.00898 + 2.90322i 0.0896722 − 0.216488i 5.93412 − 5.93412i 3.57813 − 3.57813i 1.65624 − 3.99851i
189.2 −1.79681 − 1.79681i 2.48226 − 1.02819i 4.45704i −1.34217 − 3.24029i −6.30760 − 2.61270i 0.543897 − 1.31308i 4.41484 − 4.41484i 2.98314 − 2.98314i −3.41055 + 8.23380i
189.3 −1.73990 − 1.73990i −2.20788 + 0.914534i 4.05452i −0.644366 − 1.55564i 5.43269 + 2.25030i −0.794761 + 1.91872i 3.57467 − 3.57467i 1.91704 − 1.91704i −1.58552 + 3.82779i
189.4 −1.72323 − 1.72323i −0.351603 + 0.145639i 3.93905i 1.24628 + 3.00879i 0.856863 + 0.354924i −1.75431 + 4.23529i 3.34143 − 3.34143i −2.01891 + 2.01891i 3.03721 − 7.33248i
189.5 −1.62887 − 1.62887i 0.145589 − 0.0603051i 3.30643i −1.68073 − 4.05765i −0.335375 − 0.138917i 1.01122 − 2.44130i 2.12800 − 2.12800i −2.10376 + 2.10376i −3.87168 + 9.34707i
189.6 −1.59647 − 1.59647i 2.24761 − 0.930990i 3.09745i 1.25186 + 3.02225i −5.07455 − 2.10195i 1.82436 − 4.40439i 1.75205 − 1.75205i 2.06368 − 2.06368i 2.82638 − 6.82349i
189.7 −1.46527 − 1.46527i 1.42547 − 0.590450i 2.29404i −0.0867621 − 0.209462i −2.95387 − 1.22353i −0.226040 + 0.545708i 0.430854 − 0.430854i −0.437981 + 0.437981i −0.179789 + 0.434049i
189.8 −1.31242 − 1.31242i −1.05095 + 0.435318i 1.44487i −0.145222 − 0.350596i 1.95060 + 0.807965i −1.10354 + 2.66419i −0.728565 + 0.728565i −1.20633 + 1.20633i −0.269536 + 0.650718i
189.9 −1.17603 − 1.17603i 1.90109 − 0.787456i 0.766083i −1.43295 − 3.45945i −3.16180 − 1.30966i −1.95917 + 4.72986i −1.45112 + 1.45112i 0.872726 − 0.872726i −2.38322 + 5.75360i
189.10 −1.07793 − 1.07793i −2.97316 + 1.23152i 0.323861i −1.16038 − 2.80141i 4.53235 + 1.87736i 1.27565 − 3.07968i −1.80676 + 1.80676i 5.20173 − 5.20173i −1.76891 + 4.27053i
189.11 −1.01233 − 1.01233i 1.24340 − 0.515033i 0.0496370i 1.51733 + 3.66317i −1.78012 − 0.737349i 0.127333 − 0.307409i −1.97442 + 1.97442i −0.840537 + 0.840537i 2.17230 − 5.24440i
189.12 −0.933278 − 0.933278i 1.64442 − 0.681140i − 0.257984i −0.530602 − 1.28099i −2.17039 − 0.899005i 1.21240 − 2.92700i −2.10733 + 2.10733i 0.118835 − 0.118835i −0.700317 + 1.69071i
189.13 −0.895058 − 0.895058i −0.670327 + 0.277659i − 0.397744i 0.962510 + 2.32370i 0.848502 + 0.351461i 0.00519236 − 0.0125355i −2.14612 + 2.14612i −1.74908 + 1.74908i 1.21835 − 2.94135i
189.14 −0.731234 − 0.731234i −3.03513 + 1.25719i − 0.930594i −0.154935 − 0.374047i 3.13869 + 1.30009i −0.465394 + 1.12356i −2.14295 + 2.14295i 5.51017 − 5.51017i −0.160222 + 0.386810i
189.15 −0.615538 − 0.615538i −0.644978 + 0.267159i − 1.24223i 1.32265 + 3.19316i 0.561455 + 0.232562i 0.934637 − 2.25641i −1.99571 + 1.99571i −1.77670 + 1.77670i 1.15137 − 2.77965i
189.16 −0.360653 − 0.360653i −1.16918 + 0.484289i − 1.73986i −1.32341 − 3.19499i 0.596327 + 0.247007i 1.71724 − 4.14579i −1.34879 + 1.34879i −0.988881 + 0.988881i −0.674991 + 1.62957i
189.17 −0.324479 − 0.324479i 2.74543 − 1.13719i − 1.78943i −0.380019 − 0.917447i −1.25983 − 0.521839i −0.763040 + 1.84214i −1.22959 + 1.22959i 4.12285 − 4.12285i −0.174384 + 0.421001i
189.18 −0.263449 − 0.263449i 2.84104 − 1.17680i − 1.86119i 1.22029 + 2.94604i −1.05849 − 0.438442i 0.705766 − 1.70387i −1.01722 + 1.01722i 4.56532 − 4.56532i 0.454646 − 1.09761i
189.19 −0.170154 − 0.170154i 1.26998 − 0.526042i − 1.94210i 0.289225 + 0.698250i −0.305600 − 0.126584i −1.65278 + 3.99017i −0.670764 + 0.670764i −0.785197 + 0.785197i 0.0695974 − 0.168023i
189.20 0.234025 + 0.234025i −0.150154 + 0.0621959i − 1.89046i −0.623821 − 1.50604i −0.0496952 − 0.0205844i 1.32694 − 3.20353i 0.910466 − 0.910466i −2.10264 + 2.10264i 0.206461 − 0.498440i
See next 80 embeddings (of 152 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 189.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.d even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 799.2.g.d ✓ 152
17.d even 8 1 inner 799.2.g.d ✓ 152
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
799.2.g.d ✓ 152 1.a even 1 1 trivial
799.2.g.d ✓ 152 17.d even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{152} + 4 T_{2}^{151} + 8 T_{2}^{150} + 4 T_{2}^{149} + 580 T_{2}^{148} + 2320 T_{2}^{147} + \cdots + 276147097009 \) acting on \(S_{2}^{\mathrm{new}}(799, [\chi])\). Copy content Toggle raw display