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Properties

Label 80.4.l.a
Level $80$
Weight $4$
Character orbit 80.l
Analytic conductor $4.720$
Analytic rank $0$
Dimension $48$
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] = [80,4,Mod(21,80)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(80, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.21");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 80.l (of order \(4\), degree \(2\), 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: \(4.72015280046\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 20 q^{4} + 60 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 20 q^{4} + 60 q^{6} + 60 q^{10} - 40 q^{11} - 156 q^{12} - 244 q^{14} + 120 q^{15} - 432 q^{16} - 320 q^{18} + 24 q^{19} + 40 q^{20} + 740 q^{22} + 1096 q^{24} + 272 q^{26} - 264 q^{27} - 308 q^{28} + 400 q^{29} - 1600 q^{32} - 848 q^{34} + 1452 q^{36} + 16 q^{37} + 220 q^{38} + 900 q^{42} + 808 q^{43} - 1656 q^{44} - 2188 q^{46} + 1880 q^{47} - 40 q^{48} - 2352 q^{49} + 100 q^{50} + 2144 q^{51} + 792 q^{52} + 752 q^{53} + 1696 q^{54} + 2512 q^{56} - 2036 q^{58} - 2728 q^{59} + 980 q^{60} - 912 q^{61} - 344 q^{62} - 2520 q^{63} - 1040 q^{64} - 1920 q^{66} - 2040 q^{67} + 16 q^{68} - 528 q^{69} - 1240 q^{70} + 1240 q^{72} + 1928 q^{74} - 1024 q^{76} + 1904 q^{77} + 7568 q^{78} + 2832 q^{79} + 2320 q^{80} - 3888 q^{81} + 876 q^{82} + 2440 q^{83} + 896 q^{84} - 240 q^{85} + 244 q^{86} - 8320 q^{88} - 2340 q^{90} - 4448 q^{91} + 2524 q^{92} + 4272 q^{93} + 6676 q^{94} - 3040 q^{95} - 1080 q^{96} - 1436 q^{98} - 4456 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −2.82539 − 0.130954i −1.79271 + 1.79271i 7.96570 + 0.739991i −3.53553 − 3.53553i 5.29987 − 4.83035i − 14.5354i −22.4093 − 3.13390i 20.5724i 9.52629 + 10.4523i
21.2 −2.76150 − 0.611649i −2.72881 + 2.72881i 7.25177 + 3.37814i 3.53553 + 3.53553i 9.20467 − 5.86653i − 9.20767i −17.9595 − 13.7643i 12.1072i −7.60087 − 11.9259i
21.3 −2.56890 + 1.18352i 3.98819 − 3.98819i 5.19854 − 6.08072i 3.53553 + 3.53553i −5.52517 + 14.9654i − 23.3105i −6.15787 + 21.7734i − 4.81140i −13.2668 − 4.89806i
21.4 −2.36514 + 1.55117i 0.0794032 − 0.0794032i 3.18776 − 7.33745i −3.53553 − 3.53553i −0.0646320 + 0.310967i 17.2069i 3.84211 + 22.2988i 26.9874i 13.8462 + 2.87782i
21.5 −2.31784 − 1.62100i −6.39295 + 6.39295i 2.74475 + 7.51441i −3.53553 − 3.53553i 25.1808 − 4.45489i 30.7844i 5.81894 − 21.8664i − 54.7397i 2.46371 + 13.9259i
21.6 −2.04806 − 1.95076i 3.85223 − 3.85223i 0.389099 + 7.99053i −3.53553 − 3.53553i −15.4044 + 0.374836i − 22.7583i 14.7907 − 17.1241i − 2.67940i 0.344020 + 14.1380i
21.7 −1.87695 − 2.11591i 1.30312 − 1.30312i −0.954112 + 7.94290i 3.53553 + 3.53553i −5.20319 − 0.311388i 17.9744i 18.5972 − 12.8896i 23.6037i 0.844833 − 14.1169i
21.8 −1.22450 + 2.54963i 1.69929 − 1.69929i −5.00119 − 6.24404i 3.53553 + 3.53553i 2.25177 + 6.41332i 12.7537i 22.0439 − 5.10533i 21.2249i −13.3436 + 4.68503i
21.9 −0.897318 + 2.68232i −2.65806 + 2.65806i −6.38964 − 4.81378i −3.53553 − 3.53553i −4.74464 − 9.51490i − 21.5840i 18.6456 − 12.8196i 12.8694i 12.6559 − 6.31092i
21.10 −0.848061 − 2.69829i −5.98518 + 5.98518i −6.56158 + 4.57664i 3.53553 + 3.53553i 21.2256 + 11.0740i − 30.6332i 17.9137 + 13.8238i − 44.6447i 6.54156 − 12.5383i
21.11 −0.829624 + 2.70402i 6.93104 − 6.93104i −6.62345 − 4.48664i −3.53553 − 3.53553i 12.9915 + 24.4918i 1.52434i 17.6269 − 14.1877i − 69.0786i 12.4933 − 6.62699i
21.12 −0.509224 − 2.78221i −1.80275 + 1.80275i −7.48138 + 2.83353i −3.53553 − 3.53553i 5.93363 + 4.09763i 5.36354i 11.6932 + 19.3719i 20.5002i −8.03622 + 11.6370i
21.13 −0.204644 − 2.82101i 6.86577 − 6.86577i −7.91624 + 1.15460i 3.53553 + 3.53553i −20.7735 − 17.9634i − 16.5891i 4.87716 + 22.0955i − 67.2777i 9.25027 − 10.6973i
21.14 0.357195 + 2.80578i −4.63181 + 4.63181i −7.74482 + 2.00442i 3.53553 + 3.53553i −14.6503 − 11.3414i 8.38903i −8.39038 − 21.0143i − 15.9074i −8.65706 + 11.1828i
21.15 1.29687 + 2.51359i 0.777426 − 0.777426i −4.63626 + 6.51959i −3.53553 − 3.53553i 2.96235 + 0.945909i 32.6046i −22.4002 − 3.19858i 25.7912i 4.30175 − 13.4720i
21.16 1.59762 − 2.33401i −4.91823 + 4.91823i −2.89525 − 7.45772i 3.53553 + 3.53553i 3.62178 + 19.3367i 29.5330i −22.0319 − 5.15702i − 21.3780i 13.9004 − 2.60356i
21.17 1.67184 − 2.28143i 4.03507 − 4.03507i −2.40987 − 7.62840i −3.53553 − 3.53553i −2.45973 − 15.9517i 2.50273i −21.4326 − 7.25554i − 5.56352i −13.9769 + 2.15522i
21.18 1.90546 − 2.09027i −4.76629 + 4.76629i −0.738444 − 7.96585i −3.53553 − 3.53553i 0.880849 + 19.0448i − 25.1089i −18.0578 − 13.6351i − 18.4350i −14.1270 + 0.653396i
21.19 1.99659 + 2.00340i 5.74436 − 5.74436i −0.0272178 + 7.99995i 3.53553 + 3.53553i 22.9774 + 0.0390872i 2.87918i −16.0814 + 15.9181i − 38.9954i −0.0240574 + 14.1421i
21.20 2.02261 + 1.97713i −6.54282 + 6.54282i 0.181884 + 7.99793i −3.53553 − 3.53553i −26.1696 + 0.297527i − 11.7588i −15.4451 + 16.5363i − 58.6170i −0.160774 − 14.1412i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.4.l.a ✓ 48
4.b odd 2 1 320.4.l.a 48
8.b even 2 1 640.4.l.b 48
8.d odd 2 1 640.4.l.a 48
16.e even 4 1 inner 80.4.l.a ✓ 48
16.e even 4 1 640.4.l.b 48
16.f odd 4 1 320.4.l.a 48
16.f odd 4 1 640.4.l.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.4.l.a ✓ 48 1.a even 1 1 trivial
80.4.l.a ✓ 48 16.e even 4 1 inner
320.4.l.a 48 4.b odd 2 1
320.4.l.a 48 16.f odd 4 1
640.4.l.a 48 8.d odd 2 1
640.4.l.a 48 16.f odd 4 1
640.4.l.b 48 8.b even 2 1
640.4.l.b 48 16.e even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(80, [\chi])\).