Properties

Label 61.2.k.a
Level $61$
Weight $2$
Character orbit 61.k
Analytic conductor $0.487$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [61,2,Mod(4,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.4");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 61.k (of order \(30\), degree \(8\), 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: \(0.487087452330\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 7 q^{2} - 4 q^{3} - 19 q^{4} - 18 q^{5} + 15 q^{6} - 11 q^{7} - 10 q^{8} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 7 q^{2} - 4 q^{3} - 19 q^{4} - 18 q^{5} + 15 q^{6} - 11 q^{7} - 10 q^{8} - 24 q^{9} - 13 q^{10} + 41 q^{12} - 14 q^{13} + 15 q^{14} + 10 q^{15} - 15 q^{16} + 6 q^{17} + 46 q^{18} - 35 q^{19} - 14 q^{20} - 33 q^{21} - 19 q^{22} + 30 q^{23} + 5 q^{24} + 3 q^{25} - 4 q^{26} + 11 q^{27} + 33 q^{29} + 21 q^{31} - 54 q^{32} + 5 q^{33} + 38 q^{34} + 14 q^{35} - 56 q^{36} - 10 q^{37} + 21 q^{39} + 93 q^{40} + 3 q^{41} - 75 q^{42} + 4 q^{43} + 107 q^{44} + 39 q^{45} + 87 q^{46} - 60 q^{47} + 7 q^{48} + 8 q^{49} - 47 q^{51} + 3 q^{52} - 10 q^{53} - 55 q^{54} - 32 q^{55} + 10 q^{56} + 74 q^{57} - 58 q^{58} - 41 q^{59} - 200 q^{60} + 4 q^{61} - 82 q^{62} + 42 q^{63} - 60 q^{64} - 95 q^{65} + 8 q^{66} + 31 q^{67} - 26 q^{68} - 30 q^{69} + 32 q^{70} + 23 q^{71} + 19 q^{73} - 48 q^{74} + 9 q^{75} - 29 q^{76} + 9 q^{77} + 85 q^{78} + 61 q^{79} + 73 q^{80} + 14 q^{81} - 66 q^{82} - 7 q^{83} + 190 q^{84} - 10 q^{85} + 58 q^{86} - 61 q^{87} - 6 q^{88} - 15 q^{89} + 153 q^{90} + 72 q^{91} + 59 q^{92} - 3 q^{93} + 10 q^{94} + 3 q^{95} + 100 q^{96} - 42 q^{97} - 25 q^{98} + 125 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −2.46695 0.259287i −1.14152 0.829364i 4.06232 + 0.863472i −2.05628 + 2.28373i 2.60103 + 2.34198i −1.70327 + 3.82562i −5.07938 1.65039i −0.311823 0.959694i 5.66488 5.10068i
4.2 −1.34984 0.141873i 2.48935 + 1.80862i −0.154366 0.0328114i −1.79809 + 1.99699i −3.10361 2.79451i 1.21824 2.73622i 2.78540 + 0.905030i 1.99870 + 6.15137i 2.71045 2.44050i
4.3 −0.269420 0.0283172i −2.58882 1.88089i −1.88451 0.400565i 0.677902 0.752887i 0.644220 + 0.580058i 0.612497 1.37569i 1.01167 + 0.328713i 2.23721 + 6.88542i −0.203960 + 0.183647i
4.4 0.560519 + 0.0589129i 0.925257 + 0.672238i −1.64558 0.349780i 1.41041 1.56642i 0.479020 + 0.431312i −1.58545 + 3.56099i −1.97382 0.641332i −0.522855 1.60918i 0.882846 0.794918i
4.5 1.54754 + 0.162653i −0.184257 0.133870i 0.412130 + 0.0876009i −1.91658 + 2.12858i −0.263370 0.237140i 1.50074 3.37071i −2.33627 0.759101i −0.911022 2.80384i −3.31220 + 2.98232i
5.1 −0.844477 1.89673i −1.24277 0.902925i −1.54617 + 1.71720i −1.72334 0.366308i −0.663112 + 3.11970i 3.03851 0.319360i 0.613541 + 0.199352i −0.197847 0.608912i 0.760538 + 3.57805i
5.2 −0.785340 1.76390i 1.97012 + 1.43138i −1.15633 + 1.28424i 1.14861 + 0.244144i 0.977595 4.59922i −4.17120 + 0.438411i −0.499279 0.162225i 0.905491 + 2.78681i −0.471401 2.21777i
5.3 0.0302402 + 0.0679206i −1.68685 1.22557i 1.33456 1.48218i 2.11663 + 0.449905i 0.0322307 0.151633i −1.79799 + 0.188976i 0.282447 + 0.0917726i 0.416398 + 1.28154i 0.0334497 + 0.157368i
5.4 0.320214 + 0.719213i 1.32575 + 0.963215i 0.923531 1.02569i −4.10692 0.872953i −0.268232 + 1.26193i −1.68211 + 0.176797i 2.53090 + 0.822340i −0.0972162 0.299201i −0.687255 3.23328i
5.5 0.948493 + 2.13035i −0.866253 0.629370i −2.30049 + 2.55496i −0.488412 0.103815i 0.519143 2.44238i 2.37906 0.250049i −3.18931 1.03627i −0.572763 1.76278i −0.242093 1.13896i
19.1 −0.564484 2.65569i −0.0581327 + 0.178914i −4.90695 + 2.18472i 0.269940 2.56831i 0.507955 + 0.0533882i 0.301292 + 0.271284i 5.38013 + 7.40511i 2.39842 + 1.74255i −6.97300 + 0.732892i
19.2 −0.240153 1.12983i 0.155231 0.477753i 0.608249 0.270810i −0.167336 + 1.59210i −0.577058 0.0606513i −2.62622 2.36466i −1.80991 2.49113i 2.22290 + 1.61503i 1.83898 0.193285i
19.3 −0.187984 0.884393i −1.04731 + 3.22329i 1.08028 0.480971i 0.0125746 0.119639i 3.04753 + 0.320308i 2.28414 + 2.05665i −1.69133 2.32792i −6.86567 4.98820i −0.108172 + 0.0113693i
19.4 0.422863 + 1.98941i −0.443559 + 1.36514i −1.95187 + 0.869027i 0.366719 3.48909i −2.90338 0.305158i −1.27054 1.14400i −0.163282 0.224738i 0.760202 + 0.552319i 7.09633 0.745854i
19.5 0.483303 + 2.27376i 0.893770 2.75074i −3.10932 + 1.38436i −0.281187 + 2.67531i 6.68649 + 0.702779i −2.18490 1.96729i −1.91775 2.63956i −4.34071 3.15371i −6.21892 + 0.653635i
36.1 −1.93342 1.74086i −0.935224 2.87832i 0.498468 + 4.74261i 0.269216 + 0.119863i −3.20258 + 7.19311i −0.0232368 0.109320i 4.23402 5.82763i −4.98305 + 3.62040i −0.311844 0.700413i
36.2 −1.38021 1.24274i 0.674596 + 2.07619i 0.151502 + 1.44145i 2.73442 + 1.21744i 1.64910 3.70393i −0.687078 3.23245i −0.601086 + 0.827323i −1.42845 + 1.03783i −2.26110 5.07851i
36.3 −0.144683 0.130273i −0.391997 1.20644i −0.205095 1.95135i 0.401532 + 0.178774i −0.100452 + 0.225618i 0.419277 + 1.97254i −0.453406 + 0.624060i 1.12521 0.817512i −0.0348054 0.0781742i
36.4 0.480752 + 0.432871i 0.788261 + 2.42602i −0.165312 1.57284i −2.30675 1.02703i −0.671195 + 1.50753i 0.155097 + 0.729673i 1.36186 1.87444i −2.83715 + 2.06131i −0.664404 1.49228i
36.5 1.87303 + 1.68649i −0.635636 1.95629i 0.454959 + 4.32865i −3.56305 1.58637i 2.10868 4.73618i 0.323145 + 1.52028i −3.48513 + 4.79687i −0.995971 + 0.723615i −3.99832 8.98038i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.k even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.2.k.a 40
3.b odd 2 1 549.2.bs.e 40
4.b odd 2 1 976.2.cl.c 40
61.k even 30 1 inner 61.2.k.a 40
61.l odd 60 2 3721.2.a.m 40
183.v odd 30 1 549.2.bs.e 40
244.v odd 30 1 976.2.cl.c 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.2.k.a 40 1.a even 1 1 trivial
61.2.k.a 40 61.k even 30 1 inner
549.2.bs.e 40 3.b odd 2 1
549.2.bs.e 40 183.v odd 30 1
976.2.cl.c 40 4.b odd 2 1
976.2.cl.c 40 244.v odd 30 1
3721.2.a.m 40 61.l odd 60 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(61, [\chi])\).