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Properties

Label 61.3.l.a
Level $61$
Weight $3$
Character orbit 61.l
Analytic conductor $1.662$
Analytic rank $0$
Dimension $144$
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] = [61,3,Mod(2,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.2");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 61.l (of order \(60\), degree \(16\), 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: \(1.66212961260\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(9\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 18 q^{2} - 20 q^{3} - 14 q^{4} - 6 q^{5} + 32 q^{6} - 2 q^{7} - 32 q^{8} + 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 18 q^{2} - 20 q^{3} - 14 q^{4} - 6 q^{5} + 32 q^{6} - 2 q^{7} - 32 q^{8} + 60 q^{9} - 100 q^{10} - 26 q^{11} - 36 q^{12} - 44 q^{13} - 114 q^{14} - 84 q^{15} - 26 q^{16} + 8 q^{17} + 162 q^{18} - 14 q^{19} + 156 q^{20} - 56 q^{21} - 18 q^{22} - 244 q^{23} - 172 q^{24} + 222 q^{25} + 66 q^{26} + 10 q^{27} + 84 q^{28} - 280 q^{29} - 82 q^{30} + 234 q^{31} - 316 q^{32} - 42 q^{33} + 156 q^{34} + 200 q^{35} - 500 q^{36} - 46 q^{37} - 144 q^{38} + 146 q^{39} + 438 q^{40} - 260 q^{41} - 346 q^{42} - 148 q^{43} + 1156 q^{44} + 122 q^{45} - 2 q^{46} + 20 q^{47} + 474 q^{48} + 142 q^{49} + 454 q^{50} + 284 q^{51} + 890 q^{52} + 472 q^{53} + 172 q^{54} - 4 q^{55} + 208 q^{56} + 276 q^{57} + 242 q^{58} - 20 q^{59} - 222 q^{61} - 500 q^{62} - 104 q^{63} - 660 q^{64} + 630 q^{65} - 626 q^{66} - 690 q^{67} - 114 q^{68} - 542 q^{69} - 1344 q^{70} - 802 q^{71} - 202 q^{72} - 796 q^{73} - 414 q^{74} - 990 q^{75} - 1874 q^{76} + 210 q^{77} - 122 q^{78} + 516 q^{79} - 2000 q^{80} + 320 q^{81} + 346 q^{82} + 72 q^{83} - 166 q^{84} - 282 q^{85} + 1312 q^{86} + 84 q^{87} + 140 q^{88} - 438 q^{89} - 794 q^{90} + 506 q^{91} + 456 q^{92} - 756 q^{93} - 754 q^{94} + 298 q^{95} + 2036 q^{96} - 290 q^{97} - 684 q^{98} - 70 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −3.79297 − 0.198781i −0.405961 − 0.131905i 10.3690 + 1.08983i −0.803173 − 1.80396i 1.51358 + 0.581009i −5.18078 + 3.36444i −24.1070 − 3.81818i −7.13375 − 5.18297i 2.68782 + 7.00201i
2.2 −2.45295 − 0.128553i 4.65105 + 1.51122i 2.02233 + 0.212555i 0.660289 + 1.48303i −11.2145 − 4.30484i −2.33742 + 1.51794i 4.77094 + 0.755643i 12.0673 + 8.76742i −1.42900 − 3.72268i
2.3 −2.45162 − 0.128484i −2.92464 − 0.950272i 2.01584 + 0.211873i 2.16155 + 4.85492i 7.04800 + 2.70547i 8.30313 − 5.39212i 4.78419 + 0.757741i 0.369326 + 0.268331i −4.67552 − 12.1801i
2.4 −1.28753 − 0.0674767i 0.589706 + 0.191607i −2.32490 − 0.244357i −3.29782 − 7.40703i −0.746335 − 0.286491i 3.70026 − 2.40297i 8.07060 + 1.27826i −6.97011 − 5.06408i 3.74625 + 9.75931i
2.5 −0.523521 − 0.0274366i −2.18317 − 0.709354i −3.70477 − 0.389387i 1.70841 + 3.83714i 1.12347 + 0.431260i −11.1420 + 7.23571i 3.99998 + 0.633535i −3.01812 − 2.19279i −0.789109 − 2.05570i
2.6 0.831904 + 0.0435982i 2.47572 + 0.804409i −3.28792 − 0.345575i 2.99766 + 6.73284i 2.02449 + 0.777128i 8.82856 − 5.73334i −6.01133 − 0.952101i −1.79905 − 1.30709i 2.20022 + 5.73177i
2.7 1.23992 + 0.0649816i −4.95408 − 1.60968i −2.44490 − 0.256970i −1.66304 − 3.73525i −6.03807 − 2.31780i 4.10669 − 2.66691i −7.92014 − 1.25443i 14.6707 + 10.6589i −1.81932 − 4.73948i
2.8 1.85919 + 0.0974362i 3.81541 + 1.23970i −0.530984 − 0.0558087i −1.65795 − 3.72381i 6.97279 + 2.67660i −5.01484 + 3.25668i −8.33706 − 1.32046i 5.73934 + 4.16987i −2.71961 − 7.08483i
2.9 3.17077 + 0.166173i −1.22578 − 0.398280i 6.04808 + 0.635678i 0.381835 + 0.857616i −3.82048 − 1.46654i −1.06258 + 0.690049i 6.52730 + 1.03382i −5.93725 − 4.31366i 1.06820 + 2.78275i
6.1 −3.56823 − 1.36971i 0.551331 − 0.758842i 7.88355 + 7.09838i −0.994045 + 4.67662i −3.00667 + 1.95256i 6.85155 − 8.46096i −11.4668 − 22.5048i 2.50928 + 7.72276i 9.95261 − 15.3257i
6.2 −2.29614 − 0.881407i −0.626194 + 0.861882i 1.52282 + 1.37115i −0.0682733 + 0.321201i 2.19750 − 1.42707i −6.57068 + 8.11412i 2.17830 + 4.27515i 2.43043 + 7.48010i 0.439874 − 0.677347i
6.3 −2.27225 − 0.872235i −3.07541 + 4.23293i 1.42974 + 1.28735i 1.09328 − 5.14350i 10.6802 − 6.93580i 7.21401 − 8.90857i 2.29402 + 4.50227i −5.67845 − 17.4765i −6.97055 + 10.7337i
6.4 −2.00717 − 0.770480i 3.36752 − 4.63499i 0.462510 + 0.416446i 0.00936905 − 0.0440779i −10.3304 + 6.70861i −2.02305 + 2.49826i 3.29679 + 6.47032i −7.36182 − 22.6574i −0.0527664 + 0.0812532i
6.5 −0.0821005 − 0.0315154i 0.795778 − 1.09529i −2.96683 − 2.67135i 0.924946 − 4.35153i −0.0998524 + 0.0648449i 3.10847 − 3.83864i 0.319088 + 0.626246i 2.21475 + 6.81628i −0.213079 + 0.328112i
6.6 0.0823095 + 0.0315957i −1.82855 + 2.51678i −2.96680 − 2.67132i −1.72740 + 8.12679i −0.230026 + 0.149381i −0.896421 + 1.10699i −0.319899 − 0.627837i −0.209444 − 0.644602i −0.398953 + 0.614334i
6.7 1.84888 + 0.709718i 2.32633 − 3.20191i −0.0579292 − 0.0521597i −1.42097 + 6.68514i 6.57355 − 4.26891i 1.09051 − 1.34667i −3.66645 − 7.19581i −2.05931 − 6.33790i −7.37176 + 11.3515i
6.8 2.51303 + 0.964662i 0.819320 − 1.12770i 2.41217 + 2.17193i 1.00886 − 4.74633i 3.14682 − 2.04357i −7.05097 + 8.70722i −0.921566 − 1.80868i 2.18074 + 6.71162i 7.11391 − 10.9545i
6.9 2.71825 + 1.04344i −2.00304 + 2.75694i 3.32752 + 2.99611i 0.0148924 − 0.0700631i −8.32144 + 5.40401i 3.27862 − 4.04875i 0.631338 + 1.23907i −0.807432 − 2.48502i 0.113587 − 0.174909i
7.1 −3.01123 + 1.95552i 1.11046 − 0.360811i 3.61652 − 8.12283i 5.08340 − 0.534287i −2.63829 + 3.25801i 3.47493 + 0.182113i 2.74746 + 17.3468i −6.17821 + 4.48873i −14.2625 + 11.5495i
7.2 −2.20597 + 1.43257i −5.49224 + 1.78454i 1.18708 − 2.66622i 3.89246 − 0.409114i 9.55921 − 11.8046i −4.23589 − 0.221994i −0.444990 − 2.80956i 19.6989 − 14.3121i −8.00054 + 6.47871i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.l odd 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.3.l.a ✓ 144
61.l odd 60 1 inner 61.3.l.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.3.l.a ✓ 144 1.a even 1 1 trivial
61.3.l.a ✓ 144 61.l odd 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(61, [\chi])\).