Properties

Label 59.2.c.a
Level $59$
Weight $2$
Character orbit 59.c
Analytic conductor $0.471$
Analytic rank $0$
Dimension $112$
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] = [59,2,Mod(3,59)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(59, base_ring=CyclotomicField(58))
 
chi = DirichletCharacter(H, H._module([50]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("59.3");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 59.c (of order \(29\), degree \(28\), 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: \(0.471117371926\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(4\) over \(\Q(\zeta_{29})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{29}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 26 q^{2} - 23 q^{3} - 30 q^{4} - 25 q^{5} - 13 q^{6} - 23 q^{7} - 8 q^{8} - 21 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q - 26 q^{2} - 23 q^{3} - 30 q^{4} - 25 q^{5} - 13 q^{6} - 23 q^{7} - 8 q^{8} - 21 q^{9} - 3 q^{10} - 15 q^{11} + 21 q^{12} - 23 q^{13} + 13 q^{14} + 4 q^{15} - 8 q^{16} - 10 q^{17} + 12 q^{18} - 15 q^{19} + 7 q^{20} - 12 q^{21} - q^{22} + 3 q^{23} + 25 q^{24} - 5 q^{25} + 5 q^{26} + 22 q^{27} + 29 q^{28} - 13 q^{29} + 29 q^{30} + 3 q^{31} + 36 q^{32} + 33 q^{33} + 27 q^{34} + 28 q^{35} + 20 q^{36} - 9 q^{37} + 31 q^{38} + 45 q^{39} + 79 q^{40} + 23 q^{41} + 33 q^{42} + 19 q^{43} + 43 q^{44} + 19 q^{45} - 31 q^{46} + 10 q^{47} - 25 q^{48} - 31 q^{49} - 60 q^{50} - 61 q^{51} - 75 q^{52} - 23 q^{53} - 86 q^{54} - 24 q^{55} - 103 q^{56} - 91 q^{57} + 12 q^{58} - 33 q^{59} - 210 q^{60} - 47 q^{61} - 39 q^{62} - 58 q^{63} - 152 q^{64} - 16 q^{65} - 116 q^{66} - 19 q^{67} - 5 q^{68} - 45 q^{69} + 23 q^{70} - 18 q^{71} - 36 q^{72} + 24 q^{73} + 19 q^{74} + 58 q^{75} + 97 q^{76} + 65 q^{77} + 119 q^{78} + 41 q^{79} + 107 q^{80} + 95 q^{81} + 145 q^{82} + 49 q^{83} + 167 q^{84} + 39 q^{85} + 111 q^{86} + 80 q^{87} + 175 q^{88} + 51 q^{89} + 213 q^{90} + 77 q^{91} + 135 q^{92} + 93 q^{93} + 151 q^{94} + 65 q^{95} + 181 q^{96} + 91 q^{97} + 31 q^{98} + 44 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −2.18221 + 1.00960i −1.39423 0.469770i 2.44798 2.88198i −1.93515 1.16434i 3.51678 0.382473i 0.168810 3.11351i −1.14585 + 4.12697i −0.665090 0.505588i 5.39841 + 0.587112i
3.2 −1.11469 + 0.515711i 0.0468625 + 0.0157898i −0.318193 + 0.374606i 2.81017 + 1.69082i −0.0603802 + 0.00656675i −0.144587 + 2.66676i 0.818660 2.94855i −2.38633 1.81404i −4.00445 0.435510i
3.3 −0.554859 + 0.256705i 2.89290 + 0.974731i −1.05280 + 1.23945i −2.81882 1.69603i −1.85537 + 0.201783i 0.0467597 0.862432i 0.593097 2.13614i 5.03048 + 3.82407i 1.99943 + 0.217451i
3.4 1.29680 0.599963i −0.263634 0.0888288i 0.0269577 0.0317371i −0.768757 0.462546i −0.395175 + 0.0429778i 0.00331163 0.0610794i −0.748603 + 2.69623i −2.32667 1.76869i −1.27443 0.138603i
4.1 −1.95520 0.212641i −0.330952 + 0.389626i 1.82437 + 0.401574i 3.02411 + 2.29887i 0.729929 0.691425i 0.790338 1.98360i 0.245940 + 0.0828667i 0.443066 + 2.70258i −5.42392 5.13781i
4.2 −1.36929 0.148919i −1.47847 + 1.74059i −0.100466 0.0221143i −3.18119 2.41828i 2.28366 2.16320i −1.14976 + 2.88569i 2.74480 + 0.924830i −0.358432 2.18634i 3.99584 + 3.78507i
4.3 0.670646 + 0.0729372i 0.865386 1.01881i −1.50879 0.332111i 0.331060 + 0.251665i 0.654677 0.620143i −0.928345 + 2.32997i −2.26622 0.763578i 0.196262 + 1.19715i 0.203668 + 0.192925i
4.4 1.67136 + 0.181772i −1.22878 + 1.44663i 0.807178 + 0.177673i −0.547988 0.416570i −2.31670 + 2.19450i 1.70727 4.28492i −1.86963 0.629953i −0.0974987 0.594716i −0.840167 0.795848i
5.1 −2.17862 0.734061i 1.07303 + 0.645618i 2.61533 + 1.98812i 0.194454 0.488043i −1.86379 2.19422i 3.85587 1.78392i −1.65811 2.44553i −0.670664 1.26501i −0.781893 + 0.920516i
5.2 −0.588366 0.198243i −1.87280 1.12683i −1.28531 0.977069i 0.787871 1.97741i 0.878507 + 1.03426i 0.478175 0.221227i 1.25938 + 1.85745i 0.832424 + 1.57012i −0.855565 + 1.00725i
5.3 0.0887205 + 0.0298934i 1.79944 + 1.08269i −1.58521 1.20504i −0.0780955 + 0.196005i 0.127282 + 0.149848i −1.67564 + 0.775234i −0.209696 0.309278i 0.660556 + 1.24594i −0.0127879 + 0.0150551i
5.4 1.82982 + 0.616538i −1.38105 0.830948i 1.37594 + 1.04596i −0.455265 + 1.14263i −2.01476 2.37196i −0.696774 + 0.322362i −0.294344 0.434125i −0.188410 0.355378i −1.53753 + 1.81012i
7.1 −1.04598 1.54270i 0.0724858 1.33692i −0.545581 + 1.36930i −0.914971 + 0.423310i −2.13829 + 1.28657i 0.293352 1.05656i −0.957486 + 0.210759i 1.20031 + 0.130541i 1.61008 + 0.968753i
7.2 −0.804656 1.18678i −0.142015 + 2.61930i −0.0206970 + 0.0519455i 3.22031 1.48987i 3.22281 1.93910i −0.416708 + 1.50085i −2.72235 + 0.599234i −3.85817 0.419601i −4.35939 2.62296i
7.3 0.461940 + 0.681311i −0.107820 + 1.98863i 0.489481 1.22850i −2.53352 + 1.17213i −1.40468 + 0.845168i 0.992982 3.57640i 2.67091 0.587912i −0.960605 0.104472i −1.96892 1.18466i
7.4 1.32002 + 1.94688i 0.160846 2.96664i −1.30762 + 3.28188i −2.86423 + 1.32513i 5.98801 3.60287i 0.00132718 0.00478009i −3.52113 + 0.775059i −5.79266 0.629990i −6.36072 3.82712i
9.1 −1.42070 + 1.67258i 1.16204 + 0.883363i −0.455567 2.77884i 0.717227 + 1.35283i −3.12842 + 0.688617i −2.70417 0.294096i 1.53427 + 0.923141i −0.232569 0.837638i −3.28169 0.722355i
9.2 −1.20997 + 1.42449i −2.64227 2.00860i −0.241575 1.47354i −1.05767 1.99497i 6.05831 1.33354i −1.16042 0.126203i −0.811609 0.488329i 2.14453 + 7.72389i 4.12157 + 0.907225i
9.3 0.0212776 0.0250500i 0.632796 + 0.481039i 0.323389 + 1.97259i −0.938493 1.77019i 0.0255144 0.00561614i 0.117902 + 0.0128227i 0.112619 + 0.0677605i −0.633553 2.28185i −0.0643120 0.0141561i
9.4 1.12380 1.32304i −1.60754 1.22202i −0.163940 0.999990i 0.789401 + 1.48897i −3.42332 + 0.753530i −1.11689 0.121470i 1.46757 + 0.883010i 0.288265 + 1.03824i 2.85708 + 0.628892i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
59.c even 29 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 59.2.c.a 112
3.b odd 2 1 531.2.i.a 112
4.b odd 2 1 944.2.m.c 112
59.c even 29 1 inner 59.2.c.a 112
59.c even 29 1 3481.2.a.p 56
59.d odd 58 1 3481.2.a.q 56
177.h odd 58 1 531.2.i.a 112
236.h odd 58 1 944.2.m.c 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.2.c.a 112 1.a even 1 1 trivial
59.2.c.a 112 59.c even 29 1 inner
531.2.i.a 112 3.b odd 2 1
531.2.i.a 112 177.h odd 58 1
944.2.m.c 112 4.b odd 2 1
944.2.m.c 112 236.h odd 58 1
3481.2.a.p 56 59.c even 29 1
3481.2.a.q 56 59.d odd 58 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(59, [\chi])\).