↑

Properties

Label 58.3.f.b
Level $58$
Weight $3$
Character orbit 58.f
Analytic conductor $1.580$
Analytic rank $0$
Dimension $36$
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] = [58,3,Mod(3,58)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(58, base_ring=CyclotomicField(28))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("58.3");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 58 = 2 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 58.f (of order \(28\), degree \(12\), 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: \(1.58038553329\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(3\) over \(\Q(\zeta_{28})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{28}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 6 q^{2} + 4 q^{3} - 28 q^{5} - 34 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 6 q^{2} + 4 q^{3} - 28 q^{5} - 34 q^{7} - 12 q^{8} - 4 q^{10} + 68 q^{11} - 8 q^{12} + 20 q^{14} + 62 q^{15} + 24 q^{16} + 14 q^{17} - 14 q^{18} + 28 q^{19} - 76 q^{20} - 264 q^{21} - 84 q^{22} - 184 q^{23} - 40 q^{24} + 26 q^{25} + 30 q^{26} - 188 q^{27} + 32 q^{29} + 184 q^{30} + 46 q^{31} - 24 q^{32} + 322 q^{33} + 126 q^{34} + 196 q^{35} + 140 q^{36} + 348 q^{37} + 114 q^{39} + 76 q^{40} - 30 q^{41} - 308 q^{42} - 36 q^{43} - 24 q^{44} - 258 q^{45} - 40 q^{46} + 110 q^{47} - 16 q^{48} - 514 q^{49} + 86 q^{50} + 126 q^{51} - 88 q^{52} - 86 q^{53} + 208 q^{54} - 332 q^{55} - 40 q^{56} + 142 q^{58} + 40 q^{59} + 124 q^{60} - 18 q^{61} + 56 q^{62} + 644 q^{63} + 40 q^{65} - 36 q^{66} + 70 q^{67} - 28 q^{68} + 1128 q^{69} - 208 q^{70} - 854 q^{71} + 28 q^{72} + 482 q^{73} - 360 q^{74} - 1164 q^{75} - 84 q^{76} - 1002 q^{77} - 732 q^{78} - 218 q^{79} - 898 q^{81} - 220 q^{82} + 624 q^{83} + 52 q^{84} - 260 q^{85} - 202 q^{87} + 48 q^{88} - 16 q^{89} - 148 q^{90} + 1022 q^{91} + 392 q^{92} - 644 q^{93} - 80 q^{94} + 1090 q^{95} - 52 q^{97} + 906 q^{98} + 588 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 1.19745 + 0.752407i −5.22005 + 1.82657i 0.867767 + 1.80194i −6.53903 + 1.49249i −7.62507 − 1.74037i 4.88964 + 2.35473i −0.316683 + 2.81064i 16.8761 − 13.4582i −8.95312 − 3.13283i
3.2 1.19745 + 0.752407i 0.149029 − 0.0521476i 0.867767 + 1.80194i 3.03622 − 0.692997i 0.217691 + 0.0496866i 2.89365 + 1.39351i −0.316683 + 2.81064i −7.01699 + 5.59587i 4.15713 + 1.45464i
3.3 1.19745 + 0.752407i 4.47696 − 1.56655i 0.867767 + 1.80194i −4.81404 + 1.09877i 6.53961 + 1.49262i −11.2004 − 5.39384i −0.316683 + 2.81064i 10.5526 − 8.41538i −6.59129 − 2.30639i
11.1 1.33485 − 0.467085i −0.558907 − 4.96044i 1.56366 − 1.24698i −2.91162 + 6.04603i −3.06300 − 6.36039i 4.81220 − 6.03430i 1.50481 − 2.39490i −15.5192 + 3.54216i −1.06257 + 9.43054i
11.2 1.33485 − 0.467085i −0.123918 − 1.09981i 1.56366 − 1.24698i 2.16499 − 4.49565i −0.679116 − 1.41020i −6.05241 + 7.58949i 1.50481 − 2.39490i 7.58013 − 1.73012i 0.790092 − 7.01226i
11.3 1.33485 − 0.467085i 0.397504 + 3.52794i 1.56366 − 1.24698i −0.990768 + 2.05735i 2.17846 + 4.52361i 1.68871 − 2.11758i 1.50481 − 2.39490i −3.51401 + 0.802050i −0.361571 + 3.20903i
15.1 −1.40532 + 0.158342i −2.00922 + 1.26248i 1.94986 − 0.445042i −5.85774 − 4.67140i 2.62369 − 2.09233i 2.20482 − 9.65993i −2.66971 + 0.934170i −1.46184 + 3.03554i 8.97169 + 5.63729i
15.2 −1.40532 + 0.158342i −1.32900 + 0.835065i 1.94986 − 0.445042i 2.32581 + 1.85477i 1.73544 − 1.38397i −2.63944 + 11.5641i −2.66971 + 0.934170i −2.83605 + 5.88912i −3.56220 − 2.23828i
15.3 −1.40532 + 0.158342i 4.83141 − 3.03578i 1.94986 − 0.445042i −5.08940 − 4.05866i −6.30899 + 5.03126i −1.26595 + 5.54649i −2.66971 + 0.934170i 10.2216 − 21.2254i 7.79490 + 4.89786i
19.1 −0.752407 − 1.19745i −1.24607 + 3.56107i −0.867767 + 1.80194i −7.93385 − 1.81085i 5.20176 − 1.18727i −7.28948 + 3.51043i 2.81064 − 0.316683i −4.09207 − 3.26331i 3.80108 + 10.8629i
19.2 −0.752407 − 1.19745i −0.674793 + 1.92845i −0.867767 + 1.80194i 6.79118 + 1.55004i 2.81694 − 0.642947i 3.48966 − 1.68053i 2.81064 − 0.316683i 3.77292 + 3.00880i −3.25364 − 9.29836i
19.3 −0.752407 − 1.19745i 1.71299 − 4.89546i −0.867767 + 1.80194i 1.85564 + 0.423538i −7.15093 + 1.63215i −4.77098 + 2.29758i 2.81064 − 0.316683i −13.9947 − 11.1604i −0.889032 − 2.54071i
21.1 0.467085 − 1.33485i −3.82213 − 0.430651i −1.56366 − 1.24698i −1.57560 − 3.27177i −2.36012 + 4.90083i −3.99451 − 5.00896i −2.39490 + 1.50481i 5.64887 + 1.28932i −5.10327 + 0.575001i
21.2 0.467085 − 1.33485i 3.04261 + 0.342820i −1.56366 − 1.24698i −2.27360 − 4.72119i 1.87877 − 3.90131i 6.13408 + 7.69190i −2.39490 + 1.50481i 0.365623 + 0.0834510i −7.36406 + 0.829730i
21.3 0.467085 − 1.33485i 3.31182 + 0.373152i −1.56366 − 1.24698i 4.08056 + 8.47337i 2.04500 − 4.24649i −7.80790 − 9.79080i −2.39490 + 1.50481i 2.05454 + 0.468936i 13.2167 − 1.48916i
27.1 0.158342 − 1.40532i −2.85077 + 4.53698i −1.94986 − 0.445042i −1.01211 + 0.807134i 5.92452 + 4.72465i 3.01755 + 13.2207i −0.934170 + 2.66971i −8.55233 − 17.7591i 0.974023 + 1.55015i
27.2 0.158342 − 1.40532i −0.332478 + 0.529136i −1.94986 − 0.445042i 5.90652 − 4.71029i 0.690961 + 0.551023i −1.94275 − 8.51175i −0.934170 + 2.66971i 3.73551 + 7.75687i −5.68423 − 9.04640i
27.3 0.158342 − 1.40532i 2.24502 − 3.57292i −1.94986 − 0.445042i −1.16316 + 0.927588i −4.66562 − 3.72071i 0.833531 + 3.65194i −0.934170 + 2.66971i −3.82072 − 7.93381i 1.11938 + 1.78149i
31.1 −1.40532 − 0.158342i −2.00922 − 1.26248i 1.94986 + 0.445042i −5.85774 + 4.67140i 2.62369 + 2.09233i 2.20482 + 9.65993i −2.66971 − 0.934170i −1.46184 − 3.03554i 8.97169 − 5.63729i
31.2 −1.40532 − 0.158342i −1.32900 − 0.835065i 1.94986 + 0.445042i 2.32581 − 1.85477i 1.73544 + 1.38397i −2.63944 − 11.5641i −2.66971 − 0.934170i −2.83605 − 5.88912i −3.56220 + 2.23828i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.f odd 28 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 58.3.f.b ✓ 36
29.f odd 28 1 inner 58.3.f.b ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
58.3.f.b ✓ 36 1.a even 1 1 trivial
58.3.f.b ✓ 36 29.f odd 28 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} - 4 T_{3}^{35} + 8 T_{3}^{34} - 8 T_{3}^{33} - 113 T_{3}^{32} + 2420 T_{3}^{31} + \cdots + 14\!\cdots\!24 \) acting on \(S_{3}^{\mathrm{new}}(58, [\chi])\). Copy content Toggle raw display