Properties

Label 155.2.u.a
Level $155$
Weight $2$
Character orbit 155.u
Analytic conductor $1.238$
Analytic rank $0$
Dimension $112$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [155,2,Mod(9,155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(155, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("155.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 155 = 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 155.u (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: \(1.23768123133\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(14\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 16 q^{4} - 5 q^{5} - 26 q^{6} - 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 112 q + 16 q^{4} - 5 q^{5} - 26 q^{6} - 32 q^{9} - 22 q^{10} - 10 q^{11} - 20 q^{14} - 18 q^{15} - 44 q^{16} - 46 q^{19} + 35 q^{20} + 26 q^{21} + 48 q^{24} + 19 q^{25} - 78 q^{26} + 16 q^{29} + 30 q^{30} - 26 q^{31} - 4 q^{34} - 27 q^{35} - 4 q^{36} - 63 q^{40} - 16 q^{41} - 86 q^{44} - 71 q^{45} + 4 q^{46} + 42 q^{49} - 10 q^{50} + 92 q^{51} + 32 q^{54} - 39 q^{55} - 34 q^{56} + 26 q^{59} - 71 q^{60} + 40 q^{61} + 24 q^{64} - 72 q^{65} + 252 q^{66} - 4 q^{69} - 26 q^{70} + 116 q^{71} - 18 q^{74} - 58 q^{75} + 224 q^{76} - 134 q^{79} + 158 q^{80} - 60 q^{81} - 20 q^{84} - 27 q^{85} - 126 q^{86} + 82 q^{89} + 129 q^{90} + 2 q^{91} - 244 q^{94} + 96 q^{95} + 208 q^{96} + 94 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −2.29327 0.745128i 0.184013 0.865712i 3.08583 + 2.24199i 1.31362 1.80953i −1.06706 + 1.84820i 1.16536 + 2.61745i −2.57144 3.53928i 2.02504 + 0.901606i −4.36082 + 3.17093i
9.2 −1.93608 0.629071i −0.622841 + 2.93024i 1.73465 + 1.26029i 2.22054 0.263033i 3.04920 5.28137i −0.119320 0.267996i −0.172473 0.237389i −5.45772 2.42993i −4.46462 0.887626i
9.3 −1.78210 0.579041i 0.541453 2.54734i 1.22257 + 0.888250i −2.14934 0.616727i −2.43994 + 4.22609i −0.941056 2.11365i 0.538385 + 0.741023i −3.45511 1.53831i 3.47323 + 2.34362i
9.4 −1.35835 0.441355i −0.0117953 + 0.0554924i 0.0322880 + 0.0234586i 0.797789 + 2.08891i 0.0405139 0.0701722i −0.461336 1.03618i 1.64551 + 2.26485i 2.73770 + 1.21890i −0.161727 3.18958i
9.5 −0.745432 0.242206i −0.105056 + 0.494249i −1.12103 0.814475i −2.20838 + 0.350796i 0.198022 0.342984i 1.56835 + 3.52258i 1.55979 + 2.14686i 2.50739 + 1.11636i 1.73116 + 0.273388i
9.6 −0.705654 0.229281i −0.394461 + 1.85579i −1.17266 0.851985i −0.801322 2.08755i 0.703850 1.21910i −1.25976 2.82948i 1.50438 + 2.07060i −0.547724 0.243862i 0.0868201 + 1.65682i
9.7 −0.577976 0.187796i 0.589803 2.77480i −1.31925 0.958488i 2.22899 + 0.177758i −0.861987 + 1.49301i 0.0276502 + 0.0621035i 1.29691 + 1.78504i −4.61103 2.05296i −1.25492 0.521335i
9.8 0.577976 + 0.187796i −0.589803 + 2.77480i −1.31925 0.958488i −0.960553 + 2.01924i −0.861987 + 1.49301i −0.0276502 0.0621035i −1.29691 1.78504i −4.61103 2.05296i −0.934381 + 0.986685i
9.9 0.705654 + 0.229281i 0.394461 1.85579i −1.17266 0.851985i −1.40721 1.73774i 0.703850 1.21910i 1.25976 + 2.82948i −1.50438 2.07060i −0.547724 0.243862i −0.594575 1.54889i
9.10 0.745432 + 0.242206i 0.105056 0.494249i −1.12103 0.814475i 1.40799 1.73712i 0.198022 0.342984i −1.56835 3.52258i −1.55979 2.14686i 2.50739 + 1.11636i 1.47030 0.953879i
9.11 1.35835 + 0.441355i 0.0117953 0.0554924i 0.0322880 + 0.0234586i 1.41015 + 1.73536i 0.0405139 0.0701722i 0.461336 + 1.03618i −1.64551 2.26485i 2.73770 + 1.21890i 1.14957 + 2.97960i
9.12 1.78210 + 0.579041i −0.541453 + 2.54734i 1.22257 + 0.888250i 0.540567 2.16974i −2.43994 + 4.22609i 0.941056 + 2.11365i −0.538385 0.741023i −3.45511 1.53831i 2.21972 3.55370i
9.13 1.93608 + 0.629071i 0.622841 2.93024i 1.73465 + 1.26029i −1.33807 + 1.79153i 3.04920 5.28137i 0.119320 + 0.267996i 0.172473 + 0.237389i −5.45772 2.42993i −3.71760 + 2.62681i
9.14 2.29327 + 0.745128i −0.184013 + 0.865712i 3.08583 + 2.24199i −2.22391 + 0.232862i −1.06706 + 1.84820i −1.16536 2.61745i 2.57144 + 3.53928i 2.02504 + 0.901606i −5.27354 1.12308i
14.1 −2.52575 0.820665i 2.29523 + 2.06664i 4.08788 + 2.97002i 0.752855 2.10552i −4.10116 7.10343i 1.50439 + 0.158118i −4.76557 6.55924i 0.683521 + 6.50327i −3.62945 + 4.70017i
14.2 −2.02472 0.657871i −0.0273246 0.0246032i 2.04866 + 1.48844i −2.14173 + 0.642634i 0.0391389 + 0.0677905i 3.48503 + 0.366291i −0.666072 0.916769i −0.313444 2.98222i 4.75918 + 0.107831i
14.3 −1.99742 0.649000i −1.07358 0.966660i 1.95043 + 1.41707i 2.22162 + 0.253777i 1.51703 + 2.62758i 1.57417 + 0.165452i −0.507205 0.698108i −0.0954328 0.907983i −4.27280 1.94873i
14.4 −1.38693 0.450639i 0.888812 + 0.800290i 0.102453 + 0.0744362i −2.05304 0.886010i −0.872074 1.51048i −4.30036 0.451986i 1.60578 + 2.21017i −0.164063 1.56095i 2.44815 + 2.15401i
14.5 −1.04247 0.338718i 1.69229 + 1.52374i −0.646026 0.469365i 1.68153 + 1.47393i −1.24804 2.16166i −0.717171 0.0753778i 1.80304 + 2.48167i 0.228460 + 2.17366i −1.25369 2.10609i
14.6 −0.944189 0.306786i −1.97801 1.78101i −0.820658 0.596243i −0.582501 + 2.15886i 1.32123 + 2.28844i −0.852316 0.0895820i 1.75902 + 2.42108i 0.426951 + 4.06217i 1.21230 1.85967i
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.g even 15 1 inner
155.u even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 155.2.u.a 112
5.b even 2 1 inner 155.2.u.a 112
5.c odd 4 2 775.2.bl.f 112
31.g even 15 1 inner 155.2.u.a 112
155.u even 30 1 inner 155.2.u.a 112
155.w odd 60 2 775.2.bl.f 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
155.2.u.a 112 1.a even 1 1 trivial
155.2.u.a 112 5.b even 2 1 inner
155.2.u.a 112 31.g even 15 1 inner
155.2.u.a 112 155.u even 30 1 inner
775.2.bl.f 112 5.c odd 4 2
775.2.bl.f 112 155.w odd 60 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(155, [\chi])\).