↑

Properties

Label 93.3.o.b
Level $93$
Weight $3$
Character orbit 93.o
Analytic conductor $2.534$
Analytic rank $0$
Dimension $144$
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] = [93,3,Mod(14,93)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(93, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 22]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("93.14");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 93 = 3 \cdot 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 93.o (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: \(2.53406645855\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 144 q - 8 q^{3} + 64 q^{4} - 5 q^{6} + 14 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q - 8 q^{3} + 64 q^{4} - 5 q^{6} + 14 q^{7} - 8 q^{9} - 40 q^{10} - 41 q^{12} - 36 q^{13} - 44 q^{15} - 40 q^{16} + 24 q^{18} - 84 q^{19} - 197 q^{21} - 162 q^{22} + 62 q^{24} + 226 q^{25} - 317 q^{27} + 66 q^{28} - 150 q^{30} + 112 q^{31} + 167 q^{33} + 168 q^{34} - 11 q^{36} - 28 q^{37} + 71 q^{39} + 70 q^{40} + 241 q^{42} + 120 q^{43} - 229 q^{45} - 78 q^{46} - 562 q^{48} - 332 q^{49} - 506 q^{51} + 128 q^{52} - 680 q^{54} - 286 q^{55} - 175 q^{57} + 490 q^{58} + 395 q^{60} + 460 q^{61} + 206 q^{63} + 290 q^{64} + 596 q^{66} - 18 q^{67} + 320 q^{69} - 710 q^{70} + 1783 q^{72} - 542 q^{73} + 665 q^{75} + 310 q^{76} + 671 q^{78} + 1040 q^{79} + 508 q^{81} + 120 q^{82} + 324 q^{84} + 82 q^{85} + 641 q^{87} + 184 q^{88} + 177 q^{90} + 140 q^{91} + 19 q^{93} - 1016 q^{94} + 230 q^{96} + 606 q^{97} + 388 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −3.53230 − 1.14771i 0.0100012 + 2.99998i 7.92385 + 5.75701i −0.947269 − 0.546906i 3.40780 − 10.6083i −0.0471289 + 0.448402i −12.6497 − 17.4108i −8.99980 + 0.0600068i 2.71835 + 3.01903i
14.2 −3.15472 − 1.02503i −2.50990 − 1.64328i 5.66549 + 4.11622i 6.17762 + 3.56665i 6.23363 + 7.75682i 0.929040 − 8.83923i −5.85490 − 8.05857i 3.59924 + 8.24897i −15.8327 − 17.5840i
14.3 −3.14215 − 1.02095i 2.35468 − 1.85890i 5.59469 + 4.06478i 4.27757 + 2.46965i −9.29658 + 3.43693i −1.19118 + 11.3333i −5.66161 − 7.79254i 2.08901 − 8.75420i −10.9194 − 12.1272i
14.4 −2.55828 − 0.831235i 2.98277 + 0.321110i 2.61777 + 1.90192i −4.90048 − 2.82929i −7.36382 − 3.30087i 0.609374 − 5.79780i 1.20838 + 1.66319i 8.79378 + 1.91559i 10.1850 + 11.3116i
14.5 −2.35430 − 0.764959i −2.87017 + 0.873013i 1.72150 + 1.25075i −3.48084 − 2.00967i 7.42505 + 0.140224i −0.149090 + 1.41849i 2.72398 + 3.74924i 7.47570 − 5.01138i 6.65764 + 7.39406i
14.6 −1.61402 − 0.524426i −0.785997 − 2.89520i −0.906035 − 0.658273i −2.53516 − 1.46367i −0.249707 + 5.08511i −0.329777 + 3.13761i 5.10722 + 7.02948i −7.76442 + 4.55125i 3.32420 + 3.69190i
14.7 −1.41182 − 0.458729i 1.93211 + 2.29498i −1.45326 − 1.05585i 3.91687 + 2.26141i −1.67502 − 4.12642i 0.199334 − 1.89654i 5.05761 + 6.96120i −1.53389 + 8.86832i −4.49255 − 4.98949i
14.8 −0.391776 − 0.127296i 1.58181 − 2.54909i −3.09878 − 2.25140i 5.88971 + 3.40043i −0.944204 + 0.797316i 0.468823 − 4.46055i 1.89596 + 2.60956i −3.99575 − 8.06436i −1.87459 − 2.08194i
14.9 −0.0872209 − 0.0283398i 0.337015 + 2.98101i −3.22926 − 2.34620i −6.65042 − 3.83962i 0.0550865 − 0.269557i −1.29161 + 12.2889i 0.430790 + 0.592932i −8.77284 + 2.00929i 0.471241 + 0.523366i
14.10 0.0872209 + 0.0283398i −2.99991 − 0.0235683i −3.22926 − 2.34620i 6.65042 + 3.83962i −0.260987 − 0.0870724i −1.29161 + 12.2889i −0.430790 − 0.592932i 8.99889 + 0.141405i 0.471241 + 0.523366i
14.11 0.391776 + 0.127296i 2.36978 − 1.83960i −3.09878 − 2.25140i −5.88971 − 3.40043i 1.16260 − 0.419047i 0.468823 − 4.46055i −1.89596 − 2.60956i 2.23176 − 8.71890i −1.87459 − 2.08194i
14.12 1.41182 + 0.458729i −2.48437 − 1.68164i −1.45326 − 1.05585i −3.91687 − 2.26141i −2.73608 − 3.51382i 0.199334 − 1.89654i −5.05761 − 6.96120i 3.34420 + 8.35562i −4.49255 − 4.98949i
14.13 1.61402 + 0.524426i 2.96150 + 0.479060i −0.906035 − 0.658273i 2.53516 + 1.46367i 4.52869 + 2.32630i −0.329777 + 3.13761i −5.10722 − 7.02948i 8.54100 + 2.83748i 3.32420 + 3.69190i
14.14 2.35430 + 0.764959i −0.568216 + 2.94570i 1.72150 + 1.25075i 3.48084 + 2.00967i −3.59109 + 6.50040i −0.149090 + 1.41849i −2.72398 − 3.74924i −8.35426 − 3.34759i 6.65764 + 7.39406i
14.15 2.55828 + 0.831235i −0.631134 − 2.93286i 2.61777 + 1.90192i 4.90048 + 2.82929i 0.823278 − 8.02769i 0.609374 − 5.79780i −1.20838 − 1.66319i −8.20334 + 3.70206i 10.1850 + 11.3116i
14.16 3.14215 + 1.02095i 1.60258 − 2.53609i 5.59469 + 4.06478i −4.27757 − 2.46965i 7.62475 − 6.33261i −1.19118 + 11.3333i 5.66161 + 7.79254i −3.86346 − 8.12857i −10.9194 − 12.1272i
14.17 3.15472 + 1.02503i 1.89664 + 2.32439i 5.66549 + 4.11622i −6.17762 − 3.56665i 3.60079 + 9.27689i 0.929040 − 8.83923i 5.85490 + 8.05857i −1.80553 + 8.81703i −15.8327 − 17.5840i
14.18 3.53230 + 1.14771i −2.98459 + 0.303637i 7.92385 + 5.75701i 0.947269 + 0.546906i −10.8910 − 2.35292i −0.0471289 + 0.448402i 12.6497 + 17.4108i 8.81561 − 1.81247i 2.71835 + 3.01903i
20.1 −3.53230 + 1.14771i 0.0100012 − 2.99998i 7.92385 − 5.75701i −0.947269 + 0.546906i 3.40780 + 10.6083i −0.0471289 − 0.448402i −12.6497 + 17.4108i −8.99980 − 0.0600068i 2.71835 − 3.01903i
20.2 −3.15472 + 1.02503i −2.50990 + 1.64328i 5.66549 − 4.11622i 6.17762 − 3.56665i 6.23363 − 7.75682i 0.929040 + 8.83923i −5.85490 + 8.05857i 3.59924 − 8.24897i −15.8327 + 17.5840i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
31.g even 15 1 inner
93.o odd 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 93.3.o.b ✓ 144
3.b odd 2 1 inner 93.3.o.b ✓ 144
31.g even 15 1 inner 93.3.o.b ✓ 144
93.o odd 30 1 inner 93.3.o.b ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
93.3.o.b ✓ 144 1.a even 1 1 trivial
93.3.o.b ✓ 144 3.b odd 2 1 inner
93.3.o.b ✓ 144 31.g even 15 1 inner
93.3.o.b ✓ 144 93.o odd 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{144} - 104 T_{2}^{142} + 5928 T_{2}^{140} - 245435 T_{2}^{138} + 8255636 T_{2}^{136} + \cdots + 15\!\cdots\!41 \) acting on \(S_{3}^{\mathrm{new}}(93, [\chi])\). Copy content Toggle raw display