↑

Properties

Label 380.3.bg.a
Level $380$
Weight $3$
Character orbit 380.bg
Analytic conductor $10.354$
Analytic rank $0$
Dimension $240$
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] = [380,3,Mod(17,380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(380, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([0, 9, 20]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("380.17");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.bg (of order \(36\), 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: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(20\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 18 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 18 q^{7} + 18 q^{15} + 54 q^{17} + 144 q^{21} + 132 q^{23} + 96 q^{25} + 396 q^{33} - 162 q^{35} - 24 q^{41} - 420 q^{43} - 198 q^{45} + 48 q^{47} + 24 q^{51} - 18 q^{53} + 48 q^{55} - 306 q^{57} - 312 q^{61} - 144 q^{63} + 270 q^{65} + 456 q^{67} + 162 q^{73} + 156 q^{77} + 1008 q^{81} + 390 q^{83} + 696 q^{85} + 48 q^{87} + 648 q^{91} + 384 q^{93} + 228 q^{95} + 204 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −5.79465 + 0.506966i 0 −2.78414 + 4.15314i 0 −2.22808 + 8.31532i 0 24.4577 − 4.31255i 0
17.2 0 −5.00624 + 0.437989i 0 4.11268 − 2.84357i 0 2.47440 − 9.23457i 0 16.0073 − 2.82253i 0
17.3 0 −4.87745 + 0.426722i 0 −0.627125 − 4.96052i 0 −1.06996 + 3.99316i 0 14.7441 − 2.59979i 0
17.4 0 −3.48977 + 0.305315i 0 2.05411 + 4.55858i 0 1.86871 − 6.97412i 0 3.22201 − 0.568127i 0
17.5 0 −3.09685 + 0.270939i 0 −4.50054 − 2.17834i 0 −0.0712384 + 0.265865i 0 0.653789 − 0.115281i 0
17.6 0 −3.06904 + 0.268506i 0 4.84000 − 1.25476i 0 −0.197377 + 0.736619i 0 0.483656 − 0.0852817i 0
17.7 0 −2.17441 + 0.190236i 0 2.69868 + 4.20917i 0 −2.57686 + 9.61695i 0 −4.17140 + 0.735530i 0
17.8 0 −1.75069 + 0.153166i 0 −4.53673 + 2.10192i 0 −0.521192 + 1.94511i 0 −5.82180 + 1.02654i 0
17.9 0 −0.825737 + 0.0722426i 0 −3.27492 − 3.77821i 0 2.34203 − 8.74056i 0 −8.18665 + 1.44353i 0
17.10 0 −0.0910539 + 0.00796618i 0 4.22671 − 2.67113i 0 −1.98256 + 7.39902i 0 −8.85504 + 1.56138i 0
17.11 0 0.464215 − 0.0406136i 0 −2.17422 + 4.50253i 0 2.32244 − 8.66747i 0 −8.64942 + 1.52513i 0
17.12 0 1.60296 − 0.140241i 0 1.41821 − 4.79465i 0 2.07883 − 7.75831i 0 −6.31344 + 1.11323i 0
17.13 0 1.83833 − 0.160833i 0 −3.13939 + 3.89156i 0 −1.45972 + 5.44773i 0 −5.50967 + 0.971503i 0
17.14 0 1.93580 − 0.169361i 0 −0.892923 − 4.91962i 0 −1.93494 + 7.22129i 0 −5.14462 + 0.907135i 0
17.15 0 2.45865 − 0.215104i 0 4.90537 + 0.968172i 0 2.21180 − 8.25454i 0 −2.86458 + 0.505103i 0
17.16 0 3.25459 − 0.284740i 0 −4.64190 − 1.85816i 0 −2.48680 + 9.28085i 0 1.64800 − 0.290586i 0
17.17 0 3.27984 − 0.286949i 0 1.50620 + 4.76774i 0 −0.0887155 + 0.331091i 0 1.81176 − 0.319462i 0
17.18 0 4.81952 − 0.421653i 0 −4.80329 + 1.38868i 0 1.90499 − 7.10952i 0 14.1867 − 2.50150i 0
17.19 0 4.95634 − 0.433624i 0 4.16402 + 2.76784i 0 −2.33167 + 8.70190i 0 15.5140 − 2.73554i 0
17.20 0 5.56564 − 0.486930i 0 2.38889 − 4.39240i 0 0.647833 − 2.41775i 0 21.8760 − 3.85732i 0
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
19.e even 9 1 inner
95.q odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.3.bg.a ✓ 240
5.c odd 4 1 inner 380.3.bg.a ✓ 240
19.e even 9 1 inner 380.3.bg.a ✓ 240
95.q odd 36 1 inner 380.3.bg.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.bg.a ✓ 240 1.a even 1 1 trivial
380.3.bg.a ✓ 240 5.c odd 4 1 inner
380.3.bg.a ✓ 240 19.e even 9 1 inner
380.3.bg.a ✓ 240 95.q odd 36 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(380, [\chi])\).