↑

Properties

Label 276.3.l.a
Level $276$
Weight $3$
Character orbit 276.l
Analytic conductor $7.520$
Analytic rank $0$
Dimension $480$
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] = [276,3,Mod(31,276)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(276, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 0, 6]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("276.31");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 276 = 2^{2} \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 276.l (of order \(22\), degree \(10\), 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: \(7.52045529634\)
Analytic rank: \(0\)
Dimension: \(480\)
Relative dimension: \(48\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480 q + 4 q^{2} + 8 q^{4} - 12 q^{6} - 20 q^{8} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 480 q + 4 q^{2} + 8 q^{4} - 12 q^{6} - 20 q^{8} + 144 q^{9} + 20 q^{10} - 28 q^{14} + 24 q^{16} - 12 q^{18} - 20 q^{20} - 4 q^{22} - 36 q^{24} - 240 q^{25} + 120 q^{26} + 72 q^{28} + 64 q^{29} - 48 q^{30} + 84 q^{32} + 282 q^{34} + 174 q^{36} - 192 q^{37} + 630 q^{38} + 2 q^{40} - 32 q^{41} + 270 q^{42} + 74 q^{44} + 8 q^{46} - 360 q^{48} + 432 q^{49} - 814 q^{50} - 602 q^{52} + 320 q^{53} - 162 q^{54} - 974 q^{56} - 582 q^{58} - 450 q^{60} + 64 q^{61} + 32 q^{62} - 136 q^{64} - 160 q^{65} + 144 q^{66} - 380 q^{68} - 96 q^{69} + 312 q^{70} + 60 q^{72} - 750 q^{74} - 1198 q^{76} + 2112 q^{77} - 1982 q^{80} - 432 q^{81} - 1108 q^{82} + 312 q^{84} + 1648 q^{85} - 728 q^{86} - 828 q^{88} + 368 q^{89} - 60 q^{90} + 6 q^{92} - 96 q^{93} + 554 q^{94} + 372 q^{96} - 432 q^{97} + 1556 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −1.99903 + 0.0624351i 1.57553 − 0.719520i 3.99220 − 0.249619i −5.03196 − 5.80719i −3.10460 + 1.53671i 6.36953 − 9.91118i −7.96493 + 0.748247i 1.96458 − 2.26725i 10.4216 + 11.2945i
31.2 −1.97825 − 0.294137i 1.57553 − 0.719520i 3.82697 + 1.16375i −2.76244 − 3.18803i −3.32843 + 0.959971i −6.29352 + 9.79291i −7.22841 − 3.42785i 1.96458 − 2.26725i 4.52709 + 7.11926i
31.3 −1.96873 + 0.352283i 1.57553 − 0.719520i 3.75179 − 1.38710i 3.28003 + 3.78536i −2.84832 + 1.97157i −5.39424 + 8.39359i −6.89762 + 4.05251i 1.96458 − 2.26725i −7.79102 − 6.29686i
31.4 −1.96569 + 0.368860i −1.57553 + 0.719520i 3.72788 − 1.45013i −5.51161 − 6.36074i 2.83160 − 1.99550i −2.59201 + 4.03325i −6.79297 + 4.22558i 1.96458 − 2.26725i 13.1803 + 10.4702i
31.5 −1.94750 − 0.455257i −1.57553 + 0.719520i 3.58548 + 1.77322i 2.22591 + 2.56884i 3.39590 − 0.683991i 2.96641 − 4.61582i −6.17544 − 5.08566i 1.96458 − 2.26725i −3.16547 − 6.01616i
31.6 −1.93038 + 0.523084i −1.57553 + 0.719520i 3.45277 − 2.01951i 4.23004 + 4.88172i 2.66501 − 2.21308i −4.85117 + 7.54857i −5.60880 + 5.70451i 1.96458 − 2.26725i −10.7191 − 7.21094i
31.7 −1.85282 − 0.753045i 1.57553 − 0.719520i 2.86585 + 2.79050i 5.85106 + 6.75249i −3.46099 + 0.146694i 2.79255 − 4.34529i −3.20851 − 7.32840i 1.96458 − 2.26725i −5.75602 − 16.9172i
31.8 −1.74292 + 0.980932i −1.57553 + 0.719520i 2.07555 − 3.41937i 1.00678 + 1.16188i 2.04022 − 2.79955i 5.43027 − 8.44967i −0.263340 + 7.99566i 1.96458 − 2.26725i −2.89446 − 1.03749i
31.9 −1.74210 − 0.982382i −1.57553 + 0.719520i 2.06985 + 3.42282i −0.671670 − 0.775148i 3.45158 + 0.294292i −4.37306 + 6.80461i −0.243381 − 7.99630i 1.96458 − 2.26725i 0.408627 + 2.01022i
31.10 −1.71296 − 1.03235i 1.57553 − 0.719520i 1.86850 + 3.53677i −0.305201 − 0.352221i −3.44162 − 0.393989i 2.68323 − 4.17519i 0.450523 − 7.98730i 1.96458 − 2.26725i 0.159183 + 0.918417i
31.11 −1.70646 + 1.04308i 1.57553 − 0.719520i 1.82399 − 3.55993i −0.0207148 − 0.0239061i −1.93806 + 2.87122i 0.562456 − 0.875199i 0.600718 + 7.97741i 1.96458 − 2.26725i 0.0602847 + 0.0191877i
31.12 −1.57415 − 1.23371i −1.57553 + 0.719520i 0.955913 + 3.88410i −5.16913 − 5.96549i 3.36780 + 0.811114i 2.52502 − 3.92900i 3.28711 − 7.29349i 1.96458 − 2.26725i 0.777302 + 15.7678i
31.13 −1.49258 + 1.33124i 1.57553 − 0.719520i 0.455615 − 3.97397i 4.37019 + 5.04346i −1.39376 + 3.17135i 4.16436 − 6.47988i 4.61025 + 6.53801i 1.96458 − 2.26725i −13.2369 − 1.71004i
31.14 −1.45097 + 1.37648i 1.57553 − 0.719520i 0.210608 − 3.99445i −4.27931 − 4.93859i −1.29564 + 3.21268i −2.09297 + 3.25673i 5.19270 + 6.08571i 1.96458 − 2.26725i 13.0070 + 1.27534i
31.15 −1.18110 − 1.61401i 1.57553 − 0.719520i −1.21002 + 3.81259i 0.593798 + 0.685280i −3.02216 − 1.69309i −3.80331 + 5.91807i 7.58269 − 2.55005i 1.96458 − 2.26725i 0.404712 − 1.76777i
31.16 −1.13559 + 1.64634i −1.57553 + 0.719520i −1.42086 − 3.73914i −4.22170 − 4.87210i 0.604584 − 3.41094i 0.319481 − 0.497122i 7.76941 + 1.90692i 1.96458 − 2.26725i 12.8153 − 1.41763i
31.17 −1.08832 − 1.67796i −1.57553 + 0.719520i −1.63113 + 3.65232i 4.65968 + 5.37755i 2.92201 + 1.86061i −0.0406633 + 0.0632734i 7.90364 − 1.23791i 1.96458 − 2.26725i 3.95213 − 13.6713i
31.18 −1.00708 + 1.72794i −1.57553 + 0.719520i −1.97157 − 3.48036i 0.0620186 + 0.0715732i 0.343398 − 3.44704i −6.50091 + 10.1156i 7.99940 + 0.0982449i 1.96458 − 2.26725i −0.186132 + 0.0350844i
31.19 −0.761984 + 1.84916i −1.57553 + 0.719520i −2.83876 − 2.81806i 6.06707 + 7.00177i −0.129977 − 3.46166i 3.66363 − 5.70071i 7.37412 − 3.10200i 1.96458 − 2.26725i −17.5704 + 5.88373i
31.20 −0.703593 − 1.87215i 1.57553 − 0.719520i −3.00991 + 2.63447i −0.576941 − 0.665825i −2.45558 − 2.44338i 3.99225 − 6.21207i 7.04988 + 3.78143i 1.96458 − 2.26725i −0.840596 + 1.54859i
See next 80 embeddings (of 480 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
23.c even 11 1 inner
92.g odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 276.3.l.a ✓ 480
4.b odd 2 1 inner 276.3.l.a ✓ 480
23.c even 11 1 inner 276.3.l.a ✓ 480
92.g odd 22 1 inner 276.3.l.a ✓ 480
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.3.l.a ✓ 480 1.a even 1 1 trivial
276.3.l.a ✓ 480 4.b odd 2 1 inner
276.3.l.a ✓ 480 23.c even 11 1 inner
276.3.l.a ✓ 480 92.g odd 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(276, [\chi])\).