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Properties

Label 287.2.bc.a
Level $287$
Weight $2$
Character orbit 287.bc
Analytic conductor $2.292$
Analytic rank $0$
Dimension $416$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [287,2,Mod(2,287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(287, base_ring=CyclotomicField(60))
 
chi = DirichletCharacter(H, H._module([20, 39]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("287.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 287 = 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 287.bc (of order \(60\), degree \(16\), 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: \(2.29170653801\)
Analytic rank: \(0\)
Dimension: \(416\)
Relative dimension: \(26\) over \(\Q(\zeta_{60})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 416 q - 10 q^{2} - 8 q^{3} - 54 q^{4} - 10 q^{5} - 16 q^{6} - 16 q^{7} - 40 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 416 q - 10 q^{2} - 8 q^{3} - 54 q^{4} - 10 q^{5} - 16 q^{6} - 16 q^{7} - 40 q^{8} + 18 q^{10} - 12 q^{11} - 24 q^{12} - 32 q^{13} + 10 q^{14} - 32 q^{15} + 26 q^{16} - 2 q^{17} - 30 q^{18} - 4 q^{19} + 80 q^{20} - 20 q^{21} - 32 q^{22} - 6 q^{23} + 26 q^{24} - 42 q^{25} - 18 q^{26} - 92 q^{27} - 42 q^{28} - 128 q^{29} - 38 q^{30} - 38 q^{31} + 100 q^{33} - 56 q^{34} - 2 q^{35} - 120 q^{36} + 6 q^{38} - 10 q^{39} + 20 q^{40} - 44 q^{41} + 112 q^{42} - 76 q^{44} - 106 q^{45} + 90 q^{46} + 32 q^{47} - 20 q^{48} - 48 q^{51} - 20 q^{52} - 2 q^{53} + 72 q^{54} - 16 q^{55} - 166 q^{56} - 32 q^{57} - 14 q^{58} + 54 q^{59} + 62 q^{60} - 90 q^{61} - 40 q^{62} - 100 q^{63} - 8 q^{64} + 2 q^{65} + 22 q^{66} - 24 q^{67} - 42 q^{68} + 24 q^{69} + 222 q^{70} - 92 q^{71} - 30 q^{72} - 10 q^{74} - 32 q^{75} + 348 q^{76} + 80 q^{77} + 80 q^{78} + 10 q^{79} - 90 q^{80} + 120 q^{81} - 124 q^{82} + 432 q^{83} + 76 q^{85} - 54 q^{86} - 10 q^{87} - 130 q^{88} - 50 q^{89} + 80 q^{90} - 92 q^{92} - 16 q^{93} - 50 q^{94} - 52 q^{95} - 64 q^{96} - 4 q^{97} + 66 q^{98} - 124 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −0.568103 + 2.67272i 0.0505447 + 0.0135434i −4.99358 − 2.22329i −0.549935 + 0.0578005i −0.0649123 + 0.127398i 1.14537 − 2.38498i 5.56692 − 7.66221i −2.59570 − 1.49863i 0.157936 − 1.50266i
2.2 −0.519333 + 2.44327i 2.09987 + 0.562657i −3.87276 − 1.72426i 2.28837 − 0.240517i −2.46525 + 4.83833i −0.880269 + 2.49502i 3.28769 − 4.52512i 1.49478 + 0.863010i −0.600776 + 5.71600i
2.3 −0.512813 + 2.41260i −2.70362 − 0.724432i −3.73056 − 1.66095i −1.04456 + 0.109788i 3.13421 − 6.15124i −2.52935 − 0.776139i 3.02074 − 4.15770i 4.18667 + 2.41718i 0.270791 − 2.57641i
2.4 −0.441291 + 2.07611i 2.97875 + 0.798153i −2.28841 − 1.01887i −3.71823 + 0.390802i −2.97155 + 5.83200i 2.61965 + 0.370718i 0.629992 − 0.867109i 5.63782 + 3.25500i 0.829474 − 7.89192i
2.5 −0.421114 + 1.98118i −0.975781 − 0.261460i −1.92066 − 0.855135i −1.74205 + 0.183096i 0.928914 − 1.82310i 0.892415 + 2.49070i 0.121944 − 0.167842i −1.71429 − 0.989746i 0.370852 − 3.52842i
2.6 −0.413424 + 1.94501i 0.0866205 + 0.0232099i −1.78504 − 0.794752i 2.38115 − 0.250269i −0.0809544 + 0.158882i 1.97323 − 1.76249i −0.0537962 + 0.0740442i −2.59111 − 1.49598i −0.497650 + 4.73483i
2.7 −0.328452 + 1.54525i 0.684405 + 0.183386i −0.452815 − 0.201606i −3.49854 + 0.367712i −0.508171 + 0.997341i −2.38756 − 1.13999i −1.39687 + 1.92263i −2.16330 − 1.24898i 0.580900 − 5.52689i
2.8 −0.326002 + 1.53372i −2.91195 − 0.780254i −0.418925 − 0.186518i 0.701171 − 0.0736960i 2.14599 − 4.21175i 2.48429 + 0.910098i −1.42064 + 1.95534i 5.27257 + 3.04412i −0.115554 + 1.09942i
2.9 −0.269531 + 1.26804i 2.47947 + 0.664373i 0.291806 + 0.129921i 2.01290 − 0.211565i −1.51075 + 2.96501i −1.64461 − 2.07250i −1.76737 + 2.43258i 3.10833 + 1.79459i −0.274266 + 2.60947i
2.10 −0.180323 + 0.848351i 0.677072 + 0.181421i 1.13991 + 0.507520i −0.215015 + 0.0225990i −0.276000 + 0.541680i −0.905986 + 2.48580i −1.65568 + 2.27885i −2.17256 − 1.25433i 0.0196002 − 0.186483i
2.11 −0.163542 + 0.769404i −2.22695 − 0.596709i 1.26185 + 0.561814i 2.23699 − 0.235117i 0.823309 − 1.61584i 0.257315 − 2.63321i −1.56332 + 2.15173i 2.00516 + 1.15768i −0.184941 + 1.75960i
2.12 −0.0934584 + 0.439687i −1.72460 − 0.462106i 1.64250 + 0.731288i −2.14313 + 0.225252i 0.364361 − 0.715098i −1.08772 − 2.41182i −1.00347 + 1.38116i 0.162634 + 0.0938970i 0.101253 − 0.963360i
2.13 −0.0600731 + 0.282622i 0.0649811 + 0.0174116i 1.75082 + 0.779517i 3.66510 − 0.385217i −0.00882453 + 0.0173191i 1.61810 + 2.09326i −0.665150 + 0.915501i −2.59416 − 1.49774i −0.111303 + 1.05898i
2.14 0.0378756 − 0.178191i 1.65415 + 0.443227i 1.79677 + 0.799975i −1.19075 + 0.125152i 0.141631 − 0.277966i 2.15498 − 1.53494i 0.424758 − 0.584629i −0.0583256 − 0.0336743i −0.0227992 + 0.216920i
2.15 0.0396320 − 0.186454i 2.77822 + 0.744421i 1.79390 + 0.798694i −2.60706 + 0.274013i 0.248906 − 0.488506i −1.73439 + 1.99797i 0.444101 − 0.611253i 4.56625 + 2.63632i −0.0522322 + 0.496956i
2.16 0.0732958 − 0.344830i −1.83442 − 0.491530i 1.71356 + 0.762924i −2.77801 + 0.291981i −0.303949 + 0.596534i 2.32834 + 1.25652i 0.803103 − 1.10538i 0.525401 + 0.303340i −0.102933 + 0.979341i
2.17 0.151413 − 0.712344i 0.452407 + 0.121222i 1.34258 + 0.597757i 2.32024 − 0.243867i 0.154852 − 0.303915i −2.14458 − 1.54945i 1.48521 − 2.04422i −2.40810 − 1.39032i 0.177598 − 1.68973i
2.18 0.188832 − 0.888384i −3.02660 − 0.810975i 1.07352 + 0.477963i 3.12880 − 0.328850i −1.29198 + 2.53564i −1.47874 + 2.19393i 1.69502 − 2.33299i 5.90455 + 3.40899i 0.298672 − 2.84167i
2.19 0.238308 − 1.12115i −1.34970 − 0.361651i 0.626905 + 0.279116i −1.74648 + 0.183562i −0.727110 + 1.42703i −2.23457 + 1.41657i 1.80976 − 2.49092i −0.907174 − 0.523757i −0.210399 + 2.00181i
2.20 0.340125 − 1.60016i 1.76347 + 0.472522i −0.617737 − 0.275034i −0.439959 + 0.0462415i 1.35591 − 2.66113i 1.44808 + 2.21428i 1.27292 − 1.75202i 0.288490 + 0.166560i −0.0756470 + 0.719733i
See next 80 embeddings (of 416 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.26
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
41.g even 20 1 inner
287.bc even 60 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.2.bc.a ✓ 416
7.c even 3 1 inner 287.2.bc.a ✓ 416
41.g even 20 1 inner 287.2.bc.a ✓ 416
287.bc even 60 1 inner 287.2.bc.a ✓ 416
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.2.bc.a ✓ 416 1.a even 1 1 trivial
287.2.bc.a ✓ 416 7.c even 3 1 inner
287.2.bc.a ✓ 416 41.g even 20 1 inner
287.2.bc.a ✓ 416 287.bc even 60 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(287, [\chi])\).