Properties

Label 585.2.l.a
Level $585$
Weight $2$
Character orbit 585.l
Analytic conductor $4.671$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [585,2,Mod(16,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.16");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.l (of order \(3\), degree \(2\), 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: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 56 q - q^{3} + 56 q^{4} + 28 q^{5} - q^{6} + 6 q^{7} + 6 q^{8} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - q^{3} + 56 q^{4} + 28 q^{5} - q^{6} + 6 q^{7} + 6 q^{8} - 7 q^{9} - 4 q^{11} - 10 q^{12} + q^{13} + q^{15} + 72 q^{16} - 5 q^{17} - 7 q^{18} + 19 q^{19} + 28 q^{20} + 17 q^{21} - 12 q^{23} - 30 q^{24} - 28 q^{25} + 2 q^{26} - 16 q^{27} + 21 q^{28} + 2 q^{29} + q^{30} + 20 q^{31} + 12 q^{32} - 18 q^{33} + 2 q^{34} - 6 q^{35} - 35 q^{36} + 17 q^{37} - 2 q^{38} - 51 q^{39} + 3 q^{40} - 2 q^{41} + 10 q^{42} + 4 q^{43} - 56 q^{44} - 2 q^{45} + 20 q^{46} - 15 q^{47} - 19 q^{48} - 24 q^{49} + 10 q^{51} - 2 q^{52} + 32 q^{53} - 10 q^{54} - 2 q^{55} - 20 q^{56} + 5 q^{57} - 96 q^{58} - 20 q^{59} + q^{60} + 7 q^{61} - 25 q^{62} + 24 q^{63} + 106 q^{64} + 2 q^{65} + 7 q^{66} + 48 q^{67} - 25 q^{68} + 8 q^{69} - 6 q^{71} - 36 q^{72} - 62 q^{73} - 10 q^{74} + 2 q^{75} + 44 q^{76} - 34 q^{77} + 19 q^{78} + 10 q^{79} + 36 q^{80} + 17 q^{81} - q^{82} + 9 q^{83} + 6 q^{84} - 10 q^{85} - 69 q^{86} + 65 q^{87} - 38 q^{88} + 10 q^{89} + 19 q^{90} - 26 q^{91} + 14 q^{92} + 85 q^{93} - 26 q^{94} + 38 q^{95} - 76 q^{96} + 17 q^{97} + 24 q^{98} + q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −2.76408 1.20534 1.24385i 5.64014 0.500000 + 0.866025i −3.33166 + 3.43809i 1.30005 + 2.25175i −10.0616 −0.0943127 2.99852i −1.38204 2.39376i
16.2 −2.58098 −1.21772 1.23173i 4.66146 0.500000 + 0.866025i 3.14292 + 3.17906i −1.30127 2.25386i −6.86918 −0.0343020 + 2.99980i −1.29049 2.23519i
16.3 −2.51255 −0.575022 + 1.63381i 4.31289 0.500000 + 0.866025i 1.44477 4.10503i 1.40071 + 2.42609i −5.81124 −2.33870 1.87896i −1.25627 2.17593i
16.4 −2.12815 1.35704 + 1.07630i 2.52901 0.500000 + 0.866025i −2.88799 2.29054i −0.470061 0.814170i −1.12582 0.683136 + 2.92119i −1.06407 1.84303i
16.5 −1.90412 0.150405 + 1.72551i 1.62569 0.500000 + 0.866025i −0.286390 3.28558i −1.00653 1.74336i 0.712733 −2.95476 + 0.519050i −0.952062 1.64902i
16.6 −1.85591 −1.71867 + 0.214845i 1.44441 0.500000 + 0.866025i 3.18971 0.398733i −0.968036 1.67669i 1.03113 2.90768 0.738496i −0.927956 1.60727i
16.7 −1.69989 1.65633 0.506513i 0.889617 0.500000 + 0.866025i −2.81558 + 0.861016i 1.04879 + 1.81656i 1.88753 2.48689 1.67791i −0.849944 1.47215i
16.8 −1.66452 −1.56929 0.733032i 0.770642 0.500000 + 0.866025i 2.61212 + 1.22015i 2.37650 + 4.11622i 2.04630 1.92533 + 2.30068i −0.832262 1.44152i
16.9 −1.41906 0.192139 1.72136i 0.0137247 0.500000 + 0.866025i −0.272656 + 2.44271i −0.0722641 0.125165i 2.81864 −2.92617 0.661480i −0.709529 1.22894i
16.10 −0.763785 1.64087 + 0.554555i −1.41663 0.500000 + 0.866025i −1.25328 0.423561i −2.34538 4.06232i 2.60957 2.38494 + 1.81991i −0.381893 0.661458i
16.11 −0.671737 −1.45581 + 0.938409i −1.54877 0.500000 + 0.866025i 0.977923 0.630364i −0.0474320 0.0821546i 2.38384 1.23878 2.73229i −0.335869 0.581742i
16.12 −0.657086 0.285448 + 1.70837i −1.56824 0.500000 + 0.866025i −0.187564 1.12254i 1.88943 + 3.27259i 2.34464 −2.83704 + 0.975301i −0.328543 0.569053i
16.13 −0.515516 −0.626853 1.61464i −1.73424 0.500000 + 0.866025i 0.323152 + 0.832371i −2.25375 3.90361i 1.92506 −2.21411 + 2.02428i −0.257758 0.446450i
16.14 −0.0917973 0.451768 1.67210i −1.99157 0.500000 + 0.866025i −0.0414711 + 0.153494i 2.14686 + 3.71846i 0.366416 −2.59181 1.51080i −0.0458987 0.0794988i
16.15 0.0143342 −1.64723 0.535378i −1.99979 0.500000 + 0.866025i −0.0236117 0.00767421i 0.299595 + 0.518914i −0.0573338 2.42674 + 1.76378i 0.00716709 + 0.0124138i
16.16 0.279919 1.47779 + 0.903405i −1.92165 0.500000 + 0.866025i 0.413661 + 0.252880i 0.459383 + 0.795675i −1.09774 1.36772 + 2.67008i 0.139959 + 0.242417i
16.17 0.788554 1.20224 1.24684i −1.37818 0.500000 + 0.866025i 0.948031 0.983203i −1.40210 2.42851i −2.66388 −0.109240 2.99801i 0.394277 + 0.682908i
16.18 0.809618 −1.11759 1.32325i −1.34452 0.500000 + 0.866025i −0.904819 1.07133i 0.662471 + 1.14743i −2.70778 −0.501999 + 2.95770i 0.404809 + 0.701150i
16.19 0.829649 0.333567 + 1.69963i −1.31168 0.500000 + 0.866025i 0.276744 + 1.41009i −1.11914 1.93840i −2.74753 −2.77747 + 1.13388i 0.414825 + 0.718497i
16.20 1.31077 1.72189 0.187297i −0.281879 0.500000 + 0.866025i 2.25701 0.245503i 1.76552 + 3.05796i −2.99102 2.92984 0.645010i 0.655386 + 1.13516i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.l.a yes 56
9.c even 3 1 585.2.k.a 56
13.c even 3 1 585.2.k.a 56
117.h even 3 1 inner 585.2.l.a yes 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.k.a 56 9.c even 3 1
585.2.k.a 56 13.c even 3 1
585.2.l.a yes 56 1.a even 1 1 trivial
585.2.l.a yes 56 117.h even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} - 42 T_{2}^{26} - T_{2}^{25} + 775 T_{2}^{24} + 36 T_{2}^{23} - 8277 T_{2}^{22} - 555 T_{2}^{21} + 56743 T_{2}^{20} + 4808 T_{2}^{19} - 261793 T_{2}^{18} - 25862 T_{2}^{17} + 828958 T_{2}^{16} + 90349 T_{2}^{15} - 1804468 T_{2}^{14} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display