Properties

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

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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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + q^{3} + 56 q^{4} - 28 q^{5} + 7 q^{6} - 10 q^{7} + 6 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q + q^{3} + 56 q^{4} - 28 q^{5} + 7 q^{6} - 10 q^{7} + 6 q^{8} + 9 q^{9} - 4 q^{11} - 2 q^{12} - 3 q^{13} + q^{15} + 40 q^{16} - 11 q^{17} + 3 q^{18} - 17 q^{19} - 28 q^{20} + 17 q^{21} - 12 q^{23} + 26 q^{24} - 28 q^{25} + 2 q^{26} + 16 q^{27} - 37 q^{28} + 2 q^{29} + 7 q^{30} - 12 q^{31} + 12 q^{32} + 14 q^{33} - 2 q^{34} - 10 q^{35} + 37 q^{36} - 15 q^{37} - 2 q^{38} + 25 q^{39} - 3 q^{40} - 2 q^{41} - 34 q^{42} + 4 q^{43} - 8 q^{44} - 6 q^{45} - 20 q^{46} - 21 q^{47} - 37 q^{48} - 36 q^{49} - 14 q^{51} + 12 q^{52} + 32 q^{53} - 18 q^{54} + 2 q^{55} - 20 q^{56} - 23 q^{57} + 24 q^{58} - 44 q^{59} + 7 q^{60} + 7 q^{61} - 25 q^{62} - 32 q^{63} + 42 q^{64} - 25 q^{66} - 34 q^{67} - 49 q^{68} + 72 q^{69} - 18 q^{71} + 6 q^{72} + 34 q^{73} - 22 q^{74} - 2 q^{75} - 36 q^{76} - 22 q^{77} - 105 q^{78} - 14 q^{79} - 20 q^{80} - 31 q^{81} - 17 q^{82} - 15 q^{83} + 90 q^{84} + 22 q^{85} - 69 q^{86} - 51 q^{87} + 38 q^{88} - 2 q^{89} + 15 q^{90} + 10 q^{91} + 14 q^{92} + 61 q^{93} + 62 q^{94} + 34 q^{95} + 84 q^{96} - 15 q^{97} + 24 q^{98} + 5 q^{99}+O(q^{100}) \) 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 −2.69141 −1.04767 + 1.37927i 5.24369 −0.500000 0.866025i 2.81971 3.71218i −2.38650 4.13354i −8.73010 −0.804776 2.89004i 1.34571 + 2.33083i
16.2 −2.57428 −1.67879 0.426208i 4.62691 −0.500000 0.866025i 4.32168 + 1.09718i 1.24059 + 2.14876i −6.76239 2.63669 + 1.43103i 1.28714 + 2.22939i
16.3 −2.35889 1.71934 0.209467i 3.56438 −0.500000 0.866025i −4.05574 + 0.494109i −0.253078 0.438343i −3.69022 2.91225 0.720288i 1.17945 + 2.04286i
16.4 −2.27248 0.441019 1.67496i 3.16415 −0.500000 0.866025i −1.00221 + 3.80631i 1.99843 + 3.46138i −2.64549 −2.61100 1.47738i 1.13624 + 1.96802i
16.5 −2.11041 1.03489 1.38889i 2.45385 −0.500000 0.866025i −2.18404 + 2.93113i −2.18806 3.78983i −0.957805 −0.858025 2.87468i 1.05521 + 1.82767i
16.6 −1.62135 −1.65325 0.516489i 0.628764 −0.500000 0.866025i 2.68049 + 0.837407i −1.96848 3.40951i 2.22325 2.46648 + 1.70777i 0.810673 + 1.40413i
16.7 −1.57635 −0.683981 + 1.59128i 0.484891 −0.500000 0.866025i 1.07820 2.50842i 0.939234 + 1.62680i 2.38835 −2.06434 2.17681i 0.788177 + 1.36516i
16.8 −1.54763 1.63645 + 0.567488i 0.395159 −0.500000 0.866025i −2.53261 0.878261i 1.17606 + 2.03700i 2.48370 2.35592 + 1.85733i 0.773815 + 1.34029i
16.9 −1.48791 0.736970 + 1.56744i 0.213889 −0.500000 0.866025i −1.09655 2.33222i −1.05683 1.83048i 2.65758 −1.91375 + 2.31032i 0.743957 + 1.28857i
16.10 −1.30229 −0.671232 1.59670i −0.304039 −0.500000 0.866025i 0.874139 + 2.07937i −0.414568 0.718052i 3.00053 −2.09890 + 2.14351i 0.651145 + 1.12782i
16.11 −0.704886 −1.01815 1.40121i −1.50314 −0.500000 0.866025i 0.717678 + 0.987690i 1.19422 + 2.06845i 2.46931 −0.926753 + 2.85327i 0.352443 + 0.610449i
16.12 −0.695911 −1.64450 + 0.543713i −1.51571 −0.500000 0.866025i 1.14442 0.378376i −0.108191 0.187393i 2.44662 2.40875 1.78827i 0.347956 + 0.602677i
16.13 −0.443328 1.61582 0.623809i −1.80346 −0.500000 0.866025i −0.716337 + 0.276552i −1.51185 2.61860i 1.68618 2.22172 2.01592i 0.221664 + 0.383934i
16.14 −0.0299620 0.971767 + 1.43376i −1.99910 −0.500000 0.866025i −0.0291161 0.0429584i −1.27470 2.20784i 0.119821 −1.11134 + 2.78656i 0.0149810 + 0.0259479i
16.15 0.0303319 1.64405 0.545074i −1.99908 −0.500000 0.866025i 0.0498671 0.0165331i 1.78412 + 3.09018i −0.121300 2.40579 1.79226i −0.0151659 0.0262682i
16.16 0.198180 −0.761883 + 1.55549i −1.96072 −0.500000 0.866025i −0.150990 + 0.308265i −0.807590 1.39879i −0.784935 −1.83907 2.37019i −0.0990898 0.171629i
16.17 0.915325 −0.253923 + 1.71334i −1.16218 −0.500000 0.866025i −0.232422 + 1.56826i 1.99723 + 3.45930i −2.89442 −2.87105 0.870111i −0.457662 0.792694i
16.18 0.919862 −0.842111 1.51355i −1.15385 −0.500000 0.866025i −0.774626 1.39226i 1.23537 + 2.13972i −2.90111 −1.58170 + 2.54916i −0.459931 0.796624i
16.19 1.01088 −1.70428 0.308944i −0.978128 −0.500000 0.866025i −1.72281 0.312304i −0.443377 0.767952i −3.01052 2.80911 + 1.05305i −0.505439 0.875445i
16.20 1.18191 0.368619 1.69237i −0.603097 −0.500000 0.866025i 0.435674 2.00022i −0.282481 0.489272i −3.07662 −2.72824 1.24768i −0.590953 1.02356i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.28
Significant digits:
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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.b yes 56
9.c even 3 1 585.2.k.b 56
13.c even 3 1 585.2.k.b 56
117.h even 3 1 inner 585.2.l.b yes 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.k.b 56 9.c even 3 1
585.2.k.b 56 13.c even 3 1
585.2.l.b yes 56 1.a even 1 1 trivial
585.2.l.b yes 56 117.h even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} - 42 T_{2}^{26} - T_{2}^{25} + 779 T_{2}^{24} + 36 T_{2}^{23} - 8413 T_{2}^{22} - 559 T_{2}^{21} + 58743 T_{2}^{20} + 4914 T_{2}^{19} - 278499 T_{2}^{18} - 27018 T_{2}^{17} + 916404 T_{2}^{16} + 97093 T_{2}^{15} - 2102586 T_{2}^{14} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display