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Properties

Label 585.2.w.g
Level $585$
Weight $2$
Character orbit 585.w
Analytic conductor $4.671$
Analytic rank $0$
Dimension $28$
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(73,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.73");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.w (of order \(4\), 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: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{2} + 28 q^{4} + 4 q^{5} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{2} + 28 q^{4} + 4 q^{5} + 12 q^{8} + 8 q^{11} + 28 q^{16} - 28 q^{17} - 12 q^{20} - 32 q^{22} + 8 q^{23} + 4 q^{25} - 16 q^{31} + 68 q^{32} - 28 q^{34} - 48 q^{40} - 4 q^{41} + 40 q^{44} - 16 q^{46} + 28 q^{49} - 4 q^{50} + 48 q^{52} - 20 q^{53} - 8 q^{55} + 32 q^{59} + 8 q^{61} + 72 q^{62} + 28 q^{64} - 64 q^{65} + 32 q^{67} - 60 q^{68} - 16 q^{70} - 40 q^{71} + 56 q^{73} - 40 q^{76} - 48 q^{77} + 12 q^{80} - 4 q^{82} + 44 q^{85} - 16 q^{86} - 72 q^{88} - 36 q^{89} - 56 q^{91} + 32 q^{92} + 56 q^{95} + 48 q^{97} + 12 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 −2.48675 0 4.18390 0.999208 + 2.00040i 0 0.242414i −5.43081 0 −2.48478 − 4.97448i
73.2 −2.21780 0 2.91862 1.56243 − 1.59963i 0 − 4.11325i −2.03732 0 −3.46515 + 3.54765i
73.3 −1.97160 0 1.88719 −2.14112 − 0.644677i 0 0.616758i 0.222418 0 4.22142 + 1.27104i
73.4 −1.58074 0 0.498726 −1.89581 + 1.18571i 0 − 0.974287i 2.37312 0 2.99677 − 1.87430i
73.5 −0.750656 0 −1.43651 2.19797 + 0.411007i 0 − 3.56892i 2.57964 0 −1.64992 − 0.308525i
73.6 −0.709689 0 −1.49634 −0.408252 − 2.19848i 0 3.37036i 2.48132 0 0.289732 + 1.56024i
73.7 0.147953 0 −1.97811 −0.738223 − 2.11069i 0 1.74764i −0.588572 0 −0.109222 − 0.312283i
73.8 0.470635 0 −1.77850 2.23597 − 0.0206478i 0 1.17941i −1.77829 0 1.05233 − 0.00971759i
73.9 0.792814 0 −1.37145 −0.112444 + 2.23324i 0 − 1.67222i −2.67293 0 −0.0891476 + 1.77054i
73.10 1.38150 0 −0.0914500 2.07093 + 0.843366i 0 3.94352i −2.88934 0 2.86099 + 1.16511i
73.11 1.67997 0 0.822299 −1.81863 − 1.30099i 0 − 2.35789i −1.97850 0 −3.05525 − 2.18562i
73.12 1.94332 0 1.77647 0.570984 − 2.16194i 0 − 2.33552i −0.434380 0 1.10960 − 4.20133i
73.13 2.56480 0 4.57821 1.43673 − 1.71342i 0 1.73944i 6.61261 0 3.68494 − 4.39458i
73.14 2.73623 0 5.48694 −1.95975 + 1.07675i 0 2.18253i 9.54105 0 −5.36231 + 2.94624i
577.1 −2.48675 0 4.18390 0.999208 − 2.00040i 0 − 0.242414i −5.43081 0 −2.48478 + 4.97448i
577.2 −2.21780 0 2.91862 1.56243 + 1.59963i 0 4.11325i −2.03732 0 −3.46515 − 3.54765i
577.3 −1.97160 0 1.88719 −2.14112 + 0.644677i 0 − 0.616758i 0.222418 0 4.22142 − 1.27104i
577.4 −1.58074 0 0.498726 −1.89581 − 1.18571i 0 0.974287i 2.37312 0 2.99677 + 1.87430i
577.5 −0.750656 0 −1.43651 2.19797 − 0.411007i 0 3.56892i 2.57964 0 −1.64992 + 0.308525i
577.6 −0.709689 0 −1.49634 −0.408252 + 2.19848i 0 − 3.37036i 2.48132 0 0.289732 − 1.56024i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.w.g 28
3.b odd 2 1 195.2.t.a yes 28
5.c odd 4 1 585.2.n.g 28
13.d odd 4 1 585.2.n.g 28
15.d odd 2 1 975.2.t.d 28
15.e even 4 1 195.2.k.a ✓ 28
15.e even 4 1 975.2.k.d 28
39.f even 4 1 195.2.k.a ✓ 28
65.k even 4 1 inner 585.2.w.g 28
195.j odd 4 1 195.2.t.a yes 28
195.n even 4 1 975.2.k.d 28
195.u odd 4 1 975.2.t.d 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.k.a ✓ 28 15.e even 4 1
195.2.k.a ✓ 28 39.f even 4 1
195.2.t.a yes 28 3.b odd 2 1
195.2.t.a yes 28 195.j odd 4 1
585.2.n.g 28 5.c odd 4 1
585.2.n.g 28 13.d odd 4 1
585.2.w.g 28 1.a even 1 1 trivial
585.2.w.g 28 65.k even 4 1 inner
975.2.k.d 28 15.e even 4 1
975.2.k.d 28 195.n even 4 1
975.2.t.d 28 15.d odd 2 1
975.2.t.d 28 195.u odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\):

\( T_{2}^{14} - 2 T_{2}^{13} - 19 T_{2}^{12} + 36 T_{2}^{11} + 136 T_{2}^{10} - 244 T_{2}^{9} - 452 T_{2}^{8} + \cdots + 16 \) Copy content Toggle raw display
\( T_{7}^{28} + 84 T_{7}^{26} + 3062 T_{7}^{24} + 63860 T_{7}^{22} + 846977 T_{7}^{20} + 7520800 T_{7}^{18} + \cdots + 4194304 \) Copy content Toggle raw display