Properties

Label 585.2.bu.a
Level $585$
Weight $2$
Character orbit 585.bu
Analytic conductor $4.671$
Analytic rank $0$
Dimension $4$
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(316,585)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.316");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.bu (of order \(6\), 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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 2) q^{2} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{4} - \zeta_{12}^{3} q^{5} + ( - 2 \zeta_{12}^{2} - \zeta_{12} - 2) q^{7} + (6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 2) q^{2} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{4} - \zeta_{12}^{3} q^{5} + ( - 2 \zeta_{12}^{2} - \zeta_{12} - 2) q^{7} + (6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{8} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12}) q^{10} + ( - 2 \zeta_{12}^{2} + 4) q^{11} + ( - 4 \zeta_{12}^{3} + \zeta_{12}) q^{13} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 5) q^{14} + (4 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4 \zeta_{12}) q^{16} + ( - 2 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + \zeta_{12} + 5) q^{17} + ( - 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{19} + (2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{20} + (4 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 2 \zeta_{12} + 6) q^{22} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2 \zeta_{12}) q^{23} - q^{25} + ( - 5 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{26} + ( - 10 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 10 \zeta_{12} - 4) q^{28} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12}) q^{29} + ( - 2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{31} + ( - 8 \zeta_{12}^{2} + 8 \zeta_{12} - 8) q^{32} + (2 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{34} + (4 \zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{35} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{37} - 4 q^{38} + (2 \zeta_{12}^{3} - 4 \zeta_{12} + 6) q^{40} + (7 \zeta_{12}^{3} - \zeta_{12}^{2} - 7 \zeta_{12} + 2) q^{41} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{43} + (12 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 4) q^{44} + (6 \zeta_{12}^{2} - 10 \zeta_{12} + 6) q^{46} + ( - 5 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{47} + (4 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 4 \zeta_{12}) q^{49} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 2) q^{50} + ( - 2 \zeta_{12}^{3} + 10 \zeta_{12}^{2} - 6 \zeta_{12} + 4) q^{52} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{53} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{55} + ( - 20 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 10 \zeta_{12} - 6) q^{56} - 2 \zeta_{12} q^{58} + (\zeta_{12}^{2} - 9 \zeta_{12} + 1) q^{59} + (4 \zeta_{12}^{3} + \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{61} + ( - 5 \zeta_{12}^{3} + 11 \zeta_{12}^{2} - 5 \zeta_{12}) q^{62} + ( - 8 \zeta_{12}^{3} + 16 \zeta_{12} - 16) q^{64} + ( - \zeta_{12}^{2} - 3) q^{65} + (9 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 9 \zeta_{12} - 4) q^{67} + (8 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 8 \zeta_{12}) q^{68} + (5 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{70} + (\zeta_{12}^{2} + 11 \zeta_{12} + 1) q^{71} + ( - 5 \zeta_{12}^{3} - 12 \zeta_{12}^{2} + 6) q^{73} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{74} + ( - 8 \zeta_{12}^{3} + 8 \zeta_{12}) q^{76} + (2 \zeta_{12}^{3} - 4 \zeta_{12} - 12) q^{77} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} - 5) q^{79} + (8 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 8 \zeta_{12} + 8) q^{80} + (16 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 8 \zeta_{12} + 10) q^{82} + ( - 6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{83} + ( - \zeta_{12}^{2} - 5 \zeta_{12} - 1) q^{85} + ( - 5 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{86} + (12 \zeta_{12}^{3} - 12 \zeta_{12}^{2} + 12 \zeta_{12}) q^{88} + ( - 3 \zeta_{12}^{3} + \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{89} + (14 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 10 \zeta_{12} - 4) q^{91} + (12 \zeta_{12}^{3} - 24 \zeta_{12} + 20) q^{92} + ( - 8 \zeta_{12}^{3} + 14 \zeta_{12}^{2} - 8 \zeta_{12}) q^{94} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{95} + ( - 2 \zeta_{12}^{2} - 13 \zeta_{12} - 2) q^{97} + (2 \zeta_{12}^{2} + 6 \zeta_{12} + 2) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{2} + 4 q^{4} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{2} + 4 q^{4} - 12 q^{7} + 2 q^{10} + 12 q^{11} - 20 q^{14} - 16 q^{16} + 10 q^{17} - 12 q^{19} + 12 q^{20} + 12 q^{22} + 8 q^{23} - 4 q^{25} + 4 q^{26} - 12 q^{28} - 2 q^{29} - 48 q^{32} - 2 q^{35} - 16 q^{38} + 24 q^{40} + 6 q^{41} - 4 q^{43} + 36 q^{46} + 12 q^{49} - 6 q^{50} + 36 q^{52} - 12 q^{56} + 6 q^{59} - 2 q^{61} + 22 q^{62} - 64 q^{64} - 14 q^{65} - 12 q^{67} - 8 q^{68} + 6 q^{71} - 8 q^{74} - 48 q^{77} - 20 q^{79} + 24 q^{80} + 20 q^{82} - 6 q^{85} - 24 q^{88} - 6 q^{89} - 10 q^{91} + 80 q^{92} + 28 q^{94} - 4 q^{95} - 12 q^{97} + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/585\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(1 - \zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
316.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
0.633975 + 0.366025i 0 −0.732051 1.26795i 1.00000i 0 −3.86603 + 2.23205i 2.53590i 0 −0.366025 + 0.633975i
316.2 2.36603 + 1.36603i 0 2.73205 + 4.73205i 1.00000i 0 −2.13397 + 1.23205i 9.46410i 0 1.36603 2.36603i
361.1 0.633975 0.366025i 0 −0.732051 + 1.26795i 1.00000i 0 −3.86603 2.23205i 2.53590i 0 −0.366025 0.633975i
361.2 2.36603 1.36603i 0 2.73205 4.73205i 1.00000i 0 −2.13397 1.23205i 9.46410i 0 1.36603 + 2.36603i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.bu.a 4
3.b odd 2 1 195.2.bb.a 4
13.e even 6 1 inner 585.2.bu.a 4
13.f odd 12 1 7605.2.a.y 2
13.f odd 12 1 7605.2.a.bk 2
15.d odd 2 1 975.2.bc.h 4
15.e even 4 1 975.2.w.a 4
15.e even 4 1 975.2.w.f 4
39.h odd 6 1 195.2.bb.a 4
39.k even 12 1 2535.2.a.n 2
39.k even 12 1 2535.2.a.s 2
195.y odd 6 1 975.2.bc.h 4
195.bf even 12 1 975.2.w.a 4
195.bf even 12 1 975.2.w.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bb.a 4 3.b odd 2 1
195.2.bb.a 4 39.h odd 6 1
585.2.bu.a 4 1.a even 1 1 trivial
585.2.bu.a 4 13.e even 6 1 inner
975.2.w.a 4 15.e even 4 1
975.2.w.a 4 195.bf even 12 1
975.2.w.f 4 15.e even 4 1
975.2.w.f 4 195.bf even 12 1
975.2.bc.h 4 15.d odd 2 1
975.2.bc.h 4 195.y odd 6 1
2535.2.a.n 2 39.k even 12 1
2535.2.a.s 2 39.k even 12 1
7605.2.a.y 2 13.f odd 12 1
7605.2.a.bk 2 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 6T_{2}^{3} + 14T_{2}^{2} - 12T_{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 6 T^{3} + 14 T^{2} - 12 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 12 T^{3} + 59 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 23T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 10 T^{3} + 78 T^{2} + \cdots + 484 \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + 56 T^{2} + 96 T + 64 \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} + 60 T^{2} - 32 T + 16 \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} + 6 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$31$ \( T^{4} + 62T^{2} + 529 \) Copy content Toggle raw display
$37$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$41$ \( T^{4} - 6 T^{3} - 34 T^{2} + \cdots + 2116 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} + 15 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$47$ \( T^{4} + 104T^{2} + 4 \) Copy content Toggle raw display
$53$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 6 T^{3} - 66 T^{2} + \cdots + 6084 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + 15 T^{2} - 22 T + 121 \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} - 21 T^{2} + \cdots + 4761 \) Copy content Toggle raw display
$71$ \( T^{4} - 6 T^{3} - 106 T^{2} + \cdots + 13924 \) Copy content Toggle raw display
$73$ \( T^{4} + 266T^{2} + 6889 \) Copy content Toggle raw display
$79$ \( (T^{2} + 10 T - 23)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$97$ \( T^{4} + 12 T^{3} - 109 T^{2} + \cdots + 24649 \) Copy content Toggle raw display
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