Properties

Label 585.4.a.c
Level $585$
Weight $4$
Character orbit 585.a
Self dual yes
Analytic conductor $34.516$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [585,4,Mod(1,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)

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: yes
Analytic conductor: \(34.5161173534\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 3 q^{2} + q^{4} - 5 q^{5} - 16 q^{7} - 21 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{2} + q^{4} - 5 q^{5} - 16 q^{7} - 21 q^{8} - 15 q^{10} + 36 q^{11} + 13 q^{13} - 48 q^{14} - 71 q^{16} + 30 q^{17} + 68 q^{19} - 5 q^{20} + 108 q^{22} + 120 q^{23} + 25 q^{25} + 39 q^{26} - 16 q^{28} + 186 q^{29} + 8 q^{31} - 45 q^{32} + 90 q^{34} + 80 q^{35} - 226 q^{37} + 204 q^{38} + 105 q^{40} + 342 q^{41} - 76 q^{43} + 36 q^{44} + 360 q^{46} + 552 q^{47} - 87 q^{49} + 75 q^{50} + 13 q^{52} + 738 q^{53} - 180 q^{55} + 336 q^{56} + 558 q^{58} - 780 q^{59} - 154 q^{61} + 24 q^{62} + 433 q^{64} - 65 q^{65} + 596 q^{67} + 30 q^{68} + 240 q^{70} - 1056 q^{71} - 22 q^{73} - 678 q^{74} + 68 q^{76} - 576 q^{77} - 112 q^{79} + 355 q^{80} + 1026 q^{82} + 684 q^{83} - 150 q^{85} - 228 q^{86} - 756 q^{88} - 90 q^{89} - 208 q^{91} + 120 q^{92} + 1656 q^{94} - 340 q^{95} - 334 q^{97} - 261 q^{98}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
3.00000 0 1.00000 −5.00000 0 −16.0000 −21.0000 0 −15.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.4.a.c 1
3.b odd 2 1 195.4.a.c 1
15.d odd 2 1 975.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.4.a.c 1 3.b odd 2 1
585.4.a.c 1 1.a even 1 1 trivial
975.4.a.g 1 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(585))\):

\( T_{2} - 3 \) Copy content Toggle raw display
\( T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 3 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 5 \) Copy content Toggle raw display
$7$ \( T + 16 \) Copy content Toggle raw display
$11$ \( T - 36 \) Copy content Toggle raw display
$13$ \( T - 13 \) Copy content Toggle raw display
$17$ \( T - 30 \) Copy content Toggle raw display
$19$ \( T - 68 \) Copy content Toggle raw display
$23$ \( T - 120 \) Copy content Toggle raw display
$29$ \( T - 186 \) Copy content Toggle raw display
$31$ \( T - 8 \) Copy content Toggle raw display
$37$ \( T + 226 \) Copy content Toggle raw display
$41$ \( T - 342 \) Copy content Toggle raw display
$43$ \( T + 76 \) Copy content Toggle raw display
$47$ \( T - 552 \) Copy content Toggle raw display
$53$ \( T - 738 \) Copy content Toggle raw display
$59$ \( T + 780 \) Copy content Toggle raw display
$61$ \( T + 154 \) Copy content Toggle raw display
$67$ \( T - 596 \) Copy content Toggle raw display
$71$ \( T + 1056 \) Copy content Toggle raw display
$73$ \( T + 22 \) Copy content Toggle raw display
$79$ \( T + 112 \) Copy content Toggle raw display
$83$ \( T - 684 \) Copy content Toggle raw display
$89$ \( T + 90 \) Copy content Toggle raw display
$97$ \( T + 334 \) Copy content Toggle raw display
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