Properties

Label 285.2.a.a
Level $285$
Weight $2$
Character orbit 285.a
Self dual yes
Analytic conductor $2.276$
Analytic rank $1$
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] = [285,2,Mod(1,285)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(285, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("285.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 285.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: \(2.27573645761\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{3} - q^{4} - q^{5} - q^{6} - 2 q^{7} + 3 q^{8} + q^{9} + q^{10} - 6 q^{11} - q^{12} + 2 q^{14} - q^{15} - q^{16} - 6 q^{17} - q^{18} + q^{19} + q^{20} - 2 q^{21} + 6 q^{22} - 8 q^{23} + 3 q^{24} + q^{25} + q^{27} + 2 q^{28} + 4 q^{29} + q^{30} - 5 q^{32} - 6 q^{33} + 6 q^{34} + 2 q^{35} - q^{36} + 4 q^{37} - q^{38} - 3 q^{40} + 2 q^{42} - 2 q^{43} + 6 q^{44} - q^{45} + 8 q^{46} - 8 q^{47} - q^{48} - 3 q^{49} - q^{50} - 6 q^{51} + 2 q^{53} - q^{54} + 6 q^{55} - 6 q^{56} + q^{57} - 4 q^{58} + 12 q^{59} + q^{60} + 2 q^{61} - 2 q^{63} + 7 q^{64} + 6 q^{66} - 8 q^{67} + 6 q^{68} - 8 q^{69} - 2 q^{70} + 16 q^{71} + 3 q^{72} + 14 q^{73} - 4 q^{74} + q^{75} - q^{76} + 12 q^{77} + 8 q^{79} + q^{80} + q^{81} + 2 q^{84} + 6 q^{85} + 2 q^{86} + 4 q^{87} - 18 q^{88} + q^{90} + 8 q^{92} + 8 q^{94} - q^{95} - 5 q^{96} - 12 q^{97} + 3 q^{98} - 6 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−1.00000 1.00000 −1.00000 −1.00000 −1.00000 −2.00000 3.00000 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)
\(19\) \(-1\)

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 285.2.a.a 1
3.b odd 2 1 855.2.a.c 1
4.b odd 2 1 4560.2.a.h 1
5.b even 2 1 1425.2.a.g 1
5.c odd 4 2 1425.2.c.c 2
15.d odd 2 1 4275.2.a.h 1
19.b odd 2 1 5415.2.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.2.a.a 1 1.a even 1 1 trivial
855.2.a.c 1 3.b odd 2 1
1425.2.a.g 1 5.b even 2 1
1425.2.c.c 2 5.c odd 4 2
4275.2.a.h 1 15.d odd 2 1
4560.2.a.h 1 4.b odd 2 1
5415.2.a.h 1 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(285))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{7} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T + 2 \) Copy content Toggle raw display
$11$ \( T + 6 \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T - 1 \) Copy content Toggle raw display
$23$ \( T + 8 \) Copy content Toggle raw display
$29$ \( T - 4 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T - 4 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 2 \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T - 2 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T + 8 \) Copy content Toggle raw display
$71$ \( T - 16 \) Copy content Toggle raw display
$73$ \( T - 14 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T + 12 \) Copy content Toggle raw display
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