Properties

Label 285.4.a.a
Level $285$
Weight $4$
Character orbit 285.a
Self dual yes
Analytic conductor $16.816$
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] = [285,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("285.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 285 = 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(16.8155443516\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \( q - 3 q^{2} - 3 q^{3} + q^{4} + 5 q^{5} + 9 q^{6} + 32 q^{7} + 21 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{2} - 3 q^{3} + q^{4} + 5 q^{5} + 9 q^{6} + 32 q^{7} + 21 q^{8} + 9 q^{9} - 15 q^{10} - 12 q^{11} - 3 q^{12} - 10 q^{13} - 96 q^{14} - 15 q^{15} - 71 q^{16} - 30 q^{17} - 27 q^{18} + 19 q^{19} + 5 q^{20} - 96 q^{21} + 36 q^{22} - 48 q^{23} - 63 q^{24} + 25 q^{25} + 30 q^{26} - 27 q^{27} + 32 q^{28} + 150 q^{29} + 45 q^{30} + 224 q^{31} + 45 q^{32} + 36 q^{33} + 90 q^{34} + 160 q^{35} + 9 q^{36} + 254 q^{37} - 57 q^{38} + 30 q^{39} + 105 q^{40} - 54 q^{41} + 288 q^{42} - 196 q^{43} - 12 q^{44} + 45 q^{45} + 144 q^{46} - 504 q^{47} + 213 q^{48} + 681 q^{49} - 75 q^{50} + 90 q^{51} - 10 q^{52} + 78 q^{53} + 81 q^{54} - 60 q^{55} + 672 q^{56} - 57 q^{57} - 450 q^{58} + 132 q^{59} - 15 q^{60} + 230 q^{61} - 672 q^{62} + 288 q^{63} + 433 q^{64} - 50 q^{65} - 108 q^{66} + 740 q^{67} - 30 q^{68} + 144 q^{69} - 480 q^{70} - 120 q^{71} + 189 q^{72} + 122 q^{73} - 762 q^{74} - 75 q^{75} + 19 q^{76} - 384 q^{77} - 90 q^{78} + 1184 q^{79} - 355 q^{80} + 81 q^{81} + 162 q^{82} + 612 q^{83} - 96 q^{84} - 150 q^{85} + 588 q^{86} - 450 q^{87} - 252 q^{88} + 1050 q^{89} - 135 q^{90} - 320 q^{91} - 48 q^{92} - 672 q^{93} + 1512 q^{94} + 95 q^{95} - 135 q^{96} - 1006 q^{97} - 2043 q^{98} - 108 q^{99}+O(q^{100}) \) 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
−3.00000 −3.00000 1.00000 5.00000 9.00000 32.0000 21.0000 9.00000 −15.0000
\(n\): 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 285.4.a.a 1
3.b odd 2 1 855.4.a.f 1
5.b even 2 1 1425.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
285.4.a.a 1 1.a even 1 1 trivial
855.4.a.f 1 3.b odd 2 1
1425.4.a.f 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 3 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(285))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 3 \) Copy content Toggle raw display
$3$ \( T + 3 \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T - 32 \) Copy content Toggle raw display
$11$ \( T + 12 \) Copy content Toggle raw display
$13$ \( T + 10 \) Copy content Toggle raw display
$17$ \( T + 30 \) Copy content Toggle raw display
$19$ \( T - 19 \) Copy content Toggle raw display
$23$ \( T + 48 \) Copy content Toggle raw display
$29$ \( T - 150 \) Copy content Toggle raw display
$31$ \( T - 224 \) Copy content Toggle raw display
$37$ \( T - 254 \) Copy content Toggle raw display
$41$ \( T + 54 \) Copy content Toggle raw display
$43$ \( T + 196 \) Copy content Toggle raw display
$47$ \( T + 504 \) Copy content Toggle raw display
$53$ \( T - 78 \) Copy content Toggle raw display
$59$ \( T - 132 \) Copy content Toggle raw display
$61$ \( T - 230 \) Copy content Toggle raw display
$67$ \( T - 740 \) Copy content Toggle raw display
$71$ \( T + 120 \) Copy content Toggle raw display
$73$ \( T - 122 \) Copy content Toggle raw display
$79$ \( T - 1184 \) Copy content Toggle raw display
$83$ \( T - 612 \) Copy content Toggle raw display
$89$ \( T - 1050 \) Copy content Toggle raw display
$97$ \( T + 1006 \) Copy content Toggle raw display
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