Properties

Label 495.4.a.b
Level $495$
Weight $4$
Character orbit 495.a
Self dual yes
Analytic conductor $29.206$
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] = [495,4,Mod(1,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, 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("495.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 495.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: \(29.2059454528\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
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)
 
\(f(q)\) \(=\) \( q - q^{2} - 7 q^{4} + 5 q^{5} + 36 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - 7 q^{4} + 5 q^{5} + 36 q^{7} + 15 q^{8} - 5 q^{10} - 11 q^{11} + 2 q^{13} - 36 q^{14} + 41 q^{16} - 66 q^{17} + 140 q^{19} - 35 q^{20} + 11 q^{22} + 68 q^{23} + 25 q^{25} - 2 q^{26} - 252 q^{28} - 150 q^{29} - 128 q^{31} - 161 q^{32} + 66 q^{34} + 180 q^{35} - 314 q^{37} - 140 q^{38} + 75 q^{40} + 118 q^{41} + 172 q^{43} + 77 q^{44} - 68 q^{46} + 324 q^{47} + 953 q^{49} - 25 q^{50} - 14 q^{52} - 82 q^{53} - 55 q^{55} + 540 q^{56} + 150 q^{58} + 740 q^{59} + 122 q^{61} + 128 q^{62} - 167 q^{64} + 10 q^{65} - 124 q^{67} + 462 q^{68} - 180 q^{70} + 988 q^{71} + 2 q^{73} + 314 q^{74} - 980 q^{76} - 396 q^{77} + 1100 q^{79} + 205 q^{80} - 118 q^{82} + 868 q^{83} - 330 q^{85} - 172 q^{86} - 165 q^{88} + 470 q^{89} + 72 q^{91} - 476 q^{92} - 324 q^{94} + 700 q^{95} + 1186 q^{97} - 953 q^{98}+O(q^{100}) \) 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
−1.00000 0 −7.00000 5.00000 0 36.0000 15.0000 0 −5.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(11\) \(1\)

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 495.4.a.b 1
3.b odd 2 1 165.4.a.b 1
5.b even 2 1 2475.4.a.g 1
15.d odd 2 1 825.4.a.d 1
15.e even 4 2 825.4.c.e 2
33.d even 2 1 1815.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
165.4.a.b 1 3.b odd 2 1
495.4.a.b 1 1.a even 1 1 trivial
825.4.a.d 1 15.d odd 2 1
825.4.c.e 2 15.e even 4 2
1815.4.a.c 1 33.d even 2 1
2475.4.a.g 1 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(495))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{7} - 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T - 36 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T + 66 \) Copy content Toggle raw display
$19$ \( T - 140 \) Copy content Toggle raw display
$23$ \( T - 68 \) Copy content Toggle raw display
$29$ \( T + 150 \) Copy content Toggle raw display
$31$ \( T + 128 \) Copy content Toggle raw display
$37$ \( T + 314 \) Copy content Toggle raw display
$41$ \( T - 118 \) Copy content Toggle raw display
$43$ \( T - 172 \) Copy content Toggle raw display
$47$ \( T - 324 \) Copy content Toggle raw display
$53$ \( T + 82 \) Copy content Toggle raw display
$59$ \( T - 740 \) Copy content Toggle raw display
$61$ \( T - 122 \) Copy content Toggle raw display
$67$ \( T + 124 \) Copy content Toggle raw display
$71$ \( T - 988 \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T - 1100 \) Copy content Toggle raw display
$83$ \( T - 868 \) Copy content Toggle raw display
$89$ \( T - 470 \) Copy content Toggle raw display
$97$ \( T - 1186 \) Copy content Toggle raw display
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