Properties

Label 495.2.a.f
Level $495$
Weight $2$
Character orbit 495.a
Self dual yes
Analytic conductor $3.953$
Analytic rank $0$
Dimension $4$
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] = [495,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.48704.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{2} + 2) q^{4} - q^{5} + (\beta_{3} + \beta_1 + 1) q^{7} + (\beta_{3} + 2 \beta_1 - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} + (\beta_{2} + 2) q^{4} - q^{5} + (\beta_{3} + \beta_1 + 1) q^{7} + (\beta_{3} + 2 \beta_1 - 2) q^{8} + ( - \beta_1 + 1) q^{10} + q^{11} + ( - \beta_{3} + \beta_1 + 1) q^{13} + ( - 2 \beta_{3} + \beta_{2} + 1) q^{14} + ( - 2 \beta_{3} - 2 \beta_1 + 3) q^{16} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{17} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{19} + ( - \beta_{2} - 2) q^{20} + (\beta_1 - 1) q^{22} + (2 \beta_{2} + 2) q^{23} + q^{25} + (2 \beta_{3} + \beta_{2} + 4 \beta_1 + 3) q^{26} + (3 \beta_{3} + 5 \beta_1 - 3) q^{28} + (2 \beta_{3} + 2 \beta_{2} + 2) q^{29} + (2 \beta_{3} + 2 \beta_1) q^{31} + (2 \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{32} + ( - \beta_{2} - 4 \beta_1 + 3) q^{34} + ( - \beta_{3} - \beta_1 - 1) q^{35} + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots + 2) q^{37}+ \cdots + ( - 2 \beta_{3} - 2 \beta_{2} + \cdots - 19) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 8 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 8 q^{4} - 4 q^{5} + 4 q^{7} - 6 q^{8} + 2 q^{10} + 4 q^{11} + 8 q^{13} + 8 q^{14} + 12 q^{16} - 4 q^{17} + 4 q^{19} - 8 q^{20} - 2 q^{22} + 8 q^{23} + 4 q^{25} + 16 q^{26} - 8 q^{28} + 4 q^{29} - 14 q^{32} + 4 q^{34} - 4 q^{35} + 8 q^{37} - 20 q^{38} + 6 q^{40} + 4 q^{41} + 12 q^{43} + 8 q^{44} - 16 q^{46} + 20 q^{49} - 2 q^{50} + 20 q^{52} - 16 q^{53} - 4 q^{55} + 48 q^{56} - 24 q^{58} + 24 q^{59} + 8 q^{61} + 20 q^{62} - 8 q^{65} - 48 q^{68} - 8 q^{70} + 16 q^{71} + 8 q^{73} - 12 q^{74} - 36 q^{76} + 4 q^{77} + 12 q^{79} - 12 q^{80} - 40 q^{82} - 8 q^{83} + 4 q^{85} - 16 q^{86} - 6 q^{88} + 16 q^{89} - 16 q^{91} + 72 q^{92} - 40 q^{94} - 4 q^{95} + 8 q^{97} - 62 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 6x^{2} + 4x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 3\nu^{2} - 3\nu + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 3\beta_{2} + 9\beta _1 + 4 \) 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
−1.69696
−0.852061
1.26270
3.28632
−2.69696 0 5.27358 −1.00000 0 −4.13176 −8.82872 0 2.69696
1.2 −1.85206 0 1.43013 −1.00000 0 4.90749 1.05543 0 1.85206
1.3 0.262696 0 −1.93099 −1.00000 0 0.704647 −1.03266 0 −0.262696
1.4 2.28632 0 3.22727 −1.00000 0 2.51962 2.80595 0 −2.28632
\(n\): 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 495.2.a.f 4
3.b odd 2 1 495.2.a.g yes 4
4.b odd 2 1 7920.2.a.cm 4
5.b even 2 1 2475.2.a.bj 4
5.c odd 4 2 2475.2.c.t 8
11.b odd 2 1 5445.2.a.bs 4
12.b even 2 1 7920.2.a.cn 4
15.d odd 2 1 2475.2.a.bf 4
15.e even 4 2 2475.2.c.s 8
33.d even 2 1 5445.2.a.bh 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.a.f 4 1.a even 1 1 trivial
495.2.a.g yes 4 3.b odd 2 1
2475.2.a.bf 4 15.d odd 2 1
2475.2.a.bj 4 5.b even 2 1
2475.2.c.s 8 15.e even 4 2
2475.2.c.t 8 5.c odd 4 2
5445.2.a.bh 4 33.d even 2 1
5445.2.a.bs 4 11.b odd 2 1
7920.2.a.cm 4 4.b odd 2 1
7920.2.a.cn 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 2T_{2}^{3} - 6T_{2}^{2} - 10T_{2} + 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(495))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + \cdots - 36 \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} - 8 T^{3} + \cdots - 292 \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + \cdots - 324 \) Copy content Toggle raw display
$19$ \( T^{4} - 4 T^{3} + \cdots + 288 \) Copy content Toggle raw display
$23$ \( T^{4} - 8 T^{3} + \cdots - 192 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + \cdots - 144 \) Copy content Toggle raw display
$31$ \( T^{4} - 88 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$37$ \( T^{4} - 8 T^{3} + \cdots + 976 \) Copy content Toggle raw display
$41$ \( T^{4} - 4 T^{3} + \cdots + 2160 \) Copy content Toggle raw display
$43$ \( T^{4} - 12 T^{3} + \cdots - 36 \) Copy content Toggle raw display
$47$ \( T^{4} - 128 T^{2} + \cdots - 576 \) Copy content Toggle raw display
$53$ \( T^{4} + 16 T^{3} + \cdots - 240 \) Copy content Toggle raw display
$59$ \( T^{4} - 24 T^{3} + \cdots - 8496 \) Copy content Toggle raw display
$61$ \( T^{4} - 8 T^{3} + \cdots - 2224 \) Copy content Toggle raw display
$67$ \( T^{4} - 224 T^{2} + \cdots + 6208 \) Copy content Toggle raw display
$71$ \( T^{4} - 16 T^{3} + \cdots - 240 \) Copy content Toggle raw display
$73$ \( T^{4} - 8 T^{3} + \cdots - 292 \) Copy content Toggle raw display
$79$ \( T^{4} - 12 T^{3} + \cdots + 160 \) Copy content Toggle raw display
$83$ \( T^{4} + 8 T^{3} + \cdots + 1836 \) Copy content Toggle raw display
$89$ \( T^{4} - 16 T^{3} + \cdots + 720 \) Copy content Toggle raw display
$97$ \( T^{4} - 8 T^{3} + \cdots - 2864 \) Copy content Toggle raw display
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