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Properties

Label 495.2.r.a
Level $495$
Weight $2$
Character orbit 495.r
Analytic conductor $3.953$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [495,2,Mod(164,495)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(495, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.164");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 495.r (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(3.95259490005\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (2 \beta_{2} - 2) q^{4} + (\beta_{3} + 2 \beta_{2}) q^{5} + (\beta_{3} + 3 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (2 \beta_{2} - 2) q^{4} + (\beta_{3} + 2 \beta_{2}) q^{5} + (\beta_{3} + 3 \beta_{2}) q^{9} + ( - 2 \beta_{3} - \beta_{2}) q^{11} + 2 \beta_{3} q^{12} + (\beta_{3} - \beta_1 - 3) q^{15} - 4 \beta_{2} q^{16} + ( - 2 \beta_{3} + 2 \beta_1 - 4) q^{20} + (9 \beta_{2} - 9) q^{23} + (\beta_{2} + 3 \beta_1 - 1) q^{25} + (2 \beta_{3} - 2 \beta_1 - 3) q^{27} + ( - 6 \beta_{3} + \beta_{2} + 3 \beta_1 - 4) q^{31} + (\beta_{3} - \beta_1 + 6) q^{33} + ( - 2 \beta_{3} + 2 \beta_1 - 6) q^{36} + ( - 3 \beta_{3} + 7 \beta_{2} + \cdots - 5) q^{37}+ \cdots + (3 \beta_{2} - 5 \beta_1 - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} - 4 q^{4} + 3 q^{5} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} - 4 q^{4} + 3 q^{5} + 5 q^{9} - 2 q^{12} - 14 q^{15} - 8 q^{16} - 12 q^{20} - 18 q^{23} + q^{25} - 16 q^{27} - 5 q^{31} + 22 q^{33} - 20 q^{36} - 2 q^{45} - 12 q^{47} + 8 q^{48} + 14 q^{49} - 12 q^{53} - 11 q^{55} + 45 q^{59} + 14 q^{60} + 32 q^{64} + 39 q^{67} - 9 q^{69} - 16 q^{75} + 12 q^{80} - 7 q^{81} - 36 q^{92} + 47 q^{93} + 51 q^{97} - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 2\beta _1 + 3 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/495\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(-1\) \(1 - \beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
164.1
1.68614 − 0.396143i
−1.18614 + 1.26217i
1.68614 + 0.396143i
−1.18614 − 1.26217i
0 −1.68614 + 0.396143i −1.00000 − 1.73205i 2.18614 − 0.469882i 0 0 0 2.68614 − 1.33591i 0
164.2 0 1.18614 − 1.26217i −1.00000 − 1.73205i −0.686141 − 2.12819i 0 0 0 −0.186141 − 2.99422i 0
329.1 0 −1.68614 − 0.396143i −1.00000 + 1.73205i 2.18614 + 0.469882i 0 0 0 2.68614 + 1.33591i 0
329.2 0 1.18614 + 1.26217i −1.00000 + 1.73205i −0.686141 + 2.12819i 0 0 0 −0.186141 + 2.99422i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
45.h odd 6 1 inner
495.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.2.r.a ✓ 4
5.b even 2 1 495.2.r.b yes 4
9.d odd 6 1 495.2.r.b yes 4
11.b odd 2 1 CM 495.2.r.a ✓ 4
45.h odd 6 1 inner 495.2.r.a ✓ 4
55.d odd 2 1 495.2.r.b yes 4
99.g even 6 1 495.2.r.b yes 4
495.r even 6 1 inner 495.2.r.a ✓ 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.2.r.a ✓ 4 1.a even 1 1 trivial
495.2.r.a ✓ 4 11.b odd 2 1 CM
495.2.r.a ✓ 4 45.h odd 6 1 inner
495.2.r.a ✓ 4 495.r even 6 1 inner
495.2.r.b yes 4 5.b even 2 1
495.2.r.b yes 4 9.d odd 6 1
495.2.r.b yes 4 55.d odd 2 1
495.2.r.b yes 4 99.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{23}^{2} + 9T_{23} + 81 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} - 2 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 11T^{2} + 121 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 5 T^{3} + \cdots + 4624 \) Copy content Toggle raw display
$37$ \( T^{4} + 123T^{2} + 144 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 12 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 123)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 45 T^{3} + \cdots + 27556 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} - 39 T^{3} + \cdots + 10404 \) Copy content Toggle raw display
$71$ \( T^{4} + 151T^{2} + 3844 \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 275)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 51 T^{3} + \cdots + 36864 \) Copy content Toggle raw display
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