Properties

Label 495.1.o.a
Level $495$
Weight $1$
Character orbit 495.o
Analytic conductor $0.247$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
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] = [495,1,Mod(274,495)]
 
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([2, 3, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("495.274");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 495.o (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: \(0.247037181253\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.99235125.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{6} q^{3} + \zeta_{6} q^{4} - \zeta_{6}^{2} q^{5} + \zeta_{6}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6} q^{3} + \zeta_{6} q^{4} - \zeta_{6}^{2} q^{5} + \zeta_{6}^{2} q^{9} - \zeta_{6}^{2} q^{11} - \zeta_{6}^{2} q^{12} - q^{15} + \zeta_{6}^{2} q^{16} + q^{20} - \zeta_{6} q^{25} + q^{27} - \zeta_{6} q^{31} - q^{33} - q^{36} + (\zeta_{6}^{2} + \zeta_{6}) q^{37} + q^{44} + \zeta_{6} q^{45} + ( - \zeta_{6} - 1) q^{47} + q^{48} + \zeta_{6} q^{49} + (\zeta_{6}^{2} + \zeta_{6}) q^{53} - \zeta_{6} q^{55} + \zeta_{6} q^{59} - \zeta_{6} q^{60} - q^{64} + (\zeta_{6}^{2} - 1) q^{67} - q^{71} + \zeta_{6}^{2} q^{75} + \zeta_{6} q^{80} - \zeta_{6} q^{81} - q^{89} + \zeta_{6}^{2} q^{93} + (\zeta_{6} + 1) q^{97} + \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + q^{4} + q^{5} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + q^{4} + q^{5} - q^{9} + q^{11} + q^{12} - 2 q^{15} - q^{16} + 2 q^{20} - q^{25} + 2 q^{27} - q^{31} - 2 q^{33} - 2 q^{36} + 2 q^{44} + q^{45} - 3 q^{47} + 2 q^{48} + q^{49} - q^{55} + q^{59} - q^{60} - 2 q^{64} - 3 q^{67} - 2 q^{71} - q^{75} + q^{80} - q^{81} - 4 q^{89} - q^{93} + 3 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(46\) \(56\) \(397\)
\(\chi(n)\) \(-1\) \(-\zeta_{6}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −0.500000 + 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i 0 0 0 −0.500000 0.866025i 0
439.1 0 −0.500000 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i 0 0 0 −0.500000 + 0.866025i 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.j even 6 1 inner
495.o odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 495.1.o.a 2
3.b odd 2 1 1485.1.o.a 2
5.b even 2 1 495.1.o.b yes 2
5.c odd 4 2 2475.1.y.b 4
9.c even 3 1 495.1.o.b yes 2
9.d odd 6 1 1485.1.o.b 2
11.b odd 2 1 CM 495.1.o.a 2
15.d odd 2 1 1485.1.o.b 2
33.d even 2 1 1485.1.o.a 2
45.h odd 6 1 1485.1.o.a 2
45.j even 6 1 inner 495.1.o.a 2
45.k odd 12 2 2475.1.y.b 4
55.d odd 2 1 495.1.o.b yes 2
55.e even 4 2 2475.1.y.b 4
99.g even 6 1 1485.1.o.b 2
99.h odd 6 1 495.1.o.b yes 2
165.d even 2 1 1485.1.o.b 2
495.o odd 6 1 inner 495.1.o.a 2
495.r even 6 1 1485.1.o.a 2
495.bf even 12 2 2475.1.y.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
495.1.o.a 2 1.a even 1 1 trivial
495.1.o.a 2 11.b odd 2 1 CM
495.1.o.a 2 45.j even 6 1 inner
495.1.o.a 2 495.o odd 6 1 inner
495.1.o.b yes 2 5.b even 2 1
495.1.o.b yes 2 9.c even 3 1
495.1.o.b yes 2 55.d odd 2 1
495.1.o.b yes 2 99.h odd 6 1
1485.1.o.a 2 3.b odd 2 1
1485.1.o.a 2 33.d even 2 1
1485.1.o.a 2 45.h odd 6 1
1485.1.o.a 2 495.r even 6 1
1485.1.o.b 2 9.d odd 6 1
1485.1.o.b 2 15.d odd 2 1
1485.1.o.b 2 99.g even 6 1
1485.1.o.b 2 165.d even 2 1
2475.1.y.b 4 5.c odd 4 2
2475.1.y.b 4 45.k odd 12 2
2475.1.y.b 4 55.e even 4 2
2475.1.y.b 4 495.bf even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{47}^{2} + 3T_{47} + 3 \) acting on \(S_{1}^{\mathrm{new}}(495, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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