Properties

Label 468.2.b.a
Level $468$
Weight $2$
Character orbit 468.b
Analytic conductor $3.737$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [468,2,Mod(181,468)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(468, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("468.181");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 468.b (of order \(2\), degree \(1\), 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: \(3.73699881460\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 156)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(\beta = 2\sqrt{-3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{5} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{5} - \beta q^{11} + (\beta - 1) q^{13} - 6 q^{17} - 2 \beta q^{19} - 7 q^{25} + 6 q^{29} - 2 \beta q^{31} + \beta q^{41} + 8 q^{43} - \beta q^{47} + 7 q^{49} - 6 q^{53} - 12 q^{55} + \beta q^{59} + 10 q^{61} + (\beta + 12) q^{65} + 4 \beta q^{67} - 3 \beta q^{71} - 2 \beta q^{73} - 8 q^{79} + \beta q^{83} + 6 \beta q^{85} + 5 \beta q^{89} - 24 q^{95} + 2 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{13} - 12 q^{17} - 14 q^{25} + 12 q^{29} + 16 q^{43} + 14 q^{49} - 12 q^{53} - 24 q^{55} + 20 q^{61} + 24 q^{65} - 16 q^{79} - 48 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(235\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
181.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 3.46410i 0 0 0 0 0
181.2 0 0 0 3.46410i 0 0 0 0 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
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.2.b.a 2
3.b odd 2 1 156.2.b.a 2
4.b odd 2 1 1872.2.c.c 2
12.b even 2 1 624.2.c.f 2
13.b even 2 1 inner 468.2.b.a 2
13.d odd 4 2 6084.2.a.v 2
15.d odd 2 1 3900.2.c.c 2
15.e even 4 2 3900.2.j.h 4
21.c even 2 1 7644.2.e.g 2
24.f even 2 1 2496.2.c.e 2
24.h odd 2 1 2496.2.c.l 2
39.d odd 2 1 156.2.b.a 2
39.f even 4 2 2028.2.a.g 2
39.h odd 6 1 2028.2.q.b 2
39.h odd 6 1 2028.2.q.c 2
39.i odd 6 1 2028.2.q.b 2
39.i odd 6 1 2028.2.q.c 2
39.k even 12 4 2028.2.i.i 4
52.b odd 2 1 1872.2.c.c 2
156.h even 2 1 624.2.c.f 2
156.l odd 4 2 8112.2.a.bs 2
195.e odd 2 1 3900.2.c.c 2
195.s even 4 2 3900.2.j.h 4
273.g even 2 1 7644.2.e.g 2
312.b odd 2 1 2496.2.c.l 2
312.h even 2 1 2496.2.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.b.a 2 3.b odd 2 1
156.2.b.a 2 39.d odd 2 1
468.2.b.a 2 1.a even 1 1 trivial
468.2.b.a 2 13.b even 2 1 inner
624.2.c.f 2 12.b even 2 1
624.2.c.f 2 156.h even 2 1
1872.2.c.c 2 4.b odd 2 1
1872.2.c.c 2 52.b odd 2 1
2028.2.a.g 2 39.f even 4 2
2028.2.i.i 4 39.k even 12 4
2028.2.q.b 2 39.h odd 6 1
2028.2.q.b 2 39.i odd 6 1
2028.2.q.c 2 39.h odd 6 1
2028.2.q.c 2 39.i odd 6 1
2496.2.c.e 2 24.f even 2 1
2496.2.c.e 2 312.h even 2 1
2496.2.c.l 2 24.h odd 2 1
2496.2.c.l 2 312.b odd 2 1
3900.2.c.c 2 15.d odd 2 1
3900.2.c.c 2 195.e odd 2 1
3900.2.j.h 4 15.e even 4 2
3900.2.j.h 4 195.s even 4 2
6084.2.a.v 2 13.d odd 4 2
7644.2.e.g 2 21.c even 2 1
7644.2.e.g 2 273.g even 2 1
8112.2.a.bs 2 156.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 12 \) acting on \(S_{2}^{\mathrm{new}}(468, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 12 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 12 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T + 13 \) Copy content Toggle raw display
$17$ \( (T + 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 48 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T - 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 48 \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 12 \) Copy content Toggle raw display
$43$ \( (T - 8)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 12 \) Copy content Toggle raw display
$53$ \( (T + 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 12 \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 192 \) Copy content Toggle raw display
$71$ \( T^{2} + 108 \) Copy content Toggle raw display
$73$ \( T^{2} + 48 \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 12 \) Copy content Toggle raw display
$89$ \( T^{2} + 300 \) Copy content Toggle raw display
$97$ \( T^{2} + 48 \) Copy content Toggle raw display
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