Properties

Label 468.2.bj.b
Level $468$
Weight $2$
Character orbit 468.bj
Analytic conductor $3.737$
Analytic rank $0$
Dimension $6$
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(121,468)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(468, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("468.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 468.bj (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: \(3.73699881460\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{6})\)
Coefficient field: 6.0.954288.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1) q^{3} + ( - \beta_{2} + 1) q^{5} + (\beta_{3} + \beta_1 - 1) q^{7} + ( - \beta_{5} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_1) q^{3} + ( - \beta_{2} + 1) q^{5} + (\beta_{3} + \beta_1 - 1) q^{7} + ( - \beta_{5} - \beta_{2}) q^{9} + ( - 4 \beta_{2} - 2) q^{11} + ( - 4 \beta_{2} - 3) q^{13} + (2 \beta_{4} - \beta_1) q^{15} + ( - 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 1) q^{17} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{19} + (\beta_{5} - 2 \beta_{3} + 2 \beta_{2} + 4) q^{21} + ( - \beta_{5} + 3 \beta_{4} + \beta_{2} - 2 \beta_1 + 1) q^{23} + 2 \beta_{2} q^{25} + (\beta_{5} + \beta_{4} - 5 \beta_{2} + \beta_1 - 3) q^{27} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 4 \beta_1 - 1) q^{29} + (\beta_{5} - 2 \beta_{4} - \beta_{2} + 3 \beta_1 - 1) q^{31} + (2 \beta_{4} + 2 \beta_1) q^{33} + ( - \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{35} + (\beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_1 - 1) q^{37} + (\beta_{4} + 3 \beta_1) q^{39} + (\beta_{2} + 2) q^{41} + (\beta_{5} + 4 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{43} + ( - \beta_{5} - \beta_{3} - 2 \beta_{2} - 1) q^{45} + (\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{47} + (2 \beta_{5} - 4 \beta_{4} - \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 1) q^{49} + (2 \beta_{5} - 3 \beta_{4} - \beta_{3} + 4 \beta_{2} + 5) q^{51} + ( - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} - 6 \beta_1) q^{53} + ( - 6 \beta_{2} - 6) q^{55} + ( - 3 \beta_{5} + 2 \beta_{4} + 6 \beta_{2} - 4 \beta_1 + 9) q^{57} + (4 \beta_{2} + 2) q^{59} + (\beta_{5} - 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 3) q^{61} + ( - \beta_{5} + 2 \beta_{3} + \beta_{2} - 3 \beta_1 - 7) q^{63} + ( - 5 \beta_{2} - 7) q^{65} + (\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 + 4) q^{67} + ( - \beta_{5} - \beta_{3} - 8 \beta_{2} - 4) q^{69} + ( - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 2 \beta_1 + 2) q^{71} + (3 \beta_{5} - 2 \beta_{3} + 5 \beta_{2} - 2 \beta_1 + 3) q^{73} - 2 \beta_{4} q^{75} + ( - 4 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 2) q^{77} + (\beta_{5} + \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 2 \beta_1 + 1) q^{79} + (2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 1) q^{81} + ( - 3 \beta_{5} - 3 \beta_{4} + \beta_{2} - 1) q^{83} + ( - \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + 7 \beta_{2} + 2 \beta_1 + 2) q^{85} + ( - 3 \beta_{5} - 3 \beta_{2} + 9) q^{87} + (2 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 3) q^{89} + ( - 4 \beta_{5} - 4 \beta_{4} + \beta_{3} + 4 \beta_{2} + \beta_1 + 3) q^{91} + (2 \beta_{5} - \beta_{3} + 7 \beta_{2} + 2) q^{93} + (3 \beta_{4} + 3 \beta_{3} - 3 \beta_1 + 3) q^{95} + (2 \beta_{5} - 4 \beta_{4} + 2 \beta_{3} + \beta_{2} + 8 \beta_1 - 7) q^{97} + (2 \beta_{5} - 4 \beta_{3} - 2 \beta_{2} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + 9 q^{5} - 3 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} + 9 q^{5} - 3 q^{7} + 2 q^{9} - 6 q^{13} - 3 q^{15} - 9 q^{17} + 9 q^{19} + 15 q^{21} - 3 q^{23} - 6 q^{25} - 2 q^{27} - 12 q^{29} + 3 q^{31} - 3 q^{35} - 3 q^{37} + 2 q^{39} + 9 q^{41} - 3 q^{43} - 3 q^{45} + 9 q^{47} + 12 q^{49} + 21 q^{51} - 12 q^{53} - 18 q^{55} + 27 q^{57} - 9 q^{61} - 45 q^{63} - 27 q^{65} + 21 q^{67} - 3 q^{69} + 9 q^{71} + 2 q^{75} + 6 q^{77} + 9 q^{79} - 10 q^{81} - 9 q^{83} + 60 q^{87} - 27 q^{89} + 9 q^{91} - 9 q^{93} + 18 q^{95} - 27 q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 9 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{5} + \nu^{4} + 2\nu^{3} + 27\nu^{2} - 12\nu - 36 ) / 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 2\nu^{4} + 2\nu^{3} + 6\nu + 18 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{5} + \nu^{4} + 11\nu^{3} + 15\nu + 72 ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - 5\beta_{2} + \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 4\beta_{4} - \beta_{2} + 2\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{5} + \beta_{4} - 8\beta_{2} + 4\beta _1 + 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(145\) \(209\) \(235\)
\(\chi(n)\) \(-\beta_{2}\) \(-1 - \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
0.403374 1.68443i
1.71903 + 0.211943i
−1.62241 + 0.606458i
0.403374 + 1.68443i
1.71903 0.211943i
−1.62241 0.606458i
0 −1.66044 + 0.492881i 0 1.50000 + 0.866025i 0 −3.77121 2.17731i 0 2.51414 1.63680i 0
121.2 0 −0.675970 1.59470i 0 1.50000 + 0.866025i 0 3.12920 + 1.80664i 0 −2.08613 + 2.15594i 0
121.3 0 1.33641 + 1.10182i 0 1.50000 + 0.866025i 0 −0.857990 0.495361i 0 0.571993 + 2.94497i 0
205.1 0 −1.66044 0.492881i 0 1.50000 0.866025i 0 −3.77121 + 2.17731i 0 2.51414 + 1.63680i 0
205.2 0 −0.675970 + 1.59470i 0 1.50000 0.866025i 0 3.12920 1.80664i 0 −2.08613 2.15594i 0
205.3 0 1.33641 1.10182i 0 1.50000 0.866025i 0 −0.857990 + 0.495361i 0 0.571993 2.94497i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.2.bj.b yes 6
3.b odd 2 1 1404.2.bj.b 6
9.c even 3 1 468.2.be.b 6
9.d odd 6 1 1404.2.be.b 6
13.e even 6 1 468.2.be.b 6
39.h odd 6 1 1404.2.be.b 6
117.l even 6 1 inner 468.2.bj.b yes 6
117.v odd 6 1 1404.2.bj.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
468.2.be.b 6 9.c even 3 1
468.2.be.b 6 13.e even 6 1
468.2.bj.b yes 6 1.a even 1 1 trivial
468.2.bj.b yes 6 117.l even 6 1 inner
1404.2.be.b 6 9.d odd 6 1
1404.2.be.b 6 39.h odd 6 1
1404.2.bj.b 6 3.b odd 2 1
1404.2.bj.b 6 117.v odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 3T_{5} + 3 \) acting on \(S_{2}^{\mathrm{new}}(468, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 2 T^{5} + T^{4} + 3 T^{2} + \cdots + 27 \) Copy content Toggle raw display
$5$ \( (T^{2} - 3 T + 3)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{5} - 12 T^{4} - 45 T^{3} + \cdots + 243 \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{3} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2 T + 13)^{3} \) Copy content Toggle raw display
$17$ \( T^{6} + 9 T^{5} + 78 T^{4} + \cdots + 6561 \) Copy content Toggle raw display
$19$ \( T^{6} - 9 T^{5} - 18 T^{4} + \cdots + 49923 \) Copy content Toggle raw display
$23$ \( T^{6} + 3 T^{5} + 42 T^{4} + \cdots + 13689 \) Copy content Toggle raw display
$29$ \( (T^{3} + 6 T^{2} - 72 T - 324)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 3 T^{5} - 30 T^{4} + \cdots + 2187 \) Copy content Toggle raw display
$37$ \( T^{6} + 3 T^{5} - 24 T^{4} - 81 T^{3} + \cdots + 243 \) Copy content Toggle raw display
$41$ \( (T^{2} - 3 T + 3)^{3} \) Copy content Toggle raw display
$43$ \( T^{6} + 3 T^{5} + 108 T^{4} + \cdots + 1681 \) Copy content Toggle raw display
$47$ \( T^{6} - 9 T^{5} - 18 T^{4} + \cdots + 49923 \) Copy content Toggle raw display
$53$ \( (T^{3} + 6 T^{2} - 132 T - 936)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 12)^{3} \) Copy content Toggle raw display
$61$ \( T^{6} + 9 T^{5} + 138 T^{4} + \cdots + 151321 \) Copy content Toggle raw display
$67$ \( T^{6} - 21 T^{5} + 162 T^{4} + \cdots + 9747 \) Copy content Toggle raw display
$71$ \( T^{6} - 9 T^{5} - 18 T^{4} + \cdots + 2187 \) Copy content Toggle raw display
$73$ \( T^{6} + 288 T^{4} + 25920 T^{2} + \cdots + 726192 \) Copy content Toggle raw display
$79$ \( T^{6} - 9 T^{5} + 102 T^{4} + \cdots + 22801 \) Copy content Toggle raw display
$83$ \( T^{6} + 9 T^{5} - 108 T^{4} + \cdots + 4563 \) Copy content Toggle raw display
$89$ \( T^{6} + 27 T^{5} + 108 T^{4} + \cdots + 4255443 \) Copy content Toggle raw display
$97$ \( T^{6} + 27 T^{5} + 36 T^{4} + \cdots + 8137827 \) Copy content Toggle raw display
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