Properties

Label 405.2.e.l
Level $405$
Weight $2$
Character orbit 405.e
Analytic conductor $3.234$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [405,2,Mod(136,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.136");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.e (of order \(3\), degree \(2\), not 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: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{2} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{4} + (\beta_1 - 1) q^{5} + (\beta_{2} + 3 \beta_1) q^{7} + (2 \beta_{3} - 6) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{2} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{4} + (\beta_1 - 1) q^{5} + (\beta_{2} + 3 \beta_1) q^{7} + (2 \beta_{3} - 6) q^{8} + (\beta_{3} - 1) q^{10} + (\beta_{2} + 4 \beta_1) q^{11} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots + 2) q^{13}+ \cdots + ( - \beta_{3} + 13) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 4 q^{4} - 2 q^{5} + 6 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 4 q^{4} - 2 q^{5} + 6 q^{7} - 24 q^{8} - 4 q^{10} + 8 q^{11} + 4 q^{13} - 16 q^{16} - 4 q^{17} + 4 q^{19} - 4 q^{20} - 2 q^{22} - 2 q^{25} + 32 q^{26} + 4 q^{29} + 6 q^{31} + 16 q^{32} + 4 q^{34} - 12 q^{35} - 4 q^{37} - 10 q^{38} + 12 q^{40} + 4 q^{41} + 10 q^{43} - 8 q^{44} - 24 q^{46} + 14 q^{47} - 10 q^{49} + 2 q^{50} + 32 q^{52} + 20 q^{53} - 16 q^{55} - 24 q^{56} - 22 q^{58} + 20 q^{59} - 8 q^{61} + 12 q^{62} + 64 q^{64} + 4 q^{65} - 8 q^{68} - 8 q^{71} + 4 q^{73} + 4 q^{74} + 20 q^{76} - 30 q^{77} - 24 q^{79} + 32 q^{80} - 28 q^{82} + 6 q^{83} + 2 q^{85} - 28 q^{86} - 36 q^{88} - 24 q^{92} - 8 q^{94} - 2 q^{95} + 2 q^{97} + 52 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).

\(n\) \(82\) \(326\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
136.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.366025 + 0.633975i 0 0.732051 + 1.26795i −0.500000 0.866025i 0 2.36603 4.09808i −2.53590 0 0.732051
136.2 1.36603 2.36603i 0 −2.73205 4.73205i −0.500000 0.866025i 0 0.633975 1.09808i −9.46410 0 −2.73205
271.1 −0.366025 0.633975i 0 0.732051 1.26795i −0.500000 + 0.866025i 0 2.36603 + 4.09808i −2.53590 0 0.732051
271.2 1.36603 + 2.36603i 0 −2.73205 + 4.73205i −0.500000 + 0.866025i 0 0.633975 + 1.09808i −9.46410 0 −2.73205
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.2.e.l 4
3.b odd 2 1 405.2.e.i 4
9.c even 3 1 405.2.a.g 2
9.c even 3 1 inner 405.2.e.l 4
9.d odd 6 1 405.2.a.h yes 2
9.d odd 6 1 405.2.e.i 4
36.f odd 6 1 6480.2.a.br 2
36.h even 6 1 6480.2.a.bi 2
45.h odd 6 1 2025.2.a.g 2
45.j even 6 1 2025.2.a.m 2
45.k odd 12 2 2025.2.b.g 4
45.l even 12 2 2025.2.b.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.a.g 2 9.c even 3 1
405.2.a.h yes 2 9.d odd 6 1
405.2.e.i 4 3.b odd 2 1
405.2.e.i 4 9.d odd 6 1
405.2.e.l 4 1.a even 1 1 trivial
405.2.e.l 4 9.c even 3 1 inner
2025.2.a.g 2 45.h odd 6 1
2025.2.a.m 2 45.j even 6 1
2025.2.b.g 4 45.k odd 12 2
2025.2.b.h 4 45.l even 12 2
6480.2.a.bi 2 36.h even 6 1
6480.2.a.br 2 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(405, [\chi])\):

\( T_{2}^{4} - 2T_{2}^{3} + 6T_{2}^{2} + 4T_{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{4} - 6T_{7}^{3} + 30T_{7}^{2} - 36T_{7} + 36 \) Copy content Toggle raw display
\( T_{11}^{4} - 8T_{11}^{3} + 51T_{11}^{2} - 104T_{11} + 169 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$11$ \( T^{4} - 8 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$13$ \( T^{4} - 4 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$17$ \( (T^{2} + 2 T - 2)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 11)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$31$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 4 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$43$ \( T^{4} - 10 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$47$ \( T^{4} - 14 T^{3} + \cdots + 2116 \) Copy content Toggle raw display
$53$ \( (T^{2} - 10 T + 22)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 20 T^{3} + \cdots + 9409 \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$71$ \( (T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T - 74)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 24 T^{3} + \cdots + 17424 \) Copy content Toggle raw display
$83$ \( T^{4} - 6 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$89$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 2 T^{3} + \cdots + 5476 \) Copy content Toggle raw display
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