Properties

Label 405.2.j.d
Level $405$
Weight $2$
Character orbit 405.j
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(109,405)]
 
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([2, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("405.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.j (of order \(6\), 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 \(\Q(\zeta_{6})\)
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: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1 + 1) q^{5} + (2 \beta_{2} + 2) q^{7} + (\beta_{3} + 5 \beta_{2} - \beta_1 - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{2} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{4} + ( - \beta_{2} + \beta_1 + 1) q^{5} + (2 \beta_{2} + 2) q^{7} + (\beta_{3} + 5 \beta_{2} - \beta_1 - 2) q^{8} + ( - 2 \beta_{3} - 4 \beta_{2} + \cdots + 1) q^{10}+ \cdots + (5 \beta_{3} + 5 \beta_{2} - 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} + 3 q^{4} + 3 q^{5} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} + 3 q^{4} + 3 q^{5} + 12 q^{7} - q^{10} - 3 q^{11} + 3 q^{13} + 6 q^{14} - 7 q^{16} - 10 q^{19} + 9 q^{20} + 12 q^{22} + 6 q^{23} + q^{25} - 30 q^{26} - 7 q^{31} + 3 q^{32} + 7 q^{34} + 18 q^{35} - 24 q^{38} + 28 q^{40} + 3 q^{41} + 12 q^{43} + 24 q^{44} + 28 q^{46} - 18 q^{47} + 10 q^{49} - 33 q^{50} - 45 q^{52} - 12 q^{55} - 30 q^{56} - 33 q^{58} - 9 q^{59} - 13 q^{61} - 2 q^{64} + 33 q^{65} - 6 q^{67} + 21 q^{68} + 18 q^{70} - 18 q^{71} + 21 q^{74} - 24 q^{76} - 18 q^{77} + 2 q^{79} + 27 q^{80} + 30 q^{83} - 10 q^{85} + 6 q^{86} - 6 q^{88} - 12 q^{89} + 12 q^{91} + 42 q^{92} - 20 q^{94} + 9 q^{95} + 30 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(\nu\)\(=\) \( \beta_1 \) 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

Character values

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

\(n\) \(82\) \(326\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
1.68614 0.396143i
−1.18614 + 1.26217i
1.68614 + 0.396143i
−1.18614 1.26217i
−0.686141 + 0.396143i 0 −0.686141 + 1.18843i 2.18614 + 0.469882i 0 3.00000 1.73205i 2.67181i 0 −1.68614 + 0.543620i
109.2 2.18614 1.26217i 0 2.18614 3.78651i −0.686141 + 2.12819i 0 3.00000 1.73205i 5.98844i 0 1.18614 + 5.51856i
379.1 −0.686141 0.396143i 0 −0.686141 1.18843i 2.18614 0.469882i 0 3.00000 + 1.73205i 2.67181i 0 −1.68614 0.543620i
379.2 2.18614 + 1.26217i 0 2.18614 + 3.78651i −0.686141 2.12819i 0 3.00000 + 1.73205i 5.98844i 0 1.18614 5.51856i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
45.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.2.j.d 4
3.b odd 2 1 405.2.j.b 4
5.b even 2 1 405.2.j.a 4
9.c even 3 1 405.2.b.b yes 4
9.c even 3 1 405.2.j.a 4
9.d odd 6 1 405.2.b.a 4
9.d odd 6 1 405.2.j.e 4
15.d odd 2 1 405.2.j.e 4
45.h odd 6 1 405.2.b.a 4
45.h odd 6 1 405.2.j.b 4
45.j even 6 1 405.2.b.b yes 4
45.j even 6 1 inner 405.2.j.d 4
45.k odd 12 2 2025.2.a.x 4
45.l even 12 2 2025.2.a.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.b.a 4 9.d odd 6 1
405.2.b.a 4 45.h odd 6 1
405.2.b.b yes 4 9.c even 3 1
405.2.b.b yes 4 45.j even 6 1
405.2.j.a 4 5.b even 2 1
405.2.j.a 4 9.c even 3 1
405.2.j.b 4 3.b odd 2 1
405.2.j.b 4 45.h odd 6 1
405.2.j.d 4 1.a even 1 1 trivial
405.2.j.d 4 45.j even 6 1 inner
405.2.j.e 4 9.d odd 6 1
405.2.j.e 4 15.d odd 2 1
2025.2.a.w 4 45.l even 12 2
2025.2.a.x 4 45.k odd 12 2

Hecke kernels

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

\( T_{2}^{4} - 3T_{2}^{3} + T_{2}^{2} + 6T_{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} - 6T_{7} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 3 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 3 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( T^{4} - 3 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$17$ \( T^{4} + 7T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T^{2} + 5 T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{4} + 33T^{2} + 1089 \) Copy content Toggle raw display
$31$ \( T^{4} + 7 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( T^{4} + 63T^{2} + 324 \) Copy content Toggle raw display
$41$ \( T^{4} - 3 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$43$ \( (T^{2} - 6 T + 12)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 18 T^{3} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{4} + 172T^{2} + 4096 \) Copy content Toggle raw display
$59$ \( T^{4} + 9 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$61$ \( T^{4} + 13 T^{3} + \cdots + 1156 \) Copy content Toggle raw display
$67$ \( T^{4} + 6 T^{3} + \cdots + 9216 \) Copy content Toggle raw display
$71$ \( (T^{2} + 9 T - 54)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 63T^{2} + 324 \) Copy content Toggle raw display
$79$ \( T^{4} - 2 T^{3} + \cdots + 1024 \) Copy content Toggle raw display
$83$ \( T^{4} - 30 T^{3} + \cdots + 4096 \) Copy content Toggle raw display
$89$ \( (T + 3)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} - 30 T^{3} + \cdots + 576 \) Copy content Toggle raw display
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