Properties

Label 405.2.b.d
Level $405$
Weight $2$
Character orbit 405.b
Analytic conductor $3.234$
Analytic rank $0$
Dimension $4$
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] = [405,2,Mod(244,405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(405, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("405.244");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 405.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: \(3.23394128186\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
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 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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{5} + ( - 2 \beta_{3} + \beta_1) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{5} + ( - 2 \beta_{3} + \beta_1) q^{7} + \beta_{3} q^{8} + ( - \beta_{3} + \beta_{2} + 2 \beta_1 - 1) q^{10} + ( - \beta_{2} + 3) q^{11} + ( - \beta_{3} - \beta_1) q^{13} + \beta_{2} q^{14} + (2 \beta_{2} - 1) q^{16} + ( - 3 \beta_{3} + \beta_1) q^{17} + (\beta_{2} - 1) q^{19} + ( - \beta_{3} - \beta_1 - 3) q^{20} + ( - \beta_{3} + 5 \beta_1) q^{22} + \beta_1 q^{23} + (2 \beta_{3} + 2 \beta_1 + 1) q^{25} + ( - \beta_{2} + 3) q^{26} - 3 \beta_{3} q^{28} + (2 \beta_{2} - 3) q^{29} + (\beta_{2} - 1) q^{31} + (4 \beta_{3} - 5 \beta_1) q^{32} + (\beta_{2} + 1) q^{34} + (3 \beta_{3} - \beta_{2} - 3) q^{35} + (3 \beta_{3} - 3 \beta_1) q^{37} + (\beta_{3} - 3 \beta_1) q^{38} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{40} + (\beta_{2} + 6) q^{41} + (\beta_{3} - 5 \beta_1) q^{43} + (3 \beta_{2} - 3) q^{44} + (\beta_{2} - 2) q^{46} + (\beta_{3} + 2 \beta_1) q^{47} + ( - 3 \beta_{2} + 1) q^{49} + (2 \beta_{2} + \beta_1 - 6) q^{50} + ( - 3 \beta_{3} + 3 \beta_1) q^{52} - 2 \beta_1 q^{53} + ( - 2 \beta_{3} - 3 \beta_{2} + \cdots + 3) q^{55}+ \cdots + ( - 3 \beta_{3} + 7 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{10} + 12 q^{11} - 4 q^{16} - 4 q^{19} - 12 q^{20} + 4 q^{25} + 12 q^{26} - 12 q^{29} - 4 q^{31} + 4 q^{34} - 12 q^{35} + 4 q^{40} + 24 q^{41} - 12 q^{44} - 8 q^{46} + 4 q^{49} - 24 q^{50} + 12 q^{55} + 12 q^{56} - 24 q^{59} + 8 q^{61} + 16 q^{64} - 12 q^{70} + 36 q^{71} + 12 q^{74} + 12 q^{76} - 16 q^{79} - 24 q^{80} - 16 q^{85} + 36 q^{86} - 12 q^{89} - 12 q^{91} - 20 q^{94} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(82\) \(326\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1
1.93185i
0.517638i
0.517638i
1.93185i
1.93185i 0 −1.73205 1.73205 1.41421i 0 0.896575i 0.517638i 0 −2.73205 3.34607i
244.2 0.517638i 0 1.73205 −1.73205 + 1.41421i 0 3.34607i 1.93185i 0 0.732051 + 0.896575i
244.3 0.517638i 0 1.73205 −1.73205 1.41421i 0 3.34607i 1.93185i 0 0.732051 0.896575i
244.4 1.93185i 0 −1.73205 1.73205 + 1.41421i 0 0.896575i 0.517638i 0 −2.73205 + 3.34607i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 405.2.b.d 4
3.b odd 2 1 405.2.b.c 4
5.b even 2 1 inner 405.2.b.d 4
5.c odd 4 2 2025.2.a.t 4
9.c even 3 2 45.2.j.a 8
9.d odd 6 2 135.2.j.a 8
15.d odd 2 1 405.2.b.c 4
15.e even 4 2 2025.2.a.r 4
36.f odd 6 2 720.2.by.d 8
36.h even 6 2 2160.2.by.c 8
45.h odd 6 2 135.2.j.a 8
45.j even 6 2 45.2.j.a 8
45.k odd 12 4 225.2.e.d 8
45.l even 12 4 675.2.e.d 8
180.n even 6 2 2160.2.by.c 8
180.p odd 6 2 720.2.by.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.j.a 8 9.c even 3 2
45.2.j.a 8 45.j even 6 2
135.2.j.a 8 9.d odd 6 2
135.2.j.a 8 45.h odd 6 2
225.2.e.d 8 45.k odd 12 4
405.2.b.c 4 3.b odd 2 1
405.2.b.c 4 15.d odd 2 1
405.2.b.d 4 1.a even 1 1 trivial
405.2.b.d 4 5.b even 2 1 inner
675.2.e.d 8 45.l even 12 4
720.2.by.d 8 36.f odd 6 2
720.2.by.d 8 180.p odd 6 2
2025.2.a.r 4 15.e even 4 2
2025.2.a.t 4 5.c odd 4 2
2160.2.by.c 8 36.h even 6 2
2160.2.by.c 8 180.n even 6 2

Hecke kernels

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

\( T_{2}^{4} + 4T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 2T^{2} + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 12T^{2} + 9 \) Copy content Toggle raw display
$11$ \( (T^{2} - 6 T + 6)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 28T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T - 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 4T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T - 3)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T - 2)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 18)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12 T + 33)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 84T^{2} + 36 \) Copy content Toggle raw display
$47$ \( T^{4} + 28T^{2} + 169 \) Copy content Toggle raw display
$53$ \( T^{4} + 16T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} + 12 T + 24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 71)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 84T^{2} + 1521 \) Copy content Toggle raw display
$71$ \( (T^{2} - 18 T + 54)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 172T^{2} + 6889 \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T - 99)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 336 T^{2} + 24336 \) Copy content Toggle raw display
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