Properties

Label 405.2.b.b
Level $405$
Weight $2$
Character orbit 405.b
Analytic conductor $3.234$
Analytic rank $0$
Dimension $4$
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: \(\Q(\sqrt{-3}, \sqrt{-11})\)
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: \( 2^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{3} + \beta_1 - 2) q^{4} + ( - \beta_1 + 1) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{3} + \beta_1 - 2) q^{4} + ( - \beta_1 + 1) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{3} + 2 \beta_{2} - \beta_1) q^{8} + ( - \beta_{3} - 2 \beta_{2}) q^{10} + (\beta_{3} + \beta_1 + 1) q^{11} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{13} + (2 \beta_{3} + 2 \beta_1 - 4) q^{14} + ( - \beta_{3} - \beta_1 + 4) q^{16} - \beta_{2} q^{17} + (\beta_{3} + \beta_1 - 3) q^{19} + (2 \beta_{3} - \beta_{2} + \beta_1 - 6) q^{20} + (\beta_{3} + \beta_{2} - \beta_1) q^{22} + 2 \beta_{2} q^{23} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{25} + (\beta_{3} + \beta_1 - 8) q^{26} + 6 \beta_{2} q^{28} + ( - 2 \beta_{3} - 2 \beta_1 + 1) q^{29} + (\beta_{3} + \beta_1 + 3) q^{31} + (\beta_{3} - 2 \beta_{2} - \beta_1) q^{32} + (\beta_{3} + \beta_1 - 4) q^{34} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 4) q^{35} - 3 \beta_{2} q^{37} + (\beta_{3} + 5 \beta_{2} - \beta_1) q^{38} + (5 \beta_{2} - \beta_1 - 4) q^{40} + ( - \beta_{3} - \beta_1 - 1) q^{41} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{43} + 6 q^{44} + ( - 2 \beta_{3} - 2 \beta_1 + 8) q^{46} + (\beta_{3} + 3 \beta_{2} - \beta_1) q^{47} - 5 q^{49} + ( - 2 \beta_{3} - 4 \beta_{2} + \cdots + 4) q^{50}+ \cdots + 5 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{4} + 3 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{4} + 3 q^{5} - q^{10} + 6 q^{11} - 12 q^{14} + 14 q^{16} - 10 q^{19} - 21 q^{20} + q^{25} - 30 q^{26} + 14 q^{31} - 14 q^{34} + 18 q^{35} - 17 q^{40} - 6 q^{41} + 24 q^{44} + 28 q^{46} - 20 q^{49} + 15 q^{50} - 12 q^{55} + 60 q^{56} + 18 q^{59} + 26 q^{61} - 2 q^{64} - 21 q^{65} - 42 q^{70} - 18 q^{71} - 42 q^{74} + 48 q^{76} - 4 q^{79} + 27 q^{80} - q^{85} - 12 q^{86} - 12 q^{89} + 12 q^{91} + 40 q^{94} - 24 q^{95}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu^{2} + \nu - 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} + \nu - 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{3} - \nu^{2} + 4\nu + 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{2} + \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} + 3\beta_{2} + \beta _1 + 8 ) / 2 \) 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
−1.18614 + 1.26217i
1.68614 + 0.396143i
1.68614 0.396143i
−1.18614 1.26217i
2.52434i 0 −4.37228 2.18614 + 0.469882i 0 3.46410i 5.98844i 0 1.18614 5.51856i
244.2 0.792287i 0 1.37228 −0.686141 2.12819i 0 3.46410i 2.67181i 0 −1.68614 + 0.543620i
244.3 0.792287i 0 1.37228 −0.686141 + 2.12819i 0 3.46410i 2.67181i 0 −1.68614 0.543620i
244.4 2.52434i 0 −4.37228 2.18614 0.469882i 0 3.46410i 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
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.b yes 4
3.b odd 2 1 405.2.b.a 4
5.b even 2 1 inner 405.2.b.b yes 4
5.c odd 4 2 2025.2.a.x 4
9.c even 3 1 405.2.j.a 4
9.c even 3 1 405.2.j.d 4
9.d odd 6 1 405.2.j.b 4
9.d odd 6 1 405.2.j.e 4
15.d odd 2 1 405.2.b.a 4
15.e even 4 2 2025.2.a.w 4
45.h odd 6 1 405.2.j.b 4
45.h odd 6 1 405.2.j.e 4
45.j even 6 1 405.2.j.a 4
45.j even 6 1 405.2.j.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
405.2.b.a 4 3.b odd 2 1
405.2.b.a 4 15.d odd 2 1
405.2.b.b yes 4 1.a even 1 1 trivial
405.2.b.b yes 4 5.b even 2 1 inner
405.2.j.a 4 9.c even 3 1
405.2.j.a 4 45.j even 6 1
405.2.j.b 4 9.d odd 6 1
405.2.j.b 4 45.h odd 6 1
405.2.j.d 4 9.c even 3 1
405.2.j.d 4 45.j even 6 1
405.2.j.e 4 9.d odd 6 1
405.2.j.e 4 45.h odd 6 1
2025.2.a.w 4 15.e even 4 2
2025.2.a.x 4 5.c odd 4 2

Hecke kernels

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

\( T_{2}^{4} + 7T_{2}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} - 3T_{11} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 7T^{2} + 4 \) 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
$7$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 3 T - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 51T^{2} + 576 \) 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
$23$ \( T^{4} + 28T^{2} + 64 \) Copy content Toggle raw display
$29$ \( (T^{2} - 33)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 7 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 63T^{2} + 324 \) Copy content Toggle raw display
$41$ \( (T^{2} + 3 T - 6)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 76T^{2} + 256 \) Copy content Toggle raw display
$53$ \( T^{4} + 172T^{2} + 4096 \) Copy content Toggle raw display
$59$ \( (T^{2} - 9 T + 12)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 13 T + 34)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 204T^{2} + 9216 \) 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
$79$ \( (T^{2} + 2 T - 32)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 172T^{2} + 4096 \) Copy content Toggle raw display
$89$ \( (T + 3)^{4} \) Copy content Toggle raw display
$97$ \( T^{4} + 348T^{2} + 576 \) Copy content Toggle raw display
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