Properties

Label 2280.2.bg.r
Level $2280$
Weight $2$
Character orbit 2280.bg
Analytic conductor $18.206$
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] = [2280,2,Mod(121,2280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2280, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2280.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2280.bg (of order \(3\), 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: \(18.2058916609\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{2} q^{5} + ( - \beta_{3} + 2) q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_{2} q^{5} + ( - \beta_{3} + 2) q^{7} + (\beta_{2} - 1) q^{9} + ( - 2 \beta_{3} - 2) q^{11} + (\beta_{3} + \beta_1) q^{13} + (\beta_{2} - 1) q^{15} + (5 \beta_{2} - \beta_1) q^{17} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 + 2) q^{19} + (2 \beta_{2} + \beta_1) q^{21} + ( - 2 \beta_{3} - 2 \beta_1) q^{23} + (\beta_{2} - 1) q^{25} - q^{27} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{29} - 5 q^{31} + ( - 2 \beta_{2} + 2 \beta_1) q^{33} + (2 \beta_{2} + \beta_1) q^{35} + (\beta_{3} - 4) q^{37} + \beta_{3} q^{39} + (4 \beta_{2} - \beta_1) q^{43} - q^{45} + (\beta_{3} - 3 \beta_{2} + \beta_1 + 3) q^{47} + ( - 5 \beta_{3} + 1) q^{49} + ( - \beta_{3} + 5 \beta_{2} - \beta_1 - 5) q^{51} + ( - 5 \beta_{3} + 3 \beta_{2} - 5 \beta_1 - 3) q^{53} + ( - 2 \beta_{2} + 2 \beta_1) q^{55} + (\beta_{3} - \beta_{2} + 2 \beta_1 + 3) q^{57} + ( - 2 \beta_{2} - 2 \beta_1) q^{59} + (\beta_{3} + 10 \beta_{2} + \beta_1 - 10) q^{61} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{63} + \beta_{3} q^{65} + ( - \beta_{3} + 6 \beta_{2} - \beta_1 - 6) q^{67} - 2 \beta_{3} q^{69} + 2 \beta_1 q^{71} + ( - 6 \beta_{2} + \beta_1) q^{73} - q^{75} + ( - 4 \beta_{3} + 4) q^{77} + ( - 4 \beta_{2} + \beta_1) q^{79} - \beta_{2} q^{81} + (3 \beta_{3} + 3) q^{83} + ( - \beta_{3} + 5 \beta_{2} - \beta_1 - 5) q^{85} + ( - 2 \beta_{3} - 2) q^{87} + ( - 6 \beta_{3} - 2 \beta_{2} - 6 \beta_1 + 2) q^{89} + (3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 4) q^{91} - 5 \beta_{2} q^{93} + (\beta_{3} - \beta_{2} + 2 \beta_1 + 3) q^{95} + ( - 6 \beta_{2} - 4 \beta_1) q^{97} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} + 2 q^{5} + 10 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} + 2 q^{5} + 10 q^{7} - 2 q^{9} - 4 q^{11} - q^{13} - 2 q^{15} + 9 q^{17} + 5 q^{19} + 5 q^{21} + 2 q^{23} - 2 q^{25} - 4 q^{27} - 2 q^{29} - 20 q^{31} - 2 q^{33} + 5 q^{35} - 18 q^{37} - 2 q^{39} + 7 q^{43} - 4 q^{45} + 5 q^{47} + 14 q^{49} - 9 q^{51} - q^{53} - 2 q^{55} + 10 q^{57} - 6 q^{59} - 21 q^{61} - 5 q^{63} - 2 q^{65} - 11 q^{67} + 4 q^{69} + 2 q^{71} - 11 q^{73} - 4 q^{75} + 24 q^{77} - 7 q^{79} - 2 q^{81} + 6 q^{83} - 9 q^{85} - 4 q^{87} + 10 q^{89} - 11 q^{91} - 10 q^{93} + 10 q^{95} - 16 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2280\mathbb{Z}\right)^\times\).

\(n\) \(457\) \(761\) \(1141\) \(1711\) \(1921\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−0.780776 1.35234i
1.28078 + 2.21837i
−0.780776 + 1.35234i
1.28078 2.21837i
0 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 0.438447 0 −0.500000 + 0.866025i 0
121.2 0 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 4.56155 0 −0.500000 + 0.866025i 0
961.1 0 0.500000 0.866025i 0 0.500000 0.866025i 0 0.438447 0 −0.500000 0.866025i 0
961.2 0 0.500000 0.866025i 0 0.500000 0.866025i 0 4.56155 0 −0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2280.2.bg.r 4
19.c even 3 1 inner 2280.2.bg.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2280.2.bg.r 4 1.a even 1 1 trivial
2280.2.bg.r 4 19.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2280, [\chi])\):

\( T_{7}^{2} - 5T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 5 T + 2)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T - 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + T^{3} + 5 T^{2} - 4 T + 16 \) Copy content Toggle raw display
$17$ \( T^{4} - 9 T^{3} + 65 T^{2} - 144 T + 256 \) Copy content Toggle raw display
$19$ \( T^{4} - 5 T^{3} + 6 T^{2} - 95 T + 361 \) Copy content Toggle raw display
$23$ \( T^{4} - 2 T^{3} + 20 T^{2} + 32 T + 256 \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} + 20 T^{2} - 32 T + 256 \) Copy content Toggle raw display
$31$ \( (T + 5)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 9 T + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} - 7 T^{3} + 41 T^{2} - 56 T + 64 \) Copy content Toggle raw display
$47$ \( T^{4} - 5 T^{3} + 23 T^{2} - 10 T + 4 \) Copy content Toggle raw display
$53$ \( T^{4} + T^{3} + 107 T^{2} + \cdots + 11236 \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + 44 T^{2} - 48 T + 64 \) Copy content Toggle raw display
$61$ \( T^{4} + 21 T^{3} + 335 T^{2} + \cdots + 11236 \) Copy content Toggle raw display
$67$ \( T^{4} + 11 T^{3} + 95 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$71$ \( T^{4} - 2 T^{3} + 20 T^{2} + 32 T + 256 \) Copy content Toggle raw display
$73$ \( T^{4} + 11 T^{3} + 95 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$79$ \( T^{4} + 7 T^{3} + 41 T^{2} + 56 T + 64 \) Copy content Toggle raw display
$83$ \( (T^{2} - 3 T - 36)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 10 T^{3} + 228 T^{2} + \cdots + 16384 \) Copy content Toggle raw display
$97$ \( T^{4} + 16 T^{3} + 260 T^{2} + \cdots + 16 \) Copy content Toggle raw display
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