Properties

Label 2268.2.bm.h
Level $2268$
Weight $2$
Character orbit 2268.bm
Analytic conductor $18.110$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2268,2,Mod(593,2268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2268, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.593");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.bm (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: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 756)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{5} + ( - 2 \beta_{2} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{5} + ( - 2 \beta_{2} + 1) q^{7} + (2 \beta_{2} - 2) q^{13} + \beta_1 q^{17} + ( - \beta_{2} - 2) q^{19} + (\beta_{3} + 2 \beta_1) q^{23} + 13 q^{25} + (\beta_{3} - \beta_1) q^{29} + (\beta_{2} + 2) q^{31} + (3 \beta_{3} + 2 \beta_1) q^{35} - 8 \beta_{2} q^{37} - \beta_1 q^{41} - 5 \beta_{2} q^{43} + \beta_1 q^{47} + ( - 8 \beta_{2} - 3) q^{49} + ( - 2 \beta_{3} - \beta_1) q^{53} + ( - 2 \beta_{3} - 2 \beta_1) q^{59} + ( - 3 \beta_{2} + 3) q^{61} + ( - 4 \beta_{3} - 2 \beta_1) q^{65} + 2 \beta_{2} q^{67} + ( - 2 \beta_{3} - 4 \beta_1) q^{71} + (5 \beta_{2} - 5) q^{73} + (4 \beta_{2} + 4) q^{79} + (4 \beta_{3} + 4 \beta_1) q^{83} + ( - 18 \beta_{2} - 18) q^{85} + ( - 3 \beta_{3} - 3 \beta_1) q^{89} + (10 \beta_{2} + 2) q^{91} + ( - \beta_{3} + \beta_1) q^{95} + (9 \beta_{2} + 18) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{7} - 12 q^{13} - 6 q^{19} + 52 q^{25} + 6 q^{31} + 16 q^{37} + 10 q^{43} + 4 q^{49} + 18 q^{61} - 4 q^{67} - 30 q^{73} + 8 q^{79} - 36 q^{85} - 12 q^{91} + 54 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
593.1
0.707107 + 1.22474i
−0.707107 1.22474i
0.707107 1.22474i
−0.707107 + 1.22474i
0 0 0 −4.24264 0 2.00000 1.73205i 0 0 0
593.2 0 0 0 4.24264 0 2.00000 1.73205i 0 0 0
1025.1 0 0 0 −4.24264 0 2.00000 + 1.73205i 0 0 0
1025.2 0 0 0 4.24264 0 2.00000 + 1.73205i 0 0 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
3.b odd 2 1 inner
63.k odd 6 1 inner
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.bm.h 4
3.b odd 2 1 inner 2268.2.bm.h 4
7.d odd 6 1 2268.2.w.g 4
9.c even 3 1 756.2.t.d 4
9.c even 3 1 2268.2.w.g 4
9.d odd 6 1 756.2.t.d 4
9.d odd 6 1 2268.2.w.g 4
21.g even 6 1 2268.2.w.g 4
63.g even 3 1 5292.2.f.d 4
63.i even 6 1 756.2.t.d 4
63.k odd 6 1 inner 2268.2.bm.h 4
63.k odd 6 1 5292.2.f.d 4
63.n odd 6 1 5292.2.f.d 4
63.s even 6 1 inner 2268.2.bm.h 4
63.s even 6 1 5292.2.f.d 4
63.t odd 6 1 756.2.t.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.2.t.d 4 9.c even 3 1
756.2.t.d 4 9.d odd 6 1
756.2.t.d 4 63.i even 6 1
756.2.t.d 4 63.t odd 6 1
2268.2.w.g 4 7.d odd 6 1
2268.2.w.g 4 9.c even 3 1
2268.2.w.g 4 9.d odd 6 1
2268.2.w.g 4 21.g even 6 1
2268.2.bm.h 4 1.a even 1 1 trivial
2268.2.bm.h 4 3.b odd 2 1 inner
2268.2.bm.h 4 63.k odd 6 1 inner
2268.2.bm.h 4 63.s even 6 1 inner
5292.2.f.d 4 63.g even 3 1
5292.2.f.d 4 63.k odd 6 1
5292.2.f.d 4 63.n odd 6 1
5292.2.f.d 4 63.s even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\):

\( T_{5}^{2} - 18 \) Copy content Toggle raw display
\( T_{13}^{2} + 6T_{13} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$19$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 54)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 54T^{2} + 2916 \) Copy content Toggle raw display
$31$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T + 64)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$43$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 18T^{2} + 324 \) Copy content Toggle raw display
$53$ \( T^{4} - 54T^{2} + 2916 \) Copy content Toggle raw display
$59$ \( T^{4} + 72T^{2} + 5184 \) Copy content Toggle raw display
$61$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 216)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 15 T + 75)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 288 T^{2} + 82944 \) Copy content Toggle raw display
$89$ \( T^{4} + 162 T^{2} + 26244 \) Copy content Toggle raw display
$97$ \( (T^{2} - 27 T + 243)^{2} \) Copy content Toggle raw display
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