Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2268,2,Mod(269,2268)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.269");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.w (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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 252)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + ( - 2 \beta_1 + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} + ( - 2 \beta_1 + 3) q^{7} + (\beta_{3} + \beta_{2}) q^{11} + (\beta_1 + 1) q^{13} + 2 \beta_{2} q^{17} + (3 \beta_1 + 3) q^{19} + (\beta_1 - 1) q^{25} + ( - 4 \beta_{3} + 2 \beta_{2}) q^{29} + ( - 2 \beta_1 + 1) q^{31} + ( - 2 \beta_{3} - \beta_{2}) q^{35} + ( - 5 \beta_1 + 5) q^{37} + ( - 5 \beta_{3} + 5 \beta_{2}) q^{41} + 11 \beta_1 q^{43} + \beta_{3} q^{47} + ( - 8 \beta_1 + 5) q^{49} + (4 \beta_{3} - 2 \beta_{2}) q^{53} + ( - 12 \beta_1 + 6) q^{55} + 2 \beta_{3} q^{59} + (4 \beta_1 - 2) q^{61} + (\beta_{3} - 2 \beta_{2}) q^{65} - 7 q^{67} + (3 \beta_{3} - 6 \beta_{2}) q^{71} + ( - 9 \beta_1 + 18) q^{73} + (5 \beta_{3} - \beta_{2}) q^{77} + 11 q^{79} - 5 \beta_{2} q^{83} + ( - 12 \beta_1 + 12) q^{85} + ( - 6 \beta_{3} + 6 \beta_{2}) q^{89} + ( - \beta_1 + 5) q^{91} + (3 \beta_{3} - 6 \beta_{2}) q^{95} + (2 \beta_1 - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{7} + 6 q^{13} + 18 q^{19} - 2 q^{25} + 10 q^{37} + 22 q^{43} + 4 q^{49} - 28 q^{67} + 54 q^{73} + 44 q^{79} + 24 q^{85} + 18 q^{91} - 12 q^{97}+O(q^{100}) \) 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

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
1.22474 0.707107i
−1.22474 + 0.707107i
1.22474 + 0.707107i
−1.22474 0.707107i
0 0 0 −1.22474 + 2.12132i 0 2.00000 + 1.73205i 0 0 0
269.2 0 0 0 1.22474 2.12132i 0 2.00000 + 1.73205i 0 0 0
1349.1 0 0 0 −1.22474 2.12132i 0 2.00000 1.73205i 0 0 0
1349.2 0 0 0 1.22474 + 2.12132i 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.i even 6 1 inner
63.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.w.h 4
3.b odd 2 1 inner 2268.2.w.h 4
7.d odd 6 1 2268.2.bm.g 4
9.c even 3 1 252.2.t.a 4
9.c even 3 1 2268.2.bm.g 4
9.d odd 6 1 252.2.t.a 4
9.d odd 6 1 2268.2.bm.g 4
21.g even 6 1 2268.2.bm.g 4
36.f odd 6 1 1008.2.bt.a 4
36.h even 6 1 1008.2.bt.a 4
45.h odd 6 1 6300.2.ch.a 4
45.j even 6 1 6300.2.ch.a 4
45.k odd 12 2 6300.2.dd.a 8
45.l even 12 2 6300.2.dd.a 8
63.g even 3 1 1764.2.t.a 4
63.h even 3 1 1764.2.f.a 4
63.i even 6 1 1764.2.f.a 4
63.i even 6 1 inner 2268.2.w.h 4
63.j odd 6 1 1764.2.f.a 4
63.k odd 6 1 252.2.t.a 4
63.l odd 6 1 1764.2.t.a 4
63.n odd 6 1 1764.2.t.a 4
63.o even 6 1 1764.2.t.a 4
63.s even 6 1 252.2.t.a 4
63.t odd 6 1 1764.2.f.a 4
63.t odd 6 1 inner 2268.2.w.h 4
252.n even 6 1 1008.2.bt.a 4
252.r odd 6 1 7056.2.k.a 4
252.u odd 6 1 7056.2.k.a 4
252.bb even 6 1 7056.2.k.a 4
252.bj even 6 1 7056.2.k.a 4
252.bn odd 6 1 1008.2.bt.a 4
315.u even 6 1 6300.2.ch.a 4
315.bn odd 6 1 6300.2.ch.a 4
315.bw odd 12 2 6300.2.dd.a 8
315.cg even 12 2 6300.2.dd.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.t.a 4 9.c even 3 1
252.2.t.a 4 9.d odd 6 1
252.2.t.a 4 63.k odd 6 1
252.2.t.a 4 63.s even 6 1
1008.2.bt.a 4 36.f odd 6 1
1008.2.bt.a 4 36.h even 6 1
1008.2.bt.a 4 252.n even 6 1
1008.2.bt.a 4 252.bn odd 6 1
1764.2.f.a 4 63.h even 3 1
1764.2.f.a 4 63.i even 6 1
1764.2.f.a 4 63.j odd 6 1
1764.2.f.a 4 63.t odd 6 1
1764.2.t.a 4 63.g even 3 1
1764.2.t.a 4 63.l odd 6 1
1764.2.t.a 4 63.n odd 6 1
1764.2.t.a 4 63.o even 6 1
2268.2.w.h 4 1.a even 1 1 trivial
2268.2.w.h 4 3.b odd 2 1 inner
2268.2.w.h 4 63.i even 6 1 inner
2268.2.w.h 4 63.t odd 6 1 inner
2268.2.bm.g 4 7.d odd 6 1
2268.2.bm.g 4 9.c even 3 1
2268.2.bm.g 4 9.d odd 6 1
2268.2.bm.g 4 21.g even 6 1
6300.2.ch.a 4 45.h odd 6 1
6300.2.ch.a 4 45.j even 6 1
6300.2.ch.a 4 315.u even 6 1
6300.2.ch.a 4 315.bn odd 6 1
6300.2.dd.a 8 45.k odd 12 2
6300.2.dd.a 8 45.l even 12 2
6300.2.dd.a 8 315.bw odd 12 2
6300.2.dd.a 8 315.cg even 12 2
7056.2.k.a 4 252.r odd 6 1
7056.2.k.a 4 252.u odd 6 1
7056.2.k.a 4 252.bb even 6 1
7056.2.k.a 4 252.bj even 6 1

Hecke kernels

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

\( T_{5}^{4} + 6T_{5}^{2} + 36 \) Copy content Toggle raw display
\( T_{13}^{2} - 3T_{13} + 3 \) 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
$5$ \( T^{4} + 6T^{2} + 36 \) Copy content Toggle raw display
$7$ \( (T^{2} - 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 18T^{2} + 324 \) Copy content Toggle raw display
$13$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$19$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 72T^{2} + 5184 \) Copy content Toggle raw display
$31$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 150 T^{2} + 22500 \) Copy content Toggle raw display
$43$ \( (T^{2} - 11 T + 121)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 72T^{2} + 5184 \) Copy content Toggle raw display
$59$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$67$ \( (T + 7)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 162)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 27 T + 243)^{2} \) Copy content Toggle raw display
$79$ \( (T - 11)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 150 T^{2} + 22500 \) Copy content Toggle raw display
$89$ \( T^{4} + 216 T^{2} + 46656 \) Copy content Toggle raw display
$97$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
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