Properties

Label 1440.2.d.c
Level $1440$
Weight $2$
Character orbit 1440.d
Analytic conductor $11.498$
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] = [1440,2,Mod(1009,1440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1440, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("1440.1009");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.d (of order \(2\), degree \(1\), 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: \(11.4984578911\)
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: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 40)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{5} - \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{5} - \beta_{3} q^{7} - 2 \beta_{2} q^{11} + 2 \beta_{3} q^{17} - 2 \beta_{2} q^{19} - \beta_{3} q^{23} + (2 \beta_{3} - 1) q^{25} - 4 q^{31} + (2 \beta_{2} + 3 \beta_1) q^{35} + 6 \beta_1 q^{37} + 3 \beta_1 q^{43} + 3 \beta_{3} q^{47} + q^{49} + 4 \beta_1 q^{53} + (2 \beta_{3} - 6) q^{55} - 6 \beta_{2} q^{59} + 2 \beta_{2} q^{61} - 3 \beta_1 q^{67} - 12 q^{71} - 2 \beta_{3} q^{73} + 6 \beta_1 q^{77} + 4 q^{79} + 7 \beta_1 q^{83} + ( - 4 \beta_{2} - 6 \beta_1) q^{85} - 6 q^{89} + (2 \beta_{3} - 6) q^{95} + 2 \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{25} - 16 q^{31} + 4 q^{49} - 24 q^{55} - 48 q^{71} + 16 q^{79} - 24 q^{89} - 24 q^{95}+O(q^{100}) \) 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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1
0.707107 + 1.22474i
0.707107 1.22474i
−0.707107 1.22474i
−0.707107 + 1.22474i
0 0 0 −1.41421 1.73205i 0 2.44949i 0 0 0
1009.2 0 0 0 −1.41421 + 1.73205i 0 2.44949i 0 0 0
1009.3 0 0 0 1.41421 1.73205i 0 2.44949i 0 0 0
1009.4 0 0 0 1.41421 + 1.73205i 0 2.44949i 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
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.d.c 4
3.b odd 2 1 160.2.f.a 4
4.b odd 2 1 360.2.d.b 4
5.b even 2 1 inner 1440.2.d.c 4
5.c odd 4 2 7200.2.k.l 4
8.b even 2 1 inner 1440.2.d.c 4
8.d odd 2 1 360.2.d.b 4
12.b even 2 1 40.2.f.a 4
15.d odd 2 1 160.2.f.a 4
15.e even 4 2 800.2.d.f 4
20.d odd 2 1 360.2.d.b 4
20.e even 4 2 1800.2.k.m 4
24.f even 2 1 40.2.f.a 4
24.h odd 2 1 160.2.f.a 4
40.e odd 2 1 360.2.d.b 4
40.f even 2 1 inner 1440.2.d.c 4
40.i odd 4 2 7200.2.k.l 4
40.k even 4 2 1800.2.k.m 4
48.i odd 4 2 1280.2.c.k 4
48.k even 4 2 1280.2.c.i 4
60.h even 2 1 40.2.f.a 4
60.l odd 4 2 200.2.d.e 4
120.i odd 2 1 160.2.f.a 4
120.m even 2 1 40.2.f.a 4
120.q odd 4 2 200.2.d.e 4
120.w even 4 2 800.2.d.f 4
240.t even 4 2 1280.2.c.i 4
240.z odd 4 2 6400.2.a.co 4
240.bb even 4 2 6400.2.a.cm 4
240.bd odd 4 2 6400.2.a.co 4
240.bf even 4 2 6400.2.a.cm 4
240.bm odd 4 2 1280.2.c.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.f.a 4 12.b even 2 1
40.2.f.a 4 24.f even 2 1
40.2.f.a 4 60.h even 2 1
40.2.f.a 4 120.m even 2 1
160.2.f.a 4 3.b odd 2 1
160.2.f.a 4 15.d odd 2 1
160.2.f.a 4 24.h odd 2 1
160.2.f.a 4 120.i odd 2 1
200.2.d.e 4 60.l odd 4 2
200.2.d.e 4 120.q odd 4 2
360.2.d.b 4 4.b odd 2 1
360.2.d.b 4 8.d odd 2 1
360.2.d.b 4 20.d odd 2 1
360.2.d.b 4 40.e odd 2 1
800.2.d.f 4 15.e even 4 2
800.2.d.f 4 120.w even 4 2
1280.2.c.i 4 48.k even 4 2
1280.2.c.i 4 240.t even 4 2
1280.2.c.k 4 48.i odd 4 2
1280.2.c.k 4 240.bm odd 4 2
1440.2.d.c 4 1.a even 1 1 trivial
1440.2.d.c 4 5.b even 2 1 inner
1440.2.d.c 4 8.b even 2 1 inner
1440.2.d.c 4 40.f even 2 1 inner
1800.2.k.m 4 20.e even 4 2
1800.2.k.m 4 40.k even 4 2
6400.2.a.cm 4 240.bb even 4 2
6400.2.a.cm 4 240.bf even 4 2
6400.2.a.co 4 240.z odd 4 2
6400.2.a.co 4 240.bd odd 4 2
7200.2.k.l 4 5.c odd 4 2
7200.2.k.l 4 40.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1440, [\chi])\):

\( T_{7}^{2} + 6 \) Copy content Toggle raw display
\( T_{11}^{2} + 12 \) Copy content Toggle raw display
\( T_{13} \) 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
$5$ \( T^{4} + 2T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 54)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$71$ \( (T + 12)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 98)^{2} \) Copy content Toggle raw display
$89$ \( (T + 6)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
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