Properties

Label 2880.2.f.h
Level $2880$
Weight $2$
Character orbit 2880.f
Analytic conductor $22.997$
Analytic rank $1$
Dimension $2$
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] = [2880,2,Mod(1729,2880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2880, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("2880.1729");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.f (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: \(22.9969157821\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 1) q^{5} + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 1) q^{5} + \beta q^{7} - 4 q^{11} + 2 \beta q^{13} - 4 q^{19} - \beta q^{23} + ( - 2 \beta - 3) q^{25} - 2 q^{29} + ( - \beta - 4) q^{35} - 2 \beta q^{37} - 2 q^{41} - 3 \beta q^{43} + 3 \beta q^{47} + 3 q^{49} + 2 \beta q^{53} + ( - 4 \beta + 4) q^{55} + 12 q^{59} + 10 q^{61} + ( - 2 \beta - 8) q^{65} - 7 \beta q^{67} - 8 q^{71} - 4 \beta q^{73} - 4 \beta q^{77} - 16 q^{79} - \beta q^{83} + 6 q^{89} - 8 q^{91} + ( - 4 \beta + 4) q^{95} + 8 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 8 q^{11} - 8 q^{19} - 6 q^{25} - 4 q^{29} - 8 q^{35} - 4 q^{41} + 6 q^{49} + 8 q^{55} + 24 q^{59} + 20 q^{61} - 16 q^{65} - 16 q^{71} - 32 q^{79} + 12 q^{89} - 16 q^{91} + 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1729.1
1.00000i
1.00000i
0 0 0 −1.00000 2.00000i 0 2.00000i 0 0 0
1729.2 0 0 0 −1.00000 + 2.00000i 0 2.00000i 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.f.h 2
3.b odd 2 1 320.2.c.c 2
4.b odd 2 1 2880.2.f.i 2
5.b even 2 1 inner 2880.2.f.h 2
8.b even 2 1 360.2.f.c 2
8.d odd 2 1 720.2.f.e 2
12.b even 2 1 320.2.c.b 2
15.d odd 2 1 320.2.c.c 2
15.e even 4 1 1600.2.a.f 1
15.e even 4 1 1600.2.a.v 1
20.d odd 2 1 2880.2.f.i 2
24.f even 2 1 80.2.c.a 2
24.h odd 2 1 40.2.c.a 2
40.e odd 2 1 720.2.f.e 2
40.f even 2 1 360.2.f.c 2
40.i odd 4 1 1800.2.a.j 1
40.i odd 4 1 1800.2.a.s 1
40.k even 4 1 3600.2.a.k 1
40.k even 4 1 3600.2.a.bb 1
48.i odd 4 1 1280.2.f.a 2
48.i odd 4 1 1280.2.f.f 2
48.k even 4 1 1280.2.f.b 2
48.k even 4 1 1280.2.f.e 2
60.h even 2 1 320.2.c.b 2
60.l odd 4 1 1600.2.a.d 1
60.l odd 4 1 1600.2.a.u 1
120.i odd 2 1 40.2.c.a 2
120.m even 2 1 80.2.c.a 2
120.q odd 4 1 400.2.a.b 1
120.q odd 4 1 400.2.a.g 1
120.w even 4 1 200.2.a.b 1
120.w even 4 1 200.2.a.d 1
168.i even 2 1 1960.2.g.b 2
240.t even 4 1 1280.2.f.b 2
240.t even 4 1 1280.2.f.e 2
240.bm odd 4 1 1280.2.f.a 2
240.bm odd 4 1 1280.2.f.f 2
840.u even 2 1 1960.2.g.b 2
840.bp odd 4 1 9800.2.a.d 1
840.bp odd 4 1 9800.2.a.bf 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.c.a 2 24.h odd 2 1
40.2.c.a 2 120.i odd 2 1
80.2.c.a 2 24.f even 2 1
80.2.c.a 2 120.m even 2 1
200.2.a.b 1 120.w even 4 1
200.2.a.d 1 120.w even 4 1
320.2.c.b 2 12.b even 2 1
320.2.c.b 2 60.h even 2 1
320.2.c.c 2 3.b odd 2 1
320.2.c.c 2 15.d odd 2 1
360.2.f.c 2 8.b even 2 1
360.2.f.c 2 40.f even 2 1
400.2.a.b 1 120.q odd 4 1
400.2.a.g 1 120.q odd 4 1
720.2.f.e 2 8.d odd 2 1
720.2.f.e 2 40.e odd 2 1
1280.2.f.a 2 48.i odd 4 1
1280.2.f.a 2 240.bm odd 4 1
1280.2.f.b 2 48.k even 4 1
1280.2.f.b 2 240.t even 4 1
1280.2.f.e 2 48.k even 4 1
1280.2.f.e 2 240.t even 4 1
1280.2.f.f 2 48.i odd 4 1
1280.2.f.f 2 240.bm odd 4 1
1600.2.a.d 1 60.l odd 4 1
1600.2.a.f 1 15.e even 4 1
1600.2.a.u 1 60.l odd 4 1
1600.2.a.v 1 15.e even 4 1
1800.2.a.j 1 40.i odd 4 1
1800.2.a.s 1 40.i odd 4 1
1960.2.g.b 2 168.i even 2 1
1960.2.g.b 2 840.u even 2 1
2880.2.f.h 2 1.a even 1 1 trivial
2880.2.f.h 2 5.b even 2 1 inner
2880.2.f.i 2 4.b odd 2 1
2880.2.f.i 2 20.d odd 2 1
3600.2.a.k 1 40.k even 4 1
3600.2.a.bb 1 40.k even 4 1
9800.2.a.d 1 840.bp odd 4 1
9800.2.a.bf 1 840.bp odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2880, [\chi])\):

\( T_{7}^{2} + 4 \) Copy content Toggle raw display
\( T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 16 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{19} + 4 \) Copy content Toggle raw display
\( T_{29} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 5 \) Copy content Toggle raw display
$7$ \( T^{2} + 4 \) Copy content Toggle raw display
$11$ \( (T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4 \) Copy content Toggle raw display
$29$ \( (T + 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 36 \) Copy content Toggle raw display
$47$ \( T^{2} + 36 \) Copy content Toggle raw display
$53$ \( T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 196 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 64 \) Copy content Toggle raw display
$79$ \( (T + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 4 \) Copy content Toggle raw display
$89$ \( (T - 6)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 256 \) Copy content Toggle raw display
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