Properties

Label 2880.3.e.e
Level $2880$
Weight $3$
Character orbit 2880.e
Analytic conductor $78.474$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2880,3,Mod(2431,2880)]
 
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([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2880.2431");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2880.e (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: \(78.4743161358\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 20)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + ( - \beta_{3} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + ( - \beta_{3} - \beta_1) q^{7} + (2 \beta_{3} - 2 \beta_1) q^{11} + ( - 2 \beta_{2} + 4) q^{13} + (8 \beta_{2} + 6) q^{17} + 2 \beta_{3} q^{19} + (\beta_{3} + 3 \beta_1) q^{23} + 5 q^{25} + (4 \beta_{2} - 2) q^{29} + (2 \beta_{3} + 10 \beta_1) q^{31} - 5 \beta_1 q^{35} + ( - 10 \beta_{2} - 4) q^{37} + (6 \beta_{2} + 28) q^{41} + ( - 2 \beta_{3} + 3 \beta_1) q^{43} + ( - 3 \beta_{3} + 13 \beta_1) q^{47} + ( - 10 \beta_{2} - 1) q^{49} + (10 \beta_{2} - 44) q^{53} + ( - 4 \beta_{3} + 6 \beta_1) q^{55} + (6 \beta_{3} + 12 \beta_1) q^{59} + (26 \beta_{2} - 32) q^{61} + (4 \beta_{2} - 10) q^{65} + ( - 10 \beta_{3} - 11 \beta_1) q^{67} + (10 \beta_{3} - 2 \beta_1) q^{71} + (32 \beta_{2} + 66) q^{73} + ( - 20 \beta_{2} + 60) q^{77} + ( - 8 \beta_{3} - 20 \beta_1) q^{79} + ( - 6 \beta_{3} + 13 \beta_1) q^{83} + (6 \beta_{2} + 40) q^{85} + ( - 40 \beta_{2} + 22) q^{89} + ( - 4 \beta_{3} + 6 \beta_1) q^{91} + ( - 2 \beta_{3} + 8 \beta_1) q^{95} + (12 \beta_{2} - 66) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{13} + 24 q^{17} + 20 q^{25} - 8 q^{29} - 16 q^{37} + 112 q^{41} - 4 q^{49} - 176 q^{53} - 128 q^{61} - 40 q^{65} + 264 q^{73} + 240 q^{77} + 160 q^{85} + 88 q^{89} - 264 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{10}^{3} + 2\zeta_{10}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\zeta_{10}^{3} + 2\zeta_{10}^{2} + 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\zeta_{10}^{3} - 4\zeta_{10}^{2} + 8\zeta_{10} - 4 \) Copy content Toggle raw display
\(\zeta_{10}\)\(=\) \( ( \beta_{3} + 2\beta_{2} + 2 ) / 8 \) Copy content Toggle raw display
\(\zeta_{10}^{2}\)\(=\) \( ( \beta_{2} + \beta _1 - 1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{10}^{3}\)\(=\) \( ( -\beta_{2} + \beta _1 + 1 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2431.1
−0.309017 + 0.951057i
−0.309017 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
0 0 0 −2.23607 0 5.25731i 0 0 0
2431.2 0 0 0 −2.23607 0 5.25731i 0 0 0
2431.3 0 0 0 2.23607 0 8.50651i 0 0 0
2431.4 0 0 0 2.23607 0 8.50651i 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.3.e.e 4
3.b odd 2 1 320.3.b.c 4
4.b odd 2 1 inner 2880.3.e.e 4
8.b even 2 1 180.3.c.a 4
8.d odd 2 1 180.3.c.a 4
12.b even 2 1 320.3.b.c 4
15.d odd 2 1 1600.3.b.s 4
15.e even 4 2 1600.3.h.n 8
24.f even 2 1 20.3.b.a 4
24.h odd 2 1 20.3.b.a 4
40.e odd 2 1 900.3.c.k 4
40.f even 2 1 900.3.c.k 4
40.i odd 4 2 900.3.f.e 8
40.k even 4 2 900.3.f.e 8
48.i odd 4 2 1280.3.g.e 8
48.k even 4 2 1280.3.g.e 8
60.h even 2 1 1600.3.b.s 4
60.l odd 4 2 1600.3.h.n 8
120.i odd 2 1 100.3.b.f 4
120.m even 2 1 100.3.b.f 4
120.q odd 4 2 100.3.d.b 8
120.w even 4 2 100.3.d.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.b.a 4 24.f even 2 1
20.3.b.a 4 24.h odd 2 1
100.3.b.f 4 120.i odd 2 1
100.3.b.f 4 120.m even 2 1
100.3.d.b 8 120.q odd 4 2
100.3.d.b 8 120.w even 4 2
180.3.c.a 4 8.b even 2 1
180.3.c.a 4 8.d odd 2 1
320.3.b.c 4 3.b odd 2 1
320.3.b.c 4 12.b even 2 1
900.3.c.k 4 40.e odd 2 1
900.3.c.k 4 40.f even 2 1
900.3.f.e 8 40.i odd 4 2
900.3.f.e 8 40.k even 4 2
1280.3.g.e 8 48.i odd 4 2
1280.3.g.e 8 48.k even 4 2
1600.3.b.s 4 15.d odd 2 1
1600.3.b.s 4 60.h even 2 1
1600.3.h.n 8 15.e even 4 2
1600.3.h.n 8 60.l odd 4 2
2880.3.e.e 4 1.a even 1 1 trivial
2880.3.e.e 4 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2880, [\chi])\):

\( T_{7}^{4} + 100T_{7}^{2} + 2000 \) Copy content Toggle raw display
\( T_{13}^{2} - 8T_{13} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} - 12T_{17} - 284 \) 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
$5$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 100T^{2} + 2000 \) Copy content Toggle raw display
$11$ \( T^{4} + 400T^{2} + 1280 \) Copy content Toggle raw display
$13$ \( (T^{2} - 8 T - 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 12 T - 284)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 320 T^{2} + 20480 \) Copy content Toggle raw display
$23$ \( T^{4} + 260T^{2} + 80 \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T - 76)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2320 T^{2} + 154880 \) Copy content Toggle raw display
$37$ \( (T^{2} + 8 T - 484)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 56 T + 604)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 500T^{2} + 2000 \) Copy content Toggle raw display
$47$ \( T^{4} + 4100 T^{2} + \cdots + 3561680 \) Copy content Toggle raw display
$53$ \( (T^{2} + 88 T + 1436)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 5760 T^{2} + \cdots + 1658880 \) Copy content Toggle raw display
$61$ \( (T^{2} + 64 T - 2356)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 10420 T^{2} + \cdots + 19920080 \) Copy content Toggle raw display
$71$ \( T^{4} + 8080 T^{2} + \cdots + 10138880 \) Copy content Toggle raw display
$73$ \( (T^{2} - 132 T - 764)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 13120 T^{2} + \cdots + 2478080 \) Copy content Toggle raw display
$83$ \( T^{4} + 6260 T^{2} + \cdots + 2620880 \) Copy content Toggle raw display
$89$ \( (T^{2} - 44 T - 7516)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 132 T + 3636)^{2} \) Copy content Toggle raw display
show more
show less