Properties

Label 2880.2.o.c
Level $2880$
Weight $2$
Character orbit 2880.o
Analytic conductor $22.997$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2880,2,Mod(2879,2880)]
 
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, 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("2880.2879");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.o (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: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3317760000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 180)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{5} + \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{5} + \beta_{4} q^{7} - \beta_{7} q^{11} + \beta_{5} q^{13} - \beta_{6} q^{23} + (2 \beta_{5} + 1) q^{25} + 2 \beta_1 q^{29} + \beta_{3} q^{31} + (\beta_{7} + \beta_{6}) q^{35} + 3 \beta_{5} q^{37} + \beta_1 q^{41} + 2 \beta_{4} q^{43} + 2 \beta_{6} q^{47} + 3 q^{49} + ( - 3 \beta_{4} - \beta_{3}) q^{55} + \beta_{7} q^{59} + 10 q^{61} + ( - 2 \beta_{2} + 3 \beta_1) q^{65} - 4 \beta_{4} q^{67} + 6 \beta_{5} q^{73} - 10 \beta_{2} q^{77} - \beta_{3} q^{79} - 2 \beta_{6} q^{83} + 7 \beta_1 q^{89} + \beta_{3} q^{91} - 2 \beta_{5} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{25} + 24 q^{49} + 80 q^{61}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 14\nu^{5} - 25\nu^{3} - 78\nu ) / 72 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 6\nu^{4} + 7\nu^{2} - 30 ) / 24 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{4} - 16\nu^{2} - 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 14\nu^{5} + 61\nu^{3} - 30\nu ) / 36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 6\nu^{5} + 5\nu^{3} + 18\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 6\nu^{4} + 9\nu^{2} - 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + 26\nu^{5} - 59\nu^{3} + 54\nu ) / 24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + \beta_{5} + \beta_{4} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + 6\beta_{2} + 8 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 3\beta_{5} + 7\beta_{4} + 5\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4\beta_{6} + 3\beta_{3} + 24\beta_{2} + 38 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 15\beta_{7} + 21\beta_{5} + 25\beta_{4} + 41\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 31\beta_{6} + 18\beta_{3} + 90\beta_{2} + 164 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 57\beta_{7} + 141\beta_{5} + 97\beta_{4} + 239\beta_1 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2879.1
0.578737 + 0.965926i
−0.578737 + 0.965926i
0.578737 0.965926i
−0.578737 0.965926i
−2.15988 0.258819i
2.15988 0.258819i
−2.15988 + 0.258819i
2.15988 + 0.258819i
0 0 0 −1.73205 1.41421i 0 −3.16228 0 0 0
2879.2 0 0 0 −1.73205 1.41421i 0 3.16228 0 0 0
2879.3 0 0 0 −1.73205 + 1.41421i 0 −3.16228 0 0 0
2879.4 0 0 0 −1.73205 + 1.41421i 0 3.16228 0 0 0
2879.5 0 0 0 1.73205 1.41421i 0 −3.16228 0 0 0
2879.6 0 0 0 1.73205 1.41421i 0 3.16228 0 0 0
2879.7 0 0 0 1.73205 + 1.41421i 0 −3.16228 0 0 0
2879.8 0 0 0 1.73205 + 1.41421i 0 3.16228 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2879.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h 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.o.c 8
3.b odd 2 1 inner 2880.2.o.c 8
4.b odd 2 1 inner 2880.2.o.c 8
5.b even 2 1 inner 2880.2.o.c 8
8.b even 2 1 180.2.h.b 8
8.d odd 2 1 180.2.h.b 8
12.b even 2 1 inner 2880.2.o.c 8
15.d odd 2 1 inner 2880.2.o.c 8
20.d odd 2 1 inner 2880.2.o.c 8
24.f even 2 1 180.2.h.b 8
24.h odd 2 1 180.2.h.b 8
40.e odd 2 1 180.2.h.b 8
40.f even 2 1 180.2.h.b 8
40.i odd 4 2 900.2.e.f 8
40.k even 4 2 900.2.e.f 8
60.h even 2 1 inner 2880.2.o.c 8
120.i odd 2 1 180.2.h.b 8
120.m even 2 1 180.2.h.b 8
120.q odd 4 2 900.2.e.f 8
120.w even 4 2 900.2.e.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.2.h.b 8 8.b even 2 1
180.2.h.b 8 8.d odd 2 1
180.2.h.b 8 24.f even 2 1
180.2.h.b 8 24.h odd 2 1
180.2.h.b 8 40.e odd 2 1
180.2.h.b 8 40.f even 2 1
180.2.h.b 8 120.i odd 2 1
180.2.h.b 8 120.m even 2 1
900.2.e.f 8 40.i odd 4 2
900.2.e.f 8 40.k even 4 2
900.2.e.f 8 120.q odd 4 2
900.2.e.f 8 120.w even 4 2
2880.2.o.c 8 1.a even 1 1 trivial
2880.2.o.c 8 3.b odd 2 1 inner
2880.2.o.c 8 4.b odd 2 1 inner
2880.2.o.c 8 5.b even 2 1 inner
2880.2.o.c 8 12.b even 2 1 inner
2880.2.o.c 8 15.d odd 2 1 inner
2880.2.o.c 8 20.d odd 2 1 inner
2880.2.o.c 8 60.h even 2 1 inner

Hecke kernels

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

\( T_{7}^{2} - 10 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 10)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 30)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{2} + 20)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 54)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 40)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 80)^{4} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{2} - 30)^{4} \) Copy content Toggle raw display
$61$ \( (T - 10)^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} - 160)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} + 216)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 60)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 80)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 98)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
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