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Properties

Label 2880.2.w.o
Level $2880$
Weight $2$
Character orbit 2880.w
Analytic conductor $22.997$
Analytic rank $0$
Dimension $8$
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] = [2880,2,Mod(2177,2880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2880, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2880.2177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.w (of order \(4\), 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: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.40960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7x^{4} + 1 \) 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 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + ( - \beta_{3} - \beta_{2} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + ( - \beta_{3} - \beta_{2} + 1) q^{7} + (\beta_{7} + \beta_{6} + \beta_1) q^{11} + (\beta_{5} + 2 \beta_{2} + 2) q^{13} + ( - \beta_{6} - \beta_{4} + 2 \beta_1) q^{17} + (\beta_{5} + \beta_{3} - 2 \beta_{2}) q^{19} + (2 \beta_{6} - 2 \beta_{4}) q^{23} + 5 \beta_{2} q^{25} + (\beta_{7} - 2 \beta_{4} - \beta_1) q^{29} + (\beta_{5} - \beta_{3} - 6) q^{31} + (\beta_{7} - 5 \beta_{4} - \beta_1) q^{35} + \beta_{3} q^{37} + 5 \beta_{6} q^{41} + (2 \beta_{5} - 2 \beta_{2} - 2) q^{43} + ( - \beta_{6} - \beta_{4} + 2 \beta_1) q^{47} + ( - 2 \beta_{5} - 2 \beta_{3} - 5 \beta_{2}) q^{49} + ( - \beta_{3} - 5 \beta_{2} + 5) q^{55} + (\beta_{7} + 3 \beta_{4} - \beta_1) q^{59} + ( - \beta_{5} + \beta_{3} - 6) q^{61} + ( - 2 \beta_{7} - 5 \beta_{6} - 2 \beta_1) q^{65} + ( - 2 \beta_{3} + 6 \beta_{2} - 6) q^{67} + (4 \beta_{7} + 4 \beta_1) q^{71} + (2 \beta_{5} - 5 \beta_{2} - 5) q^{73} + (6 \beta_{6} + 6 \beta_{4} + 4 \beta_1) q^{77} + (\beta_{5} + \beta_{3} + 10 \beta_{2}) q^{79} + (2 \beta_{7} - 3 \beta_{6} + 3 \beta_{4}) q^{83} + (\beta_{5} + \beta_{3} - 10 \beta_{2}) q^{85} + (2 \beta_{7} + 5 \beta_{4} - 2 \beta_1) q^{89} + (3 \beta_{5} - 3 \beta_{3} + 14) q^{91} + (2 \beta_{7} - 5 \beta_{6} + 5 \beta_{4}) q^{95} + ( - \beta_{2} + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{7} + 16 q^{13} - 48 q^{31} - 16 q^{43} + 40 q^{55} - 48 q^{61} - 48 q^{67} - 40 q^{73} + 112 q^{91} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7x^{4} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 11\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 8\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{6} + 2\nu^{4} - 18\nu^{2} + 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{7} - \nu^{5} + 13\nu^{3} - 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} - 2\nu^{4} - 18\nu^{2} - 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{7} - \nu^{5} - 13\nu^{3} - 5\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{7} + 29\nu^{3} ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{4} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + \beta_{3} + 6\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{5} + 3\beta_{3} - 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -11\beta_{6} - 11\beta_{4} - 10\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{5} - 2\beta_{3} - 9\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -26\beta_{7} - 29\beta_{6} + 29\beta_{4} ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2177.1
0.437016 − 0.437016i
1.14412 − 1.14412i
−0.437016 + 0.437016i
−1.14412 + 1.14412i
0.437016 + 0.437016i
1.14412 + 1.14412i
−0.437016 − 0.437016i
−1.14412 − 1.14412i
0 0 0 −1.58114 + 1.58114i 0 −1.23607 − 1.23607i 0 0 0
2177.2 0 0 0 −1.58114 + 1.58114i 0 3.23607 + 3.23607i 0 0 0
2177.3 0 0 0 1.58114 − 1.58114i 0 −1.23607 − 1.23607i 0 0 0
2177.4 0 0 0 1.58114 − 1.58114i 0 3.23607 + 3.23607i 0 0 0
2753.1 0 0 0 −1.58114 − 1.58114i 0 −1.23607 + 1.23607i 0 0 0
2753.2 0 0 0 −1.58114 − 1.58114i 0 3.23607 − 3.23607i 0 0 0
2753.3 0 0 0 1.58114 + 1.58114i 0 −1.23607 + 1.23607i 0 0 0
2753.4 0 0 0 1.58114 + 1.58114i 0 3.23607 − 3.23607i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2177.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2880.2.w.o 8
3.b odd 2 1 inner 2880.2.w.o 8
4.b odd 2 1 2880.2.w.m 8
5.c odd 4 1 inner 2880.2.w.o 8
8.b even 2 1 720.2.w.e 8
8.d odd 2 1 360.2.s.b ✓ 8
12.b even 2 1 2880.2.w.m 8
15.e even 4 1 inner 2880.2.w.o 8
20.e even 4 1 2880.2.w.m 8
24.f even 2 1 360.2.s.b ✓ 8
24.h odd 2 1 720.2.w.e 8
40.e odd 2 1 1800.2.s.e 8
40.f even 2 1 3600.2.w.j 8
40.i odd 4 1 720.2.w.e 8
40.i odd 4 1 3600.2.w.j 8
40.k even 4 1 360.2.s.b ✓ 8
40.k even 4 1 1800.2.s.e 8
60.l odd 4 1 2880.2.w.m 8
120.i odd 2 1 3600.2.w.j 8
120.m even 2 1 1800.2.s.e 8
120.q odd 4 1 360.2.s.b ✓ 8
120.q odd 4 1 1800.2.s.e 8
120.w even 4 1 720.2.w.e 8
120.w even 4 1 3600.2.w.j 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.s.b ✓ 8 8.d odd 2 1
360.2.s.b ✓ 8 24.f even 2 1
360.2.s.b ✓ 8 40.k even 4 1
360.2.s.b ✓ 8 120.q odd 4 1
720.2.w.e 8 8.b even 2 1
720.2.w.e 8 24.h odd 2 1
720.2.w.e 8 40.i odd 4 1
720.2.w.e 8 120.w even 4 1
1800.2.s.e 8 40.e odd 2 1
1800.2.s.e 8 40.k even 4 1
1800.2.s.e 8 120.m even 2 1
1800.2.s.e 8 120.q odd 4 1
2880.2.w.m 8 4.b odd 2 1
2880.2.w.m 8 12.b even 2 1
2880.2.w.m 8 20.e even 4 1
2880.2.w.m 8 60.l odd 4 1
2880.2.w.o 8 1.a even 1 1 trivial
2880.2.w.o 8 3.b odd 2 1 inner
2880.2.w.o 8 5.c odd 4 1 inner
2880.2.w.o 8 15.e even 4 1 inner
3600.2.w.j 8 40.f even 2 1
3600.2.w.j 8 40.i odd 4 1
3600.2.w.j 8 120.i odd 2 1
3600.2.w.j 8 120.w even 4 1

Hecke kernels

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

\( T_{7}^{4} - 4T_{7}^{3} + 8T_{7}^{2} + 32T_{7} + 64 \) Copy content Toggle raw display
\( T_{11}^{4} + 24T_{11}^{2} + 64 \) Copy content Toggle raw display
\( T_{13}^{4} - 8T_{13}^{3} + 32T_{13}^{2} + 16T_{13} + 4 \) Copy content Toggle raw display
\( T_{31}^{2} + 12T_{31} + 16 \) 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
$5$ \( (T^{4} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 64)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 24 T^{2} + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 1792 T^{4} + 65536 \) Copy content Toggle raw display
$19$ \( (T^{4} + 48 T^{2} + 256)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 256)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 36 T^{2} + 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 12 T + 16)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 100)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 50)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 8 T^{3} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 1792 T^{4} + 65536 \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} - 56 T^{2} + 64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 12 T + 16)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 24 T^{3} + \cdots + 1024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 160)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 20 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 240 T^{2} + 6400)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 12032 T^{4} + 65536 \) Copy content Toggle raw display
$89$ \( (T^{4} - 180 T^{2} + 100)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 2 T + 2)^{4} \) Copy content Toggle raw display
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