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Properties

Label 2880.3.e.h
Level $2880$
Weight $3$
Character orbit 2880.e
Analytic conductor $78.474$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 160)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + \beta_1 q^{7} + (\beta_{3} + 5 \beta_1) q^{11} + ( - 6 \beta_{2} + 8) q^{13} + ( - 4 \beta_{2} + 2) q^{17} + ( - 5 \beta_{3} - 3 \beta_1) q^{19} + (2 \beta_{3} + 7 \beta_1) q^{23} + 5 q^{25} + ( - 4 \beta_{2} + 30) q^{29} + ( - 8 \beta_{3} + 6 \beta_1) q^{31} + (\beta_{3} - 2 \beta_1) q^{35} + (14 \beta_{2} - 12) q^{37} + 26 \beta_{2} q^{41} + (16 \beta_{3} - 17 \beta_1) q^{43} + (10 \beta_{3} + 9 \beta_1) q^{47} + (2 \beta_{2} + 43) q^{49} + ( - 26 \beta_{2} - 40) q^{53} + (7 \beta_{3} - 9 \beta_1) q^{55} + (11 \beta_{3} - 31 \beta_1) q^{59} + ( - 22 \beta_{2} + 48) q^{61} + (8 \beta_{2} - 30) q^{65} + ( - 4 \beta_{3} + 21 \beta_1) q^{67} + (18 \beta_{3} - 8 \beta_1) q^{71} + ( - 20 \beta_{2} - 26) q^{73} + (8 \beta_{2} - 32) q^{77} + (20 \beta_{3} + 16 \beta_1) q^{79} + ( - 38 \beta_{3} + 31 \beta_1) q^{83} + (2 \beta_{2} - 20) q^{85} + (16 \beta_{2} - 82) q^{89} + ( - 6 \beta_{3} + 20 \beta_1) q^{91} + ( - 13 \beta_{3} + \beta_1) q^{95} + (48 \beta_{2} + 26) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{13} + 8 q^{17} + 20 q^{25} + 120 q^{29} - 48 q^{37} + 172 q^{49} - 160 q^{53} + 192 q^{61} - 120 q^{65} - 104 q^{73} - 128 q^{77} - 80 q^{85} - 328 q^{89} + 104 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{3} + 10\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} - 5\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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
− 1.61803i
1.61803i
− 0.618034i
0.618034i
0 0 0 −2.23607 0 − 3.23607i 0 0 0
2431.2 0 0 0 −2.23607 0 3.23607i 0 0 0
2431.3 0 0 0 2.23607 0 − 1.23607i 0 0 0
2431.4 0 0 0 2.23607 0 1.23607i 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.h 4
3.b odd 2 1 320.3.b.d 4
4.b odd 2 1 inner 2880.3.e.h 4
8.b even 2 1 1440.3.e.a 4
8.d odd 2 1 1440.3.e.a 4
12.b even 2 1 320.3.b.d 4
15.d odd 2 1 1600.3.b.u 4
15.e even 4 1 1600.3.h.f 4
15.e even 4 1 1600.3.h.k 4
24.f even 2 1 160.3.b.b ✓ 4
24.h odd 2 1 160.3.b.b ✓ 4
48.i odd 4 1 1280.3.g.b 4
48.i odd 4 1 1280.3.g.c 4
48.k even 4 1 1280.3.g.b 4
48.k even 4 1 1280.3.g.c 4
60.h even 2 1 1600.3.b.u 4
60.l odd 4 1 1600.3.h.f 4
60.l odd 4 1 1600.3.h.k 4
120.i odd 2 1 800.3.b.g 4
120.m even 2 1 800.3.b.g 4
120.q odd 4 1 800.3.h.e 4
120.q odd 4 1 800.3.h.h 4
120.w even 4 1 800.3.h.e 4
120.w even 4 1 800.3.h.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.b.b ✓ 4 24.f even 2 1
160.3.b.b ✓ 4 24.h odd 2 1
320.3.b.d 4 3.b odd 2 1
320.3.b.d 4 12.b even 2 1
800.3.b.g 4 120.i odd 2 1
800.3.b.g 4 120.m even 2 1
800.3.h.e 4 120.q odd 4 1
800.3.h.e 4 120.w even 4 1
800.3.h.h 4 120.q odd 4 1
800.3.h.h 4 120.w even 4 1
1280.3.g.b 4 48.i odd 4 1
1280.3.g.b 4 48.k even 4 1
1280.3.g.c 4 48.i odd 4 1
1280.3.g.c 4 48.k even 4 1
1440.3.e.a 4 8.b even 2 1
1440.3.e.a 4 8.d odd 2 1
1600.3.b.u 4 15.d odd 2 1
1600.3.b.u 4 60.h even 2 1
1600.3.h.f 4 15.e even 4 1
1600.3.h.f 4 60.l odd 4 1
1600.3.h.k 4 15.e even 4 1
1600.3.h.k 4 60.l odd 4 1
2880.3.e.h 4 1.a even 1 1 trivial
2880.3.e.h 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 \(S_{3}^{\mathrm{new}}(2880, [\chi])\):

\( T_{7}^{4} + 12T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{13}^{2} - 16T_{13} - 116 \) Copy content Toggle raw display
\( T_{17}^{2} - 4T_{17} - 76 \) 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
$7$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{4} + 368 T^{2} + 30976 \) Copy content Toggle raw display
$13$ \( (T^{2} - 16 T - 116)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T - 76)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 928 T^{2} + 30976 \) Copy content Toggle raw display
$23$ \( T^{4} + 812 T^{2} + 163216 \) Copy content Toggle raw display
$29$ \( (T^{2} - 60 T + 820)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 1840 T^{2} + 774400 \) Copy content Toggle raw display
$37$ \( (T^{2} + 24 T - 836)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 3380)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 8460 T^{2} + 17808400 \) Copy content Toggle raw display
$47$ \( T^{4} + 4492 T^{2} + 1860496 \) Copy content Toggle raw display
$53$ \( (T^{2} + 80 T - 1780)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 12192 T^{2} + 4393216 \) Copy content Toggle raw display
$61$ \( (T^{2} - 96 T - 116)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 5068 T^{2} + 126736 \) Copy content Toggle raw display
$71$ \( T^{4} + 8688 T^{2} + 11182336 \) Copy content Toggle raw display
$73$ \( (T^{2} + 52 T - 1324)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 16832 T^{2} + 20647936 \) Copy content Toggle raw display
$83$ \( T^{4} + 42540 T^{2} + 431808400 \) Copy content Toggle raw display
$89$ \( (T^{2} + 164 T + 5444)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 52 T - 10844)^{2} \) Copy content Toggle raw display
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