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Properties

Label 1440.2.q.a
Level $1440$
Weight $2$
Character orbit 1440.q
Analytic conductor $11.498$
Analytic rank $0$
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] = [1440,2,Mod(481,1440)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1440, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1440.481");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1440.q (of order \(3\), degree \(2\), 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: \(11.4984578911\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 2) q^{3} - \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 3) q^{7} + ( - 3 \zeta_{6} + 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{6} - 2) q^{3} - \zeta_{6} q^{5} + ( - 3 \zeta_{6} + 3) q^{7} + ( - 3 \zeta_{6} + 3) q^{9} + (4 \zeta_{6} - 4) q^{11} + 4 \zeta_{6} q^{13} + (\zeta_{6} + 1) q^{15} + 2 q^{17} - 2 q^{19} + (6 \zeta_{6} - 3) q^{21} - 7 \zeta_{6} q^{23} + (\zeta_{6} - 1) q^{25} + (6 \zeta_{6} - 3) q^{27} + ( - 9 \zeta_{6} + 9) q^{29} + 6 \zeta_{6} q^{31} + ( - 8 \zeta_{6} + 4) q^{33} - 3 q^{35} + 2 q^{37} + ( - 4 \zeta_{6} - 4) q^{39} - 9 \zeta_{6} q^{41} + ( - 4 \zeta_{6} + 4) q^{43} - 3 q^{45} + (3 \zeta_{6} - 3) q^{47} - 2 \zeta_{6} q^{49} + (2 \zeta_{6} - 4) q^{51} + 6 q^{53} + 4 q^{55} + ( - 2 \zeta_{6} + 4) q^{57} - 6 \zeta_{6} q^{59} + ( - 13 \zeta_{6} + 13) q^{61} - 9 \zeta_{6} q^{63} + ( - 4 \zeta_{6} + 4) q^{65} + 3 \zeta_{6} q^{67} + (7 \zeta_{6} + 7) q^{69} + 8 q^{73} + ( - 2 \zeta_{6} + 1) q^{75} + 12 \zeta_{6} q^{77} + ( - 10 \zeta_{6} + 10) q^{79} - 9 \zeta_{6} q^{81} + ( - 7 \zeta_{6} + 7) q^{83} - 2 \zeta_{6} q^{85} + (18 \zeta_{6} - 9) q^{87} + q^{89} + 12 q^{91} + ( - 6 \zeta_{6} - 6) q^{93} + 2 \zeta_{6} q^{95} + ( - 14 \zeta_{6} + 14) q^{97} + 12 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9} - 4 q^{11} + 4 q^{13} + 3 q^{15} + 4 q^{17} - 4 q^{19} - 7 q^{23} - q^{25} + 9 q^{29} + 6 q^{31} - 6 q^{35} + 4 q^{37} - 12 q^{39} - 9 q^{41} + 4 q^{43} - 6 q^{45} - 3 q^{47} - 2 q^{49} - 6 q^{51} + 12 q^{53} + 8 q^{55} + 6 q^{57} - 6 q^{59} + 13 q^{61} - 9 q^{63} + 4 q^{65} + 3 q^{67} + 21 q^{69} + 16 q^{73} + 12 q^{77} + 10 q^{79} - 9 q^{81} + 7 q^{83} - 2 q^{85} + 2 q^{89} + 24 q^{91} - 18 q^{93} + 2 q^{95} + 14 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1440\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(991\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
481.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 −1.50000 − 0.866025i 0 −0.500000 + 0.866025i 0 1.50000 + 2.59808i 0 1.50000 + 2.59808i 0
961.1 0 −1.50000 + 0.866025i 0 −0.500000 − 0.866025i 0 1.50000 − 2.59808i 0 1.50000 − 2.59808i 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
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.q.a ✓ 2
3.b odd 2 1 4320.2.q.h 2
4.b odd 2 1 1440.2.q.e yes 2
9.c even 3 1 inner 1440.2.q.a ✓ 2
9.d odd 6 1 4320.2.q.h 2
12.b even 2 1 4320.2.q.g 2
36.f odd 6 1 1440.2.q.e yes 2
36.h even 6 1 4320.2.q.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.q.a ✓ 2 1.a even 1 1 trivial
1440.2.q.a ✓ 2 9.c even 3 1 inner
1440.2.q.e yes 2 4.b odd 2 1
1440.2.q.e yes 2 36.f odd 6 1
4320.2.q.g 2 12.b even 2 1
4320.2.q.g 2 36.h even 6 1
4320.2.q.h 2 3.b odd 2 1
4320.2.q.h 2 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1440, [\chi])\):

\( T_{7}^{2} - 3T_{7} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 3 \) Copy content Toggle raw display
$5$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$11$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$29$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$31$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 8)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$83$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$89$ \( (T - 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
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