Properties

Label 1120.2.bz.a
Level $1120$
Weight $2$
Character orbit 1120.bz
Analytic conductor $8.943$
Analytic rank $0$
Dimension $4$
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] = [1120,2,Mod(271,1120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1120, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1120.271");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.bz (of order \(6\), 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: \(8.94324502638\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{2} + 2) q^{3} + (\zeta_{12}^{2} - 1) q^{5} + ( - 3 \zeta_{12}^{3} + \zeta_{12}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{2} + 2) q^{3} + (\zeta_{12}^{2} - 1) q^{5} + ( - 3 \zeta_{12}^{3} + \zeta_{12}) q^{7} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{11} + ( - \zeta_{12}^{3} + 2 \zeta_{12} - 3) q^{13} + (2 \zeta_{12}^{2} - 1) q^{15} + ( - 3 \zeta_{12}^{3} - \zeta_{12}^{2} + 3 \zeta_{12} + 2) q^{17} + (3 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{19} + ( - 4 \zeta_{12}^{3} - \zeta_{12}) q^{21} + (4 \zeta_{12}^{2} + \zeta_{12} + 4) q^{23} - \zeta_{12}^{2} q^{25} + (6 \zeta_{12}^{2} - 3) q^{27} + ( - 7 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 2) q^{29} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{31} + (2 \zeta_{12}^{2} + 6 \zeta_{12} + 2) q^{33} + (\zeta_{12}^{3} + 2 \zeta_{12}) q^{35} + 2 \zeta_{12} q^{37} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12} - 6) q^{39} + ( - 6 \zeta_{12}^{2} + 3) q^{41} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12} + 3) q^{43} + ( - 4 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 2 \zeta_{12} + 6) q^{47} + ( - 5 \zeta_{12}^{2} - 3) q^{49} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{51} + (5 \zeta_{12}^{3} + 3 \zeta_{12}^{2} - 5 \zeta_{12} - 6) q^{53} + (2 \zeta_{12}^{3} - 4 \zeta_{12} - 2) q^{55} + (3 \zeta_{12}^{3} - 6 \zeta_{12} + 9) q^{57} + ( - 3 \zeta_{12}^{3} + \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{59} + (2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - \zeta_{12} - 6) q^{61} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - \zeta_{12} + 3) q^{65} + ( - 6 \zeta_{12}^{3} + \zeta_{12}^{2} - 6 \zeta_{12}) q^{67} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 12) q^{69} + (7 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{71} + ( - 3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 3 \zeta_{12} - 10) q^{73} + ( - \zeta_{12}^{2} - 1) q^{75} + ( - 4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 6 \zeta_{12} + 10) q^{77} + ( - \zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{79} + 9 \zeta_{12}^{2} q^{81} + (6 \zeta_{12}^{3} - 10 \zeta_{12}^{2} + 5) q^{83} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{85} + ( - 7 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 7 \zeta_{12}) q^{87} + (5 \zeta_{12}^{2} - 6 \zeta_{12} + 5) q^{89} + (9 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 3 \zeta_{12} + 4) q^{91} + ( - 3 \zeta_{12}^{2} - 9 \zeta_{12} - 3) q^{93} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12} - 6) q^{95} + (4 \zeta_{12}^{2} - 2) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} - 2 q^{5} + 4 q^{11} - 12 q^{13} + 6 q^{17} + 18 q^{19} + 24 q^{23} - 2 q^{25} - 6 q^{31} + 12 q^{33} - 18 q^{39} + 12 q^{43} + 12 q^{47} - 22 q^{49} + 6 q^{51} - 18 q^{53} - 8 q^{55} + 36 q^{57} - 6 q^{59} - 12 q^{61} + 6 q^{65} + 2 q^{67} + 48 q^{69} - 30 q^{73} - 6 q^{75} + 36 q^{77} - 6 q^{79} + 18 q^{81} + 12 q^{87} + 30 q^{89} + 6 q^{91} - 18 q^{93} - 18 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1120\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(421\) \(801\) \(897\)
\(\chi(n)\) \(-1\) \(-1\) \(1 - \zeta_{12}^{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 1.50000 0.866025i 0 −0.500000 + 0.866025i 0 −0.866025 + 2.50000i 0 0 0
271.2 0 1.50000 0.866025i 0 −0.500000 + 0.866025i 0 0.866025 2.50000i 0 0 0
591.1 0 1.50000 + 0.866025i 0 −0.500000 0.866025i 0 −0.866025 2.50000i 0 0 0
591.2 0 1.50000 + 0.866025i 0 −0.500000 0.866025i 0 0.866025 + 2.50000i 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
56.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.2.bz.a 4
4.b odd 2 1 280.2.bj.a 4
7.d odd 6 1 1120.2.bz.d 4
8.b even 2 1 280.2.bj.d yes 4
8.d odd 2 1 1120.2.bz.d 4
28.f even 6 1 280.2.bj.d yes 4
56.j odd 6 1 280.2.bj.a 4
56.m even 6 1 inner 1120.2.bz.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bj.a 4 4.b odd 2 1
280.2.bj.a 4 56.j odd 6 1
280.2.bj.d yes 4 8.b even 2 1
280.2.bj.d yes 4 28.f even 6 1
1120.2.bz.a 4 1.a even 1 1 trivial
1120.2.bz.a 4 56.m even 6 1 inner
1120.2.bz.d 4 7.d odd 6 1
1120.2.bz.d 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1120, [\chi])\):

\( T_{3}^{2} - 3T_{3} + 3 \) Copy content Toggle raw display
\( T_{13}^{2} + 6T_{13} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 11T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 4 T^{3} + 24 T^{2} + 32 T + 64 \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 6 T^{3} + 6 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$19$ \( T^{4} - 18 T^{3} + 126 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$23$ \( T^{4} - 24 T^{3} + 239 T^{2} + \cdots + 2209 \) Copy content Toggle raw display
$29$ \( T^{4} + 122T^{2} + 1369 \) Copy content Toggle raw display
$31$ \( T^{4} + 6 T^{3} + 54 T^{2} - 108 T + 324 \) Copy content Toggle raw display
$37$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$41$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 6 T - 39)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + 120 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$53$ \( T^{4} + 18 T^{3} + 110 T^{2} + 36 T + 4 \) Copy content Toggle raw display
$59$ \( T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$61$ \( T^{4} + 12 T^{3} + 111 T^{2} + \cdots + 1089 \) Copy content Toggle raw display
$67$ \( T^{4} - 2 T^{3} + 111 T^{2} + \cdots + 11449 \) Copy content Toggle raw display
$71$ \( T^{4} + 104T^{2} + 2116 \) Copy content Toggle raw display
$73$ \( T^{4} + 30 T^{3} + 366 T^{2} + \cdots + 4356 \) Copy content Toggle raw display
$79$ \( T^{4} + 6 T^{3} + 6 T^{2} - 36 T + 36 \) Copy content Toggle raw display
$83$ \( T^{4} + 222T^{2} + 1521 \) Copy content Toggle raw display
$89$ \( T^{4} - 30 T^{3} + 339 T^{2} + \cdots + 1521 \) Copy content Toggle raw display
$97$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
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