Properties

Label 1120.2.w.b
Level $1120$
Weight $2$
Character orbit 1120.w
Analytic conductor $8.943$
Analytic rank $0$
Dimension $8$
CM discriminant -56
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1120,2,Mod(433,1120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1120, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1120.433");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.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: \(8.94324502638\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.40282095616.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 8x^{4} - 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 280)
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{5} q^{5} + \beta_{7} q^{7} + (\beta_{7} - 5 \beta_{4} - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{5} q^{5} + \beta_{7} q^{7} + (\beta_{7} - 5 \beta_{4} - \beta_{3}) q^{9} + \beta_1 q^{13} + ( - \beta_{7} + \beta_{4} - 2 \beta_{3} + 2) q^{15} + ( - 3 \beta_{6} + \beta_{5} + \beta_1) q^{19} + ( - \beta_{6} - 3 \beta_{5} - \beta_1) q^{21} + ( - 3 \beta_{4} + 2 \beta_{3} + 3) q^{23} + (3 \beta_{4} + \beta_{3} + 3) q^{25} + (2 \beta_{6} - 4 \beta_{5} - 3 \beta_{2} - \beta_1) q^{27} + (\beta_{5} - 2 \beta_{2} + \beta_1) q^{35} + (\beta_{7} - 8 \beta_{4} - \beta_{3}) q^{39} + (4 \beta_{6} + \beta_{5} - \beta_{2} + 3 \beta_1) q^{45} + 7 \beta_{4} q^{49} + ( - 6 \beta_{7} - \beta_{4} - 1) q^{57} + ( - 3 \beta_{6} + \beta_{5} - \beta_{2} + 2 \beta_1) q^{59} + ( - \beta_{6} - 3 \beta_{5} + 3 \beta_{2} + 2 \beta_1) q^{61} + (7 \beta_{4} + 5 \beta_{3} - 7) q^{63} + ( - \beta_{7} + \beta_{4} - 2 \beta_{3} + 2) q^{65} + ( - 6 \beta_{6} + 2 \beta_{5} - \beta_{2} + 3 \beta_1) q^{69} + ( - 3 \beta_{7} - 3 \beta_{3} - 4) q^{71} + ( - 3 \beta_{6} + \beta_{5} + 4 \beta_{2} + 3 \beta_1) q^{75} + ( - \beta_{7} + 12 \beta_{4} + \beta_{3}) q^{79} + (7 \beta_{7} + 7 \beta_{3} - 15) q^{81} + (4 \beta_{6} + 2 \beta_{5} - \beta_{2} + 2 \beta_1) q^{83} + ( - \beta_{6} - 3 \beta_{5} - \beta_1) q^{91} + (2 \beta_{7} + 13 \beta_{4} - \beta_{3} - 4) q^{95}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{15} + 24 q^{23} + 24 q^{25} - 8 q^{57} - 56 q^{63} + 16 q^{65} - 32 q^{71} - 120 q^{81} - 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 8x^{4} - 36x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 23\nu^{5} - 46\nu^{3} + 72\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -4\nu^{7} - 2\nu^{5} - 41\nu^{3} + 297\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{6} + \nu^{4} + 43\nu^{2} - 81 ) / 45 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} - \nu^{4} + 2\nu^{2} + 36 ) / 45 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7\nu^{7} - 19\nu^{5} - 7\nu^{3} - 126\nu ) / 135 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 2\nu^{5} - 9\nu^{3} + 23\nu ) / 15 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{6} + 19\nu^{4} - 38\nu^{2} + 171 ) / 45 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{6} - \beta_{5} + 3\beta_{2} + \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{6} - 9\beta_{5} + 2\beta_{2} - \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} - 5\beta_{4} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -7\beta_{6} - 16\beta_{5} - 7\beta_{2} + 21\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -\beta_{7} - 19\beta_{4} + 19 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -63\beta_{6} + 26\beta_{5} + 37\beta_{2} + 74\beta_1 ) / 5 \) Copy content Toggle raw display

Character values

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

\(n\) \(351\) \(421\) \(801\) \(897\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
−1.03179 + 1.39119i
−1.71331 0.254137i
1.71331 + 0.254137i
1.03179 1.39119i
−1.03179 1.39119i
−1.71331 + 0.254137i
1.71331 0.254137i
1.03179 + 1.39119i
0 −2.42298 2.42298i 0 −1.75060 + 1.39119i 0 −1.87083 + 1.87083i 0 8.74166i 0
433.2 0 −1.45917 1.45917i 0 2.22158 0.254137i 0 1.87083 1.87083i 0 1.25834i 0
433.3 0 1.45917 + 1.45917i 0 −2.22158 + 0.254137i 0 1.87083 1.87083i 0 1.25834i 0
433.4 0 2.42298 + 2.42298i 0 1.75060 1.39119i 0 −1.87083 + 1.87083i 0 8.74166i 0
657.1 0 −2.42298 + 2.42298i 0 −1.75060 1.39119i 0 −1.87083 1.87083i 0 8.74166i 0
657.2 0 −1.45917 + 1.45917i 0 2.22158 + 0.254137i 0 1.87083 + 1.87083i 0 1.25834i 0
657.3 0 1.45917 1.45917i 0 −2.22158 0.254137i 0 1.87083 + 1.87083i 0 1.25834i 0
657.4 0 2.42298 2.42298i 0 1.75060 + 1.39119i 0 −1.87083 1.87083i 0 8.74166i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 433.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
5.c odd 4 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
35.f even 4 1 inner
40.i odd 4 1 inner
280.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.2.w.b 8
4.b odd 2 1 280.2.s.b 8
5.c odd 4 1 inner 1120.2.w.b 8
7.b odd 2 1 inner 1120.2.w.b 8
8.b even 2 1 inner 1120.2.w.b 8
8.d odd 2 1 280.2.s.b 8
20.e even 4 1 280.2.s.b 8
28.d even 2 1 280.2.s.b 8
35.f even 4 1 inner 1120.2.w.b 8
40.i odd 4 1 inner 1120.2.w.b 8
40.k even 4 1 280.2.s.b 8
56.e even 2 1 280.2.s.b 8
56.h odd 2 1 CM 1120.2.w.b 8
140.j odd 4 1 280.2.s.b 8
280.s even 4 1 inner 1120.2.w.b 8
280.y odd 4 1 280.2.s.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.s.b 8 4.b odd 2 1
280.2.s.b 8 8.d odd 2 1
280.2.s.b 8 20.e even 4 1
280.2.s.b 8 28.d even 2 1
280.2.s.b 8 40.k even 4 1
280.2.s.b 8 56.e even 2 1
280.2.s.b 8 140.j odd 4 1
280.2.s.b 8 280.y odd 4 1
1120.2.w.b 8 1.a even 1 1 trivial
1120.2.w.b 8 5.c odd 4 1 inner
1120.2.w.b 8 7.b odd 2 1 inner
1120.2.w.b 8 8.b even 2 1 inner
1120.2.w.b 8 35.f even 4 1 inner
1120.2.w.b 8 40.i odd 4 1 inner
1120.2.w.b 8 56.h odd 2 1 CM
1120.2.w.b 8 280.s even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 156T_{3}^{4} + 2500 \) acting on \(S_{2}^{\mathrm{new}}(1120, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 156T^{4} + 2500 \) Copy content Toggle raw display
$5$ \( T^{8} - 12 T^{6} + 72 T^{4} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 49)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 156T^{4} + 2500 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 88 T^{2} + 1250)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 12 T^{3} + 72 T^{2} + 120 T + 100)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 128 T^{2} + 50)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 232 T^{2} + 6050)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8 T - 110)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} + 316 T^{2} + 16900)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 70716 T^{4} + \cdots + 71402500 \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
show more
show less