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Properties

Label 1120.2.q.h
Level $1120$
Weight $2$
Character orbit 1120.q
Analytic conductor $8.943$
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] = [1120,2,Mod(641,1120)]
 
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([0, 0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1120.641");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.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: \(8.94324502638\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1445900625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 77x^{4} + 36x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} - \beta_{3} q^{5} + ( - \beta_{7} + \beta_{5}) q^{7} + (\beta_{6} + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} - \beta_{3} q^{5} + ( - \beta_{7} + \beta_{5}) q^{7} + (\beta_{6} + \beta_{3}) q^{9} + (\beta_{5} + \beta_1) q^{11} + ( - \beta_{2} + 2) q^{13} + (\beta_{7} - \beta_{4}) q^{15} + (2 \beta_{3} + 2) q^{17} + (2 \beta_{4} - \beta_1) q^{19} + ( - \beta_{6} - 2 \beta_{3} + 2) q^{21} + (3 \beta_{4} - 3 \beta_1) q^{23} + ( - \beta_{3} - 1) q^{25} + ( - 3 \beta_{7} + 2 \beta_{5} + 3 \beta_{4}) q^{27} + ( - \beta_{2} + 3) q^{29} + 2 \beta_{7} q^{31} + 2 \beta_{3} q^{33} + ( - \beta_{7} + \beta_{4} - \beta_1) q^{35} + (\beta_{6} + 3 \beta_{3}) q^{37} + (6 \beta_{7} - 2 \beta_{5} - 2 \beta_1) q^{39} + ( - 2 \beta_{2} + 1) q^{41} + (3 \beta_{7} - 3 \beta_{4}) q^{43} + (\beta_{6} + \beta_{3} - \beta_{2} + 2) q^{45} + ( - 2 \beta_{4} - \beta_1) q^{47} + (\beta_{6} + \beta_{2}) q^{49} + 2 \beta_{4} q^{51} + ( - \beta_{6} - 7 \beta_{3} + \beta_{2} - 8) q^{53} + \beta_{5} q^{55} + (2 \beta_{2} - 8) q^{57} + ( - 4 \beta_{7} + 4 \beta_{5} + 4 \beta_1) q^{59} + ( - \beta_{6} - 4 \beta_{3}) q^{61} + (6 \beta_{7} - 2 \beta_{5} + \cdots - 3 \beta_1) q^{63}+ \cdots + ( - 2 \beta_{7} + 3 \beta_{5} + 2 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{5} - 6 q^{9} + 12 q^{13} + 8 q^{17} + 26 q^{21} - 4 q^{25} + 20 q^{29} - 8 q^{33} - 14 q^{37} + 6 q^{45} + 2 q^{49} - 30 q^{53} - 56 q^{57} + 18 q^{61} + 6 q^{65} - 60 q^{69} + 12 q^{73} + 26 q^{77} - 20 q^{81} + 16 q^{85} - 22 q^{89} - 36 q^{93} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 9x^{6} + 77x^{4} + 36x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 272 ) / 77 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9\nu^{6} - 77\nu^{4} - 693\nu^{2} - 324 ) / 308 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 657\nu ) / 154 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 9\nu^{7} + 77\nu^{5} + 693\nu^{3} + 16\nu ) / 308 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -41\nu^{6} - 385\nu^{4} - 3157\nu^{2} - 1476 ) / 308 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 9\nu^{5} + 77\nu^{3} + 36\nu ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - 5\beta_{3} - \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} + 9\beta_{5} + 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{6} + 41\beta_{3} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18\beta_{7} - 77\beta_{5} - 77\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 77\beta_{2} + 272 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -154\beta_{4} + 657\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(351\) \(421\) \(801\) \(897\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
0.342371 + 0.593004i
1.46040 + 2.52950i
−1.46040 − 2.52950i
−0.342371 − 0.593004i
0.342371 − 0.593004i
1.46040 − 2.52950i
−1.46040 + 2.52950i
−0.342371 + 0.593004i
0 −1.46040 + 2.52950i 0 0.500000 + 0.866025i 0 0.775663 − 2.52950i 0 −2.76556 − 4.79010i 0
641.2 0 −0.342371 + 0.593004i 0 0.500000 + 0.866025i 0 −2.57844 − 0.593004i 0 1.26556 + 2.19202i 0
641.3 0 0.342371 − 0.593004i 0 0.500000 + 0.866025i 0 2.57844 + 0.593004i 0 1.26556 + 2.19202i 0
641.4 0 1.46040 − 2.52950i 0 0.500000 + 0.866025i 0 −0.775663 + 2.52950i 0 −2.76556 − 4.79010i 0
961.1 0 −1.46040 − 2.52950i 0 0.500000 − 0.866025i 0 0.775663 + 2.52950i 0 −2.76556 + 4.79010i 0
961.2 0 −0.342371 − 0.593004i 0 0.500000 − 0.866025i 0 −2.57844 + 0.593004i 0 1.26556 − 2.19202i 0
961.3 0 0.342371 + 0.593004i 0 0.500000 − 0.866025i 0 2.57844 − 0.593004i 0 1.26556 − 2.19202i 0
961.4 0 1.46040 + 2.52950i 0 0.500000 − 0.866025i 0 −0.775663 − 2.52950i 0 −2.76556 + 4.79010i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 641.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.c even 3 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.2.q.h ✓ 8
4.b odd 2 1 inner 1120.2.q.h ✓ 8
7.c even 3 1 inner 1120.2.q.h ✓ 8
7.c even 3 1 7840.2.a.bs 4
7.d odd 6 1 7840.2.a.bv 4
28.f even 6 1 7840.2.a.bv 4
28.g odd 6 1 inner 1120.2.q.h ✓ 8
28.g odd 6 1 7840.2.a.bs 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.q.h ✓ 8 1.a even 1 1 trivial
1120.2.q.h ✓ 8 4.b odd 2 1 inner
1120.2.q.h ✓ 8 7.c even 3 1 inner
1120.2.q.h ✓ 8 28.g odd 6 1 inner
7840.2.a.bs 4 7.c even 3 1
7840.2.a.bs 4 28.g odd 6 1
7840.2.a.bv 4 7.d odd 6 1
7840.2.a.bv 4 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 9T_{3}^{6} + 77T_{3}^{4} + 36T_{3}^{2} + 16 \) 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
$3$ \( T^{8} + 9 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{8} + 9 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} - 3 T - 14)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} + 29 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( (T^{4} + 45 T^{2} + 2025)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 5 T - 10)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} + 36 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$37$ \( (T^{4} + 7 T^{3} + 53 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 65)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 81 T^{2} + 324)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 61 T^{6} + \cdots + 614656 \) Copy content Toggle raw display
$53$ \( (T^{4} + 15 T^{3} + \cdots + 1600)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 80 T^{2} + 6400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 9 T^{3} + 77 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 61 T^{6} + \cdots + 614656 \) Copy content Toggle raw display
$71$ \( (T^{2} - 80)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} - 6 T^{3} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + 36 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$83$ \( (T^{4} - 69 T^{2} + 784)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 11 T^{3} + \cdots + 196)^{2} \) Copy content Toggle raw display
$97$ \( (T + 2)^{8} \) Copy content Toggle raw display
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