↑

Properties

Label 1120.2.b.d
Level $1120$
Weight $2$
Character orbit 1120.b
Analytic conductor $8.943$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1120,2,Mod(561,1120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1120, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1120.561");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1120 = 2^{5} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1120.b (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: \(8.94324502638\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.8272021826830336.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 3x^{10} - 4x^{9} + 4x^{8} - 12x^{7} + 10x^{6} - 24x^{5} + 16x^{4} - 32x^{3} + 48x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 280)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - \beta_{3} q^{5} - q^{7} + (\beta_{10} - \beta_{4} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - \beta_{3} q^{5} - q^{7} + (\beta_{10} - \beta_{4} - 2) q^{9} + ( - \beta_{11} + \beta_{8} + \beta_{3}) q^{11} + ( - \beta_{11} + \beta_{3} + \beta_{2}) q^{13} + \beta_{7} q^{15} + (\beta_{7} - \beta_{4} - \beta_1) q^{17} + (\beta_{8} + \beta_{2}) q^{19} + \beta_{2} q^{21} + ( - \beta_{6} - \beta_{4} + \beta_1 - 1) q^{23} - q^{25} + (2 \beta_{11} - \beta_{9} + \cdots + 2 \beta_{2}) q^{27}+ \cdots + (4 \beta_{11} - \beta_{9} + \cdots + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{7} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{7} - 20 q^{9} - 8 q^{23} - 12 q^{25} - 24 q^{31} - 24 q^{33} + 48 q^{39} - 16 q^{41} + 16 q^{47} + 12 q^{49} + 8 q^{55} + 40 q^{57} + 20 q^{63} + 8 q^{65} + 32 q^{71} - 8 q^{73} + 60 q^{81} - 48 q^{87} - 48 q^{89} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 3x^{10} - 4x^{9} + 4x^{8} - 12x^{7} + 10x^{6} - 24x^{5} + 16x^{4} - 32x^{3} + 48x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{10} + 3\nu^{8} - 4\nu^{7} + 4\nu^{6} - 12\nu^{5} + 10\nu^{4} - 24\nu^{3} - 32\nu + 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2 \nu^{11} + 29 \nu^{10} - 10 \nu^{9} + 51 \nu^{8} - 144 \nu^{7} + 72 \nu^{6} - 248 \nu^{5} + \cdots + 1248 ) / 272 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 29 \nu^{11} + 38 \nu^{10} - 43 \nu^{9} + 102 \nu^{8} - 184 \nu^{7} + 228 \nu^{6} - 298 \nu^{5} + \cdots + 960 ) / 544 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} + \nu^{10} - 3 \nu^{9} + 7 \nu^{8} - 8 \nu^{7} + 16 \nu^{6} - 22 \nu^{5} + 34 \nu^{4} + \cdots + 32 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 30 \nu^{11} - 61 \nu^{10} + 82 \nu^{9} - 187 \nu^{8} + 324 \nu^{7} - 400 \nu^{6} + 456 \nu^{5} + \cdots - 1312 ) / 272 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 3 \nu^{10} - \nu^{9} + \nu^{8} - 8 \nu^{7} + 4 \nu^{6} - 14 \nu^{5} + 6 \nu^{4} - 40 \nu^{3} + \cdots + 88 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{11} + \nu^{10} + 2\nu^{9} - 5\nu^{8} - 4\nu^{6} - 22\nu^{4} - 16\nu^{3} + 48\nu + 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 38 \nu^{11} + 75 \nu^{10} - 54 \nu^{9} + 221 \nu^{8} - 288 \nu^{7} + 416 \nu^{6} - 496 \nu^{5} + \cdots + 1408 ) / 272 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5 \nu^{11} + 10 \nu^{10} - 11 \nu^{9} + 34 \nu^{8} - 48 \nu^{7} + 68 \nu^{6} - 90 \nu^{5} + \cdots + 256 ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3\nu^{11} + 4\nu^{10} + \nu^{9} - 4\nu^{8} - 12\nu^{7} - 18\nu^{5} - 48\nu^{3} + 40\nu^{2} + 48\nu + 176 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 113 \nu^{11} + 134 \nu^{10} - 191 \nu^{9} + 510 \nu^{8} - 656 \nu^{7} + 940 \nu^{6} - 1330 \nu^{5} + \cdots + 2240 ) / 544 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{8} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + \beta_{5} - \beta_{3} + \beta_{2} - 2\beta _1 - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{11} + 2\beta_{10} - \beta_{7} - \beta_{6} - \beta_{4} - 4\beta_{3} - 2\beta_{2} + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{11} + 2\beta_{10} - \beta_{9} + 2\beta_{8} - 4\beta_{7} + \beta_{5} + 2\beta_{4} + 3\beta_{3} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2 \beta_{11} + 2 \beta_{10} + \beta_{9} + 2 \beta_{8} + 8 \beta_{7} - 4 \beta_{6} - \beta_{5} + \cdots - 4 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2 \beta_{11} - 2 \beta_{10} + 4 \beta_{8} - \beta_{7} + 5 \beta_{6} + 10 \beta_{5} + 5 \beta_{4} + \cdots + 21 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 6 \beta_{11} - 6 \beta_{10} - \beta_{9} + 10 \beta_{8} - 8 \beta_{7} + 4 \beta_{6} - 3 \beta_{5} + \cdots + 20 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2 \beta_{11} + 10 \beta_{10} + 20 \beta_{8} + 9 \beta_{7} - 17 \beta_{6} + 10 \beta_{5} - \beta_{4} + \cdots - 41 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 14 \beta_{11} + 6 \beta_{10} + \beta_{9} + 14 \beta_{8} - 8 \beta_{7} + 12 \beta_{6} + 19 \beta_{5} + \cdots - 20 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 22 \beta_{11} + 14 \beta_{10} + 12 \beta_{9} - 29 \beta_{7} + 5 \beta_{6} - 18 \beta_{5} + 9 \beta_{4} + \cdots - 3 ) / 4 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(351\) \(421\) \(801\) \(897\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
561.1
0.832593 + 1.14315i
−0.128739 + 1.40834i
1.39608 − 0.225774i
−0.722588 − 1.21568i
−1.11909 − 0.864661i
−0.258252 + 1.39043i
−0.258252 − 1.39043i
−1.11909 + 0.864661i
−0.722588 + 1.21568i
1.39608 + 0.225774i
−0.128739 − 1.40834i
0.832593 − 1.14315i
0 − 3.42822i 0 1.00000i 0 −1.00000 0 −8.75270 0
561.2 0 − 2.83397i 0 − 1.00000i 0 −1.00000 0 −5.03141 0
561.3 0 − 2.07981i 0 − 1.00000i 0 −1.00000 0 −1.32561 0
561.4 0 − 1.52755i 0 1.00000i 0 −1.00000 0 0.666582 0
561.5 0 − 0.903031i 0 − 1.00000i 0 −1.00000 0 2.18454 0
561.6 0 − 0.861041i 0 1.00000i 0 −1.00000 0 2.25861 0
561.7 0 0.861041i 0 − 1.00000i 0 −1.00000 0 2.25861 0
561.8 0 0.903031i 0 1.00000i 0 −1.00000 0 2.18454 0
561.9 0 1.52755i 0 − 1.00000i 0 −1.00000 0 0.666582 0
561.10 0 2.07981i 0 1.00000i 0 −1.00000 0 −1.32561 0
561.11 0 2.83397i 0 1.00000i 0 −1.00000 0 −5.03141 0
561.12 0 3.42822i 0 − 1.00000i 0 −1.00000 0 −8.75270 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 561.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.2.b.d 12
4.b odd 2 1 280.2.b.d ✓ 12
8.b even 2 1 inner 1120.2.b.d 12
8.d odd 2 1 280.2.b.d ✓ 12
16.e even 4 1 8960.2.a.cd 6
16.e even 4 1 8960.2.a.cg 6
16.f odd 4 1 8960.2.a.ca 6
16.f odd 4 1 8960.2.a.cf 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.b.d ✓ 12 4.b odd 2 1
280.2.b.d ✓ 12 8.d odd 2 1
1120.2.b.d 12 1.a even 1 1 trivial
1120.2.b.d 12 8.b even 2 1 inner
8960.2.a.ca 6 16.f odd 4 1
8960.2.a.cd 6 16.e even 4 1
8960.2.a.cf 6 16.f odd 4 1
8960.2.a.cg 6 16.e even 4 1

Hecke kernels

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

\( T_{3}^{12} + 28T_{3}^{10} + 278T_{3}^{8} + 1212T_{3}^{6} + 2385T_{3}^{4} + 1984T_{3}^{2} + 576 \) Copy content Toggle raw display
\( T_{13}^{12} + 76T_{13}^{10} + 1958T_{13}^{8} + 19860T_{13}^{6} + 65713T_{13}^{4} + 77896T_{13}^{2} + 26896 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 28 T^{10} + \cdots + 576 \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$7$ \( (T + 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 76 T^{10} + \cdots + 138384 \) Copy content Toggle raw display
$13$ \( T^{12} + 76 T^{10} + \cdots + 26896 \) Copy content Toggle raw display
$17$ \( (T^{6} - 74 T^{4} + \cdots - 396)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 56 T^{10} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( (T^{6} + 4 T^{5} + \cdots - 768)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 124 T^{10} + \cdots + 123904 \) Copy content Toggle raw display
$31$ \( (T^{6} + 12 T^{5} + \cdots + 1152)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 435306496 \) Copy content Toggle raw display
$41$ \( (T^{6} + 8 T^{5} + \cdots + 7200)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 416 T^{10} + \cdots + 6718464 \) Copy content Toggle raw display
$47$ \( (T^{6} - 8 T^{5} + \cdots - 224128)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 6379536384 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 17314349056 \) Copy content Toggle raw display
$61$ \( T^{12} + 208 T^{10} + \cdots + 70829056 \) Copy content Toggle raw display
$67$ \( T^{12} + 504 T^{10} + \cdots + 18939904 \) Copy content Toggle raw display
$71$ \( (T^{6} - 16 T^{5} + \cdots + 8704)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 4 T^{5} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} - 214 T^{4} + \cdots + 90784)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 240 T^{10} + \cdots + 46022656 \) Copy content Toggle raw display
$89$ \( (T^{6} + 24 T^{5} + \cdots - 51808)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 16 T^{5} + \cdots + 38068)^{2} \) Copy content Toggle raw display
show more
show less