↑

Properties

Label 1170.2.f.f
Level $1170$
Weight $2$
Character orbit 1170.f
Analytic conductor $9.342$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1170,2,Mod(649,1170)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1170, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1170.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1170.f (of order \(2\), degree \(1\), 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: \(9.34249703649\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.41108373504.15
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 4x^{6} + 20x^{5} + 20x^{4} - 28x^{3} + 4x^{2} + 12x + 33 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} - \beta_{5} q^{5} - \beta_{4} q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} - \beta_{5} q^{5} - \beta_{4} q^{7} + q^{8} - \beta_{5} q^{10} + \beta_1 q^{11} + ( - \beta_{5} + \beta_{3} + \cdots + \beta_1) q^{13}+ \cdots + ( - \beta_{5} - \beta_{2} + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} + 8 q^{16} + 8 q^{25} + 8 q^{32} + 24 q^{49} + 8 q^{50} + 16 q^{55} + 16 q^{61} + 8 q^{64} - 24 q^{65} + 16 q^{79} - 48 q^{83} + 16 q^{91} + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 4x^{6} + 20x^{5} + 20x^{4} - 28x^{3} + 4x^{2} + 12x + 33 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -278\nu^{7} + 58\nu^{6} + 3838\nu^{5} + 6196\nu^{4} - 28046\nu^{3} - 33872\nu^{2} + 1266\nu + 10428 ) / 20097 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 398\nu^{7} - 2059\nu^{6} + 2120\nu^{5} + 3419\nu^{4} - 8524\nu^{3} + 3335\nu^{2} + 44454\nu - 5676 ) / 20097 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 112\nu^{7} - 464\nu^{6} - 245\nu^{5} + 1318\nu^{4} + 2872\nu^{3} + 1102\nu^{2} - 2307\nu - 5709 ) / 2871 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -970\nu^{7} + 3335\nu^{6} + 6548\nu^{5} - 19105\nu^{4} - 33760\nu^{3} + 28565\nu^{2} + 45768\nu - 31713 ) / 20097 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -970\nu^{7} + 3335\nu^{6} + 6548\nu^{5} - 19105\nu^{4} - 33760\nu^{3} + 28565\nu^{2} + 5574\nu - 11616 ) / 20097 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 92\nu^{7} - 580\nu^{6} + 386\nu^{5} + 2804\nu^{4} - 1444\nu^{3} - 8410\nu^{2} + 4602\nu + 1938 ) / 1827 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 44\nu^{7} - 145\nu^{6} - 292\nu^{5} + 785\nu^{4} + 1004\nu^{3} + 29\nu^{2} + 930\nu + 159 ) / 609 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} - 2\beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} - 3\beta_{6} - 11\beta_{5} + 6\beta_{4} + 4\beta_{3} + 2\beta_{2} + 5\beta _1 + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -8\beta_{7} - 14\beta_{6} - 35\beta_{5} + 16\beta_{4} + 12\beta_{3} + 13\beta_{2} + 26\beta _1 + 36 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -40\beta_{7} - 50\beta_{6} - 119\beta_{5} + 37\beta_{4} + 30\beta_{3} + 70\beta_{2} + 80\beta _1 + 91 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -130\beta_{7} - 183\beta_{6} - 383\beta_{5} + 72\beta_{4} + 46\beta_{3} + 281\beta_{2} + 272\beta _1 + 150 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -455\beta_{7} - 616\beta_{6} - 1154\beta_{5} + 38\beta_{4} + 44\beta_{3} + 1113\beta_{2} + 848\beta _1 + 127 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1170\mathbb{Z}\right)^\times\).

\(n\) \(911\) \(937\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
0.912245 − 0.707107i
−1.64430 − 0.707107i
0.912245 + 0.707107i
−1.64430 + 0.707107i
3.20070 − 0.707107i
−0.468648 − 0.707107i
3.20070 + 0.707107i
−0.468648 + 0.707107i
1.00000 0 1.00000 −1.73205 − 1.41421i 0 −2.55654 1.00000 0 −1.73205 − 1.41421i
649.2 1.00000 0 1.00000 −1.73205 − 1.41421i 0 2.55654 1.00000 0 −1.73205 − 1.41421i
649.3 1.00000 0 1.00000 −1.73205 + 1.41421i 0 −2.55654 1.00000 0 −1.73205 + 1.41421i
649.4 1.00000 0 1.00000 −1.73205 + 1.41421i 0 2.55654 1.00000 0 −1.73205 + 1.41421i
649.5 1.00000 0 1.00000 1.73205 − 1.41421i 0 −3.66935 1.00000 0 1.73205 − 1.41421i
649.6 1.00000 0 1.00000 1.73205 − 1.41421i 0 3.66935 1.00000 0 1.73205 − 1.41421i
649.7 1.00000 0 1.00000 1.73205 + 1.41421i 0 −3.66935 1.00000 0 1.73205 + 1.41421i
649.8 1.00000 0 1.00000 1.73205 + 1.41421i 0 3.66935 1.00000 0 1.73205 + 1.41421i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
39.d odd 2 1 inner
65.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1170.2.f.f yes 8
3.b odd 2 1 1170.2.f.e ✓ 8
5.b even 2 1 1170.2.f.e ✓ 8
13.b even 2 1 1170.2.f.e ✓ 8
15.d odd 2 1 inner 1170.2.f.f yes 8
39.d odd 2 1 inner 1170.2.f.f yes 8
65.d even 2 1 inner 1170.2.f.f yes 8
195.e odd 2 1 1170.2.f.e ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1170.2.f.e ✓ 8 3.b odd 2 1
1170.2.f.e ✓ 8 5.b even 2 1
1170.2.f.e ✓ 8 13.b even 2 1
1170.2.f.e ✓ 8 195.e odd 2 1
1170.2.f.f yes 8 1.a even 1 1 trivial
1170.2.f.f yes 8 15.d odd 2 1 inner
1170.2.f.f yes 8 39.d odd 2 1 inner
1170.2.f.f yes 8 65.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1170, [\chi])\):

\( T_{7}^{4} - 20T_{7}^{2} + 88 \) Copy content Toggle raw display
\( T_{83}^{2} + 12T_{83} + 24 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{2} + 25)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 20 T^{2} + 88)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 16 T^{2} + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - 4 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( (T^{4} + 52 T^{2} + 88)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 84 T^{2} + 792)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 28 T^{2} + 88)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 60 T^{2} + 792)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 120 T^{2} + 3168)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 56 T^{2} + 352)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 16 T^{2} + 16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 40 T^{2} + 352)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 112 T^{2} + 2704)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 44)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 56 T^{2} + 352)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 208 T^{2} + 7744)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 308 T^{2} + 10648)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 8)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 12 T + 24)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 304 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 212 T^{2} + 10648)^{2} \) Copy content Toggle raw display
show more
show less