↑

Properties

Label 1080.2.d.h
Level $1080$
Weight $2$
Character orbit 1080.d
Analytic conductor $8.624$
Analytic rank $0$
Dimension $16$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1080,2,Mod(109,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1080.d (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: \(8.62384341830\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} - 3 x^{14} + 36 x^{13} - 78 x^{12} - 96 x^{11} + 1194 x^{10} + 1456 x^{9} + \cdots + 45658 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} + (\beta_{5} + \beta_{3} - 1) q^{4} + \beta_{10} q^{5} - \beta_{15} q^{7} + (\beta_{14} - \beta_{12} - \beta_{2}) q^{8} + ( - \beta_{6} + \beta_{3}) q^{10} + ( - \beta_{13} - \beta_{9} + \cdots - \beta_{4}) q^{11}+ \cdots + (\beta_{14} - 2 \beta_{12} - 2 \beta_{10} + \cdots + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} - 6 q^{4} + 6 q^{5} + 2 q^{8} + 5 q^{10} - 30 q^{16} + q^{20} - 22 q^{25} - 18 q^{32} - 4 q^{34} + 2 q^{35} - 56 q^{38} + 19 q^{40} + 40 q^{46} - 44 q^{49} + 27 q^{50} - 96 q^{53} + 34 q^{55}+ \cdots + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} - 3 x^{14} + 36 x^{13} - 78 x^{12} - 96 x^{11} + 1194 x^{10} + 1456 x^{9} + \cdots + 45658 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 29\!\cdots\!50 \nu^{15} + \cdots + 26\!\cdots\!36 ) / 53\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 91\!\cdots\!99 \nu^{15} + \cdots + 52\!\cdots\!72 ) / 29\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 46\!\cdots\!13 \nu^{15} + \cdots - 13\!\cdots\!44 ) / 12\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 87\!\cdots\!94 \nu^{15} + \cdots - 16\!\cdots\!31 ) / 24\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 53\!\cdots\!17 \nu^{15} + \cdots - 38\!\cdots\!72 ) / 96\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\!\cdots\!42 \nu^{15} + \cdots + 65\!\cdots\!10 ) / 29\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 98\!\cdots\!73 \nu^{15} + \cdots - 71\!\cdots\!88 ) / 14\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!99 \nu^{15} + \cdots - 32\!\cdots\!52 ) / 14\!\cdots\!14 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 68\!\cdots\!85 \nu^{15} + \cdots + 75\!\cdots\!34 ) / 71\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 70\!\cdots\!47 \nu^{15} + \cdots - 13\!\cdots\!70 ) / 71\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 61\!\cdots\!37 \nu^{15} + \cdots - 20\!\cdots\!86 ) / 58\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 88\!\cdots\!61 \nu^{15} + \cdots - 31\!\cdots\!10 ) / 72\!\cdots\!07 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 72\!\cdots\!65 \nu^{15} + \cdots - 66\!\cdots\!86 ) / 58\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 47\!\cdots\!25 \nu^{15} + \cdots - 15\!\cdots\!04 ) / 29\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 26\!\cdots\!78 \nu^{15} + \cdots + 69\!\cdots\!74 ) / 11\!\cdots\!96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} + \beta_{11} - \beta_{5} - \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} + \beta_{11} - \beta_{8} - \beta_{7} - \beta_{6} + \cdots + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8 \beta_{15} - 5 \beta_{14} - \beta_{13} + 9 \beta_{12} + 4 \beta_{11} - 3 \beta_{10} - 4 \beta_{9} + \cdots - 8 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 30 \beta_{15} - 39 \beta_{14} + 20 \beta_{13} + 39 \beta_{12} + 8 \beta_{11} + 7 \beta_{10} + \cdots + 6 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 91 \beta_{15} - 166 \beta_{14} + 67 \beta_{13} + 134 \beta_{12} + 58 \beta_{11} + 6 \beta_{10} + \cdots + 24 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 256 \beta_{15} - 529 \beta_{14} + 262 \beta_{13} + 551 \beta_{12} + 142 \beta_{11} - 33 \beta_{10} + \cdots - 132 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 892 \beta_{15} - 1132 \beta_{14} + 1176 \beta_{13} + 2266 \beta_{12} - 75 \beta_{11} - 583 \beta_{10} + \cdots - 2482 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2010 \beta_{15} - 1747 \beta_{14} + 5656 \beta_{13} + 8167 \beta_{12} - 2868 \beta_{11} - 2173 \beta_{10} + \cdots - 15414 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 2158 \beta_{15} + 1246 \beta_{14} + 22952 \beta_{13} + 18538 \beta_{12} - 17171 \beta_{11} + \cdots - 66342 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 54552 \beta_{15} + 41459 \beta_{14} + 73850 \beta_{13} - 635 \beta_{12} - 78882 \beta_{11} + \cdots - 235586 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 344812 \beta_{15} + 348494 \beta_{14} + 175628 \beta_{13} - 276596 \beta_{12} - 329057 \beta_{11} + \cdots - 740768 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 1622020 \beta_{15} + 2015805 \beta_{14} + 149988 \beta_{13} - 1865395 \beta_{12} - 1271436 \beta_{11} + \cdots - 2002304 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 6677652 \beta_{15} + 9219268 \beta_{14} - 1552498 \beta_{13} - 9380990 \beta_{12} - 4310753 \beta_{11} + \cdots - 3501172 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 25180604 \beta_{15} + 35472515 \beta_{14} - 13780618 \beta_{13} - 41786267 \beta_{12} + \cdots + 6413714 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 84090122 \beta_{15} + 118817434 \beta_{14} - 78383728 \beta_{13} - 168880386 \beta_{12} + \cdots + 108552702 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times\).

\(n\) \(217\) \(271\) \(541\) \(1001\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
1.37578 − 2.65312i
−1.59571 + 0.665253i
−1.59571 − 0.665253i
1.37578 + 2.65312i
−2.05868 + 0.306986i
3.74844 − 1.37690i
3.74844 + 1.37690i
−2.05868 − 0.306986i
−0.315931 + 0.438405i
1.15135 + 1.37875i
1.15135 − 1.37875i
−0.315931 − 0.438405i
0.774725 + 2.71943i
−1.57997 − 0.888702i
−1.57997 + 0.888702i
0.774725 − 2.71943i
−1.05355 − 0.943417i 0 0.219928 + 1.98787i −0.463409 − 2.18752i 0 3.14971i 1.64369 − 2.30180i 0 −1.57552 + 2.74185i
109.2 −1.05355 − 0.943417i 0 0.219928 + 1.98787i −0.463409 + 2.18752i 0 − 3.14971i 1.64369 − 2.30180i 0 2.55197 − 1.86747i
109.3 −1.05355 + 0.943417i 0 0.219928 − 1.98787i −0.463409 − 2.18752i 0 3.14971i 1.64369 + 2.30180i 0 2.55197 + 1.86747i
109.4 −1.05355 + 0.943417i 0 0.219928 − 1.98787i −0.463409 + 2.18752i 0 − 3.14971i 1.64369 + 2.30180i 0 −1.57552 − 2.74185i
109.5 −0.393855 − 1.35826i 0 −1.68976 + 1.06992i 1.33104 − 1.79676i 0 − 4.27541i 2.11875 + 1.87374i 0 −2.96471 − 1.10024i
109.6 −0.393855 − 1.35826i 0 −1.68976 + 1.06992i 1.33104 + 1.79676i 0 4.27541i 2.11875 + 1.87374i 0 1.91623 − 2.51556i
109.7 −0.393855 + 1.35826i 0 −1.68976 − 1.06992i 1.33104 − 1.79676i 0 − 4.27541i 2.11875 − 1.87374i 0 1.91623 + 2.51556i
109.8 −0.393855 + 1.35826i 0 −1.68976 − 1.06992i 1.33104 + 1.79676i 0 4.27541i 2.11875 − 1.87374i 0 −2.96471 + 1.10024i
109.9 0.763079 − 1.19068i 0 −0.835421 − 1.81716i −1.27498 − 1.83696i 0 − 1.23231i −2.80114 − 0.391920i 0 −3.16014 + 0.116351i
109.10 0.763079 − 1.19068i 0 −0.835421 − 1.81716i −1.27498 + 1.83696i 0 1.23231i −2.80114 − 0.391920i 0 1.21431 + 2.91984i
109.11 0.763079 + 1.19068i 0 −0.835421 + 1.81716i −1.27498 − 1.83696i 0 − 1.23231i −2.80114 + 0.391920i 0 1.21431 − 2.91984i
109.12 0.763079 + 1.19068i 0 −0.835421 + 1.81716i −1.27498 + 1.83696i 0 1.23231i −2.80114 + 0.391920i 0 −3.16014 − 0.116351i
109.13 1.18432 − 0.772900i 0 0.805250 − 1.83073i 1.90735 − 1.16705i 0 3.04658i −0.461295 − 2.79056i 0 1.35692 − 2.85636i
109.14 1.18432 − 0.772900i 0 0.805250 − 1.83073i 1.90735 + 1.16705i 0 − 3.04658i −0.461295 − 2.79056i 0 3.16094 − 0.0920331i
109.15 1.18432 + 0.772900i 0 0.805250 + 1.83073i 1.90735 − 1.16705i 0 3.04658i −0.461295 + 2.79056i 0 3.16094 + 0.0920331i
109.16 1.18432 + 0.772900i 0 0.805250 + 1.83073i 1.90735 + 1.16705i 0 − 3.04658i −0.461295 + 2.79056i 0 1.35692 + 2.85636i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner
24.h odd 2 1 inner
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.2.d.h yes 16
3.b odd 2 1 1080.2.d.g ✓ 16
4.b odd 2 1 4320.2.d.h 16
5.b even 2 1 1080.2.d.g ✓ 16
8.b even 2 1 1080.2.d.g ✓ 16
8.d odd 2 1 4320.2.d.g 16
12.b even 2 1 4320.2.d.g 16
15.d odd 2 1 inner 1080.2.d.h yes 16
20.d odd 2 1 4320.2.d.g 16
24.f even 2 1 4320.2.d.h 16
24.h odd 2 1 inner 1080.2.d.h yes 16
40.e odd 2 1 4320.2.d.h 16
40.f even 2 1 inner 1080.2.d.h yes 16
60.h even 2 1 4320.2.d.h 16
120.i odd 2 1 1080.2.d.g ✓ 16
120.m even 2 1 4320.2.d.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.2.d.g ✓ 16 3.b odd 2 1
1080.2.d.g ✓ 16 5.b even 2 1
1080.2.d.g ✓ 16 8.b even 2 1
1080.2.d.g ✓ 16 120.i odd 2 1
1080.2.d.h yes 16 1.a even 1 1 trivial
1080.2.d.h yes 16 15.d odd 2 1 inner
1080.2.d.h yes 16 24.h odd 2 1 inner
1080.2.d.h yes 16 40.f even 2 1 inner
4320.2.d.g 16 8.d odd 2 1
4320.2.d.g 16 12.b even 2 1
4320.2.d.g 16 20.d odd 2 1
4320.2.d.g 16 120.m even 2 1
4320.2.d.h 16 4.b odd 2 1
4320.2.d.h 16 24.f even 2 1
4320.2.d.h 16 40.e odd 2 1
4320.2.d.h 16 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1080, [\chi])\):

\( T_{7}^{8} + 39T_{7}^{6} + 500T_{7}^{4} + 2356T_{7}^{2} + 2556 \) Copy content Toggle raw display
\( T_{11}^{8} + 43T_{11}^{6} + 556T_{11}^{4} + 2032T_{11}^{2} + 284 \) Copy content Toggle raw display
\( T_{13}^{8} - 88T_{13}^{6} + 2492T_{13}^{4} - 27228T_{13}^{2} + 101104 \) Copy content Toggle raw display
\( T_{53}^{4} + 24T_{53}^{3} + 206T_{53}^{2} + 754T_{53} + 999 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{7} + 2 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} - 3 T^{7} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 39 T^{6} + \cdots + 2556)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 43 T^{6} + \cdots + 284)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 88 T^{6} + \cdots + 101104)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 77 T^{6} + \cdots + 356)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 81 T^{6} + \cdots + 28836)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 73 T^{6} + \cdots + 356)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 80 T^{6} + \cdots + 72704)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 44 T^{2} + \cdots + 257)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} - 296 T^{6} + \cdots + 1617664)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 264 T^{6} + \cdots + 909936)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 216 T^{6} + \cdots + 1617664)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 140 T^{6} + \cdots + 22784)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 24 T^{3} + \cdots + 999)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + 208 T^{6} + \cdots + 3954416)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 389 T^{6} + \cdots + 44612496)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 468 T^{6} + \cdots + 6470656)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 452 T^{6} + \cdots + 73704816)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 135 T^{6} + \cdots + 207036)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - T^{3} - 165 T^{2} + \cdots + 3364)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 16 T^{3} + \cdots + 213)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} - 464 T^{6} + \cdots + 32757696)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 399 T^{6} + \cdots + 39301056)^{2} \) Copy content Toggle raw display
show more
show less