↑

Properties

Label 1080.4.f.d
Level $1080$
Weight $4$
Character orbit 1080.f
Analytic conductor $63.722$
Analytic rank $0$
Dimension $20$
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] = [1080,4,Mod(649,1080)]
 
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, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.649");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1080.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: \(63.7220628062\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 10 x^{19} + 50 x^{18} + 352 x^{17} + 21144 x^{16} - 183248 x^{15} + 837232 x^{14} + \cdots + 2209905230625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{25}\cdot 3^{12}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{7} q^{5} - \beta_{8} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{5} - \beta_{8} q^{7} - \beta_{13} q^{11} - \beta_{19} q^{13} + ( - \beta_{18} + \beta_{7} - 2 \beta_1) q^{17} + (\beta_{10} - \beta_{9} + \beta_{2} - 5) q^{19} + (\beta_{18} - \beta_{12} + \cdots + 4 \beta_1) q^{23}+ \cdots + (7 \beta_{19} + 19 \beta_{16} + \cdots - 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 100 q^{19} - 24 q^{25} + 228 q^{31} - 252 q^{49} - 64 q^{55} + 748 q^{61} - 668 q^{79} - 1500 q^{85} + 1744 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 10 x^{19} + 50 x^{18} + 352 x^{17} + 21144 x^{16} - 183248 x^{15} + 837232 x^{14} + \cdots + 2209905230625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 13\!\cdots\!52 \nu^{19} + \cdots - 66\!\cdots\!25 ) / 42\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 69\!\cdots\!29 \nu^{19} + \cdots + 19\!\cdots\!25 ) / 73\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 96\!\cdots\!37 \nu^{19} + \cdots - 28\!\cdots\!50 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 48\!\cdots\!73 \nu^{19} + \cdots + 14\!\cdots\!25 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!89 \nu^{19} + \cdots - 69\!\cdots\!75 ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 38\!\cdots\!03 \nu^{19} + \cdots - 93\!\cdots\!25 ) / 95\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 59\!\cdots\!27 \nu^{19} + \cdots + 10\!\cdots\!25 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20\!\cdots\!07 \nu^{19} + \cdots + 88\!\cdots\!25 ) / 29\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 50\!\cdots\!19 \nu^{19} + \cdots + 44\!\cdots\!25 ) / 67\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12\!\cdots\!63 \nu^{19} + \cdots + 11\!\cdots\!25 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 28\!\cdots\!97 \nu^{19} + \cdots - 13\!\cdots\!75 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 17\!\cdots\!49 \nu^{19} + \cdots + 46\!\cdots\!75 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19\!\cdots\!54 \nu^{19} + \cdots + 48\!\cdots\!75 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 25\!\cdots\!29 \nu^{19} + \cdots - 49\!\cdots\!25 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 62\!\cdots\!29 \nu^{19} + \cdots + 19\!\cdots\!25 ) / 31\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 13\!\cdots\!03 \nu^{19} + \cdots + 56\!\cdots\!50 ) / 46\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 68\!\cdots\!73 \nu^{19} + \cdots - 16\!\cdots\!75 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 21\!\cdots\!61 \nu^{19} + \cdots - 11\!\cdots\!75 ) / 57\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 70\!\cdots\!65 \nu^{19} + \cdots + 31\!\cdots\!75 ) / 14\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 5 \beta_{19} + 5 \beta_{17} + 7 \beta_{14} + 5 \beta_{8} - 35 \beta_{7} + 5 \beta_{6} + 35 \beta_{5} + \cdots + 67 ) / 120 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5 \beta_{19} - 5 \beta_{18} + 25 \beta_{15} + 7 \beta_{14} - 15 \beta_{12} - 25 \beta_{11} + 5 \beta_{8} + \cdots + 7 ) / 60 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 690 \beta_{19} - 15 \beta_{18} - 690 \beta_{17} - 275 \beta_{16} + 70 \beta_{15} + 697 \beta_{14} + \cdots - 8708 ) / 120 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1370 \beta_{17} - 10 \beta_{15} + 1822 \beta_{14} - 550 \beta_{13} - 10 \beta_{11} + 3125 \beta_{10} + \cdots - 312213 ) / 60 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 162875 \beta_{19} - 450 \beta_{18} - 162875 \beta_{17} + 64735 \beta_{16} - 33295 \beta_{15} + \cdots - 3203586 ) / 240 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 474945 \beta_{19} - 49880 \beta_{18} + 188705 \beta_{16} - 733640 \beta_{15} - 418456 \beta_{14} + \cdots - 418456 ) / 120 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 9540340 \beta_{19} - 172935 \beta_{18} + 9540340 \beta_{17} + 3314730 \beta_{16} - 2319135 \beta_{15} + \cdots + 225260737 ) / 120 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 17978855 \beta_{17} + 265125 \beta_{15} - 13013984 \beta_{14} + 6191715 \beta_{13} + 265125 \beta_{11} + \cdots + 2030570316 ) / 30 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2263215195 \beta_{19} + 64852680 \beta_{18} + 2263215195 \beta_{17} - 653584385 \beta_{16} + \cdots + 69534279154 ) / 240 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 10250533775 \beta_{19} + 1027460000 \beta_{18} - 2901658765 \beta_{16} + 10388925320 \beta_{15} + \cdots + 9014335674 ) / 120 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 272383320235 \beta_{19} + 10128270240 \beta_{18} - 272383320235 \beta_{17} - 62591147855 \beta_{16} + \cdots - 9172984781376 ) / 240 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 706725863760 \beta_{17} - 3618358790 \beta_{15} + 373334148202 \beta_{14} - 156656042690 \beta_{13} + \cdots - 58460955448863 ) / 60 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 33227317679335 \beta_{19} - 1488181753890 \beta_{18} - 33227317679335 \beta_{17} + \cdots - 13\!\cdots\!96 ) / 240 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 191044244345755 \beta_{19} - 17509851211670 \beta_{18} + 31010695791465 \beta_{16} + \cdots - 171085154629526 ) / 120 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 20\!\cdots\!25 \beta_{19} - 106172054639850 \beta_{18} + \cdots + 84\!\cdots\!17 ) / 120 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 12\!\cdots\!20 \beta_{17} - 66221869037180 \beta_{15} + \cdots + 88\!\cdots\!27 ) / 60 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 25\!\cdots\!55 \beta_{19} + \cdots + 11\!\cdots\!32 ) / 120 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 16\!\cdots\!95 \beta_{19} + \cdots + 15\!\cdots\!97 ) / 60 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 64\!\cdots\!15 \beta_{19} + \cdots - 29\!\cdots\!46 ) / 240 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
8.13699 + 8.13699i
8.13699 − 8.13699i
7.67988 − 7.67988i
7.67988 + 7.67988i
4.24611 + 4.24611i
4.24611 − 4.24611i
0.336685 + 0.336685i
0.336685 − 0.336685i
2.80557 − 2.80557i
2.80557 + 2.80557i
−1.80557 + 1.80557i
−1.80557 − 1.80557i
0.663315 + 0.663315i
0.663315 − 0.663315i
−3.24611 − 3.24611i
−3.24611 + 3.24611i
−6.67988 + 6.67988i
−6.67988 − 6.67988i
−7.13699 − 7.13699i
−7.13699 + 7.13699i
0 0 0 −11.1793 − 0.152960i 0 26.3201i 0 0 0
649.2 0 0 0 −11.1793 + 0.152960i 0 − 26.3201i 0 0 0
649.3 0 0 0 −7.99584 − 7.81451i 0 22.1308i 0 0 0
649.4 0 0 0 −7.99584 + 7.81451i 0 − 22.1308i 0 0 0
649.5 0 0 0 −6.82224 − 8.85760i 0 − 20.2612i 0 0 0
649.6 0 0 0 −6.82224 + 8.85760i 0 20.2612i 0 0 0
649.7 0 0 0 −6.23396 − 9.28104i 0 11.3256i 0 0 0
649.8 0 0 0 −6.23396 + 9.28104i 0 − 11.3256i 0 0 0
649.9 0 0 0 −5.93167 − 9.47709i 0 7.52946i 0 0 0
649.10 0 0 0 −5.93167 + 9.47709i 0 − 7.52946i 0 0 0
649.11 0 0 0 5.93167 − 9.47709i 0 − 7.52946i 0 0 0
649.12 0 0 0 5.93167 + 9.47709i 0 7.52946i 0 0 0
649.13 0 0 0 6.23396 − 9.28104i 0 − 11.3256i 0 0 0
649.14 0 0 0 6.23396 + 9.28104i 0 11.3256i 0 0 0
649.15 0 0 0 6.82224 − 8.85760i 0 20.2612i 0 0 0
649.16 0 0 0 6.82224 + 8.85760i 0 − 20.2612i 0 0 0
649.17 0 0 0 7.99584 − 7.81451i 0 − 22.1308i 0 0 0
649.18 0 0 0 7.99584 + 7.81451i 0 22.1308i 0 0 0
649.19 0 0 0 11.1793 − 0.152960i 0 − 26.3201i 0 0 0
649.20 0 0 0 11.1793 + 0.152960i 0 26.3201i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1080.4.f.d ✓ 20
3.b odd 2 1 inner 1080.4.f.d ✓ 20
5.b even 2 1 inner 1080.4.f.d ✓ 20
15.d odd 2 1 inner 1080.4.f.d ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.f.d ✓ 20 1.a even 1 1 trivial
1080.4.f.d ✓ 20 3.b odd 2 1 inner
1080.4.f.d ✓ 20 5.b even 2 1 inner
1080.4.f.d ✓ 20 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1080, [\chi])\):

\( T_{7}^{10} + 1778T_{7}^{8} + 1126657T_{7}^{6} + 303412348T_{7}^{4} + 31759502432T_{7}^{2} + 1012857062464 \) Copy content Toggle raw display
\( T_{11}^{10} - 8018T_{11}^{8} + 24335521T_{11}^{6} - 34163636860T_{11}^{4} + 21189638362400T_{11}^{2} - 4102838051560000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 1012857062464)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots - 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 21\!\cdots\!16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 134751978825984)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + 25 T^{4} + \cdots - 2408089420)^{4} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots + 44\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 89\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 57 T^{4} + \cdots - 2218318569)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 82\!\cdots\!76)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 25\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 44\!\cdots\!24)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 35\!\cdots\!25)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 66\!\cdots\!36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 2277681305552)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 41\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 60\!\cdots\!04)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 9214463451136)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots + 38\!\cdots\!25)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 35\!\cdots\!56)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 18\!\cdots\!44)^{2} \) Copy content Toggle raw display
show more
show less