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Properties

Label 1080.4.f.b
Level $1080$
Weight $4$
Character orbit 1080.f
Analytic conductor $63.722$
Analytic rank $0$
Dimension $18$
CM no
Inner twists $2$

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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: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} + 460 x^{16} + 79376 x^{14} + 6573986 x^{12} + 287860456 x^{10} + 6830463040 x^{8} + \cdots + 59049000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{10}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{6} - \beta_{5}) q^{5} + ( - \beta_{9} + 2 \beta_{5}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} - \beta_{5}) q^{5} + ( - \beta_{9} + 2 \beta_{5}) q^{7} + ( - \beta_{3} - 3) q^{11} - \beta_{13} q^{13} + ( - \beta_{13} + \beta_{12} - 6 \beta_{5}) q^{17} + (\beta_{7} - \beta_{6} - \beta_1 + 1) q^{19} + ( - \beta_{17} + \beta_{16} + \beta_{15} + \cdots - 1) q^{23}+ \cdots + (5 \beta_{17} - 5 \beta_{16} + \cdots + 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 18 q - 2 q^{5} - 62 q^{11} + 30 q^{25} + 122 q^{29} + 222 q^{31} + 256 q^{35} - 20 q^{41} - 270 q^{49} - 546 q^{55} + 120 q^{59} + 468 q^{61} + 28 q^{65} - 1268 q^{71} - 822 q^{79} - 1116 q^{85} + 348 q^{89} - 444 q^{91} + 1996 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} + 460 x^{16} + 79376 x^{14} + 6573986 x^{12} + 287860456 x^{10} + 6830463040 x^{8} + \cdots + 59049000000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 17\!\cdots\!83 \nu^{16} + \cdots - 94\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 38\!\cdots\!21 \nu^{16} + \cdots - 14\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\!\cdots\!63 \nu^{16} + \cdots - 14\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15\!\cdots\!45 \nu^{16} + \cdots - 28\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 91827850392053 \nu^{17} + \cdots - 81\!\cdots\!00 \nu ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 79\!\cdots\!09 \nu^{17} + \cdots - 45\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 79\!\cdots\!09 \nu^{17} + \cdots - 63\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10\!\cdots\!13 \nu^{17} + \cdots - 11\!\cdots\!00 ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 62\!\cdots\!57 \nu^{17} + \cdots + 26\!\cdots\!00 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 79\!\cdots\!03 \nu^{17} + \cdots + 26\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 10\!\cdots\!37 \nu^{17} + \cdots - 21\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 14\!\cdots\!83 \nu^{17} + \cdots - 31\!\cdots\!00 \nu ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12\!\cdots\!21 \nu^{17} + \cdots - 24\!\cdots\!00 \nu ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 21\!\cdots\!19 \nu^{17} + \cdots + 16\!\cdots\!00 \nu ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 66\!\cdots\!97 \nu^{17} + \cdots - 74\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 66\!\cdots\!43 \nu^{17} + \cdots + 12\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 11\!\cdots\!67 \nu^{17} + \cdots + 18\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} - \beta_{13} + \beta_{12} + \beta_{10} - 2\beta_{9} + 2\beta_{8} + \beta_{6} + \beta_{5} - 1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5 \beta_{17} + 5 \beta_{16} + 6 \beta_{11} + 9 \beta_{10} - 24 \beta_{8} + 6 \beta_{7} + 7 \beta_{6} + \cdots - 609 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 31 \beta_{17} - 31 \beta_{16} + 6 \beta_{15} - 162 \beta_{14} + 86 \beta_{13} - 112 \beta_{12} + \cdots + 130 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 746 \beta_{17} - 746 \beta_{16} - 1343 \beta_{11} - 1662 \beta_{10} + 4667 \beta_{8} - 744 \beta_{7} + \cdots + 69955 ) / 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6377 \beta_{17} + 6377 \beta_{16} - 798 \beta_{15} + 25411 \beta_{14} - 9941 \beta_{13} + \cdots - 21171 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 114807 \beta_{17} + 114807 \beta_{16} + 233687 \beta_{11} + 265239 \beta_{10} - 764165 \beta_{8} + \cdots - 9954556 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1105990 \beta_{17} - 1105990 \beta_{16} + 64020 \beta_{15} - 3959849 \beta_{14} + 1386649 \beta_{13} + \cdots + 3505773 ) / 12 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 6041467 \beta_{17} - 6041467 \beta_{16} - 12776742 \beta_{11} - 13612935 \beta_{10} + \cdots + 501591163 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 184102239 \beta_{17} + 184102239 \beta_{16} + 148458 \beta_{15} + 618534202 \beta_{14} + \cdots - 577365226 ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2890150586 \beta_{17} + 2890150586 \beta_{16} + 6182417703 \beta_{11} + 6235383582 \beta_{10} + \cdots - 231875937243 ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 30274280641 \beta_{17} - 30274280641 \beta_{16} - 1584872946 \beta_{15} - 96907677363 \beta_{14} + \cdots + 94713834835 ) / 12 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 462652030991 \beta_{17} - 462652030991 \beta_{16} - 992768474039 \beta_{11} - 951574942767 \beta_{10} + \cdots + 36005457976564 ) / 12 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 4956608301734 \beta_{17} + 4956608301734 \beta_{16} + 484288092348 \beta_{15} + \cdots - 15501801624453 ) / 12 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 74193031427913 \beta_{17} + 74193031427913 \beta_{16} + 159303415341530 \beta_{11} + \cdots - 56\!\cdots\!61 ) / 12 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 809829344153527 \beta_{17} - 809829344153527 \beta_{16} - 111448723053258 \beta_{15} + \cdots + 25\!\cdots\!18 ) / 12 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 39\!\cdots\!26 \beta_{17} + \cdots + 29\!\cdots\!25 ) / 4 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 13\!\cdots\!01 \beta_{17} + \cdots - 41\!\cdots\!95 ) / 12 \) 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
4.40631i
− 4.40631i
− 7.00251i
7.00251i
1.89926i
− 1.89926i
− 12.0345i
12.0345i
6.89843i
− 6.89843i
3.30267i
− 3.30267i
12.7197i
− 12.7197i
− 0.249200i
0.249200i
4.77118i
− 4.77118i
0 0 0 −10.7940 − 2.91364i 0 27.2986i 0 0 0
649.2 0 0 0 −10.7940 + 2.91364i 0 − 27.2986i 0 0 0
649.3 0 0 0 −9.42379 − 6.01599i 0 − 8.91736i 0 0 0
649.4 0 0 0 −9.42379 + 6.01599i 0 8.91736i 0 0 0
649.5 0 0 0 −7.85814 − 7.95297i 0 25.6640i 0 0 0
649.6 0 0 0 −7.85814 + 7.95297i 0 − 25.6640i 0 0 0
649.7 0 0 0 −3.22032 − 10.7065i 0 − 21.4734i 0 0 0
649.8 0 0 0 −3.22032 + 10.7065i 0 21.4734i 0 0 0
649.9 0 0 0 −1.79376 − 11.0355i 0 − 15.9550i 0 0 0
649.10 0 0 0 −1.79376 + 11.0355i 0 15.9550i 0 0 0
649.11 0 0 0 3.62632 − 10.5759i 0 23.9964i 0 0 0
649.12 0 0 0 3.62632 + 10.5759i 0 − 23.9964i 0 0 0
649.13 0 0 0 7.52596 − 8.26800i 0 − 4.10592i 0 0 0
649.14 0 0 0 7.52596 + 8.26800i 0 4.10592i 0 0 0
649.15 0 0 0 10.0455 − 4.90803i 0 7.13805i 0 0 0
649.16 0 0 0 10.0455 + 4.90803i 0 − 7.13805i 0 0 0
649.17 0 0 0 10.8923 − 2.52159i 0 19.4760i 0 0 0
649.18 0 0 0 10.8923 + 2.52159i 0 − 19.4760i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 649.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 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.b ✓ 18
3.b odd 2 1 1080.4.f.c yes 18
5.b even 2 1 inner 1080.4.f.b ✓ 18
15.d odd 2 1 1080.4.f.c yes 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1080.4.f.b ✓ 18 1.a even 1 1 trivial
1080.4.f.b ✓ 18 5.b even 2 1 inner
1080.4.f.c yes 18 3.b odd 2 1
1080.4.f.c yes 18 15.d odd 2 1

Hecke kernels

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

\( T_{7}^{18} + 3222 T_{7}^{16} + 4315023 T_{7}^{14} + 3106925220 T_{7}^{12} + 1298335774095 T_{7}^{10} + \cdots + 85\!\cdots\!04 \) Copy content Toggle raw display
\( T_{11}^{9} + 31 T_{11}^{8} - 6127 T_{11}^{7} - 139969 T_{11}^{6} + 10063002 T_{11}^{5} + \cdots - 3655552221032 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 74\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 85\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( (T^{9} + \cdots - 3655552221032)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 24\!\cdots\!25 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 53\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( (T^{9} + \cdots - 15\!\cdots\!40)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 55\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots - 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots + 10\!\cdots\!72)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 25\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{9} + \cdots - 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 40\!\cdots\!84 \) Copy content Toggle raw display
$59$ \( (T^{9} + \cdots + 13\!\cdots\!68)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots - 58\!\cdots\!20)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 25\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots + 13\!\cdots\!75)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots - 26\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
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