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Properties

Label 2160.4.h.g
Level $2160$
Weight $4$
Character orbit 2160.h
Analytic conductor $127.444$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2160,4,Mod(431,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2160.431");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2160.h (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: \(127.444125612\)
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: \( x^{16} - 35x^{14} + 942x^{12} - 8887x^{10} + 62174x^{8} - 137047x^{6} + 230781x^{4} - 50900x^{2} + 10000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{12} \)
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.

\(f(q)\) \(=\) \( q - 5 \beta_{4} q^{5} + (\beta_{8} - \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 5 \beta_{4} q^{5} + (\beta_{8} - \beta_{2}) q^{7} + (\beta_{9} - 3 \beta_{3}) q^{11} + \beta_1 q^{13} + (\beta_{15} + \beta_{14} - 10 \beta_{4}) q^{17} + (\beta_{10} + 4 \beta_{8} + \cdots - 2 \beta_{2}) q^{19}+ \cdots + ( - 8 \beta_{7} - 12 \beta_{5} + \cdots - 568) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{13} - 400 q^{25} - 256 q^{37} - 1328 q^{49} - 2944 q^{61} - 1016 q^{73} - 840 q^{85} - 9152 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 35x^{14} + 942x^{12} - 8887x^{10} + 62174x^{8} - 137047x^{6} + 230781x^{4} - 50900x^{2} + 10000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 172244315997 \nu^{14} + 5706064507641 \nu^{12} - 151472138397489 \nu^{10} + \cdots - 12\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 27417170667 \nu^{14} - 952557924045 \nu^{12} + 25593938220214 \nu^{10} + \cdots - 751809581795150 ) / 599636907505150 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 132094 \nu^{15} + 4699845 \nu^{13} - 127066668 \nu^{11} + 1244445153 \nu^{9} + \cdots + 15772915200 \nu ) / 3486442000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 764572 \nu^{15} - 26625645 \nu^{13} + 715394934 \nu^{11} - 6663940569 \nu^{9} + \cdots - 4253490000 \nu ) / 13759286000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 418373861361 \nu^{14} + 13904220471333 \nu^{12} - 367919156363157 \nu^{10} + \cdots - 60\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2212356222935 \nu^{14} + 78283483917333 \nu^{12} + \cdots + 56\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 55849226187 \nu^{14} + 1853402028081 \nu^{12} - 49113967386519 \nu^{10} + \cdots - 10\!\cdots\!40 ) / 119927381501030 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4099716620359 \nu^{14} + 143318191006581 \nu^{12} + \cdots + 10\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 904150581669 \nu^{15} + 31383813723260 \nu^{13} - 842786889856578 \nu^{11} + \cdots + 83\!\cdots\!00 \nu ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 6804353364363 \nu^{14} - 237560661805425 \nu^{12} + \cdots - 18\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8739736724751 \nu^{15} - 308317122842120 \nu^{13} + \cdots - 96\!\cdots\!00 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 14431261896313 \nu^{15} - 509355616442670 \nu^{13} + \cdots - 15\!\cdots\!00 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3958131261744 \nu^{15} + 137639506722615 \nu^{13} + \cdots + 21\!\cdots\!00 \nu ) / 29\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 3382410481271 \nu^{15} - 117769577700105 \nu^{13} + \cdots - 18\!\cdots\!00 \nu ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 27097769717079 \nu^{15} - 942758561268765 \nu^{13} + \cdots - 15\!\cdots\!00 \nu ) / 59\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - 2\beta_{14} + \beta_{13} - 3\beta_{12} + 6\beta_{11} + 3\beta_{9} + 8\beta_{4} + 12\beta_{3} ) / 72 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{10} + 30\beta_{8} - \beta_{7} - 15\beta_{6} + \beta_{5} + 321\beta_{2} + 14\beta _1 + 322 ) / 72 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{15} - 18\beta_{14} + 9\beta_{13} + 264\beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 45\beta_{10} + 246\beta_{8} + 15\beta_{7} - 117\beta_{6} - 19\beta_{5} + 2019\beta_{2} - 106\beta _1 - 2030 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 101 \beta_{15} - 466 \beta_{14} + 173 \beta_{13} + 303 \beta_{12} - 1206 \beta_{11} + \cdots - 8652 \beta_{3} ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1253\beta_{7} - 1829\beta_{5} - 7366\beta _1 - 135866 ) / 36 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6839 \beta_{15} + 34726 \beta_{14} - 11123 \beta_{13} + 20517 \beta_{12} - 87414 \beta_{11} + \cdots - 658788 \beta_{3} ) / 72 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 31629 \beta_{10} - 143302 \beta_{8} + 10543 \beta_{7} + 63269 \beta_{6} - 16131 \beta_{5} + \cdots - 1064046 ) / 24 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -53717\beta_{15} + 282354\beta_{14} - 85357\beta_{13} - 4824936\beta_{4} ) / 12 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 775221 \beta_{10} - 3459330 \beta_{8} - 258407 \beta_{7} + 1514745 \beta_{6} + 401687 \beta_{5} + \cdots + 25447454 ) / 24 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 3857575 \beta_{15} + 20525894 \beta_{14} - 6076147 \beta_{13} - 11572725 \beta_{12} + \cdots + 392597220 \beta_{3} ) / 72 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( -18815005\beta_{7} + 29407177\beta_{5} + 101168798\beta _1 + 1837154410 ) / 36 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 30953381 \beta_{15} - 165407826 \beta_{14} + 48602253 \beta_{13} - 92860143 \beta_{12} + \cdots + 3165940812 \beta_{3} ) / 24 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 455111493 \beta_{10} + 2016553938 \beta_{8} - 151703831 \beta_{7} - 879498681 \beta_{6} + \cdots + 14768909182 ) / 24 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 248861471\beta_{15} - 1331852854\beta_{14} + 390323867\beta_{13} + 22834392024\beta_{4} ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
0.409389 + 0.236361i
−4.25546 + 2.45689i
−2.51502 − 1.45205i
1.28380 − 0.741205i
−1.28380 − 0.741205i
2.51502 − 1.45205i
4.25546 + 2.45689i
−0.409389 + 0.236361i
−0.409389 − 0.236361i
4.25546 − 2.45689i
2.51502 + 1.45205i
−1.28380 + 0.741205i
1.28380 + 0.741205i
−2.51502 + 1.45205i
−4.25546 − 2.45689i
0.409389 − 0.236361i
0 0 0 − 5.00000i 0 − 32.4516i 0 0 0
431.2 0 0 0 − 5.00000i 0 − 23.7399i 0 0 0
431.3 0 0 0 − 5.00000i 0 − 7.39389i 0 0 0
431.4 0 0 0 − 5.00000i 0 − 5.71325i 0 0 0
431.5 0 0 0 − 5.00000i 0 5.71325i 0 0 0
431.6 0 0 0 − 5.00000i 0 7.39389i 0 0 0
431.7 0 0 0 − 5.00000i 0 23.7399i 0 0 0
431.8 0 0 0 − 5.00000i 0 32.4516i 0 0 0
431.9 0 0 0 5.00000i 0 − 32.4516i 0 0 0
431.10 0 0 0 5.00000i 0 − 23.7399i 0 0 0
431.11 0 0 0 5.00000i 0 − 7.39389i 0 0 0
431.12 0 0 0 5.00000i 0 − 5.71325i 0 0 0
431.13 0 0 0 5.00000i 0 5.71325i 0 0 0
431.14 0 0 0 5.00000i 0 7.39389i 0 0 0
431.15 0 0 0 5.00000i 0 23.7399i 0 0 0
431.16 0 0 0 5.00000i 0 32.4516i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.4.h.g ✓ 16
3.b odd 2 1 inner 2160.4.h.g ✓ 16
4.b odd 2 1 inner 2160.4.h.g ✓ 16
12.b even 2 1 inner 2160.4.h.g ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2160.4.h.g ✓ 16 1.a even 1 1 trivial
2160.4.h.g ✓ 16 3.b odd 2 1 inner
2160.4.h.g ✓ 16 4.b odd 2 1 inner
2160.4.h.g ✓ 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 1704T_{7}^{6} + 736452T_{7}^{4} + 54705024T_{7}^{2} + 1059111936 \) acting on \(S_{4}^{\mathrm{new}}(2160, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{2} + 25)^{8} \) Copy content Toggle raw display
$7$ \( (T^{8} + 1704 T^{6} + \cdots + 1059111936)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 2688 T^{6} + \cdots + 451137600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} + \cdots + 101632)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 13\!\cdots\!69)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 2639789816049)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 51976818155025)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 2084396737536)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 50\!\cdots\!61)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 64 T^{3} + \cdots - 428970800)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 93\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 44\!\cdots\!96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 26\!\cdots\!61)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 31\!\cdots\!24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 736 T^{3} + \cdots + 7694519197)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 26\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 254 T^{3} + \cdots + 30529280344)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 87\!\cdots\!09)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 57\!\cdots\!81)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 35\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 2288 T^{3} + \cdots + 96214466560)^{4} \) Copy content Toggle raw display
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