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Properties

Label 2160.4.h.f
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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Show commands: Magma / PariGP / SageMath

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: \( x^{16} - 66 x^{14} + 2839 x^{12} - 71674 x^{10} + 1316709 x^{8} - 15532736 x^{6} + 132849644 x^{4} + \cdots + 2097273616 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{20}\cdot 5^{4} \)
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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{5} - \beta_1 q^{7} + \beta_{11} q^{11} + (\beta_{5} - 5) q^{13} + (\beta_{13} + \beta_{10}) q^{17} + (\beta_{4} + \beta_1) q^{19} + ( - \beta_{14} - \beta_{11} - 2 \beta_{9}) q^{23} - 25 q^{25} + ( - \beta_{13} + \beta_{10} - 5 \beta_{7}) q^{29} + (\beta_{4} - \beta_{3} + \cdots - 5 \beta_1) q^{31}+ \cdots + ( - 3 \beta_{6} + 457) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 80 q^{13} - 400 q^{25} - 1552 q^{37} - 608 q^{49} - 688 q^{61} - 1712 q^{73} + 7312 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 66 x^{14} + 2839 x^{12} - 71674 x^{10} + 1316709 x^{8} - 15532736 x^{6} + 132849644 x^{4} + \cdots + 2097273616 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 16676902189277 \nu^{14} + \cdots + 15\!\cdots\!56 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 89217416301677 \nu^{14} + \cdots - 40\!\cdots\!96 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 565525319010943 \nu^{14} + \cdots - 27\!\cdots\!36 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 32782374973971 \nu^{14} + \cdots - 11\!\cdots\!42 ) / 56\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 80747350821 \nu^{14} - 4665390412728 \nu^{12} + 181709484211575 \nu^{10} + \cdots + 28\!\cdots\!68 ) / 98\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 72595963701 \nu^{14} - 4313550238533 \nu^{12} + 163366042177575 \nu^{10} + \cdots - 80\!\cdots\!02 ) / 24\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 222506026 \nu^{15} - 14430462833 \nu^{13} + 603718564150 \nu^{11} + \cdots - 51\!\cdots\!52 \nu ) / 71\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 143174177838 \nu^{14} - 8057138298909 \nu^{12} + 322191449537850 \nu^{10} + \cdots + 35\!\cdots\!04 ) / 24\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 194286699 \nu^{15} - 9119937717 \nu^{13} + 331196259825 \nu^{11} + \cdots + 62\!\cdots\!52 \nu ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 46\!\cdots\!74 \nu^{15} + \cdots - 10\!\cdots\!88 \nu ) / 60\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12\!\cdots\!79 \nu^{15} + \cdots + 15\!\cdots\!00 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 769404566000521 \nu^{15} + \cdots + 63\!\cdots\!88 \nu ) / 35\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 54\!\cdots\!28 \nu^{15} + \cdots - 12\!\cdots\!00 \nu ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 41\!\cdots\!43 \nu^{15} + \cdots - 12\!\cdots\!64 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 14\!\cdots\!18 \nu^{15} + \cdots + 36\!\cdots\!16 \nu ) / 12\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{12} + 12\beta_{11} + 5\beta_{10} + 6\beta_{9} - 2\beta_{7} ) / 180 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + 5\beta_{6} - 32\beta_{5} - 45\beta_{4} + 99\beta_{2} + 99\beta _1 + 4455 ) / 540 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} - 30\beta_{13} + 54\beta_{10} + 47\beta_{7} ) / 54 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -21\beta_{8} - 55\beta_{6} + 372\beta_{5} - 495\beta_{4} + 150\beta_{3} + 631\beta_{2} + 781\beta _1 - 29745 ) / 180 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 115 \beta_{15} - 4905 \beta_{14} - 5250 \beta_{13} - 8946 \beta_{12} - 13059 \beta_{11} + \cdots + 17849 \beta_{7} ) / 540 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -479\beta_{8} - 865\beta_{6} + 6508\beta_{5} - 402327 ) / 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1075 \beta_{15} - 147825 \beta_{14} + 151050 \beta_{13} - 232722 \beta_{12} - 310563 \beta_{11} + \cdots - 713297 \beta_{7} ) / 540 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 76723 \beta_{8} - 106265 \beta_{6} + 909836 \beta_{5} + 1075185 \beta_{4} - 832050 \beta_{3} + \cdots - 48401415 ) / 540 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -2159\beta_{15} + 275910\beta_{13} - 213784\beta_{10} - 1574539\beta_{7} ) / 18 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 760273 \beta_{8} + 856815 \beta_{6} - 8364836 \beta_{5} + 9414735 \beta_{4} - 8833650 \beta_{3} + \cdots + 402700185 ) / 180 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 787135 \beta_{15} + 39632955 \beta_{14} + 37271550 \beta_{13} + 51459846 \beta_{12} + \cdots - 239948059 \beta_{7} ) / 180 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 13065319\beta_{8} + 12481565\beta_{6} - 137551388\beta_{5} + 6167316051 ) / 54 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 32233475 \beta_{15} + 1099880775 \beta_{14} - 1003180350 \beta_{13} + 1336792842 \beta_{12} + \cdots + 7003201487 \beta_{7} ) / 180 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 611059793 \beta_{8} + 510999015 \beta_{6} - 6258298276 \beta_{5} - 6569175735 \beta_{4} + \cdots + 266287548465 ) / 180 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 661970321\beta_{15} - 16191511410\beta_{13} + 10649985876\beta_{10} + 119581553909\beta_{7} ) / 54 \) 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
−2.94682 − 1.70135i
3.88792 − 2.24469i
4.49984 + 2.59798i
−2.33490 + 1.34806i
2.33490 + 1.34806i
−4.49984 + 2.59798i
−3.88792 − 2.24469i
2.94682 − 1.70135i
2.94682 + 1.70135i
−3.88792 + 2.24469i
−4.49984 − 2.59798i
2.33490 − 1.34806i
−2.33490 − 1.34806i
4.49984 − 2.59798i
3.88792 + 2.24469i
−2.94682 + 1.70135i
0 0 0 − 5.00000i 0 − 32.1435i 0 0 0
431.2 0 0 0 − 5.00000i 0 − 17.4389i 0 0 0
431.3 0 0 0 − 5.00000i 0 − 13.6481i 0 0 0
431.4 0 0 0 − 5.00000i 0 − 0.639966i 0 0 0
431.5 0 0 0 − 5.00000i 0 0.639966i 0 0 0
431.6 0 0 0 − 5.00000i 0 13.6481i 0 0 0
431.7 0 0 0 − 5.00000i 0 17.4389i 0 0 0
431.8 0 0 0 − 5.00000i 0 32.1435i 0 0 0
431.9 0 0 0 5.00000i 0 − 32.1435i 0 0 0
431.10 0 0 0 5.00000i 0 − 17.4389i 0 0 0
431.11 0 0 0 5.00000i 0 − 13.6481i 0 0 0
431.12 0 0 0 5.00000i 0 − 0.639966i 0 0 0
431.13 0 0 0 5.00000i 0 0.639966i 0 0 0
431.14 0 0 0 5.00000i 0 13.6481i 0 0 0
431.15 0 0 0 5.00000i 0 17.4389i 0 0 0
431.16 0 0 0 5.00000i 0 32.1435i 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.f ✓ 16
3.b odd 2 1 inner 2160.4.h.f ✓ 16
4.b odd 2 1 inner 2160.4.h.f ✓ 16
12.b even 2 1 inner 2160.4.h.f ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2160.4.h.f ✓ 16 1.a even 1 1 trivial
2160.4.h.f ✓ 16 3.b odd 2 1 inner
2160.4.h.f ✓ 16 4.b odd 2 1 inner
2160.4.h.f ✓ 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 1524T_{7}^{6} + 563940T_{7}^{4} + 58759344T_{7}^{2} + 23970816 \) 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
$7$ \( (T^{8} + 1524 T^{6} + \cdots + 23970816)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 5201464771584)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 20 T^{3} + \cdots + 1478704)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 101878560567081)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 17\!\cdots\!29)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 39\!\cdots\!81)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 25\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 17\!\cdots\!29)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 388 T^{3} + \cdots - 1946781776)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 22\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 20\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 41\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 73\!\cdots\!29)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 43\!\cdots\!04)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 172 T^{3} + \cdots + 1216256425)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 18\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 82\!\cdots\!24)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 428 T^{3} + \cdots - 51693952256)^{4} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 12\!\cdots\!25)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 35\!\cdots\!21)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 54\!\cdots\!44)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 1828 T^{3} + \cdots - 323608678400)^{4} \) Copy content Toggle raw display
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