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Properties

Label 2160.2.bw.a
Level $2160$
Weight $2$
Character orbit 2160.bw
Analytic conductor $17.248$
Analytic rank $0$
Dimension $16$
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] = [2160,2,Mod(1151,2160)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2160, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2160.1151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2160.bw (of order \(6\), degree \(2\), not 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: \(17.2476868366\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 2 x^{14} + 14 x^{13} + 53 x^{12} - 28 x^{11} + 48 x^{10} + 288 x^{9} + 580 x^{8} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{4} \)
Twist minimal: no (minimal twist has level 720)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{10} + \beta_{4}) q^{5} + ( - \beta_{15} + \beta_{13} + \cdots - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} + \beta_{4}) q^{5} + ( - \beta_{15} + \beta_{13} + \cdots - \beta_1) q^{7}+ \cdots + (3 \beta_{15} - \beta_{14} + \cdots + 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{7} + 6 q^{11} - 2 q^{13} + 18 q^{23} + 8 q^{25} + 6 q^{29} - 4 q^{37} - 24 q^{41} + 10 q^{49} + 30 q^{59} + 14 q^{61} - 6 q^{65} - 42 q^{67} + 52 q^{73} + 66 q^{77} + 24 q^{79} + 6 q^{83} + 6 q^{85} - 12 q^{95} + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 2 x^{14} + 14 x^{13} + 53 x^{12} - 28 x^{11} + 48 x^{10} + 288 x^{9} + 580 x^{8} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1391265434888 \nu^{15} - 23925215339657 \nu^{14} + 54698012849466 \nu^{13} + \cdots - 847283712486812 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1678037506994 \nu^{15} + 10990980235985 \nu^{14} - 21493052726394 \nu^{13} + \cdots + 368209017342512 ) / 22485707154228 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5663290121522 \nu^{15} + 32144055094866 \nu^{14} - 61839420287903 \nu^{13} + \cdots + 853534722415316 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1433583634206 \nu^{15} - 3195552950691 \nu^{14} + 3650869228227 \nu^{13} + \cdots + 26512664021108 ) / 11242853577114 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 6220886839738 \nu^{15} - 16667646454028 \nu^{14} + 22498848705878 \nu^{13} + \cdots + 6693249761046 ) / 22485707154228 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 8914707052667 \nu^{15} - 19250547461728 \nu^{14} + 20190864015743 \nu^{13} + \cdots + 96366437261390 ) / 22485707154228 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18268395515561 \nu^{15} - 60976643273852 \nu^{14} + 96028134806554 \nu^{13} + \cdots - 580188391766968 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20678289830905 \nu^{15} - 8958897153587 \nu^{14} - 40001255218286 \nu^{13} + \cdots + 16\!\cdots\!36 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25768912 \nu^{15} - 48879474 \nu^{14} + 42500935 \nu^{13} + 374893060 \nu^{12} + 1392599232 \nu^{11} + \cdots + 560556516 ) / 34159128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 46107039278430 \nu^{15} + 109727910894028 \nu^{14} - 130118712721473 \nu^{13} + \cdots - 79440604929256 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 75027575592308 \nu^{15} + 148953585047323 \nu^{14} - 141259532442953 \nu^{13} + \cdots - 15\!\cdots\!60 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 75948408781307 \nu^{15} + 162728171506984 \nu^{14} - 170557155672974 \nu^{13} + \cdots - 800576101328936 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 82196008258126 \nu^{15} - 194325804891189 \nu^{14} + 229311483580237 \nu^{13} + \cdots + 99595544488368 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 88734479459158 \nu^{15} - 211217213416055 \nu^{14} + 252188506890419 \nu^{13} + \cdots + 194280474478104 ) / 44971414308456 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 106807291347049 \nu^{15} - 229184646310689 \nu^{14} + 240414111796137 \nu^{13} + \cdots + 12\!\cdots\!64 ) / 44971414308456 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{13} + \beta_{12} - 2\beta_{10} + \beta_{9} + \beta_{8} - 2\beta_{3} + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{15} + \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + 3 \beta_{9} - \beta_{7} - \beta_{6} + \cdots + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7 \beta_{15} + 7 \beta_{14} - 3 \beta_{13} - 6 \beta_{12} + 3 \beta_{11} + 3 \beta_{10} + 9 \beta_{9} + \cdots - 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8 \beta_{15} + 17 \beta_{14} - 8 \beta_{13} - 21 \beta_{12} + 25 \beta_{11} + 52 \beta_{10} + 33 \beta_{9} + \cdots - 45 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 90 \beta_{15} - 13 \beta_{13} - 59 \beta_{12} + 72 \beta_{11} + 148 \beta_{10} - 11 \beta_{9} + \cdots - 146 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 344 \beta_{15} - 77 \beta_{14} - \beta_{13} + 49 \beta_{12} + 78 \beta_{11} + 145 \beta_{10} + \cdots - 262 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 569 \beta_{15} - 569 \beta_{14} + 440 \beta_{13} + 888 \beta_{12} - 448 \beta_{11} - 852 \beta_{10} + \cdots + 433 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 1454 \beta_{15} - 1403 \beta_{14} + 1565 \beta_{13} + 3171 \beta_{12} - 2968 \beta_{11} - 6253 \beta_{10} + \cdots + 4950 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 12471 \beta_{15} - 165 \beta_{14} + 2828 \beta_{13} + 5902 \beta_{12} - 8730 \beta_{11} + \cdots + 18007 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 42942 \beta_{15} + 12011 \beta_{14} - 3243 \beta_{13} - 9143 \beta_{12} - 8768 \beta_{11} + \cdots + 31184 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 63277 \beta_{15} + 63277 \beta_{14} - 52808 \beta_{13} - 107970 \beta_{12} + 55162 \beta_{11} + \cdots - 53141 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 60644 \beta_{15} + 51787 \beta_{14} - 66583 \beta_{13} - 131821 \beta_{12} + 118370 \beta_{11} + \cdots - 197264 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1533405 \beta_{15} + 29175 \beta_{14} - 364774 \beta_{13} - 687962 \beta_{12} + 1052736 \beta_{11} + \cdots - 2180855 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 5201686 \beta_{15} - 1510245 \beta_{14} + 465235 \beta_{13} + 1161277 \beta_{12} + 1045010 \beta_{11} + \cdots - 3747514 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 7522913 \beta_{15} - 7522913 \beta_{14} + 6331626 \beta_{13} + 13002564 \beta_{12} - 6670938 \beta_{11} + \cdots + 6419725 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1 + \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1
1.43423 + 1.43423i
−0.671598 − 0.671598i
−0.922751 − 0.922751i
−0.205906 − 0.205906i
0.806564 − 0.806564i
−0.233123 + 0.233123i
−1.55038 + 1.55038i
2.34297 − 2.34297i
1.43423 − 1.43423i
−0.671598 + 0.671598i
−0.922751 + 0.922751i
−0.205906 + 0.205906i
0.806564 + 0.806564i
−0.233123 − 0.233123i
−1.55038 − 1.55038i
2.34297 + 2.34297i
0 0 0 −0.866025 − 0.500000i 0 −3.90729 + 2.25587i 0 0 0
1151.2 0 0 0 −0.866025 − 0.500000i 0 −1.35172 + 0.780414i 0 0 0
1151.3 0 0 0 −0.866025 − 0.500000i 0 0.178470 − 0.103039i 0 0 0
1151.4 0 0 0 −0.866025 − 0.500000i 0 3.58054 − 2.06722i 0 0 0
1151.5 0 0 0 0.866025 + 0.500000i 0 −3.87443 + 2.23690i 0 0 0
1151.6 0 0 0 0.866025 + 0.500000i 0 −0.465734 + 0.268892i 0 0 0
1151.7 0 0 0 0.866025 + 0.500000i 0 1.03765 − 0.599087i 0 0 0
1151.8 0 0 0 0.866025 + 0.500000i 0 1.80251 − 1.04068i 0 0 0
1871.1 0 0 0 −0.866025 + 0.500000i 0 −3.90729 − 2.25587i 0 0 0
1871.2 0 0 0 −0.866025 + 0.500000i 0 −1.35172 − 0.780414i 0 0 0
1871.3 0 0 0 −0.866025 + 0.500000i 0 0.178470 + 0.103039i 0 0 0
1871.4 0 0 0 −0.866025 + 0.500000i 0 3.58054 + 2.06722i 0 0 0
1871.5 0 0 0 0.866025 − 0.500000i 0 −3.87443 − 2.23690i 0 0 0
1871.6 0 0 0 0.866025 − 0.500000i 0 −0.465734 − 0.268892i 0 0 0
1871.7 0 0 0 0.866025 − 0.500000i 0 1.03765 + 0.599087i 0 0 0
1871.8 0 0 0 0.866025 − 0.500000i 0 1.80251 + 1.04068i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1151.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.2.bw.a 16
3.b odd 2 1 720.2.bw.a ✓ 16
4.b odd 2 1 2160.2.bw.c 16
9.c even 3 1 720.2.bw.c yes 16
9.c even 3 1 6480.2.h.a 16
9.d odd 6 1 2160.2.bw.c 16
9.d odd 6 1 6480.2.h.f 16
12.b even 2 1 720.2.bw.c yes 16
36.f odd 6 1 720.2.bw.a ✓ 16
36.f odd 6 1 6480.2.h.f 16
36.h even 6 1 inner 2160.2.bw.a 16
36.h even 6 1 6480.2.h.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.bw.a ✓ 16 3.b odd 2 1
720.2.bw.a ✓ 16 36.f odd 6 1
720.2.bw.c yes 16 9.c even 3 1
720.2.bw.c yes 16 12.b even 2 1
2160.2.bw.a 16 1.a even 1 1 trivial
2160.2.bw.a 16 36.h even 6 1 inner
2160.2.bw.c 16 4.b odd 2 1
2160.2.bw.c 16 9.d odd 6 1
6480.2.h.a 16 9.c even 3 1
6480.2.h.a 16 36.h even 6 1
6480.2.h.f 16 9.d odd 6 1
6480.2.h.f 16 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 6 T_{7}^{15} - 15 T_{7}^{14} - 162 T_{7}^{13} + 288 T_{7}^{12} + 3006 T_{7}^{11} + \cdots + 1296 \) acting on \(S_{2}^{\mathrm{new}}(2160, [\chi])\). 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
$5$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{16} + 6 T^{15} + \cdots + 1296 \) Copy content Toggle raw display
$11$ \( T^{16} - 6 T^{15} + \cdots + 2862864 \) Copy content Toggle raw display
$13$ \( T^{16} + 2 T^{15} + \cdots + 1024 \) Copy content Toggle raw display
$17$ \( T^{16} + 126 T^{14} + \cdots + 104976 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 24715612944 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 288728064 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 60156391824 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 13072720896 \) Copy content Toggle raw display
$37$ \( (T^{8} + 2 T^{7} + \cdots - 508784)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 7144495980561 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 2699225556624 \) Copy content Toggle raw display
$47$ \( T^{16} + 135 T^{14} + \cdots + 5184 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 13006946304 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 989531541504 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 12186393664 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 62361576729 \) Copy content Toggle raw display
$71$ \( (T^{8} - 144 T^{6} + \cdots + 15696)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 26 T^{7} + \cdots - 23622848)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 105936461991936 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 136048896 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 107495424 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 4014156503296 \) Copy content Toggle raw display
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