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Properties

Label 2160.3.bs.b
Level $2160$
Weight $3$
Character orbit 2160.bs
Analytic conductor $58.856$
Analytic rank $0$
Dimension $12$
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] = [2160,3,Mod(881,2160)]
 
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([0, 0, 5, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2160.881");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2160.bs (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: \(58.8557371018\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} - 11 x^{10} + 60 x^{9} - 120 x^{8} - 342 x^{7} + 2709 x^{6} - 3078 x^{5} + \cdots + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: no (minimal twist has level 180)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{5} + ( - \beta_{9} - 2 \beta_{4} + \beta_{3} + \cdots + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{5} + ( - \beta_{9} - 2 \beta_{4} + \beta_{3} + \cdots + 1) q^{7}+ \cdots + (\beta_{9} + \beta_{8} - 13 \beta_{7} + \cdots + 12) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{7} + 48 q^{11} - 30 q^{13} - 72 q^{19} - 78 q^{23} + 30 q^{25} - 150 q^{29} + 12 q^{31} - 12 q^{37} - 90 q^{41} - 114 q^{43} + 12 q^{47} + 48 q^{49} + 120 q^{55} + 48 q^{59} - 78 q^{61} + 150 q^{65} + 168 q^{67} - 24 q^{73} + 258 q^{77} - 120 q^{79} + 114 q^{83} - 30 q^{85} - 120 q^{91} + 120 q^{95} + 96 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 11 x^{10} + 60 x^{9} - 120 x^{8} - 342 x^{7} + 2709 x^{6} - 3078 x^{5} + \cdots + 531441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 239 \nu^{11} - 7615 \nu^{10} + 11402 \nu^{9} + 41574 \nu^{8} - 169188 \nu^{7} + 454671 \nu^{6} + \cdots + 109358748 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1264 \nu^{11} - 7514 \nu^{10} + 7570 \nu^{9} + 69045 \nu^{8} - 447600 \nu^{7} + 644742 \nu^{6} + \cdots + 255564072 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1654 \nu^{11} - 10768 \nu^{10} + 59306 \nu^{9} - 60720 \nu^{8} - 389148 \nu^{7} + \cdots + 1145314404 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4012 \nu^{11} + 10310 \nu^{10} - 3856 \nu^{9} - 127275 \nu^{8} + 811137 \nu^{7} + \cdots - 459460269 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 488 \nu^{11} - 3260 \nu^{10} + 14803 \nu^{9} + 16767 \nu^{8} - 186105 \nu^{7} + \cdots + 222024240 ) / 10609137 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6011 \nu^{11} + 21373 \nu^{10} - 2963 \nu^{9} - 323391 \nu^{8} + 1355442 \nu^{7} + \cdots - 817946748 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7039 \nu^{11} - 36830 \nu^{10} + 57823 \nu^{9} + 286080 \nu^{8} - 2087490 \nu^{7} + \cdots + 2411974503 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 8222 \nu^{11} + 44803 \nu^{10} - 96848 \nu^{9} - 183999 \nu^{8} + 2280102 \nu^{7} + \cdots - 1285614828 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 8998 \nu^{11} - 31784 \nu^{10} + 10651 \nu^{9} + 423195 \nu^{8} - 1888167 \nu^{7} + \cdots + 1131201693 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 9181 \nu^{11} + 94979 \nu^{10} - 184516 \nu^{9} - 661812 \nu^{8} + 4788867 \nu^{7} + \cdots - 4262097771 ) / 95482233 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9721 \nu^{11} + 68357 \nu^{10} + 18254 \nu^{9} - 630789 \nu^{8} + 2825589 \nu^{7} + \cdots - 2517199821 ) / 95482233 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{6} + \beta_{4} + \beta_{2} - \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} - 2 \beta_{10} - \beta_{9} + 2 \beta_{8} + 2 \beta_{7} + 4 \beta_{6} - \beta_{5} - 3 \beta_{4} + \cdots + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} + \beta_{10} + 2 \beta_{9} + 5 \beta_{8} - \beta_{7} + 4 \beta_{6} - \beta_{5} - 15 \beta_{4} + \cdots - 30 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{11} + 7 \beta_{10} - 10 \beta_{9} - \beta_{8} - 7 \beta_{7} + 4 \beta_{6} + 11 \beta_{5} + \cdots + 132 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2 \beta_{11} + 52 \beta_{10} + 35 \beta_{9} - 19 \beta_{8} + 65 \beta_{7} + 31 \beta_{6} + 29 \beta_{5} + \cdots - 102 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 11 \beta_{11} - 29 \beta_{10} - 46 \beta_{9} - 19 \beta_{8} + 245 \beta_{7} + 229 \beta_{6} + \cdots - 219 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 160 \beta_{11} - 38 \beta_{10} - 28 \beta_{9} + 260 \beta_{8} + 866 \beta_{7} + 22 \beta_{6} + \cdots - 4035 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 344 \beta_{11} + 412 \beta_{10} + 422 \beta_{9} - 190 \beta_{8} - 1456 \beta_{7} - 2534 \beta_{6} + \cdots + 870 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1906 \beta_{11} + 2644 \beta_{10} + 1655 \beta_{9} - 2746 \beta_{8} + 1082 \beta_{7} - 6656 \beta_{6} + \cdots + 4236 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 79 \beta_{11} + 4201 \beta_{10} + 8990 \beta_{9} - 7543 \beta_{8} + 1460 \beta_{7} - 7808 \beta_{6} + \cdots + 14406 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 22444 \beta_{11} - 23834 \beta_{10} + 1529 \beta_{9} + 323 \beta_{8} + 54110 \beta_{7} + 8590 \beta_{6} + \cdots - 98949 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
2.65605 + 1.39478i
−2.99781 + 0.114662i
0.841761 − 2.87949i
0.459278 + 2.96464i
−2.85525 + 0.920635i
2.89597 − 0.783177i
2.65605 − 1.39478i
−2.99781 − 0.114662i
0.841761 + 2.87949i
0.459278 − 2.96464i
−2.85525 − 0.920635i
2.89597 + 0.783177i
0 0 0 −1.93649 − 1.11803i 0 −1.41583 − 2.45229i 0 0 0
881.2 0 0 0 −1.93649 − 1.11803i 0 0.801399 + 1.38806i 0 0 0
881.3 0 0 0 −1.93649 − 1.11803i 0 5.98742 + 10.3705i 0 0 0
881.4 0 0 0 1.93649 + 1.11803i 0 −4.13490 − 7.16186i 0 0 0
881.5 0 0 0 1.93649 + 1.11803i 0 −0.594587 − 1.02985i 0 0 0
881.6 0 0 0 1.93649 + 1.11803i 0 2.35650 + 4.08158i 0 0 0
1601.1 0 0 0 −1.93649 + 1.11803i 0 −1.41583 + 2.45229i 0 0 0
1601.2 0 0 0 −1.93649 + 1.11803i 0 0.801399 − 1.38806i 0 0 0
1601.3 0 0 0 −1.93649 + 1.11803i 0 5.98742 − 10.3705i 0 0 0
1601.4 0 0 0 1.93649 − 1.11803i 0 −4.13490 + 7.16186i 0 0 0
1601.5 0 0 0 1.93649 − 1.11803i 0 −0.594587 + 1.02985i 0 0 0
1601.6 0 0 0 1.93649 − 1.11803i 0 2.35650 − 4.08158i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2160.3.bs.b 12
3.b odd 2 1 720.3.bs.b 12
4.b odd 2 1 540.3.o.b 12
9.c even 3 1 720.3.bs.b 12
9.d odd 6 1 inner 2160.3.bs.b 12
12.b even 2 1 180.3.o.b ✓ 12
20.d odd 2 1 2700.3.p.c 12
20.e even 4 2 2700.3.u.c 24
36.f odd 6 1 180.3.o.b ✓ 12
36.f odd 6 1 1620.3.g.b 12
36.h even 6 1 540.3.o.b 12
36.h even 6 1 1620.3.g.b 12
60.h even 2 1 900.3.p.c 12
60.l odd 4 2 900.3.u.c 24
180.n even 6 1 2700.3.p.c 12
180.p odd 6 1 900.3.p.c 12
180.v odd 12 2 2700.3.u.c 24
180.x even 12 2 900.3.u.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.o.b ✓ 12 12.b even 2 1
180.3.o.b ✓ 12 36.f odd 6 1
540.3.o.b 12 4.b odd 2 1
540.3.o.b 12 36.h even 6 1
720.3.bs.b 12 3.b odd 2 1
720.3.bs.b 12 9.c even 3 1
900.3.p.c 12 60.h even 2 1
900.3.p.c 12 180.p odd 6 1
900.3.u.c 24 60.l odd 4 2
900.3.u.c 24 180.x even 12 2
1620.3.g.b 12 36.f odd 6 1
1620.3.g.b 12 36.h even 6 1
2160.3.bs.b 12 1.a even 1 1 trivial
2160.3.bs.b 12 9.d odd 6 1 inner
2700.3.p.c 12 20.d odd 2 1
2700.3.p.c 12 180.n even 6 1
2700.3.u.c 24 20.e even 4 2
2700.3.u.c 24 180.v odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} - 6 T_{7}^{11} + 141 T_{7}^{10} + 50 T_{7}^{9} + 11340 T_{7}^{8} - 14346 T_{7}^{7} + \cdots + 6345361 \) acting on \(S_{3}^{\mathrm{new}}(2160, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{2} + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} - 6 T^{11} + \cdots + 6345361 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 416649744 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 15716943091600 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 191070131587344 \) Copy content Toggle raw display
$19$ \( (T^{6} + 36 T^{5} + \cdots - 19580204)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 13626529936569 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 3840796442025 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 177690926165776 \) Copy content Toggle raw display
$37$ \( (T^{6} + 6 T^{5} + \cdots - 2144769884)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 29\!\cdots\!25 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 67\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 47\!\cdots\!49 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 29\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 11\!\cdots\!41 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 50\!\cdots\!41 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 28\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( (T^{6} + 12 T^{5} + \cdots - 2761132736)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 15\!\cdots\!29 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 13\!\cdots\!29 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 60\!\cdots\!16 \) Copy content Toggle raw display
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