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Properties

Label 882.2.l.a
Level $882$
Weight $2$
Character orbit 882.l
Analytic conductor $7.043$
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] = [882,2,Mod(227,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.227");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.l (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: \(7.04280545828\)
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: \( x^{16} - 6x^{14} + 9x^{12} + 54x^{10} - 288x^{8} + 486x^{6} + 729x^{4} - 4374x^{2} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 126)
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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + \beta_{6} q^{3} - q^{4} + ( - \beta_{14} + \beta_{13} + \cdots - \beta_1) q^{5}+ \cdots + (\beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + \beta_{6} q^{3} - q^{4} + ( - \beta_{14} + \beta_{13} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{15} - \beta_{10} + \cdots - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} - 12 q^{9} + 12 q^{11} + 16 q^{16} + 12 q^{18} + 48 q^{23} - 8 q^{25} - 12 q^{29} + 12 q^{30} + 12 q^{36} + 4 q^{37} + 4 q^{43} - 12 q^{44} - 12 q^{46} + 60 q^{50} + 24 q^{51} + 48 q^{57} - 12 q^{58} - 16 q^{64} + 56 q^{67} - 12 q^{72} - 36 q^{74} - 24 q^{78} + 8 q^{79} - 12 q^{85} + 24 q^{86} - 48 q^{92} + 84 q^{93} - 72 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6x^{14} + 9x^{12} + 54x^{10} - 288x^{8} + 486x^{6} + 729x^{4} - 4374x^{2} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{14} - \nu^{12} + 6\nu^{10} - 36\nu^{8} + 72\nu^{6} + 234\nu^{4} + 729\nu^{2} - 243 ) / 1944 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{14} - \nu^{12} + 6\nu^{10} - 36\nu^{8} + 180\nu^{6} + 396\nu^{4} - 972\nu^{2} + 4131 ) / 1944 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{14} - 3\nu^{12} - 9\nu^{10} + 81\nu^{8} - 126\nu^{6} - 135\nu^{4} + 1458\nu^{2} - 2187 ) / 1458 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{15} - 12\nu^{13} - 144\nu^{11} + 432\nu^{9} - 468\nu^{7} - 2754\nu^{5} + 9477\nu^{3} - 13122\nu ) / 17496 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{15} + 21\nu^{13} - 18\nu^{11} - 108\nu^{9} + 576\nu^{7} - 648\nu^{5} - 972\nu^{3} + 9477\nu ) / 5832 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{15} + 6\nu^{13} - 9\nu^{11} - 54\nu^{9} + 288\nu^{7} - 486\nu^{5} - 729\nu^{3} + 4374\nu ) / 2187 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -5\nu^{14} + 18\nu^{12} - 216\nu^{8} + 792\nu^{6} + 54\nu^{4} - 4617\nu^{2} + 8748 ) / 5832 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -2\nu^{14} + 21\nu^{12} - 18\nu^{10} - 108\nu^{8} + 576\nu^{6} - 648\nu^{4} - 972\nu^{2} + 9477 ) / 5832 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{14} - 21\nu^{12} + 126\nu^{10} - 612\nu^{6} + 2862\nu^{4} - 2673\nu^{2} + 729 ) / 5832 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{15} - 3\nu^{13} + 27\nu^{9} - 45\nu^{7} + 108\nu^{5} + 324\nu^{3} ) / 1458 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -\nu^{15} + 3\nu^{13} + 9\nu^{11} - 81\nu^{9} + 126\nu^{7} + 135\nu^{5} - 1458\nu^{3} + 2187\nu ) / 1458 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -5\nu^{15} + 18\nu^{13} - 216\nu^{9} + 792\nu^{7} + 54\nu^{5} - 4617\nu^{3} + 8748\nu ) / 5832 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -8\nu^{15} + 21\nu^{13} + 18\nu^{11} - 324\nu^{9} + 684\nu^{7} - 4050\nu^{3} + 3645\nu ) / 5832 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -3\nu^{14} + 13\nu^{12} + 12\nu^{10} - 180\nu^{8} + 594\nu^{6} - 180\nu^{4} - 3321\nu^{2} + 8505 ) / 972 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{14} - 2\beta_{13} - \beta_{12} + 3\beta_{11} + \beta_{6} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{15} + \beta_{10} + 3\beta_{9} + 6\beta_{8} - \beta_{4} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{14} + 3\beta_{13} + 6\beta_{11} - 6\beta_{7} + 3\beta_{6} + 3\beta_{5} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{15} - 6\beta_{9} + 3\beta_{8} + 12\beta_{4} + 3\beta_{3} + 3\beta_{2} - 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9\beta_{14} + 3\beta_{13} - 12\beta_{12} + 18\beta_{11} + 9\beta_{7} - 3\beta_{6} - 9\beta_{5} - 12\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3\beta_{15} + 12\beta_{10} - 3\beta_{9} + 30\beta_{8} + 42\beta_{4} - 3\beta_{3} - 12\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -36\beta_{14} + 45\beta_{13} - 45\beta_{12} - 45\beta_{11} + 9\beta_{7} - 18\beta_{6} - 45\beta_{5} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 45\beta_{15} + 45\beta_{10} + 27\beta_{9} - 135\beta_{8} + 63\beta_{4} - 18\beta_{3} - 108 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -90\beta_{13} - 126\beta_{12} - 54\beta_{11} + 135\beta_{7} - 36\beta_{6} - 270\beta_{5} - 90\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -90\beta_{15} + 126\beta_{10} + 648\beta_{9} - 126\beta_{4} - 36\beta_{3} - 90\beta_{2} - 405 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -270\beta_{14} + 54\beta_{12} - 378\beta_{11} - 270\beta_{7} + 486\beta_{6} - 108\beta_{5} - 243\beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -54\beta_{10} + 621\beta_{9} - 999\beta_{8} - 378\beta_{4} + 486\beta_{3} - 243\beta_{2} - 1134 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( -729\beta_{14} - 756\beta_{13} + 1161\beta_{12} + 729\beta_{11} + 1161\beta_{6} + 162\beta_{5} - 1917\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(785\)
\(\chi(n)\) \(1 - \beta_{9}\) \(1 - \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
227.1
−0.0967785 − 1.72934i
−1.69547 + 0.354107i
1.69547 − 0.354107i
0.0967785 + 1.72934i
−1.62181 − 0.608059i
1.40917 − 1.00709i
−1.40917 + 1.00709i
1.62181 + 0.608059i
−1.62181 + 0.608059i
1.40917 + 1.00709i
−1.40917 − 1.00709i
1.62181 − 0.608059i
−0.0967785 + 1.72934i
−1.69547 − 0.354107i
1.69547 + 0.354107i
0.0967785 − 1.72934i
− 1.00000i −1.54605 − 0.780860i −1.00000 0.183299 − 0.317483i −0.780860 + 1.54605i 0 1.00000i 1.78052 + 2.41449i −0.317483 − 0.183299i
227.2 − 1.00000i −0.541068 + 1.64537i −1.00000 −0.895175 + 1.55049i 1.64537 + 0.541068i 0 1.00000i −2.41449 − 1.78052i 1.55049 + 0.895175i
227.3 − 1.00000i 0.541068 − 1.64537i −1.00000 0.895175 − 1.55049i −1.64537 − 0.541068i 0 1.00000i −2.41449 − 1.78052i −1.55049 − 0.895175i
227.4 − 1.00000i 1.54605 + 0.780860i −1.00000 −0.183299 + 0.317483i 0.780860 − 1.54605i 0 1.00000i 1.78052 + 2.41449i 0.317483 + 0.183299i
227.5 1.00000i −1.33750 + 1.10050i −1.00000 −1.94556 + 3.36980i −1.10050 − 1.33750i 0 − 1.00000i 0.577806 − 2.94383i −3.36980 − 1.94556i
227.6 1.00000i −0.167584 − 1.72392i −1.00000 −1.17468 + 2.03460i 1.72392 − 0.167584i 0 − 1.00000i −2.94383 + 0.577806i −2.03460 − 1.17468i
227.7 1.00000i 0.167584 + 1.72392i −1.00000 1.17468 − 2.03460i −1.72392 + 0.167584i 0 − 1.00000i −2.94383 + 0.577806i 2.03460 + 1.17468i
227.8 1.00000i 1.33750 − 1.10050i −1.00000 1.94556 − 3.36980i 1.10050 + 1.33750i 0 − 1.00000i 0.577806 − 2.94383i 3.36980 + 1.94556i
509.1 − 1.00000i −1.33750 − 1.10050i −1.00000 −1.94556 − 3.36980i −1.10050 + 1.33750i 0 1.00000i 0.577806 + 2.94383i −3.36980 + 1.94556i
509.2 − 1.00000i −0.167584 + 1.72392i −1.00000 −1.17468 − 2.03460i 1.72392 + 0.167584i 0 1.00000i −2.94383 − 0.577806i −2.03460 + 1.17468i
509.3 − 1.00000i 0.167584 − 1.72392i −1.00000 1.17468 + 2.03460i −1.72392 − 0.167584i 0 1.00000i −2.94383 − 0.577806i 2.03460 − 1.17468i
509.4 − 1.00000i 1.33750 + 1.10050i −1.00000 1.94556 + 3.36980i 1.10050 − 1.33750i 0 1.00000i 0.577806 + 2.94383i 3.36980 − 1.94556i
509.5 1.00000i −1.54605 + 0.780860i −1.00000 0.183299 + 0.317483i −0.780860 − 1.54605i 0 − 1.00000i 1.78052 − 2.41449i −0.317483 + 0.183299i
509.6 1.00000i −0.541068 − 1.64537i −1.00000 −0.895175 − 1.55049i 1.64537 − 0.541068i 0 − 1.00000i −2.41449 + 1.78052i 1.55049 − 0.895175i
509.7 1.00000i 0.541068 + 1.64537i −1.00000 0.895175 + 1.55049i −1.64537 + 0.541068i 0 − 1.00000i −2.41449 + 1.78052i −1.55049 + 0.895175i
509.8 1.00000i 1.54605 − 0.780860i −1.00000 −0.183299 − 0.317483i 0.780860 + 1.54605i 0 − 1.00000i 1.78052 − 2.41449i 0.317483 − 0.183299i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 227.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.i even 6 1 inner
63.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.l.a 16
3.b odd 2 1 2646.2.l.b 16
7.b odd 2 1 inner 882.2.l.a 16
7.c even 3 1 126.2.m.a ✓ 16
7.c even 3 1 882.2.t.b 16
7.d odd 6 1 126.2.m.a ✓ 16
7.d odd 6 1 882.2.t.b 16
9.c even 3 1 2646.2.t.a 16
9.d odd 6 1 882.2.t.b 16
21.c even 2 1 2646.2.l.b 16
21.g even 6 1 378.2.m.a 16
21.g even 6 1 2646.2.t.a 16
21.h odd 6 1 378.2.m.a 16
21.h odd 6 1 2646.2.t.a 16
28.f even 6 1 1008.2.cc.b 16
28.g odd 6 1 1008.2.cc.b 16
63.g even 3 1 378.2.m.a 16
63.h even 3 1 1134.2.d.a 16
63.h even 3 1 2646.2.l.b 16
63.i even 6 1 inner 882.2.l.a 16
63.i even 6 1 1134.2.d.a 16
63.j odd 6 1 inner 882.2.l.a 16
63.j odd 6 1 1134.2.d.a 16
63.k odd 6 1 378.2.m.a 16
63.l odd 6 1 2646.2.t.a 16
63.n odd 6 1 126.2.m.a ✓ 16
63.o even 6 1 882.2.t.b 16
63.s even 6 1 126.2.m.a ✓ 16
63.t odd 6 1 1134.2.d.a 16
63.t odd 6 1 2646.2.l.b 16
84.j odd 6 1 3024.2.cc.b 16
84.n even 6 1 3024.2.cc.b 16
252.n even 6 1 3024.2.cc.b 16
252.o even 6 1 1008.2.cc.b 16
252.bl odd 6 1 3024.2.cc.b 16
252.bn odd 6 1 1008.2.cc.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.m.a ✓ 16 7.c even 3 1
126.2.m.a ✓ 16 7.d odd 6 1
126.2.m.a ✓ 16 63.n odd 6 1
126.2.m.a ✓ 16 63.s even 6 1
378.2.m.a 16 21.g even 6 1
378.2.m.a 16 21.h odd 6 1
378.2.m.a 16 63.g even 3 1
378.2.m.a 16 63.k odd 6 1
882.2.l.a 16 1.a even 1 1 trivial
882.2.l.a 16 7.b odd 2 1 inner
882.2.l.a 16 63.i even 6 1 inner
882.2.l.a 16 63.j odd 6 1 inner
882.2.t.b 16 7.c even 3 1
882.2.t.b 16 7.d odd 6 1
882.2.t.b 16 9.d odd 6 1
882.2.t.b 16 63.o even 6 1
1008.2.cc.b 16 28.f even 6 1
1008.2.cc.b 16 28.g odd 6 1
1008.2.cc.b 16 252.o even 6 1
1008.2.cc.b 16 252.bn odd 6 1
1134.2.d.a 16 63.h even 3 1
1134.2.d.a 16 63.i even 6 1
1134.2.d.a 16 63.j odd 6 1
1134.2.d.a 16 63.t odd 6 1
2646.2.l.b 16 3.b odd 2 1
2646.2.l.b 16 21.c even 2 1
2646.2.l.b 16 63.h even 3 1
2646.2.l.b 16 63.t odd 6 1
2646.2.t.a 16 9.c even 3 1
2646.2.t.a 16 21.g even 6 1
2646.2.t.a 16 21.h odd 6 1
2646.2.t.a 16 63.l odd 6 1
3024.2.cc.b 16 84.j odd 6 1
3024.2.cc.b 16 84.n even 6 1
3024.2.cc.b 16 252.n even 6 1
3024.2.cc.b 16 252.bl odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{16} + 24T_{5}^{14} + 423T_{5}^{12} + 3096T_{5}^{10} + 16461T_{5}^{8} + 42336T_{5}^{6} + 77436T_{5}^{4} + 10368T_{5}^{2} + 1296 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + 6 T^{14} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} + 24 T^{14} + \cdots + 1296 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 6 T^{7} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 36 T^{14} + \cdots + 331776 \) Copy content Toggle raw display
$17$ \( T^{16} + 42 T^{14} + \cdots + 331776 \) Copy content Toggle raw display
$19$ \( T^{16} - 54 T^{14} + \cdots + 2313441 \) Copy content Toggle raw display
$23$ \( (T^{8} - 24 T^{7} + \cdots + 443556)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + 6 T^{7} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 144 T^{6} + \cdots + 746496)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 2 T^{7} + \cdots + 1784896)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 73499483897856 \) Copy content Toggle raw display
$43$ \( (T^{8} - 2 T^{7} + \cdots + 10816)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 240 T^{6} + \cdots + 1218816)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( (T^{8} - 294 T^{6} + \cdots + 36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 240 T^{6} + \cdots + 1557504)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 14 T^{3} + \cdots - 908)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + 90 T^{6} + \cdots + 82944)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 2927055626496 \) Copy content Toggle raw display
$79$ \( (T^{4} - 2 T^{3} + \cdots - 1202)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 33\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 34828517376 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 45\!\cdots\!56 \) Copy content Toggle raw display
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