Properties

Label 882.2.e.p
Level $882$
Weight $2$
Character orbit 882.e
Analytic conductor $7.043$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [882,2,Mod(373,882)]
 
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([2, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.373");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.e (of order \(3\), 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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.309123.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{3} + q^{4} + (\beta_{5} - 2 \beta_{4} - \beta_1) q^{5} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{6} + q^{8} + (\beta_{5} - 2 \beta_{4} - 2 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{3} + q^{4} + (\beta_{5} - 2 \beta_{4} - \beta_1) q^{5} + (\beta_{5} + \beta_{4} - \beta_{3}) q^{6} + q^{8} + (\beta_{5} - 2 \beta_{4} - 2 \beta_1 - 1) q^{9} + (\beta_{5} - 2 \beta_{4} - \beta_1) q^{10} + ( - \beta_{5} - \beta_{3} + \cdots - \beta_1) q^{11}+ \cdots + (3 \beta_{5} + 7 \beta_{4} - \beta_{3} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 5 q^{5} + 2 q^{6} + 6 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 5 q^{5} + 2 q^{6} + 6 q^{8} - 4 q^{9} + 5 q^{10} - q^{11} + 2 q^{12} + 2 q^{13} - 2 q^{15} + 6 q^{16} + 4 q^{17} - 4 q^{18} + 3 q^{19} + 5 q^{20} - q^{22} - 7 q^{23} + 2 q^{24} - 2 q^{25} + 2 q^{26} - 7 q^{27} - 5 q^{29} - 2 q^{30} - 28 q^{31} + 6 q^{32} + 19 q^{33} + 4 q^{34} - 4 q^{36} - 9 q^{37} + 3 q^{38} + 9 q^{39} + 5 q^{40} + 12 q^{41} + 18 q^{43} - q^{44} - 29 q^{45} - 7 q^{46} + 6 q^{47} + 2 q^{48} - 2 q^{50} + 26 q^{51} + 2 q^{52} + 9 q^{53} - 7 q^{54} - 14 q^{55} + 2 q^{57} - 5 q^{58} + 8 q^{59} - 2 q^{60} + 8 q^{61} - 28 q^{62} + 6 q^{64} + 24 q^{65} + 19 q^{66} - 10 q^{67} + 4 q^{68} + q^{69} + 14 q^{71} - 4 q^{72} + 25 q^{73} - 9 q^{74} - 44 q^{75} + 3 q^{76} + 9 q^{78} - 14 q^{79} + 5 q^{80} - 40 q^{81} + 12 q^{82} - 8 q^{83} + 14 q^{85} + 18 q^{86} - 31 q^{87} - q^{88} + 9 q^{89} - 29 q^{90} - 7 q^{92} + 6 q^{94} - 4 q^{95} + 2 q^{96} + 28 q^{97} - 41 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} + 5\nu^{3} + \nu^{2} + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} - 5\nu^{3} + 2\nu^{2} - 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} - 16\nu^{3} + 19\nu^{2} - 21\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{5} - 5\nu^{4} + 19\nu^{3} - 22\nu^{2} + 33\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 3\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 3\beta_{4} - 5\beta_{3} - 3\beta_{2} - 6\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{5} - 2\beta_{4} - 11\beta_{3} - 6\beta_{2} + 8\beta _1 + 7 \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1 - \beta_{4}\) \(-1 - \beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
373.1
0.500000 1.41036i
0.500000 + 2.05195i
0.500000 + 0.224437i
0.500000 + 1.41036i
0.500000 2.05195i
0.500000 0.224437i
1.00000 −1.09097 1.34528i 1.00000 0.880438 + 1.52496i −1.09097 1.34528i 0 1.00000 −0.619562 + 2.93533i 0.880438 + 1.52496i
373.2 1.00000 0.796790 1.53790i 1.00000 −0.230252 0.398809i 0.796790 1.53790i 0 1.00000 −1.73025 2.45076i −0.230252 0.398809i
373.3 1.00000 1.29418 + 1.15113i 1.00000 1.84981 + 3.20397i 1.29418 + 1.15113i 0 1.00000 0.349814 + 2.97954i 1.84981 + 3.20397i
655.1 1.00000 −1.09097 + 1.34528i 1.00000 0.880438 1.52496i −1.09097 + 1.34528i 0 1.00000 −0.619562 2.93533i 0.880438 1.52496i
655.2 1.00000 0.796790 + 1.53790i 1.00000 −0.230252 + 0.398809i 0.796790 + 1.53790i 0 1.00000 −1.73025 + 2.45076i −0.230252 + 0.398809i
655.3 1.00000 1.29418 1.15113i 1.00000 1.84981 3.20397i 1.29418 1.15113i 0 1.00000 0.349814 2.97954i 1.84981 3.20397i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 373.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.e.p 6
3.b odd 2 1 2646.2.e.o 6
7.b odd 2 1 126.2.e.d 6
7.c even 3 1 882.2.f.m 6
7.c even 3 1 882.2.h.o 6
7.d odd 6 1 126.2.h.c yes 6
7.d odd 6 1 882.2.f.l 6
9.c even 3 1 882.2.h.o 6
9.d odd 6 1 2646.2.h.p 6
21.c even 2 1 378.2.e.c 6
21.g even 6 1 378.2.h.d 6
21.g even 6 1 2646.2.f.o 6
21.h odd 6 1 2646.2.f.n 6
21.h odd 6 1 2646.2.h.p 6
28.d even 2 1 1008.2.q.h 6
28.f even 6 1 1008.2.t.g 6
63.g even 3 1 882.2.f.m 6
63.h even 3 1 inner 882.2.e.p 6
63.h even 3 1 7938.2.a.by 3
63.i even 6 1 378.2.e.c 6
63.i even 6 1 7938.2.a.bu 3
63.j odd 6 1 2646.2.e.o 6
63.j odd 6 1 7938.2.a.bx 3
63.k odd 6 1 882.2.f.l 6
63.k odd 6 1 1134.2.g.k 6
63.l odd 6 1 126.2.h.c yes 6
63.l odd 6 1 1134.2.g.k 6
63.n odd 6 1 2646.2.f.n 6
63.o even 6 1 378.2.h.d 6
63.o even 6 1 1134.2.g.n 6
63.s even 6 1 1134.2.g.n 6
63.s even 6 1 2646.2.f.o 6
63.t odd 6 1 126.2.e.d 6
63.t odd 6 1 7938.2.a.cb 3
84.h odd 2 1 3024.2.q.h 6
84.j odd 6 1 3024.2.t.g 6
252.r odd 6 1 3024.2.q.h 6
252.s odd 6 1 3024.2.t.g 6
252.bi even 6 1 1008.2.t.g 6
252.bj even 6 1 1008.2.q.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.e.d 6 7.b odd 2 1
126.2.e.d 6 63.t odd 6 1
126.2.h.c yes 6 7.d odd 6 1
126.2.h.c yes 6 63.l odd 6 1
378.2.e.c 6 21.c even 2 1
378.2.e.c 6 63.i even 6 1
378.2.h.d 6 21.g even 6 1
378.2.h.d 6 63.o even 6 1
882.2.e.p 6 1.a even 1 1 trivial
882.2.e.p 6 63.h even 3 1 inner
882.2.f.l 6 7.d odd 6 1
882.2.f.l 6 63.k odd 6 1
882.2.f.m 6 7.c even 3 1
882.2.f.m 6 63.g even 3 1
882.2.h.o 6 7.c even 3 1
882.2.h.o 6 9.c even 3 1
1008.2.q.h 6 28.d even 2 1
1008.2.q.h 6 252.bj even 6 1
1008.2.t.g 6 28.f even 6 1
1008.2.t.g 6 252.bi even 6 1
1134.2.g.k 6 63.k odd 6 1
1134.2.g.k 6 63.l odd 6 1
1134.2.g.n 6 63.o even 6 1
1134.2.g.n 6 63.s even 6 1
2646.2.e.o 6 3.b odd 2 1
2646.2.e.o 6 63.j odd 6 1
2646.2.f.n 6 21.h odd 6 1
2646.2.f.n 6 63.n odd 6 1
2646.2.f.o 6 21.g even 6 1
2646.2.f.o 6 63.s even 6 1
2646.2.h.p 6 9.d odd 6 1
2646.2.h.p 6 21.h odd 6 1
3024.2.q.h 6 84.h odd 2 1
3024.2.q.h 6 252.r odd 6 1
3024.2.t.g 6 84.j odd 6 1
3024.2.t.g 6 252.s odd 6 1
7938.2.a.bu 3 63.i even 6 1
7938.2.a.bx 3 63.j odd 6 1
7938.2.a.by 3 63.h even 3 1
7938.2.a.cb 3 63.t odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{6} - 5T_{5}^{5} + 21T_{5}^{4} - 26T_{5}^{3} + 31T_{5}^{2} + 12T_{5} + 9 \) Copy content Toggle raw display
\( T_{11}^{6} + T_{11}^{5} + 27T_{11}^{4} - 92T_{11}^{3} + 643T_{11}^{2} - 858T_{11} + 1089 \) Copy content Toggle raw display
\( T_{13}^{6} - 2T_{13}^{5} + 7T_{13}^{4} + 15T_{13}^{2} - 9T_{13} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 2 T^{5} + \cdots + 27 \) Copy content Toggle raw display
$5$ \( T^{6} - 5 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + T^{5} + \cdots + 1089 \) Copy content Toggle raw display
$13$ \( T^{6} - 2 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$17$ \( T^{6} - 4 T^{5} + \cdots + 28224 \) Copy content Toggle raw display
$19$ \( T^{6} - 3 T^{5} + \cdots + 49 \) Copy content Toggle raw display
$23$ \( T^{6} + 7 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$29$ \( T^{6} + 5 T^{5} + \cdots + 1089 \) Copy content Toggle raw display
$31$ \( (T^{3} + 14 T^{2} + \cdots - 27)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 9 T^{5} + \cdots + 5329 \) Copy content Toggle raw display
$41$ \( T^{6} - 12 T^{5} + \cdots + 729 \) Copy content Toggle raw display
$43$ \( T^{6} - 18 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$47$ \( (T^{3} - 3 T^{2} - 24 T - 27)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} - 9 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$59$ \( (T^{3} - 4 T^{2} + \cdots - 177)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 4 T^{2} + \cdots + 717)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 5 T^{2} + \cdots - 149)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 7 T^{2} - 50 T + 99)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 25 T^{5} + \cdots + 2401 \) Copy content Toggle raw display
$79$ \( (T^{3} + 7 T^{2} + \cdots - 771)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 8 T^{5} + \cdots + 8649 \) Copy content Toggle raw display
$89$ \( T^{6} - 9 T^{5} + \cdots + 3969 \) Copy content Toggle raw display
$97$ \( T^{6} - 28 T^{5} + \cdots + 287296 \) Copy content Toggle raw display
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