Properties

Label 882.2.e.k
Level $882$
Weight $2$
Character orbit 882.e
Analytic conductor $7.043$
Analytic rank $0$
Dimension $4$
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] = [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: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + (\beta_{3} - \beta_1) q^{3} + q^{4} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{5} + ( - \beta_{3} + \beta_1) q^{6} - q^{8} + ( - \beta_{3} + \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + (\beta_{3} - \beta_1) q^{3} + q^{4} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{5} + ( - \beta_{3} + \beta_1) q^{6} - q^{8} + ( - \beta_{3} + \beta_1 - 3) q^{9} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{10} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{11} + (\beta_{3} - \beta_1) q^{12} + ( - 2 \beta_{2} + 2) q^{13} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{15} + q^{16} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{17} + (\beta_{3} - \beta_1 + 3) q^{18} + ( - 5 \beta_{2} + 5) q^{19} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{20} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{22} + (\beta_{3} + 4 \beta_{2} - 2 \beta_1 + 1) q^{23} + ( - \beta_{3} + \beta_1) q^{24} + ( - 6 \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 7) q^{25} + (2 \beta_{2} - 2) q^{26} + ( - 2 \beta_{3} + 2 \beta_1 + 3) q^{27} + ( - 2 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 2) q^{29} + ( - 2 \beta_{3} + 3 \beta_{2} + \beta_1 - 6) q^{30} - 2 q^{31} - q^{32} + ( - \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 3) q^{33} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{34} + ( - \beta_{3} + \beta_1 - 3) q^{36} + (2 \beta_{2} - 2) q^{37} + (5 \beta_{2} - 5) q^{38} - 2 \beta_1 q^{39} + (\beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{40} + ( - 2 \beta_{3} - 8 \beta_{2} + \beta_1 + 7) q^{41} + ( - 3 \beta_{3} + \beta_{2} + 6 \beta_1 - 3) q^{43} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 1) q^{44} + (\beta_{3} - 3 \beta_{2} - 5 \beta_1 - 3) q^{45} + ( - \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 1) q^{46} + (\beta_{3} - \beta_1) q^{48} + (6 \beta_{3} - 4 \beta_{2} - 3 \beta_1 + 7) q^{50} + ( - \beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{51} + ( - 2 \beta_{2} + 2) q^{52} + ( - 2 \beta_{3} + 4 \beta_{2} + 4 \beta_1 - 2) q^{53} + (2 \beta_{3} - 2 \beta_1 - 3) q^{54} + 6 q^{55} - 5 \beta_1 q^{57} + (2 \beta_{3} - 4 \beta_{2} - 4 \beta_1 + 2) q^{58} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1) q^{59} + (2 \beta_{3} - 3 \beta_{2} - \beta_1 + 6) q^{60} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 7) q^{61} + 2 q^{62} + q^{64} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{65} + (\beta_{3} + 3 \beta_{2} - 2 \beta_1 + 3) q^{66} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 8) q^{67} + ( - \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{68} + (4 \beta_{3} + 3 \beta_{2} + \beta_1 - 6) q^{69} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{71} + (\beta_{3} - \beta_1 + 3) q^{72} + ( - 3 \beta_{3} + 5 \beta_{2} + 6 \beta_1 - 3) q^{73} + ( - 2 \beta_{2} + 2) q^{74} + (3 \beta_{3} + 9 \beta_{2} + 4 \beta_1 + 9) q^{75} + ( - 5 \beta_{2} + 5) q^{76} + 2 \beta_1 q^{78} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 5) q^{79} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{80} + (5 \beta_{3} - 5 \beta_1 + 6) q^{81} + (2 \beta_{3} + 8 \beta_{2} - \beta_1 - 7) q^{82} + ( - 4 \beta_{3} + 8 \beta_{2} + 8 \beta_1 - 4) q^{83} + (6 \beta_{2} - 6) q^{85} + (3 \beta_{3} - \beta_{2} - 6 \beta_1 + 3) q^{86} + (4 \beta_{3} - 6 \beta_{2} - 2 \beta_1 + 12) q^{87} + ( - 2 \beta_{3} - 2 \beta_{2} + \beta_1 + 1) q^{88} + (4 \beta_{3} - 8 \beta_{2} - 2 \beta_1 + 10) q^{89} + ( - \beta_{3} + 3 \beta_{2} + 5 \beta_1 + 3) q^{90} + (\beta_{3} + 4 \beta_{2} - 2 \beta_1 + 1) q^{92} + ( - 2 \beta_{3} + 2 \beta_1) q^{93} + (5 \beta_{3} + 5 \beta_{2} + 5 \beta_1 + 5) q^{95} + ( - \beta_{3} + \beta_1) q^{96} + (3 \beta_{3} - \beta_{2} - 6 \beta_1 + 3) q^{97} + ( - 5 \beta_{3} - 3 \beta_{2} + \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 3 q^{5} + 2 q^{6} - 4 q^{8} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} + 3 q^{5} + 2 q^{6} - 4 q^{8} - 10 q^{9} - 3 q^{10} - 3 q^{11} - 2 q^{12} + 4 q^{13} + 15 q^{15} + 4 q^{16} - 3 q^{17} + 10 q^{18} + 10 q^{19} + 3 q^{20} + 3 q^{22} + 9 q^{23} + 2 q^{24} - 11 q^{25} - 4 q^{26} + 16 q^{27} + 6 q^{29} - 15 q^{30} - 8 q^{31} - 4 q^{32} - 15 q^{33} + 3 q^{34} - 10 q^{36} - 4 q^{37} - 10 q^{38} - 2 q^{39} - 3 q^{40} + 15 q^{41} - q^{43} - 3 q^{44} - 24 q^{45} - 9 q^{46} - 2 q^{48} + 11 q^{50} + 18 q^{51} + 4 q^{52} + 6 q^{53} - 16 q^{54} + 24 q^{55} - 5 q^{57} - 6 q^{58} + 6 q^{59} + 15 q^{60} + 22 q^{61} + 8 q^{62} + 4 q^{64} + 12 q^{65} + 15 q^{66} + 26 q^{67} - 3 q^{68} - 21 q^{69} + 6 q^{71} + 10 q^{72} + 7 q^{73} + 4 q^{74} + 55 q^{75} + 10 q^{76} + 2 q^{78} + 14 q^{79} + 3 q^{80} + 14 q^{81} - 15 q^{82} + 12 q^{83} - 12 q^{85} + q^{86} + 30 q^{87} + 3 q^{88} + 18 q^{89} + 24 q^{90} + 9 q^{92} + 4 q^{93} + 30 q^{95} + 2 q^{96} + q^{97} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{3} + 2\beta _1 + 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1 + \beta_{2}\) \(-1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
373.1
1.68614 + 0.396143i
−1.18614 1.26217i
−1.18614 + 1.26217i
1.68614 0.396143i
−1.00000 −0.500000 1.65831i 1.00000 2.18614 + 3.78651i 0.500000 + 1.65831i 0 −1.00000 −2.50000 + 1.65831i −2.18614 3.78651i
373.2 −1.00000 −0.500000 + 1.65831i 1.00000 −0.686141 1.18843i 0.500000 1.65831i 0 −1.00000 −2.50000 1.65831i 0.686141 + 1.18843i
655.1 −1.00000 −0.500000 1.65831i 1.00000 −0.686141 + 1.18843i 0.500000 + 1.65831i 0 −1.00000 −2.50000 + 1.65831i 0.686141 1.18843i
655.2 −1.00000 −0.500000 + 1.65831i 1.00000 2.18614 3.78651i 0.500000 1.65831i 0 −1.00000 −2.50000 1.65831i −2.18614 + 3.78651i
\(n\): e.g. 2-40 or 990-1000
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.k 4
3.b odd 2 1 2646.2.e.m 4
7.b odd 2 1 882.2.e.l 4
7.c even 3 1 882.2.f.k 4
7.c even 3 1 882.2.h.n 4
7.d odd 6 1 126.2.f.d 4
7.d odd 6 1 882.2.h.m 4
9.c even 3 1 882.2.h.n 4
9.d odd 6 1 2646.2.h.l 4
21.c even 2 1 2646.2.e.n 4
21.g even 6 1 378.2.f.c 4
21.g even 6 1 2646.2.h.k 4
21.h odd 6 1 2646.2.f.j 4
21.h odd 6 1 2646.2.h.l 4
28.f even 6 1 1008.2.r.f 4
63.g even 3 1 882.2.f.k 4
63.h even 3 1 inner 882.2.e.k 4
63.h even 3 1 7938.2.a.bh 2
63.i even 6 1 1134.2.a.n 2
63.i even 6 1 2646.2.e.n 4
63.j odd 6 1 2646.2.e.m 4
63.j odd 6 1 7938.2.a.bs 2
63.k odd 6 1 126.2.f.d 4
63.l odd 6 1 882.2.h.m 4
63.n odd 6 1 2646.2.f.j 4
63.o even 6 1 2646.2.h.k 4
63.s even 6 1 378.2.f.c 4
63.t odd 6 1 882.2.e.l 4
63.t odd 6 1 1134.2.a.k 2
84.j odd 6 1 3024.2.r.f 4
252.n even 6 1 1008.2.r.f 4
252.r odd 6 1 9072.2.a.bb 2
252.bj even 6 1 9072.2.a.bm 2
252.bn odd 6 1 3024.2.r.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.f.d 4 7.d odd 6 1
126.2.f.d 4 63.k odd 6 1
378.2.f.c 4 21.g even 6 1
378.2.f.c 4 63.s even 6 1
882.2.e.k 4 1.a even 1 1 trivial
882.2.e.k 4 63.h even 3 1 inner
882.2.e.l 4 7.b odd 2 1
882.2.e.l 4 63.t odd 6 1
882.2.f.k 4 7.c even 3 1
882.2.f.k 4 63.g even 3 1
882.2.h.m 4 7.d odd 6 1
882.2.h.m 4 63.l odd 6 1
882.2.h.n 4 7.c even 3 1
882.2.h.n 4 9.c even 3 1
1008.2.r.f 4 28.f even 6 1
1008.2.r.f 4 252.n even 6 1
1134.2.a.k 2 63.t odd 6 1
1134.2.a.n 2 63.i even 6 1
2646.2.e.m 4 3.b odd 2 1
2646.2.e.m 4 63.j odd 6 1
2646.2.e.n 4 21.c even 2 1
2646.2.e.n 4 63.i even 6 1
2646.2.f.j 4 21.h odd 6 1
2646.2.f.j 4 63.n odd 6 1
2646.2.h.k 4 21.g even 6 1
2646.2.h.k 4 63.o even 6 1
2646.2.h.l 4 9.d odd 6 1
2646.2.h.l 4 21.h odd 6 1
3024.2.r.f 4 84.j odd 6 1
3024.2.r.f 4 252.bn odd 6 1
7938.2.a.bh 2 63.h even 3 1
7938.2.a.bs 2 63.j odd 6 1
9072.2.a.bb 2 252.r odd 6 1
9072.2.a.bm 2 252.bj even 6 1

Hecke kernels

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

\( T_{5}^{4} - 3T_{5}^{3} + 15T_{5}^{2} + 18T_{5} + 36 \) Copy content Toggle raw display
\( T_{11}^{4} + 3T_{11}^{3} + 15T_{11}^{2} - 18T_{11} + 36 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + 15 T^{2} + 18 T + 36 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 3 T^{3} + 15 T^{2} - 18 T + 36 \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 3 T^{3} + 15 T^{2} - 18 T + 36 \) Copy content Toggle raw display
$19$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 9 T^{3} + 69 T^{2} - 108 T + 144 \) Copy content Toggle raw display
$29$ \( T^{4} - 6 T^{3} + 60 T^{2} + 144 T + 576 \) Copy content Toggle raw display
$31$ \( (T + 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 15 T^{3} + 177 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$43$ \( T^{4} + T^{3} + 75 T^{2} - 74 T + 5476 \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + 60 T^{2} + 144 T + 576 \) Copy content Toggle raw display
$59$ \( (T^{2} - 3 T - 72)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 11 T - 44)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 13 T - 32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 3 T - 72)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 7 T^{3} + 111 T^{2} + \cdots + 3844 \) Copy content Toggle raw display
$79$ \( (T^{2} - 7 T - 62)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 12 T^{3} + 240 T^{2} + \cdots + 9216 \) Copy content Toggle raw display
$89$ \( T^{4} - 18 T^{3} + 276 T^{2} + \cdots + 2304 \) Copy content Toggle raw display
$97$ \( T^{4} - T^{3} + 75 T^{2} + 74 T + 5476 \) Copy content Toggle raw display
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