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Properties

Label 882.2.e.t
Level $882$
Weight $2$
Character orbit 882.e
Analytic conductor $7.043$
Analytic rank $0$
Dimension $8$
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(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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.3317760000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( - \beta_{6} + \beta_{5}) q^{3} + q^{4} + (\beta_{6} - 2 \beta_{5} + \cdots + 2 \beta_1) q^{5}+ \cdots + (2 \beta_{4} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + ( - \beta_{6} + \beta_{5}) q^{3} + q^{4} + (\beta_{6} - 2 \beta_{5} + \cdots + 2 \beta_1) q^{5}+ \cdots + (4 \beta_{7} - 8 \beta_{4} + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} + 8 q^{4} + 8 q^{8} + 8 q^{9} - 16 q^{11} + 32 q^{15} + 8 q^{16} + 8 q^{18} - 16 q^{22} + 4 q^{23} - 12 q^{25} - 20 q^{29} + 32 q^{30} + 8 q^{32} + 8 q^{36} + 8 q^{37} - 24 q^{39} - 20 q^{43} - 16 q^{44} + 4 q^{46} - 12 q^{50} - 4 q^{51} - 4 q^{53} - 20 q^{57} - 20 q^{58} + 32 q^{60} + 8 q^{64} + 24 q^{65} - 24 q^{67} - 48 q^{71} + 8 q^{72} + 8 q^{74} - 24 q^{78} + 16 q^{79} + 4 q^{81} + 4 q^{85} - 20 q^{86} - 16 q^{88} + 4 q^{92} + 60 q^{93} + 80 q^{95} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 7x^{4} - 36x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 14\nu^{4} + 7\nu^{2} - 36 ) / 63 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{7} + 7\nu^{5} + 35\nu^{3} + 81\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{6} + 7\nu^{4} - 28\nu^{2} + 144 ) / 63 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{7} - 7\nu^{5} - 35\nu^{3} + 180\nu ) / 189 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 4\nu^{5} + 7\nu^{3} - 36\nu ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -8\nu^{6} + 14\nu^{4} + 7\nu^{2} + 162 ) / 63 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 2\beta_{4} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + 3\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -5\beta_{6} - 7\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -7\beta_{7} + 7\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -21\beta_{5} - 21\beta_{3} + 29\beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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
−1.01575 − 1.40294i
−1.72286 − 0.178197i
1.72286 + 0.178197i
1.01575 + 1.40294i
−1.01575 + 1.40294i
−1.72286 + 0.178197i
1.72286 − 0.178197i
1.01575 − 1.40294i
1.00000 −1.72286 + 0.178197i 1.00000 −1.01575 − 1.75934i −1.72286 + 0.178197i 0 1.00000 2.93649 − 0.614017i −1.01575 − 1.75934i
373.2 1.00000 −1.01575 + 1.40294i 1.00000 −1.72286 − 2.98408i −1.01575 + 1.40294i 0 1.00000 −0.936492 − 2.85008i −1.72286 − 2.98408i
373.3 1.00000 1.01575 − 1.40294i 1.00000 1.72286 + 2.98408i 1.01575 − 1.40294i 0 1.00000 −0.936492 − 2.85008i 1.72286 + 2.98408i
373.4 1.00000 1.72286 − 0.178197i 1.00000 1.01575 + 1.75934i 1.72286 − 0.178197i 0 1.00000 2.93649 − 0.614017i 1.01575 + 1.75934i
655.1 1.00000 −1.72286 − 0.178197i 1.00000 −1.01575 + 1.75934i −1.72286 − 0.178197i 0 1.00000 2.93649 + 0.614017i −1.01575 + 1.75934i
655.2 1.00000 −1.01575 − 1.40294i 1.00000 −1.72286 + 2.98408i −1.01575 − 1.40294i 0 1.00000 −0.936492 + 2.85008i −1.72286 + 2.98408i
655.3 1.00000 1.01575 + 1.40294i 1.00000 1.72286 − 2.98408i 1.01575 + 1.40294i 0 1.00000 −0.936492 + 2.85008i 1.72286 − 2.98408i
655.4 1.00000 1.72286 + 0.178197i 1.00000 1.01575 − 1.75934i 1.72286 + 0.178197i 0 1.00000 2.93649 + 0.614017i 1.01575 − 1.75934i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 373.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
63.h even 3 1 inner
63.t 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.e.t 8
3.b odd 2 1 2646.2.e.r 8
7.b odd 2 1 inner 882.2.e.t 8
7.c even 3 1 882.2.f.p ✓ 8
7.c even 3 1 882.2.h.r 8
7.d odd 6 1 882.2.f.p ✓ 8
7.d odd 6 1 882.2.h.r 8
9.c even 3 1 882.2.h.r 8
9.d odd 6 1 2646.2.h.s 8
21.c even 2 1 2646.2.e.r 8
21.g even 6 1 2646.2.f.s 8
21.g even 6 1 2646.2.h.s 8
21.h odd 6 1 2646.2.f.s 8
21.h odd 6 1 2646.2.h.s 8
63.g even 3 1 882.2.f.p ✓ 8
63.h even 3 1 inner 882.2.e.t 8
63.h even 3 1 7938.2.a.cu 4
63.i even 6 1 2646.2.e.r 8
63.i even 6 1 7938.2.a.cd 4
63.j odd 6 1 2646.2.e.r 8
63.j odd 6 1 7938.2.a.cd 4
63.k odd 6 1 882.2.f.p ✓ 8
63.l odd 6 1 882.2.h.r 8
63.n odd 6 1 2646.2.f.s 8
63.o even 6 1 2646.2.h.s 8
63.s even 6 1 2646.2.f.s 8
63.t odd 6 1 inner 882.2.e.t 8
63.t odd 6 1 7938.2.a.cu 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.e.t 8 1.a even 1 1 trivial
882.2.e.t 8 7.b odd 2 1 inner
882.2.e.t 8 63.h even 3 1 inner
882.2.e.t 8 63.t odd 6 1 inner
882.2.f.p ✓ 8 7.c even 3 1
882.2.f.p ✓ 8 7.d odd 6 1
882.2.f.p ✓ 8 63.g even 3 1
882.2.f.p ✓ 8 63.k odd 6 1
882.2.h.r 8 7.c even 3 1
882.2.h.r 8 7.d odd 6 1
882.2.h.r 8 9.c even 3 1
882.2.h.r 8 63.l odd 6 1
2646.2.e.r 8 3.b odd 2 1
2646.2.e.r 8 21.c even 2 1
2646.2.e.r 8 63.i even 6 1
2646.2.e.r 8 63.j odd 6 1
2646.2.f.s 8 21.g even 6 1
2646.2.f.s 8 21.h odd 6 1
2646.2.f.s 8 63.n odd 6 1
2646.2.f.s 8 63.s even 6 1
2646.2.h.s 8 9.d odd 6 1
2646.2.h.s 8 21.g even 6 1
2646.2.h.s 8 21.h odd 6 1
2646.2.h.s 8 63.o even 6 1
7938.2.a.cd 4 63.i even 6 1
7938.2.a.cd 4 63.j odd 6 1
7938.2.a.cu 4 63.h even 3 1
7938.2.a.cu 4 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 \(S_{2}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{8} + 16T_{5}^{6} + 207T_{5}^{4} + 784T_{5}^{2} + 2401 \) Copy content Toggle raw display
\( T_{11}^{2} + 4T_{11} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} + 18T_{13}^{2} + 324 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} + 16 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 18 T^{2} + 324)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 40 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$23$ \( (T^{4} - 2 T^{3} + \cdots + 3481)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 10 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 30)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 4 T^{3} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 10 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 64 T^{2} + 784)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 2 T^{3} + \cdots + 17956)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 76 T^{2} + 484)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 40 T^{2} + 25)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 6 T - 6)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 21)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} + 256 T^{6} + \cdots + 21381376 \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 11)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + 76 T^{6} + \cdots + 234256 \) Copy content Toggle raw display
$89$ \( (T^{4} + 50 T^{2} + 2500)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 5158686976 \) Copy content Toggle raw display
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