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Properties

Label 882.2.d.b
Level $882$
Weight $2$
Character orbit 882.d
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(881,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(2))
 
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("882.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 882.d (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{4} - \beta_{5} q^{5} + \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{4} - \beta_{5} q^{5} + \beta_1 q^{8} + \beta_{2} q^{10} + 4 \beta_1 q^{11} + (\beta_{4} - 4 \beta_{2}) q^{13} + q^{16} - 3 \beta_{7} q^{17} + 4 \beta_{4} q^{19} + \beta_{5} q^{20} + 4 q^{22} + 4 \beta_{3} q^{23} + (\beta_{6} - 3) q^{25} + (\beta_{7} - 4 \beta_{5}) q^{26} + (4 \beta_{3} + 4 \beta_1) q^{29} + (4 \beta_{4} + 4 \beta_{2}) q^{31} - \beta_1 q^{32} + 3 \beta_{4} q^{34} + ( - 3 \beta_{6} + 4) q^{37} + 4 \beta_{7} q^{38} - \beta_{2} q^{40} + \beta_{7} q^{41} + (4 \beta_{6} + 4) q^{43} - 4 \beta_1 q^{44} + 4 \beta_{6} q^{46} + 4 \beta_{7} q^{47} + ( - \beta_{3} + 3 \beta_1) q^{50} + ( - \beta_{4} + 4 \beta_{2}) q^{52} + ( - 3 \beta_{3} + 4 \beta_1) q^{53} - 4 \beta_{2} q^{55} + (4 \beta_{6} + 4) q^{58} + ( - 4 \beta_{7} - 4 \beta_{5}) q^{59} + (8 \beta_{4} + \beta_{2}) q^{61} + (4 \beta_{7} + 4 \beta_{5}) q^{62} - q^{64} + (3 \beta_{3} + 8 \beta_1) q^{65} + 4 q^{67} + 3 \beta_{7} q^{68} + (4 \beta_{3} - 8 \beta_1) q^{71} - 5 \beta_{2} q^{73} + (3 \beta_{3} - 4 \beta_1) q^{74} - 4 \beta_{4} q^{76} - 4 \beta_{6} q^{79} - \beta_{5} q^{80} - \beta_{4} q^{82} + (4 \beta_{7} - 8 \beta_{5}) q^{83} + 3 \beta_{6} q^{85} + ( - 4 \beta_{3} - 4 \beta_1) q^{86} - 4 q^{88} + ( - 8 \beta_{7} + 5 \beta_{5}) q^{89} - 4 \beta_{3} q^{92} - 4 \beta_{4} q^{94} - 4 \beta_{3} q^{95} + 5 \beta_{2} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} + 8 q^{16} + 32 q^{22} - 24 q^{25} + 32 q^{37} + 32 q^{43} + 32 q^{58} - 8 q^{64} + 32 q^{67} - 32 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{16}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{16}^{5} + \zeta_{16}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{16}^{6} + \zeta_{16}^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{16}^{7} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{16}^{7} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{16}^{6} + \zeta_{16}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{16}^{5} + \zeta_{16}^{3} \) Copy content Toggle raw display

Display \(\zeta_{16}^j\) in terms of \(\beta_i\)

\(\zeta_{16}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{2}\)\(=\) \( ( \beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{3}\)\(=\) \( ( \beta_{7} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{16}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{6}\)\(=\) \( ( -\beta_{6} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{16}^j\)

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(-1\) \(-1\)

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
0.923880 + 0.382683i
0.382683 − 0.923880i
−0.382683 + 0.923880i
−0.923880 − 0.382683i
0.923880 − 0.382683i
0.382683 + 0.923880i
−0.382683 − 0.923880i
−0.923880 + 0.382683i
− 1.00000i 0 −1.00000 −1.84776 0 0 1.00000i 0 1.84776i
881.2 − 1.00000i 0 −1.00000 −0.765367 0 0 1.00000i 0 0.765367i
881.3 − 1.00000i 0 −1.00000 0.765367 0 0 1.00000i 0 − 0.765367i
881.4 − 1.00000i 0 −1.00000 1.84776 0 0 1.00000i 0 − 1.84776i
881.5 1.00000i 0 −1.00000 −1.84776 0 0 − 1.00000i 0 − 1.84776i
881.6 1.00000i 0 −1.00000 −0.765367 0 0 − 1.00000i 0 − 0.765367i
881.7 1.00000i 0 −1.00000 0.765367 0 0 − 1.00000i 0 0.765367i
881.8 1.00000i 0 −1.00000 1.84776 0 0 − 1.00000i 0 1.84776i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.2.d.b ✓ 8
3.b odd 2 1 inner 882.2.d.b ✓ 8
4.b odd 2 1 7056.2.k.d 8
7.b odd 2 1 inner 882.2.d.b ✓ 8
7.c even 3 2 882.2.k.b 16
7.d odd 6 2 882.2.k.b 16
12.b even 2 1 7056.2.k.d 8
21.c even 2 1 inner 882.2.d.b ✓ 8
21.g even 6 2 882.2.k.b 16
21.h odd 6 2 882.2.k.b 16
28.d even 2 1 7056.2.k.d 8
84.h odd 2 1 7056.2.k.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.2.d.b ✓ 8 1.a even 1 1 trivial
882.2.d.b ✓ 8 3.b odd 2 1 inner
882.2.d.b ✓ 8 7.b odd 2 1 inner
882.2.d.b ✓ 8 21.c even 2 1 inner
882.2.k.b 16 7.c even 3 2
882.2.k.b 16 7.d odd 6 2
882.2.k.b 16 21.g even 6 2
882.2.k.b 16 21.h odd 6 2
7056.2.k.d 8 4.b odd 2 1
7056.2.k.d 8 12.b even 2 1
7056.2.k.d 8 28.d even 2 1
7056.2.k.d 8 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 4T_{5}^{2} + 2 \) acting on \(S_{2}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 68 T^{2} + 1058)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 36 T^{2} + 162)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 64 T^{2} + 512)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 96 T^{2} + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 128 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T - 2)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 8 T - 16)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 64 T^{2} + 512)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 68 T^{2} + 4)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 128 T^{2} + 2048)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 260 T^{2} + 12482)^{2} \) Copy content Toggle raw display
$67$ \( (T - 4)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 192 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 100 T^{2} + 1250)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 32)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 320 T^{2} + 25088)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 356 T^{2} + 3362)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 100 T^{2} + 1250)^{2} \) Copy content Toggle raw display
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