Properties

Label 882.3.b.i
Level $882$
Weight $3$
Character orbit 882.b
Analytic conductor $24.033$
Analytic rank $0$
Dimension $4$
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] = [882,3,Mod(197,882)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.197");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.b (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: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - 2 q^{4} + \beta_1 q^{5} + 2 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} - 2 q^{4} + \beta_1 q^{5} + 2 \beta_{2} q^{8} + 2 \beta_{3} q^{10} + 2 \beta_{2} q^{11} - 9 \beta_{3} q^{13} + 4 q^{16} - 11 \beta_1 q^{17} + 4 \beta_{3} q^{19} - 2 \beta_1 q^{20} + 4 q^{22} - 2 \beta_{2} q^{23} + 21 q^{25} + 9 \beta_1 q^{26} + 25 \beta_{2} q^{29} + 24 \beta_{3} q^{31} - 4 \beta_{2} q^{32} - 22 \beta_{3} q^{34} + 64 q^{37} - 4 \beta_1 q^{38} - 4 \beta_{3} q^{40} + 10 \beta_1 q^{41} + 44 q^{43} - 4 \beta_{2} q^{44} - 4 q^{46} + 34 \beta_1 q^{47} - 21 \beta_{2} q^{50} + 18 \beta_{3} q^{52} - 13 \beta_{2} q^{53} - 4 \beta_{3} q^{55} + 50 q^{58} - 50 \beta_1 q^{59} - 37 \beta_{3} q^{61} - 24 \beta_1 q^{62} - 8 q^{64} - 18 \beta_{2} q^{65} + 120 q^{67} + 22 \beta_1 q^{68} - 6 \beta_{2} q^{71} + 53 \beta_{3} q^{73} - 64 \beta_{2} q^{74} - 8 \beta_{3} q^{76} + 92 q^{79} + 4 \beta_1 q^{80} + 20 \beta_{3} q^{82} + 56 \beta_1 q^{83} + 44 q^{85} - 44 \beta_{2} q^{86} - 8 q^{88} - 10 \beta_1 q^{89} + 4 \beta_{2} q^{92} + 68 \beta_{3} q^{94} + 8 \beta_{2} q^{95} + 19 \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 16 q^{16} + 16 q^{22} + 84 q^{25} + 256 q^{37} + 176 q^{43} - 16 q^{46} + 200 q^{58} - 32 q^{64} + 480 q^{67} + 368 q^{79} + 176 q^{85} - 32 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1
−0.707107 + 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
1.41421i 0 −2.00000 2.00000i 0 0 2.82843i 0 −2.82843
197.2 1.41421i 0 −2.00000 2.00000i 0 0 2.82843i 0 2.82843
197.3 1.41421i 0 −2.00000 2.00000i 0 0 2.82843i 0 2.82843
197.4 1.41421i 0 −2.00000 2.00000i 0 0 2.82843i 0 −2.82843
\(n\): e.g. 2-40 or 990-1000
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 882.3.b.i 4
3.b odd 2 1 inner 882.3.b.i 4
7.b odd 2 1 inner 882.3.b.i 4
7.c even 3 2 882.3.s.f 8
7.d odd 6 2 882.3.s.f 8
21.c even 2 1 inner 882.3.b.i 4
21.g even 6 2 882.3.s.f 8
21.h odd 6 2 882.3.s.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.3.b.i 4 1.a even 1 1 trivial
882.3.b.i 4 3.b odd 2 1 inner
882.3.b.i 4 7.b odd 2 1 inner
882.3.b.i 4 21.c even 2 1 inner
882.3.s.f 8 7.c even 3 2
882.3.s.f 8 7.d odd 6 2
882.3.s.f 8 21.g even 6 2
882.3.s.f 8 21.h odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 8 \) Copy content Toggle raw display
\( T_{13}^{2} - 162 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 484)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1250)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 1152)^{2} \) Copy content Toggle raw display
$37$ \( (T - 64)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 400)^{2} \) Copy content Toggle raw display
$43$ \( (T - 44)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 4624)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 338)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 10000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2738)^{2} \) Copy content Toggle raw display
$67$ \( (T - 120)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 72)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 5618)^{2} \) Copy content Toggle raw display
$79$ \( (T - 92)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 12544)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 400)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 722)^{2} \) Copy content Toggle raw display
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