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Properties

Label 882.3.s.g
Level $882$
Weight $3$
Character orbit 882.s
Analytic conductor $24.033$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [882,3,Mod(557,882)]
 
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([3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.557");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.s (of order \(6\), 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: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + 2 \beta_{2} q^{4} - 2 \beta_1 q^{5} + (2 \beta_{6} - 2 \beta_{5}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + 2 \beta_{2} q^{4} - 2 \beta_1 q^{5} + (2 \beta_{6} - 2 \beta_{5}) q^{8} - 4 \beta_{4} q^{10} + 2 \beta_{5} q^{11} - 9 \beta_{7} q^{13} + (4 \beta_{2} - 4) q^{16} + ( - 2 \beta_{3} + 2 \beta_1) q^{17} + (16 \beta_{7} - 16 \beta_{4}) q^{19} - 4 \beta_{3} q^{20} + 4 q^{22} + 26 \beta_{6} q^{23} - 9 \beta_{2} q^{25} - 9 \beta_1 q^{26} + ( - 23 \beta_{6} + 23 \beta_{5}) q^{29} - 36 \beta_{4} q^{31} - 4 \beta_{5} q^{32} + 4 \beta_{7} q^{34} + ( - 32 \beta_{2} + 32) q^{37} + ( - 16 \beta_{3} + 16 \beta_1) q^{38} + (8 \beta_{7} - 8 \beta_{4}) q^{40} - 19 \beta_{3} q^{41} + 20 q^{43} + 4 \beta_{6} q^{44} + 52 \beta_{2} q^{46} + 10 \beta_1 q^{47} + ( - 9 \beta_{6} + 9 \beta_{5}) q^{50} - 18 \beta_{4} q^{52} - 67 \beta_{5} q^{53} - 8 \beta_{7} q^{55} + ( - 46 \beta_{2} + 46) q^{58} + ( - 2 \beta_{3} + 2 \beta_1) q^{59} + (59 \beta_{7} - 59 \beta_{4}) q^{61} - 36 \beta_{3} q^{62} - 8 q^{64} + 36 \beta_{6} q^{65} + 48 \beta_{2} q^{67} + 4 \beta_1 q^{68} + ( - 54 \beta_{6} + 54 \beta_{5}) q^{71} - 85 \beta_{4} q^{73} + 32 \beta_{5} q^{74} + 32 \beta_{7} q^{76} + ( - 148 \beta_{2} + 148) q^{79} + ( - 8 \beta_{3} + 8 \beta_1) q^{80} + (38 \beta_{7} - 38 \beta_{4}) q^{82} + 40 \beta_{3} q^{83} - 16 q^{85} + 20 \beta_{6} q^{86} + 8 \beta_{2} q^{88} + 53 \beta_1 q^{89} + (52 \beta_{6} - 52 \beta_{5}) q^{92} + 20 \beta_{4} q^{94} - 64 \beta_{5} q^{95} - 109 \beta_{7} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} - 16 q^{16} + 32 q^{22} - 36 q^{25} + 128 q^{37} + 160 q^{43} + 208 q^{46} + 184 q^{58} - 64 q^{64} + 192 q^{67} + 592 q^{79} - 128 q^{85} + 32 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
−0.965926 + 0.258819i
−0.258819 − 0.965926i
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.965926 − 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 − 0.965926i
−1.22474 + 0.707107i 0 1.00000 − 1.73205i −3.46410 + 2.00000i 0 0 2.82843i 0 2.82843 − 4.89898i
557.2 −1.22474 + 0.707107i 0 1.00000 − 1.73205i 3.46410 − 2.00000i 0 0 2.82843i 0 −2.82843 + 4.89898i
557.3 1.22474 − 0.707107i 0 1.00000 − 1.73205i −3.46410 + 2.00000i 0 0 − 2.82843i 0 −2.82843 + 4.89898i
557.4 1.22474 − 0.707107i 0 1.00000 − 1.73205i 3.46410 − 2.00000i 0 0 − 2.82843i 0 2.82843 − 4.89898i
863.1 −1.22474 − 0.707107i 0 1.00000 + 1.73205i −3.46410 − 2.00000i 0 0 − 2.82843i 0 2.82843 + 4.89898i
863.2 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 3.46410 + 2.00000i 0 0 − 2.82843i 0 −2.82843 − 4.89898i
863.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −3.46410 − 2.00000i 0 0 2.82843i 0 −2.82843 − 4.89898i
863.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 3.46410 + 2.00000i 0 0 2.82843i 0 2.82843 + 4.89898i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 557.4
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
7.c even 3 1 inner
7.d odd 6 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.s.g 8
3.b odd 2 1 inner 882.3.s.g 8
7.b odd 2 1 inner 882.3.s.g 8
7.c even 3 1 882.3.b.h ✓ 4
7.c even 3 1 inner 882.3.s.g 8
7.d odd 6 1 882.3.b.h ✓ 4
7.d odd 6 1 inner 882.3.s.g 8
21.c even 2 1 inner 882.3.s.g 8
21.g even 6 1 882.3.b.h ✓ 4
21.g even 6 1 inner 882.3.s.g 8
21.h odd 6 1 882.3.b.h ✓ 4
21.h odd 6 1 inner 882.3.s.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.3.b.h ✓ 4 7.c even 3 1
882.3.b.h ✓ 4 7.d odd 6 1
882.3.b.h ✓ 4 21.g even 6 1
882.3.b.h ✓ 4 21.h odd 6 1
882.3.s.g 8 1.a even 1 1 trivial
882.3.s.g 8 3.b odd 2 1 inner
882.3.s.g 8 7.b odd 2 1 inner
882.3.s.g 8 7.c even 3 1 inner
882.3.s.g 8 7.d odd 6 1 inner
882.3.s.g 8 21.c even 2 1 inner
882.3.s.g 8 21.g even 6 1 inner
882.3.s.g 8 21.h odd 6 1 inner

Hecke kernels

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

\( T_{5}^{4} - 16T_{5}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{4} - 8T_{11}^{2} + 64 \) Copy content Toggle raw display
\( T_{13}^{2} - 162 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 16 T^{2} + 256)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 162)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} - 16 T^{2} + 256)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 512 T^{2} + 262144)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 1352 T^{2} + 1827904)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 1058)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 2592 T^{2} + 6718464)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 32 T + 1024)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 1444)^{4} \) Copy content Toggle raw display
$43$ \( (T - 20)^{8} \) Copy content Toggle raw display
$47$ \( (T^{4} - 400 T^{2} + 160000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 8978 T^{2} + 80604484)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 16 T^{2} + 256)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 6962 T^{2} + 48469444)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 48 T + 2304)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 5832)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 14450 T^{2} + 208802500)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 148 T + 21904)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 6400)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 11236 T^{2} + 126247696)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 23762)^{4} \) Copy content Toggle raw display
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