↑

Properties

Label 882.4.k.b
Level $882$
Weight $4$
Character orbit 882.k
Analytic conductor $52.040$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [882,4,Mod(215,882)]
 
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, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.215");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 882.k (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: \(52.0396846251\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{48})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{8} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 4 \beta_{3} q^{4} + ( - 3 \beta_{15} + 4 \beta_{8} + 3 \beta_{4}) q^{5} + 4 \beta_{5} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 4 \beta_{3} q^{4} + ( - 3 \beta_{15} + 4 \beta_{8} + 3 \beta_{4}) q^{5} + 4 \beta_{5} q^{8} + (8 \beta_{10} + 6 \beta_{7} - 6 \beta_{2}) q^{10} + ( - \beta_{5} + \beta_1) q^{11} + (4 \beta_{13} - 4 \beta_{10} + \beta_{7}) q^{13} + (16 \beta_{3} - 16) q^{16} + (39 \beta_{11} - 39 \beta_{8} - 18 \beta_{4}) q^{17} + (30 \beta_{13} - 16 \beta_{2}) q^{19} + ( - 12 \beta_{15} + 16 \beta_{11}) q^{20} + 4 q^{22} + (62 \beta_{12} - 21 \beta_1) q^{23} + (31 \beta_{6} + 75 \beta_{3}) q^{25} + ( - 2 \beta_{15} - 8 \beta_{8} + 2 \beta_{4}) q^{26} + (46 \beta_{12} - 46 \beta_{9} + 37 \beta_{5}) q^{29} + (14 \beta_{10} + 104 \beta_{7} - 104 \beta_{2}) q^{31} + (16 \beta_{5} - 16 \beta_1) q^{32} + (78 \beta_{13} - 78 \beta_{10} - 36 \beta_{7}) q^{34} + ( - 57 \beta_{14} + 57 \beta_{6} + \cdots - 64) q^{37}+ \cdots + (115 \beta_{13} - 115 \beta_{10} + 318 \beta_{7}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} - 128 q^{16} + 64 q^{22} + 600 q^{25} - 512 q^{37} + 4096 q^{43} - 672 q^{46} - 1184 q^{58} - 1024 q^{64} + 544 q^{67} - 6048 q^{79} + 6720 q^{85} + 128 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{48}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{48}^{7} + \zeta_{48} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{48}^{8} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{48}^{11} + \zeta_{48}^{5} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\zeta_{48}^{12} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{48}^{14} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \zeta_{48}^{15} + \zeta_{48}^{9} \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( -\zeta_{48}^{7} + \zeta_{48} \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( -\zeta_{48}^{14} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( -\zeta_{48}^{11} + \zeta_{48}^{5} \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( -\zeta_{48}^{15} + \zeta_{48}^{9} \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( -\zeta_{48}^{14} + \zeta_{48}^{10} + \zeta_{48}^{6} \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( \zeta_{48}^{13} - \zeta_{48}^{11} + \zeta_{48}^{3} \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( -\zeta_{48}^{10} + \zeta_{48}^{6} + \zeta_{48}^{2} \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( -\zeta_{48}^{13} + \zeta_{48}^{5} + \zeta_{48}^{3} \) Copy content Toggle raw display

Display \(\zeta_{48}^j\) in terms of \(\beta_i\)

\(\zeta_{48}\)\(=\) \( ( \beta_{8} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{2}\)\(=\) \( ( \beta_{9} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{3}\)\(=\) \( ( \beta_{15} + \beta_{13} - \beta_{10} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{4}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{5}\)\(=\) \( ( \beta_{10} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{6}\)\(=\) \( ( \beta_{14} + \beta_{12} - \beta_{9} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{7}\)\(=\) \( ( -\beta_{8} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{8}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{48}^{9}\)\(=\) \( ( \beta_{11} + \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{10}\)\(=\) \( ( -\beta_{14} + \beta_{12} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{11}\)\(=\) \( ( -\beta_{10} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{12}\)\(=\) \( ( \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{13}\)\(=\) \( ( -\beta_{15} + \beta_{13} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{14}\)\(=\) \( ( -\beta_{9} + \beta_{6} ) / 2 \) Copy content Toggle raw display
\(\zeta_{48}^{15}\)\(=\) \( ( -\beta_{11} + \beta_{7} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{48}^j\)

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(1 - \beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
215.1
−0.793353 + 0.608761i
−0.608761 − 0.793353i
0.608761 + 0.793353i
0.793353 − 0.608761i
−0.130526 + 0.991445i
0.991445 + 0.130526i
−0.991445 − 0.130526i
0.130526 − 0.991445i
−0.793353 − 0.608761i
−0.608761 + 0.793353i
0.608761 − 0.793353i
0.793353 + 0.608761i
−0.130526 − 0.991445i
0.991445 − 0.130526i
−0.991445 + 0.130526i
0.130526 + 0.991445i
−1.73205 − 1.00000i 0 2.00000 + 3.46410i −4.84357 + 8.38931i 0 0 − 8.00000i 0 16.7786 − 9.68714i
215.2 −1.73205 − 1.00000i 0 2.00000 + 3.46410i −1.24090 + 2.14931i 0 0 − 8.00000i 0 4.29862 − 2.48181i
215.3 −1.73205 − 1.00000i 0 2.00000 + 3.46410i 1.24090 − 2.14931i 0 0 − 8.00000i 0 −4.29862 + 2.48181i
215.4 −1.73205 − 1.00000i 0 2.00000 + 3.46410i 4.84357 − 8.38931i 0 0 − 8.00000i 0 −16.7786 + 9.68714i
215.5 1.73205 + 1.00000i 0 2.00000 + 3.46410i −4.84357 + 8.38931i 0 0 8.00000i 0 −16.7786 + 9.68714i
215.6 1.73205 + 1.00000i 0 2.00000 + 3.46410i −1.24090 + 2.14931i 0 0 8.00000i 0 −4.29862 + 2.48181i
215.7 1.73205 + 1.00000i 0 2.00000 + 3.46410i 1.24090 − 2.14931i 0 0 8.00000i 0 4.29862 − 2.48181i
215.8 1.73205 + 1.00000i 0 2.00000 + 3.46410i 4.84357 − 8.38931i 0 0 8.00000i 0 16.7786 − 9.68714i
521.1 −1.73205 + 1.00000i 0 2.00000 − 3.46410i −4.84357 − 8.38931i 0 0 8.00000i 0 16.7786 + 9.68714i
521.2 −1.73205 + 1.00000i 0 2.00000 − 3.46410i −1.24090 − 2.14931i 0 0 8.00000i 0 4.29862 + 2.48181i
521.3 −1.73205 + 1.00000i 0 2.00000 − 3.46410i 1.24090 + 2.14931i 0 0 8.00000i 0 −4.29862 − 2.48181i
521.4 −1.73205 + 1.00000i 0 2.00000 − 3.46410i 4.84357 + 8.38931i 0 0 8.00000i 0 −16.7786 − 9.68714i
521.5 1.73205 − 1.00000i 0 2.00000 − 3.46410i −4.84357 − 8.38931i 0 0 − 8.00000i 0 −16.7786 − 9.68714i
521.6 1.73205 − 1.00000i 0 2.00000 − 3.46410i −1.24090 − 2.14931i 0 0 − 8.00000i 0 −4.29862 − 2.48181i
521.7 1.73205 − 1.00000i 0 2.00000 − 3.46410i 1.24090 + 2.14931i 0 0 − 8.00000i 0 4.29862 + 2.48181i
521.8 1.73205 − 1.00000i 0 2.00000 − 3.46410i 4.84357 + 8.38931i 0 0 − 8.00000i 0 16.7786 + 9.68714i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 215.8
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
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 882.4.k.b 16
3.b odd 2 1 inner 882.4.k.b 16
7.b odd 2 1 inner 882.4.k.b 16
7.c even 3 1 882.4.d.a ✓ 8
7.c even 3 1 inner 882.4.k.b 16
7.d odd 6 1 882.4.d.a ✓ 8
7.d odd 6 1 inner 882.4.k.b 16
21.c even 2 1 inner 882.4.k.b 16
21.g even 6 1 882.4.d.a ✓ 8
21.g even 6 1 inner 882.4.k.b 16
21.h odd 6 1 882.4.d.a ✓ 8
21.h odd 6 1 inner 882.4.k.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.4.d.a ✓ 8 7.c even 3 1
882.4.d.a ✓ 8 7.d odd 6 1
882.4.d.a ✓ 8 21.g even 6 1
882.4.d.a ✓ 8 21.h odd 6 1
882.4.k.b 16 1.a even 1 1 trivial
882.4.k.b 16 3.b odd 2 1 inner
882.4.k.b 16 7.b odd 2 1 inner
882.4.k.b 16 7.c even 3 1 inner
882.4.k.b 16 7.d odd 6 1 inner
882.4.k.b 16 21.c even 2 1 inner
882.4.k.b 16 21.g even 6 1 inner
882.4.k.b 16 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 100T_{5}^{6} + 9422T_{5}^{4} + 57800T_{5}^{2} + 334084 \) acting on \(S_{4}^{\mathrm{new}}(882, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 16)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( (T^{8} + 100 T^{6} + \cdots + 334084)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{4} - 4 T^{2} + 16)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 68 T^{2} + 1058)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 7380 T^{6} + \cdots + 7344147204)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 4624 T^{6} + \cdots + 39884882944)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 12\!\cdots\!76)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 19416 T^{2} + 1547536)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 13\!\cdots\!04)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 128 T^{3} + \cdots + 5769604)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 177412 T^{2} + 6250290818)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 512 T + 2168)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 98\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 28\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 28\!\cdots\!24)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 21\!\cdots\!84)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 136 T^{3} + \cdots + 79041948736)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 612552 T^{2} + 24374703376)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 48\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 1512 T^{3} + \cdots + 270749552896)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1962560 T^{2} + 961584931328)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 96\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 457396 T^{2} + 435656162)^{4} \) Copy content Toggle raw display
show more
show less