Properties

Label 882.4.k.c
Level $882$
Weight $4$
Character orbit 882.k
Analytic conductor $52.040$
Analytic rank $0$
Dimension $16$
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] = [882,4,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: 16.0.721389578983833600000000.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 625x^{8} + 390625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 126)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 4 \beta_{2} q^{4} + (\beta_{6} - \beta_{4}) q^{5} + 4 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 4 \beta_{2} q^{4} + (\beta_{6} - \beta_{4}) q^{5} + 4 \beta_{3} q^{8} + ( - \beta_{11} + \beta_{10}) q^{10} + (7 \beta_{5} - 15 \beta_{3} + 15 \beta_1) q^{11} + (\beta_{12} - 2 \beta_{10}) q^{13} + (16 \beta_{2} - 16) q^{16} + (\beta_{15} + \beta_{14} + 7 \beta_{4}) q^{17} + ( - \beta_{13} + 4 \beta_{11}) q^{19} + 4 \beta_{6} q^{20} + (7 \beta_{9} + 60) q^{22} + ( - 5 \beta_{7} + 21 \beta_1) q^{23} + ( - 10 \beta_{8} + 5 \beta_{2}) q^{25} + ( - \beta_{14} - 10 \beta_{6} + 10 \beta_{4}) q^{26} + ( - 29 \beta_{7} + 29 \beta_{5} + 30 \beta_{3}) q^{29} + (2 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - 4 \beta_{10}) q^{31} + (16 \beta_{3} - 16 \beta_1) q^{32} + ( - 4 \beta_{12} - 9 \beta_{10}) q^{34} + (36 \beta_{9} - 36 \beta_{8} - 20 \beta_{2} + 20) q^{37} + (\beta_{15} + \beta_{14} - 14 \beta_{4}) q^{38} - 4 \beta_{11} q^{40} + (3 \beta_{15} + 7 \beta_{6}) q^{41} + (17 \beta_{9} + 130) q^{43} + (28 \beta_{7} + 60 \beta_1) q^{44} + ( - 5 \beta_{8} + 84 \beta_{2}) q^{46} + (3 \beta_{14} + 28 \beta_{6} - 28 \beta_{4}) q^{47} + ( - 40 \beta_{7} + 40 \beta_{5} + 5 \beta_{3}) q^{50} + ( - 4 \beta_{13} + 4 \beta_{12} + 12 \beta_{11} - 8 \beta_{10}) q^{52} + (45 \beta_{5} + 6 \beta_{3} - 6 \beta_1) q^{53} + (7 \beta_{12} + 29 \beta_{10}) q^{55} + (29 \beta_{9} - 29 \beta_{8} + 120 \beta_{2} - 120) q^{58} + ( - 8 \beta_{15} - 8 \beta_{14} + 8 \beta_{4}) q^{59} + (12 \beta_{13} - 16 \beta_{11}) q^{61} + ( - 2 \beta_{15} - 12 \beta_{6}) q^{62} - 64 q^{64} + (40 \beta_{7} + 300 \beta_1) q^{65} + (70 \beta_{8} + 460 \beta_{2}) q^{67} + (4 \beta_{14} - 28 \beta_{6} + 28 \beta_{4}) q^{68} + ( - 127 \beta_{7} + 127 \beta_{5} + 255 \beta_{3}) q^{71} + ( - 7 \beta_{13} + 7 \beta_{12} - 9 \beta_{11} + 16 \beta_{10}) q^{73} + (144 \beta_{5} - 20 \beta_{3} + 20 \beta_1) q^{74} + ( - 4 \beta_{12} + 12 \beta_{10}) q^{76} + (90 \beta_{9} - 90 \beta_{8} + 56 \beta_{2} - 56) q^{79} + 16 \beta_{4} q^{80} + ( - 12 \beta_{13} - \beta_{11}) q^{82} + ( - 9 \beta_{15} - 32 \beta_{6}) q^{83} + (130 \beta_{9} + 840) q^{85} + (68 \beta_{7} + 130 \beta_1) q^{86} + (28 \beta_{8} + 240 \beta_{2}) q^{88} + ( - 13 \beta_{14} - 33 \beta_{6} + 33 \beta_{4}) q^{89} + ( - 20 \beta_{7} + 20 \beta_{5} + 84 \beta_{3}) q^{92} + (12 \beta_{13} - 12 \beta_{12} - 34 \beta_{11} + 22 \beta_{10}) q^{94} + ( - 80 \beta_{5} + 420 \beta_{3} - 420 \beta_1) q^{95} + ( - 21 \beta_{12} - 52 \beta_{10}) 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 \( 16 q + 32 q^{4} - 128 q^{16} + 960 q^{22} + 40 q^{25} + 160 q^{37} + 2080 q^{43} + 672 q^{46} - 960 q^{58} - 1024 q^{64} + 3680 q^{67} - 448 q^{79} + 13440 q^{85} + 1920 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 625x^{8} + 390625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{4} ) / 25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} ) / 625 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{12} ) / 15625 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{15} + 250\nu^{9} + 1250\nu^{7} + 156250\nu ) / 78125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{14} + 46875\nu^{2} ) / 78125 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{15} - 250\nu^{9} + 2500\nu^{7} + 312500\nu ) / 78125 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{14} + 75\nu^{10} + 1875\nu^{6} ) / 78125 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6\nu^{14} + 93750\nu^{2} ) / 78125 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -6\nu^{10} + 150\nu^{6} + 3750\nu^{2} ) / 3125 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -4\nu^{13} - 40\nu^{11} - 2500\nu^{5} + 12500\nu^{3} ) / 15625 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 4\nu^{13} - 20\nu^{11} - 5000\nu^{5} - 12500\nu^{3} ) / 15625 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 6\nu^{15} + 10\nu^{13} + 100\nu^{11} - 750\nu^{9} - 7500\nu^{7} + 6250\nu^{5} - 31250\nu^{3} + 937500\nu ) / 78125 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12\nu^{15} + 10\nu^{13} - 50\nu^{11} - 1500\nu^{9} - 3750\nu^{7} - 12500\nu^{5} - 31250\nu^{3} + 468750\nu ) / 78125 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 12\nu^{13} + 60\nu^{11} - 15000\nu^{5} + 37500\nu^{3} ) / 15625 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 12\nu^{13} - 120\nu^{11} + 7500\nu^{5} + 37500\nu^{3} ) / 15625 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{13} + 4\beta_{12} + \beta_{11} + 2\beta_{10} + 6\beta_{6} + 6\beta_{4} ) / 72 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{8} + 10\beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{15} + 10\beta_{14} - 30\beta_{11} + 15\beta_{10} ) / 72 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 25\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 25\beta_{15} - 25\beta_{14} - 75\beta_{11} - 75\beta_{10} ) / 72 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 125\beta_{9} + 250\beta_{7} - 250\beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 250\beta_{13} - 500\beta_{12} - 125\beta_{11} - 250\beta_{10} + 750\beta_{6} + 750\beta_{4} ) / 72 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 625\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -2500\beta_{13} + 1250\beta_{12} + 1250\beta_{11} + 625\beta_{10} - 3750\beta_{6} + 7500\beta_{4} ) / 72 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -3125\beta_{9} + 3125\beta_{8} + 6250\beta_{7} ) / 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -3125\beta_{15} + 3125\beta_{14} - 9375\beta_{11} - 9375\beta_{10} ) / 72 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 15625\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 31250\beta_{15} + 15625\beta_{14} + 46875\beta_{11} - 93750\beta_{10} ) / 72 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 78125\beta_{8} - 156250\beta_{5} ) / 12 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 312500\beta_{13} - 156250\beta_{12} - 156250\beta_{11} - 78125\beta_{10} - 468750\beta_{6} + 937500\beta_{4} ) / 72 \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
215.1
−1.36123 1.77399i
−1.77399 + 1.36123i
1.77399 1.36123i
1.36123 + 1.77399i
−2.21694 0.291865i
0.291865 2.21694i
−0.291865 + 2.21694i
2.21694 + 0.291865i
−1.36123 + 1.77399i
−1.77399 1.36123i
1.77399 + 1.36123i
1.36123 1.77399i
−2.21694 + 0.291865i
0.291865 + 2.21694i
−0.291865 2.21694i
2.21694 0.291865i
−1.73205 1.00000i 0 2.00000 + 3.46410i −7.15634 + 12.3951i 0 0 8.00000i 0 24.7903 14.3127i
215.2 −1.73205 1.00000i 0 2.00000 + 3.46410i −2.96425 + 5.13424i 0 0 8.00000i 0 10.2685 5.92851i
215.3 −1.73205 1.00000i 0 2.00000 + 3.46410i 2.96425 5.13424i 0 0 8.00000i 0 −10.2685 + 5.92851i
215.4 −1.73205 1.00000i 0 2.00000 + 3.46410i 7.15634 12.3951i 0 0 8.00000i 0 −24.7903 + 14.3127i
215.5 1.73205 + 1.00000i 0 2.00000 + 3.46410i −7.15634 + 12.3951i 0 0 8.00000i 0 −24.7903 + 14.3127i
215.6 1.73205 + 1.00000i 0 2.00000 + 3.46410i −2.96425 + 5.13424i 0 0 8.00000i 0 −10.2685 + 5.92851i
215.7 1.73205 + 1.00000i 0 2.00000 + 3.46410i 2.96425 5.13424i 0 0 8.00000i 0 10.2685 5.92851i
215.8 1.73205 + 1.00000i 0 2.00000 + 3.46410i 7.15634 12.3951i 0 0 8.00000i 0 24.7903 14.3127i
521.1 −1.73205 + 1.00000i 0 2.00000 3.46410i −7.15634 12.3951i 0 0 8.00000i 0 24.7903 + 14.3127i
521.2 −1.73205 + 1.00000i 0 2.00000 3.46410i −2.96425 5.13424i 0 0 8.00000i 0 10.2685 + 5.92851i
521.3 −1.73205 + 1.00000i 0 2.00000 3.46410i 2.96425 + 5.13424i 0 0 8.00000i 0 −10.2685 5.92851i
521.4 −1.73205 + 1.00000i 0 2.00000 3.46410i 7.15634 + 12.3951i 0 0 8.00000i 0 −24.7903 14.3127i
521.5 1.73205 1.00000i 0 2.00000 3.46410i −7.15634 12.3951i 0 0 8.00000i 0 −24.7903 14.3127i
521.6 1.73205 1.00000i 0 2.00000 3.46410i −2.96425 5.13424i 0 0 8.00000i 0 −10.2685 5.92851i
521.7 1.73205 1.00000i 0 2.00000 3.46410i 2.96425 + 5.13424i 0 0 8.00000i 0 10.2685 + 5.92851i
521.8 1.73205 1.00000i 0 2.00000 3.46410i 7.15634 + 12.3951i 0 0 8.00000i 0 24.7903 + 14.3127i
\(n\): 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.c 16
3.b odd 2 1 inner 882.4.k.c 16
7.b odd 2 1 inner 882.4.k.c 16
7.c even 3 1 126.4.d.a 8
7.c even 3 1 inner 882.4.k.c 16
7.d odd 6 1 126.4.d.a 8
7.d odd 6 1 inner 882.4.k.c 16
21.c even 2 1 inner 882.4.k.c 16
21.g even 6 1 126.4.d.a 8
21.g even 6 1 inner 882.4.k.c 16
21.h odd 6 1 126.4.d.a 8
21.h odd 6 1 inner 882.4.k.c 16
28.f even 6 1 1008.4.k.c 8
28.g odd 6 1 1008.4.k.c 8
84.j odd 6 1 1008.4.k.c 8
84.n even 6 1 1008.4.k.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.d.a 8 7.c even 3 1
126.4.d.a 8 7.d odd 6 1
126.4.d.a 8 21.g even 6 1
126.4.d.a 8 21.h odd 6 1
882.4.k.c 16 1.a even 1 1 trivial
882.4.k.c 16 3.b odd 2 1 inner
882.4.k.c 16 7.b odd 2 1 inner
882.4.k.c 16 7.c even 3 1 inner
882.4.k.c 16 7.d odd 6 1 inner
882.4.k.c 16 21.c even 2 1 inner
882.4.k.c 16 21.g even 6 1 inner
882.4.k.c 16 21.h odd 6 1 inner
1008.4.k.c 8 28.f even 6 1
1008.4.k.c 8 28.g odd 6 1
1008.4.k.c 8 84.j odd 6 1
1008.4.k.c 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 240T_{5}^{6} + 50400T_{5}^{4} + 1728000T_{5}^{2} + 51840000 \) 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
$5$ \( (T^{8} + 240 T^{6} + 50400 T^{4} + \cdots + 51840000)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} - 3564 T^{6} + 12701772 T^{4} + \cdots + 104976)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 8160 T^{2} + 15235200)^{4} \) Copy content Toggle raw display
$17$ \( (T^{8} + 20400 T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 13920 T^{6} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 4428 T^{6} + \cdots + 2981133747216)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 37476 T^{2} + \cdots + 133125444)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 17280 T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 40 T^{3} + 94512 T^{2} + \cdots + 8632639744)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 89520 T^{2} + \cdots + 1999648800)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 260 T - 3908)^{8} \) Copy content Toggle raw display
$47$ \( (T^{8} + 265920 T^{6} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 73188 T^{6} + \cdots + 17\!\cdots\!96)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 568320 T^{6} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 407040 T^{6} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 920 T^{3} + \cdots + 19937440000)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 1100844 T^{2} + \cdots + 913369284)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} - 255840 T^{6} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 112 T^{3} + \cdots + 336474244096)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 945600 T^{2} + \cdots + 205927948800)^{4} \) Copy content Toggle raw display
$89$ \( (T^{8} + 1721520 T^{6} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 2605920 T^{2} + \cdots + 409114396800)^{4} \) Copy content Toggle raw display
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