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Properties

Label 882.5.c.i
Level $882$
Weight $5$
Character orbit 882.c
Analytic conductor $91.172$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [882,5,Mod(685,882)]
 
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([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.685");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 882.c (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: \(91.1723074400\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.339738624.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 294)
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.

\(f(q)\) \(=\) \( q + 2 \beta_1 q^{2} + 8 q^{4} + ( - \beta_{4} + \beta_{3} - 5 \beta_{2}) q^{5} + 16 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_1 q^{2} + 8 q^{4} + ( - \beta_{4} + \beta_{3} - 5 \beta_{2}) q^{5} + 16 \beta_1 q^{8} + (2 \beta_{6} - 10 \beta_{3} + 4 \beta_{2}) q^{10} + ( - \beta_{7} + 5 \beta_{5} + \beta_1) q^{11} + (8 \beta_{4} - 9 \beta_{3} + 23 \beta_{2}) q^{13} + 64 q^{16} + ( - 2 \beta_{6} + 27 \beta_{4} + \cdots + 21 \beta_{2}) q^{17}+ \cdots + ( - 254 \beta_{6} + \cdots + 83 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 64 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 64 q^{4} + 512 q^{16} + 32 q^{22} + 1168 q^{23} + 2568 q^{25} - 4880 q^{29} + 2048 q^{37} - 4720 q^{43} - 3744 q^{46} + 2336 q^{50} + 6864 q^{53} + 8672 q^{58} + 4096 q^{64} + 13008 q^{65} + 2624 q^{67} - 416 q^{71} + 8544 q^{74} + 35856 q^{79} + 17568 q^{85} + 33536 q^{86} + 256 q^{88} + 9344 q^{92} + 3152 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10\nu^{7} - 35\nu^{5} + 126\nu^{3} - 10\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{7} - 28\nu^{5} + 91\nu^{3} - 8\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{6} + 12\nu^{4} - 36\nu^{2} + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\nu^{7} - 84\nu^{5} + 273\nu^{3} - 288\nu ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12\nu^{6} - 42\nu^{4} + 168\nu^{2} - 54 ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24\nu^{7} - 105\nu^{5} + 378\nu^{3} - 402\nu ) / 7 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -3\beta_{7} + 2\beta_{5} - 6\beta_{3} + 18\beta_{2} ) / 84 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{4} - 3\beta _1 + 6 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{3} + 8\beta_{2} ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{6} + 4\beta_{4} + 12\beta _1 - 18 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 13\beta_{7} - 18\beta_{5} - 54\beta_{3} + 78\beta_{2} ) / 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 22\beta_{7} - 31\beta_{5} + 93\beta_{3} - 132\beta_{2} ) / 21 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
685.1
1.60021 + 0.923880i
−1.60021 + 0.923880i
−1.60021 − 0.923880i
1.60021 − 0.923880i
−0.662827 + 0.382683i
0.662827 + 0.382683i
0.662827 − 0.382683i
−0.662827 − 0.382683i
−2.82843 0 8.00000 − 26.1431i 0 0 −22.6274 0 73.9438i
685.2 −2.82843 0 8.00000 − 11.4461i 0 0 −22.6274 0 32.3746i
685.3 −2.82843 0 8.00000 11.4461i 0 0 −22.6274 0 − 32.3746i
685.4 −2.82843 0 8.00000 26.1431i 0 0 −22.6274 0 − 73.9438i
685.5 2.82843 0 8.00000 − 19.4630i 0 0 22.6274 0 − 55.0497i
685.6 2.82843 0 8.00000 − 4.76608i 0 0 22.6274 0 − 13.4805i
685.7 2.82843 0 8.00000 4.76608i 0 0 22.6274 0 13.4805i
685.8 2.82843 0 8.00000 19.4630i 0 0 22.6274 0 55.0497i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 685.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.5.c.i 8
3.b odd 2 1 294.5.c.c ✓ 8
7.b odd 2 1 inner 882.5.c.i 8
21.c even 2 1 294.5.c.c ✓ 8
21.g even 6 1 294.5.g.d 8
21.g even 6 1 294.5.g.h 8
21.h odd 6 1 294.5.g.d 8
21.h odd 6 1 294.5.g.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.5.c.c ✓ 8 3.b odd 2 1
294.5.c.c ✓ 8 21.c even 2 1
294.5.g.d 8 21.g even 6 1
294.5.g.d 8 21.h odd 6 1
294.5.g.h 8 21.g even 6 1
294.5.g.h 8 21.h odd 6 1
882.5.c.i 8 1.a even 1 1 trivial
882.5.c.i 8 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{8} + 1216T_{5}^{6} + 425180T_{5}^{4} + 42962240T_{5}^{2} + 770506564 \) Copy content Toggle raw display
\( T_{11}^{4} - 27004T_{11}^{2} + 31536T_{11} + 151117276 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 1216 T^{6} + \cdots + 770506564 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 27004 T^{2} + \cdots + 151117276)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 285452040646276 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 54\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 53\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{4} - 584 T^{3} + \cdots + 33639434332)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 1325106094244)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 48\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 8963839239556)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 80\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 14863041654512)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 6027041745392)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 39\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 34885050530944)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 267585144756316)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 11904200287504)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 36\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 83\!\cdots\!84 \) Copy content Toggle raw display
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