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Properties

Label 882.5.c.g
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: 8.0.950328623104.21
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 20x^{6} + 180x^{4} - 104x^{3} - 608x^{2} + 416x + 1262 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{4}\cdot 7^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_1 q^{2} + 8 q^{4} + ( - 4 \beta_{4} - \beta_{3}) q^{5} + 16 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_1 q^{2} + 8 q^{4} + ( - 4 \beta_{4} - \beta_{3}) q^{5} + 16 \beta_1 q^{8} + ( - 2 \beta_{6} + 8 \beta_{2}) q^{10} + ( - \beta_{5} - 20 \beta_1) q^{11} + ( - 13 \beta_{4} - 4 \beta_{3} - 16 \beta_{2}) q^{13} + 64 q^{16} + ( - 4 \beta_{6} + 24 \beta_{4} + \cdots + 36 \beta_{2}) q^{17}+ \cdots + ( - 112 \beta_{6} + \cdots - 902 \beta_{2}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 64 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 64 q^{4} + 512 q^{16} - 640 q^{22} - 2240 q^{23} - 4056 q^{25} + 448 q^{29} - 640 q^{37} - 592 q^{43} - 3072 q^{46} + 224 q^{50} - 12096 q^{53} - 5248 q^{58} + 4096 q^{64} - 36288 q^{65} - 9280 q^{67} - 26432 q^{71} + 11424 q^{74} - 27792 q^{79} + 32736 q^{85} + 4928 q^{86} - 5120 q^{88} - 17920 q^{92} + 54656 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 20x^{6} + 180x^{4} - 104x^{3} - 608x^{2} + 416x + 1262 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 850 \nu^{7} - 1872 \nu^{6} + 6084 \nu^{5} + 35477 \nu^{4} + 33824 \nu^{3} - 146198 \nu^{2} + \cdots + 761592 ) / 721777 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 96848 \nu^{7} - 107685 \nu^{6} - 1274022 \nu^{5} + 3980556 \nu^{4} + 8608250 \nu^{3} + \cdots + 100469342 ) / 16600871 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5700 \nu^{7} + 114819 \nu^{6} + 168171 \nu^{5} - 1800054 \nu^{4} - 1556394 \nu^{3} + \cdots - 37689984 ) / 721777 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 136345 \nu^{7} - 28552 \nu^{6} + 2258125 \nu^{5} + 260409 \nu^{4} - 18299666 \nu^{3} + \cdots - 19661098 ) / 16600871 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 133578 \nu^{7} + 34071 \nu^{6} + 2905266 \nu^{5} - 295545 \nu^{4} - 30015462 \nu^{3} + \cdots - 43196298 ) / 2371553 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 56412 \nu^{7} + 115221 \nu^{6} + 1249530 \nu^{5} - 1476858 \nu^{4} - 9595740 \nu^{3} + \cdots - 29913984 ) / 721777 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 307494 \nu^{7} - 1297332 \nu^{6} + 5144328 \nu^{5} + 18734694 \nu^{4} - 31446096 \nu^{3} + \cdots - 106953414 ) / 2371553 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 12\beta_{4} - 3\beta_{2} + 21\beta_1 ) / 42 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + 6\beta_{4} + 6\beta_{3} + 12\beta_{2} + 210 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 9\beta_{6} + 4\beta_{5} - 210\beta_{4} - 6\beta_{3} - 63\beta_{2} + 294\beta_1 ) / 42 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{7} - 12\beta_{6} + 2\beta_{5} + 120\beta_{4} + 36\beta_{3} + 198\beta_{2} + 273\beta _1 + 210 ) / 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 27\beta_{7} + 150\beta_{6} - 8\beta_{5} - 2094\beta_{4} - 100\beta_{3} - 534\beta_{2} + 84\beta _1 + 2730 ) / 42 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -26\beta_{7} - 69\beta_{6} + 71\beta_{5} + 1008\beta_{4} + 298\beta_{3} + 2163\beta_{2} + 5733\beta _1 - 5124 ) / 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 213\beta_{7} + 539\beta_{6} - 243\beta_{5} - 6747\beta_{4} - 147\beta_{3} - 33\beta_{2} - 17556\beta _1 + 30576 ) / 21 \) 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
−3.19851 − 1.30656i
1.78430 + 1.30656i
1.78430 − 1.30656i
−3.19851 + 1.30656i
−1.48216 − 0.541196i
2.89637 + 0.541196i
2.89637 − 0.541196i
−1.48216 + 0.541196i
−2.82843 0 8.00000 − 44.4861i 0 0 −22.6274 0 125.826i
685.2 −2.82843 0 8.00000 − 17.4581i 0 0 −22.6274 0 49.3791i
685.3 −2.82843 0 8.00000 17.4581i 0 0 −22.6274 0 − 49.3791i
685.4 −2.82843 0 8.00000 44.4861i 0 0 −22.6274 0 − 125.826i
685.5 2.82843 0 8.00000 − 43.1010i 0 0 22.6274 0 − 121.908i
685.6 2.82843 0 8.00000 − 19.6598i 0 0 22.6274 0 − 55.6062i
685.7 2.82843 0 8.00000 19.6598i 0 0 22.6274 0 55.6062i
685.8 2.82843 0 8.00000 43.1010i 0 0 22.6274 0 121.908i
\(n\): 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.g ✓ 8
3.b odd 2 1 882.5.c.h yes 8
7.b odd 2 1 inner 882.5.c.g ✓ 8
21.c even 2 1 882.5.c.h yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
882.5.c.g ✓ 8 1.a even 1 1 trivial
882.5.c.g ✓ 8 7.b odd 2 1 inner
882.5.c.h yes 8 3.b odd 2 1
882.5.c.h yes 8 21.c even 2 1

Hecke kernels

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

\( T_{5}^{8} + 4528T_{5}^{6} + 6446492T_{5}^{4} + 2993443232T_{5}^{2} + 433087712836 \) Copy content Toggle raw display
\( T_{11}^{4} - 21004T_{11}^{2} - 141120T_{11} + 77689756 \) 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
$5$ \( T^{8} + \cdots + 433087712836 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 21004 T^{2} + \cdots + 77689756)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 165576416522500 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 32975318547396 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 46\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T^{4} + 1120 T^{3} + \cdots + 2338836508)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 224 T^{3} + \cdots + 6490970908)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 1487450274236)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 13\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 18215152316176)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 21\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( (T^{4} + 6048 T^{3} + \cdots + 154780146112)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 10\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 150072069823552)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 151649835837668)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 34\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 20\!\cdots\!28)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 94\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 49\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
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