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Properties

Label 882.5.c.e
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: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 145x^{6} + 226x^{5} + 16605x^{4} + 9380x^{3} + 470596x^{2} + 93296x + 11102224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 42)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + 8 q^{4} + ( - \beta_{4} - 2 \beta_{2} - 3 \beta_1) q^{5} + 8 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + 8 q^{4} + ( - \beta_{4} - 2 \beta_{2} - 3 \beta_1) q^{5} + 8 \beta_{3} q^{8} + (\beta_{5} - \beta_{4} + \cdots + 12 \beta_1) q^{10}+ \cdots + ( - 59 \beta_{5} - 106 \beta_{4} + \cdots - 1277 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 64 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 64 q^{4} - 84 q^{11} + 512 q^{16} - 192 q^{22} - 2304 q^{23} - 6196 q^{25} + 4836 q^{29} + 1628 q^{37} - 5476 q^{43} - 672 q^{44} - 1728 q^{46} - 8640 q^{50} - 1956 q^{53} - 12384 q^{58} + 4096 q^{64} - 38568 q^{65} + 25612 q^{67} - 4392 q^{71} + 14592 q^{74} - 15440 q^{79} + 3960 q^{85} + 29952 q^{86} - 1536 q^{88} - 18432 q^{92} - 65952 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 145x^{6} + 226x^{5} + 16605x^{4} + 9380x^{3} + 470596x^{2} + 93296x + 11102224 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 3702641 \nu^{7} + 293604685 \nu^{6} - 929343755 \nu^{5} + 33785211765 \nu^{4} + \cdots + 79113712969886 ) / 56545307757694 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 883 \nu^{7} - 101726 \nu^{6} + 471231 \nu^{5} - 14394602 \nu^{4} + 1230923 \nu^{3} + \cdots - 22796084584 ) / 4362450988 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 42159 \nu^{7} - 531450 \nu^{6} + 5100043 \nu^{5} + 12734410 \nu^{4} + 181102647 \nu^{3} + \cdots + 308809845064 ) / 99908366348 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 30057338968 \nu^{7} + 128493690267 \nu^{6} - 5380303437490 \nu^{5} + \cdots + 97\!\cdots\!66 ) / 29\!\cdots\!82 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5029519629 \nu^{7} - 48369221084 \nu^{6} + 1633873460973 \nu^{5} - 5351276299948 \nu^{4} + \cdots - 44\!\cdots\!40 ) / 352576624842092 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15124218783 \nu^{7} - 27258334422 \nu^{6} + 1829601416891 \nu^{5} + 4568372184170 \nu^{4} + \cdots + 26\!\cdots\!76 ) / 352576624842092 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 39336040269 \nu^{7} + 99906288330 \nu^{6} - 4758544956513 \nu^{5} + \cdots - 82\!\cdots\!60 ) / 352576624842092 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} - 5\beta_{6} - 3\beta_{5} - 15\beta_{4} - 8\beta_{3} + 3\beta_{2} + 30\beta _1 + 23 ) / 84 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 3\beta_{6} - 6\beta_{5} - 9\beta_{4} + 303\beta_{3} + 300\beta_{2} + 1509\beta _1 - 1502 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 163\beta_{7} + 395\beta_{6} + 2942\beta_{3} - 8201 ) / 42 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 431 \beta_{7} + 748 \beta_{6} + 1293 \beta_{5} + 2244 \beta_{4} + 45553 \beta_{3} - 44805 \beta_{2} + \cdots - 149689 ) / 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 21263 \beta_{7} - 41859 \beta_{6} + 63789 \beta_{5} + 125577 \beta_{4} - 553410 \beta_{3} + \cdots + 1594349 ) / 84 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -73088\beta_{7} - 131801\beta_{6} - 5925599\beta_{3} + 17581020 ) / 21 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 2723183 \beta_{7} - 5071295 \beta_{6} - 8169549 \beta_{5} - 15213885 \beta_{4} - 89379494 \beta_{3} + \cdots + 259418973 ) / 84 \) 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
−4.66558 − 8.08102i
5.87269 − 10.1718i
5.87269 + 10.1718i
−4.66558 + 8.08102i
2.65529 − 4.59910i
−2.86240 − 4.95782i
−2.86240 + 4.95782i
2.65529 + 4.59910i
−2.82843 0 8.00000 − 42.3557i 0 0 −22.6274 0 119.800i
685.2 −2.82843 0 8.00000 − 15.5344i 0 0 −22.6274 0 43.9380i
685.3 −2.82843 0 8.00000 15.5344i 0 0 −22.6274 0 − 43.9380i
685.4 −2.82843 0 8.00000 42.3557i 0 0 −22.6274 0 − 119.800i
685.5 2.82843 0 8.00000 − 43.4726i 0 0 22.6274 0 − 122.959i
685.6 2.82843 0 8.00000 − 40.9000i 0 0 22.6274 0 − 115.683i
685.7 2.82843 0 8.00000 40.9000i 0 0 22.6274 0 115.683i
685.8 2.82843 0 8.00000 43.4726i 0 0 22.6274 0 122.959i
\(n\): 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.e 8
3.b odd 2 1 294.5.c.d 8
7.b odd 2 1 inner 882.5.c.e 8
7.c even 3 1 126.5.n.c 8
7.d odd 6 1 126.5.n.c 8
21.c even 2 1 294.5.c.d 8
21.g even 6 1 42.5.g.b ✓ 8
21.g even 6 1 294.5.g.f 8
21.h odd 6 1 42.5.g.b ✓ 8
21.h odd 6 1 294.5.g.f 8
84.j odd 6 1 336.5.bh.g 8
84.n even 6 1 336.5.bh.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.5.g.b ✓ 8 21.g even 6 1
42.5.g.b ✓ 8 21.h odd 6 1
126.5.n.c 8 7.c even 3 1
126.5.n.c 8 7.d odd 6 1
294.5.c.d 8 3.b odd 2 1
294.5.c.d 8 21.c even 2 1
294.5.g.f 8 21.g even 6 1
294.5.g.f 8 21.h odd 6 1
336.5.bh.g 8 84.j odd 6 1
336.5.bh.g 8 84.n even 6 1
882.5.c.e 8 1.a even 1 1 trivial
882.5.c.e 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 \(S_{5}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{8} + 5598T_{5}^{6} + 10845513T_{5}^{4} + 7976822328T_{5}^{2} + 1368647291664 \) Copy content Toggle raw display
\( T_{11}^{4} + 42T_{11}^{3} - 54171T_{11}^{2} - 1469124T_{11} + 644203044 \) 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
$5$ \( T^{8} + \cdots + 1368647291664 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + 42 T^{3} + \cdots + 644203044)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 71\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 46\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( (T^{4} + 1152 T^{3} + \cdots + 1357461504)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 2418 T^{3} + \cdots - 136732333824)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 15\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( (T^{4} - 814 T^{3} + \cdots - 30952860668)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 92\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 6501609230972)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 31\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 12880448648064)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 52\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 134227919838224)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 218581597268928)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 31481605471729)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 16\!\cdots\!16 \) Copy content Toggle raw display
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