Properties

Label 882.3.n.f
Level $882$
Weight $3$
Character orbit 882.n
Analytic conductor $24.033$
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] = [882,3,Mod(19,882)]
 
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([0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 882.n (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: \(24.0327593166\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 7^{2} \)
Twist minimal: no (minimal twist has level 294)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} - \beta_{2}) q^{2} + 2 \beta_{4} q^{4} + ( - 2 \beta_{5} + 2 \beta_{2} + \beta_1) q^{5} + 2 \beta_{2} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} - \beta_{2}) q^{2} + 2 \beta_{4} q^{4} + ( - 2 \beta_{5} + 2 \beta_{2} + \beta_1) q^{5} + 2 \beta_{2} q^{8} + (\beta_{7} - 4 \beta_{4} + \beta_{3} - 8) q^{10} + ( - 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{3}) q^{11} + ( - \beta_{7} - \beta_{6} - 8 \beta_{5} - 4 \beta_{2} - \beta_1) q^{13} + ( - 4 \beta_{4} - 4) q^{16} + (3 \beta_{7} - 3 \beta_{6} + 6 \beta_{5} + 4 \beta_{4} + 3 \beta_{3} + 12 \beta_{2} + \cdots + 8) q^{17}+ \cdots + (10 \beta_{7} - 9 \beta_{6} - 8 \beta_{5} - 32 \beta_{4} - 4 \beta_{2} - 9 \beta_1 - 16) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 48 q^{10} - 16 q^{16} + 48 q^{17} + 96 q^{19} + 32 q^{22} + 16 q^{23} + 36 q^{25} - 96 q^{26} + 160 q^{29} - 48 q^{31} - 64 q^{37} + 96 q^{38} + 96 q^{40} + 320 q^{43} + 48 q^{46} - 48 q^{47} - 16 q^{50} + 192 q^{53} - 40 q^{58} - 192 q^{59} - 288 q^{61} + 64 q^{64} - 144 q^{65} + 176 q^{67} - 96 q^{68} + 256 q^{71} + 384 q^{73} + 24 q^{74} - 144 q^{79} - 48 q^{82} + 768 q^{85} + 128 q^{86} - 32 q^{88} + 528 q^{89} - 64 q^{92} - 96 q^{94} - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 62\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 20 ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - 88\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 2 ) / 14 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{6} + 7\nu^{4} - 21\nu^{2} + 2 ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{7} - 35\nu^{5} + 126\nu^{3} - 72\nu ) / 14 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{7} - 28\nu^{5} + 91\nu^{3} - 8\nu ) / 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 3\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 2\beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{7} + 8\beta_{6} + 8\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{5} - 6\beta_{4} + 4\beta_{2} - 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -18\beta_{7} + 26\beta_{6} - 18\beta_{3} ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14\beta_{2} - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -62\beta_{3} - 88\beta_1 ) / 7 \) Copy content Toggle raw display

Character values

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

\(n\) \(199\) \(785\)
\(\chi(n)\) \(1 + \beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.662827 0.382683i
0.662827 + 0.382683i
−1.60021 0.923880i
1.60021 + 0.923880i
−0.662827 + 0.382683i
0.662827 0.382683i
−1.60021 + 0.923880i
1.60021 0.923880i
−0.707107 + 1.22474i 0 −1.00000 1.73205i 1.31678 + 0.760243i 0 0 2.82843 0 −1.86221 + 1.07515i
19.2 −0.707107 + 1.22474i 0 −1.00000 1.73205i 7.16850 + 4.13874i 0 0 2.82843 0 −10.1378 + 5.85306i
19.3 0.707107 1.22474i 0 −1.00000 1.73205i −6.78023 3.91457i 0 0 −2.82843 0 −9.58869 + 5.53603i
19.4 0.707107 1.22474i 0 −1.00000 1.73205i −1.70506 0.984414i 0 0 −2.82843 0 −2.41131 + 1.39217i
325.1 −0.707107 1.22474i 0 −1.00000 + 1.73205i 1.31678 0.760243i 0 0 2.82843 0 −1.86221 1.07515i
325.2 −0.707107 1.22474i 0 −1.00000 + 1.73205i 7.16850 4.13874i 0 0 2.82843 0 −10.1378 5.85306i
325.3 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −6.78023 + 3.91457i 0 0 −2.82843 0 −9.58869 5.53603i
325.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −1.70506 + 0.984414i 0 0 −2.82843 0 −2.41131 1.39217i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 882.3.n.f 8
3.b odd 2 1 294.3.g.d 8
7.b odd 2 1 882.3.n.k 8
7.c even 3 1 882.3.c.g 8
7.c even 3 1 882.3.n.k 8
7.d odd 6 1 882.3.c.g 8
7.d odd 6 1 inner 882.3.n.f 8
21.c even 2 1 294.3.g.e 8
21.g even 6 1 294.3.c.b 8
21.g even 6 1 294.3.g.d 8
21.h odd 6 1 294.3.c.b 8
21.h odd 6 1 294.3.g.e 8
84.j odd 6 1 2352.3.f.i 8
84.n even 6 1 2352.3.f.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
294.3.c.b 8 21.g even 6 1
294.3.c.b 8 21.h odd 6 1
294.3.g.d 8 3.b odd 2 1
294.3.g.d 8 21.g even 6 1
294.3.g.e 8 21.c even 2 1
294.3.g.e 8 21.h odd 6 1
882.3.c.g 8 7.c even 3 1
882.3.c.g 8 7.d odd 6 1
882.3.n.f 8 1.a even 1 1 trivial
882.3.n.f 8 7.d odd 6 1 inner
882.3.n.k 8 7.b odd 2 1
882.3.n.k 8 7.c even 3 1
2352.3.f.i 8 84.j odd 6 1
2352.3.f.i 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(882, [\chi])\):

\( T_{5}^{8} - 68T_{5}^{6} + 4430T_{5}^{4} + 3264T_{5}^{3} - 12424T_{5}^{2} - 9312T_{5} + 37636 \) Copy content Toggle raw display
\( T_{23}^{8} - 16 T_{23}^{7} + 1552 T_{23}^{6} - 40192 T_{23}^{5} + 2295616 T_{23}^{4} - 43595776 T_{23}^{3} + 761420800 T_{23}^{2} - 3916939264 T_{23} + 16531787776 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 68 T^{6} + 4430 T^{4} + \cdots + 37636 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 496 T^{6} + \cdots + 2774339584 \) Copy content Toggle raw display
$13$ \( T^{8} + 520 T^{6} + 73172 T^{4} + \cdots + 7761796 \) Copy content Toggle raw display
$17$ \( T^{8} - 48 T^{7} + \cdots + 30806568324 \) Copy content Toggle raw display
$19$ \( T^{8} - 96 T^{7} + \cdots + 79803990016 \) Copy content Toggle raw display
$23$ \( T^{8} - 16 T^{7} + \cdots + 16531787776 \) Copy content Toggle raw display
$29$ \( (T^{4} - 80 T^{3} + 1820 T^{2} + \cdots - 17852)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 48 T^{7} + \cdots + 415024562176 \) Copy content Toggle raw display
$37$ \( T^{8} + 64 T^{7} + \cdots + 6469025991184 \) Copy content Toggle raw display
$41$ \( T^{8} + 9736 T^{6} + \cdots + 187372973956 \) Copy content Toggle raw display
$43$ \( (T^{4} - 160 T^{3} + 5312 T^{2} + \cdots - 11136512)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 48 T^{7} + 448 T^{6} + \cdots + 802816 \) Copy content Toggle raw display
$53$ \( T^{8} - 192 T^{7} + \cdots + 10361600475136 \) Copy content Toggle raw display
$59$ \( T^{8} + 192 T^{7} + \cdots + 490403282944 \) Copy content Toggle raw display
$61$ \( T^{8} + 288 T^{7} + \cdots + 7215530758276 \) Copy content Toggle raw display
$67$ \( T^{8} - 176 T^{7} + \cdots + 9501781590016 \) Copy content Toggle raw display
$71$ \( (T^{4} - 128 T^{3} + 4496 T^{2} + \cdots + 252352)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 305317680648964 \) Copy content Toggle raw display
$79$ \( T^{8} + 144 T^{7} + \cdots + 1069122912256 \) Copy content Toggle raw display
$83$ \( T^{8} + 14144 T^{6} + \cdots + 264508604416 \) Copy content Toggle raw display
$89$ \( T^{8} - 528 T^{7} + \cdots + 526335364 \) Copy content Toggle raw display
$97$ \( T^{8} + 13256 T^{6} + \cdots + 22348518766084 \) Copy content Toggle raw display
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