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Properties

Label 1584.3.i.c
Level $1584$
Weight $3$
Character orbit 1584.i
Analytic conductor $43.161$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1584,3,Mod(881,1584)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1584, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1584.881"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1584.i (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,16,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(43.1608738747\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 17x^{6} + 22x^{5} + 62x^{4} + 862x^{3} + 1045x^{2} + 5238x + 13491 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 396)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{5} + (\beta_{5} + 2) q^{7} + \beta_{3} q^{11} + ( - \beta_{4} + \beta_{2} - 1) q^{13} + (2 \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{17} + ( - \beta_{5} - \beta_{4} - \beta_{2} - 5) q^{19}+ \cdots + ( - 2 \beta_{5} + 7 \beta_{4} + \cdots - 3) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{7} - 8 q^{13} - 40 q^{19} - 48 q^{25} - 40 q^{31} - 56 q^{37} - 24 q^{43} + 96 q^{49} + 120 q^{61} + 272 q^{67} - 256 q^{73} - 32 q^{79} + 432 q^{85} + 96 q^{91} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 17x^{6} + 22x^{5} + 62x^{4} + 862x^{3} + 1045x^{2} + 5238x + 13491 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -33\nu^{7} + 941\nu^{6} - 4848\nu^{5} + 25082\nu^{4} - 47292\nu^{3} + 83030\nu^{2} + 752121\nu - 808425 ) / 188352 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 49\nu^{6} - 160\nu^{5} + 586\nu^{4} + 1772\nu^{3} - 5522\nu^{2} + 31257\nu + 11763 ) / 5184 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -95\nu^{7} + 377\nu^{6} - 3968\nu^{5} + 9686\nu^{4} - 42800\nu^{3} - 55882\nu^{2} - 63873\nu - 1225413 ) / 141264 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + \nu^{6} - 16\nu^{5} - 38\nu^{4} - 100\nu^{3} - 962\nu^{2} - 279\nu - 6381 ) / 864 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} + \nu^{6} - 4\nu^{5} + 88\nu^{4} + 182\nu^{3} + 694\nu^{2} + 3582\nu + 2799 ) / 1296 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 373 \nu^{7} - 1969 \nu^{6} + 9136 \nu^{5} - 11650 \nu^{4} + 36076 \nu^{3} + 156530 \nu^{2} + \cdots + 1131741 ) / 188352 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 787 \nu^{7} + 3013 \nu^{6} - 20728 \nu^{5} + 32134 \nu^{4} - 125320 \nu^{3} - 307658 \nu^{2} + \cdots - 3856905 ) / 282528 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + 3\beta_{4} - 3\beta_{3} + 3\beta _1 + 3 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{4} - 7\beta_{3} - 2\beta_{2} + \beta _1 - 15 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 27\beta_{7} + 20\beta_{6} + 3\beta_{5} - 12\beta_{4} - 24\beta_{3} + 6\beta_{2} - 12\beta _1 - 120 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 16\beta_{7} + 41\beta_{6} - 42\beta_{5} - 55\beta_{4} + 47\beta_{3} + 30\beta_{2} - 11\beta _1 - 51 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -738\beta_{7} - 389\beta_{6} - 690\beta_{5} - 285\beta_{4} + 717\beta_{3} + 204\beta_{2} + 165\beta _1 - 1887 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -634\beta_{7} - 783\beta_{6} - 16\beta_{5} - 41\beta_{4} + 361\beta_{3} - 50\beta_{2} + 29\beta _1 - 849 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 10764 \beta_{7} - 10313 \beta_{6} + 9360 \beta_{5} + 4665 \beta_{4} + 11175 \beta_{3} - 2412 \beta_{2} + \cdots + 20793 ) / 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(353\) \(991\) \(1189\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
881.1
2.95745 + 2.85804i
−2.45745 − 0.214490i
0.792852 − 3.45790i
−0.292852 − 3.21380i
−0.292852 + 3.21380i
0.792852 + 3.45790i
−2.45745 + 0.214490i
2.95745 − 2.85804i
0 0 0 − 8.36495i 0 8.08375 0 0 0
881.2 0 0 0 − 6.95073i 0 0.606669 0 0 0
881.3 0 0 0 − 2.24252i 0 −9.78043 0 0 0
881.4 0 0 0 − 0.828310i 0 9.09001 0 0 0
881.5 0 0 0 0.828310i 0 9.09001 0 0 0
881.6 0 0 0 2.24252i 0 −9.78043 0 0 0
881.7 0 0 0 6.95073i 0 0.606669 0 0 0
881.8 0 0 0 8.36495i 0 8.08375 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 881.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1584.3.i.c 8
3.b odd 2 1 inner 1584.3.i.c 8
4.b odd 2 1 396.3.e.a ✓ 8
12.b even 2 1 396.3.e.a ✓ 8
44.c even 2 1 4356.3.e.f 8
132.d odd 2 1 4356.3.e.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
396.3.e.a ✓ 8 4.b odd 2 1
396.3.e.a ✓ 8 12.b even 2 1
1584.3.i.c 8 1.a even 1 1 trivial
1584.3.i.c 8 3.b odd 2 1 inner
4356.3.e.f 8 44.c even 2 1
4356.3.e.f 8 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 124T_{5}^{6} + 4060T_{5}^{4} + 19728T_{5}^{2} + 11664 \) acting on \(S_{3}^{\mathrm{new}}(1584, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 124 T^{6} + \cdots + 11664 \) Copy content Toggle raw display
$7$ \( (T^{4} - 8 T^{3} + \cdots - 436)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 11)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 4 T^{3} + \cdots - 2196)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 1522872576 \) Copy content Toggle raw display
$19$ \( (T^{4} + 20 T^{3} + \cdots + 22764)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 29599137936 \) Copy content Toggle raw display
$29$ \( T^{8} + 2864 T^{6} + \cdots + 136048896 \) Copy content Toggle raw display
$31$ \( (T^{4} + 20 T^{3} + \cdots + 1720368)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 28 T^{3} + \cdots + 486192)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 14135314012416 \) Copy content Toggle raw display
$43$ \( (T^{4} + 12 T^{3} + \cdots - 487604)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 38170218384 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 5449180585104 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 44929029279744 \) Copy content Toggle raw display
$61$ \( (T^{4} - 60 T^{3} + \cdots - 4232084)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 136 T^{3} + \cdots + 10712208)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 254487963024 \) Copy content Toggle raw display
$73$ \( (T^{4} + 128 T^{3} + \cdots - 12528)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 16 T^{3} + \cdots - 3386836)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 420899034398976 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 25\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{4} + 12 T^{3} + \cdots - 1916144)^{2} \) Copy content Toggle raw display
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