↑

Properties

Label 1664.4.a.j
Level $1664$
Weight $4$
Character orbit 1664.a
Self dual yes
Analytic conductor $98.179$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1664,4,Mod(1,1664)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1664.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1664 = 2^{7} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1664.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-5,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(98.1791782496\)
Analytic rank: \(1\)
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: \( x^{8} - 3x^{7} - 141x^{6} + 197x^{5} + 5748x^{4} + 714x^{3} - 74684x^{2} - 116064x + 576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + (\beta_{6} + 1) q^{5} + (\beta_{5} - 3) q^{7} + (\beta_{3} + 10) q^{9} + (\beta_{7} - \beta_{6} - \beta_1 - 6) q^{11} + 13 q^{13} + ( - \beta_{6} - \beta_{5} - \beta_{4} + \cdots - 18) q^{15}+ \cdots + ( - 5 \beta_{7} + 35 \beta_{6} + \cdots - 126) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 5 q^{3} + 5 q^{5} - 27 q^{7} + 77 q^{9} - 50 q^{11} + 104 q^{13} - 129 q^{15} - 25 q^{17} - 244 q^{19} + 161 q^{21} - 34 q^{23} + 153 q^{25} + 103 q^{27} + 134 q^{29} - 272 q^{31} - 140 q^{33} - 229 q^{35}+ \cdots - 1250 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 141x^{6} + 197x^{5} + 5748x^{4} + 714x^{3} - 74684x^{2} - 116064x + 576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{7} + 32\nu^{6} - 971\nu^{5} - 4718\nu^{4} + 26546\nu^{3} + 121276\nu^{2} + 40176\nu - 11952 ) / 9072 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - 2\nu - 36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 129\nu^{4} + 170\nu^{3} - 3558\nu^{2} - 6984\nu + 5688 ) / 252 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\nu^{7} - 44\nu^{6} - 1327\nu^{5} + 3350\nu^{4} + 38818\nu^{3} - 48172\nu^{2} - 291168\nu - 16992 ) / 9072 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 6\nu^{6} - 133\nu^{5} + 580\nu^{4} + 5202\nu^{3} - 11200\nu^{2} - 67032\nu - 27792 ) / 1512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7\nu^{7} - 88\nu^{6} - 731\nu^{5} + 9178\nu^{4} + 21602\nu^{3} - 196148\nu^{2} - 278640\nu + 197568 ) / 9072 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 2\beta _1 + 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{7} - \beta_{6} + 5\beta_{5} + 2\beta_{4} + \beta_{3} - 3\beta_{2} + 64\beta _1 + 65 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{7} - 24\beta_{6} + 15\beta_{5} + 33\beta_{4} + 93\beta_{3} + 6\beta_{2} + 215\beta _1 + 2394 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -444\beta_{7} - 264\beta_{6} + 627\beta_{5} + 303\beta_{4} + 263\beta_{3} - 315\beta_{2} + 5101\beta _1 + 8205 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1841 \beta_{7} - 3266 \beta_{6} + 2785 \beta_{5} + 4345 \beta_{4} + 8609 \beta_{3} + 264 \beta_{2} + \cdots + 197476 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 44070 \beta_{7} - 34074 \beta_{6} + 65391 \beta_{5} + 36825 \beta_{4} + 38691 \beta_{3} + \cdots + 980463 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.47084
−5.16256
−3.43860
−2.12057
0.00494703
5.77614
6.17597
10.2355
0 −9.47084 0 11.0654 0 −20.3985 0 62.6967 0
1.2 0 −6.16256 0 −4.02813 0 18.9371 0 10.9772 0
1.3 0 −4.43860 0 −6.80658 0 −24.2297 0 −7.29882 0
1.4 0 −3.12057 0 21.5539 0 11.5713 0 −17.2621 0
1.5 0 −0.995053 0 −17.6005 0 −5.03192 0 −26.0099 0
1.6 0 4.77614 0 −1.64004 0 7.71502 0 −4.18849 0
1.7 0 5.17597 0 10.9220 0 −28.3165 0 −0.209355 0
1.8 0 9.23551 0 −8.46608 0 12.7533 0 58.2947 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(13\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1664.4.a.j yes 8
4.b odd 2 1 1664.4.a.l yes 8
8.b even 2 1 1664.4.a.k yes 8
8.d odd 2 1 1664.4.a.i ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1664.4.a.i ✓ 8 8.d odd 2 1
1664.4.a.j yes 8 1.a even 1 1 trivial
1664.4.a.k yes 8 8.b even 2 1
1664.4.a.l yes 8 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1664))\):

\( T_{3}^{8} + 5T_{3}^{7} - 134T_{3}^{6} - 656T_{3}^{5} + 4583T_{3}^{4} + 22807T_{3}^{3} - 38234T_{3}^{2} - 240172T_{3} - 183656 \) Copy content Toggle raw display
\( T_{5}^{8} - 5 T_{5}^{7} - 564 T_{5}^{6} + 786 T_{5}^{5} + 82233 T_{5}^{4} + 142463 T_{5}^{3} + \cdots - 17453656 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 5 T^{7} + \cdots - 183656 \) Copy content Toggle raw display
$5$ \( T^{8} - 5 T^{7} + \cdots - 17453656 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 1518362416 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 8193540096 \) Copy content Toggle raw display
$13$ \( (T - 13)^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 8132969457400 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 18949121796096 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 54\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 65585995411456 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 29\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots - 16\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 92\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 56\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 77\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 47\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 89\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots - 12\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots - 14\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots - 94\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 39\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 64\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 67\!\cdots\!04 \) Copy content Toggle raw display
show more
show less