↑

Properties

Label 1664.2.b.k
Level $1664$
Weight $2$
Character orbit 1664.b
Analytic conductor $13.287$
Analytic rank $0$
Dimension $12$
Inner twists $4$

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] = [1664,2,Mod(833,1664)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1664.833"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1664 = 2^{7} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1664.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.2871068963\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.426337261060096.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{9} q^{3} + (\beta_{7} - \beta_{4}) q^{5} - \beta_1 q^{7} - \beta_{2} q^{9} + (\beta_{9} + \beta_{8} - \beta_{6}) q^{11} - \beta_{7} q^{13} + ( - \beta_{5} - 3 \beta_{3}) q^{15} + (\beta_{11} - 1) q^{17}+ \cdots + ( - 3 \beta_{9} + \beta_{8} - 3 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{9} - 8 q^{17} - 28 q^{25} + 32 q^{33} - 40 q^{41} - 4 q^{49} + 48 q^{57} + 16 q^{65} + 24 q^{73} - 52 q^{81} + 8 q^{89} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4x^{9} - 3x^{8} + 4x^{7} + 8x^{6} + 8x^{5} - 12x^{4} - 32x^{3} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{11} - \nu^{10} + \nu^{9} + 2 \nu^{8} + 13 \nu^{7} - 5 \nu^{6} - 19 \nu^{5} - 22 \nu^{4} + \cdots + 88 \nu ) / 80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} + 4 \nu^{10} - 4 \nu^{9} - 8 \nu^{8} + 3 \nu^{7} + 36 \nu^{5} + 28 \nu^{4} - 52 \nu^{3} + \cdots + 160 ) / 80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{11} - 6 \nu^{10} + 16 \nu^{9} + 12 \nu^{8} + 3 \nu^{7} - 50 \nu^{6} - 64 \nu^{5} + 48 \nu^{4} + \cdots - 320 ) / 160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 3 \nu^{11} + 4 \nu^{10} + 4 \nu^{8} - 15 \nu^{7} - 16 \nu^{6} + 8 \nu^{5} + 32 \nu^{4} + 28 \nu^{3} + \cdots + 64 ) / 80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} - 4\nu^{10} - \nu^{9} - 2\nu^{8} + 7\nu^{7} + 20\nu^{6} - \nu^{5} - 18\nu^{4} - 18\nu^{3} + 16\nu^{2} + 72\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{11} - 7 \nu^{10} - 15 \nu^{9} + 8 \nu^{8} + 25 \nu^{7} + 33 \nu^{6} - 19 \nu^{5} - 96 \nu^{4} + \cdots + 208 ) / 80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{11} + \nu^{10} - 4\nu^{8} - 5\nu^{7} + \nu^{6} + 12\nu^{5} + 8\nu^{4} + 12\nu^{3} - 36\nu^{2} - 16\nu - 64 ) / 80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} - \nu^{9} + 2\nu^{7} + 3\nu^{6} - 5\nu^{5} - 4\nu^{4} - 2\nu^{3} + 8\nu^{2} + 24\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9 \nu^{11} + 8 \nu^{10} - 12 \nu^{8} - 35 \nu^{7} + 28 \nu^{6} + 56 \nu^{5} + 64 \nu^{4} - 84 \nu^{3} + \cdots - 32 ) / 160 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{11} + 2 \nu^{10} + 5 \nu^{9} + 2 \nu^{8} - 5 \nu^{7} - 8 \nu^{6} - 11 \nu^{5} + 26 \nu^{4} + \cdots - 128 ) / 20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2 \nu^{11} - 2 \nu^{10} + 2 \nu^{9} + 9 \nu^{8} + 6 \nu^{7} - 10 \nu^{6} - 18 \nu^{5} + \nu^{4} + \cdots - 40 ) / 20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} + 2\beta_{9} + \beta_{8} - 2\beta_{6} - \beta_{5} - \beta_{4} - 2\beta_{3} - \beta_{2} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + \beta_{5} + 2\beta_{4} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} + \beta_{10} - 6 \beta_{9} + \beta_{8} + 8 \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 8 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{11} + 4\beta_{9} + 3\beta_{8} - 2\beta_{6} + 4\beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5 \beta_{11} - 5 \beta_{10} + 2 \beta_{9} - 7 \beta_{8} + 16 \beta_{7} - 2 \beta_{6} + 7 \beta_{5} + \cdots - 16 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4\beta_{10} - 10\beta_{7} + 7\beta_{5} + 2\beta_{4} - 8\beta_{3} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7 \beta_{11} + 7 \beta_{10} - 10 \beta_{9} - \beta_{8} + 8 \beta_{7} + 2 \beta_{6} - \beta_{5} + \cdots + 8 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 10\beta_{11} + 20\beta_{9} - 7\beta_{8} + 10\beta_{6} - 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 5 \beta_{11} + 5 \beta_{10} - 2 \beta_{9} - 33 \beta_{8} - 64 \beta_{7} - 14 \beta_{6} + 33 \beta_{5} + \cdots + 64 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 16\beta_{10} - 22\beta_{7} - 19\beta_{5} - 14\beta_{4} - 48\beta_{3} + 26\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 9 \beta_{11} + 9 \beta_{10} + 42 \beta_{9} - 7 \beta_{8} + 200 \beta_{7} + 46 \beta_{6} - 7 \beta_{5} + \cdots + 200 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1664\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(769\) \(1535\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
833.1
−1.35818 − 0.394157i
−0.394157 + 1.35818i
−0.0912546 + 1.41127i
1.41127 + 0.0912546i
−0.760198 − 1.19252i
1.19252 − 0.760198i
1.19252 + 0.760198i
−0.760198 + 1.19252i
1.41127 − 0.0912546i
−0.0912546 − 1.41127i
−0.394157 − 1.35818i
−1.35818 + 0.394157i
0 − 2.47817i 0 − 4.14134i 0 −1.96435 0 −3.14134 0
833.2 0 − 2.47817i 0 4.14134i 0 1.96435 0 −3.14134 0
833.3 0 − 1.86678i 0 − 1.48486i 0 2.55248 0 −0.484862 0
833.4 0 − 1.86678i 0 1.48486i 0 −2.55248 0 −0.484862 0
833.5 0 − 0.611393i 0 − 1.62620i 0 3.10261 0 2.62620 0
833.6 0 − 0.611393i 0 1.62620i 0 −3.10261 0 2.62620 0
833.7 0 0.611393i 0 − 1.62620i 0 −3.10261 0 2.62620 0
833.8 0 0.611393i 0 1.62620i 0 3.10261 0 2.62620 0
833.9 0 1.86678i 0 − 1.48486i 0 −2.55248 0 −0.484862 0
833.10 0 1.86678i 0 1.48486i 0 2.55248 0 −0.484862 0
833.11 0 2.47817i 0 − 4.14134i 0 1.96435 0 −3.14134 0
833.12 0 2.47817i 0 4.14134i 0 −1.96435 0 −3.14134 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 833.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1664.2.b.k ✓ 12
4.b odd 2 1 inner 1664.2.b.k ✓ 12
8.b even 2 1 inner 1664.2.b.k ✓ 12
8.d odd 2 1 inner 1664.2.b.k ✓ 12
16.e even 4 1 3328.2.a.bo 6
16.e even 4 1 3328.2.a.bp 6
16.f odd 4 1 3328.2.a.bo 6
16.f odd 4 1 3328.2.a.bp 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1664.2.b.k ✓ 12 1.a even 1 1 trivial
1664.2.b.k ✓ 12 4.b odd 2 1 inner
1664.2.b.k ✓ 12 8.b even 2 1 inner
1664.2.b.k ✓ 12 8.d odd 2 1 inner
3328.2.a.bo 6 16.e even 4 1
3328.2.a.bo 6 16.f odd 4 1
3328.2.a.bp 6 16.e even 4 1
3328.2.a.bp 6 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1664, [\chi])\):

\( T_{3}^{6} + 10T_{3}^{4} + 25T_{3}^{2} + 8 \) Copy content Toggle raw display
\( T_{5}^{6} + 22T_{5}^{4} + 89T_{5}^{2} + 100 \) Copy content Toggle raw display
\( T_{7}^{6} - 20T_{7}^{4} + 125T_{7}^{2} - 242 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + 10 T^{4} + 25 T^{2} + 8)^{2} \) Copy content Toggle raw display
$5$ \( (T^{6} + 22 T^{4} + \cdots + 100)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} - 20 T^{4} + \cdots - 242)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 42 T^{4} + \cdots + 800)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} + 2 T^{2} - 23 T - 44)^{4} \) Copy content Toggle raw display
$19$ \( (T^{6} + 66 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 40 T^{4} + \cdots - 512)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4)^{6} \) Copy content Toggle raw display
$31$ \( (T^{6} - 34 T^{4} + \cdots - 128)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 134 T^{4} + \cdots + 2500)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 10 T^{2} + \cdots - 512)^{4} \) Copy content Toggle raw display
$43$ \( (T^{6} + 34 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 52 T^{4} + \cdots - 50)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 280 T^{4} + \cdots + 350464)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 130 T^{4} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 208 T^{4} + \cdots + 16384)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 202 T^{4} + \cdots + 43808)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 196 T^{4} + \cdots - 7442)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 6 T^{2} + \cdots - 128)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} - 272 T^{4} + \cdots - 574592)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 218 T^{4} + \cdots + 161312)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 2 T^{2} - 32 T + 32)^{4} \) Copy content Toggle raw display
$97$ \( (T^{3} - 10 T^{2} + \cdots + 488)^{4} \) Copy content Toggle raw display
show more
show less