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Properties

Label 1568.3.d.l
Level $1568$
Weight $3$
Character orbit 1568.d
Analytic conductor $42.725$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1568,3,Mod(1471,1568)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1568, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 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("1568.1471"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1568.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,-6,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(42.7249054517\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.9144576.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 224)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + (\beta_{2} + 3) q^{5} - \beta_{5} q^{9} + (3 \beta_{4} + \beta_1) q^{11} - \beta_{5} q^{13} + ( - 3 \beta_{4} + \beta_{3} - \beta_1) q^{15} + ( - 5 \beta_{2} - 1) q^{17} + ( - 9 \beta_{4} + 3 \beta_{3} + 3 \beta_1) q^{19}+ \cdots + ( - 40 \beta_{4} - 11 \beta_{3} + 17 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{5} - 6 q^{17} + 186 q^{33} + 114 q^{37} + 18 q^{53} - 342 q^{57} + 318 q^{61} + 114 q^{69} + 342 q^{73} + 186 q^{81} - 498 q^{85} + 150 q^{89} + 222 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 12x^{4} + 36x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} + 6\nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 11\nu^{3} + 30\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{4} + 10\nu^{2} + 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{2} - 16 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} - 6\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{5} + 5\beta_{2} + 48 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 4\beta_{4} - 11\beta_{3} + 36\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1471.1
0.339877i
2.60168i
− 2.26180i
2.26180i
− 2.60168i
− 0.339877i
0 − 4.88448i 0 2.32025 0 0 0 −14.8582 0
1471.2 0 − 1.76873i 0 8.20336 0 0 0 5.87158 0
1471.3 0 − 0.115749i 0 −1.52360 0 0 0 8.98660 0
1471.4 0 0.115749i 0 −1.52360 0 0 0 8.98660 0
1471.5 0 1.76873i 0 8.20336 0 0 0 5.87158 0
1471.6 0 4.88448i 0 2.32025 0 0 0 −14.8582 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1471.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.3.d.l 6
4.b odd 2 1 inner 1568.3.d.l 6
7.b odd 2 1 1568.3.d.i 6
7.d odd 6 2 224.3.r.d ✓ 12
28.d even 2 1 1568.3.d.i 6
28.f even 6 2 224.3.r.d ✓ 12
56.j odd 6 2 448.3.r.f 12
56.m even 6 2 448.3.r.f 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.3.r.d ✓ 12 7.d odd 6 2
224.3.r.d ✓ 12 28.f even 6 2
448.3.r.f 12 56.j odd 6 2
448.3.r.f 12 56.m even 6 2
1568.3.d.i 6 7.b odd 2 1
1568.3.d.i 6 28.d even 2 1
1568.3.d.l 6 1.a even 1 1 trivial
1568.3.d.l 6 4.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 \(S_{3}^{\mathrm{new}}(1568, [\chi])\):

\( T_{3}^{6} + 27T_{3}^{4} + 75T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{3} - 9T_{5}^{2} + 3T_{5} + 29 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 27 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{3} - 9 T^{2} + 3 T + 29)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 363 T^{4} + \cdots + 452929 \) Copy content Toggle raw display
$13$ \( (T^{3} - 168 T + 784)^{2} \) Copy content Toggle raw display
$17$ \( (T^{3} + 3 T^{2} + \cdots + 1401)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 1755 T^{4} + \cdots + 96059601 \) Copy content Toggle raw display
$23$ \( T^{6} + 2595 T^{4} + \cdots + 163763209 \) Copy content Toggle raw display
$29$ \( (T^{3} - 1176 T - 5488)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 531 T^{4} + \cdots + 1885129 \) Copy content Toggle raw display
$37$ \( (T^{3} - 57 T^{2} + \cdots - 363)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 1848 T - 22736)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 2517630976 \) Copy content Toggle raw display
$47$ \( T^{6} + 4947 T^{4} + \cdots + 80802121 \) Copy content Toggle raw display
$53$ \( (T^{3} - 9 T^{2} + \cdots - 21531)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 60465334609 \) Copy content Toggle raw display
$61$ \( (T^{3} - 159 T^{2} + \cdots + 2323)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 3387 T^{4} + \cdots + 4068289 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 37803802624 \) Copy content Toggle raw display
$73$ \( (T^{3} - 171 T^{2} + \cdots - 134121)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 83448765625 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 14201012224 \) Copy content Toggle raw display
$89$ \( (T^{3} - 75 T^{2} + \cdots + 357111)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 168 T + 784)^{2} \) Copy content Toggle raw display
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