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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1408,2,Mod(705,1408)] 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("1408.705"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1408, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1408 = 2^{7} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1408.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.2429366046\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.3534400.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + \beta_{5} q^{5} + ( - \beta_{3} - 1) q^{7} + ( - \beta_{3} - \beta_1 - 2) q^{9} - \beta_{4} q^{11} + (\beta_{4} - \beta_{2}) q^{13} + (\beta_{3} + \beta_1 + 5) q^{15} + (\beta_{3} - 2 \beta_1 - 1) q^{17}+ \cdots + ( - \beta_{5} + 2 \beta_{4} - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{7} - 10 q^{9} + 28 q^{15} - 8 q^{17} + 4 q^{23} + 2 q^{25} + 44 q^{31} + 4 q^{39} + 24 q^{41} + 24 q^{47} + 6 q^{49} + 4 q^{57} + 48 q^{63} - 4 q^{65} - 4 q^{71} - 16 q^{73} + 40 q^{79} - 2 q^{81}+ \cdots - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 3x^{4} + 16x^{3} + x^{2} - 12x + 40 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -11\nu^{5} - 30\nu^{4} + 53\nu^{3} + 115\nu^{2} - 236\nu - 660 ) / 445 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -27\nu^{5} + 250\nu^{4} - 679\nu^{3} + 80\nu^{2} + 2293\nu - 2510 ) / 890 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -17\nu^{5} + 75\nu^{4} + \nu^{3} - 510\nu^{2} + 768\nu + 315 ) / 445 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{5} + 24\nu^{4} + 11\nu^{3} - 92\nu^{2} - 7\nu - 6 ) / 178 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -28\nu^{5} + 45\nu^{4} + 54\nu^{3} - 395\nu^{2} - 358\nu + 100 ) / 445 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{5} + 4\beta_{4} - \beta_{3} + \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{5} + 11\beta_{4} + \beta_{3} - 3\beta_{2} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -13\beta_{5} + 30\beta_{4} - 7\beta_{3} - 2\beta_{2} - 15\beta _1 - 19 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13\beta_{5} + 13\beta_{4} - 8\beta_{3} - 9\beta_{2} - 51\beta _1 - 92 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1408\mathbb{Z}\right)^\times\).

\(n\) \(133\) \(639\) \(1025\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
705.1
2.19082 − 1.44755i
0.627553 − 1.14620i
−1.81837 − 0.301352i
−1.81837 + 0.301352i
0.627553 + 1.14620i
2.19082 + 1.44755i
0 − 2.89511i 0 2.89511i 0 −1.48654 0 −5.38164 0
705.2 0 − 2.29240i 0 2.29240i 0 −3.54751 0 −2.25511 0
705.3 0 − 0.602705i 0 0.602705i 0 3.03404 0 2.63675 0
705.4 0 0.602705i 0 − 0.602705i 0 3.03404 0 2.63675 0
705.5 0 2.29240i 0 − 2.29240i 0 −3.54751 0 −2.25511 0
705.6 0 2.89511i 0 − 2.89511i 0 −1.48654 0 −5.38164 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 705.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1408.2.c.c ✓ 6
4.b odd 2 1 1408.2.c.d yes 6
8.b even 2 1 inner 1408.2.c.c ✓ 6
8.d odd 2 1 1408.2.c.d yes 6
16.e even 4 1 2816.2.a.k 3
16.e even 4 1 2816.2.a.l 3
16.f odd 4 1 2816.2.a.i 3
16.f odd 4 1 2816.2.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1408.2.c.c ✓ 6 1.a even 1 1 trivial
1408.2.c.c ✓ 6 8.b even 2 1 inner
1408.2.c.d yes 6 4.b odd 2 1
1408.2.c.d yes 6 8.d odd 2 1
2816.2.a.i 3 16.f odd 4 1
2816.2.a.j 3 16.f odd 4 1
2816.2.a.k 3 16.e even 4 1
2816.2.a.l 3 16.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1408, [\chi])\):

\( T_{3}^{6} + 14T_{3}^{4} + 49T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{3} + 2T_{7}^{2} - 10T_{7} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 14 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{6} + 14 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( (T^{3} + 2 T^{2} - 10 T - 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{6} + 24 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( (T^{3} + 4 T^{2} + \cdots - 148)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 56 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( (T^{3} - 2 T^{2} - 15 T + 20)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 96 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$31$ \( (T^{3} - 22 T^{2} + \cdots - 212)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 198 T^{4} + \cdots + 400 \) Copy content Toggle raw display
$41$ \( (T^{3} - 12 T^{2} + \cdots + 580)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 216 T^{4} + \cdots + 73984 \) Copy content Toggle raw display
$47$ \( (T^{3} - 12 T^{2} + \cdots + 80)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 104 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$59$ \( T^{6} + 382 T^{4} + \cdots + 1795600 \) Copy content Toggle raw display
$61$ \( (T^{2} + 64)^{3} \) Copy content Toggle raw display
$67$ \( T^{6} + 246 T^{4} + \cdots + 115600 \) Copy content Toggle raw display
$71$ \( (T^{3} + 2 T^{2} - 19 T + 20)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 8 T^{2} + 10 T - 20)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 20 T^{2} + \cdots + 3200)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 152 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$89$ \( (T^{3} - 8 T^{2} - 47 T - 46)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 16 T^{2} + \cdots - 422)^{2} \) Copy content Toggle raw display
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