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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,6,Mod(145,288)] 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("288.145"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 288.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(46.1905401061\)
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: \( x^{8} - 38x^{6} + 848x^{4} - 38912x^{2} + 1048576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 72)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + ( - \beta_{3} + 36) q^{7} + (2 \beta_{2} - 5 \beta_1) q^{11} + \beta_{6} q^{13} + \beta_{5} q^{17} + (\beta_{7} + 2 \beta_{6}) q^{19} + ( - 2 \beta_{5} - \beta_{4}) q^{23} + ( - 40 \beta_{3} - 931) q^{25}+ \cdots + (1840 \beta_{3} + 4962) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 288 q^{7} - 7448 q^{25} + 18272 q^{31} - 74136 q^{49} + 52736 q^{55} + 103664 q^{73} - 36768 q^{79} + 39696 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 38x^{6} + 848x^{4} - 38912x^{2} + 1048576 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 38\nu^{5} + 848\nu^{3} - 6144\nu ) / 4096 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -5\nu^{7} - 66\nu^{5} - 2704\nu^{3} + 124928\nu ) / 16384 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 38\nu^{4} + 176\nu^{2} + 19456 ) / 512 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 38\nu^{5} - 848\nu^{3} + 71680\nu ) / 256 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 6\nu^{5} + 1392\nu^{3} + 5632\nu ) / 256 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{6} + 66\nu^{4} - 368\nu^{2} - 86016 ) / 256 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} - 38\nu^{4} + 1872\nu^{2} - 38912 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + 16\beta_1 ) / 256 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 16\beta_{3} + 608 ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 16\beta_{5} + 3\beta_{4} - 128\beta_{2} + 144\beta_1 ) / 128 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{7} + 32\beta_{6} + 304\beta_{3} - 2016 ) / 32 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 48\beta_{5} + 101\beta_{4} - 4480\beta_{2} - 3216\beta_1 ) / 64 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 25\beta_{7} + 608\beta_{6} - 1712\beta_{3} + 299744 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -2480\beta_{5} + 2051\beta_{4} - 57984\beta_{2} + 51728\beta_1 ) / 32 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
5.53420 − 1.17159i
−5.53420 − 1.17159i
3.29737 + 4.59645i
−3.29737 + 4.59645i
−3.29737 − 4.59645i
3.29737 − 4.59645i
−5.53420 + 1.17159i
5.53420 + 1.17159i
0 0 0 − 84.9515i 0 −43.0190 0 0 0
145.2 0 0 0 − 84.9515i 0 −43.0190 0 0 0
145.3 0 0 0 − 29.9206i 0 115.019 0 0 0
145.4 0 0 0 − 29.9206i 0 115.019 0 0 0
145.5 0 0 0 29.9206i 0 115.019 0 0 0
145.6 0 0 0 29.9206i 0 115.019 0 0 0
145.7 0 0 0 84.9515i 0 −43.0190 0 0 0
145.8 0 0 0 84.9515i 0 −43.0190 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 145.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.6.d.c 8
3.b odd 2 1 inner 288.6.d.c 8
4.b odd 2 1 72.6.d.c ✓ 8
8.b even 2 1 inner 288.6.d.c 8
8.d odd 2 1 72.6.d.c ✓ 8
12.b even 2 1 72.6.d.c ✓ 8
24.f even 2 1 72.6.d.c ✓ 8
24.h odd 2 1 inner 288.6.d.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.6.d.c ✓ 8 4.b odd 2 1
72.6.d.c ✓ 8 8.d odd 2 1
72.6.d.c ✓ 8 12.b even 2 1
72.6.d.c ✓ 8 24.f even 2 1
288.6.d.c 8 1.a even 1 1 trivial
288.6.d.c 8 3.b odd 2 1 inner
288.6.d.c 8 8.b even 2 1 inner
288.6.d.c 8 24.h odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 8112T_{5}^{2} + 6460736 \) acting on \(S_{6}^{\mathrm{new}}(288, [\chi])\). 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
$5$ \( (T^{4} + 8112 T^{2} + 6460736)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 72 T - 4948)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 164288 T^{2} + 6587790336)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 930496 T^{2} + 193819852800)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 1964113920000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 3850448928768)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 52566322839552)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2547632 T^{2} + 16140891456)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4568 T - 24511028)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 19\!\cdots\!68)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 890541123305472)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 38\!\cdots\!52)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 25\!\cdots\!28)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 497370063716352)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 75\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 25916 T - 954611580)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 9192 T - 662975668)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 64\!\cdots\!52)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 9924 T - 21115064956)^{4} \) Copy content Toggle raw display
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