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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,6,Mod(37,72)] 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("72.37"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 72.d (of order \(2\), degree \(1\), 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: \(11.5476350265\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 10) q^{4} + ( - \beta_{4} + \beta_1) q^{5} + (\beta_{6} + 2 \beta_{2} - 35) q^{7} + (\beta_{5} + \beta_{4} + \cdots + 9 \beta_1) q^{8} + ( - \beta_{7} + 2 \beta_{6} + \cdots - 18) q^{10}+ \cdots + ( - 144 \beta_{5} - 144 \beta_{4} + \cdots - 8979 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 76 q^{4} - 288 q^{7} - 152 q^{10} - 504 q^{16} + 6896 q^{22} - 7448 q^{25} + 9752 q^{28} - 18272 q^{31} - 256 q^{34} + 29712 q^{40} + 41984 q^{46} - 74136 q^{49} - 27328 q^{52} - 52736 q^{55} + 8936 q^{58}+ \cdots + 39696 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 38\nu^{5} - 848\nu^{3} + 30720\nu ) / 4096 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{7} + 66\nu^{5} + 2704\nu^{3} - 108544\nu ) / 16384 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -9\nu^{7} + 86\nu^{5} + 10288\nu^{3} + 83968\nu ) / 16384 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{6} + 38\nu^{4} - 848\nu^{2} + 29184 ) / 512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{6} + 14\nu^{4} + 752\nu^{2} - 62464 ) / 256 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - \beta_{3} + 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{7} + 12\beta_{6} + 14\beta_{2} - 56 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 6\beta_{5} + 70\beta_{4} + 74\beta_{3} - 122\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 76\beta_{7} - 56\beta_{6} - 316\beta_{2} + 18576 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -620\beta_{5} + 1812\beta_{4} - 436\beta_{3} + 18452\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−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
−5.53420 − 1.17159i 0 29.2547 + 12.9676i 84.9515i 0 43.0190 −146.709 − 106.040i 0 99.5285 − 470.139i
37.2 −5.53420 + 1.17159i 0 29.2547 − 12.9676i − 84.9515i 0 43.0190 −146.709 + 106.040i 0 99.5285 + 470.139i
37.3 −3.29737 − 4.59645i 0 −10.2547 + 30.3124i − 29.9206i 0 −115.019 173.143 − 52.8156i 0 −137.528 + 98.6591i
37.4 −3.29737 + 4.59645i 0 −10.2547 − 30.3124i 29.9206i 0 −115.019 173.143 + 52.8156i 0 −137.528 − 98.6591i
37.5 3.29737 − 4.59645i 0 −10.2547 − 30.3124i − 29.9206i 0 −115.019 −173.143 − 52.8156i 0 −137.528 − 98.6591i
37.6 3.29737 + 4.59645i 0 −10.2547 + 30.3124i 29.9206i 0 −115.019 −173.143 + 52.8156i 0 −137.528 + 98.6591i
37.7 5.53420 − 1.17159i 0 29.2547 − 12.9676i 84.9515i 0 43.0190 146.709 − 106.040i 0 99.5285 + 470.139i
37.8 5.53420 + 1.17159i 0 29.2547 + 12.9676i − 84.9515i 0 43.0190 146.709 + 106.040i 0 99.5285 − 470.139i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 37.8
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 72.6.d.c ✓ 8
3.b odd 2 1 inner 72.6.d.c ✓ 8
4.b odd 2 1 288.6.d.c 8
8.b even 2 1 inner 72.6.d.c ✓ 8
8.d odd 2 1 288.6.d.c 8
12.b even 2 1 288.6.d.c 8
24.f even 2 1 288.6.d.c 8
24.h odd 2 1 inner 72.6.d.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.6.d.c ✓ 8 1.a even 1 1 trivial
72.6.d.c ✓ 8 3.b odd 2 1 inner
72.6.d.c ✓ 8 8.b even 2 1 inner
72.6.d.c ✓ 8 24.h odd 2 1 inner
288.6.d.c 8 4.b odd 2 1
288.6.d.c 8 8.d odd 2 1
288.6.d.c 8 12.b even 2 1
288.6.d.c 8 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 8112T_{5}^{2} + 6460736 \) acting on \(S_{6}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 38 T^{6} + \cdots + 1048576 \) 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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