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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,7,Mod(127,288)] 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("288.127"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(288, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 288.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(66.2555760825\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{17} \)
Twist minimal: no (minimal twist has level 32)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + 14) q^{5} + ( - 5 \beta_{2} + 2 \beta_1) q^{7} + (14 \beta_{2} - 17 \beta_1) q^{11} + ( - 5 \beta_{3} - 2174) q^{13} + (34 \beta_{3} + 1326) q^{17} + (102 \beta_{2} - 125 \beta_1) q^{19}+ \cdots + (4710 \beta_{3} + 729586) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 56 q^{5} - 8696 q^{13} + 5304 q^{17} + 36588 q^{25} - 77576 q^{29} + 125256 q^{37} + 209848 q^{41} + 95556 q^{49} + 68664 q^{53} + 238216 q^{61} - 613264 q^{65} + 219528 q^{73} + 1028224 q^{77} + 3416592 q^{85}+ \cdots + 2918344 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 16\nu^{3} - 12\nu^{2} + 48\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -64\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -64\nu^{3} + 192\nu ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 4\beta_{3} - 3\beta_{2} + 16\beta_1 ) / 512 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{2} ) / 64 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -12\beta_{3} - 9\beta_{2} + 48\beta_1 ) / 512 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
−1.22474 + 1.22474i
−1.22474 − 1.22474i
1.22474 − 1.22474i
1.22474 + 1.22474i
0 0 0 −142.767 0 − 217.616i 0 0 0
127.2 0 0 0 −142.767 0 217.616i 0 0 0
127.3 0 0 0 170.767 0 − 374.384i 0 0 0
127.4 0 0 0 170.767 0 374.384i 0 0 0
\(n\): e.g. 2-40 or 80-90
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 288.7.g.b 4
3.b odd 2 1 32.7.c.b ✓ 4
4.b odd 2 1 inner 288.7.g.b 4
8.b even 2 1 576.7.g.l 4
8.d odd 2 1 576.7.g.l 4
12.b even 2 1 32.7.c.b ✓ 4
24.f even 2 1 64.7.c.e 4
24.h odd 2 1 64.7.c.e 4
48.i odd 4 1 256.7.d.d 4
48.i odd 4 1 256.7.d.g 4
48.k even 4 1 256.7.d.d 4
48.k even 4 1 256.7.d.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.7.c.b ✓ 4 3.b odd 2 1
32.7.c.b ✓ 4 12.b even 2 1
64.7.c.e 4 24.f even 2 1
64.7.c.e 4 24.h odd 2 1
256.7.d.d 4 48.i odd 4 1
256.7.d.d 4 48.k even 4 1
256.7.d.g 4 48.i odd 4 1
256.7.d.g 4 48.k even 4 1
288.7.g.b 4 1.a even 1 1 trivial
288.7.g.b 4 4.b odd 2 1 inner
576.7.g.l 4 8.b even 2 1
576.7.g.l 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 28T_{5} - 24380 \) acting on \(S_{7}^{\mathrm{new}}(288, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 28 T - 24380)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 6637686784 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 1222201600 \) Copy content Toggle raw display
$13$ \( (T^{2} + 4348 T + 4111876)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 2652 T - 26651580)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 1640407654656 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{2} + 38788 T - 85139708)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( (T^{2} - 62628 T - 219749820)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 104924 T + 2638622020)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 70\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 74\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{2} - 34332 T - 659098428)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{2} - 119108 T - 30268152188)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots + 31\!\cdots\!24 \) Copy content Toggle raw display
$73$ \( (T^{2} - 109764 T - 174293877372)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{2} - 1861500 T + 848539402500)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 1459172 T - 12900710204)^{2} \) Copy content Toggle raw display
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