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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [288,9,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, 9); 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, 9, names="a")
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 288.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,1232] 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: \(117.325039698\)
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: \( x^{8} - 342x^{6} + 43937x^{4} - 2512824x^{2} + 53993104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{43}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 96)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 154) q^{5} + (\beta_{6} + \beta_{5} + \cdots + 4 \beta_1) q^{7} + ( - 9 \beta_{6} + 4 \beta_{5} + \cdots - 15 \beta_1) q^{11} + (\beta_{7} + \beta_{4} - 13 \beta_{3} - 6782) q^{13}+ \cdots + (32190 \beta_{7} - 15008 \beta_{4} + \cdots + 3478786) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 1232 q^{5} - 54256 q^{13} + 204336 q^{17} - 781416 q^{25} + 333712 q^{29} + 3945168 q^{37} - 11803088 q^{41} - 3977208 q^{49} - 788592 q^{53} - 40252336 q^{61} + 20747936 q^{65} - 56712048 q^{73}+ \cdots + 27830288 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 342x^{6} + 43937x^{4} - 2512824x^{2} + 53993104 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -27\nu^{7} + 9234\nu^{5} - 987903\nu^{3} + 33920532\nu ) / 14696 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 348\nu^{7} - 89624\nu^{5} + 7765724\nu^{3} - 226222192\nu ) / 123079 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 14562 \nu^{7} - 20207 \nu^{6} + 3716348 \nu^{5} + 5172992 \nu^{4} - 315278826 \nu^{3} + \cdots + 12452119196 ) / 492316 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1044 \nu^{7} - 44088 \nu^{6} - 268872 \nu^{5} + 11286528 \nu^{4} + 23297172 \nu^{3} + \cdots + 27168260064 ) / 123079 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 697 \nu^{7} + 411488 \nu^{6} + 253070 \nu^{5} - 121095040 \nu^{4} - 27986157 \nu^{3} + \cdots - 377413501696 ) / 984632 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 27 \nu^{7} - 352704 \nu^{6} - 23930 \nu^{5} + 90292224 \nu^{4} + 3471527 \nu^{3} + \cdots + 225427939968 ) / 492316 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14562 \nu^{7} - 773377 \nu^{6} - 3716348 \nu^{5} + 197984512 \nu^{4} + 315278826 \nu^{3} + \cdots + 476576561956 ) / 492316 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} - 9\beta_{4} + 2\beta_{3} + 27\beta_{2} ) / 3456 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 32\beta_{7} - 72\beta_{6} + 32\beta_{3} - 9\beta_{2} - 16\beta _1 + 1181952 ) / 13824 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 53\beta_{7} - 255\beta_{4} + 197\beta_{3} + 2313\beta_{2} + 64\beta_1 ) / 1152 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 5440\beta_{7} - 12744\beta_{6} - 864\beta_{5} + 5440\beta_{3} - 1539\beta_{2} - 2416\beta _1 + 100535040 ) / 13824 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 12515\beta_{7} - 64683\beta_{4} + 85523\beta_{3} + 993357\beta_{2} + 54720\beta_1 ) / 3456 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 228960 \beta_{7} - 564600 \beta_{6} - 73728 \beta_{5} + 228960 \beta_{3} - 65967 \beta_{2} + \cdots + 2834839296 ) / 4608 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 975111\beta_{7} - 5437845\beta_{4} + 10137399\beta_{3} + 119757555\beta_{2} + 9808064\beta_1 ) / 3456 \) 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
9.62695 + 0.500000i
9.62695 − 0.500000i
−8.87816 + 0.500000i
−8.87816 − 0.500000i
−9.62695 − 0.500000i
−9.62695 + 0.500000i
8.87816 − 0.500000i
8.87816 + 0.500000i
0 0 0 −455.790 0 − 1689.76i 0 0 0
127.2 0 0 0 −455.790 0 1689.76i 0 0 0
127.3 0 0 0 −261.868 0 − 1340.75i 0 0 0
127.4 0 0 0 −261.868 0 1340.75i 0 0 0
127.5 0 0 0 611.370 0 − 4427.25i 0 0 0
127.6 0 0 0 611.370 0 4427.25i 0 0 0
127.7 0 0 0 722.288 0 − 891.264i 0 0 0
127.8 0 0 0 722.288 0 891.264i 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 127.8
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 288.9.g.f 8
3.b odd 2 1 96.9.g.a ✓ 8
4.b odd 2 1 inner 288.9.g.f 8
12.b even 2 1 96.9.g.a ✓ 8
24.f even 2 1 192.9.g.f 8
24.h odd 2 1 192.9.g.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.9.g.a ✓ 8 3.b odd 2 1
96.9.g.a ✓ 8 12.b even 2 1
192.9.g.f 8 24.f even 2 1
192.9.g.f 8 24.h odd 2 1
288.9.g.f 8 1.a even 1 1 trivial
288.9.g.f 8 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 616T_{5}^{3} - 396168T_{5}^{2} + 157725920T_{5} + 52706093200 \) acting on \(S_{9}^{\mathrm{new}}(288, [\chi])\). 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
$5$ \( (T^{4} - 616 T^{3} + \cdots + 52706093200)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 79\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 65\!\cdots\!84 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots - 81\!\cdots\!44)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 28\!\cdots\!68)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 21\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 40\!\cdots\!92)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 30\!\cdots\!84)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 37\!\cdots\!12)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 34\!\cdots\!72)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 31\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 41\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 13\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 60\!\cdots\!52)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 54\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 54\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 66\!\cdots\!12)^{2} \) Copy content Toggle raw display
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