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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,-448] 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: \( x^{8} - 2x^{7} + 2x^{6} + 128x^{5} + 27955x^{4} - 38988x^{3} + 30258x^{2} + 862722x + 12299049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{6} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - 56) q^{5} + \beta_{5} q^{7} + ( - 2 \beta_{5} - \beta_{2} + 3 \beta_1) q^{11} + (\beta_{4} - 8 \beta_{3} + 358) q^{13} + ( - 3 \beta_{6} + 2 \beta_{4} + \cdots + 1392) q^{17} + (3 \beta_{7} + 2 \beta_{5} + \cdots - 64 \beta_1) q^{19}+ \cdots + ( - 2864 \beta_{6} + 1988 \beta_{4} + \cdots + 394846) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 448 q^{5} + 2864 q^{13} + 11136 q^{17} - 224904 q^{25} - 89408 q^{29} - 1539120 q^{37} + 1340032 q^{41} - 1087608 q^{49} + 6218688 q^{53} + 4895696 q^{61} + 21936512 q^{65} + 12835440 q^{73} + 82920448 q^{77}+ \cdots + 3158768 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 128x^{5} + 27955x^{4} - 38988x^{3} + 30258x^{2} + 862722x + 12299049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 2009307200 \nu^{7} + 53549948672 \nu^{6} - 70642486848 \nu^{5} - 196706999744 \nu^{4} + \cdots - 845205359173056 ) / 261434331395265 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5988599004 \nu^{7} + 457126025104 \nu^{6} + 5718382283452 \nu^{5} - 1298763831100 \nu^{4} + \cdots - 10\!\cdots\!80 ) / 161840300387545 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 433433864 \nu^{7} + 529497664 \nu^{6} + 9583709096 \nu^{5} - 156169598656 \nu^{4} + \cdots - 16\!\cdots\!64 ) / 2907310785405 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 257597024 \nu^{7} + 53467904 \nu^{6} - 75277480928 \nu^{5} - 98998120160 \nu^{4} + \cdots - 31\!\cdots\!00 ) / 969103595135 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1790471868678 \nu^{7} + 7520044684832 \nu^{6} + 33216232738478 \nu^{5} + \cdots + 82\!\cdots\!98 ) / 33\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 15306990032 \nu^{7} - 18938571136 \nu^{6} - 293278533392 \nu^{5} - 608785893824 \nu^{4} + \cdots - 25\!\cdots\!76 ) / 2907310785405 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19673279878816 \nu^{7} + 131292937996032 \nu^{6} - 519361685070880 \nu^{5} + \cdots - 86\!\cdots\!64 ) / 33\!\cdots\!45 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} - 4\beta_{6} + 32\beta_{5} + 8\beta_{4} + 56\beta_{3} - 16\beta_{2} + 73\beta _1 + 9216 ) / 36864 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -50\beta_{7} - 352\beta_{5} - 16\beta_{2} + 13453\beta_1 ) / 18432 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} + 6\beta_{6} + 48\beta_{5} - 56\beta_{4} + 108\beta_{3} - 112\beta_{2} + 570\beta _1 - 74496 ) / 1536 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -134\beta_{6} - 29\beta_{4} - 4784\beta_{3} - 4062672 ) / 288 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 14714 \beta_{7} + 6572 \beta_{6} - 52576 \beta_{5} - 225016 \beta_{4} + 483992 \beta_{3} + \cdots - 395817984 ) / 36864 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 125648\beta_{7} + 973696\beta_{5} + 63120\beta_{2} - 27586813\beta_1 ) / 1536 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1705334 \beta_{7} + 87436 \beta_{6} + 699488 \beta_{5} + 37665448 \beta_{4} - 54395816 \beta_{3} + \cdots + 89228491776 ) / 36864 \) 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
−8.97166 − 8.97166i
−8.97166 + 8.97166i
9.27000 + 9.27000i
9.27000 − 9.27000i
3.61657 − 3.61657i
3.61657 + 3.61657i
−2.91492 − 2.91492i
−2.91492 + 2.91492i
0 0 0 −564.760 0 − 2085.35i 0 0 0
127.2 0 0 0 −564.760 0 2085.35i 0 0 0
127.3 0 0 0 −424.868 0 − 1416.17i 0 0 0
127.4 0 0 0 −424.868 0 1416.17i 0 0 0
127.5 0 0 0 −190.548 0 − 4097.74i 0 0 0
127.6 0 0 0 −190.548 0 4097.74i 0 0 0
127.7 0 0 0 956.176 0 − 676.223i 0 0 0
127.8 0 0 0 956.176 0 676.223i 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.d ✓ 8
3.b odd 2 1 288.9.g.e yes 8
4.b odd 2 1 inner 288.9.g.d ✓ 8
12.b even 2 1 288.9.g.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.9.g.d ✓ 8 1.a even 1 1 trivial
288.9.g.d ✓ 8 4.b odd 2 1 inner
288.9.g.e yes 8 3.b odd 2 1
288.9.g.e yes 8 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 224T_{5}^{3} - 699936T_{5}^{2} - 364019200T_{5} - 43718086400 \) 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
$5$ \( (T^{4} + 224 T^{3} + \cdots - 43718086400)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 22\!\cdots\!04)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 16\!\cdots\!64)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 68\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 42\!\cdots\!36)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 10\!\cdots\!04)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 25\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 50\!\cdots\!20)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 81\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 18\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 85\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 42\!\cdots\!20)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 21\!\cdots\!36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 55\!\cdots\!84)^{2} \) Copy content Toggle raw display
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