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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2916,2,Mod(1,2916)] 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("2916.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2916, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2916 = 2^{2} \cdot 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2916.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,0,0,0,6] 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: yes
Analytic conductor: \(23.2843772294\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 12x^{7} + 50x^{6} - 3x^{5} - 186x^{4} + 276x^{3} - 153x^{2} + 36x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 108)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{5} + \beta_{6} q^{7} + (\beta_{6} + \beta_{3} + 1) q^{11} + ( - \beta_{6} - \beta_{4} - \beta_{2} + \cdots + 1) q^{13} + ( - \beta_{8} + \beta_{7} - \beta_{6} + \cdots + 2) q^{17}+ \cdots + ( - 2 \beta_{8} + 2 \beta_{7} + \cdots + 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 6 q^{5} + 6 q^{11} + 12 q^{17} + 3 q^{23} + 9 q^{25} + 24 q^{29} + 21 q^{35} + 33 q^{41} + 9 q^{47} + 9 q^{49} + 33 q^{53} + 30 q^{59} - 9 q^{61} + 39 q^{65} - 9 q^{67} + 12 q^{71} - 9 q^{73} + 39 q^{77}+ \cdots + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 12x^{7} + 50x^{6} - 3x^{5} - 186x^{4} + 276x^{3} - 153x^{2} + 36x - 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{7} - \nu^{6} - 15\nu^{5} + 21\nu^{4} + 53\nu^{3} - 101\nu^{2} + 30\nu + 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{8} - 2\nu^{7} - 14\nu^{6} + 36\nu^{5} + 33\nu^{4} - 154\nu^{3} + 124\nu^{2} - 21\nu - 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{8} - 6\nu^{7} - 25\nu^{6} + 101\nu^{5} + 9\nu^{4} - 393\nu^{3} + 501\nu^{2} - 204\nu + 27 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -5\nu^{8} + 12\nu^{7} + 66\nu^{6} - 210\nu^{5} - 94\nu^{4} + 858\nu^{3} - 924\nu^{2} + 300\nu - 26 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -5\nu^{8} + 12\nu^{7} + 65\nu^{6} - 209\nu^{5} - 79\nu^{4} + 837\nu^{3} - 978\nu^{2} + 399\nu - 51 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -4\nu^{8} + 13\nu^{7} + 49\nu^{6} - 216\nu^{5} + 5\nu^{4} + 829\nu^{3} - 1114\nu^{2} + 468\nu - 58 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 7\nu^{8} - 18\nu^{7} - 91\nu^{6} + 310\nu^{5} + 101\nu^{4} - 1239\nu^{3} + 1437\nu^{2} - 546\nu + 63 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( 8\nu^{8} - 21\nu^{7} - 103\nu^{6} + 360\nu^{5} + 98\nu^{4} - 1426\nu^{3} + 1714\nu^{2} - 693\nu + 88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{8} - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + 2\beta_{2} - 3\beta _1 + 8 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{8} + 8\beta_{7} - 5\beta_{6} - 6\beta_{5} + \beta_{4} + 2\beta_{3} - \beta_{2} + 6\beta _1 + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 26\beta_{8} - 15\beta_{7} + 13\beta_{6} + 5\beta_{5} + 7\beta_{4} - 4\beta_{3} + 17\beta_{2} - 27\beta _1 + 44 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -100\beta_{8} + 69\beta_{7} - 32\beta_{6} - 40\beta_{5} + 7\beta_{4} + 23\beta_{3} - 22\beta_{2} + 48\beta _1 - 22 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 257 \beta_{8} - 171 \beta_{7} + 115 \beta_{6} + 59 \beta_{5} + 40 \beta_{4} - 25 \beta_{3} + 146 \beta_{2} + \cdots + 287 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 838 \beta_{8} + 624 \beta_{7} - 242 \beta_{6} - 298 \beta_{5} + 46 \beta_{4} + 197 \beta_{3} + \cdots - 331 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 2372 \beta_{8} - 1758 \beta_{7} + 934 \beta_{6} + 560 \beta_{5} + 199 \beta_{4} - 220 \beta_{3} + \cdots + 2060 ) / 3 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.16301
2.44090
−3.01022
1.76664
0.422555
0.302126
1.48141
0.203930
−2.77036
0 0 0 −2.44631 0 −4.46504 0 0 0
1.2 0 0 0 −2.30119 0 −0.933596 0 0 0
1.3 0 0 0 −0.994440 0 −1.43919 0 0 0
1.4 0 0 0 −0.593913 0 2.85597 0 0 0
1.5 0 0 0 −0.101395 0 0.950388 0 0 0
1.6 0 0 0 1.91839 0 4.10905 0 0 0
1.7 0 0 0 2.95544 0 −2.66987 0 0 0
1.8 0 0 0 3.36302 0 −1.92237 0 0 0
1.9 0 0 0 4.20041 0 3.51465 0 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2916.2.a.d 9
3.b odd 2 1 2916.2.a.c 9
9.c even 3 2 2916.2.e.c 18
9.d odd 6 2 2916.2.e.d 18
27.e even 9 2 108.2.i.a ✓ 18
27.e even 9 2 972.2.i.a 18
27.e even 9 2 972.2.i.c 18
27.f odd 18 2 324.2.i.a 18
27.f odd 18 2 972.2.i.b 18
27.f odd 18 2 972.2.i.d 18
108.j odd 18 2 432.2.u.d 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.2.i.a ✓ 18 27.e even 9 2
324.2.i.a 18 27.f odd 18 2
432.2.u.d 18 108.j odd 18 2
972.2.i.a 18 27.e even 9 2
972.2.i.b 18 27.f odd 18 2
972.2.i.c 18 27.e even 9 2
972.2.i.d 18 27.f odd 18 2
2916.2.a.c 9 3.b odd 2 1
2916.2.a.d 9 1.a even 1 1 trivial
2916.2.e.c 18 9.c even 3 2
2916.2.e.d 18 9.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{9} - 6T_{5}^{8} - 9T_{5}^{7} + 87T_{5}^{6} + 18T_{5}^{5} - 405T_{5}^{4} - 63T_{5}^{3} + 567T_{5}^{2} + 324T_{5} + 27 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2916))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} \) Copy content Toggle raw display
$5$ \( T^{9} - 6 T^{8} + \cdots + 27 \) Copy content Toggle raw display
$7$ \( T^{9} - 36 T^{7} + \cdots + 1207 \) Copy content Toggle raw display
$11$ \( T^{9} - 6 T^{8} + \cdots - 4833 \) Copy content Toggle raw display
$13$ \( T^{9} - 63 T^{7} + \cdots + 2017 \) Copy content Toggle raw display
$17$ \( T^{9} - 12 T^{8} + \cdots - 72441 \) Copy content Toggle raw display
$19$ \( T^{9} - 90 T^{7} + \cdots - 7001 \) Copy content Toggle raw display
$23$ \( T^{9} - 3 T^{8} + \cdots + 74817 \) Copy content Toggle raw display
$29$ \( T^{9} - 24 T^{8} + \cdots + 671679 \) Copy content Toggle raw display
$31$ \( T^{9} - 144 T^{7} + \cdots - 89351 \) Copy content Toggle raw display
$37$ \( T^{9} - 171 T^{7} + \cdots + 97057 \) Copy content Toggle raw display
$41$ \( T^{9} - 33 T^{8} + \cdots + 171099 \) Copy content Toggle raw display
$43$ \( T^{9} - 171 T^{7} + \cdots + 2248471 \) Copy content Toggle raw display
$47$ \( T^{9} - 9 T^{8} + \cdots - 970299 \) Copy content Toggle raw display
$53$ \( T^{9} - 33 T^{8} + \cdots + 9249336 \) Copy content Toggle raw display
$59$ \( T^{9} - 30 T^{8} + \cdots - 3763611 \) Copy content Toggle raw display
$61$ \( T^{9} + 9 T^{8} + \cdots + 416503 \) Copy content Toggle raw display
$67$ \( T^{9} + 9 T^{8} + \cdots - 1889 \) Copy content Toggle raw display
$71$ \( T^{9} - 12 T^{8} + \cdots + 11637 \) Copy content Toggle raw display
$73$ \( T^{9} + 9 T^{8} + \cdots - 115127 \) Copy content Toggle raw display
$79$ \( T^{9} - 198 T^{7} + \cdots + 1351477 \) Copy content Toggle raw display
$83$ \( T^{9} - 36 T^{8} + \cdots + 17019963 \) Copy content Toggle raw display
$89$ \( T^{9} - 48 T^{8} + \cdots + 8774217 \) Copy content Toggle raw display
$97$ \( T^{9} - 279 T^{7} + \cdots - 27143 \) Copy content Toggle raw display
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