Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [96,9,Mod(31,96)] 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("96.31"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(96, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 96.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,560] 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: \(39.1083465659\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.3468738816.6
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 2x^{6} + 18x^{5} + 77x^{4} + 8x^{2} + 88x + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{12} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + (\beta_{4} + 70) q^{5} + ( - 7 \beta_{3} - 7 \beta_1) q^{7} - 2187 q^{9} + ( - 2 \beta_{5} - 18 \beta_{3} + \cdots - 135 \beta_1) q^{11} + ( - 4 \beta_{7} - 19 \beta_{6} + \cdots - 6782) q^{13}+ \cdots + (4374 \beta_{5} + 39366 \beta_{3} + \cdots + 295245 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 560 q^{5} - 17496 q^{9} - 54256 q^{13} + 436176 q^{17} + 54432 q^{21} + 2456472 q^{25} + 195952 q^{29} - 1065312 q^{33} - 7023408 q^{37} - 12035120 q^{41} - 1224720 q^{45} - 9602040 q^{49} + 27342192 q^{53}+ \cdots + 272647184 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 2x^{6} + 18x^{5} + 77x^{4} + 8x^{2} + 88x + 484 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 16096 \nu^{7} + 120192 \nu^{6} - 193760 \nu^{5} - 125344 \nu^{4} + 178112 \nu^{3} + \cdots - 770176 ) / 348777 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 225\nu^{7} - 846\nu^{6} + 1836\nu^{5} + 2466\nu^{4} + 7425\nu^{3} - 7920\nu^{2} + 26550\nu + 14256 ) / 572 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 272869 \nu^{7} + 1208730 \nu^{6} - 2417762 \nu^{5} - 2834248 \nu^{4} - 6832507 \nu^{3} + \cdots - 16524376 ) / 116259 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 37279 \nu^{7} - 57072 \nu^{6} - 78202 \nu^{5} + 1067344 \nu^{4} + 2474125 \nu^{3} - 149116 \nu^{2} + \cdots + 6174496 ) / 10569 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 2413799 \nu^{7} + 15867858 \nu^{6} - 33579772 \nu^{5} - 15096278 \nu^{4} + 78107953 \nu^{3} + \cdots - 97405616 ) / 348777 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 74545 \nu^{7} - 94800 \nu^{6} - 407590 \nu^{5} + 3516016 \nu^{4} + 2531875 \nu^{3} - 298180 \nu^{2} + \cdots + 43030240 ) / 10569 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 122002 \nu^{7} - 155424 \nu^{6} - 663532 \nu^{5} + 5323360 \nu^{4} + 4177750 \nu^{3} + \cdots + 46326976 ) / 10569 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 13\beta_{7} - 12\beta_{6} + 12\beta_{5} - 34\beta_{4} - 96\beta_{3} - 208\beta_{2} + 129\beta _1 + 6912 ) / 27648 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -144\beta_{3} - 704\beta_{2} + 1323\beta_1 ) / 13824 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 161 \beta_{7} + 204 \beta_{6} + 60 \beta_{5} + 122 \beta_{4} - 1056 \beta_{3} - 4880 \beta_{2} + \cdots - 200448 ) / 27648 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -355\beta_{7} + 576\beta_{6} + 10\beta_{4} - 794880 ) / 13824 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 235 \beta_{7} + 332 \beta_{6} - 52 \beta_{5} + 70 \beta_{4} + 1536 \beta_{3} + 7856 \beta_{2} + \cdots - 367872 ) / 3072 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -744\beta_{5} + 35472\beta_{3} + 179744\beta_{2} - 160413\beta_1 ) / 13824 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 28819 \beta_{7} - 42324 \beta_{6} - 5172 \beta_{5} - 5374 \beta_{4} + 189984 \beta_{3} + \cdots + 48487680 ) / 27648 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(65\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
1.10829 − 1.10829i
2.64070 + 2.64070i
−1.47432 + 1.47432i
−1.27467 − 1.27467i
1.10829 + 1.10829i
2.64070 − 2.64070i
−1.47432 − 1.47432i
−1.27467 + 1.27467i
0 − 46.7654i 0 −1159.45 0 1312.73i 0 −2187.00 0
31.2 0 − 46.7654i 0 −149.264 0 3207.68i 0 −2187.00 0
31.3 0 − 46.7654i 0 509.633 0 42.2502i 0 −2187.00 0
31.4 0 − 46.7654i 0 1079.08 0 − 3980.70i 0 −2187.00 0
31.5 0 46.7654i 0 −1159.45 0 − 1312.73i 0 −2187.00 0
31.6 0 46.7654i 0 −149.264 0 − 3207.68i 0 −2187.00 0
31.7 0 46.7654i 0 509.633 0 − 42.2502i 0 −2187.00 0
31.8 0 46.7654i 0 1079.08 0 3980.70i 0 −2187.00 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 31.8
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 96.9.g.b ✓ 8
3.b odd 2 1 288.9.g.c 8
4.b odd 2 1 inner 96.9.g.b ✓ 8
8.b even 2 1 192.9.g.d 8
8.d odd 2 1 192.9.g.d 8
12.b even 2 1 288.9.g.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.9.g.b ✓ 8 1.a even 1 1 trivial
96.9.g.b ✓ 8 4.b odd 2 1 inner
192.9.g.d 8 8.b even 2 1
192.9.g.d 8 8.d odd 2 1
288.9.g.c 8 3.b odd 2 1
288.9.g.c 8 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 280T_{5}^{3} - 1356168T_{5}^{2} + 444757280T_{5} + 95173421200 \) acting on \(S_{9}^{\mathrm{new}}(96, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 2187)^{4} \) Copy content Toggle raw display
$5$ \( (T^{4} - 280 T^{3} + \cdots + 95173421200)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 50\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 89\!\cdots\!56 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 78\!\cdots\!96)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 69\!\cdots\!44)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 81\!\cdots\!44)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 36\!\cdots\!28)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 53\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 23\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 44\!\cdots\!24)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 31\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 27\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots - 16\!\cdots\!68)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 31\!\cdots\!84 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 15\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 47\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 88\!\cdots\!36)^{2} \) Copy content Toggle raw display
show more
show less