Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6016,2,Mod(1,6016)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6016, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6016.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6016 = 2^{7} \cdot 47 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6016.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(48.0380018560\) |
| Analytic rank: | \(0\) |
| Dimension: | \(13\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{13} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} + \cdots - 832 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 163 x^{9} - 840 x^{8} - 438 x^{7} + 3358 x^{6} + 5 x^{5} + \cdots - 832 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 195435 \nu^{12} - 562332 \nu^{11} - 4303150 \nu^{10} + 11599664 \nu^{9} + 31578537 \nu^{8} + \cdots - 78029720 ) / 16452824 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 623713 \nu^{12} + 2092838 \nu^{11} + 13458666 \nu^{10} - 45195340 \nu^{9} - 97777627 \nu^{8} + \cdots - 58768544 ) / 32905648 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1685943 \nu^{12} + 3467516 \nu^{11} + 45355618 \nu^{10} - 79576928 \nu^{9} - 458324917 \nu^{8} + \cdots - 787062208 ) / 32905648 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3310971 \nu^{12} - 7613086 \nu^{11} - 84660434 \nu^{10} + 168699740 \nu^{9} + 808510705 \nu^{8} + \cdots + 1617289984 ) / 32905648 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 5277677 \nu^{12} + 12120046 \nu^{11} + 135579006 \nu^{10} - 271736444 \nu^{9} + \cdots - 2287713712 ) / 32905648 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 2662843 \nu^{12} + 5722128 \nu^{11} + 69595980 \nu^{10} - 128084672 \nu^{9} + \cdots - 1239592192 ) / 16452824 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 5558529 \nu^{12} + 12412430 \nu^{11} + 144044326 \nu^{10} - 278165356 \nu^{9} + \cdots - 2595295424 ) / 32905648 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 6218745 \nu^{12} - 14050128 \nu^{11} - 161260170 \nu^{10} + 316434536 \nu^{9} + 1563354275 \nu^{8} + \cdots + 2837565520 ) / 32905648 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 6720997 \nu^{12} - 15973050 \nu^{11} - 172617142 \nu^{10} + 361687364 \nu^{9} + 1655301055 \nu^{8} + \cdots + 3128024784 ) / 32905648 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 4185187 \nu^{12} - 9726996 \nu^{11} - 108160260 \nu^{10} + 220001744 \nu^{9} + 1045460149 \nu^{8} + \cdots + 2191670088 ) / 16452824 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - \beta_{8} - \beta_{4} - \beta_{3} + \beta_{2} + 6\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{12} - \beta_{10} + 2\beta_{9} - 2\beta_{8} + \beta_{5} + 11\beta_{2} + 39 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2 \beta_{12} - 2 \beta_{11} + 5 \beta_{10} + 17 \beta_{9} - 14 \beta_{8} + \beta_{7} - \beta_{6} + \cdots + 30 \)
|
| \(\nu^{6}\) | \(=\) |
\( 13 \beta_{12} + 3 \beta_{11} - 11 \beta_{10} + 38 \beta_{9} - 33 \beta_{8} + \beta_{7} + 3 \beta_{6} + \cdots + 351 \)
|
| \(\nu^{7}\) | \(=\) |
\( 37 \beta_{12} - 34 \beta_{11} + 89 \beta_{10} + 217 \beta_{9} - 169 \beta_{8} + 24 \beta_{7} + \cdots + 374 \)
|
| \(\nu^{8}\) | \(=\) |
\( 139 \beta_{12} + 59 \beta_{11} - 92 \beta_{10} + 513 \beta_{9} - 424 \beta_{8} + 30 \beta_{7} + \cdots + 3368 \)
|
| \(\nu^{9}\) | \(=\) |
\( 473 \beta_{12} - 396 \beta_{11} + 1134 \beta_{10} + 2495 \beta_{9} - 1925 \beta_{8} + 346 \beta_{7} + \cdots + 4367 \)
|
| \(\nu^{10}\) | \(=\) |
\( 1385 \beta_{12} + 853 \beta_{11} - 705 \beta_{10} + 6094 \beta_{9} - 4987 \beta_{8} + 477 \beta_{7} + \cdots + 33412 \)
|
| \(\nu^{11}\) | \(=\) |
\( 5193 \beta_{12} - 3906 \beta_{11} + 12697 \beta_{10} + 27252 \beta_{9} - 21274 \beta_{8} + 4102 \beta_{7} + \cdots + 49510 \)
|
| \(\nu^{12}\) | \(=\) |
\( 13150 \beta_{12} + 11091 \beta_{11} - 5337 \beta_{10} + 68027 \beta_{9} - 56328 \beta_{8} + \cdots + 337959 \)
|
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.
comment: embeddings in the coefficient field
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 |
|
0 | −3.10994 | 0 | 0.407310 | 0 | −3.05265 | 0 | 6.67175 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.44796 | 0 | 2.61725 | 0 | −0.969862 | 0 | 2.99250 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.08298 | 0 | 4.13384 | 0 | −1.67766 | 0 | 1.33879 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.49302 | 0 | −1.02737 | 0 | 3.87698 | 0 | −0.770878 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.22559 | 0 | 0.177756 | 0 | 3.50603 | 0 | −1.49794 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.319482 | 0 | −2.22340 | 0 | −0.484368 | 0 | −2.89793 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 0.904794 | 0 | 3.10347 | 0 | 1.70842 | 0 | −2.18135 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.18016 | 0 | −2.06312 | 0 | −0.279748 | 0 | −1.60723 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.73608 | 0 | −3.07388 | 0 | −4.60716 | 0 | 0.0139658 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 1.76446 | 0 | 4.26056 | 0 | 2.37797 | 0 | 0.113336 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 2.61905 | 0 | 1.63183 | 0 | −3.46539 | 0 | 3.85941 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 0 | 3.18041 | 0 | 1.45922 | 0 | 4.05725 | 0 | 7.11498 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.13 | 0 | 3.29402 | 0 | −3.40346 | 0 | 1.01019 | 0 | 7.85059 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(47\) | \(-1\) |
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 | 6016.2.a.p | yes | 13 |
| 4.b | odd | 2 | 1 | 6016.2.a.n | yes | 13 | |
| 8.b | even | 2 | 1 | 6016.2.a.m | ✓ | 13 | |
| 8.d | odd | 2 | 1 | 6016.2.a.o | yes | 13 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6016.2.a.m | ✓ | 13 | 8.b | even | 2 | 1 | |
| 6016.2.a.n | yes | 13 | 4.b | odd | 2 | 1 | |
| 6016.2.a.o | yes | 13 | 8.d | odd | 2 | 1 | |
| 6016.2.a.p | yes | 13 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6016))\):
|
\( T_{3}^{13} - 4 T_{3}^{12} - 22 T_{3}^{11} + 96 T_{3}^{10} + 163 T_{3}^{9} - 840 T_{3}^{8} - 438 T_{3}^{7} + \cdots - 832 \)
|
|
\( T_{5}^{13} - 6 T_{5}^{12} - 26 T_{5}^{11} + 190 T_{5}^{10} + 198 T_{5}^{9} - 2184 T_{5}^{8} + \cdots + 1216 \)
|
|
\( T_{13}^{13} - 4 T_{13}^{12} - 112 T_{13}^{11} + 462 T_{13}^{10} + 4540 T_{13}^{9} - 20200 T_{13}^{8} + \cdots + 6740768 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{13} \)
|
| $3$ |
\( T^{13} - 4 T^{12} + \cdots - 832 \)
|
| $5$ |
\( T^{13} - 6 T^{12} + \cdots + 1216 \)
|
| $7$ |
\( T^{13} - 2 T^{12} + \cdots + 2432 \)
|
| $11$ |
\( T^{13} - 10 T^{12} + \cdots - 851936 \)
|
| $13$ |
\( T^{13} - 4 T^{12} + \cdots + 6740768 \)
|
| $17$ |
\( T^{13} - 10 T^{12} + \cdots - 116200 \)
|
| $19$ |
\( T^{13} - 8 T^{12} + \cdots - 6533792 \)
|
| $23$ |
\( T^{13} + 18 T^{12} + \cdots + 12230656 \)
|
| $29$ |
\( T^{13} - 14 T^{12} + \cdots - 800 \)
|
| $31$ |
\( T^{13} + 4 T^{12} + \cdots - 35840 \)
|
| $37$ |
\( T^{13} - 16 T^{12} + \cdots - 9695776 \)
|
| $41$ |
\( T^{13} + \cdots + 349532032 \)
|
| $43$ |
\( T^{13} - 12 T^{12} + \cdots - 27222464 \)
|
| $47$ |
\( (T - 1)^{13} \)
|
| $53$ |
\( T^{13} + \cdots + 2093952352 \)
|
| $59$ |
\( T^{13} - 30 T^{12} + \cdots - 9040640 \)
|
| $61$ |
\( T^{13} - 18 T^{12} + \cdots + 46889312 \)
|
| $67$ |
\( T^{13} + \cdots - 543886400 \)
|
| $71$ |
\( T^{13} + \cdots + 7759724992 \)
|
| $73$ |
\( T^{13} + \cdots - 47381091968 \)
|
| $79$ |
\( T^{13} + \cdots + 81501716480 \)
|
| $83$ |
\( T^{13} + \cdots + 547205641216 \)
|
| $89$ |
\( T^{13} + \cdots - 155049684040 \)
|
| $97$ |
\( T^{13} + \cdots + 10472369024 \)
|
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