Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8016,2,Mod(1,8016)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8016, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("8016.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8016 = 2^{4} \cdot 3 \cdot 167 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8016.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: | \(64.0080822603\) |
| Analytic rank: | \(0\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + \cdots + 242 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 4008) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + \cdots + 242 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2623 \nu^{10} - 70944 \nu^{9} - 193391 \nu^{8} + 2378035 \nu^{7} + 2748114 \nu^{6} + \cdots - 7710406 ) / 10323192 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1379 \nu^{10} + 55664 \nu^{9} - 65533 \nu^{8} - 1226719 \nu^{7} + 1417390 \nu^{6} + \cdots - 2478234 ) / 1876944 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 64985 \nu^{10} - 195192 \nu^{9} - 1989097 \nu^{8} + 5477813 \nu^{7} + 21384366 \nu^{6} + \cdots + 34175614 ) / 10323192 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 101765 \nu^{10} - 492056 \nu^{9} - 2608397 \nu^{8} + 12499993 \nu^{7} + 25094846 \nu^{6} + \cdots + 39160638 ) / 10323192 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 116141 \nu^{10} - 242652 \nu^{9} - 2909401 \nu^{8} + 5468837 \nu^{7} + 25357890 \nu^{6} + \cdots + 20257270 ) / 5161596 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 522481 \nu^{10} + 1421984 \nu^{9} + 13457169 \nu^{8} - 32908293 \nu^{7} - 124595270 \nu^{6} + \cdots - 65756350 ) / 20646384 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 596041 \nu^{10} + 2015712 \nu^{9} + 14695769 \nu^{8} - 46952653 \nu^{7} - 132016230 \nu^{6} + \cdots - 13787246 ) / 20646384 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 303385 \nu^{10} - 1032256 \nu^{9} - 7769641 \nu^{8} + 25611773 \nu^{7} + 71466478 \nu^{6} + \cdots + 47745918 ) / 10323192 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - \beta_{8} + \beta_{6} - \beta_{5} + \beta_{2} + 8\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{10} + \beta_{9} + 2\beta_{7} + 3\beta_{6} + 2\beta_{5} - \beta_{4} - \beta_{3} + 11\beta_{2} + 4\beta _1 + 48 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 4 \beta_{10} + 14 \beta_{9} - 14 \beta_{8} + 3 \beta_{7} + 19 \beta_{6} - 14 \beta_{5} + 3 \beta_{3} + \cdots + 50 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 39 \beta_{10} + 17 \beta_{9} - 2 \beta_{8} + 35 \beta_{7} + 68 \beta_{6} + 20 \beta_{5} - 9 \beta_{4} + \cdots + 432 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 89 \beta_{10} + 167 \beta_{9} - 151 \beta_{8} + 77 \beta_{7} + 294 \beta_{6} - 163 \beta_{5} + \cdots + 656 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 576 \beta_{10} + 244 \beta_{9} - 31 \beta_{8} + 516 \beta_{7} + 1101 \beta_{6} + 116 \beta_{5} + \cdots + 4199 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 1462 \beta_{10} + 1922 \beta_{9} - 1489 \beta_{8} + 1378 \beta_{7} + 4200 \beta_{6} - 1860 \beta_{5} + \cdots + 8094 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 7771 \beta_{10} + 3303 \beta_{9} - 341 \beta_{8} + 7202 \beta_{7} + 15782 \beta_{6} - 172 \beta_{5} + \cdots + 43210 \)
|
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 | 1.00000 | 0 | −2.54432 | 0 | −0.525270 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | 1.00000 | 0 | −2.11942 | 0 | 0.802640 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 1.00000 | 0 | −1.80306 | 0 | 4.10861 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.00000 | 0 | −1.38119 | 0 | 0.260099 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.00000 | 0 | −0.0932775 | 0 | −3.86231 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.00000 | 0 | 0.957263 | 0 | 0.898491 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.00000 | 0 | 2.60189 | 0 | 3.58131 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.00000 | 0 | 3.17907 | 0 | −0.651548 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.00000 | 0 | 3.34947 | 0 | −3.54581 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 1.00000 | 0 | 3.83671 | 0 | −4.48179 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 1.00000 | 0 | 4.01685 | 0 | 4.41557 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
| \(167\) | \(1\) |
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 | 8016.2.a.be | 11 | |
| 4.b | odd | 2 | 1 | 4008.2.a.k | ✓ | 11 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4008.2.a.k | ✓ | 11 | 4.b | odd | 2 | 1 | |
| 8016.2.a.be | 11 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):
|
\( T_{5}^{11} - 10 T_{5}^{10} + 12 T_{5}^{9} + 155 T_{5}^{8} - 385 T_{5}^{7} - 864 T_{5}^{6} + 2665 T_{5}^{5} + \cdots + 512 \)
|
|
\( T_{7}^{11} - T_{7}^{10} - 49 T_{7}^{9} + 46 T_{7}^{8} + 800 T_{7}^{7} - 692 T_{7}^{6} - 4561 T_{7}^{5} + \cdots + 256 \)
|
|
\( T_{11}^{11} - T_{11}^{10} - 73 T_{11}^{9} + 84 T_{11}^{8} + 1865 T_{11}^{7} - 2359 T_{11}^{6} + \cdots + 86400 \)
|
|
\( T_{13}^{11} - 10 T_{13}^{10} - 27 T_{13}^{9} + 484 T_{13}^{8} - 408 T_{13}^{7} - 5699 T_{13}^{6} + \cdots + 10688 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} \)
|
| $3$ |
\( (T - 1)^{11} \)
|
| $5$ |
\( T^{11} - 10 T^{10} + \cdots + 512 \)
|
| $7$ |
\( T^{11} - T^{10} + \cdots + 256 \)
|
| $11$ |
\( T^{11} - T^{10} + \cdots + 86400 \)
|
| $13$ |
\( T^{11} - 10 T^{10} + \cdots + 10688 \)
|
| $17$ |
\( T^{11} - 17 T^{10} + \cdots + 20128 \)
|
| $19$ |
\( T^{11} + 2 T^{10} + \cdots - 858112 \)
|
| $23$ |
\( T^{11} - 3 T^{10} + \cdots - 32768 \)
|
| $29$ |
\( T^{11} - 17 T^{10} + \cdots - 30400 \)
|
| $31$ |
\( T^{11} - 15 T^{10} + \cdots - 1515520 \)
|
| $37$ |
\( T^{11} - 4 T^{10} + \cdots - 62176 \)
|
| $41$ |
\( T^{11} - 16 T^{10} + \cdots - 337072 \)
|
| $43$ |
\( T^{11} + 10 T^{10} + \cdots + 48168832 \)
|
| $47$ |
\( T^{11} + \cdots + 151369024 \)
|
| $53$ |
\( T^{11} - 42 T^{10} + \cdots + 64144928 \)
|
| $59$ |
\( T^{11} - 2 T^{10} + \cdots + 1956736 \)
|
| $61$ |
\( T^{11} - 12 T^{10} + \cdots + 61760 \)
|
| $67$ |
\( T^{11} + \cdots - 248382664 \)
|
| $71$ |
\( T^{11} + \cdots + 382383104 \)
|
| $73$ |
\( T^{11} - 24 T^{10} + \cdots - 360896 \)
|
| $79$ |
\( T^{11} + \cdots - 6033532768 \)
|
| $83$ |
\( T^{11} + 16 T^{10} + \cdots + 93972800 \)
|
| $89$ |
\( T^{11} + \cdots + 158510752 \)
|
| $97$ |
\( T^{11} - 4 T^{10} + \cdots - 20250080 \)
|
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