Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9016,2,Mod(1,9016)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9016, 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("9016.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9016 = 2^{3} \cdot 7^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9016.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: | \(71.9931224624\) |
| 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: |
\( x^{11} - 4 x^{10} - 14 x^{9} + 61 x^{8} + 71 x^{7} - 343 x^{6} - 152 x^{5} + 867 x^{4} + 102 x^{3} + \cdots + 243 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1288) |
| 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
\( x^{11} - 4 x^{10} - 14 x^{9} + 61 x^{8} + 71 x^{7} - 343 x^{6} - 152 x^{5} + 867 x^{4} + 102 x^{3} + \cdots + 243 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 30 \nu^{10} + 376 \nu^{9} - 2519 \nu^{8} - 3326 \nu^{7} + 27712 \nu^{6} + 6997 \nu^{5} - 108148 \nu^{4} + \cdots - 51801 ) / 1211 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 155 \nu^{10} + 883 \nu^{9} + 703 \nu^{8} - 10265 \nu^{7} + 4967 \nu^{6} + 38729 \nu^{5} + \cdots - 10286 ) / 1211 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 255 \nu^{10} + 1648 \nu^{9} + 219 \nu^{8} - 18958 \nu^{7} + 21180 \nu^{6} + 69497 \nu^{5} + \cdots - 44697 ) / 1211 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 278 \nu^{10} - 68 \nu^{9} - 7519 \nu^{8} + 3410 \nu^{7} + 63603 \nu^{6} - 26632 \nu^{5} + \cdots - 80877 ) / 1211 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 305 \nu^{10} - 1425 \nu^{9} - 2399 \nu^{8} + 16644 \nu^{7} + 383 \nu^{6} - 61872 \nu^{5} + \cdots + 15279 ) / 1211 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 353 \nu^{10} + 1550 \nu^{9} + 3523 \nu^{8} - 19315 \nu^{7} - 9361 \nu^{6} + 79983 \nu^{5} + \cdots - 940 ) / 1211 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 78 \nu^{10} - 268 \nu^{9} - 1048 \nu^{8} + 3324 \nu^{7} + 5550 \nu^{6} - 13709 \nu^{5} - 14661 \nu^{4} + \cdots - 5590 ) / 173 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 608 \nu^{10} - 1987 \nu^{9} - 8586 \nu^{8} + 24952 \nu^{7} + 47520 \nu^{6} - 103462 \nu^{5} + \cdots - 45188 ) / 1211 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{10} + \beta_{7} + \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{3} + 2\beta_{2} + 7\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -2\beta_{10} + \beta_{8} + \beta_{7} + 3\beta_{6} + \beta_{5} - 5\beta_{4} - 3\beta_{3} + 11\beta_{2} + 15\beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 15 \beta_{10} + \beta_{9} + 3 \beta_{8} + 12 \beta_{7} + 16 \beta_{6} + 10 \beta_{5} - 29 \beta_{4} + \cdots + 32 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 43 \beta_{10} + 7 \beta_{9} + 18 \beta_{8} + 21 \beta_{7} + 52 \beta_{6} + 15 \beta_{5} - 84 \beta_{4} + \cdots + 188 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 204 \beta_{10} + 35 \beta_{9} + 58 \beta_{8} + 129 \beta_{7} + 212 \beta_{6} + 88 \beta_{5} + \cdots + 407 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 662 \beta_{10} + 161 \beta_{9} + 253 \beta_{8} + 315 \beta_{7} + 713 \beta_{6} + 177 \beta_{5} + \cdots + 1765 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 2644 \beta_{10} + 648 \beta_{9} + 851 \beta_{8} + 1428 \beta_{7} + 2654 \beta_{6} + 796 \beta_{5} + \cdots + 4852 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 8965 \beta_{10} + 2578 \beta_{9} + 3286 \beta_{8} + 4171 \beta_{7} + 9067 \beta_{6} + 1968 \beta_{5} + \cdots + 18412 \)
|
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 | −2.32959 | 0 | −1.34069 | 0 | 0 | 0 | 2.42699 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.86046 | 0 | 3.75356 | 0 | 0 | 0 | 0.461318 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.78911 | 0 | −0.157144 | 0 | 0 | 0 | 0.200921 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.58959 | 0 | 1.03689 | 0 | 0 | 0 | −0.473209 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.560144 | 0 | −0.521201 | 0 | 0 | 0 | −2.68624 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.710583 | 0 | −1.82431 | 0 | 0 | 0 | −2.49507 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.59106 | 0 | −2.83508 | 0 | 0 | 0 | −0.468513 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.66264 | 0 | 3.34949 | 0 | 0 | 0 | −0.235633 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.99086 | 0 | 1.57733 | 0 | 0 | 0 | 0.963536 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.73452 | 0 | −3.07461 | 0 | 0 | 0 | 4.47758 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 3.43923 | 0 | 3.03575 | 0 | 0 | 0 | 8.82832 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
| \(23\) | \(1\) |
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 | 9016.2.a.br | 11 | |
| 7.b | odd | 2 | 1 | 9016.2.a.bk | 11 | ||
| 7.d | odd | 6 | 2 | 1288.2.q.d | ✓ | 22 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1288.2.q.d | ✓ | 22 | 7.d | odd | 6 | 2 | |
| 9016.2.a.bk | 11 | 7.b | odd | 2 | 1 | ||
| 9016.2.a.br | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9016))\):
|
\( T_{3}^{11} - 4 T_{3}^{10} - 14 T_{3}^{9} + 61 T_{3}^{8} + 71 T_{3}^{7} - 343 T_{3}^{6} - 152 T_{3}^{5} + \cdots + 243 \)
|
|
\( T_{5}^{11} - 3 T_{5}^{10} - 26 T_{5}^{9} + 66 T_{5}^{8} + 248 T_{5}^{7} - 466 T_{5}^{6} - 1039 T_{5}^{5} + \cdots - 109 \)
|
|
\( T_{11}^{11} - 73 T_{11}^{9} + 25 T_{11}^{8} + 1698 T_{11}^{7} - 1175 T_{11}^{6} - 12610 T_{11}^{5} + \cdots + 597 \)
|
|
\( T_{13}^{11} + 13 T_{13}^{10} - 12 T_{13}^{9} - 643 T_{13}^{8} - 528 T_{13}^{7} + 12606 T_{13}^{6} + \cdots + 475741 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} \)
|
| $3$ |
\( T^{11} - 4 T^{10} + \cdots + 243 \)
|
| $5$ |
\( T^{11} - 3 T^{10} + \cdots - 109 \)
|
| $7$ |
\( T^{11} \)
|
| $11$ |
\( T^{11} - 73 T^{9} + \cdots + 597 \)
|
| $13$ |
\( T^{11} + 13 T^{10} + \cdots + 475741 \)
|
| $17$ |
\( T^{11} - 7 T^{10} + \cdots - 492597 \)
|
| $19$ |
\( T^{11} - 8 T^{10} + \cdots - 139023 \)
|
| $23$ |
\( (T + 1)^{11} \)
|
| $29$ |
\( T^{11} + 3 T^{10} + \cdots + 650523 \)
|
| $31$ |
\( T^{11} - 12 T^{10} + \cdots + 22973031 \)
|
| $37$ |
\( T^{11} + T^{10} + \cdots + 53433 \)
|
| $41$ |
\( T^{11} - 12 T^{10} + \cdots + 109389 \)
|
| $43$ |
\( T^{11} - 9 T^{10} + \cdots + 20224211 \)
|
| $47$ |
\( T^{11} - 17 T^{10} + \cdots - 245187 \)
|
| $53$ |
\( T^{11} + \cdots + 1256162499 \)
|
| $59$ |
\( T^{11} - 33 T^{10} + \cdots - 4176519 \)
|
| $61$ |
\( T^{11} + 15 T^{10} + \cdots - 2338273 \)
|
| $67$ |
\( T^{11} + \cdots - 119435827 \)
|
| $71$ |
\( T^{11} + 9 T^{10} + \cdots - 232611 \)
|
| $73$ |
\( T^{11} - 5 T^{10} + \cdots + 48526371 \)
|
| $79$ |
\( T^{11} + \cdots - 48531651483 \)
|
| $83$ |
\( T^{11} + \cdots + 33529454091 \)
|
| $89$ |
\( T^{11} + \cdots + 2945697867 \)
|
| $97$ |
\( T^{11} - 21 T^{10} + \cdots + 25263629 \)
|
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