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: | \(1\) |
| 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: |
\( x^{11} - 4x^{10} - 14x^{9} + 63x^{8} + 51x^{7} - 305x^{6} + 16x^{5} + 429x^{4} - 234x^{3} - 42x^{2} + 39x - 3 \)
|
| 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
\( x^{11} - 4x^{10} - 14x^{9} + 63x^{8} + 51x^{7} - 305x^{6} + 16x^{5} + 429x^{4} - 234x^{3} - 42x^{2} + 39x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 5 \nu^{9} + 8 \nu^{8} + 92 \nu^{7} - 112 \nu^{6} - 562 \nu^{5} + 455 \nu^{4} + 1195 \nu^{3} + \cdots + 114 ) / 37 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 13 \nu^{10} + 43 \nu^{9} + 217 \nu^{8} - 676 \nu^{7} - 1217 \nu^{6} + 3218 \nu^{5} + 2441 \nu^{4} + \cdots + 844 \nu ) / 37 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 15 \nu^{10} - 54 \nu^{9} - 228 \nu^{8} + 851 \nu^{7} + 1051 \nu^{6} - 4145 \nu^{5} - 1188 \nu^{4} + \cdots + 129 ) / 37 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 25 \nu^{10} - 89 \nu^{9} - 389 \nu^{8} + 1395 \nu^{7} + 1890 \nu^{6} - 6643 \nu^{5} - 2515 \nu^{4} + \cdots + 318 ) / 37 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 26 \nu^{10} + 93 \nu^{9} + 408 \nu^{8} - 1466 \nu^{7} - 2033 \nu^{6} + 7060 \nu^{5} + 3061 \nu^{4} + \cdots - 130 ) / 37 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 42 \nu^{10} - 140 \nu^{9} - 680 \nu^{8} + 2193 \nu^{7} + 3577 \nu^{6} - 10430 \nu^{5} - 6129 \nu^{4} + \cdots + 94 ) / 37 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 42 \nu^{10} - 140 \nu^{9} - 680 \nu^{8} + 2193 \nu^{7} + 3577 \nu^{6} - 10430 \nu^{5} - 6129 \nu^{4} + \cdots + 205 ) / 37 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 43 \nu^{10} + 152 \nu^{9} + 664 \nu^{8} - 2389 \nu^{7} - 3156 \nu^{6} + 11502 \nu^{5} + 3912 \nu^{4} + \cdots - 414 ) / 37 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} - \beta_{8} + \beta_{2} + 8\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 9\beta_{2} + 12\beta _1 + 30 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{10} + 11\beta_{9} - 9\beta_{8} - \beta_{7} - \beta_{5} + 4\beta_{4} + \beta_{3} + 13\beta_{2} + 67\beta _1 + 22 \)
|
| \(\nu^{6}\) | \(=\) |
\( 16 \beta_{10} + 17 \beta_{9} - \beta_{8} + 9 \beta_{7} + 12 \beta_{6} + 14 \beta_{5} + 20 \beta_{4} + \cdots + 247 \)
|
| \(\nu^{7}\) | \(=\) |
\( 23 \beta_{10} + 112 \beta_{9} - 70 \beta_{8} - 10 \beta_{7} + 3 \beta_{6} - 9 \beta_{5} + 75 \beta_{4} + \cdots + 289 \)
|
| \(\nu^{8}\) | \(=\) |
\( 196 \beta_{10} + 224 \beta_{9} - 20 \beta_{8} + 67 \beta_{7} + 121 \beta_{6} + 153 \beta_{5} + \cdots + 2115 \)
|
| \(\nu^{9}\) | \(=\) |
\( 357 \beta_{10} + 1132 \beta_{9} - 525 \beta_{8} - 75 \beta_{7} + 71 \beta_{6} - 31 \beta_{5} + \cdots + 3275 \)
|
| \(\nu^{10}\) | \(=\) |
\( 2194 \beta_{10} + 2650 \beta_{9} - 236 \beta_{8} + 488 \beta_{7} + 1163 \beta_{6} + 1549 \beta_{5} + \cdots + 18607 \)
|
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 | −3.13625 | 0 | 0.831138 | 0 | 0 | 0 | 6.83609 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.93595 | 0 | −2.06839 | 0 | 0 | 0 | 5.61979 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −2.85830 | 0 | 3.84386 | 0 | 0 | 0 | 5.16987 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.947093 | 0 | −3.46093 | 0 | 0 | 0 | −2.10301 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.656305 | 0 | 2.09142 | 0 | 0 | 0 | −2.56926 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | −0.488387 | 0 | 2.64454 | 0 | 0 | 0 | −2.76148 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | −0.0889922 | 0 | −1.82516 | 0 | 0 | 0 | −2.99208 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 0.406192 | 0 | −4.43577 | 0 | 0 | 0 | −2.83501 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.59775 | 0 | 3.07107 | 0 | 0 | 0 | −0.447181 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.41250 | 0 | −0.704023 | 0 | 0 | 0 | 2.82017 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 2.69483 | 0 | −0.987745 | 0 | 0 | 0 | 4.26210 | 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.bl | 11 | |
| 7.b | odd | 2 | 1 | 9016.2.a.bq | 11 | ||
| 7.d | odd | 6 | 2 | 1288.2.q.a | ✓ | 22 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1288.2.q.a | ✓ | 22 | 7.d | odd | 6 | 2 | |
| 9016.2.a.bl | 11 | 1.a | even | 1 | 1 | trivial | |
| 9016.2.a.bq | 11 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9016))\):
|
\( T_{3}^{11} + 4 T_{3}^{10} - 14 T_{3}^{9} - 63 T_{3}^{8} + 51 T_{3}^{7} + 305 T_{3}^{6} + 16 T_{3}^{5} + \cdots + 3 \)
|
|
\( T_{5}^{11} + T_{5}^{10} - 38 T_{5}^{9} - 28 T_{5}^{8} + 498 T_{5}^{7} + 306 T_{5}^{6} - 2709 T_{5}^{5} + \cdots - 2187 \)
|
|
\( T_{11}^{11} - 71 T_{11}^{9} - 49 T_{11}^{8} + 1712 T_{11}^{7} + 1875 T_{11}^{6} - 16644 T_{11}^{5} + \cdots - 44757 \)
|
|
\( T_{13}^{11} - 3 T_{13}^{10} - 60 T_{13}^{9} + 25 T_{13}^{8} + 1312 T_{13}^{7} + 2186 T_{13}^{6} + \cdots + 4869 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} \)
|
| $3$ |
\( T^{11} + 4 T^{10} + \cdots + 3 \)
|
| $5$ |
\( T^{11} + T^{10} + \cdots - 2187 \)
|
| $7$ |
\( T^{11} \)
|
| $11$ |
\( T^{11} - 71 T^{9} + \cdots - 44757 \)
|
| $13$ |
\( T^{11} - 3 T^{10} + \cdots + 4869 \)
|
| $17$ |
\( T^{11} + 5 T^{10} + \cdots + 81 \)
|
| $19$ |
\( T^{11} + 12 T^{10} + \cdots - 29889 \)
|
| $23$ |
\( (T - 1)^{11} \)
|
| $29$ |
\( T^{11} + 15 T^{10} + \cdots + 6031539 \)
|
| $31$ |
\( T^{11} + 16 T^{10} + \cdots + 1373763 \)
|
| $37$ |
\( T^{11} - 3 T^{10} + \cdots + 1481807 \)
|
| $41$ |
\( T^{11} + 28 T^{10} + \cdots + 3620493 \)
|
| $43$ |
\( T^{11} + 9 T^{10} + \cdots - 1937263 \)
|
| $47$ |
\( T^{11} + 31 T^{10} + \cdots - 30056399 \)
|
| $53$ |
\( T^{11} - 13 T^{10} + \cdots + 5691989 \)
|
| $59$ |
\( T^{11} + 11 T^{10} + \cdots + 252513 \)
|
| $61$ |
\( T^{11} - 19 T^{10} + \cdots + 47143449 \)
|
| $67$ |
\( T^{11} + \cdots - 337063269 \)
|
| $71$ |
\( T^{11} + \cdots + 11847027177 \)
|
| $73$ |
\( T^{11} + \cdots - 457793757 \)
|
| $79$ |
\( T^{11} + 13 T^{10} + \cdots + 90133011 \)
|
| $83$ |
\( T^{11} + \cdots + 1456508193 \)
|
| $89$ |
\( T^{11} + 10 T^{10} + \cdots + 18531621 \)
|
| $97$ |
\( T^{11} + \cdots + 1138220643 \)
|
| show more | |
| show less |