Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8022,2,Mod(1,8022)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8022, 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("8022.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8022 = 2 \cdot 3 \cdot 7 \cdot 191 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8022.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: | \(64.0559925015\) |
| 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: |
\( x^{11} - 5 x^{10} - 24 x^{9} + 121 x^{8} + 208 x^{7} - 1007 x^{6} - 647 x^{5} + 3355 x^{4} - 321 x^{3} + \cdots - 486 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - 5 x^{10} - 24 x^{9} + 121 x^{8} + 208 x^{7} - 1007 x^{6} - 647 x^{5} + 3355 x^{4} - 321 x^{3} + \cdots - 486 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 14423 \nu^{10} + 329900 \nu^{9} - 2934396 \nu^{8} - 2708989 \nu^{7} + 48368051 \nu^{6} + \cdots + 25849314 ) / 7871688 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 26332 \nu^{10} - 92465 \nu^{9} - 858723 \nu^{8} + 2623519 \nu^{7} + 9471940 \nu^{6} + \cdots + 41913018 ) / 3935844 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 12597 \nu^{10} - 32627 \nu^{9} - 479339 \nu^{8} + 1046994 \nu^{7} + 5968243 \nu^{6} + \cdots + 8111484 ) / 1311948 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 4559 \nu^{10} - 19540 \nu^{9} - 128076 \nu^{8} + 493787 \nu^{7} + 1313615 \nu^{6} - 4104070 \nu^{5} + \cdots + 2444418 ) / 357804 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 155197 \nu^{10} - 721379 \nu^{9} - 3728469 \nu^{8} + 15808702 \nu^{7} + 35333941 \nu^{6} + \cdots + 80678052 ) / 3935844 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 352519 \nu^{10} + 1595210 \nu^{9} + 8745366 \nu^{8} - 35185285 \nu^{7} - 86725315 \nu^{6} + \cdots - 155302110 ) / 7871688 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 379457 \nu^{10} - 1511560 \nu^{9} - 11054952 \nu^{8} + 37689593 \nu^{7} + 119559773 \nu^{6} + \cdots + 320974542 ) / 7871688 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 428101 \nu^{10} + 1790972 \nu^{9} + 11621400 \nu^{8} - 41467249 \nu^{7} - 122534773 \nu^{6} + \cdots - 259072830 ) / 7871688 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 238631 \nu^{10} + 949513 \nu^{9} + 6778575 \nu^{8} - 22565918 \nu^{7} - 73097279 \nu^{6} + \cdots - 140334552 ) / 3935844 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} - \beta_{7} + \beta_{4} + \beta _1 + 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( - 2 \beta_{10} + 4 \beta_{9} + \beta_{8} - 2 \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + \cdots + 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 4 \beta_{10} + 21 \beta_{9} + 4 \beta_{8} - 14 \beta_{7} - 5 \beta_{5} + 19 \beta_{4} - 5 \beta_{3} + \cdots + 89 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 36 \beta_{10} + 103 \beta_{9} + 34 \beta_{8} - 51 \beta_{7} - 7 \beta_{6} - 34 \beta_{5} + 53 \beta_{4} + \cdots + 251 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 88 \beta_{10} + 478 \beta_{9} + 160 \beta_{8} - 276 \beta_{7} + 28 \beta_{6} - 175 \beta_{5} + \cdots + 1466 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 513 \beta_{10} + 2340 \beta_{9} + 938 \beta_{8} - 1224 \beta_{7} + 135 \beta_{6} - 946 \beta_{5} + \cdots + 5503 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 1392 \beta_{10} + 10788 \beta_{9} + 4519 \beta_{8} - 6100 \beta_{7} + 1376 \beta_{6} - 4750 \beta_{5} + \cdots + 27213 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 6431 \beta_{10} + 51524 \beta_{9} + 23341 \beta_{8} - 28395 \beta_{7} + 7442 \beta_{6} + \cdots + 114158 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 17343 \beta_{10} + 238096 \beta_{9} + 111908 \beta_{8} - 136727 \beta_{7} + 44381 \beta_{6} + \cdots + 532911 \)
|
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 |
|
−1.00000 | −1.00000 | 1.00000 | −3.56039 | 1.00000 | −1.00000 | −1.00000 | 1.00000 | 3.56039 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −1.00000 | 1.00000 | −2.96822 | 1.00000 | −1.00000 | −1.00000 | 1.00000 | 2.96822 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.00000 | 1.00000 | −1.90341 | 1.00000 | −1.00000 | −1.00000 | 1.00000 | 1.90341 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.00000 | 1.00000 | −0.400138 | 1.00000 | −1.00000 | −1.00000 | 1.00000 | 0.400138 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −1.00000 | 1.00000 | 0.0705504 | 1.00000 | −1.00000 | −1.00000 | 1.00000 | −0.0705504 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | −1.00000 | 1.00000 | 0.298825 | 1.00000 | −1.00000 | −1.00000 | 1.00000 | −0.298825 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | −1.00000 | 1.00000 | 0.694740 | 1.00000 | −1.00000 | −1.00000 | 1.00000 | −0.694740 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | −1.00000 | 1.00000 | 2.81270 | 1.00000 | −1.00000 | −1.00000 | 1.00000 | −2.81270 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | −1.00000 | 1.00000 | 3.25599 | 1.00000 | −1.00000 | −1.00000 | 1.00000 | −3.25599 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | −1.00000 | −1.00000 | 1.00000 | 3.83248 | 1.00000 | −1.00000 | −1.00000 | 1.00000 | −3.83248 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | −1.00000 | −1.00000 | 1.00000 | 3.86687 | 1.00000 | −1.00000 | −1.00000 | 1.00000 | −3.86687 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(7\) | \(1\) |
| \(191\) | \(-1\) |
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 | 8022.2.a.s | ✓ | 11 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8022.2.a.s | ✓ | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8022))\):
|
\( T_{5}^{11} - 6 T_{5}^{10} - 19 T_{5}^{9} + 155 T_{5}^{8} + 42 T_{5}^{7} - 1233 T_{5}^{6} + 633 T_{5}^{5} + \cdots - 16 \)
|
|
\( T_{11}^{11} + T_{11}^{10} - 66 T_{11}^{9} - 16 T_{11}^{8} + 1421 T_{11}^{7} - 931 T_{11}^{6} + \cdots - 2832 \)
|
|
\( T_{13}^{11} - 2 T_{13}^{10} - 54 T_{13}^{9} + 75 T_{13}^{8} + 893 T_{13}^{7} - 796 T_{13}^{6} + \cdots + 3288 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{11} \)
|
| $3$ |
\( (T + 1)^{11} \)
|
| $5$ |
\( T^{11} - 6 T^{10} + \cdots - 16 \)
|
| $7$ |
\( (T + 1)^{11} \)
|
| $11$ |
\( T^{11} + T^{10} + \cdots - 2832 \)
|
| $13$ |
\( T^{11} - 2 T^{10} + \cdots + 3288 \)
|
| $17$ |
\( T^{11} - 13 T^{10} + \cdots + 344 \)
|
| $19$ |
\( T^{11} + 18 T^{10} + \cdots + 5872 \)
|
| $23$ |
\( T^{11} + T^{10} + \cdots + 2511888 \)
|
| $29$ |
\( T^{11} - 9 T^{10} + \cdots - 532768 \)
|
| $31$ |
\( T^{11} + 15 T^{10} + \cdots + 4272 \)
|
| $37$ |
\( T^{11} - 17 T^{10} + \cdots - 24676544 \)
|
| $41$ |
\( T^{11} - 4 T^{10} + \cdots + 576 \)
|
| $43$ |
\( T^{11} - 3 T^{10} + \cdots + 470784 \)
|
| $47$ |
\( T^{11} + 3 T^{10} + \cdots - 7480992 \)
|
| $53$ |
\( T^{11} + \cdots - 115609752 \)
|
| $59$ |
\( T^{11} + \cdots - 296483328 \)
|
| $61$ |
\( T^{11} + 3 T^{10} + \cdots + 1118616 \)
|
| $67$ |
\( T^{11} + \cdots - 164536832 \)
|
| $71$ |
\( T^{11} + \cdots + 191979696 \)
|
| $73$ |
\( T^{11} + \cdots + 113276684 \)
|
| $79$ |
\( T^{11} + 4 T^{10} + \cdots - 12045128 \)
|
| $83$ |
\( T^{11} + \cdots + 626768166 \)
|
| $89$ |
\( T^{11} - 11 T^{10} + \cdots + 42496 \)
|
| $97$ |
\( T^{11} + 10 T^{10} + \cdots + 331456 \)
|
| show more | |
| show less |