Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [91,8,Mod(1,91)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(91, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 91.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: | \(28.4270373191\) |
| Analytic rank: | \(1\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2 x^{9} - 957 x^{8} + 1224 x^{7} + 310102 x^{6} - 241884 x^{5} - 40367312 x^{4} + \cdots - 4516262912 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2 x^{9} - 957 x^{8} + 1224 x^{7} + 310102 x^{6} - 241884 x^{5} - 40367312 x^{4} + \cdots - 4516262912 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 192 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 1176111762173 \nu^{9} - 12574063170642 \nu^{8} + 876995328306269 \nu^{7} + \cdots - 25\!\cdots\!60 ) / 59\!\cdots\!64 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2790653758009 \nu^{9} - 499024731176506 \nu^{8} + \cdots + 30\!\cdots\!76 ) / 11\!\cdots\!28 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 6214679484235 \nu^{9} - 367159236899342 \nu^{8} + \cdots - 13\!\cdots\!72 ) / 11\!\cdots\!28 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 711607147759 \nu^{9} - 687274083168 \nu^{8} + 619904788708757 \nu^{7} + \cdots - 33\!\cdots\!24 ) / 33\!\cdots\!48 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 25960034868425 \nu^{9} + 56774085353906 \nu^{8} + \cdots + 80\!\cdots\!60 ) / 11\!\cdots\!28 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 31766366680659 \nu^{9} + 48358777692646 \nu^{8} + \cdots + 32\!\cdots\!00 ) / 11\!\cdots\!28 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 13936442615075 \nu^{9} - 2356424471438 \nu^{8} + \cdots + 61\!\cdots\!84 ) / 19\!\cdots\!88 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 192 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{8} + 4\beta_{6} + \beta_{5} - \beta_{4} + 9\beta_{3} + \beta_{2} + 309\beta _1 + 145 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5 \beta_{9} + 22 \beta_{8} + 18 \beta_{7} + 44 \beta_{6} + 15 \beta_{5} + \beta_{4} - 202 \beta_{3} + \cdots + 59730 \)
|
| \(\nu^{5}\) | \(=\) |
\( 109 \beta_{9} + 2446 \beta_{8} + 118 \beta_{7} + 2192 \beta_{6} + 851 \beta_{5} - 751 \beta_{4} + \cdots + 119766 \)
|
| \(\nu^{6}\) | \(=\) |
\( 1541 \beta_{9} + 16218 \beta_{8} + 11722 \beta_{7} + 34984 \beta_{6} + 12813 \beta_{5} - 2329 \beta_{4} + \cdots + 21929496 \)
|
| \(\nu^{7}\) | \(=\) |
\( 86067 \beta_{9} + 1226770 \beta_{8} + 89618 \beta_{7} + 1040712 \beta_{6} + 516339 \beta_{5} + \cdots + 80561872 \)
|
| \(\nu^{8}\) | \(=\) |
\( 384913 \beta_{9} + 9584946 \beta_{8} + 6145098 \beta_{7} + 20408544 \beta_{6} + 7766169 \beta_{5} + \cdots + 8791834440 \)
|
| \(\nu^{9}\) | \(=\) |
\( 47808719 \beta_{9} + 580883410 \beta_{8} + 52781794 \beta_{7} + 484274136 \beta_{6} + 273774531 \beta_{5} + \cdots + 48382466052 \)
|
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 |
|
−22.0501 | −47.1151 | 358.208 | −467.623 | 1038.89 | 343.000 | −5076.12 | 32.8291 | 10311.1 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −18.8829 | 14.3398 | 228.563 | 382.689 | −270.777 | 343.000 | −1898.92 | −1981.37 | −7226.26 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −13.8703 | 79.8279 | 64.3848 | −355.678 | −1107.24 | 343.000 | 882.360 | 4185.49 | 4933.36 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −11.5114 | 23.9058 | 4.51150 | −301.396 | −275.188 | 343.000 | 1421.52 | −1615.51 | 3469.48 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −3.80018 | −76.5407 | −113.559 | −339.464 | 290.868 | 343.000 | 917.967 | 3671.47 | 1290.03 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.549314 | −7.31646 | −127.698 | 243.344 | 4.01903 | 343.000 | 140.459 | −2133.47 | −133.672 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 8.49443 | 45.4613 | −55.8447 | −130.405 | 386.168 | 343.000 | −1561.66 | −120.267 | −1107.72 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 10.2698 | −81.6519 | −22.5320 | 313.060 | −838.546 | 343.000 | −1545.93 | 4480.04 | 3215.05 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 14.1785 | 6.78261 | 73.0290 | −4.23520 | 96.1671 | 343.000 | −779.405 | −2141.00 | −60.0487 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 19.7215 | −37.6933 | 260.937 | −267.290 | −743.369 | 343.000 | 2621.72 | −766.212 | −5271.36 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(13\) | \(1\) |
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 | 91.8.a.c | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.8.a.c | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 18 T_{2}^{9} - 813 T_{2}^{8} - 13416 T_{2}^{7} + 222070 T_{2}^{6} + 3157452 T_{2}^{5} + \cdots + 3385168384 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(91))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots + 3385168384 \)
|
| $3$ |
\( T^{10} + \cdots - 685212736352256 \)
|
| $5$ |
\( T^{10} + \cdots - 73\!\cdots\!00 \)
|
| $7$ |
\( (T - 343)^{10} \)
|
| $11$ |
\( T^{10} + \cdots + 14\!\cdots\!04 \)
|
| $13$ |
\( (T + 2197)^{10} \)
|
| $17$ |
\( T^{10} + \cdots - 16\!\cdots\!56 \)
|
| $19$ |
\( T^{10} + \cdots - 12\!\cdots\!68 \)
|
| $23$ |
\( T^{10} + \cdots + 10\!\cdots\!88 \)
|
| $29$ |
\( T^{10} + \cdots + 15\!\cdots\!32 \)
|
| $31$ |
\( T^{10} + \cdots - 13\!\cdots\!68 \)
|
| $37$ |
\( T^{10} + \cdots + 91\!\cdots\!28 \)
|
| $41$ |
\( T^{10} + \cdots + 51\!\cdots\!44 \)
|
| $43$ |
\( T^{10} + \cdots + 46\!\cdots\!56 \)
|
| $47$ |
\( T^{10} + \cdots + 85\!\cdots\!08 \)
|
| $53$ |
\( T^{10} + \cdots + 63\!\cdots\!84 \)
|
| $59$ |
\( T^{10} + \cdots - 58\!\cdots\!84 \)
|
| $61$ |
\( T^{10} + \cdots - 13\!\cdots\!52 \)
|
| $67$ |
\( T^{10} + \cdots + 57\!\cdots\!04 \)
|
| $71$ |
\( T^{10} + \cdots + 80\!\cdots\!44 \)
|
| $73$ |
\( T^{10} + \cdots + 39\!\cdots\!54 \)
|
| $79$ |
\( T^{10} + \cdots + 40\!\cdots\!12 \)
|
| $83$ |
\( T^{10} + \cdots + 15\!\cdots\!36 \)
|
| $89$ |
\( T^{10} + \cdots + 13\!\cdots\!76 \)
|
| $97$ |
\( T^{10} + \cdots + 52\!\cdots\!18 \)
|
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