Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [91,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(46.8682610909\) |
| Analytic rank: | \(1\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 3 x^{11} - 4522 x^{10} + 11094 x^{9} + 7471016 x^{8} - 18339296 x^{7} - 5497728352 x^{6} + \cdots + 170905444356096 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 3 x^{11} - 4522 x^{10} + 11094 x^{9} + 7471016 x^{8} - 18339296 x^{7} - 5497728352 x^{6} + \cdots + 170905444356096 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 754 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 28\!\cdots\!39 \nu^{11} + \cdots + 20\!\cdots\!28 ) / 33\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 12\!\cdots\!51 \nu^{11} + \cdots + 50\!\cdots\!00 ) / 55\!\cdots\!48 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 10\!\cdots\!77 \nu^{11} + \cdots - 12\!\cdots\!76 ) / 27\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 11\!\cdots\!61 \nu^{11} + \cdots + 27\!\cdots\!32 ) / 61\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 20\!\cdots\!19 \nu^{11} + \cdots + 41\!\cdots\!72 ) / 82\!\cdots\!00 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 47\!\cdots\!31 \nu^{11} + \cdots + 39\!\cdots\!28 ) / 16\!\cdots\!00 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 59\!\cdots\!37 \nu^{11} + \cdots + 35\!\cdots\!56 ) / 16\!\cdots\!00 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 63\!\cdots\!13 \nu^{11} + \cdots + 10\!\cdots\!56 ) / 16\!\cdots\!00 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 42\!\cdots\!31 \nu^{11} + \cdots + 38\!\cdots\!72 ) / 82\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 754 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} - \beta_{10} + \beta_{9} - 2\beta_{8} + \beta_{7} + \beta_{5} - 17\beta_{3} + 5\beta_{2} + 1218\beta _1 + 319 \)
|
| \(\nu^{4}\) | \(=\) |
\( 18 \beta_{11} + 11 \beta_{10} + 16 \beta_{9} - 35 \beta_{8} + 23 \beta_{7} + 9 \beta_{6} - 82 \beta_{5} + \cdots + 919341 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2320 \beta_{11} - 1545 \beta_{10} + 1914 \beta_{9} - 4029 \beta_{8} + 2191 \beta_{7} - 557 \beta_{6} + \cdots + 2557937 \)
|
| \(\nu^{6}\) | \(=\) |
\( 55778 \beta_{11} + 23993 \beta_{10} + 39292 \beta_{9} - 102837 \beta_{8} + 64981 \beta_{7} + \cdots + 1277966691 \)
|
| \(\nu^{7}\) | \(=\) |
\( 4457714 \beta_{11} - 2074921 \beta_{10} + 3248736 \beta_{9} - 7455887 \beta_{8} + 4273827 \beta_{7} + \cdots + 9019267565 \)
|
| \(\nu^{8}\) | \(=\) |
\( 130964522 \beta_{11} + 41125549 \beta_{10} + 82029132 \beta_{9} - 233995353 \beta_{8} + \cdots + 1927600064999 \)
|
| \(\nu^{9}\) | \(=\) |
\( 8260490214 \beta_{11} - 2734686897 \beta_{10} + 5531449548 \beta_{9} - 13717491851 \beta_{8} + \cdots + 24056315139757 \)
|
| \(\nu^{10}\) | \(=\) |
\( 277436523322 \beta_{11} + 65113911469 \beta_{10} + 164667623548 \beta_{9} - 486291501609 \beta_{8} + \cdots + 30\!\cdots\!47 \)
|
| \(\nu^{11}\) | \(=\) |
\( 15265079776830 \beta_{11} - 3628033451641 \beta_{10} + 9648812818724 \beta_{9} - 25355854032779 \beta_{8} + \cdots + 56\!\cdots\!41 \)
|
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 |
|
−39.6810 | −201.837 | 1062.58 | 764.411 | 8009.09 | 2401.00 | −21847.5 | 21055.2 | −30332.6 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −37.8265 | 32.9997 | 918.841 | −2427.98 | −1248.26 | 2401.00 | −15389.3 | −18594.0 | 91841.7 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −35.9466 | 230.724 | 780.160 | 434.116 | −8293.74 | 2401.00 | −9639.45 | 33550.4 | −15605.0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −24.2119 | 6.29566 | 74.2177 | 2011.41 | −152.430 | 2401.00 | 10599.6 | −19643.4 | −48700.2 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −13.0043 | −81.8707 | −342.889 | −1042.73 | 1064.67 | 2401.00 | 11117.2 | −12980.2 | 13559.9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −2.85796 | 143.668 | −503.832 | −15.3125 | −410.598 | 2401.00 | 2903.21 | 957.575 | 43.7625 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −0.703483 | −141.700 | −511.505 | −2244.71 | 99.6835 | 2401.00 | 720.018 | 395.868 | 1579.12 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 11.5332 | −234.910 | −378.986 | −380.245 | −2709.25 | 2401.00 | −10275.9 | 35499.6 | −4385.43 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 22.5414 | 66.2971 | −3.88662 | −391.554 | 1494.43 | 2401.00 | −11628.8 | −15287.7 | −8826.17 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 27.7766 | 192.937 | 259.538 | −1979.40 | 5359.14 | 2401.00 | −7012.53 | 17541.9 | −54980.9 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 29.8951 | −70.1432 | 381.717 | 1113.83 | −2096.94 | 2401.00 | −3894.82 | −14762.9 | 33298.0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 41.4855 | −265.461 | 1209.05 | −1043.84 | −11012.8 | 2401.00 | 28917.3 | 50786.7 | −43304.2 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
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 | 91.10.a.a | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.10.a.a | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 21 T_{2}^{11} - 4324 T_{2}^{10} - 78246 T_{2}^{9} + 6860708 T_{2}^{8} + \cdots - 305889376665600 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(91))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots - 305889376665600 \)
|
| $3$ |
\( T^{12} + \cdots + 90\!\cdots\!08 \)
|
| $5$ |
\( T^{12} + \cdots + 19\!\cdots\!00 \)
|
| $7$ |
\( (T - 2401)^{12} \)
|
| $11$ |
\( T^{12} + \cdots - 90\!\cdots\!00 \)
|
| $13$ |
\( (T - 28561)^{12} \)
|
| $17$ |
\( T^{12} + \cdots - 31\!\cdots\!32 \)
|
| $19$ |
\( T^{12} + \cdots + 88\!\cdots\!60 \)
|
| $23$ |
\( T^{12} + \cdots - 17\!\cdots\!00 \)
|
| $29$ |
\( T^{12} + \cdots + 10\!\cdots\!64 \)
|
| $31$ |
\( T^{12} + \cdots - 81\!\cdots\!00 \)
|
| $37$ |
\( T^{12} + \cdots + 75\!\cdots\!80 \)
|
| $41$ |
\( T^{12} + \cdots - 15\!\cdots\!00 \)
|
| $43$ |
\( T^{12} + \cdots + 11\!\cdots\!88 \)
|
| $47$ |
\( T^{12} + \cdots + 85\!\cdots\!52 \)
|
| $53$ |
\( T^{12} + \cdots + 20\!\cdots\!00 \)
|
| $59$ |
\( T^{12} + \cdots - 74\!\cdots\!00 \)
|
| $61$ |
\( T^{12} + \cdots + 53\!\cdots\!00 \)
|
| $67$ |
\( T^{12} + \cdots - 25\!\cdots\!00 \)
|
| $71$ |
\( T^{12} + \cdots + 28\!\cdots\!88 \)
|
| $73$ |
\( T^{12} + \cdots + 28\!\cdots\!62 \)
|
| $79$ |
\( T^{12} + \cdots + 16\!\cdots\!56 \)
|
| $83$ |
\( T^{12} + \cdots + 37\!\cdots\!12 \)
|
| $89$ |
\( T^{12} + \cdots + 13\!\cdots\!00 \)
|
| $97$ |
\( T^{12} + \cdots - 36\!\cdots\!30 \)
|
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