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: | \(0\) |
| 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: |
\( x^{12} - 6 x^{11} - 1243 x^{10} + 5598 x^{9} + 567554 x^{8} - 1739560 x^{7} - 117081910 x^{6} + \cdots + 59402280000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 6 x^{11} - 1243 x^{10} + 5598 x^{9} + 567554 x^{8} - 1739560 x^{7} - 117081910 x^{6} + \cdots + 59402280000 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 209 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11\!\cdots\!97 \nu^{11} + \cdots + 16\!\cdots\!40 ) / 25\!\cdots\!80 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 14\!\cdots\!33 \nu^{11} + \cdots + 34\!\cdots\!20 ) / 25\!\cdots\!80 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 67\!\cdots\!59 \nu^{11} + \cdots - 91\!\cdots\!60 ) / 25\!\cdots\!80 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 44\!\cdots\!49 \nu^{11} + \cdots - 11\!\cdots\!80 ) / 12\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 45\!\cdots\!39 \nu^{11} + \cdots + 10\!\cdots\!40 ) / 93\!\cdots\!40 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 23\!\cdots\!61 \nu^{11} + \cdots - 34\!\cdots\!40 ) / 31\!\cdots\!60 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 25\!\cdots\!13 \nu^{11} + \cdots + 39\!\cdots\!68 ) / 12\!\cdots\!44 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 10\!\cdots\!89 \nu^{11} + \cdots - 14\!\cdots\!68 ) / 42\!\cdots\!48 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 32\!\cdots\!93 \nu^{11} + \cdots + 47\!\cdots\!40 ) / 12\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 209 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{8} + \beta_{4} + 15\beta_{3} + 3\beta_{2} + 345\beta _1 + 312 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4 \beta_{11} + 4 \beta_{10} - \beta_{9} + 6 \beta_{8} - \beta_{7} - 5 \beta_{6} + 13 \beta_{5} + \cdots + 71412 \)
|
| \(\nu^{5}\) | \(=\) |
\( 30 \beta_{11} + 6 \beta_{10} - 60 \beta_{9} + 636 \beta_{8} + 44 \beta_{7} - 32 \beta_{6} + \cdots + 229516 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3024 \beta_{11} + 2494 \beta_{10} - 1240 \beta_{9} + 5248 \beta_{8} - 214 \beta_{7} - 3574 \beta_{6} + \cdots + 28454111 \)
|
| \(\nu^{7}\) | \(=\) |
\( 30290 \beta_{11} + 8156 \beta_{10} - 57708 \beta_{9} + 337579 \beta_{8} + 31150 \beta_{7} + \cdots + 151525518 \)
|
| \(\nu^{8}\) | \(=\) |
\( 1739412 \beta_{11} + 1232150 \beta_{10} - 962609 \beta_{9} + 3561774 \beta_{8} + 125333 \beta_{7} + \cdots + 12316639262 \)
|
| \(\nu^{9}\) | \(=\) |
\( 20964960 \beta_{11} + 6343594 \beta_{10} - 39535284 \beta_{9} + 172280246 \beta_{8} + 17399558 \beta_{7} + \cdots + 93516704558 \)
|
| \(\nu^{10}\) | \(=\) |
\( 920147760 \beta_{11} + 573136640 \beta_{10} - 634693664 \beta_{9} + 2189379316 \beta_{8} + \cdots + 5613426256161 \)
|
| \(\nu^{11}\) | \(=\) |
\( 12687632436 \beta_{11} + 4070868840 \beta_{10} - 23812818640 \beta_{9} + 87449732997 \beta_{8} + \cdots + 55368631574220 \)
|
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 |
|
−22.0670 | 71.8770 | 358.952 | 25.2300 | −1586.11 | 343.000 | −5096.40 | 2979.31 | −556.750 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −18.0596 | −64.0898 | 198.149 | 13.5880 | 1157.44 | 343.000 | −1266.87 | 1920.50 | −245.394 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −14.8833 | −5.24717 | 93.5131 | −195.689 | 78.0952 | 343.000 | 513.280 | −2159.47 | 2912.49 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −9.77328 | −73.6575 | −32.4830 | 542.246 | 719.876 | 343.000 | 1568.45 | 3238.43 | −5299.52 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −9.59274 | 50.3591 | −35.9794 | 255.731 | −483.082 | 343.000 | 1573.01 | 349.039 | −2453.16 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.930860 | 23.4478 | −127.134 | −494.625 | 21.8266 | 343.000 | −237.493 | −1637.20 | −460.427 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 5.06937 | 91.5295 | −102.301 | −15.8892 | 463.997 | 343.000 | −1167.48 | 6190.65 | −80.5484 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 5.08138 | −25.2372 | −102.180 | 250.262 | −128.240 | 343.000 | −1169.63 | −1550.08 | 1271.68 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 13.7078 | 50.2321 | 59.9043 | 544.453 | 688.573 | 343.000 | −933.443 | 336.267 | 7463.26 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 14.1706 | −35.5734 | 72.8048 | −268.857 | −504.095 | 343.000 | −782.147 | −921.532 | −3809.85 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 19.9614 | 56.3144 | 270.459 | 1.53059 | 1124.12 | 343.000 | 2843.68 | 984.313 | 30.5528 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 21.4545 | −57.9549 | 332.295 | 368.020 | −1243.39 | 343.000 | 4383.05 | 1171.77 | 7895.68 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
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.e | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.8.a.e | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 6 T_{2}^{11} - 1243 T_{2}^{10} + 6942 T_{2}^{9} + 561506 T_{2}^{8} - 2852052 T_{2}^{7} + \cdots - 1109222885376 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(91))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots - 1109222885376 \)
|
| $3$ |
\( T^{12} + \cdots + 28\!\cdots\!40 \)
|
| $5$ |
\( T^{12} + \cdots + 15\!\cdots\!00 \)
|
| $7$ |
\( (T - 343)^{12} \)
|
| $11$ |
\( T^{12} + \cdots - 14\!\cdots\!44 \)
|
| $13$ |
\( (T - 2197)^{12} \)
|
| $17$ |
\( T^{12} + \cdots - 63\!\cdots\!24 \)
|
| $19$ |
\( T^{12} + \cdots + 62\!\cdots\!00 \)
|
| $23$ |
\( T^{12} + \cdots - 21\!\cdots\!80 \)
|
| $29$ |
\( T^{12} + \cdots + 20\!\cdots\!00 \)
|
| $31$ |
\( T^{12} + \cdots - 45\!\cdots\!28 \)
|
| $37$ |
\( T^{12} + \cdots - 54\!\cdots\!40 \)
|
| $41$ |
\( T^{12} + \cdots - 22\!\cdots\!20 \)
|
| $43$ |
\( T^{12} + \cdots - 19\!\cdots\!32 \)
|
| $47$ |
\( T^{12} + \cdots - 42\!\cdots\!72 \)
|
| $53$ |
\( T^{12} + \cdots - 49\!\cdots\!00 \)
|
| $59$ |
\( T^{12} + \cdots + 96\!\cdots\!40 \)
|
| $61$ |
\( T^{12} + \cdots - 35\!\cdots\!00 \)
|
| $67$ |
\( T^{12} + \cdots - 14\!\cdots\!40 \)
|
| $71$ |
\( T^{12} + \cdots - 16\!\cdots\!28 \)
|
| $73$ |
\( T^{12} + \cdots - 57\!\cdots\!88 \)
|
| $79$ |
\( T^{12} + \cdots + 98\!\cdots\!40 \)
|
| $83$ |
\( T^{12} + \cdots + 12\!\cdots\!40 \)
|
| $89$ |
\( T^{12} + \cdots - 86\!\cdots\!00 \)
|
| $97$ |
\( T^{12} + \cdots - 41\!\cdots\!60 \)
|
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