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: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - 4 x^{8} - 764 x^{7} + 1562 x^{6} + 176422 x^{5} + 56746 x^{4} - 13204236 x^{3} + \cdots + 176334338 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{7} \) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 4 x^{8} - 764 x^{7} + 1562 x^{6} + 176422 x^{5} + 56746 x^{4} - 13204236 x^{3} + \cdots + 176334338 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 6015827 \nu^{8} + 20549143 \nu^{7} + 4991944185 \nu^{6} - 10970858567 \nu^{5} + \cdots - 881196617033550 ) / 10791021610368 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 6015827 \nu^{8} + 20549143 \nu^{7} + 4991944185 \nu^{6} - 10970858567 \nu^{5} + \cdots - 27\!\cdots\!10 ) / 10791021610368 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 23932735 \nu^{8} + 207426657 \nu^{7} - 19557244969 \nu^{6} - 180490638053 \nu^{5} + \cdots + 19\!\cdots\!74 ) / 21582043220736 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 25387529 \nu^{8} + 253465957 \nu^{7} + 16934639179 \nu^{6} - 118481098757 \nu^{5} + \cdots + 16\!\cdots\!46 ) / 10791021610368 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 97509171 \nu^{8} + 1324540219 \nu^{7} + 60465028597 \nu^{6} - 697004753079 \nu^{5} + \cdots - 56\!\cdots\!62 ) / 21582043220736 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 28587983 \nu^{8} - 424374023 \nu^{7} - 17815070701 \nu^{6} + 242290842287 \nu^{5} + \cdots + 804364226993614 ) / 5395510805184 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 114544149 \nu^{8} + 615032077 \nu^{7} + 83581078923 \nu^{6} - 310190501481 \nu^{5} + \cdots - 13\!\cdots\!18 ) / 21582043220736 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + 3\beta _1 + 170 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{8} + 3\beta_{7} + 2\beta_{6} + \beta_{5} + 4\beta_{4} - \beta_{3} - 2\beta_{2} + 313\beta _1 + 369 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} - 16\beta_{7} - 38\beta_{6} + 47\beta_{5} + 3\beta_{4} + 420\beta_{3} - 466\beta_{2} + 1365\beta _1 + 53267 \)
|
| \(\nu^{5}\) | \(=\) |
\( 830 \beta_{8} + 1708 \beta_{7} + 1252 \beta_{6} + 706 \beta_{5} + 1970 \beta_{4} - 284 \beta_{3} + \cdots + 174164 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 1316 \beta_{8} - 6312 \beta_{7} - 19064 \beta_{6} + 28520 \beta_{5} + 4964 \beta_{4} + \cdots + 19568550 \)
|
| \(\nu^{7}\) | \(=\) |
\( 294670 \beta_{8} + 788851 \beta_{7} + 603418 \beta_{6} + 386369 \beta_{5} + 897256 \beta_{4} + \cdots + 71334873 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 1418771 \beta_{8} - 1673588 \beta_{7} - 7584502 \beta_{6} + 13920203 \beta_{5} + 3739871 \beta_{4} + \cdots + 7634119271 \)
|
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 |
|
−21.0538 | −33.4331 | 315.264 | 271.446 | 703.895 | −343.000 | −3942.62 | −1069.23 | −5714.98 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −12.7379 | −48.2951 | 34.2551 | −506.573 | 615.180 | −343.000 | 1194.12 | 145.416 | 6452.69 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −11.5479 | 14.0702 | 5.35382 | 200.806 | −162.481 | −343.000 | 1416.30 | −1989.03 | −2318.88 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −6.28525 | 79.3185 | −88.4957 | −293.470 | −498.536 | −343.000 | 1360.73 | 4104.43 | 1844.53 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.95019 | −76.8097 | −124.197 | 57.0015 | 149.793 | −343.000 | 491.831 | 3712.72 | −111.163 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.64558 | 25.8257 | −121.001 | 466.249 | 68.3242 | −343.000 | −658.753 | −1520.03 | 1233.50 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 9.68252 | 60.1290 | −34.2488 | −158.582 | 582.200 | −343.000 | −1570.98 | 1428.50 | −1535.48 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 16.5192 | −52.3583 | 144.884 | 210.689 | −864.917 | −343.000 | 278.914 | 554.387 | 3480.42 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 19.7278 | 5.55266 | 261.185 | −428.566 | 109.542 | −343.000 | 2627.45 | −2156.17 | −8454.65 | ||||||||||||||||||||||||||||||||||||||||||||||
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.b | ✓ | 9 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.8.a.b | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} + 5 T_{2}^{8} - 760 T_{2}^{7} - 3814 T_{2}^{6} + 169652 T_{2}^{5} + 935392 T_{2}^{4} + \cdots + 316890112 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(91))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + 5 T^{8} + \cdots + 316890112 \)
|
| $3$ |
\( T^{9} + \cdots - 62487429836832 \)
|
| $5$ |
\( T^{9} + \cdots - 30\!\cdots\!00 \)
|
| $7$ |
\( (T + 343)^{9} \)
|
| $11$ |
\( T^{9} + \cdots + 70\!\cdots\!36 \)
|
| $13$ |
\( (T - 2197)^{9} \)
|
| $17$ |
\( T^{9} + \cdots + 12\!\cdots\!24 \)
|
| $19$ |
\( T^{9} + \cdots - 66\!\cdots\!52 \)
|
| $23$ |
\( T^{9} + \cdots + 39\!\cdots\!59 \)
|
| $29$ |
\( T^{9} + \cdots - 15\!\cdots\!32 \)
|
| $31$ |
\( T^{9} + \cdots + 22\!\cdots\!51 \)
|
| $37$ |
\( T^{9} + \cdots + 44\!\cdots\!36 \)
|
| $41$ |
\( T^{9} + \cdots + 46\!\cdots\!76 \)
|
| $43$ |
\( T^{9} + \cdots - 12\!\cdots\!00 \)
|
| $47$ |
\( T^{9} + \cdots - 14\!\cdots\!23 \)
|
| $53$ |
\( T^{9} + \cdots + 22\!\cdots\!04 \)
|
| $59$ |
\( T^{9} + \cdots - 27\!\cdots\!92 \)
|
| $61$ |
\( T^{9} + \cdots - 56\!\cdots\!84 \)
|
| $67$ |
\( T^{9} + \cdots + 15\!\cdots\!48 \)
|
| $71$ |
\( T^{9} + \cdots - 73\!\cdots\!16 \)
|
| $73$ |
\( T^{9} + \cdots + 69\!\cdots\!79 \)
|
| $79$ |
\( T^{9} + \cdots - 13\!\cdots\!01 \)
|
| $83$ |
\( T^{9} + \cdots - 22\!\cdots\!88 \)
|
| $89$ |
\( T^{9} + \cdots + 84\!\cdots\!24 \)
|
| $97$ |
\( T^{9} + \cdots + 94\!\cdots\!47 \)
|
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