Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [722,6,Mod(1,722)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(722, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("722.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 722 = 2 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 722.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: | \(115.797117905\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 1389x^{6} - 2174x^{5} + 537201x^{4} + 2408548x^{3} - 33284256x^{2} - 41014424x + 435397616 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 19 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 1389x^{6} - 2174x^{5} + 537201x^{4} + 2408548x^{3} - 33284256x^{2} - 41014424x + 435397616 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 349577032 \nu^{7} - 921385019 \nu^{6} + 479819472152 \nu^{5} + 2923819450411 \nu^{4} + \cdots + 30\!\cdots\!38 ) / 667734333415190 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 122089113 \nu^{7} + 399252386 \nu^{6} - 164393575113 \nu^{5} - 1096318567754 \nu^{4} + \cdots - 11\!\cdots\!72 ) / 70287824570020 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 349577032 \nu^{7} - 921385019 \nu^{6} + 479819472152 \nu^{5} + 2923819450411 \nu^{4} + \cdots + 30\!\cdots\!48 ) / 70287824570020 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 44876874610503 \nu^{7} - 178919848604266 \nu^{6} + \cdots + 47\!\cdots\!92 ) / 68\!\cdots\!40 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 63975370788868 \nu^{7} - 245424053728621 \nu^{6} + \cdots + 71\!\cdots\!12 ) / 68\!\cdots\!40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 78543983222687 \nu^{7} - 171364210835849 \nu^{6} + \cdots + 73\!\cdots\!08 ) / 68\!\cdots\!40 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 165434184825993 \nu^{7} - 389504251445716 \nu^{6} + \cdots + 14\!\cdots\!52 ) / 68\!\cdots\!40 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{3} + 19\beta _1 + 1 ) / 19 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 19\beta_{7} - 38\beta_{6} - 38\beta_{5} + 38\beta_{4} + 4\beta_{3} - 19\beta_{2} + 57\beta _1 + 6610 ) / 19 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 171 \beta_{7} + 114 \beta_{6} - 456 \beta_{5} + 627 \beta_{4} - 358 \beta_{3} + 912 \beta_{2} + \cdots + 33296 ) / 19 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 14725 \beta_{7} - 23864 \beta_{6} - 25973 \beta_{5} + 27398 \beta_{4} - 20187 \beta_{3} + \cdots + 4153319 ) / 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 87343 \beta_{7} - 29013 \beta_{6} - 402477 \beta_{5} + 552235 \beta_{4} - 7418 \beta_{3} + \cdots + 39313531 ) / 19 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 10461058 \beta_{7} - 14190530 \beta_{6} - 18541055 \beta_{5} + 19065512 \beta_{4} - 22532215 \beta_{3} + \cdots + 2839422311 ) / 19 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 26593046 \beta_{7} - 84717219 \beta_{6} - 321720825 \beta_{5} + 453278060 \beta_{4} + \cdots + 36238540643 ) / 19 \)
|
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 |
|
4.00000 | −30.3182 | 16.0000 | 45.3824 | −121.273 | 233.153 | 64.0000 | 676.193 | 181.529 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 4.00000 | −21.7251 | 16.0000 | −68.1861 | −86.9004 | −190.420 | 64.0000 | 228.979 | −272.744 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 4.00000 | −12.7458 | 16.0000 | 33.8693 | −50.9833 | 193.042 | 64.0000 | −80.5442 | 135.477 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 4.00000 | −9.45753 | 16.0000 | −65.6267 | −37.8301 | −17.6814 | 64.0000 | −153.555 | −262.507 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 4.00000 | −0.500541 | 16.0000 | 95.7867 | −2.00216 | −136.636 | 64.0000 | −242.749 | 383.147 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 4.00000 | 3.82081 | 16.0000 | −15.1671 | 15.2833 | −76.4403 | 64.0000 | −228.401 | −60.6683 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 4.00000 | 22.5681 | 16.0000 | −35.6276 | 90.2723 | −16.1832 | 64.0000 | 266.318 | −142.510 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 4.00000 | 26.3583 | 16.0000 | −98.4309 | 105.433 | 105.167 | 64.0000 | 451.760 | −393.724 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 722.6.a.n | yes | 8 |
| 19.b | odd | 2 | 1 | 722.6.a.m | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 722.6.a.m | ✓ | 8 | 19.b | odd | 2 | 1 | |
| 722.6.a.n | yes | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 22 T_{3}^{7} - 1189 T_{3}^{6} - 25982 T_{3}^{5} + 308201 T_{3}^{4} + 7913248 T_{3}^{3} + \cdots - 90327024 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(722))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{8} \)
|
| $3$ |
\( T^{8} + 22 T^{7} + \cdots - 90327024 \)
|
| $5$ |
\( T^{8} + \cdots - 35042398775920 \)
|
| $7$ |
\( T^{8} + \cdots - 26\!\cdots\!44 \)
|
| $11$ |
\( T^{8} + \cdots - 26\!\cdots\!20 \)
|
| $13$ |
\( T^{8} + \cdots - 38\!\cdots\!95 \)
|
| $17$ |
\( T^{8} + \cdots - 11\!\cdots\!04 \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( T^{8} + \cdots + 23\!\cdots\!80 \)
|
| $29$ |
\( T^{8} + \cdots - 10\!\cdots\!95 \)
|
| $31$ |
\( T^{8} + \cdots - 12\!\cdots\!20 \)
|
| $37$ |
\( T^{8} + \cdots + 33\!\cdots\!81 \)
|
| $41$ |
\( T^{8} + \cdots - 21\!\cdots\!55 \)
|
| $43$ |
\( T^{8} + \cdots - 56\!\cdots\!84 \)
|
| $47$ |
\( T^{8} + \cdots - 23\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots - 60\!\cdots\!19 \)
|
| $59$ |
\( T^{8} + \cdots - 20\!\cdots\!16 \)
|
| $61$ |
\( T^{8} + \cdots + 18\!\cdots\!45 \)
|
| $67$ |
\( T^{8} + \cdots - 48\!\cdots\!80 \)
|
| $71$ |
\( T^{8} + \cdots + 14\!\cdots\!20 \)
|
| $73$ |
\( T^{8} + \cdots - 42\!\cdots\!39 \)
|
| $79$ |
\( T^{8} + \cdots - 10\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots - 27\!\cdots\!24 \)
|
| $89$ |
\( T^{8} + \cdots + 18\!\cdots\!45 \)
|
| $97$ |
\( T^{8} + \cdots + 18\!\cdots\!45 \)
|
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