Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [722,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("722.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 722 = 2 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(42.5993790241\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{9} - 165x^{7} - 28x^{6} + 8790x^{5} + 7128x^{4} - 179236x^{3} - 272553x^{2} + 1047198x + 1973017 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3\cdot 19^{2} \) |
| Twist minimal: | no (minimal twist has level 38) |
| 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
\( x^{9} - 165x^{7} - 28x^{6} + 8790x^{5} + 7128x^{4} - 179236x^{3} - 272553x^{2} + 1047198x + 1973017 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 6335134305 \nu^{8} + 934854547295 \nu^{7} - 2746648151780 \nu^{6} - 134642750962488 \nu^{5} + \cdots + 71\!\cdots\!33 ) / 24\!\cdots\!64 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2387438459 \nu^{8} - 237564972437 \nu^{7} - 304617431300 \nu^{6} + \cdots + 11\!\cdots\!85 ) / 613749763851216 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 79554421623 \nu^{8} - 234969732911 \nu^{7} - 12008855913676 \nu^{6} + 34275625099368 \nu^{5} + \cdots + 47\!\cdots\!95 ) / 24\!\cdots\!64 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 80263867713 \nu^{8} + 175840747921 \nu^{7} + 11725305911108 \nu^{6} + \cdots - 40\!\cdots\!17 ) / 24\!\cdots\!64 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 208509030407 \nu^{8} + 438012031569 \nu^{7} - 37004990909228 \nu^{6} + \cdots + 22\!\cdots\!31 ) / 24\!\cdots\!64 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 53188376307 \nu^{8} - 150897447067 \nu^{7} - 8019998851508 \nu^{6} + 21588493456776 \nu^{5} + \cdots + 19\!\cdots\!39 ) / 306874881925608 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 150111808379 \nu^{8} + 456971492163 \nu^{7} + 22746405615292 \nu^{6} + \cdots - 90\!\cdots\!19 ) / 818333018468288 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 338845236613 \nu^{8} + 1244227048585 \nu^{7} + 54754433948364 \nu^{6} + \cdots - 28\!\cdots\!29 ) / 613749763851216 \)
|
| \(\nu\) | \(=\) |
\( ( -3\beta_{7} - 2\beta_{4} - 19\beta_{3} ) / 19 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 6\beta_{8} + 4\beta_{7} + 13\beta_{6} + 12\beta_{5} - 18\beta_{4} + 8\beta_{3} + 15\beta_{2} - 6\beta _1 + 702 ) / 19 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7 \beta_{8} - 313 \beta_{7} - 7 \beta_{6} - 5 \beta_{5} + 485 \beta_{4} - 1118 \beta_{3} - 49 \beta_{2} + \cdots + 154 ) / 19 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 557 \beta_{8} + 228 \beta_{7} + 640 \beta_{6} + 1076 \beta_{5} - 2782 \beta_{4} + 1946 \beta_{3} + \cdots + 41115 ) / 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 195 \beta_{8} - 26447 \beta_{7} - 2285 \beta_{6} - 370 \beta_{5} + 54011 \beta_{4} - 79369 \beta_{3} + \cdots - 30480 ) / 19 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 48406 \beta_{8} + 24198 \beta_{7} + 31489 \beta_{6} + 87445 \beta_{5} - 318659 \beta_{4} + \cdots + 2899572 ) / 19 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 25832 \beta_{8} - 2177226 \beta_{7} - 285578 \beta_{6} - 58238 \beta_{5} + 4926932 \beta_{4} + \cdots - 6159757 ) / 19 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 4119280 \beta_{8} + 3278671 \beta_{7} + 1921694 \beta_{6} + 7123336 \beta_{5} - 32725040 \beta_{4} + \cdots + 223286606 ) / 19 \)
|
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 |
|
2.00000 | −8.16083 | 4.00000 | −0.907363 | −16.3217 | 9.36062 | 8.00000 | 39.5991 | −1.81473 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −5.18603 | 4.00000 | −14.3088 | −10.3721 | −32.1449 | 8.00000 | −0.105104 | −28.6175 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | −3.25580 | 4.00000 | −11.0544 | −6.51160 | −19.6182 | 8.00000 | −16.3998 | −22.1088 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | −3.11814 | 4.00000 | 10.7989 | −6.23628 | 27.5434 | 8.00000 | −17.2772 | 21.5977 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | −0.636547 | 4.00000 | 17.7533 | −1.27309 | 23.2072 | 8.00000 | −26.5948 | 35.5065 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.00000 | 4.43829 | 4.00000 | 21.5062 | 8.87658 | −5.97386 | 8.00000 | −7.30156 | 43.0123 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.00000 | 7.41713 | 4.00000 | −19.5988 | 14.8343 | 31.2078 | 8.00000 | 28.0138 | −39.1976 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.00000 | 7.54540 | 4.00000 | 4.50990 | 15.0908 | −8.42169 | 8.00000 | 29.9331 | 9.01980 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.00000 | 9.95653 | 4.00000 | −5.69887 | 19.9131 | 7.83973 | 8.00000 | 72.1324 | −11.3977 | ||||||||||||||||||||||||||||||||||||||||||||||
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 | 722.4.a.u | 9 | |
| 19.b | odd | 2 | 1 | 722.4.a.t | 9 | ||
| 19.f | odd | 18 | 2 | 38.4.e.b | ✓ | 18 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.4.e.b | ✓ | 18 | 19.f | odd | 18 | 2 | |
| 722.4.a.t | 9 | 19.b | odd | 2 | 1 | ||
| 722.4.a.u | 9 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(722))\):
|
\( T_{3}^{9} - 9 T_{3}^{8} - 132 T_{3}^{7} + 1043 T_{3}^{6} + 6195 T_{3}^{5} - 34233 T_{3}^{4} + \cdots + 676387 \)
|
|
\( T_{5}^{9} - 3 T_{5}^{8} - 825 T_{5}^{7} + 851 T_{5}^{6} + 212025 T_{5}^{5} + 148323 T_{5}^{4} + \cdots + 298071768 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{9} \)
|
| $3$ |
\( T^{9} - 9 T^{8} + \cdots + 676387 \)
|
| $5$ |
\( T^{9} - 3 T^{8} + \cdots + 298071768 \)
|
| $7$ |
\( T^{9} + \cdots - 46444486248 \)
|
| $11$ |
\( T^{9} + \cdots + 14065727044011 \)
|
| $13$ |
\( T^{9} + \cdots - 65320441232824 \)
|
| $17$ |
\( T^{9} + \cdots + 18784252476819 \)
|
| $19$ |
\( T^{9} \)
|
| $23$ |
\( T^{9} + \cdots + 29\!\cdots\!96 \)
|
| $29$ |
\( T^{9} + \cdots - 29\!\cdots\!84 \)
|
| $31$ |
\( T^{9} + \cdots + 20\!\cdots\!76 \)
|
| $37$ |
\( T^{9} + \cdots + 64\!\cdots\!08 \)
|
| $41$ |
\( T^{9} + \cdots + 31\!\cdots\!57 \)
|
| $43$ |
\( T^{9} + \cdots - 18\!\cdots\!27 \)
|
| $47$ |
\( T^{9} + \cdots - 39\!\cdots\!64 \)
|
| $53$ |
\( T^{9} + \cdots - 13\!\cdots\!52 \)
|
| $59$ |
\( T^{9} + \cdots - 31\!\cdots\!71 \)
|
| $61$ |
\( T^{9} + \cdots + 20\!\cdots\!04 \)
|
| $67$ |
\( T^{9} + \cdots - 15\!\cdots\!52 \)
|
| $71$ |
\( T^{9} + \cdots - 36\!\cdots\!48 \)
|
| $73$ |
\( T^{9} + \cdots + 24\!\cdots\!08 \)
|
| $79$ |
\( T^{9} + \cdots + 10\!\cdots\!08 \)
|
| $83$ |
\( T^{9} + \cdots - 20\!\cdots\!07 \)
|
| $89$ |
\( T^{9} + \cdots - 23\!\cdots\!81 \)
|
| $97$ |
\( T^{9} + \cdots + 45\!\cdots\!31 \)
|
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