Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [928,4,Mod(1,928)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(928, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("928.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 928 = 2^{5} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 928.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: | \(54.7537724853\) |
| 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: |
\( x^{9} - 4x^{8} - 138x^{7} + 394x^{6} + 5872x^{5} - 10822x^{4} - 85158x^{3} + 30654x^{2} + 439999x + 396802 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12} \) |
| 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
\( x^{9} - 4x^{8} - 138x^{7} + 394x^{6} + 5872x^{5} - 10822x^{4} - 85158x^{3} + 30654x^{2} + 439999x + 396802 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2129 \nu^{8} - 11757 \nu^{7} - 264339 \nu^{6} + 1175057 \nu^{5} + 9556105 \nu^{4} + \cdots + 369231982 ) / 4740120 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 39 \nu^{8} + 317 \nu^{7} + 4409 \nu^{6} - 34357 \nu^{5} - 127235 \nu^{4} + 1014083 \nu^{3} + \cdots - 860902 ) / 83160 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 347 \nu^{8} - 2165 \nu^{7} - 43679 \nu^{6} + 235173 \nu^{5} + 1598859 \nu^{4} - 7355187 \nu^{3} + \cdots + 78704822 ) / 316008 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3196 \nu^{8} + 17103 \nu^{7} + 412971 \nu^{6} - 1812133 \nu^{5} - 15686240 \nu^{4} + \cdots - 787744538 ) / 2370060 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 8339 \nu^{8} + 49617 \nu^{7} + 1041729 \nu^{6} - 5309147 \nu^{5} - 37053595 \nu^{4} + \cdots - 1519251202 ) / 4740120 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1333 \nu^{8} - 6999 \nu^{7} - 171603 \nu^{6} + 734299 \nu^{5} + 6418745 \nu^{4} - 22288601 \nu^{3} + \cdots + 259520594 ) / 677160 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 11597 \nu^{8} - 66681 \nu^{7} - 1481127 \nu^{6} + 7151861 \nu^{5} + 55355365 \nu^{4} + \cdots + 2604377566 ) / 4740120 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} + \beta_{5} - \beta_{4} + 2\beta _1 + 32 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{8} - 2\beta_{7} - \beta_{5} - 7\beta_{4} - 2\beta_{3} - \beta_{2} + 64\beta _1 + 37 \)
|
| \(\nu^{4}\) | \(=\) |
\( 96\beta_{8} - 17\beta_{7} - 6\beta_{6} + 73\beta_{5} - 105\beta_{4} + \beta_{3} + 5\beta_{2} + 247\beta _1 + 1882 \)
|
| \(\nu^{5}\) | \(=\) |
\( 580 \beta_{8} - 258 \beta_{7} - 96 \beta_{6} + 88 \beta_{5} - 880 \beta_{4} - 114 \beta_{3} + \cdots + 5092 \)
|
| \(\nu^{6}\) | \(=\) |
\( 8663 \beta_{8} - 2414 \beta_{7} - 1284 \beta_{6} + 5625 \beta_{5} - 10253 \beta_{4} + 250 \beta_{3} + \cdots + 134664 \)
|
| \(\nu^{7}\) | \(=\) |
\( 64308 \beta_{8} - 26950 \beta_{7} - 14772 \beta_{6} + 19853 \beta_{5} - 92853 \beta_{4} - 5062 \beta_{3} + \cdots + 611739 \)
|
| \(\nu^{8}\) | \(=\) |
\( 788310 \beta_{8} - 261219 \beta_{7} - 161082 \beta_{6} + 461973 \beta_{5} - 984441 \beta_{4} + \cdots + 10714546 \)
|
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 |
|
0 | −9.74094 | 0 | 17.5483 | 0 | −10.7391 | 0 | 67.8859 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −6.50540 | 0 | −1.74434 | 0 | −26.0581 | 0 | 15.3202 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −4.87969 | 0 | 10.4780 | 0 | 23.3159 | 0 | −3.18861 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −3.39022 | 0 | −19.3767 | 0 | 7.92881 | 0 | −15.5064 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.51339 | 0 | −6.33195 | 0 | −1.77839 | 0 | −24.7097 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 1.95108 | 0 | 18.0628 | 0 | −22.0180 | 0 | −23.1933 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 2.38920 | 0 | −3.17647 | 0 | 28.0947 | 0 | −21.2917 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 7.02483 | 0 | 2.19045 | 0 | −5.10018 | 0 | 22.3483 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 7.63776 | 0 | −7.65015 | 0 | −5.64550 | 0 | 31.3353 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(29\) | \(1\) |
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 | 928.4.a.d | ✓ | 9 |
| 4.b | odd | 2 | 1 | 928.4.a.e | yes | 9 | |
| 8.b | even | 2 | 1 | 1856.4.a.bg | 9 | ||
| 8.d | odd | 2 | 1 | 1856.4.a.bf | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 928.4.a.d | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 928.4.a.e | yes | 9 | 4.b | odd | 2 | 1 | |
| 1856.4.a.bf | 9 | 8.d | odd | 2 | 1 | ||
| 1856.4.a.bg | 9 | 8.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{9} + 4 T_{3}^{8} - 138 T_{3}^{7} - 394 T_{3}^{6} + 5872 T_{3}^{5} + 10822 T_{3}^{4} + \cdots - 396802 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(928))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} \)
|
| $3$ |
\( T^{9} + 4 T^{8} + \cdots - 396802 \)
|
| $5$ |
\( T^{9} - 10 T^{8} + \cdots + 37835002 \)
|
| $7$ |
\( T^{9} + \cdots - 1638662144 \)
|
| $11$ |
\( T^{9} + \cdots + 86950571182 \)
|
| $13$ |
\( T^{9} + \cdots - 119297288098 \)
|
| $17$ |
\( T^{9} + \cdots + 239296714517504 \)
|
| $19$ |
\( T^{9} + \cdots + 71\!\cdots\!16 \)
|
| $23$ |
\( T^{9} + \cdots + 60\!\cdots\!04 \)
|
| $29$ |
\( (T + 29)^{9} \)
|
| $31$ |
\( T^{9} + \cdots - 12\!\cdots\!34 \)
|
| $37$ |
\( T^{9} + \cdots + 25\!\cdots\!12 \)
|
| $41$ |
\( T^{9} + \cdots + 71\!\cdots\!04 \)
|
| $43$ |
\( T^{9} + \cdots + 30\!\cdots\!42 \)
|
| $47$ |
\( T^{9} + \cdots - 20\!\cdots\!22 \)
|
| $53$ |
\( T^{9} + \cdots - 25\!\cdots\!86 \)
|
| $59$ |
\( T^{9} + \cdots - 70\!\cdots\!64 \)
|
| $61$ |
\( T^{9} + \cdots + 19\!\cdots\!72 \)
|
| $67$ |
\( T^{9} + \cdots + 33\!\cdots\!96 \)
|
| $71$ |
\( T^{9} + \cdots - 55\!\cdots\!76 \)
|
| $73$ |
\( T^{9} + \cdots - 61\!\cdots\!76 \)
|
| $79$ |
\( T^{9} + \cdots + 16\!\cdots\!86 \)
|
| $83$ |
\( T^{9} + \cdots - 77\!\cdots\!36 \)
|
| $89$ |
\( T^{9} + \cdots - 37\!\cdots\!00 \)
|
| $97$ |
\( T^{9} + \cdots + 21\!\cdots\!88 \)
|
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