Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,5,Mod(48,49)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(49, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.48");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 49.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(5.06512819111\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{8} - 4x^{7} - 66x^{6} + 212x^{5} + 2021x^{4} - 4400x^{3} - 25028x^{2} + 27264x + 127778 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 7^{6} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{8} - 4x^{7} - 66x^{6} + 212x^{5} + 2021x^{4} - 4400x^{3} - 25028x^{2} + 27264x + 127778 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -12\nu^{6} + 36\nu^{5} + 475\nu^{4} - 1010\nu^{3} - 6245\nu^{2} + 6756\nu - 53158 ) / 8647 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 2912 \nu^{7} + 10192 \nu^{6} + 157016 \nu^{5} - 418020 \nu^{4} - 4500062 \nu^{3} + \cdots - 67052545 ) / 50783831 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 9708 \nu^{7} + 239533 \nu^{6} - 13483 \nu^{5} - 13961438 \nu^{4} + 11739196 \nu^{3} + \cdots - 2498696830 ) / 50783831 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 16842 \nu^{7} + 23709 \nu^{6} - 381320 \nu^{5} + 2465055 \nu^{4} + 42788326 \nu^{3} + \cdots + 383141892 ) / 50783831 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 29366 \nu^{7} + 102774 \nu^{6} - 2399398 \nu^{5} - 7654753 \nu^{4} + 68704097 \nu^{3} + \cdots - 2298466170 ) / 50783831 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 99621 \nu^{7} - 351610 \nu^{6} - 5741474 \nu^{5} + 13247636 \nu^{4} + 153295331 \nu^{3} + \cdots + 438625096 ) / 50783831 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 199363 \nu^{7} + 694834 \nu^{6} + 11495970 \nu^{5} - 32462084 \nu^{4} - 295067311 \nu^{3} + \cdots - 1159464530 ) / 50783831 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} - 4\beta_{5} + 4\beta_{3} + 49\beta_{2} + 49 ) / 49 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -10\beta_{7} - 18\beta_{6} + 2\beta_{5} + 12\beta_{3} + 49\beta_{2} + 42\beta _1 + 931 ) / 49 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -73\beta_{7} + 31\beta_{6} - 369\beta_{5} + 28\beta_{4} + 390\beta_{3} + 875\beta_{2} + 49\beta _1 + 1799 ) / 49 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -628\beta_{7} - 940\beta_{6} - 608\beta_{5} + 56\beta_{4} + 804\beta_{3} + 1701\beta_{2} + 609\beta _1 + 11389 ) / 49 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 3401 \beta_{7} - 449 \beta_{6} - 16152 \beta_{5} - 203 \beta_{4} + 16607 \beta_{3} + 406 \beta_{2} + \cdots + 24136 ) / 49 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 3468 \beta_{7} - 4462 \beta_{6} - 6394 \beta_{5} - 107 \beta_{4} + 6404 \beta_{3} - 430 \beta_{2} + \cdots - 43168 ) / 7 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 117804 \beta_{7} - 63055 \beta_{6} - 476263 \beta_{5} - 64876 \beta_{4} + 474940 \beta_{3} + \cdots - 1644783 ) / 49 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 48.1 |
|
−6.99086 | − | 7.89087i | 32.8721 | 22.0095i | 55.1639i | 0 | −117.950 | 18.7341 | − | 153.865i | ||||||||||||||||||||||||||||||||||||||||
| 48.2 | −6.99086 | 7.89087i | 32.8721 | − | 22.0095i | − | 55.1639i | 0 | −117.950 | 18.7341 | 153.865i | |||||||||||||||||||||||||||||||||||||||||
| 48.3 | −4.71722 | − | 12.5695i | 6.25217 | − | 33.6557i | 59.2933i | 0 | 45.9827 | −76.9935 | 158.761i | |||||||||||||||||||||||||||||||||||||||||
| 48.4 | −4.71722 | 12.5695i | 6.25217 | 33.6557i | − | 59.2933i | 0 | 45.9827 | −76.9935 | − | 158.761i | |||||||||||||||||||||||||||||||||||||||||
| 48.5 | 1.71722 | − | 16.1337i | −13.0512 | 7.83726i | − | 27.7052i | 0 | −49.8872 | −179.298 | 13.4583i | |||||||||||||||||||||||||||||||||||||||||
| 48.6 | 1.71722 | 16.1337i | −13.0512 | − | 7.83726i | 27.7052i | 0 | −49.8872 | −179.298 | − | 13.4583i | |||||||||||||||||||||||||||||||||||||||||
| 48.7 | 3.99086 | − | 4.40939i | −0.0730718 | − | 43.7887i | − | 17.5972i | 0 | −64.1453 | 61.5573 | − | 174.754i | |||||||||||||||||||||||||||||||||||||||
| 48.8 | 3.99086 | 4.40939i | −0.0730718 | 43.7887i | 17.5972i | 0 | −64.1453 | 61.5573 | 174.754i | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 49.5.b.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 441.5.d.g | 8 | ||
| 4.b | odd | 2 | 1 | 784.5.c.g | 8 | ||
| 7.b | odd | 2 | 1 | inner | 49.5.b.b | ✓ | 8 |
| 7.c | even | 3 | 2 | 49.5.d.c | 16 | ||
| 7.d | odd | 6 | 2 | 49.5.d.c | 16 | ||
| 21.c | even | 2 | 1 | 441.5.d.g | 8 | ||
| 28.d | even | 2 | 1 | 784.5.c.g | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 49.5.b.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 49.5.b.b | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 49.5.d.c | 16 | 7.c | even | 3 | 2 | ||
| 49.5.d.c | 16 | 7.d | odd | 6 | 2 | ||
| 441.5.d.g | 8 | 3.b | odd | 2 | 1 | ||
| 441.5.d.g | 8 | 21.c | even | 2 | 1 | ||
| 784.5.c.g | 8 | 4.b | odd | 2 | 1 | ||
| 784.5.c.g | 8 | 28.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 6T_{2}^{3} - 27T_{2}^{2} - 108T_{2} + 226 \)
acting on \(S_{5}^{\mathrm{new}}(49, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 6 T^{3} + \cdots + 226)^{2} \)
|
| $3$ |
\( T^{8} + 500 T^{6} + \cdots + 49787136 \)
|
| $5$ |
\( T^{8} + \cdots + 64623740944 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{4} - 60 T^{3} + \cdots - 27217952)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 11\!\cdots\!16 \)
|
| $17$ |
\( T^{8} + \cdots + 2775229473604 \)
|
| $19$ |
\( T^{8} + \cdots + 56\!\cdots\!56 \)
|
| $23$ |
\( (T^{4} - 876 T^{3} + \cdots - 38047796288)^{2} \)
|
| $29$ |
\( (T^{4} - 624 T^{3} + \cdots - 124444104944)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 16\!\cdots\!24 \)
|
| $37$ |
\( (T^{4} - 1184 T^{3} + \cdots + 374530303744)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 51\!\cdots\!16 \)
|
| $43$ |
\( (T^{4} + \cdots - 4296135337184)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 52\!\cdots\!64 \)
|
| $53$ |
\( (T^{4} + \cdots + 4515238365184)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 85\!\cdots\!36 \)
|
| $61$ |
\( T^{8} + \cdots + 24\!\cdots\!36 \)
|
| $67$ |
\( (T^{4} + 3720 T^{3} + \cdots + 327162240000)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 29541809698816)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 19\!\cdots\!16 \)
|
| $79$ |
\( (T^{4} + \cdots + 166340550164224)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 10\!\cdots\!96 \)
|
| $89$ |
\( T^{8} + \cdots + 43\!\cdots\!96 \)
|
| $97$ |
\( T^{8} + \cdots + 57\!\cdots\!76 \)
|
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