Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [49,9,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, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("49.48");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 49 = 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| 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: | \(19.9615518930\) |
| 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: |
\( x^{8} + 592x^{6} - 1176x^{5} + 336397x^{4} - 348096x^{3} + 8673408x^{2} + 8271396x + 197880489 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3\cdot 7^{6} \) |
| Twist minimal: | no (minimal twist has level 7) |
| 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
\( x^{8} + 592x^{6} - 1176x^{5} + 336397x^{4} - 348096x^{3} + 8673408x^{2} + 8271396x + 197880489 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 199147024 \nu^{7} - 206072832 \nu^{6} - 113162941609 \nu^{5} + 117098450112 \nu^{4} + \cdots - 17\!\cdots\!48 ) / 16\!\cdots\!75 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 405219856 \nu^{7} + 4526023767 \nu^{6} - 230261391721 \nu^{5} + 238269275328 \nu^{4} + \cdots - 45\!\cdots\!12 ) / 16\!\cdots\!75 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3985173807104 \nu^{7} - 91512438762528 \nu^{6} + \cdots - 37\!\cdots\!17 ) / 78\!\cdots\!75 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1609557045712 \nu^{7} - 15327848392416 \nu^{6} - 914611759301317 \nu^{5} + \cdots - 73\!\cdots\!99 ) / 26\!\cdots\!25 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 134480550344854 \nu^{7} + 598053298676697 \nu^{6} + \cdots + 30\!\cdots\!83 ) / 10\!\cdots\!50 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 300217716573334 \nu^{7} + \cdots - 12\!\cdots\!82 ) / 54\!\cdots\!25 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 479692272 \nu^{7} + 518353975 \nu^{6} + 272579461527 \nu^{5} - 282059055936 \nu^{4} + \cdots + 53\!\cdots\!35 ) / 14787522027926 \)
|
| \(\nu\) | \(=\) |
\( ( -7\beta_{4} + 3\beta_{3} + 49\beta_1 ) / 98 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 49\beta_{6} - 98\beta_{5} - 303\beta_{3} - 49\beta_{2} + 49\beta _1 - 14504 ) / 98 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{7} + \beta_{2} - 544\beta _1 + 441 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -29008\beta_{6} + 58016\beta_{5} - 4116\beta_{4} + 167073\beta_{3} - 29008\beta_{2} + 57820\beta _1 - 7897085 ) / 98 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 58016 \beta_{7} + 115836 \beta_{6} + 986664 \beta_{5} + 2143435 \beta_{4} - 1196235 \beta_{3} + \cdots - 21320880 ) / 98 \)
|
| \(\nu^{6}\) | \(=\) |
\( -1176\beta_{7} + 336985\beta_{2} - 1004365\beta _1 + 91505156 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 32966906 \beta_{7} - 83563767 \beta_{6} - 525177492 \beta_{5} - 1218061999 \beta_{4} + \cdots - 16961473137 ) / 98 \)
|
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 |
|
−22.2529 | − | 36.0443i | 239.191 | − | 352.395i | 802.090i | 0 | 374.058 | 5261.81 | 7841.81i | ||||||||||||||||||||||||||||||||||||||||
| 48.2 | −22.2529 | 36.0443i | 239.191 | 352.395i | − | 802.090i | 0 | 374.058 | 5261.81 | − | 7841.81i | |||||||||||||||||||||||||||||||||||||||||
| 48.3 | −4.55832 | − | 143.663i | −235.222 | 188.223i | 654.862i | 0 | 2239.15 | −14078.1 | − | 857.979i | |||||||||||||||||||||||||||||||||||||||||
| 48.4 | −4.55832 | 143.663i | −235.222 | − | 188.223i | − | 654.862i | 0 | 2239.15 | −14078.1 | 857.979i | |||||||||||||||||||||||||||||||||||||||||
| 48.5 | 5.47166 | − | 67.9930i | −226.061 | 677.279i | − | 372.035i | 0 | −2637.67 | 1937.95 | 3705.84i | |||||||||||||||||||||||||||||||||||||||||
| 48.6 | 5.47166 | 67.9930i | −226.061 | − | 677.279i | 372.035i | 0 | −2637.67 | 1937.95 | − | 3705.84i | |||||||||||||||||||||||||||||||||||||||||
| 48.7 | 25.3395 | − | 8.87176i | 386.092 | 621.635i | − | 224.806i | 0 | 3296.47 | 6482.29 | 15752.0i | |||||||||||||||||||||||||||||||||||||||||
| 48.8 | 25.3395 | 8.87176i | 386.092 | − | 621.635i | 224.806i | 0 | 3296.47 | 6482.29 | − | 15752.0i | |||||||||||||||||||||||||||||||||||||||||
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 | 49.9.b.a | 8 | |
| 7.b | odd | 2 | 1 | inner | 49.9.b.a | 8 | |
| 7.c | even | 3 | 1 | 7.9.d.a | ✓ | 8 | |
| 7.c | even | 3 | 1 | 49.9.d.c | 8 | ||
| 7.d | odd | 6 | 1 | 7.9.d.a | ✓ | 8 | |
| 7.d | odd | 6 | 1 | 49.9.d.c | 8 | ||
| 21.g | even | 6 | 1 | 63.9.m.b | 8 | ||
| 21.h | odd | 6 | 1 | 63.9.m.b | 8 | ||
| 28.f | even | 6 | 1 | 112.9.s.a | 8 | ||
| 28.g | odd | 6 | 1 | 112.9.s.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.9.d.a | ✓ | 8 | 7.c | even | 3 | 1 | |
| 7.9.d.a | ✓ | 8 | 7.d | odd | 6 | 1 | |
| 49.9.b.a | 8 | 1.a | even | 1 | 1 | trivial | |
| 49.9.b.a | 8 | 7.b | odd | 2 | 1 | inner | |
| 49.9.d.c | 8 | 7.c | even | 3 | 1 | ||
| 49.9.d.c | 8 | 7.d | odd | 6 | 1 | ||
| 63.9.m.b | 8 | 21.g | even | 6 | 1 | ||
| 63.9.m.b | 8 | 21.h | odd | 6 | 1 | ||
| 112.9.s.a | 8 | 28.f | even | 6 | 1 | ||
| 112.9.s.a | 8 | 28.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 4T_{2}^{3} - 586T_{2}^{2} + 592T_{2} + 14064 \)
acting on \(S_{9}^{\mathrm{new}}(49, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 4 T^{3} + \cdots + 14064)^{2} \)
|
| $3$ |
\( T^{8} + \cdots + 9756895701609 \)
|
| $5$ |
\( T^{8} + \cdots + 77\!\cdots\!25 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{4} + \cdots + 62\!\cdots\!23)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 13\!\cdots\!36 \)
|
| $17$ |
\( T^{8} + \cdots + 21\!\cdots\!69 \)
|
| $19$ |
\( T^{8} + \cdots + 42\!\cdots\!01 \)
|
| $23$ |
\( (T^{4} + \cdots - 99\!\cdots\!89)^{2} \)
|
| $29$ |
\( (T^{4} + \cdots - 15\!\cdots\!56)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 38\!\cdots\!09 \)
|
| $37$ |
\( (T^{4} + \cdots + 68\!\cdots\!21)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 18\!\cdots\!00 \)
|
| $43$ |
\( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 20\!\cdots\!89 \)
|
| $53$ |
\( (T^{4} + \cdots + 31\!\cdots\!25)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 20\!\cdots\!61 \)
|
| $61$ |
\( T^{8} + \cdots + 25\!\cdots\!81 \)
|
| $67$ |
\( (T^{4} + \cdots - 20\!\cdots\!93)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 63\!\cdots\!04)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 63\!\cdots\!01 \)
|
| $79$ |
\( (T^{4} + \cdots - 11\!\cdots\!17)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 44\!\cdots\!16 \)
|
| $89$ |
\( T^{8} + \cdots + 31\!\cdots\!49 \)
|
| $97$ |
\( T^{8} + \cdots + 20\!\cdots\!64 \)
|
| show more | |
| show less |