Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [40,7,Mod(17,40)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("40.17");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 40 = 2^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 40.l (of order \(4\), degree \(2\), 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: | \(9.20216334479\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 136x^{6} + 5304x^{4} + 56644x^{2} + 115600 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 5^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} + 136x^{6} + 5304x^{4} + 56644x^{2} + 115600 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} - 136\nu^{5} - 4964\nu^{3} - 33524\nu ) / 34680 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 11 \nu^{7} - 425 \nu^{6} + 2346 \nu^{5} - 45900 \nu^{4} + 155754 \nu^{3} - 1213800 \nu^{2} + \cdots - 6161480 ) / 52020 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11 \nu^{7} + 425 \nu^{6} + 2346 \nu^{5} + 45900 \nu^{4} + 155754 \nu^{3} + 1213800 \nu^{2} + \cdots + 6161480 ) / 52020 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 40\nu^{7} + 17\nu^{6} + 4998\nu^{5} + 1020\nu^{4} + 166770\nu^{3} - 48552\nu^{2} + 1204552\nu - 1433440 ) / 52020 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -40\nu^{7} + 17\nu^{6} - 4998\nu^{5} + 1020\nu^{4} - 166770\nu^{3} - 48552\nu^{2} - 1204552\nu - 1433440 ) / 52020 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11 \nu^{7} + 425 \nu^{6} - 2346 \nu^{5} + 56100 \nu^{4} - 155754 \nu^{3} + 1907400 \nu^{2} + \cdots + 9629480 ) / 52020 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 11 \nu^{7} - 425 \nu^{6} - 2346 \nu^{5} - 56100 \nu^{4} - 155754 \nu^{3} - 1907400 \nu^{2} + \cdots - 9629480 ) / 52020 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_{3} + \beta_{2} ) / 40 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{7} - 2\beta_{6} - 25\beta_{5} - 25\beta_{4} + 3\beta_{3} - 3\beta_{2} - 1348 ) / 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -35\beta_{7} - 35\beta_{6} - 25\beta_{5} + 25\beta_{4} - 22\beta_{3} - 22\beta_{2} + 1524\beta_1 ) / 20 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -119\beta_{7} + 119\beta_{6} + 850\beta_{5} + 850\beta_{4} - 153\beta_{3} + 153\beta_{2} + 39032 ) / 20 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2363\beta_{7} + 2363\beta_{6} + 2975\beta_{5} - 2975\beta_{4} + 1428\beta_{3} + 1428\beta_{2} - 172380\beta_1 ) / 20 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 2499\beta_{7} - 2499\beta_{6} - 14025\beta_{5} - 14025\beta_{4} + 3366\beta_{3} - 3366\beta_{2} - 645116 ) / 5 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 82195 \beta_{7} - 82195 \beta_{6} - 140250 \beta_{5} + 140250 \beta_{4} - 50881 \beta_{3} + \cdots + 7592472 \beta_1 ) / 10 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(21\) | \(31\) |
| \(\chi(n)\) | \(-\beta_{1}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | −26.9421 | + | 26.9421i | 0 | −101.108 | − | 73.4995i | 0 | 1.66641 | + | 1.66641i | 0 | − | 722.751i | 0 | |||||||||||||||||||||||||||||||||||
| 17.2 | 0 | −15.1555 | + | 15.1555i | 0 | 117.905 | + | 41.5133i | 0 | −90.5475 | − | 90.5475i | 0 | 269.619i | 0 | |||||||||||||||||||||||||||||||||||||
| 17.3 | 0 | 12.4737 | − | 12.4737i | 0 | −57.4843 | + | 110.998i | 0 | 181.956 | + | 181.956i | 0 | 417.813i | 0 | |||||||||||||||||||||||||||||||||||||
| 17.4 | 0 | 22.6239 | − | 22.6239i | 0 | 15.6870 | − | 124.012i | 0 | −116.075 | − | 116.075i | 0 | − | 294.682i | 0 | ||||||||||||||||||||||||||||||||||||
| 33.1 | 0 | −26.9421 | − | 26.9421i | 0 | −101.108 | + | 73.4995i | 0 | 1.66641 | − | 1.66641i | 0 | 722.751i | 0 | |||||||||||||||||||||||||||||||||||||
| 33.2 | 0 | −15.1555 | − | 15.1555i | 0 | 117.905 | − | 41.5133i | 0 | −90.5475 | + | 90.5475i | 0 | − | 269.619i | 0 | ||||||||||||||||||||||||||||||||||||
| 33.3 | 0 | 12.4737 | + | 12.4737i | 0 | −57.4843 | − | 110.998i | 0 | 181.956 | − | 181.956i | 0 | − | 417.813i | 0 | ||||||||||||||||||||||||||||||||||||
| 33.4 | 0 | 22.6239 | + | 22.6239i | 0 | 15.6870 | + | 124.012i | 0 | −116.075 | + | 116.075i | 0 | 294.682i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 40.7.l.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 360.7.v.a | 8 | ||
| 4.b | odd | 2 | 1 | 80.7.p.e | 8 | ||
| 5.b | even | 2 | 1 | 200.7.l.e | 8 | ||
| 5.c | odd | 4 | 1 | inner | 40.7.l.a | ✓ | 8 |
| 5.c | odd | 4 | 1 | 200.7.l.e | 8 | ||
| 15.e | even | 4 | 1 | 360.7.v.a | 8 | ||
| 20.e | even | 4 | 1 | 80.7.p.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.7.l.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 40.7.l.a | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
| 80.7.p.e | 8 | 4.b | odd | 2 | 1 | ||
| 80.7.p.e | 8 | 20.e | even | 4 | 1 | ||
| 200.7.l.e | 8 | 5.b | even | 2 | 1 | ||
| 200.7.l.e | 8 | 5.c | odd | 4 | 1 | ||
| 360.7.v.a | 8 | 3.b | odd | 2 | 1 | ||
| 360.7.v.a | 8 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 14 T_{3}^{7} + 98 T_{3}^{6} - 12232 T_{3}^{5} + 1555636 T_{3}^{4} + 8978616 T_{3}^{3} + \cdots + 212447246400 \)
acting on \(S_{7}^{\mathrm{new}}(40, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 212447246400 \)
|
| $5$ |
\( T^{8} + \cdots + 59\!\cdots\!25 \)
|
| $7$ |
\( T^{8} + \cdots + 162497022594624 \)
|
| $11$ |
\( (T^{4} + \cdots + 3677841205184)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 30\!\cdots\!24 \)
|
| $17$ |
\( T^{8} + \cdots + 43\!\cdots\!00 \)
|
| $19$ |
\( T^{8} + \cdots + 53\!\cdots\!36 \)
|
| $23$ |
\( T^{8} + \cdots + 36\!\cdots\!44 \)
|
| $29$ |
\( T^{8} + \cdots + 38\!\cdots\!36 \)
|
| $31$ |
\( (T^{4} + \cdots - 49\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 32\!\cdots\!56 \)
|
| $41$ |
\( (T^{4} + \cdots - 14\!\cdots\!48)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 22\!\cdots\!04 \)
|
| $47$ |
\( T^{8} + \cdots + 44\!\cdots\!36 \)
|
| $53$ |
\( T^{8} + \cdots + 69\!\cdots\!36 \)
|
| $59$ |
\( T^{8} + \cdots + 24\!\cdots\!04 \)
|
| $61$ |
\( (T^{4} + \cdots - 65\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 64\!\cdots\!64 \)
|
| $71$ |
\( (T^{4} + \cdots - 76\!\cdots\!60)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 23\!\cdots\!04 \)
|
| $79$ |
\( T^{8} + \cdots + 20\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 57\!\cdots\!04 \)
|
| $89$ |
\( T^{8} + \cdots + 21\!\cdots\!76 \)
|
| $97$ |
\( T^{8} + \cdots + 50\!\cdots\!96 \)
|
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