Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [40,8,Mod(9,40)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(40, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("40.9");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 40 = 2^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 40.c (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: | \(12.4954010194\) |
| 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: |
\( x^{8} + 196x^{6} + 7674x^{4} + 75204x^{2} + 18225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{38}\cdot 5^{3} \) |
| 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
\( x^{8} + 196x^{6} + 7674x^{4} + 75204x^{2} + 18225 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -241\nu^{7} - 47839\nu^{5} - 1942269\nu^{3} - 19327851\nu ) / 121230 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 544\nu^{7} + 106912\nu^{5} + 4276896\nu^{3} + 47739168\nu ) / 181845 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1034 \nu^{7} + 666 \nu^{6} + 194654 \nu^{5} + 115200 \nu^{4} + 6546126 \nu^{3} + 2212434 \nu^{2} + \cdots - 16016940 ) / 181845 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1034 \nu^{7} - 20070 \nu^{6} - 194654 \nu^{5} - 3366720 \nu^{4} - 6546126 \nu^{3} + \cdots - 116845740 ) / 181845 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3413 \nu^{7} - 10656 \nu^{6} + 635099 \nu^{5} - 1843200 \nu^{4} + 20357697 \nu^{3} + \cdots + 256271040 ) / 363690 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 1034 \nu^{7} - 15714 \nu^{6} - 194654 \nu^{5} - 3032640 \nu^{4} - 6546126 \nu^{3} + \cdots - 674861220 ) / 181845 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 5984 \nu^{7} - 666 \nu^{6} - 1143704 \nu^{5} - 115200 \nu^{4} - 40661076 \nu^{3} + \cdots + 16016940 ) / 181845 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{7} + \beta_{5} + 11\beta_{3} - 5\beta_{2} - 21\beta_1 ) / 640 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{6} + 11\beta_{5} - 3\beta_{4} - 26\beta_{3} - 11\beta _1 - 15680 ) / 320 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -303\beta_{7} - 117\beta_{5} - 1239\beta_{3} + 2085\beta_{2} + 4057\beta_1 ) / 640 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{6} - 1628\beta_{5} + 558\beta_{4} + 3815\beta_{3} + 1628\beta _1 + 1845440 ) / 320 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 41793\beta_{7} + 15531\beta_{5} + 166041\beta_{3} - 336105\beta_{2} - 649351\beta_1 ) / 640 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 3149\beta_{6} + 236321\beta_{5} - 86553\beta_{4} - 556046\beta_{3} - 236321\beta _1 - 259426880 ) / 320 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -6094653\beta_{7} - 2220207\beta_{5} - 23856309\beta_{3} + 50315085\beta_{2} + 97563547\beta_1 ) / 640 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).
| \(n\) | \(17\) | \(21\) | \(31\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 9.1 |
|
0 | − | 77.5195i | 0 | −184.066 | + | 210.344i | 0 | − | 1384.40i | 0 | −3822.28 | 0 | ||||||||||||||||||||||||||||||||||||||
| 9.2 | 0 | − | 65.0953i | 0 | 61.1163 | + | 272.745i | 0 | 1567.21i | 0 | −2050.39 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 9.3 | 0 | − | 52.5319i | 0 | −14.9454 | − | 279.109i | 0 | − | 338.895i | 0 | −572.598 | 0 | |||||||||||||||||||||||||||||||||||||||
| 9.4 | 0 | − | 8.10762i | 0 | −234.105 | − | 152.708i | 0 | 980.714i | 0 | 2121.27 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 9.5 | 0 | 8.10762i | 0 | −234.105 | + | 152.708i | 0 | − | 980.714i | 0 | 2121.27 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 9.6 | 0 | 52.5319i | 0 | −14.9454 | + | 279.109i | 0 | 338.895i | 0 | −572.598 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 9.7 | 0 | 65.0953i | 0 | 61.1163 | − | 272.745i | 0 | − | 1567.21i | 0 | −2050.39 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 9.8 | 0 | 77.5195i | 0 | −184.066 | − | 210.344i | 0 | 1384.40i | 0 | −3822.28 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 40.8.c.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 360.8.f.b | 8 | ||
| 4.b | odd | 2 | 1 | 80.8.c.e | 8 | ||
| 5.b | even | 2 | 1 | inner | 40.8.c.b | ✓ | 8 |
| 5.c | odd | 4 | 1 | 200.8.a.q | 4 | ||
| 5.c | odd | 4 | 1 | 200.8.a.r | 4 | ||
| 8.b | even | 2 | 1 | 320.8.c.k | 8 | ||
| 8.d | odd | 2 | 1 | 320.8.c.l | 8 | ||
| 15.d | odd | 2 | 1 | 360.8.f.b | 8 | ||
| 20.d | odd | 2 | 1 | 80.8.c.e | 8 | ||
| 20.e | even | 4 | 1 | 400.8.a.bj | 4 | ||
| 20.e | even | 4 | 1 | 400.8.a.bl | 4 | ||
| 40.e | odd | 2 | 1 | 320.8.c.l | 8 | ||
| 40.f | even | 2 | 1 | 320.8.c.k | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.8.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 40.8.c.b | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 80.8.c.e | 8 | 4.b | odd | 2 | 1 | ||
| 80.8.c.e | 8 | 20.d | odd | 2 | 1 | ||
| 200.8.a.q | 4 | 5.c | odd | 4 | 1 | ||
| 200.8.a.r | 4 | 5.c | odd | 4 | 1 | ||
| 320.8.c.k | 8 | 8.b | even | 2 | 1 | ||
| 320.8.c.k | 8 | 40.f | even | 2 | 1 | ||
| 320.8.c.l | 8 | 8.d | odd | 2 | 1 | ||
| 320.8.c.l | 8 | 40.e | odd | 2 | 1 | ||
| 360.8.f.b | 8 | 3.b | odd | 2 | 1 | ||
| 360.8.f.b | 8 | 15.d | odd | 2 | 1 | ||
| 400.8.a.bj | 4 | 20.e | even | 4 | 1 | ||
| 400.8.a.bl | 4 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 13072T_{3}^{6} + 54595296T_{3}^{4} + 73802016000T_{3}^{2} + 4619060640000 \)
acting on \(S_{8}^{\mathrm{new}}(40, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 4619060640000 \)
|
| $5$ |
\( T^{8} + \cdots + 37\!\cdots\!25 \)
|
| $7$ |
\( T^{8} + \cdots + 51\!\cdots\!36 \)
|
| $11$ |
\( (T^{4} + \cdots + 6076238622976)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 48\!\cdots\!56 \)
|
| $17$ |
\( T^{8} + \cdots + 67\!\cdots\!00 \)
|
| $19$ |
\( (T^{4} + \cdots - 50\!\cdots\!84)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 22\!\cdots\!16 \)
|
| $29$ |
\( (T^{4} + \cdots + 35\!\cdots\!96)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots - 11\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 28\!\cdots\!56 \)
|
| $41$ |
\( (T^{4} + \cdots + 36\!\cdots\!64)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 12\!\cdots\!76 \)
|
| $47$ |
\( T^{8} + \cdots + 14\!\cdots\!96 \)
|
| $53$ |
\( T^{8} + \cdots + 22\!\cdots\!56 \)
|
| $59$ |
\( (T^{4} + \cdots - 15\!\cdots\!96)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots - 10\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 11\!\cdots\!76 \)
|
| $71$ |
\( (T^{4} + \cdots - 19\!\cdots\!00)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 79\!\cdots\!16 \)
|
| $79$ |
\( (T^{4} + \cdots - 17\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 72\!\cdots\!56 \)
|
| $89$ |
\( (T^{4} + \cdots - 25\!\cdots\!84)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 55\!\cdots\!36 \)
|
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