Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [10,15,Mod(3,10)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(10, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3]))
N = Newforms(chi, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("10.3");
S:= CuspForms(chi, 15);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 10 = 2 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 15 \) |
| Character orbit: | \([\chi]\) | \(=\) | 10.c (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: | \(12.4328968152\) |
| 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: |
\( x^{8} + 513401x^{6} + 81983771116x^{4} + 4511941511282436x^{2} + 65023716741123799296 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{14}\cdot 3^{4}\cdot 5^{9} \) |
| 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
\( x^{8} + 513401x^{6} + 81983771116x^{4} + 4511941511282436x^{2} + 65023716741123799296 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 9772303 \nu^{7} + 5013078268271 \nu^{5} + \cdots + 23\!\cdots\!52 \nu ) / 15\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1104024388163 \nu^{7} - 71452148119836 \nu^{6} + \cdots - 29\!\cdots\!04 ) / 63\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1104024388163 \nu^{7} + 71452148119836 \nu^{6} + \cdots + 29\!\cdots\!04 ) / 63\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 4407175440013 \nu^{7} + \cdots - 77\!\cdots\!84 ) / 42\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1104160093943 \nu^{7} - 18699648858099 \nu^{6} + \cdots - 20\!\cdots\!86 ) / 26\!\cdots\!50 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 30820479348545 \nu^{7} + \cdots + 24\!\cdots\!72 ) / 42\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1328110165567 \nu^{7} - 135051672968520 \nu^{6} + \cdots - 50\!\cdots\!80 ) / 51\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} - 3\beta_{6} + 2\beta_{5} - 16\beta_{4} - 164\beta_{3} - 158\beta_{2} + 1661\beta _1 - 161 ) / 6000 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 27 \beta_{7} - 11 \beta_{6} + 124 \beta_{5} + 218 \beta_{4} + 25622 \beta_{3} - 25796 \beta_{2} + \cdots - 154020387 ) / 1200 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 253279 \beta_{7} + 502737 \beta_{6} + 521842 \beta_{5} + 2766964 \beta_{4} + 60221756 \beta_{3} + \cdots + 59976119 ) / 6000 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 18604191 \beta_{7} - 1084537 \beta_{6} - 94105492 \beta_{5} - 63571394 \beta_{4} + \cdots + 29883947325471 ) / 1200 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 55897578691 \beta_{7} - 107558290773 \beta_{6} - 128742623818 \beta_{5} - 593689032556 \beta_{4} + \cdots - 16613136171851 ) / 6000 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 6518835619071 \beta_{7} + 1355325485497 \beta_{6} + 33949503580852 \beta_{5} + 12522087621314 \beta_{4} + \cdots - 67\!\cdots\!51 ) / 1200 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 12\!\cdots\!39 \beta_{7} + \cdots + 44\!\cdots\!79 ) / 6000 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).
| \(n\) | \(7\) |
| \(\chi(n)\) | \(-\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
64.0000 | − | 64.0000i | −2738.23 | − | 2738.23i | − | 8192.00i | −76838.8 | − | 14117.8i | −350493. | 390848. | − | 390848.i | −524288. | − | 524288.i | 1.02128e7i | −5.82123e6 | + | 4.01414e6i | |||||||||||||||||||||||||||||
| 3.2 | 64.0000 | − | 64.0000i | −423.414 | − | 423.414i | − | 8192.00i | 72594.1 | + | 28872.4i | −54197.0 | 346999. | − | 346999.i | −524288. | − | 524288.i | − | 4.42441e6i | 6.49385e6 | − | 2.79819e6i | |||||||||||||||||||||||||||||
| 3.3 | 64.0000 | − | 64.0000i | 803.510 | + | 803.510i | − | 8192.00i | −66844.3 | + | 40439.5i | 102849. | −657264. | + | 657264.i | −524288. | − | 524288.i | − | 3.49171e6i | −1.68991e6 | + | 6.86616e6i | |||||||||||||||||||||||||||||
| 3.4 | 64.0000 | − | 64.0000i | 3060.13 | + | 3060.13i | − | 8192.00i | 20659.0 | − | 75344.0i | 391697. | 586054. | − | 586054.i | −524288. | − | 524288.i | 1.39459e7i | −3.49984e6 | − | 6.14419e6i | ||||||||||||||||||||||||||||||
| 7.1 | 64.0000 | + | 64.0000i | −2738.23 | + | 2738.23i | 8192.00i | −76838.8 | + | 14117.8i | −350493. | 390848. | + | 390848.i | −524288. | + | 524288.i | − | 1.02128e7i | −5.82123e6 | − | 4.01414e6i | ||||||||||||||||||||||||||||||
| 7.2 | 64.0000 | + | 64.0000i | −423.414 | + | 423.414i | 8192.00i | 72594.1 | − | 28872.4i | −54197.0 | 346999. | + | 346999.i | −524288. | + | 524288.i | 4.42441e6i | 6.49385e6 | + | 2.79819e6i | |||||||||||||||||||||||||||||||
| 7.3 | 64.0000 | + | 64.0000i | 803.510 | − | 803.510i | 8192.00i | −66844.3 | − | 40439.5i | 102849. | −657264. | − | 657264.i | −524288. | + | 524288.i | 3.49171e6i | −1.68991e6 | − | 6.86616e6i | |||||||||||||||||||||||||||||||
| 7.4 | 64.0000 | + | 64.0000i | 3060.13 | − | 3060.13i | 8192.00i | 20659.0 | + | 75344.0i | 391697. | 586054. | + | 586054.i | −524288. | + | 524288.i | − | 1.39459e7i | −3.49984e6 | + | 6.14419e6i | ||||||||||||||||||||||||||||||
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 | 10.15.c.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 90.15.g.b | 8 | ||
| 4.b | odd | 2 | 1 | 80.15.p.b | 8 | ||
| 5.b | even | 2 | 1 | 50.15.c.e | 8 | ||
| 5.c | odd | 4 | 1 | inner | 10.15.c.b | ✓ | 8 |
| 5.c | odd | 4 | 1 | 50.15.c.e | 8 | ||
| 15.e | even | 4 | 1 | 90.15.g.b | 8 | ||
| 20.e | even | 4 | 1 | 80.15.p.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.15.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 10.15.c.b | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
| 50.15.c.e | 8 | 5.b | even | 2 | 1 | ||
| 50.15.c.e | 8 | 5.c | odd | 4 | 1 | ||
| 80.15.p.b | 8 | 4.b | odd | 2 | 1 | ||
| 80.15.p.b | 8 | 20.e | even | 4 | 1 | ||
| 90.15.g.b | 8 | 3.b | odd | 2 | 1 | ||
| 90.15.g.b | 8 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 1404 T_{3}^{7} + 985608 T_{3}^{6} + 10963094040 T_{3}^{5} + 272841733510596 T_{3}^{4} + \cdots + 13\!\cdots\!36 \)
acting on \(S_{15}^{\mathrm{new}}(10, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 128 T + 8192)^{4} \)
|
| $3$ |
\( T^{8} + \cdots + 13\!\cdots\!36 \)
|
| $5$ |
\( T^{8} + \cdots + 13\!\cdots\!25 \)
|
| $7$ |
\( T^{8} + \cdots + 43\!\cdots\!76 \)
|
| $11$ |
\( (T^{4} + \cdots - 11\!\cdots\!24)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 12\!\cdots\!56 \)
|
| $17$ |
\( T^{8} + \cdots + 77\!\cdots\!76 \)
|
| $19$ |
\( T^{8} + \cdots + 34\!\cdots\!00 \)
|
| $23$ |
\( T^{8} + \cdots + 77\!\cdots\!56 \)
|
| $29$ |
\( T^{8} + \cdots + 45\!\cdots\!00 \)
|
| $31$ |
\( (T^{4} + \cdots + 17\!\cdots\!16)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 11\!\cdots\!16 \)
|
| $41$ |
\( (T^{4} + \cdots - 10\!\cdots\!24)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 12\!\cdots\!96 \)
|
| $47$ |
\( T^{8} + \cdots + 36\!\cdots\!56 \)
|
| $53$ |
\( T^{8} + \cdots + 10\!\cdots\!96 \)
|
| $59$ |
\( T^{8} + \cdots + 15\!\cdots\!00 \)
|
| $61$ |
\( (T^{4} + \cdots + 29\!\cdots\!16)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 45\!\cdots\!16 \)
|
| $71$ |
\( (T^{4} + \cdots - 18\!\cdots\!04)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 20\!\cdots\!16 \)
|
| $79$ |
\( T^{8} + \cdots + 45\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 55\!\cdots\!56 \)
|
| $89$ |
\( T^{8} + \cdots + 19\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 25\!\cdots\!96 \)
|
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