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: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(i)\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} - 11690x^{3} + 819025x^{2} - 12217500x + 91125000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{2}\cdot 5^{8} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 2x^{4} - 11690x^{3} + 819025x^{2} - 12217500x + 91125000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 27556 \nu^{5} - 148963 \nu^{4} - 962087 \nu^{3} + 137033615 \nu^{2} - 21376234525 \nu + 177209808750 ) / 162210633750 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -222\nu^{5} - 24026\nu^{4} - 201354\nu^{3} + 1297590\nu^{2} + 2997000\nu - 9607187345 ) / 1044835 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 43521437 \nu^{5} + 619265974 \nu^{4} + 34470074426 \nu^{3} + 991491950230 \nu^{2} + \cdots + 524538144697500 ) / 162210633750 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 91365437 \nu^{5} + 1381798526 \nu^{4} + 8924433574 \nu^{3} - 1271140130230 \nu^{2} + \cdots - 139134133350000 ) / 162210633750 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 325457492 \nu^{5} + 1767216041 \nu^{4} + 4244146309 \nu^{3} - 4855565916805 \nu^{2} + \cdots - 20\!\cdots\!50 ) / 162210633750 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - 4\beta_{4} - \beta_{2} - 201\beta _1 + 201 ) / 600 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -17\beta_{5} + 4\beta_{4} - 4\beta_{3} - 181203\beta_1 ) / 300 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 871\beta_{5} + 3604\beta_{3} + 871\beta_{2} + 3505689\beta _1 + 3505689 ) / 600 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2553\beta_{4} - 2553\beta_{3} - 3544\beta_{2} - 26521046 ) / 50 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -975227\beta_{5} + 3308348\beta_{4} + 975227\beta_{2} - 5300636493\beta _1 + 5300636493 ) / 600 \)
|
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 | −1760.67 | − | 1760.67i | − | 8192.00i | 46225.0 | − | 62982.2i | 225366. | −59627.2 | + | 59627.2i | 524288. | + | 524288.i | 1.41695e6i | 1.07246e6 | + | 6.98927e6i | |||||||||||||||||||||||
| 3.2 | −64.0000 | + | 64.0000i | 1295.91 | + | 1295.91i | − | 8192.00i | 61416.3 | + | 48286.1i | −165877. | 905655. | − | 905655.i | 524288. | + | 524288.i | − | 1.42420e6i | −7.02096e6 | + | 840331.i | |||||||||||||||||||||||
| 3.3 | −64.0000 | + | 64.0000i | 1920.76 | + | 1920.76i | − | 8192.00i | −66391.4 | − | 41178.9i | −245857. | −374464. | + | 374464.i | 524288. | + | 524288.i | 2.59566e6i | 6.88450e6 | − | 1.61360e6i | ||||||||||||||||||||||||
| 7.1 | −64.0000 | − | 64.0000i | −1760.67 | + | 1760.67i | 8192.00i | 46225.0 | + | 62982.2i | 225366. | −59627.2 | − | 59627.2i | 524288. | − | 524288.i | − | 1.41695e6i | 1.07246e6 | − | 6.98927e6i | ||||||||||||||||||||||||
| 7.2 | −64.0000 | − | 64.0000i | 1295.91 | − | 1295.91i | 8192.00i | 61416.3 | − | 48286.1i | −165877. | 905655. | + | 905655.i | 524288. | − | 524288.i | 1.42420e6i | −7.02096e6 | − | 840331.i | |||||||||||||||||||||||||
| 7.3 | −64.0000 | − | 64.0000i | 1920.76 | − | 1920.76i | 8192.00i | −66391.4 | + | 41178.9i | −245857. | −374464. | − | 374464.i | 524288. | − | 524288.i | − | 2.59566e6i | 6.88450e6 | + | 1.61360e6i | ||||||||||||||||||||||||
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.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 90.15.g.a | 6 | ||
| 4.b | odd | 2 | 1 | 80.15.p.a | 6 | ||
| 5.b | even | 2 | 1 | 50.15.c.b | 6 | ||
| 5.c | odd | 4 | 1 | inner | 10.15.c.a | ✓ | 6 |
| 5.c | odd | 4 | 1 | 50.15.c.b | 6 | ||
| 15.e | even | 4 | 1 | 90.15.g.a | 6 | ||
| 20.e | even | 4 | 1 | 80.15.p.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.15.c.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 10.15.c.a | ✓ | 6 | 5.c | odd | 4 | 1 | inner |
| 50.15.c.b | 6 | 5.b | even | 2 | 1 | ||
| 50.15.c.b | 6 | 5.c | odd | 4 | 1 | ||
| 80.15.p.a | 6 | 4.b | odd | 2 | 1 | ||
| 80.15.p.a | 6 | 20.e | even | 4 | 1 | ||
| 90.15.g.a | 6 | 3.b | odd | 2 | 1 | ||
| 90.15.g.a | 6 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 2912 T_{3}^{5} + 4239872 T_{3}^{4} + 957313944 T_{3}^{3} + 40306321823076 T_{3}^{2} + \cdots + 15\!\cdots\!12 \)
acting on \(S_{15}^{\mathrm{new}}(10, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 128 T + 8192)^{3} \)
|
| $3$ |
\( T^{6} + \cdots + 15\!\cdots\!12 \)
|
| $5$ |
\( T^{6} + \cdots + 22\!\cdots\!25 \)
|
| $7$ |
\( T^{6} + \cdots + 32\!\cdots\!72 \)
|
| $11$ |
\( (T^{3} + \cdots + 26\!\cdots\!52)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 47\!\cdots\!32 \)
|
| $17$ |
\( T^{6} + \cdots + 11\!\cdots\!92 \)
|
| $19$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 14\!\cdots\!92 \)
|
| $29$ |
\( T^{6} + \cdots + 72\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots - 37\!\cdots\!68)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 49\!\cdots\!52 \)
|
| $41$ |
\( (T^{3} + \cdots + 10\!\cdots\!92)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 10\!\cdots\!32 \)
|
| $47$ |
\( T^{6} + \cdots + 49\!\cdots\!92 \)
|
| $53$ |
\( T^{6} + \cdots + 18\!\cdots\!92 \)
|
| $59$ |
\( T^{6} + \cdots + 41\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots + 15\!\cdots\!72)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 88\!\cdots\!12 \)
|
| $71$ |
\( (T^{3} + \cdots - 37\!\cdots\!08)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 35\!\cdots\!72 \)
|
| $79$ |
\( T^{6} + \cdots + 40\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 23\!\cdots\!52 \)
|
| $89$ |
\( T^{6} + \cdots + 49\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 52\!\cdots\!12 \)
|
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