Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [90,13,Mod(37,90)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(90, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("90.37");
S:= CuspForms(chi, 13);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 13 \) |
| Character orbit: | \([\chi]\) | \(=\) | 90.g (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: | \(82.2594435549\) |
| 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: |
\( x^{6} + 2385x^{4} + 1422264x^{2} + 490000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2}\cdot 5^{6} \) |
| Twist minimal: | no (minimal twist has level 10) |
| 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
\( x^{6} + 2385x^{4} + 1422264x^{2} + 490000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} - 1685\nu^{3} - 587164\nu ) / 344400 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 17\nu^{5} - 2450\nu^{4} + 86045\nu^{3} - 2865450\nu^{2} + 77197188\nu + 44319800 ) / 91840 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 17\nu^{5} + 2450\nu^{4} + 86045\nu^{3} + 2865450\nu^{2} + 77197188\nu - 44319800 ) / 91840 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -79\nu^{5} - 9870\nu^{4} - 443075\nu^{3} - 12015990\nu^{2} - 422574076\nu - 196948920 ) / 55104 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 103\nu^{5} - 22750\nu^{4} + 575355\nu^{3} - 27657350\nu^{2} + 547516892\nu - 422891000 ) / 91840 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{5} - 4\beta_{4} - 28\beta_{3} - 29\beta_{2} - 500\beta_1 ) / 1500 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -28\beta_{5} - 21\beta_{4} - 197\beta_{3} + 204\beta_{2} - 397500 ) / 500 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3513\beta_{5} + 4684\beta_{4} + 33988\beta_{3} + 35159\beta_{2} + 738500\beta_1 ) / 1500 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 32748\beta_{5} + 24561\beta_{4} + 239777\beta_{3} - 247964\beta_{2} + 473949500 ) / 500 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 4157913\beta_{5} - 5543884\beta_{4} - 40829188\beta_{3} - 42215159\beta_{2} - 1467390500\beta_1 ) / 1500 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).
| \(n\) | \(11\) | \(37\) |
| \(\chi(n)\) | \(1\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 37.1 |
|
32.0000 | + | 32.0000i | 0 | 2048.00i | −3818.66 | + | 15151.2i | 0 | −59325.1 | − | 59325.1i | −65536.0 | + | 65536.0i | 0 | −607035. | + | 362641.i | ||||||||||||||||||||||||||
| 37.2 | 32.0000 | + | 32.0000i | 0 | 2048.00i | −2686.16 | − | 15392.4i | 0 | −6681.98 | − | 6681.98i | −65536.0 | + | 65536.0i | 0 | 406599. | − | 578513.i | |||||||||||||||||||||||||||
| 37.3 | 32.0000 | + | 32.0000i | 0 | 2048.00i | 14634.8 | − | 5473.81i | 0 | 43339.1 | + | 43339.1i | −65536.0 | + | 65536.0i | 0 | 643476. | + | 293152.i | |||||||||||||||||||||||||||
| 73.1 | 32.0000 | − | 32.0000i | 0 | − | 2048.00i | −3818.66 | − | 15151.2i | 0 | −59325.1 | + | 59325.1i | −65536.0 | − | 65536.0i | 0 | −607035. | − | 362641.i | ||||||||||||||||||||||||||
| 73.2 | 32.0000 | − | 32.0000i | 0 | − | 2048.00i | −2686.16 | + | 15392.4i | 0 | −6681.98 | + | 6681.98i | −65536.0 | − | 65536.0i | 0 | 406599. | + | 578513.i | ||||||||||||||||||||||||||
| 73.3 | 32.0000 | − | 32.0000i | 0 | − | 2048.00i | 14634.8 | + | 5473.81i | 0 | 43339.1 | − | 43339.1i | −65536.0 | − | 65536.0i | 0 | 643476. | − | 293152.i | ||||||||||||||||||||||||||
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 | 90.13.g.b | 6 | |
| 3.b | odd | 2 | 1 | 10.13.c.a | ✓ | 6 | |
| 5.c | odd | 4 | 1 | inner | 90.13.g.b | 6 | |
| 12.b | even | 2 | 1 | 80.13.p.b | 6 | ||
| 15.d | odd | 2 | 1 | 50.13.c.d | 6 | ||
| 15.e | even | 4 | 1 | 10.13.c.a | ✓ | 6 | |
| 15.e | even | 4 | 1 | 50.13.c.d | 6 | ||
| 60.l | odd | 4 | 1 | 80.13.p.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 10.13.c.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 10.13.c.a | ✓ | 6 | 15.e | even | 4 | 1 | |
| 50.13.c.d | 6 | 15.d | odd | 2 | 1 | ||
| 50.13.c.d | 6 | 15.e | even | 4 | 1 | ||
| 80.13.p.b | 6 | 12.b | even | 2 | 1 | ||
| 80.13.p.b | 6 | 60.l | odd | 4 | 1 | ||
| 90.13.g.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 90.13.g.b | 6 | 5.c | odd | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{13}^{\mathrm{new}}(90, [\chi])\):
|
\( T_{7}^{6} + 45336 T_{7}^{5} + 1027676448 T_{7}^{4} - 154721053780888 T_{7}^{3} + \cdots + 23\!\cdots\!48 \)
|
|
\( T_{11}^{3} - 1709004T_{11}^{2} - 9946911899628T_{11} + 19830278439732572768 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 64 T + 2048)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 14\!\cdots\!25 \)
|
| $7$ |
\( T^{6} + \cdots + 23\!\cdots\!48 \)
|
| $11$ |
\( (T^{3} + \cdots + 19\!\cdots\!68)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 55\!\cdots\!88 \)
|
| $17$ |
\( T^{6} + \cdots + 18\!\cdots\!48 \)
|
| $19$ |
\( T^{6} + \cdots + 76\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots + 11\!\cdots\!48 \)
|
| $29$ |
\( T^{6} + \cdots + 14\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots - 11\!\cdots\!68)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 34\!\cdots\!08 \)
|
| $41$ |
\( (T^{3} + \cdots + 19\!\cdots\!28)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 65\!\cdots\!28 \)
|
| $47$ |
\( T^{6} + \cdots + 36\!\cdots\!68 \)
|
| $53$ |
\( T^{6} + \cdots + 10\!\cdots\!88 \)
|
| $59$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots + 23\!\cdots\!72)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 55\!\cdots\!08 \)
|
| $71$ |
\( (T^{3} + \cdots - 36\!\cdots\!32)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 18\!\cdots\!88 \)
|
| $79$ |
\( T^{6} + \cdots + 43\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 54\!\cdots\!08 \)
|
| $89$ |
\( T^{6} + \cdots + 97\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 51\!\cdots\!28 \)
|
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