Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4368,2,Mod(337,4368)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4368, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4368.337");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4368.h (of order \(2\), degree \(1\), not 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: | \(34.8786556029\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 20x^{8} + 138x^{6} + 364x^{4} + 249x^{2} + 36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 2184) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} + 20x^{8} + 138x^{6} + 364x^{4} + 249x^{2} + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{8} + 11\nu^{6} + 27\nu^{4} - 11\nu^{2} - 12 ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{9} + 17\nu^{7} + 87\nu^{5} + 139\nu^{3} + 84\nu ) / 36 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{9} + 17\nu^{7} + 105\nu^{5} + 283\nu^{3} + 246\nu ) / 36 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{8} + 11\nu^{6} + 15\nu^{4} - 95\nu^{2} - 48 ) / 12 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{8} + 14\nu^{6} + 57\nu^{4} + 58\nu^{2} + 6\nu + 12 ) / 6 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{9} - 17\nu^{7} - 96\nu^{5} - 193\nu^{3} - 75\nu ) / 18 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -\nu^{8} - 14\nu^{6} - 57\nu^{4} - 58\nu^{2} + 6\nu - 12 ) / 6 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{9} + 26\nu^{7} + 213\nu^{5} + 598\nu^{3} + 264\nu ) / 18 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{8} + \beta_{6} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{8} + 2\beta_{7} - 5\beta_{6} + 2\beta_{4} + 2\beta_{3} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} + \beta_{2} - 7\beta _1 + 25 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 31\beta_{8} - 16\beta_{7} + 31\beta_{6} - 12\beta_{4} - 20\beta_{3} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -\beta_{8} + \beta_{6} + 10\beta_{5} - 14\beta_{2} + 47\beta _1 - 166 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 4\beta_{9} - 199\beta_{8} + 122\beta_{7} - 199\beta_{6} + 66\beta_{4} + 170\beta_{3} ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( 11\beta_{8} - 11\beta_{6} - 83\beta_{5} + 139\beta_{2} - 317\beta _1 + 1119 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( -68\beta_{9} + 1297\beta_{8} - 960\beta_{7} + 1297\beta_{6} - 356\beta_{4} - 1356\beta_{3} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times\).
| \(n\) | \(1093\) | \(1249\) | \(1457\) | \(2017\) | \(3823\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
0 | −1.00000 | 0 | − | 2.89917i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 337.2 | 0 | −1.00000 | 0 | − | 2.71697i | 0 | 1.00000i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 337.3 | 0 | −1.00000 | 0 | − | 2.39548i | 0 | 1.00000i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 337.4 | 0 | −1.00000 | 0 | − | 2.11295i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 337.5 | 0 | −1.00000 | 0 | − | 0.100328i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 337.6 | 0 | −1.00000 | 0 | 0.100328i | 0 | 1.00000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 337.7 | 0 | −1.00000 | 0 | 2.11295i | 0 | 1.00000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 337.8 | 0 | −1.00000 | 0 | 2.39548i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 337.9 | 0 | −1.00000 | 0 | 2.71697i | 0 | − | 1.00000i | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 337.10 | 0 | −1.00000 | 0 | 2.89917i | 0 | 1.00000i | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 4368.2.h.r | 10 | |
| 4.b | odd | 2 | 1 | 2184.2.h.f | ✓ | 10 | |
| 13.b | even | 2 | 1 | inner | 4368.2.h.r | 10 | |
| 52.b | odd | 2 | 1 | 2184.2.h.f | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2184.2.h.f | ✓ | 10 | 4.b | odd | 2 | 1 | |
| 2184.2.h.f | ✓ | 10 | 52.b | odd | 2 | 1 | |
| 4368.2.h.r | 10 | 1.a | even | 1 | 1 | trivial | |
| 4368.2.h.r | 10 | 13.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4368, [\chi])\):
|
\( T_{5}^{10} + 26T_{5}^{8} + 249T_{5}^{6} + 1040T_{5}^{4} + 1600T_{5}^{2} + 16 \)
|
|
\( T_{11}^{10} + 73T_{11}^{8} + 1976T_{11}^{6} + 24192T_{11}^{4} + 128528T_{11}^{2} + 215296 \)
|
|
\( T_{17}^{5} - 5T_{17}^{4} - 26T_{17}^{3} + 104T_{17}^{2} + 96T_{17} - 384 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( (T + 1)^{10} \)
|
| $5$ |
\( T^{10} + 26 T^{8} + \cdots + 16 \)
|
| $7$ |
\( (T^{2} + 1)^{5} \)
|
| $11$ |
\( T^{10} + 73 T^{8} + \cdots + 215296 \)
|
| $13$ |
\( T^{10} + 4 T^{9} + \cdots + 371293 \)
|
| $17$ |
\( (T^{5} - 5 T^{4} + \cdots - 384)^{2} \)
|
| $19$ |
\( T^{10} + 74 T^{8} + \cdots + 1024 \)
|
| $23$ |
\( (T^{5} + 4 T^{4} + \cdots + 5272)^{2} \)
|
| $29$ |
\( (T^{5} + 22 T^{4} + \cdots - 216)^{2} \)
|
| $31$ |
\( T^{10} + 161 T^{8} + \cdots + 4194304 \)
|
| $37$ |
\( T^{10} + 73 T^{8} + \cdots + 16384 \)
|
| $41$ |
\( T^{10} + \cdots + 707134464 \)
|
| $43$ |
\( (T^{5} + 10 T^{4} + \cdots - 2048)^{2} \)
|
| $47$ |
\( T^{10} + \cdots + 259081216 \)
|
| $53$ |
\( (T^{5} - 5 T^{4} + \cdots + 2032)^{2} \)
|
| $59$ |
\( T^{10} + \cdots + 131974144 \)
|
| $61$ |
\( (T^{5} - 9 T^{4} + \cdots - 1312)^{2} \)
|
| $67$ |
\( T^{10} + \cdots + 17314349056 \)
|
| $71$ |
\( T^{10} + 204 T^{8} + \cdots + 9216 \)
|
| $73$ |
\( T^{10} + 186 T^{8} + \cdots + 2027776 \)
|
| $79$ |
\( (T^{5} + T^{4} + \cdots + 35136)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 1802511936 \)
|
| $89$ |
\( T^{10} + 473 T^{8} + \cdots + 43243776 \)
|
| $97$ |
\( T^{10} + \cdots + 1109689344 \)
|
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