Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [111,4,Mod(73,111)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(111, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("111.73");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 111 = 3 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 111.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: | \(6.54921201064\) |
| 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: |
\( x^{10} + 55x^{8} + 989x^{6} + 6479x^{4} + 13224x^{2} + 4800 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} + 55x^{8} + 989x^{6} + 6479x^{4} + 13224x^{2} + 4800 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 19\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{8} + 33\nu^{6} + 41\nu^{4} - 4635\nu^{2} - 12900 ) / 444 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -11\nu^{9} - 585\nu^{7} - 10219\nu^{5} - 66009\nu^{3} - 127164\nu ) / 4440 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{8} + 33\nu^{6} + 485\nu^{4} + 6465\nu^{2} + 16404 ) / 444 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -11\nu^{9} - 585\nu^{7} - 9775\nu^{5} - 52689\nu^{3} - 54792\nu ) / 888 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{9} + 70\nu^{7} + 1669\nu^{5} + 15234\nu^{3} + 39344\nu ) / 370 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 2\nu^{8} + 103\nu^{6} + 1636\nu^{4} + 8009\nu^{2} + 6686 ) / 74 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 11 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 19\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} - \beta_{4} - 25\beta_{2} + 209 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{7} - 10\beta_{5} - 30\beta_{3} + 407\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{9} - 42\beta_{6} + 18\beta_{4} + 583\beta_{2} - 4519 \)
|
| \(\nu^{7}\) | \(=\) |
\( 22\beta_{8} - 88\beta_{7} + 464\beta_{5} + 771\beta_{3} - 9129\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( -66\beta_{9} + 1345\beta_{6} - 109\beta_{4} - 13579\beta_{2} + 102473 \)
|
| \(\nu^{9}\) | \(=\) |
\( -1170\beta_{8} + 2822\beta_{7} - 15790\beta_{5} - 19134\beta_{3} + 209849\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/111\mathbb{Z}\right)^\times\).
| \(n\) | \(38\) | \(76\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 73.1 |
|
− | 4.98065i | −3.00000 | −16.8069 | 15.1155i | 14.9419i | 8.03788 | 43.8639i | 9.00000 | 75.2848 | |||||||||||||||||||||||||||||||||||||||||||||||
| 73.2 | − | 4.35611i | −3.00000 | −10.9757 | − | 16.3864i | 13.0683i | −10.5327 | 12.9624i | 9.00000 | −71.3808 | |||||||||||||||||||||||||||||||||||||||||||||||
| 73.3 | − | 2.82374i | −3.00000 | 0.0264658 | − | 0.991859i | 8.47123i | 34.4802 | − | 22.6647i | 9.00000 | −2.80076 | ||||||||||||||||||||||||||||||||||||||||||||||
| 73.4 | − | 1.66873i | −3.00000 | 5.21534 | 3.99230i | 5.00619i | 2.26210 | − | 22.0528i | 9.00000 | 6.66207 | |||||||||||||||||||||||||||||||||||||||||||||||
| 73.5 | − | 0.677680i | −3.00000 | 7.54075 | 15.1026i | 2.03304i | −21.2475 | − | 10.5317i | 9.00000 | 10.2347 | |||||||||||||||||||||||||||||||||||||||||||||||
| 73.6 | 0.677680i | −3.00000 | 7.54075 | − | 15.1026i | − | 2.03304i | −21.2475 | 10.5317i | 9.00000 | 10.2347 | |||||||||||||||||||||||||||||||||||||||||||||||
| 73.7 | 1.66873i | −3.00000 | 5.21534 | − | 3.99230i | − | 5.00619i | 2.26210 | 22.0528i | 9.00000 | 6.66207 | |||||||||||||||||||||||||||||||||||||||||||||||
| 73.8 | 2.82374i | −3.00000 | 0.0264658 | 0.991859i | − | 8.47123i | 34.4802 | 22.6647i | 9.00000 | −2.80076 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 73.9 | 4.35611i | −3.00000 | −10.9757 | 16.3864i | − | 13.0683i | −10.5327 | − | 12.9624i | 9.00000 | −71.3808 | |||||||||||||||||||||||||||||||||||||||||||||||
| 73.10 | 4.98065i | −3.00000 | −16.8069 | − | 15.1155i | − | 14.9419i | 8.03788 | − | 43.8639i | 9.00000 | 75.2848 | ||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 111.4.c.a | ✓ | 10 |
| 3.b | odd | 2 | 1 | 333.4.c.f | 10 | ||
| 37.b | even | 2 | 1 | inner | 111.4.c.a | ✓ | 10 |
| 111.d | odd | 2 | 1 | 333.4.c.f | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 111.4.c.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 111.4.c.a | ✓ | 10 | 37.b | even | 2 | 1 | inner |
| 333.4.c.f | 10 | 3.b | odd | 2 | 1 | ||
| 333.4.c.f | 10 | 111.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 55T_{2}^{8} + 989T_{2}^{6} + 6479T_{2}^{4} + 13224T_{2}^{2} + 4800 \)
acting on \(S_{4}^{\mathrm{new}}(111, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 55 T^{8} + \cdots + 4800 \)
|
| $3$ |
\( (T + 3)^{10} \)
|
| $5$ |
\( T^{10} + \cdots + 219410112 \)
|
| $7$ |
\( (T^{5} - 13 T^{4} + \cdots - 140304)^{2} \)
|
| $11$ |
\( (T^{5} + 27 T^{4} + \cdots + 1150848)^{2} \)
|
| $13$ |
\( T^{10} + \cdots + 29\!\cdots\!72 \)
|
| $17$ |
\( T^{10} + \cdots + 16\!\cdots\!28 \)
|
| $19$ |
\( T^{10} + \cdots + 10\!\cdots\!12 \)
|
| $23$ |
\( T^{10} + \cdots + 12\!\cdots\!00 \)
|
| $29$ |
\( T^{10} + \cdots + 23\!\cdots\!08 \)
|
| $31$ |
\( T^{10} + \cdots + 13\!\cdots\!00 \)
|
| $37$ |
\( T^{10} + \cdots + 33\!\cdots\!93 \)
|
| $41$ |
\( (T^{5} + 174 T^{4} + \cdots - 763737328224)^{2} \)
|
| $43$ |
\( T^{10} + \cdots + 14\!\cdots\!68 \)
|
| $47$ |
\( (T^{5} + \cdots - 2054148221952)^{2} \)
|
| $53$ |
\( (T^{5} + 687 T^{4} + \cdots - 302211895536)^{2} \)
|
| $59$ |
\( T^{10} + \cdots + 25\!\cdots\!72 \)
|
| $61$ |
\( T^{10} + \cdots + 10\!\cdots\!00 \)
|
| $67$ |
\( (T^{5} + \cdots + 3827493982464)^{2} \)
|
| $71$ |
\( (T^{5} + \cdots + 4759780361472)^{2} \)
|
| $73$ |
\( (T^{5} + \cdots - 5821961006880)^{2} \)
|
| $79$ |
\( T^{10} + \cdots + 13\!\cdots\!72 \)
|
| $83$ |
\( (T^{5} + \cdots + 6295028451840)^{2} \)
|
| $89$ |
\( T^{10} + \cdots + 90\!\cdots\!12 \)
|
| $97$ |
\( T^{10} + \cdots + 98\!\cdots\!72 \)
|
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