Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1104,3,Mod(415,1104)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1104, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1104.415");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1104 = 2^{4} \cdot 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1104.k (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: | \(30.0818211854\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} + 21x^{6} - 2x^{5} + 313x^{4} - 258x^{3} + 528x^{2} + 216x + 144 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{23}]\) |
| Coefficient ring index: | \( 2^{10} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} + 21x^{6} - 2x^{5} + 313x^{4} - 258x^{3} + 528x^{2} + 216x + 144 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 17\nu^{7} - 120\nu^{6} + 309\nu^{5} + 368\nu^{4} - 471\nu^{3} + 1272\nu^{2} + 576\nu + 277848 ) / 33408 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1751 \nu^{7} - 1224 \nu^{6} + 31827 \nu^{5} + 37904 \nu^{4} + 597375 \nu^{3} + 131016 \nu^{2} + \cdots + 973224 ) / 517824 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 6325 \nu^{7} + 16152 \nu^{6} - 135273 \nu^{5} + 76304 \nu^{4} - 1903917 \nu^{3} + 2826600 \nu^{2} + \cdots - 211896 ) / 1035648 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5271 \nu^{7} - 8712 \nu^{6} + 116115 \nu^{5} + 15952 \nu^{4} + 1818047 \nu^{3} - 719096 \nu^{2} + \cdots + 936552 ) / 345216 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 19\nu^{7} - 50\nu^{6} + 411\nu^{5} - 278\nu^{4} + 5731\nu^{3} - 8502\nu^{2} + 9156\nu + 612 ) / 1116 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11809 \nu^{7} + 26968 \nu^{6} - 265413 \nu^{5} + 104816 \nu^{4} - 3836921 \nu^{3} + \cdots - 1127448 ) / 517824 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5049 \nu^{7} - 4088 \nu^{6} + 91773 \nu^{5} + 109296 \nu^{4} + 1517521 \nu^{3} + 377784 \nu^{2} + \cdots + 1690200 ) / 172608 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + 7\beta_{5} + \beta_{4} + 19\beta_{3} + \beta_{2} + 2\beta _1 - 19 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{7} + 17\beta_{2} + 2\beta _1 - 29 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -4\beta_{7} - 37\beta_{6} - 151\beta_{5} - 25\beta_{4} - 343\beta_{3} + 33\beta_{2} + 38\beta _1 - 339 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 42\beta_{7} - 367\beta_{6} - 589\beta_{5} - 241\beta_{4} - 859\beta_{3} - 325\beta_{2} - 74\beta _1 + 817 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 116\beta_{7} - 893\beta_{2} - 734\beta _1 + 6647 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 850 \beta_{7} + 7363 \beta_{6} + 13417 \beta_{5} + 4813 \beta_{4} + 21583 \beta_{3} - 6513 \beta_{2} + \cdots + 20733 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1104\mathbb{Z}\right)^\times\).
| \(n\) | \(97\) | \(277\) | \(415\) | \(737\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 415.1 |
|
0 | − | 1.73205i | 0 | −7.38013 | 0 | 8.40541i | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 415.2 | 0 | − | 1.73205i | 0 | −0.942551 | 0 | 11.6372i | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 415.3 | 0 | − | 1.73205i | 0 | 2.94255 | 0 | − | 7.77276i | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 415.4 | 0 | − | 1.73205i | 0 | 9.38013 | 0 | 5.05067i | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 415.5 | 0 | 1.73205i | 0 | −7.38013 | 0 | − | 8.40541i | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 415.6 | 0 | 1.73205i | 0 | −0.942551 | 0 | − | 11.6372i | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 415.7 | 0 | 1.73205i | 0 | 2.94255 | 0 | 7.77276i | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 415.8 | 0 | 1.73205i | 0 | 9.38013 | 0 | − | 5.05067i | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1104.3.k.b | ✓ | 8 |
| 4.b | odd | 2 | 1 | inner | 1104.3.k.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1104.3.k.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1104.3.k.b | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 4T_{5}^{3} - 68T_{5}^{2} + 144T_{5} + 192 \)
acting on \(S_{3}^{\mathrm{new}}(1104, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{2} + 3)^{4} \)
|
| $5$ |
\( (T^{4} - 4 T^{3} + \cdots + 192)^{2} \)
|
| $7$ |
\( T^{8} + 292 T^{6} + \cdots + 14745600 \)
|
| $11$ |
\( T^{8} + 620 T^{6} + \cdots + 256 \)
|
| $13$ |
\( (T - 2)^{8} \)
|
| $17$ |
\( (T^{4} + 50 T^{3} + \cdots - 13488)^{2} \)
|
| $19$ |
\( T^{8} + 824 T^{6} + \cdots + 303038464 \)
|
| $23$ |
\( (T^{2} + 23)^{4} \)
|
| $29$ |
\( (T^{4} + 56 T^{3} + \cdots - 125808)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 2415919104 \)
|
| $37$ |
\( (T^{4} - 50 T^{3} + \cdots - 8640)^{2} \)
|
| $41$ |
\( (T^{4} - 96 T^{3} + \cdots + 209680)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 262996660224 \)
|
| $47$ |
\( T^{8} + \cdots + 58042456473600 \)
|
| $53$ |
\( (T^{4} + 80 T^{3} + \cdots + 619840)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 4430233600 \)
|
| $61$ |
\( (T^{4} + 10 T^{3} + \cdots - 2370240)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 4249798742016 \)
|
| $71$ |
\( T^{8} + \cdots + 1131959812096 \)
|
| $73$ |
\( (T^{4} - 4 T^{3} + \cdots + 418800)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 570446984577024 \)
|
| $83$ |
\( T^{8} + \cdots + 296675545929984 \)
|
| $89$ |
\( (T^{4} - 222 T^{3} + \cdots - 25342640)^{2} \)
|
| $97$ |
\( (T^{4} + 72 T^{3} + \cdots + 581200)^{2} \)
|
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