Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6,17,Mod(5,6)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 17, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6.5");
S:= CuspForms(chi, 17);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6 = 2 \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 17 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6.b (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: | \(9.73947263140\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 55116x^{4} + 758395257x^{2} + 123254139008 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{27}\cdot 3^{13} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 55116x^{4} + 758395257x^{2} + 123254139008 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 16\nu^{5} + 4853824\nu^{3} + 121597790224\nu ) / 8558504955 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 199889 \nu^{5} + 15267252 \nu^{4} - 9288096866 \nu^{3} + 523452257328 \nu^{2} + \cdots + 18\!\cdots\!88 ) / 316664683335 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 8998261 \nu^{5} - 4512031524 \nu^{4} - 418952112154 \nu^{3} - 119109271737456 \nu^{2} + \cdots + 99\!\cdots\!04 ) / 316664683335 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3331582 \nu^{5} - 162850688 \nu^{4} + 154831546348 \nu^{3} - 5583490744832 \nu^{2} + \cdots - 20\!\cdots\!72 ) / 105554894445 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 176186 \nu^{5} + 5765256 \nu^{4} + 8202127844 \nu^{3} + 197667285984 \nu^{2} + \cdots + 708591367343664 ) / 2214438345 \)
|
| \(\nu\) | \(=\) |
\( ( 2\beta_{5} - 20\beta_{4} - 748\beta_{2} - 19\beta_1 ) / 62208 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 254\beta_{5} - 213\beta_{4} + 123\beta_{3} + 15819\beta_{2} - 2900\beta _1 - 253974528 ) / 13824 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -57202\beta_{5} + 550708\beta_{4} + 20711564\beta_{2} + 130082147\beta_1 ) / 62208 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 1734674 \beta_{5} + 1383243 \beta_{4} - 1054293 \beta_{3} - 101963589 \beta_{2} + \cdots + 1752172349184 ) / 3456 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2153303750\beta_{5} - 15067743932\beta_{4} - 598446208324\beta_{2} - 6042375801577\beta_1 ) / 62208 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
− | 181.019i | −4654.10 | + | 4624.51i | −32768.0 | 1539.79i | 837125. | + | 842482.i | 5.92137e6 | 5.93164e6i | 274580. | − | 4.30458e7i | 278731. | |||||||||||||||||||||||||||||
| 5.2 | − | 181.019i | 1479.09 | − | 6392.11i | −32768.0 | − | 441421.i | −1.15709e6 | − | 267744.i | 2.81464e6 | 5.93164e6i | −3.86713e7 | − | 1.89090e7i | −7.99057e7 | |||||||||||||||||||||||||||||
| 5.3 | − | 181.019i | 6178.01 | + | 2208.83i | −32768.0 | 548425.i | 399841. | − | 1.11834e6i | −8.81996e6 | 5.93164e6i | 3.32888e7 | + | 2.72924e7i | 9.92755e7 | ||||||||||||||||||||||||||||||
| 5.4 | 181.019i | −4654.10 | − | 4624.51i | −32768.0 | − | 1539.79i | 837125. | − | 842482.i | 5.92137e6 | − | 5.93164e6i | 274580. | + | 4.30458e7i | 278731. | |||||||||||||||||||||||||||||
| 5.5 | 181.019i | 1479.09 | + | 6392.11i | −32768.0 | 441421.i | −1.15709e6 | + | 267744.i | 2.81464e6 | − | 5.93164e6i | −3.86713e7 | + | 1.89090e7i | −7.99057e7 | ||||||||||||||||||||||||||||||
| 5.6 | 181.019i | 6178.01 | − | 2208.83i | −32768.0 | − | 548425.i | 399841. | + | 1.11834e6i | −8.81996e6 | − | 5.93164e6i | 3.32888e7 | − | 2.72924e7i | 9.92755e7 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 6.17.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 6.17.b.a | ✓ | 6 |
| 4.b | odd | 2 | 1 | 48.17.e.d | 6 | ||
| 12.b | even | 2 | 1 | 48.17.e.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 6.17.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 6.17.b.a | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
| 48.17.e.d | 6 | 4.b | odd | 2 | 1 | ||
| 48.17.e.d | 6 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{17}^{\mathrm{new}}(6, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 32768)^{3} \)
|
| $3$ |
\( T^{6} + \cdots + 79\!\cdots\!61 \)
|
| $5$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
| $7$ |
\( (T^{3} + \cdots + 14\!\cdots\!24)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 17\!\cdots\!08 \)
|
| $13$ |
\( (T^{3} + \cdots + 12\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 80\!\cdots\!28 \)
|
| $19$ |
\( (T^{3} + \cdots + 11\!\cdots\!92)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 10\!\cdots\!88 \)
|
| $29$ |
\( T^{6} + \cdots + 32\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots - 10\!\cdots\!52)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots + 16\!\cdots\!56)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 20\!\cdots\!28 \)
|
| $43$ |
\( (T^{3} + \cdots - 38\!\cdots\!12)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 91\!\cdots\!00 \)
|
| $53$ |
\( T^{6} + \cdots + 67\!\cdots\!28 \)
|
| $59$ |
\( T^{6} + \cdots + 98\!\cdots\!08 \)
|
| $61$ |
\( (T^{3} + \cdots - 87\!\cdots\!04)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots + 67\!\cdots\!36)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 69\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots - 19\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots + 13\!\cdots\!28)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 35\!\cdots\!72 \)
|
| $89$ |
\( T^{6} + \cdots + 48\!\cdots\!08 \)
|
| $97$ |
\( (T^{3} + \cdots + 28\!\cdots\!12)^{2} \)
|
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