Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [84,5,Mod(13,84)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(84, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("84.13");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 84 = 2^{2} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 84.d (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: | \(8.68307689904\) |
| 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: |
\( x^{6} - x^{5} + 235x^{4} + 2574x^{3} + 53586x^{2} + 273780x + 1368900 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{4} \) |
| 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
\( x^{6} - x^{5} + 235x^{4} + 2574x^{3} + 53586x^{2} + 273780x + 1368900 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 235\nu^{4} - 55225\nu^{3} + 53586\nu^{2} + 13140270\nu - 70771545 ) / 6433245 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 47\nu^{5} - 48\nu^{4} + 11280\nu^{3} + 65753\nu^{2} + 2572128\nu + 6708195 ) / 2144415 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -81\nu^{5} + 19035\nu^{4} - 184395\nu^{3} + 4340466\nu^{2} + 22176180\nu + 464149400 ) / 1429610 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -14603\nu^{5} + 66623\nu^{4} - 2789915\nu^{3} - 34010892\nu^{2} - 392037048\nu - 1217678670 ) / 38599470 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2087\nu^{5} + 9719\nu^{4} - 445895\nu^{3} - 4812768\nu^{2} - 77827464\nu - 229101210 ) / 5514210 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} + \beta _1 + 1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -24\beta_{5} + 15\beta_{4} + \beta_{3} - 153\beta_{2} - 9\beta _1 - 469 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -243\beta_{5} + 243\beta_{4} + 2\beta_{3} - 486\beta _1 - 8426 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4929\beta_{5} - 2814\beta_{4} + 235\beta_{3} + 37125\beta_{2} - 3519\beta _1 - 115129 ) / 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 151656\beta_{5} - 136905\beta_{4} - 1639\beta_{3} + 525717\beta_{2} + 70911\beta _1 + 1649701 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(43\) | \(73\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 13.1 |
|
0 | − | 5.19615i | 0 | − | 46.7016i | 0 | 26.4870 | + | 41.2243i | 0 | −27.0000 | 0 | ||||||||||||||||||||||||||||||||
| 13.2 | 0 | − | 5.19615i | 0 | 15.0883i | 0 | 39.9671 | − | 28.3484i | 0 | −27.0000 | 0 | ||||||||||||||||||||||||||||||||||
| 13.3 | 0 | − | 5.19615i | 0 | 17.7568i | 0 | −45.4541 | + | 18.3010i | 0 | −27.0000 | 0 | ||||||||||||||||||||||||||||||||||
| 13.4 | 0 | 5.19615i | 0 | − | 17.7568i | 0 | −45.4541 | − | 18.3010i | 0 | −27.0000 | 0 | ||||||||||||||||||||||||||||||||||
| 13.5 | 0 | 5.19615i | 0 | − | 15.0883i | 0 | 39.9671 | + | 28.3484i | 0 | −27.0000 | 0 | ||||||||||||||||||||||||||||||||||
| 13.6 | 0 | 5.19615i | 0 | 46.7016i | 0 | 26.4870 | − | 41.2243i | 0 | −27.0000 | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 84.5.d.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 252.5.d.d | 6 | ||
| 4.b | odd | 2 | 1 | 336.5.f.b | 6 | ||
| 7.b | odd | 2 | 1 | inner | 84.5.d.a | ✓ | 6 |
| 7.c | even | 3 | 1 | 588.5.m.b | 6 | ||
| 7.c | even | 3 | 1 | 588.5.m.c | 6 | ||
| 7.d | odd | 6 | 1 | 588.5.m.b | 6 | ||
| 7.d | odd | 6 | 1 | 588.5.m.c | 6 | ||
| 12.b | even | 2 | 1 | 1008.5.f.j | 6 | ||
| 21.c | even | 2 | 1 | 252.5.d.d | 6 | ||
| 28.d | even | 2 | 1 | 336.5.f.b | 6 | ||
| 84.h | odd | 2 | 1 | 1008.5.f.j | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.5.d.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 84.5.d.a | ✓ | 6 | 7.b | odd | 2 | 1 | inner |
| 252.5.d.d | 6 | 3.b | odd | 2 | 1 | ||
| 252.5.d.d | 6 | 21.c | even | 2 | 1 | ||
| 336.5.f.b | 6 | 4.b | odd | 2 | 1 | ||
| 336.5.f.b | 6 | 28.d | even | 2 | 1 | ||
| 588.5.m.b | 6 | 7.c | even | 3 | 1 | ||
| 588.5.m.b | 6 | 7.d | odd | 6 | 1 | ||
| 588.5.m.c | 6 | 7.c | even | 3 | 1 | ||
| 588.5.m.c | 6 | 7.d | odd | 6 | 1 | ||
| 1008.5.f.j | 6 | 12.b | even | 2 | 1 | ||
| 1008.5.f.j | 6 | 84.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(84, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} + 27)^{3} \)
|
| $5$ |
\( T^{6} + 2724 T^{4} + \cdots + 156558528 \)
|
| $7$ |
\( T^{6} + \cdots + 13841287201 \)
|
| $11$ |
\( (T^{3} + 114 T^{2} + \cdots - 755892)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 233643543552 \)
|
| $17$ |
\( T^{6} + \cdots + 682129446792192 \)
|
| $19$ |
\( T^{6} + \cdots + 134041043282688 \)
|
| $23$ |
\( (T^{3} + 570 T^{2} + \cdots + 6921612)^{2} \)
|
| $29$ |
\( (T^{3} - 750 T^{2} + \cdots - 103499208)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 20479168217088 \)
|
| $37$ |
\( (T^{3} + 2454 T^{2} + \cdots - 165822688)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 29\!\cdots\!00 \)
|
| $43$ |
\( (T^{3} - 3762 T^{2} + \cdots + 11695683152)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 16\!\cdots\!72 \)
|
| $53$ |
\( (T^{3} - 774 T^{2} + \cdots + 8533469880)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 18\!\cdots\!12 \)
|
| $61$ |
\( T^{6} + \cdots + 49\!\cdots\!68 \)
|
| $67$ |
\( (T^{3} - 1434 T^{2} + \cdots + 24550931488)^{2} \)
|
| $71$ |
\( (T^{3} - 5694 T^{2} + \cdots + 93711104172)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 76\!\cdots\!00 \)
|
| $79$ |
\( (T^{3} + 3054 T^{2} + \cdots + 134573072896)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 17\!\cdots\!52 \)
|
| $89$ |
\( T^{6} + \cdots + 31\!\cdots\!92 \)
|
| $97$ |
\( T^{6} + \cdots + 40\!\cdots\!88 \)
|
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