Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [64,7,Mod(31,64)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(64, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("64.31");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 64.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: | \(14.7234613517\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.1301023109376.9 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 45x^{6} + 1541x^{4} - 21780x^{2} + 234256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{40}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 45x^{6} + 1541x^{4} - 21780x^{2} + 234256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 171\nu^{7} + 889157\nu^{5} - 19874277\nu^{3} + 426627608\nu ) / 8204284 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -382\nu^{6} + 46230\nu^{4} + 903026\nu^{2} - 2867700 ) / 186461 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3262\nu^{6} + 144854\nu^{4} - 3535054\nu^{2} + 35991692 ) / 186461 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2315\nu^{7} + 245019\nu^{5} - 7296635\nu^{3} + 169009896\nu ) / 8204284 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -192\nu^{6} - 2475360 ) / 1541 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 720\nu^{7} - 24656\nu^{5} + 590672\nu^{3} - 3748096\nu ) / 89177 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -19125\nu^{7} + 654925\nu^{5} - 24250717\nu^{3} + 99558800\nu ) / 2051071 \)
|
| \(\nu\) | \(=\) |
\( ( -16\beta_{7} - 7\beta_{6} + 324\beta_{4} - 60\beta_1 ) / 3072 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{5} - 45\beta_{3} + 141\beta_{2} + 17280 ) / 1536 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -368\beta_{7} - 425\beta_{6} ) / 1536 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -60\beta_{5} + 293\beta_{3} + 1147\beta_{2} - 135296 ) / 512 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -8816\beta_{7} - 15737\beta_{6} - 154764\beta_{4} + 56820\beta_1 ) / 3072 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -1541\beta_{5} - 2475360 ) / 192 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 218608\beta_{7} + 502465\beta_{6} - 3613164\beta_{4} + 1633428\beta_1 ) / 3072 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(63\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0 | −46.5367 | 0 | − | 79.2169i | 0 | − | 588.831i | 0 | 1436.66 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 31.2 | 0 | −46.5367 | 0 | 79.2169i | 0 | 588.831i | 0 | 1436.66 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 31.3 | 0 | −18.8239 | 0 | − | 51.5041i | 0 | 316.831i | 0 | −374.662 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 31.4 | 0 | −18.8239 | 0 | 51.5041i | 0 | − | 316.831i | 0 | −374.662 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 31.5 | 0 | 18.8239 | 0 | − | 51.5041i | 0 | − | 316.831i | 0 | −374.662 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 31.6 | 0 | 18.8239 | 0 | 51.5041i | 0 | 316.831i | 0 | −374.662 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 31.7 | 0 | 46.5367 | 0 | − | 79.2169i | 0 | 588.831i | 0 | 1436.66 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 31.8 | 0 | 46.5367 | 0 | 79.2169i | 0 | − | 588.831i | 0 | 1436.66 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 64.7.d.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 576.7.b.e | 8 | ||
| 4.b | odd | 2 | 1 | inner | 64.7.d.b | ✓ | 8 |
| 8.b | even | 2 | 1 | inner | 64.7.d.b | ✓ | 8 |
| 8.d | odd | 2 | 1 | inner | 64.7.d.b | ✓ | 8 |
| 12.b | even | 2 | 1 | 576.7.b.e | 8 | ||
| 16.e | even | 4 | 2 | 256.7.c.k | 8 | ||
| 16.f | odd | 4 | 2 | 256.7.c.k | 8 | ||
| 24.f | even | 2 | 1 | 576.7.b.e | 8 | ||
| 24.h | odd | 2 | 1 | 576.7.b.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 64.7.d.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 64.7.d.b | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 64.7.d.b | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 64.7.d.b | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
| 256.7.c.k | 8 | 16.e | even | 4 | 2 | ||
| 256.7.c.k | 8 | 16.f | odd | 4 | 2 | ||
| 576.7.b.e | 8 | 3.b | odd | 2 | 1 | ||
| 576.7.b.e | 8 | 12.b | even | 2 | 1 | ||
| 576.7.b.e | 8 | 24.f | even | 2 | 1 | ||
| 576.7.b.e | 8 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 2520T_{3}^{2} + 767376 \)
acting on \(S_{7}^{\mathrm{new}}(64, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} - 2520 T^{2} + 767376)^{2} \)
|
| $5$ |
\( (T^{4} + 8928 T^{2} + 16646400)^{2} \)
|
| $7$ |
\( (T^{4} + 447104 T^{2} + 34804633600)^{2} \)
|
| $11$ |
\( (T^{4} - 2663640 T^{2} + 510533114256)^{2} \)
|
| $13$ |
\( (T^{4} + 5535456 T^{2} + 714579427584)^{2} \)
|
| $17$ |
\( (T^{2} - 2868 T - 18449244)^{4} \)
|
| $19$ |
\( (T^{4} + \cdots + 781841455799184)^{2} \)
|
| $23$ |
\( (T^{4} + \cdots + 21\!\cdots\!64)^{2} \)
|
| $29$ |
\( (T^{4} + \cdots + 10\!\cdots\!64)^{2} \)
|
| $31$ |
\( (T^{4} + \cdots + 15\!\cdots\!84)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots + 11\!\cdots\!64)^{2} \)
|
| $41$ |
\( (T^{2} + 30396 T - 953424252)^{4} \)
|
| $43$ |
\( (T^{4} + \cdots + 36\!\cdots\!44)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots + 16\!\cdots\!64)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots + 92\!\cdots\!44)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots + 95\!\cdots\!04)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots + 89\!\cdots\!00)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots + 10\!\cdots\!24)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 70\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{2} + 129292 T - 4522651100)^{4} \)
|
| $79$ |
\( (T^{4} + \cdots + 12\!\cdots\!76)^{2} \)
|
| $83$ |
\( (T^{4} + \cdots + 32\!\cdots\!00)^{2} \)
|
| $89$ |
\( (T^{2} - 108084 T - 585515541852)^{4} \)
|
| $97$ |
\( (T^{2} - 298996 T - 254525981660)^{4} \)
|
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