Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [63,5,Mod(8,63)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(63, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("63.8");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 63.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: | \(6.51230767428\) |
| 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: |
\( x^{8} + 92x^{6} + 2949x^{4} + 37548x^{2} + 142884 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3^{7}\cdot 7^{2} \) |
| 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
\( x^{8} + 92x^{6} + 2949x^{4} + 37548x^{2} + 142884 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 71\nu^{7} + 4642\nu^{5} + 83127\nu^{3} + 362754\nu ) / 63504 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 110\nu^{4} + 3057\nu^{2} + 19062 ) / 72 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{6} - 334\nu^{4} - 6213\nu^{2} - 25398 ) / 72 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{6} - 334\nu^{4} - 6429\nu^{2} - 30366 ) / 72 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 377\nu^{7} + 25990\nu^{5} + 550065\nu^{3} + 3503934\nu ) / 31752 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -5\nu^{7} - 320\nu^{5} - 5687\nu^{3} - 21546\nu ) / 294 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -1097\nu^{7} - 75598\nu^{5} - 1492473\nu^{3} - 7051086\nu ) / 63504 \)
|
| \(\nu\) | \(=\) |
\( ( 4\beta_{7} + 5\beta_{6} + 3\beta_{5} + 106\beta_1 ) / 126 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} - 69 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{7} - 7\beta_{6} + \beta_{5} - 148\beta_1 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 42\beta_{4} - 41\beta_{3} + 5\beta_{2} + 1927 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -62\beta_{7} + 287\beta_{6} - 107\beta_{5} + 4544\beta_1 ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -1563\beta_{4} + 1453\beta_{3} - 334\beta_{2} - 58223 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 5422\beta_{7} - 11785\beta_{6} + 5095\beta_{5} - 144232\beta_1 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).
| \(n\) | \(10\) | \(29\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 8.1 |
|
− | 7.70411i | 0 | −43.3534 | 25.4664i | 0 | −18.5203 | 210.733i | 0 | 196.196 | |||||||||||||||||||||||||||||||||||||||||
| 8.2 | − | 7.15630i | 0 | −35.2126 | − | 46.1483i | 0 | 18.5203 | 137.491i | 0 | −330.251 | |||||||||||||||||||||||||||||||||||||||||
| 8.3 | − | 3.66513i | 0 | 2.56682 | − | 14.9468i | 0 | 18.5203 | − | 68.0498i | 0 | −54.7821 | ||||||||||||||||||||||||||||||||||||||||
| 8.4 | − | 0.0296928i | 0 | 15.9991 | 28.1850i | 0 | −18.5203 | − | 0.950143i | 0 | 0.836892 | |||||||||||||||||||||||||||||||||||||||||
| 8.5 | 0.0296928i | 0 | 15.9991 | − | 28.1850i | 0 | −18.5203 | 0.950143i | 0 | 0.836892 | ||||||||||||||||||||||||||||||||||||||||||
| 8.6 | 3.66513i | 0 | 2.56682 | 14.9468i | 0 | 18.5203 | 68.0498i | 0 | −54.7821 | |||||||||||||||||||||||||||||||||||||||||||
| 8.7 | 7.15630i | 0 | −35.2126 | 46.1483i | 0 | 18.5203 | − | 137.491i | 0 | −330.251 | ||||||||||||||||||||||||||||||||||||||||||
| 8.8 | 7.70411i | 0 | −43.3534 | − | 25.4664i | 0 | −18.5203 | − | 210.733i | 0 | 196.196 | |||||||||||||||||||||||||||||||||||||||||
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 | 63.5.b.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 63.5.b.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | 1008.5.d.c | 8 | ||
| 7.b | odd | 2 | 1 | 441.5.b.d | 8 | ||
| 12.b | even | 2 | 1 | 1008.5.d.c | 8 | ||
| 21.c | even | 2 | 1 | 441.5.b.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 63.5.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 63.5.b.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 441.5.b.d | 8 | 7.b | odd | 2 | 1 | ||
| 441.5.b.d | 8 | 21.c | even | 2 | 1 | ||
| 1008.5.d.c | 8 | 4.b | odd | 2 | 1 | ||
| 1008.5.d.c | 8 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(63, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 124 T^{6} + 4525 T^{4} + \cdots + 36 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 3796 T^{6} + \cdots + 245120049216 \)
|
| $7$ |
\( (T^{2} - 343)^{4} \)
|
| $11$ |
\( T^{8} + 50308 T^{6} + \cdots + 16\!\cdots\!84 \)
|
| $13$ |
\( (T^{4} + 180 T^{3} - 64700 T^{2} + \cdots - 186689216)^{2} \)
|
| $17$ |
\( T^{8} + 268500 T^{6} + \cdots + 22\!\cdots\!84 \)
|
| $19$ |
\( (T^{4} + 444 T^{3} + 16216 T^{2} + \cdots - 1542278528)^{2} \)
|
| $23$ |
\( T^{8} + 836292 T^{6} + \cdots + 11\!\cdots\!56 \)
|
| $29$ |
\( T^{8} + 4034680 T^{6} + \cdots + 12\!\cdots\!16 \)
|
| $31$ |
\( (T^{4} - 556 T^{3} + \cdots - 63902466432)^{2} \)
|
| $37$ |
\( (T^{4} - 2140 T^{3} + \cdots + 337529037888)^{2} \)
|
| $41$ |
\( T^{8} + 6762676 T^{6} + \cdots + 67\!\cdots\!76 \)
|
| $43$ |
\( (T^{4} + 6400 T^{3} + \cdots + 3971721144832)^{2} \)
|
| $47$ |
\( T^{8} + 30135792 T^{6} + \cdots + 68\!\cdots\!24 \)
|
| $53$ |
\( T^{8} + 32836584 T^{6} + \cdots + 31\!\cdots\!84 \)
|
| $59$ |
\( T^{8} + 81221232 T^{6} + \cdots + 13\!\cdots\!96 \)
|
| $61$ |
\( (T^{4} - 2988 T^{3} + \cdots - 16126016178608)^{2} \)
|
| $67$ |
\( (T^{4} - 420 T^{3} + \cdots - 5902268493824)^{2} \)
|
| $71$ |
\( T^{8} + 54832644 T^{6} + \cdots + 40\!\cdots\!04 \)
|
| $73$ |
\( (T^{4} - 11224 T^{3} + \cdots - 14260001208768)^{2} \)
|
| $79$ |
\( (T^{4} - 16028 T^{3} + \cdots + 22129475818048)^{2} \)
|
| $83$ |
\( T^{8} + 199827072 T^{6} + \cdots + 77\!\cdots\!16 \)
|
| $89$ |
\( T^{8} + 122212756 T^{6} + \cdots + 20\!\cdots\!04 \)
|
| $97$ |
\( (T^{4} + 3776 T^{3} + \cdots + 16\!\cdots\!72)^{2} \)
|
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