Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [69,3,Mod(22,69)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(69, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("69.22");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 69.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: | \(1.88011382409\) |
| 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: |
\( x^{8} + 72x^{6} + 1545x^{4} + 9186x^{2} + 14793 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3 \) |
| 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
\( x^{8} + 72x^{6} + 1545x^{4} + 9186x^{2} + 14793 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -37\nu^{7} + 146\nu^{5} + 68564\nu^{3} + 767277\nu ) / 153401 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 37\nu^{6} - 146\nu^{4} - 68564\nu^{2} - 460475 ) / 153401 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} - 91\nu^{4} - 2103\nu^{2} - 6987 ) / 1171 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -299\nu^{7} - 23696\nu^{5} - 482422\nu^{3} - 1063317\nu ) / 153401 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 653\nu^{7} + 43029\nu^{5} + 792443\nu^{3} + 2690082\nu ) / 153401 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -821\nu^{7} - 54804\nu^{5} - 999372\nu^{3} - 3298305\nu ) / 153401 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -690\nu^{6} - 42883\nu^{4} - 723879\nu^{2} - 2076206 ) / 153401 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{6} - 2\beta_{5} + \beta_{4} + \beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{7} + 11\beta_{3} - 17\beta_{2} - 26 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 19\beta_{6} + 22\beta_{5} - 4\beta_{4} - \beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 125\beta_{7} - 483\beta_{3} + 621\beta_{2} + 674 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -597\beta_{6} - 722\beta_{5} + 57\beta_{4} + 44\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( -2533\beta_{7} + 9239\beta_{3} - 10380\beta_{2} - 10315 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 19028\beta_{6} + 24094\beta_{5} - 275\beta_{4} - 3059\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).
| \(n\) | \(28\) | \(47\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 22.1 |
|
−2.82684 | −1.73205 | 3.99102 | − | 5.36129i | 4.89623 | 13.3791i | 0.0253983 | 3.00000 | 15.1555i | |||||||||||||||||||||||||||||||||||||||||
| 22.2 | −2.82684 | −1.73205 | 3.99102 | 5.36129i | 4.89623 | − | 13.3791i | 0.0253983 | 3.00000 | − | 15.1555i | |||||||||||||||||||||||||||||||||||||||||
| 22.3 | −1.60020 | 1.73205 | −1.43937 | − | 7.14194i | −2.77162 | − | 1.56899i | 8.70406 | 3.00000 | 11.4285i | |||||||||||||||||||||||||||||||||||||||||
| 22.4 | −1.60020 | 1.73205 | −1.43937 | 7.14194i | −2.77162 | 1.56899i | 8.70406 | 3.00000 | − | 11.4285i | ||||||||||||||||||||||||||||||||||||||||||
| 22.5 | 0.0947876 | −1.73205 | −3.99102 | − | 6.23412i | −0.164177 | − | 9.32317i | −0.757449 | 3.00000 | − | 0.590917i | ||||||||||||||||||||||||||||||||||||||||
| 22.6 | 0.0947876 | −1.73205 | −3.99102 | 6.23412i | −0.164177 | 9.32317i | −0.757449 | 3.00000 | 0.590917i | |||||||||||||||||||||||||||||||||||||||||||
| 22.7 | 2.33225 | 1.73205 | 1.43937 | − | 6.11432i | 4.03957 | 7.45756i | −5.97201 | 3.00000 | − | 14.2601i | |||||||||||||||||||||||||||||||||||||||||
| 22.8 | 2.33225 | 1.73205 | 1.43937 | 6.11432i | 4.03957 | − | 7.45756i | −5.97201 | 3.00000 | 14.2601i | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 69.3.d.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 207.3.d.d | 8 | ||
| 4.b | odd | 2 | 1 | 1104.3.c.a | 8 | ||
| 23.b | odd | 2 | 1 | inner | 69.3.d.a | ✓ | 8 |
| 69.c | even | 2 | 1 | 207.3.d.d | 8 | ||
| 92.b | even | 2 | 1 | 1104.3.c.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.3.d.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 69.3.d.a | ✓ | 8 | 23.b | odd | 2 | 1 | inner |
| 207.3.d.d | 8 | 3.b | odd | 2 | 1 | ||
| 207.3.d.d | 8 | 69.c | even | 2 | 1 | ||
| 1104.3.c.a | 8 | 4.b | odd | 2 | 1 | ||
| 1104.3.c.a | 8 | 92.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(69, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 2 T^{3} - 6 T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( (T^{2} - 3)^{4} \)
|
| $5$ |
\( T^{8} + 156 T^{6} + \cdots + 2130192 \)
|
| $7$ |
\( T^{8} + 324 T^{6} + \cdots + 2130192 \)
|
| $11$ |
\( T^{8} + \cdots + 1126871568 \)
|
| $13$ |
\( (T^{4} + 4 T^{3} + \cdots - 2288)^{2} \)
|
| $17$ |
\( T^{8} + 1548 T^{6} + \cdots + 19171728 \)
|
| $19$ |
\( T^{8} + \cdots + 3240022032 \)
|
| $23$ |
\( T^{8} + \cdots + 78310985281 \)
|
| $29$ |
\( (T^{4} - 52 T^{3} + \cdots - 126512)^{2} \)
|
| $31$ |
\( (T^{4} - 8 T^{3} + \cdots + 47632)^{2} \)
|
| $37$ |
\( T^{8} + 2160 T^{6} + \cdots + 172545552 \)
|
| $41$ |
\( (T^{4} - 16 T^{3} + \cdots - 7856)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 24388568208 \)
|
| $47$ |
\( (T^{4} - 76 T^{3} + \cdots - 19136)^{2} \)
|
| $53$ |
\( T^{8} + \cdots + 3240022032 \)
|
| $59$ |
\( (T^{4} + 80 T^{3} + \cdots + 10925008)^{2} \)
|
| $61$ |
\( T^{8} + \cdots + 1070807044752 \)
|
| $67$ |
\( T^{8} + \cdots + 15402824028432 \)
|
| $71$ |
\( (T^{4} - 40 T^{3} + \cdots - 289904)^{2} \)
|
| $73$ |
\( (T^{4} - 32 T^{3} + \cdots + 571792)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 306696525522192 \)
|
| $83$ |
\( T^{8} + \cdots + 36106091910288 \)
|
| $89$ |
\( T^{8} + \cdots + 4181142987792 \)
|
| $97$ |
\( T^{8} + \cdots + 30372652449792 \)
|
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