Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [770,2,Mod(309,770)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(770, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("770.309");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 770.c (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: | \(6.14848095564\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.1698758656.6 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{8} + 18x^{6} + 97x^{4} + 176x^{2} + 64 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{6} + 14\nu^{4} + 37\nu^{2} - 8 ) / 16 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + 18\nu^{5} + 8\nu^{4} + 105\nu^{3} + 72\nu^{2} + 248\nu + 64 ) / 64 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} - 18\nu^{5} + 8\nu^{4} - 105\nu^{3} + 72\nu^{2} - 248\nu + 64 ) / 64 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} - 18\nu^{5} - 89\nu^{3} - 104\nu ) / 32 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} - 46\nu^{5} - 179\nu^{3} - 168\nu ) / 64 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 4\nu^{6} + 14\nu^{5} + 60\nu^{4} + 37\nu^{3} + 216\nu^{2} - 8\nu + 160 ) / 32 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - 4\nu^{6} + 14\nu^{5} - 60\nu^{4} + 37\nu^{3} - 216\nu^{2} - 8\nu - 160 ) / 32 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} - 4\beta _1 - 10 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -9\beta_{7} - 9\beta_{6} - 18\beta_{5} + 4\beta_{4} + 5\beta_{3} - 5\beta_{2} ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 9\beta_{7} - 9\beta_{6} + 17\beta_{3} + 17\beta_{2} + 36\beta _1 + 74 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 81\beta_{7} + 81\beta_{6} + 178\beta_{5} - 68\beta_{4} - 37\beta_{3} + 37\beta_{2} ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -89\beta_{7} + 89\beta_{6} - 201\beta_{3} - 201\beta_{2} - 324\beta _1 - 650 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -761\beta_{7} - 761\beta_{6} - 1810\beta_{5} + 804\beta_{4} + 325\beta_{3} - 325\beta_{2} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/770\mathbb{Z}\right)^\times\).
| \(n\) | \(211\) | \(617\) | \(661\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 309.1 |
|
− | 1.00000i | − | 1.41421i | −1.00000 | −2.18183 | + | 0.489528i | −1.41421 | 1.00000i | 1.00000i | 1.00000 | 0.489528 | + | 2.18183i | ||||||||||||||||||||||||||||||||||||
| 309.2 | − | 1.00000i | − | 1.41421i | −1.00000 | 1.88893 | − | 1.19663i | −1.41421 | 1.00000i | 1.00000i | 1.00000 | −1.19663 | − | 1.88893i | |||||||||||||||||||||||||||||||||||||
| 309.3 | − | 1.00000i | 1.41421i | −1.00000 | −1.63280 | − | 1.52773i | 1.41421 | 1.00000i | 1.00000i | 1.00000 | −1.52773 | + | 1.63280i | ||||||||||||||||||||||||||||||||||||||
| 309.4 | − | 1.00000i | 1.41421i | −1.00000 | −0.0743018 | + | 2.23483i | 1.41421 | 1.00000i | 1.00000i | 1.00000 | 2.23483 | + | 0.0743018i | ||||||||||||||||||||||||||||||||||||||
| 309.5 | 1.00000i | − | 1.41421i | −1.00000 | −1.63280 | + | 1.52773i | 1.41421 | − | 1.00000i | − | 1.00000i | 1.00000 | −1.52773 | − | 1.63280i | ||||||||||||||||||||||||||||||||||||
| 309.6 | 1.00000i | − | 1.41421i | −1.00000 | −0.0743018 | − | 2.23483i | 1.41421 | − | 1.00000i | − | 1.00000i | 1.00000 | 2.23483 | − | 0.0743018i | ||||||||||||||||||||||||||||||||||||
| 309.7 | 1.00000i | 1.41421i | −1.00000 | −2.18183 | − | 0.489528i | −1.41421 | − | 1.00000i | − | 1.00000i | 1.00000 | 0.489528 | − | 2.18183i | |||||||||||||||||||||||||||||||||||||
| 309.8 | 1.00000i | 1.41421i | −1.00000 | 1.88893 | + | 1.19663i | −1.41421 | − | 1.00000i | − | 1.00000i | 1.00000 | −1.19663 | + | 1.88893i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 770.2.c.e | ✓ | 8 |
| 5.b | even | 2 | 1 | inner | 770.2.c.e | ✓ | 8 |
| 5.c | odd | 4 | 1 | 3850.2.a.bx | 4 | ||
| 5.c | odd | 4 | 1 | 3850.2.a.by | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 770.2.c.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 770.2.c.e | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 3850.2.a.bx | 4 | 5.c | odd | 4 | 1 | ||
| 3850.2.a.by | 4 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\):
|
\( T_{3}^{2} + 2 \)
|
|
\( T_{13}^{8} + 60T_{13}^{6} + 1156T_{13}^{4} + 7616T_{13}^{2} + 12544 \)
|
|
\( T_{17}^{8} + 88T_{17}^{6} + 2512T_{17}^{4} + 23296T_{17}^{2} + 1024 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{4} \)
|
| $3$ |
\( (T^{2} + 2)^{4} \)
|
| $5$ |
\( T^{8} + 4 T^{7} + \cdots + 625 \)
|
| $7$ |
\( (T^{2} + 1)^{4} \)
|
| $11$ |
\( (T + 1)^{8} \)
|
| $13$ |
\( T^{8} + 60 T^{6} + \cdots + 12544 \)
|
| $17$ |
\( T^{8} + 88 T^{6} + \cdots + 1024 \)
|
| $19$ |
\( (T^{2} - 4 T - 14)^{4} \)
|
| $23$ |
\( T^{8} + 172 T^{6} + \cdots + 40000 \)
|
| $29$ |
\( (T^{4} - 12 T^{3} + \cdots - 2104)^{2} \)
|
| $31$ |
\( (T^{4} + 8 T^{3} - 48 T^{2} + \cdots - 16)^{2} \)
|
| $37$ |
\( T^{8} + 68 T^{6} + \cdots + 64 \)
|
| $41$ |
\( (T^{4} + 8 T^{3} + \cdots - 1600)^{2} \)
|
| $43$ |
\( T^{8} + 152 T^{6} + \cdots + 1024 \)
|
| $47$ |
\( (T^{4} + 72 T^{2} + 784)^{2} \)
|
| $53$ |
\( T^{8} + 212 T^{6} + \cdots + 141376 \)
|
| $59$ |
\( (T^{4} - 32 T^{3} + \cdots + 2032)^{2} \)
|
| $61$ |
\( (T^{4} + 8 T^{3} + \cdots - 248)^{2} \)
|
| $67$ |
\( T^{8} + 200 T^{6} + \cdots + 65536 \)
|
| $71$ |
\( (T^{4} - 4 T^{3} + \cdots + 224)^{2} \)
|
| $73$ |
\( T^{8} + 376 T^{6} + \cdots + 52301824 \)
|
| $79$ |
\( (T^{4} - 16 T^{3} + \cdots - 7624)^{2} \)
|
| $83$ |
\( T^{8} + 60 T^{6} + \cdots + 12544 \)
|
| $89$ |
\( (T^{4} - 24 T^{3} + \cdots - 8848)^{2} \)
|
| $97$ |
\( T^{8} + 436 T^{6} + \cdots + 40000 \)
|
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