Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [70,3,Mod(43,70)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(70, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("70.43");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 70 = 2 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 70.f (of order \(4\), degree \(2\), 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: | \(1.90736185052\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.664811929600.20 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 18x^{6} + 85x^{4} + 92x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 5 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{8} + 18x^{6} + 85x^{4} + 92x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{6} - 14\nu^{5} + 38\nu^{4} - 19\nu^{3} + 168\nu^{2} + 114\nu + 32 ) / 40 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} - 2\nu^{6} - 14\nu^{5} - 38\nu^{4} - 19\nu^{3} - 168\nu^{2} + 114\nu - 32 ) / 40 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{7} - 52\nu^{5} - 227\nu^{3} - 198\nu ) / 40 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 2\nu^{6} + 19\nu^{5} - 33\nu^{4} + 104\nu^{3} - 123\nu^{2} + 196\nu - 42 ) / 20 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{7} + \nu^{6} - 90\nu^{5} + 14\nu^{4} - 415\nu^{3} + 19\nu^{2} - 370\nu - 74 ) / 40 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 3\nu^{6} + 19\nu^{5} + 47\nu^{4} + 104\nu^{3} + 162\nu^{2} + 176\nu + 48 ) / 20 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -5\nu^{7} - \nu^{6} - 90\nu^{5} - 14\nu^{4} - 415\nu^{3} - 19\nu^{2} - 370\nu + 74 ) / 40 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + 2\beta_{4} + 3\beta_{3} - 2\beta_{2} - \beta _1 + 1 ) / 5 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{7} + 2\beta_{6} - 5\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - 2\beta _1 - 23 ) / 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{7} - 7\beta_{6} + 5\beta_{5} - 14\beta_{4} - 41\beta_{3} + 19\beta_{2} + 12\beta _1 - 7 ) / 5 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -45\beta_{7} - 26\beta_{6} + 45\beta_{5} + 13\beta_{4} - 13\beta_{3} - 23\beta_{2} + 36\beta _1 + 209 ) / 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( -17\beta_{7} + 13\beta_{6} - 17\beta_{5} + 26\beta_{4} + 103\beta_{3} - 37\beta_{2} - 24\beta _1 + 13 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 435\beta_{7} + 326\beta_{6} - 435\beta_{5} - 163\beta_{4} + 163\beta_{3} + 303\beta_{2} - 466\beta _1 - 2119 ) / 5 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 1095 \beta_{7} - 663 \beta_{6} + 1095 \beta_{5} - 1326 \beta_{4} - 6089 \beta_{3} + 1901 \beta_{2} + \cdots - 663 ) / 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(57\) |
| \(\chi(n)\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 43.1 |
|
1.00000 | − | 1.00000i | −2.48226 | − | 2.48226i | − | 2.00000i | −4.90823 | − | 0.953586i | −4.96452 | 1.87083 | − | 1.87083i | −2.00000 | − | 2.00000i | 3.32325i | −5.86181 | + | 3.95464i | |||||||||||||||||||||||||||||
| 43.2 | 1.00000 | − | 1.00000i | −1.02638 | − | 1.02638i | − | 2.00000i | 3.62753 | − | 3.44108i | −2.05275 | −1.87083 | + | 1.87083i | −2.00000 | − | 2.00000i | − | 6.89310i | 0.186445 | − | 7.06861i | |||||||||||||||||||||||||||||
| 43.3 | 1.00000 | − | 1.00000i | 1.61143 | + | 1.61143i | − | 2.00000i | 2.03740 | + | 4.56607i | 3.22287 | 1.87083 | − | 1.87083i | −2.00000 | − | 2.00000i | − | 3.80656i | 6.60347 | + | 2.52868i | |||||||||||||||||||||||||||||
| 43.4 | 1.00000 | − | 1.00000i | 3.89721 | + | 3.89721i | − | 2.00000i | −2.75670 | − | 4.17140i | 7.79441 | −1.87083 | + | 1.87083i | −2.00000 | − | 2.00000i | 21.3764i | −6.92810 | − | 1.41471i | ||||||||||||||||||||||||||||||
| 57.1 | 1.00000 | + | 1.00000i | −2.48226 | + | 2.48226i | 2.00000i | −4.90823 | + | 0.953586i | −4.96452 | 1.87083 | + | 1.87083i | −2.00000 | + | 2.00000i | − | 3.32325i | −5.86181 | − | 3.95464i | ||||||||||||||||||||||||||||||
| 57.2 | 1.00000 | + | 1.00000i | −1.02638 | + | 1.02638i | 2.00000i | 3.62753 | + | 3.44108i | −2.05275 | −1.87083 | − | 1.87083i | −2.00000 | + | 2.00000i | 6.89310i | 0.186445 | + | 7.06861i | |||||||||||||||||||||||||||||||
| 57.3 | 1.00000 | + | 1.00000i | 1.61143 | − | 1.61143i | 2.00000i | 2.03740 | − | 4.56607i | 3.22287 | 1.87083 | + | 1.87083i | −2.00000 | + | 2.00000i | 3.80656i | 6.60347 | − | 2.52868i | |||||||||||||||||||||||||||||||
| 57.4 | 1.00000 | + | 1.00000i | 3.89721 | − | 3.89721i | 2.00000i | −2.75670 | + | 4.17140i | 7.79441 | −1.87083 | − | 1.87083i | −2.00000 | + | 2.00000i | − | 21.3764i | −6.92810 | + | 1.41471i | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 70.3.f.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 630.3.o.c | 8 | ||
| 4.b | odd | 2 | 1 | 560.3.bh.c | 8 | ||
| 5.b | even | 2 | 1 | 350.3.f.d | 8 | ||
| 5.c | odd | 4 | 1 | inner | 70.3.f.b | ✓ | 8 |
| 5.c | odd | 4 | 1 | 350.3.f.d | 8 | ||
| 7.b | odd | 2 | 1 | 490.3.f.n | 8 | ||
| 15.e | even | 4 | 1 | 630.3.o.c | 8 | ||
| 20.e | even | 4 | 1 | 560.3.bh.c | 8 | ||
| 35.f | even | 4 | 1 | 490.3.f.n | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.3.f.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 70.3.f.b | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
| 350.3.f.d | 8 | 5.b | even | 2 | 1 | ||
| 350.3.f.d | 8 | 5.c | odd | 4 | 1 | ||
| 490.3.f.n | 8 | 7.b | odd | 2 | 1 | ||
| 490.3.f.n | 8 | 35.f | even | 4 | 1 | ||
| 560.3.bh.c | 8 | 4.b | odd | 2 | 1 | ||
| 560.3.bh.c | 8 | 20.e | even | 4 | 1 | ||
| 630.3.o.c | 8 | 3.b | odd | 2 | 1 | ||
| 630.3.o.c | 8 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 4T_{3}^{7} + 8T_{3}^{6} + 52T_{3}^{5} + 313T_{3}^{4} - 416T_{3}^{3} + 512T_{3}^{2} + 2048T_{3} + 4096 \)
acting on \(S_{3}^{\mathrm{new}}(70, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 2)^{4} \)
|
| $3$ |
\( T^{8} - 4 T^{7} + \cdots + 4096 \)
|
| $5$ |
\( T^{8} + 4 T^{7} + \cdots + 390625 \)
|
| $7$ |
\( (T^{4} + 49)^{2} \)
|
| $11$ |
\( (T^{4} - 10 T^{3} + \cdots + 1600)^{2} \)
|
| $13$ |
\( T^{8} + 16 T^{7} + \cdots + 522488164 \)
|
| $17$ |
\( T^{8} - 8 T^{7} + \cdots + 442429156 \)
|
| $19$ |
\( T^{8} + 1188 T^{6} + \cdots + 167961600 \)
|
| $23$ |
\( T^{8} + \cdots + 6400000000 \)
|
| $29$ |
\( T^{8} + \cdots + 1921945600 \)
|
| $31$ |
\( (T^{4} + 40 T^{3} + \cdots + 27392)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 18415575616 \)
|
| $41$ |
\( (T^{4} - 120 T^{3} + \cdots - 3003328)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 16433188864 \)
|
| $47$ |
\( T^{8} + \cdots + 138717512704 \)
|
| $53$ |
\( T^{8} + \cdots + 1675191781264 \)
|
| $59$ |
\( T^{8} + \cdots + 1057442022400 \)
|
| $61$ |
\( (T^{4} - 48 T^{3} + \cdots - 40960)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 117631898238976 \)
|
| $71$ |
\( (T^{4} - 88 T^{3} + \cdots + 2588800)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 421045254400 \)
|
| $79$ |
\( T^{8} + \cdots + 114187176505600 \)
|
| $83$ |
\( T^{8} + \cdots + 516593985519616 \)
|
| $89$ |
\( T^{8} + \cdots + 26\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 19822530062500 \)
|
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