Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [70,4,Mod(11,70)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(70, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("70.11");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 70 = 2 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 70.e (of order \(3\), 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: | \(4.13013370040\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.362560708800.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 67x^{4} - 114x^{3} + 4446x^{2} - 5940x + 8100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 67x^{4} - 114x^{3} + 4446x^{2} - 5940x + 8100 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{5} + 67\nu^{4} - 4489\nu^{3} + 4446\nu^{2} - 5940\nu + 397980 ) / 291942 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 737\nu^{5} - 722\nu^{4} + 48374\nu^{3} - 16683\nu^{2} + 3210012\nu + 90450 ) / 4379130 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2078\nu^{5} - 6745\nu^{4} - 131969\nu^{3} - 249327\nu^{2} - 8256132\nu - 13790520 ) / 875826 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2276\nu^{5} + 6521\nu^{4} - 144965\nu^{3} + 630981\nu^{2} - 9432252\nu + 25597350 ) / 875826 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{5} + \beta_{4} + 45\beta_{3} + \beta_{2} - 2\beta _1 - 45 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} - \beta_{4} - 66\beta_{2} + 45 \)
|
| \(\nu^{4}\) | \(=\) |
\( -67\beta_{5} - 134\beta_{4} - 2925\beta_{3} - 67\beta_{2} + 158\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -86\beta_{5} - 43\beta_{4} + 4095\beta_{3} + 4289\beta_{2} - 4246\beta _1 - 4095 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(57\) |
| \(\chi(n)\) | \(-1 + \beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
−1.00000 | − | 1.73205i | −4.45365 | + | 7.71395i | −2.00000 | + | 3.46410i | −2.50000 | − | 4.33013i | 17.8146 | −1.45365 | − | 18.4631i | 8.00000 | −26.1700 | − | 45.3278i | −5.00000 | + | 8.66025i | ||||||||||||||||||||||
| 11.2 | −1.00000 | − | 1.73205i | −1.18717 | + | 2.05625i | −2.00000 | + | 3.46410i | −2.50000 | − | 4.33013i | 4.74870 | 1.81283 | + | 18.4313i | 8.00000 | 10.6812 | + | 18.5004i | −5.00000 | + | 8.66025i | |||||||||||||||||||||||
| 11.3 | −1.00000 | − | 1.73205i | 3.64083 | − | 6.30609i | −2.00000 | + | 3.46410i | −2.50000 | − | 4.33013i | −14.5633 | 6.64083 | − | 17.2887i | 8.00000 | −13.0112 | − | 22.5361i | −5.00000 | + | 8.66025i | |||||||||||||||||||||||
| 51.1 | −1.00000 | + | 1.73205i | −4.45365 | − | 7.71395i | −2.00000 | − | 3.46410i | −2.50000 | + | 4.33013i | 17.8146 | −1.45365 | + | 18.4631i | 8.00000 | −26.1700 | + | 45.3278i | −5.00000 | − | 8.66025i | |||||||||||||||||||||||
| 51.2 | −1.00000 | + | 1.73205i | −1.18717 | − | 2.05625i | −2.00000 | − | 3.46410i | −2.50000 | + | 4.33013i | 4.74870 | 1.81283 | − | 18.4313i | 8.00000 | 10.6812 | − | 18.5004i | −5.00000 | − | 8.66025i | |||||||||||||||||||||||
| 51.3 | −1.00000 | + | 1.73205i | 3.64083 | + | 6.30609i | −2.00000 | − | 3.46410i | −2.50000 | + | 4.33013i | −14.5633 | 6.64083 | + | 17.2887i | 8.00000 | −13.0112 | + | 22.5361i | −5.00000 | − | 8.66025i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 70.4.e.e | ✓ | 6 |
| 3.b | odd | 2 | 1 | 630.4.k.r | 6 | ||
| 4.b | odd | 2 | 1 | 560.4.q.m | 6 | ||
| 5.b | even | 2 | 1 | 350.4.e.k | 6 | ||
| 5.c | odd | 4 | 2 | 350.4.j.i | 12 | ||
| 7.b | odd | 2 | 1 | 490.4.e.y | 6 | ||
| 7.c | even | 3 | 1 | inner | 70.4.e.e | ✓ | 6 |
| 7.c | even | 3 | 1 | 490.4.a.w | 3 | ||
| 7.d | odd | 6 | 1 | 490.4.a.v | 3 | ||
| 7.d | odd | 6 | 1 | 490.4.e.y | 6 | ||
| 21.h | odd | 6 | 1 | 630.4.k.r | 6 | ||
| 28.g | odd | 6 | 1 | 560.4.q.m | 6 | ||
| 35.i | odd | 6 | 1 | 2450.4.a.ce | 3 | ||
| 35.j | even | 6 | 1 | 350.4.e.k | 6 | ||
| 35.j | even | 6 | 1 | 2450.4.a.cb | 3 | ||
| 35.l | odd | 12 | 2 | 350.4.j.i | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.4.e.e | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 70.4.e.e | ✓ | 6 | 7.c | even | 3 | 1 | inner |
| 350.4.e.k | 6 | 5.b | even | 2 | 1 | ||
| 350.4.e.k | 6 | 35.j | even | 6 | 1 | ||
| 350.4.j.i | 12 | 5.c | odd | 4 | 2 | ||
| 350.4.j.i | 12 | 35.l | odd | 12 | 2 | ||
| 490.4.a.v | 3 | 7.d | odd | 6 | 1 | ||
| 490.4.a.w | 3 | 7.c | even | 3 | 1 | ||
| 490.4.e.y | 6 | 7.b | odd | 2 | 1 | ||
| 490.4.e.y | 6 | 7.d | odd | 6 | 1 | ||
| 560.4.q.m | 6 | 4.b | odd | 2 | 1 | ||
| 560.4.q.m | 6 | 28.g | odd | 6 | 1 | ||
| 630.4.k.r | 6 | 3.b | odd | 2 | 1 | ||
| 630.4.k.r | 6 | 21.h | odd | 6 | 1 | ||
| 2450.4.a.cb | 3 | 35.j | even | 6 | 1 | ||
| 2450.4.a.ce | 3 | 35.i | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 4T_{3}^{5} + 77T_{3}^{4} + 64T_{3}^{3} + 4337T_{3}^{2} + 9394T_{3} + 23716 \)
acting on \(S_{4}^{\mathrm{new}}(70, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 4)^{3} \)
|
| $3$ |
\( T^{6} + 4 T^{5} + \cdots + 23716 \)
|
| $5$ |
\( (T^{2} + 5 T + 25)^{3} \)
|
| $7$ |
\( T^{6} - 14 T^{5} + \cdots + 40353607 \)
|
| $11$ |
\( T^{6} + \cdots + 10489856400 \)
|
| $13$ |
\( (T^{3} + T^{2} + \cdots - 19180)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 44899914816 \)
|
| $19$ |
\( T^{6} + \cdots + 348912313344 \)
|
| $23$ |
\( T^{6} + \cdots + 5665366321209 \)
|
| $29$ |
\( (T^{3} - 268 T^{2} + \cdots + 562914)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 1155367014400 \)
|
| $37$ |
\( T^{6} + \cdots + 364355408905216 \)
|
| $41$ |
\( (T^{3} - 313 T^{2} + \cdots + 15201837)^{2} \)
|
| $43$ |
\( (T^{3} - 360 T^{2} + \cdots + 21473350)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 14\!\cdots\!56 \)
|
| $53$ |
\( T^{6} + \cdots + 89\!\cdots\!64 \)
|
| $59$ |
\( T^{6} + \cdots + 24\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 252484981330176 \)
|
| $67$ |
\( T^{6} + \cdots + 212379790441536 \)
|
| $71$ |
\( (T^{3} + 1080 T^{2} + \cdots - 45563904)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 25\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots + 696242104960000 \)
|
| $83$ |
\( (T^{3} - 626 T^{2} + \cdots + 46561032)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 22\!\cdots\!04 \)
|
| $97$ |
\( (T^{3} - 1626 T^{2} + \cdots + 35869960)^{2} \)
|
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