Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [70,5,Mod(41,70)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(70, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("70.41");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 70 = 2 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 70.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: | \(7.23589741587\) |
| 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: |
\( x^{8} + 74x^{6} - 36x^{5} + 1871x^{4} - 1152x^{3} + 18998x^{2} - 11916x + 63406 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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
\( x^{8} + 74x^{6} - 36x^{5} + 1871x^{4} - 1152x^{3} + 18998x^{2} - 11916x + 63406 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 232 \nu^{7} - 5418 \nu^{6} + 17206 \nu^{5} - 321552 \nu^{4} + 650874 \nu^{3} - 5712462 \nu^{2} + \cdots - 31107132 ) / 3382687 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 1714 \nu^{7} - 10488 \nu^{6} + 74854 \nu^{5} - 662295 \nu^{4} + 522310 \nu^{3} - 9779925 \nu^{2} + \cdots - 32555037 ) / 13574679 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 167452 \nu^{7} + 453736 \nu^{6} + 16531726 \nu^{5} + 34616629 \nu^{4} + 484847416 \nu^{3} + \cdots + 2175673339 ) / 1045250283 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 12\nu^{7} + 22\nu^{6} + 894\nu^{5} + 1516\nu^{4} + 19326\nu^{3} + 33658\nu^{2} + 103836\nu + 203728 ) / 23793 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 986796 \nu^{7} - 182791 \nu^{6} + 59537142 \nu^{5} - 41685952 \nu^{4} + 1103815734 \nu^{3} + \cdots + 1061942126 ) / 1045250283 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 4258 \nu^{7} - 27052 \nu^{6} - 227170 \nu^{5} - 1390958 \nu^{4} - 2190828 \nu^{3} + \cdots - 38321059 ) / 3382687 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 4468 \nu^{7} - 3212 \nu^{6} - 296522 \nu^{5} + 174101 \nu^{4} - 5577174 \nu^{3} + \cdots + 88008410 ) / 3382687 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + 2\beta_{3} + 2\beta_{2} - \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + 2\beta_{5} - \beta_{3} - \beta_{2} - 5\beta _1 - 37 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 6\beta_{7} - 3\beta_{6} + 18\beta_{5} + 11\beta_{4} - 49\beta_{3} - 115\beta_{2} + 58\beta _1 + 57 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{7} - 22\beta_{6} - 42\beta_{5} + 12\beta_{4} + 30\beta_{3} + 48\beta_{2} + 101\beta _1 + 438 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -370\beta_{7} + 275\beta_{6} - 774\beta_{5} + 115\beta_{4} + 1345\beta_{3} + 3847\beta_{2} - 2494\beta _1 - 3965 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( -468\beta_{7} + 740\beta_{6} + 1649\beta_{5} - 708\beta_{4} - 1450\beta_{3} - 3322\beta_{2} - 3421\beta _1 - 11462 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 16786 \beta_{7} - 15575 \beta_{6} + 26586 \beta_{5} - 10571 \beta_{4} - 37511 \beta_{3} - 112985 \beta_{2} + \cdots + 205961 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/70\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(57\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 41.1 |
|
−2.82843 | − | 7.24056i | 8.00000 | 11.1803i | 20.4794i | 47.9994 | − | 9.85182i | −22.6274 | 28.5743 | − | 31.6228i | ||||||||||||||||||||||||||||||||||||||
| 41.2 | −2.82843 | − | 3.17249i | 8.00000 | 11.1803i | 8.97314i | −48.9405 | − | 2.41364i | −22.6274 | 70.9353 | − | 31.6228i | |||||||||||||||||||||||||||||||||||||||
| 41.3 | −2.82843 | 3.17249i | 8.00000 | − | 11.1803i | − | 8.97314i | −48.9405 | + | 2.41364i | −22.6274 | 70.9353 | 31.6228i | |||||||||||||||||||||||||||||||||||||||
| 41.4 | −2.82843 | 7.24056i | 8.00000 | − | 11.1803i | − | 20.4794i | 47.9994 | + | 9.85182i | −22.6274 | 28.5743 | 31.6228i | |||||||||||||||||||||||||||||||||||||||
| 41.5 | 2.82843 | − | 15.5889i | 8.00000 | − | 11.1803i | − | 44.0921i | 45.4357 | + | 18.3467i | 22.6274 | −162.014 | − | 31.6228i | |||||||||||||||||||||||||||||||||||||
| 41.6 | 2.82843 | − | 0.703741i | 8.00000 | 11.1803i | − | 1.99048i | 21.5055 | + | 44.0286i | 22.6274 | 80.5047 | 31.6228i | |||||||||||||||||||||||||||||||||||||||
| 41.7 | 2.82843 | 0.703741i | 8.00000 | − | 11.1803i | 1.99048i | 21.5055 | − | 44.0286i | 22.6274 | 80.5047 | − | 31.6228i | |||||||||||||||||||||||||||||||||||||||
| 41.8 | 2.82843 | 15.5889i | 8.00000 | 11.1803i | 44.0921i | 45.4357 | − | 18.3467i | 22.6274 | −162.014 | 31.6228i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 70.5.b.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 630.5.f.a | 8 | ||
| 4.b | odd | 2 | 1 | 560.5.f.a | 8 | ||
| 5.b | even | 2 | 1 | 350.5.b.b | 8 | ||
| 5.c | odd | 4 | 2 | 350.5.d.b | 16 | ||
| 7.b | odd | 2 | 1 | inner | 70.5.b.a | ✓ | 8 |
| 21.c | even | 2 | 1 | 630.5.f.a | 8 | ||
| 28.d | even | 2 | 1 | 560.5.f.a | 8 | ||
| 35.c | odd | 2 | 1 | 350.5.b.b | 8 | ||
| 35.f | even | 4 | 2 | 350.5.d.b | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.5.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 70.5.b.a | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 350.5.b.b | 8 | 5.b | even | 2 | 1 | ||
| 350.5.b.b | 8 | 35.c | odd | 2 | 1 | ||
| 350.5.d.b | 16 | 5.c | odd | 4 | 2 | ||
| 350.5.d.b | 16 | 35.f | even | 4 | 2 | ||
| 560.5.f.a | 8 | 4.b | odd | 2 | 1 | ||
| 560.5.f.a | 8 | 28.d | even | 2 | 1 | ||
| 630.5.f.a | 8 | 3.b | odd | 2 | 1 | ||
| 630.5.f.a | 8 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(70, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8)^{4} \)
|
| $3$ |
\( T^{8} + 306 T^{6} + \cdots + 63504 \)
|
| $5$ |
\( (T^{2} + 125)^{4} \)
|
| $7$ |
\( T^{8} + \cdots + 33232930569601 \)
|
| $11$ |
\( (T^{4} + 150 T^{3} + \cdots - 13889756)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 71\!\cdots\!64 \)
|
| $17$ |
\( T^{8} + \cdots + 17\!\cdots\!84 \)
|
| $19$ |
\( T^{8} + \cdots + 15\!\cdots\!44 \)
|
| $23$ |
\( (T^{4} - 1332 T^{3} + \cdots + 10899863296)^{2} \)
|
| $29$ |
\( (T^{4} + 186 T^{3} + \cdots + 516529094596)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 44\!\cdots\!44 \)
|
| $37$ |
\( (T^{4} - 1236 T^{3} + \cdots + 343500544576)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 15\!\cdots\!64 \)
|
| $43$ |
\( (T^{4} + \cdots - 7604882814704)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 10\!\cdots\!04 \)
|
| $53$ |
\( (T^{4} + \cdots - 14067172624496)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 13\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots + 18\!\cdots\!04 \)
|
| $67$ |
\( (T^{4} + \cdots - 56762946241776)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots - 101846257160816)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 19\!\cdots\!84 \)
|
| $79$ |
\( (T^{4} + \cdots - 12\!\cdots\!64)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 18\!\cdots\!64 \)
|
| $89$ |
\( T^{8} + \cdots + 13\!\cdots\!04 \)
|
| $97$ |
\( T^{8} + \cdots + 30\!\cdots\!24 \)
|
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