Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [175,10,Mod(99,175)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(175, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("175.99");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 175.b (of order \(2\), degree \(1\), not 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: | \(90.1312713287\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 853x^{4} + 185508x^{2} + 4064256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 7) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 853x^{4} + 185508x^{2} + 4064256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 1163\nu^{3} + 675324\nu ) / 3205440 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -11\nu^{4} - 6287\nu^{2} - 480096 ) / 9540 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{4} - 1817\nu^{2} + 77688 ) / 1908 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 692\nu^{3} + 94501\nu ) / 11130 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 29\nu^{5} + 19697\nu^{3} + 4509828\nu ) / 320544 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{4} + 2\beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 11\beta_{3} - 25\beta_{2} - 1706 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -415\beta_{5} + 451\beta_{4} + 9538\beta_1 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -6287\beta_{3} + 9085\beta_{2} + 713186 ) / 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 192679\beta_{5} - 150811\beta_{4} - 6789298\beta_1 ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.1 |
|
− | 41.8019i | 0.232339i | −1235.40 | 0 | 9.71222 | − | 2401.00i | 30239.6i | 19682.9 | 0 | ||||||||||||||||||||||||||||||||||
| 99.2 | − | 34.1627i | 79.6469i | −655.088 | 0 | 2720.95 | 2401.00i | 4888.28i | 13339.4 | 0 | ||||||||||||||||||||||||||||||||||||
| 99.3 | − | 13.3607i | 163.415i | 333.491 | 0 | 2183.34 | − | 2401.00i | − | 11296.4i | −7021.32 | 0 | ||||||||||||||||||||||||||||||||||
| 99.4 | 13.3607i | − | 163.415i | 333.491 | 0 | 2183.34 | 2401.00i | 11296.4i | −7021.32 | 0 | ||||||||||||||||||||||||||||||||||||
| 99.5 | 34.1627i | − | 79.6469i | −655.088 | 0 | 2720.95 | − | 2401.00i | − | 4888.28i | 13339.4 | 0 | ||||||||||||||||||||||||||||||||||
| 99.6 | 41.8019i | − | 0.232339i | −1235.40 | 0 | 9.71222 | 2401.00i | − | 30239.6i | 19682.9 | 0 | |||||||||||||||||||||||||||||||||||
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 | 175.10.b.d | 6 | |
| 5.b | even | 2 | 1 | inner | 175.10.b.d | 6 | |
| 5.c | odd | 4 | 1 | 7.10.a.b | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 175.10.a.d | 3 | ||
| 15.e | even | 4 | 1 | 63.10.a.e | 3 | ||
| 20.e | even | 4 | 1 | 112.10.a.h | 3 | ||
| 35.f | even | 4 | 1 | 49.10.a.c | 3 | ||
| 35.k | even | 12 | 2 | 49.10.c.e | 6 | ||
| 35.l | odd | 12 | 2 | 49.10.c.d | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.10.a.b | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 49.10.a.c | 3 | 35.f | even | 4 | 1 | ||
| 49.10.c.d | 6 | 35.l | odd | 12 | 2 | ||
| 49.10.c.e | 6 | 35.k | even | 12 | 2 | ||
| 63.10.a.e | 3 | 15.e | even | 4 | 1 | ||
| 112.10.a.h | 3 | 20.e | even | 4 | 1 | ||
| 175.10.a.d | 3 | 5.c | odd | 4 | 1 | ||
| 175.10.b.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 175.10.b.d | 6 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 3093T_{2}^{4} + 2559636T_{2}^{2} + 364046400 \)
acting on \(S_{10}^{\mathrm{new}}(175, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 3093 T^{4} + \cdots + 364046400 \)
|
| $3$ |
\( T^{6} + 33048 T^{4} + \cdots + 9144576 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( (T^{2} + 5764801)^{3} \)
|
| $11$ |
\( (T^{3} + \cdots + 108859759460352)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 17\!\cdots\!00 \)
|
| $17$ |
\( T^{6} + \cdots + 48\!\cdots\!24 \)
|
| $19$ |
\( (T^{3} + \cdots + 43\!\cdots\!60)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 94\!\cdots\!96 \)
|
| $29$ |
\( (T^{3} + \cdots - 44\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{3} + \cdots - 74\!\cdots\!84)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 11\!\cdots\!84 \)
|
| $41$ |
\( (T^{3} + \cdots - 19\!\cdots\!12)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 47\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots + 19\!\cdots\!56 \)
|
| $53$ |
\( T^{6} + \cdots + 57\!\cdots\!84 \)
|
| $59$ |
\( (T^{3} + \cdots - 42\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots + 51\!\cdots\!08)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 41\!\cdots\!96 \)
|
| $71$ |
\( (T^{3} + \cdots + 16\!\cdots\!80)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 38\!\cdots\!04 \)
|
| $79$ |
\( (T^{3} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 36\!\cdots\!04 \)
|
| $89$ |
\( (T^{3} + \cdots - 19\!\cdots\!40)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 24\!\cdots\!56 \)
|
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