Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [175,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("175.99");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 175 = 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(28.0671684673\) |
| 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: |
\( x^{6} - 116x^{3} + 2401x^{2} - 5684x + 6728 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 35) |
| 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
\( x^{6} - 116x^{3} + 2401x^{2} - 5684x + 6728 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2401\nu^{5} + 2842\nu^{4} + 3364\nu^{3} - 139258\nu^{2} + 5599965\nu - 7018754 ) / 6628530 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 49\nu^{5} + 58\nu^{4} + 2401\nu^{3} - 2842\nu^{2} + 228570\nu - 278516 ) / 114285 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 49\nu^{5} + 58\nu^{4} + 2401\nu^{3} - 2842\nu^{2} - 278516 ) / 114285 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -52\nu^{5} - 839\nu^{4} - 2548\nu^{3} + 3016\nu^{2} - 961567 ) / 38095 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3871\nu^{5} - 4582\nu^{4} + 8816\nu^{3} + 573388\nu^{2} - 8330775\nu + 10490054 ) / 348870 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 2\beta_{2} + 33\beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 49\beta_{3} + 49\beta_{2} - 116\beta _1 + 116 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -49\beta_{4} - 156\beta_{3} - 1617 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 116\beta_{5} + 116\beta_{4} + 2633\beta_{3} - 2633\beta_{2} + 9512\beta _1 + 9512 ) / 2 \)
|
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 |
|
− | 10.3867i | − | 27.9421i | −75.8842 | 0 | −290.227 | 49.0000i | 455.813i | −537.760 | 0 | ||||||||||||||||||||||||||||||||||
| 99.2 | − | 8.91996i | 13.7828i | −47.5657 | 0 | 122.942 | − | 49.0000i | 138.845i | 53.0335 | 0 | |||||||||||||||||||||||||||||||||||
| 99.3 | − | 4.53323i | 15.7249i | 11.4498 | 0 | 71.2847 | 49.0000i | − | 196.968i | −4.27317 | 0 | |||||||||||||||||||||||||||||||||||
| 99.4 | 4.53323i | − | 15.7249i | 11.4498 | 0 | 71.2847 | − | 49.0000i | 196.968i | −4.27317 | 0 | |||||||||||||||||||||||||||||||||||
| 99.5 | 8.91996i | − | 13.7828i | −47.5657 | 0 | 122.942 | 49.0000i | − | 138.845i | 53.0335 | 0 | |||||||||||||||||||||||||||||||||||
| 99.6 | 10.3867i | 27.9421i | −75.8842 | 0 | −290.227 | − | 49.0000i | − | 455.813i | −537.760 | 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 | 175.6.b.e | 6 | |
| 5.b | even | 2 | 1 | inner | 175.6.b.e | 6 | |
| 5.c | odd | 4 | 1 | 35.6.a.c | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 175.6.a.e | 3 | ||
| 15.e | even | 4 | 1 | 315.6.a.i | 3 | ||
| 20.e | even | 4 | 1 | 560.6.a.q | 3 | ||
| 35.f | even | 4 | 1 | 245.6.a.d | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.6.a.c | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 175.6.a.e | 3 | 5.c | odd | 4 | 1 | ||
| 175.6.b.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 175.6.b.e | 6 | 5.b | even | 2 | 1 | inner | |
| 245.6.a.d | 3 | 35.f | even | 4 | 1 | ||
| 315.6.a.i | 3 | 15.e | even | 4 | 1 | ||
| 560.6.a.q | 3 | 20.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 208T_{2}^{4} + 12436T_{2}^{2} + 176400 \)
acting on \(S_{6}^{\mathrm{new}}(175, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 208 T^{4} + \cdots + 176400 \)
|
| $3$ |
\( T^{6} + 1218 T^{4} + \cdots + 36675136 \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( (T^{2} + 2401)^{3} \)
|
| $11$ |
\( (T^{3} + 194 T^{2} + \cdots - 3144468)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 29\!\cdots\!00 \)
|
| $17$ |
\( T^{6} + \cdots + 17\!\cdots\!44 \)
|
| $19$ |
\( (T^{3} + 1212 T^{2} + \cdots + 140259040)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 15\!\cdots\!56 \)
|
| $29$ |
\( (T^{3} - 11332 T^{2} + \cdots - 51669601050)^{2} \)
|
| $31$ |
\( (T^{3} - 9200 T^{2} + \cdots + 8818293376)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 10\!\cdots\!64 \)
|
| $41$ |
\( (T^{3} + \cdots + 4748673370848)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 22\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots + 11\!\cdots\!56 \)
|
| $53$ |
\( T^{6} + \cdots + 26\!\cdots\!84 \)
|
| $59$ |
\( (T^{3} + \cdots - 7000620748800)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots - 1590789613952)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 70\!\cdots\!56 \)
|
| $71$ |
\( (T^{3} + \cdots - 77522711777280)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 13\!\cdots\!24 \)
|
| $79$ |
\( (T^{3} + \cdots + 40598762145400)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 28\!\cdots\!44 \)
|
| $89$ |
\( (T^{3} + \cdots + 31660963259040)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 13\!\cdots\!56 \)
|
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