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: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 2 x^{9} + 2 x^{8} - 2016 x^{7} + 425617 x^{6} - 2170178 x^{5} + 5521250 x^{4} + \cdots + 231472080000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{9}\cdot 5^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 2 x^{9} + 2 x^{8} - 2016 x^{7} + 425617 x^{6} - 2170178 x^{5} + 5521250 x^{4} + \cdots + 231472080000 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 26\!\cdots\!89 \nu^{9} + \cdots - 13\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 33\!\cdots\!88 \nu^{9} + \cdots - 36\!\cdots\!00 ) / 12\!\cdots\!00 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 43\!\cdots\!67 \nu^{9} + \cdots + 43\!\cdots\!00 ) / 96\!\cdots\!56 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 43\!\cdots\!67 \nu^{9} + \cdots + 43\!\cdots\!00 ) / 96\!\cdots\!56 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 48\!\cdots\!79 \nu^{9} + \cdots - 52\!\cdots\!96 ) / 25\!\cdots\!52 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 52\!\cdots\!91 \nu^{9} + \cdots - 15\!\cdots\!20 ) / 24\!\cdots\!64 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 16\!\cdots\!49 \nu^{9} + \cdots + 86\!\cdots\!44 ) / 25\!\cdots\!52 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 89\!\cdots\!81 \nu^{9} + \cdots + 43\!\cdots\!00 ) / 68\!\cdots\!40 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 52\!\cdots\!83 \nu^{9} + \cdots + 25\!\cdots\!00 ) / 12\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{4} + \beta_{3} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{8} + 4\beta_{4} + 680\beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{9} - 3 \beta_{8} + 22 \beta_{7} + 3 \beta_{6} - \beta_{5} - 499 \beta_{4} - 499 \beta_{3} + \cdots + 1408 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -76\beta_{7} - 667\beta_{6} + 52\beta_{5} + 4108\beta_{3} - 338892 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 795 \beta_{9} + 3729 \beta_{8} + 18558 \beta_{7} + 3729 \beta_{6} - 795 \beta_{5} + 280069 \beta_{4} + \cdots + 1434684 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 46164\beta_{9} - 422725\beta_{8} - 3506644\beta_{4} - 108876\beta_{2} - 190186556\beta_1 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 577765 \beta_{9} + 3195135 \beta_{8} - 12541018 \beta_{7} - 3195135 \beta_{6} + 577765 \beta_{5} + \cdots - 1216877884 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 109929220\beta_{7} + 266138563\beta_{6} - 32627836\beta_{5} - 2735984620\beta_{3} + 113575944372 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 408695619 \beta_{9} - 2453336073 \beta_{8} - 7980884838 \beta_{7} - 2453336073 \beta_{6} + \cdots - 945309969492 ) / 2 \)
|
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 |
|
− | 36.4619i | 68.7273i | −817.473 | 0 | 2505.93 | 2401.00i | 11138.1i | 14959.6 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 99.2 | − | 32.5415i | 234.084i | −546.947 | 0 | 7617.44 | − | 2401.00i | 1137.23i | −35112.4 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 99.3 | − | 22.7661i | − | 239.496i | −6.29742 | 0 | −5452.41 | − | 2401.00i | − | 11512.9i | −37675.5 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 99.4 | − | 21.5262i | − | 221.976i | 48.6242 | 0 | −4778.29 | 2401.00i | − | 12068.1i | −29590.3 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 99.5 | − | 4.68049i | − | 7.83652i | 490.093 | 0 | −36.6787 | − | 2401.00i | − | 4690.29i | 19621.6 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 99.6 | 4.68049i | 7.83652i | 490.093 | 0 | −36.6787 | 2401.00i | 4690.29i | 19621.6 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 99.7 | 21.5262i | 221.976i | 48.6242 | 0 | −4778.29 | − | 2401.00i | 12068.1i | −29590.3 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 99.8 | 22.7661i | 239.496i | −6.29742 | 0 | −5452.41 | 2401.00i | 11512.9i | −37675.5 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 99.9 | 32.5415i | − | 234.084i | −546.947 | 0 | 7617.44 | 2401.00i | − | 1137.23i | −35112.4 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 99.10 | 36.4619i | − | 68.7273i | −817.473 | 0 | 2505.93 | − | 2401.00i | − | 11138.1i | 14959.6 | 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.f | 10 | |
| 5.b | even | 2 | 1 | inner | 175.10.b.f | 10 | |
| 5.c | odd | 4 | 1 | 35.10.a.d | ✓ | 5 | |
| 5.c | odd | 4 | 1 | 175.10.a.f | 5 | ||
| 15.e | even | 4 | 1 | 315.10.a.j | 5 | ||
| 35.f | even | 4 | 1 | 245.10.a.f | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.10.a.d | ✓ | 5 | 5.c | odd | 4 | 1 | |
| 175.10.a.f | 5 | 5.c | odd | 4 | 1 | ||
| 175.10.b.f | 10 | 1.a | even | 1 | 1 | trivial | |
| 175.10.b.f | 10 | 5.b | even | 2 | 1 | inner | |
| 245.10.a.f | 5 | 35.f | even | 4 | 1 | ||
| 315.10.a.j | 5 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 3392T_{2}^{8} + 4066484T_{2}^{6} + 2043125392T_{2}^{4} + 380958829056T_{2}^{2} + 7407106560000 \)
acting on \(S_{10}^{\mathrm{new}}(175, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + \cdots + 7407106560000 \)
|
| $3$ |
\( T^{10} + \cdots + 44\!\cdots\!96 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( (T^{2} + 5764801)^{5} \)
|
| $11$ |
\( (T^{5} + \cdots - 27\!\cdots\!08)^{2} \)
|
| $13$ |
\( T^{10} + \cdots + 18\!\cdots\!00 \)
|
| $17$ |
\( T^{10} + \cdots + 59\!\cdots\!56 \)
|
| $19$ |
\( (T^{5} + \cdots - 10\!\cdots\!00)^{2} \)
|
| $23$ |
\( T^{10} + \cdots + 76\!\cdots\!36 \)
|
| $29$ |
\( (T^{5} + \cdots + 10\!\cdots\!50)^{2} \)
|
| $31$ |
\( (T^{5} + \cdots + 97\!\cdots\!00)^{2} \)
|
| $37$ |
\( T^{10} + \cdots + 21\!\cdots\!64 \)
|
| $41$ |
\( (T^{5} + \cdots - 20\!\cdots\!68)^{2} \)
|
| $43$ |
\( T^{10} + \cdots + 52\!\cdots\!00 \)
|
| $47$ |
\( T^{10} + \cdots + 71\!\cdots\!84 \)
|
| $53$ |
\( T^{10} + \cdots + 13\!\cdots\!84 \)
|
| $59$ |
\( (T^{5} + \cdots - 14\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{5} + \cdots - 43\!\cdots\!12)^{2} \)
|
| $67$ |
\( T^{10} + \cdots + 10\!\cdots\!96 \)
|
| $71$ |
\( (T^{5} + \cdots + 60\!\cdots\!00)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 17\!\cdots\!76 \)
|
| $79$ |
\( (T^{5} + \cdots - 61\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 12\!\cdots\!84 \)
|
| $89$ |
\( (T^{5} + \cdots + 14\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 28\!\cdots\!36 \)
|
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