Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,8,Mod(49,75)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(75, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.49");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.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: | \(23.4288769113\) |
| 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: |
\( x^{6} - 42x^{3} + 17161x^{2} - 5502x + 882 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 5^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 42x^{3} + 17161x^{2} - 5502x + 882 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -21\nu^{5} - 17161\nu^{4} - 2751\nu^{3} + 441\nu^{2} - 196553833 ) / 1123825 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -131\nu^{5} - 21\nu^{4} - 17161\nu^{3} + 2751\nu^{2} - 4495300\nu + 720762 ) / 2247650 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -131\nu^{5} - 21\nu^{4} - 17161\nu^{3} + 2751\nu^{2} + 720762 ) / 2247650 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -17161\nu^{5} - 2751\nu^{4} - 441\nu^{3} + 360381\nu^{2} - 294442150\nu + 47219172 ) / 47200650 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -17161\nu^{5} - 2751\nu^{4} - 441\nu^{3} + 899817\nu^{2} - 294442150\nu + 47219172 ) / 269718 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 175\beta_{4} ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 42\beta_{4} - 131\beta_{3} - 131\beta_{2} + 42 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 42\beta_{3} - 131\beta _1 - 22925 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 21\beta_{5} - 9177\beta_{4} - 17161\beta_{3} + 17161\beta_{2} + 21\beta _1 + 9177 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).
| \(n\) | \(26\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 18.3444i | 27.0000i | −208.517 | 0 | 495.299 | 254.472i | 1477.04i | −729.000 | 0 | |||||||||||||||||||||||||||||||||||
| 49.2 | − | 14.0237i | − | 27.0000i | −68.6631 | 0 | −378.639 | − | 1077.72i | − | 832.121i | −729.000 | 0 | |||||||||||||||||||||||||||||||||
| 49.3 | − | 1.67926i | 27.0000i | 125.180 | 0 | 45.3401 | − | 1415.20i | − | 425.156i | −729.000 | 0 | ||||||||||||||||||||||||||||||||||
| 49.4 | 1.67926i | − | 27.0000i | 125.180 | 0 | 45.3401 | 1415.20i | 425.156i | −729.000 | 0 | ||||||||||||||||||||||||||||||||||||
| 49.5 | 14.0237i | 27.0000i | −68.6631 | 0 | −378.639 | 1077.72i | 832.121i | −729.000 | 0 | |||||||||||||||||||||||||||||||||||||
| 49.6 | 18.3444i | − | 27.0000i | −208.517 | 0 | 495.299 | − | 254.472i | − | 1477.04i | −729.000 | 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 | 75.8.b.f | 6 | |
| 3.b | odd | 2 | 1 | 225.8.b.q | 6 | ||
| 5.b | even | 2 | 1 | inner | 75.8.b.f | 6 | |
| 5.c | odd | 4 | 1 | 75.8.a.g | ✓ | 3 | |
| 5.c | odd | 4 | 1 | 75.8.a.h | yes | 3 | |
| 15.d | odd | 2 | 1 | 225.8.b.q | 6 | ||
| 15.e | even | 4 | 1 | 225.8.a.x | 3 | ||
| 15.e | even | 4 | 1 | 225.8.a.y | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.8.a.g | ✓ | 3 | 5.c | odd | 4 | 1 | |
| 75.8.a.h | yes | 3 | 5.c | odd | 4 | 1 | |
| 75.8.b.f | 6 | 1.a | even | 1 | 1 | trivial | |
| 75.8.b.f | 6 | 5.b | even | 2 | 1 | inner | |
| 225.8.a.x | 3 | 15.e | even | 4 | 1 | ||
| 225.8.a.y | 3 | 15.e | even | 4 | 1 | ||
| 225.8.b.q | 6 | 3.b | odd | 2 | 1 | ||
| 225.8.b.q | 6 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 536T_{2}^{4} + 67684T_{2}^{2} + 186624 \)
acting on \(S_{8}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 536 T^{4} + \cdots + 186624 \)
|
| $3$ |
\( (T^{2} + 729)^{3} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 15\!\cdots\!25 \)
|
| $11$ |
\( (T^{3} - 1470 T^{2} + \cdots + 102949073496)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 17\!\cdots\!61 \)
|
| $17$ |
\( T^{6} + \cdots + 48\!\cdots\!36 \)
|
| $19$ |
\( (T^{3} + \cdots - 19077937228225)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 70\!\cdots\!00 \)
|
| $29$ |
\( (T^{3} + \cdots - 853069492719000)^{2} \)
|
| $31$ |
\( (T^{3} + \cdots + 857899428441975)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 20\!\cdots\!00 \)
|
| $41$ |
\( (T^{3} + \cdots + 30\!\cdots\!20)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 27\!\cdots\!89 \)
|
| $47$ |
\( T^{6} + \cdots + 27\!\cdots\!36 \)
|
| $53$ |
\( T^{6} + \cdots + 79\!\cdots\!00 \)
|
| $59$ |
\( (T^{3} + \cdots - 15\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{3} + \cdots - 14\!\cdots\!87)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 30\!\cdots\!61 \)
|
| $71$ |
\( (T^{3} + \cdots - 21\!\cdots\!08)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 21\!\cdots\!00 \)
|
| $79$ |
\( (T^{3} + \cdots - 18\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 77\!\cdots\!96 \)
|
| $89$ |
\( (T^{3} + \cdots - 19\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 57\!\cdots\!41 \)
|
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