Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,16,Mod(1,75)]
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, 0]))
N = Newforms(chi, 16, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.1");
S:= CuspForms(chi, 16);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.a (trivial) |
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: | yes |
| Analytic conductor: | \(107.020128825\) |
| 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: |
\( x^{6} - 140297x^{4} - 1279200x^{3} + 3920349703x^{2} - 70310137200x - 19672158033999 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{13}\cdot 3^{4}\cdot 5^{7} \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{6} - 140297x^{4} - 1279200x^{3} + 3920349703x^{2} - 70310137200x - 19672158033999 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 14\nu - 46766 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9\nu^{5} + 1151\nu^{4} - 1141080\nu^{3} - 155293256\nu^{2} + 18077014855\nu + 1575300742593 ) / 35456 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 9\nu^{5} + 1151\nu^{4} - 1105624\nu^{3} - 157810632\nu^{2} + 15118566215\nu + 1670349840065 ) / 35456 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -23\nu^{5} - 2449\nu^{4} + 2901320\nu^{3} + 340699224\nu^{2} - 46373480185\nu - 3333830460751 ) / 17728 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 14\beta _1 + 46766 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} + 71\beta_{2} + 84434\beta _1 + 639624 \)
|
| \(\nu^{4}\) | \(=\) |
\( 36\beta_{5} + 30\beta_{4} + 154\beta_{3} + 116176\beta_{2} + 4488414\beta _1 + 3947554991 \)
|
| \(\nu^{5}\) | \(=\) |
\( -4604\beta_{5} + 122950\beta_{4} - 142542\beta_{3} + 11399040\beta_{2} + 8364097215\beta _1 + 208152225198 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−346.210 | 2187.00 | 87093.0 | 0 | −757160. | 4.02969e6 | −1.88078e7 | 4.78297e6 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −240.228 | 2187.00 | 24941.5 | 0 | −525379. | −3.11015e6 | 1.88015e6 | 4.78297e6 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −105.369 | 2187.00 | −21665.4 | 0 | −230442. | 764473. | 5.73558e6 | 4.78297e6 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 78.2494 | 2187.00 | −26645.0 | 0 | 171131. | 3.46184e6 | −4.64903e6 | 4.78297e6 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 82.7921 | 2187.00 | −25913.5 | 0 | 181066. | −3.04747e6 | −4.85836e6 | 4.78297e6 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 296.765 | 2187.00 | 55301.4 | 0 | 649025. | 491831. | 6.68712e6 | 4.78297e6 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 75.16.a.i | ✓ | 6 |
| 5.b | even | 2 | 1 | 75.16.a.j | yes | 6 | |
| 5.c | odd | 4 | 2 | 75.16.b.h | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.16.a.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 75.16.a.j | yes | 6 | 5.b | even | 2 | 1 | |
| 75.16.b.h | 12 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 234T_{2}^{5} - 117482T_{2}^{4} - 21979152T_{2}^{3} + 2525034496T_{2}^{2} + 196892384256T_{2} - 16848332439552 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 16848332439552 \)
|
| $3$ |
\( (T - 2187)^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots + 49\!\cdots\!25 \)
|
| $11$ |
\( T^{6} + \cdots - 41\!\cdots\!32 \)
|
| $13$ |
\( T^{6} + \cdots - 23\!\cdots\!67 \)
|
| $17$ |
\( T^{6} + \cdots - 40\!\cdots\!04 \)
|
| $19$ |
\( T^{6} + \cdots + 85\!\cdots\!25 \)
|
| $23$ |
\( T^{6} + \cdots - 40\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 39\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots - 24\!\cdots\!75 \)
|
| $37$ |
\( T^{6} + \cdots + 35\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots - 59\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots + 63\!\cdots\!61 \)
|
| $47$ |
\( T^{6} + \cdots + 26\!\cdots\!64 \)
|
| $53$ |
\( T^{6} + \cdots + 34\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 29\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 91\!\cdots\!81 \)
|
| $67$ |
\( T^{6} + \cdots + 55\!\cdots\!01 \)
|
| $71$ |
\( T^{6} + \cdots + 65\!\cdots\!64 \)
|
| $73$ |
\( T^{6} + \cdots - 35\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 10\!\cdots\!92 \)
|
| $89$ |
\( T^{6} + \cdots + 17\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 52\!\cdots\!59 \)
|
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