Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,22,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, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.1");
S:= CuspForms(chi, 22);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| 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: | \(209.608008215\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{6} - x^{5} - 530373x^{4} - 23850203x^{3} + 59940607490x^{2} + 2888057920800x - 684850637484000 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{18}\cdot 3^{6}\cdot 5^{7}\cdot 7^{2} \) |
| 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
\( x^{6} - x^{5} - 530373x^{4} - 23850203x^{3} + 59940607490x^{2} + 2888057920800x - 684850637484000 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 16\nu^{2} - 1100\nu - 2828475 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 157\nu^{4} - 456377\nu^{3} + 37318593\nu^{2} + 36666498918\nu - 909462022508 ) / 4004 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 157\nu^{4} + 712633\nu^{3} - 66403649\nu^{2} - 112675632118\nu + 2940276712412 ) / 4004 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 571\nu^{4} - 544465\nu^{3} - 249005991\nu^{2} + 48958029442\nu + 11600260895639 ) / 1001 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 275\beta _1 + 2828750 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 454\beta_{2} + 4870675\beta _1 + 781801849 ) / 64 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 88\beta_{5} + 121\beta_{4} - 231\beta_{3} + 1628146\beta_{2} + 751841447\beta _1 + 3444795464265 ) / 64 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 6908 \beta_{5} + 237687 \beta_{4} + 338183 \beta_{3} + 156769854 \beta_{2} + 856594358333 \beta _1 + 266494147996751 ) / 32 \)
|
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 |
|
−2300.75 | 59049.0 | 3.19629e6 | 0 | −1.35857e8 | 2.10374e8 | −2.52885e9 | 3.48678e9 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −1570.86 | 59049.0 | 370456. | 0 | −9.27578e7 | −1.39580e9 | 2.71240e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −594.286 | 59049.0 | −1.74398e6 | 0 | −3.50920e7 | 9.49356e8 | 2.28273e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 340.910 | 59049.0 | −1.98093e6 | 0 | 2.01304e7 | −6.73159e8 | −1.39026e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 1464.04 | 59049.0 | 46252.4 | 0 | 8.64499e7 | 1.40408e8 | −3.00259e9 | 3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2568.95 | 59049.0 | 4.50235e6 | 0 | 1.51694e8 | −5.58323e8 | 6.17885e9 | 3.48678e9 | 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 | 75.22.a.i | ✓ | 6 |
| 5.b | even | 2 | 1 | 75.22.a.j | yes | 6 | |
| 5.c | odd | 4 | 2 | 75.22.b.i | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.22.a.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 75.22.a.j | yes | 6 | 5.b | even | 2 | 1 | |
| 75.22.b.i | 12 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 92 T_{2}^{5} - 8482448 T_{2}^{4} - 2069443264 T_{2}^{3} + 15258494066176 T_{2}^{2} + \cdots - 27\!\cdots\!28 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + \cdots - 27\!\cdots\!28 \)
|
| $3$ |
\( (T - 59049)^{6} \)
|
| $5$ |
\( T^{6} \)
|
| $7$ |
\( T^{6} + \cdots - 14\!\cdots\!75 \)
|
| $11$ |
\( T^{6} + \cdots - 99\!\cdots\!52 \)
|
| $13$ |
\( T^{6} + \cdots + 45\!\cdots\!97 \)
|
| $17$ |
\( T^{6} + \cdots - 85\!\cdots\!96 \)
|
| $19$ |
\( T^{6} + \cdots - 69\!\cdots\!75 \)
|
| $23$ |
\( T^{6} + \cdots - 72\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots - 38\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 43\!\cdots\!25 \)
|
| $37$ |
\( T^{6} + \cdots - 26\!\cdots\!00 \)
|
| $41$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots - 29\!\cdots\!31 \)
|
| $47$ |
\( T^{6} + \cdots - 14\!\cdots\!04 \)
|
| $53$ |
\( T^{6} + \cdots + 61\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots - 36\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 14\!\cdots\!41 \)
|
| $67$ |
\( T^{6} + \cdots - 96\!\cdots\!51 \)
|
| $71$ |
\( T^{6} + \cdots + 54\!\cdots\!84 \)
|
| $73$ |
\( T^{6} + \cdots + 27\!\cdots\!00 \)
|
| $79$ |
\( T^{6} + \cdots - 37\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 16\!\cdots\!32 \)
|
| $89$ |
\( T^{6} + \cdots - 85\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 80\!\cdots\!91 \)
|
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