Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [75,22,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, 22, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("75.49");
S:= CuspForms(chi, 22);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| 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: | \(209.608008215\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} + 27459120 x^{14} + 298938371837796 x^{12} + \cdots + 57\!\cdots\!76 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | multiple of \( 2^{46}\cdot 3^{20}\cdot 5^{28}\cdot 7^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{15}\) 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
\( x^{16} + 27459120 x^{14} + 298938371837796 x^{12} + \cdots + 57\!\cdots\!76 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 11\!\cdots\!95 \nu^{14} + \cdots + 55\!\cdots\!60 ) / 81\!\cdots\!36 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 54\!\cdots\!55 \nu^{14} + \cdots + 90\!\cdots\!96 ) / 27\!\cdots\!12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 10\!\cdots\!11 \nu^{14} + \cdots - 16\!\cdots\!72 ) / 65\!\cdots\!64 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 14\!\cdots\!03 \nu^{14} + \cdots + 22\!\cdots\!88 ) / 19\!\cdots\!92 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 47\!\cdots\!41 \nu^{14} + \cdots - 15\!\cdots\!76 ) / 10\!\cdots\!56 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 32\!\cdots\!75 \nu^{14} + \cdots + 26\!\cdots\!20 ) / 21\!\cdots\!12 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 45\!\cdots\!43 \nu^{14} + \cdots - 47\!\cdots\!52 ) / 23\!\cdots\!68 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 15\!\cdots\!93 \nu^{15} + \cdots - 24\!\cdots\!60 \nu ) / 14\!\cdots\!84 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 12\!\cdots\!19 \nu^{15} + \cdots + 35\!\cdots\!64 \nu ) / 14\!\cdots\!84 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 65\!\cdots\!07 \nu^{15} + \cdots + 10\!\cdots\!88 \nu ) / 18\!\cdots\!48 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( - 60\!\cdots\!89 \nu^{15} + \cdots - 24\!\cdots\!72 \nu ) / 36\!\cdots\!88 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( 12\!\cdots\!77 \nu^{15} + \cdots + 39\!\cdots\!12 \nu ) / 11\!\cdots\!64 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( - 12\!\cdots\!47 \nu^{15} + \cdots - 26\!\cdots\!36 \nu ) / 40\!\cdots\!68 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 98\!\cdots\!35 \nu^{15} + \cdots + 25\!\cdots\!08 \nu ) / 24\!\cdots\!08 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - 30\!\cdots\!51 \nu^{15} + \cdots - 38\!\cdots\!68 \nu ) / 24\!\cdots\!08 \)
|
| \(\nu\) | \(=\) |
\( \beta_{9} + 83\beta_{8} \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 147\beta _1 - 3432353 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{12} + 3\beta_{11} - 279\beta_{10} - 5678217\beta_{9} - 786777187\beta_{8} \)
|
| \(\nu^{4}\) | \(=\) |
\( 4 \beta_{7} - 44 \beta_{6} - 10 \beta_{5} + 1160 \beta_{4} + 444 \beta_{3} - 7686666 \beta_{2} + \cdots + 19515369675016 \)
|
| \(\nu^{5}\) | \(=\) |
\( 28512 \beta_{15} - 6064 \beta_{14} + 46308 \beta_{13} - 9856854 \beta_{12} + \cdots + 76\!\cdots\!94 \beta_{8} \)
|
| \(\nu^{6}\) | \(=\) |
\( - 118950712 \beta_{7} + 483676616 \beta_{6} + 68912356 \beta_{5} - 15130667276 \beta_{4} + \cdots - 12\!\cdots\!00 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 380206535696 \beta_{15} + 84509567472 \beta_{14} - 634372808384 \beta_{13} + \cdots - 79\!\cdots\!68 \beta_{8} \)
|
| \(\nu^{8}\) | \(=\) |
\( 13\!\cdots\!28 \beta_{7} + \cdots + 88\!\cdots\!92 \)
|
| \(\nu^{9}\) | \(=\) |
\( 39\!\cdots\!44 \beta_{15} + \cdots + 78\!\cdots\!44 \beta_{8} \)
|
| \(\nu^{10}\) | \(=\) |
\( - 12\!\cdots\!44 \beta_{7} + \cdots - 65\!\cdots\!96 \)
|
| \(\nu^{11}\) | \(=\) |
\( - 37\!\cdots\!44 \beta_{15} + \cdots - 73\!\cdots\!52 \beta_{8} \)
|
| \(\nu^{12}\) | \(=\) |
\( 10\!\cdots\!12 \beta_{7} + \cdots + 48\!\cdots\!12 \)
|
| \(\nu^{13}\) | \(=\) |
\( 33\!\cdots\!56 \beta_{15} + \cdots + 66\!\cdots\!16 \beta_{8} \)
|
| \(\nu^{14}\) | \(=\) |
\( - 81\!\cdots\!60 \beta_{7} + \cdots - 37\!\cdots\!84 \)
|
| \(\nu^{15}\) | \(=\) |
\( - 30\!\cdots\!76 \beta_{15} + \cdots - 58\!\cdots\!20 \beta_{8} \)
|
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 |
|
− | 2845.71i | − | 59049.0i | −6.00093e6 | 0 | −1.68037e8 | − | 1.14091e9i | 1.11090e10i | −3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.2 | − | 2605.59i | 59049.0i | −4.69194e6 | 0 | 1.53857e8 | − | 1.24132e9i | 6.76095e9i | −3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.3 | − | 2127.46i | 59049.0i | −2.42895e6 | 0 | 1.25625e8 | 1.04187e9i | 7.05898e8i | −3.48678e9 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.4 | − | 1910.63i | − | 59049.0i | −1.55335e6 | 0 | −1.12821e8 | − | 1.48601e8i | − | 1.03900e9i | −3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.5 | − | 1725.33i | − | 59049.0i | −879617. | 0 | −1.01879e8 | 5.00000e8i | − | 2.10065e9i | −3.48678e9 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.6 | − | 1154.86i | 59049.0i | 763454. | 0 | 6.81932e7 | − | 5.42271e7i | − | 3.30359e9i | −3.48678e9 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.7 | − | 238.916i | − | 59049.0i | 2.04007e6 | 0 | −1.41077e7 | − | 6.22955e8i | − | 9.88448e8i | −3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.8 | − | 166.679i | 59049.0i | 2.06937e6 | 0 | 9.84223e6 | − | 1.02476e9i | − | 6.94472e8i | −3.48678e9 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.9 | 166.679i | − | 59049.0i | 2.06937e6 | 0 | 9.84223e6 | 1.02476e9i | 6.94472e8i | −3.48678e9 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.10 | 238.916i | 59049.0i | 2.04007e6 | 0 | −1.41077e7 | 6.22955e8i | 9.88448e8i | −3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.11 | 1154.86i | − | 59049.0i | 763454. | 0 | 6.81932e7 | 5.42271e7i | 3.30359e9i | −3.48678e9 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.12 | 1725.33i | 59049.0i | −879617. | 0 | −1.01879e8 | − | 5.00000e8i | 2.10065e9i | −3.48678e9 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.13 | 1910.63i | 59049.0i | −1.55335e6 | 0 | −1.12821e8 | 1.48601e8i | 1.03900e9i | −3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.14 | 2127.46i | − | 59049.0i | −2.42895e6 | 0 | 1.25625e8 | − | 1.04187e9i | − | 7.05898e8i | −3.48678e9 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.15 | 2605.59i | − | 59049.0i | −4.69194e6 | 0 | 1.53857e8 | 1.24132e9i | − | 6.76095e9i | −3.48678e9 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 49.16 | 2845.71i | 59049.0i | −6.00093e6 | 0 | −1.68037e8 | 1.14091e9i | − | 1.11090e10i | −3.48678e9 | 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 | 75.22.b.j | 16 | |
| 5.b | even | 2 | 1 | inner | 75.22.b.j | 16 | |
| 5.c | odd | 4 | 1 | 75.22.a.k | ✓ | 8 | |
| 5.c | odd | 4 | 1 | 75.22.a.l | yes | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.22.a.k | ✓ | 8 | 5.c | odd | 4 | 1 | |
| 75.22.a.l | yes | 8 | 5.c | odd | 4 | 1 | |
| 75.22.b.j | 16 | 1.a | even | 1 | 1 | trivial | |
| 75.22.b.j | 16 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 27459120 T_{2}^{14} + 298938371837796 T_{2}^{12} + \cdots + 57\!\cdots\!76 \)
acting on \(S_{22}^{\mathrm{new}}(75, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{16} + \cdots + 57\!\cdots\!76 \)
|
| $3$ |
\( (T^{2} + 3486784401)^{8} \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} + \cdots + 14\!\cdots\!25 \)
|
| $11$ |
\( (T^{8} + \cdots - 32\!\cdots\!04)^{2} \)
|
| $13$ |
\( T^{16} + \cdots + 26\!\cdots\!41 \)
|
| $17$ |
\( T^{16} + \cdots + 16\!\cdots\!56 \)
|
| $19$ |
\( (T^{8} + \cdots + 10\!\cdots\!25)^{2} \)
|
| $23$ |
\( T^{16} + \cdots + 99\!\cdots\!00 \)
|
| $29$ |
\( (T^{8} + \cdots - 81\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{8} + \cdots + 65\!\cdots\!25)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 34\!\cdots\!00 \)
|
| $41$ |
\( (T^{8} + \cdots + 39\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{16} + \cdots + 31\!\cdots\!01 \)
|
| $47$ |
\( T^{16} + \cdots + 31\!\cdots\!96 \)
|
| $53$ |
\( T^{16} + \cdots + 28\!\cdots\!00 \)
|
| $59$ |
\( (T^{8} + \cdots + 72\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{8} + \cdots - 14\!\cdots\!39)^{2} \)
|
| $67$ |
\( T^{16} + \cdots + 30\!\cdots\!61 \)
|
| $71$ |
\( (T^{8} + \cdots - 43\!\cdots\!44)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 30\!\cdots\!00 \)
|
| $79$ |
\( (T^{8} + \cdots - 13\!\cdots\!00)^{2} \)
|
| $83$ |
\( T^{16} + \cdots + 84\!\cdots\!56 \)
|
| $89$ |
\( (T^{8} + \cdots - 35\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{16} + \cdots + 91\!\cdots\!81 \)
|
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