Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [750,3,Mod(193,750)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(750, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("750.193");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 750 = 2 \cdot 3 \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 750.f (of order \(4\), degree \(2\), 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: | \(20.4360198270\) |
| Analytic rank: | \(0\) |
| Dimension: | \(16\) |
| Relative dimension: | \(8\) over \(\Q(i)\) |
| Coefficient field: | 16.0.6879707136000000000000.9 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{16} - 9x^{12} + 81x^{8} - 729x^{4} + 6561 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 5^{8} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{16} - 9x^{12} + 81x^{8} - 729x^{4} + 6561 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} ) / 9 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{10} ) / 243 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{15} ) / 2187 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{12} + 9\nu^{9} + 27\nu^{8} + 27\nu^{7} - 162\nu^{4} - 243\nu^{3} + 729\nu + 729 ) / 729 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{15} + \nu^{14} - 18 \nu^{10} - 27 \nu^{9} + 81 \nu^{7} + 243 \nu^{6} + 243 \nu^{5} + \cdots - 2187 \nu ) / 2187 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{14} + 6\nu^{13} - 3\nu^{12} + 27\nu^{9} + 27\nu^{8} - 81\nu^{6} + 243\nu^{5} - 6561\nu + 2187 ) / 2187 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 2 \nu^{14} + 12 \nu^{13} - 6 \nu^{12} + 9 \nu^{10} - 81 \nu^{9} + 54 \nu^{8} - 162 \nu^{6} + \cdots + 2187 ) / 2187 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -2\nu^{15} - \nu^{14} + 3\nu^{12} + 36\nu^{11} - 27\nu^{8} - 81\nu^{7} - 81\nu^{6} + 2187\nu^{3} - 2187 ) / 2187 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - \nu^{15} + 2 \nu^{14} - 6 \nu^{12} + 18 \nu^{11} - 9 \nu^{10} + 54 \nu^{8} - 243 \nu^{7} + \cdots + 2187 ) / 2187 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - \nu^{15} + 12 \nu^{12} - 27 \nu^{9} + 54 \nu^{8} - 81 \nu^{7} + 243 \nu^{5} + 486 \nu^{4} + \cdots - 2187 ) / 2187 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 4\nu^{14} - 27\nu^{10} + 27\nu^{9} - 81\nu^{7} + 162\nu^{6} + 729\nu^{3} - 4374\nu^{2} + 2187\nu ) / 2187 \)
|
| \(\beta_{12}\) | \(=\) |
\( ( - \nu^{14} + 3 \nu^{12} + 3 \nu^{10} + 9 \nu^{9} - 9 \nu^{8} + 27 \nu^{6} - 81 \nu^{5} + 324 \nu^{4} + \cdots - 1458 ) / 729 \)
|
| \(\beta_{13}\) | \(=\) |
\( ( -5\nu^{12} + 45\nu^{8} + 2187 ) / 729 \)
|
| \(\beta_{14}\) | \(=\) |
\( ( 5\nu^{14} - 18\nu^{10} + 405\nu^{6} ) / 2187 \)
|
| \(\beta_{15}\) | \(=\) |
\( ( - \nu^{15} - 3 \nu^{14} - 9 \nu^{12} + 9 \nu^{10} + 27 \nu^{8} - 81 \nu^{7} + 81 \nu^{6} + \cdots + 4374 ) / 2187 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{12} + \beta_{11} - \beta_{10} + \beta_{7} - 2\beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + 3\beta _1 + 1 ) / 10 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{14} - 3\beta_{11} - 3\beta_{5} + 3\beta_{3} - 9\beta_{2} + 3\beta_1 ) / 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3 \beta_{15} + 3 \beta_{14} + 3 \beta_{13} + 3 \beta_{11} + 3 \beta_{10} - 6 \beta_{9} + \cdots + 9 \beta_{3} ) / 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -9\beta_{15} + 9\beta_{13} + 9\beta_{12} - 9\beta_{4} - 9\beta_{3} + 9\beta _1 + 18 ) / 10 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 27\beta_{15} + 27\beta_{14} + 27\beta_{12} + 27\beta_{5} + 54\beta_{2} ) / 10 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 27 \beta_{15} + 27 \beta_{14} + 27 \beta_{13} - 27 \beta_{11} - 27 \beta_{10} - 54 \beta_{9} + \cdots - 81 \beta_{3} ) / 10 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 81\beta_{13} + 81\beta_{10} + 81\beta_{4} + 81\beta_{3} - 81\beta _1 - 243 ) / 10 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 81 \beta_{12} + 81 \beta_{11} - 81 \beta_{10} - 81 \beta_{7} + 162 \beta_{6} - 81 \beta_{5} + 81 \beta_{4} + \cdots - 81 ) / 10 \)
|
| \(\nu^{10}\) | \(=\) |
\( 243\beta_{2} \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 243 \beta_{15} - 243 \beta_{11} - 243 \beta_{10} + 243 \beta_{9} + 486 \beta_{8} + 243 \beta_{5} + \cdots + 243 ) / 10 \)
|
| \(\nu^{12}\) | \(=\) |
\( ( -729\beta_{13} + 729\beta_{10} + 729\beta_{4} + 729\beta_{3} - 729\beta _1 + 2187 ) / 10 \)
|
| \(\nu^{13}\) | \(=\) |
\( ( 729 \beta_{14} - 729 \beta_{13} + 729 \beta_{12} + 729 \beta_{11} - 729 \beta_{10} + 1458 \beta_{7} + \cdots - 1458 \beta_1 ) / 10 \)
|
| \(\nu^{14}\) | \(=\) |
\( ( -2187\beta_{15} + 2187\beta_{14} - 2187\beta_{12} - 2187\beta_{5} + 4374\beta_{2} ) / 10 \)
|
| \(\nu^{15}\) | \(=\) |
\( 2187\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/750\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(251\) |
| \(\chi(n)\) | \(-\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 193.1 |
|
−1.00000 | + | 1.00000i | −1.22474 | − | 1.22474i | − | 2.00000i | 0 | 2.44949 | −7.22047 | + | 7.22047i | 2.00000 | + | 2.00000i | 3.00000i | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.2 | −1.00000 | + | 1.00000i | −1.22474 | − | 1.22474i | − | 2.00000i | 0 | 2.44949 | −0.758620 | + | 0.758620i | 2.00000 | + | 2.00000i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.3 | −1.00000 | + | 1.00000i | −1.22474 | − | 1.22474i | − | 2.00000i | 0 | 2.44949 | 3.19783 | − | 3.19783i | 2.00000 | + | 2.00000i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.4 | −1.00000 | + | 1.00000i | −1.22474 | − | 1.22474i | − | 2.00000i | 0 | 2.44949 | 5.68025 | − | 5.68025i | 2.00000 | + | 2.00000i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.5 | −1.00000 | + | 1.00000i | 1.22474 | + | 1.22474i | − | 2.00000i | 0 | −2.44949 | −9.66996 | + | 9.66996i | 2.00000 | + | 2.00000i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.6 | −1.00000 | + | 1.00000i | 1.22474 | + | 1.22474i | − | 2.00000i | 0 | −2.44949 | −3.20811 | + | 3.20811i | 2.00000 | + | 2.00000i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.7 | −1.00000 | + | 1.00000i | 1.22474 | + | 1.22474i | − | 2.00000i | 0 | −2.44949 | 0.748338 | − | 0.748338i | 2.00000 | + | 2.00000i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 193.8 | −1.00000 | + | 1.00000i | 1.22474 | + | 1.22474i | − | 2.00000i | 0 | −2.44949 | 3.23076 | − | 3.23076i | 2.00000 | + | 2.00000i | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 307.1 | −1.00000 | − | 1.00000i | −1.22474 | + | 1.22474i | 2.00000i | 0 | 2.44949 | −7.22047 | − | 7.22047i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 307.2 | −1.00000 | − | 1.00000i | −1.22474 | + | 1.22474i | 2.00000i | 0 | 2.44949 | −0.758620 | − | 0.758620i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 307.3 | −1.00000 | − | 1.00000i | −1.22474 | + | 1.22474i | 2.00000i | 0 | 2.44949 | 3.19783 | + | 3.19783i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 307.4 | −1.00000 | − | 1.00000i | −1.22474 | + | 1.22474i | 2.00000i | 0 | 2.44949 | 5.68025 | + | 5.68025i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 307.5 | −1.00000 | − | 1.00000i | 1.22474 | − | 1.22474i | 2.00000i | 0 | −2.44949 | −9.66996 | − | 9.66996i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 307.6 | −1.00000 | − | 1.00000i | 1.22474 | − | 1.22474i | 2.00000i | 0 | −2.44949 | −3.20811 | − | 3.20811i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 307.7 | −1.00000 | − | 1.00000i | 1.22474 | − | 1.22474i | 2.00000i | 0 | −2.44949 | 0.748338 | + | 0.748338i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 307.8 | −1.00000 | − | 1.00000i | 1.22474 | − | 1.22474i | 2.00000i | 0 | −2.44949 | 3.23076 | + | 3.23076i | 2.00000 | − | 2.00000i | − | 3.00000i | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 750.3.f.a | ✓ | 16 |
| 5.b | even | 2 | 1 | 750.3.f.d | yes | 16 | |
| 5.c | odd | 4 | 1 | inner | 750.3.f.a | ✓ | 16 |
| 5.c | odd | 4 | 1 | 750.3.f.d | yes | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 750.3.f.a | ✓ | 16 | 1.a | even | 1 | 1 | trivial |
| 750.3.f.a | ✓ | 16 | 5.c | odd | 4 | 1 | inner |
| 750.3.f.d | yes | 16 | 5.b | even | 2 | 1 | |
| 750.3.f.d | yes | 16 | 5.c | odd | 4 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{16} + 16 T_{7}^{15} + 128 T_{7}^{14} - 736 T_{7}^{13} + 5678 T_{7}^{12} + 80816 T_{7}^{11} + \cdots + 14256598801 \)
acting on \(S_{3}^{\mathrm{new}}(750, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 2)^{8} \)
|
| $3$ |
\( (T^{4} + 9)^{4} \)
|
| $5$ |
\( T^{16} \)
|
| $7$ |
\( T^{16} + \cdots + 14256598801 \)
|
| $11$ |
\( (T^{8} - 8 T^{7} + \cdots + 1944025)^{2} \)
|
| $13$ |
\( T^{16} + \cdots + 58\!\cdots\!61 \)
|
| $17$ |
\( T^{16} + \cdots + 13\!\cdots\!21 \)
|
| $19$ |
\( T^{16} + \cdots + 12\!\cdots\!25 \)
|
| $23$ |
\( T^{16} + \cdots + 20\!\cdots\!25 \)
|
| $29$ |
\( T^{16} + \cdots + 17\!\cdots\!25 \)
|
| $31$ |
\( (T^{8} - 20 T^{7} + \cdots - 23698978319)^{2} \)
|
| $37$ |
\( T^{16} + \cdots + 15\!\cdots\!21 \)
|
| $41$ |
\( (T^{8} - 64 T^{7} + \cdots + 552505735921)^{2} \)
|
| $43$ |
\( T^{16} + \cdots + 34\!\cdots\!41 \)
|
| $47$ |
\( T^{16} + \cdots + 35\!\cdots\!25 \)
|
| $53$ |
\( T^{16} + \cdots + 11\!\cdots\!01 \)
|
| $59$ |
\( T^{16} + \cdots + 86\!\cdots\!25 \)
|
| $61$ |
\( (T^{8} + \cdots + 22865848332025)^{2} \)
|
| $67$ |
\( T^{16} + \cdots + 67\!\cdots\!76 \)
|
| $71$ |
\( (T^{8} + \cdots + 53117427107025)^{2} \)
|
| $73$ |
\( T^{16} + \cdots + 23\!\cdots\!81 \)
|
| $79$ |
\( T^{16} + \cdots + 23\!\cdots\!00 \)
|
| $83$ |
\( T^{16} + \cdots + 68\!\cdots\!01 \)
|
| $89$ |
\( T^{16} + \cdots + 25\!\cdots\!25 \)
|
| $97$ |
\( T^{16} + \cdots + 67\!\cdots\!01 \)
|
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