Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [300,5,Mod(157,300)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(300, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("300.157");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 300.k (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: | \(31.0109889252\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(i)\) |
| Coefficient field: | 8.0.77720518656.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 41x^{4} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{2}\cdot 5^{4} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 41x^{4} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 31\nu ) / 14 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 45\nu^{2} ) / 28 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{7} + 197\nu^{3} ) / 56 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 135\nu^{7} + 60\nu^{5} + 5655\nu^{3} + 3540\nu ) / 28 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -135\nu^{7} + 60\nu^{5} - 5655\nu^{3} + 3540\nu ) / 28 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -35\nu^{6} + 40\nu^{4} - 1295\nu^{2} + 820 ) / 14 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -35\nu^{6} - 40\nu^{4} - 1295\nu^{2} - 820 ) / 14 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - 60\beta_1 ) / 120 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + 140\beta_{2} ) / 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} - 108\beta_{3} ) / 24 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -7\beta_{7} + 7\beta_{6} - 820 ) / 40 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -31\beta_{5} - 31\beta_{4} + 3540\beta_1 ) / 120 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -9\beta_{7} - 9\beta_{6} - 1036\beta_{2} ) / 8 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 197\beta_{5} - 197\beta_{4} + 22620\beta_{3} ) / 120 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(151\) | \(277\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 157.1 |
|
0 | −3.67423 | + | 3.67423i | 0 | 0 | 0 | −58.6704 | − | 58.6704i | 0 | − | 27.0000i | 0 | |||||||||||||||||||||||||||||||||||||
| 157.2 | 0 | −3.67423 | + | 3.67423i | 0 | 0 | 0 | 56.2209 | + | 56.2209i | 0 | − | 27.0000i | 0 | ||||||||||||||||||||||||||||||||||||||
| 157.3 | 0 | 3.67423 | − | 3.67423i | 0 | 0 | 0 | −56.2209 | − | 56.2209i | 0 | − | 27.0000i | 0 | ||||||||||||||||||||||||||||||||||||||
| 157.4 | 0 | 3.67423 | − | 3.67423i | 0 | 0 | 0 | 58.6704 | + | 58.6704i | 0 | − | 27.0000i | 0 | ||||||||||||||||||||||||||||||||||||||
| 193.1 | 0 | −3.67423 | − | 3.67423i | 0 | 0 | 0 | −58.6704 | + | 58.6704i | 0 | 27.0000i | 0 | |||||||||||||||||||||||||||||||||||||||
| 193.2 | 0 | −3.67423 | − | 3.67423i | 0 | 0 | 0 | 56.2209 | − | 56.2209i | 0 | 27.0000i | 0 | |||||||||||||||||||||||||||||||||||||||
| 193.3 | 0 | 3.67423 | + | 3.67423i | 0 | 0 | 0 | −56.2209 | + | 56.2209i | 0 | 27.0000i | 0 | |||||||||||||||||||||||||||||||||||||||
| 193.4 | 0 | 3.67423 | + | 3.67423i | 0 | 0 | 0 | 58.6704 | − | 58.6704i | 0 | 27.0000i | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 5.c | odd | 4 | 2 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 300.5.k.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | 900.5.l.i | 8 | ||
| 5.b | even | 2 | 1 | inner | 300.5.k.c | ✓ | 8 |
| 5.c | odd | 4 | 2 | inner | 300.5.k.c | ✓ | 8 |
| 15.d | odd | 2 | 1 | 900.5.l.i | 8 | ||
| 15.e | even | 4 | 2 | 900.5.l.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 300.5.k.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 300.5.k.c | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 300.5.k.c | ✓ | 8 | 5.c | odd | 4 | 2 | inner |
| 900.5.l.i | 8 | 3.b | odd | 2 | 1 | ||
| 900.5.l.i | 8 | 15.d | odd | 2 | 1 | ||
| 900.5.l.i | 8 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 87357618T_{7}^{4} + 1894025999527281 \)
acting on \(S_{5}^{\mathrm{new}}(300, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} + 729)^{2} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 18\!\cdots\!81 \)
|
| $11$ |
\( (T^{2} + 108 T - 16884)^{4} \)
|
| $13$ |
\( T^{8} + \cdots + 82\!\cdots\!25 \)
|
| $17$ |
\( T^{8} + \cdots + 97\!\cdots\!56 \)
|
| $19$ |
\( (T^{4} + 39650 T^{2} + 391050625)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 51\!\cdots\!76 \)
|
| $29$ |
\( (T^{4} + 1223928 T^{2} + 142911217296)^{2} \)
|
| $31$ |
\( (T^{2} + 1642 T + 495841)^{4} \)
|
| $37$ |
\( T^{8} + \cdots + 31\!\cdots\!96 \)
|
| $41$ |
\( (T^{2} + 1080 T - 2104200)^{4} \)
|
| $43$ |
\( T^{8} + \cdots + 17\!\cdots\!01 \)
|
| $47$ |
\( T^{8} + \cdots + 50\!\cdots\!76 \)
|
| $53$ |
\( T^{8} + \cdots + 79\!\cdots\!36 \)
|
| $59$ |
\( (T^{4} + \cdots + 880228436798736)^{2} \)
|
| $61$ |
\( (T^{2} + 3146 T + 2157529)^{4} \)
|
| $67$ |
\( T^{8} + \cdots + 93\!\cdots\!61 \)
|
| $71$ |
\( (T^{2} + 3960 T - 55974600)^{4} \)
|
| $73$ |
\( T^{8} + \cdots + 23\!\cdots\!56 \)
|
| $79$ |
\( (T^{4} + \cdots + 51442690590736)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 60\!\cdots\!16 \)
|
| $89$ |
\( (T^{4} + \cdots + 34\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 13\!\cdots\!81 \)
|
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