Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1150,4,Mod(599,1150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1150.599");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1150 = 2 \cdot 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1150.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: | \(67.8521965066\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 168x^{6} + 9540x^{4} + 208777x^{2} + 1542564 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 230) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 168x^{6} + 9540x^{4} + 208777x^{2} + 1542564 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + 168\nu^{5} + 8298\nu^{3} + 104449\nu ) / 13662 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 16\nu^{7} + 2274\nu^{5} + 100269\nu^{3} + 1270846\nu ) / 34155 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + 84\nu^{2} + 1242 ) / 11 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} - 134\nu^{4} - 5299\nu^{2} - 56226 ) / 165 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} + 134\nu^{4} + 5464\nu^{2} + 63156 ) / 165 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -89\nu^{7} - 12261\nu^{5} - 510201\nu^{3} - 5839889\nu ) / 34155 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} - 42 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} + 13\beta_{3} - 12\beta_{2} - 50\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -84\beta_{6} - 84\beta_{5} + 11\beta_{4} + 2286 \)
|
| \(\nu^{5}\) | \(=\) |
\( -157\beta_{7} - 1103\beta_{3} + 1470\beta_{2} + 2958\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 5957\beta_{6} + 5792\beta_{5} - 1474\beta_{4} - 139992 \)
|
| \(\nu^{7}\) | \(=\) |
\( 9780\beta_{7} + 77430\beta_{3} - 133722\beta_{2} - 186493\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).
| \(n\) | \(51\) | \(277\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 599.1 |
|
− | 2.00000i | − | 7.16920i | −4.00000 | 0 | −14.3384 | 25.3945i | 8.00000i | −24.3974 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 599.2 | − | 2.00000i | − | 3.26018i | −4.00000 | 0 | −6.52037 | − | 27.7921i | 8.00000i | 16.3712 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 599.3 | − | 2.00000i | 5.50148i | −4.00000 | 0 | 11.0030 | − | 0.0500526i | 8.00000i | −3.26627 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 599.4 | − | 2.00000i | 8.92791i | −4.00000 | 0 | 17.8558 | − | 23.5524i | 8.00000i | −52.7075 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 599.5 | 2.00000i | − | 8.92791i | −4.00000 | 0 | 17.8558 | 23.5524i | − | 8.00000i | −52.7075 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 599.6 | 2.00000i | − | 5.50148i | −4.00000 | 0 | 11.0030 | 0.0500526i | − | 8.00000i | −3.26627 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 599.7 | 2.00000i | 3.26018i | −4.00000 | 0 | −6.52037 | 27.7921i | − | 8.00000i | 16.3712 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 599.8 | 2.00000i | 7.16920i | −4.00000 | 0 | −14.3384 | − | 25.3945i | − | 8.00000i | −24.3974 | 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 | 1150.4.b.m | 8 | |
| 5.b | even | 2 | 1 | inner | 1150.4.b.m | 8 | |
| 5.c | odd | 4 | 1 | 230.4.a.i | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 1150.4.a.o | 4 | ||
| 15.e | even | 4 | 1 | 2070.4.a.bi | 4 | ||
| 20.e | even | 4 | 1 | 1840.4.a.l | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 230.4.a.i | ✓ | 4 | 5.c | odd | 4 | 1 | |
| 1150.4.a.o | 4 | 5.c | odd | 4 | 1 | ||
| 1150.4.b.m | 8 | 1.a | even | 1 | 1 | trivial | |
| 1150.4.b.m | 8 | 5.b | even | 2 | 1 | inner | |
| 1840.4.a.l | 4 | 20.e | even | 4 | 1 | ||
| 2070.4.a.bi | 4 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1150, [\chi])\):
|
\( T_{3}^{8} + 172T_{3}^{6} + 9780T_{3}^{4} + 209713T_{3}^{2} + 1317904 \)
|
|
\( T_{7}^{8} + 1972T_{7}^{6} + 1284300T_{7}^{4} + 276310753T_{7}^{2} + 692224 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{4} \)
|
| $3$ |
\( T^{8} + 172 T^{6} + \cdots + 1317904 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + 1972 T^{6} + \cdots + 692224 \)
|
| $11$ |
\( (T^{4} - 93 T^{3} + \cdots - 41700)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 465890883844 \)
|
| $17$ |
\( T^{8} + \cdots + 9239054400 \)
|
| $19$ |
\( (T^{4} + 185 T^{3} + \cdots + 1404820)^{2} \)
|
| $23$ |
\( (T^{2} + 529)^{4} \)
|
| $29$ |
\( (T^{4} + 294 T^{3} + \cdots - 1735042500)^{2} \)
|
| $31$ |
\( (T^{4} + 211 T^{3} + \cdots - 311259365)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 201273919906816 \)
|
| $41$ |
\( (T^{4} + 369 T^{3} + \cdots - 114199911)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 86\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots + 94\!\cdots\!76 \)
|
| $53$ |
\( T^{8} + \cdots + 65\!\cdots\!96 \)
|
| $59$ |
\( (T^{4} - 33 T^{3} + \cdots - 623752800)^{2} \)
|
| $61$ |
\( (T^{4} + 307 T^{3} + \cdots - 5277943032)^{2} \)
|
| $67$ |
\( T^{8} + \cdots + 24\!\cdots\!00 \)
|
| $71$ |
\( (T^{4} + 1257 T^{3} + \cdots - 3703975749)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 65\!\cdots\!44 \)
|
| $79$ |
\( (T^{4} + 1202 T^{3} + \cdots - 498954065920)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 18\!\cdots\!04 \)
|
| $89$ |
\( (T^{4} - 984 T^{3} + \cdots + 92383027200)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 53\!\cdots\!84 \)
|
| show more | |
| show less |