Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [153,4,Mod(118,153)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(153, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("153.118");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 153 = 3^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 153.d (of order \(2\), degree \(1\), 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: | \(9.02729223088\) |
| 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: |
\( x^{8} + 50x^{6} + 685x^{4} + 1728x^{2} + 324 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 51) |
| 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
\( x^{8} + 50x^{6} + 685x^{4} + 1728x^{2} + 324 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{6} + 35\nu^{4} + 262\nu^{2} + 348 ) / 204 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} + 35\nu^{4} + 58\nu^{2} - 2304 ) / 204 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{7} - 191\nu^{5} - 2425\nu^{3} - 3942\nu ) / 612 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 13\nu^{7} + 659\nu^{5} + 8302\nu^{3} + 1872\nu ) / 1836 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 13\nu^{7} + 659\nu^{5} + 9220\nu^{3} + 24822\nu ) / 918 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -4\nu^{7} - 191\nu^{5} - 2425\nu^{3} - 5166\nu ) / 306 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 7\nu^{6} + 347\nu^{4} + 4384\nu^{2} + 4680 ) / 204 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{6} + 2\beta_{3} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{2} + \beta _1 - 13 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 25\beta_{6} + 4\beta_{5} - 8\beta_{4} - 50\beta_{3} ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} + 25\beta_{2} - 39\beta _1 + 303 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -561\beta_{6} - 44\beta_{5} + 280\beta_{4} + 1330\beta_{3} ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( -70\beta_{7} - 613\beta_{2} + 1307\beta _1 - 7547 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 12617\beta_{6} - 324\beta_{5} - 8520\beta_{4} - 35778\beta_{3} ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/153\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(137\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 118.1 |
|
−4.46083 | 0 | 11.8990 | − | 0.805870i | 0 | − | 7.44837i | −17.3927 | 0 | 3.59485i | ||||||||||||||||||||||||||||||||||||||||
| 118.2 | −4.46083 | 0 | 11.8990 | 0.805870i | 0 | 7.44837i | −17.3927 | 0 | − | 3.59485i | ||||||||||||||||||||||||||||||||||||||||||
| 118.3 | −0.451338 | 0 | −7.79629 | − | 8.74515i | 0 | 20.7311i | 7.12947 | 0 | 3.94702i | ||||||||||||||||||||||||||||||||||||||||||
| 118.4 | −0.451338 | 0 | −7.79629 | 8.74515i | 0 | − | 20.7311i | 7.12947 | 0 | − | 3.94702i | |||||||||||||||||||||||||||||||||||||||||
| 118.5 | 1.72284 | 0 | −5.03184 | − | 10.2055i | 0 | 20.3112i | −22.4517 | 0 | − | 17.5823i | |||||||||||||||||||||||||||||||||||||||||
| 118.6 | 1.72284 | 0 | −5.03184 | 10.2055i | 0 | − | 20.3112i | −22.4517 | 0 | 17.5823i | ||||||||||||||||||||||||||||||||||||||||||
| 118.7 | 5.18933 | 0 | 18.9291 | − | 11.3455i | 0 | − | 21.0285i | 56.7149 | 0 | − | 58.8758i | ||||||||||||||||||||||||||||||||||||||||
| 118.8 | 5.18933 | 0 | 18.9291 | 11.3455i | 0 | 21.0285i | 56.7149 | 0 | 58.8758i | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 153.4.d.d | 8 | |
| 3.b | odd | 2 | 1 | 51.4.d.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 816.4.c.f | 8 | ||
| 17.b | even | 2 | 1 | inner | 153.4.d.d | 8 | |
| 51.c | odd | 2 | 1 | 51.4.d.a | ✓ | 8 | |
| 51.f | odd | 4 | 1 | 867.4.a.n | 4 | ||
| 51.f | odd | 4 | 1 | 867.4.a.o | 4 | ||
| 204.h | even | 2 | 1 | 816.4.c.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 51.4.d.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 51.4.d.a | ✓ | 8 | 51.c | odd | 2 | 1 | |
| 153.4.d.d | 8 | 1.a | even | 1 | 1 | trivial | |
| 153.4.d.d | 8 | 17.b | even | 2 | 1 | inner | |
| 816.4.c.f | 8 | 12.b | even | 2 | 1 | ||
| 816.4.c.f | 8 | 204.h | even | 2 | 1 | ||
| 867.4.a.n | 4 | 51.f | odd | 4 | 1 | ||
| 867.4.a.o | 4 | 51.f | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 2T_{2}^{3} - 23T_{2}^{2} + 30T_{2} + 18 \)
acting on \(S_{4}^{\mathrm{new}}(153, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 2 T^{3} - 23 T^{2} + \cdots + 18)^{2} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 310 T^{6} + \cdots + 665856 \)
|
| $7$ |
\( T^{8} + \cdots + 4349666304 \)
|
| $11$ |
\( T^{8} + \cdots + 5341855744 \)
|
| $13$ |
\( (T^{4} + 118 T^{3} + \cdots - 1094828)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 582622237229761 \)
|
| $19$ |
\( (T^{4} - 38 T^{3} + \cdots + 3560224)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 17\!\cdots\!84 \)
|
| $29$ |
\( T^{8} + \cdots + 36\!\cdots\!64 \)
|
| $31$ |
\( T^{8} + \cdots + 15\!\cdots\!44 \)
|
| $37$ |
\( T^{8} + \cdots + 34\!\cdots\!36 \)
|
| $41$ |
\( T^{8} + \cdots + 30\!\cdots\!76 \)
|
| $43$ |
\( (T^{4} - 62 T^{3} + \cdots - 884788304)^{2} \)
|
| $47$ |
\( (T^{4} - 236 T^{3} + \cdots - 49049856)^{2} \)
|
| $53$ |
\( (T^{4} + 972 T^{3} + \cdots - 23669953488)^{2} \)
|
| $59$ |
\( (T^{4} + 612 T^{3} + \cdots + 2682851328)^{2} \)
|
| $61$ |
\( T^{8} + \cdots + 16\!\cdots\!96 \)
|
| $67$ |
\( (T^{4} - 460 T^{3} + \cdots - 16007052288)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 26\!\cdots\!16 \)
|
| $73$ |
\( T^{8} + \cdots + 25\!\cdots\!84 \)
|
| $79$ |
\( T^{8} + \cdots + 19\!\cdots\!04 \)
|
| $83$ |
\( (T^{4} - 168 T^{3} + \cdots - 20547582336)^{2} \)
|
| $89$ |
\( (T^{4} - 172 T^{3} + \cdots + 138200373936)^{2} \)
|
| $97$ |
\( T^{8} + \cdots + 22\!\cdots\!44 \)
|
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