Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [156,3,Mod(53,156)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(156, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("156.53");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 156 = 2^{2} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 156.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: | \(4.25069212402\) |
| 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: |
\( x^{8} - 2x^{7} + 15x^{6} + 10x^{5} + 97x^{4} + 252x^{3} + 700x^{2} + 1696x + 3792 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} + 15x^{6} + 10x^{5} + 97x^{4} + 252x^{3} + 700x^{2} + 1696x + 3792 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{7} + 28\nu^{6} - 95\nu^{5} + 516\nu^{4} - 553\nu^{3} + 3758\nu^{2} + 736\nu + 13836 ) / 2916 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{7} - 4\nu^{6} + 11\nu^{5} - 408\nu^{4} + 511\nu^{3} - 1676\nu^{2} - 1972\nu - 1560 ) / 2916 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -5\nu^{7} + 14\nu^{6} + 11\nu^{5} - 156\nu^{4} - 263\nu^{3} - 272\nu^{2} - 1144\nu - 6204 ) / 2916 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + 4\nu^{6} - 14\nu^{5} + 18\nu^{4} - 7\nu^{3} - 139\nu^{2} + 280\nu + 102 ) / 486 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 7\nu^{7} - 52\nu^{6} + 179\nu^{5} - 138\nu^{4} + 109\nu^{3} - 494\nu^{2} + 2444\nu + 132 ) / 2916 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 13\nu^{7} - 40\nu^{6} + 263\nu^{5} - 228\nu^{4} + 2005\nu^{3} + 718\nu^{2} + 9224\nu + 13560 ) / 2916 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{3} + \beta_{2} + \beta _1 - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 2\beta_{5} - 2\beta_{4} + 2\beta_{3} + \beta_{2} - 4\beta _1 - 13 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + 6\beta_{6} + 13\beta_{5} - 2\beta_{4} - 3\beta_{3} + 2\beta_{2} - 19\beta _1 - 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( 8\beta_{7} + 12\beta_{6} + 19\beta_{5} + 32\beta_{4} - 17\beta_{3} - \beta_{2} - 45\beta _1 + 22 \)
|
| \(\nu^{6}\) | \(=\) |
\( 29\beta_{7} - 84\beta_{6} - 18\beta_{5} + 104\beta_{4} - 112\beta_{3} - 63\beta_{2} - 30\beta _1 + 333 \)
|
| \(\nu^{7}\) | \(=\) |
\( 15\beta_{7} - 396\beta_{6} - 465\beta_{5} - 54\beta_{4} - 417\beta_{3} - 348\beta_{2} + 337\beta _1 + 1359 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(79\) | \(145\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 53.1 |
|
0 | −1.33932 | − | 2.68444i | 0 | 5.74349i | 0 | −9.51181 | 0 | −5.41242 | + | 7.19067i | 0 | ||||||||||||||||||||||||||||||||||||||
| 53.2 | 0 | −1.33932 | + | 2.68444i | 0 | − | 5.74349i | 0 | −9.51181 | 0 | −5.41242 | − | 7.19067i | 0 | ||||||||||||||||||||||||||||||||||||||
| 53.3 | 0 | −0.121622 | − | 2.99753i | 0 | − | 0.347906i | 0 | 10.5746 | 0 | −8.97042 | + | 0.729130i | 0 | ||||||||||||||||||||||||||||||||||||||
| 53.4 | 0 | −0.121622 | + | 2.99753i | 0 | 0.347906i | 0 | 10.5746 | 0 | −8.97042 | − | 0.729130i | 0 | |||||||||||||||||||||||||||||||||||||||
| 53.5 | 0 | 1.62162 | − | 2.52395i | 0 | − | 7.04754i | 0 | −8.96906 | 0 | −3.74069 | − | 8.18580i | 0 | ||||||||||||||||||||||||||||||||||||||
| 53.6 | 0 | 1.62162 | + | 2.52395i | 0 | 7.04754i | 0 | −8.96906 | 0 | −3.74069 | + | 8.18580i | 0 | |||||||||||||||||||||||||||||||||||||||
| 53.7 | 0 | 2.83932 | − | 0.968627i | 0 | 5.58779i | 0 | 3.90626 | 0 | 7.12352 | − | 5.50049i | 0 | |||||||||||||||||||||||||||||||||||||||
| 53.8 | 0 | 2.83932 | + | 0.968627i | 0 | − | 5.58779i | 0 | 3.90626 | 0 | 7.12352 | + | 5.50049i | 0 | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 156.3.d.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 156.3.d.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | 624.3.f.a | 8 | ||
| 12.b | even | 2 | 1 | 624.3.f.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 156.3.d.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 156.3.d.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 624.3.f.a | 8 | 4.b | odd | 2 | 1 | ||
| 624.3.f.a | 8 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(156, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 6 T^{7} + \cdots + 6561 \)
|
| $5$ |
\( T^{8} + 114 T^{6} + \cdots + 6192 \)
|
| $7$ |
\( (T^{4} + 4 T^{3} + \cdots + 3524)^{2} \)
|
| $11$ |
\( T^{8} + 612 T^{6} + \cdots + 128397312 \)
|
| $13$ |
\( (T^{2} - 13)^{4} \)
|
| $17$ |
\( T^{8} + 2342 T^{6} + \cdots + 72223488 \)
|
| $19$ |
\( (T^{4} + 12 T^{3} + \cdots - 46224)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 2054356992 \)
|
| $29$ |
\( T^{8} + \cdots + 120750538752 \)
|
| $31$ |
\( (T^{2} - 24 T - 1156)^{4} \)
|
| $37$ |
\( (T^{4} + 48 T^{3} + \cdots - 3708)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 406257120000 \)
|
| $43$ |
\( (T^{4} - 38 T^{3} + \cdots - 4083628)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 181027138608 \)
|
| $53$ |
\( T^{8} + \cdots + 722234880000 \)
|
| $59$ |
\( T^{8} + \cdots + 717961310208 \)
|
| $61$ |
\( (T^{4} + 52 T^{3} + \cdots + 7856)^{2} \)
|
| $67$ |
\( (T^{4} + 192 T^{3} + \cdots + 953488)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 12770012803248 \)
|
| $73$ |
\( (T^{4} + 4 T^{3} + \cdots + 68032)^{2} \)
|
| $79$ |
\( (T^{4} - 164 T^{3} + \cdots - 966848)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 28\!\cdots\!12 \)
|
| $89$ |
\( T^{8} + \cdots + 10668283884288 \)
|
| $97$ |
\( (T^{4} - 208 T^{3} + \cdots + 57925568)^{2} \)
|
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