Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [42,5,Mod(29,42)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(42, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("42.29");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 42 = 2 \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 42.b (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: | \(4.34153844952\) |
| 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: |
\( x^{8} - 44x^{6} + 646x^{4} - 3060x^{2} + 6561 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{3}\cdot 7^{2} \) |
| 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
\( x^{8} - 44x^{6} + 646x^{4} - 3060x^{2} + 6561 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 14\nu^{7} - 535\nu^{5} + 6452\nu^{3} - 14571\nu ) / 7047 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7\nu^{6} - 224\nu^{4} + 1225\nu^{2} + 6678 ) / 522 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -56\nu^{7} - 297\nu^{6} + 2140\nu^{5} + 11853\nu^{4} - 25808\nu^{3} - 145935\nu^{2} + 114660\nu + 386127 ) / 28188 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 10\nu^{7} - 54\nu^{6} - 494\nu^{5} + 1728\nu^{4} + 8188\nu^{3} - 18846\nu^{2} - 44748\nu + 51840 ) / 2349 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -112\nu^{7} + 108\nu^{6} + 4280\nu^{5} - 1107\nu^{4} - 42220\nu^{3} - 4590\nu^{2} - 5580\nu - 16767 ) / 14094 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 172\nu^{7} - 351\nu^{6} - 7244\nu^{5} + 22977\nu^{4} + 91348\nu^{3} - 362097\nu^{2} - 262908\nu + 1067499 ) / 28188 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 472 \nu^{7} + 567 \nu^{6} - 22064 \nu^{5} - 25191 \nu^{4} + 308800 \nu^{3} + 352917 \nu^{2} + \cdots - 1101033 ) / 28188 \)
|
| \(\nu\) | \(=\) |
\( ( 7\beta_{7} - 14\beta_{6} + 7\beta_{5} - 14\beta_{4} + 77\beta_{3} + 9\beta_{2} + 119\beta _1 - 49 ) / 252 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{7} + 2\beta_{6} + 5\beta_{5} - 7\beta_{4} - 14\beta_{3} - 27\beta_{2} - 2\beta _1 + 700 ) / 63 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 73\beta_{7} - 326\beta_{6} + 361\beta_{5} - 56\beta_{4} + 1127\beta_{3} + 171\beta_{2} + 3077\beta _1 - 763 ) / 252 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 38\beta_{7} + 164\beta_{6} + 158\beta_{5} - 196\beta_{4} - 266\beta_{3} - 540\beta_{2} - 38\beta _1 + 10339 ) / 63 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 485 \beta_{7} - 6806 \beta_{6} + 7723 \beta_{5} - 434 \beta_{4} + 17129 \beta_{3} + 2709 \beta_{2} + \cdots - 11725 ) / 252 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 866 \beta_{7} + 4898 \beta_{6} + 4181 \beta_{5} - 5047 \beta_{4} - 6062 \beta_{3} - 7857 \beta_{2} + \cdots + 148246 ) / 63 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 6413 \beta_{7} - 17774 \beta_{6} + 19435 \beta_{5} - 764 \beta_{4} + 30761 \beta_{3} + 4869 \beta_{2} + \cdots - 21061 ) / 36 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(31\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 29.1 |
|
− | 2.82843i | −8.64042 | − | 2.51857i | −8.00000 | 34.6139i | −7.12360 | + | 24.4388i | 18.5203 | 22.6274i | 68.3136 | + | 43.5230i | 97.9029 | |||||||||||||||||||||||||||||||||||
| 29.2 | − | 2.82843i | −0.952172 | − | 8.94949i | −8.00000 | − | 10.3985i | −25.3130 | + | 2.69315i | −18.5203 | 22.6274i | −79.1867 | + | 17.0429i | −29.4113 | |||||||||||||||||||||||||||||||||||
| 29.3 | − | 2.82843i | 6.59792 | + | 6.12106i | −8.00000 | 47.9925i | 17.3130 | − | 18.6617i | −18.5203 | 22.6274i | 6.06518 | + | 80.7726i | 135.743 | ||||||||||||||||||||||||||||||||||||
| 29.4 | − | 2.82843i | 8.99466 | − | 0.309853i | −8.00000 | − | 26.9531i | −0.876398 | − | 25.4408i | 18.5203 | 22.6274i | 80.8080 | − | 5.57406i | −76.2349 | |||||||||||||||||||||||||||||||||||
| 29.5 | 2.82843i | −8.64042 | + | 2.51857i | −8.00000 | − | 34.6139i | −7.12360 | − | 24.4388i | 18.5203 | − | 22.6274i | 68.3136 | − | 43.5230i | 97.9029 | |||||||||||||||||||||||||||||||||||
| 29.6 | 2.82843i | −0.952172 | + | 8.94949i | −8.00000 | 10.3985i | −25.3130 | − | 2.69315i | −18.5203 | − | 22.6274i | −79.1867 | − | 17.0429i | −29.4113 | ||||||||||||||||||||||||||||||||||||
| 29.7 | 2.82843i | 6.59792 | − | 6.12106i | −8.00000 | − | 47.9925i | 17.3130 | + | 18.6617i | −18.5203 | − | 22.6274i | 6.06518 | − | 80.7726i | 135.743 | |||||||||||||||||||||||||||||||||||
| 29.8 | 2.82843i | 8.99466 | + | 0.309853i | −8.00000 | 26.9531i | −0.876398 | + | 25.4408i | 18.5203 | − | 22.6274i | 80.8080 | + | 5.57406i | −76.2349 | ||||||||||||||||||||||||||||||||||||
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 | 42.5.b.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 42.5.b.a | ✓ | 8 |
| 4.b | odd | 2 | 1 | 336.5.d.a | 8 | ||
| 7.b | odd | 2 | 1 | 294.5.b.d | 8 | ||
| 12.b | even | 2 | 1 | 336.5.d.a | 8 | ||
| 21.c | even | 2 | 1 | 294.5.b.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.5.b.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 42.5.b.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 294.5.b.d | 8 | 7.b | odd | 2 | 1 | ||
| 294.5.b.d | 8 | 21.c | even | 2 | 1 | ||
| 336.5.d.a | 8 | 4.b | odd | 2 | 1 | ||
| 336.5.d.a | 8 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(42, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8)^{4} \)
|
| $3$ |
\( T^{8} - 12 T^{7} + \cdots + 43046721 \)
|
| $5$ |
\( T^{8} + \cdots + 216772185744 \)
|
| $7$ |
\( (T^{2} - 343)^{4} \)
|
| $11$ |
\( T^{8} + \cdots + 17\!\cdots\!16 \)
|
| $13$ |
\( (T^{4} + 260 T^{3} + \cdots - 628257852)^{2} \)
|
| $17$ |
\( T^{8} + \cdots + 26\!\cdots\!56 \)
|
| $19$ |
\( (T^{4} - 668 T^{3} + \cdots - 3943244156)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 55\!\cdots\!04 \)
|
| $29$ |
\( T^{8} + \cdots + 24\!\cdots\!76 \)
|
| $31$ |
\( (T^{4} - 2440 T^{3} + \cdots - 407095948224)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots + 4270226378896)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 10\!\cdots\!64 \)
|
| $43$ |
\( (T^{4} + \cdots + 3649641123088)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 71\!\cdots\!64 \)
|
| $53$ |
\( T^{8} + \cdots + 21\!\cdots\!36 \)
|
| $59$ |
\( T^{8} + \cdots + 91\!\cdots\!96 \)
|
| $61$ |
\( (T^{4} + \cdots + 43195048412292)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots - 26829784150256)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 16\!\cdots\!96 \)
|
| $73$ |
\( (T^{4} + \cdots - 479820270553328)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots + 343544312164608)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 36\!\cdots\!76 \)
|
| $89$ |
\( T^{8} + \cdots + 17\!\cdots\!64 \)
|
| $97$ |
\( (T^{4} + \cdots - 47\!\cdots\!88)^{2} \)
|
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