Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [7,18,Mod(1,7)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(7, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 18, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("7.1");
S:= CuspForms(chi, 18);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 7 \) |
| Weight: | \( k \) | \(=\) | \( 18 \) |
| Character orbit: | \([\chi]\) | \(=\) | 7.a (trivial) |
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: | yes |
| Analytic conductor: | \(12.8255461141\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 559376x^{3} + 70948970x^{2} + 30882981215x + 584478460232 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{3}\cdot 7^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 559376x^{3} + 70948970x^{2} + 30882981215x + 584478460232 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 431\nu^{4} - 30823\nu^{3} - 199426527\nu^{2} + 52955726287\nu + 61684487176 ) / 46343808 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 57\nu^{4} + 7871\nu^{3} - 28704009\nu^{2} + 1193532057\nu + 1032327997240 ) / 46343808 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -29\nu^{4} + 8901\nu^{3} + 18707757\nu^{2} - 5631802621\nu - 900760238232 ) / 6620544 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - 4\beta_{3} + \beta_{2} - 189\beta _1 + 223826 \)
|
| \(\nu^{3}\) | \(=\) |
\( 195\beta_{4} + 3099\beta_{3} - 318\beta_{2} + 449436\beta _1 - 42077489 \)
|
| \(\nu^{4}\) | \(=\) |
\( 476652\beta_{4} - 1629201\beta_{3} + 547491\beta_{2} - 178177202\beta _1 + 100413463809 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−655.082 | 15677.8 | 298061. | 1.36270e6 | −1.02702e7 | 5.76480e6 | −1.09391e8 | 1.16652e8 | −8.92682e8 | |||||||||||||||||||||||||||||||||
| 1.2 | −51.2830 | 1738.66 | −128442. | 190715. | −89163.6 | 5.76480e6 | 1.33087e7 | −1.26117e8 | −9.78045e6 | ||||||||||||||||||||||||||||||||||
| 1.3 | 99.0121 | −21601.2 | −121269. | −991914. | −2.13878e6 | 5.76480e6 | −2.49848e7 | 3.37472e8 | −9.82115e7 | ||||||||||||||||||||||||||||||||||
| 1.4 | 494.387 | 16699.6 | 113346. | 421257. | 8.25605e6 | 5.76480e6 | −8.76337e6 | 1.49736e8 | 2.08264e8 | ||||||||||||||||||||||||||||||||||
| 1.5 | 709.967 | −14284.8 | 372981. | 630065. | −1.01417e7 | 5.76480e6 | 1.71747e8 | 7.49154e7 | 4.47325e8 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 7.18.a.b | ✓ | 5 |
| 3.b | odd | 2 | 1 | 63.18.a.e | 5 | ||
| 4.b | odd | 2 | 1 | 112.18.a.h | 5 | ||
| 7.b | odd | 2 | 1 | 49.18.a.d | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.18.a.b | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 49.18.a.d | 5 | 7.b | odd | 2 | 1 | ||
| 63.18.a.e | 5 | 3.b | odd | 2 | 1 | ||
| 112.18.a.h | 5 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 597T_{2}^{4} - 416814T_{2}^{3} + 253624680T_{2}^{2} - 8750693376T_{2} - 1167515043840 \)
acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(7))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + \cdots - 1167515043840 \)
|
| $3$ |
\( T^{5} + \cdots - 14\!\cdots\!96 \)
|
| $5$ |
\( T^{5} + \cdots + 68\!\cdots\!00 \)
|
| $7$ |
\( (T - 5764801)^{5} \)
|
| $11$ |
\( T^{5} + \cdots - 10\!\cdots\!68 \)
|
| $13$ |
\( T^{5} + \cdots - 83\!\cdots\!00 \)
|
| $17$ |
\( T^{5} + \cdots - 24\!\cdots\!64 \)
|
| $19$ |
\( T^{5} + \cdots + 86\!\cdots\!60 \)
|
| $23$ |
\( T^{5} + \cdots + 14\!\cdots\!44 \)
|
| $29$ |
\( T^{5} + \cdots - 15\!\cdots\!00 \)
|
| $31$ |
\( T^{5} + \cdots - 18\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots - 20\!\cdots\!08 \)
|
| $41$ |
\( T^{5} + \cdots - 87\!\cdots\!48 \)
|
| $43$ |
\( T^{5} + \cdots + 19\!\cdots\!00 \)
|
| $47$ |
\( T^{5} + \cdots - 25\!\cdots\!68 \)
|
| $53$ |
\( T^{5} + \cdots + 60\!\cdots\!92 \)
|
| $59$ |
\( T^{5} + \cdots + 70\!\cdots\!00 \)
|
| $61$ |
\( T^{5} + \cdots + 26\!\cdots\!48 \)
|
| $67$ |
\( T^{5} + \cdots + 19\!\cdots\!56 \)
|
| $71$ |
\( T^{5} + \cdots + 22\!\cdots\!40 \)
|
| $73$ |
\( T^{5} + \cdots - 53\!\cdots\!16 \)
|
| $79$ |
\( T^{5} + \cdots + 18\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 19\!\cdots\!88 \)
|
| $89$ |
\( T^{5} + \cdots + 25\!\cdots\!60 \)
|
| $97$ |
\( T^{5} + \cdots - 81\!\cdots\!44 \)
|
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