Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [605,4,Mod(1,605)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(605, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("605.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 605 = 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 605.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: | \(35.6961555535\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 33x^{4} + 67x^{3} + 256x^{2} - 236x + 44 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{5} - 33x^{4} + 67x^{3} + 256x^{2} - 236x + 44 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 19\nu + 14 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 35\nu^{3} + 40\nu^{2} + 288\nu - 108 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -5\nu^{5} + 14\nu^{4} + 163\nu^{3} - 280\nu^{2} - 1264\nu + 596 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 21\beta _1 + 10 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} + 10\beta_{4} + 3\beta_{3} + 26\beta_{2} + 39\beta _1 + 256 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{5} + 28\beta_{4} + 41\beta_{3} + 82\beta_{2} + 485\beta _1 + 490 \)
|
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 |
|
−4.49294 | 5.00860 | 12.1865 | 5.00000 | −22.5034 | −9.87650 | −18.8098 | −1.91388 | −22.4647 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −3.05333 | −8.55397 | 1.32283 | 5.00000 | 26.1181 | −6.83056 | 20.3876 | 46.1704 | −15.2667 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.445796 | 9.67111 | −7.80127 | 5.00000 | 4.31134 | 30.4607 | −7.04413 | 66.5304 | 2.22898 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.729042 | −0.737592 | −7.46850 | 5.00000 | −0.537736 | −19.3354 | −11.2772 | −26.4560 | 3.64521 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 4.03437 | −1.78381 | 8.27611 | 5.00000 | −7.19655 | 26.2015 | 1.11392 | −23.8180 | 20.1718 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 5.33707 | 8.39566 | 20.4843 | 5.00000 | 44.8082 | −30.6198 | 66.6296 | 43.4871 | 26.6853 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(11\) | \( -1 \) |
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 | 605.4.a.n | yes | 6 |
| 11.b | odd | 2 | 1 | 605.4.a.m | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 605.4.a.m | ✓ | 6 | 11.b | odd | 2 | 1 | |
| 605.4.a.n | yes | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(605))\):
|
\( T_{2}^{6} - 3T_{2}^{5} - 33T_{2}^{4} + 75T_{2}^{3} + 244T_{2}^{2} - 336T_{2} + 96 \)
|
|
\( T_{3}^{6} - 12T_{3}^{5} - 61T_{3}^{4} + 978T_{3}^{3} - 835T_{3}^{2} - 7374T_{3} - 4577 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 96 \)
|
| $3$ |
\( T^{6} - 12 T^{5} + \cdots - 4577 \)
|
| $5$ |
\( (T - 5)^{6} \)
|
| $7$ |
\( T^{6} + 10 T^{5} + \cdots + 31877147 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} - 80 T^{5} + \cdots - 9114688 \)
|
| $17$ |
\( T^{6} + \cdots - 3820861440 \)
|
| $19$ |
\( T^{6} + \cdots - 111603969216 \)
|
| $23$ |
\( T^{6} + \cdots + 4087676116032 \)
|
| $29$ |
\( T^{6} + \cdots - 11701424064 \)
|
| $31$ |
\( T^{6} + \cdots - 15269134498272 \)
|
| $37$ |
\( T^{6} + \cdots - 5305026789376 \)
|
| $41$ |
\( T^{6} + \cdots - 255688255776375 \)
|
| $43$ |
\( T^{6} + \cdots - 588984378067305 \)
|
| $47$ |
\( T^{6} + \cdots + 38794898412339 \)
|
| $53$ |
\( T^{6} + \cdots - 20\!\cdots\!92 \)
|
| $59$ |
\( T^{6} + \cdots - 160905290704224 \)
|
| $61$ |
\( T^{6} + \cdots - 19\!\cdots\!87 \)
|
| $67$ |
\( T^{6} + \cdots - 17\!\cdots\!89 \)
|
| $71$ |
\( T^{6} + \cdots + 30\!\cdots\!12 \)
|
| $73$ |
\( T^{6} + \cdots - 64\!\cdots\!36 \)
|
| $79$ |
\( T^{6} + \cdots - 21\!\cdots\!60 \)
|
| $83$ |
\( T^{6} + \cdots + 39\!\cdots\!12 \)
|
| $89$ |
\( T^{6} + \cdots + 88\!\cdots\!97 \)
|
| $97$ |
\( T^{6} + \cdots - 46\!\cdots\!36 \)
|
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