Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [605,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("605.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 605 = 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(97.0322109869\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{8} - 178x^{6} + 9081x^{4} - 116760x^{2} + 248400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 178x^{6} + 9081x^{4} - 116760x^{2} + 248400 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} + 148\nu^{4} - 5391\nu^{2} + 24780 ) / 1200 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{7} + 148\nu^{5} - 5391\nu^{3} + 23580\nu ) / 1200 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{2} - 45 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{3} - 76\nu \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} + 223\nu^{4} - 12366\nu^{2} + 81480 ) / 600 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} + 178\nu^{5} - 8661\nu^{3} + 79140\nu ) / 480 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + 45 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 76\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{6} + 93\beta_{4} - 16\beta_{2} + 3429 \)
|
| \(\nu^{5}\) | \(=\) |
\( 16\beta_{7} + 109\beta_{5} - 40\beta_{3} + 6432\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 1184\beta_{6} + 8373\beta_{4} - 3568\beta_{2} + 289677 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2368\beta_{7} + 10741\beta_{5} - 7120\beta_{3} + 565800\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−9.75834 | 8.35464 | 63.2252 | −25.0000 | −81.5274 | 91.5480 | −304.706 | −173.200 | 243.958 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −8.05803 | −20.7998 | 32.9318 | −25.0000 | 167.605 | −117.565 | −7.50858 | 189.631 | 201.451 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −3.89872 | 22.0676 | −16.8000 | −25.0000 | −86.0353 | −124.872 | 190.257 | 243.978 | 97.4680 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.62573 | −9.62244 | −29.3570 | −25.0000 | 15.6435 | 193.035 | 99.7501 | −150.409 | 40.6433 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.62573 | −9.62244 | −29.3570 | −25.0000 | −15.6435 | −193.035 | −99.7501 | −150.409 | −40.6433 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 3.89872 | 22.0676 | −16.8000 | −25.0000 | 86.0353 | 124.872 | −190.257 | 243.978 | −97.4680 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 8.05803 | −20.7998 | 32.9318 | −25.0000 | −167.605 | 117.565 | 7.50858 | 189.631 | −201.451 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 9.75834 | 8.35464 | 63.2252 | −25.0000 | 81.5274 | −91.5480 | 304.706 | −173.200 | −243.958 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(11\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 605.6.a.h | ✓ | 8 |
| 11.b | odd | 2 | 1 | inner | 605.6.a.h | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 605.6.a.h | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 605.6.a.h | ✓ | 8 | 11.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 178T_{2}^{6} + 9081T_{2}^{4} - 116760T_{2}^{2} + 248400 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(605))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 178 T^{6} + \cdots + 248400 \)
|
| $3$ |
\( (T^{4} - 541 T^{2} + \cdots + 36900)^{2} \)
|
| $5$ |
\( (T + 25)^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 67\!\cdots\!00 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + \cdots + 21\!\cdots\!00 \)
|
| $17$ |
\( T^{8} + \cdots + 11\!\cdots\!00 \)
|
| $19$ |
\( T^{8} + \cdots + 40\!\cdots\!00 \)
|
| $23$ |
\( (T^{4} + 3680 T^{3} + \cdots + 108513345600)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 13\!\cdots\!00 \)
|
| $31$ |
\( (T^{4} + \cdots + 31496179622000)^{2} \)
|
| $37$ |
\( (T^{4} + \cdots + 456845130660900)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 20\!\cdots\!00 \)
|
| $43$ |
\( T^{8} + \cdots + 52\!\cdots\!00 \)
|
| $47$ |
\( (T^{4} + \cdots + 12\!\cdots\!00)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots - 12\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots - 57\!\cdots\!00)^{2} \)
|
| $61$ |
\( T^{8} + \cdots + 21\!\cdots\!00 \)
|
| $67$ |
\( (T^{4} + \cdots + 77\!\cdots\!00)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 52\!\cdots\!36)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 49\!\cdots\!00 \)
|
| $79$ |
\( T^{8} + \cdots + 25\!\cdots\!00 \)
|
| $83$ |
\( T^{8} + \cdots + 30\!\cdots\!00 \)
|
| $89$ |
\( (T^{4} + \cdots - 11\!\cdots\!80)^{2} \)
|
| $97$ |
\( (T^{4} + \cdots + 14\!\cdots\!00)^{2} \)
|
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