Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [365,2,Mod(1,365)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(365, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("365.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 365 = 5 \cdot 73 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 365.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: | \(2.91453967378\) |
| 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: |
\( x^{8} - 2x^{7} - 11x^{6} + 19x^{5} + 36x^{4} - 46x^{3} - 41x^{2} + 25x + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{8} - 2x^{7} - 11x^{6} + 19x^{5} + 36x^{4} - 46x^{3} - 41x^{2} + 25x + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 9\nu^{4} + 6\nu^{3} + 16\nu^{2} - 3\nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} - 9\nu^{3} + 6\nu^{2} + 16\nu - 3 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} - 12\nu^{5} - 13\nu^{4} + 37\nu^{3} + 37\nu^{2} - 18\nu - 9 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} - 8\nu^{5} + 27\nu^{4} + 13\nu^{3} - 55\nu^{2} - 10\nu + 15 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 10\nu^{5} + 7\nu^{4} + 23\nu^{3} - 9\nu^{2} - 6\nu + 5 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{7} + \beta_{6} + \beta_{5} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} - \beta_{5} + 2\beta_{3} + 7\beta_{2} + \beta _1 + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( -9\beta_{7} + 10\beta_{6} + 8\beta_{5} + 2\beta_{4} + 2\beta_{3} + \beta_{2} + 21\beta _1 + 9 \)
|
| \(\nu^{6}\) | \(=\) |
\( -3\beta_{7} + 13\beta_{6} - 7\beta_{5} + 2\beta_{4} + 22\beta_{3} + 48\beta_{2} + 9\beta _1 + 90 \)
|
| \(\nu^{7}\) | \(=\) |
\( -68\beta_{7} + 83\beta_{6} + 57\beta_{5} + 22\beta_{4} + 28\beta_{3} + 18\beta_{2} + 126\beta _1 + 74 \)
|
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 |
|
−2.48488 | 1.24302 | 4.17463 | −1.00000 | −3.08876 | 3.66126 | −5.40368 | −1.45490 | 2.48488 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.52323 | −1.38707 | 0.320226 | −1.00000 | 2.11282 | 1.06956 | 2.55868 | −1.07604 | 1.52323 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.21721 | 2.11107 | −0.518406 | −1.00000 | −2.56960 | −2.81771 | 3.06542 | 1.45660 | 1.21721 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.104384 | 3.29733 | −1.98910 | −1.00000 | −0.344190 | 3.75152 | 0.416399 | 7.87240 | 0.104384 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.568487 | −2.16815 | −1.67682 | −1.00000 | −1.23256 | 2.51444 | −2.09023 | 1.70086 | −0.568487 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.89253 | 1.39280 | 1.58168 | −1.00000 | 2.63593 | 3.67013 | −0.791688 | −1.06010 | −1.89253 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.07672 | 3.23834 | 2.31278 | −1.00000 | 6.72513 | −3.98796 | 0.649559 | 7.48684 | −2.07672 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.79196 | 0.272652 | 5.79502 | −1.00000 | 0.761232 | −0.861234 | 10.5955 | −2.92566 | −2.79196 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(73\) | \(-1\) |
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 | 365.2.a.e | ✓ | 8 |
| 3.b | odd | 2 | 1 | 3285.2.a.q | 8 | ||
| 4.b | odd | 2 | 1 | 5840.2.a.be | 8 | ||
| 5.b | even | 2 | 1 | 1825.2.a.h | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 365.2.a.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1825.2.a.h | 8 | 5.b | even | 2 | 1 | ||
| 3285.2.a.q | 8 | 3.b | odd | 2 | 1 | ||
| 5840.2.a.be | 8 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 2T_{2}^{7} - 11T_{2}^{6} + 19T_{2}^{5} + 36T_{2}^{4} - 46T_{2}^{3} - 41T_{2}^{2} + 25T_{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(365))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 2 T^{7} + \cdots + 3 \)
|
| $3$ |
\( T^{8} - 8 T^{7} + \cdots + 32 \)
|
| $5$ |
\( (T + 1)^{8} \)
|
| $7$ |
\( T^{8} - 7 T^{7} + \cdots - 1312 \)
|
| $11$ |
\( T^{8} + 7 T^{7} + \cdots + 9702 \)
|
| $13$ |
\( T^{8} - 11 T^{7} + \cdots - 74 \)
|
| $17$ |
\( T^{8} - 5 T^{7} + \cdots - 1494 \)
|
| $19$ |
\( T^{8} - 21 T^{7} + \cdots - 11264 \)
|
| $23$ |
\( T^{8} - 5 T^{7} + \cdots + 16212 \)
|
| $29$ |
\( T^{8} + 3 T^{7} + \cdots - 6144 \)
|
| $31$ |
\( T^{8} - 3 T^{7} + \cdots - 322 \)
|
| $37$ |
\( T^{8} - 18 T^{7} + \cdots - 27104 \)
|
| $41$ |
\( T^{8} + 7 T^{7} + \cdots - 18714 \)
|
| $43$ |
\( T^{8} - 14 T^{7} + \cdots - 53344 \)
|
| $47$ |
\( T^{8} - 11 T^{7} + \cdots - 1152 \)
|
| $53$ |
\( T^{8} - 122 T^{6} + \cdots - 8106 \)
|
| $59$ |
\( T^{8} - 13 T^{7} + \cdots + 1626912 \)
|
| $61$ |
\( T^{8} - 5 T^{7} + \cdots - 489214 \)
|
| $67$ |
\( T^{8} - 29 T^{7} + \cdots - 2333800 \)
|
| $71$ |
\( T^{8} + 46 T^{7} + \cdots - 1600512 \)
|
| $73$ |
\( (T - 1)^{8} \)
|
| $79$ |
\( T^{8} + 9 T^{7} + \cdots - 178496 \)
|
| $83$ |
\( T^{8} + 11 T^{7} + \cdots + 130641312 \)
|
| $89$ |
\( T^{8} + 28 T^{7} + \cdots - 13970706 \)
|
| $97$ |
\( T^{8} - 10 T^{7} + \cdots - 2473984 \)
|
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