Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [637,6,Mod(1,637)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(637, 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("637.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 637 = 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 637.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: | \(102.164493221\) |
| 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: |
\( x^{6} - x^{5} - 106x^{4} + 222x^{3} + 2264x^{2} - 7384x + 5760 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 91) |
| 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
\( x^{6} - x^{5} - 106x^{4} + 222x^{3} + 2264x^{2} - 7384x + 5760 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2\nu - 36 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{5} + \nu^{4} - 427\nu^{3} + 316\nu^{2} + 9774\nu - 15372 ) / 68 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{5} + 6\nu^{4} - 709\nu^{3} + 230\nu^{2} + 14886\nu - 22056 ) / 68 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} - 104\nu^{3} + 22\nu^{2} + 2300\nu - 3264 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2\beta _1 + 36 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{5} + 3\beta_{4} - \beta_{3} + 62\beta _1 - 69 \)
|
| \(\nu^{4}\) | \(=\) |
\( 18\beta_{5} - 11\beta_{4} - 19\beta_{3} + 76\beta_{2} - 188\beta _1 + 2217 \)
|
| \(\nu^{5}\) | \(=\) |
\( -218\beta_{5} + 323\beta_{4} - 85\beta_{3} - 98\beta_{2} + 4380\beta _1 - 6921 \)
|
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 |
|
−9.98515 | −5.69627 | 67.7031 | 34.3361 | 56.8781 | 0 | −356.501 | −210.552 | −342.851 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −6.89296 | 30.3014 | 15.5129 | 74.0864 | −208.866 | 0 | 113.645 | 675.173 | −510.675 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −3.79757 | 16.1157 | −17.5784 | −29.5515 | −61.2005 | 0 | 188.278 | 16.7152 | 112.224 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −3.48936 | −17.9190 | −19.8244 | 15.2661 | 62.5258 | 0 | 180.834 | 78.0906 | −53.2688 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 4.02405 | 6.15211 | −15.8070 | −48.2758 | 24.7564 | 0 | −192.378 | −205.152 | −194.264 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 7.14099 | −2.95388 | 18.9938 | 84.1388 | −21.0936 | 0 | −92.8774 | −234.275 | 600.835 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(13\) | \(1\) |
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 | 637.6.a.c | 6 | |
| 7.b | odd | 2 | 1 | 91.6.a.a | ✓ | 6 | |
| 21.c | even | 2 | 1 | 819.6.a.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.6.a.a | ✓ | 6 | 7.b | odd | 2 | 1 | |
| 637.6.a.c | 6 | 1.a | even | 1 | 1 | trivial | |
| 819.6.a.e | 6 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(637))\):
|
\( T_{2}^{6} + 13T_{2}^{5} - 36T_{2}^{4} - 870T_{2}^{3} - 1292T_{2}^{2} + 10656T_{2} + 26208 \)
|
|
\( T_{3}^{6} - 26T_{3}^{5} - 451T_{3}^{4} + 8826T_{3}^{3} + 37306T_{3}^{2} - 282880T_{3} - 905800 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 13 T^{5} + \cdots + 26208 \)
|
| $3$ |
\( T^{6} - 26 T^{5} + \cdots - 905800 \)
|
| $5$ |
\( T^{6} + \cdots + 4661471808 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + \cdots + 304915315723200 \)
|
| $13$ |
\( (T + 169)^{6} \)
|
| $17$ |
\( T^{6} + \cdots - 19\!\cdots\!68 \)
|
| $19$ |
\( T^{6} + \cdots - 27\!\cdots\!80 \)
|
| $23$ |
\( T^{6} + \cdots - 15\!\cdots\!85 \)
|
| $29$ |
\( T^{6} + \cdots + 49\!\cdots\!60 \)
|
| $31$ |
\( T^{6} + \cdots - 92\!\cdots\!75 \)
|
| $37$ |
\( T^{6} + \cdots + 53\!\cdots\!48 \)
|
| $41$ |
\( T^{6} + \cdots - 18\!\cdots\!48 \)
|
| $43$ |
\( T^{6} + \cdots - 19\!\cdots\!72 \)
|
| $47$ |
\( T^{6} + \cdots + 44\!\cdots\!81 \)
|
| $53$ |
\( T^{6} + \cdots - 36\!\cdots\!48 \)
|
| $59$ |
\( T^{6} + \cdots + 34\!\cdots\!88 \)
|
| $61$ |
\( T^{6} + \cdots + 70\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots - 52\!\cdots\!08 \)
|
| $71$ |
\( T^{6} + \cdots - 13\!\cdots\!76 \)
|
| $73$ |
\( T^{6} + \cdots - 14\!\cdots\!45 \)
|
| $79$ |
\( T^{6} + \cdots + 28\!\cdots\!35 \)
|
| $83$ |
\( T^{6} + \cdots - 68\!\cdots\!60 \)
|
| $89$ |
\( T^{6} + \cdots + 72\!\cdots\!88 \)
|
| $97$ |
\( T^{6} + \cdots - 19\!\cdots\!49 \)
|
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