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: | \(1\) |
| 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: |
\( x^{8} - x^{7} - 250x^{6} + 210x^{5} + 20076x^{4} - 12252x^{3} - 544784x^{2} + 65648x + 2393792 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 250x^{6} + 210x^{5} + 20076x^{4} - 12252x^{3} - 544784x^{2} + 65648x + 2393792 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 63 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 2501 \nu^{7} + 35467 \nu^{6} + 1254008 \nu^{5} - 8124970 \nu^{4} - 138426240 \nu^{3} + \cdots - 2702698464 ) / 99699232 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 29683 \nu^{7} - 461047 \nu^{6} + 5659660 \nu^{5} + 89483730 \nu^{4} - 226396840 \nu^{3} + \cdots + 27309929696 ) / 598195392 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 109811 \nu^{7} - 52255 \nu^{6} + 21329812 \nu^{5} + 32657314 \nu^{4} - 1151406808 \nu^{3} + \cdots + 15868421856 ) / 598195392 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 91705 \nu^{7} + 702673 \nu^{6} - 17428720 \nu^{5} - 131929806 \nu^{4} + 849745264 \nu^{3} + \cdots - 30023872448 ) / 299097696 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 190073 \nu^{7} + 1176317 \nu^{6} - 34535804 \nu^{5} - 218923206 \nu^{4} + 1412408360 \nu^{3} + \cdots - 14628902368 ) / 598195392 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 63 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{7} + 4\beta_{6} + 5\beta_{4} + \beta_{3} - 2\beta_{2} + 88\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 9\beta_{7} + 8\beta_{6} + 18\beta_{5} + 43\beta_{4} - 5\beta_{3} + 114\beta_{2} - 2\beta _1 + 5629 \)
|
| \(\nu^{5}\) | \(=\) |
\( -389\beta_{7} + 578\beta_{6} + 18\beta_{5} + 885\beta_{4} + 255\beta_{3} - 234\beta_{2} + 8650\beta _1 - 203 \)
|
| \(\nu^{6}\) | \(=\) |
\( 1147 \beta_{7} + 2066 \beta_{6} + 3606 \beta_{5} + 7161 \beta_{4} - 773 \beta_{3} + 12090 \beta_{2} + \cdots + 553569 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 41973 \beta_{7} + 71726 \beta_{6} + 1686 \beta_{5} + 128857 \beta_{4} + 37927 \beta_{3} + \cdots + 17009 \)
|
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 |
|
−10.9127 | 26.7530 | 87.0880 | −91.1424 | −291.949 | 0 | −601.161 | 472.725 | 994.614 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −8.37806 | 5.43213 | 38.1920 | 5.58680 | −45.5107 | 0 | −51.8766 | −213.492 | −46.8066 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −7.87682 | −28.4534 | 30.0443 | −12.9458 | 224.122 | 0 | 15.4049 | 566.593 | 101.971 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −2.33047 | −3.45560 | −26.5689 | −94.0769 | 8.05316 | 0 | 136.493 | −231.059 | 219.243 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 2.31470 | 16.6968 | −26.6421 | 48.4304 | 38.6483 | 0 | −135.739 | 35.7848 | 112.102 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 6.17463 | −24.2913 | 6.12609 | −68.1144 | −149.990 | 0 | −159.762 | 347.066 | −420.581 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 9.46332 | 0.254216 | 57.5544 | −54.0690 | 2.40573 | 0 | 241.829 | −242.935 | −511.672 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 10.5454 | −20.9360 | 79.2064 | 47.3313 | −220.779 | 0 | 497.812 | 195.317 | 499.129 | |||||||||||||||||||||||||||||||||||||||||||
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.e | 8 | |
| 7.b | odd | 2 | 1 | 91.6.a.c | ✓ | 8 | |
| 21.c | even | 2 | 1 | 819.6.a.j | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.6.a.c | ✓ | 8 | 7.b | odd | 2 | 1 | |
| 637.6.a.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 819.6.a.j | 8 | 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(637))\):
|
\( T_{2}^{8} + T_{2}^{7} - 250T_{2}^{6} - 210T_{2}^{5} + 20076T_{2}^{4} + 12252T_{2}^{3} - 544784T_{2}^{2} - 65648T_{2} + 2393792 \)
|
|
\( T_{3}^{8} + 28 T_{3}^{7} - 1045 T_{3}^{6} - 29036 T_{3}^{5} + 259264 T_{3}^{4} + 6632208 T_{3}^{3} + \cdots + 30844800 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + T^{7} + \cdots + 2393792 \)
|
| $3$ |
\( T^{8} + 28 T^{7} + \cdots + 30844800 \)
|
| $5$ |
\( T^{8} + \cdots - 5235380906112 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} + \cdots + 96\!\cdots\!00 \)
|
| $13$ |
\( (T - 169)^{8} \)
|
| $17$ |
\( T^{8} + \cdots - 98\!\cdots\!72 \)
|
| $19$ |
\( T^{8} + \cdots - 75\!\cdots\!80 \)
|
| $23$ |
\( T^{8} + \cdots + 65\!\cdots\!40 \)
|
| $29$ |
\( T^{8} + \cdots + 77\!\cdots\!52 \)
|
| $31$ |
\( T^{8} + \cdots - 75\!\cdots\!60 \)
|
| $37$ |
\( T^{8} + \cdots - 59\!\cdots\!00 \)
|
| $41$ |
\( T^{8} + \cdots - 33\!\cdots\!80 \)
|
| $43$ |
\( T^{8} + \cdots - 13\!\cdots\!96 \)
|
| $47$ |
\( T^{8} + \cdots + 17\!\cdots\!80 \)
|
| $53$ |
\( T^{8} + \cdots - 31\!\cdots\!76 \)
|
| $59$ |
\( T^{8} + \cdots + 10\!\cdots\!40 \)
|
| $61$ |
\( T^{8} + \cdots + 34\!\cdots\!20 \)
|
| $67$ |
\( T^{8} + \cdots - 20\!\cdots\!92 \)
|
| $71$ |
\( T^{8} + \cdots + 42\!\cdots\!04 \)
|
| $73$ |
\( T^{8} + \cdots + 29\!\cdots\!22 \)
|
| $79$ |
\( T^{8} + \cdots + 25\!\cdots\!36 \)
|
| $83$ |
\( T^{8} + \cdots + 19\!\cdots\!96 \)
|
| $89$ |
\( T^{8} + \cdots + 77\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots - 28\!\cdots\!50 \)
|
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