Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [520,6,Mod(1,520)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(520, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 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("520.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 520 = 2^{3} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 520.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: | \(83.3995863027\) |
| 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: |
\( x^{8} - 1392x^{6} - 960x^{5} + 541704x^{4} + 955392x^{3} - 49992640x^{2} - 201007872x + 5544720 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{15}\cdot 3 \) |
| 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
\( x^{8} - 1392x^{6} - 960x^{5} + 541704x^{4} + 955392x^{3} - 49992640x^{2} - 201007872x + 5544720 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 163 \nu^{7} + 574 \nu^{6} + 250262 \nu^{5} - 902522 \nu^{4} - 114612364 \nu^{3} + \cdots + 17405786520 ) / 283757760 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 163 \nu^{7} + 574 \nu^{6} + 250262 \nu^{5} - 902522 \nu^{4} - 114612364 \nu^{3} + \cdots - 81341913960 ) / 283757760 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 20 \nu^{7} + 167 \nu^{6} + 21286 \nu^{5} - 160462 \nu^{4} - 6054464 \nu^{3} + \cdots + 2326668840 ) / 21827520 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 406 \nu^{7} + 12310 \nu^{6} + 547190 \nu^{5} - 14254391 \nu^{4} - 198999712 \nu^{3} + \cdots - 70426804860 ) / 141878880 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 250 \nu^{7} + 1119 \nu^{6} - 359938 \nu^{5} - 2053654 \nu^{4} + 152417592 \nu^{3} + \cdots - 89164763640 ) / 94585920 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 1376 \nu^{7} - 5543 \nu^{6} - 1880389 \nu^{5} + 6171619 \nu^{4} + 696868448 \nu^{3} + \cdots - 73845429300 ) / 141878880 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - \beta_{2} + \beta _1 + 348 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{7} + \beta_{6} - 7\beta_{4} - 4\beta_{3} - 14\beta_{2} + 620\beta _1 + 360 \)
|
| \(\nu^{4}\) | \(=\) |
\( 16\beta_{7} - 104\beta_{6} + 32\beta_{5} - 40\beta_{4} + 760\beta_{3} - 1064\beta_{2} + 776\beta _1 + 213564 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 1552 \beta_{7} + 152 \beta_{6} + 352 \beta_{5} - 8552 \beta_{4} - 2928 \beta_{3} - 10688 \beta_{2} + \cdots + 238080 \)
|
| \(\nu^{6}\) | \(=\) |
\( 22688 \beta_{7} - 121648 \beta_{6} + 50816 \beta_{5} - 48368 \beta_{4} + 567420 \beta_{3} + \cdots + 146608176 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 985272 \beta_{7} - 322308 \beta_{6} + 542208 \beta_{5} - 8157156 \beta_{4} - 2077200 \beta_{3} + \cdots + 184820832 \)
|
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 |
|
0 | −29.7866 | 0 | −25.0000 | 0 | 138.753 | 0 | 644.243 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −23.5235 | 0 | −25.0000 | 0 | −122.557 | 0 | 310.356 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −11.0745 | 0 | −25.0000 | 0 | 109.586 | 0 | −120.355 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −6.68805 | 0 | −25.0000 | 0 | 140.231 | 0 | −198.270 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −1.97260 | 0 | −25.0000 | 0 | −39.7415 | 0 | −239.109 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 10.7436 | 0 | −25.0000 | 0 | −213.035 | 0 | −127.575 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 20.2538 | 0 | −25.0000 | 0 | 32.4777 | 0 | 167.214 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 26.0480 | 0 | −25.0000 | 0 | 138.286 | 0 | 435.496 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 520.6.a.e | ✓ | 8 |
| 4.b | odd | 2 | 1 | 1040.6.a.z | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 520.6.a.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1040.6.a.z | 8 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 16 T_{3}^{7} - 1280 T_{3}^{6} - 17216 T_{3}^{5} + 449704 T_{3}^{4} + 5029696 T_{3}^{3} + \cdots - 580250736 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(520))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + 16 T^{7} + \cdots - 580250736 \)
|
| $5$ |
\( (T + 25)^{8} \)
|
| $7$ |
\( T^{8} + \cdots - 99\!\cdots\!80 \)
|
| $11$ |
\( T^{8} + \cdots - 22\!\cdots\!08 \)
|
| $13$ |
\( (T - 169)^{8} \)
|
| $17$ |
\( T^{8} + \cdots + 22\!\cdots\!40 \)
|
| $19$ |
\( T^{8} + \cdots + 11\!\cdots\!36 \)
|
| $23$ |
\( T^{8} + \cdots + 41\!\cdots\!12 \)
|
| $29$ |
\( T^{8} + \cdots + 23\!\cdots\!36 \)
|
| $31$ |
\( T^{8} + \cdots + 35\!\cdots\!40 \)
|
| $37$ |
\( T^{8} + \cdots + 58\!\cdots\!00 \)
|
| $41$ |
\( T^{8} + \cdots - 49\!\cdots\!64 \)
|
| $43$ |
\( T^{8} + \cdots + 73\!\cdots\!88 \)
|
| $47$ |
\( T^{8} + \cdots + 51\!\cdots\!28 \)
|
| $53$ |
\( T^{8} + \cdots + 12\!\cdots\!32 \)
|
| $59$ |
\( T^{8} + \cdots + 64\!\cdots\!12 \)
|
| $61$ |
\( T^{8} + \cdots - 22\!\cdots\!00 \)
|
| $67$ |
\( T^{8} + \cdots - 89\!\cdots\!04 \)
|
| $71$ |
\( T^{8} + \cdots - 21\!\cdots\!52 \)
|
| $73$ |
\( T^{8} + \cdots - 39\!\cdots\!76 \)
|
| $79$ |
\( T^{8} + \cdots - 95\!\cdots\!32 \)
|
| $83$ |
\( T^{8} + \cdots + 47\!\cdots\!92 \)
|
| $89$ |
\( T^{8} + \cdots + 88\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots - 18\!\cdots\!00 \)
|
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