Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [475,8,Mod(1,475)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(475, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("475.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 475 = 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 475.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: | \(148.382887105\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} - 3 x^{11} - 1257 x^{10} + 3085 x^{9} + 579313 x^{8} - 912319 x^{7} - 120570671 x^{6} + \cdots - 640955565180 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 95) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} - 3 x^{11} - 1257 x^{10} + 3085 x^{9} + 579313 x^{8} - 912319 x^{7} - 120570671 x^{6} + \cdots - 640955565180 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 43\!\cdots\!83 \nu^{11} + \cdots + 20\!\cdots\!04 ) / 13\!\cdots\!64 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 43\!\cdots\!83 \nu^{11} + \cdots - 49\!\cdots\!44 ) / 13\!\cdots\!64 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 56\!\cdots\!19 \nu^{11} + \cdots - 72\!\cdots\!12 ) / 28\!\cdots\!68 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 28\!\cdots\!33 \nu^{11} + \cdots + 43\!\cdots\!28 ) / 13\!\cdots\!64 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 21\!\cdots\!07 \nu^{11} + \cdots - 18\!\cdots\!68 ) / 98\!\cdots\!76 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 45\!\cdots\!77 \nu^{11} + \cdots - 56\!\cdots\!12 ) / 13\!\cdots\!64 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 19\!\cdots\!77 \nu^{11} + \cdots + 38\!\cdots\!24 ) / 46\!\cdots\!88 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 12\!\cdots\!97 \nu^{11} + \cdots - 10\!\cdots\!68 ) / 24\!\cdots\!44 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 43\!\cdots\!53 \nu^{11} + \cdots + 15\!\cdots\!76 ) / 69\!\cdots\!32 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 60\!\cdots\!93 \nu^{11} + \cdots - 28\!\cdots\!80 ) / 69\!\cdots\!32 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + \beta _1 + 210 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{11} + \beta_{10} - \beta_{9} - 3\beta_{8} - \beta_{7} - 2\beta_{6} + 2\beta_{5} - 6\beta_{2} + 337\beta _1 + 87 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3 \beta_{11} - 21 \beta_{10} + 8 \beta_{9} + 34 \beta_{8} + 12 \beta_{7} - 20 \beta_{6} + 3 \beta_{5} + \cdots + 70932 \)
|
| \(\nu^{5}\) | \(=\) |
\( 658 \beta_{11} + 935 \beta_{10} - 662 \beta_{9} - 2017 \beta_{8} - 476 \beta_{7} - 922 \beta_{6} + \cdots - 15685 \)
|
| \(\nu^{6}\) | \(=\) |
\( 3166 \beta_{11} - 13688 \beta_{10} + 5068 \beta_{9} + 26497 \beta_{8} + 8184 \beta_{7} - 13706 \beta_{6} + \cdots + 27438351 \)
|
| \(\nu^{7}\) | \(=\) |
\( 351935 \beta_{11} + 576602 \beta_{10} - 370994 \beta_{9} - 1092876 \beta_{8} - 194808 \beta_{7} + \cdots - 39069400 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2053843 \beta_{11} - 7256849 \beta_{10} + 2883446 \beta_{9} + 16108590 \beta_{8} + 4578230 \beta_{7} + \cdots + 11414005028 \)
|
| \(\nu^{9}\) | \(=\) |
\( 175234832 \beta_{11} + 313015489 \beta_{10} - 195416636 \beta_{9} - 559612561 \beta_{8} + \cdots - 34377176725 \)
|
| \(\nu^{10}\) | \(=\) |
\( 1090449380 \beta_{11} - 3702237112 \beta_{10} + 1595134786 \beta_{9} + 8965335399 \beta_{8} + \cdots + 4972782215673 \)
|
| \(\nu^{11}\) | \(=\) |
\( 84444303501 \beta_{11} + 161135111510 \beta_{10} - 99593481444 \beta_{9} - 281137843876 \beta_{8} + \cdots - 23735872986684 \)
|
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 |
|
−21.3647 | −39.4680 | 328.452 | 0 | 843.224 | 71.1859 | −4282.61 | −629.277 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −16.3705 | −80.9045 | 139.993 | 0 | 1324.45 | −1597.54 | −196.331 | 4358.54 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −13.9989 | 87.7184 | 67.9686 | 0 | −1227.96 | 704.481 | 840.373 | 5507.51 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −10.6916 | 18.7478 | −13.6889 | 0 | −200.444 | −494.873 | 1514.89 | −1835.52 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −4.49416 | 14.6076 | −107.803 | 0 | −65.6491 | 1398.86 | 1059.73 | −1973.62 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.33527 | 39.1949 | −126.217 | 0 | −52.3358 | 413.297 | 339.449 | −650.761 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.0203668 | −55.2048 | −128.000 | 0 | −1.12434 | −880.140 | −5.21389 | 860.572 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 10.5946 | −28.8801 | −15.7535 | 0 | −305.974 | 1047.78 | −1523.02 | −1352.94 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 12.1833 | 16.6057 | 20.4323 | 0 | 202.312 | −1767.82 | −1310.53 | −1911.25 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 18.9242 | 87.1729 | 230.124 | 0 | 1649.67 | 38.0369 | 1932.61 | 5412.11 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 19.5400 | −29.0395 | 253.813 | 0 | −567.432 | 1767.74 | 2458.38 | −1343.71 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.12 | 21.9927 | −50.5503 | 355.679 | 0 | −1111.74 | −1421.00 | 5007.26 | 368.337 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(19\) | \(1\) |
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 | 475.8.a.e | 12 | |
| 5.b | even | 2 | 1 | 95.8.a.c | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.8.a.c | ✓ | 12 | 5.b | even | 2 | 1 | |
| 475.8.a.e | 12 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} - 15 T_{2}^{11} - 1158 T_{2}^{10} + 15270 T_{2}^{9} + 495973 T_{2}^{8} - 5286705 T_{2}^{7} + \cdots + 6715963136 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(475))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} + \cdots + 6715963136 \)
|
| $3$ |
\( T^{12} + \cdots + 10\!\cdots\!36 \)
|
| $5$ |
\( T^{12} \)
|
| $7$ |
\( T^{12} + \cdots - 35\!\cdots\!96 \)
|
| $11$ |
\( T^{12} + \cdots + 15\!\cdots\!64 \)
|
| $13$ |
\( T^{12} + \cdots - 33\!\cdots\!04 \)
|
| $17$ |
\( T^{12} + \cdots + 20\!\cdots\!44 \)
|
| $19$ |
\( (T + 6859)^{12} \)
|
| $23$ |
\( T^{12} + \cdots - 23\!\cdots\!96 \)
|
| $29$ |
\( T^{12} + \cdots + 16\!\cdots\!00 \)
|
| $31$ |
\( T^{12} + \cdots + 93\!\cdots\!04 \)
|
| $37$ |
\( T^{12} + \cdots - 65\!\cdots\!64 \)
|
| $41$ |
\( T^{12} + \cdots - 32\!\cdots\!04 \)
|
| $43$ |
\( T^{12} + \cdots + 37\!\cdots\!96 \)
|
| $47$ |
\( T^{12} + \cdots - 13\!\cdots\!96 \)
|
| $53$ |
\( T^{12} + \cdots + 80\!\cdots\!56 \)
|
| $59$ |
\( T^{12} + \cdots - 46\!\cdots\!00 \)
|
| $61$ |
\( T^{12} + \cdots - 22\!\cdots\!36 \)
|
| $67$ |
\( T^{12} + \cdots + 36\!\cdots\!16 \)
|
| $71$ |
\( T^{12} + \cdots - 49\!\cdots\!16 \)
|
| $73$ |
\( T^{12} + \cdots - 10\!\cdots\!04 \)
|
| $79$ |
\( T^{12} + \cdots + 12\!\cdots\!00 \)
|
| $83$ |
\( T^{12} + \cdots - 11\!\cdots\!16 \)
|
| $89$ |
\( T^{12} + \cdots - 65\!\cdots\!00 \)
|
| $97$ |
\( T^{12} + \cdots - 49\!\cdots\!76 \)
|
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