Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [575,8,Mod(1,575)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(575, 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("575.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 575 = 5^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 575.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: | \(179.621389653\) |
| 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: |
\( x^{8} - 832x^{6} - 1059x^{5} + 203052x^{4} + 678328x^{3} - 13424272x^{2} - 73308944x - 37372224 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{4}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 23) |
| 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
\( x^{8} - 832x^{6} - 1059x^{5} + 203052x^{4} + 678328x^{3} - 13424272x^{2} - 73308944x - 37372224 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 70454057 \nu^{7} + 418096898 \nu^{6} + 53704469932 \nu^{5} - 276596270045 \nu^{4} + \cdots + 209986539079392 ) / 3315013604640 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 70454057 \nu^{7} - 418096898 \nu^{6} - 53704469932 \nu^{5} + 276596270045 \nu^{4} + \cdots - 899509368844512 ) / 3315013604640 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 99425533 \nu^{7} + 632516202 \nu^{6} + 81664904028 \nu^{5} - 385294374305 \nu^{4} + \cdots + 10\!\cdots\!08 ) / 3315013604640 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 11355547 \nu^{7} - 155754618 \nu^{6} - 9395550780 \nu^{5} + 95325842855 \nu^{4} + \cdots + 20717221503936 ) / 331501360464 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 66289297 \nu^{7} + 447046798 \nu^{6} + 51731639972 \nu^{5} - 294409331845 \nu^{4} + \cdots + 178479916668432 ) / 1657506802320 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 169439777 \nu^{7} - 1061508378 \nu^{6} - 133842936252 \nu^{5} + 651322391125 \nu^{4} + \cdots - 12\!\cdots\!72 ) / 3315013604640 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 2\beta _1 + 208 \)
|
| \(\nu^{3}\) | \(=\) |
\( -5\beta_{7} + 8\beta_{6} + \beta_{5} - 6\beta_{4} + \beta_{3} - 16\beta_{2} + 345\beta _1 + 395 \)
|
| \(\nu^{4}\) | \(=\) |
\( -72\beta_{7} - 110\beta_{6} - 10\beta_{5} - 25\beta_{4} + 389\beta_{3} + 442\beta_{2} + 672\beta _1 + 71549 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 2255 \beta_{7} + 4809 \beta_{6} + 610 \beta_{5} - 2433 \beta_{4} + 546 \beta_{3} - 9510 \beta_{2} + \cdots + 125645 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 40845 \beta_{7} - 74022 \beta_{6} - 10925 \beta_{5} - 16040 \beta_{4} + 153621 \beta_{3} + \cdots + 27779983 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 865180 \beta_{7} + 2356790 \beta_{6} + 276718 \beta_{5} - 875491 \beta_{4} + 141203 \beta_{3} + \cdots + 23127043 \)
|
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 |
|
−19.9556 | 76.4323 | 270.224 | 0 | −1525.25 | −653.513 | −2838.16 | 3654.90 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −19.4241 | −36.3647 | 249.296 | 0 | 706.352 | 461.175 | −2356.08 | −864.611 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −11.0962 | −60.8046 | −4.87416 | 0 | 674.701 | −952.148 | 1474.40 | 1510.20 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.570902 | 37.8447 | −127.674 | 0 | 21.6056 | −1733.23 | −145.965 | −754.777 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 6.60982 | 84.4445 | −84.3103 | 0 | 558.163 | 780.885 | −1403.33 | 4943.87 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 7.41631 | −65.3727 | −72.9984 | 0 | −484.824 | 902.074 | −1490.67 | 2086.59 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 14.5712 | 10.7680 | 84.3195 | 0 | 156.902 | 387.911 | −636.476 | −2071.05 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 21.3077 | −86.9475 | 326.017 | 0 | −1852.65 | −639.155 | 4219.28 | 5372.88 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(23\) | \(-1\) |
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 | 575.8.a.b | 8 | |
| 5.b | even | 2 | 1 | 23.8.a.b | ✓ | 8 | |
| 15.d | odd | 2 | 1 | 207.8.a.f | 8 | ||
| 20.d | odd | 2 | 1 | 368.8.a.h | 8 | ||
| 115.c | odd | 2 | 1 | 529.8.a.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 23.8.a.b | ✓ | 8 | 5.b | even | 2 | 1 | |
| 207.8.a.f | 8 | 15.d | odd | 2 | 1 | ||
| 368.8.a.h | 8 | 20.d | odd | 2 | 1 | ||
| 529.8.a.c | 8 | 115.c | odd | 2 | 1 | ||
| 575.8.a.b | 8 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 832T_{2}^{6} + 1059T_{2}^{5} + 203052T_{2}^{4} - 678328T_{2}^{3} - 13424272T_{2}^{2} + 73308944T_{2} - 37372224 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(575))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 832 T^{6} + \cdots - 37372224 \)
|
| $3$ |
\( T^{8} + \cdots + 33056528652000 \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} + \cdots + 86\!\cdots\!92 \)
|
| $11$ |
\( T^{8} + \cdots + 23\!\cdots\!20 \)
|
| $13$ |
\( T^{8} + \cdots + 39\!\cdots\!68 \)
|
| $17$ |
\( T^{8} + \cdots - 38\!\cdots\!00 \)
|
| $19$ |
\( T^{8} + \cdots + 22\!\cdots\!00 \)
|
| $23$ |
\( (T - 12167)^{8} \)
|
| $29$ |
\( T^{8} + \cdots - 13\!\cdots\!00 \)
|
| $31$ |
\( T^{8} + \cdots + 80\!\cdots\!00 \)
|
| $37$ |
\( T^{8} + \cdots + 16\!\cdots\!76 \)
|
| $41$ |
\( T^{8} + \cdots - 38\!\cdots\!40 \)
|
| $43$ |
\( T^{8} + \cdots - 10\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots + 82\!\cdots\!00 \)
|
| $53$ |
\( T^{8} + \cdots + 18\!\cdots\!88 \)
|
| $59$ |
\( T^{8} + \cdots - 15\!\cdots\!60 \)
|
| $61$ |
\( T^{8} + \cdots + 59\!\cdots\!88 \)
|
| $67$ |
\( T^{8} + \cdots + 34\!\cdots\!00 \)
|
| $71$ |
\( T^{8} + \cdots + 13\!\cdots\!00 \)
|
| $73$ |
\( T^{8} + \cdots + 90\!\cdots\!68 \)
|
| $79$ |
\( T^{8} + \cdots + 16\!\cdots\!60 \)
|
| $83$ |
\( T^{8} + \cdots + 20\!\cdots\!16 \)
|
| $89$ |
\( T^{8} + \cdots - 38\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 17\!\cdots\!92 \)
|
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