Newspace parameters
Level: | \( N \) | \(=\) | \( 20 = 2^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 5 \) |
Character orbit: | \([\chi]\) | \(=\) | 20.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.06739926168\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.1816805376000000.2 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 6x^{6} + 31x^{4} + 96x^{2} + 256 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{14} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
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 \( x^{8} + 6x^{6} + 31x^{4} + 96x^{2} + 256 \) :
\(\beta_{1}\) | \(=\) | \( 2\nu \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{7} + 10\nu^{5} + \nu^{3} + 80\nu ) / 32 \) |
\(\beta_{3}\) | \(=\) | \( 4\nu^{2} + 6 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{7} + 6\nu^{5} + 31\nu^{3} + 80\nu ) / 8 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{6} + 2\nu^{4} + \nu^{2} + 28 ) / 2 \) |
\(\beta_{6}\) | \(=\) | \( ( -7\nu^{7} + 8\nu^{6} - 26\nu^{5} + 48\nu^{4} - 57\nu^{3} + 120\nu^{2} - 176\nu + 384 ) / 32 \) |
\(\beta_{7}\) | \(=\) | \( ( -\nu^{6} - 6\nu^{4} - 23\nu^{2} - 60 ) / 2 \) |
\(\nu\) | \(=\) | \( ( \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{3} - 6 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( \beta_{7} + 2\beta_{6} + 3\beta_{4} + \beta_{3} - 2\beta_{2} - 7\beta_1 ) / 8 \) |
\(\nu^{4}\) | \(=\) | \( ( -\beta_{7} + \beta_{5} - 3\beta_{3} - 26 ) / 4 \) |
\(\nu^{5}\) | \(=\) | \( ( -\beta_{7} - 2\beta_{6} - \beta_{4} - \beta_{3} + 10\beta_{2} - 13\beta_1 ) / 4 \) |
\(\nu^{6}\) | \(=\) | \( ( -2\beta_{7} - 6\beta_{5} - 5\beta_{3} + 54 ) / 4 \) |
\(\nu^{7}\) | \(=\) | \( ( -19\beta_{7} - 38\beta_{6} - 17\beta_{4} - 19\beta_{3} - 58\beta_{2} + 53\beta_1 ) / 8 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).
\(n\) | \(11\) | \(17\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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.
Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 |
|
−2.85697 | − | 2.79959i | −6.64118 | 0.324555 | + | 15.9967i | −17.6491 | + | 17.7062i | 18.9737 | + | 18.5926i | −39.0703 | 43.8570 | − | 46.6107i | −36.8947 | 99.9931 | − | 1.17570i | ||||||||||||||||||||||||||||||
19.2 | −2.85697 | + | 2.79959i | −6.64118 | 0.324555 | − | 15.9967i | −17.6491 | − | 17.7062i | 18.9737 | − | 18.5926i | −39.0703 | 43.8570 | + | 46.6107i | −36.8947 | 99.9931 | + | 1.17570i | |||||||||||||||||||||||||||||||
19.3 | −1.35563 | − | 3.76328i | 13.9962 | −12.3246 | + | 10.2032i | 7.64911 | − | 23.8011i | −18.9737 | − | 52.6718i | −35.6863 | 55.1050 | + | 32.5490i | 114.895 | −99.9394 | + | 3.47961i | |||||||||||||||||||||||||||||||
19.4 | −1.35563 | + | 3.76328i | 13.9962 | −12.3246 | − | 10.2032i | 7.64911 | + | 23.8011i | −18.9737 | + | 52.6718i | −35.6863 | 55.1050 | − | 32.5490i | 114.895 | −99.9394 | − | 3.47961i | |||||||||||||||||||||||||||||||
19.5 | 1.35563 | − | 3.76328i | −13.9962 | −12.3246 | − | 10.2032i | 7.64911 | − | 23.8011i | −18.9737 | + | 52.6718i | 35.6863 | −55.1050 | + | 32.5490i | 114.895 | −79.2008 | − | 61.0511i | |||||||||||||||||||||||||||||||
19.6 | 1.35563 | + | 3.76328i | −13.9962 | −12.3246 | + | 10.2032i | 7.64911 | + | 23.8011i | −18.9737 | − | 52.6718i | 35.6863 | −55.1050 | − | 32.5490i | 114.895 | −79.2008 | + | 61.0511i | |||||||||||||||||||||||||||||||
19.7 | 2.85697 | − | 2.79959i | 6.64118 | 0.324555 | − | 15.9967i | −17.6491 | + | 17.7062i | 18.9737 | − | 18.5926i | 39.0703 | −43.8570 | − | 46.6107i | −36.8947 | −0.852871 | + | 99.9964i | |||||||||||||||||||||||||||||||
19.8 | 2.85697 | + | 2.79959i | 6.64118 | 0.324555 | + | 15.9967i | −17.6491 | − | 17.7062i | 18.9737 | + | 18.5926i | 39.0703 | −43.8570 | + | 46.6107i | −36.8947 | −0.852871 | − | 99.9964i | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
20.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 20.5.d.c | ✓ | 8 |
3.b | odd | 2 | 1 | 180.5.f.g | 8 | ||
4.b | odd | 2 | 1 | inner | 20.5.d.c | ✓ | 8 |
5.b | even | 2 | 1 | inner | 20.5.d.c | ✓ | 8 |
5.c | odd | 4 | 2 | 100.5.b.e | 8 | ||
8.b | even | 2 | 1 | 320.5.h.f | 8 | ||
8.d | odd | 2 | 1 | 320.5.h.f | 8 | ||
12.b | even | 2 | 1 | 180.5.f.g | 8 | ||
15.d | odd | 2 | 1 | 180.5.f.g | 8 | ||
20.d | odd | 2 | 1 | inner | 20.5.d.c | ✓ | 8 |
20.e | even | 4 | 2 | 100.5.b.e | 8 | ||
40.e | odd | 2 | 1 | 320.5.h.f | 8 | ||
40.f | even | 2 | 1 | 320.5.h.f | 8 | ||
60.h | even | 2 | 1 | 180.5.f.g | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
20.5.d.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
20.5.d.c | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
20.5.d.c | ✓ | 8 | 5.b | even | 2 | 1 | inner |
20.5.d.c | ✓ | 8 | 20.d | odd | 2 | 1 | inner |
100.5.b.e | 8 | 5.c | odd | 4 | 2 | ||
100.5.b.e | 8 | 20.e | even | 4 | 2 | ||
180.5.f.g | 8 | 3.b | odd | 2 | 1 | ||
180.5.f.g | 8 | 12.b | even | 2 | 1 | ||
180.5.f.g | 8 | 15.d | odd | 2 | 1 | ||
180.5.f.g | 8 | 60.h | even | 2 | 1 | ||
320.5.h.f | 8 | 8.b | even | 2 | 1 | ||
320.5.h.f | 8 | 8.d | odd | 2 | 1 | ||
320.5.h.f | 8 | 40.e | odd | 2 | 1 | ||
320.5.h.f | 8 | 40.f | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 240T_{3}^{2} + 8640 \)
acting on \(S_{5}^{\mathrm{new}}(20, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 24 T^{6} + 496 T^{4} + \cdots + 65536 \)
$3$
\( (T^{4} - 240 T^{2} + 8640)^{2} \)
$5$
\( (T^{4} + 20 T^{3} + 710 T^{2} + \cdots + 390625)^{2} \)
$7$
\( (T^{4} - 2800 T^{2} + 1944000)^{2} \)
$11$
\( (T^{4} + 48000 T^{2} + \cdots + 552407040)^{2} \)
$13$
\( (T^{4} + 20928 T^{2} + 82861056)^{2} \)
$17$
\( (T^{4} + 26368 T^{2} + 16367616)^{2} \)
$19$
\( (T^{4} + 447360 T^{2} + \cdots + 10372976640)^{2} \)
$23$
\( (T^{4} - 350320 T^{2} + \cdots + 30678152640)^{2} \)
$29$
\( (T^{2} - 676 T + 104004)^{4} \)
$31$
\( (T^{4} + 1013760 T^{2} + \cdots + 1745879040)^{2} \)
$37$
\( (T^{4} + 3008448 T^{2} + \cdots + 126031666176)^{2} \)
$41$
\( (T^{2} - 364 T - 3961116)^{4} \)
$43$
\( (T^{4} - 10034800 T^{2} + \cdots + 325194264000)^{2} \)
$47$
\( (T^{4} - 751600 T^{2} + \cdots + 49724936640)^{2} \)
$53$
\( (T^{4} + 24538048 T^{2} + \cdots + 20248378776576)^{2} \)
$59$
\( (T^{4} + 30276480 T^{2} + \cdots + 225559394979840)^{2} \)
$61$
\( (T^{2} - 1644 T - 10307356)^{4} \)
$67$
\( (T^{4} - 11037040 T^{2} + \cdots + 7632038946240)^{2} \)
$71$
\( (T^{4} + 64765440 T^{2} + \cdots + 13443377725440)^{2} \)
$73$
\( (T^{4} + 68179968 T^{2} + \cdots + 11\!\cdots\!16)^{2} \)
$79$
\( (T^{4} + 27279360 T^{2} + \cdots + 152966937968640)^{2} \)
$83$
\( (T^{4} - 118219120 T^{2} + \cdots + 10\!\cdots\!40)^{2} \)
$89$
\( (T^{2} - 6916 T - 62026236)^{4} \)
$97$
\( (T^{4} + 86769408 T^{2} + \cdots + 13\!\cdots\!76)^{2} \)
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