Newspace parameters
Level: | \( N \) | \(=\) | \( 100 = 2^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 100.h (of order \(10\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.72480264360\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
Coefficient field: | \(\Q(\zeta_{20})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{4} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{10}]$ |
$q$-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
\(\beta_{1}\) | \(=\) | \( 2\zeta_{20} \) |
\(\beta_{2}\) | \(=\) | \( \zeta_{20}^{2} \) |
\(\beta_{3}\) | \(=\) | \( 2\zeta_{20}^{3} \) |
\(\beta_{4}\) | \(=\) | \( \zeta_{20}^{4} \) |
\(\beta_{5}\) | \(=\) | \( 2\zeta_{20}^{5} \) |
\(\beta_{6}\) | \(=\) | \( \zeta_{20}^{6} \) |
\(\beta_{7}\) | \(=\) | \( 2\zeta_{20}^{7} \) |
\(\zeta_{20}\) | \(=\) | \( ( \beta_1 ) / 2 \) |
\(\zeta_{20}^{2}\) | \(=\) | \( \beta_{2} \) |
\(\zeta_{20}^{3}\) | \(=\) | \( ( \beta_{3} ) / 2 \) |
\(\zeta_{20}^{4}\) | \(=\) | \( \beta_{4} \) |
\(\zeta_{20}^{5}\) | \(=\) | \( ( \beta_{5} ) / 2 \) |
\(\zeta_{20}^{6}\) | \(=\) | \( \beta_{6} \) |
\(\zeta_{20}^{7}\) | \(=\) | \( ( \beta_{7} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).
\(n\) | \(51\) | \(77\) |
\(\chi(n)\) | \(-1\) | \(\beta_{2}\) |
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 |
|
−1.90211 | + | 0.618034i | 0 | 3.23607 | − | 2.35114i | −4.77819 | + | 1.47271i | 0 | 0 | −4.70228 | + | 6.47214i | −2.78115 | + | 8.55951i | 8.17848 | − | 5.75435i | ||||||||||||||||||||||||||||||
19.2 | 1.90211 | − | 0.618034i | 0 | 3.23607 | − | 2.35114i | −0.0759100 | − | 4.99942i | 0 | 0 | 4.70228 | − | 6.47214i | −2.78115 | + | 8.55951i | −3.23420 | − | 9.46255i | |||||||||||||||||||||||||||||||
39.1 | −1.17557 | − | 1.61803i | 0 | −1.23607 | + | 3.80423i | 4.73128 | + | 1.61710i | 0 | 0 | 7.60845 | − | 2.47214i | 7.28115 | + | 5.29007i | −2.94542 | − | 9.55638i | |||||||||||||||||||||||||||||||
39.2 | 1.17557 | + | 1.61803i | 0 | −1.23607 | + | 3.80423i | −2.87718 | + | 4.08924i | 0 | 0 | −7.60845 | + | 2.47214i | 7.28115 | + | 5.29007i | −9.99885 | + | 0.151820i | |||||||||||||||||||||||||||||||
59.1 | −1.17557 | + | 1.61803i | 0 | −1.23607 | − | 3.80423i | 4.73128 | − | 1.61710i | 0 | 0 | 7.60845 | + | 2.47214i | 7.28115 | − | 5.29007i | −2.94542 | + | 9.55638i | |||||||||||||||||||||||||||||||
59.2 | 1.17557 | − | 1.61803i | 0 | −1.23607 | − | 3.80423i | −2.87718 | − | 4.08924i | 0 | 0 | −7.60845 | − | 2.47214i | 7.28115 | − | 5.29007i | −9.99885 | − | 0.151820i | |||||||||||||||||||||||||||||||
79.1 | −1.90211 | − | 0.618034i | 0 | 3.23607 | + | 2.35114i | −4.77819 | − | 1.47271i | 0 | 0 | −4.70228 | − | 6.47214i | −2.78115 | − | 8.55951i | 8.17848 | + | 5.75435i | |||||||||||||||||||||||||||||||
79.2 | 1.90211 | + | 0.618034i | 0 | 3.23607 | + | 2.35114i | −0.0759100 | + | 4.99942i | 0 | 0 | 4.70228 | + | 6.47214i | −2.78115 | − | 8.55951i | −3.23420 | + | 9.46255i | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-1}) \) |
25.e | even | 10 | 1 | inner |
100.h | odd | 10 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 100.3.h.a | ✓ | 8 |
4.b | odd | 2 | 1 | CM | 100.3.h.a | ✓ | 8 |
25.e | even | 10 | 1 | inner | 100.3.h.a | ✓ | 8 |
100.h | odd | 10 | 1 | inner | 100.3.h.a | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
100.3.h.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
100.3.h.a | ✓ | 8 | 4.b | odd | 2 | 1 | CM |
100.3.h.a | ✓ | 8 | 25.e | even | 10 | 1 | inner |
100.3.h.a | ✓ | 8 | 100.h | odd | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} \)
acting on \(S_{3}^{\mathrm{new}}(100, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - 4 T^{6} + 16 T^{4} - 64 T^{2} + \cdots + 256 \)
$3$
\( T^{8} \)
$5$
\( T^{8} + 6 T^{7} + 11 T^{6} + \cdots + 390625 \)
$7$
\( T^{8} \)
$11$
\( T^{8} \)
$13$
\( T^{8} - 576 T^{6} + \cdots + 147355321 \)
$17$
\( T^{8} - 256 T^{6} + \cdots + 12818994841 \)
$19$
\( T^{8} \)
$23$
\( T^{8} \)
$29$
\( T^{8} + 84 T^{7} + \cdots + 592159491361 \)
$31$
\( T^{8} \)
$37$
\( T^{8} - 350 T^{7} + \cdots + 33175998739321 \)
$41$
\( T^{8} + \cdots + 132493014534721 \)
$43$
\( T^{8} \)
$47$
\( T^{8} \)
$53$
\( T^{8} + 450 T^{7} + \cdots + 27476268051961 \)
$59$
\( T^{8} \)
$61$
\( T^{8} + 44 T^{7} + \cdots + 36\!\cdots\!81 \)
$67$
\( T^{8} \)
$71$
\( T^{8} \)
$73$
\( T^{8} + \cdots + 347243062222681 \)
$79$
\( T^{8} \)
$83$
\( T^{8} \)
$89$
\( T^{8} + 234 T^{7} + \cdots + 12\!\cdots\!81 \)
$97$
\( T^{8} - 20736 T^{6} + \cdots + 10\!\cdots\!61 \)
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