Newspace parameters
| Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 50.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(139.738672144\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 2) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 1024.00i | 59316.0i | −1.04858e6 | 0 | 6.07396e7 | − | 1.42743e9i | 1.07374e9i | 6.94197e9 | 0 | ||||||||||||||||||||||
| 49.2 | 1024.00i | − | 59316.0i | −1.04858e6 | 0 | 6.07396e7 | 1.42743e9i | − | 1.07374e9i | 6.94197e9 | 0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 50.22.b.c | 2 | |
| 5.b | even | 2 | 1 | inner | 50.22.b.c | 2 | |
| 5.c | odd | 4 | 1 | 2.22.a.b | ✓ | 1 | |
| 5.c | odd | 4 | 1 | 50.22.a.a | 1 | ||
| 15.e | even | 4 | 1 | 18.22.a.b | 1 | ||
| 20.e | even | 4 | 1 | 16.22.a.b | 1 | ||
| 40.i | odd | 4 | 1 | 64.22.a.c | 1 | ||
| 40.k | even | 4 | 1 | 64.22.a.e | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.22.a.b | ✓ | 1 | 5.c | odd | 4 | 1 | |
| 16.22.a.b | 1 | 20.e | even | 4 | 1 | ||
| 18.22.a.b | 1 | 15.e | even | 4 | 1 | ||
| 50.22.a.a | 1 | 5.c | odd | 4 | 1 | ||
| 50.22.b.c | 2 | 1.a | even | 1 | 1 | trivial | |
| 50.22.b.c | 2 | 5.b | even | 2 | 1 | inner | |
| 64.22.a.c | 1 | 40.i | odd | 4 | 1 | ||
| 64.22.a.e | 1 | 40.k | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 3518387856 \)
acting on \(S_{22}^{\mathrm{new}}(50, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 1048576 \)
$3$
\( T^{2} + 3518387856 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 20\!\cdots\!24 \)
$11$
\( (T + 106767894948)^{2} \)
$13$
\( T^{2} + 22\!\cdots\!76 \)
$17$
\( T^{2} + 12\!\cdots\!64 \)
$19$
\( (T + 11024055955460)^{2} \)
$23$
\( T^{2} + 16\!\cdots\!16 \)
$29$
\( (T + 23\!\cdots\!10)^{2} \)
$31$
\( (T + 878552957377888)^{2} \)
$37$
\( T^{2} + 96\!\cdots\!84 \)
$41$
\( (T + 24\!\cdots\!38)^{2} \)
$43$
\( T^{2} + 17\!\cdots\!56 \)
$47$
\( T^{2} + 37\!\cdots\!64 \)
$53$
\( T^{2} + 35\!\cdots\!96 \)
$59$
\( (T - 29\!\cdots\!80)^{2} \)
$61$
\( (T - 79\!\cdots\!22)^{2} \)
$67$
\( T^{2} + 23\!\cdots\!04 \)
$71$
\( (T - 88\!\cdots\!32)^{2} \)
$73$
\( T^{2} + 13\!\cdots\!56 \)
$79$
\( (T + 33\!\cdots\!20)^{2} \)
$83$
\( T^{2} + 41\!\cdots\!56 \)
$89$
\( (T - 41\!\cdots\!10)^{2} \)
$97$
\( T^{2} + 52\!\cdots\!04 \)
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