Newspace parameters
Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 50.c (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.36240132180\) |
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: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).
\(n\) | \(27\) |
\(\chi(n)\) | \(i\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 |
|
−1.00000 | − | 1.00000i | 3.00000 | − | 3.00000i | 2.00000i | 0 | −6.00000 | −3.00000 | − | 3.00000i | 2.00000 | − | 2.00000i | − | 9.00000i | 0 | |||||||||||||||
43.1 | −1.00000 | + | 1.00000i | 3.00000 | + | 3.00000i | − | 2.00000i | 0 | −6.00000 | −3.00000 | + | 3.00000i | 2.00000 | + | 2.00000i | 9.00000i | 0 | ||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 50.3.c.a | ✓ | 2 |
3.b | odd | 2 | 1 | 450.3.g.e | 2 | ||
4.b | odd | 2 | 1 | 400.3.p.a | 2 | ||
5.b | even | 2 | 1 | 50.3.c.b | yes | 2 | |
5.c | odd | 4 | 1 | inner | 50.3.c.a | ✓ | 2 |
5.c | odd | 4 | 1 | 50.3.c.b | yes | 2 | |
15.d | odd | 2 | 1 | 450.3.g.c | 2 | ||
15.e | even | 4 | 1 | 450.3.g.c | 2 | ||
15.e | even | 4 | 1 | 450.3.g.e | 2 | ||
20.d | odd | 2 | 1 | 400.3.p.g | 2 | ||
20.e | even | 4 | 1 | 400.3.p.a | 2 | ||
20.e | even | 4 | 1 | 400.3.p.g | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
50.3.c.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
50.3.c.a | ✓ | 2 | 5.c | odd | 4 | 1 | inner |
50.3.c.b | yes | 2 | 5.b | even | 2 | 1 | |
50.3.c.b | yes | 2 | 5.c | odd | 4 | 1 | |
400.3.p.a | 2 | 4.b | odd | 2 | 1 | ||
400.3.p.a | 2 | 20.e | even | 4 | 1 | ||
400.3.p.g | 2 | 20.d | odd | 2 | 1 | ||
400.3.p.g | 2 | 20.e | even | 4 | 1 | ||
450.3.g.c | 2 | 15.d | odd | 2 | 1 | ||
450.3.g.c | 2 | 15.e | even | 4 | 1 | ||
450.3.g.e | 2 | 3.b | odd | 2 | 1 | ||
450.3.g.e | 2 | 15.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 6T_{3} + 18 \)
acting on \(S_{3}^{\mathrm{new}}(50, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 2T + 2 \)
$3$
\( T^{2} - 6T + 18 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 6T + 18 \)
$11$
\( (T - 12)^{2} \)
$13$
\( T^{2} + 24T + 288 \)
$17$
\( T^{2} - 24T + 288 \)
$19$
\( T^{2} + 400 \)
$23$
\( T^{2} - 6T + 18 \)
$29$
\( T^{2} + 900 \)
$31$
\( (T + 8)^{2} \)
$37$
\( T^{2} + 96T + 4608 \)
$41$
\( (T + 48)^{2} \)
$43$
\( T^{2} + 54T + 1458 \)
$47$
\( T^{2} - 54T + 1458 \)
$53$
\( T^{2} + 24T + 288 \)
$59$
\( T^{2} + 3600 \)
$61$
\( (T - 32)^{2} \)
$67$
\( T^{2} + 6T + 18 \)
$71$
\( (T + 48)^{2} \)
$73$
\( T^{2} + 24T + 288 \)
$79$
\( T^{2} + 1600 \)
$83$
\( T^{2} - 186T + 17298 \)
$89$
\( T^{2} + 900 \)
$97$
\( T^{2} - 24T + 288 \)
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