Newspace parameters
Level: | \( N \) | \(=\) | \( 740 = 2^{2} \cdot 5 \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 740.t (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.369308109348\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(i)\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(S_{4}\) |
Projective field: | Galois closure of 4.0.5065300.1 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/740\mathbb{Z}\right)^\times\).
\(n\) | \(261\) | \(297\) | \(371\) |
\(\chi(n)\) | \(-i\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
549.1 |
|
0 | 1.00000 | 0 | − | 1.00000i | 0 | − | 1.00000i | 0 | 0 | 0 | ||||||||||||||||||||||
709.1 | 0 | 1.00000 | 0 | 1.00000i | 0 | 1.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
185.j | odd | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 740.1.t.b | yes | 2 |
4.b | odd | 2 | 1 | 2960.1.cj.a | 2 | ||
5.b | even | 2 | 1 | 740.1.t.a | ✓ | 2 | |
5.c | odd | 4 | 1 | 3700.1.j.a | 2 | ||
5.c | odd | 4 | 1 | 3700.1.j.b | 2 | ||
20.d | odd | 2 | 1 | 2960.1.cj.b | 2 | ||
37.d | odd | 4 | 1 | 740.1.t.a | ✓ | 2 | |
148.g | even | 4 | 1 | 2960.1.cj.b | 2 | ||
185.f | even | 4 | 1 | 3700.1.j.a | 2 | ||
185.j | odd | 4 | 1 | inner | 740.1.t.b | yes | 2 |
185.k | even | 4 | 1 | 3700.1.j.b | 2 | ||
740.k | even | 4 | 1 | 2960.1.cj.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
740.1.t.a | ✓ | 2 | 5.b | even | 2 | 1 | |
740.1.t.a | ✓ | 2 | 37.d | odd | 4 | 1 | |
740.1.t.b | yes | 2 | 1.a | even | 1 | 1 | trivial |
740.1.t.b | yes | 2 | 185.j | odd | 4 | 1 | inner |
2960.1.cj.a | 2 | 4.b | odd | 2 | 1 | ||
2960.1.cj.a | 2 | 740.k | even | 4 | 1 | ||
2960.1.cj.b | 2 | 20.d | odd | 2 | 1 | ||
2960.1.cj.b | 2 | 148.g | even | 4 | 1 | ||
3700.1.j.a | 2 | 5.c | odd | 4 | 1 | ||
3700.1.j.a | 2 | 185.f | even | 4 | 1 | ||
3700.1.j.b | 2 | 5.c | odd | 4 | 1 | ||
3700.1.j.b | 2 | 185.k | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 1 \)
acting on \(S_{1}^{\mathrm{new}}(740, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( (T - 1)^{2} \)
$5$
\( T^{2} + 1 \)
$7$
\( T^{2} + 1 \)
$11$
\( T^{2} + 1 \)
$13$
\( T^{2} \)
$17$
\( T^{2} \)
$19$
\( T^{2} \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} - 2T + 2 \)
$37$
\( T^{2} + 1 \)
$41$
\( T^{2} + 1 \)
$43$
\( T^{2} - 2T + 2 \)
$47$
\( T^{2} + 1 \)
$53$
\( T^{2} + 1 \)
$59$
\( T^{2} \)
$61$
\( T^{2} + 2T + 2 \)
$67$
\( T^{2} \)
$71$
\( (T + 1)^{2} \)
$73$
\( (T + 1)^{2} \)
$79$
\( T^{2} - 2T + 2 \)
$83$
\( T^{2} + 1 \)
$89$
\( T^{2} + 2T + 2 \)
$97$
\( T^{2} + 2T + 2 \)
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