Newspace parameters
Level: | \( N \) | \(=\) | \( 2601 = 3^{2} \cdot 17^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2601.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.29806809786\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-2}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} + 2 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Projective image: | \(S_{4}\) |
Projective field: | Galois closure of 4.2.7803.1 |
Artin image: | $\GL(2,3)$ |
Artin field: | Galois closure of 8.2.52788863403.1 |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2601\mathbb{Z}\right)^\times\).
\(n\) | \(290\) | \(2026\) |
\(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2024.1 |
|
− | 1.41421i | 0 | −1.00000 | − | 1.41421i | 0 | −1.00000 | 0 | 0 | −2.00000 | ||||||||||||||||||||||
2024.2 | 1.41421i | 0 | −1.00000 | 1.41421i | 0 | −1.00000 | 0 | 0 | −2.00000 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2601.1.b.a | ✓ | 2 |
3.b | odd | 2 | 1 | inner | 2601.1.b.a | ✓ | 2 |
17.b | even | 2 | 1 | 2601.1.b.b | yes | 2 | |
17.c | even | 4 | 2 | 2601.1.c.a | 4 | ||
17.d | even | 8 | 2 | 2601.1.g.a | 4 | ||
17.d | even | 8 | 2 | 2601.1.g.b | 4 | ||
17.e | odd | 16 | 4 | 2601.1.k.a | 8 | ||
17.e | odd | 16 | 4 | 2601.1.k.b | 8 | ||
51.c | odd | 2 | 1 | 2601.1.b.b | yes | 2 | |
51.f | odd | 4 | 2 | 2601.1.c.a | 4 | ||
51.g | odd | 8 | 2 | 2601.1.g.a | 4 | ||
51.g | odd | 8 | 2 | 2601.1.g.b | 4 | ||
51.i | even | 16 | 4 | 2601.1.k.a | 8 | ||
51.i | even | 16 | 4 | 2601.1.k.b | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2601.1.b.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
2601.1.b.a | ✓ | 2 | 3.b | odd | 2 | 1 | inner |
2601.1.b.b | yes | 2 | 17.b | even | 2 | 1 | |
2601.1.b.b | yes | 2 | 51.c | odd | 2 | 1 | |
2601.1.c.a | 4 | 17.c | even | 4 | 2 | ||
2601.1.c.a | 4 | 51.f | odd | 4 | 2 | ||
2601.1.g.a | 4 | 17.d | even | 8 | 2 | ||
2601.1.g.a | 4 | 51.g | odd | 8 | 2 | ||
2601.1.g.b | 4 | 17.d | even | 8 | 2 | ||
2601.1.g.b | 4 | 51.g | odd | 8 | 2 | ||
2601.1.k.a | 8 | 17.e | odd | 16 | 4 | ||
2601.1.k.a | 8 | 51.i | even | 16 | 4 | ||
2601.1.k.b | 8 | 17.e | odd | 16 | 4 | ||
2601.1.k.b | 8 | 51.i | even | 16 | 4 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2601, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 2 \)
$3$
\( T^{2} \)
$5$
\( T^{2} + 2 \)
$7$
\( (T + 1)^{2} \)
$11$
\( T^{2} \)
$13$
\( (T - 1)^{2} \)
$17$
\( T^{2} \)
$19$
\( (T + 1)^{2} \)
$23$
\( T^{2} + 2 \)
$29$
\( T^{2} \)
$31$
\( (T + 1)^{2} \)
$37$
\( (T + 1)^{2} \)
$41$
\( T^{2} \)
$43$
\( (T - 1)^{2} \)
$47$
\( T^{2} \)
$53$
\( T^{2} \)
$59$
\( T^{2} \)
$61$
\( (T - 1)^{2} \)
$67$
\( (T - 1)^{2} \)
$71$
\( T^{2} + 2 \)
$73$
\( T^{2} \)
$79$
\( T^{2} \)
$83$
\( T^{2} + 2 \)
$89$
\( T^{2} + 2 \)
$97$
\( (T - 1)^{2} \)
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