Newspace parameters
Level: | \( N \) | \(=\) | \( 2700 = 2^{2} \cdot 3^{3} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2700.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.34747553411\) |
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: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 108) |
Projective image: | \(D_{3}\) |
Projective field: | Galois closure of 3.1.108.1 |
Artin image: | $C_4\times S_3$ |
Artin field: | Galois closure of 12.0.265720500000000.3 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2700\mathbb{Z}\right)^\times\).
\(n\) | \(1001\) | \(1351\) | \(2377\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1349.1 |
|
0 | 0 | 0 | 0 | 0 | − | 1.00000i | 0 | 0 | 0 | |||||||||||||||||||||||
1349.2 | 0 | 0 | 0 | 0 | 0 | 1.00000i | 0 | 0 | 0 | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
5.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2700.1.b.b | 2 | |
3.b | odd | 2 | 1 | CM | 2700.1.b.b | 2 | |
5.b | even | 2 | 1 | inner | 2700.1.b.b | 2 | |
5.c | odd | 4 | 1 | 108.1.c.a | ✓ | 1 | |
5.c | odd | 4 | 1 | 2700.1.g.b | 1 | ||
15.d | odd | 2 | 1 | inner | 2700.1.b.b | 2 | |
15.e | even | 4 | 1 | 108.1.c.a | ✓ | 1 | |
15.e | even | 4 | 1 | 2700.1.g.b | 1 | ||
20.e | even | 4 | 1 | 432.1.e.a | 1 | ||
40.i | odd | 4 | 1 | 1728.1.e.a | 1 | ||
40.k | even | 4 | 1 | 1728.1.e.b | 1 | ||
45.k | odd | 12 | 2 | 324.1.g.a | 2 | ||
45.l | even | 12 | 2 | 324.1.g.a | 2 | ||
60.l | odd | 4 | 1 | 432.1.e.a | 1 | ||
120.q | odd | 4 | 1 | 1728.1.e.b | 1 | ||
120.w | even | 4 | 1 | 1728.1.e.a | 1 | ||
135.q | even | 36 | 6 | 2916.1.k.c | 6 | ||
135.r | odd | 36 | 6 | 2916.1.k.c | 6 | ||
180.v | odd | 12 | 2 | 1296.1.q.a | 2 | ||
180.x | even | 12 | 2 | 1296.1.q.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
108.1.c.a | ✓ | 1 | 5.c | odd | 4 | 1 | |
108.1.c.a | ✓ | 1 | 15.e | even | 4 | 1 | |
324.1.g.a | 2 | 45.k | odd | 12 | 2 | ||
324.1.g.a | 2 | 45.l | even | 12 | 2 | ||
432.1.e.a | 1 | 20.e | even | 4 | 1 | ||
432.1.e.a | 1 | 60.l | odd | 4 | 1 | ||
1296.1.q.a | 2 | 180.v | odd | 12 | 2 | ||
1296.1.q.a | 2 | 180.x | even | 12 | 2 | ||
1728.1.e.a | 1 | 40.i | odd | 4 | 1 | ||
1728.1.e.a | 1 | 120.w | even | 4 | 1 | ||
1728.1.e.b | 1 | 40.k | even | 4 | 1 | ||
1728.1.e.b | 1 | 120.q | odd | 4 | 1 | ||
2700.1.b.b | 2 | 1.a | even | 1 | 1 | trivial | |
2700.1.b.b | 2 | 3.b | odd | 2 | 1 | CM | |
2700.1.b.b | 2 | 5.b | even | 2 | 1 | inner | |
2700.1.b.b | 2 | 15.d | odd | 2 | 1 | inner | |
2700.1.g.b | 1 | 5.c | odd | 4 | 1 | ||
2700.1.g.b | 1 | 15.e | even | 4 | 1 | ||
2916.1.k.c | 6 | 135.q | even | 36 | 6 | ||
2916.1.k.c | 6 | 135.r | odd | 36 | 6 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + 1 \)
acting on \(S_{1}^{\mathrm{new}}(2700, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 1 \)
$11$
\( T^{2} \)
$13$
\( T^{2} + 1 \)
$17$
\( T^{2} \)
$19$
\( (T - 1)^{2} \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( (T - 2)^{2} \)
$37$
\( T^{2} + 1 \)
$41$
\( T^{2} \)
$43$
\( T^{2} + 4 \)
$47$
\( T^{2} \)
$53$
\( T^{2} \)
$59$
\( T^{2} \)
$61$
\( (T + 1)^{2} \)
$67$
\( T^{2} + 1 \)
$71$
\( T^{2} \)
$73$
\( T^{2} + 1 \)
$79$
\( (T - 1)^{2} \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} + 1 \)
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