Newspace parameters
Level: | \( N \) | \(=\) | \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2340.h (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(18.6849940730\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\zeta_{8})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 780) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times\).
\(n\) | \(937\) | \(1081\) | \(1171\) | \(2081\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
469.1 |
|
0 | 0 | 0 | −2.12132 | − | 0.707107i | 0 | 0.828427i | 0 | 0 | 0 | ||||||||||||||||||||||||||||
469.2 | 0 | 0 | 0 | −2.12132 | + | 0.707107i | 0 | − | 0.828427i | 0 | 0 | 0 | ||||||||||||||||||||||||||||
469.3 | 0 | 0 | 0 | 2.12132 | − | 0.707107i | 0 | 4.82843i | 0 | 0 | 0 | |||||||||||||||||||||||||||||
469.4 | 0 | 0 | 0 | 2.12132 | + | 0.707107i | 0 | − | 4.82843i | 0 | 0 | 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 | 2340.2.h.c | 4 | |
3.b | odd | 2 | 1 | 780.2.h.d | ✓ | 4 | |
5.b | even | 2 | 1 | inner | 2340.2.h.c | 4 | |
12.b | even | 2 | 1 | 3120.2.l.l | 4 | ||
15.d | odd | 2 | 1 | 780.2.h.d | ✓ | 4 | |
15.e | even | 4 | 1 | 3900.2.a.q | 2 | ||
15.e | even | 4 | 1 | 3900.2.a.r | 2 | ||
60.h | even | 2 | 1 | 3120.2.l.l | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
780.2.h.d | ✓ | 4 | 3.b | odd | 2 | 1 | |
780.2.h.d | ✓ | 4 | 15.d | odd | 2 | 1 | |
2340.2.h.c | 4 | 1.a | even | 1 | 1 | trivial | |
2340.2.h.c | 4 | 5.b | even | 2 | 1 | inner | |
3120.2.l.l | 4 | 12.b | even | 2 | 1 | ||
3120.2.l.l | 4 | 60.h | even | 2 | 1 | ||
3900.2.a.q | 2 | 15.e | even | 4 | 1 | ||
3900.2.a.r | 2 | 15.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 24T_{7}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(2340, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} \)
$5$
\( T^{4} - 8T^{2} + 25 \)
$7$
\( T^{4} + 24T^{2} + 16 \)
$11$
\( (T^{2} - 2)^{2} \)
$13$
\( (T^{2} + 1)^{2} \)
$17$
\( T^{4} + 72T^{2} + 784 \)
$19$
\( (T^{2} - 4 T - 28)^{2} \)
$23$
\( T^{4} + 48T^{2} + 64 \)
$29$
\( (T + 6)^{4} \)
$31$
\( (T + 10)^{4} \)
$37$
\( T^{4} + 24T^{2} + 16 \)
$41$
\( (T^{2} - 8 T - 34)^{2} \)
$43$
\( T^{4} + 72T^{2} + 784 \)
$47$
\( T^{4} + 132T^{2} + 3844 \)
$53$
\( T^{4} + 216T^{2} + 8464 \)
$59$
\( (T^{2} + 8 T - 34)^{2} \)
$61$
\( (T^{2} - 8 T - 16)^{2} \)
$67$
\( T^{4} + 216T^{2} + 1296 \)
$71$
\( (T^{2} - 8 T + 14)^{2} \)
$73$
\( T^{4} + 264T^{2} + 4624 \)
$79$
\( (T + 2)^{4} \)
$83$
\( T^{4} + 228T^{2} + 6724 \)
$89$
\( (T^{2} - 16 T + 62)^{2} \)
$97$
\( (T^{2} + 36)^{2} \)
show more
show less