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: | \(\Q(i, \sqrt{6})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| 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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} \)
|
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 | −0.224745 | − | 2.22474i | 0 | 2.00000i | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 469.2 | 0 | 0 | 0 | −0.224745 | + | 2.22474i | 0 | − | 2.00000i | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 469.3 | 0 | 0 | 0 | 2.22474 | − | 0.224745i | 0 | − | 2.00000i | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 469.4 | 0 | 0 | 0 | 2.22474 | + | 0.224745i | 0 | 2.00000i | 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.d | 4 | |
| 3.b | odd | 2 | 1 | 780.2.h.c | ✓ | 4 | |
| 5.b | even | 2 | 1 | inner | 2340.2.h.d | 4 | |
| 12.b | even | 2 | 1 | 3120.2.l.j | 4 | ||
| 15.d | odd | 2 | 1 | 780.2.h.c | ✓ | 4 | |
| 15.e | even | 4 | 1 | 3900.2.a.o | 2 | ||
| 15.e | even | 4 | 1 | 3900.2.a.t | 2 | ||
| 60.h | even | 2 | 1 | 3120.2.l.j | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 780.2.h.c | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 780.2.h.c | ✓ | 4 | 15.d | odd | 2 | 1 | |
| 2340.2.h.d | 4 | 1.a | even | 1 | 1 | trivial | |
| 2340.2.h.d | 4 | 5.b | even | 2 | 1 | inner | |
| 3120.2.l.j | 4 | 12.b | even | 2 | 1 | ||
| 3120.2.l.j | 4 | 60.h | even | 2 | 1 | ||
| 3900.2.a.o | 2 | 15.e | even | 4 | 1 | ||
| 3900.2.a.t | 2 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(2340, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 4 T^{3} + \cdots + 25 \)
|
| $7$ |
\( (T^{2} + 4)^{2} \)
|
| $11$ |
\( (T^{2} + 4 T - 2)^{2} \)
|
| $13$ |
\( (T^{2} + 1)^{2} \)
|
| $17$ |
\( (T^{2} + 4)^{2} \)
|
| $19$ |
\( (T^{2} + 4 T - 20)^{2} \)
|
| $23$ |
\( (T^{2} + 24)^{2} \)
|
| $29$ |
\( (T + 2)^{4} \)
|
| $31$ |
\( (T^{2} - 4 T - 20)^{2} \)
|
| $37$ |
\( T^{4} + 120T^{2} + 144 \)
|
| $41$ |
\( (T^{2} + 12 T + 30)^{2} \)
|
| $43$ |
\( T^{4} + 200T^{2} + 8464 \)
|
| $47$ |
\( T^{4} + 116T^{2} + 2500 \)
|
| $53$ |
\( T^{4} + 56T^{2} + 400 \)
|
| $59$ |
\( (T^{2} + 4 T - 2)^{2} \)
|
| $61$ |
\( (T - 4)^{4} \)
|
| $67$ |
\( T^{4} + 200T^{2} + 8464 \)
|
| $71$ |
\( (T^{2} + 4 T - 146)^{2} \)
|
| $73$ |
\( T^{4} + 200T^{2} + 8464 \)
|
| $79$ |
\( (T + 10)^{4} \)
|
| $83$ |
\( T^{4} + 84T^{2} + 900 \)
|
| $89$ |
\( (T^{2} + 12 T - 18)^{2} \)
|
| $97$ |
\( (T^{2} + 36)^{2} \)
|
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