Newspace parameters
| Level: | \( N \) | \(=\) | \( 2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2340.u (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(18.6849940730\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{5})\) |
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|
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| Defining polynomial: |
\( x^{4} + 3x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 260) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
\( x^{4} + 3x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + \nu + 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + 2\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{2} + \nu - 1 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta _1 - 2 ) / 2 \)
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| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + \beta_{2} - \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times\).
| \(n\) | \(937\) | \(1081\) | \(1171\) | \(2081\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(-\beta_{2}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 73.1 |
|
0 | 0 | 0 | − | 2.23607i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 73.2 | 0 | 0 | 0 | 2.23607i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 577.1 | 0 | 0 | 0 | − | 2.23607i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 577.2 | 0 | 0 | 0 | 2.23607i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.k | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2340.2.u.f | 4 | |
| 3.b | odd | 2 | 1 | 260.2.m.b | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 2340.2.bp.e | 4 | ||
| 12.b | even | 2 | 1 | 1040.2.bg.j | 4 | ||
| 13.d | odd | 4 | 1 | 2340.2.bp.e | 4 | ||
| 15.d | odd | 2 | 1 | 1300.2.m.b | 4 | ||
| 15.e | even | 4 | 1 | 260.2.r.b | yes | 4 | |
| 15.e | even | 4 | 1 | 1300.2.r.b | 4 | ||
| 39.f | even | 4 | 1 | 260.2.r.b | yes | 4 | |
| 60.l | odd | 4 | 1 | 1040.2.cd.j | 4 | ||
| 65.k | even | 4 | 1 | inner | 2340.2.u.f | 4 | |
| 156.l | odd | 4 | 1 | 1040.2.cd.j | 4 | ||
| 195.j | odd | 4 | 1 | 260.2.m.b | ✓ | 4 | |
| 195.n | even | 4 | 1 | 1300.2.r.b | 4 | ||
| 195.u | odd | 4 | 1 | 1300.2.m.b | 4 | ||
| 780.bn | even | 4 | 1 | 1040.2.bg.j | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 260.2.m.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 260.2.m.b | ✓ | 4 | 195.j | odd | 4 | 1 | |
| 260.2.r.b | yes | 4 | 15.e | even | 4 | 1 | |
| 260.2.r.b | yes | 4 | 39.f | even | 4 | 1 | |
| 1040.2.bg.j | 4 | 12.b | even | 2 | 1 | ||
| 1040.2.bg.j | 4 | 780.bn | even | 4 | 1 | ||
| 1040.2.cd.j | 4 | 60.l | odd | 4 | 1 | ||
| 1040.2.cd.j | 4 | 156.l | odd | 4 | 1 | ||
| 1300.2.m.b | 4 | 15.d | odd | 2 | 1 | ||
| 1300.2.m.b | 4 | 195.u | odd | 4 | 1 | ||
| 1300.2.r.b | 4 | 15.e | even | 4 | 1 | ||
| 1300.2.r.b | 4 | 195.n | even | 4 | 1 | ||
| 2340.2.u.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 2340.2.u.f | 4 | 65.k | even | 4 | 1 | inner | |
| 2340.2.bp.e | 4 | 5.c | odd | 4 | 1 | ||
| 2340.2.bp.e | 4 | 13.d | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2340, [\chi])\):
|
\( T_{7} \)
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\( T_{11}^{4} - 6T_{11}^{3} + 18T_{11}^{2} - 12T_{11} + 4 \)
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\( T_{17}^{4} + 100 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
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| $5$ |
\( (T^{2} + 5)^{2} \)
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| $7$ |
\( T^{4} \)
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| $11$ |
\( T^{4} - 6 T^{3} + \cdots + 4 \)
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| $13$ |
\( (T^{2} + 4 T + 13)^{2} \)
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| $17$ |
\( T^{4} + 100 \)
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| $19$ |
\( T^{4} - 10 T^{3} + \cdots + 100 \)
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| $23$ |
\( T^{4} + 18 T^{3} + \cdots + 1444 \)
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| $29$ |
\( T^{4} + 28T^{2} + 16 \)
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| $31$ |
\( T^{4} - 10 T^{3} + \cdots + 100 \)
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| $37$ |
\( T^{4} + 108T^{2} + 1296 \)
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| $41$ |
\( T^{4} + 24 T^{3} + \cdots + 3844 \)
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| $43$ |
\( T^{4} + 2 T^{3} + \cdots + 484 \)
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| $47$ |
\( (T^{2} + 80)^{2} \)
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| $53$ |
\( T^{4} - 12 T^{3} + \cdots + 484 \)
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| $59$ |
\( T^{4} - 18 T^{3} + \cdots + 1444 \)
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| $61$ |
\( (T^{2} + 2 T - 44)^{2} \)
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| $67$ |
\( (T^{2} + 14 T + 4)^{2} \)
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| $71$ |
\( T^{4} + 6 T^{3} + \cdots + 4 \)
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| $73$ |
\( (T^{2} - 10 T - 20)^{2} \)
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| $79$ |
\( T^{4} + 108T^{2} + 1296 \)
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| $83$ |
\( T^{4} + 112T^{2} + 256 \)
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| $89$ |
\( T^{4} + 100 \)
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| $97$ |
\( (T^{2} - 4 T - 176)^{2} \)
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