Newspace parameters
| Level: | \( N \) | \(=\) | \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 360.v (of order \(4\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(37.2131867102\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{29})\) |
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|
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| Defining polynomial: |
\( x^{4} + 15x^{2} + 49 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 40) |
| 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
\( x^{4} + 15x^{2} + 49 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 8\nu ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 4\nu^{3} - 28\nu^{2} + 88\nu - 210 ) / 7 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} + 56\nu^{2} + 44\nu + 420 ) / 7 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} - 10\beta_1 ) / 20 \)
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| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{3} - \beta_{2} - 150 ) / 20 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -2\beta_{3} - 4\beta_{2} + 55\beta_1 ) / 5 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).
| \(n\) | \(181\) | \(217\) | \(271\) | \(281\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{1}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 73.1 |
|
0 | 0 | 0 | −18.5407 | − | 16.7703i | 0 | 64.8516 | − | 64.8516i | 0 | 0 | 0 | ||||||||||||||||||||||||||
| 73.2 | 0 | 0 | 0 | 24.5407 | + | 4.77033i | 0 | −42.8516 | + | 42.8516i | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 217.1 | 0 | 0 | 0 | −18.5407 | + | 16.7703i | 0 | 64.8516 | + | 64.8516i | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 217.2 | 0 | 0 | 0 | 24.5407 | − | 4.77033i | 0 | −42.8516 | − | 42.8516i | 0 | 0 | 0 | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.c | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 360.5.v.b | 4 | |
| 3.b | odd | 2 | 1 | 40.5.l.b | ✓ | 4 | |
| 5.c | odd | 4 | 1 | inner | 360.5.v.b | 4 | |
| 12.b | even | 2 | 1 | 80.5.p.f | 4 | ||
| 15.d | odd | 2 | 1 | 200.5.l.c | 4 | ||
| 15.e | even | 4 | 1 | 40.5.l.b | ✓ | 4 | |
| 15.e | even | 4 | 1 | 200.5.l.c | 4 | ||
| 24.f | even | 2 | 1 | 320.5.p.k | 4 | ||
| 24.h | odd | 2 | 1 | 320.5.p.n | 4 | ||
| 60.h | even | 2 | 1 | 400.5.p.g | 4 | ||
| 60.l | odd | 4 | 1 | 80.5.p.f | 4 | ||
| 60.l | odd | 4 | 1 | 400.5.p.g | 4 | ||
| 120.q | odd | 4 | 1 | 320.5.p.k | 4 | ||
| 120.w | even | 4 | 1 | 320.5.p.n | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.5.l.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 40.5.l.b | ✓ | 4 | 15.e | even | 4 | 1 | |
| 80.5.p.f | 4 | 12.b | even | 2 | 1 | ||
| 80.5.p.f | 4 | 60.l | odd | 4 | 1 | ||
| 200.5.l.c | 4 | 15.d | odd | 2 | 1 | ||
| 200.5.l.c | 4 | 15.e | even | 4 | 1 | ||
| 320.5.p.k | 4 | 24.f | even | 2 | 1 | ||
| 320.5.p.k | 4 | 120.q | odd | 4 | 1 | ||
| 320.5.p.n | 4 | 24.h | odd | 2 | 1 | ||
| 320.5.p.n | 4 | 120.w | even | 4 | 1 | ||
| 360.5.v.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 360.5.v.b | 4 | 5.c | odd | 4 | 1 | inner | |
| 400.5.p.g | 4 | 60.h | even | 2 | 1 | ||
| 400.5.p.g | 4 | 60.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(360, [\chi])\):
|
\( T_{7}^{4} - 44T_{7}^{3} + 968T_{7}^{2} + 244552T_{7} + 30891364 \)
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\( T_{11}^{2} - 148T_{11} - 6124 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
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| $3$ |
\( T^{4} \)
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| $5$ |
\( T^{4} - 12 T^{3} + \cdots + 390625 \)
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| $7$ |
\( T^{4} - 44 T^{3} + \cdots + 30891364 \)
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| $11$ |
\( (T^{2} - 148 T - 6124)^{2} \)
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| $13$ |
\( T^{4} - 316 T^{3} + \cdots + 44649124 \)
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| $17$ |
\( T^{4} + \cdots + 7525562500 \)
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| $19$ |
\( T^{4} + 93888 T^{2} + 563777536 \)
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| $23$ |
\( T^{4} + 652 T^{3} + \cdots + 879844 \)
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| $29$ |
\( T^{4} + \cdots + 167195938816 \)
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| $31$ |
\( (T^{2} + 1380 T + 186100)^{2} \)
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| $37$ |
\( T^{4} + \cdots + 1322182619044 \)
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| $41$ |
\( (T^{2} - 2156 T - 2596316)^{2} \)
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| $43$ |
\( T^{4} + \cdots + 5932527319684 \)
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| $47$ |
\( T^{4} + \cdots + 3139055540644 \)
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| $53$ |
\( T^{4} + \cdots + 21974737549284 \)
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| $59$ |
\( T^{4} + \cdots + 242185305788416 \)
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| $61$ |
\( (T^{2} + 6060 T - 4228700)^{2} \)
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| $67$ |
\( T^{4} + \cdots + 2926898229124 \)
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| $71$ |
\( (T^{2} + 1116 T - 15569036)^{2} \)
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| $73$ |
\( T^{4} + \cdots + 12\!\cdots\!24 \)
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| $79$ |
\( T^{4} + \cdots + 618613200388096 \)
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| $83$ |
\( T^{4} + \cdots + 14274961481284 \)
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| $89$ |
\( T^{4} + \cdots + 17\!\cdots\!36 \)
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| $97$ |
\( T^{4} + \cdots + 98478194209924 \)
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