Newspace parameters
| Level: | \( N \) | \(=\) | \( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1452.e (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(39.5641343851\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.46805967296.1 |
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| Defining polynomial: |
\( x^{6} - 2x^{5} + 6x^{4} + 2x^{3} + 62x^{2} + 174x + 486 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 6x^{4} + 2x^{3} + 62x^{2} + 174x + 486 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{5} + 20\nu^{4} - 91\nu^{3} + 157\nu^{2} - 246\nu + 243 ) / 243 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} + 5\nu^{3} + 7\nu^{2} + 69\nu + 162 ) / 81 \)
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| \(\beta_{4}\) | \(=\) |
\( ( -4\nu^{5} + 13\nu^{4} - 20\nu^{3} + 179\nu^{2} - 492\nu ) / 243 \)
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| \(\beta_{5}\) | \(=\) |
\( ( -4\nu^{5} + 13\nu^{4} - 20\nu^{3} - 64\nu^{2} - 6\nu - 486 ) / 243 \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + \beta_{4} + 2\beta _1 - 2 \)
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| \(\nu^{3}\) | \(=\) |
\( 3\beta_{5} + 3\beta_{4} + 6\beta_{3} - 3\beta_{2} - 2\beta _1 - 3 \)
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| \(\nu^{4}\) | \(=\) |
\( 23\beta_{5} + 4\beta_{4} + 36\beta_{3} - 22\beta _1 - 26 \)
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| \(\nu^{5}\) | \(=\) |
\( 15\beta_{5} - 18\beta_{4} + 87\beta_{3} + 15\beta_{2} - 95\beta _1 - 159 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1452\mathbb{Z}\right)^\times\).
| \(n\) | \(485\) | \(727\) | \(1333\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 485.1 |
|
0 | −1.84805 | − | 2.36320i | 0 | − | 2.88401i | 0 | −6.13687 | 0 | −2.16942 | + | 8.73462i | 0 | |||||||||||||||||||||||||||||||
| 485.2 | 0 | −1.84805 | + | 2.36320i | 0 | 2.88401i | 0 | −6.13687 | 0 | −2.16942 | − | 8.73462i | 0 | |||||||||||||||||||||||||||||||||
| 485.3 | 0 | 1.15560 | − | 2.76850i | 0 | 4.29893i | 0 | 5.41679 | 0 | −6.32920 | − | 6.39853i | 0 | |||||||||||||||||||||||||||||||||
| 485.4 | 0 | 1.15560 | + | 2.76850i | 0 | − | 4.29893i | 0 | 5.41679 | 0 | −6.32920 | + | 6.39853i | 0 | ||||||||||||||||||||||||||||||||
| 485.5 | 0 | 2.69245 | − | 1.32314i | 0 | − | 8.37865i | 0 | −7.27992 | 0 | 5.49862 | − | 7.12497i | 0 | ||||||||||||||||||||||||||||||||
| 485.6 | 0 | 2.69245 | + | 1.32314i | 0 | 8.37865i | 0 | −7.27992 | 0 | 5.49862 | + | 7.12497i | 0 | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1452.3.e.i | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 1452.3.e.i | ✓ | 6 |
| 11.b | odd | 2 | 1 | 1452.3.e.j | yes | 6 | |
| 33.d | even | 2 | 1 | 1452.3.e.j | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1452.3.e.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1452.3.e.i | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
| 1452.3.e.j | yes | 6 | 11.b | odd | 2 | 1 | |
| 1452.3.e.j | yes | 6 | 33.d | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1452, [\chi])\):
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\( T_{5}^{6} + 97T_{5}^{4} + 2035T_{5}^{2} + 10791 \)
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\( T_{7}^{3} + 8T_{7}^{2} - 28T_{7} - 242 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} - 4 T^{5} + \cdots + 729 \)
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| $5$ |
\( T^{6} + 97 T^{4} + \cdots + 10791 \)
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| $7$ |
\( (T^{3} + 8 T^{2} + \cdots - 242)^{2} \)
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| $11$ |
\( T^{6} \)
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| $13$ |
\( (T^{3} - 3 T^{2} - 259 T + 99)^{2} \)
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| $17$ |
\( T^{6} + 781 T^{4} + \cdots + 1305711 \)
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| $19$ |
\( (T^{3} + 6 T^{2} + \cdots + 4686)^{2} \)
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| $23$ |
\( T^{6} + 2852 T^{4} + \cdots + 858143484 \)
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| $29$ |
\( T^{6} + 3509 T^{4} + \cdots + 951863319 \)
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| $31$ |
\( (T^{3} - 24 T^{2} + \cdots + 1746)^{2} \)
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| $37$ |
\( (T^{3} - 51 T^{2} + \cdots + 2763)^{2} \)
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| $41$ |
\( T^{6} + \cdots + 1421919279 \)
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| $43$ |
\( (T^{3} - 26 T^{2} + \cdots - 8404)^{2} \)
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| $47$ |
\( T^{6} + 8888 T^{4} + \cdots + 423050364 \)
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| $53$ |
\( T^{6} + 7249 T^{4} + \cdots + 11751399 \)
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| $59$ |
\( T^{6} + \cdots + 232726648176 \)
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| $61$ |
\( (T^{3} + 76 T^{2} + \cdots - 495088)^{2} \)
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| $67$ |
\( (T^{3} + 12 T^{2} + \cdots - 5886)^{2} \)
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| $71$ |
\( T^{6} + \cdots + 69181359984 \)
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| $73$ |
\( (T^{3} + 140 T^{2} + \cdots - 445500)^{2} \)
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| $79$ |
\( (T^{3} - 136 T^{2} + \cdots + 320254)^{2} \)
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| $83$ |
\( T^{6} + \cdots + 103835361564 \)
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| $89$ |
\( T^{6} + \cdots + 423278798679 \)
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| $97$ |
\( (T^{3} - 127 T^{2} + \cdots - 17729)^{2} \)
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