Newspace parameters
| Level: | \( N \) | \(=\) | \( 375 = 3 \cdot 5^{3} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 375.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(22.1257162522\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.920588125.1 |
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| Defining polynomial: |
\( x^{6} - 2x^{5} - 31x^{4} + 12x^{3} + 231x^{2} + 20x - 400 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 5^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{6} - 2x^{5} - 31x^{4} + 12x^{3} + 231x^{2} + 20x - 400 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -4\nu^{5} + 43\nu^{4} - 41\nu^{3} - 583\nu^{2} + 586\nu + 1390 ) / 130 \)
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| \(\beta_{2}\) | \(=\) |
\( ( 33\nu^{5} - 176\nu^{4} - 523\nu^{3} + 2356\nu^{2} + 1893\nu - 6300 ) / 260 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -37\nu^{5} + 154\nu^{4} + 677\nu^{3} - 1574\nu^{2} - 2217\nu + 3140 ) / 260 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 29\nu^{5} - 133\nu^{4} - 499\nu^{3} + 1448\nu^{2} + 2024\nu - 2960 ) / 130 \)
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| \(\beta_{5}\) | \(=\) |
\( ( 79\nu^{5} - 378\nu^{4} - 1319\nu^{3} + 4218\nu^{2} + 4579\nu - 7920 ) / 260 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + 2\beta_{4} + \beta_{3} + 3 ) / 5 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -6\beta_{5} + 2\beta_{4} - 5\beta_{3} + 5\beta_{2} - \beta _1 + 55 ) / 5 \)
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| \(\nu^{3}\) | \(=\) |
\( ( -37\beta_{5} + 34\beta_{4} - 18\beta_{3} + 5\beta_{2} - 15\beta _1 + 146 ) / 5 \)
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| \(\nu^{4}\) | \(=\) |
\( ( -223\beta_{5} + 116\beta_{4} - 193\beta_{3} + 100\beta_{2} - 56\beta _1 + 1201 ) / 5 \)
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| \(\nu^{5}\) | \(=\) |
\( -258\beta_{5} + 180\beta_{4} - 203\beta_{3} + 59\beta_{2} - 93\beta _1 + 1115 \)
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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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−3.63787 | −3.00000 | 5.23409 | 0 | 10.9136 | 34.3410 | 10.0620 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.77928 | −3.00000 | −0.275604 | 0 | 8.33784 | 13.3180 | 23.0002 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −0.341110 | −3.00000 | −7.88364 | 0 | 1.02333 | −10.9396 | 5.41808 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.03190 | −3.00000 | −3.87136 | 0 | −6.09571 | −17.8006 | −24.1215 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 4.36541 | −3.00000 | 11.0568 | 0 | −13.0962 | 27.4269 | 13.3442 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 5.36094 | −3.00000 | 20.7397 | 0 | −16.0828 | −18.3456 | 68.2970 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(5\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 375.4.a.f | yes | 6 |
| 3.b | odd | 2 | 1 | 1125.4.a.g | 6 | ||
| 5.b | even | 2 | 1 | 375.4.a.c | ✓ | 6 | |
| 5.c | odd | 4 | 2 | 375.4.b.c | 12 | ||
| 15.d | odd | 2 | 1 | 1125.4.a.j | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 375.4.a.c | ✓ | 6 | 5.b | even | 2 | 1 | |
| 375.4.a.f | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 375.4.b.c | 12 | 5.c | odd | 4 | 2 | ||
| 1125.4.a.g | 6 | 3.b | odd | 2 | 1 | ||
| 1125.4.a.j | 6 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 5T_{2}^{5} - 24T_{2}^{4} + 103T_{2}^{3} + 169T_{2}^{2} - 436T_{2} - 164 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(375))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 5 T^{5} + \cdots - 164 \)
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| $3$ |
\( (T + 3)^{6} \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} - 28 T^{5} + \cdots - 44812100 \)
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| $11$ |
\( T^{6} + 77 T^{5} + \cdots + 77405399 \)
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| $13$ |
\( T^{6} + \cdots + 58038672499 \)
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| $17$ |
\( T^{6} + \cdots - 26364344519 \)
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| $19$ |
\( T^{6} + \cdots - 19782508100 \)
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| $23$ |
\( T^{6} + \cdots + 267264399475 \)
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| $29$ |
\( T^{6} + \cdots + 5051630101905 \)
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| $31$ |
\( T^{6} + \cdots + 37626542730625 \)
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| $37$ |
\( T^{6} + \cdots + 14253391256875 \)
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| $41$ |
\( T^{6} + \cdots - 11669349046900 \)
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| $43$ |
\( T^{6} + \cdots + 154714460589 \)
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| $47$ |
\( T^{6} + \cdots + 11\!\cdots\!01 \)
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| $53$ |
\( T^{6} + \cdots - 675158380993900 \)
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| $59$ |
\( T^{6} + \cdots + 14\!\cdots\!05 \)
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| $61$ |
\( T^{6} + \cdots + 24\!\cdots\!04 \)
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| $67$ |
\( T^{6} + \cdots + 10\!\cdots\!36 \)
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| $71$ |
\( T^{6} + \cdots - 12\!\cdots\!36 \)
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| $73$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
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| $79$ |
\( T^{6} + \cdots + 413522946922000 \)
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| $83$ |
\( T^{6} + \cdots - 14\!\cdots\!76 \)
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| $89$ |
\( T^{6} + \cdots - 36\!\cdots\!80 \)
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| $97$ |
\( T^{6} + \cdots + 42\!\cdots\!16 \)
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