Newspace parameters
| Level: | \( N \) | \(=\) | \( 50 = 2 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 50.d (of order \(5\), degree \(4\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.399252010106\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{10})\) |
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| Defining polynomial: |
\( x^{4} - x^{3} + x^{2} - x + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) |
| \(\chi(n)\) | \(-\zeta_{10}^{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 11.1 |
|
0.809017 | − | 0.587785i | −0.809017 | + | 2.48990i | 0.309017 | − | 0.951057i | 0.690983 | − | 2.12663i | 0.809017 | + | 2.48990i | −3.00000 | −0.309017 | − | 0.951057i | −3.11803 | − | 2.26538i | −0.690983 | − | 2.12663i | ||||||||||||||
| 21.1 | −0.309017 | + | 0.951057i | 0.309017 | + | 0.224514i | −0.809017 | − | 0.587785i | 1.80902 | + | 1.31433i | −0.309017 | + | 0.224514i | −3.00000 | 0.809017 | − | 0.587785i | −0.881966 | − | 2.71441i | −1.80902 | + | 1.31433i | |||||||||||||||
| 31.1 | −0.309017 | − | 0.951057i | 0.309017 | − | 0.224514i | −0.809017 | + | 0.587785i | 1.80902 | − | 1.31433i | −0.309017 | − | 0.224514i | −3.00000 | 0.809017 | + | 0.587785i | −0.881966 | + | 2.71441i | −1.80902 | − | 1.31433i | |||||||||||||||
| 41.1 | 0.809017 | + | 0.587785i | −0.809017 | − | 2.48990i | 0.309017 | + | 0.951057i | 0.690983 | + | 2.12663i | 0.809017 | − | 2.48990i | −3.00000 | −0.309017 | + | 0.951057i | −3.11803 | + | 2.26538i | −0.690983 | + | 2.12663i | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 25.d | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 50.2.d.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 450.2.h.a | 4 | ||
| 4.b | odd | 2 | 1 | 400.2.u.c | 4 | ||
| 5.b | even | 2 | 1 | 250.2.d.a | 4 | ||
| 5.c | odd | 4 | 2 | 250.2.e.b | 8 | ||
| 25.d | even | 5 | 1 | inner | 50.2.d.a | ✓ | 4 |
| 25.d | even | 5 | 1 | 1250.2.a.a | 2 | ||
| 25.e | even | 10 | 1 | 250.2.d.a | 4 | ||
| 25.e | even | 10 | 1 | 1250.2.a.d | 2 | ||
| 25.f | odd | 20 | 2 | 250.2.e.b | 8 | ||
| 25.f | odd | 20 | 2 | 1250.2.b.b | 4 | ||
| 75.j | odd | 10 | 1 | 450.2.h.a | 4 | ||
| 100.h | odd | 10 | 1 | 10000.2.a.a | 2 | ||
| 100.j | odd | 10 | 1 | 400.2.u.c | 4 | ||
| 100.j | odd | 10 | 1 | 10000.2.a.n | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 50.2.d.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 50.2.d.a | ✓ | 4 | 25.d | even | 5 | 1 | inner |
| 250.2.d.a | 4 | 5.b | even | 2 | 1 | ||
| 250.2.d.a | 4 | 25.e | even | 10 | 1 | ||
| 250.2.e.b | 8 | 5.c | odd | 4 | 2 | ||
| 250.2.e.b | 8 | 25.f | odd | 20 | 2 | ||
| 400.2.u.c | 4 | 4.b | odd | 2 | 1 | ||
| 400.2.u.c | 4 | 100.j | odd | 10 | 1 | ||
| 450.2.h.a | 4 | 3.b | odd | 2 | 1 | ||
| 450.2.h.a | 4 | 75.j | odd | 10 | 1 | ||
| 1250.2.a.a | 2 | 25.d | even | 5 | 1 | ||
| 1250.2.a.d | 2 | 25.e | even | 10 | 1 | ||
| 1250.2.b.b | 4 | 25.f | odd | 20 | 2 | ||
| 10000.2.a.a | 2 | 100.h | odd | 10 | 1 | ||
| 10000.2.a.n | 2 | 100.j | odd | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + T_{3}^{3} + 6T_{3}^{2} - 4T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(50, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} + T^{2} + \cdots + 1 \)
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| $3$ |
\( T^{4} + T^{3} + 6 T^{2} + \cdots + 1 \)
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| $5$ |
\( T^{4} - 5 T^{3} + \cdots + 25 \)
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| $7$ |
\( (T + 3)^{4} \)
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| $11$ |
\( T^{4} - 3 T^{3} + \cdots + 1 \)
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| $13$ |
\( T^{4} + T^{3} + T^{2} + \cdots + 1 \)
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| $17$ |
\( T^{4} - 3 T^{3} + \cdots + 81 \)
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| $19$ |
\( T^{4} + 10 T^{3} + \cdots + 25 \)
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| $23$ |
\( T^{4} - 9 T^{3} + \cdots + 121 \)
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| $29$ |
\( T^{4} - 15 T^{3} + \cdots + 25 \)
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| $31$ |
\( T^{4} - 3 T^{3} + \cdots + 1 \)
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| $37$ |
\( T^{4} + 17 T^{3} + \cdots + 3721 \)
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| $41$ |
\( T^{4} - 13 T^{3} + \cdots + 121 \)
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| $43$ |
\( (T^{2} + 8 T + 11)^{2} \)
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| $47$ |
\( T^{4} - 23 T^{3} + \cdots + 5041 \)
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| $53$ |
\( T^{4} + 16 T^{3} + \cdots + 256 \)
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| $59$ |
\( T^{4} + 10 T^{3} + \cdots + 400 \)
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| $61$ |
\( T^{4} + 2 T^{3} + \cdots + 361 \)
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| $67$ |
\( T^{4} - 13 T^{3} + \cdots + 3481 \)
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| $71$ |
\( T^{4} - 3 T^{3} + \cdots + 81 \)
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| $73$ |
\( T^{4} - 14 T^{3} + \cdots + 1936 \)
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| $79$ |
\( T^{4} - 10 T^{3} + \cdots + 400 \)
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| $83$ |
\( T^{4} + T^{3} + \cdots + 3721 \)
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| $89$ |
\( T^{4} - 10 T^{3} + \cdots + 400 \)
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| $97$ |
\( T^{4} + 22 T^{3} + \cdots + 10201 \)
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