Newspace parameters
| Level: | \( N \) | \(=\) | \( 58 = 2 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 58.d (of order \(7\), degree \(6\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.463132331723\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{14})\) |
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|
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| Defining polynomial: |
\( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{7}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{14}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/58\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) |
| \(\chi(n)\) | \(-\zeta_{14}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
0.222521 | + | 0.974928i | 0.500000 | + | 0.240787i | −0.900969 | + | 0.433884i | −0.0440730 | − | 0.193096i | −0.123490 | + | 0.541044i | −0.0990311 | − | 0.0476909i | −0.623490 | − | 0.781831i | −1.67845 | − | 2.10471i | 0.178448 | − | 0.0859360i | ||||||||||||||||||
| 23.1 | −0.623490 | + | 0.781831i | 0.500000 | − | 2.19064i | −0.222521 | − | 0.974928i | 0.969501 | − | 1.21572i | 1.40097 | + | 1.75676i | −0.777479 | + | 3.40636i | 0.900969 | + | 0.433884i | −1.84601 | − | 0.888992i | 0.346011 | + | 1.51597i | |||||||||||||||||||
| 25.1 | 0.222521 | − | 0.974928i | 0.500000 | − | 0.240787i | −0.900969 | − | 0.433884i | −0.0440730 | + | 0.193096i | −0.123490 | − | 0.541044i | −0.0990311 | + | 0.0476909i | −0.623490 | + | 0.781831i | −1.67845 | + | 2.10471i | 0.178448 | + | 0.0859360i | |||||||||||||||||||
| 45.1 | 0.900969 | + | 0.433884i | 0.500000 | − | 0.626980i | 0.623490 | + | 0.781831i | −2.92543 | − | 1.40881i | 0.722521 | − | 0.347948i | −1.62349 | + | 2.03579i | 0.222521 | + | 0.974928i | 0.524459 | + | 2.29780i | −2.02446 | − | 2.53859i | |||||||||||||||||||
| 49.1 | 0.900969 | − | 0.433884i | 0.500000 | + | 0.626980i | 0.623490 | − | 0.781831i | −2.92543 | + | 1.40881i | 0.722521 | + | 0.347948i | −1.62349 | − | 2.03579i | 0.222521 | − | 0.974928i | 0.524459 | − | 2.29780i | −2.02446 | + | 2.53859i | |||||||||||||||||||
| 53.1 | −0.623490 | − | 0.781831i | 0.500000 | + | 2.19064i | −0.222521 | + | 0.974928i | 0.969501 | + | 1.21572i | 1.40097 | − | 1.75676i | −0.777479 | − | 3.40636i | 0.900969 | − | 0.433884i | −1.84601 | + | 0.888992i | 0.346011 | − | 1.51597i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 29.d | even | 7 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 58.2.d.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 522.2.k.c | 6 | ||
| 4.b | odd | 2 | 1 | 464.2.u.b | 6 | ||
| 29.d | even | 7 | 1 | inner | 58.2.d.a | ✓ | 6 |
| 29.d | even | 7 | 1 | 1682.2.a.m | 3 | ||
| 29.e | even | 14 | 1 | 1682.2.a.n | 3 | ||
| 29.f | odd | 28 | 2 | 1682.2.b.g | 6 | ||
| 87.j | odd | 14 | 1 | 522.2.k.c | 6 | ||
| 116.j | odd | 14 | 1 | 464.2.u.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 58.2.d.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 58.2.d.a | ✓ | 6 | 29.d | even | 7 | 1 | inner |
| 464.2.u.b | 6 | 4.b | odd | 2 | 1 | ||
| 464.2.u.b | 6 | 116.j | odd | 14 | 1 | ||
| 522.2.k.c | 6 | 3.b | odd | 2 | 1 | ||
| 522.2.k.c | 6 | 87.j | odd | 14 | 1 | ||
| 1682.2.a.m | 3 | 29.d | even | 7 | 1 | ||
| 1682.2.a.n | 3 | 29.e | even | 14 | 1 | ||
| 1682.2.b.g | 6 | 29.f | odd | 28 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 3T_{3}^{5} + 9T_{3}^{4} - 13T_{3}^{3} + 11T_{3}^{2} - 5T_{3} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(58, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{5} + T^{4} + \cdots + 1 \)
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| $3$ |
\( T^{6} - 3 T^{5} + \cdots + 1 \)
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| $5$ |
\( T^{6} + 4 T^{5} + \cdots + 1 \)
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| $7$ |
\( T^{6} + 5 T^{5} + \cdots + 1 \)
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| $11$ |
\( T^{6} + 6 T^{5} + \cdots + 729 \)
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| $13$ |
\( T^{6} - 11 T^{5} + \cdots + 841 \)
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| $17$ |
\( (T^{3} + 5 T^{2} - 8 T - 41)^{2} \)
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| $19$ |
\( T^{6} + 6 T^{5} + \cdots + 1681 \)
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| $23$ |
\( T^{6} + 7 T^{5} + \cdots + 8281 \)
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| $29$ |
\( T^{6} - 15 T^{5} + \cdots + 24389 \)
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| $31$ |
\( T^{6} - 4 T^{5} + \cdots + 1849 \)
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| $37$ |
\( T^{6} + 13 T^{5} + \cdots + 19321 \)
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| $41$ |
\( (T^{3} + 6 T^{2} - 37 T + 1)^{2} \)
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| $43$ |
\( T^{6} + 7 T^{5} + \cdots + 41209 \)
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| $47$ |
\( T^{6} - 3 T^{5} + \cdots + 1 \)
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| $53$ |
\( T^{6} + 22 T^{5} + \cdots + 113569 \)
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| $59$ |
\( (T^{3} + 19 T^{2} + \cdots + 169)^{2} \)
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| $61$ |
\( T^{6} - 31 T^{5} + \cdots + 169 \)
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| $67$ |
\( T^{6} - 11 T^{5} + \cdots + 1771561 \)
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| $71$ |
\( T^{6} - 23 T^{5} + \cdots + 63001 \)
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| $73$ |
\( T^{6} - 5 T^{5} + \cdots + 241081 \)
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| $79$ |
\( T^{6} + 56 T^{4} + \cdots + 3136 \)
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| $83$ |
\( T^{6} - 35 T^{5} + \cdots + 790321 \)
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| $89$ |
\( T^{6} - 13 T^{5} + \cdots + 253009 \)
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| $97$ |
\( T^{6} - 20 T^{5} + \cdots + 10816 \)
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