Newspace parameters
| Level: | \( N \) | \(=\) | \( 174 = 2 \cdot 3 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 174.g (of order \(7\), degree \(6\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.38939699517\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{14})\) |
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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 \(\left(\mathbb{Z}/174\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(59\) |
| \(\chi(n)\) | \(-\zeta_{14}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
0.222521 | + | 0.974928i | 0.900969 | + | 0.433884i | −0.900969 | + | 0.433884i | 0.722521 | + | 3.16557i | −0.222521 | + | 0.974928i | −2.24698 | − | 1.08209i | −0.623490 | − | 0.781831i | 0.623490 | + | 0.781831i | −2.92543 | + | 1.40881i | ||||||||||||||||||
| 25.1 | 0.222521 | − | 0.974928i | 0.900969 | − | 0.433884i | −0.900969 | − | 0.433884i | 0.722521 | − | 3.16557i | −0.222521 | − | 0.974928i | −2.24698 | + | 1.08209i | −0.623490 | + | 0.781831i | 0.623490 | − | 0.781831i | −2.92543 | − | 1.40881i | |||||||||||||||||||
| 49.1 | 0.900969 | − | 0.433884i | −0.623490 | − | 0.781831i | 0.623490 | − | 0.781831i | 1.40097 | − | 0.674671i | −0.900969 | − | 0.433884i | −0.554958 | − | 0.695895i | 0.222521 | − | 0.974928i | −0.222521 | + | 0.974928i | 0.969501 | − | 1.21572i | |||||||||||||||||||
| 103.1 | 0.900969 | + | 0.433884i | −0.623490 | + | 0.781831i | 0.623490 | + | 0.781831i | 1.40097 | + | 0.674671i | −0.900969 | + | 0.433884i | −0.554958 | + | 0.695895i | 0.222521 | + | 0.974928i | −0.222521 | − | 0.974928i | 0.969501 | + | 1.21572i | |||||||||||||||||||
| 139.1 | −0.623490 | + | 0.781831i | 0.222521 | − | 0.974928i | −0.222521 | − | 0.974928i | −0.123490 | + | 0.154851i | 0.623490 | + | 0.781831i | 0.801938 | − | 3.51352i | 0.900969 | + | 0.433884i | −0.900969 | − | 0.433884i | −0.0440730 | − | 0.193096i | |||||||||||||||||||
| 169.1 | −0.623490 | − | 0.781831i | 0.222521 | + | 0.974928i | −0.222521 | + | 0.974928i | −0.123490 | − | 0.154851i | 0.623490 | − | 0.781831i | 0.801938 | + | 3.51352i | 0.900969 | − | 0.433884i | −0.900969 | + | 0.433884i | −0.0440730 | + | 0.193096i | |||||||||||||||||||
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 | 174.2.g.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 522.2.k.a | 6 | ||
| 29.d | even | 7 | 1 | inner | 174.2.g.b | ✓ | 6 |
| 29.d | even | 7 | 1 | 5046.2.a.z | 3 | ||
| 29.e | even | 14 | 1 | 5046.2.a.bc | 3 | ||
| 87.j | odd | 14 | 1 | 522.2.k.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 174.2.g.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 174.2.g.b | ✓ | 6 | 29.d | even | 7 | 1 | inner |
| 522.2.k.a | 6 | 3.b | odd | 2 | 1 | ||
| 522.2.k.a | 6 | 87.j | odd | 14 | 1 | ||
| 5046.2.a.z | 3 | 29.d | even | 7 | 1 | ||
| 5046.2.a.bc | 3 | 29.e | even | 14 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 4T_{5}^{5} + 16T_{5}^{4} - 29T_{5}^{3} + 18T_{5}^{2} + 5T_{5} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(174, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - T^{5} + T^{4} + \cdots + 1 \)
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| $3$ |
\( T^{6} - T^{5} + T^{4} + \cdots + 1 \)
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| $5$ |
\( T^{6} - 4 T^{5} + \cdots + 1 \)
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| $7$ |
\( T^{6} + 4 T^{5} + \cdots + 64 \)
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| $11$ |
\( T^{6} - 8 T^{5} + \cdots + 64 \)
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| $13$ |
\( T^{6} - 35 T^{3} + \cdots + 49 \)
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| $17$ |
\( (T^{3} - 7 T^{2} + 14 T - 7)^{2} \)
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| $19$ |
\( T^{6} - 2 T^{5} + \cdots + 10816 \)
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| $23$ |
\( T^{6} - 8 T^{5} + \cdots + 64 \)
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| $29$ |
\( T^{6} + T^{5} + \cdots + 24389 \)
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| $31$ |
\( T^{6} + 6 T^{5} + \cdots + 64 \)
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| $37$ |
\( T^{6} + 32 T^{5} + \cdots + 703921 \)
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| $41$ |
\( (T^{3} + T^{2} - 72 T + 41)^{2} \)
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| $43$ |
\( T^{6} - 4 T^{5} + \cdots + 10816 \)
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| $47$ |
\( T^{6} + 2 T^{5} + \cdots + 53824 \)
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| $53$ |
\( T^{6} - 23 T^{5} + \cdots + 49729 \)
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| $59$ |
\( (T^{3} - 2 T^{2} - 8 T + 8)^{2} \)
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| $61$ |
\( T^{6} + 36 T^{5} + \cdots + 613089 \)
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| $67$ |
\( T^{6} - 784 T^{3} + \cdots + 153664 \)
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| $71$ |
\( T^{6} + 56 T^{4} + \cdots + 2637376 \)
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| $73$ |
\( T^{6} - 17 T^{5} + \cdots + 6889 \)
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| $79$ |
\( T^{6} - 2 T^{5} + \cdots + 64 \)
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| $83$ |
\( T^{6} + 14 T^{5} + \cdots + 3136 \)
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| $89$ |
\( T^{6} - 4 T^{5} + \cdots + 284089 \)
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| $97$ |
\( T^{6} + 7 T^{5} + \cdots + 3869089 \)
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