Newspace parameters
| Level: | \( N \) | \(=\) | \( 174 = 2 \cdot 3 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 174.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(10.2663323410\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
|
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| Defining polynomial: |
\( x^{6} + 98x^{4} + 2401x^{2} + 9216 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{6} + 98x^{4} + 2401x^{2} + 9216 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 49\nu ) / 96 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 145\nu^{2} + 3072 ) / 192 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{4} - 341\nu^{2} - 3072 ) / 192 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 81\nu^{3} + 1472\nu ) / 192 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} - 81\nu^{3} - 1088\nu ) / 192 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 5\beta_{2} - 64 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -49\beta_{5} - 49\beta_{4} + 192\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -145\beta_{3} - 341\beta_{2} + 3136 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 2497\beta_{5} + 2881\beta_{4} - 15552\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/174\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(59\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 115.1 |
|
− | 2.00000i | − | 3.00000i | −4.00000 | −10.1710 | −6.00000 | −3.29956 | 8.00000i | −9.00000 | 20.3420i | ||||||||||||||||||||||||||||||||||
| 115.2 | − | 2.00000i | − | 3.00000i | −4.00000 | 8.93551 | −6.00000 | 12.4377 | 8.00000i | −9.00000 | − | 17.8710i | ||||||||||||||||||||||||||||||||||
| 115.3 | − | 2.00000i | − | 3.00000i | −4.00000 | 12.2355 | −6.00000 | −34.1382 | 8.00000i | −9.00000 | − | 24.4710i | ||||||||||||||||||||||||||||||||||
| 115.4 | 2.00000i | 3.00000i | −4.00000 | −10.1710 | −6.00000 | −3.29956 | − | 8.00000i | −9.00000 | − | 20.3420i | |||||||||||||||||||||||||||||||||||
| 115.5 | 2.00000i | 3.00000i | −4.00000 | 8.93551 | −6.00000 | 12.4377 | − | 8.00000i | −9.00000 | 17.8710i | ||||||||||||||||||||||||||||||||||||
| 115.6 | 2.00000i | 3.00000i | −4.00000 | 12.2355 | −6.00000 | −34.1382 | − | 8.00000i | −9.00000 | 24.4710i | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 29.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 174.4.d.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 522.4.d.a | 6 | ||
| 4.b | odd | 2 | 1 | 1392.4.o.a | 6 | ||
| 29.b | even | 2 | 1 | inner | 174.4.d.a | ✓ | 6 |
| 87.d | odd | 2 | 1 | 522.4.d.a | 6 | ||
| 116.d | odd | 2 | 1 | 1392.4.o.a | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 174.4.d.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 174.4.d.a | ✓ | 6 | 29.b | even | 2 | 1 | inner |
| 522.4.d.a | 6 | 3.b | odd | 2 | 1 | ||
| 522.4.d.a | 6 | 87.d | odd | 2 | 1 | ||
| 1392.4.o.a | 6 | 4.b | odd | 2 | 1 | ||
| 1392.4.o.a | 6 | 116.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 11T_{5}^{2} - 106T_{5} + 1112 \)
acting on \(S_{4}^{\mathrm{new}}(174, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{3} \)
|
| $3$ |
\( (T^{2} + 9)^{3} \)
|
| $5$ |
\( (T^{3} - 11 T^{2} + \cdots + 1112)^{2} \)
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| $7$ |
\( (T^{3} + 25 T^{2} + \cdots - 1401)^{2} \)
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| $11$ |
\( T^{6} + 5210 T^{4} + \cdots + 596922624 \)
|
| $13$ |
\( (T^{3} + 10 T^{2} + \cdots + 30500)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 123574043961 \)
|
| $19$ |
\( T^{6} + \cdots + 3957919744 \)
|
| $23$ |
\( (T^{3} + 264 T^{2} + \cdots - 610416)^{2} \)
|
| $29$ |
\( T^{6} + \cdots + 14507145975869 \)
|
| $31$ |
\( T^{6} + \cdots + 25247938816 \)
|
| $37$ |
\( T^{6} + \cdots + 22587663011904 \)
|
| $41$ |
\( T^{6} + \cdots + 3248371800976 \)
|
| $43$ |
\( T^{6} + \cdots + 16\!\cdots\!44 \)
|
| $47$ |
\( T^{6} + \cdots + 99\!\cdots\!41 \)
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| $53$ |
\( (T^{3} - 408 T^{2} + \cdots + 42052512)^{2} \)
|
| $59$ |
\( (T^{3} + 595 T^{2} + \cdots - 161932048)^{2} \)
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| $61$ |
\( T^{6} + \cdots + 33\!\cdots\!00 \)
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| $67$ |
\( (T^{3} + 172 T^{2} + \cdots + 41447078)^{2} \)
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| $71$ |
\( (T^{3} + 150 T^{2} + \cdots + 36888000)^{2} \)
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| $73$ |
\( T^{6} + \cdots + 40\!\cdots\!36 \)
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| $79$ |
\( T^{6} + \cdots + 25247938816 \)
|
| $83$ |
\( (T^{3} + 2044 T^{2} + \cdots - 351351856)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 74\!\cdots\!76 \)
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| $97$ |
\( T^{6} + \cdots + 2591636340736 \)
|
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