Newspace parameters
| Level: | \( N \) | \(=\) | \( 147 = 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 147.e (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(8.67328077084\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
|
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| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).
| \(n\) | \(50\) | \(52\) |
| \(\chi(n)\) | \(1\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
0.500000 | + | 0.866025i | −1.50000 | + | 2.59808i | 3.50000 | − | 6.06218i | 6.00000 | + | 10.3923i | −3.00000 | 0 | 15.0000 | −4.50000 | − | 7.79423i | −6.00000 | + | 10.3923i | ||||||||||||
| 79.1 | 0.500000 | − | 0.866025i | −1.50000 | − | 2.59808i | 3.50000 | + | 6.06218i | 6.00000 | − | 10.3923i | −3.00000 | 0 | 15.0000 | −4.50000 | + | 7.79423i | −6.00000 | − | 10.3923i | |||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 147.4.e.e | 2 | |
| 3.b | odd | 2 | 1 | 441.4.e.f | 2 | ||
| 7.b | odd | 2 | 1 | 147.4.e.f | 2 | ||
| 7.c | even | 3 | 1 | 147.4.a.e | yes | 1 | |
| 7.c | even | 3 | 1 | inner | 147.4.e.e | 2 | |
| 7.d | odd | 6 | 1 | 147.4.a.d | ✓ | 1 | |
| 7.d | odd | 6 | 1 | 147.4.e.f | 2 | ||
| 21.c | even | 2 | 1 | 441.4.e.g | 2 | ||
| 21.g | even | 6 | 1 | 441.4.a.g | 1 | ||
| 21.g | even | 6 | 1 | 441.4.e.g | 2 | ||
| 21.h | odd | 6 | 1 | 441.4.a.h | 1 | ||
| 21.h | odd | 6 | 1 | 441.4.e.f | 2 | ||
| 28.f | even | 6 | 1 | 2352.4.a.bi | 1 | ||
| 28.g | odd | 6 | 1 | 2352.4.a.b | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 147.4.a.d | ✓ | 1 | 7.d | odd | 6 | 1 | |
| 147.4.a.e | yes | 1 | 7.c | even | 3 | 1 | |
| 147.4.e.e | 2 | 1.a | even | 1 | 1 | trivial | |
| 147.4.e.e | 2 | 7.c | even | 3 | 1 | inner | |
| 147.4.e.f | 2 | 7.b | odd | 2 | 1 | ||
| 147.4.e.f | 2 | 7.d | odd | 6 | 1 | ||
| 441.4.a.g | 1 | 21.g | even | 6 | 1 | ||
| 441.4.a.h | 1 | 21.h | odd | 6 | 1 | ||
| 441.4.e.f | 2 | 3.b | odd | 2 | 1 | ||
| 441.4.e.f | 2 | 21.h | odd | 6 | 1 | ||
| 441.4.e.g | 2 | 21.c | even | 2 | 1 | ||
| 441.4.e.g | 2 | 21.g | even | 6 | 1 | ||
| 2352.4.a.b | 1 | 28.g | odd | 6 | 1 | ||
| 2352.4.a.bi | 1 | 28.f | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(147, [\chi])\):
|
\( T_{2}^{2} - T_{2} + 1 \)
|
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\( T_{5}^{2} - 12T_{5} + 144 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - T + 1 \)
|
| $3$ |
\( T^{2} + 3T + 9 \)
|
| $5$ |
\( T^{2} - 12T + 144 \)
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| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + 20T + 400 \)
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| $13$ |
\( (T - 84)^{2} \)
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| $17$ |
\( T^{2} + 96T + 9216 \)
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| $19$ |
\( T^{2} - 12T + 144 \)
|
| $23$ |
\( T^{2} - 176T + 30976 \)
|
| $29$ |
\( (T - 58)^{2} \)
|
| $31$ |
\( T^{2} + 264T + 69696 \)
|
| $37$ |
\( T^{2} + 258T + 66564 \)
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| $41$ |
\( T^{2} \)
|
| $43$ |
\( (T - 156)^{2} \)
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| $47$ |
\( T^{2} + 408T + 166464 \)
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| $53$ |
\( T^{2} - 722T + 521284 \)
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| $59$ |
\( T^{2} - 492T + 242064 \)
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| $61$ |
\( T^{2} + 492T + 242064 \)
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| $67$ |
\( T^{2} + 412T + 169744 \)
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| $71$ |
\( (T - 296)^{2} \)
|
| $73$ |
\( T^{2} - 240T + 57600 \)
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| $79$ |
\( T^{2} + 776T + 602176 \)
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| $83$ |
\( (T + 924)^{2} \)
|
| $89$ |
\( T^{2} + 744T + 553536 \)
|
| $97$ |
\( (T - 168)^{2} \)
|
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