Newspace parameters
| Level: | \( N \) | \(=\) | \( 1470 = 2 \cdot 3 \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1470.i (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(11.7380090971\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
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| Defining polynomial: |
\( x^{2} - x + 1 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 210) |
| 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 \(\left(\mathbb{Z}/1470\mathbb{Z}\right)^\times\).
| \(n\) | \(491\) | \(1081\) | \(1177\) |
| \(\chi(n)\) | \(1\) | \(-\zeta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 361.1 |
|
−0.500000 | − | 0.866025i | −0.500000 | + | 0.866025i | −0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | 1.00000 | 0 | 1.00000 | −0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | ||||||||||||
| 961.1 | −0.500000 | + | 0.866025i | −0.500000 | − | 0.866025i | −0.500000 | − | 0.866025i | 0.500000 | − | 0.866025i | 1.00000 | 0 | 1.00000 | −0.500000 | + | 0.866025i | 0.500000 | + | 0.866025i | |||||||||||||
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 | 1470.2.i.b | 2 | |
| 7.b | odd | 2 | 1 | 1470.2.i.f | 2 | ||
| 7.c | even | 3 | 1 | 1470.2.a.q | 1 | ||
| 7.c | even | 3 | 1 | inner | 1470.2.i.b | 2 | |
| 7.d | odd | 6 | 1 | 210.2.a.c | ✓ | 1 | |
| 7.d | odd | 6 | 1 | 1470.2.i.f | 2 | ||
| 21.g | even | 6 | 1 | 630.2.a.b | 1 | ||
| 21.h | odd | 6 | 1 | 4410.2.a.l | 1 | ||
| 28.f | even | 6 | 1 | 1680.2.a.q | 1 | ||
| 35.i | odd | 6 | 1 | 1050.2.a.h | 1 | ||
| 35.j | even | 6 | 1 | 7350.2.a.p | 1 | ||
| 35.k | even | 12 | 2 | 1050.2.g.d | 2 | ||
| 56.j | odd | 6 | 1 | 6720.2.a.bp | 1 | ||
| 56.m | even | 6 | 1 | 6720.2.a.k | 1 | ||
| 84.j | odd | 6 | 1 | 5040.2.a.i | 1 | ||
| 105.p | even | 6 | 1 | 3150.2.a.w | 1 | ||
| 105.w | odd | 12 | 2 | 3150.2.g.e | 2 | ||
| 140.s | even | 6 | 1 | 8400.2.a.p | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 210.2.a.c | ✓ | 1 | 7.d | odd | 6 | 1 | |
| 630.2.a.b | 1 | 21.g | even | 6 | 1 | ||
| 1050.2.a.h | 1 | 35.i | odd | 6 | 1 | ||
| 1050.2.g.d | 2 | 35.k | even | 12 | 2 | ||
| 1470.2.a.q | 1 | 7.c | even | 3 | 1 | ||
| 1470.2.i.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 1470.2.i.b | 2 | 7.c | even | 3 | 1 | inner | |
| 1470.2.i.f | 2 | 7.b | odd | 2 | 1 | ||
| 1470.2.i.f | 2 | 7.d | odd | 6 | 1 | ||
| 1680.2.a.q | 1 | 28.f | even | 6 | 1 | ||
| 3150.2.a.w | 1 | 105.p | even | 6 | 1 | ||
| 3150.2.g.e | 2 | 105.w | odd | 12 | 2 | ||
| 4410.2.a.l | 1 | 21.h | odd | 6 | 1 | ||
| 5040.2.a.i | 1 | 84.j | odd | 6 | 1 | ||
| 6720.2.a.k | 1 | 56.m | even | 6 | 1 | ||
| 6720.2.a.bp | 1 | 56.j | odd | 6 | 1 | ||
| 7350.2.a.p | 1 | 35.j | even | 6 | 1 | ||
| 8400.2.a.p | 1 | 140.s | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1470, [\chi])\):
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\( T_{11}^{2} + 4T_{11} + 16 \)
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\( T_{13} - 2 \)
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\( T_{17}^{2} - 2T_{17} + 4 \)
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\( T_{19}^{2} + 4T_{19} + 16 \)
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\( T_{31}^{2} + 8T_{31} + 64 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + T + 1 \)
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| $3$ |
\( T^{2} + T + 1 \)
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| $5$ |
\( T^{2} - T + 1 \)
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| $7$ |
\( T^{2} \)
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| $11$ |
\( T^{2} + 4T + 16 \)
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| $13$ |
\( (T - 2)^{2} \)
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| $17$ |
\( T^{2} - 2T + 4 \)
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| $19$ |
\( T^{2} + 4T + 16 \)
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| $23$ |
\( T^{2} - 8T + 64 \)
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| $29$ |
\( (T - 6)^{2} \)
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| $31$ |
\( T^{2} + 8T + 64 \)
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| $37$ |
\( T^{2} - 2T + 4 \)
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| $41$ |
\( (T + 2)^{2} \)
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| $43$ |
\( (T + 12)^{2} \)
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| $47$ |
\( T^{2} + 8T + 64 \)
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| $53$ |
\( T^{2} + 6T + 36 \)
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| $59$ |
\( T^{2} - 4T + 16 \)
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| $61$ |
\( T^{2} + 2T + 4 \)
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| $67$ |
\( T^{2} + 12T + 144 \)
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| $71$ |
\( (T - 8)^{2} \)
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| $73$ |
\( T^{2} + 14T + 196 \)
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| $79$ |
\( T^{2} \)
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| $83$ |
\( (T + 12)^{2} \)
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| $89$ |
\( T^{2} - 2T + 4 \)
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| $97$ |
\( (T + 10)^{2} \)
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