Newspace parameters
| Level: | \( N \) | \(=\) | \( 405 = 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 405.e (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(3.23394128186\) |
| 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: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 135) |
| 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 \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).
| \(n\) | \(82\) | \(326\) |
| \(\chi(n)\) | \(1\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 136.1 |
|
1.00000 | − | 1.73205i | 0 | −1.00000 | − | 1.73205i | 0.500000 | + | 0.866025i | 0 | 1.50000 | − | 2.59808i | 0 | 0 | 2.00000 | ||||||||||||||||
| 271.1 | 1.00000 | + | 1.73205i | 0 | −1.00000 | + | 1.73205i | 0.500000 | − | 0.866025i | 0 | 1.50000 | + | 2.59808i | 0 | 0 | 2.00000 | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 405.2.e.h | 2 | |
| 3.b | odd | 2 | 1 | 405.2.e.b | 2 | ||
| 9.c | even | 3 | 1 | 135.2.a.a | ✓ | 1 | |
| 9.c | even | 3 | 1 | inner | 405.2.e.h | 2 | |
| 9.d | odd | 6 | 1 | 135.2.a.b | yes | 1 | |
| 9.d | odd | 6 | 1 | 405.2.e.b | 2 | ||
| 36.f | odd | 6 | 1 | 2160.2.a.j | 1 | ||
| 36.h | even | 6 | 1 | 2160.2.a.v | 1 | ||
| 45.h | odd | 6 | 1 | 675.2.a.a | 1 | ||
| 45.j | even | 6 | 1 | 675.2.a.i | 1 | ||
| 45.k | odd | 12 | 2 | 675.2.b.a | 2 | ||
| 45.l | even | 12 | 2 | 675.2.b.b | 2 | ||
| 63.l | odd | 6 | 1 | 6615.2.a.a | 1 | ||
| 63.o | even | 6 | 1 | 6615.2.a.j | 1 | ||
| 72.j | odd | 6 | 1 | 8640.2.a.c | 1 | ||
| 72.l | even | 6 | 1 | 8640.2.a.bb | 1 | ||
| 72.n | even | 6 | 1 | 8640.2.a.bh | 1 | ||
| 72.p | odd | 6 | 1 | 8640.2.a.ce | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 135.2.a.a | ✓ | 1 | 9.c | even | 3 | 1 | |
| 135.2.a.b | yes | 1 | 9.d | odd | 6 | 1 | |
| 405.2.e.b | 2 | 3.b | odd | 2 | 1 | ||
| 405.2.e.b | 2 | 9.d | odd | 6 | 1 | ||
| 405.2.e.h | 2 | 1.a | even | 1 | 1 | trivial | |
| 405.2.e.h | 2 | 9.c | even | 3 | 1 | inner | |
| 675.2.a.a | 1 | 45.h | odd | 6 | 1 | ||
| 675.2.a.i | 1 | 45.j | even | 6 | 1 | ||
| 675.2.b.a | 2 | 45.k | odd | 12 | 2 | ||
| 675.2.b.b | 2 | 45.l | even | 12 | 2 | ||
| 2160.2.a.j | 1 | 36.f | odd | 6 | 1 | ||
| 2160.2.a.v | 1 | 36.h | even | 6 | 1 | ||
| 6615.2.a.a | 1 | 63.l | odd | 6 | 1 | ||
| 6615.2.a.j | 1 | 63.o | even | 6 | 1 | ||
| 8640.2.a.c | 1 | 72.j | odd | 6 | 1 | ||
| 8640.2.a.bb | 1 | 72.l | even | 6 | 1 | ||
| 8640.2.a.bh | 1 | 72.n | even | 6 | 1 | ||
| 8640.2.a.ce | 1 | 72.p | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(405, [\chi])\):
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\( T_{2}^{2} - 2T_{2} + 4 \)
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\( T_{7}^{2} - 3T_{7} + 9 \)
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\( T_{11}^{2} - 2T_{11} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 2T + 4 \)
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| $3$ |
\( T^{2} \)
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| $5$ |
\( T^{2} - T + 1 \)
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| $7$ |
\( T^{2} - 3T + 9 \)
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| $11$ |
\( T^{2} - 2T + 4 \)
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| $13$ |
\( T^{2} - 5T + 25 \)
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| $17$ |
\( (T + 8)^{2} \)
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| $19$ |
\( (T - 1)^{2} \)
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| $23$ |
\( T^{2} + 6T + 36 \)
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| $29$ |
\( T^{2} + 2T + 4 \)
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| $31$ |
\( T^{2} \)
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| $37$ |
\( (T - 5)^{2} \)
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| $41$ |
\( T^{2} - 10T + 100 \)
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| $43$ |
\( T^{2} + 4T + 16 \)
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| $47$ |
\( T^{2} + 4T + 16 \)
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| $53$ |
\( (T + 2)^{2} \)
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| $59$ |
\( T^{2} - 8T + 64 \)
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| $61$ |
\( T^{2} + 7T + 49 \)
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| $67$ |
\( T^{2} - 9T + 81 \)
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| $71$ |
\( (T - 2)^{2} \)
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| $73$ |
\( (T + 5)^{2} \)
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| $79$ |
\( T^{2} - 3T + 9 \)
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| $83$ |
\( T^{2} + 6T + 36 \)
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| $89$ |
\( (T + 12)^{2} \)
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| $97$ |
\( T^{2} - 13T + 169 \)
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