Newspace parameters
| Level: | \( N \) | \(=\) | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 882.g (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(52.0396846251\) |
| 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: | \(\Z[a_1, \ldots, a_{25}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 14) |
| 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 \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).
| \(n\) | \(199\) | \(785\) |
| \(\chi(n)\) | \(-\zeta_{6}\) | \(1\) |
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 |
|
−1.00000 | − | 1.73205i | 0 | −2.00000 | + | 3.46410i | 7.00000 | + | 12.1244i | 0 | 0 | 8.00000 | 0 | 14.0000 | − | 24.2487i | ||||||||||||||||
| 667.1 | −1.00000 | + | 1.73205i | 0 | −2.00000 | − | 3.46410i | 7.00000 | − | 12.1244i | 0 | 0 | 8.00000 | 0 | 14.0000 | + | 24.2487i | |||||||||||||||||
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 | 882.4.g.k | 2 | |
| 3.b | odd | 2 | 1 | 98.4.c.f | 2 | ||
| 7.b | odd | 2 | 1 | 882.4.g.b | 2 | ||
| 7.c | even | 3 | 1 | 882.4.a.i | 1 | ||
| 7.c | even | 3 | 1 | inner | 882.4.g.k | 2 | |
| 7.d | odd | 6 | 1 | 126.4.a.h | 1 | ||
| 7.d | odd | 6 | 1 | 882.4.g.b | 2 | ||
| 21.c | even | 2 | 1 | 98.4.c.d | 2 | ||
| 21.g | even | 6 | 1 | 14.4.a.a | ✓ | 1 | |
| 21.g | even | 6 | 1 | 98.4.c.d | 2 | ||
| 21.h | odd | 6 | 1 | 98.4.a.a | 1 | ||
| 21.h | odd | 6 | 1 | 98.4.c.f | 2 | ||
| 28.f | even | 6 | 1 | 1008.4.a.s | 1 | ||
| 84.j | odd | 6 | 1 | 112.4.a.a | 1 | ||
| 84.n | even | 6 | 1 | 784.4.a.s | 1 | ||
| 105.o | odd | 6 | 1 | 2450.4.a.bo | 1 | ||
| 105.p | even | 6 | 1 | 350.4.a.l | 1 | ||
| 105.w | odd | 12 | 2 | 350.4.c.b | 2 | ||
| 168.ba | even | 6 | 1 | 448.4.a.b | 1 | ||
| 168.be | odd | 6 | 1 | 448.4.a.o | 1 | ||
| 231.k | odd | 6 | 1 | 1694.4.a.g | 1 | ||
| 273.ba | even | 6 | 1 | 2366.4.a.h | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.4.a.a | ✓ | 1 | 21.g | even | 6 | 1 | |
| 98.4.a.a | 1 | 21.h | odd | 6 | 1 | ||
| 98.4.c.d | 2 | 21.c | even | 2 | 1 | ||
| 98.4.c.d | 2 | 21.g | even | 6 | 1 | ||
| 98.4.c.f | 2 | 3.b | odd | 2 | 1 | ||
| 98.4.c.f | 2 | 21.h | odd | 6 | 1 | ||
| 112.4.a.a | 1 | 84.j | odd | 6 | 1 | ||
| 126.4.a.h | 1 | 7.d | odd | 6 | 1 | ||
| 350.4.a.l | 1 | 105.p | even | 6 | 1 | ||
| 350.4.c.b | 2 | 105.w | odd | 12 | 2 | ||
| 448.4.a.b | 1 | 168.ba | even | 6 | 1 | ||
| 448.4.a.o | 1 | 168.be | odd | 6 | 1 | ||
| 784.4.a.s | 1 | 84.n | even | 6 | 1 | ||
| 882.4.a.i | 1 | 7.c | even | 3 | 1 | ||
| 882.4.g.b | 2 | 7.b | odd | 2 | 1 | ||
| 882.4.g.b | 2 | 7.d | odd | 6 | 1 | ||
| 882.4.g.k | 2 | 1.a | even | 1 | 1 | trivial | |
| 882.4.g.k | 2 | 7.c | even | 3 | 1 | inner | |
| 1008.4.a.s | 1 | 28.f | even | 6 | 1 | ||
| 1694.4.a.g | 1 | 231.k | odd | 6 | 1 | ||
| 2366.4.a.h | 1 | 273.ba | even | 6 | 1 | ||
| 2450.4.a.bo | 1 | 105.o | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(882, [\chi])\):
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\( T_{5}^{2} - 14T_{5} + 196 \)
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\( T_{11}^{2} + 28T_{11} + 784 \)
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\( T_{13} + 18 \)
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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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| $5$ |
\( T^{2} - 14T + 196 \)
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| $7$ |
\( T^{2} \)
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| $11$ |
\( T^{2} + 28T + 784 \)
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| $13$ |
\( (T + 18)^{2} \)
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| $17$ |
\( T^{2} + 74T + 5476 \)
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| $19$ |
\( T^{2} - 80T + 6400 \)
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| $23$ |
\( T^{2} + 112T + 12544 \)
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| $29$ |
\( (T + 190)^{2} \)
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| $31$ |
\( T^{2} - 72T + 5184 \)
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| $37$ |
\( T^{2} - 346T + 119716 \)
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| $41$ |
\( (T - 162)^{2} \)
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| $43$ |
\( (T + 412)^{2} \)
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| $47$ |
\( T^{2} + 24T + 576 \)
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| $53$ |
\( T^{2} - 318T + 101124 \)
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| $59$ |
\( T^{2} - 200T + 40000 \)
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| $61$ |
\( T^{2} + 198T + 39204 \)
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| $67$ |
\( T^{2} - 716T + 512656 \)
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| $71$ |
\( (T + 392)^{2} \)
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| $73$ |
\( T^{2} - 538T + 289444 \)
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| $79$ |
\( T^{2} + 240T + 57600 \)
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| $83$ |
\( (T + 1072)^{2} \)
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| $89$ |
\( T^{2} + 810T + 656100 \)
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| $97$ |
\( (T + 1354)^{2} \)
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