Newspace parameters
| Level: | \( N \) | \(=\) | \( 3168 = 2^{5} \cdot 3^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3168.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(25.2966073603\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{17}) \) |
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| Defining polynomial: |
\( x^{2} - x - 4 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 352) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 0 | 0 | −3.56155 | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 0.561553 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(11\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3168.2.a.bd | 2 | |
| 3.b | odd | 2 | 1 | 352.2.a.g | ✓ | 2 | |
| 4.b | odd | 2 | 1 | 3168.2.a.bc | 2 | ||
| 8.b | even | 2 | 1 | 6336.2.a.cv | 2 | ||
| 8.d | odd | 2 | 1 | 6336.2.a.cw | 2 | ||
| 12.b | even | 2 | 1 | 352.2.a.h | yes | 2 | |
| 15.d | odd | 2 | 1 | 8800.2.a.be | 2 | ||
| 24.f | even | 2 | 1 | 704.2.a.n | 2 | ||
| 24.h | odd | 2 | 1 | 704.2.a.o | 2 | ||
| 33.d | even | 2 | 1 | 3872.2.a.p | 2 | ||
| 48.i | odd | 4 | 2 | 2816.2.c.t | 4 | ||
| 48.k | even | 4 | 2 | 2816.2.c.s | 4 | ||
| 60.h | even | 2 | 1 | 8800.2.a.bd | 2 | ||
| 132.d | odd | 2 | 1 | 3872.2.a.ba | 2 | ||
| 264.m | even | 2 | 1 | 7744.2.a.cm | 2 | ||
| 264.p | odd | 2 | 1 | 7744.2.a.bw | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 352.2.a.g | ✓ | 2 | 3.b | odd | 2 | 1 | |
| 352.2.a.h | yes | 2 | 12.b | even | 2 | 1 | |
| 704.2.a.n | 2 | 24.f | even | 2 | 1 | ||
| 704.2.a.o | 2 | 24.h | odd | 2 | 1 | ||
| 2816.2.c.s | 4 | 48.k | even | 4 | 2 | ||
| 2816.2.c.t | 4 | 48.i | odd | 4 | 2 | ||
| 3168.2.a.bc | 2 | 4.b | odd | 2 | 1 | ||
| 3168.2.a.bd | 2 | 1.a | even | 1 | 1 | trivial | |
| 3872.2.a.p | 2 | 33.d | even | 2 | 1 | ||
| 3872.2.a.ba | 2 | 132.d | odd | 2 | 1 | ||
| 6336.2.a.cv | 2 | 8.b | even | 2 | 1 | ||
| 6336.2.a.cw | 2 | 8.d | odd | 2 | 1 | ||
| 7744.2.a.bw | 2 | 264.p | odd | 2 | 1 | ||
| 7744.2.a.cm | 2 | 264.m | even | 2 | 1 | ||
| 8800.2.a.bd | 2 | 60.h | even | 2 | 1 | ||
| 8800.2.a.be | 2 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3168))\):
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\( T_{5}^{2} + 3T_{5} - 2 \)
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\( T_{7} \)
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\( T_{13} - 2 \)
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\( T_{17}^{2} + 6T_{17} - 8 \)
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\( T_{19}^{2} + 6T_{19} - 8 \)
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\( T_{47} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
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| $3$ |
\( T^{2} \)
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| $5$ |
\( T^{2} + 3T - 2 \)
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| $7$ |
\( T^{2} \)
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| $11$ |
\( (T - 1)^{2} \)
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| $13$ |
\( (T - 2)^{2} \)
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| $17$ |
\( T^{2} + 6T - 8 \)
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| $19$ |
\( T^{2} + 6T - 8 \)
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| $23$ |
\( T^{2} + 3T - 36 \)
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| $29$ |
\( T^{2} + 6T - 8 \)
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| $31$ |
\( T^{2} - 15T + 52 \)
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| $37$ |
\( T^{2} - T - 38 \)
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| $41$ |
\( T^{2} - 68 \)
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| $43$ |
\( T^{2} + 6T - 8 \)
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| $47$ |
\( (T + 4)^{2} \)
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| $53$ |
\( T^{2} - 68 \)
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| $59$ |
\( T^{2} + 13T + 4 \)
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| $61$ |
\( T^{2} + 6T - 144 \)
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| $67$ |
\( T^{2} + 3T - 36 \)
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| $71$ |
\( T^{2} - 19T + 52 \)
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| $73$ |
\( (T + 6)^{2} \)
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| $79$ |
\( T^{2} - 18T + 64 \)
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| $83$ |
\( T^{2} - 2T - 152 \)
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| $89$ |
\( T^{2} - 3T - 2 \)
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| $97$ |
\( T^{2} + T - 38 \)
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