Newspace parameters
| Level: | \( N \) | \(=\) | \( 361 = 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 361.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(2.88259951297\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\Q(\zeta_{18})^+\) |
|
|
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| Defining polynomial: |
\( x^{3} - 3x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 19) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
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 |
|
−2.53209 | −0.652704 | 4.41147 | −1.34730 | 1.65270 | 1.53209 | −6.10607 | −2.57398 | 3.41147 | |||||||||||||||||||||||||||
| 1.2 | −1.34730 | −2.87939 | −0.184793 | 0.879385 | 3.87939 | 0.347296 | 2.94356 | 5.29086 | −1.18479 | ||||||||||||||||||||||||||||
| 1.3 | 0.879385 | 0.532089 | −1.22668 | −2.53209 | 0.467911 | −1.87939 | −2.83750 | −2.71688 | −2.22668 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(19\) | \( +1 \) |
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 | 361.2.a.g | 3 | |
| 3.b | odd | 2 | 1 | 3249.2.a.z | 3 | ||
| 4.b | odd | 2 | 1 | 5776.2.a.br | 3 | ||
| 5.b | even | 2 | 1 | 9025.2.a.bd | 3 | ||
| 19.b | odd | 2 | 1 | 361.2.a.h | 3 | ||
| 19.c | even | 3 | 2 | 361.2.c.i | 6 | ||
| 19.d | odd | 6 | 2 | 361.2.c.h | 6 | ||
| 19.e | even | 9 | 2 | 19.2.e.a | ✓ | 6 | |
| 19.e | even | 9 | 2 | 361.2.e.f | 6 | ||
| 19.e | even | 9 | 2 | 361.2.e.g | 6 | ||
| 19.f | odd | 18 | 2 | 361.2.e.a | 6 | ||
| 19.f | odd | 18 | 2 | 361.2.e.b | 6 | ||
| 19.f | odd | 18 | 2 | 361.2.e.h | 6 | ||
| 57.d | even | 2 | 1 | 3249.2.a.s | 3 | ||
| 57.l | odd | 18 | 2 | 171.2.u.c | 6 | ||
| 76.d | even | 2 | 1 | 5776.2.a.bi | 3 | ||
| 76.l | odd | 18 | 2 | 304.2.u.b | 6 | ||
| 95.d | odd | 2 | 1 | 9025.2.a.x | 3 | ||
| 95.p | even | 18 | 2 | 475.2.l.a | 6 | ||
| 95.q | odd | 36 | 4 | 475.2.u.a | 12 | ||
| 133.u | even | 9 | 2 | 931.2.v.b | 6 | ||
| 133.w | even | 9 | 2 | 931.2.x.a | 6 | ||
| 133.x | odd | 18 | 2 | 931.2.v.a | 6 | ||
| 133.y | odd | 18 | 2 | 931.2.w.a | 6 | ||
| 133.z | odd | 18 | 2 | 931.2.x.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 19.2.e.a | ✓ | 6 | 19.e | even | 9 | 2 | |
| 171.2.u.c | 6 | 57.l | odd | 18 | 2 | ||
| 304.2.u.b | 6 | 76.l | odd | 18 | 2 | ||
| 361.2.a.g | 3 | 1.a | even | 1 | 1 | trivial | |
| 361.2.a.h | 3 | 19.b | odd | 2 | 1 | ||
| 361.2.c.h | 6 | 19.d | odd | 6 | 2 | ||
| 361.2.c.i | 6 | 19.c | even | 3 | 2 | ||
| 361.2.e.a | 6 | 19.f | odd | 18 | 2 | ||
| 361.2.e.b | 6 | 19.f | odd | 18 | 2 | ||
| 361.2.e.f | 6 | 19.e | even | 9 | 2 | ||
| 361.2.e.g | 6 | 19.e | even | 9 | 2 | ||
| 361.2.e.h | 6 | 19.f | odd | 18 | 2 | ||
| 475.2.l.a | 6 | 95.p | even | 18 | 2 | ||
| 475.2.u.a | 12 | 95.q | odd | 36 | 4 | ||
| 931.2.v.a | 6 | 133.x | odd | 18 | 2 | ||
| 931.2.v.b | 6 | 133.u | even | 9 | 2 | ||
| 931.2.w.a | 6 | 133.y | odd | 18 | 2 | ||
| 931.2.x.a | 6 | 133.w | even | 9 | 2 | ||
| 931.2.x.b | 6 | 133.z | odd | 18 | 2 | ||
| 3249.2.a.s | 3 | 57.d | even | 2 | 1 | ||
| 3249.2.a.z | 3 | 3.b | odd | 2 | 1 | ||
| 5776.2.a.bi | 3 | 76.d | even | 2 | 1 | ||
| 5776.2.a.br | 3 | 4.b | odd | 2 | 1 | ||
| 9025.2.a.x | 3 | 95.d | odd | 2 | 1 | ||
| 9025.2.a.bd | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(361))\):
|
\( T_{2}^{3} + 3T_{2}^{2} - 3 \)
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\( T_{3}^{3} + 3T_{3}^{2} - 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 3T^{2} - 3 \)
|
| $3$ |
\( T^{3} + 3T^{2} - 1 \)
|
| $5$ |
\( T^{3} + 3T^{2} - 3 \)
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| $7$ |
\( T^{3} - 3T + 1 \)
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| $11$ |
\( T^{3} - 9T - 9 \)
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| $13$ |
\( T^{3} - 21T + 37 \)
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| $17$ |
\( T^{3} - 6 T^{2} + \cdots - 3 \)
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| $19$ |
\( T^{3} \)
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| $23$ |
\( T^{3} + 6T^{2} - 24 \)
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| $29$ |
\( T^{3} + 15 T^{2} + \cdots + 111 \)
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| $31$ |
\( T^{3} + 9 T^{2} + \cdots - 53 \)
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| $37$ |
\( T^{3} - 21T - 17 \)
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| $41$ |
\( T^{3} + 12 T^{2} + \cdots - 111 \)
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| $43$ |
\( T^{3} - 57T + 163 \)
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| $47$ |
\( T^{3} + 6 T^{2} + \cdots + 3 \)
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| $53$ |
\( T^{3} + 6 T^{2} + \cdots - 51 \)
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| $59$ |
\( T^{3} + 21 T^{2} + \cdots + 267 \)
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| $61$ |
\( T^{3} - 9 T^{2} + \cdots + 181 \)
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| $67$ |
\( T^{3} - 18 T^{2} + \cdots + 424 \)
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| $71$ |
\( T^{3} + 30 T^{2} + \cdots + 888 \)
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| $73$ |
\( T^{3} - 48T + 64 \)
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| $79$ |
\( T^{3} + 9 T^{2} + \cdots - 809 \)
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| $83$ |
\( T^{3} - 189T + 459 \)
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| $89$ |
\( T^{3} + 15 T^{2} + \cdots + 57 \)
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| $97$ |
\( T^{3} - 15 T^{2} + \cdots + 127 \)
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