Newspace parameters
| Level: | \( N \) | \(=\) | \( 3249 = 3^{2} \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3249.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(25.9433956167\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.21415104.1 |
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| Defining polynomial: |
\( x^{6} - 12x^{4} + 45x^{2} - 51 \)
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| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 171) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{6} - 12x^{4} + 45x^{2} - 51 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
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| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 8\nu^{2} + 14 \)
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| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 8\nu^{3} + 14\nu \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 8\beta_{2} + 18 \)
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| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 8\beta_{3} + 26\beta_1 \)
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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 |
|
−2.35204 | 0 | 3.53209 | 3.60353 | 0 | 0.184793 | −3.60353 | 0 | −8.47565 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.08502 | 0 | 2.34730 | 0.724119 | 0 | 1.22668 | −0.724119 | 0 | −1.50980 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.45623 | 0 | 0.120615 | −2.73682 | 0 | −4.41147 | 2.73682 | 0 | 3.98545 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.45623 | 0 | 0.120615 | 2.73682 | 0 | −4.41147 | −2.73682 | 0 | 3.98545 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.08502 | 0 | 2.34730 | −0.724119 | 0 | 1.22668 | 0.724119 | 0 | −1.50980 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.35204 | 0 | 3.53209 | −3.60353 | 0 | 0.184793 | 3.60353 | 0 | −8.47565 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(19\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3249.2.a.bi | 6 | |
| 3.b | odd | 2 | 1 | inner | 3249.2.a.bi | 6 | |
| 19.b | odd | 2 | 1 | 3249.2.a.bj | 6 | ||
| 19.e | even | 9 | 2 | 171.2.u.d | ✓ | 12 | |
| 57.d | even | 2 | 1 | 3249.2.a.bj | 6 | ||
| 57.l | odd | 18 | 2 | 171.2.u.d | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 171.2.u.d | ✓ | 12 | 19.e | even | 9 | 2 | |
| 171.2.u.d | ✓ | 12 | 57.l | odd | 18 | 2 | |
| 3249.2.a.bi | 6 | 1.a | even | 1 | 1 | trivial | |
| 3249.2.a.bi | 6 | 3.b | odd | 2 | 1 | inner | |
| 3249.2.a.bj | 6 | 19.b | odd | 2 | 1 | ||
| 3249.2.a.bj | 6 | 57.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3249))\):
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\( T_{2}^{6} - 12T_{2}^{4} + 45T_{2}^{2} - 51 \)
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\( T_{5}^{6} - 21T_{5}^{4} + 108T_{5}^{2} - 51 \)
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\( T_{13}^{3} + 9T_{13}^{2} + 15T_{13} - 17 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 12 T^{4} + \cdots - 51 \)
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| $3$ |
\( T^{6} \)
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| $5$ |
\( T^{6} - 21 T^{4} + \cdots - 51 \)
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| $7$ |
\( (T^{3} + 3 T^{2} - 6 T + 1)^{2} \)
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| $11$ |
\( T^{6} - 27 T^{4} + \cdots - 459 \)
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| $13$ |
\( (T^{3} + 9 T^{2} + 15 T - 17)^{2} \)
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| $17$ |
\( T^{6} - 93 T^{4} + \cdots - 18411 \)
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| $19$ |
\( T^{6} \)
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| $23$ |
\( T^{6} - 30 T^{4} + \cdots - 51 \)
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| $29$ |
\( T^{6} - 93 T^{4} + \cdots - 14739 \)
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| $31$ |
\( (T^{3} + 12 T^{2} + \cdots + 37)^{2} \)
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| $37$ |
\( (T^{3} + 12 T^{2} + \cdots + 37)^{2} \)
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| $41$ |
\( T^{6} - 246 T^{4} + \cdots - 69819 \)
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| $43$ |
\( (T^{3} - 9 T^{2} + 15 T + 1)^{2} \)
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| $47$ |
\( T^{6} - 183 T^{4} + \cdots - 143259 \)
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| $53$ |
\( T^{6} - 120 T^{4} + \cdots - 51 \)
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| $59$ |
\( T^{6} - 201 T^{4} + \cdots - 271779 \)
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| $61$ |
\( (T^{3} + 9 T^{2} + \cdots - 179)^{2} \)
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| $67$ |
\( (T^{3} - 48 T + 64)^{2} \)
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| $71$ |
\( T^{6} - 309 T^{4} + \cdots - 143259 \)
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| $73$ |
\( (T^{3} - 57 T + 163)^{2} \)
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| $79$ |
\( (T^{3} + 27 T^{2} + \cdots + 361)^{2} \)
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| $83$ |
\( T^{6} - 207 T^{4} + \cdots - 459 \)
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| $89$ |
\( T^{6} - 300 T^{4} + \cdots - 605931 \)
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| $97$ |
\( (T^{3} - 21 T - 17)^{2} \)
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