Newspace parameters
| Level: | \( N \) | \(=\) | \( 3025 = 5^{2} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3025.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(24.1547466114\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{18 +2 \sqrt{21}})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 9x^{2} + 15 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 9x^{2} + 15 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
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| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.60601 | 2.79129 | 4.79129 | 0 | −7.27412 | 2.60601 | −7.27412 | 4.79129 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −1.48617 | −1.79129 | 0.208712 | 0 | 2.66216 | 1.48617 | 2.66216 | 0.208712 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 1.48617 | −1.79129 | 0.208712 | 0 | −2.66216 | −1.48617 | −2.66216 | 0.208712 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 2.60601 | 2.79129 | 4.79129 | 0 | 7.27412 | −2.60601 | 7.27412 | 4.79129 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(11\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3025.2.a.bc | yes | 4 |
| 5.b | even | 2 | 1 | 3025.2.a.x | ✓ | 4 | |
| 11.b | odd | 2 | 1 | inner | 3025.2.a.bc | yes | 4 |
| 55.d | odd | 2 | 1 | 3025.2.a.x | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3025.2.a.x | ✓ | 4 | 5.b | even | 2 | 1 | |
| 3025.2.a.x | ✓ | 4 | 55.d | odd | 2 | 1 | |
| 3025.2.a.bc | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 3025.2.a.bc | yes | 4 | 11.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3025))\):
|
\( T_{2}^{4} - 9T_{2}^{2} + 15 \)
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\( T_{3}^{2} - T_{3} - 5 \)
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\( T_{19}^{4} - 60T_{19}^{2} + 375 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 9T^{2} + 15 \)
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| $3$ |
\( (T^{2} - T - 5)^{2} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} - 9T^{2} + 15 \)
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| $11$ |
\( T^{4} \)
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| $13$ |
\( T^{4} - 36T^{2} + 135 \)
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| $17$ |
\( T^{4} - 39T^{2} + 375 \)
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| $19$ |
\( T^{4} - 60T^{2} + 375 \)
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| $23$ |
\( (T^{2} - 9 T + 15)^{2} \)
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| $29$ |
\( T^{4} - 135T^{2} + 3375 \)
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| $31$ |
\( (T + 5)^{4} \)
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| $37$ |
\( (T^{2} - 8 T - 5)^{2} \)
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| $41$ |
\( T^{4} - 60T^{2} + 375 \)
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| $43$ |
\( T^{4} - 36T^{2} + 240 \)
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| $47$ |
\( (T^{2} - 12 T + 15)^{2} \)
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| $53$ |
\( (T^{2} - 9 T + 15)^{2} \)
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| $59$ |
\( (T^{2} + 6 T - 75)^{2} \)
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| $61$ |
\( T^{4} - 165T^{2} + 375 \)
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| $67$ |
\( (T^{2} - 2 T - 20)^{2} \)
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| $71$ |
\( (T^{2} - 84)^{2} \)
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| $73$ |
\( T^{4} - 141T^{2} + 4335 \)
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| $79$ |
\( T^{4} - 315 T^{2} + 18375 \)
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| $83$ |
\( T^{4} - 51T^{2} + 15 \)
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| $89$ |
\( (T^{2} - 21 T + 105)^{2} \)
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| $97$ |
\( (T^{2} - 13 T - 5)^{2} \)
|
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