Newspace parameters
| Level: | \( N \) | \(=\) | \( 2450 = 2 \cdot 5^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2450.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(144.554679514\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.238585.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - 67x - 189 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 350) |
| 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 a root \(\nu\) of
\( x^{3} - 67x - 189 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4\nu - 45 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
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| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4\beta _1 + 45 \)
|
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 |
|
2.00000 | −6.92433 | 4.00000 | 0 | −13.8487 | 0 | 8.00000 | 20.9463 | 0 | |||||||||||||||||||||||||||
| 1.2 | 2.00000 | −4.41567 | 4.00000 | 0 | −8.83134 | 0 | 8.00000 | −7.50184 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 8.34000 | 4.00000 | 0 | 16.6800 | 0 | 8.00000 | 42.5556 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(7\) | \( -1 \) |
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 | 2450.4.a.ch | 3 | |
| 5.b | even | 2 | 1 | 2450.4.a.cd | 3 | ||
| 7.b | odd | 2 | 1 | 2450.4.a.ci | 3 | ||
| 7.d | odd | 6 | 2 | 350.4.e.i | ✓ | 6 | |
| 35.c | odd | 2 | 1 | 2450.4.a.cc | 3 | ||
| 35.i | odd | 6 | 2 | 350.4.e.j | yes | 6 | |
| 35.k | even | 12 | 4 | 350.4.j.h | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 350.4.e.i | ✓ | 6 | 7.d | odd | 6 | 2 | |
| 350.4.e.j | yes | 6 | 35.i | odd | 6 | 2 | |
| 350.4.j.h | 12 | 35.k | even | 12 | 4 | ||
| 2450.4.a.cc | 3 | 35.c | odd | 2 | 1 | ||
| 2450.4.a.cd | 3 | 5.b | even | 2 | 1 | ||
| 2450.4.a.ch | 3 | 1.a | even | 1 | 1 | trivial | |
| 2450.4.a.ci | 3 | 7.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2450))\):
|
\( T_{3}^{3} + 3T_{3}^{2} - 64T_{3} - 255 \)
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\( T_{11}^{3} + 26T_{11}^{2} - 1393T_{11} - 36407 \)
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\( T_{19}^{3} - 18T_{19}^{2} - 12709T_{19} - 95205 \)
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\( T_{23}^{3} + 53T_{23}^{2} - 14822T_{23} - 94749 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{3} \)
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| $3$ |
\( T^{3} + 3 T^{2} + \cdots - 255 \)
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| $5$ |
\( T^{3} \)
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| $7$ |
\( T^{3} \)
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| $11$ |
\( T^{3} + 26 T^{2} + \cdots - 36407 \)
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| $13$ |
\( T^{3} - 80 T^{2} + \cdots + 160725 \)
|
| $17$ |
\( T^{3} - 30 T^{2} + \cdots - 6341 \)
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| $19$ |
\( T^{3} - 18 T^{2} + \cdots - 95205 \)
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| $23$ |
\( T^{3} + 53 T^{2} + \cdots - 94749 \)
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| $29$ |
\( T^{3} + 404 T^{2} + \cdots + 1808541 \)
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| $31$ |
\( T^{3} - 615 T^{2} + \cdots - 7562577 \)
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| $37$ |
\( T^{3} - 426 T^{2} + \cdots + 11343957 \)
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| $41$ |
\( T^{3} - 101 T^{2} + \cdots + 7167 \)
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| $43$ |
\( T^{3} + 249 T^{2} + \cdots - 27999735 \)
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| $47$ |
\( T^{3} - 402 T^{2} + \cdots + 52820631 \)
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| $53$ |
\( T^{3} - 390 T^{2} + \cdots + 93437133 \)
|
| $59$ |
\( T^{3} - 91 T^{2} + \cdots + 92928465 \)
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| $61$ |
\( T^{3} - 647 T^{2} + \cdots + 929141 \)
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| $67$ |
\( T^{3} - 708 T^{2} + \cdots + 2673864 \)
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| $71$ |
\( T^{3} - 533 T^{2} + \cdots + 74350109 \)
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| $73$ |
\( T^{3} - 772 T^{2} + \cdots + 209497408 \)
|
| $79$ |
\( T^{3} - 1421 T^{2} + \cdots - 54979815 \)
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| $83$ |
\( T^{3} + 520 T^{2} + \cdots + 294469047 \)
|
| $89$ |
\( T^{3} - 194 T^{2} + \cdots + 843604613 \)
|
| $97$ |
\( T^{3} + 1027 T^{2} + \cdots - 883021 \)
|
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