Newspace parameters
| Level: | \( N \) | \(=\) | \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1872.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(110.451575531\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.18257.1 |
|
|
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| Defining polynomial: |
\( x^{3} - x^{2} - 26x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 104) |
| 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
\( x^{3} - x^{2} - 26x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + \nu - 18 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{2} + 3\nu + 16 ) / 2 \)
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| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 2 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + 3\beta _1 + 35 ) / 2 \)
|
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 |
|
0 | 0 | 0 | −7.51634 | 0 | −12.0519 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −6.19042 | 0 | −3.19918 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 21.7068 | 0 | −20.7489 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(13\) | \( -1 \) |
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 | 1872.4.a.bm | 3 | |
| 3.b | odd | 2 | 1 | 208.4.a.l | 3 | ||
| 4.b | odd | 2 | 1 | 936.4.a.m | 3 | ||
| 12.b | even | 2 | 1 | 104.4.a.e | ✓ | 3 | |
| 24.f | even | 2 | 1 | 832.4.a.bc | 3 | ||
| 24.h | odd | 2 | 1 | 832.4.a.bb | 3 | ||
| 156.h | even | 2 | 1 | 1352.4.a.h | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.4.a.e | ✓ | 3 | 12.b | even | 2 | 1 | |
| 208.4.a.l | 3 | 3.b | odd | 2 | 1 | ||
| 832.4.a.bb | 3 | 24.h | odd | 2 | 1 | ||
| 832.4.a.bc | 3 | 24.f | even | 2 | 1 | ||
| 936.4.a.m | 3 | 4.b | odd | 2 | 1 | ||
| 1352.4.a.h | 3 | 156.h | even | 2 | 1 | ||
| 1872.4.a.bm | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1872))\):
|
\( T_{5}^{3} - 8T_{5}^{2} - 251T_{5} - 1010 \)
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\( T_{7}^{3} + 36T_{7}^{2} + 355T_{7} + 800 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
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| $3$ |
\( T^{3} \)
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| $5$ |
\( T^{3} - 8 T^{2} + \cdots - 1010 \)
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| $7$ |
\( T^{3} + 36 T^{2} + \cdots + 800 \)
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| $11$ |
\( T^{3} - 52 T^{2} + \cdots + 59184 \)
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| $13$ |
\( (T - 13)^{3} \)
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| $17$ |
\( T^{3} - 116 T^{2} + \cdots + 1048898 \)
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| $19$ |
\( T^{3} + 124 T^{2} + \cdots + 34864 \)
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| $23$ |
\( T^{3} - 232 T^{2} + \cdots + 65536 \)
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| $29$ |
\( T^{3} - 30 T^{2} + \cdots - 1387112 \)
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| $31$ |
\( T^{3} + 240 T^{2} + \cdots - 311936 \)
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| $37$ |
\( T^{3} + 264 T^{2} + \cdots + 99730 \)
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| $41$ |
\( T^{3} - 374 T^{2} + \cdots + 29246464 \)
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| $43$ |
\( T^{3} + 248 T^{2} + \cdots - 52832740 \)
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| $47$ |
\( T^{3} + 412 T^{2} + \cdots - 2411208 \)
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| $53$ |
\( T^{3} - 386 T^{2} + \cdots + 4448256 \)
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| $59$ |
\( T^{3} + 940 T^{2} + \cdots - 37508496 \)
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| $61$ |
\( T^{3} - 1206 T^{2} + \cdots - 46097792 \)
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| $67$ |
\( T^{3} - 564 T^{2} + \cdots - 2603408 \)
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| $71$ |
\( T^{3} - 1260 T^{2} + \cdots + 198899280 \)
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| $73$ |
\( T^{3} - 142 T^{2} + \cdots - 146317608 \)
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| $79$ |
\( T^{3} + 1040 T^{2} + \cdots - 373329920 \)
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| $83$ |
\( T^{3} - 756 T^{2} + \cdots + 64918848 \)
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| $89$ |
\( T^{3} - 18 T^{2} + \cdots - 121968344 \)
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| $97$ |
\( T^{3} - 2174 T^{2} + \cdots + 7072888 \)
|
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