Newspace parameters
| Level: | \( N \) | \(=\) | \( 75 = 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 75.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(38.6276877123\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{4} - 2x^{3} - 1546x^{2} + 152x + 559560 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3\cdot 5^{2}\cdot 23 \) |
| 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
\( x^{4} - 2x^{3} - 1546x^{2} + 152x + 559560 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 24\nu^{2} + 640\nu - 16752 ) / 126 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -25\nu^{3} + 915\nu^{2} + 18520\nu - 662295 ) / 63 \)
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| \(\beta_{3}\) | \(=\) |
\( ( -629\nu^{3} + 15726\nu^{2} + 494540\nu - 11031054 ) / 126 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} - 579\beta _1 + 56 ) / 690 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -4\beta_{3} + 73\beta_{2} - 1134\beta _1 + 266461 ) / 345 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 224\beta_{3} + 1432\beta_{2} - 255966\beta _1 + 633544 ) / 345 \)
|
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 |
|
−44.2736 | 81.0000 | 1448.15 | 0 | −3586.16 | 3879.20 | −41447.0 | 6561.00 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −11.8931 | 81.0000 | −370.553 | 0 | −963.344 | 10946.0 | 10496.3 | 6561.00 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 15.4348 | 81.0000 | −273.767 | 0 | 1250.22 | −8162.66 | −12128.2 | 6561.00 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 38.7320 | 81.0000 | 988.165 | 0 | 3137.29 | 6373.44 | 18442.8 | 6561.00 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
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 | 75.10.a.j | ✓ | 4 |
| 3.b | odd | 2 | 1 | 225.10.a.t | 4 | ||
| 5.b | even | 2 | 1 | 75.10.a.k | yes | 4 | |
| 5.c | odd | 4 | 2 | 75.10.b.h | 8 | ||
| 15.d | odd | 2 | 1 | 225.10.a.r | 4 | ||
| 15.e | even | 4 | 2 | 225.10.b.n | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 75.10.a.j | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 75.10.a.k | yes | 4 | 5.b | even | 2 | 1 | |
| 75.10.b.h | 8 | 5.c | odd | 4 | 2 | ||
| 225.10.a.r | 4 | 15.d | odd | 2 | 1 | ||
| 225.10.a.t | 4 | 3.b | odd | 2 | 1 | ||
| 225.10.b.n | 8 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 2T_{2}^{3} - 1918T_{2}^{2} + 5056T_{2} + 314784 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(75))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 314784 \)
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| $3$ |
\( (T - 81)^{4} \)
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| $5$ |
\( T^{4} \)
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| $7$ |
\( T^{4} + \cdots - 22\!\cdots\!75 \)
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| $11$ |
\( T^{4} + \cdots + 18\!\cdots\!24 \)
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| $13$ |
\( T^{4} + \cdots - 92\!\cdots\!71 \)
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| $17$ |
\( T^{4} + \cdots - 11\!\cdots\!64 \)
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| $19$ |
\( T^{4} + \cdots + 76\!\cdots\!25 \)
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| $23$ |
\( T^{4} + \cdots - 17\!\cdots\!00 \)
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| $29$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
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| $31$ |
\( T^{4} + \cdots + 98\!\cdots\!25 \)
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| $37$ |
\( T^{4} + \cdots + 71\!\cdots\!00 \)
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| $41$ |
\( T^{4} + \cdots - 14\!\cdots\!00 \)
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| $43$ |
\( T^{4} + \cdots + 38\!\cdots\!21 \)
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| $47$ |
\( T^{4} + \cdots + 14\!\cdots\!76 \)
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| $53$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
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| $59$ |
\( T^{4} + \cdots + 28\!\cdots\!00 \)
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| $61$ |
\( T^{4} + \cdots + 28\!\cdots\!21 \)
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| $67$ |
\( T^{4} + \cdots + 35\!\cdots\!81 \)
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| $71$ |
\( T^{4} + \cdots + 35\!\cdots\!56 \)
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| $73$ |
\( T^{4} + \cdots + 42\!\cdots\!00 \)
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| $79$ |
\( T^{4} + \cdots + 45\!\cdots\!00 \)
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| $83$ |
\( T^{4} + \cdots + 20\!\cdots\!44 \)
|
| $89$ |
\( T^{4} + \cdots - 31\!\cdots\!00 \)
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| $97$ |
\( T^{4} + \cdots - 52\!\cdots\!19 \)
|
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