Newspace parameters
| Level: | \( N \) | \(=\) | \( 735 = 3 \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 735.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(5.86900454856\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.4352.1 |
|
|
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| Defining polynomial: |
\( x^{4} - 6x^{2} - 4x + 2 \)
|
| 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
\( x^{4} - 6x^{2} - 4x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 5\beta _1 + 3 \)
|
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 |
|
−1.68554 | 1.00000 | 0.841058 | −1.00000 | −1.68554 | 0 | 1.95345 | 1.00000 | 1.68554 | ||||||||||||||||||||||||||||||
| 1.2 | 0.665096 | 1.00000 | −1.55765 | −1.00000 | 0.665096 | 0 | −2.36618 | 1.00000 | −0.665096 | |||||||||||||||||||||||||||||||
| 1.3 | 2.27133 | 1.00000 | 3.15894 | −1.00000 | 2.27133 | 0 | 2.63234 | 1.00000 | −2.27133 | |||||||||||||||||||||||||||||||
| 1.4 | 2.74912 | 1.00000 | 5.55765 | −1.00000 | 2.74912 | 0 | 9.78039 | 1.00000 | −2.74912 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
| \(7\) | \( +1 \) |
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 | 735.2.a.o | yes | 4 |
| 3.b | odd | 2 | 1 | 2205.2.a.bg | 4 | ||
| 5.b | even | 2 | 1 | 3675.2.a.bk | 4 | ||
| 7.b | odd | 2 | 1 | 735.2.a.n | ✓ | 4 | |
| 7.c | even | 3 | 2 | 735.2.i.m | 8 | ||
| 7.d | odd | 6 | 2 | 735.2.i.n | 8 | ||
| 21.c | even | 2 | 1 | 2205.2.a.bf | 4 | ||
| 35.c | odd | 2 | 1 | 3675.2.a.bl | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 735.2.a.n | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 735.2.a.o | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 735.2.i.m | 8 | 7.c | even | 3 | 2 | ||
| 735.2.i.n | 8 | 7.d | odd | 6 | 2 | ||
| 2205.2.a.bf | 4 | 21.c | even | 2 | 1 | ||
| 2205.2.a.bg | 4 | 3.b | odd | 2 | 1 | ||
| 3675.2.a.bk | 4 | 5.b | even | 2 | 1 | ||
| 3675.2.a.bl | 4 | 35.c | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(735))\):
|
\( T_{2}^{4} - 4T_{2}^{3} + 12T_{2} - 7 \)
|
|
\( T_{13}^{4} - 32T_{13}^{2} - 64T_{13} + 16 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 4 T^{3} + \cdots - 7 \)
|
| $3$ |
\( (T - 1)^{4} \)
|
| $5$ |
\( (T + 1)^{4} \)
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| $7$ |
\( T^{4} \)
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| $11$ |
\( T^{4} - 8 T^{3} + \cdots - 224 \)
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| $13$ |
\( T^{4} - 32 T^{2} + \cdots + 16 \)
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| $17$ |
\( T^{4} - 8 T^{3} + \cdots + 128 \)
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| $19$ |
\( T^{4} + 8 T^{3} + \cdots - 284 \)
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| $23$ |
\( T^{4} - 44 T^{2} + \cdots + 196 \)
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| $29$ |
\( T^{4} - 8 T^{3} + \cdots - 368 \)
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| $31$ |
\( T^{4} + 8 T^{3} + \cdots - 28 \)
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| $37$ |
\( T^{4} - 8 T^{3} + \cdots + 784 \)
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| $41$ |
\( T^{4} - 56 T^{2} + \cdots + 16 \)
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| $43$ |
\( T^{4} + 8 T^{3} + \cdots + 64 \)
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| $47$ |
\( (T^{2} - 4 T - 68)^{2} \)
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| $53$ |
\( T^{4} - 8 T^{3} + \cdots - 28 \)
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| $59$ |
\( T^{4} - 16 T^{3} + \cdots - 4544 \)
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| $61$ |
\( (T^{2} + 16 T + 62)^{2} \)
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| $67$ |
\( T^{4} - 176 T^{2} + \cdots + 3136 \)
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| $71$ |
\( T^{4} + 8 T^{3} + \cdots + 32 \)
|
| $73$ |
\( T^{4} - 128 T^{2} + \cdots + 3088 \)
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| $79$ |
\( T^{4} - 184 T^{2} + \cdots + 2576 \)
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| $83$ |
\( (T^{2} - 20 T + 92)^{2} \)
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| $89$ |
\( (T^{2} + 4 T - 4)^{2} \)
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| $97$ |
\( T^{4} - 192 T^{2} + \cdots - 2032 \)
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