Newspace parameters
| Level: | \( N \) | \(=\) | \( 70 = 2 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 16 \) |
| Character orbit: | \([\chi]\) | \(=\) | 70.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(99.8854535699\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 26976507x^{2} + 23999135013x + 94436136779094 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3\cdot 5^{3}\cdot 7 \) |
| 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
\( x^{4} - x^{3} - 26976507x^{2} + 23999135013x + 94436136779094 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} - 2612\nu^{2} + 16491933\nu + 17248077216 ) / 1134 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 3746\nu^{2} - 14979177\nu - 32544134874 ) / 1134 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} - 1334\beta _1 + 13488587 \)
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| \(\nu^{3}\) | \(=\) |
\( -2612\beta_{3} - 3746\beta_{2} + 19976341\beta _1 - 17984112028 \)
|
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 |
|
128.000 | −2140.31 | 16384.0 | 78125.0 | −273959. | −823543. | 2.09715e6 | −9.76800e6 | 1.00000e7 | ||||||||||||||||||||||||||||||
| 1.2 | 128.000 | −1683.91 | 16384.0 | 78125.0 | −215540. | −823543. | 2.09715e6 | −1.15134e7 | 1.00000e7 | |||||||||||||||||||||||||||||||
| 1.3 | 128.000 | 3036.56 | 16384.0 | 78125.0 | 388680. | −823543. | 2.09715e6 | −5.12822e6 | 1.00000e7 | |||||||||||||||||||||||||||||||
| 1.4 | 128.000 | 6810.65 | 16384.0 | 78125.0 | 871764. | −823543. | 2.09715e6 | 3.20361e7 | 1.00000e7 | |||||||||||||||||||||||||||||||
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 | 70.16.a.h | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 70.16.a.h | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 6023T_{3}^{3} - 13372809T_{3}^{2} + 43598259315T_{3} + 74535716713500 \)
acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(70))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 128)^{4} \)
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| $3$ |
\( T^{4} + \cdots + 74535716713500 \)
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| $5$ |
\( (T - 78125)^{4} \)
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| $7$ |
\( (T + 823543)^{4} \)
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| $11$ |
\( T^{4} + \cdots - 63\!\cdots\!00 \)
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| $13$ |
\( T^{4} + \cdots - 11\!\cdots\!54 \)
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| $17$ |
\( T^{4} + \cdots + 11\!\cdots\!54 \)
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| $19$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
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| $23$ |
\( T^{4} + \cdots + 42\!\cdots\!00 \)
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| $29$ |
\( T^{4} + \cdots - 79\!\cdots\!34 \)
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| $31$ |
\( T^{4} + \cdots + 46\!\cdots\!44 \)
|
| $37$ |
\( T^{4} + \cdots + 67\!\cdots\!00 \)
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| $41$ |
\( T^{4} + \cdots - 12\!\cdots\!96 \)
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| $43$ |
\( T^{4} + \cdots + 30\!\cdots\!24 \)
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| $47$ |
\( T^{4} + \cdots + 74\!\cdots\!56 \)
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| $53$ |
\( T^{4} + \cdots + 52\!\cdots\!04 \)
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| $59$ |
\( T^{4} + \cdots + 14\!\cdots\!44 \)
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| $61$ |
\( T^{4} + \cdots - 31\!\cdots\!76 \)
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| $67$ |
\( T^{4} + \cdots - 14\!\cdots\!36 \)
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| $71$ |
\( T^{4} + \cdots - 18\!\cdots\!00 \)
|
| $73$ |
\( T^{4} + \cdots + 54\!\cdots\!76 \)
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| $79$ |
\( T^{4} + \cdots + 45\!\cdots\!36 \)
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| $83$ |
\( T^{4} + \cdots - 27\!\cdots\!76 \)
|
| $89$ |
\( T^{4} + \cdots - 27\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 24\!\cdots\!50 \)
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