Newspace parameters
Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 350.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(109.334758919\) |
Analytic rank: | \(1\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 70) |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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 |
|
−8.00000 | 63.0000 | 64.0000 | 0 | −504.000 | −343.000 | −512.000 | 1782.00 | 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 | 350.8.a.c | 1 | |
5.b | even | 2 | 1 | 350.8.a.f | 1 | ||
5.c | odd | 4 | 2 | 70.8.c.a | ✓ | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
70.8.c.a | ✓ | 2 | 5.c | odd | 4 | 2 | |
350.8.a.c | 1 | 1.a | even | 1 | 1 | trivial | |
350.8.a.f | 1 | 5.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} - 63 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(350))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T + 8 \)
$3$
\( T - 63 \)
$5$
\( T \)
$7$
\( T + 343 \)
$11$
\( T + 2727 \)
$13$
\( T + 5269 \)
$17$
\( T - 17701 \)
$19$
\( T - 712 \)
$23$
\( T - 29330 \)
$29$
\( T - 68491 \)
$31$
\( T - 185026 \)
$37$
\( T - 250046 \)
$41$
\( T + 125814 \)
$43$
\( T + 747476 \)
$47$
\( T + 317317 \)
$53$
\( T + 1623246 \)
$59$
\( T + 1519262 \)
$61$
\( T + 3308640 \)
$67$
\( T - 2272366 \)
$71$
\( T + 4963104 \)
$73$
\( T + 2351750 \)
$79$
\( T + 2524249 \)
$83$
\( T + 6051492 \)
$89$
\( T - 8043880 \)
$97$
\( T - 2337645 \)
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