Newspace parameters
Level: | \( N \) | \(=\) | \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 700.m (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.58952814149\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(i)\) |
Coefficient field: | \(\Q(\zeta_{24})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{8} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
\(\beta_{1}\) | \(=\) | \( 2\zeta_{24}^{3} \) |
\(\beta_{2}\) | \(=\) | \( \zeta_{24}^{5} + \zeta_{24} \) |
\(\beta_{3}\) | \(=\) | \( \zeta_{24}^{6} \) |
\(\beta_{4}\) | \(=\) | \( 2\zeta_{24}^{7} - \zeta_{24}^{3} \) |
\(\beta_{5}\) | \(=\) | \( 4\zeta_{24}^{4} - 2 \) |
\(\beta_{6}\) | \(=\) | \( -2\zeta_{24}^{5} + 2\zeta_{24} \) |
\(\beta_{7}\) | \(=\) | \( -2\zeta_{24}^{6} + 4\zeta_{24}^{2} \) |
\(\zeta_{24}\) | \(=\) | \( ( \beta_{6} + 2\beta_{2} ) / 4 \) |
\(\zeta_{24}^{2}\) | \(=\) | \( ( \beta_{7} + 2\beta_{3} ) / 4 \) |
\(\zeta_{24}^{3}\) | \(=\) | \( ( \beta_1 ) / 2 \) |
\(\zeta_{24}^{4}\) | \(=\) | \( ( \beta_{5} + 2 ) / 4 \) |
\(\zeta_{24}^{5}\) | \(=\) | \( ( -\beta_{6} + 2\beta_{2} ) / 4 \) |
\(\zeta_{24}^{6}\) | \(=\) | \( \beta_{3} \) |
\(\zeta_{24}^{7}\) | \(=\) | \( ( 2\beta_{4} + \beta_1 ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(351\) | \(477\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
293.1 |
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0 | −1.41421 | + | 1.41421i | 0 | 0 | 0 | −2.63896 | − | 0.189469i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||
293.2 | 0 | −1.41421 | + | 1.41421i | 0 | 0 | 0 | −0.189469 | − | 2.63896i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
293.3 | 0 | 1.41421 | − | 1.41421i | 0 | 0 | 0 | 0.189469 | + | 2.63896i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
293.4 | 0 | 1.41421 | − | 1.41421i | 0 | 0 | 0 | 2.63896 | + | 0.189469i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||||||||||||||||
657.1 | 0 | −1.41421 | − | 1.41421i | 0 | 0 | 0 | −2.63896 | + | 0.189469i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
657.2 | 0 | −1.41421 | − | 1.41421i | 0 | 0 | 0 | −0.189469 | + | 2.63896i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
657.3 | 0 | 1.41421 | + | 1.41421i | 0 | 0 | 0 | 0.189469 | − | 2.63896i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
657.4 | 0 | 1.41421 | + | 1.41421i | 0 | 0 | 0 | 2.63896 | − | 0.189469i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
7.b | odd | 2 | 1 | inner |
35.c | odd | 2 | 1 | inner |
35.f | even | 4 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 700.2.m.b | ✓ | 8 |
5.b | even | 2 | 1 | inner | 700.2.m.b | ✓ | 8 |
5.c | odd | 4 | 2 | inner | 700.2.m.b | ✓ | 8 |
7.b | odd | 2 | 1 | inner | 700.2.m.b | ✓ | 8 |
35.c | odd | 2 | 1 | inner | 700.2.m.b | ✓ | 8 |
35.f | even | 4 | 2 | inner | 700.2.m.b | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
700.2.m.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
700.2.m.b | ✓ | 8 | 5.b | even | 2 | 1 | inner |
700.2.m.b | ✓ | 8 | 5.c | odd | 4 | 2 | inner |
700.2.m.b | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
700.2.m.b | ✓ | 8 | 35.c | odd | 2 | 1 | inner |
700.2.m.b | ✓ | 8 | 35.f | even | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(700, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{4} + 16)^{2} \)
$5$
\( T^{8} \)
$7$
\( T^{8} - 94T^{4} + 2401 \)
$11$
\( (T - 3)^{8} \)
$13$
\( (T^{4} + 16)^{2} \)
$17$
\( T^{8} \)
$19$
\( (T^{2} - 48)^{4} \)
$23$
\( (T^{4} + 729)^{2} \)
$29$
\( (T^{2} + 9)^{4} \)
$31$
\( (T^{2} + 12)^{4} \)
$37$
\( (T^{4} + 729)^{2} \)
$41$
\( (T^{2} + 108)^{4} \)
$43$
\( (T^{4} + 5625)^{2} \)
$47$
\( (T^{4} + 20736)^{2} \)
$53$
\( T^{8} \)
$59$
\( (T^{2} - 108)^{4} \)
$61$
\( (T^{2} + 12)^{4} \)
$67$
\( (T^{4} + 5625)^{2} \)
$71$
\( (T + 15)^{8} \)
$73$
\( (T^{4} + 16)^{2} \)
$79$
\( (T^{2} + 1)^{4} \)
$83$
\( (T^{4} + 1296)^{2} \)
$89$
\( (T^{2} - 108)^{4} \)
$97$
\( (T^{4} + 38416)^{2} \)
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