Newspace parameters
Level: | \( N \) | \(=\) | \( 1600 = 2^{6} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1600.j (of order \(4\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(12.7760643234\) |
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: | \(\Z[a_1, \ldots, a_{17}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | no (minimal twist has level 400) |
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
\(\beta_{1}\) | \(=\) | \( \zeta_{24}^{3} \) |
\(\beta_{2}\) | \(=\) | \( \zeta_{24}^{4} + \zeta_{24}^{2} \) |
\(\beta_{3}\) | \(=\) | \( \zeta_{24}^{6} \) |
\(\beta_{4}\) | \(=\) | \( \zeta_{24}^{7} + \zeta_{24} \) |
\(\beta_{5}\) | \(=\) | \( -\zeta_{24}^{5} + \zeta_{24} \) |
\(\beta_{6}\) | \(=\) | \( -\zeta_{24}^{4} + \zeta_{24}^{2} \) |
\(\beta_{7}\) | \(=\) | \( -\zeta_{24}^{7} + \zeta_{24}^{5} \) |
\(\zeta_{24}\) | \(=\) | \( ( \beta_{7} + \beta_{5} + \beta_{4} ) / 2 \) |
\(\zeta_{24}^{2}\) | \(=\) | \( ( \beta_{6} + \beta_{2} ) / 2 \) |
\(\zeta_{24}^{3}\) | \(=\) | \( \beta_1 \) |
\(\zeta_{24}^{4}\) | \(=\) | \( ( -\beta_{6} + \beta_{2} ) / 2 \) |
\(\zeta_{24}^{5}\) | \(=\) | \( ( \beta_{7} - \beta_{5} + \beta_{4} ) / 2 \) |
\(\zeta_{24}^{6}\) | \(=\) | \( \beta_{3} \) |
\(\zeta_{24}^{7}\) | \(=\) | \( ( -\beta_{7} - \beta_{5} + \beta_{4} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).
\(n\) | \(577\) | \(901\) | \(1151\) |
\(\chi(n)\) | \(\beta_{3}\) | \(-\beta_{3}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
143.1 |
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0 | − | 1.93185i | 0 | 0 | 0 | −0.896575 | + | 0.896575i | 0 | −0.732051 | 0 | |||||||||||||||||||||||||||||||||||||||
143.2 | 0 | − | 0.517638i | 0 | 0 | 0 | 3.34607 | − | 3.34607i | 0 | 2.73205 | 0 | ||||||||||||||||||||||||||||||||||||||||
143.3 | 0 | 0.517638i | 0 | 0 | 0 | −3.34607 | + | 3.34607i | 0 | 2.73205 | 0 | |||||||||||||||||||||||||||||||||||||||||
143.4 | 0 | 1.93185i | 0 | 0 | 0 | 0.896575 | − | 0.896575i | 0 | −0.732051 | 0 | |||||||||||||||||||||||||||||||||||||||||
1007.1 | 0 | − | 1.93185i | 0 | 0 | 0 | 0.896575 | + | 0.896575i | 0 | −0.732051 | 0 | ||||||||||||||||||||||||||||||||||||||||
1007.2 | 0 | − | 0.517638i | 0 | 0 | 0 | −3.34607 | − | 3.34607i | 0 | 2.73205 | 0 | ||||||||||||||||||||||||||||||||||||||||
1007.3 | 0 | 0.517638i | 0 | 0 | 0 | 3.34607 | + | 3.34607i | 0 | 2.73205 | 0 | |||||||||||||||||||||||||||||||||||||||||
1007.4 | 0 | 1.93185i | 0 | 0 | 0 | −0.896575 | − | 0.896575i | 0 | −0.732051 | 0 | |||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
80.j | even | 4 | 1 | inner |
80.s | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1600.2.j.b | 8 | |
4.b | odd | 2 | 1 | 400.2.j.b | ✓ | 8 | |
5.b | even | 2 | 1 | inner | 1600.2.j.b | 8 | |
5.c | odd | 4 | 2 | 1600.2.s.b | 8 | ||
16.e | even | 4 | 1 | 400.2.s.b | yes | 8 | |
16.f | odd | 4 | 1 | 1600.2.s.b | 8 | ||
20.d | odd | 2 | 1 | 400.2.j.b | ✓ | 8 | |
20.e | even | 4 | 2 | 400.2.s.b | yes | 8 | |
80.i | odd | 4 | 1 | 400.2.j.b | ✓ | 8 | |
80.j | even | 4 | 1 | inner | 1600.2.j.b | 8 | |
80.k | odd | 4 | 1 | 1600.2.s.b | 8 | ||
80.q | even | 4 | 1 | 400.2.s.b | yes | 8 | |
80.s | even | 4 | 1 | inner | 1600.2.j.b | 8 | |
80.t | odd | 4 | 1 | 400.2.j.b | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
400.2.j.b | ✓ | 8 | 4.b | odd | 2 | 1 | |
400.2.j.b | ✓ | 8 | 20.d | odd | 2 | 1 | |
400.2.j.b | ✓ | 8 | 80.i | odd | 4 | 1 | |
400.2.j.b | ✓ | 8 | 80.t | odd | 4 | 1 | |
400.2.s.b | yes | 8 | 16.e | even | 4 | 1 | |
400.2.s.b | yes | 8 | 20.e | even | 4 | 2 | |
400.2.s.b | yes | 8 | 80.q | even | 4 | 1 | |
1600.2.j.b | 8 | 1.a | even | 1 | 1 | trivial | |
1600.2.j.b | 8 | 5.b | even | 2 | 1 | inner | |
1600.2.j.b | 8 | 80.j | even | 4 | 1 | inner | |
1600.2.j.b | 8 | 80.s | even | 4 | 1 | inner | |
1600.2.s.b | 8 | 5.c | odd | 4 | 2 | ||
1600.2.s.b | 8 | 16.f | odd | 4 | 1 | ||
1600.2.s.b | 8 | 80.k | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 4T_{3}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(1600, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{4} + 4 T^{2} + 1)^{2} \)
$5$
\( T^{8} \)
$7$
\( T^{8} + 504T^{4} + 1296 \)
$11$
\( (T^{4} + 6 T^{3} + 18 T^{2} - 54 T + 81)^{2} \)
$13$
\( (T^{2} - 24)^{4} \)
$17$
\( (T^{4} + 1)^{2} \)
$19$
\( (T^{4} - 2 T^{3} + 2 T^{2} + 26 T + 169)^{2} \)
$23$
\( T^{8} + 1784 T^{4} + 456976 \)
$29$
\( (T^{4} + 12 T^{3} + 72 T^{2} + 144 T + 144)^{2} \)
$31$
\( (T^{4} + 56 T^{2} + 676)^{2} \)
$37$
\( (T^{4} - 156 T^{2} + 4356)^{2} \)
$41$
\( (T^{4} + 42 T^{2} + 9)^{2} \)
$43$
\( (T^{4} - 84 T^{2} + 36)^{2} \)
$47$
\( (T^{4} + 16)^{2} \)
$53$
\( (T^{2} + 98)^{4} \)
$59$
\( (T^{4} + 24 T^{3} + 288 T^{2} + 1584 T + 4356)^{2} \)
$61$
\( (T^{4} - 8 T^{3} + 32 T^{2} + 368 T + 2116)^{2} \)
$67$
\( (T^{4} - 36 T^{2} + 81)^{2} \)
$71$
\( (T^{2} - 12 T - 12)^{4} \)
$73$
\( T^{8} + 25074 T^{4} + \cdots + 22667121 \)
$79$
\( (T^{2} + 6 T - 18)^{4} \)
$83$
\( (T^{4} + 324 T^{2} + 6561)^{2} \)
$89$
\( (T^{2} + 24 T + 141)^{4} \)
$97$
\( T^{8} + 72576 T^{4} + \cdots + 26873856 \)
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