Newspace parameters
Level: | \( N \) | \(=\) | \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1950.y (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(15.5708283941\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{24})\) |
Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).
\(n\) | \(301\) | \(1301\) | \(1327\) |
\(\chi(n)\) | \(1 - \zeta_{24}^{4}\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
|
0.500000 | + | 0.866025i | −0.866025 | + | 0.500000i | −0.500000 | + | 0.866025i | 0 | −0.866025 | − | 0.500000i | −0.465926 | + | 0.807007i | −1.00000 | 0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||||||||||||
49.2 | 0.500000 | + | 0.866025i | −0.866025 | + | 0.500000i | −0.500000 | + | 0.866025i | 0 | −0.866025 | − | 0.500000i | 1.46593 | − | 2.53906i | −1.00000 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||
49.3 | 0.500000 | + | 0.866025i | 0.866025 | − | 0.500000i | −0.500000 | + | 0.866025i | 0 | 0.866025 | + | 0.500000i | 0.241181 | − | 0.417738i | −1.00000 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||
49.4 | 0.500000 | + | 0.866025i | 0.866025 | − | 0.500000i | −0.500000 | + | 0.866025i | 0 | 0.866025 | + | 0.500000i | 0.758819 | − | 1.31431i | −1.00000 | 0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||||||||||
199.1 | 0.500000 | − | 0.866025i | −0.866025 | − | 0.500000i | −0.500000 | − | 0.866025i | 0 | −0.866025 | + | 0.500000i | −0.465926 | − | 0.807007i | −1.00000 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||
199.2 | 0.500000 | − | 0.866025i | −0.866025 | − | 0.500000i | −0.500000 | − | 0.866025i | 0 | −0.866025 | + | 0.500000i | 1.46593 | + | 2.53906i | −1.00000 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||
199.3 | 0.500000 | − | 0.866025i | 0.866025 | + | 0.500000i | −0.500000 | − | 0.866025i | 0 | 0.866025 | − | 0.500000i | 0.241181 | + | 0.417738i | −1.00000 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||
199.4 | 0.500000 | − | 0.866025i | 0.866025 | + | 0.500000i | −0.500000 | − | 0.866025i | 0 | 0.866025 | − | 0.500000i | 0.758819 | + | 1.31431i | −1.00000 | 0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
65.l | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1950.2.y.l | 8 | |
5.b | even | 2 | 1 | 1950.2.y.i | 8 | ||
5.c | odd | 4 | 1 | 1950.2.bc.e | ✓ | 8 | |
5.c | odd | 4 | 1 | 1950.2.bc.f | yes | 8 | |
13.e | even | 6 | 1 | 1950.2.y.i | 8 | ||
65.l | even | 6 | 1 | inner | 1950.2.y.l | 8 | |
65.r | odd | 12 | 1 | 1950.2.bc.e | ✓ | 8 | |
65.r | odd | 12 | 1 | 1950.2.bc.f | yes | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1950.2.y.i | 8 | 5.b | even | 2 | 1 | ||
1950.2.y.i | 8 | 13.e | even | 6 | 1 | ||
1950.2.y.l | 8 | 1.a | even | 1 | 1 | trivial | |
1950.2.y.l | 8 | 65.l | even | 6 | 1 | inner | |
1950.2.bc.e | ✓ | 8 | 5.c | odd | 4 | 1 | |
1950.2.bc.e | ✓ | 8 | 65.r | odd | 12 | 1 | |
1950.2.bc.f | yes | 8 | 5.c | odd | 4 | 1 | |
1950.2.bc.f | yes | 8 | 65.r | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} - 4T_{7}^{7} + 14T_{7}^{6} - 16T_{7}^{5} + 22T_{7}^{4} - 8T_{7}^{3} + 20T_{7}^{2} - 8T_{7} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - T + 1)^{4} \)
$3$
\( (T^{4} - T^{2} + 1)^{2} \)
$5$
\( T^{8} \)
$7$
\( T^{8} - 4 T^{7} + 14 T^{6} - 16 T^{5} + \cdots + 4 \)
$11$
\( T^{8} + 12 T^{7} + 54 T^{6} + \cdots + 529 \)
$13$
\( T^{8} + 191 T^{4} + 28561 \)
$17$
\( T^{8} - 38 T^{6} + 1398 T^{4} + \cdots + 2116 \)
$19$
\( T^{8} + 12 T^{7} + 54 T^{6} + \cdots + 324 \)
$23$
\( T^{8} - 12 T^{7} + 12 T^{6} + \cdots + 58081 \)
$29$
\( T^{8} + 4 T^{7} + 100 T^{6} + \cdots + 341056 \)
$31$
\( T^{8} + 204 T^{6} + 12128 T^{4} + \cdots + 145924 \)
$37$
\( T^{8} - 16 T^{7} + 260 T^{6} + \cdots + 16056049 \)
$41$
\( T^{8} - 102 T^{6} + 10550 T^{4} + \cdots + 21316 \)
$43$
\( T^{8} + 24 T^{7} + 186 T^{6} + \cdots + 386884 \)
$47$
\( (T^{4} + 16 T^{3} + 20 T^{2} - 544 T - 1724)^{2} \)
$53$
\( T^{8} + 76 T^{6} + 1824 T^{4} + \cdots + 9604 \)
$59$
\( T^{8} - 12 T^{7} - 64 T^{6} + \cdots + 5740816 \)
$61$
\( T^{8} + 16 T^{7} + 292 T^{6} + \cdots + 8567329 \)
$67$
\( T^{8} + 24 T^{7} + 448 T^{6} + \cdots + 20647936 \)
$71$
\( T^{8} - 60 T^{7} + 1596 T^{6} + \cdots + 6115729 \)
$73$
\( (T^{4} - 12 T^{3} - 34 T^{2} + 804 T - 2231)^{2} \)
$79$
\( (T^{4} - 4 T^{3} - 102 T^{2} + 212 T + 622)^{2} \)
$83$
\( (T^{4} - 4 T^{3} - 70 T^{2} + 340 T - 263)^{2} \)
$89$
\( T^{8} + 24 T^{7} + 138 T^{6} + \cdots + 5080516 \)
$97$
\( T^{8} + 214 T^{6} + \cdots + 61606801 \)
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