Newspace parameters
Level: | \( N \) | \(=\) | \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2400.k (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(19.1640964851\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Coefficient field: | 12.0.180227832610816.1 |
Defining polynomial: |
\( x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{31}]\) |
Coefficient ring index: | \( 2^{14} \) |
Twist minimal: | no (minimal twist has level 120) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} + x^{10} - 8x^{6} + 16x^{2} + 64 \)
:
\(\beta_{1}\) | \(=\) |
\( ( \nu^{9} + \nu^{7} + 8\nu ) / 8 \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{8} + \nu^{6} + 4\nu^{4} - 4\nu^{2} ) / 8 \)
|
\(\beta_{3}\) | \(=\) |
\( ( \nu^{11} - \nu^{9} + 2\nu^{7} + 4\nu^{5} + 8\nu^{3} ) / 64 \)
|
\(\beta_{4}\) | \(=\) |
\( ( -\nu^{8} - \nu^{6} + 4\nu^{4} + 12\nu^{2} ) / 8 \)
|
\(\beta_{5}\) | \(=\) |
\( ( -\nu^{11} - 3\nu^{9} + 2\nu^{7} + 4\nu^{5} + 24\nu^{3} ) / 32 \)
|
\(\beta_{6}\) | \(=\) |
\( ( -\nu^{8} + 3\nu^{6} + 4\nu^{2} - 16 ) / 8 \)
|
\(\beta_{7}\) | \(=\) |
\( ( \nu^{10} + \nu^{8} - 4\nu^{6} - 12\nu^{4} + 16\nu^{2} + 32 ) / 16 \)
|
\(\beta_{8}\) | \(=\) |
\( ( -\nu^{11} + \nu^{9} - 2\nu^{7} + 12\nu^{5} - 24\nu^{3} + 32\nu ) / 32 \)
|
\(\beta_{9}\) | \(=\) |
\( ( \nu^{11} - \nu^{9} - 6\nu^{7} - 4\nu^{5} + 40\nu^{3} + 32\nu ) / 32 \)
|
\(\beta_{10}\) | \(=\) |
\( ( -\nu^{10} + \nu^{6} + 4\nu^{4} + 4\nu^{2} - 8 ) / 8 \)
|
\(\beta_{11}\) | \(=\) |
\( ( 3\nu^{11} - 3\nu^{9} - 10\nu^{7} - 36\nu^{5} - 8\nu^{3} + 128\nu ) / 64 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{11} + \beta_{8} + \beta_{5} + \beta_{3} + \beta_1 ) / 4 \)
|
\(\nu^{2}\) | \(=\) |
\( ( \beta_{10} + 2\beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} - 1 ) / 4 \)
|
\(\nu^{3}\) | \(=\) |
\( ( -\beta_{11} + 2\beta_{9} - \beta_{8} + \beta_{5} - \beta_{3} + \beta_1 ) / 4 \)
|
\(\nu^{4}\) | \(=\) |
\( ( -\beta_{10} - 2\beta_{7} - \beta_{6} + 3\beta_{4} + 3\beta_{2} + 1 ) / 4 \)
|
\(\nu^{5}\) | \(=\) |
\( ( -3\beta_{11} + 2\beta_{9} + 5\beta_{8} - \beta_{5} + 13\beta_{3} - \beta_1 ) / 4 \)
|
\(\nu^{6}\) | \(=\) |
\( ( \beta_{10} + 2\beta_{7} + 9\beta_{6} - 3\beta_{4} + 5\beta_{2} + 15 ) / 4 \)
|
\(\nu^{7}\) | \(=\) |
\( ( 3\beta_{11} - 10\beta_{9} - 5\beta_{8} + 9\beta_{5} + 19\beta_{3} + 9\beta_1 ) / 4 \)
|
\(\nu^{8}\) | \(=\) |
\( ( 7\beta_{10} + 14\beta_{7} - \beta_{6} - 5\beta_{4} + 19\beta_{2} - 23 ) / 4 \)
|
\(\nu^{9}\) | \(=\) |
\( ( -11\beta_{11} + 10\beta_{9} - 3\beta_{8} - 17\beta_{5} - 27\beta_{3} + 15\beta_1 ) / 4 \)
|
\(\nu^{10}\) | \(=\) |
\( ( -31\beta_{10} + 2\beta_{7} + 9\beta_{6} + 13\beta_{4} + 21\beta_{2} - 17 ) / 4 \)
|
\(\nu^{11}\) | \(=\) |
\( ( 3\beta_{11} + 6\beta_{9} - 5\beta_{8} - 39\beta_{5} + 147\beta_{3} - 7\beta_1 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2400\mathbb{Z}\right)^\times\).
\(n\) | \(577\) | \(901\) | \(1601\) | \(1951\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1201.1 |
|
0 | − | 1.00000i | 0 | 0 | 0 | −4.05705 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
1201.2 | 0 | − | 1.00000i | 0 | 0 | 0 | −2.64265 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
1201.3 | 0 | − | 1.00000i | 0 | 0 | 0 | −0.746175 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
1201.4 | 0 | − | 1.00000i | 0 | 0 | 0 | 0.746175 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
1201.5 | 0 | − | 1.00000i | 0 | 0 | 0 | 2.64265 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
1201.6 | 0 | − | 1.00000i | 0 | 0 | 0 | 4.05705 | 0 | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
1201.7 | 0 | 1.00000i | 0 | 0 | 0 | −4.05705 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
1201.8 | 0 | 1.00000i | 0 | 0 | 0 | −2.64265 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
1201.9 | 0 | 1.00000i | 0 | 0 | 0 | −0.746175 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
1201.10 | 0 | 1.00000i | 0 | 0 | 0 | 0.746175 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
1201.11 | 0 | 1.00000i | 0 | 0 | 0 | 2.64265 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
1201.12 | 0 | 1.00000i | 0 | 0 | 0 | 4.05705 | 0 | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
40.f | even | 2 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} - 24T_{7}^{4} + 128T_{7}^{2} - 64 \)
acting on \(S_{2}^{\mathrm{new}}(2400, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} \)
$3$
\( (T^{2} + 1)^{6} \)
$5$
\( T^{12} \)
$7$
\( (T^{6} - 24 T^{4} + 128 T^{2} - 64)^{2} \)
$11$
\( (T^{6} + 32 T^{4} + 96 T^{2} + 64)^{2} \)
$13$
\( (T^{6} + 48 T^{4} + 704 T^{2} + 3136)^{2} \)
$17$
\( (T^{6} - 36 T^{4} + 368 T^{2} - 1024)^{2} \)
$19$
\( (T^{6} + 60 T^{4} + 512 T^{2} + 1024)^{2} \)
$23$
\( (T^{6} - 92 T^{4} + 2304 T^{2} + \cdots - 16384)^{2} \)
$29$
\( (T^{6} + 108 T^{4} + 3120 T^{2} + \cdots + 12544)^{2} \)
$31$
\( (T^{3} - 8 T^{2} - 4 T + 64)^{4} \)
$37$
\( (T^{6} + 64 T^{4} + 128 T^{2} + 64)^{2} \)
$41$
\( (T^{3} + 2 T^{2} - 100 T + 56)^{4} \)
$43$
\( (T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 4096)^{2} \)
$47$
\( (T^{6} - 60 T^{4} + 512 T^{2} - 1024)^{2} \)
$53$
\( (T^{6} + 80 T^{4} + 1216 T^{2} + 64)^{2} \)
$59$
\( (T^{6} + 176 T^{4} + 9888 T^{2} + \cdots + 179776)^{2} \)
$61$
\( (T^{6} + 176 T^{4} + 7168 T^{2} + \cdots + 65536)^{2} \)
$67$
\( (T^{6} + 128 T^{4} + 4096 T^{2} + \cdots + 4096)^{2} \)
$71$
\( (T^{3} + 8 T^{2} - 80 T - 128)^{4} \)
$73$
\( (T^{6} - 384 T^{4} + 34560 T^{2} + \cdots - 16384)^{2} \)
$79$
\( (T^{3} - 8 T^{2} - 4 T + 64)^{4} \)
$83$
\( (T^{6} + 192 T^{4} + 11264 T^{2} + \cdots + 200704)^{2} \)
$89$
\( (T^{3} - 10 T^{2} - 164 T + 1384)^{4} \)
$97$
\( (T^{6} - 336 T^{4} + 28416 T^{2} + \cdots - 262144)^{2} \)
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