Newspace parameters
Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 600.k (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.79102412128\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.214798336.3 |
Defining polynomial: |
\( x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 2x^{5} + 9x^{4} - 4x^{3} - 16x + 16 \)
:
\(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} + 2\nu^{4} + 3\nu^{3} - 6\nu^{2} - 4\nu ) / 8 \)
|
\(\beta_{2}\) | \(=\) |
\( ( -5\nu^{7} + 2\nu^{6} + 4\nu^{5} + 18\nu^{4} - 21\nu^{3} - 12\nu^{2} - 20\nu + 56 ) / 8 \)
|
\(\beta_{3}\) | \(=\) |
\( ( 5\nu^{7} - 4\nu^{6} - 4\nu^{5} - 18\nu^{4} + 25\nu^{3} + 10\nu^{2} + 24\nu - 64 ) / 8 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 3\nu^{7} - 2\nu^{6} - 2\nu^{5} - 10\nu^{4} + 15\nu^{3} + 8\nu^{2} + 10\nu - 36 ) / 4 \)
|
\(\beta_{5}\) | \(=\) |
\( ( -3\nu^{7} + 2\nu^{6} + 4\nu^{5} + 10\nu^{4} - 15\nu^{3} - 12\nu^{2} - 8\nu + 36 ) / 4 \)
|
\(\beta_{6}\) | \(=\) |
\( ( -4\nu^{7} + 3\nu^{6} + 4\nu^{5} + 12\nu^{4} - 18\nu^{3} - 9\nu^{2} - 10\nu + 44 ) / 4 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 7\nu^{7} - 4\nu^{6} - 6\nu^{5} - 22\nu^{4} + 31\nu^{3} + 18\nu^{2} + 26\nu - 80 ) / 4 \)
|
\(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + 1 ) / 2 \)
|
\(\nu^{2}\) | \(=\) |
\( ( \beta_{6} - \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 ) / 2 \)
|
\(\nu^{3}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} + 2\beta _1 + 3 ) / 2 \)
|
\(\nu^{4}\) | \(=\) |
\( ( 4\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 2\beta_{3} + 5\beta_{2} + \beta_1 ) / 2 \)
|
\(\nu^{5}\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} + 2\beta_{5} + 7\beta_{4} - \beta_{3} + \beta_{2} - 2\beta _1 - 1 ) / 2 \)
|
\(\nu^{6}\) | \(=\) |
\( ( 4\beta_{7} + 3\beta_{6} + \beta_{5} - \beta_{4} - 8\beta_{3} - 5\beta_{2} + 5\beta_1 ) / 2 \)
|
\(\nu^{7}\) | \(=\) |
\( ( 7\beta_{7} - 5\beta_{6} + 8\beta_{5} - \beta_{4} - 11\beta_{3} + 3\beta_{2} - 2\beta _1 + 5 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).
\(n\) | \(151\) | \(301\) | \(401\) | \(577\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
301.1 |
|
−1.29150 | − | 0.576222i | 1.00000i | 1.33594 | + | 1.48838i | 0 | 0.576222 | − | 1.29150i | −1.97676 | −0.867721 | − | 2.69204i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||
301.2 | −1.29150 | + | 0.576222i | − | 1.00000i | 1.33594 | − | 1.48838i | 0 | 0.576222 | + | 1.29150i | −1.97676 | −0.867721 | + | 2.69204i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||
301.3 | −1.16863 | − | 0.796431i | − | 1.00000i | 0.731395 | + | 1.86147i | 0 | −0.796431 | + | 1.16863i | 4.72294 | 0.627801 | − | 2.75787i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||
301.4 | −1.16863 | + | 0.796431i | 1.00000i | 0.731395 | − | 1.86147i | 0 | −0.796431 | − | 1.16863i | 4.72294 | 0.627801 | + | 2.75787i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
301.5 | 0.0591148 | − | 1.41298i | 1.00000i | −1.99301 | − | 0.167056i | 0 | 1.41298 | + | 0.0591148i | 1.33411 | −0.353863 | + | 2.80620i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
301.6 | 0.0591148 | + | 1.41298i | − | 1.00000i | −1.99301 | + | 0.167056i | 0 | 1.41298 | − | 0.0591148i | 1.33411 | −0.353863 | − | 2.80620i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||
301.7 | 1.40101 | − | 0.192769i | − | 1.00000i | 1.92568 | − | 0.540143i | 0 | −0.192769 | − | 1.40101i | −0.0802864 | 2.59378 | − | 1.12796i | −1.00000 | 0 | ||||||||||||||||||||||||||||||||||
301.8 | 1.40101 | + | 0.192769i | 1.00000i | 1.92568 | + | 0.540143i | 0 | −0.192769 | + | 1.40101i | −0.0802864 | 2.59378 | + | 1.12796i | −1.00000 | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
8.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 600.2.k.d | ✓ | 8 |
3.b | odd | 2 | 1 | 1800.2.k.t | 8 | ||
4.b | odd | 2 | 1 | 2400.2.k.d | 8 | ||
5.b | even | 2 | 1 | 600.2.k.e | yes | 8 | |
5.c | odd | 4 | 1 | 600.2.d.g | 8 | ||
5.c | odd | 4 | 1 | 600.2.d.h | 8 | ||
8.b | even | 2 | 1 | inner | 600.2.k.d | ✓ | 8 |
8.d | odd | 2 | 1 | 2400.2.k.d | 8 | ||
12.b | even | 2 | 1 | 7200.2.k.r | 8 | ||
15.d | odd | 2 | 1 | 1800.2.k.q | 8 | ||
15.e | even | 4 | 1 | 1800.2.d.s | 8 | ||
15.e | even | 4 | 1 | 1800.2.d.t | 8 | ||
20.d | odd | 2 | 1 | 2400.2.k.e | 8 | ||
20.e | even | 4 | 1 | 2400.2.d.g | 8 | ||
20.e | even | 4 | 1 | 2400.2.d.h | 8 | ||
24.f | even | 2 | 1 | 7200.2.k.r | 8 | ||
24.h | odd | 2 | 1 | 1800.2.k.t | 8 | ||
40.e | odd | 2 | 1 | 2400.2.k.e | 8 | ||
40.f | even | 2 | 1 | 600.2.k.e | yes | 8 | |
40.i | odd | 4 | 1 | 600.2.d.g | 8 | ||
40.i | odd | 4 | 1 | 600.2.d.h | 8 | ||
40.k | even | 4 | 1 | 2400.2.d.g | 8 | ||
40.k | even | 4 | 1 | 2400.2.d.h | 8 | ||
60.h | even | 2 | 1 | 7200.2.k.s | 8 | ||
60.l | odd | 4 | 1 | 7200.2.d.s | 8 | ||
60.l | odd | 4 | 1 | 7200.2.d.t | 8 | ||
120.i | odd | 2 | 1 | 1800.2.k.q | 8 | ||
120.m | even | 2 | 1 | 7200.2.k.s | 8 | ||
120.q | odd | 4 | 1 | 7200.2.d.s | 8 | ||
120.q | odd | 4 | 1 | 7200.2.d.t | 8 | ||
120.w | even | 4 | 1 | 1800.2.d.s | 8 | ||
120.w | even | 4 | 1 | 1800.2.d.t | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
600.2.d.g | 8 | 5.c | odd | 4 | 1 | ||
600.2.d.g | 8 | 40.i | odd | 4 | 1 | ||
600.2.d.h | 8 | 5.c | odd | 4 | 1 | ||
600.2.d.h | 8 | 40.i | odd | 4 | 1 | ||
600.2.k.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
600.2.k.d | ✓ | 8 | 8.b | even | 2 | 1 | inner |
600.2.k.e | yes | 8 | 5.b | even | 2 | 1 | |
600.2.k.e | yes | 8 | 40.f | even | 2 | 1 | |
1800.2.d.s | 8 | 15.e | even | 4 | 1 | ||
1800.2.d.s | 8 | 120.w | even | 4 | 1 | ||
1800.2.d.t | 8 | 15.e | even | 4 | 1 | ||
1800.2.d.t | 8 | 120.w | even | 4 | 1 | ||
1800.2.k.q | 8 | 15.d | odd | 2 | 1 | ||
1800.2.k.q | 8 | 120.i | odd | 2 | 1 | ||
1800.2.k.t | 8 | 3.b | odd | 2 | 1 | ||
1800.2.k.t | 8 | 24.h | odd | 2 | 1 | ||
2400.2.d.g | 8 | 20.e | even | 4 | 1 | ||
2400.2.d.g | 8 | 40.k | even | 4 | 1 | ||
2400.2.d.h | 8 | 20.e | even | 4 | 1 | ||
2400.2.d.h | 8 | 40.k | even | 4 | 1 | ||
2400.2.k.d | 8 | 4.b | odd | 2 | 1 | ||
2400.2.k.d | 8 | 8.d | odd | 2 | 1 | ||
2400.2.k.e | 8 | 20.d | odd | 2 | 1 | ||
2400.2.k.e | 8 | 40.e | odd | 2 | 1 | ||
7200.2.d.s | 8 | 60.l | odd | 4 | 1 | ||
7200.2.d.s | 8 | 120.q | odd | 4 | 1 | ||
7200.2.d.t | 8 | 60.l | odd | 4 | 1 | ||
7200.2.d.t | 8 | 120.q | odd | 4 | 1 | ||
7200.2.k.r | 8 | 12.b | even | 2 | 1 | ||
7200.2.k.r | 8 | 24.f | even | 2 | 1 | ||
7200.2.k.s | 8 | 60.h | even | 2 | 1 | ||
7200.2.k.s | 8 | 120.m | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 4T_{7}^{3} - 6T_{7}^{2} + 12T_{7} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} + 2 T^{7} - 4 T^{5} - 6 T^{4} + \cdots + 16 \)
$3$
\( (T^{2} + 1)^{4} \)
$5$
\( T^{8} \)
$7$
\( (T^{4} - 4 T^{3} - 6 T^{2} + 12 T + 1)^{2} \)
$11$
\( T^{8} + 32 T^{6} + 336 T^{4} + \cdots + 1600 \)
$13$
\( T^{8} + 44 T^{6} + 502 T^{4} + \cdots + 81 \)
$17$
\( (T^{4} - 40 T^{2} + 104 T - 24)^{2} \)
$19$
\( T^{8} + 116 T^{6} + 4662 T^{4} + \cdots + 380689 \)
$23$
\( (T^{4} - 4 T^{3} - 56 T^{2} + 152 T - 88)^{2} \)
$29$
\( T^{8} + 144 T^{6} + 6288 T^{4} + \cdots + 627264 \)
$31$
\( (T^{4} - 4 T^{3} - 70 T^{2} + 204 T + 673)^{2} \)
$37$
\( T^{8} + 128 T^{6} + 4224 T^{4} + \cdots + 4096 \)
$41$
\( (T^{4} - 64 T^{2} + 56 T + 328)^{2} \)
$43$
\( T^{8} + 244 T^{6} + 18614 T^{4} + \cdots + 4363921 \)
$47$
\( (T^{4} - 72 T^{2} - 256 T - 176)^{2} \)
$53$
\( T^{8} + 256 T^{6} + 19536 T^{4} + \cdots + 23104 \)
$59$
\( T^{8} + 432 T^{6} + \cdots + 31181056 \)
$61$
\( T^{8} + 236 T^{6} + 14838 T^{4} + \cdots + 3025 \)
$67$
\( T^{8} + 372 T^{6} + \cdots + 25979409 \)
$71$
\( (T^{4} + 20 T^{3} + 96 T^{2} - 72 T - 536)^{2} \)
$73$
\( (T^{4} + 8 T^{3} - 168 T^{2} - 864 T - 432)^{2} \)
$79$
\( (T^{4} + 8 T^{3} - 184 T^{2} - 864 T + 8080)^{2} \)
$83$
\( T^{8} + 368 T^{6} + 44256 T^{4} + \cdots + 3873024 \)
$89$
\( (T^{4} - 224 T^{2} + 64 T + 10880)^{2} \)
$97$
\( (T^{4} + 4 T^{3} - 202 T^{2} - 348 T + 8881)^{2} \)
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