Newspace parameters
Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 600.f (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(96.2302918878\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-1}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 24) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).
\(n\) | \(151\) | \(301\) | \(401\) | \(577\) |
\(\chi(n)\) | \(1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
|
0 | − | 9.00000i | 0 | 0 | 0 | 120.000i | 0 | −81.0000 | 0 | |||||||||||||||||||||||
49.2 | 0 | 9.00000i | 0 | 0 | 0 | − | 120.000i | 0 | −81.0000 | 0 | ||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 600.6.f.h | 2 | |
5.b | even | 2 | 1 | inner | 600.6.f.h | 2 | |
5.c | odd | 4 | 1 | 24.6.a.c | ✓ | 1 | |
5.c | odd | 4 | 1 | 600.6.a.a | 1 | ||
15.e | even | 4 | 1 | 72.6.a.b | 1 | ||
20.e | even | 4 | 1 | 48.6.a.b | 1 | ||
40.i | odd | 4 | 1 | 192.6.a.b | 1 | ||
40.k | even | 4 | 1 | 192.6.a.j | 1 | ||
60.l | odd | 4 | 1 | 144.6.a.d | 1 | ||
80.i | odd | 4 | 1 | 768.6.d.f | 2 | ||
80.j | even | 4 | 1 | 768.6.d.m | 2 | ||
80.s | even | 4 | 1 | 768.6.d.m | 2 | ||
80.t | odd | 4 | 1 | 768.6.d.f | 2 | ||
120.q | odd | 4 | 1 | 576.6.a.ba | 1 | ||
120.w | even | 4 | 1 | 576.6.a.bb | 1 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
24.6.a.c | ✓ | 1 | 5.c | odd | 4 | 1 | |
48.6.a.b | 1 | 20.e | even | 4 | 1 | ||
72.6.a.b | 1 | 15.e | even | 4 | 1 | ||
144.6.a.d | 1 | 60.l | odd | 4 | 1 | ||
192.6.a.b | 1 | 40.i | odd | 4 | 1 | ||
192.6.a.j | 1 | 40.k | even | 4 | 1 | ||
576.6.a.ba | 1 | 120.q | odd | 4 | 1 | ||
576.6.a.bb | 1 | 120.w | even | 4 | 1 | ||
600.6.a.a | 1 | 5.c | odd | 4 | 1 | ||
600.6.f.h | 2 | 1.a | even | 1 | 1 | trivial | |
600.6.f.h | 2 | 5.b | even | 2 | 1 | inner | |
768.6.d.f | 2 | 80.i | odd | 4 | 1 | ||
768.6.d.f | 2 | 80.t | odd | 4 | 1 | ||
768.6.d.m | 2 | 80.j | even | 4 | 1 | ||
768.6.d.m | 2 | 80.s | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + 14400 \)
acting on \(S_{6}^{\mathrm{new}}(600, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} + 81 \)
$5$
\( T^{2} \)
$7$
\( T^{2} + 14400 \)
$11$
\( (T - 524)^{2} \)
$13$
\( T^{2} + 925444 \)
$17$
\( T^{2} + 1844164 \)
$19$
\( (T - 2284)^{2} \)
$23$
\( T^{2} + 6512704 \)
$29$
\( (T + 3966)^{2} \)
$31$
\( (T + 2992)^{2} \)
$37$
\( T^{2} + 174398436 \)
$41$
\( (T + 15126)^{2} \)
$43$
\( T^{2} + 53523856 \)
$47$
\( T^{2} + 48441600 \)
$53$
\( T^{2} + 305620324 \)
$59$
\( (T + 33884)^{2} \)
$61$
\( (T - 39118)^{2} \)
$67$
\( T^{2} + 1088736016 \)
$71$
\( (T - 14248)^{2} \)
$73$
\( T^{2} + 1295280100 \)
$79$
\( (T - 29888)^{2} \)
$83$
\( T^{2} + 2691949456 \)
$89$
\( (T + 30714)^{2} \)
$97$
\( T^{2} + 2350116484 \)
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