Newspace parameters
Level: | \( N \) | \(=\) | \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 300.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(2.39551206064\) |
Analytic rank: | \(0\) |
Dimension: | \(1\) |
Coefficient field: | \(\mathbb{Q}\) |
Coefficient ring: | \(\mathbb{Z}\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 60) |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1.1 |
|
0 | 1.00000 | 0 | 0 | 0 | 4.00000 | 0 | 1.00000 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(3\) | \(-1\) |
\(5\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 300.2.a.d | 1 | |
3.b | odd | 2 | 1 | 900.2.a.h | 1 | ||
4.b | odd | 2 | 1 | 1200.2.a.a | 1 | ||
5.b | even | 2 | 1 | 300.2.a.a | 1 | ||
5.c | odd | 4 | 2 | 60.2.d.a | ✓ | 2 | |
8.b | even | 2 | 1 | 4800.2.a.bj | 1 | ||
8.d | odd | 2 | 1 | 4800.2.a.bk | 1 | ||
12.b | even | 2 | 1 | 3600.2.a.d | 1 | ||
15.d | odd | 2 | 1 | 900.2.a.a | 1 | ||
15.e | even | 4 | 2 | 180.2.d.a | 2 | ||
20.d | odd | 2 | 1 | 1200.2.a.s | 1 | ||
20.e | even | 4 | 2 | 240.2.f.b | 2 | ||
35.f | even | 4 | 2 | 2940.2.k.c | 2 | ||
35.k | even | 12 | 4 | 2940.2.bb.e | 4 | ||
35.l | odd | 12 | 4 | 2940.2.bb.d | 4 | ||
40.e | odd | 2 | 1 | 4800.2.a.bf | 1 | ||
40.f | even | 2 | 1 | 4800.2.a.bn | 1 | ||
40.i | odd | 4 | 2 | 960.2.f.f | 2 | ||
40.k | even | 4 | 2 | 960.2.f.c | 2 | ||
45.k | odd | 12 | 4 | 1620.2.r.c | 4 | ||
45.l | even | 12 | 4 | 1620.2.r.d | 4 | ||
60.h | even | 2 | 1 | 3600.2.a.bm | 1 | ||
60.l | odd | 4 | 2 | 720.2.f.c | 2 | ||
80.i | odd | 4 | 2 | 3840.2.d.o | 2 | ||
80.j | even | 4 | 2 | 3840.2.d.b | 2 | ||
80.s | even | 4 | 2 | 3840.2.d.be | 2 | ||
80.t | odd | 4 | 2 | 3840.2.d.r | 2 | ||
120.q | odd | 4 | 2 | 2880.2.f.p | 2 | ||
120.w | even | 4 | 2 | 2880.2.f.l | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
60.2.d.a | ✓ | 2 | 5.c | odd | 4 | 2 | |
180.2.d.a | 2 | 15.e | even | 4 | 2 | ||
240.2.f.b | 2 | 20.e | even | 4 | 2 | ||
300.2.a.a | 1 | 5.b | even | 2 | 1 | ||
300.2.a.d | 1 | 1.a | even | 1 | 1 | trivial | |
720.2.f.c | 2 | 60.l | odd | 4 | 2 | ||
900.2.a.a | 1 | 15.d | odd | 2 | 1 | ||
900.2.a.h | 1 | 3.b | odd | 2 | 1 | ||
960.2.f.c | 2 | 40.k | even | 4 | 2 | ||
960.2.f.f | 2 | 40.i | odd | 4 | 2 | ||
1200.2.a.a | 1 | 4.b | odd | 2 | 1 | ||
1200.2.a.s | 1 | 20.d | odd | 2 | 1 | ||
1620.2.r.c | 4 | 45.k | odd | 12 | 4 | ||
1620.2.r.d | 4 | 45.l | even | 12 | 4 | ||
2880.2.f.l | 2 | 120.w | even | 4 | 2 | ||
2880.2.f.p | 2 | 120.q | odd | 4 | 2 | ||
2940.2.k.c | 2 | 35.f | even | 4 | 2 | ||
2940.2.bb.d | 4 | 35.l | odd | 12 | 4 | ||
2940.2.bb.e | 4 | 35.k | even | 12 | 4 | ||
3600.2.a.d | 1 | 12.b | even | 2 | 1 | ||
3600.2.a.bm | 1 | 60.h | even | 2 | 1 | ||
3840.2.d.b | 2 | 80.j | even | 4 | 2 | ||
3840.2.d.o | 2 | 80.i | odd | 4 | 2 | ||
3840.2.d.r | 2 | 80.t | odd | 4 | 2 | ||
3840.2.d.be | 2 | 80.s | even | 4 | 2 | ||
4800.2.a.bf | 1 | 40.e | odd | 2 | 1 | ||
4800.2.a.bj | 1 | 8.b | even | 2 | 1 | ||
4800.2.a.bk | 1 | 8.d | odd | 2 | 1 | ||
4800.2.a.bn | 1 | 40.f | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(300))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T \)
$3$
\( T - 1 \)
$5$
\( T \)
$7$
\( T - 4 \)
$11$
\( T + 4 \)
$13$
\( T \)
$17$
\( T - 4 \)
$19$
\( T \)
$23$
\( T - 4 \)
$29$
\( T + 6 \)
$31$
\( T - 4 \)
$37$
\( T + 8 \)
$41$
\( T + 10 \)
$43$
\( T - 4 \)
$47$
\( T + 4 \)
$53$
\( T + 12 \)
$59$
\( T - 4 \)
$61$
\( T - 2 \)
$67$
\( T + 4 \)
$71$
\( T \)
$73$
\( T + 8 \)
$79$
\( T + 12 \)
$83$
\( T - 4 \)
$89$
\( T + 10 \)
$97$
\( T - 8 \)
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