Newspace parameters
Level: | \( N \) | \(=\) | \( 63 = 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 63.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(3.71712033036\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{57}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{2} - x - 14 \)
|
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 21) |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−2.27492 | 0 | −2.82475 | 4.54983 | 0 | 7.00000 | 24.6254 | 0 | −10.3505 | ||||||||||||||||||||||||
1.2 | 5.27492 | 0 | 19.8248 | −10.5498 | 0 | 7.00000 | 62.3746 | 0 | −55.6495 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(7\) | \(-1\) |
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 | 63.4.a.e | 2 | |
3.b | odd | 2 | 1 | 21.4.a.c | ✓ | 2 | |
4.b | odd | 2 | 1 | 1008.4.a.ba | 2 | ||
5.b | even | 2 | 1 | 1575.4.a.p | 2 | ||
7.b | odd | 2 | 1 | 441.4.a.r | 2 | ||
7.c | even | 3 | 2 | 441.4.e.q | 4 | ||
7.d | odd | 6 | 2 | 441.4.e.p | 4 | ||
12.b | even | 2 | 1 | 336.4.a.m | 2 | ||
15.d | odd | 2 | 1 | 525.4.a.n | 2 | ||
15.e | even | 4 | 2 | 525.4.d.g | 4 | ||
21.c | even | 2 | 1 | 147.4.a.i | 2 | ||
21.g | even | 6 | 2 | 147.4.e.m | 4 | ||
21.h | odd | 6 | 2 | 147.4.e.l | 4 | ||
24.f | even | 2 | 1 | 1344.4.a.bo | 2 | ||
24.h | odd | 2 | 1 | 1344.4.a.bg | 2 | ||
84.h | odd | 2 | 1 | 2352.4.a.bz | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
21.4.a.c | ✓ | 2 | 3.b | odd | 2 | 1 | |
63.4.a.e | 2 | 1.a | even | 1 | 1 | trivial | |
147.4.a.i | 2 | 21.c | even | 2 | 1 | ||
147.4.e.l | 4 | 21.h | odd | 6 | 2 | ||
147.4.e.m | 4 | 21.g | even | 6 | 2 | ||
336.4.a.m | 2 | 12.b | even | 2 | 1 | ||
441.4.a.r | 2 | 7.b | odd | 2 | 1 | ||
441.4.e.p | 4 | 7.d | odd | 6 | 2 | ||
441.4.e.q | 4 | 7.c | even | 3 | 2 | ||
525.4.a.n | 2 | 15.d | odd | 2 | 1 | ||
525.4.d.g | 4 | 15.e | even | 4 | 2 | ||
1008.4.a.ba | 2 | 4.b | odd | 2 | 1 | ||
1344.4.a.bg | 2 | 24.h | odd | 2 | 1 | ||
1344.4.a.bo | 2 | 24.f | even | 2 | 1 | ||
1575.4.a.p | 2 | 5.b | even | 2 | 1 | ||
2352.4.a.bz | 2 | 84.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 3T_{2} - 12 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(63))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 3T - 12 \)
$3$
\( T^{2} \)
$5$
\( T^{2} + 6T - 48 \)
$7$
\( (T - 7)^{2} \)
$11$
\( T^{2} - 6T - 1416 \)
$13$
\( T^{2} - 16T - 1988 \)
$17$
\( T^{2} - 6T - 48 \)
$19$
\( T^{2} - 64T - 7184 \)
$23$
\( T^{2} + 6T - 16464 \)
$29$
\( T^{2} - 252T + 7668 \)
$31$
\( T^{2} - 40T - 73472 \)
$37$
\( T^{2} + 248T - 3092 \)
$41$
\( T^{2} - 450T + 37800 \)
$43$
\( T^{2} - 376T + 2512 \)
$47$
\( T^{2} - 12T - 65856 \)
$53$
\( T^{2} - 1104 T + 304476 \)
$59$
\( T^{2} + 804T - 30144 \)
$61$
\( T^{2} + 428T - 28076 \)
$67$
\( T^{2} - 148T - 160736 \)
$71$
\( T^{2} + 954T + 214704 \)
$73$
\( T^{2} - 1072 T + 285244 \)
$79$
\( T^{2} + 572T - 84416 \)
$83$
\( T^{2} + 1944 T + 813456 \)
$89$
\( T^{2} + 366T - 253848 \)
$97$
\( T^{2} - 808T - 922292 \)
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