Newspace parameters
Level: | \( N \) | \(=\) | \( 73 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 73.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(0.582907934755\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{13}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x - 3 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{13})\). 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 |
|
−1.30278 | 2.30278 | −0.302776 | 1.30278 | −3.00000 | −1.00000 | 3.00000 | 2.30278 | −1.69722 | ||||||||||||||||||||||||
1.2 | 2.30278 | −1.30278 | 3.30278 | −2.30278 | −3.00000 | −1.00000 | 3.00000 | −1.30278 | −5.30278 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(73\) | \(-1\) |
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 | 73.2.a.c | ✓ | 2 |
3.b | odd | 2 | 1 | 657.2.a.e | 2 | ||
4.b | odd | 2 | 1 | 1168.2.a.c | 2 | ||
5.b | even | 2 | 1 | 1825.2.a.b | 2 | ||
7.b | odd | 2 | 1 | 3577.2.a.c | 2 | ||
8.b | even | 2 | 1 | 4672.2.a.f | 2 | ||
8.d | odd | 2 | 1 | 4672.2.a.i | 2 | ||
11.b | odd | 2 | 1 | 8833.2.a.d | 2 | ||
73.b | even | 2 | 1 | 5329.2.a.c | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
73.2.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
657.2.a.e | 2 | 3.b | odd | 2 | 1 | ||
1168.2.a.c | 2 | 4.b | odd | 2 | 1 | ||
1825.2.a.b | 2 | 5.b | even | 2 | 1 | ||
3577.2.a.c | 2 | 7.b | odd | 2 | 1 | ||
4672.2.a.f | 2 | 8.b | even | 2 | 1 | ||
4672.2.a.i | 2 | 8.d | odd | 2 | 1 | ||
5329.2.a.c | 2 | 73.b | even | 2 | 1 | ||
8833.2.a.d | 2 | 11.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} - 3 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(73))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - T - 3 \)
$3$
\( T^{2} - T - 3 \)
$5$
\( T^{2} + T - 3 \)
$7$
\( (T + 1)^{2} \)
$11$
\( T^{2} - 7T + 9 \)
$13$
\( T^{2} + T - 3 \)
$17$
\( T^{2} + 4T - 9 \)
$19$
\( (T + 7)^{2} \)
$23$
\( T^{2} - 13T + 39 \)
$29$
\( T^{2} - 2T - 51 \)
$31$
\( T^{2} - 6T - 4 \)
$37$
\( T^{2} - 8T + 3 \)
$41$
\( (T + 6)^{2} \)
$43$
\( T^{2} - 6T - 43 \)
$47$
\( (T - 9)^{2} \)
$53$
\( T^{2} + 2T - 51 \)
$59$
\( T^{2} \)
$61$
\( T^{2} + 9T + 17 \)
$67$
\( T^{2} - 4T - 113 \)
$71$
\( T^{2} - 3T - 27 \)
$73$
\( (T - 1)^{2} \)
$79$
\( T^{2} - T - 29 \)
$83$
\( T^{2} - 7T - 69 \)
$89$
\( T^{2} - 12T - 81 \)
$97$
\( T^{2} + 5T - 23 \)
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