Newspace parameters
Level: | \( N \) | \(=\) | \( 261 = 3^{2} \cdot 29 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 261.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(15.3994985115\) |
Analytic rank: | \(1\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{2}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - 2 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 29) |
Fricke sign: | \(-1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). 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 |
|
−0.414214 | 0 | −7.82843 | −0.656854 | 0 | 6.14214 | 6.55635 | 0 | 0.272078 | ||||||||||||||||||||||||
1.2 | 2.41421 | 0 | −2.17157 | 10.6569 | 0 | −22.1421 | −24.5563 | 0 | 25.7279 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(29\) | \(1\) |
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 | 261.4.a.b | 2 | |
3.b | odd | 2 | 1 | 29.4.a.a | ✓ | 2 | |
12.b | even | 2 | 1 | 464.4.a.f | 2 | ||
15.d | odd | 2 | 1 | 725.4.a.b | 2 | ||
21.c | even | 2 | 1 | 1421.4.a.c | 2 | ||
24.f | even | 2 | 1 | 1856.4.a.h | 2 | ||
24.h | odd | 2 | 1 | 1856.4.a.n | 2 | ||
87.d | odd | 2 | 1 | 841.4.a.a | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
29.4.a.a | ✓ | 2 | 3.b | odd | 2 | 1 | |
261.4.a.b | 2 | 1.a | even | 1 | 1 | trivial | |
464.4.a.f | 2 | 12.b | even | 2 | 1 | ||
725.4.a.b | 2 | 15.d | odd | 2 | 1 | ||
841.4.a.a | 2 | 87.d | odd | 2 | 1 | ||
1421.4.a.c | 2 | 21.c | even | 2 | 1 | ||
1856.4.a.h | 2 | 24.f | even | 2 | 1 | ||
1856.4.a.n | 2 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 2T_{2} - 1 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(261))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 2T - 1 \)
$3$
\( T^{2} \)
$5$
\( T^{2} - 10T - 7 \)
$7$
\( T^{2} + 16T - 136 \)
$11$
\( T^{2} - 26T - 2569 \)
$13$
\( T^{2} + 26T - 1183 \)
$17$
\( T^{2} + 60T + 252 \)
$19$
\( T^{2} + 220T + 10052 \)
$23$
\( T^{2} + 52T - 3932 \)
$29$
\( (T + 29)^{2} \)
$31$
\( T^{2} + 294T + 13671 \)
$37$
\( T^{2} - 312T + 18064 \)
$41$
\( T^{2} + 40T - 37688 \)
$43$
\( T^{2} + 322T - 32561 \)
$47$
\( T^{2} - 130T - 81473 \)
$53$
\( T^{2} + 1002 T + 221233 \)
$59$
\( T^{2} - 900T + 79492 \)
$61$
\( T^{2} + 948T + 161308 \)
$67$
\( T^{2} - 320T - 442912 \)
$71$
\( T^{2} - 660T + 106588 \)
$73$
\( T^{2} - 648T - 714224 \)
$79$
\( T^{2} - 258T - 215921 \)
$83$
\( T^{2} + 1212 T + 359044 \)
$89$
\( T^{2} + 760T - 400568 \)
$97$
\( T^{2} - 24T - 668024 \)
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