Newspace parameters
Level: | \( N \) | \(=\) | \( 15 = 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 8 \) |
Character orbit: | \([\chi]\) | \(=\) | 15.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(4.68577538226\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{601}) \) |
Defining polynomial: |
\( x^{2} - x - 150 \)
|
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 \(\beta = \frac{1}{2}(1 + \sqrt{601})\). 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 |
|
−8.75765 | 27.0000 | −51.3036 | 125.000 | −236.457 | 1338.43 | 1570.28 | 729.000 | −1094.71 | ||||||||||||||||||||||||
1.2 | 15.7577 | 27.0000 | 120.304 | 125.000 | 425.457 | −34.4284 | −121.278 | 729.000 | 1969.71 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(-1\) |
\(5\) | \(-1\) |
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 | 15.8.a.c | ✓ | 2 |
3.b | odd | 2 | 1 | 45.8.a.i | 2 | ||
4.b | odd | 2 | 1 | 240.8.a.p | 2 | ||
5.b | even | 2 | 1 | 75.8.a.e | 2 | ||
5.c | odd | 4 | 2 | 75.8.b.d | 4 | ||
15.d | odd | 2 | 1 | 225.8.a.t | 2 | ||
15.e | even | 4 | 2 | 225.8.b.n | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
15.8.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
45.8.a.i | 2 | 3.b | odd | 2 | 1 | ||
75.8.a.e | 2 | 5.b | even | 2 | 1 | ||
75.8.b.d | 4 | 5.c | odd | 4 | 2 | ||
225.8.a.t | 2 | 15.d | odd | 2 | 1 | ||
225.8.b.n | 4 | 15.e | even | 4 | 2 | ||
240.8.a.p | 2 | 4.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 7T_{2} - 138 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(15))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} - 7T - 138 \)
$3$
\( (T - 27)^{2} \)
$5$
\( (T - 125)^{2} \)
$7$
\( T^{2} - 1304T - 46080 \)
$11$
\( T^{2} - 3448 T - 29376048 \)
$13$
\( T^{2} + 8988 T - 81820108 \)
$17$
\( T^{2} + 5492 T - 286949484 \)
$19$
\( T^{2} + 49584 T - 20483920 \)
$23$
\( T^{2} - 91848 T - 1966256640 \)
$29$
\( T^{2} - 181772 T - 2060549340 \)
$31$
\( T^{2} - 304232 T + 22433068800 \)
$37$
\( T^{2} + 502316 T + 31820561620 \)
$41$
\( T^{2} - 631172 T + 30837469380 \)
$43$
\( T^{2} - 353640 T + 23614207376 \)
$47$
\( T^{2} + 467480 T + 49382888064 \)
$53$
\( T^{2} + 568052 T + 54364123620 \)
$59$
\( T^{2} - 287224 T - 3349911332400 \)
$61$
\( T^{2} + 2514180 T + 1579956747716 \)
$67$
\( T^{2} + 5073832 T + 3899842029456 \)
$71$
\( T^{2} + 3748816 T + 2634564492864 \)
$73$
\( T^{2} + 1477212 T - 22097955229180 \)
$79$
\( T^{2} + 4627720 T + 4381741411200 \)
$83$
\( T^{2} + 6072936 T + 7951958141328 \)
$89$
\( T^{2} - 16516356 T + 68134385955780 \)
$97$
\( T^{2} - 2723428 T - 25751505484604 \)
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