Newspace parameters
Level: | \( N \) | \(=\) | \( 2368 = 2^{6} \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 2368.e (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(18.9085751986\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Coefficient field: | 8.0.8540717056.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{8} + 31x^{4} + 625 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{6} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 31x^{4} + 625 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{6} + 56\nu^{2} ) / 225 \) |
\(\beta_{2}\) | \(=\) | \( ( 2\nu^{4} + 31 ) / 9 \) |
\(\beta_{3}\) | \(=\) | \( ( -4\nu^{7} + 25\nu^{5} + \nu^{3} + 275\nu ) / 1125 \) |
\(\beta_{4}\) | \(=\) | \( ( -4\nu^{7} - 25\nu^{5} + \nu^{3} - 275\nu ) / 1125 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{6} - 6\nu^{2} ) / 25 \) |
\(\beta_{6}\) | \(=\) | \( ( 14\nu^{7} + 25\nu^{5} + 559\nu^{3} + 2525\nu ) / 1125 \) |
\(\beta_{7}\) | \(=\) | \( ( -14\nu^{7} + 25\nu^{5} - 559\nu^{3} + 2525\nu ) / 1125 \) |
\(\nu\) | \(=\) | \( ( \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} ) / 4 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{5} + 9\beta_1 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( ( -2\beta_{7} + 2\beta_{6} + 7\beta_{4} + 7\beta_{3} ) / 2 \) |
\(\nu^{4}\) | \(=\) | \( ( 9\beta_{2} - 31 ) / 2 \) |
\(\nu^{5}\) | \(=\) | \( ( -11\beta_{7} - 11\beta_{6} - 101\beta_{4} + 101\beta_{3} ) / 4 \) |
\(\nu^{6}\) | \(=\) | \( -28\beta_{5} - 27\beta_1 \) |
\(\nu^{7}\) | \(=\) | \( ( -\beta_{7} + \beta_{6} - 559\beta_{4} - 559\beta_{3} ) / 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2368\mathbb{Z}\right)^\times\).
\(n\) | \(705\) | \(1407\) | \(1925\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2145.1 |
|
0 | − | 1.00000i | 0 | −1.41421 | 0 | −4.35890 | 0 | 2.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||
2145.2 | 0 | − | 1.00000i | 0 | −1.41421 | 0 | 4.35890 | 0 | 2.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
2145.3 | 0 | − | 1.00000i | 0 | 1.41421 | 0 | −4.35890 | 0 | 2.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
2145.4 | 0 | − | 1.00000i | 0 | 1.41421 | 0 | 4.35890 | 0 | 2.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||
2145.5 | 0 | 1.00000i | 0 | −1.41421 | 0 | −4.35890 | 0 | 2.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
2145.6 | 0 | 1.00000i | 0 | −1.41421 | 0 | 4.35890 | 0 | 2.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
2145.7 | 0 | 1.00000i | 0 | 1.41421 | 0 | −4.35890 | 0 | 2.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
2145.8 | 0 | 1.00000i | 0 | 1.41421 | 0 | 4.35890 | 0 | 2.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
8.b | even | 2 | 1 | inner |
8.d | odd | 2 | 1 | inner |
37.b | even | 2 | 1 | inner |
148.b | odd | 2 | 1 | inner |
296.e | even | 2 | 1 | inner |
296.h | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2368.2.e.a | ✓ | 8 |
4.b | odd | 2 | 1 | inner | 2368.2.e.a | ✓ | 8 |
8.b | even | 2 | 1 | inner | 2368.2.e.a | ✓ | 8 |
8.d | odd | 2 | 1 | inner | 2368.2.e.a | ✓ | 8 |
37.b | even | 2 | 1 | inner | 2368.2.e.a | ✓ | 8 |
148.b | odd | 2 | 1 | inner | 2368.2.e.a | ✓ | 8 |
296.e | even | 2 | 1 | inner | 2368.2.e.a | ✓ | 8 |
296.h | odd | 2 | 1 | inner | 2368.2.e.a | ✓ | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2368.2.e.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
2368.2.e.a | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
2368.2.e.a | ✓ | 8 | 8.b | even | 2 | 1 | inner |
2368.2.e.a | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
2368.2.e.a | ✓ | 8 | 37.b | even | 2 | 1 | inner |
2368.2.e.a | ✓ | 8 | 148.b | odd | 2 | 1 | inner |
2368.2.e.a | ✓ | 8 | 296.e | even | 2 | 1 | inner |
2368.2.e.a | ✓ | 8 | 296.h | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(2368, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( (T^{2} + 1)^{4} \)
$5$
\( (T^{2} - 2)^{4} \)
$7$
\( (T^{2} - 19)^{4} \)
$11$
\( (T^{2} + 1)^{4} \)
$13$
\( (T^{2} - 18)^{4} \)
$17$
\( (T^{2} + 38)^{4} \)
$19$
\( (T^{2} - 38)^{4} \)
$23$
\( (T^{2} + 32)^{4} \)
$29$
\( (T^{2} - 8)^{4} \)
$31$
\( (T^{2} + 72)^{4} \)
$37$
\( (T^{4} + 2 T^{2} + 1369)^{2} \)
$41$
\( (T + 9)^{8} \)
$43$
\( (T^{2} - 38)^{4} \)
$47$
\( (T^{2} - 19)^{4} \)
$53$
\( (T^{2} + 19)^{4} \)
$59$
\( (T^{2} - 152)^{4} \)
$61$
\( (T^{2} - 32)^{4} \)
$67$
\( (T^{2} + 100)^{4} \)
$71$
\( (T^{2} - 171)^{4} \)
$73$
\( (T + 3)^{8} \)
$79$
\( (T^{2} + 2)^{4} \)
$83$
\( (T^{2} + 169)^{4} \)
$89$
\( (T^{2} + 342)^{4} \)
$97$
\( T^{8} \)
show more
show less