Newspace parameters
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.bh (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.51528766367\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\zeta_{12})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(136\) | \(281\) |
\(\chi(n)\) | \(-1\) | \(1\) | \(-\zeta_{12}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
169.1 |
|
−1.73205 | − | 1.00000i | −0.866025 | + | 1.50000i | 1.00000 | + | 1.73205i | 0.133975 | + | 2.23205i | 3.00000 | − | 1.73205i | 0.866025 | + | 0.500000i | 0 | −1.50000 | − | 2.59808i | 2.00000 | − | 4.00000i | ||||||||||||||
169.2 | 1.73205 | + | 1.00000i | 0.866025 | − | 1.50000i | 1.00000 | + | 1.73205i | 1.86603 | + | 1.23205i | 3.00000 | − | 1.73205i | −0.866025 | − | 0.500000i | 0 | −1.50000 | − | 2.59808i | 2.00000 | + | 4.00000i | |||||||||||||||
274.1 | −1.73205 | + | 1.00000i | −0.866025 | − | 1.50000i | 1.00000 | − | 1.73205i | 0.133975 | − | 2.23205i | 3.00000 | + | 1.73205i | 0.866025 | − | 0.500000i | 0 | −1.50000 | + | 2.59808i | 2.00000 | + | 4.00000i | |||||||||||||||
274.2 | 1.73205 | − | 1.00000i | 0.866025 | + | 1.50000i | 1.00000 | − | 1.73205i | 1.86603 | − | 1.23205i | 3.00000 | + | 1.73205i | −0.866025 | + | 0.500000i | 0 | −1.50000 | + | 2.59808i | 2.00000 | − | 4.00000i | |||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
9.c | even | 3 | 1 | inner |
45.j | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.2.bh.b | ✓ | 4 |
3.b | odd | 2 | 1 | 945.2.bh.b | 4 | ||
5.b | even | 2 | 1 | inner | 315.2.bh.b | ✓ | 4 |
9.c | even | 3 | 1 | inner | 315.2.bh.b | ✓ | 4 |
9.d | odd | 6 | 1 | 945.2.bh.b | 4 | ||
15.d | odd | 2 | 1 | 945.2.bh.b | 4 | ||
45.h | odd | 6 | 1 | 945.2.bh.b | 4 | ||
45.j | even | 6 | 1 | inner | 315.2.bh.b | ✓ | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
315.2.bh.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
315.2.bh.b | ✓ | 4 | 5.b | even | 2 | 1 | inner |
315.2.bh.b | ✓ | 4 | 9.c | even | 3 | 1 | inner |
315.2.bh.b | ✓ | 4 | 45.j | even | 6 | 1 | inner |
945.2.bh.b | 4 | 3.b | odd | 2 | 1 | ||
945.2.bh.b | 4 | 9.d | odd | 6 | 1 | ||
945.2.bh.b | 4 | 15.d | odd | 2 | 1 | ||
945.2.bh.b | 4 | 45.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 4T_{2}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - 4T^{2} + 16 \)
$3$
\( T^{4} + 3T^{2} + 9 \)
$5$
\( T^{4} - 4 T^{3} + 11 T^{2} - 20 T + 25 \)
$7$
\( T^{4} - T^{2} + 1 \)
$11$
\( (T^{2} + 3 T + 9)^{2} \)
$13$
\( T^{4} - 4T^{2} + 16 \)
$17$
\( (T^{2} + 4)^{2} \)
$19$
\( (T + 6)^{4} \)
$23$
\( T^{4} - 36T^{2} + 1296 \)
$29$
\( (T^{2} + 5 T + 25)^{2} \)
$31$
\( (T^{2} - 4 T + 16)^{2} \)
$37$
\( (T^{2} + 4)^{2} \)
$41$
\( (T^{2} + 8 T + 64)^{2} \)
$43$
\( T^{4} - 4T^{2} + 16 \)
$47$
\( T^{4} - 9T^{2} + 81 \)
$53$
\( T^{4} \)
$59$
\( (T^{2} + 14 T + 196)^{2} \)
$61$
\( (T^{2} + 10 T + 100)^{2} \)
$67$
\( T^{4} - 100 T^{2} + 10000 \)
$71$
\( (T - 7)^{4} \)
$73$
\( (T^{2} + 81)^{2} \)
$79$
\( (T^{2} - T + 1)^{2} \)
$83$
\( T^{4} - 289 T^{2} + 83521 \)
$89$
\( (T - 12)^{4} \)
$97$
\( T^{4} - 49T^{2} + 2401 \)
show more
show less