Newspace parameters
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.j (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(2.51528766367\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 105) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(136\) | \(281\) |
\(\chi(n)\) | \(1\) | \(-\zeta_{6}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
46.1 |
|
−1.00000 | − | 1.73205i | 0 | −1.00000 | + | 1.73205i | 0.500000 | + | 0.866025i | 0 | −0.500000 | + | 2.59808i | 0 | 0 | 1.00000 | − | 1.73205i | ||||||||||||||
226.1 | −1.00000 | + | 1.73205i | 0 | −1.00000 | − | 1.73205i | 0.500000 | − | 0.866025i | 0 | −0.500000 | − | 2.59808i | 0 | 0 | 1.00000 | + | 1.73205i | |||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 315.2.j.a | 2 | |
3.b | odd | 2 | 1 | 105.2.i.b | ✓ | 2 | |
7.c | even | 3 | 1 | inner | 315.2.j.a | 2 | |
7.c | even | 3 | 1 | 2205.2.a.k | 1 | ||
7.d | odd | 6 | 1 | 2205.2.a.m | 1 | ||
12.b | even | 2 | 1 | 1680.2.bg.l | 2 | ||
15.d | odd | 2 | 1 | 525.2.i.a | 2 | ||
15.e | even | 4 | 2 | 525.2.r.d | 4 | ||
21.c | even | 2 | 1 | 735.2.i.f | 2 | ||
21.g | even | 6 | 1 | 735.2.a.a | 1 | ||
21.g | even | 6 | 1 | 735.2.i.f | 2 | ||
21.h | odd | 6 | 1 | 105.2.i.b | ✓ | 2 | |
21.h | odd | 6 | 1 | 735.2.a.b | 1 | ||
84.n | even | 6 | 1 | 1680.2.bg.l | 2 | ||
105.o | odd | 6 | 1 | 525.2.i.a | 2 | ||
105.o | odd | 6 | 1 | 3675.2.a.o | 1 | ||
105.p | even | 6 | 1 | 3675.2.a.p | 1 | ||
105.x | even | 12 | 2 | 525.2.r.d | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
105.2.i.b | ✓ | 2 | 3.b | odd | 2 | 1 | |
105.2.i.b | ✓ | 2 | 21.h | odd | 6 | 1 | |
315.2.j.a | 2 | 1.a | even | 1 | 1 | trivial | |
315.2.j.a | 2 | 7.c | even | 3 | 1 | inner | |
525.2.i.a | 2 | 15.d | odd | 2 | 1 | ||
525.2.i.a | 2 | 105.o | odd | 6 | 1 | ||
525.2.r.d | 4 | 15.e | even | 4 | 2 | ||
525.2.r.d | 4 | 105.x | even | 12 | 2 | ||
735.2.a.a | 1 | 21.g | even | 6 | 1 | ||
735.2.a.b | 1 | 21.h | odd | 6 | 1 | ||
735.2.i.f | 2 | 21.c | even | 2 | 1 | ||
735.2.i.f | 2 | 21.g | even | 6 | 1 | ||
1680.2.bg.l | 2 | 12.b | even | 2 | 1 | ||
1680.2.bg.l | 2 | 84.n | even | 6 | 1 | ||
2205.2.a.k | 1 | 7.c | even | 3 | 1 | ||
2205.2.a.m | 1 | 7.d | odd | 6 | 1 | ||
3675.2.a.o | 1 | 105.o | odd | 6 | 1 | ||
3675.2.a.p | 1 | 105.p | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 2T_{2} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + 2T + 4 \)
$3$
\( T^{2} \)
$5$
\( T^{2} - T + 1 \)
$7$
\( T^{2} + T + 7 \)
$11$
\( T^{2} + 6T + 36 \)
$13$
\( (T + 3)^{2} \)
$17$
\( T^{2} + 4T + 16 \)
$19$
\( T^{2} + T + 1 \)
$23$
\( T^{2} + 4T + 16 \)
$29$
\( (T - 8)^{2} \)
$31$
\( T^{2} + T + 1 \)
$37$
\( T^{2} + 7T + 49 \)
$41$
\( (T - 6)^{2} \)
$43$
\( (T - 1)^{2} \)
$47$
\( T^{2} - 2T + 4 \)
$53$
\( T^{2} - 4T + 16 \)
$59$
\( T^{2} + 8T + 64 \)
$61$
\( T^{2} - 14T + 196 \)
$67$
\( T^{2} + 7T + 49 \)
$71$
\( (T + 6)^{2} \)
$73$
\( T^{2} + T + 1 \)
$79$
\( T^{2} - T + 1 \)
$83$
\( (T + 2)^{2} \)
$89$
\( T^{2} + 12T + 144 \)
$97$
\( (T + 6)^{2} \)
show more
show less