Newspace parameters
Level: | \( N \) | \(=\) | \( 315 = 3^{2} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 315.bf (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
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Defining polynomial: | \( x^{4} - x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 35) |
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 \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).
\(n\) | \(127\) | \(136\) | \(281\) |
\(\chi(n)\) | \(-1\) | \(-1 + \zeta_{12}^{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
109.1 |
|
−0.866025 | + | 0.500000i | 0 | −0.500000 | + | 0.866025i | 2.23205 | − | 0.133975i | 0 | 2.59808 | − | 0.500000i | − | 3.00000i | 0 | −1.86603 | + | 1.23205i | |||||||||||||||||||
109.2 | 0.866025 | − | 0.500000i | 0 | −0.500000 | + | 0.866025i | −1.23205 | + | 1.86603i | 0 | −2.59808 | + | 0.500000i | 3.00000i | 0 | −0.133975 | + | 2.23205i | |||||||||||||||||||||
289.1 | −0.866025 | − | 0.500000i | 0 | −0.500000 | − | 0.866025i | 2.23205 | + | 0.133975i | 0 | 2.59808 | + | 0.500000i | 3.00000i | 0 | −1.86603 | − | 1.23205i | |||||||||||||||||||||
289.2 | 0.866025 | + | 0.500000i | 0 | −0.500000 | − | 0.866025i | −1.23205 | − | 1.86603i | 0 | −2.59808 | − | 0.500000i | − | 3.00000i | 0 | −0.133975 | − | 2.23205i | ||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
35.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.bf.a | 4 | |
3.b | odd | 2 | 1 | 35.2.j.a | ✓ | 4 | |
5.b | even | 2 | 1 | inner | 315.2.bf.a | 4 | |
7.c | even | 3 | 1 | inner | 315.2.bf.a | 4 | |
7.c | even | 3 | 1 | 2205.2.d.d | 2 | ||
7.d | odd | 6 | 1 | 2205.2.d.e | 2 | ||
12.b | even | 2 | 1 | 560.2.bw.b | 4 | ||
15.d | odd | 2 | 1 | 35.2.j.a | ✓ | 4 | |
15.e | even | 4 | 1 | 175.2.e.a | 2 | ||
15.e | even | 4 | 1 | 175.2.e.b | 2 | ||
21.c | even | 2 | 1 | 245.2.j.c | 4 | ||
21.g | even | 6 | 1 | 245.2.b.b | 2 | ||
21.g | even | 6 | 1 | 245.2.j.c | 4 | ||
21.h | odd | 6 | 1 | 35.2.j.a | ✓ | 4 | |
21.h | odd | 6 | 1 | 245.2.b.c | 2 | ||
35.i | odd | 6 | 1 | 2205.2.d.e | 2 | ||
35.j | even | 6 | 1 | inner | 315.2.bf.a | 4 | |
35.j | even | 6 | 1 | 2205.2.d.d | 2 | ||
60.h | even | 2 | 1 | 560.2.bw.b | 4 | ||
84.n | even | 6 | 1 | 560.2.bw.b | 4 | ||
105.g | even | 2 | 1 | 245.2.j.c | 4 | ||
105.o | odd | 6 | 1 | 35.2.j.a | ✓ | 4 | |
105.o | odd | 6 | 1 | 245.2.b.c | 2 | ||
105.p | even | 6 | 1 | 245.2.b.b | 2 | ||
105.p | even | 6 | 1 | 245.2.j.c | 4 | ||
105.w | odd | 12 | 1 | 1225.2.a.b | 1 | ||
105.w | odd | 12 | 1 | 1225.2.a.g | 1 | ||
105.x | even | 12 | 1 | 175.2.e.a | 2 | ||
105.x | even | 12 | 1 | 175.2.e.b | 2 | ||
105.x | even | 12 | 1 | 1225.2.a.d | 1 | ||
105.x | even | 12 | 1 | 1225.2.a.f | 1 | ||
420.ba | even | 6 | 1 | 560.2.bw.b | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
35.2.j.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
35.2.j.a | ✓ | 4 | 15.d | odd | 2 | 1 | |
35.2.j.a | ✓ | 4 | 21.h | odd | 6 | 1 | |
35.2.j.a | ✓ | 4 | 105.o | odd | 6 | 1 | |
175.2.e.a | 2 | 15.e | even | 4 | 1 | ||
175.2.e.a | 2 | 105.x | even | 12 | 1 | ||
175.2.e.b | 2 | 15.e | even | 4 | 1 | ||
175.2.e.b | 2 | 105.x | even | 12 | 1 | ||
245.2.b.b | 2 | 21.g | even | 6 | 1 | ||
245.2.b.b | 2 | 105.p | even | 6 | 1 | ||
245.2.b.c | 2 | 21.h | odd | 6 | 1 | ||
245.2.b.c | 2 | 105.o | odd | 6 | 1 | ||
245.2.j.c | 4 | 21.c | even | 2 | 1 | ||
245.2.j.c | 4 | 21.g | even | 6 | 1 | ||
245.2.j.c | 4 | 105.g | even | 2 | 1 | ||
245.2.j.c | 4 | 105.p | even | 6 | 1 | ||
315.2.bf.a | 4 | 1.a | even | 1 | 1 | trivial | |
315.2.bf.a | 4 | 5.b | even | 2 | 1 | inner | |
315.2.bf.a | 4 | 7.c | even | 3 | 1 | inner | |
315.2.bf.a | 4 | 35.j | even | 6 | 1 | inner | |
560.2.bw.b | 4 | 12.b | even | 2 | 1 | ||
560.2.bw.b | 4 | 60.h | even | 2 | 1 | ||
560.2.bw.b | 4 | 84.n | even | 6 | 1 | ||
560.2.bw.b | 4 | 420.ba | even | 6 | 1 | ||
1225.2.a.b | 1 | 105.w | odd | 12 | 1 | ||
1225.2.a.d | 1 | 105.x | even | 12 | 1 | ||
1225.2.a.f | 1 | 105.x | even | 12 | 1 | ||
1225.2.a.g | 1 | 105.w | odd | 12 | 1 | ||
2205.2.d.d | 2 | 7.c | even | 3 | 1 | ||
2205.2.d.d | 2 | 35.j | even | 6 | 1 | ||
2205.2.d.e | 2 | 7.d | odd | 6 | 1 | ||
2205.2.d.e | 2 | 35.i | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - T^{2} + 1 \)
$3$
\( T^{4} \)
$5$
\( T^{4} - 2 T^{3} - T^{2} - 10 T + 25 \)
$7$
\( T^{4} - 13T^{2} + 49 \)
$11$
\( T^{4} \)
$13$
\( (T^{2} + 4)^{2} \)
$17$
\( T^{4} - 4T^{2} + 16 \)
$19$
\( (T^{2} - 6 T + 36)^{2} \)
$23$
\( T^{4} - 9T^{2} + 81 \)
$29$
\( (T - 7)^{4} \)
$31$
\( (T^{2} + 2 T + 4)^{2} \)
$37$
\( T^{4} - 64T^{2} + 4096 \)
$41$
\( (T + 5)^{4} \)
$43$
\( (T^{2} + 49)^{2} \)
$47$
\( T^{4} \)
$53$
\( T^{4} - 36T^{2} + 1296 \)
$59$
\( (T^{2} + 10 T + 100)^{2} \)
$61$
\( (T^{2} + 7 T + 49)^{2} \)
$67$
\( T^{4} - 25T^{2} + 625 \)
$71$
\( (T - 2)^{4} \)
$73$
\( T^{4} - 36T^{2} + 1296 \)
$79$
\( (T^{2} + 2 T + 4)^{2} \)
$83$
\( (T^{2} + 121)^{2} \)
$89$
\( (T^{2} + 9 T + 81)^{2} \)
$97$
\( (T^{2} + 256)^{2} \)
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