Newspace parameters
Level: | \( N \) | \(=\) | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 90.i (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.718653618192\) |
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: | 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 \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).
\(n\) | \(11\) | \(37\) |
\(\chi(n)\) | \(-\zeta_{12}^{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 |
|
−0.866025 | + | 0.500000i | 0.866025 | − | 1.50000i | 0.500000 | − | 0.866025i | −1.23205 | − | 1.86603i | 1.73205i | 0.866025 | − | 0.500000i | 1.00000i | −1.50000 | − | 2.59808i | 2.00000 | + | 1.00000i | ||||||||||||||||
49.2 | 0.866025 | − | 0.500000i | −0.866025 | + | 1.50000i | 0.500000 | − | 0.866025i | 2.23205 | + | 0.133975i | 1.73205i | −0.866025 | + | 0.500000i | − | 1.00000i | −1.50000 | − | 2.59808i | 2.00000 | − | 1.00000i | ||||||||||||||||
79.1 | −0.866025 | − | 0.500000i | 0.866025 | + | 1.50000i | 0.500000 | + | 0.866025i | −1.23205 | + | 1.86603i | − | 1.73205i | 0.866025 | + | 0.500000i | − | 1.00000i | −1.50000 | + | 2.59808i | 2.00000 | − | 1.00000i | |||||||||||||||
79.2 | 0.866025 | + | 0.500000i | −0.866025 | − | 1.50000i | 0.500000 | + | 0.866025i | 2.23205 | − | 0.133975i | − | 1.73205i | −0.866025 | − | 0.500000i | 1.00000i | −1.50000 | + | 2.59808i | 2.00000 | + | 1.00000i | ||||||||||||||||
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 | 90.2.i.a | ✓ | 4 |
3.b | odd | 2 | 1 | 270.2.i.a | 4 | ||
4.b | odd | 2 | 1 | 720.2.by.a | 4 | ||
5.b | even | 2 | 1 | inner | 90.2.i.a | ✓ | 4 |
5.c | odd | 4 | 1 | 450.2.e.b | 2 | ||
5.c | odd | 4 | 1 | 450.2.e.g | 2 | ||
9.c | even | 3 | 1 | inner | 90.2.i.a | ✓ | 4 |
9.c | even | 3 | 1 | 810.2.c.b | 2 | ||
9.d | odd | 6 | 1 | 270.2.i.a | 4 | ||
9.d | odd | 6 | 1 | 810.2.c.c | 2 | ||
12.b | even | 2 | 1 | 2160.2.by.b | 4 | ||
15.d | odd | 2 | 1 | 270.2.i.a | 4 | ||
15.e | even | 4 | 1 | 1350.2.e.a | 2 | ||
15.e | even | 4 | 1 | 1350.2.e.i | 2 | ||
20.d | odd | 2 | 1 | 720.2.by.a | 4 | ||
36.f | odd | 6 | 1 | 720.2.by.a | 4 | ||
36.h | even | 6 | 1 | 2160.2.by.b | 4 | ||
45.h | odd | 6 | 1 | 270.2.i.a | 4 | ||
45.h | odd | 6 | 1 | 810.2.c.c | 2 | ||
45.j | even | 6 | 1 | inner | 90.2.i.a | ✓ | 4 |
45.j | even | 6 | 1 | 810.2.c.b | 2 | ||
45.k | odd | 12 | 1 | 450.2.e.b | 2 | ||
45.k | odd | 12 | 1 | 450.2.e.g | 2 | ||
45.k | odd | 12 | 1 | 4050.2.a.j | 1 | ||
45.k | odd | 12 | 1 | 4050.2.a.x | 1 | ||
45.l | even | 12 | 1 | 1350.2.e.a | 2 | ||
45.l | even | 12 | 1 | 1350.2.e.i | 2 | ||
45.l | even | 12 | 1 | 4050.2.a.g | 1 | ||
45.l | even | 12 | 1 | 4050.2.a.be | 1 | ||
60.h | even | 2 | 1 | 2160.2.by.b | 4 | ||
180.n | even | 6 | 1 | 2160.2.by.b | 4 | ||
180.p | odd | 6 | 1 | 720.2.by.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
90.2.i.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
90.2.i.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
90.2.i.a | ✓ | 4 | 9.c | even | 3 | 1 | inner |
90.2.i.a | ✓ | 4 | 45.j | even | 6 | 1 | inner |
270.2.i.a | 4 | 3.b | odd | 2 | 1 | ||
270.2.i.a | 4 | 9.d | odd | 6 | 1 | ||
270.2.i.a | 4 | 15.d | odd | 2 | 1 | ||
270.2.i.a | 4 | 45.h | odd | 6 | 1 | ||
450.2.e.b | 2 | 5.c | odd | 4 | 1 | ||
450.2.e.b | 2 | 45.k | odd | 12 | 1 | ||
450.2.e.g | 2 | 5.c | odd | 4 | 1 | ||
450.2.e.g | 2 | 45.k | odd | 12 | 1 | ||
720.2.by.a | 4 | 4.b | odd | 2 | 1 | ||
720.2.by.a | 4 | 20.d | odd | 2 | 1 | ||
720.2.by.a | 4 | 36.f | odd | 6 | 1 | ||
720.2.by.a | 4 | 180.p | odd | 6 | 1 | ||
810.2.c.b | 2 | 9.c | even | 3 | 1 | ||
810.2.c.b | 2 | 45.j | even | 6 | 1 | ||
810.2.c.c | 2 | 9.d | odd | 6 | 1 | ||
810.2.c.c | 2 | 45.h | odd | 6 | 1 | ||
1350.2.e.a | 2 | 15.e | even | 4 | 1 | ||
1350.2.e.a | 2 | 45.l | even | 12 | 1 | ||
1350.2.e.i | 2 | 15.e | even | 4 | 1 | ||
1350.2.e.i | 2 | 45.l | even | 12 | 1 | ||
2160.2.by.b | 4 | 12.b | even | 2 | 1 | ||
2160.2.by.b | 4 | 36.h | even | 6 | 1 | ||
2160.2.by.b | 4 | 60.h | even | 2 | 1 | ||
2160.2.by.b | 4 | 180.n | even | 6 | 1 | ||
4050.2.a.g | 1 | 45.l | even | 12 | 1 | ||
4050.2.a.j | 1 | 45.k | odd | 12 | 1 | ||
4050.2.a.x | 1 | 45.k | odd | 12 | 1 | ||
4050.2.a.be | 1 | 45.l | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - T_{7}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(90, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - T^{2} + 1 \)
$3$
\( T^{4} + 3T^{2} + 9 \)
$5$
\( T^{4} - 2 T^{3} - T^{2} - 10 T + 25 \)
$7$
\( T^{4} - T^{2} + 1 \)
$11$
\( (T^{2} - 2 T + 4)^{2} \)
$13$
\( T^{4} - 36T^{2} + 1296 \)
$17$
\( (T^{2} + 4)^{2} \)
$19$
\( (T + 6)^{4} \)
$23$
\( T^{4} - T^{2} + 1 \)
$29$
\( (T^{2} - 9 T + 81)^{2} \)
$31$
\( (T^{2} - 2 T + 4)^{2} \)
$37$
\( (T^{2} + 4)^{2} \)
$41$
\( (T^{2} - 11 T + 121)^{2} \)
$43$
\( T^{4} - 16T^{2} + 256 \)
$47$
\( T^{4} - 49T^{2} + 2401 \)
$53$
\( T^{4} \)
$59$
\( (T^{2} + 4 T + 16)^{2} \)
$61$
\( (T^{2} - 7 T + 49)^{2} \)
$67$
\( T^{4} - 121 T^{2} + 14641 \)
$71$
\( (T + 6)^{4} \)
$73$
\( (T^{2} + 16)^{2} \)
$79$
\( (T^{2} + 12 T + 144)^{2} \)
$83$
\( T^{4} - 121 T^{2} + 14641 \)
$89$
\( (T + 1)^{4} \)
$97$
\( T^{4} - 64T^{2} + 4096 \)
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