Newspace parameters
Level: | \( N \) | \(=\) | \( 90 = 2 \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 90.l (of order \(12\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.718653618192\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
Coefficient field: | \(\Q(\zeta_{24})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} - x^{4} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).
\(n\) | \(11\) | \(37\) |
\(\chi(n)\) | \(1 - \zeta_{24}^{4}\) | \(-\zeta_{24}^{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
23.1 |
|
−0.258819 | + | 0.965926i | 1.62484 | + | 0.599900i | −0.866025 | − | 0.500000i | 2.20711 | − | 0.358719i | −1.00000 | + | 1.41421i | −4.40508 | − | 1.18034i | 0.707107 | − | 0.707107i | 2.28024 | + | 1.94949i | −0.224745 | + | 2.22474i | ||||||||||||||||||||||||
23.2 | 0.258819 | − | 0.965926i | 1.10721 | − | 1.33195i | −0.866025 | − | 0.500000i | 0.792893 | + | 2.09077i | −1.00000 | − | 1.41421i | −1.05902 | − | 0.283763i | −0.707107 | + | 0.707107i | −0.548188 | − | 2.94949i | 2.22474 | − | 0.224745i | |||||||||||||||||||||||||
47.1 | −0.258819 | − | 0.965926i | 1.62484 | − | 0.599900i | −0.866025 | + | 0.500000i | 2.20711 | + | 0.358719i | −1.00000 | − | 1.41421i | −4.40508 | + | 1.18034i | 0.707107 | + | 0.707107i | 2.28024 | − | 1.94949i | −0.224745 | − | 2.22474i | |||||||||||||||||||||||||
47.2 | 0.258819 | + | 0.965926i | 1.10721 | + | 1.33195i | −0.866025 | + | 0.500000i | 0.792893 | − | 2.09077i | −1.00000 | + | 1.41421i | −1.05902 | + | 0.283763i | −0.707107 | − | 0.707107i | −0.548188 | + | 2.94949i | 2.22474 | + | 0.224745i | |||||||||||||||||||||||||
77.1 | −0.965926 | − | 0.258819i | 0.599900 | − | 1.62484i | 0.866025 | + | 0.500000i | 0.792893 | + | 2.09077i | −1.00000 | + | 1.41421i | 1.18034 | − | 4.40508i | −0.707107 | − | 0.707107i | −2.28024 | − | 1.94949i | −0.224745 | − | 2.22474i | |||||||||||||||||||||||||
77.2 | 0.965926 | + | 0.258819i | −1.33195 | − | 1.10721i | 0.866025 | + | 0.500000i | 2.20711 | − | 0.358719i | −1.00000 | − | 1.41421i | 0.283763 | − | 1.05902i | 0.707107 | + | 0.707107i | 0.548188 | + | 2.94949i | 2.22474 | + | 0.224745i | |||||||||||||||||||||||||
83.1 | −0.965926 | + | 0.258819i | 0.599900 | + | 1.62484i | 0.866025 | − | 0.500000i | 0.792893 | − | 2.09077i | −1.00000 | − | 1.41421i | 1.18034 | + | 4.40508i | −0.707107 | + | 0.707107i | −2.28024 | + | 1.94949i | −0.224745 | + | 2.22474i | |||||||||||||||||||||||||
83.2 | 0.965926 | − | 0.258819i | −1.33195 | + | 1.10721i | 0.866025 | − | 0.500000i | 2.20711 | + | 0.358719i | −1.00000 | + | 1.41421i | 0.283763 | + | 1.05902i | 0.707107 | − | 0.707107i | 0.548188 | − | 2.94949i | 2.22474 | − | 0.224745i | |||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
9.d | odd | 6 | 1 | inner |
45.l | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 90.2.l.a | ✓ | 8 |
3.b | odd | 2 | 1 | 270.2.m.a | 8 | ||
4.b | odd | 2 | 1 | 720.2.cu.a | 8 | ||
5.b | even | 2 | 1 | 450.2.p.a | 8 | ||
5.c | odd | 4 | 1 | inner | 90.2.l.a | ✓ | 8 |
5.c | odd | 4 | 1 | 450.2.p.a | 8 | ||
9.c | even | 3 | 1 | 270.2.m.a | 8 | ||
9.c | even | 3 | 1 | 810.2.f.b | 8 | ||
9.d | odd | 6 | 1 | inner | 90.2.l.a | ✓ | 8 |
9.d | odd | 6 | 1 | 810.2.f.b | 8 | ||
15.d | odd | 2 | 1 | 1350.2.q.g | 8 | ||
15.e | even | 4 | 1 | 270.2.m.a | 8 | ||
15.e | even | 4 | 1 | 1350.2.q.g | 8 | ||
20.e | even | 4 | 1 | 720.2.cu.a | 8 | ||
36.h | even | 6 | 1 | 720.2.cu.a | 8 | ||
45.h | odd | 6 | 1 | 450.2.p.a | 8 | ||
45.j | even | 6 | 1 | 1350.2.q.g | 8 | ||
45.k | odd | 12 | 1 | 270.2.m.a | 8 | ||
45.k | odd | 12 | 1 | 810.2.f.b | 8 | ||
45.k | odd | 12 | 1 | 1350.2.q.g | 8 | ||
45.l | even | 12 | 1 | inner | 90.2.l.a | ✓ | 8 |
45.l | even | 12 | 1 | 450.2.p.a | 8 | ||
45.l | even | 12 | 1 | 810.2.f.b | 8 | ||
180.v | odd | 12 | 1 | 720.2.cu.a | 8 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
90.2.l.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
90.2.l.a | ✓ | 8 | 5.c | odd | 4 | 1 | inner |
90.2.l.a | ✓ | 8 | 9.d | odd | 6 | 1 | inner |
90.2.l.a | ✓ | 8 | 45.l | even | 12 | 1 | inner |
270.2.m.a | 8 | 3.b | odd | 2 | 1 | ||
270.2.m.a | 8 | 9.c | even | 3 | 1 | ||
270.2.m.a | 8 | 15.e | even | 4 | 1 | ||
270.2.m.a | 8 | 45.k | odd | 12 | 1 | ||
450.2.p.a | 8 | 5.b | even | 2 | 1 | ||
450.2.p.a | 8 | 5.c | odd | 4 | 1 | ||
450.2.p.a | 8 | 45.h | odd | 6 | 1 | ||
450.2.p.a | 8 | 45.l | even | 12 | 1 | ||
720.2.cu.a | 8 | 4.b | odd | 2 | 1 | ||
720.2.cu.a | 8 | 20.e | even | 4 | 1 | ||
720.2.cu.a | 8 | 36.h | even | 6 | 1 | ||
720.2.cu.a | 8 | 180.v | odd | 12 | 1 | ||
810.2.f.b | 8 | 9.c | even | 3 | 1 | ||
810.2.f.b | 8 | 9.d | odd | 6 | 1 | ||
810.2.f.b | 8 | 45.k | odd | 12 | 1 | ||
810.2.f.b | 8 | 45.l | even | 12 | 1 | ||
1350.2.q.g | 8 | 15.d | odd | 2 | 1 | ||
1350.2.q.g | 8 | 15.e | even | 4 | 1 | ||
1350.2.q.g | 8 | 45.j | even | 6 | 1 | ||
1350.2.q.g | 8 | 45.k | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} + 8T_{7}^{7} + 32T_{7}^{6} + 176T_{7}^{5} + 679T_{7}^{4} + 880T_{7}^{3} + 800T_{7}^{2} + 1000T_{7} + 625 \)
acting on \(S_{2}^{\mathrm{new}}(90, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - T^{4} + 1 \)
$3$
\( T^{8} - 4 T^{7} + 8 T^{6} - 8 T^{5} + \cdots + 81 \)
$5$
\( (T^{4} - 6 T^{3} + 17 T^{2} - 30 T + 25)^{2} \)
$7$
\( T^{8} + 8 T^{7} + 32 T^{6} + 176 T^{5} + \cdots + 625 \)
$11$
\( (T^{4} + 12 T^{3} + 52 T^{2} + 48 T + 16)^{2} \)
$13$
\( T^{8} - 144 T^{4} + 20736 \)
$17$
\( T^{8} + 392T^{4} + 16 \)
$19$
\( (T^{4} + 44 T^{2} + 100)^{2} \)
$23$
\( T^{8} - T^{4} + 1 \)
$29$
\( T^{8} + 10 T^{6} + 99 T^{4} + 10 T^{2} + \cdots + 1 \)
$31$
\( (T^{4} - 4 T^{3} + 18 T^{2} + 8 T + 4)^{2} \)
$37$
\( (T^{2} + 6 T + 18)^{4} \)
$41$
\( (T^{4} - 6 T^{3} - 17 T^{2} + 174 T + 841)^{2} \)
$43$
\( T^{8} - 144 T^{4} + 20736 \)
$47$
\( T^{8} - 6561 T^{4} + \cdots + 43046721 \)
$53$
\( T^{8} + 8456 T^{4} + \cdots + 6250000 \)
$59$
\( T^{8} + 220 T^{6} + \cdots + 126247696 \)
$61$
\( (T^{4} - 6 T^{3} + 33 T^{2} - 18 T + 9)^{2} \)
$67$
\( T^{8} - 4 T^{7} + 8 T^{6} + \cdots + 390625 \)
$71$
\( (T^{4} + 40 T^{2} + 16)^{2} \)
$73$
\( (T^{4} + 8 T^{3} + 32 T^{2} - 320 T + 1600)^{2} \)
$79$
\( (T^{4} - 6 T^{2} + 36)^{2} \)
$83$
\( T^{8} - 12 T^{7} + 72 T^{6} - 288 T^{5} + \cdots + 81 \)
$89$
\( (T^{4} - 70 T^{2} + 361)^{2} \)
$97$
\( T^{8} - 12 T^{7} + 72 T^{6} + \cdots + 810000 \)
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