Newspace parameters
Level: | \( N \) | \(=\) | \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 450.k (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(12.2616118962\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
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: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | no (minimal twist has level 18) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
\(\beta_{1}\) | \(=\) | \( \zeta_{24}^{2} \) |
\(\beta_{2}\) | \(=\) | \( \zeta_{24}^{4} \) |
\(\beta_{3}\) | \(=\) | \( \zeta_{24}^{6} \) |
\(\beta_{4}\) | \(=\) | \( \zeta_{24}^{7} + \zeta_{24} \) |
\(\beta_{5}\) | \(=\) | \( -\zeta_{24}^{7} + \zeta_{24} \) |
\(\beta_{6}\) | \(=\) | \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) |
\(\beta_{7}\) | \(=\) | \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) |
\(\zeta_{24}\) | \(=\) | \( ( \beta_{5} + \beta_{4} ) / 2 \) |
\(\zeta_{24}^{2}\) | \(=\) | \( \beta_1 \) |
\(\zeta_{24}^{3}\) | \(=\) | \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) |
\(\zeta_{24}^{4}\) | \(=\) | \( \beta_{2} \) |
\(\zeta_{24}^{5}\) | \(=\) | \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) |
\(\zeta_{24}^{6}\) | \(=\) | \( \beta_{3} \) |
\(\zeta_{24}^{7}\) | \(=\) | \( ( -\beta_{5} + \beta_{4} ) / 2 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(127\) |
\(\chi(n)\) | \(\beta_{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
149.1 |
|
−0.707107 | − | 1.22474i | −1.73205 | − | 2.44949i | −1.00000 | + | 1.73205i | 0 | −1.77526 | + | 3.85337i | 7.22999 | − | 4.17423i | 2.82843 | −3.00000 | + | 8.48528i | 0 | ||||||||||||||||||||||||||||||
149.2 | −0.707107 | − | 1.22474i | 1.73205 | − | 2.44949i | −1.00000 | + | 1.73205i | 0 | −4.22474 | − | 0.389270i | 5.49794 | − | 3.17423i | 2.82843 | −3.00000 | − | 8.48528i | 0 | |||||||||||||||||||||||||||||||
149.3 | 0.707107 | + | 1.22474i | −1.73205 | + | 2.44949i | −1.00000 | + | 1.73205i | 0 | −4.22474 | − | 0.389270i | −5.49794 | + | 3.17423i | −2.82843 | −3.00000 | − | 8.48528i | 0 | |||||||||||||||||||||||||||||||
149.4 | 0.707107 | + | 1.22474i | 1.73205 | + | 2.44949i | −1.00000 | + | 1.73205i | 0 | −1.77526 | + | 3.85337i | −7.22999 | + | 4.17423i | −2.82843 | −3.00000 | + | 8.48528i | 0 | |||||||||||||||||||||||||||||||
299.1 | −0.707107 | + | 1.22474i | −1.73205 | + | 2.44949i | −1.00000 | − | 1.73205i | 0 | −1.77526 | − | 3.85337i | 7.22999 | + | 4.17423i | 2.82843 | −3.00000 | − | 8.48528i | 0 | |||||||||||||||||||||||||||||||
299.2 | −0.707107 | + | 1.22474i | 1.73205 | + | 2.44949i | −1.00000 | − | 1.73205i | 0 | −4.22474 | + | 0.389270i | 5.49794 | + | 3.17423i | 2.82843 | −3.00000 | + | 8.48528i | 0 | |||||||||||||||||||||||||||||||
299.3 | 0.707107 | − | 1.22474i | −1.73205 | − | 2.44949i | −1.00000 | − | 1.73205i | 0 | −4.22474 | + | 0.389270i | −5.49794 | − | 3.17423i | −2.82843 | −3.00000 | + | 8.48528i | 0 | |||||||||||||||||||||||||||||||
299.4 | 0.707107 | − | 1.22474i | 1.73205 | − | 2.44949i | −1.00000 | − | 1.73205i | 0 | −1.77526 | − | 3.85337i | −7.22999 | − | 4.17423i | −2.82843 | −3.00000 | − | 8.48528i | 0 | |||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
9.d | odd | 6 | 1 | inner |
45.h | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 450.3.k.a | 8 | |
3.b | odd | 2 | 1 | 1350.3.k.a | 8 | ||
5.b | even | 2 | 1 | inner | 450.3.k.a | 8 | |
5.c | odd | 4 | 1 | 18.3.d.a | ✓ | 4 | |
5.c | odd | 4 | 1 | 450.3.i.b | 4 | ||
9.c | even | 3 | 1 | 1350.3.k.a | 8 | ||
9.d | odd | 6 | 1 | inner | 450.3.k.a | 8 | |
15.d | odd | 2 | 1 | 1350.3.k.a | 8 | ||
15.e | even | 4 | 1 | 54.3.d.a | 4 | ||
15.e | even | 4 | 1 | 1350.3.i.b | 4 | ||
20.e | even | 4 | 1 | 144.3.q.c | 4 | ||
40.i | odd | 4 | 1 | 576.3.q.f | 4 | ||
40.k | even | 4 | 1 | 576.3.q.e | 4 | ||
45.h | odd | 6 | 1 | inner | 450.3.k.a | 8 | |
45.j | even | 6 | 1 | 1350.3.k.a | 8 | ||
45.k | odd | 12 | 1 | 54.3.d.a | 4 | ||
45.k | odd | 12 | 1 | 162.3.b.a | 4 | ||
45.k | odd | 12 | 1 | 1350.3.i.b | 4 | ||
45.l | even | 12 | 1 | 18.3.d.a | ✓ | 4 | |
45.l | even | 12 | 1 | 162.3.b.a | 4 | ||
45.l | even | 12 | 1 | 450.3.i.b | 4 | ||
60.l | odd | 4 | 1 | 432.3.q.d | 4 | ||
120.q | odd | 4 | 1 | 1728.3.q.c | 4 | ||
120.w | even | 4 | 1 | 1728.3.q.d | 4 | ||
180.v | odd | 12 | 1 | 144.3.q.c | 4 | ||
180.v | odd | 12 | 1 | 1296.3.e.g | 4 | ||
180.x | even | 12 | 1 | 432.3.q.d | 4 | ||
180.x | even | 12 | 1 | 1296.3.e.g | 4 | ||
360.bo | even | 12 | 1 | 1728.3.q.c | 4 | ||
360.br | even | 12 | 1 | 576.3.q.f | 4 | ||
360.bt | odd | 12 | 1 | 576.3.q.e | 4 | ||
360.bu | odd | 12 | 1 | 1728.3.q.d | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
18.3.d.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
18.3.d.a | ✓ | 4 | 45.l | even | 12 | 1 | |
54.3.d.a | 4 | 15.e | even | 4 | 1 | ||
54.3.d.a | 4 | 45.k | odd | 12 | 1 | ||
144.3.q.c | 4 | 20.e | even | 4 | 1 | ||
144.3.q.c | 4 | 180.v | odd | 12 | 1 | ||
162.3.b.a | 4 | 45.k | odd | 12 | 1 | ||
162.3.b.a | 4 | 45.l | even | 12 | 1 | ||
432.3.q.d | 4 | 60.l | odd | 4 | 1 | ||
432.3.q.d | 4 | 180.x | even | 12 | 1 | ||
450.3.i.b | 4 | 5.c | odd | 4 | 1 | ||
450.3.i.b | 4 | 45.l | even | 12 | 1 | ||
450.3.k.a | 8 | 1.a | even | 1 | 1 | trivial | |
450.3.k.a | 8 | 5.b | even | 2 | 1 | inner | |
450.3.k.a | 8 | 9.d | odd | 6 | 1 | inner | |
450.3.k.a | 8 | 45.h | odd | 6 | 1 | inner | |
576.3.q.e | 4 | 40.k | even | 4 | 1 | ||
576.3.q.e | 4 | 360.bt | odd | 12 | 1 | ||
576.3.q.f | 4 | 40.i | odd | 4 | 1 | ||
576.3.q.f | 4 | 360.br | even | 12 | 1 | ||
1296.3.e.g | 4 | 180.v | odd | 12 | 1 | ||
1296.3.e.g | 4 | 180.x | even | 12 | 1 | ||
1350.3.i.b | 4 | 15.e | even | 4 | 1 | ||
1350.3.i.b | 4 | 45.k | odd | 12 | 1 | ||
1350.3.k.a | 8 | 3.b | odd | 2 | 1 | ||
1350.3.k.a | 8 | 9.c | even | 3 | 1 | ||
1350.3.k.a | 8 | 15.d | odd | 2 | 1 | ||
1350.3.k.a | 8 | 45.j | even | 6 | 1 | ||
1728.3.q.c | 4 | 120.q | odd | 4 | 1 | ||
1728.3.q.c | 4 | 360.bo | even | 12 | 1 | ||
1728.3.q.d | 4 | 120.w | even | 4 | 1 | ||
1728.3.q.d | 4 | 360.bu | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} - 110T_{7}^{6} + 9291T_{7}^{4} - 308990T_{7}^{2} + 7890481 \)
acting on \(S_{3}^{\mathrm{new}}(450, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{4} + 2 T^{2} + 4)^{2} \)
$3$
\( (T^{4} + 6 T^{2} + 81)^{2} \)
$5$
\( T^{8} \)
$7$
\( T^{8} - 110 T^{6} + 9291 T^{4} + \cdots + 7890481 \)
$11$
\( (T^{4} - 18 T^{3} + 117 T^{2} - 162 T + 81)^{2} \)
$13$
\( T^{8} - 482 T^{6} + \cdots + 1330863361 \)
$17$
\( (T^{4} - 360 T^{2} + 1296)^{2} \)
$19$
\( (T^{2} - 20 T - 116)^{4} \)
$23$
\( T^{8} + 90 T^{6} + 8019 T^{4} + \cdots + 6561 \)
$29$
\( (T^{4} + 18 T^{3} + 63 T^{2} - 810 T + 2025)^{2} \)
$31$
\( (T^{4} - 38 T^{3} + 1569 T^{2} + \cdots + 15625)^{2} \)
$37$
\( (T^{4} + 2480 T^{2} + 652864)^{2} \)
$41$
\( (T^{4} + 126 T^{3} + 5967 T^{2} + \cdots + 455625)^{2} \)
$43$
\( T^{8} - 2030 T^{6} + \cdots + 3418801 \)
$47$
\( T^{8} + 2250 T^{6} + \cdots + 166726039041 \)
$53$
\( (T^{4} - 9000 T^{2} + 810000)^{2} \)
$59$
\( (T^{4} + 126 T^{3} + 3573 T^{2} + \cdots + 2954961)^{2} \)
$61$
\( (T^{4} - 62 T^{3} + 4827 T^{2} + \cdots + 966289)^{2} \)
$67$
\( T^{8} - 6590 T^{6} + \cdots + 29120366676241 \)
$71$
\( (T^{4} + 7704 T^{2} + 2396304)^{2} \)
$73$
\( (T^{4} + 9296 T^{2} + 577600)^{2} \)
$79$
\( (T^{4} + 14 T^{3} + 1497 T^{2} + \cdots + 1692601)^{2} \)
$83$
\( T^{8} + 24714 T^{6} + \cdots + 17\!\cdots\!01 \)
$89$
\( (T^{4} + 22824 T^{2} + 36144144)^{2} \)
$97$
\( T^{8} - 21266 T^{6} + \cdots + 12\!\cdots\!25 \)
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