Newspace parameters
Level: | \( N \) | \(=\) | \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 900.u (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(24.5232237924\) |
Analytic rank: | \(0\) |
Dimension: | \(8\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 8.0.303595776.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 3^{2} \) |
Twist minimal: | no (minimal twist has level 36) |
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 in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 16x^{4} + 45x^{2} + 81 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{7} + 32\nu^{5} + 16\nu^{3} + 45\nu ) / 432 \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{6} - 16\nu^{4} - 32\nu^{2} - 51 ) / 48 \) |
\(\beta_{3}\) | \(=\) | \( ( -5\nu^{6} - 16\nu^{4} - 80\nu^{2} - 225 ) / 144 \) |
\(\beta_{4}\) | \(=\) | \( ( \nu^{7} + 8\nu^{5} + 40\nu^{3} + 165\nu ) / 72 \) |
\(\beta_{5}\) | \(=\) | \( ( \nu^{7} + 13\nu ) / 48 \) |
\(\beta_{6}\) | \(=\) | \( ( -\nu^{6} - 5\nu^{4} - 7\nu^{2} - 27 ) / 9 \) |
\(\beta_{7}\) | \(=\) | \( ( \nu^{7} + 2\nu^{5} + 10\nu^{3} - 3\nu ) / 18 \) |
\(\nu\) | \(=\) | \( ( -\beta_{7} + 2\beta_{5} + \beta_{4} ) / 3 \) |
\(\nu^{2}\) | \(=\) | \( ( 2\beta_{6} - 7\beta_{3} - \beta_{2} - 6 ) / 3 \) |
\(\nu^{3}\) | \(=\) | \( ( 4\beta_{7} - 11\beta_{5} + 2\beta_{4} - 9\beta_1 ) / 3 \) |
\(\nu^{4}\) | \(=\) | \( ( -5\beta_{6} + 13\beta_{3} - 5\beta_{2} ) / 3 \) |
\(\nu^{5}\) | \(=\) | \( ( -\beta_{7} - \beta_{5} - 2\beta_{4} + 45\beta_1 ) / 3 \) |
\(\nu^{6}\) | \(=\) | \( ( -16\beta_{6} - 16\beta_{3} + 32\beta_{2} - 39 ) / 3 \) |
\(\nu^{7}\) | \(=\) | \( ( 13\beta_{7} + 118\beta_{5} - 13\beta_{4} ) / 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(451\) | \(577\) |
\(\chi(n)\) | \(1 + \beta_{3}\) | \(1\) | \(-1\) |
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 |
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0 | −2.92048 | − | 0.686141i | 0 | 0 | 0 | −7.89542 | + | 4.55842i | 0 | 8.05842 | + | 4.00772i | 0 | ||||||||||||||||||||||||||||||||||||
149.2 | 0 | −2.05446 | − | 2.18614i | 0 | 0 | 0 | −7.02939 | + | 4.05842i | 0 | −0.558422 | + | 8.98266i | 0 | |||||||||||||||||||||||||||||||||||||
149.3 | 0 | 2.05446 | + | 2.18614i | 0 | 0 | 0 | 7.02939 | − | 4.05842i | 0 | −0.558422 | + | 8.98266i | 0 | |||||||||||||||||||||||||||||||||||||
149.4 | 0 | 2.92048 | + | 0.686141i | 0 | 0 | 0 | 7.89542 | − | 4.55842i | 0 | 8.05842 | + | 4.00772i | 0 | |||||||||||||||||||||||||||||||||||||
749.1 | 0 | −2.92048 | + | 0.686141i | 0 | 0 | 0 | −7.89542 | − | 4.55842i | 0 | 8.05842 | − | 4.00772i | 0 | |||||||||||||||||||||||||||||||||||||
749.2 | 0 | −2.05446 | + | 2.18614i | 0 | 0 | 0 | −7.02939 | − | 4.05842i | 0 | −0.558422 | − | 8.98266i | 0 | |||||||||||||||||||||||||||||||||||||
749.3 | 0 | 2.05446 | − | 2.18614i | 0 | 0 | 0 | 7.02939 | + | 4.05842i | 0 | −0.558422 | − | 8.98266i | 0 | |||||||||||||||||||||||||||||||||||||
749.4 | 0 | 2.92048 | − | 0.686141i | 0 | 0 | 0 | 7.89542 | + | 4.55842i | 0 | 8.05842 | − | 4.00772i | 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 | 900.3.u.a | 8 | |
3.b | odd | 2 | 1 | 2700.3.u.b | 8 | ||
5.b | even | 2 | 1 | inner | 900.3.u.a | 8 | |
5.c | odd | 4 | 1 | 36.3.g.a | ✓ | 4 | |
5.c | odd | 4 | 1 | 900.3.p.a | 4 | ||
9.c | even | 3 | 1 | 2700.3.u.b | 8 | ||
9.d | odd | 6 | 1 | inner | 900.3.u.a | 8 | |
15.d | odd | 2 | 1 | 2700.3.u.b | 8 | ||
15.e | even | 4 | 1 | 108.3.g.a | 4 | ||
15.e | even | 4 | 1 | 2700.3.p.b | 4 | ||
20.e | even | 4 | 1 | 144.3.q.b | 4 | ||
40.i | odd | 4 | 1 | 576.3.q.d | 4 | ||
40.k | even | 4 | 1 | 576.3.q.g | 4 | ||
45.h | odd | 6 | 1 | inner | 900.3.u.a | 8 | |
45.j | even | 6 | 1 | 2700.3.u.b | 8 | ||
45.k | odd | 12 | 1 | 108.3.g.a | 4 | ||
45.k | odd | 12 | 1 | 324.3.c.b | 4 | ||
45.k | odd | 12 | 1 | 2700.3.p.b | 4 | ||
45.l | even | 12 | 1 | 36.3.g.a | ✓ | 4 | |
45.l | even | 12 | 1 | 324.3.c.b | 4 | ||
45.l | even | 12 | 1 | 900.3.p.a | 4 | ||
60.l | odd | 4 | 1 | 432.3.q.b | 4 | ||
120.q | odd | 4 | 1 | 1728.3.q.h | 4 | ||
120.w | even | 4 | 1 | 1728.3.q.g | 4 | ||
180.v | odd | 12 | 1 | 144.3.q.b | 4 | ||
180.v | odd | 12 | 1 | 1296.3.e.e | 4 | ||
180.x | even | 12 | 1 | 432.3.q.b | 4 | ||
180.x | even | 12 | 1 | 1296.3.e.e | 4 | ||
360.bo | even | 12 | 1 | 1728.3.q.h | 4 | ||
360.br | even | 12 | 1 | 576.3.q.d | 4 | ||
360.bt | odd | 12 | 1 | 576.3.q.g | 4 | ||
360.bu | odd | 12 | 1 | 1728.3.q.g | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
36.3.g.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
36.3.g.a | ✓ | 4 | 45.l | even | 12 | 1 | |
108.3.g.a | 4 | 15.e | even | 4 | 1 | ||
108.3.g.a | 4 | 45.k | odd | 12 | 1 | ||
144.3.q.b | 4 | 20.e | even | 4 | 1 | ||
144.3.q.b | 4 | 180.v | odd | 12 | 1 | ||
324.3.c.b | 4 | 45.k | odd | 12 | 1 | ||
324.3.c.b | 4 | 45.l | even | 12 | 1 | ||
432.3.q.b | 4 | 60.l | odd | 4 | 1 | ||
432.3.q.b | 4 | 180.x | even | 12 | 1 | ||
576.3.q.d | 4 | 40.i | odd | 4 | 1 | ||
576.3.q.d | 4 | 360.br | even | 12 | 1 | ||
576.3.q.g | 4 | 40.k | even | 4 | 1 | ||
576.3.q.g | 4 | 360.bt | odd | 12 | 1 | ||
900.3.p.a | 4 | 5.c | odd | 4 | 1 | ||
900.3.p.a | 4 | 45.l | even | 12 | 1 | ||
900.3.u.a | 8 | 1.a | even | 1 | 1 | trivial | |
900.3.u.a | 8 | 5.b | even | 2 | 1 | inner | |
900.3.u.a | 8 | 9.d | odd | 6 | 1 | inner | |
900.3.u.a | 8 | 45.h | odd | 6 | 1 | inner | |
1296.3.e.e | 4 | 180.v | odd | 12 | 1 | ||
1296.3.e.e | 4 | 180.x | even | 12 | 1 | ||
1728.3.q.g | 4 | 120.w | even | 4 | 1 | ||
1728.3.q.g | 4 | 360.bu | odd | 12 | 1 | ||
1728.3.q.h | 4 | 120.q | odd | 4 | 1 | ||
1728.3.q.h | 4 | 360.bo | even | 12 | 1 | ||
2700.3.p.b | 4 | 15.e | even | 4 | 1 | ||
2700.3.p.b | 4 | 45.k | odd | 12 | 1 | ||
2700.3.u.b | 8 | 3.b | odd | 2 | 1 | ||
2700.3.u.b | 8 | 9.c | even | 3 | 1 | ||
2700.3.u.b | 8 | 15.d | odd | 2 | 1 | ||
2700.3.u.b | 8 | 45.j | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{8} - 149T_{7}^{6} + 16725T_{7}^{4} - 815924T_{7}^{2} + 29986576 \)
acting on \(S_{3}^{\mathrm{new}}(900, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} \)
$3$
\( T^{8} - 15 T^{6} + 144 T^{4} + \cdots + 6561 \)
$5$
\( T^{8} \)
$7$
\( T^{8} - 149 T^{6} + \cdots + 29986576 \)
$11$
\( (T^{4} + 36 T^{3} + 441 T^{2} + 324 T + 81)^{2} \)
$13$
\( T^{8} - 161 T^{6} + \cdots + 21381376 \)
$17$
\( (T^{4} - 387 T^{2} + 20736)^{2} \)
$19$
\( (T^{2} + T - 74)^{4} \)
$23$
\( T^{8} + 1683 T^{6} + \cdots + 393460125696 \)
$29$
\( (T^{4} - 63 T^{3} + 441 T^{2} + \cdots + 777924)^{2} \)
$31$
\( (T^{4} + 7 T^{3} + 705 T^{2} + \cdots + 430336)^{2} \)
$37$
\( (T^{4} + 2888 T^{2} + 868624)^{2} \)
$41$
\( (T^{4} + 18 T^{3} - 1449 T^{2} + \cdots + 2424249)^{2} \)
$43$
\( (T^{4} - 529 T^{2} + 279841)^{2} \)
$47$
\( T^{8} + 1539 T^{6} + \cdots + 11019960576 \)
$53$
\( (T^{4} - 4032 T^{2} + 1327104)^{2} \)
$59$
\( (T^{4} + 126 T^{3} + 5031 T^{2} + \cdots + 68121)^{2} \)
$61$
\( (T^{4} - 41 T^{3} + 1929 T^{2} + \cdots + 61504)^{2} \)
$67$
\( T^{8} - 12074 T^{6} + \cdots + 227988105361 \)
$71$
\( (T^{4} + 1548 T^{2} + 331776)^{2} \)
$73$
\( (T^{4} + 2261 T^{2} + 42436)^{2} \)
$79$
\( (T^{4} + 83 T^{3} + 7023 T^{2} + \cdots + 17956)^{2} \)
$83$
\( T^{8} + 1539 T^{6} + \cdots + 11019960576 \)
$89$
\( (T^{4} + 24768 T^{2} + 84934656)^{2} \)
$97$
\( T^{8} - 19802 T^{6} + \cdots + 75\!\cdots\!01 \)
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