Newspace parameters
Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 225.k (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(13.2754297513\) |
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: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 3^{2} \) |
Twist minimal: | no (minimal twist has level 45) |
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 \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).
\(n\) | \(101\) | \(127\) |
\(\chi(n)\) | \(\beta_{3}\) | \(-1\) |
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 |
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−2.92048 | + | 1.68614i | 4.97494 | + | 1.50000i | 1.68614 | − | 2.92048i | 0 | −17.0584 | + | 4.00772i | 1.40965 | − | 0.813859i | − | 15.6060i | 22.5000 | + | 14.9248i | 0 | |||||||||||||||||||||||||||||
49.2 | −2.05446 | + | 1.18614i | 4.97494 | − | 1.50000i | −1.18614 | + | 2.05446i | 0 | −8.44158 | + | 8.98266i | −6.38458 | + | 3.68614i | − | 24.6060i | 22.5000 | − | 14.9248i | 0 | ||||||||||||||||||||||||||||||
49.3 | 2.05446 | − | 1.18614i | −4.97494 | + | 1.50000i | −1.18614 | + | 2.05446i | 0 | −8.44158 | + | 8.98266i | 6.38458 | − | 3.68614i | 24.6060i | 22.5000 | − | 14.9248i | 0 | |||||||||||||||||||||||||||||||
49.4 | 2.92048 | − | 1.68614i | −4.97494 | − | 1.50000i | 1.68614 | − | 2.92048i | 0 | −17.0584 | + | 4.00772i | −1.40965 | + | 0.813859i | 15.6060i | 22.5000 | + | 14.9248i | 0 | |||||||||||||||||||||||||||||||
124.1 | −2.92048 | − | 1.68614i | 4.97494 | − | 1.50000i | 1.68614 | + | 2.92048i | 0 | −17.0584 | − | 4.00772i | 1.40965 | + | 0.813859i | 15.6060i | 22.5000 | − | 14.9248i | 0 | |||||||||||||||||||||||||||||||
124.2 | −2.05446 | − | 1.18614i | 4.97494 | + | 1.50000i | −1.18614 | − | 2.05446i | 0 | −8.44158 | − | 8.98266i | −6.38458 | − | 3.68614i | 24.6060i | 22.5000 | + | 14.9248i | 0 | |||||||||||||||||||||||||||||||
124.3 | 2.05446 | + | 1.18614i | −4.97494 | − | 1.50000i | −1.18614 | − | 2.05446i | 0 | −8.44158 | − | 8.98266i | 6.38458 | + | 3.68614i | − | 24.6060i | 22.5000 | + | 14.9248i | 0 | ||||||||||||||||||||||||||||||
124.4 | 2.92048 | + | 1.68614i | −4.97494 | + | 1.50000i | 1.68614 | + | 2.92048i | 0 | −17.0584 | − | 4.00772i | −1.40965 | − | 0.813859i | − | 15.6060i | 22.5000 | − | 14.9248i | 0 | ||||||||||||||||||||||||||||||
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 | 225.4.k.a | 8 | |
5.b | even | 2 | 1 | inner | 225.4.k.a | 8 | |
5.c | odd | 4 | 1 | 45.4.e.a | ✓ | 4 | |
5.c | odd | 4 | 1 | 225.4.e.a | 4 | ||
9.c | even | 3 | 1 | inner | 225.4.k.a | 8 | |
15.e | even | 4 | 1 | 135.4.e.a | 4 | ||
45.j | even | 6 | 1 | inner | 225.4.k.a | 8 | |
45.k | odd | 12 | 1 | 45.4.e.a | ✓ | 4 | |
45.k | odd | 12 | 1 | 225.4.e.a | 4 | ||
45.k | odd | 12 | 1 | 405.4.a.d | 2 | ||
45.k | odd | 12 | 1 | 2025.4.a.l | 2 | ||
45.l | even | 12 | 1 | 135.4.e.a | 4 | ||
45.l | even | 12 | 1 | 405.4.a.e | 2 | ||
45.l | even | 12 | 1 | 2025.4.a.j | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
45.4.e.a | ✓ | 4 | 5.c | odd | 4 | 1 | |
45.4.e.a | ✓ | 4 | 45.k | odd | 12 | 1 | |
135.4.e.a | 4 | 15.e | even | 4 | 1 | ||
135.4.e.a | 4 | 45.l | even | 12 | 1 | ||
225.4.e.a | 4 | 5.c | odd | 4 | 1 | ||
225.4.e.a | 4 | 45.k | odd | 12 | 1 | ||
225.4.k.a | 8 | 1.a | even | 1 | 1 | trivial | |
225.4.k.a | 8 | 5.b | even | 2 | 1 | inner | |
225.4.k.a | 8 | 9.c | even | 3 | 1 | inner | |
225.4.k.a | 8 | 45.j | even | 6 | 1 | inner | |
405.4.a.d | 2 | 45.k | odd | 12 | 1 | ||
405.4.a.e | 2 | 45.l | even | 12 | 1 | ||
2025.4.a.j | 2 | 45.l | even | 12 | 1 | ||
2025.4.a.l | 2 | 45.k | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 17T_{2}^{6} + 225T_{2}^{4} - 1088T_{2}^{2} + 4096 \)
acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{8} - 17 T^{6} + 225 T^{4} + \cdots + 4096 \)
$3$
\( (T^{4} - 45 T^{2} + 729)^{2} \)
$5$
\( T^{8} \)
$7$
\( T^{8} - 57 T^{6} + 3105 T^{4} + \cdots + 20736 \)
$11$
\( (T^{4} - 37 T^{3} + 1233 T^{2} + \cdots + 18496)^{2} \)
$13$
\( T^{8} - 7328 T^{6} + \cdots + 46262633168896 \)
$17$
\( (T^{4} + 13277 T^{2} + 13498276)^{2} \)
$19$
\( (T^{2} + 35 T - 4850)^{4} \)
$23$
\( T^{8} - 44373 T^{6} + \cdots + 32\!\cdots\!96 \)
$29$
\( (T^{4} - 325 T^{3} + 91767 T^{2} + \cdots + 192044164)^{2} \)
$31$
\( (T^{2} - 6 T + 36)^{4} \)
$37$
\( (T^{4} + 205172 T^{2} + \cdots + 10188076096)^{2} \)
$41$
\( (T^{4} + 238 T^{3} + 72183 T^{2} + \cdots + 241460521)^{2} \)
$43$
\( T^{8} - 129593 T^{6} + \cdots + 13\!\cdots\!96 \)
$47$
\( T^{8} - 407897 T^{6} + \cdots + 16\!\cdots\!16 \)
$53$
\( (T^{4} + 190088 T^{2} + \cdots + 4893841936)^{2} \)
$59$
\( (T^{4} + 85 T^{3} + 25227 T^{2} + \cdots + 324072004)^{2} \)
$61$
\( (T^{4} - 247 T^{3} + \cdots + 89074790116)^{2} \)
$67$
\( T^{8} - 742242 T^{6} + \cdots + 12\!\cdots\!81 \)
$71$
\( (T^{2} - 394 T - 61016)^{4} \)
$73$
\( (T^{4} + 1022273 T^{2} + \cdots + 33224540176)^{2} \)
$79$
\( (T^{4} + 840 T^{3} + \cdots + 16576047504)^{2} \)
$83$
\( T^{8} - 3460833 T^{6} + \cdots + 75\!\cdots\!76 \)
$89$
\( (T^{2} - 1065 T - 535050)^{4} \)
$97$
\( T^{8} - 814637 T^{6} + \cdots + 23\!\cdots\!36 \)
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