Newspace parameters
Level: | \( N \) | \(=\) | \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 468.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.73699881460\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Coefficient field: | 12.0.1279179096064000000.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{12} + 5x^{8} - 4x^{6} + 20x^{4} + 64 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
Coefficient ring index: | \( 2^{4} \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) 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 \( x^{12} + 5x^{8} - 4x^{6} + 20x^{4} + 64 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} \) |
\(\beta_{3}\) | \(=\) | \( \nu^{3} \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{9} + 2\nu^{7} - \nu^{5} + 6\nu^{3} - 8\nu ) / 16 \) |
\(\beta_{5}\) | \(=\) | \( ( -\nu^{11} + 3\nu^{7} - 12\nu^{5} - 12\nu^{3} - 48\nu ) / 32 \) |
\(\beta_{6}\) | \(=\) | \( ( \nu^{10} - 2\nu^{8} + \nu^{6} - 6\nu^{4} + 8\nu^{2} ) / 16 \) |
\(\beta_{7}\) | \(=\) | \( ( -\nu^{8} + 2\nu^{6} - \nu^{4} + 6\nu^{2} - 8 ) / 8 \) |
\(\beta_{8}\) | \(=\) | \( ( \nu^{10} + 5\nu^{6} - 4\nu^{4} + 12\nu^{2} ) / 8 \) |
\(\beta_{9}\) | \(=\) | \( ( \nu^{11} + 5\nu^{7} - 4\nu^{5} + 20\nu^{3} ) / 16 \) |
\(\beta_{10}\) | \(=\) | \( ( -\nu^{11} + 4\nu^{9} - 5\nu^{7} + 24\nu^{5} - 36\nu^{3} + 48\nu ) / 32 \) |
\(\beta_{11}\) | \(=\) | \( ( \nu^{8} - 2\nu^{6} + 9\nu^{4} - 6\nu^{2} + 24 ) / 8 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( \beta_{3} \) |
\(\nu^{4}\) | \(=\) | \( \beta_{11} + \beta_{7} - 2 \) |
\(\nu^{5}\) | \(=\) | \( \beta_{10} - \beta_{5} + 2\beta_{4} - 2\beta_1 \) |
\(\nu^{6}\) | \(=\) | \( \beta_{8} + 2\beta_{7} - 2\beta_{6} - 2\beta_{2} + 2 \) |
\(\nu^{7}\) | \(=\) | \( 2\beta_{10} + 2\beta_{9} + 2\beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_1 \) |
\(\nu^{8}\) | \(=\) | \( -\beta_{11} + 2\beta_{8} - 5\beta_{7} - 4\beta_{6} + 2\beta_{2} - 2 \) |
\(\nu^{9}\) | \(=\) | \( 3\beta_{10} + 4\beta_{9} + 5\beta_{5} - 10\beta_{4} + 4\beta_{3} - 2\beta_1 \) |
\(\nu^{10}\) | \(=\) | \( 4\beta_{11} + 3\beta_{8} - 6\beta_{7} + 10\beta_{6} - 2\beta_{2} - 18 \) |
\(\nu^{11}\) | \(=\) | \( -6\beta_{10} + 6\beta_{9} - 14\beta_{5} - 12\beta_{4} - 15\beta_{3} - 18\beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).
\(n\) | \(145\) | \(209\) | \(235\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
287.1 |
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−1.31293 | − | 0.525570i | 0 | 1.44755 | + | 1.38007i | − | 4.09430i | 0 | − | 3.71352i | −1.17521 | − | 2.57272i | 0 | −2.15184 | + | 5.37551i | ||||||||||||||||||||||||||||||||||||||||||||
287.2 | −1.31293 | + | 0.525570i | 0 | 1.44755 | − | 1.38007i | 4.09430i | 0 | 3.71352i | −1.17521 | + | 2.57272i | 0 | −2.15184 | − | 5.37551i | |||||||||||||||||||||||||||||||||||||||||||||||
287.3 | −0.921588 | − | 1.07270i | 0 | −0.301352 | + | 1.97717i | − | 0.852353i | 0 | 2.60664i | 2.39862 | − | 1.49887i | 0 | −0.914316 | + | 0.785518i | ||||||||||||||||||||||||||||||||||||||||||||||
287.4 | −0.921588 | + | 1.07270i | 0 | −0.301352 | − | 1.97717i | 0.852353i | 0 | − | 2.60664i | 2.39862 | + | 1.49887i | 0 | −0.914316 | − | 0.785518i | ||||||||||||||||||||||||||||||||||||||||||||||
287.5 | −0.653376 | − | 1.25423i | 0 | −1.14620 | + | 1.63897i | 3.24195i | 0 | − | 1.84803i | 2.80455 | + | 0.366740i | 0 | 4.06615 | − | 2.11821i | ||||||||||||||||||||||||||||||||||||||||||||||
287.6 | −0.653376 | + | 1.25423i | 0 | −1.14620 | − | 1.63897i | − | 3.24195i | 0 | 1.84803i | 2.80455 | − | 0.366740i | 0 | 4.06615 | + | 2.11821i | ||||||||||||||||||||||||||||||||||||||||||||||
287.7 | 0.653376 | − | 1.25423i | 0 | −1.14620 | − | 1.63897i | 3.24195i | 0 | 1.84803i | −2.80455 | + | 0.366740i | 0 | 4.06615 | + | 2.11821i | |||||||||||||||||||||||||||||||||||||||||||||||
287.8 | 0.653376 | + | 1.25423i | 0 | −1.14620 | + | 1.63897i | − | 3.24195i | 0 | − | 1.84803i | −2.80455 | − | 0.366740i | 0 | 4.06615 | − | 2.11821i | |||||||||||||||||||||||||||||||||||||||||||||
287.9 | 0.921588 | − | 1.07270i | 0 | −0.301352 | − | 1.97717i | − | 0.852353i | 0 | − | 2.60664i | −2.39862 | − | 1.49887i | 0 | −0.914316 | − | 0.785518i | |||||||||||||||||||||||||||||||||||||||||||||
287.10 | 0.921588 | + | 1.07270i | 0 | −0.301352 | + | 1.97717i | 0.852353i | 0 | 2.60664i | −2.39862 | + | 1.49887i | 0 | −0.914316 | + | 0.785518i | |||||||||||||||||||||||||||||||||||||||||||||||
287.11 | 1.31293 | − | 0.525570i | 0 | 1.44755 | − | 1.38007i | − | 4.09430i | 0 | 3.71352i | 1.17521 | − | 2.57272i | 0 | −2.15184 | − | 5.37551i | ||||||||||||||||||||||||||||||||||||||||||||||
287.12 | 1.31293 | + | 0.525570i | 0 | 1.44755 | + | 1.38007i | 4.09430i | 0 | − | 3.71352i | 1.17521 | + | 2.57272i | 0 | −2.15184 | + | 5.37551i | ||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
4.b | odd | 2 | 1 | inner |
12.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 468.2.c.c | ✓ | 12 |
3.b | odd | 2 | 1 | inner | 468.2.c.c | ✓ | 12 |
4.b | odd | 2 | 1 | inner | 468.2.c.c | ✓ | 12 |
8.b | even | 2 | 1 | 7488.2.d.m | 12 | ||
8.d | odd | 2 | 1 | 7488.2.d.m | 12 | ||
12.b | even | 2 | 1 | inner | 468.2.c.c | ✓ | 12 |
24.f | even | 2 | 1 | 7488.2.d.m | 12 | ||
24.h | odd | 2 | 1 | 7488.2.d.m | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
468.2.c.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
468.2.c.c | ✓ | 12 | 3.b | odd | 2 | 1 | inner |
468.2.c.c | ✓ | 12 | 4.b | odd | 2 | 1 | inner |
468.2.c.c | ✓ | 12 | 12.b | even | 2 | 1 | inner |
7488.2.d.m | 12 | 8.b | even | 2 | 1 | ||
7488.2.d.m | 12 | 8.d | odd | 2 | 1 | ||
7488.2.d.m | 12 | 24.f | even | 2 | 1 | ||
7488.2.d.m | 12 | 24.h | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 28T_{5}^{4} + 196T_{5}^{2} + 128 \)
acting on \(S_{2}^{\mathrm{new}}(468, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{12} + 5 T^{8} - 4 T^{6} + 20 T^{4} + \cdots + 64 \)
$3$
\( T^{12} \)
$5$
\( (T^{6} + 28 T^{4} + 196 T^{2} + 128)^{2} \)
$7$
\( (T^{6} + 24 T^{4} + 164 T^{2} + 320)^{2} \)
$11$
\( (T^{6} - 38 T^{4} + 156 T^{2} - 40)^{2} \)
$13$
\( (T + 1)^{12} \)
$17$
\( (T^{6} + 46 T^{4} + 444 T^{2} + 200)^{2} \)
$19$
\( (T^{6} + 76 T^{4} + 1444 T^{2} + 80)^{2} \)
$23$
\( (T^{6} - 88 T^{4} + 656 T^{2} - 640)^{2} \)
$29$
\( (T^{2} + 18)^{6} \)
$31$
\( (T^{6} + 124 T^{4} + 4164 T^{2} + \cdots + 23120)^{2} \)
$37$
\( (T^{3} - 2 T^{2} - 80 T + 320)^{4} \)
$41$
\( (T^{6} + 52 T^{4} + 580 T^{2} + 800)^{2} \)
$43$
\( (T^{6} + 104 T^{4} + 3344 T^{2} + \cdots + 32000)^{2} \)
$47$
\( (T^{6} - 198 T^{4} + 8156 T^{2} + \cdots - 88360)^{2} \)
$53$
\( (T^{6} + 46 T^{4} + 444 T^{2} + 200)^{2} \)
$59$
\( (T^{2} - 10)^{6} \)
$61$
\( (T^{3} - 24 T^{2} + 164 T - 320)^{4} \)
$67$
\( (T^{6} + 136 T^{4} + 2404 T^{2} + \cdots + 5120)^{2} \)
$71$
\( (T^{6} - 222 T^{4} + 14636 T^{2} + \cdots - 289000)^{2} \)
$73$
\( (T^{3} + 10 T^{2} - 32 T - 256)^{4} \)
$79$
\( (T^{6} + 304 T^{4} + 23104 T^{2} + \cdots + 5120)^{2} \)
$83$
\( (T^{6} - 702 T^{4} + 162476 T^{2} + \cdots - 12409960)^{2} \)
$89$
\( (T^{6} + 124 T^{4} + 3652 T^{2} + \cdots + 8192)^{2} \)
$97$
\( (T^{3} + 14 T^{2} - 128)^{4} \)
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