Newspace parameters
Level: | \( N \) | \(=\) | \( 15 = 3 \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 9 \) |
Character orbit: | \([\chi]\) | \(=\) | 15.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.11067915092\) |
Analytic rank: | \(0\) |
Dimension: | \(12\) |
Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
Defining polynomial: |
\( x^{12} + 2096 x^{10} + 1966565 x^{8} + 942732880 x^{6} + 407378811520 x^{4} + 58185108489824 x^{2} + 32\!\cdots\!64 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 2^{13}\cdot 3^{18}\cdot 5^{7} \) |
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
\( x^{12} + 2096 x^{10} + 1966565 x^{8} + 942732880 x^{6} + 407378811520 x^{4} + 58185108489824 x^{2} + 32\!\cdots\!64 \)
:
\(\beta_{1}\) | \(=\) |
\( ( - 27046273969224 \nu^{10} + \cdots + 20\!\cdots\!06 ) / 53\!\cdots\!25 \)
|
\(\beta_{2}\) | \(=\) |
\( ( - 18\!\cdots\!08 \nu^{11} + \cdots - 55\!\cdots\!98 \nu ) / 60\!\cdots\!00 \)
|
\(\beta_{3}\) | \(=\) |
\( ( - 18\!\cdots\!47 \nu^{11} + \cdots - 82\!\cdots\!32 \nu ) / 46\!\cdots\!00 \)
|
\(\beta_{4}\) | \(=\) |
\( ( 10\!\cdots\!58 \nu^{11} + \cdots + 46\!\cdots\!98 \nu ) / 10\!\cdots\!00 \)
|
\(\beta_{5}\) | \(=\) |
\( ( 18\!\cdots\!70 \nu^{11} + \cdots + 52\!\cdots\!56 ) / 79\!\cdots\!00 \)
|
\(\beta_{6}\) | \(=\) |
\( ( 45\!\cdots\!50 \nu^{11} + \cdots + 98\!\cdots\!52 ) / 19\!\cdots\!00 \)
|
\(\beta_{7}\) | \(=\) |
\( ( 45\!\cdots\!50 \nu^{11} + \cdots + 15\!\cdots\!48 ) / 19\!\cdots\!00 \)
|
\(\beta_{8}\) | \(=\) |
\( ( 40\!\cdots\!82 \nu^{11} + \cdots + 44\!\cdots\!42 \nu ) / 11\!\cdots\!25 \)
|
\(\beta_{9}\) | \(=\) |
\( ( - 16\!\cdots\!61 \nu^{11} + \cdots - 47\!\cdots\!36 ) / 39\!\cdots\!00 \)
|
\(\beta_{10}\) | \(=\) |
\( ( 32\!\cdots\!35 \nu^{11} + \cdots + 60\!\cdots\!32 ) / 19\!\cdots\!00 \)
|
\(\beta_{11}\) | \(=\) |
\( ( - 27\!\cdots\!27 \nu^{11} + \cdots + 86\!\cdots\!00 ) / 13\!\cdots\!00 \)
|
\(\nu\) | \(=\) |
\( ( 2\beta_{10} - 4\beta_{8} - \beta_{7} - \beta_{6} - 6\beta_{4} - 20\beta_{3} + 546\beta_{2} - \beta_1 ) / 1080 \)
|
\(\nu^{2}\) | \(=\) |
\( ( 36 \beta_{11} + 16 \beta_{9} - 55 \beta_{7} + 43 \beta_{6} + 368 \beta_{5} + 72 \beta_{4} + 196 \beta_{3} - 196 \beta_{2} - 147 \beta _1 - 188640 ) / 540 \)
|
\(\nu^{3}\) | \(=\) |
\( ( - 405 \beta_{11} - 146 \beta_{10} + 1210 \beta_{8} + 73 \beta_{7} + 73 \beta_{6} + 405 \beta_{5} - 3780 \beta_{4} + 34403 \beta_{3} - 177525 \beta_{2} + 73 \beta_1 ) / 270 \)
|
\(\nu^{4}\) | \(=\) |
\( ( - 7308 \beta_{11} - 6100 \beta_{9} - 11710 \beta_{7} - 8630 \beta_{6} - 57632 \beta_{5} - 14616 \beta_{4} - 42640 \beta_{3} + 42640 \beta_{2} + 38485 \beta _1 + 6901290 ) / 90 \)
|
\(\nu^{5}\) | \(=\) |
\( ( 1838250 \beta_{11} + 8006 \beta_{10} - 786322 \beta_{8} - 4003 \beta_{7} - 4003 \beta_{6} - 1838250 \beta_{5} + 24871512 \beta_{4} - 78239210 \beta_{3} + 278421738 \beta_{2} - 4003 \beta_1 ) / 540 \)
|
\(\nu^{6}\) | \(=\) |
\( ( 22828914 \beta_{11} + 21596458 \beta_{9} + 76448465 \beta_{7} + 12420259 \beta_{6} + 159784418 \beta_{5} + 45657828 \beta_{4} + 135741028 \beta_{3} + \cdots + 14822160480 ) / 270 \)
|
\(\nu^{7}\) | \(=\) |
\( ( - 908939475 \beta_{11} + 106657574 \beta_{10} - 673007455 \beta_{8} - 53328787 \beta_{7} - 53328787 \beta_{6} + 908939475 \beta_{5} - 16128952095 \beta_{4} + \cdots - 53328787 \beta_1 ) / 270 \)
|
\(\nu^{8}\) | \(=\) |
\( ( - 6416158896 \beta_{11} - 5896584400 \beta_{9} - 25482970880 \beta_{7} + 2881127600 \beta_{6} - 46936755584 \beta_{5} - 12832317792 \beta_{4} + \cdots - 18730791768210 ) / 90 \)
|
\(\nu^{9}\) | \(=\) |
\( ( 545120161605 \beta_{11} - 240444715972 \beta_{10} + 1519724010869 \beta_{8} + 120222357986 \beta_{7} + 120222357986 \beta_{6} - 545120161605 \beta_{5} + \cdots + 120222357986 \beta_1 ) / 270 \)
|
\(\nu^{10}\) | \(=\) |
\( ( 9248043682710 \beta_{11} + 8079802159058 \beta_{9} + 53791826718790 \beta_{7} - 26480028306886 \beta_{6} + 72080199050602 \beta_{5} + \cdots + 94\!\cdots\!80 ) / 270 \)
|
\(\nu^{11}\) | \(=\) |
\( ( - 60068117290725 \beta_{11} + 324679499790154 \beta_{10} + \cdots - 162339749895077 \beta_1 ) / 270 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/15\mathbb{Z}\right)^\times\).
\(n\) | \(7\) | \(11\) |
\(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
14.1 |
|
−29.2008 | −12.3596 | − | 80.0515i | 596.686 | 218.046 | − | 585.731i | 360.909 | + | 2337.57i | − | 2074.09i | −9948.30 | −6255.48 | + | 1978.80i | −6367.13 | + | 17103.8i | |||||||||||||||||||||||||||||||||||||||||||
14.2 | −29.2008 | −12.3596 | + | 80.0515i | 596.686 | 218.046 | + | 585.731i | 360.909 | − | 2337.57i | 2074.09i | −9948.30 | −6255.48 | − | 1978.80i | −6367.13 | − | 17103.8i | |||||||||||||||||||||||||||||||||||||||||||||
14.3 | −17.5844 | 74.7380 | − | 31.2287i | 53.2123 | −31.8709 | + | 624.187i | −1314.22 | + | 549.140i | − | 3426.92i | 3565.91 | 4610.53 | − | 4667.95i | 560.432 | − | 10976.0i | ||||||||||||||||||||||||||||||||||||||||||||
14.4 | −17.5844 | 74.7380 | + | 31.2287i | 53.2123 | −31.8709 | − | 624.187i | −1314.22 | − | 549.140i | 3426.92i | 3565.91 | 4610.53 | + | 4667.95i | 560.432 | + | 10976.0i | |||||||||||||||||||||||||||||||||||||||||||||
14.5 | −5.66585 | −35.3550 | − | 72.8768i | −223.898 | 531.836 | + | 328.291i | 200.316 | + | 412.909i | 1674.62i | 2719.03 | −4061.05 | + | 5153.12i | −3013.30 | − | 1860.05i | |||||||||||||||||||||||||||||||||||||||||||||
14.6 | −5.66585 | −35.3550 | + | 72.8768i | −223.898 | 531.836 | − | 328.291i | 200.316 | − | 412.909i | − | 1674.62i | 2719.03 | −4061.05 | − | 5153.12i | −3013.30 | + | 1860.05i | ||||||||||||||||||||||||||||||||||||||||||||
14.7 | 5.66585 | 35.3550 | − | 72.8768i | −223.898 | −531.836 | − | 328.291i | 200.316 | − | 412.909i | 1674.62i | −2719.03 | −4061.05 | − | 5153.12i | −3013.30 | − | 1860.05i | |||||||||||||||||||||||||||||||||||||||||||||
14.8 | 5.66585 | 35.3550 | + | 72.8768i | −223.898 | −531.836 | + | 328.291i | 200.316 | + | 412.909i | − | 1674.62i | −2719.03 | −4061.05 | + | 5153.12i | −3013.30 | + | 1860.05i | ||||||||||||||||||||||||||||||||||||||||||||
14.9 | 17.5844 | −74.7380 | − | 31.2287i | 53.2123 | 31.8709 | − | 624.187i | −1314.22 | − | 549.140i | − | 3426.92i | −3565.91 | 4610.53 | + | 4667.95i | 560.432 | − | 10976.0i | ||||||||||||||||||||||||||||||||||||||||||||
14.10 | 17.5844 | −74.7380 | + | 31.2287i | 53.2123 | 31.8709 | + | 624.187i | −1314.22 | + | 549.140i | 3426.92i | −3565.91 | 4610.53 | − | 4667.95i | 560.432 | + | 10976.0i | |||||||||||||||||||||||||||||||||||||||||||||
14.11 | 29.2008 | 12.3596 | − | 80.0515i | 596.686 | −218.046 | + | 585.731i | 360.909 | − | 2337.57i | − | 2074.09i | 9948.30 | −6255.48 | − | 1978.80i | −6367.13 | + | 17103.8i | ||||||||||||||||||||||||||||||||||||||||||||
14.12 | 29.2008 | 12.3596 | + | 80.0515i | 596.686 | −218.046 | − | 585.731i | 360.909 | + | 2337.57i | 2074.09i | 9948.30 | −6255.48 | + | 1978.80i | −6367.13 | − | 17103.8i | |||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
15.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 15.9.d.c | ✓ | 12 |
3.b | odd | 2 | 1 | inner | 15.9.d.c | ✓ | 12 |
4.b | odd | 2 | 1 | 240.9.c.c | 12 | ||
5.b | even | 2 | 1 | inner | 15.9.d.c | ✓ | 12 |
5.c | odd | 4 | 2 | 75.9.c.h | 12 | ||
12.b | even | 2 | 1 | 240.9.c.c | 12 | ||
15.d | odd | 2 | 1 | inner | 15.9.d.c | ✓ | 12 |
15.e | even | 4 | 2 | 75.9.c.h | 12 | ||
20.d | odd | 2 | 1 | 240.9.c.c | 12 | ||
60.h | even | 2 | 1 | 240.9.c.c | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
15.9.d.c | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
15.9.d.c | ✓ | 12 | 3.b | odd | 2 | 1 | inner |
15.9.d.c | ✓ | 12 | 5.b | even | 2 | 1 | inner |
15.9.d.c | ✓ | 12 | 15.d | odd | 2 | 1 | inner |
75.9.c.h | 12 | 5.c | odd | 4 | 2 | ||
75.9.c.h | 12 | 15.e | even | 4 | 2 | ||
240.9.c.c | 12 | 4.b | odd | 2 | 1 | ||
240.9.c.c | 12 | 12.b | even | 2 | 1 | ||
240.9.c.c | 12 | 20.d | odd | 2 | 1 | ||
240.9.c.c | 12 | 60.h | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 1194T_{2}^{4} + 300960T_{2}^{2} - 8464000 \)
acting on \(S_{9}^{\mathrm{new}}(15, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{6} - 1194 T^{4} + 300960 T^{2} + \cdots - 8464000)^{2} \)
$3$
\( T^{12} + 11412 T^{10} + \cdots + 79\!\cdots\!61 \)
$5$
\( T^{12} + 1018110 T^{10} + \cdots + 35\!\cdots\!25 \)
$7$
\( (T^{6} + 18849978 T^{4} + \cdots + 14\!\cdots\!00)^{2} \)
$11$
\( (T^{6} + 849216240 T^{4} + \cdots + 16\!\cdots\!00)^{2} \)
$13$
\( (T^{6} + 1541733408 T^{4} + \cdots + 20\!\cdots\!00)^{2} \)
$17$
\( (T^{6} - 6047295144 T^{4} + \cdots - 10\!\cdots\!00)^{2} \)
$19$
\( (T^{3} - 69048 T^{2} + \cdots - 466181174169488)^{4} \)
$23$
\( (T^{6} - 82938208434 T^{4} + \cdots - 21\!\cdots\!00)^{2} \)
$29$
\( (T^{6} + 2476179315360 T^{4} + \cdots + 46\!\cdots\!00)^{2} \)
$31$
\( (T^{3} + 69804 T^{2} + \cdots + 37\!\cdots\!32)^{4} \)
$37$
\( (T^{6} + 8623600521408 T^{4} + \cdots + 75\!\cdots\!00)^{2} \)
$41$
\( (T^{6} + 1559710996740 T^{4} + \cdots + 29\!\cdots\!00)^{2} \)
$43$
\( (T^{6} + 50066687266938 T^{4} + \cdots + 16\!\cdots\!00)^{2} \)
$47$
\( (T^{6} - 17398060199634 T^{4} + \cdots - 10\!\cdots\!00)^{2} \)
$53$
\( (T^{6} - 177874005292584 T^{4} + \cdots - 13\!\cdots\!00)^{2} \)
$59$
\( (T^{6} + 206536910649840 T^{4} + \cdots + 24\!\cdots\!00)^{2} \)
$61$
\( (T^{3} + 22989564 T^{2} + \cdots - 34\!\cdots\!28)^{4} \)
$67$
\( (T^{6} + \cdots + 28\!\cdots\!00)^{2} \)
$71$
\( (T^{6} + \cdots + 29\!\cdots\!00)^{2} \)
$73$
\( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \)
$79$
\( (T^{3} - 105100668 T^{2} + \cdots - 37\!\cdots\!68)^{4} \)
$83$
\( (T^{6} + \cdots - 21\!\cdots\!00)^{2} \)
$89$
\( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \)
$97$
\( (T^{6} + \cdots + 28\!\cdots\!00)^{2} \)
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