# Properties

 Label 1512.2.s.l Level 1512 Weight 2 Character orbit 1512.s Analytic conductor 12.073 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1512 = 2^{3} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1512.s (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.0733807856$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.309123.1 Defining polynomial: $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + \beta_{5} ) q^{5} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{7} +O(q^{10})$$ $$q + ( -\beta_{1} + \beta_{5} ) q^{5} + ( -1 - \beta_{1} + \beta_{2} + \beta_{5} ) q^{7} + ( -1 + \beta_{2} + \beta_{4} - \beta_{5} ) q^{11} + ( 1 + \beta_{1} ) q^{13} + ( -\beta_{2} + \beta_{5} ) q^{17} + ( \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{19} + ( \beta_{2} + \beta_{3} + \beta_{4} ) q^{23} + ( 2 + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{25} + ( -1 - 3 \beta_{1} - \beta_{3} ) q^{29} + ( -1 + 2 \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{31} + ( -2 + 2 \beta_{1} + \beta_{2} + 3 \beta_{4} ) q^{35} + ( \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - \beta_{5} ) q^{37} + ( 1 + 3 \beta_{3} ) q^{41} + ( 1 + 5 \beta_{1} + \beta_{3} ) q^{43} + ( 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 5 \beta_{5} ) q^{47} + ( 3 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{49} + ( 3 - 2 \beta_{2} - 3 \beta_{4} - \beta_{5} ) q^{53} + ( 4 + \beta_{1} + \beta_{3} ) q^{55} + ( 4 + \beta_{2} - 4 \beta_{4} + 4 \beta_{5} ) q^{59} + ( 2 \beta_{2} + 2 \beta_{3} + 5 \beta_{4} ) q^{61} + ( -\beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{65} + ( 5 + 3 \beta_{2} - 5 \beta_{4} + 2 \beta_{5} ) q^{67} + ( -2 - 2 \beta_{1} - 3 \beta_{3} ) q^{71} + ( -1 + 3 \beta_{2} + \beta_{4} + 3 \beta_{5} ) q^{73} + ( 5 - \beta_{1} + 2 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} ) q^{77} + ( 3 \beta_{2} + 3 \beta_{3} + 5 \beta_{4} ) q^{79} + ( 3 \beta_{1} - 4 \beta_{3} ) q^{83} + ( -4 - \beta_{3} ) q^{85} + ( 2 \beta_{1} - 5 \beta_{4} - 2 \beta_{5} ) q^{89} + ( -2 - \beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{91} + ( 1 - \beta_{4} + 3 \beta_{5} ) q^{95} + ( 4 - 2 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + q^{5} - 4q^{7} + O(q^{10})$$ $$6q + q^{5} - 4q^{7} - q^{11} + 4q^{13} - 2q^{17} + 5q^{19} + 2q^{23} + 6q^{25} + 2q^{29} - 4q^{31} - 6q^{35} + 8q^{37} - 6q^{43} - 3q^{47} - 12q^{49} + 8q^{53} + 20q^{55} + 9q^{59} + 13q^{61} - 8q^{65} + 16q^{67} - 2q^{71} - 3q^{73} + 7q^{77} + 12q^{79} + 2q^{83} - 22q^{85} - 17q^{89} - 13q^{91} + 28q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3 x^{5} + 10 x^{4} - 15 x^{3} + 19 x^{2} - 12 x + 3$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + \nu^{4} - 8 \nu^{3} + 5 \nu^{2} - 18 \nu + 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 2 \nu^{3} + 6 \nu^{2} - 5 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{5} + 5 \nu^{4} - 16 \nu^{3} + 19 \nu^{2} - 21 \nu + 9$$$$)/3$$ $$\beta_{5}$$ $$=$$ $$($$$$2 \nu^{5} - 5 \nu^{4} + 19 \nu^{3} - 22 \nu^{2} + 30 \nu - 9$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \beta_{1} + 2$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-2 \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 4 \beta_{1} - 4$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$7 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \beta_{1} - 10$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$($$$$16 \beta_{5} + 11 \beta_{4} + 8 \beta_{3} + 10 \beta_{2} - 17 \beta_{1} + 5$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-14 \beta_{5} - 16 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} - 23 \beta_{1} + 47$$$$)/3$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1512\mathbb{Z}\right)^\times$$.

 $$n$$ $$757$$ $$785$$ $$1081$$ $$1135$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{4}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
865.1
 0.5 − 0.224437i 0.5 + 1.41036i 0.5 − 2.05195i 0.5 + 0.224437i 0.5 − 1.41036i 0.5 + 2.05195i
0 0 0 −0.849814 1.47192i 0 −2.64400 0.0963576i 0 0 0
865.2 0 0 0 0.119562 + 0.207087i 0 0.710533 2.54856i 0 0 0
865.3 0 0 0 1.23025 + 2.13086i 0 −0.0665372 + 2.64491i 0 0 0
1297.1 0 0 0 −0.849814 + 1.47192i 0 −2.64400 + 0.0963576i 0 0 0
1297.2 0 0 0 0.119562 0.207087i 0 0.710533 + 2.54856i 0 0 0
1297.3 0 0 0 1.23025 2.13086i 0 −0.0665372 2.64491i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1297.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1512.2.s.l yes 6
3.b odd 2 1 1512.2.s.k 6
7.c even 3 1 inner 1512.2.s.l yes 6
21.h odd 6 1 1512.2.s.k 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1512.2.s.k 6 3.b odd 2 1
1512.2.s.k 6 21.h odd 6 1
1512.2.s.l yes 6 1.a even 1 1 trivial
1512.2.s.l yes 6 7.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1512, [\chi])$$:

 $$T_{5}^{6} - T_{5}^{5} + 5 T_{5}^{4} + 2 T_{5}^{3} + 17 T_{5}^{2} - 4 T_{5} + 1$$ $$T_{11}^{6} + T_{11}^{5} + 13 T_{11}^{4} - 30 T_{11}^{3} + 135 T_{11}^{2} - 108 T_{11} + 81$$ $$T_{13}^{3} - 2 T_{13}^{2} - 3 T_{13} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - T - 10 T^{2} + 7 T^{3} + 57 T^{4} - 14 T^{5} - 299 T^{6} - 70 T^{7} + 1425 T^{8} + 875 T^{9} - 6250 T^{10} - 3125 T^{11} + 15625 T^{12}$$
$7$ $$1 + 4 T + 14 T^{2} + 55 T^{3} + 98 T^{4} + 196 T^{5} + 343 T^{6}$$
$11$ $$1 + T - 20 T^{2} - 41 T^{3} + 179 T^{4} + 310 T^{5} - 1349 T^{6} + 3410 T^{7} + 21659 T^{8} - 54571 T^{9} - 292820 T^{10} + 161051 T^{11} + 1771561 T^{12}$$
$13$ $$( 1 - 2 T + 36 T^{2} - 49 T^{3} + 468 T^{4} - 338 T^{5} + 2197 T^{6} )^{2}$$
$17$ $$1 + 2 T - 36 T^{2} - 14 T^{3} + 826 T^{4} - 262 T^{5} - 16321 T^{6} - 4454 T^{7} + 238714 T^{8} - 68782 T^{9} - 3006756 T^{10} + 2839714 T^{11} + 24137569 T^{12}$$
$19$ $$1 - 5 T - 34 T^{2} + 63 T^{3} + 1465 T^{4} - 1156 T^{5} - 29405 T^{6} - 21964 T^{7} + 528865 T^{8} + 432117 T^{9} - 4430914 T^{10} - 12380495 T^{11} + 47045881 T^{12}$$
$23$ $$1 - 2 T - 60 T^{2} + 38 T^{3} + 2458 T^{4} - 482 T^{5} - 65101 T^{6} - 11086 T^{7} + 1300282 T^{8} + 462346 T^{9} - 16790460 T^{10} - 12872686 T^{11} + 148035889 T^{12}$$
$29$ $$( 1 - T + 37 T^{2} + 71 T^{3} + 1073 T^{4} - 841 T^{5} + 24389 T^{6} )^{2}$$
$31$ $$1 + 4 T - 28 T^{2} - 402 T^{3} - 584 T^{4} + 5648 T^{5} + 67711 T^{6} + 175088 T^{7} - 561224 T^{8} - 11975982 T^{9} - 25858588 T^{10} + 114516604 T^{11} + 887503681 T^{12}$$
$37$ $$1 - 8 T - 2 T^{2} + 254 T^{3} - 1214 T^{4} + 2314 T^{5} + 5399 T^{6} + 85618 T^{7} - 1661966 T^{8} + 12865862 T^{9} - 3748322 T^{10} - 554751656 T^{11} + 2565726409 T^{12}$$
$41$ $$( 1 + 66 T^{2} + 137 T^{3} + 2706 T^{4} + 68921 T^{6} )^{2}$$
$43$ $$( 1 + 3 T + 9 T^{2} - 143 T^{3} + 387 T^{4} + 5547 T^{5} + 79507 T^{6} )^{2}$$
$47$ $$1 + 3 T - 18 T^{2} - 1225 T^{3} - 2499 T^{4} + 16644 T^{5} + 579911 T^{6} + 782268 T^{7} - 5520291 T^{8} - 127183175 T^{9} - 87834258 T^{10} + 688035021 T^{11} + 10779215329 T^{12}$$
$53$ $$1 - 8 T - 90 T^{2} + 278 T^{3} + 9514 T^{4} - 8150 T^{5} - 568945 T^{6} - 431950 T^{7} + 26724826 T^{8} + 41387806 T^{9} - 710143290 T^{10} - 3345563944 T^{11} + 22164361129 T^{12}$$
$59$ $$1 - 9 T - 54 T^{2} + 171 T^{3} + 4023 T^{4} + 18486 T^{5} - 466229 T^{6} + 1090674 T^{7} + 14004063 T^{8} + 35119809 T^{9} - 654337494 T^{10} - 6434318691 T^{11} + 42180533641 T^{12}$$
$61$ $$1 - 13 T - 45 T^{2} + 252 T^{3} + 13021 T^{4} - 33607 T^{5} - 667154 T^{6} - 2050027 T^{7} + 48451141 T^{8} + 57199212 T^{9} - 623062845 T^{10} - 10979751913 T^{11} + 51520374361 T^{12}$$
$67$ $$1 - 16 T + 34 T^{2} + 562 T^{3} + 1498 T^{4} - 52510 T^{5} + 381563 T^{6} - 3518170 T^{7} + 6724522 T^{8} + 169028806 T^{9} + 685138114 T^{10} - 21602001712 T^{11} + 90458382169 T^{12}$$
$71$ $$( 1 + T + 129 T^{2} + 235 T^{3} + 9159 T^{4} + 5041 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$1 + 3 T - 132 T^{2} - 347 T^{3} + 8433 T^{4} + 8514 T^{5} - 573015 T^{6} + 621522 T^{7} + 44939457 T^{8} - 134988899 T^{9} - 3748567812 T^{10} + 6219214779 T^{11} + 151334226289 T^{12}$$
$79$ $$1 - 12 T - 84 T^{2} + 454 T^{3} + 14832 T^{4} - 6264 T^{5} - 1475337 T^{6} - 494856 T^{7} + 92566512 T^{8} + 223839706 T^{9} - 3271806804 T^{10} - 36924676788 T^{11} + 243087455521 T^{12}$$
$83$ $$( 1 - T + 129 T^{2} - 313 T^{3} + 10707 T^{4} - 6889 T^{5} + 571787 T^{6} )^{2}$$
$89$ $$1 + 17 T - 57 T^{2} - 344 T^{3} + 36001 T^{4} + 164759 T^{5} - 1843186 T^{6} + 14663551 T^{7} + 285163921 T^{8} - 242509336 T^{9} - 3576307737 T^{10} + 94929010633 T^{11} + 496981290961 T^{12}$$
$97$ $$( 1 - 14 T + 331 T^{2} - 2740 T^{3} + 32107 T^{4} - 131726 T^{5} + 912673 T^{6} )^{2}$$