# Properties

 Label 1764.2.e.j Level $1764$ Weight $2$ Character orbit 1764.e Analytic conductor $14.086$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1764.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.0856109166$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 8 x^{14} + 49 x^{12} - 104 x^{10} + 160 x^{8} - 104 x^{6} + 49 x^{4} - 8 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{14}$$ 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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{3} q^{2} + ( 1 - \beta_{8} ) q^{4} + \beta_{2} q^{5} + ( -\beta_{3} + \beta_{12} + \beta_{14} ) q^{8} +O(q^{10})$$ $$q -\beta_{3} q^{2} + ( 1 - \beta_{8} ) q^{4} + \beta_{2} q^{5} + ( -\beta_{3} + \beta_{12} + \beta_{14} ) q^{8} + ( -\beta_{5} + \beta_{10} + \beta_{11} ) q^{10} + ( \beta_{3} + 2 \beta_{12} + \beta_{14} ) q^{11} -\beta_{10} q^{13} + ( \beta_{4} + \beta_{6} - 2 \beta_{8} ) q^{16} + ( -2 \beta_{2} - \beta_{9} ) q^{17} + ( -\beta_{5} - \beta_{10} + 2 \beta_{15} ) q^{19} + ( -\beta_{1} + \beta_{2} - \beta_{7} ) q^{20} + ( -1 + 2 \beta_{4} + \beta_{6} ) q^{22} + ( \beta_{3} + 2 \beta_{12} + \beta_{14} ) q^{23} + ( 1 + \beta_{4} - \beta_{8} ) q^{25} + \beta_{7} q^{26} + ( -\beta_{3} + 2 \beta_{13} + \beta_{14} ) q^{29} + ( -\beta_{5} + \beta_{10} + 4 \beta_{11} - 2 \beta_{15} ) q^{31} + ( 2 \beta_{12} + 3 \beta_{13} + \beta_{14} ) q^{32} + ( 2 \beta_{5} - 3 \beta_{10} - 3 \beta_{11} + \beta_{15} ) q^{34} + ( \beta_{4} - \beta_{8} ) q^{37} + ( -\beta_{1} + 3 \beta_{2} - 4 \beta_{9} ) q^{38} + ( 3 \beta_{10} - \beta_{11} - \beta_{15} ) q^{40} + ( -3 \beta_{2} + 4 \beta_{9} ) q^{41} + ( -2 - \beta_{4} - 4 \beta_{6} + \beta_{8} ) q^{43} + ( \beta_{3} + \beta_{12} + 5 \beta_{13} - 2 \beta_{14} ) q^{44} + ( -1 + 2 \beta_{4} + \beta_{6} ) q^{46} + ( 2 \beta_{1} + \beta_{2} - 4 \beta_{7} + \beta_{9} ) q^{47} + ( -\beta_{3} + 2 \beta_{12} + 2 \beta_{13} ) q^{50} + ( -\beta_{5} - \beta_{10} + \beta_{11} ) q^{52} + ( -2 \beta_{3} + 3 \beta_{13} + 2 \beta_{14} ) q^{53} + ( 3 \beta_{5} + \beta_{10} - 4 \beta_{11} - 2 \beta_{15} ) q^{55} + ( 1 + 3 \beta_{6} - 2 \beta_{8} ) q^{58} + ( 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{7} + \beta_{9} ) q^{59} + ( 5 \beta_{5} - 2 \beta_{10} ) q^{61} + ( -\beta_{1} + 3 \beta_{2} - 2 \beta_{7} + 4 \beta_{9} ) q^{62} + ( 3 + 2 \beta_{4} + 4 \beta_{6} - \beta_{8} ) q^{64} + ( -\beta_{3} - \beta_{13} + \beta_{14} ) q^{65} + ( \beta_{4} + 3 \beta_{8} ) q^{67} + ( 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{7} - 2 \beta_{9} ) q^{68} + ( 7 \beta_{3} + 2 \beta_{12} - 2 \beta_{13} + 3 \beta_{14} ) q^{71} + ( -3 \beta_{5} + 4 \beta_{10} ) q^{73} + ( 2 \beta_{12} + 2 \beta_{13} ) q^{74} + ( -3 \beta_{5} - 2 \beta_{11} + 3 \beta_{15} ) q^{76} + ( -2 - 2 \beta_{4} - 4 \beta_{6} - 2 \beta_{8} ) q^{79} + ( -4 \beta_{2} - \beta_{7} + 2 \beta_{9} ) q^{80} + ( 3 \beta_{5} + \beta_{10} + \beta_{11} - 4 \beta_{15} ) q^{82} + ( -4 \beta_{1} + 4 \beta_{7} - 2 \beta_{9} ) q^{83} + ( 10 - 2 \beta_{4} + 2 \beta_{8} ) q^{85} + ( 2 \beta_{3} + 2 \beta_{12} - 6 \beta_{13} ) q^{86} + ( 9 + \beta_{4} + 3 \beta_{6} + 3 \beta_{8} ) q^{88} + ( 3 \beta_{2} - 4 \beta_{9} ) q^{89} + ( \beta_{3} + \beta_{12} + 5 \beta_{13} - 2 \beta_{14} ) q^{92} + ( 3 \beta_{5} + 4 \beta_{10} + \beta_{15} ) q^{94} + ( 2 \beta_{3} + 4 \beta_{12} + 2 \beta_{14} ) q^{95} + ( -5 \beta_{5} - 2 \beta_{10} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 16 q^{4} + O(q^{10})$$ $$16 q + 16 q^{4} - 8 q^{16} - 24 q^{22} + 16 q^{25} - 24 q^{46} - 8 q^{58} + 16 q^{64} + 160 q^{85} + 120 q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{14} + 49 x^{12} - 104 x^{10} + 160 x^{8} - 104 x^{6} + 49 x^{4} - 8 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-52 \nu^{14} + 655 \nu^{12} - 4512 \nu^{10} + 17312 \nu^{8} - 33992 \nu^{6} + 38784 \nu^{4} - 17044 \nu^{2} - 305$$$$)/1584$$ $$\beta_{2}$$ $$=$$ $$($$$$116 \nu^{14} - 791 \nu^{12} + 4704 \nu^{10} - 6208 \nu^{8} + 9544 \nu^{6} + 480 \nu^{4} + 7316 \nu^{2} - 599$$$$)/1584$$ $$\beta_{3}$$ $$=$$ $$($$$$-13 \nu^{15} + 227 \nu^{13} - 1568 \nu^{11} + 6968 \nu^{9} - 12392 \nu^{7} + 16208 \nu^{5} - 6637 \nu^{3} + 2451 \nu$$$$)/528$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{14} - 32 \nu^{12} + 184 \nu^{10} - 152 \nu^{8} + 112 \nu^{6} + 464 \nu^{4} - 195 \nu^{2} + 88$$$$)/24$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{15} + 13 \nu^{13} - 88 \nu^{11} + 340 \nu^{9} - 624 \nu^{7} + 760 \nu^{5} - 357 \nu^{3} + 101 \nu$$$$)/12$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{14} - 56 \nu^{12} + 344 \nu^{10} - 736 \nu^{8} + 1168 \nu^{6} - 824 \nu^{4} + 447 \nu^{2} - 80$$$$)/24$$ $$\beta_{7}$$ $$=$$ $$($$$$284 \nu^{14} - 2105 \nu^{12} + 12600 \nu^{10} - 21544 \nu^{8} + 29320 \nu^{6} - 6408 \nu^{4} + 2900 \nu^{2} + 1351$$$$)/792$$ $$\beta_{8}$$ $$=$$ $$($$$$-5 \nu^{14} + 40 \nu^{12} - 244 \nu^{10} + 512 \nu^{8} - 752 \nu^{6} + 424 \nu^{4} - 129 \nu^{2} + 4$$$$)/12$$ $$\beta_{9}$$ $$=$$ $$($$$$1060 \nu^{14} - 8407 \nu^{12} + 51360 \nu^{10} - 106688 \nu^{8} + 162152 \nu^{6} - 98592 \nu^{4} + 44932 \nu^{2} - 4759$$$$)/1584$$ $$\beta_{10}$$ $$=$$ $$($$$$-41 \nu^{15} + 329 \nu^{13} - 2016 \nu^{11} + 4312 \nu^{9} - 6664 \nu^{7} + 4608 \nu^{5} - 2345 \nu^{3} + 665 \nu$$$$)/144$$ $$\beta_{11}$$ $$=$$ $$($$$$-47 \nu^{15} + 383 \nu^{13} - 2352 \nu^{11} + 5176 \nu^{9} - 7912 \nu^{7} + 5328 \nu^{5} - 2015 \nu^{3} + 287 \nu$$$$)/48$$ $$\beta_{12}$$ $$=$$ $$\nu^{15} - 8 \nu^{13} + 49 \nu^{11} - 104 \nu^{9} + 160 \nu^{7} - 104 \nu^{5} + 49 \nu^{3} - 9 \nu$$ $$\beta_{13}$$ $$=$$ $$($$$$-183 \nu^{15} + 1449 \nu^{13} - 8832 \nu^{11} + 18184 \nu^{9} - 27048 \nu^{7} + 15456 \nu^{5} - 5943 \nu^{3} + 345 \nu$$$$)/176$$ $$\beta_{14}$$ $$=$$ $$($$$$-689 \nu^{15} + 5519 \nu^{13} - 33824 \nu^{11} + 72040 \nu^{9} - 111176 \nu^{7} + 72656 \nu^{5} - 33377 \nu^{3} + 3711 \nu$$$$)/528$$ $$\beta_{15}$$ $$=$$ $$($$$$97 \nu^{15} - 745 \nu^{13} + 4512 \nu^{11} - 8624 \nu^{9} + 12632 \nu^{7} - 5808 \nu^{5} + 2569 \nu^{3} + 263 \nu$$$$)/72$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} - 2 \beta_{12} - 2 \beta_{10} - \beta_{5}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{9} + 2 \beta_{8} - 2 \beta_{7} - 4 \beta_{6} + \beta_{2} + \beta_{1} + 2$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{15} + 3 \beta_{14} + 4 \beta_{13} - 2 \beta_{11} - 2 \beta_{10} - \beta_{5} - 3 \beta_{3}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$18 \beta_{9} + 20 \beta_{8} + 12 \beta_{7} - 22 \beta_{6} - 6 \beta_{4} - 7 \beta_{2} - 5 \beta_{1} - 28$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$21 \beta_{15} + 12 \beta_{14} + 42 \beta_{13} + 28 \beta_{12} - 14 \beta_{11} + 36 \beta_{10} + 21 \beta_{5} - 54 \beta_{3}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-13 \beta_{9} + 24 \beta_{8} + 68 \beta_{7} - 24 \beta_{4} - 21 \beta_{2} - 26 \beta_{1} - 88$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-115 \beta_{15} - 108 \beta_{14} - 42 \beta_{13} + 148 \beta_{12} + 82 \beta_{11} + 308 \beta_{10} + 157 \beta_{5} - 138 \beta_{3}$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$-718 \beta_{9} - 388 \beta_{8} + 380 \beta_{7} + 662 \beta_{6} - 90 \beta_{4} + 49 \beta_{2} - 141 \beta_{1} - 148$$$$)/4$$ $$\nu^{9}$$ $$=$$ $$($$$$-636 \beta_{15} - 459 \beta_{14} - 788 \beta_{13} + 462 \beta_{11} + 318 \beta_{10} + 87 \beta_{5} + 459 \beta_{3}$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-3212 \beta_{9} - 3682 \beta_{8} - 2114 \beta_{7} + 3668 \beta_{6} + 1008 \beta_{4} + 1625 \beta_{2} + 777 \beta_{1} + 4478$$$$)/4$$ $$\nu^{11}$$ $$=$$ $$($$$$-3527 \beta_{15} - 1728 \beta_{14} - 7440 \beta_{13} - 4478 \beta_{12} + 2576 \beta_{11} - 5858 \beta_{10} - 3913 \beta_{5} + 9456 \beta_{3}$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$1076 \beta_{9} - 2112 \beta_{8} - 5872 \beta_{7} + 2112 \beta_{4} + 1860 \beta_{2} + 2152 \beta_{1} + 7327$$ $$\nu^{13}$$ $$=$$ $$($$$$19575 \beta_{15} + 18624 \beta_{14} + 7440 \beta_{13} - 24830 \beta_{12} - 14320 \beta_{11} - 52062 \beta_{10} - 27015 \beta_{5} + 24336 \beta_{3}$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$122988 \beta_{9} + 66782 \beta_{8} - 65214 \beta_{7} - 112972 \beta_{6} + 15888 \beta_{4} - 9065 \beta_{2} + 23879 \beta_{1} + 24830$$$$)/4$$ $$\nu^{15}$$ $$=$$ $$($$$$108668 \beta_{15} + 78213 \beta_{14} + 135436 \beta_{13} - 79534 \beta_{11} - 54334 \beta_{10} - 14567 \beta_{5} - 78213 \beta_{3}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1079.1
 2.04058 − 1.17813i 0.367543 + 0.212201i 0.367543 − 0.212201i 2.04058 + 1.17813i −0.670418 + 0.387066i −1.11871 − 0.645885i −1.11871 + 0.645885i −0.670418 − 0.387066i 1.11871 − 0.645885i 0.670418 + 0.387066i 0.670418 − 0.387066i 1.11871 + 0.645885i −0.367543 + 0.212201i −2.04058 − 1.17813i −2.04058 + 1.17813i −0.367543 − 0.212201i
−1.39033 0.258819i 0 1.86603 + 0.719687i 0.732051i 0 0 −2.40812 1.48356i 0 −0.189469 + 1.01779i
1079.2 −1.39033 0.258819i 0 1.86603 + 0.719687i 0.732051i 0 0 −2.40812 1.48356i 0 0.189469 1.01779i
1079.3 −1.39033 + 0.258819i 0 1.86603 0.719687i 0.732051i 0 0 −2.40812 + 1.48356i 0 0.189469 + 1.01779i
1079.4 −1.39033 + 0.258819i 0 1.86603 0.719687i 0.732051i 0 0 −2.40812 + 1.48356i 0 −0.189469 1.01779i
1079.5 −1.03295 0.965926i 0 0.133975 + 1.99551i 2.73205i 0 0 1.78912 2.19067i 0 −2.63896 + 2.82207i
1079.6 −1.03295 0.965926i 0 0.133975 + 1.99551i 2.73205i 0 0 1.78912 2.19067i 0 2.63896 2.82207i
1079.7 −1.03295 + 0.965926i 0 0.133975 1.99551i 2.73205i 0 0 1.78912 + 2.19067i 0 2.63896 + 2.82207i
1079.8 −1.03295 + 0.965926i 0 0.133975 1.99551i 2.73205i 0 0 1.78912 + 2.19067i 0 −2.63896 2.82207i
1079.9 1.03295 0.965926i 0 0.133975 1.99551i 2.73205i 0 0 −1.78912 2.19067i 0 −2.63896 2.82207i
1079.10 1.03295 0.965926i 0 0.133975 1.99551i 2.73205i 0 0 −1.78912 2.19067i 0 2.63896 + 2.82207i
1079.11 1.03295 + 0.965926i 0 0.133975 + 1.99551i 2.73205i 0 0 −1.78912 + 2.19067i 0 2.63896 2.82207i
1079.12 1.03295 + 0.965926i 0 0.133975 + 1.99551i 2.73205i 0 0 −1.78912 + 2.19067i 0 −2.63896 + 2.82207i
1079.13 1.39033 0.258819i 0 1.86603 0.719687i 0.732051i 0 0 2.40812 1.48356i 0 −0.189469 1.01779i
1079.14 1.39033 0.258819i 0 1.86603 0.719687i 0.732051i 0 0 2.40812 1.48356i 0 0.189469 + 1.01779i
1079.15 1.39033 + 0.258819i 0 1.86603 + 0.719687i 0.732051i 0 0 2.40812 + 1.48356i 0 0.189469 1.01779i
1079.16 1.39033 + 0.258819i 0 1.86603 + 0.719687i 0.732051i 0 0 2.40812 + 1.48356i 0 −0.189469 + 1.01779i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1079.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 2 1 inner
28.d even 2 1 inner
84.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.e.j 16
3.b odd 2 1 inner 1764.2.e.j 16
4.b odd 2 1 inner 1764.2.e.j 16
7.b odd 2 1 inner 1764.2.e.j 16
12.b even 2 1 inner 1764.2.e.j 16
21.c even 2 1 inner 1764.2.e.j 16
28.d even 2 1 inner 1764.2.e.j 16
84.h odd 2 1 inner 1764.2.e.j 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.e.j 16 1.a even 1 1 trivial
1764.2.e.j 16 3.b odd 2 1 inner
1764.2.e.j 16 4.b odd 2 1 inner
1764.2.e.j 16 7.b odd 2 1 inner
1764.2.e.j 16 12.b even 2 1 inner
1764.2.e.j 16 21.c even 2 1 inner
1764.2.e.j 16 28.d even 2 1 inner
1764.2.e.j 16 84.h odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1764, [\chi])$$:

 $$T_{5}^{4} + 8 T_{5}^{2} + 4$$ $$T_{13}^{2} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 - 16 T^{2} + 9 T^{4} - 4 T^{6} + T^{8} )^{2}$$
$3$ $$T^{16}$$
$5$ $$( 4 + 8 T^{2} + T^{4} )^{4}$$
$7$ $$T^{16}$$
$11$ $$( 132 - 36 T^{2} + T^{4} )^{4}$$
$13$ $$( -2 + T^{2} )^{8}$$
$17$ $$( 676 + 56 T^{2} + T^{4} )^{4}$$
$19$ $$( 528 + 48 T^{2} + T^{4} )^{4}$$
$23$ $$( 132 - 36 T^{2} + T^{4} )^{4}$$
$29$ $$( 4 + 28 T^{2} + T^{4} )^{4}$$
$31$ $$( 4752 + 144 T^{2} + T^{4} )^{4}$$
$37$ $$( -12 + T^{2} )^{8}$$
$41$ $$( 2116 + 104 T^{2} + T^{4} )^{4}$$
$43$ $$( 2112 + 96 T^{2} + T^{4} )^{4}$$
$47$ $$( 528 - 120 T^{2} + T^{4} )^{4}$$
$53$ $$( 36 + 84 T^{2} + T^{4} )^{4}$$
$59$ $$( 4752 - 216 T^{2} + T^{4} )^{4}$$
$61$ $$( 20164 - 316 T^{2} + T^{4} )^{4}$$
$67$ $$( 528 + 72 T^{2} + T^{4} )^{4}$$
$71$ $$( 15972 - 276 T^{2} + T^{4} )^{4}$$
$73$ $$( 484 - 172 T^{2} + T^{4} )^{4}$$
$79$ $$( 4752 + 216 T^{2} + T^{4} )^{4}$$
$83$ $$( 19008 - 288 T^{2} + T^{4} )^{4}$$
$89$ $$( 2116 + 104 T^{2} + T^{4} )^{4}$$
$97$ $$( 20164 - 316 T^{2} + T^{4} )^{4}$$