# Properties

 Label 560.2.x.a Level 560 Weight 2 Character orbit 560.x Analytic conductor 4.472 Analytic rank 0 Dimension 12 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 560.x (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: 12.0.426337261060096.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{4} - \beta_{8} ) q^{3} + ( -\beta_{6} + \beta_{9} ) q^{5} + \beta_{1} q^{7} + ( -\beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{9} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{4} - \beta_{8} ) q^{3} + ( -\beta_{6} + \beta_{9} ) q^{5} + \beta_{1} q^{7} + ( -\beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{9} ) q^{9} + ( \beta_{1} + \beta_{4} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{11} + ( 2 + \beta_{2} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{13} + ( -3 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{8} - \beta_{10} - 2 \beta_{11} ) q^{15} + ( \beta_{6} - \beta_{7} + \beta_{9} ) q^{17} + ( -\beta_{1} - 4 \beta_{3} + \beta_{4} + \beta_{8} - \beta_{10} ) q^{19} + ( 1 + \beta_{7} ) q^{21} + ( 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{8} - 2 \beta_{11} ) q^{23} + ( -2 + \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{9} ) q^{25} + ( -\beta_{1} - 2 \beta_{3} - \beta_{10} - 2 \beta_{11} ) q^{27} + ( 3 \beta_{2} - \beta_{5} + \beta_{6} - 3 \beta_{9} ) q^{29} + ( 3 \beta_{1} + 3 \beta_{4} - \beta_{8} - \beta_{10} - 2 \beta_{11} ) q^{31} + ( -2 - 3 \beta_{2} + 2 \beta_{5} ) q^{33} + ( \beta_{3} - \beta_{8} ) q^{35} + ( -4 - 4 \beta_{5} + \beta_{6} - \beta_{7} ) q^{37} + ( 5 \beta_{1} + 3 \beta_{3} - 5 \beta_{4} - 3 \beta_{8} + 3 \beta_{10} ) q^{39} + ( -2 - \beta_{2} + 2 \beta_{7} - \beta_{9} ) q^{41} + ( -3 \beta_{3} - 2 \beta_{4} + 2 \beta_{8} + 3 \beta_{11} ) q^{43} + ( -4 - \beta_{2} + 2 \beta_{5} - 3 \beta_{7} - \beta_{9} ) q^{45} + ( -9 \beta_{1} - 2 \beta_{3} - \beta_{10} - 2 \beta_{11} ) q^{47} -\beta_{5} q^{49} + ( 5 \beta_{1} + 5 \beta_{4} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{51} + ( 3 + 2 \beta_{2} - 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{53} + ( -4 \beta_{1} + 3 \beta_{4} - \beta_{8} + 2 \beta_{11} ) q^{55} + ( -1 - \beta_{5} + 5 \beta_{6} - 5 \beta_{7} - 6 \beta_{9} ) q^{57} + ( 3 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} + \beta_{8} - \beta_{10} ) q^{59} + ( 2 \beta_{2} + 4 \beta_{7} + 2 \beta_{9} ) q^{61} + ( \beta_{3} - 2 \beta_{4} - \beta_{8} - \beta_{11} ) q^{63} + ( 4 + \beta_{2} + 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} + 3 \beta_{9} ) q^{65} + ( \beta_{3} - 4 \beta_{10} + \beta_{11} ) q^{67} + ( -4 \beta_{2} + 6 \beta_{5} - 6 \beta_{6} + 4 \beta_{9} ) q^{69} + ( 2 \beta_{1} + 2 \beta_{4} + 2 \beta_{8} + 2 \beta_{10} ) q^{71} + ( 2 \beta_{2} - \beta_{6} - \beta_{7} ) q^{73} + ( -7 \beta_{1} - 2 \beta_{3} + 4 \beta_{4} + 4 \beta_{8} + \beta_{10} + 2 \beta_{11} ) q^{75} + ( -1 - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} ) q^{77} + ( 7 \beta_{1} + \beta_{3} - 7 \beta_{4} - \beta_{8} + \beta_{10} ) q^{79} + ( 5 - 2 \beta_{7} ) q^{81} -8 \beta_{4} q^{83} + ( 5 + 3 \beta_{2} + \beta_{6} - 2 \beta_{7} - \beta_{9} ) q^{85} + ( -\beta_{1} + 4 \beta_{3} + 7 \beta_{10} + 4 \beta_{11} ) q^{87} + ( -2 \beta_{2} - 8 \beta_{5} + 2 \beta_{9} ) q^{89} + ( 2 \beta_{1} + 2 \beta_{4} + \beta_{8} + \beta_{10} + \beta_{11} ) q^{91} + ( 1 - 4 \beta_{2} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{93} + ( 3 \beta_{3} + 10 \beta_{4} + 2 \beta_{8} + 4 \beta_{10} - \beta_{11} ) q^{95} + ( 2 + 2 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} + 7 \beta_{9} ) q^{97} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{8} - 2 \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + O(q^{10})$$ $$12q + 20q^{13} + 4q^{17} + 8q^{21} - 20q^{25} - 24q^{33} - 44q^{37} - 32q^{41} - 36q^{45} + 28q^{53} + 8q^{57} - 16q^{61} + 36q^{65} + 4q^{73} - 16q^{77} + 68q^{81} + 68q^{85} + 8q^{93} + 12q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{9} - 3 x^{8} + 4 x^{7} + 8 x^{6} + 8 x^{5} - 12 x^{4} - 32 x^{3} + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{11} + \nu^{10} + 4 \nu^{9} - 2 \nu^{8} - 3 \nu^{7} + 5 \nu^{6} + 24 \nu^{5} + 2 \nu^{4} - 8 \nu^{3} - 24 \nu^{2} - 48 \nu$$$$)/160$$ $$\beta_{2}$$ $$=$$ $$($$$$-3 \nu^{11} + 16 \nu^{9} + 28 \nu^{8} - 7 \nu^{7} - 44 \nu^{6} - 72 \nu^{5} + 56 \nu^{4} + 180 \nu^{3} + 128 \nu^{2} - 128 \nu - 384$$$$)/160$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{11} + 6 \nu^{10} - 16 \nu^{9} - 12 \nu^{8} - 3 \nu^{7} + 50 \nu^{6} + 64 \nu^{5} - 48 \nu^{4} - 68 \nu^{3} - 104 \nu^{2} + 112 \nu + 320$$$$)/160$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{11} + 9 \nu^{10} + 6 \nu^{9} + 2 \nu^{8} - 17 \nu^{7} - 35 \nu^{6} + 26 \nu^{5} + 38 \nu^{4} + 28 \nu^{3} - 56 \nu^{2} - 192 \nu$$$$)/160$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{11} - \nu^{10} + 4 \nu^{8} + 5 \nu^{7} - \nu^{6} - 12 \nu^{5} - 8 \nu^{4} - 12 \nu^{3} + 36 \nu^{2} + 16 \nu + 64$$$$)/80$$ $$\beta_{6}$$ $$=$$ $$($$$$-7 \nu^{11} - 4 \nu^{10} - 20 \nu^{9} - 4 \nu^{8} + 5 \nu^{7} + 16 \nu^{6} + 52 \nu^{5} - 72 \nu^{4} - 108 \nu^{3} + 64 \nu^{2} + 144 \nu + 576$$$$)/160$$ $$\beta_{7}$$ $$=$$ $$($$$$9 \nu^{11} + 4 \nu^{10} - 4 \nu^{9} - 28 \nu^{8} - 27 \nu^{7} + 40 \nu^{6} + 36 \nu^{5} - 32 \nu^{4} - 92 \nu^{3} - 176 \nu^{2} + 48 \nu$$$$)/160$$ $$\beta_{8}$$ $$=$$ $$($$$$-6 \nu^{11} - 9 \nu^{10} - 22 \nu^{9} + 10 \nu^{8} + 54 \nu^{7} + 39 \nu^{6} - 34 \nu^{5} - 174 \nu^{4} - 88 \nu^{3} + 248 \nu^{2} + 320 \nu + 384$$$$)/160$$ $$\beta_{9}$$ $$=$$ $$($$$$-\nu^{11} - 3 \nu^{10} - \nu^{9} + 4 \nu^{8} + 7 \nu^{7} + \nu^{6} - 9 \nu^{5} - 20 \nu^{4} + 4 \nu^{3} + 20 \nu^{2} + 76 \nu + 16$$$$)/40$$ $$\beta_{10}$$ $$=$$ $$($$$$-5 \nu^{11} - 13 \nu^{10} - 8 \nu^{9} + 18 \nu^{8} + 31 \nu^{7} - \nu^{6} - 60 \nu^{5} - 82 \nu^{4} + 24 \nu^{3} + 104 \nu^{2} + 32 \nu + 64$$$$)/160$$ $$\beta_{11}$$ $$=$$ $$($$$$9 \nu^{11} + 8 \nu^{10} - 12 \nu^{8} - 35 \nu^{7} + 28 \nu^{6} + 56 \nu^{5} + 64 \nu^{4} - 84 \nu^{3} - 208 \nu^{2} - 48 \nu - 32$$$$)/160$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{10} + \beta_{9} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{10} - \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-2 \beta_{11} - \beta_{8} - 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + \beta_{2} + 2$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} - 3 \beta_{7} - 3 \beta_{4} - \beta_{2} - 3 \beta_{1} + 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{10} + \beta_{9} - 2 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} + 7 \beta_{1} - 4$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$\beta_{10} - 3 \beta_{9} - \beta_{8} - \beta_{6} + 5 \beta_{5} - 7 \beta_{4} + 5 \beta_{3} + 3 \beta_{2} + 7 \beta_{1}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-2 \beta_{11} + \beta_{8} - 8 \beta_{7} - 8 \beta_{6} - 2 \beta_{5} + \beta_{4} + 2 \beta_{3} - 9 \beta_{2} + 2$$$$)/2$$ $$\nu^{8}$$ $$=$$ $$($$$$15 \beta_{11} + 5 \beta_{10} + 5 \beta_{9} + 5 \beta_{8} - 5 \beta_{7} + 7 \beta_{4} + 5 \beta_{2} + 7 \beta_{1} - 1$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-4 \beta_{11} - 7 \beta_{10} + 7 \beta_{9} + 6 \beta_{7} - 6 \beta_{6} + 16 \beta_{5} - 4 \beta_{3} + 33 \beta_{1} + 16$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-13 \beta_{10} - \beta_{9} + 13 \beta_{8} - 15 \beta_{6} + 11 \beta_{5} + 19 \beta_{4} + 11 \beta_{3} + \beta_{2} - 19 \beta_{1}$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$22 \beta_{11} + 23 \beta_{8} - 4 \beta_{7} - 4 \beta_{6} - 50 \beta_{5} + 7 \beta_{4} - 22 \beta_{3} + \beta_{2} + 50$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$351$$ $$421$$ $$\chi(n)$$ $$1$$ $$\beta_{5}$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 1.19252 − 0.760198i 1.41127 − 0.0912546i −0.394157 − 1.35818i −1.35818 − 0.394157i −0.0912546 + 1.41127i −0.760198 + 1.19252i 1.19252 + 0.760198i 1.41127 + 0.0912546i −0.394157 + 1.35818i −1.35818 + 0.394157i −0.0912546 − 1.41127i −0.760198 − 1.19252i
0 −1.95272 1.95272i 0 0.432320 + 2.19388i 0 −0.707107 + 0.707107i 0 4.62620i 0
127.2 0 −1.50252 1.50252i 0 1.32001 1.80487i 0 0.707107 0.707107i 0 1.51514i 0
127.3 0 −0.964019 0.964019i 0 −1.75233 1.38900i 0 −0.707107 + 0.707107i 0 1.14134i 0
127.4 0 0.964019 + 0.964019i 0 −1.75233 1.38900i 0 0.707107 0.707107i 0 1.14134i 0
127.5 0 1.50252 + 1.50252i 0 1.32001 1.80487i 0 −0.707107 + 0.707107i 0 1.51514i 0
127.6 0 1.95272 + 1.95272i 0 0.432320 + 2.19388i 0 0.707107 0.707107i 0 4.62620i 0
463.1 0 −1.95272 + 1.95272i 0 0.432320 2.19388i 0 −0.707107 0.707107i 0 4.62620i 0
463.2 0 −1.50252 + 1.50252i 0 1.32001 + 1.80487i 0 0.707107 + 0.707107i 0 1.51514i 0
463.3 0 −0.964019 + 0.964019i 0 −1.75233 + 1.38900i 0 −0.707107 0.707107i 0 1.14134i 0
463.4 0 0.964019 0.964019i 0 −1.75233 + 1.38900i 0 0.707107 + 0.707107i 0 1.14134i 0
463.5 0 1.50252 1.50252i 0 1.32001 + 1.80487i 0 −0.707107 0.707107i 0 1.51514i 0
463.6 0 1.95272 1.95272i 0 0.432320 2.19388i 0 0.707107 + 0.707107i 0 4.62620i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 463.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.x.a 12
4.b odd 2 1 inner 560.2.x.a 12
5.c odd 4 1 inner 560.2.x.a 12
20.e even 4 1 inner 560.2.x.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.x.a 12 1.a even 1 1 trivial
560.2.x.a 12 4.b odd 2 1 inner
560.2.x.a 12 5.c odd 4 1 inner
560.2.x.a 12 20.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 82 T_{3}^{8} + 1457 T_{3}^{4} + 4096$$ acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 32 T^{4} + 512 T^{8} - 5390 T^{12} + 41472 T^{16} - 209952 T^{20} + 531441 T^{24}$$
$5$ $$( 1 + 5 T^{2} + 8 T^{3} + 25 T^{4} + 125 T^{6} )^{2}$$
$7$ $$( 1 + T^{4} )^{3}$$
$11$ $$( 1 - 36 T^{2} + 752 T^{4} - 9982 T^{6} + 90992 T^{8} - 527076 T^{10} + 1771561 T^{12} )^{2}$$
$13$ $$( 1 - 10 T + 50 T^{2} - 198 T^{3} + 920 T^{4} - 4382 T^{5} + 17422 T^{6} - 56966 T^{7} + 155480 T^{8} - 435006 T^{9} + 1428050 T^{10} - 3712930 T^{11} + 4826809 T^{12} )^{2}$$
$17$ $$( 1 - 2 T + 2 T^{2} + 66 T^{3} - 208 T^{4} - 1606 T^{5} + 5806 T^{6} - 27302 T^{7} - 60112 T^{8} + 324258 T^{9} + 167042 T^{10} - 2839714 T^{11} + 24137569 T^{12} )^{2}$$
$19$ $$( 1 - 10 T^{2} + 635 T^{4} - 6132 T^{6} + 229235 T^{8} - 1303210 T^{10} + 47045881 T^{12} )^{2}$$
$23$ $$1 - 346 T^{4} - 120337 T^{8} + 92134740 T^{12} - 33675226417 T^{16} - 27095600907226 T^{20} + 21914624432020321 T^{24}$$
$29$ $$( 1 - 28 T^{2} + 1856 T^{4} - 28674 T^{6} + 1560896 T^{8} - 19803868 T^{10} + 594823321 T^{12} )^{2}$$
$31$ $$( 1 - 86 T^{2} + 3875 T^{4} - 134028 T^{6} + 3723875 T^{8} - 79422806 T^{10} + 887503681 T^{12} )^{2}$$
$37$ $$( 1 + 22 T + 242 T^{2} + 2222 T^{3} + 18795 T^{4} + 131340 T^{5} + 809732 T^{6} + 4859580 T^{7} + 25730355 T^{8} + 112550966 T^{9} + 453546962 T^{10} + 1525567054 T^{11} + 2565726409 T^{12} )^{2}$$
$41$ $$( 1 + 8 T + 93 T^{2} + 672 T^{3} + 3813 T^{4} + 13448 T^{5} + 68921 T^{6} )^{4}$$
$43$ $$1 + 3854 T^{4} + 9941327 T^{8} + 22828080804 T^{12} + 33987418688927 T^{16} + 45046323869874254 T^{20} + 39959630797262576401 T^{24}$$
$47$ $$1 - 656 T^{4} + 2791888 T^{8} + 394809730 T^{12} + 13623522827728 T^{16} - 15620204050115216 T^{20} +$$$$11\!\cdots\!41$$$$T^{24}$$
$53$ $$( 1 - 14 T + 98 T^{2} - 662 T^{3} + 7191 T^{4} - 73588 T^{5} + 544636 T^{6} - 3900164 T^{7} + 20199519 T^{8} - 98556574 T^{9} + 773267138 T^{10} - 5854736902 T^{11} + 22164361129 T^{12} )^{2}$$
$59$ $$( 1 + 190 T^{2} + 15115 T^{4} + 869500 T^{6} + 52615315 T^{8} + 2302298590 T^{10} + 42180533641 T^{12} )^{2}$$
$61$ $$( 1 + 4 T + 111 T^{2} + 136 T^{3} + 6771 T^{4} + 14884 T^{5} + 226981 T^{6} )^{4}$$
$67$ $$1 + 3278 T^{4} + 61319599 T^{8} + 132161723300 T^{12} + 1235658659120479 T^{16} + 1331089847030669198 T^{20} +$$$$81\!\cdots\!61$$$$T^{24}$$
$71$ $$( 1 - 314 T^{2} + 47455 T^{4} - 4255916 T^{6} + 239220655 T^{8} - 7979267834 T^{10} + 128100283921 T^{12} )^{2}$$
$73$ $$( 1 - 2 T + 2 T^{2} + 30 T^{3} + 4275 T^{4} - 27988 T^{5} + 47876 T^{6} - 2043124 T^{7} + 22781475 T^{8} + 11670510 T^{9} + 56796482 T^{10} - 4146143186 T^{11} + 151334226289 T^{12} )^{2}$$
$79$ $$( 1 + 188 T^{2} + 26408 T^{4} + 2237334 T^{6} + 164812328 T^{8} + 7322615228 T^{10} + 243087455521 T^{12} )^{2}$$
$83$ $$( 1 - 3374 T^{4} + 47458321 T^{8} )^{3}$$
$89$ $$( 1 - 262 T^{2} + 32799 T^{4} - 3032020 T^{6} + 259800879 T^{8} - 16438467142 T^{10} + 496981290961 T^{12} )^{2}$$
$97$ $$( 1 - 6 T + 18 T^{2} - 178 T^{3} - 9328 T^{4} + 69598 T^{5} - 233842 T^{6} + 6751006 T^{7} - 87767152 T^{8} - 162455794 T^{9} + 1593527058 T^{10} - 51524041542 T^{11} + 832972004929 T^{12} )^{2}$$