# Properties

 Label 4560.2.d.l Level $4560$ Weight $2$ Character orbit 4560.d Analytic conductor $36.412$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4560.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$36.4117833217$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 35 x^{10} + 202 x^{8} + 362 x^{6} + 245 x^{4} + 63 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{11}\cdot 3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + \beta_{2} q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} + q^{5} + \beta_{2} q^{7} + q^{9} + \beta_{6} q^{11} -\beta_{4} q^{13} + q^{15} -\beta_{5} q^{17} + \beta_{8} q^{19} + \beta_{2} q^{21} + ( \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{23} + q^{25} + q^{27} + ( -\beta_{6} + \beta_{7} ) q^{29} + ( -1 + \beta_{3} ) q^{31} + \beta_{6} q^{33} + \beta_{2} q^{35} + ( -\beta_{6} + \beta_{7} - \beta_{10} ) q^{37} -\beta_{4} q^{39} + ( -\beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{41} + ( -\beta_{6} + \beta_{10} ) q^{43} + q^{45} + ( -\beta_{4} + \beta_{6} ) q^{47} + ( -2 + \beta_{5} - \beta_{11} ) q^{49} -\beta_{5} q^{51} + ( \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{53} + \beta_{6} q^{55} + \beta_{8} q^{57} + ( 4 + \beta_{3} - \beta_{5} + \beta_{11} ) q^{59} + ( -5 - \beta_{5} + \beta_{11} ) q^{61} + \beta_{2} q^{63} -\beta_{4} q^{65} + ( -3 + \beta_{1} + \beta_{3} ) q^{67} + ( \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{69} + ( 1 + \beta_{3} + \beta_{5} ) q^{71} + ( 3 + \beta_{3} + \beta_{5} - \beta_{8} + \beta_{9} ) q^{73} + q^{75} + ( 2 + \beta_{3} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{77} + ( -2 - \beta_{1} + \beta_{3} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{79} + q^{81} + ( -\beta_{2} + \beta_{4} - \beta_{10} ) q^{83} -\beta_{5} q^{85} + ( -\beta_{6} + \beta_{7} ) q^{87} + ( -\beta_{2} + \beta_{7} + \beta_{10} ) q^{89} + ( 3 + \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + \beta_{11} ) q^{91} + ( -1 + \beta_{3} ) q^{93} + \beta_{8} q^{95} + ( \beta_{6} - \beta_{7} - \beta_{10} ) q^{97} + \beta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 12q^{3} + 12q^{5} + 12q^{9} + O(q^{10})$$ $$12q + 12q^{3} + 12q^{5} + 12q^{9} + 12q^{15} + 4q^{17} + 12q^{25} + 12q^{27} - 12q^{31} + 12q^{45} - 28q^{49} + 4q^{51} + 52q^{59} - 56q^{61} - 32q^{67} + 8q^{71} + 32q^{73} + 12q^{75} + 24q^{77} - 28q^{79} + 12q^{81} + 4q^{85} + 32q^{91} - 12q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 35 x^{10} + 202 x^{8} + 362 x^{6} + 245 x^{4} + 63 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-27 \nu^{10} - 946 \nu^{8} - 5482 \nu^{6} - 9742 \nu^{4} - 5871 \nu^{2} - 580$$$$)/76$$ $$\beta_{2}$$ $$=$$ $$($$$$-110 \nu^{11} - 3783 \nu^{9} - 19907 \nu^{7} - 27391 \nu^{5} - 8645 \nu^{3} + 556 \nu$$$$)/76$$ $$\beta_{3}$$ $$=$$ $$($$$$27 \nu^{10} + 927 \nu^{8} + 4836 \nu^{6} + 6550 \nu^{4} + 2242 \nu^{2} + 105$$$$)/19$$ $$\beta_{4}$$ $$=$$ $$($$$$-121 \nu^{11} - 4205 \nu^{9} - 23393 \nu^{7} - 37789 \nu^{5} - 19380 \nu^{3} - 2208 \nu$$$$)/76$$ $$\beta_{5}$$ $$=$$ $$($$$$-199 \nu^{10} - 6864 \nu^{8} - 36724 \nu^{6} - 53724 \nu^{4} - 22933 \nu^{2} - 2220$$$$)/76$$ $$\beta_{6}$$ $$=$$ $$($$$$245 \nu^{11} + 8437 \nu^{9} + 44733 \nu^{7} + 63333 \nu^{5} + 23560 \nu^{3} + 1052 \nu$$$$)/76$$ $$\beta_{7}$$ $$=$$ $$($$$$-148 \nu^{11} - 5113 \nu^{9} - 27583 \nu^{7} - 41147 \nu^{5} - 17955 \nu^{3} - 1762 \nu$$$$)/38$$ $$\beta_{8}$$ $$=$$ $$($$$$-277 \nu^{11} - 116 \nu^{10} - 9542 \nu^{9} - 4008 \nu^{8} - 50682 \nu^{7} - 21634 \nu^{6} - 72224 \nu^{5} - 32256 \nu^{4} - 27569 \nu^{3} - 13946 \nu^{2} - 1358 \nu - 1304$$$$)/76$$ $$\beta_{9}$$ $$=$$ $$($$$$-277 \nu^{11} + 116 \nu^{10} - 9542 \nu^{9} + 4008 \nu^{8} - 50682 \nu^{7} + 21634 \nu^{6} - 72224 \nu^{5} + 32256 \nu^{4} - 27569 \nu^{3} + 13946 \nu^{2} - 1358 \nu + 1304$$$$)/76$$ $$\beta_{10}$$ $$=$$ $$($$$$-479 \nu^{11} - 16566 \nu^{9} - 89894 \nu^{7} - 136674 \nu^{5} - 63631 \nu^{3} - 7396 \nu$$$$)/76$$ $$\beta_{11}$$ $$=$$ $$($$$$-581 \nu^{10} - 20030 \nu^{8} - 106846 \nu^{6} - 154190 \nu^{4} - 61085 \nu^{2} - 4056$$$$)/76$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{10} + 2 \beta_{9} + 2 \beta_{8} - \beta_{7} + \beta_{6} + 2 \beta_{4} - 3 \beta_{2}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{11} + 3 \beta_{9} - 3 \beta_{8} - 2 \beta_{5} - 3 \beta_{3} + 7 \beta_{1} - 38$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$29 \beta_{10} - 40 \beta_{9} - 40 \beta_{8} - 13 \beta_{7} - 35 \beta_{6} - 28 \beta_{4} + 63 \beta_{2}$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$-13 \beta_{11} - 57 \beta_{9} + 57 \beta_{8} + 62 \beta_{5} + 117 \beta_{3} - 199 \beta_{1} + 908$$$$)/6$$ $$\nu^{5}$$ $$=$$ $$($$$$-811 \beta_{10} + 1112 \beta_{9} + 1112 \beta_{8} + 449 \beta_{7} + 1075 \beta_{6} + 722 \beta_{4} - 1677 \beta_{2}$$$$)/6$$ $$\nu^{6}$$ $$=$$ $$($$$$103 \beta_{11} + 491 \beta_{9} - 491 \beta_{8} - 606 \beta_{5} - 1149 \beta_{3} + 1873 \beta_{1} - 8410$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$22889 \beta_{10} - 31438 \beta_{9} - 31438 \beta_{8} - 12955 \beta_{7} - 30827 \beta_{6} - 20164 \beta_{4} + 47043 \beta_{2}$$$$)/6$$ $$\nu^{8}$$ $$=$$ $$($$$$-8513 \beta_{11} - 41079 \beta_{9} + 41079 \beta_{8} + 51778 \beta_{5} + 98109 \beta_{3} - 158975 \beta_{1} + 712384$$$$)/6$$ $$\nu^{9}$$ $$=$$ $$($$$$-647551 \beta_{10} + 889742 \beta_{9} + 889742 \beta_{8} + 367595 \beta_{7} + 874255 \beta_{6} + 569606 \beta_{4} - 1329801 \beta_{2}$$$$)/6$$ $$\nu^{10}$$ $$=$$ $$($$$$240005 \beta_{11} + 1160127 \beta_{9} - 1160127 \beta_{8} - 1466962 \beta_{5} - 2779143 \beta_{3} + 4499411 \beta_{1} - 20156686$$$$)/6$$ $$\nu^{11}$$ $$=$$ $$($$$$18327245 \beta_{10} - 25183360 \beta_{9} - 25183360 \beta_{8} - 10408213 \beta_{7} - 24752507 \beta_{6} - 16117708 \beta_{4} + 37632183 \beta_{2}$$$$)/6$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times$$.

 $$n$$ $$1141$$ $$1711$$ $$1921$$ $$2737$$ $$3041$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2431.1
 − 0.303940i − 0.687758i 5.32019i 2.03924i 1.19434i − 0.738386i 0.738386i − 1.19434i − 2.03924i − 5.32019i 0.687758i 0.303940i
0 1.00000 0 1.00000 0 4.54425i 0 1.00000 0
2431.2 0 1.00000 0 1.00000 0 3.95229i 0 1.00000 0
2431.3 0 1.00000 0 1.00000 0 3.21222i 0 1.00000 0
2431.4 0 1.00000 0 1.00000 0 2.79053i 0 1.00000 0
2431.5 0 1.00000 0 1.00000 0 1.25182i 0 1.00000 0
2431.6 0 1.00000 0 1.00000 0 0.238175i 0 1.00000 0
2431.7 0 1.00000 0 1.00000 0 0.238175i 0 1.00000 0
2431.8 0 1.00000 0 1.00000 0 1.25182i 0 1.00000 0
2431.9 0 1.00000 0 1.00000 0 2.79053i 0 1.00000 0
2431.10 0 1.00000 0 1.00000 0 3.21222i 0 1.00000 0
2431.11 0 1.00000 0 1.00000 0 3.95229i 0 1.00000 0
2431.12 0 1.00000 0 1.00000 0 4.54425i 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2431.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4560.2.d.l yes 12
4.b odd 2 1 4560.2.d.j 12
19.b odd 2 1 4560.2.d.j 12
76.d even 2 1 inner 4560.2.d.l yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4560.2.d.j 12 4.b odd 2 1
4560.2.d.j 12 19.b odd 2 1
4560.2.d.l yes 12 1.a even 1 1 trivial
4560.2.d.l yes 12 76.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4560, [\chi])$$:

 $$T_{7}^{12} + 56 T_{7}^{10} + 1148 T_{7}^{8} + 10480 T_{7}^{6} + 40228 T_{7}^{4} + 42864 T_{7}^{2} + 2304$$ $$T_{31}^{6} + 6 T_{31}^{5} - 52 T_{31}^{4} - 256 T_{31}^{3} + 724 T_{31}^{2} + 2168 T_{31} - 3392$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$( -1 + T )^{12}$$
$5$ $$( -1 + T )^{12}$$
$7$ $$2304 + 42864 T^{2} + 40228 T^{4} + 10480 T^{6} + 1148 T^{8} + 56 T^{10} + T^{12}$$
$11$ $$46656 + 146016 T^{2} + 78148 T^{4} + 16720 T^{6} + 1628 T^{8} + 68 T^{10} + T^{12}$$
$13$ $$5308416 + 3923136 T^{2} + 915844 T^{4} + 95872 T^{6} + 4892 T^{8} + 116 T^{10} + T^{12}$$
$17$ $$( -2592 - 504 T + 820 T^{2} + 104 T^{3} - 64 T^{4} - 2 T^{5} + T^{6} )^{2}$$
$19$ $$47045881 + 3388346 T^{2} - 438976 T^{3} + 170031 T^{4} - 44992 T^{5} + 4652 T^{6} - 2368 T^{7} + 471 T^{8} - 64 T^{9} + 26 T^{10} + T^{12}$$
$23$ $$7485696 + 37862784 T^{2} + 7555984 T^{4} + 499072 T^{6} + 14744 T^{8} + 200 T^{10} + T^{12}$$
$29$ $$36864 + 102336 T^{2} + 94180 T^{4} + 32128 T^{6} + 3596 T^{8} + 116 T^{10} + T^{12}$$
$31$ $$( -3392 + 2168 T + 724 T^{2} - 256 T^{3} - 52 T^{4} + 6 T^{5} + T^{6} )^{2}$$
$37$ $$2657608704 + 1107646848 T^{2} + 92838340 T^{4} + 3093712 T^{6} + 48332 T^{8} + 356 T^{10} + T^{12}$$
$41$ $$50466816 + 35407536 T^{2} + 7112260 T^{4} + 503264 T^{6} + 15452 T^{8} + 208 T^{10} + T^{12}$$
$43$ $$7043909184 + 2716191456 T^{2} + 174292708 T^{4} + 4700032 T^{6} + 62588 T^{8} + 404 T^{10} + T^{12}$$
$47$ $$20736 + 316800 T^{2} + 1177360 T^{4} + 200000 T^{6} + 10616 T^{8} + 184 T^{10} + T^{12}$$
$53$ $$603979776 + 209750784 T^{2} + 24686224 T^{4} + 1204480 T^{6} + 26984 T^{8} + 272 T^{10} + T^{12}$$
$59$ $$( 36864 + 192 T - 17120 T^{2} + 2560 T^{3} + 72 T^{4} - 26 T^{5} + T^{6} )^{2}$$
$61$ $$( 42736 - 7808 T - 10284 T^{2} - 1072 T^{3} + 168 T^{4} + 28 T^{5} + T^{6} )^{2}$$
$67$ $$( 205696 + 75712 T - 1512 T^{2} - 2440 T^{3} - 120 T^{4} + 16 T^{5} + T^{6} )^{2}$$
$71$ $$( 9216 - 9408 T + 1624 T^{2} + 440 T^{3} - 96 T^{4} - 4 T^{5} + T^{6} )^{2}$$
$73$ $$( -10784 + 24960 T - 12616 T^{2} + 2072 T^{3} - 44 T^{4} - 16 T^{5} + T^{6} )^{2}$$
$79$ $$( -622592 + 263168 T + 21504 T^{2} - 4736 T^{3} - 336 T^{4} + 14 T^{5} + T^{6} )^{2}$$
$83$ $$8617780224 + 1827805632 T^{2} + 127443088 T^{4} + 3821920 T^{6} + 55208 T^{8} + 380 T^{10} + T^{12}$$
$89$ $$1201038336 + 367093680 T^{2} + 40092004 T^{4} + 2009296 T^{6} + 47516 T^{8} + 464 T^{10} + T^{12}$$
$97$ $$7481558016 + 3424592256 T^{2} + 293339044 T^{4} + 7933936 T^{6} + 93308 T^{8} + 500 T^{10} + T^{12}$$