# Properties

 Label 4560.2.d.k 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: $$x^{12} - 4 x^{11} + 8 x^{10} + 4 x^{9} + 36 x^{8} - 128 x^{7} + 232 x^{6} + 104 x^{5} + 324 x^{4} - 784 x^{3} + 800 x^{2} - 320 x + 64$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{3} - q^{5} + \beta_{5} q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} - q^{5} + \beta_{5} q^{7} + q^{9} + ( \beta_{3} - \beta_{5} - \beta_{8} ) q^{11} + \beta_{2} q^{13} - q^{15} + ( -1 + \beta_{1} ) q^{17} + ( 1 - \beta_{3} - \beta_{4} ) q^{19} + \beta_{5} q^{21} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{23} + q^{25} + q^{27} + ( \beta_{3} - \beta_{5} + \beta_{8} - \beta_{11} ) q^{29} -\beta_{7} q^{31} + ( \beta_{3} - \beta_{5} - \beta_{8} ) q^{33} -\beta_{5} q^{35} + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{9} ) q^{37} + \beta_{2} q^{39} + ( \beta_{2} - \beta_{3} - \beta_{4} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{41} + ( \beta_{2} - \beta_{3} - \beta_{4} - \beta_{9} + \beta_{11} ) q^{43} - q^{45} + ( -\beta_{2} + \beta_{3} + \beta_{5} + \beta_{8} ) q^{47} + ( -3 + \beta_{1} + 2 \beta_{4} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} ) q^{49} + ( -1 + \beta_{1} ) q^{51} + ( \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{53} + ( -\beta_{3} + \beta_{5} + \beta_{8} ) q^{55} + ( 1 - \beta_{3} - \beta_{4} ) q^{57} + ( \beta_{1} + \beta_{7} + \beta_{10} ) q^{59} + ( \beta_{1} - \beta_{10} ) q^{61} + \beta_{5} q^{63} -\beta_{2} q^{65} + ( -1 - \beta_{6} - 2 \beta_{7} ) q^{67} + ( -\beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{69} + ( 3 + \beta_{1} + \beta_{7} - 2 \beta_{10} ) q^{71} + ( -1 - \beta_{1} - \beta_{4} - \beta_{7} + \beta_{9} ) q^{73} + q^{75} + ( 5 - 2 \beta_{1} - \beta_{4} - 2 \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} ) q^{77} + ( 2 - 2 \beta_{1} - \beta_{4} - \beta_{6} + \beta_{9} + \beta_{10} ) q^{79} + q^{81} + ( 2 \beta_{2} - 2 \beta_{3} - \beta_{4} + 3 \beta_{8} - \beta_{9} - \beta_{11} ) q^{83} + ( 1 - \beta_{1} ) q^{85} + ( \beta_{3} - \beta_{5} + \beta_{8} - \beta_{11} ) q^{87} + ( -3 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{8} + \beta_{9} ) q^{89} + ( -2 - \beta_{6} + \beta_{7} + \beta_{10} ) q^{91} -\beta_{7} q^{93} + ( -1 + \beta_{3} + \beta_{4} ) q^{95} + ( -\beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{8} + \beta_{9} ) q^{97} + ( \beta_{3} - \beta_{5} - \beta_{8} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 12q^{3} - 12q^{5} + 12q^{9} + O(q^{10})$$ $$12q + 12q^{3} - 12q^{5} + 12q^{9} - 12q^{15} - 8q^{17} + 8q^{19} + 12q^{25} + 12q^{27} - 12q^{45} - 20q^{49} - 8q^{51} + 8q^{57} + 8q^{59} - 8q^{67} + 32q^{71} - 24q^{73} + 12q^{75} + 56q^{77} + 16q^{79} + 12q^{81} + 8q^{85} - 16q^{91} - 8q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 4 x^{11} + 8 x^{10} + 4 x^{9} + 36 x^{8} - 128 x^{7} + 232 x^{6} + 104 x^{5} + 324 x^{4} - 784 x^{3} + 800 x^{2} - 320 x + 64$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2101877 \nu^{11} - 9077802 \nu^{10} + 23703524 \nu^{9} - 16487388 \nu^{8} + 110538924 \nu^{7} - 276648264 \nu^{6} + 705909736 \nu^{5} - 913655768 \nu^{4} + 2051812292 \nu^{3} - 1394812968 \nu^{2} + 473528496 \nu - 8758151872$$$$)/ 2555640224$$ $$\beta_{2}$$ $$=$$ $$($$$$-3940714 \nu^{11} + 13398997 \nu^{10} - 25107184 \nu^{9} - 24324212 \nu^{8} - 172368288 \nu^{7} + 403872146 \nu^{6} - 770790480 \nu^{5} - 647605528 \nu^{4} - 2002990488 \nu^{3} + 1256839928 \nu^{2} - 3217347232 \nu + 792458944$$$$)/ 638910056$$ $$\beta_{3}$$ $$=$$ $$($$$$13468838 \nu^{11} - 49457227 \nu^{10} + 93612964 \nu^{9} + 77899386 \nu^{8} + 520946664 \nu^{7} - 1529364412 \nu^{6} + 2700529808 \nu^{5} + 2085279108 \nu^{4} + 5438016344 \nu^{3} - 7939856812 \nu^{2} + 9480711840 \nu - 2348307072$$$$)/ 638910056$$ $$\beta_{4}$$ $$=$$ $$($$$$-27908578 \nu^{11} + 58054709 \nu^{10} - 50596168 \nu^{9} - 384771120 \nu^{8} - 1515562492 \nu^{7} + 1427235004 \nu^{6} - 1224384256 \nu^{5} - 10495968688 \nu^{4} - 22613183792 \nu^{3} - 2051473260 \nu^{2} + 4138399968 \nu - 5758911088$$$$)/ 1277820112$$ $$\beta_{5}$$ $$=$$ $$($$$$72213471 \nu^{11} - 276433066 \nu^{10} + 528675316 \nu^{9} + 398111644 \nu^{8} + 2588973828 \nu^{7} - 8645816016 \nu^{6} + 15077086984 \nu^{5} + 10709015192 \nu^{4} + 22961507788 \nu^{3} - 49363607352 \nu^{2} + 44433332048 \nu - 11077367072$$$$)/ 2555640224$$ $$\beta_{6}$$ $$=$$ $$($$$$-19601935 \nu^{11} + 59711704 \nu^{10} - 92253612 \nu^{9} - 182811978 \nu^{8} - 863678932 \nu^{7} + 1785177452 \nu^{6} - 2446008600 \nu^{5} - 4875049852 \nu^{4} - 10479489340 \nu^{3} + 7930481560 \nu^{2} - 2838067024 \nu - 944943624$$$$)/ 638910056$$ $$\beta_{7}$$ $$=$$ $$($$$$95220313 \nu^{11} - 282494178 \nu^{10} + 403947252 \nu^{9} + 1041253324 \nu^{8} + 4059889564 \nu^{7} - 8435149128 \nu^{6} + 11008990088 \nu^{5} + 28520993784 \nu^{4} + 48690475668 \nu^{3} - 37124235336 \nu^{2} + 13331224688 \nu + 17856922464$$$$)/ 2555640224$$ $$\beta_{8}$$ $$=$$ $$($$$$101159245 \nu^{11} - 373781562 \nu^{10} + 708970492 \nu^{9} + 580599556 \nu^{8} + 3871197868 \nu^{7} - 11577045064 \nu^{6} + 20456241944 \nu^{5} + 15554634344 \nu^{4} + 39327243748 \nu^{3} - 63031865576 \nu^{2} + 69664703472 \nu - 17273372000$$$$)/ 2555640224$$ $$\beta_{9}$$ $$=$$ $$($$$$75048949 \nu^{11} - 253862169 \nu^{10} + 423270284 \nu^{9} + 625404004 \nu^{8} + 2978874760 \nu^{7} - 7896232908 \nu^{6} + 11475993800 \nu^{5} + 17019026280 \nu^{4} + 31981959420 \nu^{3} - 43407293348 \nu^{2} + 18945108080 \nu - 68354224$$$$)/ 1277820112$$ $$\beta_{10}$$ $$=$$ $$($$$$-155955553 \nu^{11} + 464790034 \nu^{10} - 676865564 \nu^{9} - 1628227252 \nu^{8} - 6736017884 \nu^{7} + 13878896504 \nu^{6} - 18054944664 \nu^{5} - 43724741864 \nu^{4} - 80129584308 \nu^{3} + 61113087112 \nu^{2} - 21947888880 \nu - 9750878144$$$$)/ 2555640224$$ $$\beta_{11}$$ $$=$$ $$($$$$-7567 \nu^{11} + 29316 \nu^{10} - 55332 \nu^{9} - 41748 \nu^{8} - 272116 \nu^{7} + 956992 \nu^{6} - 1582408 \nu^{5} - 1122984 \nu^{4} - 2406636 \nu^{3} + 6211120 \nu^{2} - 4658896 \nu + 1161504$$$$)/123664$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{10} - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_{2} - \beta_{1} + 3$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$-3 \beta_{8} + \beta_{5} + 4 \beta_{3} - \beta_{2}$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$6 \beta_{10} - 3 \beta_{9} - 6 \beta_{8} + 9 \beta_{7} - 9 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} + 16 \beta_{3} + 8 \beta_{2} + 3 \beta_{1} - 30$$$$)/6$$ $$\nu^{4}$$ $$=$$ $$($$$$10 \beta_{10} - 3 \beta_{9} + 10 \beta_{7} - 16 \beta_{6} + 3 \beta_{4} - 14 \beta_{1} - 90$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$3 \beta_{11} + 22 \beta_{10} - 12 \beta_{9} + 36 \beta_{8} + 37 \beta_{7} - 46 \beta_{6} - 10 \beta_{5} + 24 \beta_{4} - 67 \beta_{3} - 38 \beta_{2} + \beta_{1} - 159$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$($$$$36 \beta_{11} + 48 \beta_{9} + 210 \beta_{8} - 94 \beta_{5} + 48 \beta_{4} - 280 \beta_{3} - 110 \beta_{2}$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$54 \beta_{11} - 190 \beta_{10} + 291 \beta_{9} + 390 \beta_{8} - 325 \beta_{7} + 481 \beta_{6} - 194 \beta_{5} - 87 \beta_{4} - 608 \beta_{3} - 388 \beta_{2} + 65 \beta_{1} + 1638$$$$)/3$$ $$\nu^{8}$$ $$=$$ $$($$$$-836 \beta_{10} + 702 \beta_{9} - 1160 \beta_{7} + 2120 \beta_{6} - 702 \beta_{4} + 832 \beta_{1} + 7776$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$-702 \beta_{11} - 1800 \beta_{10} + 672 \beta_{9} - 4116 \beta_{8} - 3054 \beta_{7} + 5040 \beta_{6} + 2440 \beta_{5} - 3240 \beta_{4} + 5854 \beta_{3} + 4040 \beta_{2} + 1062 \beta_{1} + 16854$$$$)/3$$ $$\nu^{10}$$ $$=$$ $$($$$$-3912 \beta_{11} - 6480 \beta_{9} - 19788 \beta_{8} + 12052 \beta_{5} - 6480 \beta_{4} + 25984 \beta_{3} + 17084 \beta_{2}$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-8088 \beta_{11} + 17876 \beta_{10} - 34854 \beta_{9} - 43068 \beta_{8} + 30074 \beta_{7} - 52730 \beta_{6} + 27476 \beta_{5} + 5718 \beta_{4} + 58448 \beta_{3} + 42232 \beta_{2} - 12994 \beta_{1} - 174132$$$$)/3$$

## 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
 −1.15765 + 1.15765i 2.28268 − 2.28268i 1.62658 + 1.62658i 0.280289 − 0.280289i −1.56297 + 1.56297i 0.531065 − 0.531065i 0.531065 + 0.531065i −1.56297 − 1.56297i 0.280289 + 0.280289i 1.62658 − 1.62658i 2.28268 + 2.28268i −1.15765 − 1.15765i
0 1.00000 0 −1.00000 0 5.16275i 0 1.00000 0
2431.2 0 1.00000 0 −1.00000 0 3.67104i 0 1.00000 0
2431.3 0 1.00000 0 −1.00000 0 2.44391i 0 1.00000 0
2431.4 0 1.00000 0 −1.00000 0 2.20519i 0 1.00000 0
2431.5 0 1.00000 0 −1.00000 0 0.855829i 0 1.00000 0
2431.6 0 1.00000 0 −1.00000 0 0.549104i 0 1.00000 0
2431.7 0 1.00000 0 −1.00000 0 0.549104i 0 1.00000 0
2431.8 0 1.00000 0 −1.00000 0 0.855829i 0 1.00000 0
2431.9 0 1.00000 0 −1.00000 0 2.20519i 0 1.00000 0
2431.10 0 1.00000 0 −1.00000 0 2.44391i 0 1.00000 0
2431.11 0 1.00000 0 −1.00000 0 3.67104i 0 1.00000 0
2431.12 0 1.00000 0 −1.00000 0 5.16275i 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.k yes 12
4.b odd 2 1 4560.2.d.i 12
19.b odd 2 1 4560.2.d.i 12
76.d even 2 1 inner 4560.2.d.k yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4560.2.d.i 12 4.b odd 2 1
4560.2.d.i 12 19.b odd 2 1
4560.2.d.k yes 12 1.a even 1 1 trivial
4560.2.d.k 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 $$S_{2}^{\mathrm{new}}(4560, [\chi])$$:

 $$T_{7}^{12} + 52 T_{7}^{10} + 876 T_{7}^{8} + 5920 T_{7}^{6} + 15844 T_{7}^{4} + 11904 T_{7}^{2} + 2304$$ $$T_{31}^{6} - 48 T_{31}^{4} + 88 T_{31}^{3} + 276 T_{31}^{2} - 576 T_{31} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$( -1 + T )^{12}$$
$5$ $$( 1 + T )^{12}$$
$7$ $$2304 + 11904 T^{2} + 15844 T^{4} + 5920 T^{6} + 876 T^{8} + 52 T^{10} + T^{12}$$
$11$ $$3968064 + 2706144 T^{2} + 660100 T^{4} + 71776 T^{6} + 3852 T^{8} + 100 T^{10} + T^{12}$$
$13$ $$82944 + 146304 T^{2} + 85828 T^{4} + 19456 T^{6} + 1884 T^{8} + 76 T^{10} + T^{12}$$
$17$ $$( -288 + 240 T + 148 T^{2} - 80 T^{3} - 24 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$19$ $$47045881 - 19808792 T + 7558618 T^{2} - 1262056 T^{3} + 123823 T^{4} + 53200 T^{5} - 13588 T^{6} + 2800 T^{7} + 343 T^{8} - 184 T^{9} + 58 T^{10} - 8 T^{11} + T^{12}$$
$23$ $$39538944 + 19243392 T^{2} + 3517456 T^{4} + 296512 T^{6} + 11448 T^{8} + 184 T^{10} + T^{12}$$
$29$ $$418120704 + 140529024 T^{2} + 16469092 T^{4} + 865840 T^{6} + 21564 T^{8} + 244 T^{10} + T^{12}$$
$31$ $$( 16 - 576 T + 276 T^{2} + 88 T^{3} - 48 T^{4} + T^{6} )^{2}$$
$37$ $$2985984 + 2999808 T^{2} + 847300 T^{4} + 100048 T^{6} + 5388 T^{8} + 124 T^{10} + T^{12}$$
$41$ $$26132544 + 56874528 T^{2} + 10090692 T^{4} + 681984 T^{6} + 20460 T^{8} + 252 T^{10} + T^{12}$$
$43$ $$1272384 + 1899744 T^{2} + 662308 T^{4} + 89632 T^{6} + 5244 T^{8} + 124 T^{10} + T^{12}$$
$47$ $$1285652736 + 495220608 T^{2} + 50776720 T^{4} + 2116096 T^{6} + 39384 T^{8} + 328 T^{10} + T^{12}$$
$53$ $$20846739456 + 4036337664 T^{2} + 249497872 T^{4} + 6804928 T^{6} + 87432 T^{8} + 496 T^{10} + T^{12}$$
$59$ $$( -1152 - 3072 T - 800 T^{2} + 1040 T^{3} - 156 T^{4} - 4 T^{5} + T^{6} )^{2}$$
$61$ $$( 1744 - 2544 T + 564 T^{2} + 304 T^{3} - 96 T^{4} + T^{6} )^{2}$$
$67$ $$( 175488 + 94848 T + 11064 T^{2} - 1512 T^{3} - 272 T^{4} + 4 T^{5} + T^{6} )^{2}$$
$71$ $$( 497664 - 172800 T - 104 T^{2} + 4424 T^{3} - 240 T^{4} - 16 T^{5} + T^{6} )^{2}$$
$73$ $$( -32 + 408 T^{2} - 296 T^{3} - 12 T^{4} + 12 T^{5} + T^{6} )^{2}$$
$79$ $$( -8192 - 14336 T + 6656 T^{2} + 1600 T^{3} - 220 T^{4} - 8 T^{5} + T^{6} )^{2}$$
$83$ $$103859241984 + 32300176896 T^{2} + 1321989264 T^{4} + 21839616 T^{6} + 174504 T^{8} + 672 T^{10} + T^{12}$$
$89$ $$627264 + 598934304 T^{2} + 137946276 T^{4} + 7732080 T^{6} + 112908 T^{8} + 588 T^{10} + T^{12}$$
$97$ $$1214383104 + 3532443264 T^{2} + 327848292 T^{4} + 10174896 T^{6} + 119964 T^{8} + 588 T^{10} + T^{12}$$