# Properties

 Label 1728.4.c.j Level $1728$ Weight $4$ Character orbit 1728.c Analytic conductor $101.955$ Analytic rank $0$ Dimension $12$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1728 = 2^{6} \cdot 3^{3}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1728.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$101.955300490$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 3 x^{10} - 12 x^{9} + 73 x^{8} - 12 x^{7} + 589 x^{6} + 84 x^{5} + 2452 x^{4} + 852 x^{3} + 6854 x^{2} - 888 x + 9496$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{32}\cdot 3^{12}$$ Twist minimal: no (minimal twist has level 108) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} -\beta_{1} q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} -\beta_{1} q^{7} -\beta_{6} q^{11} + ( 6 - \beta_{5} ) q^{13} + \beta_{10} q^{17} + ( -\beta_{1} + \beta_{9} ) q^{19} + ( 2 \beta_{3} + \beta_{6} - \beta_{8} ) q^{23} + ( -32 - \beta_{4} + 2 \beta_{5} ) q^{25} + ( -3 \beta_{2} - \beta_{7} + \beta_{10} ) q^{29} + ( -4 \beta_{1} - \beta_{9} - \beta_{11} ) q^{31} + ( 5 \beta_{3} + 2 \beta_{6} - 2 \beta_{8} ) q^{35} + ( 20 - 5 \beta_{5} ) q^{37} + ( -2 \beta_{2} - 4 \beta_{7} - \beta_{10} ) q^{41} + ( 8 \beta_{1} + 2 \beta_{9} - 2 \beta_{11} ) q^{43} + ( 2 \beta_{3} - 6 \beta_{6} + 2 \beta_{8} ) q^{47} + ( 24 + \beta_{4} + 4 \beta_{5} ) q^{49} + ( 18 \beta_{2} - 5 \beta_{7} + 3 \beta_{10} ) q^{53} + ( -13 \beta_{1} + 2 \beta_{9} - 2 \beta_{11} ) q^{55} + ( 6 \beta_{6} + 8 \beta_{8} ) q^{59} + ( -12 + 2 \beta_{4} + 6 \beta_{5} ) q^{61} + ( -26 \beta_{2} + 2 \beta_{7} + 3 \beta_{10} ) q^{65} -22 \beta_{1} q^{67} + ( 10 \beta_{3} + 7 \beta_{6} + 7 \beta_{8} ) q^{71} + ( 13 + \beta_{4} - 8 \beta_{5} ) q^{73} + ( -27 \beta_{2} + 14 \beta_{7} - 4 \beta_{10} ) q^{77} + ( 29 \beta_{1} + 3 \beta_{9} + \beta_{11} ) q^{79} + ( 32 \beta_{3} - 13 \beta_{6} - 6 \beta_{8} ) q^{83} + ( 14 - 2 \beta_{4} + 11 \beta_{5} ) q^{85} + ( 64 \beta_{2} + 14 \beta_{7} - 4 \beta_{10} ) q^{89} + ( -31 \beta_{1} - 3 \beta_{9} - 2 \beta_{11} ) q^{91} + ( 50 \beta_{3} + 11 \beta_{6} - 19 \beta_{8} ) q^{95} + ( 43 + 4 \beta_{4} - 8 \beta_{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + O(q^{10})$$ $$12q + 72q^{13} - 384q^{25} + 240q^{37} + 288q^{49} - 144q^{61} + 156q^{73} + 168q^{85} + 516q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 3 x^{10} - 12 x^{9} + 73 x^{8} - 12 x^{7} + 589 x^{6} + 84 x^{5} + 2452 x^{4} + 852 x^{3} + 6854 x^{2} - 888 x + 9496$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2849350 \nu^{11} + 6585588 \nu^{10} - 7900308 \nu^{9} + 8937888 \nu^{8} - 274261062 \nu^{7} + 514828262 \nu^{6} - 1348583642 \nu^{5} + 577023966 \nu^{4} - 3610618898 \nu^{3} - 4417697498 \nu^{2} - 20418808208 \nu - 3774567143$$$$)/ 3046053151$$ $$\beta_{2}$$ $$=$$ $$($$$$454920367261 \nu^{11} + 699408466494 \nu^{10} + 909094946747 \nu^{9} - 6055960107450 \nu^{8} + 16604234843593 \nu^{7} + 44620628131914 \nu^{6} + 261918488231045 \nu^{5} + 346379732140290 \nu^{4} + 372084361441392 \nu^{3} + 4408588811940 \nu^{2} - 876761148010650 \nu - 873663409442004$$$$)/ 449256287134688$$ $$\beta_{3}$$ $$=$$ $$($$$$-2201987655 \nu^{11} + 7037103366 \nu^{10} + 321745311 \nu^{9} + 50820234894 \nu^{8} - 233951534667 \nu^{7} + 379443106530 \nu^{6} - 994923167391 \nu^{5} + 4318536234906 \nu^{4} - 450964003536 \nu^{3} + 12604127190228 \nu^{2} + 566117025390 \nu + 33640324703676$$$$)/ 1072210709152$$ $$\beta_{4}$$ $$=$$ $$($$$$-3668351532 \nu^{11} - 35323691472 \nu^{10} - 36152667432 \nu^{9} - 188138665632 \nu^{8} + 50069628876 \nu^{7} - 1565811304500 \nu^{6} - 2253785545884 \nu^{5} - 29394681445056 \nu^{4} - 43298645321028 \nu^{3} - 89923171899804 \nu^{2} - 56510573448384 \nu - 171807057509552$$$$)/ 1276296270269$$ $$\beta_{5}$$ $$=$$ $$($$$$-9151812708 \nu^{11} + 12155753232 \nu^{10} + 35817493464 \nu^{9} + 135939078600 \nu^{8} - 789050014620 \nu^{7} + 300610219236 \nu^{6} - 1310314725588 \nu^{5} + 7296602696088 \nu^{4} + 3399104212836 \nu^{3} + 28479977397228 \nu^{2} + 5390654550912 \nu + 120796778623888$$$$)/ 1276296270269$$ $$\beta_{6}$$ $$=$$ $$($$$$-3222436257455 \nu^{11} + 5388050787734 \nu^{10} + 2723330832455 \nu^{9} + 52384868035038 \nu^{8} - 299880875358483 \nu^{7} + 246749489170834 \nu^{6} - 1028976838142151 \nu^{5} + 2676763977698314 \nu^{4} - 1992011057883600 \nu^{3} + 8892862433464884 \nu^{2} - 1768324514742466 \nu + 20917972597278300$$$$)/ 449256287134688$$ $$\beta_{7}$$ $$=$$ $$($$$$-18642750728255 \nu^{11} - 54280349116842 \nu^{10} - 38930579155849 \nu^{9} + 292758720471294 \nu^{8} - 854630055113507 \nu^{7} - 3576388112829486 \nu^{6} - 11660381327420215 \nu^{5} - 16340881815026454 \nu^{4} - 49527457079318736 \nu^{3} - 51882006104080140 \nu^{2} - 134713599325199970 \nu - 45157448521252068$$$$)/ 2246281435673440$$ $$\beta_{8}$$ $$=$$ $$($$$$-4893277942309 \nu^{11} + 8469535734322 \nu^{10} - 13632883283795 \nu^{9} + 76780435438410 \nu^{8} - 443280641637105 \nu^{7} + 639207165622022 \nu^{6} - 2619966859658157 \nu^{5} + 3875655823133870 \nu^{4} - 9916828921048944 \nu^{3} + 9414663721373244 \nu^{2} - 10536224050201622 \nu + 41995323116010612$$$$)/ 449256287134688$$ $$\beta_{9}$$ $$=$$ $$($$$$82140767042 \nu^{11} + 128260169172 \nu^{10} + 66239186292 \nu^{9} - 613299928464 \nu^{8} + 4292059269954 \nu^{7} + 11981142736814 \nu^{6} + 31453542055774 \nu^{5} + 69841926412242 \nu^{4} + 128188443328414 \nu^{3} + 430011600480190 \nu^{2} + 361115104649488 \nu + 579514515914887$$$$)/ 6381481351345$$ $$\beta_{10}$$ $$=$$ $$($$$$-24096087148205 \nu^{11} - 30785315288286 \nu^{10} - 65693209087147 \nu^{9} + 322701238859802 \nu^{8} - 1328933424436121 \nu^{7} - 1800413902806378 \nu^{6} - 14997619440051925 \nu^{5} - 12281584417815522 \nu^{4} - 53711753834599728 \nu^{3} - 25304246119887780 \nu^{2} - 157744121239963590 \nu + 4917658014297876$$$$)/ 1123140717836720$$ $$\beta_{11}$$ $$=$$ $$($$$$426968239932 \nu^{11} + 457010627112 \nu^{10} + 737602108632 \nu^{9} - 4752896618904 \nu^{8} + 22258024812924 \nu^{7} + 33730962757524 \nu^{6} + 222741323689284 \nu^{5} + 267153193701852 \nu^{4} + 654755938274244 \nu^{3} + 913427570025060 \nu^{2} + 1505040883452768 \nu + 749667835341342$$$$)/ 6381481351345$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-2 \beta_{10} + 2 \beta_{7} - \beta_{5} + \beta_{4} + 8 \beta_{3} - 14 \beta_{2}$$$$)/144$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{11} + 6 \beta_{10} - 2 \beta_{9} - 3 \beta_{8} - 6 \beta_{7} + 6 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - 30 \beta_{2} - 10 \beta_{1} - 72$$$$)/144$$ $$\nu^{3}$$ $$=$$ $$($$$$6 \beta_{11} - 2 \beta_{10} - 6 \beta_{9} - 9 \beta_{8} + 17 \beta_{7} + 18 \beta_{6} - 8 \beta_{5} - 10 \beta_{4} - 45 \beta_{3} - 197 \beta_{2} + 186 \beta_{1} + 864$$$$)/288$$ $$\nu^{4}$$ $$=$$ $$($$$$13 \beta_{11} + 8 \beta_{10} - 28 \beta_{9} + 21 \beta_{8} + 7 \beta_{7} - 96 \beta_{6} + 41 \beta_{5} + \beta_{4} + 91 \beta_{3} - 199 \beta_{2} + 58 \beta_{1} - 3288$$$$)/144$$ $$\nu^{5}$$ $$=$$ $$($$$$55 \beta_{11} + 122 \beta_{10} - 130 \beta_{9} + 240 \beta_{8} - 152 \beta_{7} + 150 \beta_{6} + 93 \beta_{5} - 129 \beta_{4} - 1666 \beta_{3} - 436 \beta_{2} - 380 \beta_{1} - 2880$$$$)/288$$ $$\nu^{6}$$ $$=$$ $$($$$$-93 \beta_{11} - 480 \beta_{10} + 408 \beta_{9} + 711 \beta_{8} + 795 \beta_{7} - 936 \beta_{6} + 745 \beta_{5} - 223 \beta_{4} - 2655 \beta_{3} + 213 \beta_{2} + 2850 \beta_{1} - 44208$$$$)/288$$ $$\nu^{7}$$ $$=$$ $$($$$$-469 \beta_{11} + 714 \beta_{10} + 154 \beta_{9} + 1554 \beta_{8} - 2454 \beta_{7} - 4242 \beta_{6} + 2261 \beta_{5} - 317 \beta_{4} - 1836 \beta_{3} + 11754 \beta_{2} - 11200 \beta_{1} - 149184$$$$)/288$$ $$\nu^{8}$$ $$=$$ $$($$$$-634 \beta_{11} - 800 \beta_{10} + 1114 \beta_{9} + 654 \beta_{8} + 470 \beta_{7} + 2256 \beta_{6} - 769 \beta_{5} - 713 \beta_{4} - 7518 \beta_{3} + 11530 \beta_{2} - 3682 \beta_{1} + 50628$$$$)/72$$ $$\nu^{9}$$ $$=$$ $$($$$$-8343 \beta_{11} - 13446 \beta_{10} + 17118 \beta_{9} - 6588 \beta_{8} + 11856 \beta_{7} - 22626 \beta_{6} + 10267 \beta_{5} + 7301 \beta_{4} + 73170 \beta_{3} + 143076 \beta_{2} - 14688 \beta_{1} - 632448$$$$)/288$$ $$\nu^{10}$$ $$=$$ $$($$$$-14603 \beta_{11} + 22552 \beta_{10} + 3248 \beta_{9} - 42141 \beta_{8} - 81157 \beta_{7} + 72240 \beta_{6} - 54397 \beta_{5} + 12451 \beta_{4} + 131133 \beta_{3} + 389797 \beta_{2} - 347450 \beta_{1} + 3303696$$$$)/288$$ $$\nu^{11}$$ $$=$$ $$($$$$-24805 \beta_{11} - 117994 \beta_{10} + 96910 \beta_{9} - 176418 \beta_{8} + 208474 \beta_{7} + 345906 \beta_{6} - 246703 \beta_{5} + 46507 \beta_{4} + 467516 \beta_{3} + 225074 \beta_{2} + 475244 \beta_{1} + 15174720$$$$)/288$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$703$$ $$1217$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1727.1
 0.886307 − 1.60260i 2.61836 − 1.60260i 0.433705 − 1.36719i −1.29835 − 1.36719i −0.453986 − 2.07664i −2.18604 − 2.07664i −2.18604 + 2.07664i −0.453986 + 2.07664i −1.29835 + 1.36719i 0.433705 + 1.36719i 2.61836 + 1.60260i 0.886307 + 1.60260i
0 0 0 20.8488i 0 13.9048i 0 0 0
1727.2 0 0 0 20.8488i 0 13.9048i 0 0 0
1727.3 0 0 0 5.83890i 0 8.83113i 0 0 0
1727.4 0 0 0 5.83890i 0 8.83113i 0 0 0
1727.5 0 0 0 1.49508i 0 26.1852i 0 0 0
1727.6 0 0 0 1.49508i 0 26.1852i 0 0 0
1727.7 0 0 0 1.49508i 0 26.1852i 0 0 0
1727.8 0 0 0 1.49508i 0 26.1852i 0 0 0
1727.9 0 0 0 5.83890i 0 8.83113i 0 0 0
1727.10 0 0 0 5.83890i 0 8.83113i 0 0 0
1727.11 0 0 0 20.8488i 0 13.9048i 0 0 0
1727.12 0 0 0 20.8488i 0 13.9048i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1727.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.4.c.j 12
3.b odd 2 1 inner 1728.4.c.j 12
4.b odd 2 1 inner 1728.4.c.j 12
8.b even 2 1 108.4.b.b 12
8.d odd 2 1 108.4.b.b 12
12.b even 2 1 inner 1728.4.c.j 12
24.f even 2 1 108.4.b.b 12
24.h odd 2 1 108.4.b.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.4.b.b 12 8.b even 2 1
108.4.b.b 12 8.d odd 2 1
108.4.b.b 12 24.f even 2 1
108.4.b.b 12 24.h odd 2 1
1728.4.c.j 12 1.a even 1 1 trivial
1728.4.c.j 12 3.b odd 2 1 inner
1728.4.c.j 12 4.b odd 2 1 inner
1728.4.c.j 12 12.b 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 $$S_{4}^{\mathrm{new}}(1728, [\chi])$$:

 $$T_{5}^{6} + 471 T_{5}^{4} + 15867 T_{5}^{2} + 33125$$ $$T_{7}^{6} + 957 T_{7}^{4} + 201123 T_{7}^{2} + 10338975$$ $$T_{11}^{6} - 4929 T_{11}^{4} + 6225579 T_{11}^{2} - 2112318675$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$( 33125 + 15867 T^{2} + 471 T^{4} + T^{6} )^{2}$$
$7$ $$( 10338975 + 201123 T^{2} + 957 T^{4} + T^{6} )^{2}$$
$11$ $$( -2112318675 + 6225579 T^{2} - 4929 T^{4} + T^{6} )^{2}$$
$13$ $$( 70760 - 2676 T - 18 T^{2} + T^{3} )^{4}$$
$17$ $$( 42531205952 + 55212720 T^{2} + 17532 T^{4} + T^{6} )^{2}$$
$19$ $$( 1093873434624 + 341148672 T^{2} + 32832 T^{4} + T^{6} )^{2}$$
$23$ $$( -15885545472 + 36142848 T^{2} - 20688 T^{4} + T^{6} )^{2}$$
$29$ $$( 251748219200 + 199904688 T^{2} + 28380 T^{4} + T^{6} )^{2}$$
$31$ $$( 37183780589679 + 4493907459 T^{2} + 133005 T^{4} + T^{6} )^{2}$$
$37$ $$( 8168000 - 68400 T - 60 T^{2} + T^{3} )^{4}$$
$41$ $$( 855375564800 + 12008894208 T^{2} + 223920 T^{4} + T^{6} )^{2}$$
$43$ $$( 4142014821374400 + 81389094192 T^{2} + 504084 T^{4} + T^{6} )^{2}$$
$47$ $$( -62306621744832 + 15265401264 T^{2} - 245796 T^{4} + T^{6} )^{2}$$
$53$ $$( 820610118691973 + 44456955195 T^{2} + 494583 T^{4} + T^{6} )^{2}$$
$59$ $$( -9877489598635200 + 183762093744 T^{2} - 956292 T^{4} + T^{6} )^{2}$$
$61$ $$( 68226752 - 329808 T + 36 T^{2} + T^{3} )^{4}$$
$67$ $$( 1172231992958400 + 47114269488 T^{2} + 463188 T^{4} + T^{6} )^{2}$$
$71$ $$( -467529795115200 + 54101959344 T^{2} - 943812 T^{4} + T^{6} )^{2}$$
$73$ $$( 14711755 - 211941 T - 39 T^{2} + T^{3} )^{4}$$
$79$ $$( 4976229809870784 + 153820502064 T^{2} + 1242228 T^{4} + T^{6} )^{2}$$
$83$ $$( -39034497811277043 + 537347415915 T^{2} - 1829985 T^{4} + T^{6} )^{2}$$
$89$ $$( 1976574935229320000 + 5694295453872 T^{2} + 4353660 T^{4} + T^{6} )^{2}$$
$97$ $$( -298155155 - 923733 T - 129 T^{2} + T^{3} )^{4}$$