# Properties

 Label 2178.4.a.bt Level $2178$ Weight $4$ Character orbit 2178.a Self dual yes Analytic conductor $128.506$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2178 = 2 \cdot 3^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 2178.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$128.506159993$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.978025.2 Defining polynomial: $$x^{4} - 2 x^{3} - 99 x^{2} + 100 x + 2420$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$11$$ Twist minimal: no (minimal twist has level 22) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + 4 q^{4} + ( -6 + \beta_{2} ) q^{5} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{7} -8 q^{8} +O(q^{10})$$ $$q -2 q^{2} + 4 q^{4} + ( -6 + \beta_{2} ) q^{5} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{7} -8 q^{8} + ( 12 - 2 \beta_{2} ) q^{10} + ( -10 - 6 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{13} + ( -2 - 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{14} + 16 q^{16} + ( -14 + 3 \beta_{1} + 4 \beta_{3} ) q^{17} + ( 7 - 6 \beta_{1} + 7 \beta_{2} - 5 \beta_{3} ) q^{19} + ( -24 + 4 \beta_{2} ) q^{20} + ( -76 + 6 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} ) q^{23} + ( -6 - 7 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{25} + ( 20 + 12 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{26} + ( 4 + 4 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} ) q^{28} + ( -143 + 9 \beta_{1} - 6 \beta_{2} + 5 \beta_{3} ) q^{29} + ( 48 - 4 \beta_{1} + 7 \beta_{2} + 14 \beta_{3} ) q^{31} -32 q^{32} + ( 28 - 6 \beta_{1} - 8 \beta_{3} ) q^{34} + ( 181 - 19 \beta_{1} - 3 \beta_{2} - 13 \beta_{3} ) q^{35} + ( 93 - 27 \beta_{1} + 10 \beta_{2} - 5 \beta_{3} ) q^{37} + ( -14 + 12 \beta_{1} - 14 \beta_{2} + 10 \beta_{3} ) q^{38} + ( 48 - 8 \beta_{2} ) q^{40} + ( -3 + 7 \beta_{1} + 22 \beta_{2} + 2 \beta_{3} ) q^{41} + ( -188 + 14 \beta_{1} - 12 \beta_{2} + 19 \beta_{3} ) q^{43} + ( 152 - 12 \beta_{1} - 4 \beta_{2} + 16 \beta_{3} ) q^{46} + ( -93 + 21 \beta_{1} + 20 \beta_{2} - 11 \beta_{3} ) q^{47} + ( 209 - 2 \beta_{1} + 9 \beta_{2} - 4 \beta_{3} ) q^{49} + ( 12 + 14 \beta_{1} + 12 \beta_{2} + 6 \beta_{3} ) q^{50} + ( -40 - 24 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} ) q^{52} + ( 34 - 8 \beta_{1} + 21 \beta_{2} + 18 \beta_{3} ) q^{53} + ( -8 - 8 \beta_{1} - 16 \beta_{2} - 8 \beta_{3} ) q^{56} + ( 286 - 18 \beta_{1} + 12 \beta_{2} - 10 \beta_{3} ) q^{58} + ( -255 - 26 \beta_{1} + 5 \beta_{2} + 9 \beta_{3} ) q^{59} + ( -230 + 28 \beta_{1} + 13 \beta_{2} + 14 \beta_{3} ) q^{61} + ( -96 + 8 \beta_{1} - 14 \beta_{2} - 28 \beta_{3} ) q^{62} + 64 q^{64} + ( -13 + 53 \beta_{1} + 22 \beta_{2} + 5 \beta_{3} ) q^{65} + ( 72 + 51 \beta_{1} + 29 \beta_{2} + 30 \beta_{3} ) q^{67} + ( -56 + 12 \beta_{1} + 16 \beta_{3} ) q^{68} + ( -362 + 38 \beta_{1} + 6 \beta_{2} + 26 \beta_{3} ) q^{70} + ( 300 + 32 \beta_{1} + 23 \beta_{2} + 28 \beta_{3} ) q^{71} + ( -297 + 45 \beta_{1} + 54 \beta_{2} + 8 \beta_{3} ) q^{73} + ( -186 + 54 \beta_{1} - 20 \beta_{2} + 10 \beta_{3} ) q^{74} + ( 28 - 24 \beta_{1} + 28 \beta_{2} - 20 \beta_{3} ) q^{76} + ( -144 + 36 \beta_{1} + 41 \beta_{2} + 14 \beta_{3} ) q^{79} + ( -96 + 16 \beta_{2} ) q^{80} + ( 6 - 14 \beta_{1} - 44 \beta_{2} - 4 \beta_{3} ) q^{82} + ( -384 - 92 \beta_{1} - 5 \beta_{2} + 17 \beta_{3} ) q^{83} + ( 181 - 11 \beta_{1} - 24 \beta_{2} - 31 \beta_{3} ) q^{85} + ( 376 - 28 \beta_{1} + 24 \beta_{2} - 38 \beta_{3} ) q^{86} + ( 298 + 26 \beta_{1} + 76 \beta_{2} - 15 \beta_{3} ) q^{89} + ( -547 - 31 \beta_{1} + 45 \beta_{2} - 73 \beta_{3} ) q^{91} + ( -304 + 24 \beta_{1} + 8 \beta_{2} - 32 \beta_{3} ) q^{92} + ( 186 - 42 \beta_{1} - 40 \beta_{2} + 22 \beta_{3} ) q^{94} + ( 447 - 15 \beta_{1} + 33 \beta_{2} + 11 \beta_{3} ) q^{95} + ( 566 - 20 \beta_{1} + 69 \beta_{2} - 79 \beta_{3} ) q^{97} + ( -418 + 4 \beta_{1} - 18 \beta_{2} + 8 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{2} + 16q^{4} - 25q^{5} + 3q^{7} - 32q^{8} + O(q^{10})$$ $$4q - 8q^{2} + 16q^{4} - 25q^{5} + 3q^{7} - 32q^{8} + 50q^{10} - 41q^{13} - 6q^{14} + 64q^{16} - 52q^{17} + 16q^{19} - 100q^{20} - 314q^{23} - 21q^{25} + 82q^{26} + 12q^{28} - 561q^{29} + 199q^{31} - 128q^{32} + 104q^{34} + 714q^{35} + 357q^{37} - 32q^{38} + 200q^{40} - 32q^{41} - 721q^{43} + 628q^{46} - 403q^{47} + 823q^{49} + 42q^{50} - 164q^{52} + 133q^{53} - 24q^{56} + 1122q^{58} - 1016q^{59} - 919q^{61} - 398q^{62} + 256q^{64} - 69q^{65} + 289q^{67} - 208q^{68} - 1428q^{70} + 1205q^{71} - 1234q^{73} - 714q^{74} + 64q^{76} - 603q^{79} - 400q^{80} + 64q^{82} - 1514q^{83} + 717q^{85} + 1442q^{86} + 1101q^{89} - 2306q^{91} - 1256q^{92} + 806q^{94} + 1766q^{95} + 2116q^{97} - 1646q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 99 x^{2} + 100 x + 2420$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{2} + 9 \nu + 46$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 5 \nu^{2} - 50 \nu + 200$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 6 \nu^{2} - 61 \nu - 346$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} + 11 \beta_{1} + 5$$$$)/11$$ $$\nu^{2}$$ $$=$$ $$($$$$9 \beta_{3} - 9 \beta_{2} + 11 \beta_{1} + 551$$$$)/11$$ $$\nu^{3}$$ $$=$$ $$($$$$95 \beta_{3} - 7 \beta_{2} + 605 \beta_{1} + 805$$$$)/11$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −7.19378 6.92695 8.19378 −5.92695
−2.00000 0 4.00000 −14.9181 0 −21.7679 −8.00000 0 29.8363
1.2 −2.00000 0 4.00000 −12.7359 0 −23.4611 −8.00000 0 25.4718
1.3 −2.00000 0 4.00000 −5.40810 0 22.1498 −8.00000 0 10.8162
1.4 −2.00000 0 4.00000 8.06215 0 26.0792 −8.00000 0 −16.1243
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$-1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2178.4.a.bt 4
3.b odd 2 1 242.4.a.o 4
11.b odd 2 1 2178.4.a.by 4
11.d odd 10 2 198.4.f.d 8
12.b even 2 1 1936.4.a.bm 4
33.d even 2 1 242.4.a.n 4
33.f even 10 2 22.4.c.b 8
33.f even 10 2 242.4.c.r 8
33.h odd 10 2 242.4.c.n 8
33.h odd 10 2 242.4.c.q 8
132.d odd 2 1 1936.4.a.bn 4
132.n odd 10 2 176.4.m.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.c.b 8 33.f even 10 2
176.4.m.b 8 132.n odd 10 2
198.4.f.d 8 11.d odd 10 2
242.4.a.n 4 33.d even 2 1
242.4.a.o 4 3.b odd 2 1
242.4.c.n 8 33.h odd 10 2
242.4.c.q 8 33.h odd 10 2
242.4.c.r 8 33.f even 10 2
1936.4.a.bm 4 12.b even 2 1
1936.4.a.bn 4 132.d odd 2 1
2178.4.a.bt 4 1.a even 1 1 trivial
2178.4.a.by 4 11.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(2178))$$:

 $$T_{5}^{4} + 25 T_{5}^{3} + 73 T_{5}^{2} - 1710 T_{5} - 8284$$ $$T_{7}^{4} - 3 T_{7}^{3} - 1093 T_{7}^{2} + 1496 T_{7} + 295004$$ $$T_{17}^{4} + 52 T_{17}^{3} - 3079 T_{17}^{2} - 174580 T_{17} - 2019455$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T )^{4}$$
$3$ 1
$5$ $$1 + 25 T + 573 T^{2} + 7665 T^{3} + 103716 T^{4} + 958125 T^{5} + 8953125 T^{6} + 48828125 T^{7} + 244140625 T^{8}$$
$7$ $$1 - 3 T + 279 T^{2} - 1591 T^{3} + 251100 T^{4} - 545713 T^{5} + 32824071 T^{6} - 121060821 T^{7} + 13841287201 T^{8}$$
$11$ 1
$13$ $$1 + 41 T + 5637 T^{2} + 241261 T^{3} + 15241940 T^{4} + 530050417 T^{5} + 27208722333 T^{6} + 434784474293 T^{7} + 23298085122481 T^{8}$$
$17$ $$1 + 52 T + 16573 T^{2} + 591848 T^{3} + 112551705 T^{4} + 2907749224 T^{5} + 400031931037 T^{6} + 6166569577844 T^{7} + 582622237229761 T^{8}$$
$19$ $$1 - 16 T + 14825 T^{2} - 500832 T^{3} + 111187933 T^{4} - 3435206688 T^{5} + 697455185825 T^{6} - 5163003164464 T^{7} + 2213314919066161 T^{8}$$
$23$ $$1 + 314 T + 61816 T^{2} + 8042722 T^{3} + 950275950 T^{4} + 97855798574 T^{5} + 9150986514424 T^{6} + 565561935699382 T^{7} + 21914624432020321 T^{8}$$
$29$ $$1 + 561 T + 200635 T^{2} + 47945667 T^{3} + 8699420208 T^{4} + 1169346872463 T^{5} + 119342377008835 T^{6} + 8138508892462509 T^{7} + 353814783205469041 T^{8}$$
$31$ $$1 - 199 T + 51813 T^{2} + 256853 T^{3} + 389222024 T^{4} + 7651907723 T^{5} + 45984228223653 T^{6} - 5261484809973529 T^{7} + 787662783788549761 T^{8}$$
$37$ $$1 - 357 T + 160529 T^{2} - 42781809 T^{3} + 12141822140 T^{4} - 2167026971277 T^{5} + 411873494710361 T^{6} - 46396341106842489 T^{7} + 6582952005840035281 T^{8}$$
$41$ $$1 + 32 T + 194093 T^{2} + 3202364 T^{3} + 18031305245 T^{4} + 220710129244 T^{5} + 921961982448413 T^{6} + 10476221900606752 T^{7} + 22563490300366186081 T^{8}$$
$43$ $$1 + 721 T + 420117 T^{2} + 154459221 T^{3} + 51447883420 T^{4} + 12280589284047 T^{5} + 2655712080056733 T^{6} + 362369273206463803 T^{7} + 39959630797262576401 T^{8}$$
$47$ $$1 + 403 T + 357463 T^{2} + 121226987 T^{3} + 52982120160 T^{4} + 12586149471301 T^{5} + 3853170649150327 T^{6} + 451009580660415101 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$
$53$ $$1 - 133 T + 365887 T^{2} - 13770175 T^{3} + 65244311076 T^{4} - 2050062343475 T^{5} + 8109651600406423 T^{6} - 438868557709683689 T^{7} +$$$$49\!\cdots\!41$$$$T^{8}$$
$59$ $$1 + 1016 T + 1078585 T^{2} + 639078152 T^{3} + 358860597453 T^{4} + 131253231779608 T^{5} + 45495290877177985 T^{6} + 8801603751753418024 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$
$61$ $$1 + 919 T + 1093783 T^{2} + 606559837 T^{3} + 388669851044 T^{4} + 137677558362097 T^{5} + 56352109629697663 T^{6} + 10746920259314575579 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$
$67$ $$1 - 289 T + 686735 T^{2} - 243308709 T^{3} + 267199450216 T^{4} - 73178257244967 T^{5} + 62120937078828215 T^{6} - 7862688440529239683 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$
$71$ $$1 - 1205 T + 1608131 T^{2} - 1210556495 T^{3} + 899763338876 T^{4} - 433271485681945 T^{5} + 206002037682161651 T^{6} - 55247443365731082355 T^{7} +$$$$16\!\cdots\!41$$$$T^{8}$$
$73$ $$1 + 1234 T + 1514091 T^{2} + 1290056552 T^{3} + 893250631145 T^{4} + 501853929689384 T^{5} + 229133790016138299 T^{6} + 72647537998002604642 T^{7} +$$$$22\!\cdots\!21$$$$T^{8}$$
$79$ $$1 + 603 T + 1688657 T^{2} + 808010451 T^{3} + 1205184633704 T^{4} + 398380664750589 T^{5} + 410491333377725297 T^{6} + 72270512377518846357 T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$
$83$ $$1 + 1514 T + 2070047 T^{2} + 2046213464 T^{3} + 1753896205685 T^{4} + 1169998257940168 T^{5} + 676781939071378343 T^{6} +$$$$28\!\cdots\!42$$$$T^{7} +$$$$10\!\cdots\!61$$$$T^{8}$$
$89$ $$1 - 1101 T + 2406895 T^{2} - 1914547747 T^{3} + 2334666485008 T^{4} - 1349696810654843 T^{5} + 1196181784307576095 T^{6} -$$$$38\!\cdots\!09$$$$T^{7} +$$$$24\!\cdots\!21$$$$T^{8}$$
$97$ $$1 - 2116 T + 3632073 T^{2} - 4516910940 T^{3} + 5180506457941 T^{4} - 4122462658342620 T^{5} + 3025415128858487817 T^{6} -$$$$16\!\cdots\!72$$$$T^{7} +$$$$69\!\cdots\!41$$$$T^{8}$$