# Properties

 Label 242.4.a.o Level $242$ Weight $4$ Character orbit 242.a Self dual yes Analytic conductor $14.278$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$242 = 2 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 242.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$14.2784622214$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + ( 1 - \beta_{1} ) q^{3} + 4 q^{4} + ( 6 - \beta_{2} ) q^{5} + ( 2 - 2 \beta_{1} ) q^{6} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{7} + 8 q^{8} + ( 25 - \beta_{1} - 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + 2 q^{2} + ( 1 - \beta_{1} ) q^{3} + 4 q^{4} + ( 6 - \beta_{2} ) q^{5} + ( 2 - 2 \beta_{1} ) q^{6} + ( 1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{7} + 8 q^{8} + ( 25 - \beta_{1} - 2 \beta_{2} ) q^{9} + ( 12 - 2 \beta_{2} ) q^{10} + ( 4 - 4 \beta_{1} ) q^{12} + ( -10 + 6 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{13} + ( 2 - 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{14} + ( 19 - 9 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{15} + 16 q^{16} + ( 14 + 3 \beta_{1} + 4 \beta_{3} ) q^{17} + ( 50 - 2 \beta_{1} - 4 \beta_{2} ) q^{18} + ( 7 + 6 \beta_{1} + 7 \beta_{2} + 5 \beta_{3} ) q^{19} + ( 24 - 4 \beta_{2} ) q^{20} + ( 2 \beta_{1} - 11 \beta_{2} - 14 \beta_{3} ) q^{21} + ( 76 + 6 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} ) q^{23} + ( 8 - 8 \beta_{1} ) q^{24} + ( -6 + 7 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{25} + ( -20 + 12 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{26} + ( 75 - 4 \beta_{1} + 8 \beta_{2} + 6 \beta_{3} ) q^{27} + ( 4 - 4 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{28} + ( 143 + 9 \beta_{1} + 6 \beta_{2} + 5 \beta_{3} ) q^{29} + ( 38 - 18 \beta_{1} + 10 \beta_{2} + 6 \beta_{3} ) q^{30} + ( 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} + ( 100 - 4 \beta_{1} - 8 \beta_{2} ) q^{36} + ( 93 + 27 \beta_{1} + 10 \beta_{2} + 5 \beta_{3} ) q^{37} + ( 14 + 12 \beta_{1} + 14 \beta_{2} + 10 \beta_{3} ) q^{38} + ( -251 + 13 \beta_{1} + 15 \beta_{2} + 19 \beta_{3} ) q^{39} + ( 48 - 8 \beta_{2} ) q^{40} + ( 3 + 7 \beta_{1} - 22 \beta_{2} + 2 \beta_{3} ) q^{41} + ( 4 \beta_{1} - 22 \beta_{2} - 28 \beta_{3} ) q^{42} + ( -188 - 14 \beta_{1} - 12 \beta_{2} - 19 \beta_{3} ) q^{43} + ( 329 + 5 \beta_{1} - 19 \beta_{2} + 9 \beta_{3} ) q^{45} + ( 152 + 12 \beta_{1} - 4 \beta_{2} - 16 \beta_{3} ) q^{46} + ( 93 + 21 \beta_{1} - 20 \beta_{2} - 11 \beta_{3} ) q^{47} + ( 16 - 16 \beta_{1} ) q^{48} + ( 209 + 2 \beta_{1} + 9 \beta_{2} + 4 \beta_{3} ) q^{49} + ( -12 + 14 \beta_{1} - 12 \beta_{2} + 6 \beta_{3} ) q^{50} + ( -35 - 2 \beta_{1} + 2 \beta_{2} + 32 \beta_{3} ) q^{51} + ( -40 + 24 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{52} + ( -34 - 8 \beta_{1} - 21 \beta_{2} + 18 \beta_{3} ) q^{53} + ( 150 - 8 \beta_{1} + 16 \beta_{2} + 12 \beta_{3} ) q^{54} + ( 8 - 8 \beta_{1} + 16 \beta_{2} - 8 \beta_{3} ) q^{56} + ( -260 + 29 \beta_{1} - 28 \beta_{2} + 19 \beta_{3} ) q^{57} + ( 286 + 18 \beta_{1} + 12 \beta_{2} + 10 \beta_{3} ) q^{58} + ( 255 - 26 \beta_{1} - 5 \beta_{2} + 9 \beta_{3} ) q^{59} + ( 76 - 36 \beta_{1} + 20 \beta_{2} + 12 \beta_{3} ) q^{60} + ( -230 - 28 \beta_{1} + 13 \beta_{2} - 14 \beta_{3} ) q^{61} + ( 96 + 8 \beta_{1} + 14 \beta_{2} - 28 \beta_{3} ) q^{62} + ( -350 - 48 \beta_{1} + 19 \beta_{2} - 52 \beta_{3} ) q^{63} + 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} + ( -412 - 106 \beta_{1} + 30 \beta_{2} - 58 \beta_{3} ) q^{69} + ( -362 - 38 \beta_{1} + 6 \beta_{2} - 26 \beta_{3} ) q^{70} + ( -300 + 32 \beta_{1} - 23 \beta_{2} + 28 \beta_{3} ) q^{71} + ( 200 - 8 \beta_{1} - 16 \beta_{2} ) q^{72} + ( -297 - 45 \beta_{1} + 54 \beta_{2} - 8 \beta_{3} ) q^{73} + ( 186 + 54 \beta_{1} + 20 \beta_{2} + 10 \beta_{3} ) q^{74} + ( -207 - 3 \beta_{1} + 41 \beta_{2} + 42 \beta_{3} ) q^{75} + ( 28 + 24 \beta_{1} + 28 \beta_{2} + 20 \beta_{3} ) q^{76} + ( -502 + 26 \beta_{1} + 30 \beta_{2} + 38 \beta_{3} ) q^{78} + ( -144 - 36 \beta_{1} + 41 \beta_{2} - 14 \beta_{3} ) q^{79} + ( 96 - 16 \beta_{2} ) q^{80} + ( -344 - 6 \beta_{1} + 24 \beta_{3} ) q^{81} + ( 6 + 14 \beta_{1} - 44 \beta_{2} + 4 \beta_{3} ) q^{82} + ( 384 - 92 \beta_{1} + 5 \beta_{2} + 17 \beta_{3} ) q^{83} + ( 8 \beta_{1} - 44 \beta_{2} - 56 \beta_{3} ) q^{84} + ( 181 + 11 \beta_{1} - 24 \beta_{2} + 31 \beta_{3} ) q^{85} + ( -376 - 28 \beta_{1} - 24 \beta_{2} - 38 \beta_{3} ) q^{86} + ( -264 - 110 \beta_{1} - 17 \beta_{2} + 22 \beta_{3} ) q^{87} + ( -298 + 26 \beta_{1} - 76 \beta_{2} - 15 \beta_{3} ) q^{89} + ( 658 + 10 \beta_{1} - 38 \beta_{2} + 18 \beta_{3} ) q^{90} + ( -547 + 31 \beta_{1} + 45 \beta_{2} + 73 \beta_{3} ) q^{91} + ( 304 + 24 \beta_{1} - 8 \beta_{2} - 32 \beta_{3} ) q^{92} + ( -611 - 69 \beta_{1} - 13 \beta_{2} - 133 \beta_{3} ) q^{93} + ( 186 + 42 \beta_{1} - 40 \beta_{2} - 22 \beta_{3} ) q^{94} + ( -447 - 15 \beta_{1} - 33 \beta_{2} + 11 \beta_{3} ) q^{95} + ( 32 - 32 \beta_{1} ) q^{96} + ( 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{2} + 4q^{3} + 16q^{4} + 25q^{5} + 8q^{6} + 3q^{7} + 32q^{8} + 102q^{9} + O(q^{10})$$ $$4q + 8q^{2} + 4q^{3} + 16q^{4} + 25q^{5} + 8q^{6} + 3q^{7} + 32q^{8} + 102q^{9} + 50q^{10} + 16q^{12} - 41q^{13} + 6q^{14} + 68q^{15} + 64q^{16} + 52q^{17} + 204q^{18} + 16q^{19} + 100q^{20} + 25q^{21} + 314q^{23} + 32q^{24} - 21q^{25} - 82q^{26} + 286q^{27} + 12q^{28} + 561q^{29} + 136q^{30} + 199q^{31} + 128q^{32} + 104q^{34} - 714q^{35} + 408q^{36} + 357q^{37} + 32q^{38} - 1038q^{39} + 200q^{40} + 32q^{41} + 50q^{42} - 721q^{43} + 1326q^{45} + 628q^{46} + 403q^{47} + 64q^{48} + 823q^{49} - 42q^{50} - 174q^{51} - 164q^{52} - 133q^{53} + 572q^{54} + 24q^{56} - 1031q^{57} + 1122q^{58} + 1016q^{59} + 272q^{60} - 919q^{61} + 398q^{62} - 1367q^{63} + 256q^{64} + 69q^{65} + 289q^{67} + 208q^{68} - 1620q^{69} - 1428q^{70} - 1205q^{71} + 816q^{72} - 1234q^{73} + 714q^{74} - 911q^{75} + 64q^{76} - 2076q^{78} - 603q^{79} + 400q^{80} - 1400q^{81} + 64q^{82} + 1514q^{83} + 100q^{84} + 717q^{85} - 1442q^{86} - 1061q^{87} - 1101q^{89} + 2652q^{90} - 2306q^{91} + 1256q^{92} - 2298q^{93} + 806q^{94} - 1766q^{95} + 128q^{96} + 2116q^{97} + 1646q^{98} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{2} + 7 \nu - 54$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} - 2 \nu^{2} + 57 \nu + 146$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - 9 \nu^{2} - 46 \nu + 400$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 11 \beta_{1} + 6$$$$)/11$$ $$\nu^{2}$$ $$=$$ $$($$$$-7 \beta_{3} - 7 \beta_{2} + 11 \beta_{1} + 552$$$$)/11$$ $$\nu^{3}$$ $$=$$ $$($$$$71 \beta_{3} - 17 \beta_{2} + 605 \beta_{1} + 844$$$$)/11$$

## 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
 8.19378 6.92695 −7.19378 −5.92695
2.00000 −7.81182 4.00000 14.9181 −15.6236 −21.7679 8.00000 34.0245 29.8363
1.2 2.00000 −4.30892 4.00000 −8.06215 −8.61784 26.0792 8.00000 −8.43321 −16.1243
1.3 2.00000 7.57575 4.00000 5.40810 15.1515 22.1498 8.00000 30.3919 10.8162
1.4 2.00000 8.54499 4.00000 12.7359 17.0900 −23.4611 8.00000 46.0168 25.4718
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 242.4.a.o 4
3.b odd 2 1 2178.4.a.bt 4
4.b odd 2 1 1936.4.a.bm 4
11.b odd 2 1 242.4.a.n 4
11.c even 5 2 242.4.c.n 8
11.c even 5 2 242.4.c.q 8
11.d odd 10 2 22.4.c.b 8
11.d odd 10 2 242.4.c.r 8
33.d even 2 1 2178.4.a.by 4
33.f even 10 2 198.4.f.d 8
44.c even 2 1 1936.4.a.bn 4
44.g even 10 2 176.4.m.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.4.c.b 8 11.d odd 10 2
176.4.m.b 8 44.g even 10 2
198.4.f.d 8 33.f even 10 2
242.4.a.n 4 11.b odd 2 1
242.4.a.o 4 1.a even 1 1 trivial
242.4.c.n 8 11.c even 5 2
242.4.c.q 8 11.c even 5 2
242.4.c.r 8 11.d odd 10 2
1936.4.a.bm 4 4.b odd 2 1
1936.4.a.bn 4 44.c even 2 1
2178.4.a.bt 4 3.b odd 2 1
2178.4.a.by 4 33.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(242))$$:

 $$T_{3}^{4} - 4 T_{3}^{3} - 97 T_{3}^{2} + 242 T_{3} + 2179$$ $$T_{5}^{4} - 25 T_{5}^{3} + 73 T_{5}^{2} + 1710 T_{5} - 8284$$ $$T_{7}^{4} - 3 T_{7}^{3} - 1093 T_{7}^{2} + 1496 T_{7} + 295004$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T )^{4}$$
$3$ $$1 - 4 T + 11 T^{2} - 82 T^{3} + 1315 T^{4} - 2214 T^{5} + 8019 T^{6} - 78732 T^{7} + 531441 T^{8}$$
$5$ $$1 - 25 T + 573 T^{2} - 7665 T^{3} + 103716 T^{4} - 958125 T^{5} + 8953125 T^{6} - 48828125 T^{7} + 244140625 T^{8}$$
$7$ $$1 - 3 T + 279 T^{2} - 1591 T^{3} + 251100 T^{4} - 545713 T^{5} + 32824071 T^{6} - 121060821 T^{7} + 13841287201 T^{8}$$
$11$ 1
$13$ $$1 + 41 T + 5637 T^{2} + 241261 T^{3} + 15241940 T^{4} + 530050417 T^{5} + 27208722333 T^{6} + 434784474293 T^{7} + 23298085122481 T^{8}$$
$17$ $$1 - 52 T + 16573 T^{2} - 591848 T^{3} + 112551705 T^{4} - 2907749224 T^{5} + 400031931037 T^{6} - 6166569577844 T^{7} + 582622237229761 T^{8}$$
$19$ $$1 - 16 T + 14825 T^{2} - 500832 T^{3} + 111187933 T^{4} - 3435206688 T^{5} + 697455185825 T^{6} - 5163003164464 T^{7} + 2213314919066161 T^{8}$$
$23$ $$1 - 314 T + 61816 T^{2} - 8042722 T^{3} + 950275950 T^{4} - 97855798574 T^{5} + 9150986514424 T^{6} - 565561935699382 T^{7} + 21914624432020321 T^{8}$$
$29$ $$1 - 561 T + 200635 T^{2} - 47945667 T^{3} + 8699420208 T^{4} - 1169346872463 T^{5} + 119342377008835 T^{6} - 8138508892462509 T^{7} + 353814783205469041 T^{8}$$
$31$ $$1 - 199 T + 51813 T^{2} + 256853 T^{3} + 389222024 T^{4} + 7651907723 T^{5} + 45984228223653 T^{6} - 5261484809973529 T^{7} + 787662783788549761 T^{8}$$
$37$ $$1 - 357 T + 160529 T^{2} - 42781809 T^{3} + 12141822140 T^{4} - 2167026971277 T^{5} + 411873494710361 T^{6} - 46396341106842489 T^{7} + 6582952005840035281 T^{8}$$
$41$ $$1 - 32 T + 194093 T^{2} - 3202364 T^{3} + 18031305245 T^{4} - 220710129244 T^{5} + 921961982448413 T^{6} - 10476221900606752 T^{7} + 22563490300366186081 T^{8}$$
$43$ $$1 + 721 T + 420117 T^{2} + 154459221 T^{3} + 51447883420 T^{4} + 12280589284047 T^{5} + 2655712080056733 T^{6} + 362369273206463803 T^{7} + 39959630797262576401 T^{8}$$
$47$ $$1 - 403 T + 357463 T^{2} - 121226987 T^{3} + 52982120160 T^{4} - 12586149471301 T^{5} + 3853170649150327 T^{6} - 451009580660415101 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$
$53$ $$1 + 133 T + 365887 T^{2} + 13770175 T^{3} + 65244311076 T^{4} + 2050062343475 T^{5} + 8109651600406423 T^{6} + 438868557709683689 T^{7} +$$$$49\!\cdots\!41$$$$T^{8}$$
$59$ $$1 - 1016 T + 1078585 T^{2} - 639078152 T^{3} + 358860597453 T^{4} - 131253231779608 T^{5} + 45495290877177985 T^{6} - 8801603751753418024 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$
$61$ $$1 + 919 T + 1093783 T^{2} + 606559837 T^{3} + 388669851044 T^{4} + 137677558362097 T^{5} + 56352109629697663 T^{6} + 10746920259314575579 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$
$67$ $$1 - 289 T + 686735 T^{2} - 243308709 T^{3} + 267199450216 T^{4} - 73178257244967 T^{5} + 62120937078828215 T^{6} - 7862688440529239683 T^{7} +$$$$81\!\cdots\!61$$$$T^{8}$$
$71$ $$1 + 1205 T + 1608131 T^{2} + 1210556495 T^{3} + 899763338876 T^{4} + 433271485681945 T^{5} + 206002037682161651 T^{6} + 55247443365731082355 T^{7} +$$$$16\!\cdots\!41$$$$T^{8}$$
$73$ $$1 + 1234 T + 1514091 T^{2} + 1290056552 T^{3} + 893250631145 T^{4} + 501853929689384 T^{5} + 229133790016138299 T^{6} + 72647537998002604642 T^{7} +$$$$22\!\cdots\!21$$$$T^{8}$$
$79$ $$1 + 603 T + 1688657 T^{2} + 808010451 T^{3} + 1205184633704 T^{4} + 398380664750589 T^{5} + 410491333377725297 T^{6} + 72270512377518846357 T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$
$83$ $$1 - 1514 T + 2070047 T^{2} - 2046213464 T^{3} + 1753896205685 T^{4} - 1169998257940168 T^{5} + 676781939071378343 T^{6} -$$$$28\!\cdots\!42$$$$T^{7} +$$$$10\!\cdots\!61$$$$T^{8}$$
$89$ $$1 + 1101 T + 2406895 T^{2} + 1914547747 T^{3} + 2334666485008 T^{4} + 1349696810654843 T^{5} + 1196181784307576095 T^{6} +$$$$38\!\cdots\!09$$$$T^{7} +$$$$24\!\cdots\!21$$$$T^{8}$$
$97$ $$1 - 2116 T + 3632073 T^{2} - 4516910940 T^{3} + 5180506457941 T^{4} - 4122462658342620 T^{5} + 3025415128858487817 T^{6} -$$$$16\!\cdots\!72$$$$T^{7} +$$$$69\!\cdots\!41$$$$T^{8}$$