Properties

 Label 242.4.c.n Level $242$ Weight $4$ Character orbit 242.c Analytic conductor $14.278$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$242 = 2 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 242.c (of order $$5$$, degree $$4$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$14.2784622214$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} + 71 x^{6} - 141 x^{5} + 2911 x^{4} + 2710 x^{3} + 75340 x^{2} + 169400 x + 5856400$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 22) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 \beta_{4} q^{2} + ( -1 + \beta_{4} + \beta_{7} ) q^{3} + 4 \beta_{3} q^{4} + ( \beta_{1} + 8 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{5} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{6} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{5} - 3 \beta_{7} ) q^{7} + ( -8 - 8 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} ) q^{8} + ( -5 - 2 \beta_{1} - 7 \beta_{3} - 21 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{9} +O(q^{10})$$ $$q -2 \beta_{4} q^{2} + ( -1 + \beta_{4} + \beta_{7} ) q^{3} + 4 \beta_{3} q^{4} + ( \beta_{1} + 8 \beta_{2} + 4 \beta_{3} - 4 \beta_{4} + \beta_{5} ) q^{5} + ( -2 + 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{6} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{5} - 3 \beta_{7} ) q^{7} + ( -8 - 8 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} ) q^{8} + ( -5 - 2 \beta_{1} - 7 \beta_{3} - 21 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} ) q^{9} + ( 8 + 2 \beta_{1} - 8 \beta_{2} + 8 \beta_{4} - 2 \beta_{7} ) q^{10} + ( 8 + 4 \beta_{2} - 4 \beta_{4} + 4 \beta_{6} ) q^{12} + ( 7 - 3 \beta_{1} + 4 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} + 3 \beta_{6} ) q^{13} + ( 2 + 4 \beta_{4} - 6 \beta_{5} + 6 \beta_{6} + 2 \beta_{7} ) q^{14} + ( -50 - 7 \beta_{1} - 50 \beta_{2} - 12 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{15} + 16 \beta_{2} q^{16} + ( 3 + \beta_{1} + \beta_{2} - 25 \beta_{3} + 25 \beta_{4} + \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{17} + ( 14 - 6 \beta_{1} + 14 \beta_{2} + 56 \beta_{3} - 4 \beta_{5} + 4 \beta_{7} ) q^{18} + ( -34 + 21 \beta_{2} + 21 \beta_{3} + 36 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 6 \beta_{7} ) q^{19} + ( -12 - 4 \beta_{1} - 16 \beta_{3} - 16 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} ) q^{20} + ( -66 - 3 \beta_{1} - 144 \beta_{2} + 144 \beta_{4} - 2 \beta_{6} + 3 \beta_{7} ) q^{21} + ( 42 - 6 \beta_{1} - 70 \beta_{2} + 70 \beta_{4} - 6 \beta_{6} + 6 \beta_{7} ) q^{23} + ( 8 + 8 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} ) q^{24} + ( -\beta_{2} - \beta_{3} - 9 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} - 7 \beta_{7} ) q^{25} + ( -8 + 6 \beta_{1} - 8 \beta_{2} - 20 \beta_{3} - 6 \beta_{5} + 6 \beta_{7} ) q^{26} + ( -4 + 2 \beta_{1} + 31 \beta_{2} - 78 \beta_{3} + 78 \beta_{4} + 2 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{27} + ( -4 - 8 \beta_{1} - 8 \beta_{3} + 8 \beta_{4} - 8 \beta_{5} - 4 \beta_{6} - 4 \beta_{7} ) q^{28} + ( -50 + 10 \beta_{1} - 50 \beta_{2} + 120 \beta_{3} + \beta_{5} - \beta_{7} ) q^{29} + ( -72 + 24 \beta_{2} + 24 \beta_{3} + 76 \beta_{4} - 4 \beta_{5} + 4 \beta_{6} + 18 \beta_{7} ) q^{30} + ( -95 + 21 \beta_{1} - 74 \beta_{3} - 20 \beta_{4} + 25 \beta_{5} - 21 \beta_{6} ) q^{31} + 32 q^{32} + ( 46 + 8 \beta_{1} + 50 \beta_{2} - 50 \beta_{4} - 6 \beta_{6} - 8 \beta_{7} ) q^{34} + ( -76 + 16 \beta_{1} - 60 \beta_{3} + 214 \beta_{4} - 3 \beta_{5} - 16 \beta_{6} ) q^{35} + ( -92 - 112 \beta_{2} - 112 \beta_{3} + 84 \beta_{4} + 8 \beta_{5} - 8 \beta_{6} + 4 \beta_{7} ) q^{36} + ( -48 + 32 \beta_{1} - 48 \beta_{2} + 80 \beta_{3} + 5 \beta_{5} - 5 \beta_{7} ) q^{37} + ( 12 - 16 \beta_{1} - 42 \beta_{2} - 114 \beta_{3} + 114 \beta_{4} - 16 \beta_{5} + 12 \beta_{6} + 12 \beta_{7} ) q^{38} + ( 13 - 9 \beta_{1} - 344 \beta_{2} - 180 \beta_{3} + 180 \beta_{4} - 9 \beta_{5} + 13 \beta_{6} + 13 \beta_{7} ) q^{39} + ( 32 - 8 \beta_{1} + 32 \beta_{2} + 64 \beta_{3} - 8 \beta_{5} + 8 \beta_{7} ) q^{40} + ( 11 - 46 \beta_{2} - 46 \beta_{3} - 35 \beta_{4} + 24 \beta_{5} - 24 \beta_{6} - 7 \beta_{7} ) q^{41} + ( -294 + 6 \beta_{1} - 288 \beta_{3} + 128 \beta_{4} + 10 \beta_{5} - 6 \beta_{6} ) q^{42} + ( -255 - 7 \beta_{1} - 167 \beta_{2} + 167 \beta_{4} + 14 \beta_{6} + 7 \beta_{7} ) q^{43} + ( 313 + 28 \beta_{1} - 18 \beta_{2} + 18 \beta_{4} - 5 \beta_{6} - 28 \beta_{7} ) q^{45} + ( -152 + 12 \beta_{1} - 140 \beta_{3} - 96 \beta_{4} + 24 \beta_{5} - 12 \beta_{6} ) q^{46} + ( -29 - 198 \beta_{2} - 198 \beta_{3} + 20 \beta_{4} + 9 \beta_{5} - 9 \beta_{6} - 21 \beta_{7} ) q^{47} + ( -16 - 16 \beta_{1} - 16 \beta_{2} ) q^{48} + ( 2 - 7 \beta_{1} + 177 \beta_{2} - 62 \beta_{3} + 62 \beta_{4} - 7 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{49} + ( 14 + 4 \beta_{1} + 2 \beta_{2} + 20 \beta_{3} - 20 \beta_{4} + 4 \beta_{5} + 14 \beta_{6} + 14 \beta_{7} ) q^{50} + ( -234 - 32 \beta_{1} - 234 \beta_{2} - 169 \beta_{3} - 30 \beta_{5} + 30 \beta_{7} ) q^{51} + ( 12 + 40 \beta_{2} + 40 \beta_{3} - 24 \beta_{4} + 12 \beta_{5} - 12 \beta_{6} - 24 \beta_{7} ) q^{52} + ( 89 - 39 \beta_{1} + 50 \beta_{3} + 22 \beta_{4} - 47 \beta_{5} + 39 \beta_{6} ) q^{53} + ( 226 - 4 \beta_{1} + 156 \beta_{2} - 156 \beta_{4} + 8 \beta_{6} + 4 \beta_{7} ) q^{54} + ( 24 - 24 \beta_{1} + 16 \beta_{2} - 16 \beta_{4} + 8 \beta_{6} + 24 \beta_{7} ) q^{56} + ( 39 - 47 \beta_{1} - 8 \beta_{3} + 259 \beta_{4} - 18 \beta_{5} + 47 \beta_{6} ) q^{57} + ( -338 - 240 \beta_{2} - 240 \beta_{3} + 340 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - 18 \beta_{7} ) q^{58} + ( -69 - 40 \beta_{1} - 69 \beta_{2} + 203 \beta_{3} - 14 \beta_{5} + 14 \beta_{7} ) q^{59} + ( -36 + 28 \beta_{1} - 48 \beta_{2} - 200 \beta_{3} + 200 \beta_{4} + 28 \beta_{5} - 36 \beta_{6} - 36 \beta_{7} ) q^{60} + ( -28 + \beta_{1} - 228 \beta_{2} + 18 \beta_{3} - 18 \beta_{4} + \beta_{5} - 28 \beta_{6} - 28 \beta_{7} ) q^{61} + ( 148 + 50 \beta_{1} + 148 \beta_{2} + 188 \beta_{3} + 42 \beta_{5} - 42 \beta_{7} ) q^{62} + ( 539 + 228 \beta_{2} + 228 \beta_{3} - 468 \beta_{4} - 71 \beta_{5} + 71 \beta_{6} + 48 \beta_{7} ) q^{63} -64 \beta_{4} q^{64} + ( -69 + 27 \beta_{1} - 106 \beta_{2} + 106 \beta_{4} - 53 \beta_{6} - 27 \beta_{7} ) q^{65} + ( 91 - 59 \beta_{1} - 43 \beta_{2} + 43 \beta_{4} + 51 \beta_{6} + 59 \beta_{7} ) q^{67} + ( 116 - 16 \beta_{1} + 100 \beta_{3} - 104 \beta_{4} - 4 \beta_{5} + 16 \beta_{6} ) q^{68} + ( 614 + 346 \beta_{2} + 346 \beta_{3} - 526 \beta_{4} - 88 \beta_{5} + 88 \beta_{6} + 106 \beta_{7} ) q^{69} + ( 120 - 6 \beta_{1} + 120 \beta_{2} - 308 \beta_{3} + 32 \beta_{5} - 32 \beta_{7} ) q^{70} + ( 32 + 19 \beta_{1} - 334 \beta_{2} - 72 \beta_{3} + 72 \beta_{4} + 19 \beta_{5} + 32 \beta_{6} + 32 \beta_{7} ) q^{71} + ( -8 + 24 \beta_{1} + 224 \beta_{2} + 56 \beta_{3} - 56 \beta_{4} + 24 \beta_{5} - 8 \beta_{6} - 8 \beta_{7} ) q^{72} + ( -205 + 17 \beta_{1} - 205 \beta_{2} - 418 \beta_{3} + 62 \beta_{5} - 62 \beta_{7} ) q^{73} + ( -246 - 160 \beta_{2} - 160 \beta_{3} + 256 \beta_{4} - 10 \beta_{5} + 10 \beta_{6} - 54 \beta_{7} ) q^{74} + ( 462 - \beta_{1} + 461 \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{6} ) q^{75} + ( 120 - 8 \beta_{1} + 228 \beta_{2} - 228 \beta_{4} - 24 \beta_{6} + 8 \beta_{7} ) q^{76} + ( -354 + 8 \beta_{1} + 360 \beta_{2} - 360 \beta_{4} - 26 \beta_{6} - 8 \beta_{7} ) q^{78} + ( 47 + 55 \beta_{1} + 102 \beta_{3} + 104 \beta_{4} + 19 \beta_{5} - 55 \beta_{6} ) q^{79} + ( -80 - 128 \beta_{2} - 128 \beta_{3} + 64 \beta_{4} + 16 \beta_{5} - 16 \beta_{6} ) q^{80} + ( -174 - 30 \beta_{1} - 174 \beta_{2} - 446 \beta_{3} - 24 \beta_{5} + 24 \beta_{7} ) q^{81} + ( 14 + 34 \beta_{1} + 92 \beta_{2} + 162 \beta_{3} - 162 \beta_{4} + 34 \beta_{5} + 14 \beta_{6} + 14 \beta_{7} ) q^{82} + ( -92 + 104 \beta_{1} + 214 \beta_{2} - 231 \beta_{3} + 231 \beta_{4} + 104 \beta_{5} - 92 \beta_{6} - 92 \beta_{7} ) q^{83} + ( 576 + 20 \beta_{1} + 576 \beta_{2} + 320 \beta_{3} + 12 \beta_{5} - 12 \beta_{7} ) q^{84} + ( -281 - 116 \beta_{2} - 116 \beta_{3} + 226 \beta_{4} + 55 \beta_{5} - 55 \beta_{6} - 11 \beta_{7} ) q^{85} + ( -348 + 14 \beta_{1} - 334 \beta_{3} + 538 \beta_{4} - 14 \beta_{5} - 14 \beta_{6} ) q^{86} + ( -122 + 39 \beta_{1} + 196 \beta_{2} - 196 \beta_{4} + 110 \beta_{6} - 39 \beta_{7} ) q^{87} + ( -521 + 61 \beta_{1} - 435 \beta_{2} + 435 \beta_{4} - 26 \beta_{6} - 61 \beta_{7} ) q^{89} + ( 20 - 56 \beta_{1} - 36 \beta_{3} - 636 \beta_{4} - 46 \beta_{5} + 56 \beta_{6} ) q^{90} + ( 210 + 898 \beta_{2} + 898 \beta_{3} - 238 \beta_{4} + 28 \beta_{5} - 28 \beta_{6} - 31 \beta_{7} ) q^{91} + ( 280 + 48 \beta_{1} + 280 \beta_{2} + 472 \beta_{3} + 24 \beta_{5} - 24 \beta_{7} ) q^{92} + ( -69 - 51 \beta_{1} - 122 \beta_{2} + 914 \beta_{3} - 914 \beta_{4} - 51 \beta_{5} - 69 \beta_{6} - 69 \beta_{7} ) q^{93} + ( 42 - 24 \beta_{1} + 396 \beta_{2} + 356 \beta_{3} - 356 \beta_{4} - 24 \beta_{5} + 42 \beta_{6} + 42 \beta_{7} ) q^{94} + ( 40 - 59 \beta_{1} + 40 \beta_{2} - 440 \beta_{3} - 44 \beta_{5} + 44 \beta_{7} ) q^{95} + ( -32 + 32 \beta_{4} + 32 \beta_{7} ) q^{96} + ( 819 - 10 \beta_{1} + 809 \beta_{3} - 941 \beta_{4} + 10 \beta_{5} + 10 \beta_{6} ) q^{97} + ( 474 - 10 \beta_{1} + 124 \beta_{2} - 124 \beta_{4} - 4 \beta_{6} + 10 \beta_{7} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{2} - 7q^{3} - 8q^{4} - 30q^{5} + 6q^{6} - 4q^{7} - 16q^{8} - 81q^{9} + O(q^{10})$$ $$8q - 4q^{2} - 7q^{3} - 8q^{4} - 30q^{5} + 6q^{6} - 4q^{7} - 16q^{8} - 81q^{9} + 100q^{10} + 32q^{12} + 48q^{13} - 8q^{14} - 279q^{15} - 32q^{16} + 109q^{17} - 42q^{18} - 288q^{19} - 120q^{20} + 50q^{21} + 628q^{23} + 24q^{24} + 38q^{25} - 14q^{26} + 242q^{27} + 4q^{28} - 528q^{29} - 558q^{30} - 522q^{31} + 256q^{32} + 208q^{34} + 17q^{35} - 84q^{36} - 406q^{37} + 544q^{38} + 1429q^{39} + 40q^{40} + 329q^{41} - 1480q^{42} - 1442q^{43} + 2652q^{45} - 1044q^{46} + 666q^{47} - 112q^{48} - 114q^{49} - 34q^{50} - 1158q^{51} - 28q^{52} + 414q^{53} + 1144q^{54} + 48q^{56} + 593q^{57} - 1056q^{58} - 888q^{59} + 844q^{60} + 302q^{61} + 646q^{62} + 2061q^{63} - 128q^{64} + 138q^{65} + 578q^{67} + 436q^{68} + 1930q^{69} + 1394q^{70} + 1090q^{71} - 648q^{72} - 253q^{73} - 812q^{74} + 2763q^{75} + 128q^{76} - 4152q^{78} + 674q^{79} + 80q^{80} - 230q^{81} - 722q^{82} + 428q^{83} + 2860q^{84} - 1046q^{85} - 984q^{86} - 2122q^{87} - 2202q^{89} - 1366q^{90} - 2217q^{91} + 832q^{92} - 3721q^{93} - 2138q^{94} + 973q^{95} - 224q^{96} + 3012q^{97} + 3292q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 71 x^{6} - 141 x^{5} + 2911 x^{4} + 2710 x^{3} + 75340 x^{2} + 169400 x + 5856400$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$22554025143 \nu^{7} + 151013926876 \nu^{6} - 1356924294556 \nu^{5} + 28230146638036 \nu^{4} - 111071208987556 \nu^{3} + 8011043329394989 \nu^{2} - 4059658693406480 \nu + 48229141584676800$$$$)/ 342693986325717620$$ $$\beta_{3}$$ $$=$$ $$($$$$14112299722 \nu^{7} + 955320989289 \nu^{6} + 1256420838616 \nu^{5} - 5250112809736 \nu^{4} + 12754848436776 \nu^{3} - 1313471371492594 \nu^{2} + 8498101996878245 \nu - 66484077419717320$$$$)/ 171346993162858810$$ $$\beta_{4}$$ $$=$$ $$($$$$-123788106187 \nu^{7} + 5732748190805 \nu^{6} - 11662655347575 \nu^{5} + 374269556401525 \nu^{4} - 772584075715295 \nu^{3} + 13504262211521188 \nu^{2} + 13624366122837560 \nu + 426480737760227920$$$$)/ 685387972651435240$$ $$\beta_{5}$$ $$=$$ $$($$$$15778904729 \nu^{7} - 268932734519 \nu^{6} + 2855478562109 \nu^{5} - 16065997834439 \nu^{4} + 722720174659769 \nu^{3} - 523534449789100 \nu^{2} + 4037135429586600 \nu - 12007762986133200$$$$)/ 31153998756883420$$ $$\beta_{6}$$ $$=$$ $$($$$$-15728307097 \nu^{7} - 23011705883 \nu^{6} - 3489045673392 \nu^{5} - 619537249643 \nu^{4} - 38031215360567 \nu^{3} - 53736302216310 \nu^{2} + 582451195267480 \nu - 22785233088495405$$$$)/ 7788499689220855$$ $$\beta_{7}$$ $$=$$ $$($$$$254952731119 \nu^{7} - 130622718559 \nu^{6} + 16218883337689 \nu^{5} - 18738040845679 \nu^{4} + 629078544513089 \nu^{3} + 1043207365589370 \nu^{2} + 20338656497650260 \nu + 32952393866979400$$$$)/ 31153998756883420$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{6} + \beta_{5} - 8 \beta_{4} + 8 \beta_{3} + 54 \beta_{2} + \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$8 \beta_{6} + 47 \beta_{5} + 46 \beta_{4} + 8 \beta_{3} - 8 \beta_{1} + 16$$ $$\nu^{4}$$ $$=$$ $$109 \beta_{7} + 16 \beta_{6} - 16 \beta_{5} + 2226 \beta_{4} - 3034 \beta_{3} - 3034 \beta_{2} - 2210$$ $$\nu^{5}$$ $$=$$ $$-824 \beta_{7} - 3143 \beta_{6} - 1608 \beta_{4} + 1608 \beta_{2} + 824 \beta_{1} - 7549$$ $$\nu^{6}$$ $$=$$ $$-2432 \beta_{7} + 2432 \beta_{5} + 176314 \beta_{3} + 63048 \beta_{2} - 6725 \beta_{1} + 63048$$ $$\nu^{7}$$ $$=$$ $$185471 \beta_{7} + 185471 \beta_{6} - 119991 \beta_{5} + 185128 \beta_{4} - 185128 \beta_{3} - 513934 \beta_{2} - 119991 \beta_{1} + 185471$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/242\mathbb{Z}\right)^\times$$.

 $$n$$ $$123$$ $$\chi(n)$$ $$\beta_{3}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 2.22300 − 6.84169i −2.53202 + 7.79275i 5.60402 + 4.07156i −4.79501 − 3.48378i 5.60402 − 4.07156i −4.79501 + 3.48378i 2.22300 + 6.84169i −2.53202 − 7.79275i
−1.61803 + 1.17557i −2.41398 7.42948i 1.23607 3.80423i −12.0690 8.76866i 12.6398 + 9.18334i −6.72664 + 20.7025i 2.47214 + 7.60845i −27.5264 + 19.9991i 29.8363
3.2 −1.61803 + 1.17557i 2.34103 + 7.20496i 1.23607 3.80423i −4.37525 3.17880i −12.2578 8.90583i 6.84467 21.0657i 2.47214 + 7.60845i −24.5876 + 17.8639i 10.8162
9.1 0.618034 1.90211i −6.91304 + 5.02262i −3.23607 2.35114i 3.93561 + 12.1126i 5.28109 + 16.2535i 18.9804 + 13.7901i −6.47214 + 4.70228i 14.2200 43.7646i 25.4718
9.2 0.618034 1.90211i 3.48599 2.53272i −3.23607 2.35114i −2.49134 7.66756i −2.66306 8.19605i −21.0985 15.3289i −6.47214 + 4.70228i −2.60601 + 8.02046i −16.1243
27.1 0.618034 + 1.90211i −6.91304 5.02262i −3.23607 + 2.35114i 3.93561 12.1126i 5.28109 16.2535i 18.9804 13.7901i −6.47214 4.70228i 14.2200 + 43.7646i 25.4718
27.2 0.618034 + 1.90211i 3.48599 + 2.53272i −3.23607 + 2.35114i −2.49134 + 7.66756i −2.66306 + 8.19605i −21.0985 + 15.3289i −6.47214 4.70228i −2.60601 8.02046i −16.1243
81.1 −1.61803 1.17557i −2.41398 + 7.42948i 1.23607 + 3.80423i −12.0690 + 8.76866i 12.6398 9.18334i −6.72664 20.7025i 2.47214 7.60845i −27.5264 19.9991i 29.8363
81.2 −1.61803 1.17557i 2.34103 7.20496i 1.23607 + 3.80423i −4.37525 + 3.17880i −12.2578 + 8.90583i 6.84467 + 21.0657i 2.47214 7.60845i −24.5876 17.8639i 10.8162
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 81.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

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

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.d odd 10 1
242.4.a.o 4 11.c even 5 1
242.4.c.n 8 1.a even 1 1 trivial
242.4.c.n 8 11.c even 5 1 inner
242.4.c.q 8 11.c even 5 2
242.4.c.r 8 11.b odd 2 1
242.4.c.r 8 11.d odd 10 1
1936.4.a.bm 4 44.h odd 10 1
1936.4.a.bn 4 44.g even 10 1
2178.4.a.bt 4 33.h odd 10 1
2178.4.a.by 4 33.f even 10 1

Hecke kernels

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

 $$T_{3}^{8} + \cdots$$ $$T_{5}^{8} + \cdots$$ $$T_{7}^{8} + \cdots$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T + 4 T^{2} + 8 T^{3} + 16 T^{4} )^{2}$$
$3$ $$1 + 7 T + 38 T^{2} + 98 T^{3} + 62 T^{4} + 4223 T^{5} + 33219 T^{6} + 213996 T^{7} + 1147672 T^{8} + 5777892 T^{9} + 24216651 T^{10} + 83121309 T^{11} + 32949342 T^{12} + 1406192886 T^{13} + 14721978582 T^{14} + 73222472421 T^{15} + 282429536481 T^{16}$$
$5$ $$1 + 30 T + 306 T^{2} + 920 T^{3} - 569 T^{4} + 85310 T^{5} + 1402636 T^{6} + 18705240 T^{7} + 272699281 T^{8} + 2338155000 T^{9} + 21916187500 T^{10} + 166621093750 T^{11} - 138916015625 T^{12} + 28076171875000 T^{13} + 1167297363281250 T^{14} + 14305114746093750 T^{15} + 59604644775390625 T^{16}$$
$7$ $$1 + 4 T - 278 T^{2} - 1486 T^{3} - 85843 T^{4} + 588616 T^{5} + 34111706 T^{6} - 31711312 T^{7} + 2545738687 T^{8} - 10876980016 T^{9} + 4013208099194 T^{10} + 23752778737912 T^{11} - 1188177617195443 T^{12} - 7054876403775298 T^{13} - 452698980219104822 T^{14} + 2234183456333136028 T^{15} +$$$$19\!\cdots\!01$$$$T^{16}$$
$11$ 1
$13$ $$1 - 48 T - 5550 T^{2} + 362926 T^{3} + 4851591 T^{4} - 1163685872 T^{5} + 49432845496 T^{6} + 1273897050480 T^{7} - 189168222289331 T^{8} + 2798751819904560 T^{9} + 238602903535702264 T^{10} - 12340306099992958256 T^{11} +$$$$11\!\cdots\!71$$$$T^{12} +$$$$18\!\cdots\!82$$$$T^{13} -$$$$62\!\cdots\!50$$$$T^{14} -$$$$11\!\cdots\!24$$$$T^{15} +$$$$54\!\cdots\!61$$$$T^{16}$$
$17$ $$1 - 109 T - 1340 T^{2} - 17028 T^{3} + 61854916 T^{4} - 3258771851 T^{5} + 82359392839 T^{6} - 11257525677000 T^{7} + 1063382720825544 T^{8} - 55308223651101000 T^{9} + 1987955527449468391 T^{10} -$$$$38\!\cdots\!47$$$$T^{11} +$$$$36\!\cdots\!76$$$$T^{12} -$$$$48\!\cdots\!04$$$$T^{13} -$$$$18\!\cdots\!60$$$$T^{14} -$$$$75\!\cdots\!53$$$$T^{15} +$$$$33\!\cdots\!21$$$$T^{16}$$
$19$ $$1 + 288 T + 30566 T^{2} + 1822176 T^{3} + 176552535 T^{4} + 24586907808 T^{5} + 2222669375260 T^{6} + 119753076108480 T^{7} + 6107895510065021 T^{8} + 821386349028064320 T^{9} +$$$$10\!\cdots\!60$$$$T^{10} +$$$$79\!\cdots\!32$$$$T^{11} +$$$$39\!\cdots\!35$$$$T^{12} +$$$$27\!\cdots\!24$$$$T^{13} +$$$$31\!\cdots\!06$$$$T^{14} +$$$$20\!\cdots\!72$$$$T^{15} +$$$$48\!\cdots\!21$$$$T^{16}$$
$23$ $$( 1 - 314 T + 61816 T^{2} - 8042722 T^{3} + 950275950 T^{4} - 97855798574 T^{5} + 9150986514424 T^{6} - 565561935699382 T^{7} + 21914624432020321 T^{8} )^{2}$$
$29$ $$1 + 528 T + 82486 T^{2} - 8724144 T^{3} - 5230024245 T^{4} - 769204136712 T^{5} - 8600199101200 T^{6} + 17285931767133480 T^{7} + 3853264451262183881 T^{8} +$$$$42\!\cdots\!20$$$$T^{9} -$$$$51\!\cdots\!00$$$$T^{10} -$$$$11\!\cdots\!28$$$$T^{11} -$$$$18\!\cdots\!45$$$$T^{12} -$$$$75\!\cdots\!56$$$$T^{13} +$$$$17\!\cdots\!46$$$$T^{14} +$$$$27\!\cdots\!12$$$$T^{15} +$$$$12\!\cdots\!81$$$$T^{16}$$
$31$ $$1 + 522 T + 160492 T^{2} + 35181052 T^{3} + 6294650003 T^{4} + 763981141902 T^{5} + 33910403803870 T^{6} - 9279992614495420 T^{7} - 2496218287452452949 T^{8} -$$$$27\!\cdots\!20$$$$T^{9} +$$$$30\!\cdots\!70$$$$T^{10} +$$$$20\!\cdots\!42$$$$T^{11} +$$$$49\!\cdots\!83$$$$T^{12} +$$$$82\!\cdots\!52$$$$T^{13} +$$$$11\!\cdots\!72$$$$T^{14} +$$$$10\!\cdots\!82$$$$T^{15} +$$$$62\!\cdots\!21$$$$T^{16}$$
$37$ $$1 + 406 T - 5318 T^{2} - 27663524 T^{3} - 3521516073 T^{4} + 1758227311414 T^{5} + 498571849360436 T^{6} - 45032760859024008 T^{7} - 32601870976696136623 T^{8} -$$$$22\!\cdots\!24$$$$T^{9} +$$$$12\!\cdots\!24$$$$T^{10} +$$$$22\!\cdots\!78$$$$T^{11} -$$$$23\!\cdots\!13$$$$T^{12} -$$$$92\!\cdots\!32$$$$T^{13} -$$$$89\!\cdots\!22$$$$T^{14} +$$$$34\!\cdots\!22$$$$T^{15} +$$$$43\!\cdots\!61$$$$T^{16}$$
$41$ $$1 - 329 T - 55160 T^{2} + 17291460 T^{3} + 3867540040 T^{4} - 810769263487 T^{5} + 94427755341103 T^{6} + 42504597447646080 T^{7} - 34767930814290379080 T^{8} +$$$$29\!\cdots\!80$$$$T^{9} +$$$$44\!\cdots\!23$$$$T^{10} -$$$$26\!\cdots\!07$$$$T^{11} +$$$$87\!\cdots\!40$$$$T^{12} +$$$$26\!\cdots\!60$$$$T^{13} -$$$$59\!\cdots\!60$$$$T^{14} -$$$$24\!\cdots\!89$$$$T^{15} +$$$$50\!\cdots\!61$$$$T^{16}$$
$43$ $$( 1 + 721 T + 420117 T^{2} + 154459221 T^{3} + 51447883420 T^{4} + 12280589284047 T^{5} + 2655712080056733 T^{6} + 362369273206463803 T^{7} + 39959630797262576401 T^{8} )^{2}$$
$47$ $$1 - 666 T + 215830 T^{2} - 88278972 T^{3} + 49867598541 T^{4} - 15263379264354 T^{5} + 2694775262754914 T^{6} - 1173324858240942840 T^{7} +$$$$55\!\cdots\!99$$$$T^{8} -$$$$12\!\cdots\!20$$$$T^{9} +$$$$29\!\cdots\!06$$$$T^{10} -$$$$17\!\cdots\!18$$$$T^{11} +$$$$57\!\cdots\!81$$$$T^{12} -$$$$10\!\cdots\!96$$$$T^{13} +$$$$27\!\cdots\!70$$$$T^{14} -$$$$86\!\cdots\!02$$$$T^{15} +$$$$13\!\cdots\!81$$$$T^{16}$$
$53$ $$1 - 414 T + 111938 T^{2} - 31720518 T^{3} + 42174716831 T^{4} - 4836347208574 T^{5} - 2624686559799020 T^{6} + 993826932968752496 T^{7} +$$$$38\!\cdots\!49$$$$T^{8} +$$$$14\!\cdots\!92$$$$T^{9} -$$$$58\!\cdots\!80$$$$T^{10} -$$$$15\!\cdots\!42$$$$T^{11} +$$$$20\!\cdots\!71$$$$T^{12} -$$$$23\!\cdots\!26$$$$T^{13} +$$$$12\!\cdots\!82$$$$T^{14} -$$$$67\!\cdots\!42$$$$T^{15} +$$$$24\!\cdots\!81$$$$T^{16}$$
$59$ $$1 + 888 T - 72874 T^{2} - 366469944 T^{3} - 125594098825 T^{4} + 29314994105528 T^{5} + 26622351545153740 T^{6} + 1700698924437048800 T^{7} -$$$$25\!\cdots\!59$$$$T^{8} +$$$$34\!\cdots\!00$$$$T^{9} +$$$$11\!\cdots\!40$$$$T^{10} +$$$$25\!\cdots\!92$$$$T^{11} -$$$$22\!\cdots\!25$$$$T^{12} -$$$$13\!\cdots\!56$$$$T^{13} -$$$$54\!\cdots\!54$$$$T^{14} +$$$$13\!\cdots\!92$$$$T^{15} +$$$$31\!\cdots\!61$$$$T^{16}$$
$61$ $$1 - 302 T - 374758 T^{2} + 265079458 T^{3} + 17172817983 T^{4} - 53588613835342 T^{5} + 17468090773098500 T^{6} + 4210498173089967520 T^{7} -$$$$41\!\cdots\!19$$$$T^{8} +$$$$95\!\cdots\!20$$$$T^{9} +$$$$89\!\cdots\!00$$$$T^{10} -$$$$62\!\cdots\!22$$$$T^{11} +$$$$45\!\cdots\!43$$$$T^{12} +$$$$15\!\cdots\!58$$$$T^{13} -$$$$51\!\cdots\!98$$$$T^{14} -$$$$93\!\cdots\!22$$$$T^{15} +$$$$70\!\cdots\!41$$$$T^{16}$$
$67$ $$( 1 - 289 T + 686735 T^{2} - 243308709 T^{3} + 267199450216 T^{4} - 73178257244967 T^{5} + 62120937078828215 T^{6} - 7862688440529239683 T^{7} +$$$$81\!\cdots\!61$$$$T^{8} )^{2}$$
$71$ $$1 - 1090 T - 143028 T^{2} + 705054350 T^{3} - 262816671537 T^{4} - 195394750802610 T^{5} + 164095529988407974 T^{6} + 23390610332557696900 T^{7} -$$$$62\!\cdots\!45$$$$T^{8} +$$$$83\!\cdots\!00$$$$T^{9} +$$$$21\!\cdots\!54$$$$T^{10} -$$$$89\!\cdots\!10$$$$T^{11} -$$$$43\!\cdots\!17$$$$T^{12} +$$$$41\!\cdots\!50$$$$T^{13} -$$$$30\!\cdots\!08$$$$T^{14} -$$$$82\!\cdots\!90$$$$T^{15} +$$$$26\!\cdots\!81$$$$T^{16}$$
$73$ $$1 + 253 T + 242148 T^{2} - 153484248 T^{3} + 27413114052 T^{4} - 97831236100713 T^{5} + 36243442544494539 T^{6} - 8871101046105798936 T^{7} +$$$$23\!\cdots\!32$$$$T^{8} -$$$$34\!\cdots\!12$$$$T^{9} +$$$$54\!\cdots\!71$$$$T^{10} -$$$$57\!\cdots\!69$$$$T^{11} +$$$$62\!\cdots\!92$$$$T^{12} -$$$$13\!\cdots\!36$$$$T^{13} +$$$$83\!\cdots\!12$$$$T^{14} +$$$$34\!\cdots\!69$$$$T^{15} +$$$$52\!\cdots\!41$$$$T^{16}$$
$79$ $$1 - 674 T - 776972 T^{2} + 654255424 T^{3} + 174326887083 T^{4} - 318069877959854 T^{5} + 120607404735943670 T^{6} + 80049450610837979100 T^{7} -$$$$13\!\cdots\!89$$$$T^{8} +$$$$39\!\cdots\!00$$$$T^{9} +$$$$29\!\cdots\!70$$$$T^{10} -$$$$38\!\cdots\!26$$$$T^{11} +$$$$10\!\cdots\!03$$$$T^{12} +$$$$19\!\cdots\!76$$$$T^{13} -$$$$11\!\cdots\!92$$$$T^{14} -$$$$47\!\cdots\!46$$$$T^{15} +$$$$34\!\cdots\!81$$$$T^{16}$$
$83$ $$1 - 428 T + 177930 T^{2} + 134660776 T^{3} - 247636118949 T^{4} + 379957559050908 T^{5} + 90010500467785276 T^{6} -$$$$15\!\cdots\!00$$$$T^{7} +$$$$16\!\cdots\!89$$$$T^{8} -$$$$86\!\cdots\!00$$$$T^{9} +$$$$29\!\cdots\!44$$$$T^{10} +$$$$71\!\cdots\!24$$$$T^{11} -$$$$26\!\cdots\!89$$$$T^{12} +$$$$82\!\cdots\!32$$$$T^{13} +$$$$62\!\cdots\!70$$$$T^{14} -$$$$85\!\cdots\!24$$$$T^{15} +$$$$11\!\cdots\!21$$$$T^{16}$$
$89$ $$( 1 + 1101 T + 2406895 T^{2} + 1914547747 T^{3} + 2334666485008 T^{4} + 1349696810654843 T^{5} + 1196181784307576095 T^{6} +$$$$38\!\cdots\!09$$$$T^{7} +$$$$24\!\cdots\!21$$$$T^{8} )^{2}$$
$97$ $$1 - 3012 T + 5947742 T^{2} - 10226856716 T^{3} + 16448850705271 T^{4} - 21915763249625172 T^{5} + 26024459406245646260 T^{6} -$$$$28\!\cdots\!48$$$$T^{7} +$$$$29\!\cdots\!09$$$$T^{8} -$$$$26\!\cdots\!04$$$$T^{9} +$$$$21\!\cdots\!40$$$$T^{10} -$$$$16\!\cdots\!24$$$$T^{11} +$$$$11\!\cdots\!11$$$$T^{12} -$$$$64\!\cdots\!88$$$$T^{13} +$$$$34\!\cdots\!38$$$$T^{14} -$$$$15\!\cdots\!64$$$$T^{15} +$$$$48\!\cdots\!81$$$$T^{16}$$