# Properties

 Label 289.10.a.c Level $289$ Weight $10$ Character orbit 289.a Self dual yes Analytic conductor $148.845$ Analytic rank $1$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$289 = 17^{2}$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 289.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$148.845356651$$ Analytic rank: $$1$$ Dimension: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ Defining polynomial: $$x^{12} - 122690 x^{10} + 5157152560 x^{8} - 87983684680032 x^{6} + 612743619071665152 x^{4} - 1335826553351738886144 x^{2} + 203949399568932198678528$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{17}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 17) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + \beta_{2} ) q^{2} -\beta_{1} q^{3} + ( 154 - 4 \beta_{2} - \beta_{3} ) q^{4} + ( -\beta_{1} - \beta_{8} ) q^{5} + \beta_{6} q^{6} + ( \beta_{1} + \beta_{9} ) q^{7} + ( -1904 + 117 \beta_{2} + 3 \beta_{3} - \beta_{5} ) q^{8} + ( 801 + 69 \beta_{2} + 8 \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( -2 + \beta_{2} ) q^{2} -\beta_{1} q^{3} + ( 154 - 4 \beta_{2} - \beta_{3} ) q^{4} + ( -\beta_{1} - \beta_{8} ) q^{5} + \beta_{6} q^{6} + ( \beta_{1} + \beta_{9} ) q^{7} + ( -1904 + 117 \beta_{2} + 3 \beta_{3} - \beta_{5} ) q^{8} + ( 801 + 69 \beta_{2} + 8 \beta_{3} + \beta_{4} ) q^{9} + ( 2 \beta_{1} + 8 \beta_{8} - \beta_{9} + \beta_{11} ) q^{10} + ( 26 \beta_{1} - 11 \beta_{8} - \beta_{9} - \beta_{10} ) q^{11} + ( -26 \beta_{1} + 4 \beta_{8} + 3 \beta_{9} + \beta_{10} - \beta_{11} ) q^{12} + ( -5362 - 194 \beta_{2} + 16 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{13} + ( -50 \beta_{1} + 7 \beta_{6} + 16 \beta_{8} - 3 \beta_{9} + 2 \beta_{10} - \beta_{11} ) q^{14} + ( 20300 + 3 \beta_{2} + 36 \beta_{3} + 6 \beta_{4} + 8 \beta_{5} - \beta_{7} ) q^{15} + ( 2624 - 1239 \beta_{2} - 57 \beta_{3} - 8 \beta_{4} + 7 \beta_{5} ) q^{16} + ( 44140 - 2978 \beta_{2} - 166 \beta_{3} - 12 \beta_{4} + 27 \beta_{5} - 2 \beta_{7} ) q^{18} + ( -90936 + 3253 \beta_{2} + 40 \beta_{3} - 20 \beta_{4} - 32 \beta_{5} - \beta_{7} ) q^{19} + ( 418 \beta_{1} - 78 \beta_{6} - 312 \beta_{8} - \beta_{9} - 13 \beta_{10} - 13 \beta_{11} ) q^{20} + ( -13676 + 1525 \beta_{2} - 276 \beta_{3} - 13 \beta_{4} - 24 \beta_{5} - 2 \beta_{7} ) q^{21} + ( 76 \beta_{1} - 161 \beta_{6} + 432 \beta_{8} - 118 \beta_{9} - 4 \beta_{10} + 30 \beta_{11} ) q^{22} + ( 834 \beta_{1} - 76 \beta_{6} + 105 \beta_{8} - 120 \beta_{9} - 5 \beta_{10} - 20 \beta_{11} ) q^{23} + ( -34 \beta_{1} - 266 \beta_{6} + 400 \beta_{8} + 117 \beta_{9} + 19 \beta_{10} - 19 \beta_{11} ) q^{24} + ( 360571 - 18985 \beta_{2} - 1012 \beta_{3} - 12 \beta_{4} + 162 \beta_{5} - \beta_{7} ) q^{25} + ( -117282 - 11691 \beta_{2} - 626 \beta_{3} - 116 \beta_{4} + 17 \beta_{5} + 2 \beta_{7} ) q^{26} + ( 781 \beta_{1} - 392 \beta_{6} - 721 \beta_{8} - 117 \beta_{9} + 17 \beta_{10} + 16 \beta_{11} ) q^{27} + ( -4040 \beta_{1} + 350 \beta_{6} - 124 \beta_{8} - 234 \beta_{9} + 16 \beta_{10} - 50 \beta_{11} ) q^{28} + ( 2677 \beta_{1} - 84 \beta_{6} + 801 \beta_{8} + 578 \beta_{9} + 2 \beta_{10} - 28 \beta_{11} ) q^{29} + ( -40240 + 1880 \beta_{2} + 2864 \beta_{3} + 80 \beta_{4} + 144 \beta_{5} - 16 \beta_{7} ) q^{30} + ( 6220 \beta_{1} + 1428 \beta_{6} + 797 \beta_{8} - 692 \beta_{9} + 3 \beta_{10} + 4 \beta_{11} ) q^{31} + ( 146616 - 30707 \beta_{2} + 3671 \beta_{3} + 152 \beta_{4} + 275 \beta_{5} + 16 \beta_{7} ) q^{32} + ( -548356 - 5660 \beta_{2} + 4604 \beta_{3} - 123 \beta_{4} + 274 \beta_{5} - \beta_{7} ) q^{33} + ( 265112 - 56658 \beta_{2} - 228 \beta_{3} - 30 \beta_{4} + 268 \beta_{5} + 36 \beta_{7} ) q^{35} + ( -2476562 + 84034 \beta_{2} + 11903 \beta_{3} + 24 \beta_{4} - 450 \beta_{5} + 16 \beta_{7} ) q^{36} + ( 8457 \beta_{1} - 2360 \beta_{6} - 1091 \beta_{8} - 116 \beta_{9} + 164 \beta_{10} + 32 \beta_{11} ) q^{37} + ( 2337892 - 110114 \beta_{2} - 15852 \beta_{3} + 72 \beta_{4} - 186 \beta_{5} + 36 \beta_{7} ) q^{38} + ( -12867 \beta_{1} - 4004 \beta_{6} + 6823 \beta_{8} - 849 \beta_{9} + 73 \beta_{10} - 268 \beta_{11} ) q^{39} + ( 43330 \beta_{1} - 1258 \beta_{6} + 12424 \beta_{8} - 1305 \beta_{9} + 37 \beta_{10} + 191 \beta_{11} ) q^{40} + ( 22880 \beta_{1} - 4076 \beta_{6} + 8384 \beta_{8} + 102 \beta_{9} - 262 \beta_{10} + 236 \beta_{11} ) q^{41} + ( 1047954 + 116667 \beta_{2} - 11714 \beta_{3} + 140 \beta_{4} - 375 \beta_{5} + 18 \beta_{7} ) q^{42} + ( -827392 + 7479 \beta_{2} + 16644 \beta_{3} + 834 \beta_{4} - 60 \beta_{5} - 37 \beta_{7} ) q^{43} + ( 74518 \beta_{1} - 3332 \beta_{6} - 21236 \beta_{8} + 15 \beta_{9} - 223 \beta_{10} - 261 \beta_{11} ) q^{44} + ( -99203 \beta_{1} + 2828 \beta_{6} + 5357 \beta_{8} - 714 \beta_{9} - 82 \beta_{10} + 508 \beta_{11} ) q^{45} + ( 49258 \beta_{1} - 749 \beta_{6} + 13616 \beta_{8} - 73 \beta_{9} - 106 \beta_{10} + 301 \beta_{11} ) q^{46} + ( -9381820 - 17509 \beta_{2} + 39300 \beta_{3} - 822 \beta_{4} - 864 \beta_{5} + 99 \beta_{7} ) q^{47} + ( 152126 \beta_{1} + 5018 \beta_{6} + 2856 \beta_{8} + 33 \beta_{9} - 297 \beta_{10} + 33 \beta_{11} ) q^{48} + ( -10141779 - 69042 \beta_{2} + 37716 \beta_{3} - 1583 \beta_{4} - 1730 \beta_{5} - 99 \beta_{7} ) q^{49} + ( -13311954 + 829025 \beta_{2} + 92764 \beta_{3} + 1528 \beta_{4} - 1862 \beta_{5} + 20 \beta_{7} ) q^{50} + ( -4780424 + 282778 \beta_{2} + 23146 \beta_{3} + 840 \beta_{4} - 1926 \beta_{5} - 272 \beta_{7} ) q^{52} + ( -6408638 - 48119 \beta_{2} + 84412 \beta_{3} + 658 \beta_{4} + 3734 \beta_{5} - 97 \beta_{7} ) q^{53} + ( 216072 \beta_{1} - 1682 \beta_{6} - 15696 \beta_{8} + 132 \beta_{9} - 768 \beta_{10} + 828 \beta_{11} ) q^{54} + ( 24337572 - 1405615 \beta_{2} - 128788 \beta_{3} - 74 \beta_{4} + 1840 \beta_{5} - 271 \beta_{7} ) q^{55} + ( -144416 \beta_{1} + 4116 \beta_{6} + 21896 \beta_{8} + 5524 \beta_{9} - 560 \beta_{10} + 532 \beta_{11} ) q^{56} + ( 373688 \beta_{1} + 2376 \beta_{6} + 20052 \beta_{8} + 6732 \beta_{9} + 144 \beta_{10} - 576 \beta_{11} ) q^{57} + ( 18266 \beta_{1} + 5108 \beta_{6} + 22616 \beta_{8} - 629 \beta_{9} + 1384 \beta_{10} - 1163 \beta_{11} ) q^{58} + ( -192488 + 1578741 \beta_{2} - 74844 \beta_{3} - 906 \beta_{4} - 2452 \beta_{5} - 135 \beta_{7} ) q^{59} + ( -9167952 - 1414208 \beta_{2} + 36808 \beta_{3} - 1472 \beta_{4} + 128 \beta_{5} + 288 \beta_{7} ) q^{60} + ( 585715 \beta_{1} + 8036 \beta_{6} + 3407 \beta_{8} + 1242 \beta_{9} + 1138 \beta_{10} + 1180 \beta_{11} ) q^{61} + ( -735782 \beta_{1} - 10915 \beta_{6} - 15792 \beta_{8} + 7439 \beta_{9} + 6 \beta_{10} - 1611 \beta_{11} ) q^{62} + ( 206858 \beta_{1} + 4452 \beta_{6} + 6445 \beta_{8} - 12702 \beta_{9} + 439 \beta_{10} - 412 \beta_{11} ) q^{63} + ( -22104712 - 924323 \beta_{2} + 175591 \beta_{3} + 3064 \beta_{4} + 1459 \beta_{5} - 240 \beta_{7} ) q^{64} + ( 775200 \beta_{1} - 24452 \beta_{6} + 28272 \beta_{8} + 18174 \beta_{9} + 2258 \beta_{10} - 140 \beta_{11} ) q^{65} + ( -2895094 - 2767993 \beta_{2} + 147670 \beta_{3} + 3756 \beta_{4} + 1197 \beta_{5} + 242 \beta_{7} ) q^{66} + ( -24264096 + 2114305 \beta_{2} + 166768 \beta_{3} - 4636 \beta_{4} + 2000 \beta_{5} - 173 \beta_{7} ) q^{67} + ( -18400100 - 1580886 \beta_{2} - 376 \beta_{3} - 2027 \beta_{4} - 2882 \beta_{5} + 1057 \beta_{7} ) q^{69} + ( -38119084 + 420134 \beta_{2} + 187964 \beta_{3} - 664 \beta_{4} - 2806 \beta_{5} + 204 \beta_{7} ) q^{70} + ( 844405 \beta_{1} + 42648 \beta_{6} - 4950 \beta_{8} + 1343 \beta_{9} + 1298 \beta_{10} - 6040 \beta_{11} ) q^{71} + ( 37789784 - 6560419 \beta_{2} - 191545 \beta_{3} + 848 \beta_{4} - 81 \beta_{5} + 1040 \beta_{7} ) q^{72} + ( -821622 \beta_{1} + 38500 \beta_{6} - 14898 \beta_{8} - 722 \beta_{9} + 2246 \beta_{10} + 2004 \beta_{11} ) q^{73} + ( 1272910 \beta_{1} + 6116 \beta_{6} - 85416 \beta_{8} + 9833 \beta_{9} - 2616 \beta_{10} + 423 \beta_{11} ) q^{74} + ( 217609 \beta_{1} + 52888 \beta_{6} - 61180 \beta_{8} - 15300 \beta_{9} - 2872 \beta_{10} + 2704 \beta_{11} ) q^{75} + ( -30492960 + 8354796 \beta_{2} - 11964 \beta_{3} + 4720 \beta_{4} + 1708 \beta_{5} + 512 \beta_{7} ) q^{76} + ( -14552412 - 6047384 \beta_{2} + 19152 \beta_{3} - 1281 \beta_{4} - 3458 \beta_{5} - 1155 \beta_{7} ) q^{77} + ( 2216968 \beta_{1} + 41734 \beta_{6} + 88928 \beta_{8} + 8604 \beta_{9} - 2608 \beta_{10} - 1676 \beta_{11} ) q^{78} + ( -2172530 \beta_{1} - 6484 \beta_{6} + 27441 \beta_{8} + 5796 \beta_{9} + 1507 \beta_{10} + 60 \beta_{11} ) q^{79} + ( 478594 \beta_{1} - 11634 \beta_{6} - 120984 \beta_{8} + 14855 \beta_{9} + 761 \beta_{10} - 5017 \beta_{11} ) q^{80} + ( -34349691 - 8118834 \beta_{2} - 73732 \beta_{3} - 6203 \beta_{4} - 1818 \beta_{5} - 255 \beta_{7} ) q^{81} + ( 2141860 \beta_{1} - 62008 \beta_{6} - 163008 \beta_{8} - 35802 \beta_{9} - 6992 \beta_{10} - 1110 \beta_{11} ) q^{82} + ( 63349312 - 14153191 \beta_{2} + 34052 \beta_{3} + 5498 \beta_{4} + 1724 \beta_{5} + 217 \beta_{7} ) q^{83} + ( 82618212 + 5614866 \beta_{2} - 169172 \beta_{3} + 392 \beta_{4} + 4266 \beta_{5} + 816 \beta_{7} ) q^{84} + ( 6397904 - 8526520 \beta_{2} - 112328 \beta_{3} - 7232 \beta_{4} + 33692 \beta_{5} - 1816 \beta_{7} ) q^{86} + ( -51310188 - 696995 \beta_{2} - 272684 \beta_{3} - 15362 \beta_{4} - 28024 \beta_{5} + 525 \beta_{7} ) q^{87} + ( 1829726 \beta_{1} - 19234 \beta_{6} + 210080 \beta_{8} + 7741 \beta_{9} + 1171 \beta_{10} + 14773 \beta_{11} ) q^{88} + ( -85219534 - 14165234 \beta_{2} + 372588 \beta_{3} - 9093 \beta_{4} - 28726 \beta_{5} - 1701 \beta_{7} ) q^{89} + ( -1561174 \beta_{1} + 50728 \beta_{6} - 391400 \beta_{8} + 867 \beta_{9} - 4352 \beta_{10} - 9043 \beta_{11} ) q^{90} + ( 1518839 \beta_{1} + 37520 \beta_{6} - 239211 \beta_{8} - 53837 \beta_{9} + 11851 \beta_{10} - 1848 \beta_{11} ) q^{91} + ( -85732 \beta_{1} - 32486 \beta_{6} - 352636 \beta_{8} + 57756 \beta_{9} - 1858 \beta_{10} - 2452 \beta_{11} ) q^{92} + ( -125112620 + 22392088 \beta_{2} - 125032 \beta_{3} - 18105 \beta_{4} + 49038 \beta_{5} - 803 \beta_{7} ) q^{93} + ( 6088968 - 27544820 \beta_{2} - 232280 \beta_{3} - 5760 \beta_{4} + 24564 \beta_{5} + 2040 \beta_{7} ) q^{94} + ( 2538588 \beta_{1} - 63784 \beta_{6} + 325260 \beta_{8} - 48840 \beta_{9} + 1936 \beta_{10} - 1672 \beta_{11} ) q^{95} + ( -2692674 \beta_{1} - 49774 \beta_{6} - 124680 \beta_{8} - 75159 \beta_{9} - 5601 \beta_{10} + 6969 \beta_{11} ) q^{96} + ( 1032878 \beta_{1} + 65312 \beta_{6} - 139982 \beta_{8} + 44488 \beta_{9} - 9548 \beta_{10} + 23240 \beta_{11} ) q^{97} + ( -26424984 - 27346936 \beta_{2} - 531898 \beta_{3} + 13868 \beta_{4} + 17133 \beta_{5} + 2770 \beta_{7} ) q^{98} + ( 1952069 \beta_{1} + 132920 \beta_{6} + 163204 \beta_{8} - 16254 \beta_{9} + 5956 \beta_{10} + 9896 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q - 30q^{2} + 1874q^{4} - 23550q^{8} + 9184q^{9} + O(q^{10})$$ $$12q - 30q^{2} + 1874q^{4} - 23550q^{8} + 9184q^{9} - 63204q^{13} + 243480q^{15} + 38978q^{16} + 547706q^{18} - 1110672q^{19} - 172580q^{21} + 4441796q^{25} - 1336332q^{26} - 500496q^{30} + 1934850q^{32} - 6557404q^{33} + 3519864q^{35} - 30244102q^{36} + 28748136q^{38} + 11901296q^{42} - 10004616q^{43} - 112552440q^{47} - 121354720q^{49} - 164889018q^{50} - 59093180q^{52} - 76804272q^{53} + 300732568q^{55} - 11618904q^{59} - 101609232q^{60} - 260062974q^{64} - 18429632q^{66} - 304208752q^{67} - 211308236q^{69} - 460311456q^{70} + 493218954q^{72} - 416024248q^{76} - 138357828q^{77} - 363335792q^{81} + 845042136q^{83} + 958037984q^{84} + 127952904q^{86} - 610860648q^{87} - 938223804q^{89} - 1635779524q^{93} + 238629952q^{94} - 152046078q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 122690 x^{10} + 5157152560 x^{8} - 87983684680032 x^{6} + 612743619071665152 x^{4} - 1335826553351738886144 x^{2} + 203949399568932198678528$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$175241908055785943 \nu^{10} - 19735045019476798563382 \nu^{8} + 707005003891973818402957088 \nu^{6} - 8952673593172053232807179016608 \nu^{4} + 51661184146782169407310710985943808 \nu^{2} - 131897513893675299396877649325524557824$$$$)/$$$$33\!\cdots\!80$$ $$\beta_{3}$$ $$=$$ $$($$$$-1593590262544328399 \nu^{10} + 154489881220918895337766 \nu^{8} - 3787560181122413150958339104 \nu^{6} - 6676284565900242669040343109216 \nu^{4} + 489376099599415519082531407571149056 \nu^{2} - 882724314836126559093043439962154999808$$$$)/$$$$11\!\cdots\!60$$ $$\beta_{4}$$ $$=$$ $$($$$$968684246119061167 \nu^{10} - 86890334924376088416038 \nu^{8} + 1559929595495989709377633312 \nu^{6} + 28813529907795462856320873583968 \nu^{4} - 444088256471534748461416956225047808 \nu^{2} - 1396500583854154726008019874857755303936$$$$)/$$$$12\!\cdots\!40$$ $$\beta_{5}$$ $$=$$ $$($$$$-54817148412898972805 \nu^{10} + 6226553365634794437059746 \nu^{8} - 228914720897391276633280343648 \nu^{6} + 2993923695173799872752867504622304 \nu^{4} - 13018623888515762407136192171474266368 \nu^{2} + 9909358480784700705611617758441276162048$$$$)/$$$$66\!\cdots\!56$$ $$\beta_{6}$$ $$=$$ $$($$$$-175241908055785943 \nu^{11} + 19735045019476798563382 \nu^{9} - 707005003891973818402957088 \nu^{7} + 8952673593172053232807179016608 \nu^{5} - 51661184146782169407310710985943808 \nu^{3} + 138536044237240307274131519368847376384 \nu$$$$)/$$$$33\!\cdots\!80$$ $$\beta_{7}$$ $$=$$ $$($$$$-2879264693276268104777 \nu^{10} + 329435767917512426060932618 \nu^{8} - 12088968068904757696235803965152 \nu^{6} + 152165834653280262812376122788686432 \nu^{4} - 544947798042533004615283215559408342272 \nu^{2} + 39032488563303557944481663656465529020416$$$$)/$$$$33\!\cdots\!80$$ $$\beta_{8}$$ $$=$$ $$($$$$-682121394157411195889 \nu^{11} + 84599136211278102293652706 \nu^{9} - 3623163864366082195709832446384 \nu^{7} + 63776666810768784596995686770627424 \nu^{5} - 456534446722028713200462398063208520704 \nu^{3} + 913834302933669450884956241493859416655872 \nu$$$$)/$$$$44\!\cdots\!60$$ $$\beta_{9}$$ $$=$$ $$($$$$-1942670331239272223513 \nu^{11} + 219956883212419537819652722 \nu^{9} - 7942468264882114951021174302128 \nu^{7} + 96037585979614477301684268041267808 \nu^{5} - 267270207601698508678899591797268231168 \nu^{3} - 769038416385690078965724519671041097005056 \nu$$$$)/$$$$44\!\cdots\!60$$ $$\beta_{10}$$ $$=$$ $$($$$$-42420627941316180088439 \nu^{11} + 4975628272489906759548662926 \nu^{9} - 193806903803732535662297520360464 \nu^{7} + 2892748476468367786780029318620789664 \nu^{5} - 17445414874604406996300223283383915413504 \nu^{3} + 39791181275180337521161030435955451275876352 \nu$$$$)/$$$$82\!\cdots\!60$$ $$\beta_{11}$$ $$=$$ $$($$$$-287888801331355798024579 \nu^{11} + 35108296002539943174974741606 \nu^{9} - 1457263921015624790451129091781584 \nu^{7} + 24163467085675543102357471383402821664 \nu^{5} - 158520641797210764142811229341053875001344 \nu^{3} + 307198095444760634976265501122940707914545152 \nu$$$$)/$$$$49\!\cdots\!60$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + 8 \beta_{3} + 69 \beta_{2} + 20484$$ $$\nu^{3}$$ $$=$$ $$-16 \beta_{11} - 17 \beta_{10} + 117 \beta_{9} + 721 \beta_{8} + 392 \beta_{6} + 38585 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-255 \beta_{7} - 1818 \beta_{5} + 52846 \beta_{4} + 398660 \beta_{3} - 4044453 \beta_{2} + 787789536$$ $$\nu^{5}$$ $$=$$ $$-844672 \beta_{11} - 957386 \beta_{10} + 5636790 \beta_{9} + 38978278 \beta_{8} + 28327880 \beta_{6} + 1729287230 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-669498 \beta_{7} - 317420460 \beta_{5} + 2581679884 \beta_{4} + 19478780120 \beta_{3} - 361368938934 \beta_{2} + 35253701400576$$ $$\nu^{7}$$ $$=$$ $$-33315004576 \beta_{11} - 52033077140 \beta_{10} + 257894873532 \beta_{9} + 1730435781532 \beta_{8} + 1662863762960 \beta_{6} + 81193337780012 \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$828626755356 \beta_{7} - 27163756037592 \beta_{5} + 125092695479896 \beta_{4} + 917219353093424 \beta_{3} - 22517905432046268 \beta_{2} + 1653714802741687488$$ $$\nu^{9}$$ $$=$$ $$-1114858848777280 \beta_{11} - 2824273201838120 \beta_{10} + 11758127163315768 \beta_{9} + 73071389258370424 \beta_{8} + 92063016590283680 \beta_{6} + 3879723595102465304 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$82990382565914232 \beta_{7} - 1871333512473873456 \beta_{5} + 6076749544904463280 \beta_{4} + 42715544905955900768 \beta_{3} - 1285972379408325685560 \beta_{2} + 78965615775605233070976$$ $$\nu^{11}$$ $$=$$ $$-29578589114419661440 \beta_{11} - 152032338642399181904 \beta_{10} + 537167087085552394608 \beta_{9} + 3026402145838710044656 \beta_{8} + 4971725082031200235328 \beta_{6} + 187109107952876417656112 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 12.8394 −12.8394 225.146 −225.146 59.5904 −59.5904 105.759 −105.759 206.667 −206.667 119.947 −119.947
−39.2436 −12.8394 1028.06 −2413.73 503.863 −2302.99 −20252.0 −19518.2 94723.3
1.2 −39.2436 12.8394 1028.06 2413.73 −503.863 2302.99 −20252.0 −19518.2 −94723.3
1.3 −25.8215 −225.146 154.751 −96.4328 5813.61 1385.77 9224.72 31007.8 2490.04
1.4 −25.8215 225.146 154.751 96.4328 −5813.61 −1385.77 9224.72 31007.8 −2490.04
1.5 −11.8575 −59.5904 −371.399 633.821 706.595 −10932.1 10474.9 −16132.0 −7515.55
1.6 −11.8575 59.5904 −371.399 −633.821 −706.595 10932.1 10474.9 −16132.0 7515.55
1.7 9.15386 −105.759 −428.207 1752.99 −968.101 5869.78 −8606.52 −8498.07 16046.7
1.8 9.15386 105.759 −428.207 −1752.99 968.101 −5869.78 −8606.52 −8498.07 −16046.7
1.9 16.7531 −206.667 −231.335 −1946.29 −3462.30 1633.30 −12453.1 23028.2 −32606.3
1.10 16.7531 206.667 −231.335 1946.29 3462.30 −1633.30 −12453.1 23028.2 32606.3
1.11 36.0157 −119.947 785.131 917.335 −4319.97 −4193.73 9837.00 −5295.73 33038.5
1.12 36.0157 119.947 785.131 −917.335 4319.97 4193.73 9837.00 −5295.73 −33038.5
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$17$$ $$1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 289.10.a.c 12
17.b even 2 1 inner 289.10.a.c 12
17.c even 4 2 17.10.b.a 12
51.f odd 4 2 153.10.d.b 12
68.f odd 4 2 272.10.b.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.10.b.a 12 17.c even 4 2
153.10.d.b 12 51.f odd 4 2
272.10.b.c 12 68.f odd 4 2
289.10.a.c 12 1.a even 1 1 trivial
289.10.a.c 12 17.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 $$S_{10}^{\mathrm{new}}(\Gamma_0(289))$$:

 $$T_{2}^{6} + 15 T_{2}^{5} - 1892 T_{2}^{4} - 20460 T_{2}^{3} + 770176 T_{2}^{2} + 3195840 T_{2} - 66364416$$ $$T_{3}^{12} - 122690 T_{3}^{10} + 5157152560 T_{3}^{8} -$$$$87\!\cdots\!32$$$$T_{3}^{6} +$$$$61\!\cdots\!52$$$$T_{3}^{4} -$$$$13\!\cdots\!44$$$$T_{3}^{2} +$$$$20\!\cdots\!28$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -66364416 + 3195840 T + 770176 T^{2} - 20460 T^{3} - 1892 T^{4} + 15 T^{5} + T^{6} )^{2}$$
$3$ $$20\!\cdots\!28$$$$-$$$$13\!\cdots\!44$$$$T^{2} + 612743619071665152 T^{4} - 87983684680032 T^{6} + 5157152560 T^{8} - 122690 T^{10} + T^{12}$$
$5$ $$21\!\cdots\!00$$$$-$$$$23\!\cdots\!00$$$$T^{2} +$$$$10\!\cdots\!00$$$$T^{4} -$$$$13\!\cdots\!96$$$$T^{6} + 67854209805568 T^{8} - 13939648 T^{10} + T^{12}$$
$7$ $$19\!\cdots\!00$$$$-$$$$23\!\cdots\!48$$$$T^{2} +$$$$92\!\cdots\!48$$$$T^{4} -$$$$14\!\cdots\!32$$$$T^{6} + 8551923317087424 T^{8} - 181444282 T^{10} + T^{12}$$
$11$ $$72\!\cdots\!00$$$$-$$$$75\!\cdots\!68$$$$T^{2} +$$$$18\!\cdots\!44$$$$T^{4} -$$$$18\!\cdots\!48$$$$T^{6} + 79438649347703918384 T^{8} - 15005870978 T^{10} + T^{12}$$
$13$ $$( -$$$$63\!\cdots\!00$$$$+$$$$31\!\cdots\!72$$$$T +$$$$58\!\cdots\!04$$$$T^{2} - 2221336902733840 T^{3} - 49809987216 T^{4} + 31602 T^{5} + T^{6} )^{2}$$
$17$ $$T^{12}$$
$19$ $$($$$$83\!\cdots\!00$$$$+$$$$51\!\cdots\!76$$$$T -$$$$89\!\cdots\!80$$$$T^{2} - 490191387006094592 T^{3} - 677476788432 T^{4} + 555336 T^{5} + T^{6} )^{2}$$
$23$ $$16\!\cdots\!88$$$$-$$$$36\!\cdots\!80$$$$T^{2} +$$$$23\!\cdots\!12$$$$T^{4} -$$$$38\!\cdots\!16$$$$T^{6} +$$$$27\!\cdots\!44$$$$T^{8} - 8617041961994 T^{10} + T^{12}$$
$29$ $$89\!\cdots\!52$$$$-$$$$46\!\cdots\!12$$$$T^{2} +$$$$47\!\cdots\!16$$$$T^{4} -$$$$17\!\cdots\!88$$$$T^{6} +$$$$18\!\cdots\!12$$$$T^{8} - 76963053057856 T^{10} + T^{12}$$
$31$ $$53\!\cdots\!00$$$$-$$$$14\!\cdots\!68$$$$T^{2} +$$$$15\!\cdots\!76$$$$T^{4} -$$$$78\!\cdots\!44$$$$T^{6} +$$$$19\!\cdots\!32$$$$T^{8} - 227940215668474 T^{10} + T^{12}$$
$37$ $$81\!\cdots\!00$$$$-$$$$66\!\cdots\!12$$$$T^{2} +$$$$21\!\cdots\!60$$$$T^{4} -$$$$31\!\cdots\!84$$$$T^{6} +$$$$23\!\cdots\!28$$$$T^{8} - 816609714898240 T^{10} + T^{12}$$
$41$ $$55\!\cdots\!00$$$$-$$$$98\!\cdots\!08$$$$T^{2} +$$$$63\!\cdots\!96$$$$T^{4} -$$$$20\!\cdots\!68$$$$T^{6} +$$$$34\!\cdots\!20$$$$T^{8} - 2925882681979232 T^{10} + T^{12}$$
$43$ $$($$$$77\!\cdots\!24$$$$+$$$$34\!\cdots\!04$$$$T +$$$$26\!\cdots\!56$$$$T^{2} -$$$$90\!\cdots\!12$$$$T^{3} - 1059207796303296 T^{4} + 5002308 T^{5} + T^{6} )^{2}$$
$47$ $$($$$$48\!\cdots\!36$$$$+$$$$52\!\cdots\!00$$$$T -$$$$18\!\cdots\!60$$$$T^{2} -$$$$11\!\cdots\!60$$$$T^{3} - 1459509409573760 T^{4} + 56276220 T^{5} + T^{6} )^{2}$$
$53$ $$($$$$11\!\cdots\!28$$$$+$$$$52\!\cdots\!52$$$$T +$$$$18\!\cdots\!00$$$$T^{2} -$$$$31\!\cdots\!00$$$$T^{3} - 12314076983638092 T^{4} + 38402136 T^{5} + T^{6} )^{2}$$
$59$ $$($$$$19\!\cdots\!80$$$$+$$$$11\!\cdots\!84$$$$T -$$$$17\!\cdots\!56$$$$T^{2} -$$$$92\!\cdots\!16$$$$T^{3} - 13956799765468320 T^{4} + 5809452 T^{5} + T^{6} )^{2}$$
$61$ $$12\!\cdots\!00$$$$-$$$$27\!\cdots\!28$$$$T^{2} +$$$$17\!\cdots\!84$$$$T^{4} -$$$$38\!\cdots\!20$$$$T^{6} +$$$$29\!\cdots\!48$$$$T^{8} - 91863601051868224 T^{10} + T^{12}$$
$67$ $$( -$$$$20\!\cdots\!40$$$$+$$$$40\!\cdots\!48$$$$T +$$$$75\!\cdots\!40$$$$T^{2} -$$$$50\!\cdots\!84$$$$T^{3} - 52635985628963408 T^{4} + 152104376 T^{5} + T^{6} )^{2}$$
$71$ $$27\!\cdots\!00$$$$-$$$$96\!\cdots\!72$$$$T^{2} +$$$$50\!\cdots\!96$$$$T^{4} -$$$$10\!\cdots\!96$$$$T^{6} +$$$$10\!\cdots\!32$$$$T^{8} - 522268204322304826 T^{10} + T^{12}$$
$73$ $$15\!\cdots\!32$$$$-$$$$86\!\cdots\!20$$$$T^{2} +$$$$22\!\cdots\!64$$$$T^{4} -$$$$15\!\cdots\!96$$$$T^{6} +$$$$35\!\cdots\!08$$$$T^{8} - 338151556053131904 T^{10} + T^{12}$$
$79$ $$23\!\cdots\!92$$$$-$$$$48\!\cdots\!08$$$$T^{2} +$$$$35\!\cdots\!60$$$$T^{4} -$$$$10\!\cdots\!60$$$$T^{6} +$$$$12\!\cdots\!32$$$$T^{8} - 628759136375969242 T^{10} + T^{12}$$
$83$ $$( -$$$$80\!\cdots\!24$$$$-$$$$12\!\cdots\!72$$$$T +$$$$21\!\cdots\!68$$$$T^{2} +$$$$11\!\cdots\!68$$$$T^{3} - 356920811451371808 T^{4} - 422521068 T^{5} + T^{6} )^{2}$$
$89$ $$($$$$17\!\cdots\!80$$$$+$$$$52\!\cdots\!76$$$$T +$$$$44\!\cdots\!92$$$$T^{2} -$$$$66\!\cdots\!24$$$$T^{3} - 959717591512590944 T^{4} + 469111902 T^{5} + T^{6} )^{2}$$
$97$ $$48\!\cdots\!52$$$$-$$$$35\!\cdots\!20$$$$T^{2} +$$$$10\!\cdots\!92$$$$T^{4} -$$$$15\!\cdots\!08$$$$T^{6} +$$$$13\!\cdots\!20$$$$T^{8} - 5703274637121469792 T^{10} + T^{12}$$