Properties

Label 960.3.l.j
Level $960$
Weight $3$
Character orbit 960.l
Analytic conductor $26.158$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 960.l (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.1581053786\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 8 x^{15} + 56 x^{14} - 252 x^{13} + 1094 x^{12} - 3652 x^{11} + 5452 x^{10} + 1164 x^{9} - 5879 x^{8} - 15060 x^{7} + 42772 x^{6} - 39536 x^{5} + 64024 x^{4} - 97600 x^{3} + 22080 x^{2} + 25344 x + 20736\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{28}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 480)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{3} + \beta_{4} q^{5} + ( \beta_{1} + \beta_{5} ) q^{7} + ( 1 + \beta_{6} ) q^{9} +O(q^{10})\) \( q + \beta_{5} q^{3} + \beta_{4} q^{5} + ( \beta_{1} + \beta_{5} ) q^{7} + ( 1 + \beta_{6} ) q^{9} + ( \beta_{1} + \beta_{2} - \beta_{5} + \beta_{7} - \beta_{13} ) q^{11} + ( 1 - \beta_{12} ) q^{13} + \beta_{3} q^{15} + ( \beta_{6} - \beta_{9} + \beta_{15} ) q^{17} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{7} ) q^{19} + ( 7 + 2 \beta_{4} + \beta_{6} - \beta_{10} - \beta_{12} ) q^{21} + ( -\beta_{2} + \beta_{3} - \beta_{5} - \beta_{11} + 2 \beta_{13} ) q^{23} -5 q^{25} + ( \beta_{1} + \beta_{5} + 2 \beta_{7} + \beta_{8} + \beta_{11} - \beta_{13} ) q^{27} + ( -1 - 4 \beta_{4} - \beta_{6} - \beta_{9} + \beta_{14} - \beta_{15} ) q^{29} + ( 4 \beta_{1} - \beta_{2} + \beta_{5} - \beta_{7} - \beta_{8} ) q^{31} + ( 11 + 2 \beta_{4} - \beta_{6} - \beta_{9} + \beta_{10} - \beta_{12} ) q^{33} + ( -\beta_{2} + \beta_{3} + \beta_{5} + \beta_{11} ) q^{35} + ( -9 + 2 \beta_{10} + \beta_{12} + \beta_{15} ) q^{37} + ( \beta_{1} - 5 \beta_{2} + \beta_{3} + \beta_{5} - 2 \beta_{7} + 3 \beta_{13} ) q^{39} + ( -1 - 6 \beta_{4} - 3 \beta_{6} + \beta_{9} + \beta_{14} - 2 \beta_{15} ) q^{41} + ( -3 \beta_{1} - \beta_{2} - 4 \beta_{3} + 2 \beta_{5} + 3 \beta_{7} ) q^{43} + ( -3 + \beta_{4} - \beta_{9} + \beta_{12} ) q^{45} + ( -2 \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{5} - 2 \beta_{7} - 3 \beta_{11} + 2 \beta_{13} ) q^{47} + ( -6 - 2 \beta_{4} + 3 \beta_{6} + \beta_{9} - 4 \beta_{10} - 4 \beta_{12} + \beta_{14} - 2 \beta_{15} ) q^{49} + ( -2 \beta_{1} - 4 \beta_{2} - \beta_{3} - 2 \beta_{5} + 3 \beta_{7} - \beta_{8} + 2 \beta_{11} - 2 \beta_{13} ) q^{51} + ( 2 - 6 \beta_{4} + 4 \beta_{6} - 2 \beta_{14} + \beta_{15} ) q^{53} + ( -\beta_{2} - \beta_{3} - \beta_{5} - \beta_{8} ) q^{55} + ( -5 + 10 \beta_{4} - 2 \beta_{6} + 2 \beta_{9} + 3 \beta_{10} + \beta_{12} ) q^{57} + ( \beta_{1} + \beta_{2} + 3 \beta_{5} + \beta_{7} - 2 \beta_{11} - 5 \beta_{13} ) q^{59} + ( -7 - 2 \beta_{4} + 3 \beta_{6} + \beta_{9} + 2 \beta_{10} + \beta_{14} + \beta_{15} ) q^{61} + ( 6 \beta_{1} - 3 \beta_{2} + 5 \beta_{3} + 7 \beta_{5} + 3 \beta_{11} ) q^{63} + ( 1 + 2 \beta_{6} - \beta_{14} ) q^{65} + ( \beta_{1} - 5 \beta_{2} - 6 \beta_{3} - 2 \beta_{5} + \beta_{7} + 2 \beta_{8} ) q^{67} + ( 16 + 10 \beta_{4} - 4 \beta_{6} + \beta_{9} - 3 \beta_{10} + 3 \beta_{12} + \beta_{14} - 3 \beta_{15} ) q^{69} + ( 4 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} + 4 \beta_{7} + 2 \beta_{11} ) q^{71} + ( -4 + 4 \beta_{4} - 6 \beta_{6} - 2 \beta_{9} - 2 \beta_{10} + 4 \beta_{12} - 2 \beta_{14} - \beta_{15} ) q^{73} -5 \beta_{5} q^{75} + ( -4 \beta_{4} + \beta_{15} ) q^{77} + ( 2 \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 9 \beta_{5} - 3 \beta_{7} + 3 \beta_{8} ) q^{79} + ( 12 + 10 \beta_{4} + \beta_{6} - \beta_{9} + 4 \beta_{10} + \beta_{14} + 6 \beta_{15} ) q^{81} + ( 7 \beta_{1} - \beta_{2} + 8 \beta_{3} - 2 \beta_{5} + 7 \beta_{7} - 4 \beta_{13} ) q^{83} + ( -2 + 2 \beta_{4} - 3 \beta_{6} - \beta_{9} + 2 \beta_{10} + \beta_{12} - \beta_{14} + \beta_{15} ) q^{85} + ( 5 \beta_{1} + \beta_{2} - 7 \beta_{3} + 2 \beta_{5} + 6 \beta_{7} - 2 \beta_{8} - 5 \beta_{11} - 4 \beta_{13} ) q^{87} + ( 2 - 20 \beta_{4} + 2 \beta_{6} + 2 \beta_{9} - 2 \beta_{14} + 2 \beta_{15} ) q^{89} + ( 10 \beta_{1} + 2 \beta_{2} + 16 \beta_{5} + 2 \beta_{7} ) q^{91} + ( 7 + 14 \beta_{4} + 2 \beta_{6} - 6 \beta_{10} - 5 \beta_{12} - 2 \beta_{14} - 3 \beta_{15} ) q^{93} + ( -3 \beta_{1} - \beta_{2} - 2 \beta_{3} + 4 \beta_{5} - 3 \beta_{7} - 3 \beta_{11} ) q^{95} + ( -10 - 2 \beta_{10} + 4 \beta_{12} - \beta_{15} ) q^{97} + ( -6 \beta_{1} - 11 \beta_{2} - \beta_{3} + 9 \beta_{5} - 4 \beta_{7} - \beta_{8} - 4 \beta_{11} + 4 \beta_{13} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{9} + O(q^{10}) \) \( 16q + 8q^{9} + 16q^{13} + 104q^{21} - 80q^{25} + 192q^{33} - 144q^{37} - 40q^{45} - 128q^{49} - 80q^{57} - 144q^{61} + 280q^{69} + 192q^{81} + 96q^{93} - 160q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{15} + 56 x^{14} - 252 x^{13} + 1094 x^{12} - 3652 x^{11} + 5452 x^{10} + 1164 x^{9} - 5879 x^{8} - 15060 x^{7} + 42772 x^{6} - 39536 x^{5} + 64024 x^{4} - 97600 x^{3} + 22080 x^{2} + 25344 x + 20736\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(265120597 \nu^{15} + 942387 \nu^{14} + 1401383345 \nu^{13} + 23583208255 \nu^{12} - 49086984965 \nu^{11} + 474613304305 \nu^{10} - 2548177422345 \nu^{9} - 639742489323 \nu^{8} + 18201626058604 \nu^{7} - 5806041972976 \nu^{6} - 44484736653028 \nu^{5} + 15056300199704 \nu^{4} + 61515846591008 \nu^{3} - 4821416354400 \nu^{2} + 79102524876192 \nu - 219855865256640\)\()/ 24362570035008 \)
\(\beta_{2}\)\(=\)\((\)\(1388522339 \nu^{15} - 15286379448 \nu^{14} + 109021438216 \nu^{13} - 566220766030 \nu^{12} + 2453365747430 \nu^{11} - 9102684804970 \nu^{10} + 20579384485884 \nu^{9} - 13528608519354 \nu^{8} - 22747577303665 \nu^{7} - 4783747131134 \nu^{6} + 132847932873724 \nu^{5} - 189838434339752 \nu^{4} + 170570503443016 \nu^{3} - 283982134939536 \nu^{2} + 305717048465472 \nu + 75251029162752\)\()/ 24362570035008 \)
\(\beta_{3}\)\(=\)\((\)\(-505236429 \nu^{15} + 4979134639 \nu^{14} - 34906315777 \nu^{13} + 175567630407 \nu^{12} - 757312464791 \nu^{11} + 2774270333757 \nu^{10} - 5696660018879 \nu^{9} + 3471163007573 \nu^{8} + 2640699196336 \nu^{7} + 6083732187264 \nu^{6} - 26079192921292 \nu^{5} + 46974061102584 \nu^{4} - 76130303986624 \nu^{3} + 112713741559584 \nu^{2} - 53596515079584 \nu + 20673942144192\)\()/ 8120856678336 \)
\(\beta_{4}\)\(=\)\((\)\(815120 \nu^{15} - 6113400 \nu^{14} + 43546246 \nu^{13} - 190330699 \nu^{12} + 843073652 \nu^{11} - 2747251177 \nu^{10} + 3906813126 \nu^{9} + 320376327 \nu^{8} - 2322071872 \nu^{7} - 9023272175 \nu^{6} + 28286792176 \nu^{5} - 36351856388 \nu^{4} + 61910332048 \nu^{3} - 64548897240 \nu^{2} + 23496792480 \nu - 1809374112\)\()/ 11502629856 \)
\(\beta_{5}\)\(=\)\((\)\(-798796512 \nu^{15} + 5835652835 \nu^{14} - 40881356273 \nu^{13} + 174012166641 \nu^{12} - 761071384117 \nu^{11} + 2416339281171 \nu^{10} - 2805502329259 \nu^{9} - 2556353203277 \nu^{8} + 3072229607225 \nu^{7} + 12921360743286 \nu^{6} - 25024105501664 \nu^{5} + 14645490513600 \nu^{4} - 41010538226504 \nu^{3} + 52845911615088 \nu^{2} + 6483143119968 \nu - 13495915113408\)\()/ 8120856678336 \)
\(\beta_{6}\)\(=\)\((\)\(1237882115 \nu^{15} - 11984663208 \nu^{14} + 82872054007 \nu^{13} - 408200868067 \nu^{12} + 1734325986665 \nu^{11} - 6226029855523 \nu^{10} + 11794006077933 \nu^{9} - 2356345733685 \nu^{8} - 15280788194728 \nu^{7} - 19715265361865 \nu^{6} + 91088336879464 \nu^{5} - 94875776113004 \nu^{4} + 77556324860368 \nu^{3} - 219987819107112 \nu^{2} + 110027655717984 \nu + 66289229426880\)\()/ 12181285017504 \)
\(\beta_{7}\)\(=\)\((\)\(364118865 \nu^{15} - 2121631763 \nu^{14} + 14855155525 \nu^{13} - 52668235763 \nu^{12} + 235966807483 \nu^{11} - 612508005025 \nu^{10} - 240939896141 \nu^{9} + 2734680760327 \nu^{8} + 435302218040 \nu^{7} - 7729491804616 \nu^{6} + 5091833560196 \nu^{5} + 7352027732392 \nu^{4} + 6085363602560 \nu^{3} + 1144642961056 \nu^{2} - 23853082568352 \nu + 3345747297984\)\()/ 2706952226112 \)
\(\beta_{8}\)\(=\)\((\)\(289826899 \nu^{15} - 1960713902 \nu^{14} + 13581507538 \nu^{13} - 55261425500 \nu^{12} + 242402461184 \nu^{11} - 745469493164 \nu^{10} + 603250657906 \nu^{9} + 1084864250156 \nu^{8} + 1116407321501 \nu^{7} - 6743045317870 \nu^{6} + 3020785905748 \nu^{5} - 41504631832 \nu^{4} + 21814552257304 \nu^{3} - 25368703235472 \nu^{2} - 19574644394304 \nu - 11609626651776\)\()/ 2030214169584 \)
\(\beta_{9}\)\(=\)\((\)\(1800067744 \nu^{15} - 10199009661 \nu^{14} + 70944100568 \nu^{13} - 244437213062 \nu^{12} + 1092247683292 \nu^{11} - 2722348964984 \nu^{10} - 2270472657744 \nu^{9} + 15037802021562 \nu^{8} + 3063681912700 \nu^{7} - 33366749620267 \nu^{6} + 4415142727856 \nu^{5} + 37130753454692 \nu^{4} + 33420745579472 \nu^{3} + 121136142659304 \nu^{2} - 240939827503200 \nu - 24853781473920\)\()/ 12181285017504 \)
\(\beta_{10}\)\(=\)\((\)\(-562185629 \nu^{15} + 4214896266 \nu^{14} - 30075895252 \nu^{13} + 131275899592 \nu^{12} - 582108991226 \nu^{11} + 1890725793160 \nu^{10} - 2686929542796 \nu^{9} - 341714336808 \nu^{8} + 2204342895583 \nu^{7} + 4863267971270 \nu^{6} - 17574822135376 \nu^{5} + 22772555764952 \nu^{4} - 40022412577624 \nu^{3} + 42966251413392 \nu^{2} - 15986684646432 \nu - 37610128370880\)\()/ 3045321254376 \)
\(\beta_{11}\)\(=\)\((\)\(-2262094677 \nu^{15} + 18447021626 \nu^{14} - 129698062940 \nu^{13} + 591401105904 \nu^{12} - 2575743277690 \nu^{11} + 8700548675700 \nu^{10} - 13867014600736 \nu^{9} + 102727111432 \nu^{8} + 12234397174895 \nu^{7} + 32231169622818 \nu^{6} - 100728387519980 \nu^{5} + 113221214769864 \nu^{4} - 185862178865912 \nu^{3} + 253699981662384 \nu^{2} - 121857601480704 \nu - 8951407210368\)\()/ 8120856678336 \)
\(\beta_{12}\)\(=\)\((\)\(-3877174285 \nu^{15} + 29080303089 \nu^{14} - 200800728677 \nu^{13} + 864444965729 \nu^{12} - 3716683796203 \nu^{11} + 11910654837647 \nu^{10} - 13348001972103 \nu^{9} - 16962661053285 \nu^{8} + 23367049560524 \nu^{7} + 64427265086812 \nu^{6} - 134392025542280 \nu^{5} + 88438960046440 \nu^{4} - 140099557664864 \nu^{3} + 178677070972800 \nu^{2} + 125413640805888 \nu - 53395813114272\)\()/ 12181285017504 \)
\(\beta_{13}\)\(=\)\((\)\(8508808957 \nu^{15} - 59838095547 \nu^{14} + 422583847199 \nu^{13} - 1764447726707 \nu^{12} + 7809900580657 \nu^{11} - 24420961756685 \nu^{10} + 26806167012297 \nu^{9} + 22705296840783 \nu^{8} - 9392641929242 \nu^{7} - 139605532067020 \nu^{6} + 232358610102932 \nu^{5} - 150709076210824 \nu^{4} + 451857005790416 \nu^{3} - 471751746360000 \nu^{2} + 136155276435744 \nu - 83338409891520\)\()/ 24362570035008 \)
\(\beta_{14}\)\(=\)\((\)\(4423480295 \nu^{15} - 28375958595 \nu^{14} + 198651799903 \nu^{13} - 768982727755 \nu^{12} + 3400584183545 \nu^{11} - 9795050415421 \nu^{10} + 4298002420341 \nu^{9} + 26055884793639 \nu^{8} - 5225655376876 \nu^{7} - 74846244646436 \nu^{6} + 83230916271256 \nu^{5} - 32291422181048 \nu^{4} + 172246515880288 \nu^{3} + 17428691721984 \nu^{2} - 308931716107008 \nu - 3906025371936\)\()/ 12181285017504 \)
\(\beta_{15}\)\(=\)\((\)\(15402346 \nu^{15} - 115517595 \nu^{14} + 824284025 \nu^{13} - 3605829305 \nu^{12} + 15999837805 \nu^{11} - 52189422659 \nu^{10} + 75091501575 \nu^{9} + 2272237365 \nu^{8} - 40339333115 \nu^{7} - 166977133630 \nu^{6} + 518172613580 \nu^{5} - 662475798640 \nu^{4} + 1133246792600 \nu^{3} - 1185313524720 \nu^{2} + 430946648640 \nu - 32776379136\)\()/ 41716729512 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(3 \beta_{15} + 3 \beta_{13} - 6 \beta_{11} - 3 \beta_{8} - 5 \beta_{7} + 4 \beta_{5} - 24 \beta_{4} + 3 \beta_{3} - 8 \beta_{2} + \beta_{1} + 24\)\()/48\)
\(\nu^{2}\)\(=\)\((\)\(-4 \beta_{14} - 9 \beta_{13} - 6 \beta_{11} - 6 \beta_{10} + 8 \beta_{9} - 3 \beta_{8} + 23 \beta_{7} + 20 \beta_{5} - 16 \beta_{4} - 9 \beta_{3} + 8 \beta_{2} + 5 \beta_{1} - 68\)\()/24\)
\(\nu^{3}\)\(=\)\((\)\(-15 \beta_{15} - 12 \beta_{14} - 9 \beta_{13} - 18 \beta_{12} + 3 \beta_{11} - 9 \beta_{10} - 3 \beta_{9} + 15 \beta_{8} + 38 \beta_{7} - 27 \beta_{6} + 53 \beta_{5} + 72 \beta_{4} + 44 \beta_{2} + 2 \beta_{1} - 72\)\()/12\)
\(\nu^{4}\)\(=\)\((\)\(-30 \beta_{15} + 19 \beta_{14} + 96 \beta_{13} - 18 \beta_{12} + 30 \beta_{11} - 12 \beta_{10} - 53 \beta_{9} + 15 \beta_{8} - 72 \beta_{7} - 39 \beta_{6} + 156 \beta_{5} + 106 \beta_{4} - 30 \beta_{3} - 42 \beta_{2} - 114 \beta_{1} - 151\)\()/6\)
\(\nu^{5}\)\(=\)\((\)\(-417 \beta_{15} + 310 \beta_{14} + 405 \beta_{13} + 360 \beta_{12} + 435 \beta_{11} + 60 \beta_{10} - 65 \beta_{9} - 336 \beta_{8} - 580 \beta_{7} + 435 \beta_{6} - 1069 \beta_{5} + 1780 \beta_{4} - 648 \beta_{3} - 538 \beta_{2} - 256 \beta_{1} - 244\)\()/12\)
\(\nu^{6}\)\(=\)\((\)\(279 \beta_{15} - 70 \beta_{14} - 1167 \beta_{13} + 585 \beta_{12} + 201 \beta_{11} + 1362 \beta_{10} + 806 \beta_{9} - 372 \beta_{8} - 1094 \beta_{7} + 954 \beta_{6} - 7241 \beta_{5} + 1916 \beta_{4} + 1254 \beta_{3} + 232 \beta_{2} + 2260 \beta_{1} + 16243\)\()/6\)
\(\nu^{7}\)\(=\)\((\)\(20175 \beta_{15} - 3122 \beta_{14} - 4965 \beta_{13} - 504 \beta_{12} - 15225 \beta_{11} + 8484 \beta_{10} + 4333 \beta_{9} + 1716 \beta_{8} - 13172 \beta_{7} + 525 \beta_{6} - 9905 \beta_{5} - 76712 \beta_{4} + 23952 \beta_{3} - 12086 \beta_{2} + 5302 \beta_{1} + 111404\)\()/12\)
\(\nu^{8}\)\(=\)\((\)\(6646 \beta_{15} - 2565 \beta_{14} - 3364 \beta_{13} - 1260 \beta_{12} - 10916 \beta_{11} - 9684 \beta_{10} + 2555 \beta_{9} - 40 \beta_{8} + 16856 \beta_{7} - 911 \beta_{6} + 43164 \beta_{5} - 54646 \beta_{4} - 3808 \beta_{3} - 3288 \beta_{2} - 5040 \beta_{1} - 117649\)\()/2\)
\(\nu^{9}\)\(=\)\((\)\(-348639 \beta_{15} - 110442 \beta_{14} - 163941 \beta_{13} - 161856 \beta_{12} + 140007 \beta_{11} - 345204 \beta_{10} - 18525 \beta_{9} + 110382 \beta_{8} + 882526 \beta_{7} - 209601 \beta_{6} + 1014331 \beta_{5} + 1081692 \beta_{4} - 390750 \beta_{3} + 664678 \beta_{2} + 30970 \beta_{1} - 3968316\)\()/12\)
\(\nu^{10}\)\(=\)\((\)\(-858231 \beta_{15} + 132980 \beta_{14} + 775377 \beta_{13} - 372105 \beta_{12} + 870879 \beta_{11} - 13110 \beta_{10} - 654400 \beta_{9} + 357882 \beta_{8} - 436204 \beta_{7} - 644040 \beta_{6} + 651569 \beta_{5} + 4298600 \beta_{4} - 314244 \beta_{3} + 641972 \beta_{2} - 682282 \beta_{1} + 361867\)\()/6\)
\(\nu^{11}\)\(=\)\((\)\(-1968045 \beta_{15} + 5502750 \beta_{14} + 10839051 \beta_{13} + 4009104 \beta_{12} + 5295855 \beta_{11} + 4769820 \beta_{10} - 3914823 \beta_{9} - 3213840 \beta_{8} - 20687248 \beta_{7} + 4033953 \beta_{6} - 17229289 \beta_{5} + 14074704 \beta_{4} - 2416356 \beta_{3} - 13554886 \beta_{2} - 7840090 \beta_{1} + 47715300\)\()/12\)
\(\nu^{12}\)\(=\)\((\)\(9733056 \beta_{15} + 3633163 \beta_{14} - 12022230 \beta_{13} + 16044930 \beta_{12} - 3025842 \beta_{11} + 16045440 \beta_{10} + 13408459 \beta_{9} - 13289268 \beta_{8} - 20611704 \beta_{7} + 23847297 \beta_{6} - 96133770 \beta_{5} - 19093046 \beta_{4} + 8410236 \beta_{3} - 18686268 \beta_{2} + 24089940 \beta_{1} + 170848733\)\()/6\)
\(\nu^{13}\)\(=\)\((\)\(247038333 \beta_{15} - 98372378 \beta_{14} - 267689277 \beta_{13} - 2123784 \beta_{12} - 212580609 \beta_{11} + 99319740 \beta_{10} + 153004735 \beta_{9} + 15426330 \beta_{8} + 80410490 \beta_{7} + 37845795 \beta_{6} - 341397949 \beta_{5} - 883505708 \beta_{4} + 317078982 \beta_{3} + 43774646 \beta_{2} + 297232082 \beta_{1} + 1358442572\)\()/12\)
\(\nu^{14}\)\(=\)\((\)\(376491231 \beta_{15} - 206037562 \beta_{14} - 129411177 \beta_{13} - 262867059 \beta_{12} - 520024479 \beta_{11} - 353926554 \beta_{10} + 11985722 \beta_{9} + 205602774 \beta_{8} + 682576516 \beta_{7} - 329328378 \beta_{6} + 2229600115 \beta_{5} - 2458967020 \beta_{4} + 175456848 \beta_{3} + 85552744 \beta_{2} - 249449870 \beta_{1} - 3804271997\)\()/6\)
\(\nu^{15}\)\(=\)\((\)\(-5022518529 \beta_{15} - 416959010 \beta_{14} + 1509604131 \beta_{13} - 2213887320 \beta_{12} + 1839103527 \beta_{11} - 6414414420 \beta_{10} - 1983476795 \beta_{9} + 1269720636 \beta_{8} + 10453941868 \beta_{7} - 3325814475 \beta_{6} + 23896036687 \beta_{5} + 10527176200 \beta_{4} - 8001672432 \beta_{3} + 5369974426 \beta_{2} - 4933575026 \beta_{1} - 76790358052\)\()/12\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
641.1
2.45733 4.45426i
2.45733 + 4.45426i
−0.317839 + 0.355703i
−0.317839 0.355703i
−0.317839 + 1.17756i
−0.317839 1.17756i
−1.45733 4.45426i
−1.45733 + 4.45426i
2.45733 0.803980i
2.45733 + 0.803980i
1.31784 + 0.355703i
1.31784 0.355703i
1.31784 + 1.17756i
1.31784 1.17756i
−1.45733 0.803980i
−1.45733 + 0.803980i
0 −2.99655 0.143843i 0 2.23607i 0 2.24307 0 8.95862 + 0.862068i 0
641.2 0 −2.99655 + 0.143843i 0 2.23607i 0 2.24307 0 8.95862 0.862068i 0
641.3 0 −2.65305 1.40048i 0 2.23607i 0 −12.5386 0 5.07731 + 7.43108i 0
641.4 0 −2.65305 + 1.40048i 0 2.23607i 0 −12.5386 0 5.07731 7.43108i 0
641.5 0 −1.56887 2.55708i 0 2.23607i 0 0.823421 0 −4.07731 + 8.02344i 0
641.6 0 −1.56887 + 2.55708i 0 2.23607i 0 0.823421 0 −4.07731 8.02344i 0
641.7 0 −0.721589 2.91193i 0 2.23607i 0 −1.03633 0 −7.95862 + 4.20243i 0
641.8 0 −0.721589 + 2.91193i 0 2.23607i 0 −1.03633 0 −7.95862 4.20243i 0
641.9 0 0.721589 2.91193i 0 2.23607i 0 1.03633 0 −7.95862 4.20243i 0
641.10 0 0.721589 + 2.91193i 0 2.23607i 0 1.03633 0 −7.95862 + 4.20243i 0
641.11 0 1.56887 2.55708i 0 2.23607i 0 −0.823421 0 −4.07731 8.02344i 0
641.12 0 1.56887 + 2.55708i 0 2.23607i 0 −0.823421 0 −4.07731 + 8.02344i 0
641.13 0 2.65305 1.40048i 0 2.23607i 0 12.5386 0 5.07731 7.43108i 0
641.14 0 2.65305 + 1.40048i 0 2.23607i 0 12.5386 0 5.07731 + 7.43108i 0
641.15 0 2.99655 0.143843i 0 2.23607i 0 −2.24307 0 8.95862 0.862068i 0
641.16 0 2.99655 + 0.143843i 0 2.23607i 0 −2.24307 0 8.95862 + 0.862068i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 641.16
Significant digits:
Format:

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 960.3.l.j 16
3.b odd 2 1 inner 960.3.l.j 16
4.b odd 2 1 inner 960.3.l.j 16
8.b even 2 1 480.3.l.b 16
8.d odd 2 1 480.3.l.b 16
12.b even 2 1 inner 960.3.l.j 16
24.f even 2 1 480.3.l.b 16
24.h odd 2 1 480.3.l.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.3.l.b 16 8.b even 2 1
480.3.l.b 16 8.d odd 2 1
480.3.l.b 16 24.f even 2 1
480.3.l.b 16 24.h odd 2 1
960.3.l.j 16 1.a even 1 1 trivial
960.3.l.j 16 3.b odd 2 1 inner
960.3.l.j 16 4.b odd 2 1 inner
960.3.l.j 16 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 164 T_{7}^{6} + 1076 T_{7}^{4} - 1504 T_{7}^{2} + 576 \) acting on \(S_{3}^{\mathrm{new}}(960, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \)
$3$ \( 43046721 - 2125764 T^{2} - 262440 T^{4} - 19116 T^{6} + 4014 T^{8} - 236 T^{10} - 40 T^{12} - 4 T^{14} + T^{16} \)
$5$ \( ( 5 + T^{2} )^{8} \)
$7$ \( ( 576 - 1504 T^{2} + 1076 T^{4} - 164 T^{6} + T^{8} )^{2} \)
$11$ \( ( 9216 + 259328 T^{2} + 35024 T^{4} + 472 T^{6} + T^{8} )^{2} \)
$13$ \( ( 20576 + 736 T - 324 T^{2} - 4 T^{3} + T^{4} )^{4} \)
$17$ \( ( 2005606656 + 73338368 T^{2} + 755424 T^{4} + 1632 T^{6} + T^{8} )^{2} \)
$19$ \( ( 13604889600 - 172627200 T^{2} + 778320 T^{4} - 1480 T^{6} + T^{8} )^{2} \)
$23$ \( ( 28237441600 + 376572000 T^{2} + 1636180 T^{4} + 2580 T^{6} + T^{8} )^{2} \)
$29$ \( ( 115883053056 + 3108782592 T^{2} + 6992784 T^{4} + 4808 T^{6} + T^{8} )^{2} \)
$31$ \( ( 243104219136 - 4814761984 T^{2} + 8632976 T^{4} - 5144 T^{6} + T^{8} )^{2} \)
$37$ \( ( 19296 - 19744 T - 804 T^{2} + 36 T^{3} + T^{4} )^{4} \)
$41$ \( ( 15336345600 + 385574400 T^{2} + 2849680 T^{4} + 6280 T^{6} + T^{8} )^{2} \)
$43$ \( ( 2103149650176 - 8835666624 T^{2} + 12186756 T^{4} - 6124 T^{6} + T^{8} )^{2} \)
$47$ \( ( 2975293809216 + 14549866848 T^{2} + 20265364 T^{4} + 8372 T^{6} + T^{8} )^{2} \)
$53$ \( ( 82199318835456 + 115437581568 T^{2} + 58683744 T^{4} + 12752 T^{6} + T^{8} )^{2} \)
$59$ \( ( 435726729216 + 9394906112 T^{2} + 37731344 T^{4} + 12328 T^{6} + T^{8} )^{2} \)
$61$ \( ( -975104 - 172224 T - 4124 T^{2} + 36 T^{3} + T^{4} )^{4} \)
$67$ \( ( 189747360000 - 3197304000 T^{2} + 13620100 T^{4} - 15340 T^{6} + T^{8} )^{2} \)
$71$ \( ( 32895088459776 + 291103096832 T^{2} + 140853824 T^{4} + 21808 T^{6} + T^{8} )^{2} \)
$73$ \( ( 13055760 - 637440 T - 16840 T^{2} + T^{4} )^{4} \)
$79$ \( ( 44014414160896 - 154305981696 T^{2} + 133654096 T^{4} - 23496 T^{6} + T^{8} )^{2} \)
$83$ \( ( 1478619673574976 + 1278034742432 T^{2} + 355339604 T^{4} + 34948 T^{6} + T^{8} )^{2} \)
$89$ \( ( 4467034423296 + 74003775488 T^{2} + 81778944 T^{4} + 23712 T^{6} + T^{8} )^{2} \)
$97$ \( ( 143760 - 168160 T - 7240 T^{2} + 40 T^{3} + T^{4} )^{4} \)
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