Properties

Label 315.2.bf.b
Level 315
Weight 2
Character orbit 315.bf
Analytic conductor 2.515
Analytic rank 0
Dimension 16
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.bf (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( -\beta_{5} + \beta_{6} - \beta_{15} ) q^{2} + ( 1 - \beta_{2} + \beta_{7} + \beta_{9} + \beta_{10} ) q^{4} + ( -1 + \beta_{2} - \beta_{4} + \beta_{8} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{5} + ( 1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{13} ) q^{7} + ( 1 - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + \beta_{12} - \beta_{13} ) q^{8} +O(q^{10})\) \( q + ( -\beta_{5} + \beta_{6} - \beta_{15} ) q^{2} + ( 1 - \beta_{2} + \beta_{7} + \beta_{9} + \beta_{10} ) q^{4} + ( -1 + \beta_{2} - \beta_{4} + \beta_{8} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{5} + ( 1 - \beta_{2} + \beta_{4} - \beta_{5} - \beta_{13} ) q^{7} + ( 1 - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + \beta_{12} - \beta_{13} ) q^{8} + ( 2 \beta_{3} - \beta_{7} - \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{10} + ( 2 - 2 \beta_{2} - \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{14} ) q^{11} + ( -2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} - 2 \beta_{12} + \beta_{15} ) q^{13} + ( 1 + 2 \beta_{1} + \beta_{2} + 3 \beta_{7} + \beta_{8} + 3 \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{14} + ( 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{8} + 2 \beta_{10} - 2 \beta_{12} + 2 \beta_{13} + 2 \beta_{14} ) q^{16} + ( -\beta_{3} - \beta_{5} - 2 \beta_{11} - 2 \beta_{14} ) q^{17} + ( -3 + \beta_{1} + 4 \beta_{7} - \beta_{9} + \beta_{10} ) q^{19} + ( -1 + \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} - 3 \beta_{6} - \beta_{12} + \beta_{13} + \beta_{15} ) q^{20} + ( -1 + \beta_{2} + 4 \beta_{3} - \beta_{4} + 4 \beta_{6} + \beta_{13} - \beta_{15} ) q^{22} + ( 2 - 2 \beta_{2} + 3 \beta_{5} - 5 \beta_{6} - 2 \beta_{8} - 2 \beta_{12} - 2 \beta_{13} + 2 \beta_{14} + 3 \beta_{15} ) q^{23} + ( 3 - 3 \beta_{2} - 2 \beta_{3} - \beta_{5} - 2 \beta_{7} - \beta_{8} + 3 \beta_{9} - \beta_{10} - \beta_{11} - 2 \beta_{14} ) q^{25} + ( -3 \beta_{1} - 2 \beta_{2} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{26} + ( 1 - \beta_{2} + 4 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{11} - \beta_{13} + 2 \beta_{14} - \beta_{15} ) q^{28} + ( -2 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{12} - 2 \beta_{13} ) q^{29} + ( \beta_{7} - 2 \beta_{10} ) q^{31} + ( 2 \beta_{3} + 2 \beta_{11} + 2 \beta_{14} ) q^{32} + ( 1 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{12} - 2 \beta_{13} ) q^{34} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{11} + 2 \beta_{13} + \beta_{14} - \beta_{15} ) q^{35} + ( -1 + \beta_{2} - \beta_{4} - 6 \beta_{6} + \beta_{8} + \beta_{11} + \beta_{13} ) q^{37} + ( -\beta_{3} + 3 \beta_{5} + \beta_{8} - \beta_{14} ) q^{38} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{6} - 3 \beta_{7} + \beta_{8} - 3 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} + \beta_{15} ) q^{40} + ( 1 + 3 \beta_{1} - \beta_{2} - \beta_{12} + \beta_{13} ) q^{41} + ( 1 - \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{6} + 2 \beta_{12} - \beta_{13} + 2 \beta_{15} ) q^{43} + ( -3 + \beta_{2} + 2 \beta_{7} + \beta_{8} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{44} + ( -5 + 5 \beta_{2} - \beta_{7} - 5 \beta_{9} + \beta_{10} ) q^{46} + ( -1 + \beta_{2} - 2 \beta_{6} + \beta_{8} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{47} + ( -3 + \beta_{2} + 2 \beta_{7} + \beta_{9} + 3 \beta_{10} ) q^{49} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 7 \beta_{3} + 7 \beta_{6} + 2 \beta_{12} - \beta_{15} ) q^{50} + ( -10 \beta_{3} - \beta_{5} - 3 \beta_{8} - 5 \beta_{11} - 2 \beta_{14} ) q^{52} + ( -6 \beta_{3} - 2 \beta_{11} - 2 \beta_{14} ) q^{53} + ( -1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{6} + \beta_{13} - 2 \beta_{15} ) q^{55} + ( -5 + 2 \beta_{1} + 3 \beta_{2} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{56} + ( 3 - 3 \beta_{2} + 7 \beta_{4} + \beta_{5} + 4 \beta_{6} - 3 \beta_{8} - 7 \beta_{11} + 4 \beta_{12} - 3 \beta_{13} - 4 \beta_{14} + \beta_{15} ) q^{58} + ( -\beta_{7} - \beta_{10} ) q^{59} + ( 4 - 3 \beta_{1} - 2 \beta_{2} - 5 \beta_{7} - 2 \beta_{8} + 5 \beta_{9} - 3 \beta_{10} + 2 \beta_{12} - 2 \beta_{13} - 2 \beta_{14} ) q^{61} + ( -2 + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} + 3 \beta_{6} + 2 \beta_{13} - \beta_{15} ) q^{62} + ( -2 \beta_{1} + 2 \beta_{12} - 2 \beta_{13} ) q^{64} + ( -3 + \beta_{1} + \beta_{2} + \beta_{4} - 4 \beta_{5} + 3 \beta_{7} + \beta_{8} + 2 \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} - 4 \beta_{15} ) q^{65} + ( \beta_{5} + 2 \beta_{8} + 4 \beta_{11} + 2 \beta_{14} ) q^{67} + ( 3 - 3 \beta_{2} + 4 \beta_{4} + 8 \beta_{6} - 3 \beta_{8} - 4 \beta_{11} + \beta_{12} - 3 \beta_{13} - \beta_{14} ) q^{68} + ( -2 + 5 \beta_{2} - 8 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} + 2 \beta_{13} + \beta_{15} ) q^{70} + ( -5 \beta_{1} + 2 \beta_{2} + 2 \beta_{12} - 2 \beta_{13} ) q^{71} + ( 4 \beta_{3} + 4 \beta_{5} - \beta_{8} - 3 \beta_{11} - 2 \beta_{14} ) q^{73} + ( -6 + 6 \beta_{2} - 3 \beta_{7} - \beta_{8} - 6 \beta_{9} - 3 \beta_{10} - \beta_{14} ) q^{74} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{76} + ( -\beta_{3} + 3 \beta_{6} + \beta_{8} + 4 \beta_{11} + 3 \beta_{14} + \beta_{15} ) q^{77} + ( -3 + 3 \beta_{1} + 4 \beta_{2} + 2 \beta_{7} + 4 \beta_{8} - \beta_{9} + 3 \beta_{10} - 4 \beta_{12} + 4 \beta_{13} + 4 \beta_{14} ) q^{79} + ( 2 - 2 \beta_{2} - 8 \beta_{3} - 4 \beta_{5} - 8 \beta_{7} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{14} ) q^{80} + ( -3 + 3 \beta_{2} - 3 \beta_{4} + \beta_{5} + 4 \beta_{6} + 3 \beta_{8} + 3 \beta_{11} + 3 \beta_{13} + \beta_{15} ) q^{82} + ( 2 - 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{6} - 2 \beta_{13} - \beta_{15} ) q^{83} + ( -6 + 3 \beta_{2} + \beta_{4} + 2 \beta_{12} + \beta_{13} - 4 \beta_{15} ) q^{85} + ( 5 + 3 \beta_{1} + 3 \beta_{2} - 5 \beta_{7} + 3 \beta_{8} - 2 \beta_{9} + 3 \beta_{10} - 3 \beta_{12} + 3 \beta_{13} + 3 \beta_{14} ) q^{86} + ( -2 \beta_{3} + 2 \beta_{8} + 4 \beta_{11} + 2 \beta_{14} ) q^{88} + ( 3 + 3 \beta_{1} - \beta_{2} + \beta_{7} - \beta_{8} + 6 \beta_{9} + 3 \beta_{10} + \beta_{12} - \beta_{13} - \beta_{14} ) q^{89} + ( 2 - 2 \beta_{1} - 5 \beta_{2} - 2 \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{14} ) q^{91} + ( -3 + 3 \beta_{2} - 12 \beta_{3} - 4 \beta_{4} - 12 \beta_{6} - \beta_{12} + 3 \beta_{13} ) q^{92} + ( -2 + 2 \beta_{2} - 4 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} ) q^{94} + ( -2 + 2 \beta_{2} - \beta_{3} - \beta_{5} + 5 \beta_{7} - 2 \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{14} ) q^{95} + ( -1 + \beta_{2} - \beta_{4} + \beta_{13} - \beta_{15} ) q^{97} + ( 3 - 3 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 7 \beta_{6} + 2 \beta_{11} - \beta_{12} - 3 \beta_{13} + 2 \beta_{14} + \beta_{15} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 8q^{4} - 2q^{5} + O(q^{10}) \) \( 16q + 8q^{4} - 2q^{5} - 4q^{10} + 24q^{14} - 24q^{19} + 8q^{20} - 4q^{25} + 12q^{26} - 24q^{29} + 16q^{31} + 16q^{34} + 10q^{35} + 32q^{40} - 16q^{41} - 20q^{44} - 32q^{46} - 40q^{49} + 40q^{50} + 8q^{55} - 84q^{56} - 4q^{59} + 16q^{61} + 16q^{64} - 30q^{65} + 16q^{70} + 56q^{71} - 40q^{74} - 64q^{76} - 16q^{79} - 52q^{80} - 64q^{85} + 48q^{86} - 16q^{89} + 8q^{91} - 32q^{94} + 22q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 2 x^{15} + 2 x^{14} - 4 x^{13} - 14 x^{12} + 38 x^{11} - 40 x^{10} + 64 x^{9} + 291 x^{8} - 630 x^{7} + 584 x^{6} - 800 x^{5} + 734 x^{4} - 188 x^{3} + 32 x^{2} - 8 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(-38414168495 \nu^{15} + 46999484762 \nu^{14} - 56685616158 \nu^{13} + 131873366880 \nu^{12} + 642956778028 \nu^{11} - 939694854143 \nu^{10} + 1093318205564 \nu^{9} - 2230952782978 \nu^{8} - 12780049004340 \nu^{7} + 14111475820386 \nu^{6} - 17131346947484 \nu^{5} + 27207219807940 \nu^{4} - 7455199259279 \nu^{3} + 1191125364091 \nu^{2} - 321416591346 \nu - 33746305280705\)\()/ 11431248166457 \)
\(\beta_{2}\)\(=\)\((\)\(145306983361 \nu^{15} - 177522986678 \nu^{14} + 213949914242 \nu^{13} - 520543533819 \nu^{12} - 2426061321940 \nu^{11} + 3549177780569 \nu^{10} - 4097251614084 \nu^{9} + 8418948255726 \nu^{8} + 48225232348268 \nu^{7} - 53280290889126 \nu^{6} + 64651815055380 \nu^{5} - 102672086774396 \nu^{4} + 28133974246241 \nu^{3} - 4494960282621 \nu^{2} + 1212926356606 \nu + 31632456480199\)\()/ 11431248166457 \)
\(\beta_{3}\)\(=\)\((\)\(760172030748 \nu^{15} - 826140524550 \nu^{14} + 311805690170 \nu^{13} - 1979665829518 \nu^{12} - 13114010371696 \nu^{11} + 18515621292532 \nu^{10} - 6663971281019 \nu^{9} + 27229891583895 \nu^{8} + 259493064521626 \nu^{7} - 266716860192172 \nu^{6} + 60835212617387 \nu^{5} - 306079675769764 \nu^{4} + 90642215592396 \nu^{3} + 246128680839936 \nu^{2} + 3861979127591 \nu - 575089767434\)\()/ 11431248166457 \)
\(\beta_{4}\)\(=\)\((\)\(-1159675947687 \nu^{15} + 2123382923210 \nu^{14} - 1790860341485 \nu^{13} + 4132499467647 \nu^{12} + 17082017216494 \nu^{11} - 41638199149391 \nu^{10} + 36484081923634 \nu^{9} - 63733593693012 \nu^{8} - 351285551153804 \nu^{7} + 677606774159518 \nu^{6} - 505106942012337 \nu^{5} + 778126373623118 \nu^{4} - 678461323755244 \nu^{3} + 34074325119351 \nu^{2} - 9164323089196 \nu + 29672440598565\)\()/ 11431248166457 \)
\(\beta_{5}\)\(=\)\((\)\(-2241001108620 \nu^{15} + 2440547176294 \nu^{14} - 921260080696 \nu^{13} + 5836557871079 \nu^{12} + 38651783264660 \nu^{11} - 54581263789178 \nu^{10} + 19684393272760 \nu^{9} - 80280420418569 \nu^{8} - 764989544114834 \nu^{7} + 786254645863424 \nu^{6} - 179383356485812 \nu^{5} + 902375388086909 \nu^{4} - 267226659358818 \nu^{3} - 682424450829525 \nu^{2} - 11385720733444 \nu + 1695440450074\)\()/ 11431248166457 \)
\(\beta_{6}\)\(=\)\((\)\(4058097157982 \nu^{15} - 7568755097298 \nu^{14} + 7059730047127 \nu^{13} - 15184509679358 \nu^{12} - 58946431470254 \nu^{11} + 146411669575112 \nu^{10} - 142164152788000 \nu^{9} + 238760344218351 \nu^{8} + 1214849064761822 \nu^{7} - 2395065444113876 \nu^{6} + 2037822081506248 \nu^{5} - 2940884698505042 \nu^{4} + 2557823285973472 \nu^{3} - 388309099007712 \nu^{2} + 34395735205495 \nu - 16137847630635\)\()/ 11431248166457 \)
\(\beta_{7}\)\(=\)\((\)\(-2575730834 \nu^{15} + 4699915310 \nu^{14} - 4379029892 \nu^{13} + 9614611682 \nu^{12} + 37669504970 \nu^{11} - 91091321784 \nu^{10} + 87843289680 \nu^{9} - 151013347873 \nu^{8} - 774495170890 \nu^{7} + 1484105110236 \nu^{6} - 1259913976952 \nu^{5} + 1863977222846 \nu^{4} - 1585945295448 \nu^{3} + 240744226192 \nu^{2} - 69083699784 \nu + 17015828908\)\()/ 7008735847 \)
\(\beta_{8}\)\(=\)\((\)\(1063280569755 \nu^{15} - 2138087245431 \nu^{14} + 2140603616666 \nu^{13} - 4270022226214 \nu^{12} - 14845214238260 \nu^{11} + 40596196036702 \nu^{10} - 42811945511246 \nu^{9} + 68369173152460 \nu^{8} + 308749841689177 \nu^{7} - 673674444501418 \nu^{6} + 625167345362715 \nu^{5} - 855730190069746 \nu^{4} + 788555489457606 \nu^{3} - 202118395383557 \nu^{2} + 34379925210470 \nu - 8602022998880\)\()/ 1633035452351 \)
\(\beta_{9}\)\(=\)\((\)\(7502690467678 \nu^{15} - 13694199839130 \nu^{14} + 12758334474904 \nu^{13} - 28102531513444 \nu^{12} - 109693802936905 \nu^{11} + 265404776190093 \nu^{10} - 255814696983230 \nu^{9} + 441714091122816 \nu^{8} + 2255188372760505 \nu^{7} - 4324366285850692 \nu^{6} + 3669645519916604 \nu^{5} - 5466340585268175 \nu^{4} + 4623119858303861 \nu^{3} - 701766532319694 \nu^{2} + 201383747371308 \nu - 29172119880252\)\()/ 11431248166457 \)
\(\beta_{10}\)\(=\)\((\)\(-8183751109447 \nu^{15} + 14956739476040 \nu^{14} - 13922497684986 \nu^{13} + 30578923581156 \nu^{12} + 119598143979065 \nu^{11} - 289853173711402 \nu^{10} + 279305469940010 \nu^{9} - 480196298050966 \nu^{8} - 2459380223502305 \nu^{7} + 4723682477021618 \nu^{6} - 4004669271289776 \nu^{5} + 5932582107617908 \nu^{4} - 5055851615402594 \nu^{3} + 767408313770491 \nu^{2} - 220236155839242 \nu + 54263991142284\)\()/ 11431248166457 \)
\(\beta_{11}\)\(=\)\((\)\(-9494932205673 \nu^{15} + 17203785936492 \nu^{14} - 15830728042254 \nu^{13} + 35235059970569 \nu^{12} + 139295605560736 \nu^{11} - 334146095435292 \nu^{10} + 317763568748586 \nu^{9} - 552101744717173 \nu^{8} - 2861708932288753 \nu^{7} + 5435608190228966 \nu^{6} - 4540481060164001 \nu^{5} + 6816437052589699 \nu^{4} - 5764590891006364 \nu^{3} + 806110661364879 \nu^{2} - 251085559700414 \nu + 61766681609962\)\()/ 11431248166457 \)
\(\beta_{12}\)\(=\)\((\)\(10583106332007 \nu^{15} - 18562866254784 \nu^{14} + 16212768119001 \nu^{13} - 37702760224023 \nu^{12} - 157985234866350 \nu^{11} + 364596213484152 \nu^{10} - 327711794585876 \nu^{9} + 583962180212718 \nu^{8} + 3234466606384584 \nu^{7} - 5891738412332556 \nu^{6} + 4610113536311955 \nu^{5} - 7128621954756198 \nu^{4} + 5858692599566580 \nu^{3} - 312154621077708 \nu^{2} + 83988991866438 \nu - 62361843284805\)\()/ 11431248166457 \)
\(\beta_{13}\)\(=\)\((\)\(10663789073454 \nu^{15} - 18661163594876 \nu^{14} + 16331067749359 \nu^{13} - 37987756392193 \nu^{12} - 159325975568434 \nu^{11} + 366561272531474 \nu^{10} - 329946311402856 \nu^{9} + 588615808979414 \nu^{8} + 3261120405274960 \nu^{7} - 5921218860713121 \nu^{6} + 4645853676172177 \nu^{5} - 7185374813558250 \nu^{4} + 5874244137453899 \nu^{3} - 314639249925878 \nu^{2} + 84659440952318 \nu - 32056382963606\)\()/ 11431248166457 \)
\(\beta_{14}\)\(=\)\((\)\(11015276267169 \nu^{15} - 18856066985592 \nu^{14} + 16454339422594 \nu^{13} - 39194391629605 \nu^{12} - 165523626304128 \nu^{11} + 371177338020356 \nu^{10} - 331091511310624 \nu^{9} + 606561527884963 \nu^{8} + 3380695061332005 \nu^{7} - 5969041910613310 \nu^{6} + 4662151485398775 \nu^{5} - 7428596404129227 \nu^{4} + 5945875322191156 \nu^{3} - 325284547851464 \nu^{2} + 258809517955596 \nu - 62916861144830\)\()/ 11431248166457 \)
\(\beta_{15}\)\(=\)\((\)\(13399447651346 \nu^{15} - 23430444102746 \nu^{14} + 20519536606234 \nu^{13} - 47737126077397 \nu^{12} - 200273669782708 \nu^{11} + 460259833552415 \nu^{10} - 414511080319944 \nu^{9} + 739757034059786 \nu^{8} + 4098887977021204 \nu^{7} - 7433685405000834 \nu^{6} + 5838337123809904 \nu^{5} - 9030367022903148 \nu^{4} + 7367017819324544 \nu^{3} - 395429090524619 \nu^{2} + 106398295849754 \nu - 46488804269431\)\()/ 11431248166457 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-\beta_{15} - \beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{7} - \beta_{5} - \beta_{4} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\(-\beta_{14} - \beta_{11} + 2 \beta_{3}\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{15} - 8 \beta_{13} + 3 \beta_{12} - 2 \beta_{6} + 3 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} - 3 \beta_{1} + 3\)\()/2\)
\(\nu^{4}\)\(=\)\(5 \beta_{10} + \beta_{9} - 8 \beta_{7} + 5 \beta_{1} + 13\)
\(\nu^{5}\)\(=\)\((\)\(-23 \beta_{14} + 11 \beta_{11} - 11 \beta_{10} + 23 \beta_{9} - 13 \beta_{7} - 23 \beta_{5} - 10 \beta_{3} - 23 \beta_{2} + 23\)\()/2\)
\(\nu^{6}\)\(=\)\(7 \beta_{15} - \beta_{13} - 23 \beta_{12} + 28 \beta_{6} - 22 \beta_{4} + 28 \beta_{3} - \beta_{2} + 1\)
\(\nu^{7}\)\(=\)\((\)\(103 \beta_{15} - 43 \beta_{14} - 148 \beta_{13} + 43 \beta_{12} - 43 \beta_{11} - 43 \beta_{10} + 103 \beta_{9} - 148 \beta_{8} - 61 \beta_{7} - 42 \beta_{6} + 103 \beta_{5} + 43 \beta_{4} - 148 \beta_{2} - 43 \beta_{1} + 166\)\()/2\)
\(\nu^{8}\)\(=\)\(-10 \beta_{14} + 94 \beta_{10} + 40 \beta_{9} - 10 \beta_{8} - 157 \beta_{7} - 40 \beta_{2} + 40\)
\(\nu^{9}\)\(=\)\((\)\(459 \beta_{15} - 479 \beta_{12} - 168 \beta_{6} + 171 \beta_{4} - 168 \beta_{3} - 459 \beta_{2} + 171 \beta_{1} + 339\)\()/2\)
\(\nu^{10}\)\(=\)\(214 \beta_{15} + 469 \beta_{14} - 70 \beta_{13} - 469 \beta_{12} + 399 \beta_{11} - 70 \beta_{8} + 496 \beta_{6} + 214 \beta_{5} - 399 \beta_{4} - 70 \beta_{2} + 70\)
\(\nu^{11}\)\(=\)\((\)\(-681 \beta_{14} - 681 \beta_{11} - 681 \beta_{10} + 2047 \beta_{9} - 2868 \beta_{8} - 1389 \beta_{7} + 2047 \beta_{5} + 658 \beta_{3} - 2047 \beta_{2} + 2047\)\()/2\)
\(\nu^{12}\)\(=\)\(424 \beta_{13} - 424 \beta_{12} - 683 \beta_{2} - 1693 \beta_{1} - 4232\)
\(\nu^{13}\)\(=\)\((\)\(9149 \beta_{15} + 9997 \beta_{14} - 9997 \beta_{12} - 2701 \beta_{11} + 2701 \beta_{10} - 9149 \beta_{9} + 6611 \beta_{7} - 2538 \beta_{6} + 9149 \beta_{5} + 2701 \beta_{4} + 2701 \beta_{1} - 3910\)\()/2\)
\(\nu^{14}\)\(=\)\(9573 \beta_{14} + 7194 \beta_{11} - 2379 \beta_{8} + 5603 \beta_{5} - 9050 \beta_{3}\)
\(\nu^{15}\)\(=\)\((\)\(-40993 \beta_{15} + 56392 \beta_{13} - 10641 \beta_{12} + 9630 \beta_{6} - 10641 \beta_{4} + 9630 \beta_{3} + 15399 \beta_{2} + 10641 \beta_{1} - 36121\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(-1 + \beta_{7}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
1.05078 + 0.281555i
−0.0543258 + 0.202747i
−1.96595 0.526774i
−0.556918 + 2.07845i
2.07845 + 0.556918i
0.526774 1.96595i
0.202747 + 0.0543258i
−0.281555 + 1.05078i
1.05078 0.281555i
−0.0543258 0.202747i
−1.96595 + 0.526774i
−0.556918 2.07845i
2.07845 0.556918i
0.526774 + 1.96595i
0.202747 0.0543258i
−0.281555 1.05078i
−2.17942 + 1.25829i 0 2.16659 3.75264i 1.11685 1.93717i 0 −1.31340 + 2.29673i 5.87162i 0 0.00343282 + 5.62724i
109.2 −1.54296 + 0.890827i 0 0.587145 1.01696i 1.33920 + 1.79069i 0 −2.40898 + 1.09398i 1.47113i 0 −3.66151 1.56996i
109.3 −1.34443 + 0.776205i 0 0.204988 0.355049i −1.88899 + 1.19655i 0 −0.478401 2.60214i 2.46837i 0 1.61083 3.07491i
109.4 −0.248840 + 0.143668i 0 −0.958719 + 1.66055i −0.717839 2.11771i 0 −1.11487 2.39939i 1.12562i 0 0.482874 + 0.423842i
109.5 0.248840 0.143668i 0 −0.958719 + 1.66055i −1.47507 1.68052i 0 1.11487 + 2.39939i 1.12562i 0 −0.608495 0.206261i
109.6 1.34443 0.776205i 0 0.204988 0.355049i 1.98074 1.03763i 0 0.478401 + 2.60214i 2.46837i 0 1.85754 2.93248i
109.7 1.54296 0.890827i 0 0.587145 1.01696i 0.881181 + 2.05512i 0 2.40898 1.09398i 1.47113i 0 3.19038 + 2.38598i
109.8 2.17942 1.25829i 0 2.16659 3.75264i −2.23607 0.00136408i 0 1.31340 2.29673i 5.87162i 0 −4.87505 + 2.81065i
289.1 −2.17942 1.25829i 0 2.16659 + 3.75264i 1.11685 + 1.93717i 0 −1.31340 2.29673i 5.87162i 0 0.00343282 5.62724i
289.2 −1.54296 0.890827i 0 0.587145 + 1.01696i 1.33920 1.79069i 0 −2.40898 1.09398i 1.47113i 0 −3.66151 + 1.56996i
289.3 −1.34443 0.776205i 0 0.204988 + 0.355049i −1.88899 1.19655i 0 −0.478401 + 2.60214i 2.46837i 0 1.61083 + 3.07491i
289.4 −0.248840 0.143668i 0 −0.958719 1.66055i −0.717839 + 2.11771i 0 −1.11487 + 2.39939i 1.12562i 0 0.482874 0.423842i
289.5 0.248840 + 0.143668i 0 −0.958719 1.66055i −1.47507 + 1.68052i 0 1.11487 2.39939i 1.12562i 0 −0.608495 + 0.206261i
289.6 1.34443 + 0.776205i 0 0.204988 + 0.355049i 1.98074 + 1.03763i 0 0.478401 2.60214i 2.46837i 0 1.85754 + 2.93248i
289.7 1.54296 + 0.890827i 0 0.587145 + 1.01696i 0.881181 2.05512i 0 2.40898 + 1.09398i 1.47113i 0 3.19038 2.38598i
289.8 2.17942 + 1.25829i 0 2.16659 + 3.75264i −2.23607 + 0.00136408i 0 1.31340 + 2.29673i 5.87162i 0 −4.87505 2.81065i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.bf.b 16
3.b odd 2 1 105.2.q.a 16
5.b even 2 1 inner 315.2.bf.b 16
7.c even 3 1 inner 315.2.bf.b 16
7.c even 3 1 2205.2.d.s 8
7.d odd 6 1 2205.2.d.o 8
12.b even 2 1 1680.2.di.d 16
15.d odd 2 1 105.2.q.a 16
15.e even 4 1 525.2.i.h 8
15.e even 4 1 525.2.i.k 8
21.c even 2 1 735.2.q.g 16
21.g even 6 1 735.2.d.e 8
21.g even 6 1 735.2.q.g 16
21.h odd 6 1 105.2.q.a 16
21.h odd 6 1 735.2.d.d 8
35.i odd 6 1 2205.2.d.o 8
35.j even 6 1 inner 315.2.bf.b 16
35.j even 6 1 2205.2.d.s 8
60.h even 2 1 1680.2.di.d 16
84.n even 6 1 1680.2.di.d 16
105.g even 2 1 735.2.q.g 16
105.o odd 6 1 105.2.q.a 16
105.o odd 6 1 735.2.d.d 8
105.p even 6 1 735.2.d.e 8
105.p even 6 1 735.2.q.g 16
105.w odd 12 1 3675.2.a.bn 4
105.w odd 12 1 3675.2.a.cb 4
105.x even 12 1 525.2.i.h 8
105.x even 12 1 525.2.i.k 8
105.x even 12 1 3675.2.a.bp 4
105.x even 12 1 3675.2.a.bz 4
420.ba even 6 1 1680.2.di.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.q.a 16 3.b odd 2 1
105.2.q.a 16 15.d odd 2 1
105.2.q.a 16 21.h odd 6 1
105.2.q.a 16 105.o odd 6 1
315.2.bf.b 16 1.a even 1 1 trivial
315.2.bf.b 16 5.b even 2 1 inner
315.2.bf.b 16 7.c even 3 1 inner
315.2.bf.b 16 35.j even 6 1 inner
525.2.i.h 8 15.e even 4 1
525.2.i.h 8 105.x even 12 1
525.2.i.k 8 15.e even 4 1
525.2.i.k 8 105.x even 12 1
735.2.d.d 8 21.h odd 6 1
735.2.d.d 8 105.o odd 6 1
735.2.d.e 8 21.g even 6 1
735.2.d.e 8 105.p even 6 1
735.2.q.g 16 21.c even 2 1
735.2.q.g 16 21.g even 6 1
735.2.q.g 16 105.g even 2 1
735.2.q.g 16 105.p even 6 1
1680.2.di.d 16 12.b even 2 1
1680.2.di.d 16 60.h even 2 1
1680.2.di.d 16 84.n even 6 1
1680.2.di.d 16 420.ba even 6 1
2205.2.d.o 8 7.d odd 6 1
2205.2.d.o 8 35.i odd 6 1
2205.2.d.s 8 7.c even 3 1
2205.2.d.s 8 35.j even 6 1
3675.2.a.bn 4 105.w odd 12 1
3675.2.a.bp 4 105.x even 12 1
3675.2.a.bz 4 105.x even 12 1
3675.2.a.cb 4 105.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{16} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T^{2} + 4 T^{4} - 8 T^{6} - 20 T^{8} + 32 T^{10} + 96 T^{12} - 208 T^{14} - 944 T^{16} - 832 T^{18} + 1536 T^{20} + 2048 T^{22} - 5120 T^{24} - 8192 T^{26} + 16384 T^{28} + 65536 T^{30} + 65536 T^{32} \)
$3$ \( \)
$5$ \( 1 + 2 T + 4 T^{2} + 20 T^{3} + 62 T^{4} + 118 T^{5} + 200 T^{6} + 670 T^{7} + 1639 T^{8} + 3350 T^{9} + 5000 T^{10} + 14750 T^{11} + 38750 T^{12} + 62500 T^{13} + 62500 T^{14} + 156250 T^{15} + 390625 T^{16} \)
$7$ \( 1 + 20 T^{2} + 202 T^{4} + 1244 T^{6} + 7267 T^{8} + 60956 T^{10} + 485002 T^{12} + 2352980 T^{14} + 5764801 T^{16} \)
$11$ \( ( 1 - 26 T^{2} + 28 T^{3} + 316 T^{4} - 518 T^{5} - 2872 T^{6} + 2968 T^{7} + 31007 T^{8} + 32648 T^{9} - 347512 T^{10} - 689458 T^{11} + 4626556 T^{12} + 4509428 T^{13} - 46060586 T^{14} + 214358881 T^{16} )^{2} \)
$13$ \( ( 1 - 44 T^{2} + 1234 T^{4} - 23988 T^{6} + 358939 T^{8} - 4053972 T^{10} + 35244274 T^{12} - 212379596 T^{14} + 815730721 T^{16} )^{2} \)
$17$ \( 1 + 76 T^{2} + 2664 T^{4} + 67392 T^{6} + 1583990 T^{8} + 33141472 T^{10} + 603483136 T^{12} + 11039990476 T^{14} + 198790801859 T^{16} + 3190557247564 T^{18} + 50403515001856 T^{20} + 799954567161568 T^{22} + 11049530028969590 T^{24} + 135861860939059008 T^{26} + 1552105639980083304 T^{28} + 12796714818514470604 T^{30} + 48661191875666868481 T^{32} \)
$19$ \( ( 1 + 12 T + 30 T^{2} - 80 T^{3} + 689 T^{4} + 9436 T^{5} + 28846 T^{6} + 59944 T^{7} + 291332 T^{8} + 1138936 T^{9} + 10413406 T^{10} + 64721524 T^{11} + 89791169 T^{12} - 198087920 T^{13} + 1411376430 T^{14} + 10726460868 T^{15} + 16983563041 T^{16} )^{2} \)
$23$ \( 1 + 20 T^{2} - 1512 T^{4} - 18448 T^{6} + 1612902 T^{8} + 10149224 T^{10} - 1248424544 T^{12} - 1840459676 T^{14} + 770565238915 T^{16} - 973603168604 T^{18} - 349360372817504 T^{20} + 1502449397500136 T^{22} + 126307944781695462 T^{24} - 764236278869396752 T^{26} - 33134912141214725352 T^{28} + \)\(23\!\cdots\!80\)\( T^{30} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( ( 1 + 6 T + 78 T^{2} + 332 T^{3} + 2820 T^{4} + 9628 T^{5} + 65598 T^{6} + 146334 T^{7} + 707281 T^{8} )^{4} \)
$31$ \( ( 1 - 8 T - 54 T^{2} + 352 T^{3} + 3285 T^{4} - 9272 T^{5} - 157694 T^{6} + 60416 T^{7} + 6248764 T^{8} + 1872896 T^{9} - 151543934 T^{10} - 276222152 T^{11} + 3033766485 T^{12} + 10077461152 T^{13} - 47925198774 T^{14} - 220100912888 T^{15} + 852891037441 T^{16} )^{2} \)
$37$ \( 1 + 128 T^{2} + 6574 T^{4} + 187656 T^{6} + 5183961 T^{8} + 158877780 T^{10} + 5690568750 T^{12} + 459641012708 T^{14} + 25029766954564 T^{16} + 629248546397252 T^{18} + 10665042019068750 T^{20} + 407636915949292020 T^{22} + 18208556502427761081 T^{24} + \)\(90\!\cdots\!44\)\( T^{26} + \)\(43\!\cdots\!94\)\( T^{28} + \)\(11\!\cdots\!92\)\( T^{30} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( ( 1 + 4 T + 114 T^{2} + 346 T^{3} + 5976 T^{4} + 14186 T^{5} + 191634 T^{6} + 275684 T^{7} + 2825761 T^{8} )^{4} \)
$43$ \( ( 1 - 216 T^{2} + 22138 T^{4} - 1464812 T^{6} + 71619331 T^{8} - 2708437388 T^{10} + 75685416538 T^{12} - 1365414418584 T^{14} + 11688200277601 T^{16} )^{2} \)
$47$ \( 1 + 308 T^{2} + 50892 T^{4} + 5950136 T^{6} + 547353834 T^{8} + 41768130380 T^{10} + 2726163659344 T^{12} + 155042751818668 T^{14} + 7757667166613347 T^{16} + 342489438767437612 T^{18} + 13302809011391389264 T^{20} + \)\(45\!\cdots\!20\)\( T^{22} + \)\(13\!\cdots\!74\)\( T^{24} + \)\(31\!\cdots\!64\)\( T^{26} + \)\(59\!\cdots\!72\)\( T^{28} + \)\(79\!\cdots\!52\)\( T^{30} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 + 264 T^{2} + 35652 T^{4} + 3177712 T^{6} + 212585578 T^{8} + 12279652216 T^{10} + 710817442960 T^{12} + 42611108403128 T^{14} + 2398437218467955 T^{16} + 119694603504386552 T^{18} + 5608691528144463760 T^{20} + \)\(27\!\cdots\!64\)\( T^{22} + \)\(13\!\cdots\!58\)\( T^{24} + \)\(55\!\cdots\!88\)\( T^{26} + \)\(17\!\cdots\!32\)\( T^{28} + \)\(36\!\cdots\!16\)\( T^{30} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( ( 1 + 2 T - 226 T^{2} - 236 T^{3} + 32016 T^{4} + 19820 T^{5} - 2961940 T^{6} - 407278 T^{7} + 205790979 T^{8} - 24029402 T^{9} - 10310513140 T^{10} + 4070611780 T^{11} + 387949429776 T^{12} - 168722134564 T^{13} - 9532800602866 T^{14} + 4977302969638 T^{15} + 146830437604321 T^{16} )^{2} \)
$61$ \( ( 1 - 8 T - 80 T^{2} + 376 T^{3} + 3998 T^{4} + 13396 T^{5} - 247632 T^{6} - 798752 T^{7} + 15355491 T^{8} - 48723872 T^{9} - 921438672 T^{10} + 3040637476 T^{11} + 55355672318 T^{12} + 317568209176 T^{13} - 4121629948880 T^{14} - 25141942688168 T^{15} + 191707312997281 T^{16} )^{2} \)
$67$ \( 1 + 356 T^{2} + 63502 T^{4} + 8090208 T^{6} + 858425049 T^{8} + 80231005476 T^{10} + 6774300927054 T^{12} + 525083738078264 T^{14} + 37066264561921156 T^{16} + 2357100900233327096 T^{18} + \)\(13\!\cdots\!34\)\( T^{20} + \)\(72\!\cdots\!44\)\( T^{22} + \)\(34\!\cdots\!09\)\( T^{24} + \)\(14\!\cdots\!92\)\( T^{26} + \)\(51\!\cdots\!22\)\( T^{28} + \)\(13\!\cdots\!24\)\( T^{30} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( ( 1 - 14 T + 194 T^{2} - 1648 T^{3} + 14264 T^{4} - 117008 T^{5} + 977954 T^{6} - 5010754 T^{7} + 25411681 T^{8} )^{4} \)
$73$ \( 1 + 312 T^{2} + 46894 T^{4} + 4662008 T^{6} + 358067385 T^{8} + 22613492772 T^{10} + 1186535858958 T^{12} + 57781939528716 T^{14} + 3463839236542756 T^{16} + 307919955748527564 T^{18} + 33695531277831292878 T^{20} + \)\(34\!\cdots\!08\)\( T^{22} + \)\(28\!\cdots\!85\)\( T^{24} + \)\(20\!\cdots\!92\)\( T^{26} + \)\(10\!\cdots\!74\)\( T^{28} + \)\(38\!\cdots\!08\)\( T^{30} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( ( 1 + 8 T - 98 T^{2} - 1128 T^{3} - 3647 T^{4} - 31028 T^{5} - 546546 T^{6} + 5486636 T^{7} + 152161556 T^{8} + 433444244 T^{9} - 3410993586 T^{10} - 15298014092 T^{11} - 142050945407 T^{12} - 3470919618072 T^{13} - 23822570641058 T^{14} + 153631271889272 T^{15} + 1517108809906561 T^{16} )^{2} \)
$83$ \( ( 1 - 516 T^{2} + 125904 T^{4} - 18859016 T^{6} + 1891189930 T^{8} - 129919761224 T^{10} + 5975192447184 T^{12} - 168701232658404 T^{14} + 2252292232139041 T^{16} )^{2} \)
$89$ \( ( 1 + 8 T - 98 T^{2} - 1004 T^{3} + 4584 T^{4} + 89290 T^{5} + 1051584 T^{6} - 2927032 T^{7} - 142835833 T^{8} - 260505848 T^{9} + 8329596864 T^{10} + 62946682010 T^{11} + 287610432744 T^{12} - 5606395686796 T^{13} - 48704166514178 T^{14} + 353850679164232 T^{15} + 3936588805702081 T^{16} )^{2} \)
$97$ \( ( 1 - 756 T^{2} + 251932 T^{4} - 48333356 T^{6} + 5838717718 T^{8} - 454768546604 T^{10} + 22303358820892 T^{12} - 629726835726324 T^{14} + 7837433594376961 T^{16} )^{2} \)
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