Properties

 Label 350.2.g.b Level 350 Weight 2 Character orbit 350.g Analytic conductor 2.795 Analytic rank 0 Dimension 16 CM no Inner twists 8

Related objects

Newspace parameters

 Level: $$N$$ = $$350 = 2 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 350.g (of order $$4$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$2.79476407074$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + \beta_{3} q^{3} + \beta_{10} q^{4} + \beta_{15} q^{6} + ( 2 \beta_{2} + \beta_{6} + \beta_{12} ) q^{7} -\beta_{8} q^{8} + ( -2 \beta_{1} + 4 \beta_{10} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{2} + \beta_{3} q^{3} + \beta_{10} q^{4} + \beta_{15} q^{6} + ( 2 \beta_{2} + \beta_{6} + \beta_{12} ) q^{7} -\beta_{8} q^{8} + ( -2 \beta_{1} + 4 \beta_{10} ) q^{9} + \beta_{13} q^{11} -\beta_{14} q^{12} + ( 2 \beta_{5} + \beta_{8} - \beta_{11} ) q^{13} + ( \beta_{9} - \beta_{10} ) q^{14} - q^{16} + ( \beta_{2} + \beta_{6} + 2 \beta_{12} + \beta_{14} ) q^{17} + ( -4 \beta_{8} - 2 \beta_{11} ) q^{18} + ( -\beta_{1} + \beta_{7} + 2 \beta_{9} + \beta_{10} ) q^{19} + ( -1 - \beta_{4} - 3 \beta_{13} - 2 \beta_{15} ) q^{21} + \beta_{6} q^{22} + ( 3 \beta_{8} - 3 \beta_{11} ) q^{23} -\beta_{7} q^{24} + ( -1 - 2 \beta_{4} - \beta_{13} ) q^{26} + ( -2 \beta_{2} - 2 \beta_{6} - 4 \beta_{12} - 3 \beta_{14} ) q^{27} + ( -\beta_{5} + \beta_{8} + \beta_{11} ) q^{28} + ( -\beta_{1} - 3 \beta_{10} ) q^{29} + ( 1 + 2 \beta_{4} + \beta_{13} - 2 \beta_{15} ) q^{31} + \beta_{2} q^{32} + ( -\beta_{3} + 2 \beta_{5} + \beta_{8} - \beta_{11} ) q^{33} + ( -\beta_{1} + \beta_{7} + 2 \beta_{9} + \beta_{10} ) q^{34} + ( -4 + 2 \beta_{13} ) q^{36} + ( 3 \beta_{2} + 3 \beta_{6} ) q^{37} + ( \beta_{3} - 2 \beta_{5} - \beta_{8} + \beta_{11} ) q^{38} + ( 5 \beta_{1} + \beta_{10} ) q^{39} + 3 \beta_{15} q^{41} + ( \beta_{2} - 2 \beta_{6} + \beta_{12} + 2 \beta_{14} ) q^{42} + 2 \beta_{11} q^{43} + \beta_{1} q^{44} + ( 3 + 3 \beta_{13} ) q^{46} + ( -\beta_{2} - \beta_{6} - 2 \beta_{12} + 2 \beta_{14} ) q^{47} -\beta_{3} q^{48} + ( -2 \beta_{1} - \beta_{7} - 2 \beta_{9} - 2 \beta_{10} ) q^{49} + ( 6 - 7 \beta_{13} ) q^{51} + ( \beta_{2} + \beta_{6} + 2 \beta_{12} ) q^{52} + ( -9 \beta_{8} + 3 \beta_{11} ) q^{53} + ( 2 \beta_{1} - 3 \beta_{7} - 4 \beta_{9} - 2 \beta_{10} ) q^{54} + ( 2 + \beta_{4} ) q^{56} + ( -6 \beta_{2} - 7 \beta_{6} ) q^{57} + ( 3 \beta_{8} - \beta_{11} ) q^{58} + ( 2 + 4 \beta_{4} + 2 \beta_{13} + 2 \beta_{15} ) q^{61} + ( -\beta_{2} - \beta_{6} - 2 \beta_{12} + 2 \beta_{14} ) q^{62} + ( 2 \beta_{3} - 2 \beta_{5} + 8 \beta_{8} + 6 \beta_{11} ) q^{63} -\beta_{10} q^{64} + ( -1 - 2 \beta_{4} - \beta_{13} - \beta_{15} ) q^{66} + ( -3 \beta_{2} + 4 \beta_{6} ) q^{67} + ( \beta_{3} - 2 \beta_{5} - \beta_{8} + \beta_{11} ) q^{68} + ( 3 \beta_{1} - 6 \beta_{9} - 3 \beta_{10} ) q^{69} + ( 3 + 3 \beta_{13} ) q^{71} + ( 4 \beta_{2} + 2 \beta_{6} ) q^{72} + ( -3 \beta_{3} - 2 \beta_{5} - \beta_{8} + \beta_{11} ) q^{73} + ( 3 \beta_{1} - 3 \beta_{10} ) q^{74} + ( 1 + 2 \beta_{4} + \beta_{13} + \beta_{15} ) q^{76} + ( -\beta_{2} - \beta_{6} + \beta_{12} - \beta_{14} ) q^{77} + ( -\beta_{8} + 5 \beta_{11} ) q^{78} -2 \beta_{10} q^{79} + ( -7 + 10 \beta_{13} ) q^{81} -3 \beta_{14} q^{82} -3 \beta_{3} q^{83} + ( -3 \beta_{1} + 2 \beta_{7} + \beta_{9} ) q^{84} -2 \beta_{13} q^{86} + ( -\beta_{2} - \beta_{6} - 2 \beta_{12} + 2 \beta_{14} ) q^{87} + \beta_{11} q^{88} + 3 \beta_{7} q^{89} + ( -7 + 2 \beta_{4} - 2 \beta_{13} + \beta_{15} ) q^{91} + ( -3 \beta_{2} + 3 \beta_{6} ) q^{92} + ( 15 \beta_{8} - \beta_{11} ) q^{93} + ( \beta_{1} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} ) q^{94} -\beta_{15} q^{96} + ( 2 \beta_{2} + 2 \beta_{6} + 4 \beta_{12} ) q^{97} + ( -\beta_{3} + 2 \beta_{5} + 2 \beta_{8} - 4 \beta_{11} ) q^{98} + ( 4 \beta_{1} - 6 \beta_{10} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 16q^{16} - 8q^{21} - 64q^{36} + 48q^{46} + 96q^{51} + 24q^{56} + 48q^{71} - 112q^{81} - 128q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 4 x^{15} + 22 x^{14} - 52 x^{13} + 72 x^{12} - 32 x^{11} + 148 x^{10} + 268 x^{9} - 461 x^{8} - 1548 x^{7} - 840 x^{6} + 1800 x^{5} + 2772 x^{4} + 1296 x^{3} + 54 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-38872 \nu^{15} - 2381588 \nu^{14} + 12272984 \nu^{13} - 69438992 \nu^{12} + 212969856 \nu^{11} - 434697520 \nu^{10} + 570478520 \nu^{9} - 992796022 \nu^{8} + 314494856 \nu^{7} + 1091417064 \nu^{6} + 2291730528 \nu^{5} - 458726904 \nu^{4} - 3471322176 \nu^{3} - 2544609708 \nu^{2} - 972706320 \nu - 2919969$$$$)/ 345215871$$ $$\beta_{2}$$ $$=$$ $$($$$$164256682774 \nu^{15} + 204536771036 \nu^{14} + 325719164833 \nu^{13} + 8692516945889 \nu^{12} - 23516385009300 \nu^{11} + 15906345107914 \nu^{10} + 111290307836611 \nu^{9} - 55211894373164 \nu^{8} + 497892445866616 \nu^{7} - 1133837927643444 \nu^{6} - 1268541200481000 \nu^{5} - 299275361605782 \nu^{4} + 2590246950620463 \nu^{3} + 2454748202917620 \nu^{2} + 273108810467583 \nu - 284144487711858$$$$)/ 549627118237197$$ $$\beta_{3}$$ $$=$$ $$($$$$43638140738 \nu^{15} - 108501139208 \nu^{14} + 1089181275053 \nu^{13} - 3087532903250 \nu^{12} + 11959778511348 \nu^{11} - 36325148486737 \nu^{10} + 89956571259893 \nu^{9} - 100487090964100 \nu^{8} + 186320651064800 \nu^{7} - 211665977308953 \nu^{6} - 288058576947906 \nu^{5} - 220128912256230 \nu^{4} + 618585338637621 \nu^{3} + 637410758877801 \nu^{2} + 82388130747813 \nu - 82638681727491$$$$)/ 78097150848717$$ $$\beta_{4}$$ $$=$$ $$($$$$163615281827 \nu^{15} - 1924117603277 \nu^{14} + 9820060318151 \nu^{13} - 42082757802527 \nu^{12} + 108099545116668 \nu^{11} - 183794062239982 \nu^{10} + 225298025529227 \nu^{9} - 319652233046497 \nu^{8} - 102895505752852 \nu^{7} + 371431297830933 \nu^{6} + 1219336153638045 \nu^{5} + 200951791653015 \nu^{4} - 1607767040263761 \nu^{3} - 1253417372476425 \nu^{2} - 533327597537166 \nu - 146731242111066$$$$)/ 120872353803201$$ $$\beta_{5}$$ $$=$$ $$($$$$-70174138296448 \nu^{15} + 577544903020459 \nu^{14} - 2934740679409645 \nu^{13} + 11028779203463398 \nu^{12} - 25069276072999551 \nu^{11} + 34770605390794244 \nu^{10} - 35376058055538736 \nu^{9} + 31294434914848703 \nu^{8} + 84855619162436663 \nu^{7} - 78749415441458469 \nu^{6} - 298416311024553075 \nu^{5} - 64985603648594967 \nu^{4} + 409553940671029440 \nu^{3} + 302818878889737561 \nu^{2} - 60669250048180440 \nu - 46094509753176879$$$$)/ 29130237266571441$$ $$\beta_{6}$$ $$=$$ $$($$$$2609331078951 \nu^{15} - 16870152732110 \nu^{14} + 88009506904826 \nu^{13} - 299991731407889 \nu^{12} + 644449878874712 \nu^{11} - 880207550804925 \nu^{10} + 1155109294634038 \nu^{9} - 756579156568964 \nu^{8} - 1920313255528394 \nu^{7} - 433288765014029 \nu^{6} + 5076366605029425 \nu^{5} + 4061177041351872 \nu^{4} - 3935681025672654 \nu^{3} - 6196762966975137 \nu^{2} - 486446367946239 \nu + 743111712710100$$$$)/ 746929160681319$$ $$\beta_{7}$$ $$=$$ $$($$$$-54279338604888 \nu^{15} + 236515227234986 \nu^{14} - 1273873716269762 \nu^{13} + 3252351796040789 \nu^{12} - 4956727004433128 \nu^{11} + 3206056427189835 \nu^{10} - 8962430797065358 \nu^{9} - 10642004692598344 \nu^{8} + 27384995181843716 \nu^{7} + 76730197022533115 \nu^{6} + 11939512158957732 \nu^{5} - 112098722037237624 \nu^{4} - 89041748984658522 \nu^{3} - 13941349662908187 \nu^{2} + 8382124735449354 \nu - 24869721363328422$$$$)/ 9710079088857147$$ $$\beta_{8}$$ $$=$$ $$($$$$3142423666847 \nu^{15} - 16131464631308 \nu^{14} + 87273661483760 \nu^{13} - 260336341738865 \nu^{12} + 510756735067080 \nu^{11} - 632955095385907 \nu^{10} + 1055357401698752 \nu^{9} - 133202435424376 \nu^{8} - 1557584687050456 \nu^{7} - 2769882069123531 \nu^{6} + 610256463229419 \nu^{5} + 4212499684952934 \nu^{4} + 2724647867499006 \nu^{3} + 1531564756202619 \nu^{2} + 777478465736487 \nu + 421448942618136$$$$)/ 549627118237197$$ $$\beta_{9}$$ $$=$$ $$($$$$-235914283929869 \nu^{15} + 1086467692342127 \nu^{14} - 5844236930008145 \nu^{13} + 15708745163468318 \nu^{12} - 25945308398381394 \nu^{11} + 20511041018220835 \nu^{10} - 38904643127564015 \nu^{9} - 57297344904804641 \nu^{8} + 167337934618115974 \nu^{7} + 231800181017805096 \nu^{6} + 65405734199691825 \nu^{5} - 421558023784791933 \nu^{4} - 383413014826398765 \nu^{3} - 127530458017118910 \nu^{2} + 25202589053652882 \nu + 47379199593598020$$$$)/ 29130237266571441$$ $$\beta_{10}$$ $$=$$ $$($$$$-1220621110 \nu^{15} + 5538431044 \nu^{14} - 29916467584 \nu^{13} + 79884634924 \nu^{12} - 132656080236 \nu^{11} + 114721479509 \nu^{10} - 248411543284 \nu^{9} - 187820132338 \nu^{8} + 639264717188 \nu^{7} + 1547056145433 \nu^{6} + 203026460946 \nu^{5} - 2103594948162 \nu^{4} - 2157242899680 \nu^{3} - 655544414871 \nu^{2} - 78497093142 \nu - 104638235589$$$$)/ 113348549853$$ $$\beta_{11}$$ $$=$$ $$($$$$9183033380386 \nu^{15} - 42272449001952 \nu^{14} + 226004099288307 \nu^{13} - 608902798973025 \nu^{12} + 1000731748673054 \nu^{11} - 850245848383358 \nu^{10} + 1856538403429967 \nu^{9} + 1197376668202520 \nu^{8} - 4945484611719772 \nu^{7} - 12018853180199480 \nu^{6} + 208522897581678 \nu^{5} + 19287495565584420 \nu^{4} + 15629706233463195 \nu^{3} - 390934421357616 \nu^{2} - 3695943675002571 \nu + 614103236946774$$$$)/ 746929160681319$$ $$\beta_{12}$$ $$=$$ $$($$$$-369366743509877 \nu^{15} + 1760194066318517 \nu^{14} - 9356450740323629 \nu^{13} + 25876155838080746 \nu^{12} - 43731935946834513 \nu^{11} + 38712940016522782 \nu^{10} - 74882249931414506 \nu^{9} - 45784465615227785 \nu^{8} + 218914197525848155 \nu^{7} + 441973470675802617 \nu^{6} - 103695898874018280 \nu^{5} - 738129670285227165 \nu^{4} - 523968980224927176 \nu^{3} + 133474711682638197 \nu^{2} + 182327929422619221 \nu - 15473205872688150$$$$)/ 29130237266571441$$ $$\beta_{13}$$ $$=$$ $$($$$$-126187710 \nu^{15} + 593368666 \nu^{14} - 3184365676 \nu^{13} + 8758095976 \nu^{12} - 15009614260 \nu^{11} + 13934786871 \nu^{10} - 27150242048 \nu^{9} - 16487642192 \nu^{8} + 73000370560 \nu^{7} + 143462152855 \nu^{6} + 4244412486 \nu^{5} - 240140451714 \nu^{4} - 184773085128 \nu^{3} - 33770571183 \nu^{2} + 18801253062 \nu + 2771443701$$$$)/ 9819835389$$ $$\beta_{14}$$ $$=$$ $$($$$$-1457028738787 \nu^{15} + 7164351711370 \nu^{14} - 38812705709614 \nu^{13} + 111962508475924 \nu^{12} - 211062170853936 \nu^{11} + 246082436553464 \nu^{10} - 444281345205412 \nu^{9} + 427598325680 \nu^{8} + 674868713943308 \nu^{7} + 1521773137852152 \nu^{6} - 78631195057815 \nu^{5} - 2261087359388622 \nu^{4} - 1667564364864600 \nu^{3} - 549328181757480 \nu^{2} - 121813985328993 \nu - 181099278294957$$$$)/ 78097150848717$$ $$\beta_{15}$$ $$=$$ $$($$$$3123343272944 \nu^{15} - 14410051312652 \nu^{14} + 78052801664720 \nu^{13} - 212540358636827 \nu^{12} + 367646793364128 \nu^{11} - 358625051054356 \nu^{10} + 741529550053700 \nu^{9} + 321059535220034 \nu^{8} - 1506565032423148 \nu^{7} - 3857353147330062 \nu^{6} - 443680272047280 \nu^{5} + 5307285636786978 \nu^{4} + 5273386081423200 \nu^{3} + 1447629997747284 \nu^{2} + 73689678228060 \nu + 277416603956664$$$$)/ 120872353803201$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{15} - \beta_{14} - \beta_{13} + \beta_{11} - 2 \beta_{8} + \beta_{7} - \beta_{6} - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{14} + 2 \beta_{13} + 2 \beta_{11} - \beta_{9} + 3 \beta_{8} - \beta_{7} - 4 \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} - 3 \beta_{2} + 2 \beta_{1} - 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$9 \beta_{15} + 9 \beta_{14} + 3 \beta_{13} + 3 \beta_{12} + 4 \beta_{11} + 16 \beta_{10} - \beta_{9} + 17 \beta_{8} - 3 \beta_{7} + 3 \beta_{6} + \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 7 \beta_{2} - 4 \beta_{1} - 4$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$15 \beta_{15} - 4 \beta_{14} - 36 \beta_{13} - 12 \beta_{12} - 26 \beta_{11} + 51 \beta_{10} + 24 \beta_{9} + 6 \beta_{8} + 21 \beta_{7} + 30 \beta_{6} + 20 \beta_{5} + 6 \beta_{4} - 24 \beta_{3} + 50 \beta_{2} - 27 \beta_{1} + 48$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-185 \beta_{15} - 221 \beta_{14} - 259 \beta_{13} - 194 \beta_{12} - 279 \beta_{11} - 282 \beta_{10} + 146 \beta_{9} - 390 \beta_{8} + 129 \beta_{7} - 103 \beta_{6} + 8 \beta_{5} - 128 \beta_{4} + 45 \beta_{3} - 212 \beta_{2} + 121 \beta_{1} + 142$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-315 \beta_{15} + 245 \beta_{13} + 130 \beta_{11} - 798 \beta_{10} - 182 \beta_{9} - 130 \beta_{8} - 315 \beta_{7} - 480 \beta_{6} - 260 \beta_{5} - 182 \beta_{4} + 450 \beta_{3} - 1010 \beta_{2} + 427 \beta_{1} - 798$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$2007 \beta_{15} + 3365 \beta_{14} + 4053 \beta_{13} + 2688 \beta_{12} + 4189 \beta_{11} + 2782 \beta_{10} - 1968 \beta_{9} + 6534 \beta_{8} - 2755 \beta_{7} + 1029 \beta_{6} - 106 \beta_{5} + 1834 \beta_{4} + 543 \beta_{3} + 50 \beta_{2} - 1731 \beta_{1} - 4640$$$$)/4$$ $$\nu^{8}$$ $$=$$ $$($$$$5644 \beta_{15} + 1584 \beta_{14} - 1530 \beta_{13} + 1632 \beta_{12} + 12 \beta_{11} + 12546 \beta_{10} + 2244 \beta_{9} + 5364 \beta_{8} + 3400 \beta_{7} + 8604 \beta_{6} + 4800 \beta_{5} + 4556 \beta_{4} - 6384 \beta_{3} + 13980 \beta_{2} - 8347 \beta_{1} + 9503$$$$)/2$$ $$\nu^{9}$$ $$=$$ $$($$$$-10188 \beta_{15} - 24621 \beta_{14} - 31146 \beta_{13} - 16820 \beta_{12} - 31146 \beta_{11} - 12698 \beta_{10} + 16820 \beta_{9} - 47531 \beta_{8} + 24621 \beta_{7} + 3038 \beta_{6} + 6960 \beta_{5} - 6960 \beta_{4} - 10188 \beta_{3} + 12698 \beta_{2} + 3038 \beta_{1} + 47531$$$$)/2$$ $$\nu^{10}$$ $$=$$ $$($$$$-98523 \beta_{15} - 51649 \beta_{14} - 2583 \beta_{13} - 36076 \beta_{12} - 25868 \beta_{11} - 213118 \beta_{10} - 22017 \beta_{9} - 141085 \beta_{8} - 25502 \beta_{7} - 125512 \beta_{6} - 67222 \beta_{5} - 73021 \beta_{4} + 87725 \beta_{3} - 196475 \beta_{2} + 129027 \beta_{1} - 86674$$$$)/2$$ $$\nu^{11}$$ $$=$$ $$($$$$109009 \beta_{15} + 637253 \beta_{14} + 981177 \beta_{13} + 464092 \beta_{12} + 936605 \beta_{11} - 62304 \beta_{10} - 591228 \beta_{9} + 1120218 \beta_{8} - 792287 \beta_{7} - 358907 \beta_{6} - 372042 \beta_{5} + 65162 \beta_{4} + 483229 \beta_{3} - 744524 \beta_{2} + 182621 \beta_{1} - 1581580$$$$)/4$$ $$\nu^{12}$$ $$=$$ $$787347 \beta_{15} + 556710 \beta_{14} + 282051 \beta_{13} + 398860 \beta_{12} + 450940 \beta_{11} + 1624689 \beta_{10} + 1348200 \beta_{8} + 849800 \beta_{6} + 398860 \beta_{5} + 564102 \beta_{4} - 556710 \beta_{3} + 1348200 \beta_{2} - 919809 \beta_{1} + 282051$$ $$\nu^{13}$$ $$=$$ $$($$$$1632245 \beta_{15} - 7435165 \beta_{14} - 13915707 \beta_{13} - 5572470 \beta_{12} - 12497147 \beta_{11} + 7903392 \beta_{10} + 8971670 \beta_{9} - 11593074 \beta_{8} + 12146865 \beta_{7} + 8725773 \beta_{6} + 7115420 \beta_{5} + 1091220 \beta_{4} - 9743145 \beta_{3} + 17196962 \beta_{2} - 6305223 \beta_{1} + 25389014$$$$)/4$$ $$\nu^{14}$$ $$=$$ $$($$$$-23172484 \beta_{15} - 20744243 \beta_{14} - 15986471 \beta_{13} - 15336412 \beta_{12} - 20372612 \beta_{11} - 45163113 \beta_{10} + 4680337 \beta_{9} - 46807761 \beta_{8} + 6164049 \beta_{7} - 21817900 \beta_{6} - 8717358 \beta_{5} - 17008435 \beta_{4} + 12026885 \beta_{3} - 32399497 \beta_{2} + 25152599 \beta_{1} + 4024043$$$$)/2$$ $$\nu^{15}$$ $$=$$ $$($$$$-35996985 \beta_{15} + 35996985 \beta_{14} + 89333392 \beta_{13} + 26174563 \beta_{12} + 74019469 \beta_{11} - 105530663 \beta_{10} - 63191309 \beta_{9} + 42339354 \beta_{8} - 86904855 \beta_{7} - 89333392 \beta_{6} - 63191309 \beta_{5} - 26174563 \beta_{4} + 86904855 \beta_{3} - 165408534 \beta_{2} + 74019469 \beta_{1} - 191583097$$$$)/2$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/350\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$\beta_{10}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
293.1
 0.510680 + 3.87900i 2.11123 − 1.62000i −0.645299 + 0.495156i −0.269499 − 2.04705i −0.847914 + 0.111630i 0.205100 + 0.267292i −0.671026 − 0.874498i 1.60673 − 0.211530i 0.510680 − 3.87900i 2.11123 + 1.62000i −0.645299 − 0.495156i −0.269499 + 2.04705i −0.847914 − 0.111630i 0.205100 − 0.267292i −0.671026 + 0.874498i 1.60673 + 0.211530i
−0.707107 + 0.707107i −2.28737 + 2.28737i 1.00000i 0 3.23483i 0.835798 2.51027i 0.707107 + 0.707107i 7.46410i 0
293.2 −0.707107 + 0.707107i −1.32964 + 1.32964i 1.00000i 0 1.88040i 2.26461 + 1.36804i 0.707107 + 0.707107i 0.535898i 0
293.3 −0.707107 + 0.707107i 1.32964 1.32964i 1.00000i 0 1.88040i −1.36804 2.26461i 0.707107 + 0.707107i 0.535898i 0
293.4 −0.707107 + 0.707107i 2.28737 2.28737i 1.00000i 0 3.23483i 2.51027 0.835798i 0.707107 + 0.707107i 7.46410i 0
293.5 0.707107 0.707107i −2.28737 + 2.28737i 1.00000i 0 3.23483i −2.51027 + 0.835798i −0.707107 0.707107i 7.46410i 0
293.6 0.707107 0.707107i −1.32964 + 1.32964i 1.00000i 0 1.88040i 1.36804 + 2.26461i −0.707107 0.707107i 0.535898i 0
293.7 0.707107 0.707107i 1.32964 1.32964i 1.00000i 0 1.88040i −2.26461 1.36804i −0.707107 0.707107i 0.535898i 0
293.8 0.707107 0.707107i 2.28737 2.28737i 1.00000i 0 3.23483i −0.835798 + 2.51027i −0.707107 0.707107i 7.46410i 0
307.1 −0.707107 0.707107i −2.28737 2.28737i 1.00000i 0 3.23483i 0.835798 + 2.51027i 0.707107 0.707107i 7.46410i 0
307.2 −0.707107 0.707107i −1.32964 1.32964i 1.00000i 0 1.88040i 2.26461 1.36804i 0.707107 0.707107i 0.535898i 0
307.3 −0.707107 0.707107i 1.32964 + 1.32964i 1.00000i 0 1.88040i −1.36804 + 2.26461i 0.707107 0.707107i 0.535898i 0
307.4 −0.707107 0.707107i 2.28737 + 2.28737i 1.00000i 0 3.23483i 2.51027 + 0.835798i 0.707107 0.707107i 7.46410i 0
307.5 0.707107 + 0.707107i −2.28737 2.28737i 1.00000i 0 3.23483i −2.51027 0.835798i −0.707107 + 0.707107i 7.46410i 0
307.6 0.707107 + 0.707107i −1.32964 1.32964i 1.00000i 0 1.88040i 1.36804 2.26461i −0.707107 + 0.707107i 0.535898i 0
307.7 0.707107 + 0.707107i 1.32964 + 1.32964i 1.00000i 0 1.88040i −2.26461 + 1.36804i −0.707107 + 0.707107i 0.535898i 0
307.8 0.707107 + 0.707107i 2.28737 + 2.28737i 1.00000i 0 3.23483i −0.835798 2.51027i −0.707107 + 0.707107i 7.46410i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 307.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.b odd 2 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.g.b 16
5.b even 2 1 inner 350.2.g.b 16
5.c odd 4 2 inner 350.2.g.b 16
7.b odd 2 1 inner 350.2.g.b 16
35.c odd 2 1 inner 350.2.g.b 16
35.f even 4 2 inner 350.2.g.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.g.b 16 1.a even 1 1 trivial
350.2.g.b 16 5.b even 2 1 inner
350.2.g.b 16 5.c odd 4 2 inner
350.2.g.b 16 7.b odd 2 1 inner
350.2.g.b 16 35.c odd 2 1 inner
350.2.g.b 16 35.f even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 122 T_{3}^{4} + 1369$$ acting on $$S_{2}^{\mathrm{new}}(350, [\chi])$$.

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{4}$$
$3$ $$( 1 - 10 T^{4} + 139 T^{8} - 810 T^{12} + 6561 T^{16} )^{2}$$
$5$ 
$7$ $$1 + 28 T^{4} + 3270 T^{8} + 67228 T^{12} + 5764801 T^{16}$$
$11$ $$( 1 + 19 T^{2} + 121 T^{4} )^{8}$$
$13$ $$( 1 - 260 T^{4} + 30822 T^{8} - 7425860 T^{12} + 815730721 T^{16} )^{2}$$
$17$ $$( 1 - 602 T^{4} + 184635 T^{8} - 50279642 T^{12} + 6975757441 T^{16} )^{2}$$
$19$ $$( 1 + 34 T^{2} + 903 T^{4} + 12274 T^{6} + 130321 T^{8} )^{4}$$
$23$ $$( 1 + 28 T^{4} + 171078 T^{8} + 7835548 T^{12} + 78310985281 T^{16} )^{2}$$
$29$ $$( 1 - 92 T^{2} + 3690 T^{4} - 77372 T^{6} + 707281 T^{8} )^{4}$$
$31$ $$( 1 - 28 T^{2} + 1146 T^{4} - 26908 T^{6} + 923521 T^{8} )^{4}$$
$37$ $$( 1 - 644 T^{4} - 1762266 T^{8} - 1206959684 T^{12} + 3512479453921 T^{16} )^{2}$$
$41$ $$( 1 - 38 T^{2} + 2751 T^{4} - 63878 T^{6} + 2825761 T^{8} )^{4}$$
$43$ $$( 1 + 1778 T^{4} + 3418801 T^{8} )^{4}$$
$47$ $$( 1 - 2660 T^{4} + 3301254 T^{8} - 12979951460 T^{12} + 23811286661761 T^{16} )^{2}$$
$53$ $$( 1 + 6268 T^{4} + 25462950 T^{8} + 49457534908 T^{12} + 62259690411361 T^{16} )^{2}$$
$59$ $$( 1 + 59 T^{2} )^{16}$$
$61$ $$( 1 - 76 T^{2} + 7158 T^{4} - 282796 T^{6} + 13845841 T^{8} )^{4}$$
$67$ $$( 1 - 2642 T^{4} + 1066035 T^{8} - 53239261682 T^{12} + 406067677556641 T^{16} )^{2}$$
$71$ $$( 1 - 6 T + 124 T^{2} - 426 T^{3} + 5041 T^{4} )^{8}$$
$73$ $$( 1 - 11930 T^{4} + 77910459 T^{8} - 338791015130 T^{12} + 806460091894081 T^{16} )^{2}$$
$79$ $$( 1 - 154 T^{2} + 6241 T^{4} )^{8}$$
$83$ $$( 1 - 4394 T^{4} + 58495659 T^{8} - 208531862474 T^{12} + 2252292232139041 T^{16} )^{2}$$
$89$ $$( 1 + 230 T^{2} + 28095 T^{4} + 1821830 T^{6} + 62742241 T^{8} )^{4}$$
$97$ $$( 1 - 380 T^{4} + 60281862 T^{8} - 33641126780 T^{12} + 7837433594376961 T^{16} )^{2}$$