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

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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:

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

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