Properties

Label 1050.3.f.e
Level $1050$
Weight $3$
Character orbit 1050.f
Analytic conductor $28.610$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.6104277578\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 8 x^{15} + 88 x^{14} - 476 x^{13} + 2744 x^{12} - 10640 x^{11} + 39126 x^{10} - 108488 x^{9} + 259514 x^{8} - 486436 x^{7} + 690168 x^{6} - 725188 x^{5} + 619745 x^{4} - 430504 x^{3} + 108130 x^{2} + 42224 x + 33124\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: no (minimal twist has level 210)
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.

\(f(q)\) \(=\) \( q -\beta_{5} q^{2} -\beta_{1} q^{3} + 2 q^{4} -\beta_{8} q^{6} + ( \beta_{1} - \beta_{4} - \beta_{6} ) q^{7} -2 \beta_{5} q^{8} -3 q^{9} +O(q^{10})\) \( q -\beta_{5} q^{2} -\beta_{1} q^{3} + 2 q^{4} -\beta_{8} q^{6} + ( \beta_{1} - \beta_{4} - \beta_{6} ) q^{7} -2 \beta_{5} q^{8} -3 q^{9} + ( 6 + \beta_{10} - \beta_{11} ) q^{11} -2 \beta_{1} q^{12} + ( 3 \beta_{1} - \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{13} + ( -1 + \beta_{2} + \beta_{8} + \beta_{13} ) q^{14} + 4 q^{16} + ( -\beta_{1} - \beta_{3} - 2 \beta_{4} ) q^{17} + 3 \beta_{5} q^{18} + ( -1 - \beta_{9} + \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{19} + ( 1 + \beta_{2} + \beta_{10} - \beta_{13} ) q^{21} + ( -6 \beta_{5} - \beta_{14} - \beta_{15} ) q^{22} + ( -\beta_{1} + 5 \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{14} + \beta_{15} ) q^{23} -2 \beta_{8} q^{24} + ( 1 + 2 \beta_{8} + \beta_{9} - \beta_{11} - 2 \beta_{12} ) q^{26} + 3 \beta_{1} q^{27} + ( 2 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} ) q^{28} + ( -4 - 4 \beta_{2} + 2 \beta_{9} + 2 \beta_{11} ) q^{29} + ( 2 - 4 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} + \beta_{12} + 4 \beta_{13} ) q^{31} -4 \beta_{5} q^{32} + ( -6 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{33} + ( 1 + 2 \beta_{8} + \beta_{9} - \beta_{11} + 3 \beta_{12} + 3 \beta_{13} ) q^{34} -6 q^{36} + ( 3 \beta_{1} + 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} - 3 \beta_{14} + \beta_{15} ) q^{37} + ( \beta_{1} + 2 \beta_{3} - 4 \beta_{4} - 3 \beta_{6} + 3 \beta_{7} ) q^{38} + ( 9 + 3 \beta_{2} - 2 \beta_{9} - 2 \beta_{10} ) q^{39} + ( 1 + 5 \beta_{8} + \beta_{9} - \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{41} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} - 3 \beta_{5} - 2 \beta_{7} - \beta_{14} ) q^{42} + ( 6 \beta_{1} - 18 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} + 2 \beta_{14} ) q^{43} + ( 12 + 2 \beta_{10} - 2 \beta_{11} ) q^{44} + ( -11 - 2 \beta_{2} + 3 \beta_{9} + 4 \beta_{10} - \beta_{11} ) q^{46} + ( -13 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} - 5 \beta_{6} + 5 \beta_{7} ) q^{47} -4 \beta_{1} q^{48} + ( -14 - 2 \beta_{2} + 10 \beta_{8} - \beta_{9} + 2 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} ) q^{49} + ( -3 + \beta_{2} + 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{51} + ( 6 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{52} + ( \beta_{1} - 10 \beta_{5} - \beta_{6} - \beta_{7} - 4 \beta_{14} - 5 \beta_{15} ) q^{53} + 3 \beta_{8} q^{54} + ( -2 + 2 \beta_{2} + 2 \beta_{8} + 2 \beta_{13} ) q^{56} + ( 2 \beta_{1} - 4 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - 3 \beta_{14} - 2 \beta_{15} ) q^{57} + ( -4 \beta_{1} + 6 \beta_{5} + 4 \beta_{6} + 4 \beta_{7} + 4 \beta_{15} ) q^{58} + ( 4 - 8 \beta_{8} + 4 \beta_{9} - 4 \beta_{11} + 8 \beta_{12} + 2 \beta_{13} ) q^{59} + ( 12 \beta_{8} + 6 \beta_{12} + 6 \beta_{13} ) q^{61} + ( -13 \beta_{1} - 4 \beta_{3} - 4 \beta_{4} - 3 \beta_{6} + 3 \beta_{7} ) q^{62} + ( -3 \beta_{1} + 3 \beta_{4} + 3 \beta_{6} ) q^{63} + 8 q^{64} + ( 2 - 4 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} + 3 \beta_{12} + \beta_{13} ) q^{66} + ( 3 \beta_{1} - 22 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} - 2 \beta_{14} + 4 \beta_{15} ) q^{67} + ( -2 \beta_{1} - 2 \beta_{3} - 4 \beta_{4} ) q^{68} + ( 2 + 3 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} - 3 \beta_{12} - 2 \beta_{13} ) q^{69} + ( -24 + 10 \beta_{2} - 4 \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{71} + 6 \beta_{5} q^{72} + ( 3 \beta_{1} - 7 \beta_{3} + 7 \beta_{4} + 7 \beta_{6} - 7 \beta_{7} ) q^{73} + ( -14 + 6 \beta_{2} + 4 \beta_{9} + 6 \beta_{10} - 2 \beta_{11} ) q^{74} + ( -2 - 2 \beta_{9} + 2 \beta_{11} - 4 \beta_{12} + 2 \beta_{13} ) q^{76} + ( 22 \beta_{1} - 5 \beta_{3} + 2 \beta_{5} - 9 \beta_{6} - 5 \beta_{7} + 2 \beta_{14} + \beta_{15} ) q^{77} + ( 3 \beta_{1} - 10 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} + 2 \beta_{14} - 2 \beta_{15} ) q^{78} + ( 38 - 8 \beta_{2} + 2 \beta_{9} + 2 \beta_{11} ) q^{79} + 9 q^{81} + ( 7 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{82} + ( 5 \beta_{1} + 4 \beta_{3} - 10 \beta_{4} - 3 \beta_{6} + 3 \beta_{7} ) q^{83} + ( 2 + 2 \beta_{2} + 2 \beta_{10} - 2 \beta_{13} ) q^{84} + ( 26 + 12 \beta_{2} - 2 \beta_{9} - 4 \beta_{10} + 2 \beta_{11} ) q^{86} + ( 4 \beta_{1} - 6 \beta_{6} + 6 \beta_{7} ) q^{87} + ( -12 \beta_{5} - 2 \beta_{14} - 2 \beta_{15} ) q^{88} + ( -6 - 17 \beta_{8} - 6 \beta_{9} + 6 \beta_{11} - 5 \beta_{12} - 12 \beta_{13} ) q^{89} + ( 13 - 10 \beta_{2} + 8 \beta_{8} + 3 \beta_{9} - 4 \beta_{10} + 9 \beta_{11} - 8 \beta_{12} + \beta_{13} ) q^{91} + ( -2 \beta_{1} + 10 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} - 4 \beta_{14} + 2 \beta_{15} ) q^{92} + ( 5 \beta_{1} + 14 \beta_{5} - 5 \beta_{6} - 5 \beta_{7} + \beta_{15} ) q^{93} + ( 4 - 12 \beta_{8} + 4 \beta_{9} - 4 \beta_{11} - 4 \beta_{12} + 6 \beta_{13} ) q^{94} -4 \beta_{8} q^{96} + ( -33 \beta_{1} - 3 \beta_{3} - 5 \beta_{4} - 5 \beta_{6} + 5 \beta_{7} ) q^{97} + ( 20 \beta_{1} + 17 \beta_{5} + 4 \beta_{7} - 2 \beta_{14} - 4 \beta_{15} ) q^{98} + ( -18 - 3 \beta_{10} + 3 \beta_{11} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 48 q^{9} + O(q^{10}) \) \( 16 q + 32 q^{4} - 48 q^{9} + 96 q^{11} - 16 q^{14} + 64 q^{16} + 24 q^{21} - 64 q^{29} - 96 q^{36} + 144 q^{39} + 192 q^{44} - 176 q^{46} - 224 q^{49} - 48 q^{51} - 32 q^{56} + 128 q^{64} - 384 q^{71} - 224 q^{74} + 608 q^{79} + 144 q^{81} + 48 q^{84} + 416 q^{86} + 224 q^{91} - 288 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16} - 8 x^{15} + 88 x^{14} - 476 x^{13} + 2744 x^{12} - 10640 x^{11} + 39126 x^{10} - 108488 x^{9} + 259514 x^{8} - 486436 x^{7} + 690168 x^{6} - 725188 x^{5} + 619745 x^{4} - 430504 x^{3} + 108130 x^{2} + 42224 x + 33124\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\(92936 \nu^{14} - 650552 \nu^{13} + 7629434 \nu^{12} - 37319428 \nu^{11} + 225649370 \nu^{10} - 801656916 \nu^{9} + 3054133421 \nu^{8} - 7790526200 \nu^{7} + 19009619662 \nu^{6} - 32746371306 \nu^{5} + 45399734980 \nu^{4} - 43898979412 \nu^{3} + 35060551363 \nu^{2} - 17481907352 \nu - 649676357\)\()/ 5030871765 \)
\(\beta_{2}\)\(=\)\((\)\(-4788335 \nu^{14} + 33518345 \nu^{13} - 384267955 \nu^{12} + 1869869245 \nu^{11} - 10965281907 \nu^{10} + 38484795345 \nu^{9} - 139565919575 \nu^{8} + 346554004115 \nu^{7} - 765787653189 \nu^{6} + 1226929460919 \nu^{5} - 1236294585205 \nu^{4} + 764608199695 \nu^{3} - 162963123242 \nu^{2} - 62514228256 \nu + 205883061930\)\()/ 119428521030 \)
\(\beta_{3}\)\(=\)\((\)\(-116229757 \nu^{14} + 813608299 \nu^{13} - 9262265428 \nu^{12} + 44996684681 \nu^{11} - 261565786159 \nu^{10} + 914750319012 \nu^{9} - 3276860586835 \nu^{8} + 8080662254257 \nu^{7} - 17579924497940 \nu^{6} + 27839347208307 \nu^{5} - 29101821255449 \nu^{4} + 19632522741596 \nu^{3} - 27051359900108 \nu^{2} + 20767817705524 \nu + 2568347439658\)\()/ 2746855983690 \)
\(\beta_{4}\)\(=\)\((\)\(-1093349498 \nu^{14} + 7653446486 \nu^{13} - 88689952367 \nu^{12} + 432644909884 \nu^{11} - 2578996359596 \nu^{10} + 9111477265293 \nu^{9} - 34039488965360 \nu^{8} + 85935488321978 \nu^{7} - 203322728917885 \nu^{6} + 343031420986488 \nu^{5} - 448503864698116 \nu^{4} + 409537531927639 \nu^{3} - 359029814503882 \nu^{2} + 199508459888936 \nu + 12465433464302\)\()/ 8240567951070 \)
\(\beta_{5}\)\(=\)\((\)\(279842740 \nu^{15} - 2098820550 \nu^{14} + 23452089504 \nu^{13} - 120606470101 \nu^{12} + 697811667584 \nu^{11} - 2581323646286 \nu^{10} + 9387854016075 \nu^{9} - 24725356812534 \nu^{8} + 56950285171984 \nu^{7} - 99567014762287 \nu^{6} + 126105936855176 \nu^{5} - 112964032428330 \nu^{4} + 89991465893173 \nu^{3} - 58640398536476 \nu^{2} - 24686735907128 \nu + 20065240923728\)\()/ 21993425639458 \)
\(\beta_{6}\)\(=\)\((\)\(1064447881686 \nu^{15} - 3209377373879 \nu^{14} + 57853449494147 \nu^{13} - 83261406982037 \nu^{12} + 909308118211537 \nu^{11} + 858731238749983 \nu^{10} - 531851674789797 \nu^{9} + 44319669382131871 \nu^{8} - 122069337632357551 \nu^{7} + 437800427292617549 \nu^{6} - 888806142074331045 \nu^{5} + 1400448210877820969 \nu^{4} - 1535846687594913197 \nu^{3} + 1536062304795605372 \nu^{2} - 923159515366322356 \nu - 22682224994288314\)\()/ 68289586610517090 \)
\(\beta_{7}\)\(=\)\((\)\(1064447881686 \nu^{15} - 11495817736195 \nu^{14} + 115858532030359 \nu^{13} - 757541135750011 \nu^{12} + 4200920417848625 \nu^{11} - 18832916003825449 \nu^{10} + 69135726258527109 \nu^{9} - 217209573670629559 \nu^{8} + 539879980331172325 \nu^{7} - 1140018364997652341 \nu^{6} + 1786712131988850591 \nu^{5} - 2148896267184962303 \nu^{4} + 1753431284375887721 \nu^{3} - 1280080729563355862 \nu^{2} + 602287486497156556 \nu + 63905884783233220\)\()/ 68289586610517090 \)
\(\beta_{8}\)\(=\)\((\)\(12448 \nu^{15} - 93360 \nu^{14} + 1065318 \nu^{13} - 5508607 \nu^{12} + 32749646 \nu^{11} - 122643488 \nu^{10} + 464891353 \nu^{9} - 1256401704 \nu^{8} + 3117962578 \nu^{7} - 5796321469 \nu^{6} + 8742397202 \nu^{5} - 9767388084 \nu^{4} + 9261694875 \nu^{3} - 6344286416 \nu^{2} + 2030131612 \nu - 179130952\)\()/ 520340730 \)
\(\beta_{9}\)\(=\)\((\)\(-3854937063308 \nu^{15} + 23962636724121 \nu^{14} - 295302862411617 \nu^{13} + 1309538863202951 \nu^{12} - 8216025180516379 \nu^{11} + 26694378874412479 \nu^{10} - 104460706623912221 \nu^{9} + 245952950235553953 \nu^{8} - 612048024969048539 \nu^{7} + 1016038863804619877 \nu^{6} - 1473931130917122289 \nu^{5} + 1803061234163058501 \nu^{4} - 2138724181565629263 \nu^{3} + 1836991252719575194 \nu^{2} - 705321532058718680 \nu + 744756406728367124\)\()/ 68289586610517090 \)
\(\beta_{10}\)\(=\)\((\)\(3854937063308 \nu^{15} - 30212752287741 \nu^{14} + 339053671356957 \nu^{13} - 1809683908646036 \nu^{12} + 10648134936885469 \nu^{11} - 40914836644345741 \nu^{10} + 154311383653392476 \nu^{9} - 426117375260218293 \nu^{8} + 1058567337159689039 \nu^{7} - 1998914880969722516 \nu^{6} + 3044238135652700623 \nu^{5} - 3384302188261511751 \nu^{4} + 3117694261499572908 \nu^{3} - 2045914723451540566 \nu^{2} + 625130317554887084 \nu + 531749434692057946\)\()/ 68289586610517090 \)
\(\beta_{11}\)\(=\)\((\)\(3854937063308 \nu^{15} - 33861419225499 \nu^{14} + 364594339921263 \nu^{13} - 2102794770058199 \nu^{12} + 12074771414022469 \nu^{11} - 49298728429864189 \nu^{10} + 183762060808011509 \nu^{9} - 533270283374438967 \nu^{8} + 1325124389193165749 \nu^{7} - 2590358094781911509 \nu^{6} + 3994713632454400783 \nu^{5} - 4342698800407316979 \nu^{4} + 3709736650323528063 \nu^{3} - 2171916157642537894 \nu^{2} + 576827345172515312 \nu + 700117415521608994\)\()/ 68289586610517090 \)
\(\beta_{12}\)\(=\)\((\)\(5472163425770 \nu^{15} - 41041225693275 \nu^{14} + 464858064633033 \nu^{13} - 2399118830433377 \nu^{12} + 14125631930262397 \nu^{11} - 52670077378974979 \nu^{10} + 196742572634383223 \nu^{9} - 526967009577493719 \nu^{8} + 1274658163466793299 \nu^{7} - 2322142348180659911 \nu^{6} + 3317271373015585711 \nu^{5} - 3490535031960335649 \nu^{4} + 3178048469640185847 \nu^{3} - 2154252345020041282 \nu^{2} + 693477650854020896 \nu - 62893609797828992\)\()/ 68289586610517090 \)
\(\beta_{13}\)\(=\)\((\)\(7209986841170 \nu^{15} - 54074901308775 \nu^{14} + 610495540452873 \nu^{13} - 3148085009760587 \nu^{12} + 18459042385959037 \nu^{11} - 68700097222411039 \nu^{10} + 255041146074208973 \nu^{9} - 680511475383329859 \nu^{8} + 1628319434384813939 \nu^{7} - 2940453509854462181 \nu^{6} + 4100389240886228671 \nu^{5} - 4192041673340264949 \nu^{4} + 3736895472836790177 \nu^{3} - 2518409219931557242 \nu^{2} + 813331367312824376 \nu - 74867636882512292\)\()/ 68289586610517090 \)
\(\beta_{14}\)\(=\)\((\)\(11652755758 \nu^{15} - 87395668185 \nu^{14} + 980333632691 \nu^{13} - 5046667645019 \nu^{12} + 29328141084055 \nu^{11} - 108707533995533 \nu^{10} + 397669560517609 \nu^{9} - 1051227743230329 \nu^{8} + 2442760651969745 \nu^{7} - 4302607169649537 \nu^{6} + 5537445103727985 \nu^{5} - 5073064410728007 \nu^{4} + 3851913596232853 \nu^{3} - 2295132049284554 \nu^{2} - 959984408973388 \nu + 767874169626928\)\()/ 109967128197290 \)
\(\beta_{15}\)\(=\)\((\)\(-165988958814 \nu^{15} + 1244917191105 \nu^{14} - 14017718745515 \nu^{13} + 72233927780755 \nu^{12} - 421896015444175 \nu^{11} + 1567393616297861 \nu^{10} - 5777614529073753 \nu^{9} + 15346661487816381 \nu^{8} - 36122988388313897 \nu^{7} + 64319930863026889 \nu^{6} - 84744155655241521 \nu^{5} + 80129924292009159 \nu^{4} - 56716001622329941 \nu^{3} + 28891531292633078 \nu^{2} + 11928324887229772 \nu - 9230202682938692\)\()/ 989704153775610 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{13} - \beta_{12} - 2 \beta_{5} + 2\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{13} - \beta_{12} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{2} - 28\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(3 \beta_{15} - 3 \beta_{14} - 9 \beta_{13} + 17 \beta_{12} - \beta_{11} + 3 \beta_{10} + 4 \beta_{9} - 11 \beta_{8} + 9 \beta_{7} + 3 \beta_{6} + 47 \beta_{5} - 3 \beta_{4} - 6 \beta_{2} - 6 \beta_{1} - 42\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(6 \beta_{15} - 6 \beta_{14} - 19 \beta_{13} + 35 \beta_{12} - 12 \beta_{11} - 12 \beta_{10} - 20 \beta_{9} - 22 \beta_{8} - 44 \beta_{7} + 68 \beta_{6} + 96 \beta_{5} + 44 \beta_{4} - 4 \beta_{3} + 64 \beta_{2} - 72 \beta_{1} + 330\)\()/4\)
\(\nu^{5}\)\(=\)\((\)\(-90 \beta_{15} + 40 \beta_{14} + 101 \beta_{13} - 185 \beta_{12} - 37 \beta_{11} - 35 \beta_{10} - 48 \beta_{9} + 233 \beta_{8} - 325 \beta_{7} - 35 \beta_{6} - 1083 \beta_{5} + 115 \beta_{4} - 10 \beta_{3} + 170 \beta_{2} + 30 \beta_{1} + 904\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(-285 \beta_{15} + 135 \beta_{14} + 351 \beta_{13} - 643 \beta_{12} + 174 \beta_{11} - 20 \beta_{10} + 216 \beta_{9} + 754 \beta_{8} + 572 \beta_{7} - 1712 \beta_{6} - 3490 \beta_{5} - 758 \beta_{4} - 52 \beta_{3} - 690 \beta_{2} + 2520 \beta_{1} - 2742\)\()/4\)
\(\nu^{7}\)\(=\)\((\)\(2009 \beta_{15} - 224 \beta_{14} - 619 \beta_{13} + 1156 \beta_{12} + 1081 \beta_{11} + 56 \beta_{10} + 585 \beta_{9} - 2444 \beta_{8} + 8211 \beta_{7} - 805 \beta_{6} + 19276 \beta_{5} - 3059 \beta_{4} - 147 \beta_{3} - 3017 \beta_{2} + 3647 \beta_{1} - 13154\)\()/4\)
\(\nu^{8}\)\(=\)\((\)\(9380 \beta_{15} - 1540 \beta_{14} - 4159 \beta_{13} + 7707 \beta_{12} - 186 \beta_{11} + 1340 \beta_{10} - 1334 \beta_{9} - 13346 \beta_{8} - 792 \beta_{7} + 35792 \beta_{6} + 93616 \beta_{5} + 10240 \beta_{4} + 2656 \beta_{3} + 2140 \beta_{2} - 57960 \beta_{1} + 272\)\()/4\)
\(\nu^{9}\)\(=\)\((\)\(-32508 \beta_{15} - 2622 \beta_{14} - 6786 \beta_{13} + 12086 \beta_{12} - 12235 \beta_{11} + 5541 \beta_{10} - 4964 \beta_{9} - 6575 \beta_{8} - 167163 \beta_{7} + 52791 \beta_{6} - 266315 \beta_{5} + 64923 \beta_{4} + 12792 \beta_{3} + 28458 \beta_{2} - 169614 \beta_{1} + 88788\)\()/4\)
\(\nu^{10}\)\(=\)\((\)\(-234915 \beta_{15} - 585 \beta_{14} - 184 \beta_{13} - 2050 \beta_{12} - 42672 \beta_{11} - 8352 \beta_{10} - 23300 \beta_{9} + 72608 \beta_{8} - 206946 \beta_{7} - 635514 \beta_{6} - 2058608 \beta_{5} - 100122 \beta_{4} - 49786 \beta_{3} + 121954 \beta_{2} + 992880 \beta_{1} + 569434\)\()/4\)
\(\nu^{11}\)\(=\)\((\)\(329714 \beta_{15} + 80366 \beta_{14} + 305101 \beta_{13} - 561877 \beta_{12} - 115963 \beta_{11} - 95711 \beta_{10} - 69948 \beta_{9} + 872117 \beta_{8} + 2794847 \beta_{7} - 1680921 \beta_{6} + 2308077 \beta_{5} - 1180729 \beta_{4} - 392590 \beta_{3} + 374792 \beta_{2} + 4748612 \beta_{1} + 2168840\)\()/4\)
\(\nu^{12}\)\(=\)\((\)\(2361744 \beta_{15} + 230076 \beta_{14} + 877983 \beta_{13} - 1603409 \beta_{12} + 435912 \beta_{11} - 118765 \beta_{10} + 568668 \beta_{9} + 2098427 \beta_{8} + 3670524 \beta_{7} + 4606272 \beta_{6} + 19058094 \beta_{5} + 106506 \beta_{4} + 214164 \beta_{3} - 1751265 \beta_{2} - 5938218 \beta_{1} - 6519638\)\()/2\)
\(\nu^{13}\)\(=\)\((\)\(667368 \beta_{15} - 692783 \beta_{14} - 5309960 \beta_{13} + 9895207 \beta_{12} + 9475281 \beta_{11} - 171496 \beta_{10} + 5519413 \beta_{9} - 18651514 \beta_{8} - 36490831 \beta_{7} + 39482053 \beta_{6} + 10923252 \beta_{5} + 18411367 \beta_{4} + 8198255 \beta_{3} - 26829803 \beta_{2} - 99577881 \beta_{1} - 112895206\)\()/4\)
\(\nu^{14}\)\(=\)\((\)\(-75505976 \beta_{15} - 11723894 \beta_{14} - 63474043 \beta_{13} + 117216695 \beta_{12} - 249204 \beta_{11} + 6881616 \beta_{10} - 26378810 \beta_{9} - 190758814 \beta_{8} - 167447536 \beta_{7} - 93405238 \beta_{6} - 577371860 \beta_{5} + 20843508 \beta_{4} + 5260408 \beta_{3} + 44813578 \beta_{2} + 49105602 \beta_{1} + 132095548\)\()/4\)
\(\nu^{15}\)\(=\)\((\)\(-124496877 \beta_{15} - 11320543 \beta_{14} + 24283611 \beta_{13} - 46833803 \beta_{12} - 276574061 \beta_{11} + 49618467 \beta_{10} - 191575552 \beta_{9} + 132896429 \beta_{8} + 284533417 \beta_{7} - 721534045 \beta_{6} - 1010528001 \beta_{5} - 228015103 \beta_{4} - 123432884 \beta_{3} + 818598666 \beta_{2} + 1595075514 \beta_{1} + 3156761778\)\()/4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(701\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
601.1
−0.207107 4.25887i
−0.207107 0.264184i
−0.207107 + 3.39361i
−0.207107 + 1.12945i
−0.207107 + 4.25887i
−0.207107 + 0.264184i
−0.207107 3.39361i
−0.207107 1.12945i
1.20711 1.12945i
1.20711 3.39361i
1.20711 + 0.264184i
1.20711 + 4.25887i
1.20711 + 1.12945i
1.20711 + 3.39361i
1.20711 0.264184i
1.20711 4.25887i
−1.41421 1.73205i 2.00000 0 2.44949i −4.63050 + 5.24961i −2.82843 −3.00000 0
601.2 −1.41421 1.73205i 2.00000 0 2.44949i −0.853218 + 6.94781i −2.82843 −3.00000 0
601.3 −1.41421 1.73205i 2.00000 0 2.44949i 1.57881 6.81963i −2.82843 −3.00000 0
601.4 −1.41421 1.73205i 2.00000 0 2.44949i 6.73333 1.91369i −2.82843 −3.00000 0
601.5 −1.41421 1.73205i 2.00000 0 2.44949i −4.63050 5.24961i −2.82843 −3.00000 0
601.6 −1.41421 1.73205i 2.00000 0 2.44949i −0.853218 6.94781i −2.82843 −3.00000 0
601.7 −1.41421 1.73205i 2.00000 0 2.44949i 1.57881 + 6.81963i −2.82843 −3.00000 0
601.8 −1.41421 1.73205i 2.00000 0 2.44949i 6.73333 + 1.91369i −2.82843 −3.00000 0
601.9 1.41421 1.73205i 2.00000 0 2.44949i −6.73333 1.91369i 2.82843 −3.00000 0
601.10 1.41421 1.73205i 2.00000 0 2.44949i −1.57881 6.81963i 2.82843 −3.00000 0
601.11 1.41421 1.73205i 2.00000 0 2.44949i 0.853218 + 6.94781i 2.82843 −3.00000 0
601.12 1.41421 1.73205i 2.00000 0 2.44949i 4.63050 + 5.24961i 2.82843 −3.00000 0
601.13 1.41421 1.73205i 2.00000 0 2.44949i −6.73333 + 1.91369i 2.82843 −3.00000 0
601.14 1.41421 1.73205i 2.00000 0 2.44949i −1.57881 + 6.81963i 2.82843 −3.00000 0
601.15 1.41421 1.73205i 2.00000 0 2.44949i 0.853218 6.94781i 2.82843 −3.00000 0
601.16 1.41421 1.73205i 2.00000 0 2.44949i 4.63050 5.24961i 2.82843 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 601.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1050.3.f.e 16
5.b even 2 1 inner 1050.3.f.e 16
5.c odd 4 2 210.3.h.a 16
7.b odd 2 1 inner 1050.3.f.e 16
15.e even 4 2 630.3.h.e 16
20.e even 4 2 1680.3.bd.a 16
35.c odd 2 1 inner 1050.3.f.e 16
35.f even 4 2 210.3.h.a 16
105.k odd 4 2 630.3.h.e 16
140.j odd 4 2 1680.3.bd.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.3.h.a 16 5.c odd 4 2
210.3.h.a 16 35.f even 4 2
630.3.h.e 16 15.e even 4 2
630.3.h.e 16 105.k odd 4 2
1050.3.f.e 16 1.a even 1 1 trivial
1050.3.f.e 16 5.b even 2 1 inner
1050.3.f.e 16 7.b odd 2 1 inner
1050.3.f.e 16 35.c odd 2 1 inner
1680.3.bd.a 16 20.e even 4 2
1680.3.bd.a 16 140.j odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1050, [\chi])\):

\( T_{11}^{4} - 24 T_{11}^{3} + 28 T_{11}^{2} + 1776 T_{11} - 4844 \)
\( T_{23}^{8} - 2932 T_{23}^{6} + 2745076 T_{23}^{4} - 843688800 T_{23}^{2} + 11186869824 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -2 + T^{2} )^{8} \)
$3$ \( ( 3 + T^{2} )^{8} \)
$5$ \( T^{16} \)
$7$ \( 33232930569601 + 1550224166512 T^{2} + 22667197532 T^{4} + 59352720 T^{6} - 1183546 T^{8} + 24720 T^{10} + 3932 T^{12} + 112 T^{14} + T^{16} \)
$11$ \( ( -4844 + 1776 T + 28 T^{2} - 24 T^{3} + T^{4} )^{4} \)
$13$ \( ( 586802176 + 31875456 T^{2} + 313732 T^{4} + 1044 T^{6} + T^{8} )^{2} \)
$17$ \( ( 338560000 + 18057600 T^{2} + 270436 T^{4} + 996 T^{6} + T^{8} )^{2} \)
$19$ \( ( 2149991424 + 82078848 T^{2} + 817828 T^{4} + 1796 T^{6} + T^{8} )^{2} \)
$23$ \( ( 11186869824 - 843688800 T^{2} + 2745076 T^{4} - 2932 T^{6} + T^{8} )^{2} \)
$29$ \( ( 19600 + 15040 T - 2088 T^{2} + 16 T^{3} + T^{4} )^{4} \)
$31$ \( ( 747256896 + 135657120 T^{2} + 1609684 T^{4} + 3740 T^{6} + T^{8} )^{2} \)
$37$ \( ( 3617786594304 - 22298057472 T^{2} + 23474128 T^{4} - 8536 T^{6} + T^{8} )^{2} \)
$41$ \( ( 1141899264 + 31862784 T^{2} + 280612 T^{4} + 932 T^{6} + T^{8} )^{2} \)
$43$ \( ( 9151544623104 - 58526429184 T^{2} + 49800256 T^{4} - 12880 T^{6} + T^{8} )^{2} \)
$47$ \( ( 14545741676544 + 43335462912 T^{2} + 36173968 T^{4} + 10904 T^{6} + T^{8} )^{2} \)
$53$ \( ( 38389325511744 - 151688109792 T^{2} + 81835828 T^{4} - 15604 T^{6} + T^{8} )^{2} \)
$59$ \( ( 203235065856 + 93596602368 T^{2} + 77016640 T^{4} + 16112 T^{6} + T^{8} )^{2} \)
$61$ \( ( 72666906624 + 3009871872 T^{2} + 17376768 T^{4} + 13824 T^{6} + T^{8} )^{2} \)
$67$ \( ( 404129746944 - 17186990592 T^{2} + 41031568 T^{4} - 17224 T^{6} + T^{8} )^{2} \)
$71$ \( ( -18715436 - 805728 T - 6116 T^{2} + 96 T^{3} + T^{4} )^{4} \)
$73$ \( ( 105841627140096 + 293584216320 T^{2} + 150294724 T^{4} + 24020 T^{6} + T^{8} )^{2} \)
$79$ \( ( -12612096 + 304896 T + 3952 T^{2} - 152 T^{3} + T^{4} )^{4} \)
$83$ \( ( 13129665286144 + 61332269568 T^{2} + 67029904 T^{4} + 15480 T^{6} + T^{8} )^{2} \)
$89$ \( ( 135150669186624 + 565049618976 T^{2} + 203551636 T^{4} + 24860 T^{6} + T^{8} )^{2} \)
$97$ \( ( 5898136817664 + 159955263744 T^{2} + 116506372 T^{4} + 20468 T^{6} + T^{8} )^{2} \)
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