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 $4$

Related objects

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: $$x^{16} - 8 x^{15} + 88 x^{14} - 476 x^{13} + 2744 x^{12} - 10640 x^{11} + 39126 x^{10} - 108488 x^{9} + 259514 x^{8} - 486436 x^{7} + 690168 x^{6} - 725188 x^{5} + \cdots + 33124$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} - \beta_1 q^{3} + 2 q^{4} - \beta_{8} q^{6} + ( - \beta_{6} - \beta_{4} + \beta_1) q^{7} - 2 \beta_{5} q^{8} - 3 q^{9}+O(q^{10})$$ q - b5 * q^2 - b1 * q^3 + 2 * q^4 - b8 * q^6 + (-b6 - b4 + b1) * q^7 - 2*b5 * q^8 - 3 * q^9 $$q - \beta_{5} q^{2} - \beta_1 q^{3} + 2 q^{4} - \beta_{8} q^{6} + ( - \beta_{6} - \beta_{4} + \beta_1) q^{7} - 2 \beta_{5} q^{8} - 3 q^{9} + ( - \beta_{11} + \beta_{10} + 6) q^{11} - 2 \beta_1 q^{12} + (\beta_{7} - \beta_{6} + \beta_{4} - \beta_{3} + 3 \beta_1) q^{13} + (\beta_{13} + \beta_{8} + \beta_{2} - 1) q^{14} + 4 q^{16} + ( - 2 \beta_{4} - \beta_{3} - \beta_1) q^{17} + 3 \beta_{5} q^{18} + (\beta_{13} - 2 \beta_{12} + \beta_{11} - \beta_{9} - 1) q^{19} + ( - \beta_{13} + \beta_{10} + \beta_{2} + 1) q^{21} + ( - \beta_{15} - \beta_{14} - 6 \beta_{5}) q^{22} + (\beta_{15} - 2 \beta_{14} + \beta_{7} + \beta_{6} + 5 \beta_{5} - \beta_1) q^{23} - 2 \beta_{8} q^{24} + ( - 2 \beta_{12} - \beta_{11} + \beta_{9} + 2 \beta_{8} + 1) q^{26} + 3 \beta_1 q^{27} + ( - 2 \beta_{6} - 2 \beta_{4} + 2 \beta_1) q^{28} + (2 \beta_{11} + 2 \beta_{9} - 4 \beta_{2} - 4) q^{29} + (4 \beta_{13} + \beta_{12} - 2 \beta_{11} + 2 \beta_{9} - 4 \beta_{8} + 2) q^{31} - 4 \beta_{5} q^{32} + ( - \beta_{7} + \beta_{6} + \beta_{4} - 2 \beta_{3} - 6 \beta_1) q^{33} + (3 \beta_{13} + 3 \beta_{12} - \beta_{11} + \beta_{9} + 2 \beta_{8} + 1) q^{34} - 6 q^{36} + (\beta_{15} - 3 \beta_{14} - 3 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + 3 \beta_1) q^{37} + (3 \beta_{7} - 3 \beta_{6} - 4 \beta_{4} + 2 \beta_{3} + \beta_1) q^{38} + ( - 2 \beta_{10} - 2 \beta_{9} + 3 \beta_{2} + 9) q^{39} + (2 \beta_{13} + \beta_{12} - \beta_{11} + \beta_{9} + 5 \beta_{8} + 1) q^{41} + ( - \beta_{14} - 2 \beta_{7} - 3 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_1) q^{42} + (2 \beta_{14} - 6 \beta_{7} - 6 \beta_{6} - 18 \beta_{5} + 6 \beta_1) q^{43} + ( - 2 \beta_{11} + 2 \beta_{10} + 12) q^{44} + ( - \beta_{11} + 4 \beta_{10} + 3 \beta_{9} - 2 \beta_{2} - 11) q^{46} + (5 \beta_{7} - 5 \beta_{6} - 2 \beta_{4} - 4 \beta_{3} - 13 \beta_1) q^{47} - 4 \beta_1 q^{48} + ( - 2 \beta_{12} - 3 \beta_{11} + 2 \beta_{10} - \beta_{9} + 10 \beta_{8} - 2 \beta_{2} + \cdots - 14) q^{49}+ \cdots + (3 \beta_{11} - 3 \beta_{10} - 18) q^{99}+O(q^{100})$$ q - b5 * q^2 - b1 * q^3 + 2 * q^4 - b8 * q^6 + (-b6 - b4 + b1) * q^7 - 2*b5 * q^8 - 3 * q^9 + (-b11 + b10 + 6) * q^11 - 2*b1 * q^12 + (b7 - b6 + b4 - b3 + 3*b1) * q^13 + (b13 + b8 + b2 - 1) * q^14 + 4 * q^16 + (-2*b4 - b3 - b1) * q^17 + 3*b5 * q^18 + (b13 - 2*b12 + b11 - b9 - 1) * q^19 + (-b13 + b10 + b2 + 1) * q^21 + (-b15 - b14 - 6*b5) * q^22 + (b15 - 2*b14 + b7 + b6 + 5*b5 - b1) * q^23 - 2*b8 * q^24 + (-2*b12 - b11 + b9 + 2*b8 + 1) * q^26 + 3*b1 * q^27 + (-2*b6 - 2*b4 + 2*b1) * q^28 + (2*b11 + 2*b9 - 4*b2 - 4) * q^29 + (4*b13 + b12 - 2*b11 + 2*b9 - 4*b8 + 2) * q^31 - 4*b5 * q^32 + (-b7 + b6 + b4 - 2*b3 - 6*b1) * q^33 + (3*b13 + 3*b12 - b11 + b9 + 2*b8 + 1) * q^34 - 6 * q^36 + (b15 - 3*b14 - 3*b7 - 3*b6 + 2*b5 + 3*b1) * q^37 + (3*b7 - 3*b6 - 4*b4 + 2*b3 + b1) * q^38 + (-2*b10 - 2*b9 + 3*b2 + 9) * q^39 + (2*b13 + b12 - b11 + b9 + 5*b8 + 1) * q^41 + (-b14 - 2*b7 - 3*b5 + b4 + b3 + 2*b1) * q^42 + (2*b14 - 6*b7 - 6*b6 - 18*b5 + 6*b1) * q^43 + (-2*b11 + 2*b10 + 12) * q^44 + (-b11 + 4*b10 + 3*b9 - 2*b2 - 11) * q^46 + (5*b7 - 5*b6 - 2*b4 - 4*b3 - 13*b1) * q^47 - 4*b1 * q^48 + (-2*b12 - 3*b11 + 2*b10 - b9 + 10*b8 - 2*b2 - 14) * q^49 + (b11 + b10 + 2*b9 + b2 - 3) * q^51 + (2*b7 - 2*b6 + 2*b4 - 2*b3 + 6*b1) * q^52 + (-5*b15 - 4*b14 - b7 - b6 - 10*b5 + b1) * q^53 + 3*b8 * q^54 + (2*b13 + 2*b8 + 2*b2 - 2) * q^56 + (-2*b15 - 3*b14 - 2*b7 - 2*b6 - 4*b5 + 2*b1) * q^57 + (4*b15 + 4*b7 + 4*b6 + 6*b5 - 4*b1) * q^58 + (2*b13 + 8*b12 - 4*b11 + 4*b9 - 8*b8 + 4) * q^59 + (6*b13 + 6*b12 + 12*b8) * q^61 + (3*b7 - 3*b6 - 4*b4 - 4*b3 - 13*b1) * q^62 + (3*b6 + 3*b4 - 3*b1) * q^63 + 8 * q^64 + (b13 + 3*b12 - 2*b11 + 2*b9 - 4*b8 + 2) * q^66 + (4*b15 - 2*b14 - 3*b7 - 3*b6 - 22*b5 + 3*b1) * q^67 + (-4*b4 - 2*b3 - 2*b1) * q^68 + (-2*b13 - 3*b12 - 2*b11 + 2*b9 + 3*b8 + 2) * q^69 + (-3*b11 - b10 - 4*b9 + 10*b2 - 24) * q^71 + 6*b5 * q^72 + (-7*b7 + 7*b6 + 7*b4 - 7*b3 + 3*b1) * q^73 + (-2*b11 + 6*b10 + 4*b9 + 6*b2 - 14) * q^74 + (2*b13 - 4*b12 + 2*b11 - 2*b9 - 2) * q^76 + (b15 + 2*b14 - 5*b7 - 9*b6 + 2*b5 - 5*b3 + 22*b1) * q^77 + (-2*b15 + 2*b14 - 3*b7 - 3*b6 - 10*b5 + 3*b1) * q^78 + (2*b11 + 2*b9 - 8*b2 + 38) * q^79 + 9 * q^81 + (b7 - b6 - 2*b4 - 2*b3 + 7*b1) * q^82 + (3*b7 - 3*b6 - 10*b4 + 4*b3 + 5*b1) * q^83 + (-2*b13 + 2*b10 + 2*b2 + 2) * q^84 + (2*b11 - 4*b10 - 2*b9 + 12*b2 + 26) * q^86 + (6*b7 - 6*b6 + 4*b1) * q^87 + (-2*b15 - 2*b14 - 12*b5) * q^88 + (-12*b13 - 5*b12 + 6*b11 - 6*b9 - 17*b8 - 6) * q^89 + (b13 - 8*b12 + 9*b11 - 4*b10 + 3*b9 + 8*b8 - 10*b2 + 13) * q^91 + (2*b15 - 4*b14 + 2*b7 + 2*b6 + 10*b5 - 2*b1) * q^92 + (b15 - 5*b7 - 5*b6 + 14*b5 + 5*b1) * q^93 + (6*b13 - 4*b12 - 4*b11 + 4*b9 - 12*b8 + 4) * q^94 - 4*b8 * q^96 + (5*b7 - 5*b6 - 5*b4 - 3*b3 - 33*b1) * q^97 + (-4*b15 - 2*b14 + 4*b7 + 17*b5 + 20*b1) * q^98 + (3*b11 - 3*b10 - 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 32 q^{4} - 48 q^{9}+O(q^{10})$$ 16 * q + 32 * q^4 - 48 * q^9 $$16 q + 32 q^{4} - 48 q^{9} + 96 q^{11} - 16 q^{14} + 64 q^{16} + 24 q^{21} - 64 q^{29} - 96 q^{36} + 144 q^{39} + 192 q^{44} - 176 q^{46} - 224 q^{49} - 48 q^{51} - 32 q^{56} + 128 q^{64} - 384 q^{71} - 224 q^{74} + 608 q^{79} + 144 q^{81} + 48 q^{84} + 416 q^{86} + 224 q^{91} - 288 q^{99}+O(q^{100})$$ 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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 88 x^{14} - 476 x^{13} + 2744 x^{12} - 10640 x^{11} + 39126 x^{10} - 108488 x^{9} + 259514 x^{8} - 486436 x^{7} + 690168 x^{6} - 725188 x^{5} + \cdots + 33124$$ :

 $$\beta_{1}$$ $$=$$ $$( 92936 \nu^{14} - 650552 \nu^{13} + 7629434 \nu^{12} - 37319428 \nu^{11} + 225649370 \nu^{10} - 801656916 \nu^{9} + 3054133421 \nu^{8} + \cdots - 649676357 ) / 5030871765$$ (92936*v^14 - 650552*v^13 + 7629434*v^12 - 37319428*v^11 + 225649370*v^10 - 801656916*v^9 + 3054133421*v^8 - 7790526200*v^7 + 19009619662*v^6 - 32746371306*v^5 + 45399734980*v^4 - 43898979412*v^3 + 35060551363*v^2 - 17481907352*v - 649676357) / 5030871765 $$\beta_{2}$$ $$=$$ $$( - 4788335 \nu^{14} + 33518345 \nu^{13} - 384267955 \nu^{12} + 1869869245 \nu^{11} - 10965281907 \nu^{10} + 38484795345 \nu^{9} + \cdots + 205883061930 ) / 119428521030$$ (-4788335*v^14 + 33518345*v^13 - 384267955*v^12 + 1869869245*v^11 - 10965281907*v^10 + 38484795345*v^9 - 139565919575*v^8 + 346554004115*v^7 - 765787653189*v^6 + 1226929460919*v^5 - 1236294585205*v^4 + 764608199695*v^3 - 162963123242*v^2 - 62514228256*v + 205883061930) / 119428521030 $$\beta_{3}$$ $$=$$ $$( - 116229757 \nu^{14} + 813608299 \nu^{13} - 9262265428 \nu^{12} + 44996684681 \nu^{11} - 261565786159 \nu^{10} + \cdots + 2568347439658 ) / 2746855983690$$ (-116229757*v^14 + 813608299*v^13 - 9262265428*v^12 + 44996684681*v^11 - 261565786159*v^10 + 914750319012*v^9 - 3276860586835*v^8 + 8080662254257*v^7 - 17579924497940*v^6 + 27839347208307*v^5 - 29101821255449*v^4 + 19632522741596*v^3 - 27051359900108*v^2 + 20767817705524*v + 2568347439658) / 2746855983690 $$\beta_{4}$$ $$=$$ $$( - 1093349498 \nu^{14} + 7653446486 \nu^{13} - 88689952367 \nu^{12} + 432644909884 \nu^{11} - 2578996359596 \nu^{10} + \cdots + 12465433464302 ) / 8240567951070$$ (-1093349498*v^14 + 7653446486*v^13 - 88689952367*v^12 + 432644909884*v^11 - 2578996359596*v^10 + 9111477265293*v^9 - 34039488965360*v^8 + 85935488321978*v^7 - 203322728917885*v^6 + 343031420986488*v^5 - 448503864698116*v^4 + 409537531927639*v^3 - 359029814503882*v^2 + 199508459888936*v + 12465433464302) / 8240567951070 $$\beta_{5}$$ $$=$$ $$( 279842740 \nu^{15} - 2098820550 \nu^{14} + 23452089504 \nu^{13} - 120606470101 \nu^{12} + 697811667584 \nu^{11} + \cdots + 20065240923728 ) / 21993425639458$$ (279842740*v^15 - 2098820550*v^14 + 23452089504*v^13 - 120606470101*v^12 + 697811667584*v^11 - 2581323646286*v^10 + 9387854016075*v^9 - 24725356812534*v^8 + 56950285171984*v^7 - 99567014762287*v^6 + 126105936855176*v^5 - 112964032428330*v^4 + 89991465893173*v^3 - 58640398536476*v^2 - 24686735907128*v + 20065240923728) / 21993425639458 $$\beta_{6}$$ $$=$$ $$( 1064447881686 \nu^{15} - 3209377373879 \nu^{14} + 57853449494147 \nu^{13} - 83261406982037 \nu^{12} + 909308118211537 \nu^{11} + \cdots - 22\!\cdots\!14 ) / 68\!\cdots\!90$$ (1064447881686*v^15 - 3209377373879*v^14 + 57853449494147*v^13 - 83261406982037*v^12 + 909308118211537*v^11 + 858731238749983*v^10 - 531851674789797*v^9 + 44319669382131871*v^8 - 122069337632357551*v^7 + 437800427292617549*v^6 - 888806142074331045*v^5 + 1400448210877820969*v^4 - 1535846687594913197*v^3 + 1536062304795605372*v^2 - 923159515366322356*v - 22682224994288314) / 68289586610517090 $$\beta_{7}$$ $$=$$ $$( 1064447881686 \nu^{15} - 11495817736195 \nu^{14} + 115858532030359 \nu^{13} - 757541135750011 \nu^{12} + \cdots + 63\!\cdots\!20 ) / 68\!\cdots\!90$$ (1064447881686*v^15 - 11495817736195*v^14 + 115858532030359*v^13 - 757541135750011*v^12 + 4200920417848625*v^11 - 18832916003825449*v^10 + 69135726258527109*v^9 - 217209573670629559*v^8 + 539879980331172325*v^7 - 1140018364997652341*v^6 + 1786712131988850591*v^5 - 2148896267184962303*v^4 + 1753431284375887721*v^3 - 1280080729563355862*v^2 + 602287486497156556*v + 63905884783233220) / 68289586610517090 $$\beta_{8}$$ $$=$$ $$( 12448 \nu^{15} - 93360 \nu^{14} + 1065318 \nu^{13} - 5508607 \nu^{12} + 32749646 \nu^{11} - 122643488 \nu^{10} + 464891353 \nu^{9} - 1256401704 \nu^{8} + \cdots - 179130952 ) / 520340730$$ (12448*v^15 - 93360*v^14 + 1065318*v^13 - 5508607*v^12 + 32749646*v^11 - 122643488*v^10 + 464891353*v^9 - 1256401704*v^8 + 3117962578*v^7 - 5796321469*v^6 + 8742397202*v^5 - 9767388084*v^4 + 9261694875*v^3 - 6344286416*v^2 + 2030131612*v - 179130952) / 520340730 $$\beta_{9}$$ $$=$$ $$( - 3854937063308 \nu^{15} + 23962636724121 \nu^{14} - 295302862411617 \nu^{13} + \cdots + 74\!\cdots\!24 ) / 68\!\cdots\!90$$ (-3854937063308*v^15 + 23962636724121*v^14 - 295302862411617*v^13 + 1309538863202951*v^12 - 8216025180516379*v^11 + 26694378874412479*v^10 - 104460706623912221*v^9 + 245952950235553953*v^8 - 612048024969048539*v^7 + 1016038863804619877*v^6 - 1473931130917122289*v^5 + 1803061234163058501*v^4 - 2138724181565629263*v^3 + 1836991252719575194*v^2 - 705321532058718680*v + 744756406728367124) / 68289586610517090 $$\beta_{10}$$ $$=$$ $$( 3854937063308 \nu^{15} - 30212752287741 \nu^{14} + 339053671356957 \nu^{13} + \cdots + 53\!\cdots\!46 ) / 68\!\cdots\!90$$ (3854937063308*v^15 - 30212752287741*v^14 + 339053671356957*v^13 - 1809683908646036*v^12 + 10648134936885469*v^11 - 40914836644345741*v^10 + 154311383653392476*v^9 - 426117375260218293*v^8 + 1058567337159689039*v^7 - 1998914880969722516*v^6 + 3044238135652700623*v^5 - 3384302188261511751*v^4 + 3117694261499572908*v^3 - 2045914723451540566*v^2 + 625130317554887084*v + 531749434692057946) / 68289586610517090 $$\beta_{11}$$ $$=$$ $$( 3854937063308 \nu^{15} - 33861419225499 \nu^{14} + 364594339921263 \nu^{13} + \cdots + 70\!\cdots\!94 ) / 68\!\cdots\!90$$ (3854937063308*v^15 - 33861419225499*v^14 + 364594339921263*v^13 - 2102794770058199*v^12 + 12074771414022469*v^11 - 49298728429864189*v^10 + 183762060808011509*v^9 - 533270283374438967*v^8 + 1325124389193165749*v^7 - 2590358094781911509*v^6 + 3994713632454400783*v^5 - 4342698800407316979*v^4 + 3709736650323528063*v^3 - 2171916157642537894*v^2 + 576827345172515312*v + 700117415521608994) / 68289586610517090 $$\beta_{12}$$ $$=$$ $$( 5472163425770 \nu^{15} - 41041225693275 \nu^{14} + 464858064633033 \nu^{13} + \cdots - 62\!\cdots\!92 ) / 68\!\cdots\!90$$ (5472163425770*v^15 - 41041225693275*v^14 + 464858064633033*v^13 - 2399118830433377*v^12 + 14125631930262397*v^11 - 52670077378974979*v^10 + 196742572634383223*v^9 - 526967009577493719*v^8 + 1274658163466793299*v^7 - 2322142348180659911*v^6 + 3317271373015585711*v^5 - 3490535031960335649*v^4 + 3178048469640185847*v^3 - 2154252345020041282*v^2 + 693477650854020896*v - 62893609797828992) / 68289586610517090 $$\beta_{13}$$ $$=$$ $$( 7209986841170 \nu^{15} - 54074901308775 \nu^{14} + 610495540452873 \nu^{13} + \cdots - 74\!\cdots\!92 ) / 68\!\cdots\!90$$ (7209986841170*v^15 - 54074901308775*v^14 + 610495540452873*v^13 - 3148085009760587*v^12 + 18459042385959037*v^11 - 68700097222411039*v^10 + 255041146074208973*v^9 - 680511475383329859*v^8 + 1628319434384813939*v^7 - 2940453509854462181*v^6 + 4100389240886228671*v^5 - 4192041673340264949*v^4 + 3736895472836790177*v^3 - 2518409219931557242*v^2 + 813331367312824376*v - 74867636882512292) / 68289586610517090 $$\beta_{14}$$ $$=$$ $$( 11652755758 \nu^{15} - 87395668185 \nu^{14} + 980333632691 \nu^{13} - 5046667645019 \nu^{12} + 29328141084055 \nu^{11} + \cdots + 767874169626928 ) / 109967128197290$$ (11652755758*v^15 - 87395668185*v^14 + 980333632691*v^13 - 5046667645019*v^12 + 29328141084055*v^11 - 108707533995533*v^10 + 397669560517609*v^9 - 1051227743230329*v^8 + 2442760651969745*v^7 - 4302607169649537*v^6 + 5537445103727985*v^5 - 5073064410728007*v^4 + 3851913596232853*v^3 - 2295132049284554*v^2 - 959984408973388*v + 767874169626928) / 109967128197290 $$\beta_{15}$$ $$=$$ $$( - 165988958814 \nu^{15} + 1244917191105 \nu^{14} - 14017718745515 \nu^{13} + 72233927780755 \nu^{12} + \cdots - 92\!\cdots\!92 ) / 989704153775610$$ (-165988958814*v^15 + 1244917191105*v^14 - 14017718745515*v^13 + 72233927780755*v^12 - 421896015444175*v^11 + 1567393616297861*v^10 - 5777614529073753*v^9 + 15346661487816381*v^8 - 36122988388313897*v^7 + 64319930863026889*v^6 - 84744155655241521*v^5 + 80129924292009159*v^4 - 56716001622329941*v^3 + 28891531292633078*v^2 + 11928324887229772*v - 9230202682938692) / 989704153775610
 $$\nu$$ $$=$$ $$( \beta_{13} - \beta_{12} - 2\beta_{5} + 2 ) / 4$$ (b13 - b12 - 2*b5 + 2) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{13} - \beta_{12} + 2 \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{2} - 28 ) / 4$$ (b13 - b12 + 2*b10 + 2*b9 + 2*b7 - 2*b6 - 2*b5 - 2*b4 - 4*b2 - 28) / 4 $$\nu^{3}$$ $$=$$ $$( 3 \beta_{15} - 3 \beta_{14} - 9 \beta_{13} + 17 \beta_{12} - \beta_{11} + 3 \beta_{10} + 4 \beta_{9} - 11 \beta_{8} + 9 \beta_{7} + 3 \beta_{6} + 47 \beta_{5} - 3 \beta_{4} - 6 \beta_{2} - 6 \beta _1 - 42 ) / 4$$ (3*b15 - 3*b14 - 9*b13 + 17*b12 - b11 + 3*b10 + 4*b9 - 11*b8 + 9*b7 + 3*b6 + 47*b5 - 3*b4 - 6*b2 - 6*b1 - 42) / 4 $$\nu^{4}$$ $$=$$ $$( 6 \beta_{15} - 6 \beta_{14} - 19 \beta_{13} + 35 \beta_{12} - 12 \beta_{11} - 12 \beta_{10} - 20 \beta_{9} - 22 \beta_{8} - 44 \beta_{7} + 68 \beta_{6} + 96 \beta_{5} + 44 \beta_{4} - 4 \beta_{3} + 64 \beta_{2} - 72 \beta _1 + 330 ) / 4$$ (6*b15 - 6*b14 - 19*b13 + 35*b12 - 12*b11 - 12*b10 - 20*b9 - 22*b8 - 44*b7 + 68*b6 + 96*b5 + 44*b4 - 4*b3 + 64*b2 - 72*b1 + 330) / 4 $$\nu^{5}$$ $$=$$ $$( - 90 \beta_{15} + 40 \beta_{14} + 101 \beta_{13} - 185 \beta_{12} - 37 \beta_{11} - 35 \beta_{10} - 48 \beta_{9} + 233 \beta_{8} - 325 \beta_{7} - 35 \beta_{6} - 1083 \beta_{5} + 115 \beta_{4} - 10 \beta_{3} + 170 \beta_{2} + \cdots + 904 ) / 4$$ (-90*b15 + 40*b14 + 101*b13 - 185*b12 - 37*b11 - 35*b10 - 48*b9 + 233*b8 - 325*b7 - 35*b6 - 1083*b5 + 115*b4 - 10*b3 + 170*b2 + 30*b1 + 904) / 4 $$\nu^{6}$$ $$=$$ $$( - 285 \beta_{15} + 135 \beta_{14} + 351 \beta_{13} - 643 \beta_{12} + 174 \beta_{11} - 20 \beta_{10} + 216 \beta_{9} + 754 \beta_{8} + 572 \beta_{7} - 1712 \beta_{6} - 3490 \beta_{5} - 758 \beta_{4} - 52 \beta_{3} + \cdots - 2742 ) / 4$$ (-285*b15 + 135*b14 + 351*b13 - 643*b12 + 174*b11 - 20*b10 + 216*b9 + 754*b8 + 572*b7 - 1712*b6 - 3490*b5 - 758*b4 - 52*b3 - 690*b2 + 2520*b1 - 2742) / 4 $$\nu^{7}$$ $$=$$ $$( 2009 \beta_{15} - 224 \beta_{14} - 619 \beta_{13} + 1156 \beta_{12} + 1081 \beta_{11} + 56 \beta_{10} + 585 \beta_{9} - 2444 \beta_{8} + 8211 \beta_{7} - 805 \beta_{6} + 19276 \beta_{5} - 3059 \beta_{4} - 147 \beta_{3} + \cdots - 13154 ) / 4$$ (2009*b15 - 224*b14 - 619*b13 + 1156*b12 + 1081*b11 + 56*b10 + 585*b9 - 2444*b8 + 8211*b7 - 805*b6 + 19276*b5 - 3059*b4 - 147*b3 - 3017*b2 + 3647*b1 - 13154) / 4 $$\nu^{8}$$ $$=$$ $$( 9380 \beta_{15} - 1540 \beta_{14} - 4159 \beta_{13} + 7707 \beta_{12} - 186 \beta_{11} + 1340 \beta_{10} - 1334 \beta_{9} - 13346 \beta_{8} - 792 \beta_{7} + 35792 \beta_{6} + 93616 \beta_{5} + 10240 \beta_{4} + \cdots + 272 ) / 4$$ (9380*b15 - 1540*b14 - 4159*b13 + 7707*b12 - 186*b11 + 1340*b10 - 1334*b9 - 13346*b8 - 792*b7 + 35792*b6 + 93616*b5 + 10240*b4 + 2656*b3 + 2140*b2 - 57960*b1 + 272) / 4 $$\nu^{9}$$ $$=$$ $$( - 32508 \beta_{15} - 2622 \beta_{14} - 6786 \beta_{13} + 12086 \beta_{12} - 12235 \beta_{11} + 5541 \beta_{10} - 4964 \beta_{9} - 6575 \beta_{8} - 167163 \beta_{7} + 52791 \beta_{6} - 266315 \beta_{5} + \cdots + 88788 ) / 4$$ (-32508*b15 - 2622*b14 - 6786*b13 + 12086*b12 - 12235*b11 + 5541*b10 - 4964*b9 - 6575*b8 - 167163*b7 + 52791*b6 - 266315*b5 + 64923*b4 + 12792*b3 + 28458*b2 - 169614*b1 + 88788) / 4 $$\nu^{10}$$ $$=$$ $$( - 234915 \beta_{15} - 585 \beta_{14} - 184 \beta_{13} - 2050 \beta_{12} - 42672 \beta_{11} - 8352 \beta_{10} - 23300 \beta_{9} + 72608 \beta_{8} - 206946 \beta_{7} - 635514 \beta_{6} - 2058608 \beta_{5} + \cdots + 569434 ) / 4$$ (-234915*b15 - 585*b14 - 184*b13 - 2050*b12 - 42672*b11 - 8352*b10 - 23300*b9 + 72608*b8 - 206946*b7 - 635514*b6 - 2058608*b5 - 100122*b4 - 49786*b3 + 121954*b2 + 992880*b1 + 569434) / 4 $$\nu^{11}$$ $$=$$ $$( 329714 \beta_{15} + 80366 \beta_{14} + 305101 \beta_{13} - 561877 \beta_{12} - 115963 \beta_{11} - 95711 \beta_{10} - 69948 \beta_{9} + 872117 \beta_{8} + 2794847 \beta_{7} - 1680921 \beta_{6} + \cdots + 2168840 ) / 4$$ (329714*b15 + 80366*b14 + 305101*b13 - 561877*b12 - 115963*b11 - 95711*b10 - 69948*b9 + 872117*b8 + 2794847*b7 - 1680921*b6 + 2308077*b5 - 1180729*b4 - 392590*b3 + 374792*b2 + 4748612*b1 + 2168840) / 4 $$\nu^{12}$$ $$=$$ $$( 2361744 \beta_{15} + 230076 \beta_{14} + 877983 \beta_{13} - 1603409 \beta_{12} + 435912 \beta_{11} - 118765 \beta_{10} + 568668 \beta_{9} + 2098427 \beta_{8} + 3670524 \beta_{7} + \cdots - 6519638 ) / 2$$ (2361744*b15 + 230076*b14 + 877983*b13 - 1603409*b12 + 435912*b11 - 118765*b10 + 568668*b9 + 2098427*b8 + 3670524*b7 + 4606272*b6 + 19058094*b5 + 106506*b4 + 214164*b3 - 1751265*b2 - 5938218*b1 - 6519638) / 2 $$\nu^{13}$$ $$=$$ $$( 667368 \beta_{15} - 692783 \beta_{14} - 5309960 \beta_{13} + 9895207 \beta_{12} + 9475281 \beta_{11} - 171496 \beta_{10} + 5519413 \beta_{9} - 18651514 \beta_{8} - 36490831 \beta_{7} + \cdots - 112895206 ) / 4$$ (667368*b15 - 692783*b14 - 5309960*b13 + 9895207*b12 + 9475281*b11 - 171496*b10 + 5519413*b9 - 18651514*b8 - 36490831*b7 + 39482053*b6 + 10923252*b5 + 18411367*b4 + 8198255*b3 - 26829803*b2 - 99577881*b1 - 112895206) / 4 $$\nu^{14}$$ $$=$$ $$( - 75505976 \beta_{15} - 11723894 \beta_{14} - 63474043 \beta_{13} + 117216695 \beta_{12} - 249204 \beta_{11} + 6881616 \beta_{10} - 26378810 \beta_{9} - 190758814 \beta_{8} + \cdots + 132095548 ) / 4$$ (-75505976*b15 - 11723894*b14 - 63474043*b13 + 117216695*b12 - 249204*b11 + 6881616*b10 - 26378810*b9 - 190758814*b8 - 167447536*b7 - 93405238*b6 - 577371860*b5 + 20843508*b4 + 5260408*b3 + 44813578*b2 + 49105602*b1 + 132095548) / 4 $$\nu^{15}$$ $$=$$ $$( - 124496877 \beta_{15} - 11320543 \beta_{14} + 24283611 \beta_{13} - 46833803 \beta_{12} - 276574061 \beta_{11} + 49618467 \beta_{10} - 191575552 \beta_{9} + \cdots + 3156761778 ) / 4$$ (-124496877*b15 - 11320543*b14 + 24283611*b13 - 46833803*b12 - 276574061*b11 + 49618467*b10 - 191575552*b9 + 132896429*b8 + 284533417*b7 - 721534045*b6 - 1010528001*b5 - 228015103*b4 - 123432884*b3 + 818598666*b2 + 1595075514*b1 + 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: 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
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 $$S_{3}^{\mathrm{new}}(1050, [\chi])$$:

 $$T_{11}^{4} - 24T_{11}^{3} + 28T_{11}^{2} + 1776T_{11} - 4844$$ T11^4 - 24*T11^3 + 28*T11^2 + 1776*T11 - 4844 $$T_{23}^{8} - 2932T_{23}^{6} + 2745076T_{23}^{4} - 843688800T_{23}^{2} + 11186869824$$ T23^8 - 2932*T23^6 + 2745076*T23^4 - 843688800*T23^2 + 11186869824

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{8}$$
$3$ $$(T^{2} + 3)^{8}$$
$5$ $$T^{16}$$
$7$ $$T^{16} + 112 T^{14} + \cdots + 33232930569601$$
$11$ $$(T^{4} - 24 T^{3} + 28 T^{2} + 1776 T - 4844)^{4}$$
$13$ $$(T^{8} + 1044 T^{6} + 313732 T^{4} + \cdots + 586802176)^{2}$$
$17$ $$(T^{8} + 996 T^{6} + 270436 T^{4} + \cdots + 338560000)^{2}$$
$19$ $$(T^{8} + 1796 T^{6} + \cdots + 2149991424)^{2}$$
$23$ $$(T^{8} - 2932 T^{6} + \cdots + 11186869824)^{2}$$
$29$ $$(T^{4} + 16 T^{3} - 2088 T^{2} + \cdots + 19600)^{4}$$
$31$ $$(T^{8} + 3740 T^{6} + 1609684 T^{4} + \cdots + 747256896)^{2}$$
$37$ $$(T^{8} - 8536 T^{6} + \cdots + 3617786594304)^{2}$$
$41$ $$(T^{8} + 932 T^{6} + 280612 T^{4} + \cdots + 1141899264)^{2}$$
$43$ $$(T^{8} - 12880 T^{6} + \cdots + 9151544623104)^{2}$$
$47$ $$(T^{8} + 10904 T^{6} + \cdots + 14545741676544)^{2}$$
$53$ $$(T^{8} - 15604 T^{6} + \cdots + 38389325511744)^{2}$$
$59$ $$(T^{8} + 16112 T^{6} + \cdots + 203235065856)^{2}$$
$61$ $$(T^{8} + 13824 T^{6} + \cdots + 72666906624)^{2}$$
$67$ $$(T^{8} - 17224 T^{6} + \cdots + 404129746944)^{2}$$
$71$ $$(T^{4} + 96 T^{3} - 6116 T^{2} + \cdots - 18715436)^{4}$$
$73$ $$(T^{8} + 24020 T^{6} + \cdots + 105841627140096)^{2}$$
$79$ $$(T^{4} - 152 T^{3} + 3952 T^{2} + \cdots - 12612096)^{4}$$
$83$ $$(T^{8} + 15480 T^{6} + \cdots + 13129665286144)^{2}$$
$89$ $$(T^{8} + 24860 T^{6} + \cdots + 135150669186624)^{2}$$
$97$ $$(T^{8} + 20468 T^{6} + \cdots + 5898136817664)^{2}$$