Properties

 Label 1134.2.d.b Level $1134$ Weight $2$ Character orbit 1134.d Analytic conductor $9.055$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1134 = 2 \cdot 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1134.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$9.05503558921$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 8 x^{15} + 56 x^{14} - 252 x^{13} + 962 x^{12} - 2860 x^{11} + 7240 x^{10} - 15036 x^{9} + 26533 x^{8} - 38796 x^{7} + 47500 x^{6} - 47396 x^{5} + 38144 x^{4} + \cdots + 457$$ x^16 - 8*x^15 + 56*x^14 - 252*x^13 + 962*x^12 - 2860*x^11 + 7240*x^10 - 15036*x^9 + 26533*x^8 - 38796*x^7 + 47500*x^6 - 47396*x^5 + 38144*x^4 - 23676*x^3 + 10748*x^2 - 3160*x + 457 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{10}\cdot 3^{6}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - \beta_{11} q^{2} - q^{4} - \beta_{2} q^{5} + \beta_{14} q^{7} + \beta_{11} q^{8}+O(q^{10})$$ q - b11 * q^2 - q^4 - b2 * q^5 + b14 * q^7 + b11 * q^8 $$q - \beta_{11} q^{2} - q^{4} - \beta_{2} q^{5} + \beta_{14} q^{7} + \beta_{11} q^{8} - \beta_{12} q^{10} + \beta_{9} q^{11} + (\beta_{15} + \beta_{10}) q^{13} + \beta_{7} q^{14} + q^{16} + ( - \beta_{5} + \beta_{2}) q^{17} + (\beta_{15} - \beta_{12} - \beta_{10}) q^{19} + \beta_{2} q^{20} - \beta_{4} q^{22} + \beta_{8} q^{23} + (\beta_{14} - \beta_{13} + \beta_{4} - \beta_1 + 1) q^{25} + ( - \beta_{5} + \beta_{3}) q^{26} - \beta_{14} q^{28} + ( - 3 \beta_{11} - \beta_{9} + \beta_{8}) q^{29} + (\beta_{15} + \beta_{14} + \beta_{13} + 2 \beta_{12} - 1) q^{31} - \beta_{11} q^{32} + ( - \beta_{15} + \beta_{12}) q^{34} + ( - 2 \beta_{11} + \beta_{8} - \beta_{6} - \beta_{3} - \beta_{2}) q^{35} + ( - 2 \beta_{14} + 2 \beta_{13} + \beta_{4} + 1) q^{37} + ( - \beta_{5} - \beta_{3} + \beta_{2}) q^{38} + \beta_{12} q^{40} + (\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{3} + \beta_{2}) q^{41} + ( - 2 \beta_1 + 4) q^{43} - \beta_{9} q^{44} + \beta_1 q^{46} + (2 \beta_{7} - 2 \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2}) q^{47} + (\beta_{14} - \beta_{13} + \beta_{12} + \beta_{10} + \beta_{4} - \beta_1 - 2) q^{49} + (\beta_{9} + \beta_{8} + \beta_{7} + \beta_{6}) q^{50} + ( - \beta_{15} - \beta_{10}) q^{52} + ( - 4 \beta_{11} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6}) q^{53} + (\beta_{15} - 2 \beta_{12}) q^{55} - \beta_{7} q^{56} + (\beta_{4} + \beta_1 - 3) q^{58} - \beta_{5} q^{59} + ( - \beta_{15} + \beta_{14} + \beta_{13} + \beta_{12} - 2 \beta_{10} - 1) q^{61} + (\beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{2}) q^{62} - q^{64} + (3 \beta_{11} + \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + 3 \beta_{6}) q^{65} + ( - \beta_{14} + \beta_{13} - 2 \beta_{4} - \beta_1 - 1) q^{67} + (\beta_{5} - \beta_{2}) q^{68} + ( - \beta_{13} - \beta_{12} + \beta_{10} + \beta_1 - 1) q^{70} + ( - 2 \beta_{11} + 2 \beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{6}) q^{71} + ( - \beta_{14} - \beta_{13} - \beta_{10} + 1) q^{73} + ( - 3 \beta_{11} + \beta_{9} - 2 \beta_{7} - 2 \beta_{6}) q^{74} + ( - \beta_{15} + \beta_{12} + \beta_{10}) q^{76} + (\beta_{11} + \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{2}) q^{77} + (\beta_{4} + 2 \beta_1 + 2) q^{79} - \beta_{2} q^{80} + ( - 2 \beta_{15} - \beta_{14} - \beta_{13} + \beta_{12} - \beta_{10} + 1) q^{82} + ( - \beta_{7} + \beta_{6} - \beta_{5} + 2 \beta_{3} - 2 \beta_{2}) q^{83} + ( - 2 \beta_{4} + \beta_1 - 3) q^{85} + ( - 4 \beta_{11} + 2 \beta_{8}) q^{86} + \beta_{4} q^{88} + ( - \beta_{5} + 5 \beta_{2}) q^{89} + (\beta_{15} - \beta_{14} + 2 \beta_{13} - 2 \beta_{12} + \beta_{10} - 2 \beta_{4} - 2 \beta_1 + 1) q^{91} - \beta_{8} q^{92} + ( - \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - \beta_{12} + \beta_{10} + 2) q^{94} + ( - 4 \beta_{11} + 2 \beta_{9} - \beta_{8} - 4 \beta_{7} - 4 \beta_{6}) q^{95} + ( - \beta_{12} + \beta_{10}) q^{97} + (3 \beta_{11} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{3} - \beta_{2}) q^{98}+O(q^{100})$$ q - b11 * q^2 - q^4 - b2 * q^5 + b14 * q^7 + b11 * q^8 - b12 * q^10 + b9 * q^11 + (b15 + b10) * q^13 + b7 * q^14 + q^16 + (-b5 + b2) * q^17 + (b15 - b12 - b10) * q^19 + b2 * q^20 - b4 * q^22 + b8 * q^23 + (b14 - b13 + b4 - b1 + 1) * q^25 + (-b5 + b3) * q^26 - b14 * q^28 + (-3*b11 - b9 + b8) * q^29 + (b15 + b14 + b13 + 2*b12 - 1) * q^31 - b11 * q^32 + (-b15 + b12) * q^34 + (-2*b11 + b8 - b6 - b3 - b2) * q^35 + (-2*b14 + 2*b13 + b4 + 1) * q^37 + (-b5 - b3 + b2) * q^38 + b12 * q^40 + (b7 - b6 - 2*b5 + b3 + b2) * q^41 + (-2*b1 + 4) * q^43 - b9 * q^44 + b1 * q^46 + (2*b7 - 2*b6 - b5 - b3 - b2) * q^47 + (b14 - b13 + b12 + b10 + b4 - b1 - 2) * q^49 + (b9 + b8 + b7 + b6) * q^50 + (-b15 - b10) * q^52 + (-4*b11 + b9 - b8 - b7 - b6) * q^53 + (b15 - 2*b12) * q^55 - b7 * q^56 + (b4 + b1 - 3) * q^58 - b5 * q^59 + (-b15 + b14 + b13 + b12 - 2*b10 - 1) * q^61 + (b7 - b6 - b5 - 2*b2) * q^62 - q^64 + (3*b11 + b9 + 2*b8 + 3*b7 + 3*b6) * q^65 + (-b14 + b13 - 2*b4 - b1 - 1) * q^67 + (b5 - b2) * q^68 + (-b13 - b12 + b10 + b1 - 1) * q^70 + (-2*b11 + 2*b9 + 2*b8 + b7 + b6) * q^71 + (-b14 - b13 - b10 + 1) * q^73 + (-3*b11 + b9 - 2*b7 - 2*b6) * q^74 + (-b15 + b12 + b10) * q^76 + (b11 + b7 + b6 - 2*b5 + b2) * q^77 + (b4 + 2*b1 + 2) * q^79 - b2 * q^80 + (-2*b15 - b14 - b13 + b12 - b10 + 1) * q^82 + (-b7 + b6 - b5 + 2*b3 - 2*b2) * q^83 + (-2*b4 + b1 - 3) * q^85 + (-4*b11 + 2*b8) * q^86 + b4 * q^88 + (-b5 + 5*b2) * q^89 + (b15 - b14 + 2*b13 - 2*b12 + b10 - 2*b4 - 2*b1 + 1) * q^91 - b8 * q^92 + (-b15 - 2*b14 - 2*b13 - b12 + b10 + 2) * q^94 + (-4*b11 + 2*b9 - b8 - 4*b7 - 4*b6) * q^95 + (-b12 + b10) * q^97 + (3*b11 + b9 + b8 + b7 + b6 + b3 - b2) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 16 q^{4} + 8 q^{7}+O(q^{10})$$ 16 * q - 16 * q^4 + 8 * q^7 $$16 q - 16 q^{4} + 8 q^{7} + 16 q^{16} + 16 q^{25} - 8 q^{28} + 16 q^{37} + 64 q^{43} - 32 q^{49} - 48 q^{58} - 16 q^{64} - 16 q^{67} - 24 q^{70} + 32 q^{79} - 48 q^{85} + 24 q^{91}+O(q^{100})$$ 16 * q - 16 * q^4 + 8 * q^7 + 16 * q^16 + 16 * q^25 - 8 * q^28 + 16 * q^37 + 64 * q^43 - 32 * q^49 - 48 * q^58 - 16 * q^64 - 16 * q^67 - 24 * q^70 + 32 * q^79 - 48 * q^85 + 24 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 56 x^{14} - 252 x^{13} + 962 x^{12} - 2860 x^{11} + 7240 x^{10} - 15036 x^{9} + 26533 x^{8} - 38796 x^{7} + 47500 x^{6} - 47396 x^{5} + 38144 x^{4} + \cdots + 457$$ :

 $$\beta_{1}$$ $$=$$ $$( 54 \nu^{14} - 378 \nu^{13} + 2506 \nu^{12} - 10122 \nu^{11} + 35295 \nu^{10} - 92699 \nu^{9} + 205329 \nu^{8} - 361020 \nu^{7} + 520549 \nu^{6} - 592287 \nu^{5} + 520258 \nu^{4} + \cdots + 1872 ) / 655$$ (54*v^14 - 378*v^13 + 2506*v^12 - 10122*v^11 + 35295*v^10 - 92699*v^9 + 205329*v^8 - 361020*v^7 + 520549*v^6 - 592287*v^5 + 520258*v^4 - 336644*v^3 + 140708*v^2 - 31549*v + 1872) / 655 $$\beta_{2}$$ $$=$$ $$( - 321 \nu^{14} + 2247 \nu^{13} - 15086 \nu^{12} + 61305 \nu^{11} - 217429 \nu^{10} + 578736 \nu^{9} - 1315127 \nu^{8} + 2370641 \nu^{7} - 3566011 \nu^{6} + 4253522 \nu^{5} + \cdots - 76759 ) / 1965$$ (-321*v^14 + 2247*v^13 - 15086*v^12 + 61305*v^11 - 217429*v^10 + 578736*v^9 - 1315127*v^8 + 2370641*v^7 - 3566011*v^6 + 4253522*v^5 - 4047300*v^4 + 2902165*v^3 - 1488709*v^2 + 481367*v - 76759) / 1965 $$\beta_{3}$$ $$=$$ $$( 259 \nu^{14} - 1813 \nu^{13} + 12233 \nu^{12} - 49829 \nu^{11} + 177662 \nu^{10} - 474754 \nu^{9} + 1084910 \nu^{8} - 1965161 \nu^{7} + 2973244 \nu^{6} - 3565294 \nu^{5} + \cdots + 62645 ) / 393$$ (259*v^14 - 1813*v^13 + 12233*v^12 - 49829*v^11 + 177662*v^10 - 474754*v^9 + 1084910*v^8 - 1965161*v^7 + 2973244*v^6 - 3565294*v^5 + 3411739*v^4 - 2459279*v^3 + 1265008*v^2 - 408925*v + 62645) / 393 $$\beta_{4}$$ $$=$$ $$( - 588 \nu^{14} + 4116 \nu^{13} - 27797 \nu^{12} + 113274 \nu^{11} - 404410 \nu^{10} + 1081803 \nu^{9} - 2477718 \nu^{8} + 4497900 \nu^{7} - 6833518 \nu^{6} + \cdots - 164484 ) / 655$$ (-588*v^14 + 4116*v^13 - 27797*v^12 + 113274*v^11 - 404410*v^10 + 1081803*v^9 - 2477718*v^8 + 4497900*v^7 - 6833518*v^6 + 8232039*v^5 - 7943631*v^4 + 5784968*v^3 - 3034651*v^2 + 1008213*v - 164484) / 655 $$\beta_{5}$$ $$=$$ $$( 395 \nu^{14} - 2765 \nu^{13} + 18656 \nu^{12} - 75991 \nu^{11} + 270978 \nu^{10} - 724205 \nu^{9} + 1655774 \nu^{8} - 3000797 \nu^{7} + 4545742 \nu^{6} - 5458815 \nu^{5} + \cdots + 96267 ) / 393$$ (395*v^14 - 2765*v^13 + 18656*v^12 - 75991*v^11 + 270978*v^10 - 724205*v^9 + 1655774*v^8 - 3000797*v^7 + 4545742*v^6 - 5458815*v^5 + 5237615*v^4 - 3787767*v^3 + 1957354*v^2 - 636174*v + 96267) / 393 $$\beta_{6}$$ $$=$$ $$( 325218 \nu^{15} - 2510469 \nu^{14} + 16986696 \nu^{13} - 73571231 \nu^{12} + 267766992 \nu^{11} - 755616592 \nu^{10} + 1787650380 \nu^{9} + \cdots - 30378190 ) / 2595765$$ (325218*v^15 - 2510469*v^14 + 16986696*v^13 - 73571231*v^12 + 267766992*v^11 - 755616592*v^10 + 1787650380*v^9 - 3438485903*v^8 + 5486496404*v^7 - 7112671903*v^6 + 7417334981*v^5 - 6025679763*v^4 + 3648052981*v^3 - 1498869805*v^2 + 362310557*v - 30378190) / 2595765 $$\beta_{7}$$ $$=$$ $$( 325218 \nu^{15} - 2367801 \nu^{14} + 15988020 \nu^{13} - 66777328 \nu^{12} + 239986362 \nu^{11} - 655964315 \nu^{10} + 1520242992 \nu^{9} + \cdots - 49140353 ) / 2595765$$ (325218*v^15 - 2367801*v^14 + 15988020*v^13 - 66777328*v^12 + 239986362*v^11 - 655964315*v^10 + 1520242992*v^9 - 2827132387*v^8 + 4380742786*v^7 - 5464826120*v^6 + 5483567995*v^5 - 4290519843*v^4 + 2514291521*v^3 - 1068108878*v^2 + 300070321*v - 49140353) / 2595765 $$\beta_{8}$$ $$=$$ $$( - 143764 \nu^{15} + 1078230 \nu^{14} - 7240864 \nu^{13} + 30712461 \nu^{12} - 110181896 \nu^{11} + 304140298 \nu^{10} - 704002373 \nu^{9} + \cdots + 4929077 ) / 865255$$ (-143764*v^15 + 1078230*v^14 - 7240864*v^13 + 30712461*v^12 - 110181896*v^11 + 304140298*v^10 - 704002373*v^9 + 1316620605*v^8 - 2031718950*v^7 + 2528025602*v^6 - 2493491829*v^5 + 1890068654*v^4 - 1028814578*v^3 + 366902317*v^2 - 71812067*v + 4929077) / 865255 $$\beta_{9}$$ $$=$$ $$( 144812 \nu^{15} - 1086090 \nu^{14} + 7295622 \nu^{13} - 30949178 \nu^{12} + 110985188 \nu^{11} - 306216779 \nu^{10} + 707350209 \nu^{9} - 1319456100 \nu^{8} + \cdots + 11072049 ) / 865255$$ (144812*v^15 - 1086090*v^14 + 7295622*v^13 - 30949178*v^12 + 110985188*v^11 - 306216779*v^10 + 707350209*v^9 - 1319456100*v^8 + 2023238010*v^7 - 2495224381*v^6 + 2413011717*v^5 - 1770414957*v^4 + 884700954*v^3 - 254206816*v^2 + 8683691*v + 11072049) / 865255 $$\beta_{10}$$ $$=$$ $$( - 96874 \nu^{15} + 726555 \nu^{14} - 4887955 \nu^{13} + 20752290 \nu^{12} - 74578887 \nu^{11} + 206151407 \nu^{10} - 477998494 \nu^{9} + 895321773 \nu^{8} + \cdots + 3067531 ) / 519153$$ (-96874*v^15 + 726555*v^14 - 4887955*v^13 + 20752290*v^12 - 74578887*v^11 + 206151407*v^10 - 477998494*v^9 + 895321773*v^8 - 1384310135*v^7 + 1726154823*v^6 - 1709227692*v^5 + 1303324972*v^4 - 717486533*v^3 + 261301701*v^2 - 51282013*v + 3067531) / 519153 $$\beta_{11}$$ $$=$$ $$( 32580 \nu^{15} - 244350 \nu^{14} + 1650786 \nu^{13} - 7024134 \nu^{12} + 25400574 \nu^{11} - 70590828 \nu^{10} + 165381778 \nu^{9} - 313213977 \nu^{8} + \cdots - 3363404 ) / 173051$$ (32580*v^15 - 244350*v^14 + 1650786*v^13 - 7024134*v^12 + 25400574*v^11 - 70590828*v^10 + 165381778*v^9 - 313213977*v^8 + 493311054*v^7 - 628987746*v^6 + 645331338*v^5 - 516105789*v^4 + 307360314*v^3 - 127013004*v^2 + 31438212*v - 3363404) / 173051 $$\beta_{12}$$ $$=$$ $$( - 847558 \nu^{15} + 6356685 \nu^{14} - 42971171 \nu^{13} + 182902889 \nu^{12} - 661951717 \nu^{11} + 1840904054 \nu^{10} - 4318389740 \nu^{9} + \cdots + 117878701 ) / 2595765$$ (-847558*v^15 + 6356685*v^14 - 42971171*v^13 + 182902889*v^12 - 661951717*v^11 + 1840904054*v^10 - 4318389740*v^9 + 8189368116*v^8 - 12928279637*v^7 + 16530696510*v^6 - 17049504748*v^5 + 13736874985*v^4 - 8310030777*v^3 + 3527197972*v^2 - 938083265*v + 117878701) / 2595765 $$\beta_{13}$$ $$=$$ $$( 1265468 \nu^{15} - 9990348 \nu^{14} + 67896127 \nu^{13} - 298263571 \nu^{12} + 1094893661 \nu^{11} - 3131539819 \nu^{10} + 7501129528 \nu^{9} + \cdots - 347093285 ) / 2595765$$ (1265468*v^15 - 9990348*v^14 + 67896127*v^13 - 298263571*v^12 + 1094893661*v^11 - 3131539819*v^10 + 7501129528*v^9 - 14664774819*v^8 + 23833677907*v^7 - 31646865318*v^6 + 34027921247*v^5 - 28885056971*v^4 + 18622064190*v^3 - 8561202083*v^2 + 2498153638*v - 347093285) / 2595765 $$\beta_{14}$$ $$=$$ $$( 1265468 \nu^{15} - 8991672 \nu^{14} + 60905395 \nu^{13} - 251052352 \nu^{12} + 902505863 \nu^{11} - 2445049144 \nu^{10} + 5665618522 \nu^{9} + \cdots - 99619787 ) / 2595765$$ (1265468*v^15 - 8991672*v^14 + 60905395*v^13 - 251052352*v^12 + 902505863*v^11 - 2445049144*v^10 + 5665618522*v^9 - 10467672483*v^8 + 16227690157*v^7 - 20134441467*v^6 + 20218823969*v^5 - 15662214209*v^4 + 9082243404*v^3 - 3639199676*v^2 + 898877062*v - 99619787) / 2595765 $$\beta_{15}$$ $$=$$ $$( - 312314 \nu^{15} + 2342355 \nu^{14} - 15898989 \nu^{13} + 67817711 \nu^{12} - 246733441 \nu^{11} + 689195683 \nu^{10} - 1628974479 \nu^{9} + \cdots + 64404140 ) / 519153$$ (-312314*v^15 + 2342355*v^14 - 15898989*v^13 + 67817711*v^12 - 246733441*v^11 + 689195683*v^10 - 1628974479*v^9 + 3112931382*v^8 - 4972565887*v^7 + 6443817363*v^6 - 6780921079*v^5 + 5601651077*v^4 - 3521575313*v^3 + 1575990199*v^2 - 455572548*v + 64404140) / 519153
 $$\nu$$ $$=$$ $$( -2\beta_{15} - \beta_{14} - \beta_{13} + 3\beta_{12} + 3\beta_{11} - \beta_{10} + 4 ) / 6$$ (-2*b15 - b14 - b13 + 3*b12 + 3*b11 - b10 + 4) / 6 $$\nu^{2}$$ $$=$$ $$( - 2 \beta_{15} - 4 \beta_{14} + 2 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} - \beta_{10} + \beta_{7} - \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} + 3 \beta_{2} + 3 \beta _1 - 17 ) / 6$$ (-2*b15 - 4*b14 + 2*b13 + 3*b12 + 3*b11 - b10 + b7 - b6 - 2*b5 - 3*b4 + b3 + 3*b2 + 3*b1 - 17) / 6 $$\nu^{3}$$ $$=$$ $$( 4 \beta_{15} - 2 \beta_{14} + 4 \beta_{13} - 13 \beta_{12} - 14 \beta_{11} + 5 \beta_{10} + 3 \beta_{9} + 3 \beta_{8} + 4 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 3 \beta_{4} + \beta_{3} + 3 \beta_{2} + 3 \beta _1 - 20 ) / 4$$ (4*b15 - 2*b14 + 4*b13 - 13*b12 - 14*b11 + 5*b10 + 3*b9 + 3*b8 + 4*b7 + 2*b6 - 2*b5 - 3*b4 + b3 + 3*b2 + 3*b1 - 20) / 4 $$\nu^{4}$$ $$=$$ $$( 14 \beta_{15} + 34 \beta_{14} - 26 \beta_{13} - 42 \beta_{12} - 45 \beta_{11} + 16 \beta_{10} + 9 \beta_{9} + 9 \beta_{8} + 7 \beta_{7} + 11 \beta_{6} + 10 \beta_{5} + 18 \beta_{4} - 14 \beta_{3} - 36 \beta_{2} - 24 \beta _1 + 89 ) / 6$$ (14*b15 + 34*b14 - 26*b13 - 42*b12 - 45*b11 + 16*b10 + 9*b9 + 9*b8 + 7*b7 + 11*b6 + 10*b5 + 18*b4 - 14*b3 - 36*b2 - 24*b1 + 89) / 6 $$\nu^{5}$$ $$=$$ $$( - 58 \beta_{15} + 172 \beta_{14} - 158 \beta_{13} + 249 \beta_{12} + 246 \beta_{11} - 119 \beta_{10} - 75 \beta_{9} - 105 \beta_{8} - 150 \beta_{7} - 120 \beta_{6} + 60 \beta_{5} + 105 \beta_{4} - 75 \beta_{3} - 195 \beta_{2} + \cdots + 554 ) / 12$$ (-58*b15 + 172*b14 - 158*b13 + 249*b12 + 246*b11 - 119*b10 - 75*b9 - 105*b8 - 150*b7 - 120*b6 + 60*b5 + 105*b4 - 75*b3 - 195*b2 - 135*b1 + 554) / 12 $$\nu^{6}$$ $$=$$ $$( - 41 \beta_{15} - 77 \beta_{14} + 77 \beta_{13} + 160 \beta_{12} + 161 \beta_{11} - 73 \beta_{10} - 45 \beta_{9} - 60 \beta_{8} - 80 \beta_{7} - 70 \beta_{6} - 21 \beta_{5} - 34 \beta_{4} + 48 \beta_{3} + 95 \beta_{2} + 53 \beta _1 - 180 ) / 2$$ (-41*b15 - 77*b14 + 77*b13 + 160*b12 + 161*b11 - 73*b10 - 45*b9 - 60*b8 - 80*b7 - 70*b6 - 21*b5 - 34*b4 + 48*b3 + 95*b2 + 53*b1 - 180) / 2 $$\nu^{7}$$ $$=$$ $$( 150 \beta_{15} - 1164 \beta_{14} + 1041 \beta_{13} - 708 \beta_{12} - 675 \beta_{11} + 399 \beta_{10} + 231 \beta_{9} + 378 \beta_{8} + 553 \beta_{7} + 602 \beta_{6} - 329 \beta_{5} - 546 \beta_{4} + 637 \beta_{3} + 1344 \beta_{2} + \cdots - 2844 ) / 6$$ (150*b15 - 1164*b14 + 1041*b13 - 708*b12 - 675*b11 + 399*b10 + 231*b9 + 378*b8 + 553*b7 + 602*b6 - 329*b5 - 546*b4 + 637*b3 + 1344*b2 + 798*b1 - 2844) / 6 $$\nu^{8}$$ $$=$$ $$( 1208 \beta_{15} + 1066 \beta_{14} - 1538 \beta_{13} - 5172 \beta_{12} - 5061 \beta_{11} + 2656 \beta_{10} + 1575 \beta_{9} + 2373 \beta_{8} + 3553 \beta_{7} + 3209 \beta_{6} + 328 \beta_{5} + 450 \beta_{4} + \cdots + 2771 ) / 6$$ (1208*b15 + 1066*b14 - 1538*b13 - 5172*b12 - 5061*b11 + 2656*b10 + 1575*b9 + 2373*b8 + 3553*b7 + 3209*b6 + 328*b5 + 450*b4 - 956*b3 - 1584*b2 - 786*b1 + 2771) / 6 $$\nu^{9}$$ $$=$$ $$( - 270 \beta_{15} + 8796 \beta_{14} - 8310 \beta_{13} + 1431 \beta_{12} + 1316 \beta_{11} - 1017 \beta_{10} - 525 \beta_{9} - 987 \beta_{8} - 1188 \beta_{7} - 2370 \beta_{6} + 2388 \beta_{5} + 3687 \beta_{4} + \cdots + 19808 ) / 4$$ (-270*b15 + 8796*b14 - 8310*b13 + 1431*b12 + 1316*b11 - 1017*b10 - 525*b9 - 987*b8 - 1188*b7 - 2370*b6 + 2388*b5 + 3687*b4 - 5523*b3 - 10407*b2 - 5745*b1 + 19808) / 4 $$\nu^{10}$$ $$=$$ $$( - 12019 \beta_{15} + 2860 \beta_{14} + 4282 \beta_{13} + 53097 \beta_{12} + 51438 \beta_{11} - 29162 \beta_{10} - 16740 \beta_{9} - 26505 \beta_{8} - 40966 \beta_{7} - 39674 \beta_{6} + 41 \beta_{5} + \cdots - 1384 ) / 6$$ (-12019*b15 + 2860*b14 + 4282*b13 + 53097*b12 + 51438*b11 - 29162*b10 - 16740*b9 - 26505*b8 - 40966*b7 - 39674*b6 + 41*b5 + 576*b4 + 1397*b3 + 672*b2 - 315*b1 - 1384) / 6 $$\nu^{11}$$ $$=$$ $$( - 13518 \beta_{15} - 267924 \beta_{14} + 270372 \beta_{13} + 56313 \beta_{12} + 55698 \beta_{11} - 26361 \beta_{10} - 16467 \beta_{9} - 23397 \beta_{8} - 53548 \beta_{7} - 6116 \beta_{6} + \cdots - 598980 ) / 12$$ (-13518*b15 - 267924*b14 + 270372*b13 + 56313*b12 + 55698*b11 - 26361*b10 - 16467*b9 - 23397*b8 - 53548*b7 - 6116*b6 - 73150*b5 - 108273*b4 + 182105*b3 + 325347*b2 + 173613*b1 - 598980) / 12 $$\nu^{12}$$ $$=$$ $$( 38178 \beta_{15} - 55687 \beta_{14} + 29399 \beta_{13} - 170578 \beta_{12} - 164538 \beta_{11} + 96870 \beta_{10} + 54615 \beta_{9} + 88209 \beta_{8} + 135537 \beta_{7} + 141069 \beta_{6} + \cdots - 87901 ) / 2$$ (38178*b15 - 55687*b14 + 29399*b13 - 170578*b12 - 164538*b11 + 96870*b10 + 54615*b9 + 88209*b8 + 135537*b7 + 141069*b6 - 11974*b5 - 18639*b4 + 27206*b3 + 51962*b2 + 28881*b1 - 87901) / 2 $$\nu^{13}$$ $$=$$ $$( 176273 \beta_{15} + 1221805 \beta_{14} - 1326845 \beta_{13} - 778395 \beta_{12} - 754452 \beta_{11} + 428080 \beta_{10} + 245583 \beta_{9} + 389103 \beta_{8} + 714870 \beta_{7} + \cdots + 2815751 ) / 6$$ (176273*b15 + 1221805*b14 - 1326845*b13 - 778395*b12 - 754452*b11 + 428080*b10 + 245583*b9 + 389103*b8 + 714870*b7 + 470886*b6 + 342147*b5 + 496782*b4 - 878280*b3 - 1532739*b2 - 806286*b1 + 2815751) / 6 $$\nu^{14}$$ $$=$$ $$( - 1004014 \beta_{15} + 2973496 \beta_{14} - 2241272 \beta_{13} + 4505304 \beta_{12} + 4335321 \beta_{11} - 2597126 \beta_{10} - 1451268 \beta_{9} - 2362542 \beta_{8} + \cdots + 5415443 ) / 6$$ (-1004014*b15 + 2973496*b14 - 2241272*b13 + 4505304*b12 + 4335321*b11 - 2597126*b10 - 1451268*b9 - 2362542*b8 - 3522031*b7 - 3983831*b6 + 708320*b5 + 1046343*b4 - 1767052*b3 - 3149460*b2 - 1678611*b1 + 5415443) / 6 $$\nu^{15}$$ $$=$$ $$( - 1855568 \beta_{15} - 6465538 \beta_{14} + 7738524 \beta_{13} + 8283611 \beta_{12} + 7991044 \beta_{11} - 4704505 \beta_{10} - 2651201 \beta_{9} - 4279555 \beta_{8} + \cdots - 15882042 ) / 4$$ (-1855568*b15 - 6465538*b14 + 7738524*b13 + 8283611*b12 + 7991044*b11 - 4704505*b10 - 2651201*b9 - 4279555*b8 - 7412880*b7 - 6010658*b6 - 1895954*b5 - 2731969*b4 + 4928317*b3 + 8516577*b2 + 4456055*b1 - 15882042) / 4

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1134\mathbb{Z}\right)^\times$$.

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$-1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1133.1
 0.5 − 2.18553i 0.5 − 1.24085i 0.5 + 1.61238i 0.5 + 1.22108i 0.5 − 0.221083i 0.5 − 0.612384i 0.5 + 2.24085i 0.5 + 3.18553i 0.5 + 2.18553i 0.5 + 1.24085i 0.5 − 1.61238i 0.5 − 1.22108i 0.5 + 0.221083i 0.5 + 0.612384i 0.5 − 2.24085i 0.5 − 3.18553i
1.00000i 0 −1.00000 −3.79792 0 2.59077 + 0.536572i 1.00000i 0 3.79792i
1133.2 1.00000i 0 −1.00000 −2.46193 0 −1.59077 + 2.11411i 1.00000i 0 2.46193i
1133.3 1.00000i 0 −1.00000 −1.57315 0 0.858719 2.50252i 1.00000i 0 1.57315i
1133.4 1.00000i 0 −1.00000 −1.01976 0 0.141281 + 2.64198i 1.00000i 0 1.01976i
1133.5 1.00000i 0 −1.00000 1.01976 0 0.141281 2.64198i 1.00000i 0 1.01976i
1133.6 1.00000i 0 −1.00000 1.57315 0 0.858719 + 2.50252i 1.00000i 0 1.57315i
1133.7 1.00000i 0 −1.00000 2.46193 0 −1.59077 2.11411i 1.00000i 0 2.46193i
1133.8 1.00000i 0 −1.00000 3.79792 0 2.59077 0.536572i 1.00000i 0 3.79792i
1133.9 1.00000i 0 −1.00000 −3.79792 0 2.59077 0.536572i 1.00000i 0 3.79792i
1133.10 1.00000i 0 −1.00000 −2.46193 0 −1.59077 2.11411i 1.00000i 0 2.46193i
1133.11 1.00000i 0 −1.00000 −1.57315 0 0.858719 + 2.50252i 1.00000i 0 1.57315i
1133.12 1.00000i 0 −1.00000 −1.01976 0 0.141281 2.64198i 1.00000i 0 1.01976i
1133.13 1.00000i 0 −1.00000 1.01976 0 0.141281 + 2.64198i 1.00000i 0 1.01976i
1133.14 1.00000i 0 −1.00000 1.57315 0 0.858719 2.50252i 1.00000i 0 1.57315i
1133.15 1.00000i 0 −1.00000 2.46193 0 −1.59077 + 2.11411i 1.00000i 0 2.46193i
1133.16 1.00000i 0 −1.00000 3.79792 0 2.59077 + 0.536572i 1.00000i 0 3.79792i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1133.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1134.2.d.b 16
3.b odd 2 1 inner 1134.2.d.b 16
7.b odd 2 1 inner 1134.2.d.b 16
9.c even 3 1 1134.2.m.h 16
9.c even 3 1 1134.2.m.i 16
9.d odd 6 1 1134.2.m.h 16
9.d odd 6 1 1134.2.m.i 16
21.c even 2 1 inner 1134.2.d.b 16
63.l odd 6 1 1134.2.m.h 16
63.l odd 6 1 1134.2.m.i 16
63.o even 6 1 1134.2.m.h 16
63.o even 6 1 1134.2.m.i 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1134.2.d.b 16 1.a even 1 1 trivial
1134.2.d.b 16 3.b odd 2 1 inner
1134.2.d.b 16 7.b odd 2 1 inner
1134.2.d.b 16 21.c even 2 1 inner
1134.2.m.h 16 9.c even 3 1
1134.2.m.h 16 9.d odd 6 1
1134.2.m.h 16 63.l odd 6 1
1134.2.m.h 16 63.o even 6 1
1134.2.m.i 16 9.c even 3 1
1134.2.m.i 16 9.d odd 6 1
1134.2.m.i 16 63.l odd 6 1
1134.2.m.i 16 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 24T_{5}^{6} + 162T_{5}^{4} - 360T_{5}^{2} + 225$$ acting on $$S_{2}^{\mathrm{new}}(1134, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{8}$$
$3$ $$T^{16}$$
$5$ $$(T^{8} - 24 T^{6} + 162 T^{4} - 360 T^{2} + \cdots + 225)^{2}$$
$7$ $$(T^{8} - 4 T^{7} + 16 T^{6} - 52 T^{5} + \cdots + 2401)^{2}$$
$11$ $$(T^{8} + 36 T^{6} + 288 T^{4} + 648 T^{2} + \cdots + 324)^{2}$$
$13$ $$(T^{8} + 84 T^{6} + 2502 T^{4} + \cdots + 119025)^{2}$$
$17$ $$(T^{8} - 60 T^{6} + 918 T^{4} - 3420 T^{2} + \cdots + 225)^{2}$$
$19$ $$(T^{8} + 156 T^{6} + 7632 T^{4} + \cdots + 476100)^{2}$$
$23$ $$(T^{8} + 36 T^{6} + 288 T^{4} + 648 T^{2} + \cdots + 324)^{2}$$
$29$ $$(T^{8} + 108 T^{6} + 2502 T^{4} + \cdots + 42849)^{2}$$
$31$ $$(T^{8} + 180 T^{6} + 9072 T^{4} + \cdots + 72900)^{2}$$
$37$ $$(T^{4} - 4 T^{3} - 84 T^{2} - 148 T + 73)^{4}$$
$41$ $$(T^{8} - 240 T^{6} + 20232 T^{4} + \cdots + 8643600)^{2}$$
$43$ $$(T^{4} - 16 T^{3} + 24 T^{2} + 608 T - 2336)^{4}$$
$47$ $$(T^{8} - 348 T^{6} + 39888 T^{4} + \cdots + 476100)^{2}$$
$53$ $$(T^{4} + 72 T^{2} + 324)^{4}$$
$59$ $$(T^{8} - 60 T^{6} + 432 T^{4} - 1080 T^{2} + \cdots + 900)^{2}$$
$61$ $$(T^{8} + 348 T^{6} + 34038 T^{4} + \cdots + 497025)^{2}$$
$67$ $$(T^{4} + 4 T^{3} - 102 T^{2} - 572 T - 242)^{4}$$
$71$ $$(T^{8} + 360 T^{6} + 32256 T^{4} + \cdots + 11451456)^{2}$$
$73$ $$(T^{8} + 168 T^{6} + 9378 T^{4} + \cdots + 225)^{2}$$
$79$ $$(T^{4} - 8 T^{3} - 66 T^{2} + 4 T + 142)^{4}$$
$83$ $$(T^{8} - 420 T^{6} + 46512 T^{4} + \cdots + 8468100)^{2}$$
$89$ $$(T^{8} - 540 T^{6} + 79542 T^{4} + \cdots + 9641025)^{2}$$
$97$ $$(T^{8} + 120 T^{6} + 1368 T^{4} + \cdots + 3600)^{2}$$