# Properties

 Label 29.6.a.b Level $29$ Weight $6$ Character orbit 29.a Self dual yes Analytic conductor $4.651$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$29$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 29.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.65113077458$$ Analytic rank: $$0$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ Defining polynomial: $$x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902$$ x^7 - 3*x^6 - 184*x^5 + 584*x^4 + 10145*x^3 - 34491*x^2 - 149754*x + 524902 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + (\beta_{5} + 4) q^{3} + (\beta_{2} - 3 \beta_1 + 23) q^{4} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 6) q^{5} + (\beta_{6} - \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 2 \beta_1 + 3) q^{6} + ( - 4 \beta_{6} - 4 \beta_{4} + \beta_{3} + \beta_{2} + 6 \beta_1 + 23) q^{7} + (2 \beta_{6} - 6 \beta_{5} + 8 \beta_{4} + \beta_{3} - \beta_{2} - 13 \beta_1 + 140) q^{8} + (7 \beta_{6} + 5 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 15 \beta_1 + 139) q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^2 + (b5 + 4) * q^3 + (b2 - 3*b1 + 23) * q^4 + (-b5 + b4 + b3 - 2*b2 - 2*b1 + 6) * q^5 + (b6 - b5 - 2*b4 - 3*b3 - 2*b1 + 3) * q^6 + (-4*b6 - 4*b4 + b3 + b2 + 6*b1 + 23) * q^7 + (2*b6 - 6*b5 + 8*b4 + b3 - b2 - 13*b1 + 140) * q^8 + (7*b6 + 5*b4 + 2*b3 - 3*b2 + 15*b1 + 139) * q^9 $$q + ( - \beta_1 + 1) q^{2} + (\beta_{5} + 4) q^{3} + (\beta_{2} - 3 \beta_1 + 23) q^{4} + ( - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 6) q^{5} + (\beta_{6} - \beta_{5} - 2 \beta_{4} - 3 \beta_{3} - 2 \beta_1 + 3) q^{6} + ( - 4 \beta_{6} - 4 \beta_{4} + \beta_{3} + \beta_{2} + 6 \beta_1 + 23) q^{7} + (2 \beta_{6} - 6 \beta_{5} + 8 \beta_{4} + \beta_{3} - \beta_{2} - 13 \beta_1 + 140) q^{8} + (7 \beta_{6} + 5 \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 15 \beta_1 + 139) q^{9} + ( - 2 \beta_{6} - 4 \beta_{5} - 10 \beta_{4} + 6 \beta_{3} + 8 \beta_{2} + \cdots + 123) q^{10}+ \cdots + (2061 \beta_{6} + 2369 \beta_{5} + 3564 \beta_{4} + 1740 \beta_{3} + \cdots + 43573) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^2 + (b5 + 4) * q^3 + (b2 - 3*b1 + 23) * q^4 + (-b5 + b4 + b3 - 2*b2 - 2*b1 + 6) * q^5 + (b6 - b5 - 2*b4 - 3*b3 - 2*b1 + 3) * q^6 + (-4*b6 - 4*b4 + b3 + b2 + 6*b1 + 23) * q^7 + (2*b6 - 6*b5 + 8*b4 + b3 - b2 - 13*b1 + 140) * q^8 + (7*b6 + 5*b4 + 2*b3 - 3*b2 + 15*b1 + 139) * q^9 + (-2*b6 - 4*b5 - 10*b4 + 6*b3 + 8*b2 + 11*b1 + 123) * q^10 + (-3*b6 + 10*b5 + 8*b4 - 4*b3 - 2*b2 + 21*b1 + 153) * q^11 + (-14*b6 + 22*b5 - 8*b4 - 15*b3 + 11*b2 + 15*b1 + 24) * q^12 + (-7*b6 + 20*b5 - 15*b4 - 7*b3 + 8*b2 + 55*b1 + 35) * q^13 + (20*b6 - 28*b5 + 20*b3 - 26*b2 - 20*b1 - 276) * q^14 + (39*b6 - 6*b5 + 8*b4 + 5*b3 - 25*b2 + 41*b1 - 98) * q^15 + (-8*b6 - 16*b5 + 32*b4 + 6*b3 - b2 - 101*b1 + 79) * q^16 + (-43*b6 + 21*b5 + 2*b4 - 17*b3 + 15*b2 - 45*b1 - 106) * q^17 + (-32*b6 + 10*b5 - 18*b4 - 6*b3 + 6*b2 - 84*b1 - 766) * q^18 + (35*b6 - 55*b5 + 36*b4 + 8*b3 + 18*b2 - 77*b1 + 631) * q^19 + (24*b6 - 76*b5 + 4*b4 + 52*b3 - 25*b2 - 165*b1 - 843) * q^20 + (-33*b6 + 39*b5 - 114*b4 - 7*b3 + 65*b2 + 129*b1 - 680) * q^21 + (21*b6 + 19*b5 + 2*b4 - 73*b3 + 16*b2 - 74*b1 - 851) * q^22 + (-24*b6 - 14*b5 - 52*b4 + 5*b3 + 13*b2 + 26*b1 - 673) * q^23 + (44*b6 + 108*b5 + 104*b4 - 34*b3 - 29*b2 + 31*b1 - 581) * q^24 + (-30*b6 + 5*b5 + 31*b4 + 10*b3 + 21*b2 + 114*b1 + 742) * q^25 + (42*b6 + 40*b5 - 22*b4 - 64*b3 - 96*b2 + 75*b1 - 2835) * q^26 + (45*b6 - 66*b5 + 204*b4 + 100*b3 - 150*b2 + 101*b1 + 873) * q^27 + (-12*b6 + 4*b5 - 64*b4 + 146*b3 + 32*b2 + 324*b1 - 206) * q^28 + 841 * q^29 + (-233*b6 + 257*b5 - 234*b4 - 27*b3 + 94*b2 + 550*b1 - 2617) * q^30 + (-32*b6 - 157*b5 - 76*b4 - 87*b3 - 23*b2 + 170*b1 + 979) * q^31 + (2*b6 - 6*b5 - 88*b4 + b3 + 169*b2 - 47*b1 + 1086) * q^32 + (154*b6 + 75*b5 + 209*b4 + 76*b3 + 17*b2 - 42*b1 + 4991) * q^33 + (234*b6 - 94*b5 + 172*b4 - 128*b3 + 14*b2 - 148*b1 + 3016) * q^34 + (-18*b6 - 270*b5 + 48*b4 - 105*b3 - 105*b2 - 844*b1 + 1319) * q^35 + (-72*b6 - 68*b5 - 140*b4 - 62*b3 + 112*b2 + 112*b1 - 210) * q^36 + (94*b6 - 56*b5 - 246*b4 - 110*b3 - 96*b2 + 26*b1 + 676) * q^37 + (-142*b6 - 98*b5 + 328*b4 + 132*b3 + 80*b2 - 1332*b1 + 3976) * q^38 + (-150*b6 + 117*b5 - 124*b4 + 115*b3 + 175*b2 - 136*b1 + 4811) * q^39 + (-74*b6 - 266*b5 + 240*b4 + 343*b3 - 167*b2 + 157*b1 + 3260) * q^40 + (139*b6 + 221*b5 - 256*b4 - 177*b3 + 169*b2 - 543*b1 + 3018) * q^41 + (206*b6 - 154*b5 + 52*b4 + 160*b3 - 622*b2 + 434*b1 - 7942) * q^42 + (-405*b6 + 196*b5 - 184*b4 + 8*b3 - 58*b2 + 331*b1 + 2691) * q^43 + (-102*b6 + 686*b5 - 200*b4 - 449*b3 + 391*b2 + 191*b1 - 958) * q^44 + (221*b6 + 335*b5 + 482*b4 + 218*b3 - 386*b2 - 119*b1 + 1479) * q^45 + (90*b6 - 98*b5 - 28*b4 + 218*b3 - 226*b2 + 648*b1 - 1986) * q^46 + (11*b6 - 176*b5 - 32*b4 + 159*b3 + 105*b2 - 1247*b1 + 2512) * q^47 + (314*b6 - 254*b5 + 136*b4 - 363*b3 + 121*b2 + 641*b1 - 2758) * q^48 + (-340*b6 + 564*b5 + 80*b4 - 176*b3 + 280*b2 + 188*b1 + 5477) * q^49 + (248*b6 - 514*b5 + 342*b4 + 44*b3 - 206*b2 - 1044*b1 - 5452) * q^50 + (134*b6 - 524*b5 - 472*b4 - 415*b3 + 849*b2 - 240*b1 + 2997) * q^51 + (-288*b6 + 1068*b5 - 540*b4 - 438*b3 + 307*b2 + 3291*b1 - 6233) * q^52 + (77*b6 + 430*b5 - 125*b4 + 447*b3 - 16*b2 - 801*b1 - 7861) * q^53 + (-187*b6 - 821*b5 - 342*b4 + 295*b3 + 552*b2 + 326*b1 - 4995) * q^54 + (248*b6 - 345*b5 + 452*b4 + 73*b3 - 871*b2 + 1906*b1 - 4245) * q^55 + (-284*b6 - 1212*b5 + 16*b4 + 542*b3 - 344*b2 + 504*b1 - 11750) * q^56 + (-485*b6 + 841*b5 + 570*b4 - 266*b3 - 210*b2 + 47*b1 - 12325) * q^57 + (-841*b1 + 841) * q^58 + (614*b6 - 304*b5 + 408*b4 + 545*b3 - 143*b2 + 976*b1 + 1513) * q^59 + (74*b6 - 714*b5 - 488*b4 - 325*b3 - 719*b2 + 2253*b1 - 26912) * q^60 + (159*b6 - 861*b5 + 798*b4 + 135*b3 + 279*b2 - 1235*b1 + 15032) * q^61 + (-293*b6 + 1581*b5 - 110*b4 + 23*b3 + 86*b2 + 122*b1 - 5731) * q^62 + (-750*b6 + 98*b5 - 1044*b4 - 340*b3 + 892*b2 - 934*b1 - 12686) * q^63 + (492*b6 - 236*b5 - 16*b4 + 176*b3 - 775*b2 - 315*b1 - 1141) * q^64 + (399*b6 - 34*b5 + 67*b4 - 494*b3 - 837*b2 - 529*b1 - 20920) * q^65 + (-300*b6 - 1098*b5 + 514*b4 - 248*b3 + 242*b2 - 6259*b1 + 3439) * q^66 + (1190*b6 + 306*b5 - 244*b4 + 364*b3 - 60*b2 + 974*b1 + 14418) * q^67 + (56*b6 + 1784*b5 + 456*b4 - 634*b3 + 702*b2 - 2098*b1 + 13340) * q^68 + (-631*b6 - 369*b5 - 1260*b4 - 143*b3 + 515*b2 + 211*b1 - 15864) * q^69 + (-552*b6 + 2136*b5 - 72*b4 - 108*b3 + 1762*b2 - 1552*b1 + 50672) * q^70 + (-500*b6 - 626*b5 + 900*b4 - 164*b3 - 1208*b2 - 428*b1 - 6916) * q^71 + (1276*b6 + 4*b5 + 1192*b4 + 426*b3 - 856*b2 + 1992*b1 + 18990) * q^72 + (128*b6 + 230*b5 - 1170*b4 + 656*b3 + 78*b2 - 2024*b1 - 1672) * q^73 + (-1184*b6 + 3380*b5 - 1828*b4 - 300*b3 + 400*b2 + 2428*b1 + 1240) * q^74 + (-18*b6 + 750*b5 + 308*b4 + 86*b3 + 606*b2 + 2538*b1 + 4632) * q^75 + (244*b6 - 2164*b5 + 1280*b4 + 528*b3 + 422*b2 - 7766*b1 + 54382) * q^76 + (-1607*b6 - 279*b5 - 1406*b4 - 43*b3 + 453*b2 + 6147*b1 - 16494) * q^77 + (1323*b6 - 3155*b5 + 970*b4 + 1027*b3 - 1422*b2 - 7878*b1 + 9793) * q^78 + (-545*b6 - 910*b5 + 368*b4 - 807*b3 + 707*b2 - 375*b1 + 29930) * q^79 + (-72*b6 - 2192*b5 + 176*b4 + 962*b3 + 345*b2 + 1725*b1 + 19057) * q^80 + (1893*b6 + 139*b5 + 2610*b4 + 910*b3 - 1998*b2 + 3353*b1 - 18650) * q^81 + (-746*b6 + 2706*b5 - 392*b4 - 1360*b3 + 202*b2 - 4318*b1 + 31162) * q^82 + (620*b6 - 1810*b5 + 228*b4 - 1663*b3 - 7*b2 + 1222*b1 - 18805) * q^83 + (-1000*b6 + 1272*b5 - 1224*b4 + 874*b3 - 356*b2 + 13524*b1 - 7142) * q^84 + (-255*b6 - 3409*b5 - 908*b4 - 401*b3 - 567*b2 + 4087*b1 - 18370) * q^85 + (1937*b6 - 1433*b5 - 782*b4 + 183*b3 - 904*b2 - 418*b1 - 9099) * q^86 + (841*b5 + 3364) * q^87 + (208*b6 + 2736*b5 + 1096*b4 - 1972*b3 - 973*b2 + 1767*b1 + 19701) * q^88 + (697*b6 + 83*b5 + 1892*b4 + 687*b3 + 169*b2 + 9087*b1 - 13052) * q^89 + (-624*b6 - 1412*b5 - 2272*b4 - 1050*b3 + 1976*b2 + 2332*b1 + 5570) * q^90 + (-1576*b6 + 3492*b5 - 1852*b4 - 701*b3 + 1867*b2 + 7078*b1 + 24833) * q^91 + (176*b6 - 616*b5 - 240*b4 + 1400*b3 - 998*b2 + 4982*b1 - 17590) * q^92 + (-1500*b6 + 1759*b5 - 1759*b4 + 87*b3 + 1220*b2 - 10706*b1 - 63250) * q^93 + (265*b6 - 2529*b5 + 1042*b4 + 1799*b3 + 138*b2 - 7694*b1 + 65645) * q^94 + (-279*b6 - 329*b5 + 1148*b4 + 2342*b3 + 140*b2 - 4987*b1 - 16309) * q^95 + (-3580*b6 + 2316*b5 - 1936*b4 - 1156*b3 + 1841*b2 + 2189*b1 - 17961) * q^96 + (865*b6 + 475*b5 - 1176*b4 + 535*b3 - 43*b2 - 7193*b1 - 13744) * q^97 + (2552*b6 - 1720*b5 + 2112*b4 - 2464*b3 - 784*b2 - 8233*b1 - 951) * q^98 + (2061*b6 + 2369*b5 + 3564*b4 + 1740*b3 - 2318*b2 + 3445*b1 + 43573) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + 4 q^{2} + 26 q^{3} + 154 q^{4} + 32 q^{5} + 22 q^{6} + 184 q^{7} + 942 q^{8} + 1005 q^{9}+O(q^{10})$$ 7 * q + 4 * q^2 + 26 * q^3 + 154 * q^4 + 32 * q^5 + 22 * q^6 + 184 * q^7 + 942 * q^8 + 1005 * q^9 $$7 q + 4 q^{2} + 26 q^{3} + 154 q^{4} + 32 q^{5} + 22 q^{6} + 184 q^{7} + 942 q^{8} + 1005 q^{9} + 922 q^{10} + 1106 q^{11} + 214 q^{12} + 408 q^{13} - 2008 q^{14} - 614 q^{15} + 242 q^{16} - 874 q^{17} - 5598 q^{18} + 4288 q^{19} - 6350 q^{20} - 4200 q^{21} - 6114 q^{22} - 4532 q^{23} - 4318 q^{24} + 5527 q^{25} - 19806 q^{26} + 5942 q^{27} - 496 q^{28} + 5887 q^{29} - 16734 q^{30} + 7794 q^{31} + 7898 q^{32} + 34410 q^{33} + 20840 q^{34} + 7088 q^{35} - 572 q^{36} + 5086 q^{37} + 23732 q^{38} + 33394 q^{39} + 22906 q^{40} + 19826 q^{41} - 55440 q^{42} + 19498 q^{43} - 6074 q^{44} + 7854 q^{45} - 12404 q^{46} + 14278 q^{47} - 16406 q^{48} + 38431 q^{49} - 41066 q^{50} + 23892 q^{51} - 34302 q^{52} - 58644 q^{53} - 31194 q^{54} - 25574 q^{55} - 79560 q^{56} - 88540 q^{57} + 3364 q^{58} + 12888 q^{59} - 180822 q^{60} + 102866 q^{61} - 42654 q^{62} - 88632 q^{63} - 10170 q^{64} - 149206 q^{65} + 7710 q^{66} + 102996 q^{67} + 85100 q^{68} - 107244 q^{69} + 349480 q^{70} - 51596 q^{71} + 135568 q^{72} - 17566 q^{73} + 12132 q^{74} + 39356 q^{75} + 360740 q^{76} - 94104 q^{77} + 46386 q^{78} + 212058 q^{79} + 142510 q^{80} - 128285 q^{81} + 201924 q^{82} - 122928 q^{83} - 12328 q^{84} - 109336 q^{85} - 63290 q^{86} + 21866 q^{87} + 136666 q^{88} - 66510 q^{89} + 56084 q^{90} + 194368 q^{91} - 110108 q^{92} - 474274 q^{93} + 438926 q^{94} - 131676 q^{95} - 117018 q^{96} - 118182 q^{97} - 29132 q^{98} + 300668 q^{99}+O(q^{100})$$ 7 * q + 4 * q^2 + 26 * q^3 + 154 * q^4 + 32 * q^5 + 22 * q^6 + 184 * q^7 + 942 * q^8 + 1005 * q^9 + 922 * q^10 + 1106 * q^11 + 214 * q^12 + 408 * q^13 - 2008 * q^14 - 614 * q^15 + 242 * q^16 - 874 * q^17 - 5598 * q^18 + 4288 * q^19 - 6350 * q^20 - 4200 * q^21 - 6114 * q^22 - 4532 * q^23 - 4318 * q^24 + 5527 * q^25 - 19806 * q^26 + 5942 * q^27 - 496 * q^28 + 5887 * q^29 - 16734 * q^30 + 7794 * q^31 + 7898 * q^32 + 34410 * q^33 + 20840 * q^34 + 7088 * q^35 - 572 * q^36 + 5086 * q^37 + 23732 * q^38 + 33394 * q^39 + 22906 * q^40 + 19826 * q^41 - 55440 * q^42 + 19498 * q^43 - 6074 * q^44 + 7854 * q^45 - 12404 * q^46 + 14278 * q^47 - 16406 * q^48 + 38431 * q^49 - 41066 * q^50 + 23892 * q^51 - 34302 * q^52 - 58644 * q^53 - 31194 * q^54 - 25574 * q^55 - 79560 * q^56 - 88540 * q^57 + 3364 * q^58 + 12888 * q^59 - 180822 * q^60 + 102866 * q^61 - 42654 * q^62 - 88632 * q^63 - 10170 * q^64 - 149206 * q^65 + 7710 * q^66 + 102996 * q^67 + 85100 * q^68 - 107244 * q^69 + 349480 * q^70 - 51596 * q^71 + 135568 * q^72 - 17566 * q^73 + 12132 * q^74 + 39356 * q^75 + 360740 * q^76 - 94104 * q^77 + 46386 * q^78 + 212058 * q^79 + 142510 * q^80 - 128285 * q^81 + 201924 * q^82 - 122928 * q^83 - 12328 * q^84 - 109336 * q^85 - 63290 * q^86 + 21866 * q^87 + 136666 * q^88 - 66510 * q^89 + 56084 * q^90 + 194368 * q^91 - 110108 * q^92 - 474274 * q^93 + 438926 * q^94 - 131676 * q^95 - 117018 * q^96 - 118182 * q^97 - 29132 * q^98 + 300668 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 3x^{6} - 184x^{5} + 584x^{4} + 10145x^{3} - 34491x^{2} - 149754x + 524902$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 54$$ v^2 + v - 54 $$\beta_{3}$$ $$=$$ $$( \nu^{6} - 6\nu^{5} - 114\nu^{4} + 618\nu^{3} + 2727\nu^{2} - 10724\nu - 11462 ) / 240$$ (v^6 - 6*v^5 - 114*v^4 + 618*v^3 + 2727*v^2 - 10724*v - 11462) / 240 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - 5\nu^{4} - 119\nu^{3} + 547\nu^{2} + 2890\nu - 11242 ) / 96$$ (v^5 - 5*v^4 - 119*v^3 + 547*v^2 + 2890*v - 11242) / 96 $$\beta_{5}$$ $$=$$ $$( \nu^{6} + 10\nu^{5} - 218\nu^{4} - 1094\nu^{3} + 13231\nu^{2} + 21356\nu - 185766 ) / 960$$ (v^6 + 10*v^5 - 218*v^4 - 1094*v^3 + 13231*v^2 + 21356*v - 185766) / 960 $$\beta_{6}$$ $$=$$ $$( \nu^{6} + 2\nu^{5} - 226\nu^{4} - 238\nu^{3} + 14279\nu^{2} + 5916\nu - 208054 ) / 960$$ (v^6 + 2*v^5 - 226*v^4 - 238*v^3 + 14279*v^2 + 5916*v - 208054) / 960
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta _1 + 54$$ b2 - b1 + 54 $$\nu^{3}$$ $$=$$ $$-2\beta_{6} + 6\beta_{5} - 8\beta_{4} - \beta_{3} + 4\beta_{2} + 71\beta _1 - 41$$ -2*b6 + 6*b5 - 8*b4 - b3 + 4*b2 + 71*b1 - 41 $$\nu^{4}$$ $$=$$ $$-16\beta_{6} + 8\beta_{5} + 2\beta_{3} + 105\beta_{2} - 95\beta _1 + 3846$$ -16*b6 + 8*b5 + 2*b3 + 105*b2 - 95*b1 + 3846 $$\nu^{5}$$ $$=$$ $$-318\beta_{6} + 754\beta_{5} - 856\beta_{4} - 109\beta_{3} + 454\beta_{2} + 5631\beta _1 - 3945$$ -318*b6 + 754*b5 - 856*b4 - 109*b3 + 454*b2 + 5631*b1 - 3945 $$\nu^{6}$$ $$=$$ $$-2496\beta_{6} + 1728\beta_{5} - 192\beta_{4} + 432\beta_{3} + 9495\beta_{2} - 7471\beta _1 + 304316$$ -2496*b6 + 1728*b5 - 192*b4 + 432*b3 + 9495*b2 - 7471*b1 + 304316

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 9.56883 7.83842 4.90786 3.60554 −4.83960 −8.92709 −9.15396
−8.56883 18.3274 41.4249 −84.3249 −157.045 216.816 −80.7606 92.8952 722.566
1.2 −6.83842 −24.5656 14.7640 −54.3066 167.990 −37.6697 117.867 360.469 371.372
1.3 −3.90786 29.3989 −16.7287 64.0801 −114.887 −138.793 190.425 621.298 −250.416
1.4 −2.60554 −13.2844 −25.2112 69.0035 34.6131 156.573 149.066 −66.5241 −179.792
1.5 5.83960 15.9679 2.10095 31.5616 93.2461 106.304 −174.599 11.9736 184.307
1.6 9.92709 15.4219 66.5471 −58.0818 153.094 −210.388 342.952 −5.16616 −576.583
1.7 10.1540 −15.2661 71.1029 64.0682 −155.012 91.1564 397.049 −9.94539 650.545
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$29$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 29.6.a.b 7
3.b odd 2 1 261.6.a.e 7
4.b odd 2 1 464.6.a.k 7
5.b even 2 1 725.6.a.b 7
29.b even 2 1 841.6.a.b 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.6.a.b 7 1.a even 1 1 trivial
261.6.a.e 7 3.b odd 2 1
464.6.a.k 7 4.b odd 2 1
725.6.a.b 7 5.b even 2 1
841.6.a.b 7 29.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{7} - 4T_{2}^{6} - 181T_{2}^{5} + 346T_{2}^{4} + 10616T_{2}^{3} + 2416T_{2}^{2} - 186896T_{2} - 351200$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(29))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7} - 4 T^{6} - 181 T^{5} + \cdots - 351200$$
$3$ $$T^{7} - 26 T^{6} + \cdots + 661023756$$
$5$ $$T^{7} - 32 T^{6} + \cdots + 2378174390186$$
$7$ $$T^{7} + \cdots + 361848785235968$$
$11$ $$T^{7} - 1106 T^{6} + \cdots + 51\!\cdots\!44$$
$13$ $$T^{7} - 408 T^{6} + \cdots + 10\!\cdots\!34$$
$17$ $$T^{7} + 874 T^{6} + \cdots - 17\!\cdots\!28$$
$19$ $$T^{7} - 4288 T^{6} + \cdots + 15\!\cdots\!92$$
$23$ $$T^{7} + 4532 T^{6} + \cdots - 15\!\cdots\!56$$
$29$ $$(T - 841)^{7}$$
$31$ $$T^{7} - 7794 T^{6} + \cdots + 64\!\cdots\!48$$
$37$ $$T^{7} - 5086 T^{6} + \cdots - 16\!\cdots\!56$$
$41$ $$T^{7} - 19826 T^{6} + \cdots - 56\!\cdots\!72$$
$43$ $$T^{7} - 19498 T^{6} + \cdots - 58\!\cdots\!60$$
$47$ $$T^{7} - 14278 T^{6} + \cdots - 14\!\cdots\!36$$
$53$ $$T^{7} + 58644 T^{6} + \cdots + 84\!\cdots\!94$$
$59$ $$T^{7} - 12888 T^{6} + \cdots + 15\!\cdots\!00$$
$61$ $$T^{7} - 102866 T^{6} + \cdots + 45\!\cdots\!80$$
$67$ $$T^{7} - 102996 T^{6} + \cdots - 11\!\cdots\!60$$
$71$ $$T^{7} + 51596 T^{6} + \cdots - 20\!\cdots\!52$$
$73$ $$T^{7} + 17566 T^{6} + \cdots + 43\!\cdots\!96$$
$79$ $$T^{7} - 212058 T^{6} + \cdots - 28\!\cdots\!00$$
$83$ $$T^{7} + 122928 T^{6} + \cdots - 97\!\cdots\!52$$
$89$ $$T^{7} + 66510 T^{6} + \cdots - 79\!\cdots\!20$$
$97$ $$T^{7} + 118182 T^{6} + \cdots + 62\!\cdots\!52$$