# Properties

 Label 1225.4.a.bj Level $1225$ Weight $4$ Character orbit 1225.a Self dual yes Analytic conductor $72.277$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1225 = 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.2773397570$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.6.1163891200.1 Defining polynomial: $$x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28$$ x^6 - 2*x^5 - 23*x^4 + 12*x^3 + 154*x^2 + 152*x + 28 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2\cdot 7$$ Twist minimal: no (minimal twist has level 245) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{5} + 3) q^{3} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 2) q^{4} + ( - \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 6) q^{6} + ( - 2 \beta_{5} - 3 \beta_{4} - 5 \beta_{3} + \beta_{2} + 9 \beta_1 + 8) q^{8} + ( - 6 \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} - 5 \beta_1 + 16) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b5 + 3) * q^3 + (-b5 - b4 - b3 + 2*b2 + 3*b1 + 2) * q^4 + (-b3 - 2*b2 + 4*b1 - 6) * q^6 + (-2*b5 - 3*b4 - 5*b3 + b2 + 9*b1 + 8) * q^8 + (-6*b5 + 3*b4 + b3 - b2 - 5*b1 + 16) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{5} + 3) q^{3} + ( - \beta_{5} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 2) q^{4} + ( - \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 6) q^{6} + ( - 2 \beta_{5} - 3 \beta_{4} - 5 \beta_{3} + \beta_{2} + 9 \beta_1 + 8) q^{8} + ( - 6 \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} - 5 \beta_1 + 16) q^{9} + ( - 2 \beta_{5} - 5 \beta_{4} - \beta_{3} - 5 \beta_{2} + 5 \beta_1 - 7) q^{11} + ( - 2 \beta_{5} - 5 \beta_{4} - 8 \beta_{3} + 13 \beta_{2} + 6 \beta_1 + 28) q^{12} + ( - 5 \beta_{4} - 8 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 27) q^{13} + ( - 12 \beta_{5} - 3 \beta_{4} - 15 \beta_{3} + \beta_{2} + 27 \beta_1 + 44) q^{16} + ( - \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 10 \beta_{2} - 16 \beta_1 + 1) q^{17} + (8 \beta_{5} + 3 \beta_{4} + 3 \beta_{3} - 21 \beta_{2} + \beta_1 - 68) q^{18} + (3 \beta_{5} - 3 \beta_{4} - 4 \beta_{3} + 10 \beta_{2} + 6 \beta_1 - 52) q^{19} + ( - 22 \beta_{5} - \beta_{4} - 19 \beta_{3} + 17 \beta_{2} + 12 \beta_1 + 48) q^{22} + (6 \beta_{5} - 8 \beta_{4} + 9 \beta_{3} + 11 \beta_{2} - 3 \beta_1 + 56) q^{23} + ( - \beta_{5} - 9 \beta_{4} - 8 \beta_{3} + 6 \beta_{2} + 50 \beta_1 - 2) q^{24} + ( - 15 \beta_{5} - 5 \beta_{4} - 19 \beta_{3} + 4 \beta_{2} + 58 \beta_1 - 24) q^{26} + ( - 12 \beta_{5} + 21 \beta_{4} + 16 \beta_{3} - 10 \beta_{2} - 50 \beta_1 + 185) q^{27} + ( - 8 \beta_{5} - \beta_{4} + 13 \beta_{3} + 7 \beta_{2} + 39 \beta_1 + 21) q^{29} + ( - 10 \beta_{5} + 17 \beta_{3} - 29 \beta_{2} - 11 \beta_1 - 68) q^{31} + ( - 42 \beta_{5} - 15 \beta_{4} - 31 \beta_{3} + 35 \beta_{2} + 99 \beta_1 + 116) q^{32} + (8 \beta_{5} + 23 \beta_{4} + 8 \beta_{3} - 44 \beta_{2} + 26 \beta_1 + 13) q^{33} + (2 \beta_{5} + 22 \beta_{4} + 11 \beta_{3} - 18 \beta_{2} - 62 \beta_1 - 114) q^{34} + (14 \beta_{5} - 25 \beta_{4} - 13 \beta_{3} + 65 \beta_{2} - 63 \beta_1 + 68) q^{36} + ( - 4 \beta_{5} - 28 \beta_{4} + 9 \beta_{3} + 11 \beta_{2} + 33 \beta_1 + 8) q^{37} + (3 \beta_{5} - 7 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} - 16 \beta_1 + 6) q^{38} + ( - 6 \beta_{5} - 31 \beta_{4} - 11 \beta_{3} + 57 \beta_{2} + 39 \beta_1 + 21) q^{39} + ( - 37 \beta_{5} + 15 \beta_{4} + 3 \beta_{3} + 7 \beta_{2} - 11 \beta_1 - 86) q^{41} + ( - 2 \beta_{5} + 34 \beta_{4} + 16 \beta_{3} - 8 \beta_{2} + 74) q^{43} + ( - \beta_{5} + 10 \beta_{4} - 48 \beta_{3} + 5 \beta_{2} + 122 \beta_1 - 62) q^{44} + (35 \beta_{5} + 20 \beta_{4} + 30 \beta_{3} - 25 \beta_{2} + 42 \beta_1 - 92) q^{46} + ( - 2 \beta_{5} - 9 \beta_{4} + 32 \beta_{3} - 14 \beta_{2} + 50 \beta_1 + 81) q^{47} + ( - 47 \beta_{5} - 9 \beta_{4} - 6 \beta_{3} - 10 \beta_{2} + 132 \beta_1 + 198) q^{48} + ( - 10 \beta_{5} + 47 \beta_{4} + 41 \beta_{3} - 59 \beta_{2} - 65 \beta_1 + 145) q^{51} + ( - 93 \beta_{5} - 32 \beta_{4} - 48 \beta_{3} + 65 \beta_{2} + 196 \beta_1 + 230) q^{52} + ( - 2 \beta_{5} + 60 \beta_{4} + 19 \beta_{3} - 35 \beta_{2} - 73 \beta_1 + 146) q^{53} + (83 \beta_{5} + 45 \beta_{4} + 81 \beta_{3} - 120 \beta_{2} - 16 \beta_1 - 428) q^{54} + (72 \beta_{5} - 48 \beta_{4} - 38 \beta_{3} + 86 \beta_{2} + 30 \beta_1 - 266) q^{57} + ( - 25 \beta_{4} - 15 \beta_{3} + 35 \beta_{2} + 128 \beta_1 + 296) q^{58} + (11 \beta_{5} - 59 \beta_{4} - 47 \beta_{3} + 33 \beta_{2} - 49 \beta_1 - 162) q^{59} + ( - 24 \beta_{5} + 10 \beta_{4} + 4 \beta_{3} - 48 \beta_{2} - 132 \beta_1 - 84) q^{61} + ( - 13 \beta_{5} + 28 \beta_{4} + 6 \beta_{3} - \beta_{2} - 154 \beta_1 + 4) q^{62} + ( - 10 \beta_{5} - 91 \beta_{4} - 63 \beta_{3} + 67 \beta_{2} + 351 \beta_1 + 116) q^{64} + ( - 75 \beta_{5} - 41 \beta_{4} - 23 \beta_{3} + 148 \beta_{2} + \cdots + 664) q^{66}+ \cdots + (26 \beta_{5} + 244 \beta_{4} + 140 \beta_{3} - 308 \beta_{2} + 116 \beta_1 - 536) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b5 + 3) * q^3 + (-b5 - b4 - b3 + 2*b2 + 3*b1 + 2) * q^4 + (-b3 - 2*b2 + 4*b1 - 6) * q^6 + (-2*b5 - 3*b4 - 5*b3 + b2 + 9*b1 + 8) * q^8 + (-6*b5 + 3*b4 + b3 - b2 - 5*b1 + 16) * q^9 + (-2*b5 - 5*b4 - b3 - 5*b2 + 5*b1 - 7) * q^11 + (-2*b5 - 5*b4 - 8*b3 + 13*b2 + 6*b1 + 28) * q^12 + (-5*b4 - 8*b3 + 4*b2 + 2*b1 + 27) * q^13 + (-12*b5 - 3*b4 - 15*b3 + b2 + 27*b1 + 44) * q^16 + (-b5 - 2*b4 + 4*b3 - 10*b2 - 16*b1 + 1) * q^17 + (8*b5 + 3*b4 + 3*b3 - 21*b2 + b1 - 68) * q^18 + (3*b5 - 3*b4 - 4*b3 + 10*b2 + 6*b1 - 52) * q^19 + (-22*b5 - b4 - 19*b3 + 17*b2 + 12*b1 + 48) * q^22 + (6*b5 - 8*b4 + 9*b3 + 11*b2 - 3*b1 + 56) * q^23 + (-b5 - 9*b4 - 8*b3 + 6*b2 + 50*b1 - 2) * q^24 + (-15*b5 - 5*b4 - 19*b3 + 4*b2 + 58*b1 - 24) * q^26 + (-12*b5 + 21*b4 + 16*b3 - 10*b2 - 50*b1 + 185) * q^27 + (-8*b5 - b4 + 13*b3 + 7*b2 + 39*b1 + 21) * q^29 + (-10*b5 + 17*b3 - 29*b2 - 11*b1 - 68) * q^31 + (-42*b5 - 15*b4 - 31*b3 + 35*b2 + 99*b1 + 116) * q^32 + (8*b5 + 23*b4 + 8*b3 - 44*b2 + 26*b1 + 13) * q^33 + (2*b5 + 22*b4 + 11*b3 - 18*b2 - 62*b1 - 114) * q^34 + (14*b5 - 25*b4 - 13*b3 + 65*b2 - 63*b1 + 68) * q^36 + (-4*b5 - 28*b4 + 9*b3 + 11*b2 + 33*b1 + 8) * q^37 + (3*b5 - 7*b4 - 4*b3 + 2*b2 - 16*b1 + 6) * q^38 + (-6*b5 - 31*b4 - 11*b3 + 57*b2 + 39*b1 + 21) * q^39 + (-37*b5 + 15*b4 + 3*b3 + 7*b2 - 11*b1 - 86) * q^41 + (-2*b5 + 34*b4 + 16*b3 - 8*b2 + 74) * q^43 + (-b5 + 10*b4 - 48*b3 + 5*b2 + 122*b1 - 62) * q^44 + (35*b5 + 20*b4 + 30*b3 - 25*b2 + 42*b1 - 92) * q^46 + (-2*b5 - 9*b4 + 32*b3 - 14*b2 + 50*b1 + 81) * q^47 + (-47*b5 - 9*b4 - 6*b3 - 10*b2 + 132*b1 + 198) * q^48 + (-10*b5 + 47*b4 + 41*b3 - 59*b2 - 65*b1 + 145) * q^51 + (-93*b5 - 32*b4 - 48*b3 + 65*b2 + 196*b1 + 230) * q^52 + (-2*b5 + 60*b4 + 19*b3 - 35*b2 - 73*b1 + 146) * q^53 + (83*b5 + 45*b4 + 81*b3 - 120*b2 - 16*b1 - 428) * q^54 + (72*b5 - 48*b4 - 38*b3 + 86*b2 + 30*b1 - 266) * q^57 + (-25*b4 - 15*b3 + 35*b2 + 128*b1 + 296) * q^58 + (11*b5 - 59*b4 - 47*b3 + 33*b2 - 49*b1 - 162) * q^59 + (-24*b5 + 10*b4 + 4*b3 - 48*b2 - 132*b1 - 84) * q^61 + (-13*b5 + 28*b4 + 6*b3 - b2 - 154*b1 + 4) * q^62 + (-10*b5 - 91*b4 - 63*b3 + 67*b2 + 351*b1 + 116) * q^64 + (-75*b5 - 41*b4 - 23*b3 + 148*b2 + 664) * q^66 + (12*b5 - 12*b4 + 35*b3 - 31*b2 + 51*b1 - 330) * q^67 + (78*b5 + 67*b4 + 58*b3 - 15*b2 - 236*b1 - 420) * q^68 + (-48*b5 - 16*b4 - 53*b3 + 81*b2 - 77*b1 + 238) * q^69 + (-14*b5 - 92*b3 + 60*b2 - 116*b1 + 26) * q^71 + (78*b5 + 51*b4 + 67*b3 - 47*b2 - 27*b1 - 492) * q^72 + (-11*b5 - 85*b4 + 46*b3 + 60*b2 + 64*b1 + 414) * q^73 + (-21*b5 + 4*b4 - 36*b3 + 27*b2 + 132*b1 + 128) * q^74 + (-19*b5 + 43*b4 + 38*b3 - 106*b2 - 76*b1 + 234) * q^76 + (22*b5 - 19*b4 - 41*b3 - 37*b2 + 254*b1 - 112) * q^78 + (10*b5 - 55*b4 + 65*b3 + 25*b2 + 67*b1 + 237) * q^79 + (-112*b5 + 12*b4 + 70*b3 - 34*b2 - 170*b1 + 827) * q^81 + (46*b5 - b4 + 2*b3 - 113*b2 - 96*b1 - 314) * q^82 + (-69*b5 - 41*b4 - 154*b3 + 80*b2 - 32*b1 + 70) * q^83 + (50*b5 - 18*b4 + 56*b3 - 4*b2 + 2*b1 + 172) * q^86 + (-70*b5 + 43*b4 - 66*b3 - 68*b2 + 16*b1 + 333) * q^87 + (-22*b5 - 172*b4 - 52*b3 + 144*b2 + 300*b1 + 840) * q^88 + (33*b5 + 79*b4 + 80*b3 + 62*b2 - 74*b1 + 378) * q^89 + (-60*b5 + 32*b4 - 24*b3 + 86*b2 - 82*b1 + 412) * q^92 + (4*b5 + 168*b4 + 91*b3 - 271*b2 - 109*b1 + 266) * q^93 + (-23*b5 - 9*b4 - 11*b3 + 92*b2 + 164*b1 + 536) * q^94 + (-165*b5 - 57*b4 - 146*b3 + 148*b2 + 252*b1 + 1078) * q^96 + (-77*b5 - 84*b4 + 14*b3 + 72*b2 + 134*b1 + 63) * q^97 + (26*b5 + 244*b4 + 140*b3 - 308*b2 + 116*b1 - 536) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{2} + 16 q^{3} + 14 q^{4} - 24 q^{6} + 66 q^{8} + 70 q^{9}+O(q^{10})$$ 6 * q + 2 * q^2 + 16 * q^3 + 14 * q^4 - 24 * q^6 + 66 * q^8 + 70 * q^9 $$6 q + 2 q^{2} + 16 q^{3} + 14 q^{4} - 24 q^{6} + 66 q^{8} + 70 q^{9} - 16 q^{11} + 160 q^{12} + 168 q^{13} + 298 q^{16} - 4 q^{17} - 354 q^{18} - 308 q^{19} + 236 q^{22} + 336 q^{23} + 92 q^{24} - 56 q^{26} + 964 q^{27} + 176 q^{29} - 392 q^{31} + 770 q^{32} + 188 q^{33} - 812 q^{34} + 230 q^{36} + 140 q^{37} + 20 q^{38} + 140 q^{39} - 656 q^{41} + 388 q^{43} - 160 q^{44} - 388 q^{46} + 628 q^{47} + 1396 q^{48} + 744 q^{51} + 1520 q^{52} + 676 q^{53} - 2284 q^{54} - 1468 q^{57} + 2012 q^{58} - 996 q^{59} - 740 q^{61} - 364 q^{62} + 1426 q^{64} + 3620 q^{66} - 1768 q^{67} - 2940 q^{68} + 1048 q^{69} - 224 q^{71} - 2858 q^{72} + 2640 q^{73} + 928 q^{74} + 1340 q^{76} - 8 q^{78} + 1636 q^{79} + 4442 q^{81} - 1756 q^{82} + 140 q^{83} + 1180 q^{86} + 1940 q^{87} + 5652 q^{88} + 1904 q^{89} + 1952 q^{92} + 1592 q^{93} + 3332 q^{94} + 6460 q^{96} + 516 q^{97} - 2804 q^{99}+O(q^{100})$$ 6 * q + 2 * q^2 + 16 * q^3 + 14 * q^4 - 24 * q^6 + 66 * q^8 + 70 * q^9 - 16 * q^11 + 160 * q^12 + 168 * q^13 + 298 * q^16 - 4 * q^17 - 354 * q^18 - 308 * q^19 + 236 * q^22 + 336 * q^23 + 92 * q^24 - 56 * q^26 + 964 * q^27 + 176 * q^29 - 392 * q^31 + 770 * q^32 + 188 * q^33 - 812 * q^34 + 230 * q^36 + 140 * q^37 + 20 * q^38 + 140 * q^39 - 656 * q^41 + 388 * q^43 - 160 * q^44 - 388 * q^46 + 628 * q^47 + 1396 * q^48 + 744 * q^51 + 1520 * q^52 + 676 * q^53 - 2284 * q^54 - 1468 * q^57 + 2012 * q^58 - 996 * q^59 - 740 * q^61 - 364 * q^62 + 1426 * q^64 + 3620 * q^66 - 1768 * q^67 - 2940 * q^68 + 1048 * q^69 - 224 * q^71 - 2858 * q^72 + 2640 * q^73 + 928 * q^74 + 1340 * q^76 - 8 * q^78 + 1636 * q^79 + 4442 * q^81 - 1756 * q^82 + 140 * q^83 + 1180 * q^86 + 1940 * q^87 + 5652 * q^88 + 1904 * q^89 + 1952 * q^92 + 1592 * q^93 + 3332 * q^94 + 6460 * q^96 + 516 * q^97 - 2804 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 23x^{4} + 12x^{3} + 154x^{2} + 152x + 28$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{5} + 23\nu^{3} + 8\nu^{2} - 86\nu - 64 ) / 26$$ (-v^5 + 23*v^3 + 8*v^2 - 86*v - 64) / 26 $$\beta_{2}$$ $$=$$ $$( \nu^{5} - 23\nu^{3} + 18\nu^{2} + 60\nu - 144 ) / 26$$ (v^5 - 23*v^3 + 18*v^2 + 60*v - 144) / 26 $$\beta_{3}$$ $$=$$ $$( -5\nu^{5} + 26\nu^{4} + 89\nu^{3} - 298\nu^{2} - 638\nu - 164 ) / 26$$ (-5*v^5 + 26*v^4 + 89*v^3 - 298*v^2 - 638*v - 164) / 26 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} + 49\nu^{3} + 8\nu^{2} - 476\nu - 428 ) / 26$$ (-v^5 + 49*v^3 + 8*v^2 - 476*v - 428) / 26 $$\beta_{5}$$ $$=$$ $$( 9\nu^{5} - 26\nu^{4} - 155\nu^{3} + 240\nu^{2} + 800\nu + 264 ) / 26$$ (9*v^5 - 26*v^4 - 155*v^3 + 240*v^2 + 800*v + 264) / 26
 $$\nu$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + 6\beta_1 ) / 7$$ (b5 - b4 + b3 + b2 + 6*b1) / 7 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - \beta_{4} + \beta_{3} + 8\beta_{2} + 13\beta _1 + 56 ) / 7$$ (b5 - b4 + b3 + 8*b2 + 13*b1 + 56) / 7 $$\nu^{3}$$ $$=$$ $$( 15\beta_{5} - 8\beta_{4} + 15\beta_{3} + 15\beta_{2} + 83\beta _1 + 98 ) / 7$$ (15*b5 - 8*b4 + 15*b3 + 15*b2 + 83*b1 + 98) / 7 $$\nu^{4}$$ $$=$$ $$( 36\beta_{5} - 29\beta_{4} + 43\beta_{3} + 127\beta_{2} + 265\beta _1 + 784 ) / 7$$ (36*b5 - 29*b4 + 43*b3 + 127*b2 + 265*b1 + 784) / 7 $$\nu^{5}$$ $$=$$ $$( 267\beta_{5} - 106\beta_{4} + 267\beta_{3} + 323\beta_{2} + 1315\beta _1 + 2254 ) / 7$$ (267*b5 - 106*b4 + 267*b3 + 323*b2 + 1315*b1 + 2254) / 7

## 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
 −3.05323 −0.241849 −2.05886 −1.04490 4.29508 4.10376
−4.46745 9.80920 11.9581 0 −43.8221 0 −17.6824 69.2204 0
1.2 −1.65606 −0.332888 −5.25746 0 0.551283 0 21.9552 −26.8892 0
1.3 −0.644648 −4.18687 −7.58443 0 2.69906 0 10.0465 −9.47008 0
1.4 0.369315 9.74070 −7.86361 0 3.59738 0 −5.85867 67.8812 0
1.5 2.88087 −2.89052 0.299392 0 −8.32721 0 −22.1844 −18.6449 0
1.6 5.51797 3.86039 22.4480 0 21.3015 0 79.7239 −12.0974 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$7$$ $$1$$

## 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 1225.4.a.bj 6
5.b even 2 1 245.4.a.o 6
7.b odd 2 1 1225.4.a.bi 6
15.d odd 2 1 2205.4.a.bz 6
35.c odd 2 1 245.4.a.p yes 6
35.i odd 6 2 245.4.e.p 12
35.j even 6 2 245.4.e.q 12
105.g even 2 1 2205.4.a.ca 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.a.o 6 5.b even 2 1
245.4.a.p yes 6 35.c odd 2 1
245.4.e.p 12 35.i odd 6 2
245.4.e.q 12 35.j even 6 2
1225.4.a.bi 6 7.b odd 2 1
1225.4.a.bj 6 1.a even 1 1 trivial
2205.4.a.bz 6 15.d odd 2 1
2205.4.a.ca 6 105.g even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1225))$$:

 $$T_{2}^{6} - 2T_{2}^{5} - 29T_{2}^{4} + 28T_{2}^{3} + 134T_{2}^{2} + 24T_{2} - 28$$ T2^6 - 2*T2^5 - 29*T2^4 + 28*T2^3 + 134*T2^2 + 24*T2 - 28 $$T_{3}^{6} - 16T_{3}^{5} + 12T_{3}^{4} + 564T_{3}^{3} - 355T_{3}^{2} - 4644T_{3} - 1486$$ T3^6 - 16*T3^5 + 12*T3^4 + 564*T3^3 - 355*T3^2 - 4644*T3 - 1486 $$T_{19}^{6} + 308T_{19}^{5} + 32250T_{19}^{4} + 1514512T_{19}^{3} + 34989568T_{19}^{2} + 386280192T_{19} + 1617325344$$ T19^6 + 308*T19^5 + 32250*T19^4 + 1514512*T19^3 + 34989568*T19^2 + 386280192*T19 + 1617325344

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 2 T^{5} - 29 T^{4} + 28 T^{3} + \cdots - 28$$
$3$ $$T^{6} - 16 T^{5} + 12 T^{4} + \cdots - 1486$$
$5$ $$T^{6}$$
$7$ $$T^{6}$$
$11$ $$T^{6} + 16 T^{5} + \cdots - 9225436100$$
$13$ $$T^{6} - 168 T^{5} + \cdots + 12513937372$$
$17$ $$T^{6} + 4 T^{5} + \cdots - 3807091762$$
$19$ $$T^{6} + 308 T^{5} + \cdots + 1617325344$$
$23$ $$T^{6} - 336 T^{5} + \cdots + 298897696$$
$29$ $$T^{6} - 176 T^{5} + \cdots - 544215793700$$
$31$ $$T^{6} + 392 T^{5} + \cdots + 16539893268192$$
$37$ $$T^{6} - 140 T^{5} + \cdots + 271258136464$$
$41$ $$T^{6} + \cdots + 144691772208184$$
$43$ $$T^{6} - 388 T^{5} + \cdots + 440374360000$$
$47$ $$T^{6} + \cdots + 251564448569400$$
$53$ $$T^{6} + \cdots + 590408333736048$$
$59$ $$T^{6} + 996 T^{5} + \cdots + 14\!\cdots\!04$$
$61$ $$T^{6} + \cdots - 842000334839552$$
$67$ $$T^{6} + 1768 T^{5} + \cdots + 885207397312$$
$71$ $$T^{6} + 224 T^{5} + \cdots + 12\!\cdots\!88$$
$73$ $$T^{6} - 2640 T^{5} + \cdots - 85\!\cdots\!92$$
$79$ $$T^{6} - 1636 T^{5} + \cdots - 12\!\cdots\!68$$
$83$ $$T^{6} - 140 T^{5} + \cdots + 22\!\cdots\!00$$
$89$ $$T^{6} - 1904 T^{5} + \cdots + 45\!\cdots\!76$$
$97$ $$T^{6} + \cdots - 510868966648482$$