# Properties

 Label 825.6.a.s Level $825$ Weight $6$ Character orbit 825.a Self dual yes Analytic conductor $132.317$ Analytic rank $0$ Dimension $9$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$825 = 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 825.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$132.316651346$$ Analytic rank: $$0$$ Dimension: $$9$$ Coefficient field: $$\mathbb{Q}[x]/(x^{9} - \cdots)$$ Defining polynomial: $$x^{9} - x^{8} - 229 x^{7} + 267 x^{6} + 16434 x^{5} - 16568 x^{4} - 405504 x^{3} + 202288 x^{2} + 2184608 x + 1190400$$ x^9 - x^8 - 229*x^7 + 267*x^6 + 16434*x^5 - 16568*x^4 - 405504*x^3 + 202288*x^2 + 2184608*x + 1190400 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}\cdot 5^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{8}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - 9 q^{3} + (\beta_{2} + 19) q^{4} - 9 \beta_1 q^{6} + (\beta_{4} + 2 \beta_1 - 7) q^{7} + (\beta_{4} + \beta_{3} - 2 \beta_{2} + 21 \beta_1 - 22) q^{8} + 81 q^{9}+O(q^{10})$$ q + b1 * q^2 - 9 * q^3 + (b2 + 19) * q^4 - 9*b1 * q^6 + (b4 + 2*b1 - 7) * q^7 + (b4 + b3 - 2*b2 + 21*b1 - 22) * q^8 + 81 * q^9 $$q + \beta_1 q^{2} - 9 q^{3} + (\beta_{2} + 19) q^{4} - 9 \beta_1 q^{6} + (\beta_{4} + 2 \beta_1 - 7) q^{7} + (\beta_{4} + \beta_{3} - 2 \beta_{2} + 21 \beta_1 - 22) q^{8} + 81 q^{9} - 121 q^{11} + ( - 9 \beta_{2} - 171) q^{12} + ( - \beta_{7} - 2 \beta_{6} + \beta_{2} - 4 \beta_1 + 82) q^{13} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 5 \beta_{2} + 8 \beta_1 + 80) q^{14} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + 21 \beta_{2} - 62 \beta_1 + 467) q^{16} + ( - \beta_{8} + \beta_{7} - 2 \beta_{6} - 2 \beta_{3} - 5 \beta_{2} - 8 \beta_1 - 311) q^{17} + 81 \beta_1 q^{18} + (\beta_{8} - 2 \beta_{6} - \beta_{4} + 3 \beta_{3} + 14 \beta_{2} + 34 \beta_1 - 179) q^{19} + ( - 9 \beta_{4} - 18 \beta_1 + 63) q^{21} - 121 \beta_1 q^{22} + ( - \beta_{8} - 2 \beta_{7} - 3 \beta_{6} + 6 \beta_{4} + 4 \beta_{3} - 21 \beta_{2} + \cdots - 284) q^{23}+ \cdots - 9801 q^{99}+O(q^{100})$$ q + b1 * q^2 - 9 * q^3 + (b2 + 19) * q^4 - 9*b1 * q^6 + (b4 + 2*b1 - 7) * q^7 + (b4 + b3 - 2*b2 + 21*b1 - 22) * q^8 + 81 * q^9 - 121 * q^11 + (-9*b2 - 171) * q^12 + (-b7 - 2*b6 + b2 - 4*b1 + 82) * q^13 + (-2*b6 + b5 + b4 + b3 + 5*b2 + 8*b1 + 80) * q^14 + (2*b7 - 2*b6 + 2*b5 - 2*b3 + 21*b2 - 62*b1 + 467) * q^16 + (-b8 + b7 - 2*b6 - 2*b3 - 5*b2 - 8*b1 - 311) * q^17 + 81*b1 * q^18 + (b8 - 2*b6 - b4 + 3*b3 + 14*b2 + 34*b1 - 179) * q^19 + (-9*b4 - 18*b1 + 63) * q^21 - 121*b1 * q^22 + (-b8 - 2*b7 - 3*b6 + 6*b4 + 4*b3 - 21*b2 + 174*b1 - 284) * q^23 + (-9*b4 - 9*b3 + 18*b2 - 189*b1 + 198) * q^24 + (-b7 - 5*b6 - 5*b5 + 25*b4 - 4*b3 - 26*b2 + 134*b1 - 246) * q^26 - 729 * q^27 + (2*b8 + b7 + b6 - b5 + 22*b4 + 13*b3 + 235*b1 + 462) * q^28 + (-3*b8 + 3*b7 + 9*b6 + 6*b5 + 2*b4 + 20*b3 + 12*b2 - 10*b1 + 667) * q^29 + (-3*b7 + 18*b6 + 6*b5 + 7*b4 + b3 + 31*b2 - 202*b1 + 2410) * q^31 + (4*b8 - 6*b7 + 10*b6 - 2*b5 + 55*b4 + 21*b3 - 112*b2 + 521*b1 - 2908) * q^32 + 1089 * q^33 + (-12*b7 - 14*b6 - 15*b5 + 9*b4 + 5*b3 - 59*b2 - 466*b1 - 284) * q^34 + (81*b2 + 1539) * q^36 + (4*b8 + 2*b7 - 3*b6 + 10*b5 + 42*b4 + 7*b3 - 21*b2 + 366*b1 - 1192) * q^37 + (9*b7 + 3*b6 + 8*b5 + 42*b4 + 19*b3 + 79*b2 + 358*b1 + 1364) * q^38 + (9*b7 + 18*b6 - 9*b2 + 36*b1 - 738) * q^39 + (-3*b8 - 6*b7 + 9*b6 - 10*b5 - 37*b4 - 16*b3 + 71*b2 - 112*b1 + 1498) * q^41 + (18*b6 - 9*b5 - 9*b4 - 9*b3 - 45*b2 - 72*b1 - 720) * q^42 + (-2*b8 + 8*b7 + 17*b6 - 19*b4 + 161*b2 + 600*b1 - 1605) * q^43 + (-121*b2 - 2299) * q^44 + (2*b7 - 26*b6 - 8*b5 + 22*b4 - 36*b3 + 288*b2 - 841*b1 + 9110) * q^46 + (-6*b8 - 13*b7 - 28*b6 - 18*b5 + 34*b4 + 30*b3 - 113*b2 - 84*b1 - 1531) * q^47 + (-18*b7 + 18*b6 - 18*b5 + 18*b3 - 189*b2 + 558*b1 - 4203) * q^48 + (5*b8 - 5*b7 + 33*b6 - 8*b4 + 14*b3 + 8*b2 - 30*b1 - 2163) * q^49 + (9*b8 - 9*b7 + 18*b6 + 18*b3 + 45*b2 + 72*b1 + 2799) * q^51 + (-10*b8 + 15*b7 - 45*b6 + 5*b5 - 73*b4 - 16*b3 + 174*b2 - 558*b1 + 4228) * q^52 + (-6*b8 + 2*b7 - 20*b6 + 6*b5 - 43*b4 - 55*b3 + 298*b2 + 302*b1 - 4938) * q^53 - 729*b1 * q^54 + (-2*b8 + 35*b7 + 27*b6 + 25*b5 - 27*b4 - 10*b3 + 519*b2 + 724*b1 + 8699) * q^56 + (-9*b8 + 18*b6 + 9*b4 - 27*b3 - 126*b2 - 306*b1 + 1611) * q^57 + (12*b8 + 37*b7 + 53*b6 + 19*b5 + 67*b4 + 12*b3 + 418*b2 + 1182*b1 - 1040) * q^58 + (-4*b8 + 10*b7 - 69*b6 - 34*b5 - 76*b4 - 11*b3 + 137*b2 - 1026*b1 + 1462) * q^59 + (7*b8 - 26*b7 - 105*b6 - 24*b5 + 35*b4 + 15*b3 + 285*b2 + 984*b1 + 5435) * q^61 + (12*b8 + 29*b7 + 97*b6 + 47*b5 - 3*b4 - 2*b3 - 254*b2 + 3186*b1 - 10904) * q^62 + (81*b4 + 162*b1 - 567) * q^63 + (-4*b8 + 12*b7 + 64*b5 - 168*b4 - 76*b3 + 829*b2 - 3836*b1 + 12615) * q^64 + 1089*b1 * q^66 + (5*b8 + 15*b7 - 52*b6 + 22*b5 - 76*b4 - 135*b3 + 3*b2 + 520*b1 + 2886) * q^67 + (2*b8 - 39*b7 - 119*b6 - 41*b5 - 242*b4 - 131*b3 + 14*b2 - 1613*b1 - 12984) * q^68 + (9*b8 + 18*b7 + 27*b6 - 54*b4 - 36*b3 + 189*b2 - 1566*b1 + 2556) * q^69 + (-15*b8 - 15*b7 - 18*b6 - 10*b5 - 27*b4 + 152*b3 + 191*b2 + 2922*b1 + 8485) * q^71 + (81*b4 + 81*b3 - 162*b2 + 1701*b1 - 1782) * q^72 + (-12*b8 - 33*b7 - 33*b6 - 44*b5 + 116*b4 + 31*b3 + 306*b2 + 1262*b1 - 11273) * q^73 + (20*b8 + 39*b7 + 23*b6 + 95*b5 + 323*b4 + 98*b3 + 598*b2 - 1199*b1 + 18050) * q^74 + (-16*b8 + 40*b7 + 52*b6 + 84*b5 + 321*b4 + 113*b3 + 330*b2 + 3829*b1 + 21014) * q^76 + (-121*b4 - 242*b1 + 847) * q^77 + (9*b7 + 45*b6 + 45*b5 - 225*b4 + 36*b3 + 234*b2 - 1206*b1 + 2214) * q^78 + (-7*b8 - 93*b7 - 28*b6 - 68*b5 + 222*b4 - 64*b3 - 109*b2 - 1896*b1 - 815) * q^79 + 6561 * q^81 + (-20*b8 - 42*b7 - 4*b6 - 81*b5 - 331*b4 - 77*b3 - 653*b2 + 3016*b1 - 6062) * q^82 + (3*b8 - 57*b7 - 9*b6 + 50*b5 - 171*b4 - 35*b3 + 502*b2 - 1628*b1 - 17233) * q^83 + (-18*b8 - 9*b7 - 9*b6 + 9*b5 - 198*b4 - 117*b3 - 2115*b1 - 4158) * q^84 + (-b7 + 67*b6 + 3*b5 - 69*b4 + 170*b3 + 364*b2 + 3414*b1 + 27660) * q^86 + (27*b8 - 27*b7 - 81*b6 - 54*b5 - 18*b4 - 180*b3 - 108*b2 + 90*b1 - 6003) * q^87 + (-121*b4 - 121*b3 + 242*b2 - 2541*b1 + 2662) * q^88 + (5*b8 + 8*b7 + 48*b6 - 36*b5 + 384*b4 + 65*b3 - 640*b2 + 3016*b1 - 7784) * q^89 + (15*b8 + 49*b7 + 86*b6 + 40*b5 - 251*b4 - 282*b3 - 173*b2 + 7538*b1 - 12778) * q^91 + (16*b8 - 44*b7 - 100*b6 - 72*b5 + 164*b4 + 256*b3 - 1561*b2 + 13728*b1 - 40327) * q^92 + (27*b7 - 162*b6 - 54*b5 - 63*b4 - 9*b3 - 279*b2 + 1818*b1 - 21690) * q^93 + (-36*b8 - 3*b7 - 343*b6 - 83*b5 - 189*b4 - 272*b3 + 946*b2 - 4039*b1 - 3406) * q^94 + (-36*b8 + 54*b7 - 90*b6 + 18*b5 - 495*b4 - 189*b3 + 1008*b2 - 4689*b1 + 26172) * q^96 + (30*b8 - 119*b7 + 115*b6 - 38*b5 - 323*b4 + 108*b3 + 92*b2 + 4708*b1 + 12630) * q^97 + (91*b7 + 155*b6 + 117*b5 - 319*b4 - 100*b3 + 450*b2 - 2402*b1 - 1396) * q^98 - 9801 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9}+O(q^{10})$$ 9 * q + q^2 - 81 * q^3 + 171 * q^4 - 9 * q^6 - 57 * q^7 - 177 * q^8 + 729 * q^9 $$9 q + q^{2} - 81 q^{3} + 171 q^{4} - 9 q^{6} - 57 q^{7} - 177 q^{8} + 729 q^{9} - 1089 q^{11} - 1539 q^{12} + 723 q^{13} + 720 q^{14} + 4147 q^{16} - 2804 q^{17} + 81 q^{18} - 1601 q^{19} + 513 q^{21} - 121 q^{22} - 2392 q^{23} + 1593 q^{24} - 1987 q^{26} - 6561 q^{27} + 4436 q^{28} + 5966 q^{29} + 21575 q^{31} - 25493 q^{32} + 9801 q^{33} - 3098 q^{34} + 13851 q^{36} - 10228 q^{37} + 12765 q^{38} - 6507 q^{39} + 13304 q^{41} - 6480 q^{42} - 13829 q^{43} - 20691 q^{44} + 81283 q^{46} - 13998 q^{47} - 37323 q^{48} - 19468 q^{49} + 25236 q^{51} + 37131 q^{52} - 44166 q^{53} - 729 q^{54} + 79160 q^{56} + 14409 q^{57} - 7635 q^{58} + 11626 q^{59} + 49481 q^{61} - 94479 q^{62} - 4617 q^{63} + 109367 q^{64} + 1089 q^{66} + 26567 q^{67} - 119506 q^{68} + 21528 q^{69} + 78454 q^{71} - 14337 q^{72} - 100086 q^{73} + 162360 q^{74} + 194115 q^{76} + 6897 q^{77} + 17883 q^{78} - 8478 q^{79} + 59049 q^{81} - 52700 q^{82} - 157476 q^{83} - 39924 q^{84} + 251663 q^{86} - 53694 q^{87} + 21417 q^{88} - 65548 q^{89} - 106849 q^{91} - 350115 q^{92} - 194175 q^{93} - 35742 q^{94} + 229437 q^{96} + 116757 q^{97} - 14949 q^{98} - 88209 q^{99}+O(q^{100})$$ 9 * q + q^2 - 81 * q^3 + 171 * q^4 - 9 * q^6 - 57 * q^7 - 177 * q^8 + 729 * q^9 - 1089 * q^11 - 1539 * q^12 + 723 * q^13 + 720 * q^14 + 4147 * q^16 - 2804 * q^17 + 81 * q^18 - 1601 * q^19 + 513 * q^21 - 121 * q^22 - 2392 * q^23 + 1593 * q^24 - 1987 * q^26 - 6561 * q^27 + 4436 * q^28 + 5966 * q^29 + 21575 * q^31 - 25493 * q^32 + 9801 * q^33 - 3098 * q^34 + 13851 * q^36 - 10228 * q^37 + 12765 * q^38 - 6507 * q^39 + 13304 * q^41 - 6480 * q^42 - 13829 * q^43 - 20691 * q^44 + 81283 * q^46 - 13998 * q^47 - 37323 * q^48 - 19468 * q^49 + 25236 * q^51 + 37131 * q^52 - 44166 * q^53 - 729 * q^54 + 79160 * q^56 + 14409 * q^57 - 7635 * q^58 + 11626 * q^59 + 49481 * q^61 - 94479 * q^62 - 4617 * q^63 + 109367 * q^64 + 1089 * q^66 + 26567 * q^67 - 119506 * q^68 + 21528 * q^69 + 78454 * q^71 - 14337 * q^72 - 100086 * q^73 + 162360 * q^74 + 194115 * q^76 + 6897 * q^77 + 17883 * q^78 - 8478 * q^79 + 59049 * q^81 - 52700 * q^82 - 157476 * q^83 - 39924 * q^84 + 251663 * q^86 - 53694 * q^87 + 21417 * q^88 - 65548 * q^89 - 106849 * q^91 - 350115 * q^92 - 194175 * q^93 - 35742 * q^94 + 229437 * q^96 + 116757 * q^97 - 14949 * q^98 - 88209 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{9} - x^{8} - 229 x^{7} + 267 x^{6} + 16434 x^{5} - 16568 x^{4} - 405504 x^{3} + 202288 x^{2} + 2184608 x + 1190400$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 51$$ v^2 - 51 $$\beta_{3}$$ $$=$$ $$( \nu^{8} - \nu^{7} - 213\nu^{6} + 243\nu^{5} + 13338\nu^{4} - 8432\nu^{3} - 242664\nu^{2} - 234544\nu + 288000 ) / 4800$$ (v^8 - v^7 - 213*v^6 + 243*v^5 + 13338*v^4 - 8432*v^3 - 242664*v^2 - 234544*v + 288000) / 4800 $$\beta_{4}$$ $$=$$ $$( - \nu^{8} + \nu^{7} + 213 \nu^{6} - 243 \nu^{5} - 13338 \nu^{4} + 13232 \nu^{3} + 252264 \nu^{2} - 173456 \nu - 672000 ) / 4800$$ (-v^8 + v^7 + 213*v^6 - 243*v^5 - 13338*v^4 + 13232*v^3 + 252264*v^2 - 173456*v - 672000) / 4800 $$\beta_{5}$$ $$=$$ $$( - \nu^{8} - \nu^{7} + 291 \nu^{6} + 219 \nu^{5} - 25980 \nu^{4} - 12448 \nu^{3} + 693568 \nu^{2} + 210320 \nu - 1473600 ) / 4800$$ (-v^8 - v^7 + 291*v^6 + 219*v^5 - 25980*v^4 - 12448*v^3 + 693568*v^2 + 210320*v - 1473600) / 4800 $$\beta_{6}$$ $$=$$ $$( - \nu^{8} + 15 \nu^{7} + 267 \nu^{6} - 2877 \nu^{5} - 22644 \nu^{4} + 145592 \nu^{3} + 688736 \nu^{2} - 1638288 \nu - 3888000 ) / 9600$$ (-v^8 + 15*v^7 + 267*v^6 - 2877*v^5 - 22644*v^4 + 145592*v^3 + 688736*v^2 - 1638288*v - 3888000) / 9600 $$\beta_{7}$$ $$=$$ $$( \nu^{8} + 5 \nu^{7} - 247 \nu^{6} - 943 \nu^{5} + 20264 \nu^{4} + 51208 \nu^{3} - 581776 \nu^{2} - 743472 \nu + 2483200 ) / 3200$$ (v^8 + 5*v^7 - 247*v^6 - 943*v^5 + 20264*v^4 + 51208*v^3 - 581776*v^2 - 743472*v + 2483200) / 3200 $$\beta_{8}$$ $$=$$ $$( 23 \nu^{8} - 33 \nu^{7} - 5109 \nu^{6} + 9699 \nu^{5} + 348564 \nu^{4} - 728536 \nu^{3} - 7766752 \nu^{2} + 14315568 \nu + 28147200 ) / 9600$$ (23*v^8 - 33*v^7 - 5109*v^6 + 9699*v^5 + 348564*v^4 - 728536*v^3 - 7766752*v^2 + 14315568*v + 28147200) / 9600
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 51$$ b2 + 51 $$\nu^{3}$$ $$=$$ $$\beta_{4} + \beta_{3} - 2\beta_{2} + 85\beta _1 - 22$$ b4 + b3 - 2*b2 + 85*b1 - 22 $$\nu^{4}$$ $$=$$ $$2\beta_{7} - 2\beta_{6} + 2\beta_{5} - 2\beta_{3} + 117\beta_{2} - 62\beta _1 + 4339$$ 2*b7 - 2*b6 + 2*b5 - 2*b3 + 117*b2 - 62*b1 + 4339 $$\nu^{5}$$ $$=$$ $$4\beta_{8} - 6\beta_{7} + 10\beta_{6} - 2\beta_{5} + 183\beta_{4} + 149\beta_{3} - 368\beta_{2} + 8329\beta _1 - 5724$$ 4*b8 - 6*b7 + 10*b6 - 2*b5 + 183*b4 + 149*b3 - 368*b2 + 8329*b1 - 5724 $$\nu^{6}$$ $$=$$ $$- 4 \beta_{8} + 332 \beta_{7} - 320 \beta_{6} + 384 \beta_{5} - 168 \beta_{4} - 396 \beta_{3} + 13405 \beta_{2} - 13756 \beta _1 + 426279$$ -4*b8 + 332*b7 - 320*b6 + 384*b5 - 168*b4 - 396*b3 + 13405*b2 - 13756*b1 + 426279 $$\nu^{7}$$ $$=$$ $$768 \beta_{8} - 1080 \beta_{7} + 2472 \beta_{6} - 528 \beta_{5} + 25281 \beta_{4} + 18777 \beta_{3} - 55438 \beta_{2} + 879905 \beta _1 - 989250$$ 768*b8 - 1080*b7 + 2472*b6 - 528*b5 + 25281*b4 + 18777*b3 - 55438*b2 + 879905*b1 - 989250 $$\nu^{8}$$ $$=$$ $$- 1056 \beta_{8} + 44418 \beta_{7} - 41442 \beta_{6} + 55074 \beta_{5} - 46540 \beta_{4} - 61870 \beta_{3} + 1554505 \beta_{2} - 2295850 \beta _1 + 45227887$$ -1056*b8 + 44418*b7 - 41442*b6 + 55074*b5 - 46540*b4 - 61870*b3 + 1554505*b2 - 2295850*b1 + 45227887

## 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
 −11.1064 −8.18565 −5.77674 −1.92452 −0.624305 3.27244 6.71211 8.36252 10.2706
−11.1064 −9.00000 91.3531 0 99.9580 −76.1461 −659.202 81.0000 0
1.2 −8.18565 −9.00000 35.0048 0 73.6708 163.661 −24.5962 81.0000 0
1.3 −5.77674 −9.00000 1.37069 0 51.9906 −150.201 176.937 81.0000 0
1.4 −1.92452 −9.00000 −28.2962 0 17.3207 59.1120 116.041 81.0000 0
1.5 −0.624305 −9.00000 −31.6102 0 5.61875 −106.290 39.7122 81.0000 0
1.6 3.27244 −9.00000 −21.2911 0 −29.4519 115.628 −174.392 81.0000 0
1.7 6.71211 −9.00000 13.0525 0 −60.4090 −177.172 −127.178 81.0000 0
1.8 8.36252 −9.00000 37.9318 0 −75.2627 −15.4419 49.6048 81.0000 0
1.9 10.2706 −9.00000 73.4848 0 −92.4352 129.851 426.072 81.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$-1$$
$$11$$ $$1$$

## 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 825.6.a.s yes 9
5.b even 2 1 825.6.a.r 9

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
825.6.a.r 9 5.b even 2 1
825.6.a.s yes 9 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{9} - T_{2}^{8} - 229 T_{2}^{7} + 267 T_{2}^{6} + 16434 T_{2}^{5} - 16568 T_{2}^{4} - 405504 T_{2}^{3} + 202288 T_{2}^{2} + 2184608 T_{2} + 1190400$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(825))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{9} - T^{8} - 229 T^{7} + \cdots + 1190400$$
$3$ $$(T + 9)^{9}$$
$5$ $$T^{9}$$
$7$ $$T^{9} + 57 T^{8} + \cdots + 48\!\cdots\!44$$
$11$ $$(T + 121)^{9}$$
$13$ $$T^{9} - 723 T^{8} + \cdots + 71\!\cdots\!64$$
$17$ $$T^{9} + 2804 T^{8} + \cdots - 16\!\cdots\!00$$
$19$ $$T^{9} + 1601 T^{8} + \cdots - 83\!\cdots\!77$$
$23$ $$T^{9} + 2392 T^{8} + \cdots - 55\!\cdots\!82$$
$29$ $$T^{9} - 5966 T^{8} + \cdots + 27\!\cdots\!52$$
$31$ $$T^{9} - 21575 T^{8} + \cdots - 53\!\cdots\!00$$
$37$ $$T^{9} + 10228 T^{8} + \cdots - 14\!\cdots\!88$$
$41$ $$T^{9} - 13304 T^{8} + \cdots + 28\!\cdots\!52$$
$43$ $$T^{9} + 13829 T^{8} + \cdots + 12\!\cdots\!88$$
$47$ $$T^{9} + 13998 T^{8} + \cdots + 44\!\cdots\!60$$
$53$ $$T^{9} + 44166 T^{8} + \cdots - 24\!\cdots\!28$$
$59$ $$T^{9} - 11626 T^{8} + \cdots + 17\!\cdots\!20$$
$61$ $$T^{9} - 49481 T^{8} + \cdots + 49\!\cdots\!20$$
$67$ $$T^{9} - 26567 T^{8} + \cdots - 13\!\cdots\!20$$
$71$ $$T^{9} - 78454 T^{8} + \cdots - 52\!\cdots\!72$$
$73$ $$T^{9} + 100086 T^{8} + \cdots - 37\!\cdots\!92$$
$79$ $$T^{9} + 8478 T^{8} + \cdots + 26\!\cdots\!44$$
$83$ $$T^{9} + 157476 T^{8} + \cdots + 62\!\cdots\!00$$
$89$ $$T^{9} + 65548 T^{8} + \cdots + 30\!\cdots\!56$$
$97$ $$T^{9} - 116757 T^{8} + \cdots - 14\!\cdots\!25$$