# Properties

 Label 75.9.c.e Level $75$ Weight $9$ Character orbit 75.c Analytic conductor $30.553$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 75.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.5533957546$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ Defining polynomial: $$x^{10} + 1634x^{8} + 776307x^{6} + 148116566x^{4} + 10575941812x^{2} + 105274575720$$ x^10 + 1634*x^8 + 776307*x^6 + 148116566*x^4 + 10575941812*x^2 + 105274575720 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{8}\cdot 3^{11}\cdot 5^{5}$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + (\beta_{4} - \beta_{2} - 2) q^{3} + ( - \beta_1 - 155) q^{4} + (\beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_1 + 224) q^{6} + ( - \beta_{8} - \beta_{6} + 12 \beta_{4} - 5 \beta_{2} - 5 \beta_1 + 204) q^{7} + (\beta_{9} + \beta_{7} + 11 \beta_{4} + \beta_{3} - 113 \beta_{2} + 5) q^{8} + ( - \beta_{8} - \beta_{7} - 3 \beta_{6} + \beta_{5} + 2 \beta_{3} + 55 \beta_{2} + \cdots - 1119) q^{9}+O(q^{10})$$ q + b2 * q^2 + (b4 - b2 - 2) * q^3 + (-b1 - 155) * q^4 + (b6 + b5 - b4 + 2*b1 + 224) * q^6 + (-b8 - b6 + 12*b4 - 5*b2 - 5*b1 + 204) * q^7 + (b9 + b7 + 11*b4 + b3 - 113*b2 + 5) * q^8 + (-b8 - b7 - 3*b6 + b5 + 2*b3 + 55*b2 - 3*b1 - 1119) * q^9 $$q + \beta_{2} q^{2} + (\beta_{4} - \beta_{2} - 2) q^{3} + ( - \beta_1 - 155) q^{4} + (\beta_{6} + \beta_{5} - \beta_{4} + 2 \beta_1 + 224) q^{6} + ( - \beta_{8} - \beta_{6} + 12 \beta_{4} - 5 \beta_{2} - 5 \beta_1 + 204) q^{7} + (\beta_{9} + \beta_{7} + 11 \beta_{4} + \beta_{3} - 113 \beta_{2} + 5) q^{8} + ( - \beta_{8} - \beta_{7} - 3 \beta_{6} + \beta_{5} + 2 \beta_{3} + 55 \beta_{2} + \cdots - 1119) q^{9}+ \cdots + ( - 20700 \beta_{9} + 43198 \beta_{8} - 4208 \beta_{7} + \cdots - 38563731) q^{99}+O(q^{100})$$ q + b2 * q^2 + (b4 - b2 - 2) * q^3 + (-b1 - 155) * q^4 + (b6 + b5 - b4 + 2*b1 + 224) * q^6 + (-b8 - b6 + 12*b4 - 5*b2 - 5*b1 + 204) * q^7 + (b9 + b7 + 11*b4 + b3 - 113*b2 + 5) * q^8 + (-b8 - b7 - 3*b6 + b5 + 2*b3 + 55*b2 - 3*b1 - 1119) * q^9 + (5*b9 + 2*b8 - b7 + 3*b5 + 62*b4 + b3 - 227*b2 + 3*b1 + 27) * q^11 + (-6*b9 + b8 - b7 - 2*b6 + 3*b5 - 145*b4 - 8*b3 + 453*b2 + 20*b1 - 671) * q^12 + (-9*b6 + 14*b5 - 133*b4 + 9*b3 + 55*b2 + 55*b1 - 1782) * q^13 + (5*b9 - 5*b8 + 11*b7 + 9*b6 - 3*b5 + 24*b4 + 15*b3 - 821*b2 - 3*b1 + 11) * q^14 + (29*b8 + 62*b6 + 34*b5 - 287*b4 - 33*b3 + 114*b2 + 157*b1 + 4237) * q^16 + (11*b9 + 3*b8 + 11*b7 - 45*b6 - 12*b5 - 119*b4 - 31*b3 - 1146*b2 - 65) * q^17 + (9*b9 - 23*b8 + 19*b7 - 48*b6 + 20*b5 + 189*b4 + 19*b3 - 1882*b2 - 225*b1 - 22311) * q^18 + (-18*b8 - 69*b6 - 83*b5 + 274*b4 + 51*b3 - 103*b2 + 7*b1 - 14219) * q^19 + (-15*b9 - 47*b8 - 19*b7 - 47*b6 + 30*b5 + 315*b4 - 101*b3 + 872*b2 + 56*b1 + 67460) * q^21 + (-38*b8 + 79*b6 + 253*b5 + 10*b4 - 117*b3 - 35*b2 + 600*b1 + 81553) * q^22 + (-43*b9 + 29*b8 - 13*b7 - 90*b6 - 6*b5 + 2*b4 - 74*b3 + 3399*b2 - 15*b1 + 30) * q^23 + (-45*b9 - 117*b8 - 18*b7 - 30*b6 - 195*b5 + 396*b4 + 9*b3 + 3126*b2 - 1023*b1 - 101001) * q^24 + (-70*b9 - 34*b8 + 14*b7 - 207*b6 - 117*b5 - 2566*b4 - 227*b3 + 9652*b2 - 42*b1 - 1227) * q^26 + (63*b9 + 73*b8 - 17*b7 + 123*b6 + 215*b5 - 1206*b4 - 317*b3 + 2714*b2 + 360*b1 - 26991) * q^27 + (189*b8 + 294*b6 + 210*b5 - 2583*b4 - 105*b3 + 1050*b2 + 2450*b1 + 378644) * q^28 + (5*b9 - 179*b8 - 61*b7 + 198*b6 - 102*b5 - 4446*b4 - 42*b3 + 13887*b2 + 33*b1 - 2242) * q^29 + (-149*b8 - 47*b6 + 236*b5 + 1322*b4 - 102*b3 - 579*b2 - 1371*b1 - 300298) * q^31 + (-102*b9 - 157*b8 - 72*b7 - 270*b6 - 252*b5 - 7547*b4 - 499*b3 + 7554*b2 - 15*b1 - 3715) * q^32 + (-63*b9 + 432*b8 - 198*b7 + 373*b6 + 349*b5 - 28*b4 - 234*b3 - 2658*b2 - 3340*b1 - 466504) * q^33 + (-314*b8 + 313*b6 - 829*b5 + 12302*b4 - 627*b3 - 5109*b2 + 2268*b1 + 503019) * q^34 + (180*b9 - 50*b8 + 217*b7 + 36*b6 - 43*b5 - 3330*b4 + 145*b3 - 54427*b2 + 5109*b1 + 446244) * q^36 + (-122*b8 + 532*b6 - 794*b5 + 10012*b4 - 654*b3 - 4160*b2 + 2860*b1 + 305580) * q^37 + (395*b9 + 529*b8 + 5*b7 + 450*b6 + 744*b5 + 20070*b4 + 984*b3 - 11380*b2 + 195*b1 + 9740) * q^38 + (465*b9 - 370*b8 + 217*b7 + 461*b6 - 228*b5 - 3380*b4 - 820*b3 - 46302*b2 - 992*b1 - 753076) * q^39 + (485*b9 - 357*b8 - 19*b7 + 945*b6 + 42*b5 - 2897*b4 + 569*b3 + 73222*b2 + 252*b1 - 1817) * q^41 + (-306*b9 - 1764*b8 + 234*b7 - 945*b6 - 2139*b5 + 5664*b4 + 405*b3 + 73554*b2 - 8730*b1 - 393105) * q^42 + (389*b8 - 781*b6 + 510*b5 - 15408*b4 + 1170*b3 + 6475*b2 + 5535*b1 + 1165014) * q^43 + (-430*b9 - 1411*b8 - 220*b7 - 882*b6 - 1740*b5 - 51673*b4 - 2513*b3 + 151394*b2 - 105*b1 - 25569) * q^44 + (-1466*b8 - 1358*b6 - 2666*b5 + 31678*b4 - 108*b3 - 12986*b2 - 12908*b1 - 1372094) * q^46 + (272*b9 - 1372*b8 - 148*b7 + 2250*b6 - 552*b5 - 25378*b4 + 730*b3 - 74702*b2 + 210*b1 - 12930) * q^47 + (12*b9 + 970*b8 + 737*b7 - 668*b6 + 93*b5 - 3622*b4 + 3805*b3 - 190923*b2 - 6100*b1 - 1473503) * q^48 + (-3094*b8 - 847*b6 + 2296*b5 + 41377*b4 - 2247*b3 - 17619*b2 + 7833*b1 + 911547) * q^49 + (-900*b9 - 153*b8 - 954*b7 + 1981*b6 - 3290*b5 - 4003*b4 - 954*b3 - 181614*b2 - 1123*b1 + 620600) * q^51 + (-2459*b8 - 5642*b6 - 6462*b5 + 39537*b4 + 3183*b3 - 15670*b2 - 12080*b1 - 3864338) * q^52 + (-1419*b9 + 1179*b8 + 471*b7 - 900*b6 + 564*b5 + 29856*b4 + 750*b3 + 205525*b2 - 945*b1 + 16110) * q^53 + (-1575*b9 - 4063*b8 - 553*b7 - 1824*b6 - 3362*b5 - 16020*b4 - 4744*b3 + 18430*b2 - 6723*b1 - 893214) * q^54 + (-2115*b9 - 1805*b8 - 453*b7 - 342*b6 - 2196*b5 - 65242*b4 - 2600*b3 + 682583*b2 - 831*b1 - 31148) * q^56 + (1077*b9 + 194*b8 - 788*b7 + 2798*b6 - 1569*b5 - 3221*b4 - 877*b3 + 335133*b2 + 6985*b1 + 1471202) * q^57 + (166*b8 - 5678*b6 - 2826*b5 - 28770*b4 + 5844*b3 + 12710*b2 - 27220*b1 - 4841686) * q^58 + (2160*b9 - 2678*b8 + 2508*b7 - 2214*b6 - 3474*b5 - 69742*b4 - 2384*b3 + 112314*b2 - 174*b1 - 35864) * q^59 + (1544*b8 + 2297*b6 - 4506*b5 + 9273*b4 - 753*b3 - 3551*b2 + 21081*b1 + 149352) * q^61 + (357*b9 - 2333*b8 + 2667*b7 + 135*b6 - 2673*b5 - 44668*b4 + 469*b3 - 589099*b2 - 1155*b1 - 21935) * q^62 + (828*b9 + 411*b8 - 942*b7 - 2361*b6 + 3762*b5 + 50670*b4 - 2640*b3 - 546747*b2 + 15375*b1 + 7492404) * q^63 + (923*b8 + 4034*b6 - 10482*b5 + 63111*b4 - 3111*b3 - 25682*b2 - 15955*b1 - 413551) * q^64 + (2415*b9 - 6311*b8 + 1877*b7 - 8684*b6 + 3504*b5 + 48865*b4 - 713*b3 - 1230833*b2 - 34651*b1 + 1082283) * q^66 + (-1498*b8 + 10745*b6 + 7897*b5 + 64192*b4 - 12243*b3 - 28425*b2 + 685*b1 - 1601149) * q^67 + (1154*b9 + 835*b8 - 1936*b7 + 4410*b6 + 2820*b5 + 42789*b4 + 3309*b3 + 730338*b2 + 1545*b1 + 20045) * q^68 + (315*b9 + 1323*b8 - 1089*b7 + 2112*b6 + 1854*b5 - 25194*b4 + 684*b3 - 425973*b2 + 55677*b1 + 1104192) * q^69 + (5385*b9 + 1939*b8 + 2403*b7 + 180*b6 + 2496*b5 + 92384*b4 + 4522*b3 + 851161*b2 + 1491*b1 + 42754) * q^71 + (-2943*b9 + 240*b8 - 318*b7 - 4824*b6 + 4629*b5 - 66393*b4 - 5664*b3 + 1076376*b2 + 51300*b1 + 17168922) * q^72 + (12628*b8 + 31267*b6 + 20866*b5 - 125393*b4 - 18639*b3 + 48955*b2 + 32675*b1 - 5296635) * q^73 + (-2440*b9 + 744*b8 - 6472*b7 + 14130*b6 + 6126*b5 + 63484*b4 + 8402*b3 + 944176*b2 + 2016*b1 + 31954) * q^74 + (2998*b8 + 7084*b6 + 30628*b5 - 160514*b4 - 4086*b3 + 63988*b2 + 107653*b1 - 3086637) * q^76 + (-3721*b9 + 6171*b8 - 9211*b7 - 18000*b6 + 1086*b5 - 36296*b4 - 21040*b3 - 322501*b2 + 2745*b1 - 17660) * q^77 + (-981*b9 - 297*b8 - 171*b7 + 12169*b6 - 5531*b5 - 43216*b4 - 2637*b3 - 985791*b2 + 105275*b1 + 19757273) * q^78 + (1892*b8 - 6484*b6 + 3242*b5 - 97546*b4 + 8376*b3 + 41072*b2 - 98948*b1 + 847336) * q^79 + (-3600*b9 + 10013*b8 - 1489*b7 + 8220*b6 + 9427*b5 - 91917*b4 + 1529*b3 - 1709768*b2 - 32508*b1 + 2228094) * q^81 + (10138*b8 + 5941*b6 + 26087*b5 - 281470*b4 + 4197*b3 + 115455*b2 - 6220*b1 - 29716233) * q^82 + (6708*b9 + 537*b8 + 408*b7 - 18630*b6 - 4623*b5 - 102372*b4 - 17685*b3 + 946122*b2 + 3150*b1 - 56115) * q^83 + (10290*b9 + 4333*b8 + 6566*b7 + 4438*b6 - 6090*b5 + 318255*b4 + 29029*b3 - 1833748*b2 - 58744*b1 - 13504540) * q^84 + (-3555*b9 + 4075*b8 - 2829*b7 - 18621*b6 - 2253*b5 - 46996*b4 - 17375*b3 + 2307969*b2 - 363*b1 - 21539) * q^86 + (3339*b9 + 10683*b8 + 7119*b7 + 13544*b6 + 16454*b5 + 13774*b4 - 8532*b3 + 485979*b2 - 9155*b1 + 31890688) * q^87 + (-33799*b8 - 43378*b6 - 44926*b5 + 563165*b4 + 9579*b3 - 230110*b2 - 219625*b1 - 30681541) * q^88 + (-8700*b9 + 16758*b8 + 1002*b7 - 14031*b6 + 10464*b5 + 389847*b4 + 3729*b3 + 626361*b2 - 4851*b1 + 201699) * q^89 + (-6429*b8 - 19197*b6 - 25074*b5 + 113142*b4 + 12768*b3 - 43989*b2 - 117377*b1 - 10607552) * q^91 + (9574*b9 + 12992*b8 + 10054*b7 + 15120*b6 + 17952*b5 + 610194*b4 + 38166*b3 - 3156790*b2 - 240*b1 + 300430) * q^92 + (-7521*b9 - 9911*b8 - 1741*b7 - 15785*b6 + 10614*b5 - 304299*b4 - 23537*b3 - 233674*b2 + 1880*b1 + 7811816) * q^93 + (21280*b8 - 17630*b6 + 8470*b5 - 570080*b4 + 38910*b3 + 240070*b2 + 99144*b1 + 35296294) * q^94 + (1350*b9 + 18108*b8 + 2295*b7 + 17552*b6 - 8179*b5 - 70382*b4 - 14751*b3 - 1768437*b2 + 149368*b1 + 52445887) * q^96 + (12206*b8 - 12727*b6 + 1828*b5 - 330143*b4 + 24933*b3 + 139485*b2 + 158945*b1 - 40438498) * q^97 + (-21462*b9 - 35938*b8 + 10878*b7 + 27405*b6 - 32193*b5 - 827582*b4 + 2345*b3 + 2724057*b2 - 16170*b1 - 394975) * q^98 + (-20700*b9 + 43198*b8 - 4208*b7 - 14151*b6 + 42029*b5 - 538083*b4 - 9311*b3 + 178466*b2 + 137529*b1 - 38563731) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 25 q^{3} - 1554 q^{4} + 2257 q^{6} + 1960 q^{7} - 11207 q^{9}+O(q^{10})$$ 10 * q - 25 * q^3 - 1554 * q^4 + 2257 * q^6 + 1960 * q^7 - 11207 * q^9 $$10 q - 25 q^{3} - 1554 q^{4} + 2257 q^{6} + 1960 q^{7} - 11207 q^{9} - 5915 q^{12} - 16920 q^{13} + 44634 q^{16} - 224875 q^{18} - 143934 q^{19} + 673428 q^{21} + 818990 q^{22} - 1016859 q^{24} - 260830 q^{27} + 3810100 q^{28} - 3014060 q^{31} - 4677515 q^{33} + 4977146 q^{34} + 4500527 q^{36} + 3016760 q^{37} - 7513282 q^{39} - 4001760 q^{42} + 11747340 q^{43} - 13938636 q^{46} - 14748755 q^{48} + 8953546 q^{49} + 6209287 q^{51} - 38918320 q^{52} - 8886272 q^{54} + 14759525 q^{57} - 48407900 q^{58} + 1520220 q^{61} + 74748240 q^{63} - 4536998 q^{64} + 10465295 q^{66} - 16269290 q^{67} + 11394978 q^{69} + 172231185 q^{72} - 52090170 q^{73} - 29529046 q^{76} + 198205810 q^{78} + 8549896 q^{79} + 22612945 q^{81} - 295714190 q^{82} - 136883292 q^{84} + 318901610 q^{87} - 310673250 q^{88} - 107224264 q^{91} + 79679130 q^{93} + 356118596 q^{94} + 525424001 q^{96} - 402167800 q^{97} - 382421335 q^{99}+O(q^{100})$$ 10 * q - 25 * q^3 - 1554 * q^4 + 2257 * q^6 + 1960 * q^7 - 11207 * q^9 - 5915 * q^12 - 16920 * q^13 + 44634 * q^16 - 224875 * q^18 - 143934 * q^19 + 673428 * q^21 + 818990 * q^22 - 1016859 * q^24 - 260830 * q^27 + 3810100 * q^28 - 3014060 * q^31 - 4677515 * q^33 + 4977146 * q^34 + 4500527 * q^36 + 3016760 * q^37 - 7513282 * q^39 - 4001760 * q^42 + 11747340 * q^43 - 13938636 * q^46 - 14748755 * q^48 + 8953546 * q^49 + 6209287 * q^51 - 38918320 * q^52 - 8886272 * q^54 + 14759525 * q^57 - 48407900 * q^58 + 1520220 * q^61 + 74748240 * q^63 - 4536998 * q^64 + 10465295 * q^66 - 16269290 * q^67 + 11394978 * q^69 + 172231185 * q^72 - 52090170 * q^73 - 29529046 * q^76 + 198205810 * q^78 + 8549896 * q^79 + 22612945 * q^81 - 295714190 * q^82 - 136883292 * q^84 + 318901610 * q^87 - 310673250 * q^88 - 107224264 * q^91 + 79679130 * q^93 + 356118596 * q^94 + 525424001 * q^96 - 402167800 * q^97 - 382421335 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 1634x^{8} + 776307x^{6} + 148116566x^{4} + 10575941812x^{2} + 105274575720$$ :

 $$\beta_{1}$$ $$=$$ $$( - 872557 \nu^{8} - 1265751468 \nu^{6} - 444620825535 \nu^{4} - 46859095322612 \nu^{2} - 420291052310412 ) / 157481360352$$ (-872557*v^8 - 1265751468*v^6 - 444620825535*v^4 - 46859095322612*v^2 - 420291052310412) / 157481360352 $$\beta_{2}$$ $$=$$ $$( 8400715 \nu^{9} + 12128076852 \nu^{7} + 4207972253529 \nu^{5} + 435449159046332 \nu^{3} + 32\!\cdots\!04 \nu ) / 153386844982848$$ (8400715*v^9 + 12128076852*v^7 + 4207972253529*v^5 + 435449159046332*v^3 + 3248388872882004*v) / 153386844982848 $$\beta_{3}$$ $$=$$ $$( 57764587 \nu^{9} + 16283051488 \nu^{8} + 95509278940 \nu^{7} + 23429801734080 \nu^{6} + 45386771554953 \nu^{5} + \cdots + 85\!\cdots\!12 ) / 818063173241856$$ (57764587*v^9 + 16283051488*v^8 + 95509278940*v^7 + 23429801734080*v^6 + 45386771554953*v^5 + 8070520521247776*v^4 + 7688439646206644*v^3 + 838390872436304960*v^2 + 333149360354096228*v + 8500823024651488512) / 818063173241856 $$\beta_{4}$$ $$=$$ $$( 472531299 \nu^{9} - 5511595228 \nu^{8} + 683931862716 \nu^{7} - 7973245106928 \nu^{6} + 239395694736177 \nu^{5} + \cdots - 31\!\cdots\!36 ) / 24\!\cdots\!68$$ (472531299*v^9 - 5511595228*v^8 + 683931862716*v^7 - 7973245106928*v^6 + 239395694736177*v^5 - 2784656217757620*v^4 + 25589598548147796*v^3 - 294739647408790352*v^2 + 302148258954712068*v - 3166797504195456336) / 2454189519725568 $$\beta_{5}$$ $$=$$ $$( 2093833615 \nu^{9} + 135239021452 \nu^{8} + 3031560854316 \nu^{7} + 195884281212336 \nu^{6} + \cdots + 78\!\cdots\!24 ) / 24\!\cdots\!68$$ (2093833615*v^9 + 135239021452*v^8 + 3031560854316*v^7 + 195884281212336*v^6 + 1062323361567957*v^5 + 68650021977427524*v^4 + 114013619651256356*v^3 + 7306496448374739344*v^2 + 1406792850841336212*v + 78686308946433819024) / 2454189519725568 $$\beta_{6}$$ $$=$$ $$( - 227599251 \nu^{9} - 4459205273 \nu^{8} - 326570626108 \nu^{7} - 6426141790740 \nu^{6} - 111468906693249 \nu^{5} + \cdots - 23\!\cdots\!08 ) / 204515793310464$$ (-227599251*v^9 - 4459205273*v^8 - 326570626108*v^7 - 6426141790740*v^6 - 111468906693249*v^5 - 2222840829944163*v^4 - 11263359272015892*v^3 - 232437279369669340*v^2 - 79972255476863300*v - 2376421310019837708) / 204515793310464 $$\beta_{7}$$ $$=$$ $$( 2367048033 \nu^{9} + 7788738074 \nu^{8} + 3397055634708 \nu^{7} + 11462785669512 \nu^{6} + \cdots + 61\!\cdots\!24 ) / 12\!\cdots\!84$$ (2367048033*v^9 + 7788738074*v^8 + 3397055634708*v^7 + 11462785669512*v^6 + 1168104544393851*v^5 + 4181709941725758*v^4 + 126525776892571548*v^3 + 470492136055365208*v^2 + 2502732354612363948*v + 6149466470074792824) / 1227094759862784 $$\beta_{8}$$ $$=$$ $$( 1932377350 \nu^{9} - 4685079769 \nu^{8} + 2788945929432 \nu^{7} - 6744083301684 \nu^{6} + 968434359217698 \nu^{5} + \cdots - 23\!\cdots\!00 ) / 613547379931392$$ (1932377350*v^9 - 4685079769*v^8 + 2788945929432*v^7 - 6744083301684*v^6 + 968434359217698*v^5 - 2330474252110563*v^4 + 101849890279564424*v^3 - 244869910854985916*v^2 + 1081393765837086024*v - 2305416434330119500) / 613547379931392 $$\beta_{9}$$ $$=$$ $$( 3538913459 \nu^{9} - 949770776 \nu^{8} + 5148112932156 \nu^{7} - 1377320091264 \nu^{6} + \cdots - 74\!\cdots\!32 ) / 613547379931392$$ (3538913459*v^9 - 949770776*v^8 + 5148112932156*v^7 - 1377320091264*v^6 + 1824598860588897*v^5 - 485940762965256*v^4 + 197852659386244756*v^3 - 53505191980737856*v^2 + 2509334301004612548*v - 744725103888164832) / 613547379931392
 $$\nu$$ $$=$$ $$( 9 \beta_{9} - 17 \beta_{8} + 15 \beta_{7} - 18 \beta_{5} - 358 \beta_{4} - 2 \beta_{3} + 1043 \beta_{2} - 3 \beta _1 - 182 ) / 4320$$ (9*b9 - 17*b8 + 15*b7 - 18*b5 - 358*b4 - 2*b3 + 1043*b2 - 3*b1 - 182) / 4320 $$\nu^{2}$$ $$=$$ $$( 39\beta_{8} + 24\beta_{6} - 2\beta_{5} - 563\beta_{4} + 15\beta_{3} + 236\beta_{2} + 114\beta _1 - 88558 ) / 270$$ (39*b8 + 24*b6 - 2*b5 - 563*b4 + 15*b3 + 236*b2 + 114*b1 - 88558) / 270 $$\nu^{3}$$ $$=$$ $$( - 375 \beta_{9} + 3967 \beta_{8} - 3153 \beta_{7} - 3696 \beta_{6} + 3198 \beta_{5} + 63242 \beta_{4} - 2882 \beta_{3} - 421117 \beta_{2} + 1389 \beta _1 + 31114 ) / 1440$$ (-375*b9 + 3967*b8 - 3153*b7 - 3696*b6 + 3198*b5 + 63242*b4 - 2882*b3 - 421117*b2 + 1389*b1 + 31114) / 1440 $$\nu^{4}$$ $$=$$ $$( - 8805 \beta_{8} - 4944 \beta_{6} + 1894 \beta_{5} + 123217 \beta_{4} - 3861 \beta_{3} - 51820 \beta_{2} - 13386 \beta _1 + 12132530 ) / 54$$ (-8805*b8 - 4944*b6 + 1894*b5 + 123217*b4 - 3861*b3 - 51820*b2 - 13386*b1 + 12132530) / 54 $$\nu^{5}$$ $$=$$ $$( - 587295 \beta_{9} - 10471033 \beta_{8} + 8476887 \beta_{7} + 13073904 \beta_{6} - 7623762 \beta_{5} - 144648998 \beta_{4} + 11079758 \beta_{3} + 1244093803 \beta_{2} + \cdots - 69764806 ) / 4320$$ (-587295*b9 - 10471033*b8 + 8476887*b7 + 13073904*b6 - 7623762*b5 - 144648998*b4 + 11079758*b3 + 1244093803*b2 - 4532091*b1 - 69764806) / 4320 $$\nu^{6}$$ $$=$$ $$( 4991021 \beta_{8} + 2823976 \beta_{6} - 1189238 \beta_{5} - 69115377 \beta_{4} + 2167045 \beta_{3} + 29077764 \beta_{2} + 5980126 \beta _1 - 6045548182 ) / 30$$ (4991021*b8 + 2823976*b6 - 1189238*b5 - 69115377*b4 + 2167045*b3 + 29077764*b2 + 5980126*b1 - 6045548182) / 30 $$\nu^{7}$$ $$=$$ $$( 1002570867 \beta_{9} + 10039223749 \beta_{8} - 8231449035 \beta_{7} - 13312393200 \beta_{6} + 7140762666 \beta_{5} + 133013599406 \beta_{4} + \cdots + 63697009294 ) / 4320$$ (1002570867*b9 + 10039223749*b8 - 8231449035*b7 - 13312393200*b6 + 7140762666*b5 + 133013599406*b4 - 11504618486*b3 - 1227510975631*b2 + 4617009951*b1 + 63697009294) / 4320 $$\nu^{8}$$ $$=$$ $$( - 8964366177 \beta_{8} - 5112252240 \beta_{6} + 2161618478 \beta_{5} + 123729099293 \beta_{4} - 3852113937 \beta_{3} - 52054935740 \beta_{2} + \cdots + 10528560319474 ) / 54$$ (-8964366177*b8 - 5112252240*b6 + 2161618478*b5 + 123729099293*b4 - 3852113937*b3 - 52054935740*b2 - 10028035842*b1 + 10528560319474) / 54 $$\nu^{9}$$ $$=$$ $$( - 366131162877 \beta_{9} - 3286296411019 \beta_{8} + 2707361971365 \beta_{7} + 4414995739920 \beta_{6} - 2326880020086 \beta_{5} + \cdots - 20593816306834 ) / 1440$$ (-366131162877*b9 - 3286296411019*b8 + 2707361971365*b7 + 4414995739920*b6 - 2326880020086*b5 - 43090510343666*b4 + 3836061300266*b3 + 404712969369121*b2 - 1536746567121*b1 - 20593816306834) / 1440

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$$.

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 3.43232i 13.5890i − 31.4704i − 13.7835i − 16.0371i 16.0371i 13.7835i 31.4704i − 13.5890i − 3.43232i
29.4751i −29.7353 + 75.3446i −612.784 0 2220.79 + 876.451i −3164.62 10516.3i −4792.63 4480.79i 0
26.2 24.1370i 39.5746 70.6742i −326.594 0 −1705.86 955.212i −1042.22 1703.92i −3428.70 5593.81i 0
26.3 18.2182i 69.9805 + 40.7889i −75.9021 0 743.099 1274.92i 4676.68 3281.06i 3233.54 + 5708.85i 0
26.4 16.2639i −77.7167 22.8278i −8.51375 0 −371.269 + 1263.98i 569.895 4025.09i 5518.78 + 3548.20i 0
26.5 3.03414i −14.6031 + 79.6728i 246.794 0 241.739 + 44.3080i −59.7348 1525.55i −6134.50 2326.94i 0
26.6 3.03414i −14.6031 79.6728i 246.794 0 241.739 44.3080i −59.7348 1525.55i −6134.50 + 2326.94i 0
26.7 16.2639i −77.7167 + 22.8278i −8.51375 0 −371.269 1263.98i 569.895 4025.09i 5518.78 3548.20i 0
26.8 18.2182i 69.9805 40.7889i −75.9021 0 743.099 + 1274.92i 4676.68 3281.06i 3233.54 5708.85i 0
26.9 24.1370i 39.5746 + 70.6742i −326.594 0 −1705.86 + 955.212i −1042.22 1703.92i −3428.70 + 5593.81i 0
26.10 29.4751i −29.7353 75.3446i −612.784 0 2220.79 876.451i −3164.62 10516.3i −4792.63 + 4480.79i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 26.10 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.9.c.e 10
3.b odd 2 1 inner 75.9.c.e 10
5.b even 2 1 75.9.c.f yes 10
5.c odd 4 2 75.9.d.d 20
15.d odd 2 1 75.9.c.f yes 10
15.e even 4 2 75.9.d.d 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.9.c.e 10 1.a even 1 1 trivial
75.9.c.e 10 3.b odd 2 1 inner
75.9.c.f yes 10 5.b even 2 1
75.9.c.f yes 10 15.d odd 2 1
75.9.d.d 20 5.c odd 4 2
75.9.d.d 20 15.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{9}^{\mathrm{new}}(75, [\chi])$$:

 $$T_{2}^{10} + 2057T_{2}^{8} + 1478418T_{2}^{6} + 442732064T_{2}^{4} + 48388216960T_{2}^{2} + 409079808000$$ T2^10 + 2057*T2^8 + 1478418*T2^6 + 442732064*T2^4 + 48388216960*T2^2 + 409079808000 $$T_{7}^{5} - 980T_{7}^{4} - 16170189T_{7}^{3} - 7054500600T_{7}^{2} + 8426583900000T_{7} + 525098700000000$$ T7^5 - 980*T7^4 - 16170189*T7^3 - 7054500600*T7^2 + 8426583900000*T7 + 525098700000000

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 2057 T^{8} + \cdots + 409079808000$$
$3$ $$T^{10} + 25 T^{9} + \cdots + 12\!\cdots\!01$$
$5$ $$T^{10}$$
$7$ $$(T^{5} - 980 T^{4} + \cdots + 525098700000000)^{2}$$
$11$ $$T^{10} + 1951248065 T^{8} + \cdots + 20\!\cdots\!00$$
$13$ $$(T^{5} + 8460 T^{4} + \cdots + 59\!\cdots\!00)^{2}$$
$17$ $$T^{10} + 45135371657 T^{8} + \cdots + 16\!\cdots\!00$$
$19$ $$(T^{5} + 71967 T^{4} + \cdots + 52\!\cdots\!67)^{2}$$
$23$ $$T^{10} + 280193128092 T^{8} + \cdots + 11\!\cdots\!00$$
$29$ $$T^{10} + 2261313689660 T^{8} + \cdots + 85\!\cdots\!00$$
$31$ $$(T^{5} + 1507030 T^{4} + \cdots + 96\!\cdots\!08)^{2}$$
$37$ $$(T^{5} - 1508380 T^{4} + \cdots - 19\!\cdots\!00)^{2}$$
$41$ $$T^{10} + 38200056696065 T^{8} + \cdots + 40\!\cdots\!00$$
$43$ $$(T^{5} - 5873670 T^{4} + \cdots - 35\!\cdots\!00)^{2}$$
$47$ $$T^{10} + 143634526671812 T^{8} + \cdots + 10\!\cdots\!00$$
$53$ $$T^{10} + 307776345854012 T^{8} + \cdots + 15\!\cdots\!00$$
$59$ $$T^{10} + 924675900073040 T^{8} + \cdots + 84\!\cdots\!00$$
$61$ $$(T^{5} - 760110 T^{4} + \cdots + 41\!\cdots\!68)^{2}$$
$67$ $$(T^{5} + 8134645 T^{4} + \cdots - 15\!\cdots\!75)^{2}$$
$71$ $$T^{10} + \cdots + 61\!\cdots\!00$$
$73$ $$(T^{5} + 26045085 T^{4} + \cdots + 11\!\cdots\!00)^{2}$$
$79$ $$(T^{5} - 4274948 T^{4} + \cdots + 23\!\cdots\!52)^{2}$$
$83$ $$T^{10} + \cdots + 20\!\cdots\!00$$
$89$ $$T^{10} + \cdots + 63\!\cdots\!00$$
$97$ $$(T^{5} + 201083900 T^{4} + \cdots + 82\!\cdots\!00)^{2}$$