# Properties

 Label 1075.6.a.a Level $1075$ Weight $6$ Character orbit 1075.a Self dual yes Analytic conductor $172.413$ Analytic rank $0$ Dimension $8$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1075 = 5^{2} \cdot 43$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 1075.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$172.412606299$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984$$ x^8 - 4*x^7 - 173*x^6 + 462*x^5 + 9118*x^4 - 14192*x^3 - 167688*x^2 + 106368*x + 681984 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 43) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 2) q^{2} + (\beta_{7} - \beta_{4} + \beta_1 + 2) q^{3} + (\beta_{7} + \beta_{5} - 2 \beta_{4} - 2 \beta_1 + 15) q^{4} + (4 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - \beta_{3} - 10 \beta_1 - 5) q^{6} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 4 \beta_{3} + \beta_{2} + 13 \beta_1 + 10) q^{7} + (6 \beta_{7} - 8 \beta_{6} - \beta_{5} - 8 \beta_{4} + \beta_{3} - 5 \beta_{2} - 7 \beta_1 + 82) q^{8} + ( - \beta_{7} + \beta_{6} + 8 \beta_{5} - 15 \beta_{4} - 9 \beta_{3} - 4 \beta_{2} + \cdots + 62) q^{9}+O(q^{10})$$ q + (-b1 + 2) * q^2 + (b7 - b4 + b1 + 2) * q^3 + (b7 + b5 - 2*b4 - 2*b1 + 15) * q^4 + (4*b7 - 3*b6 - 3*b5 - 2*b4 - b3 - 10*b1 - 5) * q^6 + (-b7 + 2*b6 - 2*b5 - 4*b3 + b2 + 13*b1 + 10) * q^7 + (6*b7 - 8*b6 - b5 - 8*b4 + b3 - 5*b2 - 7*b1 + 82) * q^8 + (-b7 + b6 + 8*b5 - 15*b4 - 9*b3 - 4*b2 - 8*b1 + 62) * q^9 $$q + ( - \beta_1 + 2) q^{2} + (\beta_{7} - \beta_{4} + \beta_1 + 2) q^{3} + (\beta_{7} + \beta_{5} - 2 \beta_{4} - 2 \beta_1 + 15) q^{4} + (4 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - \beta_{3} - 10 \beta_1 - 5) q^{6} + ( - \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 4 \beta_{3} + \beta_{2} + 13 \beta_1 + 10) q^{7} + (6 \beta_{7} - 8 \beta_{6} - \beta_{5} - 8 \beta_{4} + \beta_{3} - 5 \beta_{2} - 7 \beta_1 + 82) q^{8} + ( - \beta_{7} + \beta_{6} + 8 \beta_{5} - 15 \beta_{4} - 9 \beta_{3} - 4 \beta_{2} + \cdots + 62) q^{9}+ \cdots + (1884 \beta_{7} + 2105 \beta_{6} + 1126 \beta_{5} - 3870 \beta_{4} + \cdots + 23821) q^{99}+O(q^{100})$$ q + (-b1 + 2) * q^2 + (b7 - b4 + b1 + 2) * q^3 + (b7 + b5 - 2*b4 - 2*b1 + 15) * q^4 + (4*b7 - 3*b6 - 3*b5 - 2*b4 - b3 - 10*b1 - 5) * q^6 + (-b7 + 2*b6 - 2*b5 - 4*b3 + b2 + 13*b1 + 10) * q^7 + (6*b7 - 8*b6 - b5 - 8*b4 + b3 - 5*b2 - 7*b1 + 82) * q^8 + (-b7 + b6 + 8*b5 - 15*b4 - 9*b3 - 4*b2 - 8*b1 + 62) * q^9 + (-3*b7 - b6 - 2*b5 - 9*b3 + 19*b2 + 28*b1 - 74) * q^11 + (20*b7 - 7*b6 + 3*b5 - 52*b4 - 19*b3 - 10*b1 + 499) * q^12 + (b7 + 19*b6 - 2*b5 - 24*b4 - 13*b3 - b2 + 4*b1 + 292) * q^13 + (-6*b7 + 4*b6 - 18*b5 + 36*b4 - 24*b3 - 8*b2 + 24*b1 - 534) * q^14 + (22*b7 - 36*b6 - 29*b5 - 88*b4 - 15*b3 - 9*b2 - 53*b1 + 226) * q^16 + (-32*b7 - 9*b6 + 8*b5 + 67*b4 + 12*b3 + 17*b2 - 109*b1 + 419) * q^17 + (10*b7 - 63*b6 - 12*b5 - 50*b4 - 40*b3 - 71*b2 - 182*b1 + 513) * q^18 + (38*b7 + 9*b6 + 64*b5 - 9*b4 - 62*b3 - 105*b2 - 17*b1 - 284) * q^19 + (23*b7 + 30*b6 + 24*b5 - 4*b4 + 87*b3 + 65*b2 + 189*b1 - 351) * q^21 + (76*b7 - 22*b6 - 57*b5 + 8*b4 - 37*b3 - 107*b2 + 84*b1 - 1542) * q^22 + (-20*b7 + 26*b6 + 70*b5 + 125*b4 - 112*b3 + 21*b2 + 2*b1 + 329) * q^23 + (90*b7 - 109*b6 + 25*b5 - 268*b4 - 103*b3 - 190*b2 - 488*b1 + 2531) * q^24 + (-44*b7 - 54*b6 - 29*b5 + 168*b4 - 65*b3 - 55*b2 - 394*b1 + 830) * q^26 + (-28*b7 - 93*b6 - 6*b5 - 150*b4 + 74*b3 + 98*b2 - 125*b1 + 1216) * q^27 + (40*b7 + 108*b6 + 30*b5 + 56*b4 - 50*b3 + 102*b2 + 678*b1 - 2668) * q^28 + (-139*b7 + 12*b6 - 122*b5 + 19*b4 + 54*b3 + 52*b2 + 103*b1 - 492) * q^29 + (56*b7 + 39*b6 - 202*b5 - 83*b4 - 140*b3 + 155*b2 + 137*b1 + 631) * q^31 + (204*b7 - 108*b6 + 49*b5 - 692*b4 - 247*b3 - 177*b2 - 241*b1 + 1964) * q^32 + (-133*b7 + 146*b6 + 118*b5 - 164*b4 + 34*b3 + 161*b2 + 61*b1 + 1582) * q^33 + (6*b7 + 193*b6 + 172*b5 - 82*b4 + 128*b3 + 61*b2 - 92*b1 + 3749) * q^34 + (467*b7 - 265*b6 - 91*b5 - 1070*b4 - 120*b3 - 5*b2 - 33*b1 + 7982) * q^36 + (398*b7 - 98*b6 - 280*b5 - 217*b4 - 114*b3 + 307*b2 + 422*b1 + 176) * q^37 + (-74*b7 + 61*b6 - 142*b5 - 122*b4 - 378*b3 + 375*b2 + 223*b1 + 739) * q^38 + (477*b7 - 30*b6 - 2*b5 - 418*b4 + 242*b3 + 445*b2 + 1023*b1 + 778) * q^39 + (-245*b7 - 214*b6 + 138*b5 - b4 - 173*b3 - 274*b2 - 295*b1 - 1208) * q^41 + (-336*b7 - 22*b6 - 150*b5 + 1140*b4 + 642*b3 + 56*b2 - 576*b1 - 8790) * q^42 + 1849 * q^43 + (279*b7 + 260*b6 - 82*b5 - 538*b4 - 253*b3 + 73*b2 + 1667*b1 - 1275) * q^44 + (230*b7 + 441*b6 - 277*b5 + 350*b4 - 465*b3 + 120*b2 + 407*b1 - 2729) * q^46 + (99*b7 - 108*b6 + 300*b5 + 25*b4 - 193*b3 - 8*b2 - 439*b1 + 9863) * q^47 + (470*b7 - 811*b6 + 171*b5 - 1916*b4 - 349*b3 - 510*b2 - 3208*b1 + 15453) * q^48 + (-741*b7 + 162*b6 + 516*b5 - 254*b4 + 205*b3 - 61*b2 - 1383*b1 + 1696) * q^49 + (419*b7 - 21*b6 - 482*b5 + 269*b4 - 54*b3 - 80*b2 - 398*b1 - 9866) * q^51 + (481*b7 - 4*b6 + 500*b5 - 670*b4 - 97*b3 + 485*b2 + 1067*b1 + 6723) * q^52 + (348*b7 + 423*b6 - 406*b5 - 568*b4 + 100*b3 + 48*b2 + 1303*b1 + 6787) * q^53 + (552*b7 - 906*b6 + 237*b5 - 1512*b4 + 465*b3 - 1229*b2 - 2755*b1 + 8084) * q^54 + (-248*b7 + 220*b6 - 414*b5 + 1608*b4 + 698*b3 + 426*b2 + 1818*b1 - 17948) * q^56 + (910*b7 - 617*b6 - 350*b5 + 84*b4 + 1038*b3 - 342*b2 + 17*b1 + 112) * q^57 + (-744*b7 + 31*b6 + 353*b5 + 662*b4 + 283*b3 - 422*b2 + 1424*b1 - 6943) * q^58 + (179*b7 + 388*b6 + 728*b5 + 226*b4 - 445*b3 + 1163*b2 + 815*b1 - 3531) * q^59 + (161*b7 + 204*b6 - 646*b5 + 1356*b4 - 296*b3 + 1067*b2 - 1315*b1 - 8856) * q^61 + (1330*b7 - 151*b6 - 482*b5 - 422*b4 - 1106*b3 - 689*b2 + 746*b1 - 339) * q^62 + (-1639*b7 + 1076*b6 + 1414*b5 + 894*b4 - 484*b3 + 97*b2 + 877*b1 + 7702) * q^63 + (1328*b7 - 1500*b6 + 395*b5 - 1860*b4 - 1393*b3 - 1823*b2 - 3883*b1 + 19272) * q^64 + (-1066*b7 - 776*b6 - 6*b5 + 2476*b4 + 880*b3 - 1444*b2 - 4458*b1 - 2634) * q^66 + (-1283*b7 - 159*b6 + 874*b5 + 948*b4 - 77*b3 - 1093*b2 - 50*b1 - 3110) * q^67 + (-259*b7 + 119*b6 - 153*b5 + 1054*b4 + 976*b3 - 429*b2 - 3405*b1 - 2796) * q^68 + (1377*b7 + 449*b6 - 578*b5 + 142*b4 + 1728*b3 + 963*b2 + 1856*b1 - 12360) * q^69 + (1198*b7 + 612*b6 + 1644*b5 - 1340*b4 + 816*b3 - 1682*b2 - 838*b1 - 2506) * q^71 + (3924*b7 - 2231*b6 - 661*b5 - 5912*b4 - 359*b3 - 1064*b2 - 8416*b1 + 24007) * q^72 + (463*b7 + 32*b6 + 714*b5 + 908*b4 + 280*b3 + 65*b2 - 1677*b1 - 780) * q^73 + (4022*b7 - 693*b6 - 1473*b5 - 2774*b4 - 1433*b3 - 746*b2 + 174*b1 - 4821) * q^74 + (1040*b7 - 961*b6 - 3112*b5 - 304*b4 - 58*b3 + 645*b2 + 1007*b1 + 9) * q^76 + (-935*b7 + 328*b6 + 1842*b5 - 544*b4 + 1714*b3 + 907*b2 - 1371*b1 + 28556) * q^77 + (2326*b7 - 1698*b6 - 1936*b5 + 800*b4 + 1386*b3 - 1148*b2 - 6678*b1 - 29672) * q^78 + (275*b7 + 195*b6 + 1676*b5 + 1839*b4 - 639*b3 + 60*b2 + 1226*b1 + 19609) * q^79 + (493*b7 - 701*b6 - 1210*b5 - 1522*b4 - 467*b3 + 249*b2 - 1448*b1 + 20406) * q^81 + (-402*b7 - 541*b6 + 575*b5 - 3514*b4 - 1005*b3 - 1014*b2 + 2105*b1 + 3185) * q^82 + (546*b7 + 1885*b6 + 2078*b5 - 384*b4 - 884*b3 + 502*b2 + 1607*b1 + 7731) * q^83 + (-3924*b7 + 3314*b6 + 1858*b5 + 3888*b4 + 1138*b3 + 2280*b2 + 9052*b1 + 530) * q^84 + (-1849*b1 + 3698) * q^86 + (-1337*b7 + 327*b6 + 80*b5 + 1871*b4 + 639*b3 + 1040*b2 + 2474*b1 - 33093) * q^87 + (-2618*b7 - 640*b6 - 898*b5 + 3908*b4 - 444*b3 + 2452*b2 - 3934*b1 - 10698) * q^88 + (1507*b7 + 1516*b6 - 1342*b5 - 430*b4 + 1528*b3 + 1847*b2 + 4015*b1 - 35082) * q^89 + (-2601*b7 + 2052*b6 + 894*b5 - 1056*b4 - 2070*b3 - 3*b2 + 2775*b1 + 48192) * q^91 + (1795*b7 + 1837*b6 - 3880*b5 + 814*b4 + 571*b3 + 1838*b2 + 8902*b1 - 27760) * q^92 + (829*b7 - 25*b6 + 638*b5 - 1783*b4 + 1312*b3 + 1948*b2 - 3476*b1 + 17226) * q^93 + (1914*b7 - 309*b6 - 437*b5 - 2710*b4 - 773*b3 - 482*b2 - 11390*b1 + 34811) * q^94 + (5918*b7 - 4989*b6 + 1109*b5 - 15020*b4 - 247*b3 - 966*b2 - 11172*b1 + 117271) * q^96 + (2298*b7 + 760*b6 - 2578*b5 - 4143*b4 - 1116*b3 - 1891*b2 + 1322*b1 - 22025) * q^97 + (-4712*b7 - 1888*b6 + 2820*b5 + 2328*b4 + 3104*b3 - 3356*b2 - 8381*b1 + 42998) * q^98 + (1884*b7 + 2105*b6 + 1126*b5 - 3870*b4 + 1668*b3 + 202*b2 + 7573*b1 + 23821) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9}+O(q^{10})$$ 8 * q + 12 * q^2 + 26 * q^3 + 122 * q^4 - 69 * q^6 + 136 * q^7 + 666 * q^8 + 546 * q^9 $$8 q + 12 q^{2} + 26 q^{3} + 122 q^{4} - 69 q^{6} + 136 q^{7} + 666 q^{8} + 546 q^{9} - 532 q^{11} + 4195 q^{12} + 2492 q^{13} - 4240 q^{14} + 1882 q^{16} + 2534 q^{17} + 3711 q^{18} - 1678 q^{19} - 2256 q^{21} - 11502 q^{22} + 2488 q^{23} + 19953 q^{24} + 4586 q^{26} + 8960 q^{27} - 18640 q^{28} - 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 12852 q^{33} + 30007 q^{34} + 67969 q^{36} + 3772 q^{37} + 6559 q^{38} + 11120 q^{39} - 10698 q^{41} - 78698 q^{42} + 14792 q^{43} - 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 118727 q^{48} + 7188 q^{49} - 80246 q^{51} + 60736 q^{52} + 62352 q^{53} + 61026 q^{54} - 144528 q^{56} + 808 q^{57} - 52951 q^{58} - 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 61768 q^{63} + 153858 q^{64} - 48516 q^{66} - 27784 q^{67} - 40507 q^{68} - 93776 q^{69} - 9504 q^{71} + 186687 q^{72} - 14260 q^{73} - 15239 q^{74} + 1279 q^{76} + 218140 q^{77} - 264170 q^{78} + 160248 q^{79} + 161076 q^{81} + 47781 q^{82} + 77176 q^{83} + 16382 q^{84} + 22188 q^{86} - 268136 q^{87} - 129544 q^{88} - 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 123860 q^{93} + 248737 q^{94} + 950817 q^{96} - 144742 q^{97} + 292244 q^{98} + 239516 q^{99}+O(q^{100})$$ 8 * q + 12 * q^2 + 26 * q^3 + 122 * q^4 - 69 * q^6 + 136 * q^7 + 666 * q^8 + 546 * q^9 - 532 * q^11 + 4195 * q^12 + 2492 * q^13 - 4240 * q^14 + 1882 * q^16 + 2534 * q^17 + 3711 * q^18 - 1678 * q^19 - 2256 * q^21 - 11502 * q^22 + 2488 * q^23 + 19953 * q^24 + 4586 * q^26 + 8960 * q^27 - 18640 * q^28 - 4360 * q^29 + 5704 * q^31 + 18294 * q^32 + 12852 * q^33 + 30007 * q^34 + 67969 * q^36 + 3772 * q^37 + 6559 * q^38 + 11120 * q^39 - 10698 * q^41 - 78698 * q^42 + 14792 * q^43 - 356 * q^44 - 19389 * q^46 + 77864 * q^47 + 118727 * q^48 + 7188 * q^49 - 80246 * q^51 + 60736 * q^52 + 62352 * q^53 + 61026 * q^54 - 144528 * q^56 + 808 * q^57 - 52951 * q^58 - 26224 * q^59 - 82540 * q^61 + 9023 * q^62 + 61768 * q^63 + 153858 * q^64 - 48516 * q^66 - 27784 * q^67 - 40507 * q^68 - 93776 * q^69 - 9504 * q^71 + 186687 * q^72 - 14260 * q^73 - 15239 * q^74 + 1279 * q^76 + 218140 * q^77 - 264170 * q^78 + 160248 * q^79 + 161076 * q^81 + 47781 * q^82 + 77176 * q^83 + 16382 * q^84 + 22188 * q^86 - 268136 * q^87 - 129544 * q^88 - 265692 * q^89 + 401148 * q^91 - 190391 * q^92 + 123860 * q^93 + 248737 * q^94 + 950817 * q^96 - 144742 * q^97 + 292244 * q^98 + 239516 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 8125 \nu^{7} + 510320 \nu^{6} - 2147429 \nu^{5} - 76756386 \nu^{4} + 85006462 \nu^{3} + 2758106176 \nu^{2} - 206135304 \nu - 20471935488 ) / 547265856$$ (8125*v^7 + 510320*v^6 - 2147429*v^5 - 76756386*v^4 + 85006462*v^3 + 2758106176*v^2 - 206135304*v - 20471935488) / 547265856 $$\beta_{3}$$ $$=$$ $$( 12949 \nu^{7} - 681148 \nu^{6} + 2601727 \nu^{5} + 80176782 \nu^{4} - 378607946 \nu^{3} - 1837549088 \nu^{2} + 7674423864 \nu + 1709840448 ) / 547265856$$ (12949*v^7 - 681148*v^6 + 2601727*v^5 + 80176782*v^4 - 378607946*v^3 - 1837549088*v^2 + 7674423864*v + 1709840448) / 547265856 $$\beta_{4}$$ $$=$$ $$( - 17485 \nu^{7} + 269956 \nu^{6} + 1611305 \nu^{5} - 37491046 \nu^{4} + 6237658 \nu^{3} + 1274965584 \nu^{2} - 1052986520 \nu - 6710235008 ) / 364843904$$ (-17485*v^7 + 269956*v^6 + 1611305*v^5 - 37491046*v^4 + 6237658*v^3 + 1274965584*v^2 - 1052986520*v - 6710235008) / 364843904 $$\beta_{5}$$ $$=$$ $$( 38033 \nu^{7} - 126518 \nu^{6} - 5886547 \nu^{5} + 8711358 \nu^{4} + 252968474 \nu^{3} + 152252096 \nu^{2} - 3214801896 \nu - 7712512416 ) / 273632928$$ (38033*v^7 - 126518*v^6 - 5886547*v^5 + 8711358*v^4 + 252968474*v^3 + 152252096*v^2 - 3214801896*v - 7712512416) / 273632928 $$\beta_{6}$$ $$=$$ $$( - 83131 \nu^{7} + 19780 \nu^{6} + 13177295 \nu^{5} + 14631678 \nu^{4} - 470063386 \nu^{3} - 1263726976 \nu^{2} + 1067316504 \nu + 17642049216 ) / 547265856$$ (-83131*v^7 + 19780*v^6 + 13177295*v^5 + 14631678*v^4 - 470063386*v^3 - 1263726976*v^2 + 1067316504*v + 17642049216) / 547265856 $$\beta_{7}$$ $$=$$ $$( - 128521 \nu^{7} + 1062904 \nu^{6} + 16607009 \nu^{5} - 129895854 \nu^{4} - 487223974 \nu^{3} + 4067658416 \nu^{2} + 2176112520 \nu - 28238112000 ) / 547265856$$ (-128521*v^7 + 1062904*v^6 + 16607009*v^5 - 129895854*v^4 - 487223974*v^3 + 4067658416*v^2 + 2176112520*v - 28238112000) / 547265856
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{5} - 2\beta_{4} + 2\beta _1 + 43$$ b7 + b5 - 2*b4 + 2*b1 + 43 $$\nu^{3}$$ $$=$$ $$8\beta_{6} + 7\beta_{5} - 4\beta_{4} - \beta_{3} + 5\beta_{2} + 71\beta _1 + 56$$ 8*b6 + 7*b5 - 4*b4 - b3 + 5*b2 + 71*b1 + 56 $$\nu^{4}$$ $$=$$ $$94\beta_{7} + 28\beta_{6} + 99\beta_{5} - 264\beta_{4} - 23\beta_{3} + 31\beta_{2} + 307\beta _1 + 3114$$ 94*b7 + 28*b6 + 99*b5 - 264*b4 - 23*b3 + 31*b2 + 307*b1 + 3114 $$\nu^{5}$$ $$=$$ $$48\beta_{7} + 1092\beta_{6} + 869\beta_{5} - 924\beta_{4} - 71\beta_{3} + 927\beta_{2} + 6567\beta _1 + 9672$$ 48*b7 + 1092*b6 + 869*b5 - 924*b4 - 71*b3 + 927*b2 + 6567*b1 + 9672 $$\nu^{6}$$ $$=$$ $$8760 \beta_{7} + 5444 \beta_{6} + 10339 \beta_{5} - 29780 \beta_{4} - 3425 \beta_{3} + 6801 \beta_{2} + 40661 \beta _1 + 285312$$ 8760*b7 + 5444*b6 + 10339*b5 - 29780*b4 - 3425*b3 + 6801*b2 + 40661*b1 + 285312 $$\nu^{7}$$ $$=$$ $$11036 \beta_{7} + 127500 \beta_{6} + 102849 \beta_{5} - 146996 \beta_{4} - 10463 \beta_{3} + 125743 \beta_{2} + 685627 \beta _1 + 1391004$$ 11036*b7 + 127500*b6 + 102849*b5 - 146996*b4 - 10463*b3 + 125743*b2 + 685627*b1 + 1391004

## 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
 10.9591 7.21373 5.65705 2.58275 −2.08717 −5.06235 −6.09504 −9.16809
−8.95911 25.1057 48.2657 0 −224.924 166.001 −145.726 387.294 0
1.2 −5.21373 11.2683 −4.81697 0 −58.7500 11.1041 191.954 −116.025 0
1.3 −3.65705 −7.84314 −18.6260 0 28.6827 25.5214 185.142 −181.485 0
1.4 −0.582753 −3.05838 −31.6604 0 1.78228 103.690 37.0983 −233.646 0
1.5 4.08717 −25.6605 −15.2950 0 −104.879 184.774 −193.303 415.460 0
1.6 7.06235 9.19190 17.8767 0 64.9164 4.24720 −99.7434 −158.509 0
1.7 8.09504 −11.1803 33.5297 0 −90.5052 −223.489 12.3830 −118.000 0
1.8 11.1681 28.1764 92.7262 0 314.677 −135.849 678.196 550.912 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$43$$ $$-1$$

## 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 1075.6.a.a 8
5.b even 2 1 43.6.a.a 8
15.d odd 2 1 387.6.a.c 8
20.d odd 2 1 688.6.a.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.6.a.a 8 5.b even 2 1
387.6.a.c 8 15.d odd 2 1
688.6.a.e 8 20.d odd 2 1
1075.6.a.a 8 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - 12T_{2}^{7} - 117T_{2}^{6} + 1502T_{2}^{5} + 3358T_{2}^{4} - 49104T_{2}^{3} - 39464T_{2}^{2} + 439936T_{2} + 259776$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(1075))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 12 T^{7} - 117 T^{6} + \cdots + 259776$$
$3$ $$T^{8} - 26 T^{7} + \cdots + 504223128$$
$5$ $$T^{8}$$
$7$ $$T^{8} + \cdots + 116222354316288$$
$11$ $$T^{8} + 532 T^{7} + \cdots + 16\!\cdots\!76$$
$13$ $$T^{8} - 2492 T^{7} + \cdots - 20\!\cdots\!44$$
$17$ $$T^{8} - 2534 T^{7} + \cdots + 37\!\cdots\!33$$
$19$ $$T^{8} + 1678 T^{7} + \cdots + 20\!\cdots\!68$$
$23$ $$T^{8} - 2488 T^{7} + \cdots - 18\!\cdots\!83$$
$29$ $$T^{8} + 4360 T^{7} + \cdots - 39\!\cdots\!36$$
$31$ $$T^{8} - 5704 T^{7} + \cdots + 10\!\cdots\!13$$
$37$ $$T^{8} - 3772 T^{7} + \cdots - 10\!\cdots\!04$$
$41$ $$T^{8} + 10698 T^{7} + \cdots + 17\!\cdots\!57$$
$43$ $$(T - 1849)^{8}$$
$47$ $$T^{8} - 77864 T^{7} + \cdots - 19\!\cdots\!04$$
$53$ $$T^{8} - 62352 T^{7} + \cdots + 55\!\cdots\!84$$
$59$ $$T^{8} + 26224 T^{7} + \cdots - 68\!\cdots\!68$$
$61$ $$T^{8} + 82540 T^{7} + \cdots + 23\!\cdots\!84$$
$67$ $$T^{8} + 27784 T^{7} + \cdots + 69\!\cdots\!48$$
$71$ $$T^{8} + 9504 T^{7} + \cdots - 15\!\cdots\!64$$
$73$ $$T^{8} + 14260 T^{7} + \cdots + 80\!\cdots\!92$$
$79$ $$T^{8} - 160248 T^{7} + \cdots + 19\!\cdots\!12$$
$83$ $$T^{8} - 77176 T^{7} + \cdots - 19\!\cdots\!08$$
$89$ $$T^{8} + 265692 T^{7} + \cdots - 13\!\cdots\!64$$
$97$ $$T^{8} + 144742 T^{7} + \cdots + 38\!\cdots\!17$$