# Properties

 Label 75.4.e.d Level $75$ Weight $4$ Character orbit 75.e Analytic conductor $4.425$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 75.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.42514325043$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 36 x^{14} + 562 x^{12} - 3672 x^{10} + 16413 x^{8} - 6588 x^{6} + 43024 x^{4} + 499896 x^{2} + 532900$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}\cdot 3^{8}\cdot 5^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} -\beta_{8} q^{3} + ( 6 \beta_{1} + \beta_{5} ) q^{4} + ( -6 + \beta_{4} - \beta_{6} ) q^{6} + ( -\beta_{3} + 2 \beta_{7} - \beta_{9} - \beta_{12} ) q^{7} + ( 3 \beta_{2} - 4 \beta_{8} - \beta_{14} ) q^{8} + ( -11 \beta_{1} + \beta_{5} + \beta_{11} - \beta_{15} ) q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{2} -\beta_{8} q^{3} + ( 6 \beta_{1} + \beta_{5} ) q^{4} + ( -6 + \beta_{4} - \beta_{6} ) q^{6} + ( -\beta_{3} + 2 \beta_{7} - \beta_{9} - \beta_{12} ) q^{7} + ( 3 \beta_{2} - 4 \beta_{8} - \beta_{14} ) q^{8} + ( -11 \beta_{1} + \beta_{5} + \beta_{11} - \beta_{15} ) q^{9} + ( -\beta_{4} - 2 \beta_{6} - 2 \beta_{10} ) q^{11} + ( -11 \beta_{3} - 3 \beta_{7} + 3 \beta_{9} + \beta_{12} ) q^{12} + ( 3 \beta_{2} + 6 \beta_{8} + 3 \beta_{13} - 3 \beta_{14} ) q^{13} + ( -4 \beta_{5} - 8 \beta_{11} + \beta_{15} ) q^{14} + ( -16 + 3 \beta_{4} ) q^{16} + ( 19 \beta_{3} - 4 \beta_{7} - \beta_{12} ) q^{17} + ( -6 \beta_{2} + 3 \beta_{8} - 9 \beta_{13} - 3 \beta_{14} ) q^{18} + ( 53 \beta_{1} - 4 \beta_{5} ) q^{19} + ( 53 - 7 \beta_{4} - 6 \beta_{6} + 2 \beta_{10} ) q^{21} + ( 5 \beta_{3} - 10 \beta_{7} - 6 \beta_{9} + 5 \beta_{12} ) q^{22} + ( -6 \beta_{2} + 8 \beta_{8} + 2 \beta_{14} ) q^{23} + ( -126 \beta_{1} + \beta_{5} + 5 \beta_{11} - 3 \beta_{15} ) q^{24} + ( 6 \beta_{4} + 12 \beta_{6} + 3 \beta_{10} ) q^{26} + ( -20 \beta_{3} + 16 \beta_{7} + 3 \beta_{9} - 8 \beta_{12} ) q^{27} + ( -12 \beta_{2} - 24 \beta_{8} + 22 \beta_{13} + 12 \beta_{14} ) q^{28} + ( 8 \beta_{5} + 16 \beta_{11} + 4 \beta_{15} ) q^{29} + ( -73 - 5 \beta_{4} ) q^{31} + ( -7 \beta_{3} + 20 \beta_{7} + 5 \beta_{12} ) q^{32} + ( 13 \beta_{2} - \beta_{8} - 21 \beta_{13} + 11 \beta_{14} ) q^{33} + ( 244 \beta_{1} + 11 \beta_{5} ) q^{34} + ( 20 + 4 \beta_{4} + 25 \beta_{6} - \beta_{10} ) q^{36} + ( -6 \beta_{3} + 12 \beta_{7} - 8 \beta_{9} - 6 \beta_{12} ) q^{37} + ( 33 \beta_{2} + 16 \beta_{8} + 4 \beta_{14} ) q^{38} + ( -129 \beta_{1} + 3 \beta_{5} + 12 \beta_{15} ) q^{39} + ( \beta_{4} + 2 \beta_{6} + 2 \beta_{10} ) q^{41} + ( 88 \beta_{3} - 14 \beta_{7} + 30 \beta_{9} + 19 \beta_{12} ) q^{42} + ( 3 \beta_{2} + 6 \beta_{8} + 17 \beta_{13} - 3 \beta_{14} ) q^{43} + ( \beta_{5} + 2 \beta_{11} - 10 \beta_{15} ) q^{44} + ( 128 - 22 \beta_{4} ) q^{46} + ( -80 \beta_{3} - 32 \beta_{7} - 8 \beta_{12} ) q^{47} + ( -33 \beta_{2} + 7 \beta_{8} - 9 \beta_{13} - 3 \beta_{14} ) q^{48} + ( -80 \beta_{1} - 39 \beta_{5} ) q^{49} + ( -6 + 23 \beta_{4} - 17 \beta_{6} - 3 \beta_{10} ) q^{51} + ( -6 \beta_{3} + 12 \beta_{7} - 42 \beta_{9} - 6 \beta_{12} ) q^{52} + ( -138 \beta_{2} - 8 \beta_{8} - 2 \beta_{14} ) q^{53} + ( -168 \beta_{1} - 33 \beta_{5} - 42 \beta_{11} - 3 \beta_{15} ) q^{54} + ( -26 \beta_{4} - 52 \beta_{6} + 14 \beta_{10} ) q^{56} + ( 44 \beta_{3} - 65 \beta_{7} - 12 \beta_{9} - 4 \beta_{12} ) q^{57} + ( 40 \beta_{2} + 80 \beta_{8} - 24 \beta_{13} - 40 \beta_{14} ) q^{58} + ( 6 \beta_{5} + 12 \beta_{11} - 12 \beta_{15} ) q^{59} + ( -113 + 35 \beta_{4} ) q^{61} + ( -48 \beta_{3} + 20 \beta_{7} + 5 \beta_{12} ) q^{62} + ( 115 \beta_{2} - 28 \beta_{8} - 3 \beta_{13} - 19 \beta_{14} ) q^{63} + ( 140 \beta_{1} + 57 \beta_{5} ) q^{64} + ( -210 - 9 \beta_{4} - 3 \beta_{6} - 21 \beta_{10} ) q^{66} + ( 14 \beta_{3} - 28 \beta_{7} + 75 \beta_{9} + 14 \beta_{12} ) q^{67} + ( 147 \beta_{2} - 76 \beta_{8} - 19 \beta_{14} ) q^{68} + ( 252 \beta_{1} - 2 \beta_{5} - 10 \beta_{11} + 6 \beta_{15} ) q^{69} + ( 8 \beta_{4} + 16 \beta_{6} - 44 \beta_{10} ) q^{71} + ( -48 \beta_{3} + 135 \beta_{7} - 9 \beta_{9} - 30 \beta_{12} ) q^{72} + ( -37 \beta_{2} - 74 \beta_{8} + 26 \beta_{13} + 37 \beta_{14} ) q^{73} + ( -26 \beta_{5} - 52 \beta_{11} + 8 \beta_{15} ) q^{74} + ( 50 + 33 \beta_{4} ) q^{76} + ( 78 \beta_{3} - 24 \beta_{7} - 6 \beta_{12} ) q^{77} + ( -114 \beta_{2} - 12 \beta_{8} + 72 \beta_{13} - 3 \beta_{14} ) q^{78} + ( -212 \beta_{1} + 46 \beta_{5} ) q^{79} + ( 441 - 24 \beta_{4} - 3 \beta_{6} + 27 \beta_{10} ) q^{81} + ( -5 \beta_{3} + 10 \beta_{7} + 6 \beta_{9} - 5 \beta_{12} ) q^{82} + ( 115 \beta_{2} - 44 \beta_{8} - 11 \beta_{14} ) q^{83} + ( 686 \beta_{1} + 124 \beta_{5} + 102 \beta_{11} - 14 \beta_{15} ) q^{84} + ( -8 \beta_{4} - 16 \beta_{6} + 17 \beta_{10} ) q^{86} + ( -152 \beta_{3} + 16 \beta_{7} - 96 \beta_{9} + 4 \beta_{12} ) q^{87} + ( -35 \beta_{2} - 70 \beta_{8} - 114 \beta_{13} + 35 \beta_{14} ) q^{88} + ( 19 \beta_{5} + 38 \beta_{11} + 2 \beta_{15} ) q^{89} + ( -639 + 27 \beta_{4} ) q^{91} + ( 190 \beta_{3} + 24 \beta_{7} + 6 \beta_{12} ) q^{92} + ( 55 \beta_{2} + 88 \beta_{8} + 15 \beta_{13} + 5 \beta_{14} ) q^{93} + ( -1296 \beta_{1} - 144 \beta_{5} ) q^{94} + ( -498 - 27 \beta_{4} - 3 \beta_{6} + 15 \beta_{10} ) q^{96} + ( -\beta_{3} + 2 \beta_{7} + 13 \beta_{9} - \beta_{12} ) q^{97} + ( -275 \beta_{2} + 156 \beta_{8} + 39 \beta_{14} ) q^{98} + ( 510 \beta_{1} - 96 \beta_{5} - 27 \beta_{11} - 33 \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 84q^{6} + O(q^{10})$$ $$16q - 84q^{6} - 232q^{16} + 816q^{21} - 1208q^{31} + 252q^{36} + 1872q^{46} + 156q^{51} - 1528q^{61} - 3420q^{66} + 1064q^{76} + 6876q^{81} - 10008q^{91} - 8172q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 36 x^{14} + 562 x^{12} - 3672 x^{10} + 16413 x^{8} - 6588 x^{6} + 43024 x^{4} + 499896 x^{2} + 532900$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-166 \nu^{14} + 6345 \nu^{12} - 107207 \nu^{10} + 837000 \nu^{8} - 4351513 \nu^{6} + 8384355 \nu^{4} - 17177814 \nu^{2} - 61944300$$$$)/17721150$$ $$\beta_{2}$$ $$=$$ $$($$$$1476838 \nu^{14} - 56632185 \nu^{12} + 969443611 \nu^{10} - 7877375550 \nu^{8} + 44264156989 \nu^{6} - 104226630615 \nu^{4} + 170768020012 \nu^{2} + 850813760250$$$$)/ 135306887300$$ $$\beta_{3}$$ $$=$$ $$($$$$5782554 \nu^{14} - 226662200 \nu^{12} + 3896596638 \nu^{10} - 30968795985 \nu^{8} + 152916400962 \nu^{6} - 292898382315 \nu^{4} + 315982675746 \nu^{2} + 1184712292450$$$$)/ 405920661900$$ $$\beta_{4}$$ $$=$$ $$($$$$1236 \nu^{14} - 46710 \nu^{12} + 752616 \nu^{10} - 4977045 \nu^{8} + 16234140 \nu^{6} + 23885280 \nu^{4} - 121409592 \nu^{2} + 507816635$$$$)/65682955$$ $$\beta_{5}$$ $$=$$ $$($$$$21947581 \nu^{14} - 815488875 \nu^{12} + 13314880712 \nu^{10} - 97200412890 \nu^{8} + 489483870733 \nu^{6} - 810490185435 \nu^{4} + 2529236647674 \nu^{2} + 7590938337300$$$$)/ 202960330950$$ $$\beta_{6}$$ $$=$$ $$($$$$602367 \nu^{15} - 721824 \nu^{14} - 24352340 \nu^{13} + 27278640 \nu^{12} + 427554484 \nu^{11} - 439527744 \nu^{10} - 3444009860 \nu^{9} + 2906594280 \nu^{8} + 15244862711 \nu^{7} - 9480737760 \nu^{6} - 13937691940 \nu^{5} - 13949003520 \nu^{4} - 103082514062 \nu^{3} + 70903201728 \nu^{2} + 230005815740 \nu - 296564914840$$$$)/ 76717691440$$ $$\beta_{7}$$ $$=$$ $$($$$$1529999985 \nu^{15} - 3594093388 \nu^{14} - 56466906750 \nu^{13} + 134283386850 \nu^{12} + 913861053180 \nu^{11} - 2202776644836 \nu^{10} - 6571776199200 \nu^{9} + 16265489440320 \nu^{8} + 33371139972885 \nu^{7} - 83697411098964 \nu^{6} - 59601552530850 \nu^{5} + 172537945734330 \nu^{4} + 200332462815450 \nu^{3} - 520573183999612 \nu^{2} + 398036650336800 \nu - 1112759128092900$$$$)/ 118528833274800$$ $$\beta_{8}$$ $$=$$ $$($$$$-2076350345 \nu^{15} + 5205196818 \nu^{14} + 77769418120 \nu^{13} - 191292621510 \nu^{12} - 1277152417140 \nu^{11} + 3066501922596 \nu^{10} + 9368399257860 \nu^{9} - 21425533302600 \nu^{8} - 45792427387785 \nu^{7} + 103077458167554 \nu^{6} + 67372557930120 \nu^{5} - 137380724700990 \nu^{4} - 154793304903230 \nu^{3} + 406430962301232 \nu^{2} - 719198436052100 \nu + 1732212184810500$$$$)/ 118528833274800$$ $$\beta_{9}$$ $$=$$ $$($$$$155639007 \nu^{15} - 5942535720 \nu^{13} + 100420254144 \nu^{11} - 788497463630 \nu^{9} + 4203565203591 \nu^{7} - 9115443347310 \nu^{5} + 19518721769358 \nu^{3} + 45288096872360 \nu$$$$)/ 3950961109160$$ $$\beta_{10}$$ $$=$$ $$($$$$801486 \nu^{15} - 28294900 \nu^{13} + 429690357 \nu^{11} - 2628310960 \nu^{9} + 11370098493 \nu^{7} - 281947220 \nu^{5} + 14646759714 \nu^{3} + 482698562980 \nu$$$$)/ 19179422860$$ $$\beta_{11}$$ $$=$$ $$($$$$5705218035 \nu^{15} - 6408693652 \nu^{14} - 211546219050 \nu^{13} + 238122751500 \nu^{12} + 3420653524020 \nu^{11} - 3887945167904 \nu^{10} - 24206016760200 \nu^{9} + 28382520563880 \nu^{8} + 114761159757855 \nu^{7} - 142929290254036 \nu^{6} - 157253447154150 \nu^{5} + 236663134147020 \nu^{4} + 491451941776590 \nu^{3} - 738537101120808 \nu^{2} + 1533773911163400 \nu - 2216553994491600$$$$)/ 118528833274800$$ $$\beta_{12}$$ $$=$$ $$($$$$-1529999985 \nu^{15} - 2219173136 \nu^{14} + 56466906750 \nu^{13} + 83688034025 \nu^{12} - 913861053180 \nu^{11} - 1385839876992 \nu^{10} + 6571776199200 \nu^{9} + 10393466827065 \nu^{8} - 33371139972885 \nu^{7} - 53011602819708 \nu^{6} + 59601552530850 \nu^{5} + 107650554776160 \nu^{4} - 200332462815450 \nu^{3} - 283353327329264 \nu^{2} - 398036650336800 \nu - 642863561395300$$$$)/ 29632208318700$$ $$\beta_{13}$$ $$=$$ $$($$$$-658212131 \nu^{15} + 24539889600 \nu^{13} - 403007898822 \nu^{11} + 2985858054180 \nu^{9} - 15324679919913 \nu^{7} + 26662096059480 \nu^{5} - 71082082542134 \nu^{3} - 282904140779460 \nu$$$$)/ 11852883327480$$ $$\beta_{14}$$ $$=$$ $$($$$$2076350345 \nu^{15} + 2926025931 \nu^{14} - 77769418120 \nu^{13} - 108048759270 \nu^{12} + 1277152417140 \nu^{11} + 1745559112107 \nu^{10} - 9368399257860 \nu^{9} - 12437911896750 \nu^{8} + 45792427387785 \nu^{7} + 61232579464368 \nu^{6} - 67372557930120 \nu^{5} - 91515994455180 \nu^{4} + 154793304903230 \nu^{3} + 240613677533244 \nu^{2} + 719198436052100 \nu + 1052434305900000$$$$)/ 29632208318700$$ $$\beta_{15}$$ $$=$$ $$($$$$84726489 \nu^{15} - 3193068843 \nu^{13} + 52930433070 \nu^{11} - 397922684157 \nu^{9} + 2026143959463 \nu^{7} - 3782989318554 \nu^{5} + 9131962809318 \nu^{3} + 29738030133204 \nu$$$$)/ 395096110916$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} - 2 \beta_{11} - 3 \beta_{9} - \beta_{5}$$$$)/15$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{4} - 6 \beta_{2} - 9 \beta_{1} + 14$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{15} - 9 \beta_{13} - 15 \beta_{12} - 18 \beta_{11} - 9 \beta_{10} - 36 \beta_{9} + 30 \beta_{7} + 18 \beta_{6} - 9 \beta_{5} + 9 \beta_{4} - 15 \beta_{3}$$$$)/15$$ $$\nu^{4}$$ $$=$$ $$($$$$4 \beta_{14} + 16 \beta_{8} - 18 \beta_{5} - 9 \beta_{4} - 36 \beta_{3} - 76 \beta_{2} - 252 \beta_{1} + 69$$$$)/3$$ $$\nu^{5}$$ $$=$$ $$($$$$-19 \beta_{15} - 150 \beta_{14} - 360 \beta_{13} - 225 \beta_{12} - 52 \beta_{11} - 120 \beta_{10} - 363 \beta_{9} + 300 \beta_{8} + 450 \beta_{7} + 540 \beta_{6} - 26 \beta_{5} + 270 \beta_{4} - 225 \beta_{3} + 150 \beta_{2}$$$$)/15$$ $$\nu^{6}$$ $$=$$ $$($$$$54 \beta_{14} - 60 \beta_{12} + 216 \beta_{8} - 240 \beta_{7} - 405 \beta_{5} + 64 \beta_{4} - 1140 \beta_{3} - 684 \beta_{2} - 4239 \beta_{1} - 1166$$$$)/3$$ $$\nu^{7}$$ $$=$$ $$($$$$-1066 \beta_{15} - 4725 \beta_{14} - 8109 \beta_{13} - 1540 \beta_{12} + 4572 \beta_{11} - 1449 \beta_{10} - 1806 \beta_{9} + 9450 \beta_{8} + 3080 \beta_{7} + 8568 \beta_{6} + 2286 \beta_{5} + 4284 \beta_{4} - 1540 \beta_{3} + 4725 \beta_{2}$$$$)/15$$ $$\nu^{8}$$ $$=$$ $$($$$$64 \beta_{14} - 1512 \beta_{12} + 256 \beta_{8} - 6048 \beta_{7} - 5208 \beta_{5} + 5121 \beta_{4} - 23040 \beta_{3} + 944 \beta_{2} - 50232 \beta_{1} - 54681$$$$)/3$$ $$\nu^{9}$$ $$=$$ $$($$$$-27269 \beta_{15} - 84600 \beta_{14} - 135000 \beta_{13} + 26325 \beta_{12} + 151108 \beta_{11} - 11880 \beta_{10} + 48087 \beta_{9} + 169200 \beta_{8} - 52650 \beta_{7} + 77760 \beta_{6} + 75554 \beta_{5} + 38880 \beta_{4} + 26325 \beta_{3} + 84600 \beta_{2}$$$$)/15$$ $$\nu^{10}$$ $$=$$ $$($$$$-14922 \beta_{14} - 22320 \beta_{12} - 59688 \beta_{8} - 89280 \beta_{7} - 23085 \beta_{5} + 126314 \beta_{4} - 326880 \beta_{3} + 234348 \beta_{2} - 206469 \beta_{1} - 1274566$$$$)/3$$ $$\nu^{11}$$ $$=$$ $$($$$$-511286 \beta_{15} - 957825 \beta_{14} - 1512459 \beta_{13} + 1221110 \beta_{12} + 2955312 \beta_{11} + 76461 \beta_{10} + 1989174 \beta_{9} + 1915650 \beta_{8} - 2442220 \beta_{7} - 369732 \beta_{6} + 1477656 \beta_{5} - 184866 \beta_{4} + 1221110 \beta_{3} + 957825 \beta_{2}$$$$)/15$$ $$\nu^{12}$$ $$=$$ $$-145832 \beta_{14} - 56628 \beta_{12} - 583328 \beta_{8} - 226512 \beta_{7} + 305844 \beta_{5} + 684117 \beta_{4} - 824088 \beta_{3} + 2195168 \beta_{2} + 3123516 \beta_{1} - 6858577$$ $$\nu^{13}$$ $$=$$ $$($$$$-6773419 \beta_{15} - 1797900 \beta_{14} - 2693340 \beta_{13} + 27070875 \beta_{12} + 39166868 \beta_{11} + 6034860 \beta_{10} + 43688037 \beta_{9} + 3595800 \beta_{8} - 54141750 \beta_{7} - 34369920 \beta_{6} + 19583434 \beta_{5} - 17184960 \beta_{4} + 27070875 \beta_{3} + 1797900 \beta_{2}$$$$)/15$$ $$\nu^{14}$$ $$=$$ $$($$$$-7976178 \beta_{14} + 1870440 \beta_{12} - 31904712 \beta_{8} + 7481760 \beta_{7} + 33427485 \beta_{5} + 20662174 \beta_{4} + 28504320 \beta_{3} + 119841540 \beta_{2} + 337238001 \beta_{1} - 207545486$$$$)/3$$ $$\nu^{15}$$ $$=$$ $$($$$$-39995806 \beta_{15} + 239583825 \beta_{14} + 388141191 \beta_{13} + 409395810 \beta_{12} + 230500152 \beta_{11} + 154135971 \beta_{10} + 661539654 \beta_{9} - 479167650 \beta_{8} - 818791620 \beta_{7} - 881832132 \beta_{6} + 115250076 \beta_{5} - 440916066 \beta_{4} + 409395810 \beta_{3} - 239583825 \beta_{2}$$$$)/15$$

## Character values

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

 $$n$$ $$26$$ $$52$$ $$\chi(n)$$ $$-1$$ $$-\beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 1.55042 − 1.22474i −1.55042 + 1.22474i 0.0852547 + 1.22474i −0.0852547 − 1.22474i 2.36424 + 1.22474i −2.36424 − 1.22474i −3.99991 − 1.22474i 3.99991 + 1.22474i 1.55042 + 1.22474i −1.55042 − 1.22474i 0.0852547 − 1.22474i −0.0852547 + 1.22474i 2.36424 − 1.22474i −2.36424 + 1.22474i −3.99991 + 1.22474i 3.99991 − 1.22474i
−3.39887 + 3.39887i 0.624963 + 5.15843i 15.1047i 0 −19.6570 15.4087i −19.7241 19.7241i 24.1479 + 24.1479i −26.2188 + 6.44766i 0
32.2 −3.39887 + 3.39887i 5.15843 + 0.624963i 15.1047i 0 −19.6570 + 15.4087i 19.7241 + 19.7241i 24.1479 + 24.1479i 26.2188 + 6.44766i 0
32.3 −1.39558 + 1.39558i −4.93508 1.62635i 4.10469i 0 9.15703 4.61761i −3.80245 3.80245i −16.8931 16.8931i 21.7100 + 16.0523i 0
32.4 −1.39558 + 1.39558i −1.62635 4.93508i 4.10469i 0 9.15703 + 4.61761i 3.80245 + 3.80245i −16.8931 16.8931i −21.7100 + 16.0523i 0
32.5 1.39558 1.39558i 1.62635 + 4.93508i 4.10469i 0 9.15703 + 4.61761i −3.80245 3.80245i 16.8931 + 16.8931i −21.7100 + 16.0523i 0
32.6 1.39558 1.39558i 4.93508 + 1.62635i 4.10469i 0 9.15703 4.61761i 3.80245 + 3.80245i 16.8931 + 16.8931i 21.7100 + 16.0523i 0
32.7 3.39887 3.39887i −5.15843 0.624963i 15.1047i 0 −19.6570 + 15.4087i −19.7241 19.7241i −24.1479 24.1479i 26.2188 + 6.44766i 0
32.8 3.39887 3.39887i −0.624963 5.15843i 15.1047i 0 −19.6570 15.4087i 19.7241 + 19.7241i −24.1479 24.1479i −26.2188 + 6.44766i 0
68.1 −3.39887 3.39887i 0.624963 5.15843i 15.1047i 0 −19.6570 + 15.4087i −19.7241 + 19.7241i 24.1479 24.1479i −26.2188 6.44766i 0
68.2 −3.39887 3.39887i 5.15843 0.624963i 15.1047i 0 −19.6570 15.4087i 19.7241 19.7241i 24.1479 24.1479i 26.2188 6.44766i 0
68.3 −1.39558 1.39558i −4.93508 + 1.62635i 4.10469i 0 9.15703 + 4.61761i −3.80245 + 3.80245i −16.8931 + 16.8931i 21.7100 16.0523i 0
68.4 −1.39558 1.39558i −1.62635 + 4.93508i 4.10469i 0 9.15703 4.61761i 3.80245 3.80245i −16.8931 + 16.8931i −21.7100 16.0523i 0
68.5 1.39558 + 1.39558i 1.62635 4.93508i 4.10469i 0 9.15703 4.61761i −3.80245 + 3.80245i 16.8931 16.8931i −21.7100 16.0523i 0
68.6 1.39558 + 1.39558i 4.93508 1.62635i 4.10469i 0 9.15703 + 4.61761i 3.80245 3.80245i 16.8931 16.8931i 21.7100 16.0523i 0
68.7 3.39887 + 3.39887i −5.15843 + 0.624963i 15.1047i 0 −19.6570 15.4087i −19.7241 + 19.7241i −24.1479 + 24.1479i 26.2188 6.44766i 0
68.8 3.39887 + 3.39887i −0.624963 + 5.15843i 15.1047i 0 −19.6570 + 15.4087i 19.7241 19.7241i −24.1479 + 24.1479i −26.2188 6.44766i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 68.8 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
5.b even 2 1 inner
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.4.e.d 16
3.b odd 2 1 inner 75.4.e.d 16
5.b even 2 1 inner 75.4.e.d 16
5.c odd 4 2 inner 75.4.e.d 16
15.d odd 2 1 inner 75.4.e.d 16
15.e even 4 2 inner 75.4.e.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.e.d 16 1.a even 1 1 trivial
75.4.e.d 16 3.b odd 2 1 inner
75.4.e.d 16 5.b even 2 1 inner
75.4.e.d 16 5.c odd 4 2 inner
75.4.e.d 16 15.d odd 2 1 inner
75.4.e.d 16 15.e even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 549 T_{2}^{4} + 8100$$ acting on $$S_{4}^{\mathrm{new}}(75, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 8100 + 549 T^{4} + T^{8} )^{2}$$
$3$ $$282429536481 - 913547079 T^{4} + 1614816 T^{8} - 1719 T^{12} + T^{16}$$
$5$ $$T^{16}$$
$7$ $$( 506250000 + 606249 T^{4} + T^{8} )^{2}$$
$11$ $$( 324000 + 3915 T^{2} + T^{4} )^{4}$$
$13$ $$( 21767823360000 + 22512249 T^{4} + T^{8} )^{2}$$
$17$ $$( 539435927577600 + 46806129 T^{4} + T^{8} )^{2}$$
$19$ $$( 2399401 + 9002 T^{2} + T^{4} )^{4}$$
$23$ $$( 113427612057600 + 26974224 T^{4} + T^{8} )^{2}$$
$29$ $$( 1327104000 - 73440 T^{2} + T^{4} )^{4}$$
$31$ $$( 3394 + 151 T + T^{2} )^{8}$$
$37$ $$( 58027829760000 + 1163451024 T^{4} + T^{8} )^{2}$$
$41$ $$( 324000 + 3915 T^{2} + T^{4} )^{4}$$
$43$ $$( 82542726506250000 + 1349357769 T^{4} + T^{8} )^{2}$$
$47$ $$( 3652034743605657600 + 86581592064 T^{4} + T^{8} )^{2}$$
$53$ $$($$$$18\!\cdots\!00$$$$+ 163633806864 T^{4} + T^{8} )^{2}$$
$59$ $$( 6718464000 - 166860 T^{2} + T^{4} )^{4}$$
$61$ $$( -103886 + 191 T + T^{2} )^{8}$$
$67$ $$($$$$30\!\cdots\!25$$$$+ 754749377874 T^{4} + T^{8} )^{2}$$
$71$ $$( 613453824000 + 1697760 T^{2} + T^{4} )^{4}$$
$73$ $$($$$$57\!\cdots\!00$$$$+ 777732606009 T^{4} + T^{8} )^{2}$$
$79$ $$( 19593280576 + 500852 T^{2} + T^{4} )^{4}$$
$83$ $$( 36479156981701017600 + 351279805329 T^{4} + T^{8} )^{2}$$
$89$ $$( 16548624000 - 370035 T^{2} + T^{4} )^{4}$$
$97$ $$( 19519554713760000 + 332989209 T^{4} + T^{8} )^{2}$$