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 $8$

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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:

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

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