# Properties

 Label 76.6.e.a Level $76$ Weight $6$ Character orbit 76.e Analytic conductor $12.189$ Analytic rank $0$ Dimension $18$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 76.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.1891703058$$ Analytic rank: $$0$$ Dimension: $$18$$ Relative dimension: $$9$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{18} - \cdots)$$ Defining polynomial: $$x^{18} - 2 x^{17} + 1540 x^{16} - 768 x^{15} + 1608492 x^{14} - 1027368 x^{13} + 897054160 x^{12} - 1275481376 x^{11} + 361098181456 x^{10} - 863969476320 x^{9} + 79755165392064 x^{8} - 375077568148992 x^{7} + 12736924096193536 x^{6} - 57314532742553600 x^{5} + 977121800205220864 x^{4} - 4977732006498379776 x^{3} + 53672321824823513088 x^{2} - 185653809995679793152 x + 804303742853852430336$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{16}\cdot 3^{3}\cdot 5^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{17}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{3} + ( \beta_{2} + \beta_{6} + \beta_{8} ) q^{5} + ( 18 - 2 \beta_{3} + \beta_{4} - \beta_{9} ) q^{7} + ( -99 - 3 \beta_{1} + 99 \beta_{2} + \beta_{6} - \beta_{7} - \beta_{10} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{3} + ( \beta_{2} + \beta_{6} + \beta_{8} ) q^{5} + ( 18 - 2 \beta_{3} + \beta_{4} - \beta_{9} ) q^{7} + ( -99 - 3 \beta_{1} + 99 \beta_{2} + \beta_{6} - \beta_{7} - \beta_{10} ) q^{9} + ( -18 - \beta_{3} - \beta_{13} ) q^{11} + ( 23 + 5 \beta_{1} - 23 \beta_{2} - 3 \beta_{6} + \beta_{7} + \beta_{10} + \beta_{14} - \beta_{16} ) q^{13} + ( -15 + 11 \beta_{1} + 15 \beta_{2} - \beta_{5} - 3 \beta_{6} + \beta_{7} + \beta_{10} ) q^{15} + ( -1 - 15 \beta_{1} + 17 \beta_{2} - 16 \beta_{3} + \beta_{6} + \beta_{8} + \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} - \beta_{16} - 2 \beta_{17} ) q^{17} + ( -172 + 13 \beta_{1} + 247 \beta_{2} + 6 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} - 3 \beta_{8} + 2 \beta_{9} - \beta_{11} + 2 \beta_{13} + \beta_{14} - \beta_{15} - 2 \beta_{16} ) q^{19} + ( -1 + 30 \beta_{1} - 621 \beta_{2} + 29 \beta_{3} - 2 \beta_{5} - 3 \beta_{6} - 3 \beta_{8} + 6 \beta_{9} + 8 \beta_{10} - \beta_{11} - 2 \beta_{12} - 5 \beta_{13} + \beta_{14} + 5 \beta_{15} - \beta_{16} - 2 \beta_{17} ) q^{21} + ( -378 - 17 \beta_{1} + 377 \beta_{2} - \beta_{3} + 6 \beta_{4} + \beta_{5} - 2 \beta_{7} - 2 \beta_{10} - 2 \beta_{11} + 3 \beta_{14} - \beta_{15} - 4 \beta_{16} - \beta_{17} ) q^{23} + ( -775 - 20 \beta_{1} + 775 \beta_{2} - 10 \beta_{4} - 2 \beta_{5} + 10 \beta_{6} - 2 \beta_{7} - 2 \beta_{10} + \beta_{14} - 4 \beta_{15} - \beta_{16} ) q^{25} + ( 789 + \beta_{1} + \beta_{2} - 79 \beta_{3} + 15 \beta_{4} + 8 \beta_{7} + 18 \beta_{8} - 15 \beta_{9} + \beta_{11} + \beta_{12} - 4 \beta_{13} + 4 \beta_{14} - \beta_{17} ) q^{27} + ( -832 + 60 \beta_{1} + 833 \beta_{2} + \beta_{3} + 6 \beta_{4} - 2 \beta_{5} - 12 \beta_{6} - 7 \beta_{7} - 7 \beta_{10} + 2 \beta_{11} + 3 \beta_{14} + 3 \beta_{15} - 2 \beta_{16} + \beta_{17} ) q^{29} + ( -560 - \beta_{1} - \beta_{2} - 19 \beta_{3} + 11 \beta_{4} - 7 \beta_{7} + 13 \beta_{8} - 11 \beta_{9} - \beta_{11} + 2 \beta_{12} - 9 \beta_{13} - 6 \beta_{14} + \beta_{17} ) q^{31} + ( -1 + 46 \beta_{1} - 317 \beta_{2} + 45 \beta_{3} - 28 \beta_{6} - 28 \beta_{8} - 24 \beta_{9} + 12 \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} - 7 \beta_{16} - 2 \beta_{17} ) q^{33} + ( -41 \beta_{1} + 1728 \beta_{2} - 41 \beta_{3} + \beta_{5} + 97 \beta_{6} + 97 \beta_{8} + 15 \beta_{9} - 27 \beta_{10} + \beta_{12} - 5 \beta_{13} + 5 \beta_{15} + 6 \beta_{16} ) q^{35} + ( 1454 - 81 \beta_{3} + 20 \beta_{4} + 9 \beta_{7} - 42 \beta_{8} - 20 \beta_{9} - 4 \beta_{12} - 8 \beta_{13} + 3 \beta_{14} ) q^{37} + ( -1633 - \beta_{1} - \beta_{2} + 192 \beta_{3} + 9 \beta_{4} - 20 \beta_{7} + 6 \beta_{8} - 9 \beta_{9} - \beta_{11} - 3 \beta_{12} + 15 \beta_{13} + 2 \beta_{14} + \beta_{17} ) q^{39} + ( -47 \beta_{1} - 830 \beta_{2} - 47 \beta_{3} + 55 \beta_{6} + 55 \beta_{8} + 15 \beta_{10} + 16 \beta_{13} - 16 \beta_{15} ) q^{41} + ( 1 + 148 \beta_{1} - 882 \beta_{2} + 149 \beta_{3} + 5 \beta_{5} - 16 \beta_{6} - 16 \beta_{8} + 29 \beta_{9} - 6 \beta_{10} + \beta_{11} + 5 \beta_{12} + 11 \beta_{13} - \beta_{14} - 11 \beta_{15} + 6 \beta_{16} + 2 \beta_{17} ) q^{43} + ( -3340 + \beta_{1} + \beta_{2} + 273 \beta_{3} - 90 \beta_{4} - 27 \beta_{7} - 103 \beta_{8} + 90 \beta_{9} + \beta_{11} - 2 \beta_{12} - 5 \beta_{13} - 8 \beta_{14} - \beta_{17} ) q^{45} + ( 4173 - 53 \beta_{1} - 4173 \beta_{2} + 30 \beta_{4} - \beta_{5} - 19 \beta_{6} + 21 \beta_{7} + 21 \beta_{10} - 12 \beta_{14} + 32 \beta_{15} + 12 \beta_{16} ) q^{47} + ( 6810 - \beta_{1} - \beta_{2} - 474 \beta_{3} + 14 \beta_{4} + 66 \beta_{7} - 44 \beta_{8} - 14 \beta_{9} - \beta_{11} + 2 \beta_{12} + 9 \beta_{13} - 8 \beta_{14} + \beta_{17} ) q^{49} + ( 5230 - 70 \beta_{1} - 5229 \beta_{2} + \beta_{3} - 96 \beta_{4} + 2 \beta_{5} - 163 \beta_{6} + 59 \beta_{7} + 59 \beta_{10} + 2 \beta_{11} - 3 \beta_{14} + 16 \beta_{15} + 4 \beta_{16} + \beta_{17} ) q^{51} + ( -2212 - 323 \beta_{1} + 2213 \beta_{2} + \beta_{3} - 66 \beta_{4} + 10 \beta_{5} - 228 \beta_{6} - 22 \beta_{7} - 22 \beta_{10} + 2 \beta_{11} - 6 \beta_{14} - 13 \beta_{15} + 7 \beta_{16} + \beta_{17} ) q^{53} + ( 10 - 172 \beta_{1} + 60 \beta_{2} - 162 \beta_{3} + 4 \beta_{5} - 202 \beta_{6} - 202 \beta_{8} - 71 \beta_{9} - 14 \beta_{10} + 10 \beta_{11} + 4 \beta_{12} + 30 \beta_{13} - 10 \beta_{14} - 30 \beta_{15} + 20 \beta_{16} + 20 \beta_{17} ) q^{55} + ( -4137 + 127 \beta_{1} + 1971 \beta_{2} - 97 \beta_{3} - 48 \beta_{4} + 6 \beta_{5} + 92 \beta_{6} - 55 \beta_{7} - 175 \beta_{8} + 126 \beta_{9} - 54 \beta_{10} + 11 \beta_{11} - 2 \beta_{12} + 13 \beta_{13} - \beta_{14} - 16 \beta_{15} + 9 \beta_{16} + \beta_{17} ) q^{57} + ( 11 + 167 \beta_{1} - 8230 \beta_{2} + 178 \beta_{3} + 17 \beta_{5} + 102 \beta_{6} + 102 \beta_{8} + 129 \beta_{9} + 28 \beta_{10} + 11 \beta_{11} + 17 \beta_{12} + 10 \beta_{13} - 11 \beta_{14} - 10 \beta_{15} + 14 \beta_{16} + 22 \beta_{17} ) q^{59} + ( -849 - 32 \beta_{1} + 859 \beta_{2} + 10 \beta_{3} + 120 \beta_{4} - 4 \beta_{5} - 19 \beta_{6} - 30 \beta_{7} - 30 \beta_{10} + 20 \beta_{11} - 14 \beta_{14} - 2 \beta_{15} + 24 \beta_{16} + 10 \beta_{17} ) q^{61} + ( -5288 - 1660 \beta_{1} + 5288 \beta_{2} - 240 \beta_{4} + 16 \beta_{5} + 504 \beta_{6} - 28 \beta_{7} - 28 \beta_{10} - 12 \beta_{14} + 12 \beta_{16} ) q^{63} + ( 10487 - 10 \beta_{1} - 10 \beta_{2} + 129 \beta_{3} + 186 \beta_{4} + 55 \beta_{7} - 29 \beta_{8} - 186 \beta_{9} - 10 \beta_{11} - 10 \beta_{12} + 10 \beta_{13} - 25 \beta_{14} + 10 \beta_{17} ) q^{65} + ( -3029 + 532 \beta_{1} + 3019 \beta_{2} - 10 \beta_{3} + 29 \beta_{4} + 19 \beta_{5} - 103 \beta_{6} - 71 \beta_{7} - 71 \beta_{10} - 20 \beta_{11} - 24 \beta_{14} + 27 \beta_{15} + 14 \beta_{16} - 10 \beta_{17} ) q^{67} + ( 6091 + 10 \beta_{1} + 10 \beta_{2} - 994 \beta_{3} + 108 \beta_{4} - 2 \beta_{7} + 547 \beta_{8} - 108 \beta_{9} + 10 \beta_{11} - 16 \beta_{12} + 14 \beta_{13} + 28 \beta_{14} - 10 \beta_{17} ) q^{69} + ( 11 + 1382 \beta_{1} - 5644 \beta_{2} + 1393 \beta_{3} + \beta_{5} - 172 \beta_{6} - 172 \beta_{8} - 201 \beta_{9} + 10 \beta_{10} + 11 \beta_{11} + \beta_{12} + 13 \beta_{13} - 11 \beta_{14} - 13 \beta_{15} + 44 \beta_{16} + 22 \beta_{17} ) q^{71} + ( 1 + 1134 \beta_{1} - 1431 \beta_{2} + 1135 \beta_{3} - 6 \beta_{5} + 700 \beta_{6} + 700 \beta_{8} + 154 \beta_{9} + 2 \beta_{10} + \beta_{11} - 6 \beta_{12} + 17 \beta_{13} - \beta_{14} - 17 \beta_{15} - 26 \beta_{16} + 2 \beta_{17} ) q^{73} + ( 6562 - 1181 \beta_{3} + 60 \beta_{4} + 28 \beta_{7} - 620 \beta_{8} - 60 \beta_{9} + 32 \beta_{12} + 31 \beta_{13} - 24 \beta_{14} ) q^{75} + ( -1540 + 11 \beta_{1} + 11 \beta_{2} + 1448 \beta_{3} + 6 \beta_{4} - 2 \beta_{7} + 325 \beta_{8} - 6 \beta_{9} + 11 \beta_{11} + 26 \beta_{12} - 39 \beta_{13} - 13 \beta_{14} - 11 \beta_{17} ) q^{77} + ( 1 - 1476 \beta_{1} + 3094 \beta_{2} - 1475 \beta_{3} + 3 \beta_{5} + 540 \beta_{6} + 540 \beta_{8} + 21 \beta_{9} + 10 \beta_{10} + \beta_{11} + 3 \beta_{12} - 57 \beta_{13} - \beta_{14} + 57 \beta_{15} + 8 \beta_{16} + 2 \beta_{17} ) q^{79} + ( -10 + 2235 \beta_{1} - 1686 \beta_{2} + 2225 \beta_{3} - 44 \beta_{5} - 457 \beta_{6} - 457 \beta_{8} + 60 \beta_{9} + 89 \beta_{10} - 10 \beta_{11} - 44 \beta_{12} - 34 \beta_{13} + 10 \beta_{14} + 34 \beta_{15} - 40 \beta_{16} - 20 \beta_{17} ) q^{81} + ( 4516 - 11 \beta_{1} - 11 \beta_{2} - 291 \beta_{3} - 117 \beta_{4} + 134 \beta_{7} - 68 \beta_{8} + 117 \beta_{9} - 11 \beta_{11} + 15 \beta_{12} + 41 \beta_{13} + 52 \beta_{14} + 11 \beta_{17} ) q^{83} + ( -5427 - 1275 \beta_{1} + 5427 \beta_{2} + 12 \beta_{5} - 303 \beta_{6} - 15 \beta_{7} - 15 \beta_{10} + 81 \beta_{14} - 144 \beta_{15} - 81 \beta_{16} ) q^{85} + ( -19223 + 11 \beta_{1} + 11 \beta_{2} - 2340 \beta_{3} - 24 \beta_{4} - 124 \beta_{7} - 598 \beta_{8} + 24 \beta_{9} + 11 \beta_{11} - 19 \beta_{12} - 51 \beta_{13} + 62 \beta_{14} - 11 \beta_{17} ) q^{87} + ( -17077 - 894 \beta_{1} + 17066 \beta_{2} - 11 \beta_{3} - 24 \beta_{4} - 16 \beta_{5} - 347 \beta_{6} - 151 \beta_{7} - 151 \beta_{10} - 22 \beta_{11} - 12 \beta_{14} - 81 \beta_{15} + \beta_{16} - 11 \beta_{17} ) q^{89} + ( -606 - 104 \beta_{1} + 596 \beta_{2} - 10 \beta_{3} - 50 \beta_{4} - 40 \beta_{5} - 1170 \beta_{6} + 138 \beta_{7} + 138 \beta_{10} - 20 \beta_{11} + 38 \beta_{14} + 86 \beta_{15} - 48 \beta_{16} - 10 \beta_{17} ) q^{91} + ( -34 - 765 \beta_{1} - 5050 \beta_{2} - 799 \beta_{3} - 36 \beta_{5} - 886 \beta_{6} - 886 \beta_{8} + 72 \beta_{9} + 111 \beta_{10} - 34 \beta_{11} - 36 \beta_{12} - 154 \beta_{13} + 34 \beta_{14} + 154 \beta_{15} - 115 \beta_{16} - 68 \beta_{17} ) q^{93} + ( 10183 + 471 \beta_{1} - 15995 \beta_{2} - 2678 \beta_{3} + 123 \beta_{4} + \beta_{5} + 334 \beta_{6} + 129 \beta_{7} - 541 \beta_{8} - 36 \beta_{9} + 76 \beta_{10} - 46 \beta_{11} + 21 \beta_{12} - 98 \beta_{13} - 23 \beta_{14} + 97 \beta_{15} - 4 \beta_{16} - 11 \beta_{17} ) q^{95} + ( -45 - 1061 \beta_{1} + 12028 \beta_{2} - 1106 \beta_{3} - 50 \beta_{5} + 721 \beta_{6} + 721 \beta_{8} + 134 \beta_{9} + \beta_{10} - 45 \beta_{11} - 50 \beta_{12} - 5 \beta_{13} + 45 \beta_{14} + 5 \beta_{15} - 64 \beta_{16} - 90 \beta_{17} ) q^{97} + ( -21070 - 2225 \beta_{1} + 21036 \beta_{2} - 34 \beta_{3} + 27 \beta_{4} - 17 \beta_{5} - 99 \beta_{6} + 21 \beta_{7} + 21 \beta_{10} - 68 \beta_{11} + 3 \beta_{15} - 34 \beta_{16} - 34 \beta_{17} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$18q - 11q^{3} + 11q^{5} + 336q^{7} - 902q^{9} + O(q^{10})$$ $$18q - 11q^{3} + 11q^{5} + 336q^{7} - 902q^{9} - 320q^{11} + 227q^{13} - 101q^{15} + 179q^{17} - 868q^{19} - 5700q^{21} - 3425q^{23} - 7054q^{25} + 14722q^{27} - 7349q^{29} - 9960q^{31} - 2998q^{33} + 15888q^{35} + 26444q^{37} - 30246q^{39} - 7311q^{41} - 8283q^{43} - 62164q^{45} + 37603q^{47} + 124738q^{49} + 47227q^{51} - 20337q^{53} + 716q^{55} - 57555q^{57} - 74455q^{59} - 7569q^{61} - 52544q^{63} + 188998q^{65} - 26177q^{67} + 116282q^{69} - 53463q^{71} - 14103q^{73} + 120912q^{75} - 31960q^{77} + 31825q^{79} - 21137q^{81} + 82600q^{83} - 50787q^{85} - 339766q^{87} - 155197q^{89} - 2800q^{91} - 46460q^{93} + 49315q^{95} + 111241q^{97} - 193544q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{18} - 2 x^{17} + 1540 x^{16} - 768 x^{15} + 1608492 x^{14} - 1027368 x^{13} + 897054160 x^{12} - 1275481376 x^{11} + 361098181456 x^{10} - 863969476320 x^{9} + 79755165392064 x^{8} - 375077568148992 x^{7} + 12736924096193536 x^{6} - 57314532742553600 x^{5} + 977121800205220864 x^{4} - 4977732006498379776 x^{3} + 53672321824823513088 x^{2} - 185653809995679793152 x + 804303742853852430336$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-$$$$47\!\cdots\!39$$$$\nu^{17} +$$$$17\!\cdots\!42$$$$\nu^{16} -$$$$72\!\cdots\!28$$$$\nu^{15} +$$$$13\!\cdots\!12$$$$\nu^{14} -$$$$75\!\cdots\!08$$$$\nu^{13} +$$$$15\!\cdots\!68$$$$\nu^{12} -$$$$41\!\cdots\!16$$$$\nu^{11} +$$$$10\!\cdots\!56$$$$\nu^{10} -$$$$16\!\cdots\!24$$$$\nu^{9} +$$$$55\!\cdots\!96$$$$\nu^{8} -$$$$36\!\cdots\!56$$$$\nu^{7} +$$$$18\!\cdots\!88$$$$\nu^{6} -$$$$58\!\cdots\!80$$$$\nu^{5} +$$$$25\!\cdots\!96$$$$\nu^{4} -$$$$40\!\cdots\!08$$$$\nu^{3} +$$$$11\!\cdots\!36$$$$\nu^{2} -$$$$21\!\cdots\!64$$$$\nu +$$$$71\!\cdots\!56$$$$)/$$$$80\!\cdots\!68$$ $$\beta_{3}$$ $$=$$ $$($$$$-$$$$21\!\cdots\!33$$$$\nu^{17} -$$$$94\!\cdots\!04$$$$\nu^{16} -$$$$27\!\cdots\!20$$$$\nu^{15} -$$$$20\!\cdots\!60$$$$\nu^{14} -$$$$27\!\cdots\!52$$$$\nu^{13} -$$$$19\!\cdots\!28$$$$\nu^{12} -$$$$12\!\cdots\!24$$$$\nu^{11} -$$$$94\!\cdots\!20$$$$\nu^{10} -$$$$39\!\cdots\!52$$$$\nu^{9} -$$$$32\!\cdots\!80$$$$\nu^{8} -$$$$20\!\cdots\!00$$$$\nu^{7} -$$$$48\!\cdots\!28$$$$\nu^{6} +$$$$41\!\cdots\!88$$$$\nu^{5} -$$$$15\!\cdots\!36$$$$\nu^{4} +$$$$32\!\cdots\!16$$$$\nu^{3} -$$$$11\!\cdots\!96$$$$\nu^{2} +$$$$42\!\cdots\!84$$$$\nu -$$$$10\!\cdots\!88$$$$)/$$$$21\!\cdots\!96$$ $$\beta_{4}$$ $$=$$ $$($$$$10\!\cdots\!29$$$$\nu^{17} +$$$$48\!\cdots\!10$$$$\nu^{16} +$$$$17\!\cdots\!12$$$$\nu^{15} +$$$$75\!\cdots\!16$$$$\nu^{14} +$$$$20\!\cdots\!44$$$$\nu^{13} +$$$$78\!\cdots\!44$$$$\nu^{12} +$$$$12\!\cdots\!12$$$$\nu^{11} +$$$$41\!\cdots\!40$$$$\nu^{10} +$$$$51\!\cdots\!16$$$$\nu^{9} +$$$$15\!\cdots\!00$$$$\nu^{8} +$$$$11\!\cdots\!92$$$$\nu^{7} +$$$$27\!\cdots\!36$$$$\nu^{6} +$$$$64\!\cdots\!08$$$$\nu^{5} +$$$$30\!\cdots\!40$$$$\nu^{4} +$$$$53\!\cdots\!32$$$$\nu^{3} +$$$$12\!\cdots\!00$$$$\nu^{2} -$$$$10\!\cdots\!36$$$$\nu +$$$$28\!\cdots\!72$$$$)/$$$$71\!\cdots\!72$$ $$\beta_{5}$$ $$=$$ $$($$$$-$$$$23\!\cdots\!79$$$$\nu^{17} +$$$$40\!\cdots\!58$$$$\nu^{16} -$$$$20\!\cdots\!12$$$$\nu^{15} +$$$$59\!\cdots\!24$$$$\nu^{14} -$$$$66\!\cdots\!92$$$$\nu^{13} +$$$$61\!\cdots\!16$$$$\nu^{12} +$$$$83\!\cdots\!48$$$$\nu^{11} +$$$$32\!\cdots\!72$$$$\nu^{10} +$$$$30\!\cdots\!24$$$$\nu^{9} +$$$$12\!\cdots\!60$$$$\nu^{8} +$$$$75\!\cdots\!92$$$$\nu^{7} +$$$$24\!\cdots\!64$$$$\nu^{6} -$$$$73\!\cdots\!68$$$$\nu^{5} +$$$$29\!\cdots\!28$$$$\nu^{4} -$$$$80\!\cdots\!32$$$$\nu^{3} +$$$$11\!\cdots\!80$$$$\nu^{2} -$$$$75\!\cdots\!56$$$$\nu +$$$$19\!\cdots\!88$$$$)/$$$$79\!\cdots\!08$$ $$\beta_{6}$$ $$=$$ $$($$$$53\!\cdots\!01$$$$\nu^{17} +$$$$75\!\cdots\!70$$$$\nu^{16} +$$$$85\!\cdots\!32$$$$\nu^{15} +$$$$12\!\cdots\!80$$$$\nu^{14} +$$$$92\!\cdots\!84$$$$\nu^{13} +$$$$13\!\cdots\!20$$$$\nu^{12} +$$$$53\!\cdots\!16$$$$\nu^{11} +$$$$66\!\cdots\!00$$$$\nu^{10} +$$$$21\!\cdots\!64$$$$\nu^{9} +$$$$23\!\cdots\!08$$$$\nu^{8} +$$$$47\!\cdots\!44$$$$\nu^{7} +$$$$35\!\cdots\!88$$$$\nu^{6} +$$$$54\!\cdots\!88$$$$\nu^{5} +$$$$40\!\cdots\!96$$$$\nu^{4} +$$$$30\!\cdots\!12$$$$\nu^{3} +$$$$13\!\cdots\!92$$$$\nu^{2} -$$$$36\!\cdots\!32$$$$\nu +$$$$57\!\cdots\!00$$$$)/$$$$15\!\cdots\!16$$ $$\beta_{7}$$ $$=$$ $$($$$$-$$$$18\!\cdots\!65$$$$\nu^{17} +$$$$33\!\cdots\!46$$$$\nu^{16} -$$$$25\!\cdots\!12$$$$\nu^{15} +$$$$28\!\cdots\!52$$$$\nu^{14} -$$$$25\!\cdots\!00$$$$\nu^{13} +$$$$46\!\cdots\!68$$$$\nu^{12} -$$$$11\!\cdots\!80$$$$\nu^{11} +$$$$52\!\cdots\!16$$$$\nu^{10} -$$$$42\!\cdots\!52$$$$\nu^{9} +$$$$28\!\cdots\!44$$$$\nu^{8} -$$$$56\!\cdots\!12$$$$\nu^{7} +$$$$10\!\cdots\!12$$$$\nu^{6} -$$$$67\!\cdots\!12$$$$\nu^{5} +$$$$87\!\cdots\!08$$$$\nu^{4} +$$$$15\!\cdots\!92$$$$\nu^{3} +$$$$57\!\cdots\!52$$$$\nu^{2} -$$$$22\!\cdots\!48$$$$\nu -$$$$15\!\cdots\!36$$$$)/$$$$51\!\cdots\!24$$ $$\beta_{8}$$ $$=$$ $$($$$$20\!\cdots\!93$$$$\nu^{17} -$$$$14\!\cdots\!22$$$$\nu^{16} +$$$$29\!\cdots\!08$$$$\nu^{15} -$$$$19\!\cdots\!88$$$$\nu^{14} +$$$$29\!\cdots\!80$$$$\nu^{13} -$$$$21\!\cdots\!64$$$$\nu^{12} +$$$$14\!\cdots\!56$$$$\nu^{11} -$$$$14\!\cdots\!20$$$$\nu^{10} +$$$$53\!\cdots\!24$$$$\nu^{9} -$$$$64\!\cdots\!92$$$$\nu^{8} +$$$$86\!\cdots\!28$$$$\nu^{7} -$$$$17\!\cdots\!88$$$$\nu^{6} +$$$$13\!\cdots\!68$$$$\nu^{5} -$$$$21\!\cdots\!24$$$$\nu^{4} +$$$$44\!\cdots\!88$$$$\nu^{3} -$$$$14\!\cdots\!72$$$$\nu^{2} +$$$$57\!\cdots\!20$$$$\nu -$$$$28\!\cdots\!92$$$$)/$$$$51\!\cdots\!24$$ $$\beta_{9}$$ $$=$$ $$($$$$45\!\cdots\!89$$$$\nu^{17} +$$$$27\!\cdots\!98$$$$\nu^{16} +$$$$68\!\cdots\!12$$$$\nu^{15} +$$$$48\!\cdots\!92$$$$\nu^{14} +$$$$70\!\cdots\!64$$$$\nu^{13} +$$$$46\!\cdots\!88$$$$\nu^{12} +$$$$37\!\cdots\!12$$$$\nu^{11} +$$$$19\!\cdots\!48$$$$\nu^{10} +$$$$14\!\cdots\!40$$$$\nu^{9} +$$$$53\!\cdots\!72$$$$\nu^{8} +$$$$25\!\cdots\!36$$$$\nu^{7} -$$$$22\!\cdots\!44$$$$\nu^{6} +$$$$27\!\cdots\!72$$$$\nu^{5} -$$$$32\!\cdots\!96$$$$\nu^{4} +$$$$73\!\cdots\!56$$$$\nu^{3} -$$$$11\!\cdots\!64$$$$\nu^{2} -$$$$11\!\cdots\!00$$$$\nu +$$$$13\!\cdots\!64$$$$)/$$$$71\!\cdots\!72$$ $$\beta_{10}$$ $$=$$ $$($$$$-$$$$24\!\cdots\!04$$$$\nu^{17} +$$$$18\!\cdots\!15$$$$\nu^{16} -$$$$38\!\cdots\!94$$$$\nu^{15} +$$$$20\!\cdots\!44$$$$\nu^{14} -$$$$40\!\cdots\!80$$$$\nu^{13} +$$$$20\!\cdots\!36$$$$\nu^{12} -$$$$22\!\cdots\!56$$$$\nu^{11} +$$$$11\!\cdots\!24$$$$\nu^{10} -$$$$92\!\cdots\!92$$$$\nu^{9} +$$$$52\!\cdots\!24$$$$\nu^{8} -$$$$21\!\cdots\!32$$$$\nu^{7} +$$$$13\!\cdots\!60$$$$\nu^{6} -$$$$37\!\cdots\!76$$$$\nu^{5} +$$$$18\!\cdots\!96$$$$\nu^{4} -$$$$28\!\cdots\!96$$$$\nu^{3} +$$$$77\!\cdots\!36$$$$\nu^{2} -$$$$16\!\cdots\!40$$$$\nu +$$$$57\!\cdots\!84$$$$)/$$$$19\!\cdots\!52$$ $$\beta_{11}$$ $$=$$ $$($$$$-$$$$28\!\cdots\!65$$$$\nu^{17} -$$$$48\!\cdots\!58$$$$\nu^{16} -$$$$63\!\cdots\!32$$$$\nu^{15} -$$$$75\!\cdots\!44$$$$\nu^{14} -$$$$86\!\cdots\!24$$$$\nu^{13} -$$$$78\!\cdots\!20$$$$\nu^{12} -$$$$66\!\cdots\!52$$$$\nu^{11} -$$$$43\!\cdots\!36$$$$\nu^{10} -$$$$28\!\cdots\!08$$$$\nu^{9} -$$$$16\!\cdots\!16$$$$\nu^{8} -$$$$76\!\cdots\!28$$$$\nu^{7} -$$$$34\!\cdots\!76$$$$\nu^{6} -$$$$47\!\cdots\!52$$$$\nu^{5} -$$$$47\!\cdots\!72$$$$\nu^{4} -$$$$39\!\cdots\!24$$$$\nu^{3} -$$$$27\!\cdots\!28$$$$\nu^{2} +$$$$46\!\cdots\!24$$$$\nu -$$$$96\!\cdots\!76$$$$)/$$$$14\!\cdots\!44$$ $$\beta_{12}$$ $$=$$ $$($$$$14\!\cdots\!81$$$$\nu^{17} -$$$$14\!\cdots\!86$$$$\nu^{16} +$$$$20\!\cdots\!04$$$$\nu^{15} -$$$$19\!\cdots\!64$$$$\nu^{14} +$$$$20\!\cdots\!52$$$$\nu^{13} -$$$$20\!\cdots\!00$$$$\nu^{12} +$$$$10\!\cdots\!20$$$$\nu^{11} -$$$$13\!\cdots\!48$$$$\nu^{10} +$$$$38\!\cdots\!84$$$$\nu^{9} -$$$$57\!\cdots\!64$$$$\nu^{8} +$$$$68\!\cdots\!44$$$$\nu^{7} -$$$$15\!\cdots\!08$$$$\nu^{6} +$$$$11\!\cdots\!16$$$$\nu^{5} -$$$$20\!\cdots\!92$$$$\nu^{4} +$$$$82\!\cdots\!20$$$$\nu^{3} -$$$$13\!\cdots\!92$$$$\nu^{2} +$$$$53\!\cdots\!40$$$$\nu -$$$$11\!\cdots\!68$$$$)/$$$$51\!\cdots\!24$$ $$\beta_{13}$$ $$=$$ $$($$$$96\!\cdots\!23$$$$\nu^{17} -$$$$13\!\cdots\!02$$$$\nu^{16} +$$$$14\!\cdots\!80$$$$\nu^{15} -$$$$18\!\cdots\!12$$$$\nu^{14} +$$$$14\!\cdots\!52$$$$\nu^{13} -$$$$19\!\cdots\!68$$$$\nu^{12} +$$$$72\!\cdots\!04$$$$\nu^{11} -$$$$11\!\cdots\!56$$$$\nu^{10} +$$$$27\!\cdots\!48$$$$\nu^{9} -$$$$50\!\cdots\!04$$$$\nu^{8} +$$$$54\!\cdots\!32$$$$\nu^{7} -$$$$12\!\cdots\!52$$$$\nu^{6} +$$$$98\!\cdots\!44$$$$\nu^{5} -$$$$17\!\cdots\!52$$$$\nu^{4} +$$$$59\!\cdots\!16$$$$\nu^{3} -$$$$12\!\cdots\!76$$$$\nu^{2} +$$$$47\!\cdots\!44$$$$\nu -$$$$36\!\cdots\!56$$$$)/$$$$17\!\cdots\!08$$ $$\beta_{14}$$ $$=$$ $$($$$$-$$$$26\!\cdots\!83$$$$\nu^{17} +$$$$23\!\cdots\!62$$$$\nu^{16} -$$$$37\!\cdots\!68$$$$\nu^{15} +$$$$29\!\cdots\!96$$$$\nu^{14} -$$$$37\!\cdots\!04$$$$\nu^{13} +$$$$32\!\cdots\!16$$$$\nu^{12} -$$$$18\!\cdots\!32$$$$\nu^{11} +$$$$19\!\cdots\!92$$$$\nu^{10} -$$$$69\!\cdots\!84$$$$\nu^{9} +$$$$86\!\cdots\!56$$$$\nu^{8} -$$$$11\!\cdots\!96$$$$\nu^{7} +$$$$22\!\cdots\!68$$$$\nu^{6} -$$$$17\!\cdots\!60$$$$\nu^{5} +$$$$28\!\cdots\!20$$$$\nu^{4} -$$$$34\!\cdots\!72$$$$\nu^{3} +$$$$19\!\cdots\!80$$$$\nu^{2} -$$$$76\!\cdots\!08$$$$\nu +$$$$65\!\cdots\!36$$$$)/$$$$46\!\cdots\!16$$ $$\beta_{15}$$ $$=$$ $$($$$$-$$$$53\!\cdots\!31$$$$\nu^{17} -$$$$45\!\cdots\!98$$$$\nu^{16} -$$$$81\!\cdots\!20$$$$\nu^{15} -$$$$80\!\cdots\!84$$$$\nu^{14} -$$$$86\!\cdots\!04$$$$\nu^{13} -$$$$82\!\cdots\!16$$$$\nu^{12} -$$$$48\!\cdots\!60$$$$\nu^{11} -$$$$40\!\cdots\!80$$$$\nu^{10} -$$$$19\!\cdots\!96$$$$\nu^{9} -$$$$14\!\cdots\!52$$$$\nu^{8} -$$$$40\!\cdots\!68$$$$\nu^{7} -$$$$18\!\cdots\!56$$$$\nu^{6} -$$$$52\!\cdots\!72$$$$\nu^{5} -$$$$25\!\cdots\!60$$$$\nu^{4} -$$$$32\!\cdots\!40$$$$\nu^{3} -$$$$54\!\cdots\!00$$$$\nu^{2} -$$$$83\!\cdots\!56$$$$\nu -$$$$40\!\cdots\!32$$$$)/$$$$53\!\cdots\!72$$ $$\beta_{16}$$ $$=$$ $$($$$$-$$$$10\!\cdots\!37$$$$\nu^{17} +$$$$32\!\cdots\!58$$$$\nu^{16} -$$$$16\!\cdots\!16$$$$\nu^{15} -$$$$19\!\cdots\!88$$$$\nu^{14} -$$$$16\!\cdots\!80$$$$\nu^{13} -$$$$17\!\cdots\!48$$$$\nu^{12} -$$$$91\!\cdots\!48$$$$\nu^{11} -$$$$13\!\cdots\!76$$$$\nu^{10} -$$$$35\!\cdots\!00$$$$\nu^{9} +$$$$32\!\cdots\!16$$$$\nu^{8} -$$$$71\!\cdots\!00$$$$\nu^{7} +$$$$26\!\cdots\!40$$$$\nu^{6} -$$$$10\!\cdots\!48$$$$\nu^{5} +$$$$34\!\cdots\!52$$$$\nu^{4} -$$$$53\!\cdots\!24$$$$\nu^{3} +$$$$31\!\cdots\!68$$$$\nu^{2} -$$$$20\!\cdots\!44$$$$\nu +$$$$59\!\cdots\!76$$$$)/$$$$71\!\cdots\!72$$ $$\beta_{17}$$ $$=$$ $$($$$$-$$$$30\!\cdots\!91$$$$\nu^{17} +$$$$12\!\cdots\!26$$$$\nu^{16} -$$$$45\!\cdots\!80$$$$\nu^{15} +$$$$13\!\cdots\!04$$$$\nu^{14} -$$$$46\!\cdots\!56$$$$\nu^{13} +$$$$14\!\cdots\!92$$$$\nu^{12} -$$$$24\!\cdots\!68$$$$\nu^{11} +$$$$10\!\cdots\!64$$$$\nu^{10} -$$$$95\!\cdots\!72$$$$\nu^{9} +$$$$54\!\cdots\!16$$$$\nu^{8} -$$$$19\!\cdots\!16$$$$\nu^{7} +$$$$17\!\cdots\!20$$$$\nu^{6} -$$$$29\!\cdots\!36$$$$\nu^{5} +$$$$25\!\cdots\!52$$$$\nu^{4} -$$$$17\!\cdots\!12$$$$\nu^{3} +$$$$20\!\cdots\!28$$$$\nu^{2} -$$$$10\!\cdots\!32$$$$\nu +$$$$57\!\cdots\!52$$$$)/$$$$71\!\cdots\!72$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{10} - \beta_{8} - \beta_{6} + \beta_{3} - 341 \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{17} - 4 \beta_{14} + 4 \beta_{13} - \beta_{12} - \beta_{11} + 15 \beta_{9} - 15 \beta_{8} - 5 \beta_{7} - 15 \beta_{4} + 559 \beta_{3} - \beta_{2} - \beta_{1} - 251$$ $$\nu^{4}$$ $$=$$ $$14 \beta_{17} + 32 \beta_{16} - 50 \beta_{15} - 18 \beta_{14} + 28 \beta_{11} - 792 \beta_{10} - 792 \beta_{7} + 1120 \beta_{6} + 40 \beta_{5} + 14 \beta_{3} + 188914 \beta_{2} - 2162 \beta_{1} - 188900$$ $$\nu^{5}$$ $$=$$ $$-1468 \beta_{17} + 1540 \beta_{16} + 3260 \beta_{15} + 734 \beta_{14} - 3260 \beta_{13} + 1082 \beta_{12} - 734 \beta_{11} - 5122 \beta_{10} - 16446 \beta_{9} + 22734 \beta_{8} + 22734 \beta_{6} + 1082 \beta_{5} - 363372 \beta_{3} + 665088 \beta_{2} - 362638 \beta_{1} - 734$$ $$\nu^{6}$$ $$=$$ $$15280 \beta_{17} + 5816 \beta_{14} + 61060 \beta_{13} + 51260 \beta_{12} - 15280 \beta_{11} - 65556 \beta_{9} + 936436 \beta_{8} + 589628 \beta_{7} + 65556 \beta_{4} - 2515492 \beta_{3} - 15280 \beta_{2} - 15280 \beta_{1} + 121678584$$ $$\nu^{7}$$ $$=$$ $$452052 \beta_{17} - 1202040 \beta_{16} - 2091648 \beta_{15} + 1654092 \beta_{14} + 904104 \beta_{11} + 5022516 \beta_{10} + 5022516 \beta_{7} - 23315004 \beta_{6} - 1010868 \beta_{5} + 14846796 \beta_{4} + 452052 \beta_{3} - 792652188 \beta_{2} + 252368224 \beta_{1} + 793104240$$ $$\nu^{8}$$ $$=$$ $$-26067120 \beta_{17} - 32425248 \beta_{16} + 54202584 \beta_{15} + 13033560 \beta_{14} - 54202584 \beta_{13} - 49630704 \beta_{12} - 13033560 \beta_{11} + 440992048 \beta_{10} + 109722096 \beta_{9} - 753386032 \beta_{8} - 753386032 \beta_{6} - 49630704 \beta_{5} + 2461235128 \beta_{3} - 84010832432 \beta_{2} + 2474268688 \beta_{1} - 13033560$$ $$\nu^{9}$$ $$=$$ $$260144008 \beta_{17} - 1498148416 \beta_{14} + 1233764272 \beta_{13} - 912669496 \beta_{12} - 260144008 \beta_{11} + 12464168424 \beta_{9} - 21188341896 \beta_{8} - 4745336696 \beta_{7} - 12464168424 \beta_{4} + 182526700168 \beta_{3} - 260144008 \beta_{2} - 260144008 \beta_{1} - 789149237528$$ $$\nu^{10}$$ $$=$$ $$10257646976 \beta_{17} + 27145364768 \beta_{16} - 43264584944 \beta_{15} - 16887717792 \beta_{14} + 20515293952 \beta_{11} - 333790714704 \beta_{10} - 333790714704 \beta_{7} + 608404029808 \beta_{6} + 43510907728 \beta_{5} - 127800447984 \beta_{4} + 10257646976 \beta_{3} + 60589902258016 \beta_{2} - 2266378801712 \beta_{1} - 60579644611040$$ $$\nu^{11}$$ $$=$$ $$-283435484512 \beta_{17} + 804464195872 \beta_{16} + 689690631680 \beta_{15} + 141717742256 \beta_{14} - 689690631680 \beta_{13} + 807756051248 \beta_{12} - 141717742256 \beta_{11} - 4334729496880 \beta_{10} - 10126911729360 \beta_{9} + 18247113689616 \beta_{8} + 18247113689616 \beta_{6} + 807756051248 \beta_{5} - 136286910698880 \beta_{3} + 730933193488704 \beta_{2} - 136145192956624 \beta_{1} - 141717742256$$ $$\nu^{12}$$ $$=$$ $$7813949482144 \beta_{17} + 6701339085632 \beta_{14} + 33095720229280 \beta_{13} + 36460271819072 \beta_{12} - 7813949482144 \beta_{11} - 128898351959616 \beta_{9} + 495185617648192 \beta_{8} + 255685661148992 \beta_{7} + 128898351959616 \beta_{4} - 2022646578125248 \beta_{3} - 7813949482144 \beta_{2} - 7813949482144 \beta_{1} + 45026323201499616$$ $$\nu^{13}$$ $$=$$ $$71334586816224 \beta_{17} - 661186383087936 \beta_{16} - 361921497433920 \beta_{15} + 732520969904160 \beta_{14} + 142669173632448 \beta_{11} + 3857188942012704 \beta_{10} + 3857188942012704 \beta_{7} - 15284782337319648 \beta_{6} - 703503626375712 \beta_{5} + 8105622543821664 \beta_{4} + 71334586816224 \beta_{3} - 651900198119851680 \beta_{2} + 103762848846013888 \beta_{1} + 651971532706667904$$ $$\nu^{14}$$ $$=$$ $$-11749386356723712 \beta_{17} - 18283156437031296 \beta_{16} + 24881682369339456 \beta_{15} + 5874693178361856 \beta_{14} - 24881682369339456 \beta_{13} - 29875016411762880 \beta_{12} - 5874693178361856 \beta_{11} + 197922480679257280 \beta_{10} + 120816028162838592 \beta_{9} - 405157813143676480 \beta_{8} - 405157813143676480 \beta_{6} - 29875016411762880 \beta_{5} + 1752836058837053248 \beta_{3} - 34190074514373223040 \beta_{2} + 1758710752015415104 \beta_{1} - 5874693178361856$$ $$\nu^{15}$$ $$=$$ $$30488715705508672 \beta_{17} - 602605078257306112 \beta_{14} + 169612719742993408 \beta_{13} - 604663916180913472 \beta_{12} - 30488715705508672 \beta_{11} + 6447758351707974336 \beta_{9} - 12613080966473609664 \beta_{8} - 3365487569465355584 \beta_{7} - 6447758351707974336 \beta_{4} + 80192826374568078016 \beta_{3} - 30488715705508672 \beta_{2} - 30488715705508672 \beta_{1} - 568436877878031117248$$ $$\nu^{16}$$ $$=$$ $$4400233902936421760 \beta_{17} + 14954609594780295680 \beta_{16} - 18600138050483480960 \beta_{15} - 10554375691843873920 \beta_{14} + 8800467805872843520 \beta_{11} - 154571159988299440896 \beta_{10} - 154571159988299440896 \beta_{7} + 332352420046733800192 \beta_{6} + 24207803690411205376 \beta_{5} - 108531544243799741184 \beta_{4} + 4400233902936421760 \beta_{3} + 26389199755685801673856 \beta_{2} - 1502573236504362308480 \beta_{1} - 26384799521782865252096$$ $$\nu^{17}$$ $$=$$ $$-14847357216575852800 \beta_{17} + 442456568770888155904 \beta_{16} + 59457667164048423680 \beta_{15} + 7423678608287926400 \beta_{14} - 59457667164048423680 \beta_{13} + 514200602862320137088 \beta_{12} - 7423678608287926400 \beta_{11} - 2893523935569151666048 \beta_{10} - 5120722026899057331840 \beta_{9} + 10324339976509485070464 \beta_{8} + 10324339976509485070464 \beta_{6} + 514200602862320137088 \beta_{5} - 62737975810372379906304 \beta_{3} + 488093528470496191351296 \beta_{2} - 62730552131764091979904 \beta_{1} - 7423678608287926400$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$-1 + \beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
45.1
 14.2764 − 24.7275i 11.1685 − 19.3444i 6.29505 − 10.9033i 3.49628 − 6.05573i 2.80322 − 4.85531i −4.70426 + 8.14802i −8.45017 + 14.6361i −10.9539 + 18.9727i −12.9311 + 22.3974i 14.2764 + 24.7275i 11.1685 + 19.3444i 6.29505 + 10.9033i 3.49628 + 6.05573i 2.80322 + 4.85531i −4.70426 − 8.14802i −8.45017 − 14.6361i −10.9539 − 18.9727i −12.9311 − 22.3974i
0 −14.7764 25.5935i 0 35.6401 + 61.7304i 0 252.315 0 −315.186 + 545.918i 0
45.2 0 −11.6685 20.2105i 0 −25.2470 43.7291i 0 −187.942 0 −150.808 + 261.208i 0
45.3 0 −6.79505 11.7694i 0 −47.4871 82.2500i 0 189.860 0 29.1546 50.4972i 0
45.4 0 −3.99628 6.92176i 0 31.6056 + 54.7425i 0 −80.2775 0 89.5595 155.122i 0
45.5 0 −3.30322 5.72134i 0 12.6095 + 21.8402i 0 −40.7176 0 99.6775 172.647i 0
45.6 0 4.20426 + 7.28199i 0 −18.6530 32.3080i 0 15.8772 0 86.1484 149.213i 0
45.7 0 7.95017 + 13.7701i 0 −15.4645 26.7852i 0 132.225 0 −4.91049 + 8.50521i 0
45.8 0 10.4539 + 18.1067i 0 50.5928 + 87.6293i 0 95.5451 0 −97.0688 + 168.128i 0
45.9 0 12.4311 + 21.5314i 0 −18.0963 31.3437i 0 −208.885 0 −187.566 + 324.875i 0
49.1 0 −14.7764 + 25.5935i 0 35.6401 61.7304i 0 252.315 0 −315.186 545.918i 0
49.2 0 −11.6685 + 20.2105i 0 −25.2470 + 43.7291i 0 −187.942 0 −150.808 261.208i 0
49.3 0 −6.79505 + 11.7694i 0 −47.4871 + 82.2500i 0 189.860 0 29.1546 + 50.4972i 0
49.4 0 −3.99628 + 6.92176i 0 31.6056 54.7425i 0 −80.2775 0 89.5595 + 155.122i 0
49.5 0 −3.30322 + 5.72134i 0 12.6095 21.8402i 0 −40.7176 0 99.6775 + 172.647i 0
49.6 0 4.20426 7.28199i 0 −18.6530 + 32.3080i 0 15.8772 0 86.1484 + 149.213i 0
49.7 0 7.95017 13.7701i 0 −15.4645 + 26.7852i 0 132.225 0 −4.91049 8.50521i 0
49.8 0 10.4539 18.1067i 0 50.5928 87.6293i 0 95.5451 0 −97.0688 168.128i 0
49.9 0 12.4311 21.5314i 0 −18.0963 + 31.3437i 0 −208.885 0 −187.566 324.875i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.6.e.a 18
3.b odd 2 1 684.6.k.f 18
4.b odd 2 1 304.6.i.d 18
19.c even 3 1 inner 76.6.e.a 18
57.h odd 6 1 684.6.k.f 18
76.g odd 6 1 304.6.i.d 18

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.6.e.a 18 1.a even 1 1 trivial
76.6.e.a 18 19.c even 3 1 inner
304.6.i.d 18 4.b odd 2 1
304.6.i.d 18 76.g odd 6 1
684.6.k.f 18 3.b odd 2 1
684.6.k.f 18 57.h odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{6}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{18}$$
$3$ $$11\!\cdots\!89$$$$+$$$$18\!\cdots\!43$$$$T + 55353296119122128061 T^{2} + 4135023977940829116 T^{3} + 1064365845873830586 T^{4} + 64277359976697138 T^{5} + 13339239476598918 T^{6} + 391884723575136 T^{7} + 82262584011775 T^{8} + 1520828506301 T^{9} + 365497827591 T^{10} + 3216311392 T^{11} + 919778846 T^{12} + 7740906 T^{13} + 1629002 T^{14} + 8740 T^{15} + 1605 T^{16} + 11 T^{17} + T^{18}$$
$5$ $$53\!\cdots\!96$$$$+$$$$19\!\cdots\!64$$$$T +$$$$17\!\cdots\!60$$$$T^{2} +$$$$51\!\cdots\!56$$$$T^{3} +$$$$32\!\cdots\!64$$$$T^{4} +$$$$85\!\cdots\!40$$$$T^{5} +$$$$33\!\cdots\!96$$$$T^{6} +$$$$56\!\cdots\!24$$$$T^{7} + 16796878410512090940 T^{8} + 182235863003254716 T^{9} + 5792633935619680 T^{10} + 38635011538024 T^{11} + 1306014959701 T^{12} + 3086592365 T^{13} + 211569494 T^{14} + 174581 T^{15} + 17650 T^{16} - 11 T^{17} + T^{18}$$
$7$ $$( -1233055145631744000 + 53167521425264640 T + 1928158954033152 T^{2} - 19652044114368 T^{3} - 334655586976 T^{4} + 2542736868 T^{5} + 13902400 T^{6} - 92704 T^{7} - 168 T^{8} + T^{9} )^{2}$$
$11$ $$($$$$15\!\cdots\!96$$$$+ 30973025188546606224 T - 2007063245270803884 T^{2} - 8001493002002373 T^{3} + 31276588367368 T^{4} + 134453704519 T^{5} - 146426924 T^{6} - 687587 T^{7} + 160 T^{8} + T^{9} )^{2}$$
$13$ $$60\!\cdots\!64$$$$+$$$$71\!\cdots\!76$$$$T +$$$$81\!\cdots\!20$$$$T^{2} +$$$$18\!\cdots\!84$$$$T^{3} +$$$$98\!\cdots\!00$$$$T^{4} +$$$$65\!\cdots\!04$$$$T^{5} +$$$$74\!\cdots\!24$$$$T^{6} +$$$$17\!\cdots\!76$$$$T^{7} +$$$$30\!\cdots\!40$$$$T^{8} +$$$$19\!\cdots\!40$$$$T^{9} +$$$$91\!\cdots\!68$$$$T^{10} + 15606742310908401088 T^{11} + 1766913125495305309 T^{12} - 49775665263507 T^{13} + 2463225262058 T^{14} - 100940479 T^{15} + 1961430 T^{16} - 227 T^{17} + T^{18}$$
$17$ $$16\!\cdots\!84$$$$-$$$$28\!\cdots\!92$$$$T +$$$$83\!\cdots\!04$$$$T^{2} -$$$$63\!\cdots\!00$$$$T^{3} +$$$$15\!\cdots\!04$$$$T^{4} -$$$$10\!\cdots\!76$$$$T^{5} +$$$$19\!\cdots\!16$$$$T^{6} -$$$$84\!\cdots\!84$$$$T^{7} +$$$$13\!\cdots\!40$$$$T^{8} -$$$$46\!\cdots\!68$$$$T^{9} +$$$$71\!\cdots\!40$$$$T^{10} -$$$$16\!\cdots\!60$$$$T^{11} +$$$$24\!\cdots\!29$$$$T^{12} - 37790308727158707 T^{13} + 63161431950210 T^{14} - 5114803983 T^{15} + 9862822 T^{16} - 179 T^{17} + T^{18}$$
$19$ $$34\!\cdots\!99$$$$+$$$$12\!\cdots\!68$$$$T +$$$$41\!\cdots\!72$$$$T^{2} +$$$$24\!\cdots\!24$$$$T^{3} +$$$$30\!\cdots\!48$$$$T^{4} +$$$$20\!\cdots\!76$$$$T^{5} +$$$$14\!\cdots\!03$$$$T^{6} +$$$$12\!\cdots\!80$$$$T^{7} +$$$$60\!\cdots\!84$$$$T^{8} +$$$$54\!\cdots\!88$$$$T^{9} +$$$$24\!\cdots\!16$$$$T^{10} +$$$$20\!\cdots\!80$$$$T^{11} + 96465687567916357097 T^{12} + 54428798269344676 T^{13} + 32274638969252 T^{14} + 10562267824 T^{15} + 7248128 T^{16} + 868 T^{17} + T^{18}$$
$23$ $$75\!\cdots\!00$$$$+$$$$45\!\cdots\!00$$$$T +$$$$32\!\cdots\!00$$$$T^{2} -$$$$24\!\cdots\!00$$$$T^{3} +$$$$50\!\cdots\!96$$$$T^{4} -$$$$59\!\cdots\!96$$$$T^{5} +$$$$22\!\cdots\!32$$$$T^{6} +$$$$75\!\cdots\!08$$$$T^{7} +$$$$75\!\cdots\!28$$$$T^{8} +$$$$10\!\cdots\!04$$$$T^{9} +$$$$13\!\cdots\!72$$$$T^{10} +$$$$16\!\cdots\!72$$$$T^{11} +$$$$15\!\cdots\!81$$$$T^{12} + 1959012574736536137 T^{13} + 1113768359604122 T^{14} + 89398709041 T^{15} + 44751186 T^{16} + 3425 T^{17} + T^{18}$$
$29$ $$96\!\cdots\!24$$$$+$$$$14\!\cdots\!80$$$$T +$$$$18\!\cdots\!96$$$$T^{2} +$$$$10\!\cdots\!80$$$$T^{3} +$$$$69\!\cdots\!64$$$$T^{4} +$$$$30\!\cdots\!76$$$$T^{5} +$$$$14\!\cdots\!88$$$$T^{6} +$$$$52\!\cdots\!68$$$$T^{7} +$$$$18\!\cdots\!24$$$$T^{8} +$$$$43\!\cdots\!64$$$$T^{9} +$$$$11\!\cdots\!88$$$$T^{10} +$$$$20\!\cdots\!64$$$$T^{11} +$$$$44\!\cdots\!49$$$$T^{12} + 61471755887734205509 T^{13} + 10053122334757166 T^{14} + 861197973469 T^{15} + 125209066 T^{16} + 7349 T^{17} + T^{18}$$
$31$ $$( -$$$$91\!\cdots\!60$$$$-$$$$78\!\cdots\!92$$$$T -$$$$49\!\cdots\!88$$$$T^{2} -$$$$44\!\cdots\!00$$$$T^{3} + 49426332686599495056 T^{4} + 6591076477990692 T^{5} - 989114300688 T^{6} - 167296276 T^{7} + 4980 T^{8} + T^{9} )^{2}$$
$37$ $$( -$$$$70\!\cdots\!00$$$$-$$$$14\!\cdots\!80$$$$T +$$$$38\!\cdots\!28$$$$T^{2} +$$$$31\!\cdots\!88$$$$T^{3} - 91986213539390854248 T^{4} - 4630799241237276 T^{5} + 3092826215800 T^{6} - 174702136 T^{7} - 13222 T^{8} + T^{9} )^{2}$$
$41$ $$33\!\cdots\!25$$$$-$$$$74\!\cdots\!25$$$$T +$$$$16\!\cdots\!25$$$$T^{2} -$$$$26\!\cdots\!00$$$$T^{3} +$$$$31\!\cdots\!14$$$$T^{4} -$$$$72\!\cdots\!30$$$$T^{5} +$$$$45\!\cdots\!06$$$$T^{6} -$$$$69\!\cdots\!44$$$$T^{7} +$$$$22\!\cdots\!23$$$$T^{8} -$$$$12\!\cdots\!27$$$$T^{9} +$$$$55\!\cdots\!99$$$$T^{10} -$$$$64\!\cdots\!52$$$$T^{11} +$$$$11\!\cdots\!54$$$$T^{12} +$$$$45\!\cdots\!54$$$$T^{13} + 102037756916691790 T^{14} + 1992308609992 T^{15} + 386322313 T^{16} + 7311 T^{17} + T^{18}$$
$43$ $$51\!\cdots\!00$$$$+$$$$14\!\cdots\!00$$$$T +$$$$38\!\cdots\!00$$$$T^{2} +$$$$62\!\cdots\!00$$$$T^{3} +$$$$90\!\cdots\!00$$$$T^{4} +$$$$41\!\cdots\!40$$$$T^{5} +$$$$18\!\cdots\!76$$$$T^{6} +$$$$47\!\cdots\!96$$$$T^{7} +$$$$10\!\cdots\!80$$$$T^{8} +$$$$14\!\cdots\!84$$$$T^{9} +$$$$18\!\cdots\!84$$$$T^{10} +$$$$17\!\cdots\!12$$$$T^{11} +$$$$17\!\cdots\!89$$$$T^{12} +$$$$12\!\cdots\!91$$$$T^{13} + 112961728986806062 T^{14} + 4760975065791 T^{15} + 379698838 T^{16} + 8283 T^{17} + T^{18}$$
$47$ $$16\!\cdots\!16$$$$+$$$$23\!\cdots\!44$$$$T +$$$$96\!\cdots\!24$$$$T^{2} +$$$$78\!\cdots\!88$$$$T^{3} +$$$$21\!\cdots\!36$$$$T^{4} -$$$$67\!\cdots\!76$$$$T^{5} +$$$$40\!\cdots\!32$$$$T^{6} -$$$$26\!\cdots\!32$$$$T^{7} +$$$$39\!\cdots\!12$$$$T^{8} -$$$$20\!\cdots\!40$$$$T^{9} +$$$$20\!\cdots\!08$$$$T^{10} -$$$$95\!\cdots\!48$$$$T^{11} +$$$$70\!\cdots\!29$$$$T^{12} -$$$$25\!\cdots\!99$$$$T^{13} + 1438596564851851442 T^{14} - 41903660395535 T^{15} + 1912969374 T^{16} - 37603 T^{17} + T^{18}$$
$53$ $$50\!\cdots\!64$$$$+$$$$39\!\cdots\!44$$$$T +$$$$17\!\cdots\!60$$$$T^{2} +$$$$65\!\cdots\!96$$$$T^{3} +$$$$33\!\cdots\!16$$$$T^{4} +$$$$11\!\cdots\!92$$$$T^{5} +$$$$37\!\cdots\!08$$$$T^{6} +$$$$73\!\cdots\!92$$$$T^{7} +$$$$27\!\cdots\!96$$$$T^{8} +$$$$69\!\cdots\!00$$$$T^{9} +$$$$10\!\cdots\!88$$$$T^{10} +$$$$24\!\cdots\!56$$$$T^{11} +$$$$25\!\cdots\!69$$$$T^{12} +$$$$59\!\cdots\!49$$$$T^{13} + 3460270021546904626 T^{14} + 43357881060629 T^{15} + 2267852878 T^{16} + 20337 T^{17} + T^{18}$$
$59$ $$21\!\cdots\!89$$$$+$$$$40\!\cdots\!55$$$$T +$$$$71\!\cdots\!53$$$$T^{2} +$$$$47\!\cdots\!00$$$$T^{3} +$$$$44\!\cdots\!82$$$$T^{4} +$$$$25\!\cdots\!38$$$$T^{5} +$$$$17\!\cdots\!26$$$$T^{6} +$$$$79\!\cdots\!68$$$$T^{7} +$$$$39\!\cdots\!87$$$$T^{8} +$$$$13\!\cdots\!77$$$$T^{9} +$$$$53\!\cdots\!07$$$$T^{10} +$$$$15\!\cdots\!52$$$$T^{11} +$$$$47\!\cdots\!62$$$$T^{12} +$$$$10\!\cdots\!18$$$$T^{13} + 23327287366044483930 T^{14} + 357651739695276 T^{15} + 7129477477 T^{16} + 74455 T^{17} + T^{18}$$
$61$ $$21\!\cdots\!56$$$$-$$$$20\!\cdots\!04$$$$T +$$$$11\!\cdots\!76$$$$T^{2} -$$$$16\!\cdots\!40$$$$T^{3} +$$$$57\!\cdots\!80$$$$T^{4} -$$$$63\!\cdots\!04$$$$T^{5} +$$$$81\!\cdots\!92$$$$T^{6} -$$$$33\!\cdots\!76$$$$T^{7} +$$$$26\!\cdots\!64$$$$T^{8} -$$$$57\!\cdots\!12$$$$T^{9} +$$$$57\!\cdots\!12$$$$T^{10} -$$$$69\!\cdots\!56$$$$T^{11} +$$$$74\!\cdots\!81$$$$T^{12} -$$$$16\!\cdots\!47$$$$T^{13} + 6930515824571977902 T^{14} - 1910399499327 T^{15} + 3182325786 T^{16} + 7569 T^{17} + T^{18}$$
$67$ $$92\!\cdots\!81$$$$+$$$$49\!\cdots\!29$$$$T +$$$$58\!\cdots\!85$$$$T^{2} +$$$$13\!\cdots\!60$$$$T^{3} +$$$$15\!\cdots\!62$$$$T^{4} +$$$$27\!\cdots\!90$$$$T^{5} +$$$$25\!\cdots\!86$$$$T^{6} +$$$$31\!\cdots\!92$$$$T^{7} +$$$$27\!\cdots\!35$$$$T^{8} +$$$$30\!\cdots\!31$$$$T^{9} +$$$$21\!\cdots\!79$$$$T^{10} +$$$$17\!\cdots\!60$$$$T^{11} +$$$$10\!\cdots\!10$$$$T^{12} +$$$$82\!\cdots\!10$$$$T^{13} + 38408940894285316426 T^{14} + 181133911740404 T^{15} + 7768368661 T^{16} + 26177 T^{17} + T^{18}$$
$71$ $$17\!\cdots\!00$$$$-$$$$39\!\cdots\!80$$$$T +$$$$13\!\cdots\!96$$$$T^{2} -$$$$43\!\cdots\!60$$$$T^{3} +$$$$28\!\cdots\!72$$$$T^{4} -$$$$21\!\cdots\!92$$$$T^{5} +$$$$27\!\cdots\!00$$$$T^{6} -$$$$71\!\cdots\!28$$$$T^{7} +$$$$42\!\cdots\!72$$$$T^{8} -$$$$65\!\cdots\!32$$$$T^{9} +$$$$41\!\cdots\!64$$$$T^{10} -$$$$44\!\cdots\!68$$$$T^{11} +$$$$19\!\cdots\!33$$$$T^{12} -$$$$39\!\cdots\!97$$$$T^{13} + 53973233604256978870 T^{14} + 53391804438599 T^{15} + 10847901010 T^{16} + 53463 T^{17} + T^{18}$$
$73$ $$43\!\cdots\!25$$$$+$$$$74\!\cdots\!75$$$$T +$$$$29\!\cdots\!25$$$$T^{2} +$$$$13\!\cdots\!20$$$$T^{3} +$$$$17\!\cdots\!06$$$$T^{4} +$$$$62\!\cdots\!86$$$$T^{5} +$$$$33\!\cdots\!42$$$$T^{6} +$$$$67\!\cdots\!72$$$$T^{7} +$$$$29\!\cdots\!47$$$$T^{8} +$$$$45\!\cdots\!81$$$$T^{9} +$$$$17\!\cdots\!35$$$$T^{10} +$$$$15\!\cdots\!60$$$$T^{11} +$$$$50\!\cdots\!46$$$$T^{12} +$$$$28\!\cdots\!26$$$$T^{13} +$$$$11\!\cdots\!74$$$$T^{14} + 307313940776492 T^{15} + 12536846917 T^{16} + 14103 T^{17} + T^{18}$$
$79$ $$12\!\cdots\!44$$$$+$$$$90\!\cdots\!32$$$$T +$$$$33\!\cdots\!84$$$$T^{2} -$$$$34\!\cdots\!16$$$$T^{3} +$$$$49\!\cdots\!56$$$$T^{4} -$$$$22\!\cdots\!72$$$$T^{5} +$$$$11\!\cdots\!08$$$$T^{6} -$$$$26\!\cdots\!68$$$$T^{7} +$$$$12\!\cdots\!16$$$$T^{8} -$$$$20\!\cdots\!36$$$$T^{9} +$$$$86\!\cdots\!68$$$$T^{10} -$$$$76\!\cdots\!32$$$$T^{11} +$$$$31\!\cdots\!41$$$$T^{12} -$$$$18\!\cdots\!25$$$$T^{13} + 83272908481639449190 T^{14} - 246647107958493 T^{15} + 11209186574 T^{16} - 31825 T^{17} + T^{18}$$
$83$ $$($$$$24\!\cdots\!00$$$$+$$$$20\!\cdots\!20$$$$T +$$$$98\!\cdots\!28$$$$T^{2} -$$$$39\!\cdots\!37$$$$T^{3} -$$$$15\!\cdots\!20$$$$T^{4} + 38020731823258649847 T^{5} + 511917348889560 T^{6} - 11706552339 T^{7} - 41300 T^{8} + T^{9} )^{2}$$
$89$ $$86\!\cdots\!44$$$$+$$$$57\!\cdots\!40$$$$T +$$$$64\!\cdots\!92$$$$T^{2} +$$$$27\!\cdots\!72$$$$T^{3} +$$$$26\!\cdots\!88$$$$T^{4} +$$$$11\!\cdots\!40$$$$T^{5} +$$$$49\!\cdots\!48$$$$T^{6} +$$$$13\!\cdots\!36$$$$T^{7} +$$$$35\!\cdots\!68$$$$T^{8} +$$$$69\!\cdots\!20$$$$T^{9} +$$$$14\!\cdots\!24$$$$T^{10} +$$$$22\!\cdots\!88$$$$T^{11} +$$$$35\!\cdots\!61$$$$T^{12} +$$$$38\!\cdots\!45$$$$T^{13} +$$$$41\!\cdots\!70$$$$T^{14} + 3121899045395049 T^{15} + 29541644542 T^{16} + 155197 T^{17} + T^{18}$$
$97$ $$93\!\cdots\!25$$$$+$$$$95\!\cdots\!75$$$$T +$$$$10\!\cdots\!25$$$$T^{2} +$$$$26\!\cdots\!00$$$$T^{3} +$$$$17\!\cdots\!14$$$$T^{4} +$$$$38\!\cdots\!90$$$$T^{5} +$$$$22\!\cdots\!50$$$$T^{6} +$$$$28\!\cdots\!48$$$$T^{7} +$$$$71\!\cdots\!71$$$$T^{8} +$$$$31\!\cdots\!17$$$$T^{9} +$$$$10\!\cdots\!83$$$$T^{10} +$$$$19\!\cdots\!24$$$$T^{11} +$$$$12\!\cdots\!86$$$$T^{12} -$$$$12\!\cdots\!34$$$$T^{13} +$$$$85\!\cdots\!22$$$$T^{14} - 1646469577269280 T^{15} + 42695596097 T^{16} - 111241 T^{17} + T^{18}$$