# Properties

 Label 60.7.k.a Level $60$ Weight $7$ Character orbit 60.k Analytic conductor $13.803$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 60.k (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.8032450172$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ Defining polynomial: $$x^{12} + 66 x^{10} + 1601 x^{8} + 17520 x^{6} + 84208 x^{4} + 136704 x^{2} + 14400$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{19}\cdot 3^{10}\cdot 5^{7}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( 26 - 4 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{5} + ( 10 + 10 \beta_{1} - \beta_{2} - 6 \beta_{3} + \beta_{4} + \beta_{8} - \beta_{11} ) q^{7} -243 \beta_{1} q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( 26 - 4 \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{5} + ( 10 + 10 \beta_{1} - \beta_{2} - 6 \beta_{3} + \beta_{4} + \beta_{8} - \beta_{11} ) q^{7} -243 \beta_{1} q^{9} + ( -269 - 2 \beta_{1} - 11 \beta_{2} - 11 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{11} + ( -167 + 171 \beta_{1} + 28 \beta_{2} + 6 \beta_{4} - 3 \beta_{5} - 4 \beta_{7} - 2 \beta_{8} + 10 \beta_{9} + 7 \beta_{10} - 5 \beta_{11} ) q^{13} + ( 378 - 213 \beta_{1} + 24 \beta_{2} - 8 \beta_{3} + 6 \beta_{6} - 3 \beta_{7} + 3 \beta_{9} - 6 \beta_{10} ) q^{15} + ( -447 - 469 \beta_{1} + 2 \beta_{2} - 38 \beta_{3} - 14 \beta_{4} - 6 \beta_{5} - 10 \beta_{6} + 2 \beta_{7} - 13 \beta_{8} + 5 \beta_{10} - 17 \beta_{11} ) q^{17} + ( 16 - 2096 \beta_{1} + 6 \beta_{2} - 16 \beta_{3} - 16 \beta_{4} - 5 \beta_{5} + 4 \beta_{6} - 10 \beta_{7} - 11 \beta_{8} + 4 \beta_{9} - 22 \beta_{10} - 22 \beta_{11} ) q^{19} + ( -1281 - 9 \beta_{1} - \beta_{2} + 17 \beta_{3} + 15 \beta_{4} - 12 \beta_{6} + 3 \beta_{7} - 3 \beta_{8} + 12 \beta_{9} + 21 \beta_{10} - 21 \beta_{11} ) q^{21} + ( -1938 + 1964 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} - 6 \beta_{4} - 9 \beta_{5} - 34 \beta_{7} - 13 \beta_{8} + 60 \beta_{9} + 6 \beta_{10} - 30 \beta_{11} ) q^{23} + ( 889 - 5166 \beta_{1} + 115 \beta_{2} - 217 \beta_{3} - 39 \beta_{4} - 50 \beta_{5} + 2 \beta_{6} - 47 \beta_{7} - 25 \beta_{8} + 36 \beta_{9} - 2 \beta_{10} ) q^{25} -243 \beta_{3} q^{27} + ( 25 - 7767 \beta_{1} + 314 \beta_{2} - 396 \beta_{3} + 2 \beta_{4} - 41 \beta_{5} + 50 \beta_{6} - 20 \beta_{7} + 16 \beta_{8} + 50 \beta_{9} - 30 \beta_{10} - 30 \beta_{11} ) q^{29} + ( -10506 - 84 \beta_{1} + 263 \beta_{2} + 343 \beta_{3} - 55 \beta_{4} - 44 \beta_{5} - 62 \beta_{6} - 29 \beta_{7} - 22 \beta_{8} + 62 \beta_{9} + 51 \beta_{10} - 51 \beta_{11} ) q^{31} + ( -2976 + 2958 \beta_{1} - 257 \beta_{2} + 3 \beta_{3} - 87 \beta_{4} - 6 \beta_{5} + 15 \beta_{7} + 9 \beta_{8} - 66 \beta_{10} ) q^{33} + ( 8587 - 18340 \beta_{1} - 204 \beta_{2} - 314 \beta_{3} + 64 \beta_{4} + 100 \beta_{5} - 59 \beta_{6} + 35 \beta_{7} + 25 \beta_{8} - 7 \beta_{9} + 59 \beta_{10} - 125 \beta_{11} ) q^{35} + ( 23651 + 23537 \beta_{1} - 100 \beta_{2} + 16 \beta_{3} - 64 \beta_{4} - 82 \beta_{5} + 50 \beta_{6} - 156 \beta_{7} + 43 \beta_{8} - 25 \beta_{10} - 83 \beta_{11} ) q^{37} + ( -39 - 8106 \beta_{1} - 117 \beta_{2} + 321 \beta_{3} - 120 \beta_{4} + 102 \beta_{5} - 21 \beta_{6} + 15 \beta_{7} - 63 \beta_{8} - 21 \beta_{9} + 63 \beta_{10} + 63 \beta_{11} ) q^{39} + ( -26820 + 88 \beta_{1} + 828 \beta_{2} + 670 \beta_{3} + 30 \beta_{4} + 9 \beta_{5} + 64 \beta_{6} + 76 \beta_{7} - 15 \beta_{8} - 64 \beta_{9} - 82 \beta_{10} + 82 \beta_{11} ) q^{41} + ( -5332 + 5376 \beta_{1} + 198 \beta_{2} - 174 \beta_{3} + 246 \beta_{4} - 72 \beta_{5} + 130 \beta_{7} - 22 \beta_{8} - 160 \beta_{9} - 52 \beta_{10} + 80 \beta_{11} ) q^{43} + ( -972 - 6318 \beta_{1} + 486 \beta_{2} - 243 \beta_{3} + 243 \beta_{7} ) q^{45} + ( 31556 + 31922 \beta_{1} + 290 \beta_{2} - 356 \beta_{3} + 56 \beta_{4} + 173 \beta_{5} + 20 \beta_{6} + 234 \beta_{7} - 107 \beta_{8} - 10 \beta_{10} + 412 \beta_{11} ) q^{47} + ( 82 + 18991 \beta_{1} - 3150 \beta_{2} + 2876 \beta_{3} + 512 \beta_{4} - 137 \beta_{5} - 2 \beta_{6} - 400 \beta_{7} + 55 \beta_{8} - 2 \beta_{9} + 236 \beta_{10} + 236 \beta_{11} ) q^{49} + ( -12261 + 252 \beta_{1} - 638 \beta_{2} - 560 \beta_{3} + 375 \beta_{4} + 291 \beta_{5} + 81 \beta_{6} - 126 \beta_{7} + 120 \beta_{8} - 81 \beta_{9} - 48 \beta_{10} + 48 \beta_{11} ) q^{51} + ( -33795 + 33445 \beta_{1} - 5717 \beta_{2} + 325 \beta_{3} - 315 \beta_{4} + 390 \beta_{5} + 25 \beta_{7} + 175 \beta_{8} - 480 \beta_{9} + 120 \beta_{10} + 240 \beta_{11} ) q^{53} + ( 41736 - 12254 \beta_{1} + 3541 \beta_{2} - 618 \beta_{3} + 169 \beta_{4} - 225 \beta_{5} - 50 \beta_{6} - 120 \beta_{7} - 350 \beta_{9} - 200 \beta_{10} + 375 \beta_{11} ) q^{55} + ( -3624 - 3048 \beta_{1} + 39 \beta_{2} - 2357 \beta_{3} + 507 \beta_{4} + 273 \beta_{5} + 30 \beta_{6} + 729 \beta_{7} + 249 \beta_{8} - 15 \beta_{10} - 129 \beta_{11} ) q^{57} + ( -597 - 5660 \beta_{1} - 2944 \beta_{2} + 3450 \beta_{3} - 114 \beta_{4} + 253 \beta_{5} - 533 \beta_{6} + 1335 \beta_{7} + 344 \beta_{8} - 533 \beta_{9} - 141 \beta_{10} - 141 \beta_{11} ) q^{59} + ( 66652 + 18 \beta_{1} + 8118 \beta_{2} + 7520 \beta_{3} - 920 \beta_{4} - 281 \beta_{5} + 594 \beta_{6} + 236 \beta_{7} + 295 \beta_{8} - 594 \beta_{9} - 362 \beta_{10} + 362 \beta_{11} ) q^{61} + ( 2430 - 2430 \beta_{1} - 1458 \beta_{2} + 243 \beta_{3} + 243 \beta_{5} - 243 \beta_{7} + 243 \beta_{10} ) q^{63} + ( 9735 + 37645 \beta_{1} + 1630 \beta_{2} - 11040 \beta_{3} + 10 \beta_{4} + 550 \beta_{5} + 600 \beta_{6} - 330 \beta_{7} + 425 \beta_{8} - 50 \beta_{9} + 275 \beta_{10} + 625 \beta_{11} ) q^{65} + ( 63648 + 62196 \beta_{1} - 758 \beta_{2} + 936 \beta_{3} - 254 \beta_{4} - 506 \beta_{5} - 440 \beta_{6} - 2148 \beta_{7} + 32 \beta_{8} + 220 \beta_{10} + 158 \beta_{11} ) q^{67} + ( -492 - 9270 \beta_{1} - 1583 \beta_{2} + 2501 \beta_{3} - 951 \beta_{4} + 459 \beta_{5} + 102 \beta_{6} + 915 \beta_{7} + 33 \beta_{8} + 102 \beta_{9} + 69 \beta_{10} + 69 \beta_{11} ) q^{69} + ( 17140 + 350 \beta_{1} + 11348 \beta_{2} + 11272 \beta_{3} + 1060 \beta_{4} + 312 \beta_{5} - 390 \beta_{6} + 46 \beta_{7} - 428 \beta_{8} + 390 \beta_{9} - 30 \beta_{10} + 30 \beta_{11} ) q^{71} + ( -26455 + 26315 \beta_{1} - 11804 \beta_{2} - 1086 \beta_{3} + 780 \beta_{4} - 1476 \beta_{5} + 1226 \beta_{7} + 70 \beta_{8} + 640 \beta_{9} + 154 \beta_{10} - 320 \beta_{11} ) q^{73} + ( -54033 - 28248 \beta_{1} + 1205 \beta_{2} - 4776 \beta_{3} - 492 \beta_{4} + 525 \beta_{5} - 249 \beta_{6} + 189 \beta_{7} - 300 \beta_{8} - 207 \beta_{9} - 501 \beta_{10} + 375 \beta_{11} ) q^{75} + ( -204544 - 203778 \beta_{1} + 370 \beta_{2} - 13130 \beta_{3} + 36 \beta_{4} + 203 \beta_{5} + 360 \beta_{6} + 1904 \beta_{7} + 13 \beta_{8} - 180 \beta_{10} - 948 \beta_{11} ) q^{77} + ( 1854 + 74212 \beta_{1} - 8345 \beta_{2} + 6657 \beta_{3} + 1789 \beta_{4} - 844 \beta_{5} - 234 \beta_{6} - 3575 \beta_{7} - 1010 \beta_{8} - 234 \beta_{9} - 133 \beta_{10} - 133 \beta_{11} ) q^{79} -59049 q^{81} + ( 159042 - 157696 \beta_{1} - 17481 \beta_{2} + 573 \beta_{3} + 179 \beta_{4} + 711 \beta_{5} - 1919 \beta_{7} - 673 \beta_{8} + 1070 \beta_{9} - 59 \beta_{10} - 535 \beta_{11} ) q^{83} + ( -69543 + 231025 \beta_{1} + 14056 \beta_{2} - 21244 \beta_{3} - 266 \beta_{4} - 1925 \beta_{5} + 216 \beta_{6} - 400 \beta_{7} - 200 \beta_{8} + 1218 \beta_{9} + 409 \beta_{10} - 2125 \beta_{11} ) q^{85} + ( -84246 - 83832 \beta_{1} + 762 \beta_{2} - 7421 \beta_{3} + 102 \beta_{4} + 432 \beta_{5} - 450 \beta_{6} + 1641 \beta_{7} - 555 \beta_{8} + 225 \beta_{10} - 240 \beta_{11} ) q^{87} + ( 382 - 73322 \beta_{1} - 10742 \beta_{2} + 10084 \beta_{3} - 1904 \beta_{4} - 329 \beta_{5} + 2088 \beta_{6} - 290 \beta_{7} - 53 \beta_{8} + 2088 \beta_{9} - 474 \beta_{10} - 474 \beta_{11} ) q^{89} + ( 194170 - 2664 \beta_{1} + 37940 \beta_{2} + 40216 \beta_{3} - 1700 \beta_{4} - 1526 \beta_{5} - 2082 \beta_{6} - 730 \beta_{7} - 944 \beta_{8} + 2082 \beta_{9} + 1546 \beta_{10} - 1546 \beta_{11} ) q^{91} + ( 69768 - 69744 \beta_{1} - 11147 \beta_{2} + 951 \beta_{3} - 1269 \beta_{4} + 888 \beta_{5} - 975 \beta_{7} - 12 \beta_{8} + 150 \beta_{9} - 267 \beta_{10} - 75 \beta_{11} ) q^{93} + ( -216400 + 59500 \beta_{1} + 29650 \beta_{2} - 32000 \beta_{3} - 200 \beta_{4} - 375 \beta_{5} - 2150 \beta_{6} + 1300 \beta_{7} - 1475 \beta_{8} + 850 \beta_{9} - 600 \beta_{10} - 2250 \beta_{11} ) q^{95} + ( -261779 - 264323 \beta_{1} - 440 \beta_{2} - 25592 \beta_{3} - 3684 \beta_{4} - 2062 \beta_{5} + 1580 \beta_{6} - 4116 \beta_{7} - 832 \beta_{8} - 790 \beta_{10} + 342 \beta_{11} ) q^{97} + ( -243 + 65610 \beta_{1} - 2916 \beta_{2} + 2430 \beta_{3} - 486 \beta_{4} - 243 \beta_{5} + 243 \beta_{6} + 1215 \beta_{7} + 486 \beta_{8} + 243 \beta_{9} - 729 \beta_{10} - 729 \beta_{11} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12q + 312q^{5} + 120q^{7} + O(q^{10})$$ $$12q + 312q^{5} + 120q^{7} - 3248q^{11} - 2100q^{13} + 4536q^{15} - 5540q^{17} - 15552q^{21} - 23840q^{23} + 10044q^{25} - 127152q^{31} - 35640q^{33} + 102976q^{35} + 282900q^{37} - 320720q^{41} - 62880q^{43} - 10692q^{45} + 381600q^{47} - 145152q^{51} - 400300q^{53} + 502152q^{55} - 38880q^{57} + 807024q^{61} + 29160q^{63} + 124500q^{65} + 752160q^{67} + 202400q^{71} - 322020q^{73} - 645408q^{75} - 2448400q^{77} - 708588q^{81} + 1894560q^{83} - 857124q^{85} - 1007640q^{87} + 2294400q^{91} + 835920q^{93} - 2620000q^{95} - 3161700q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 66 x^{10} + 1601 x^{8} + 17520 x^{6} + 84208 x^{4} + 136704 x^{2} + 14400$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-19 \nu^{11} - 1185 \nu^{9} - 26441 \nu^{7} - 259203 \nu^{5} - 1142320 \nu^{3} - 2087592 \nu$$$$)/659520$$ $$\beta_{2}$$ $$=$$ $$($$$$687 \nu^{11} + 1380 \nu^{10} + 53586 \nu^{9} + 79560 \nu^{8} + 1528575 \nu^{7} + 1638420 \nu^{6} + 19233252 \nu^{5} + 15253200 \nu^{4} + 99384168 \nu^{3} + 63616800 \nu^{2} + 135927072 \nu + 49000320$$$$)/3810560$$ $$\beta_{3}$$ $$=$$ $$($$$$-687 \nu^{11} + 1380 \nu^{10} - 53586 \nu^{9} + 79560 \nu^{8} - 1528575 \nu^{7} + 1638420 \nu^{6} - 19233252 \nu^{5} + 15253200 \nu^{4} - 99384168 \nu^{3} + 63616800 \nu^{2} - 135927072 \nu + 49000320$$$$)/3810560$$ $$\beta_{4}$$ $$=$$ $$($$$$-1169 \nu^{11} + 13980 \nu^{10} - 77103 \nu^{9} + 860340 \nu^{8} - 1891927 \nu^{7} + 18298680 \nu^{6} - 21572817 \nu^{5} + 154179840 \nu^{4} - 113809904 \nu^{3} + 392658240 \nu^{2} - 133704120 \nu - 145394880$$$$)/2857920$$ $$\beta_{5}$$ $$=$$ $$($$$$7261 \nu^{11} + 13950 \nu^{10} + 478023 \nu^{9} + 1177020 \nu^{8} + 11695295 \nu^{7} + 35573670 \nu^{6} + 132806541 \nu^{5} + 436005720 \nu^{4} + 697709584 \nu^{3} + 1591415280 \nu^{2} + 829363416 \nu - 888773760$$$$)/8573760$$ $$\beta_{6}$$ $$=$$ $$($$$$-10756 \nu^{11} - 114660 \nu^{10} - 737763 \nu^{9} - 6587100 \nu^{8} - 20154950 \nu^{7} - 131727960 \nu^{6} - 277362471 \nu^{5} - 1082540160 \nu^{4} - 1821510184 \nu^{3} - 3284498880 \nu^{2} - 3929581896 \nu - 2593618560$$$$)/8573760$$ $$\beta_{7}$$ $$=$$ $$($$$$-11789 \nu^{11} + 31860 \nu^{10} - 645879 \nu^{9} + 1813500 \nu^{8} - 10996579 \nu^{7} + 33422760 \nu^{6} - 45632625 \nu^{5} + 210216960 \nu^{4} + 306902128 \nu^{3} + 239129280 \nu^{2} + 1711375944 \nu - 97761600$$$$)/8573760$$ $$\beta_{8}$$ $$=$$ $$($$$$-55495 \nu^{11} - 127440 \nu^{10} - 3057288 \nu^{9} - 7254000 \nu^{8} - 53619731 \nu^{7} - 133691040 \nu^{6} - 275044314 \nu^{5} - 840867840 \nu^{4} + 726646856 \nu^{3} - 956517120 \nu^{2} + 6410120592 \nu + 408193920$$$$)/17147520$$ $$\beta_{9}$$ $$=$$ $$($$$$-7777 \nu^{11} + 34800 \nu^{10} - 452244 \nu^{9} + 2037360 \nu^{8} - 9175341 \nu^{7} + 41472000 \nu^{6} - 77814706 \nu^{5} + 343351040 \nu^{4} - 248168232 \nu^{3} + 991660800 \nu^{2} - 71868720 \nu + 479429760$$$$)/1905280$$ $$\beta_{10}$$ $$=$$ $$($$$$-380345 \nu^{11} - 239580 \nu^{10} - 24313458 \nu^{9} - 16328520 \nu^{8} - 565989601 \nu^{7} - 399910140 \nu^{6} - 5862158304 \nu^{5} - 4138804080 \nu^{4} - 25991693624 \nu^{3} - 15850954080 \nu^{2} - 36650106048 \nu - 11332494720$$$$)/34295040$$ $$\beta_{11}$$ $$=$$ $$($$$$-441529 \nu^{11} + 199260 \nu^{10} - 27822210 \nu^{9} + 13258440 \nu^{8} - 632679041 \nu^{7} + 314017020 \nu^{6} - 6303562608 \nu^{5} + 3129513840 \nu^{4} - 26129803960 \nu^{3} + 12095572320 \nu^{2} - 31409051712 \nu + 12686186880$$$$)/34295040$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{9} + 3 \beta_{8} - 4 \beta_{7} - \beta_{6} - 5 \beta_{5} + 11 \beta_{4} - 7 \beta_{3} - 3 \beta_{2} - 239 \beta_{1} + 2$$$$)/720$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{11} - 2 \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{7} - \beta_{6} - 3 \beta_{5} - 7 \beta_{4} - 25 \beta_{3} - 25 \beta_{2} - 3 \beta_{1} - 1978$$$$)/180$$ $$\nu^{3}$$ $$=$$ $$($$$$-9 \beta_{11} - 9 \beta_{10} + 28 \beta_{9} - 12 \beta_{8} + 49 \beta_{7} + 28 \beta_{6} + 32 \beta_{5} - 101 \beta_{4} + 203 \beta_{3} - 139 \beta_{2} + 1544 \beta_{1} - 20$$$$)/360$$ $$\nu^{4}$$ $$=$$ $$($$$$-6 \beta_{11} + 6 \beta_{10} - 6 \beta_{9} + 3 \beta_{8} + 12 \beta_{7} + 6 \beta_{6} + 9 \beta_{5} + 24 \beta_{4} + 152 \beta_{3} + 158 \beta_{2} + 12 \beta_{1} + 5760$$$$)/30$$ $$\nu^{5}$$ $$=$$ $$($$$$84 \beta_{11} + 84 \beta_{10} - 199 \beta_{9} + 24 \beta_{8} - 268 \beta_{7} - 199 \beta_{6} - 116 \beta_{5} + 515 \beta_{4} - 1419 \beta_{3} + 1187 \beta_{2} - 17333 \beta_{1} + 92$$$$)/90$$ $$\nu^{6}$$ $$=$$ $$($$$$332 \beta_{11} - 332 \beta_{10} + 514 \beta_{9} - 121 \beta_{8} - 1118 \beta_{7} - 514 \beta_{6} - 483 \beta_{5} - 1570 \beta_{4} - 11742 \beta_{3} - 12528 \beta_{2} - 876 \beta_{1} - 345064$$$$)/90$$ $$\nu^{7}$$ $$=$$ $$($$$$-2460 \beta_{11} - 2460 \beta_{10} + 4934 \beta_{9} + 27 \beta_{8} + 6320 \beta_{7} + 4934 \beta_{6} + 1903 \beta_{5} - 11200 \beta_{4} + 37004 \beta_{3} - 33198 \beta_{2} + 603382 \beta_{1} - 1930$$$$)/90$$ $$\nu^{8}$$ $$=$$ $$($$$$-1956 \beta_{11} + 1956 \beta_{10} - 4626 \beta_{9} + 351 \beta_{8} + 10506 \beta_{7} + 4626 \beta_{6} + 3093 \beta_{5} + 12066 \beta_{4} + 93614 \beta_{3} + 102164 \beta_{2} + 7368 \beta_{1} + 2461728$$$$)/30$$ $$\nu^{9}$$ $$=$$ $$($$$$66186 \beta_{11} + 66186 \beta_{10} - 117470 \beta_{9} - 9363 \beta_{8} - 154286 \beta_{7} - 117470 \beta_{6} - 34687 \beta_{5} + 253030 \beta_{4} - 955318 \beta_{3} + 885944 \beta_{2} - 17971750 \beta_{1} + 44050$$$$)/90$$ $$\nu^{10}$$ $$=$$ $$($$$$96988 \beta_{11} - 96988 \beta_{10} + 365750 \beta_{9} + 6523 \beta_{8} - 841534 \beta_{7} - 365750 \beta_{6} - 190791 \beta_{5} - 857366 \beta_{4} - 6590026 \beta_{3} - 7334572 \beta_{2} - 563064 \beta_{1} - 164691152$$$$)/90$$ $$\nu^{11}$$ $$=$$ $$($$$$-1715178 \beta_{11} - 1715178 \beta_{10} + 2767798 \beta_{9} + 358131 \beta_{8} + 3802030 \beta_{7} + 2767798 \beta_{6} + 685295 \beta_{5} - 5853566 \beta_{4} + 24489054 \beta_{3} - 23118464 \beta_{2} + 494596862 \beta_{1} - 1043426$$$$)/90$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$37$$ $$41$$ $$\chi(n)$$ $$1$$ $$\beta_{1}$$ $$1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 2.81042i − 3.51909i 1.70867i 4.93976i − 4.27595i 0.336188i − 2.81042i 3.51909i − 1.70867i − 4.93976i 4.27595i − 0.336188i
0 −11.0227 11.0227i 0 −102.872 71.0089i 0 −79.3849 + 79.3849i 0 243.000i 0
13.2 0 −11.0227 11.0227i 0 −23.5531 + 122.761i 0 312.733 312.733i 0 243.000i 0
13.3 0 −11.0227 11.0227i 0 123.592 18.7066i 0 −26.9845 + 26.9845i 0 243.000i 0
13.4 0 11.0227 + 11.0227i 0 −41.3794 117.952i 0 −49.0142 + 49.0142i 0 243.000i 0
13.5 0 11.0227 + 11.0227i 0 75.4215 + 99.6825i 0 337.242 337.242i 0 243.000i 0
13.6 0 11.0227 + 11.0227i 0 124.791 + 7.22440i 0 −434.591 + 434.591i 0 243.000i 0
37.1 0 −11.0227 + 11.0227i 0 −102.872 + 71.0089i 0 −79.3849 79.3849i 0 243.000i 0
37.2 0 −11.0227 + 11.0227i 0 −23.5531 122.761i 0 312.733 + 312.733i 0 243.000i 0
37.3 0 −11.0227 + 11.0227i 0 123.592 + 18.7066i 0 −26.9845 26.9845i 0 243.000i 0
37.4 0 11.0227 11.0227i 0 −41.3794 + 117.952i 0 −49.0142 49.0142i 0 243.000i 0
37.5 0 11.0227 11.0227i 0 75.4215 99.6825i 0 337.242 + 337.242i 0 243.000i 0
37.6 0 11.0227 11.0227i 0 124.791 7.22440i 0 −434.591 434.591i 0 243.000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 37.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.7.k.a 12
3.b odd 2 1 180.7.l.b 12
4.b odd 2 1 240.7.bg.d 12
5.b even 2 1 300.7.k.d 12
5.c odd 4 1 inner 60.7.k.a 12
5.c odd 4 1 300.7.k.d 12
15.e even 4 1 180.7.l.b 12
20.e even 4 1 240.7.bg.d 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.k.a 12 1.a even 1 1 trivial
60.7.k.a 12 5.c odd 4 1 inner
180.7.l.b 12 3.b odd 2 1
180.7.l.b 12 15.e even 4 1
240.7.bg.d 12 4.b odd 2 1
240.7.bg.d 12 20.e even 4 1
300.7.k.d 12 5.b even 2 1
300.7.k.d 12 5.c odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(60, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$( 59049 + T^{4} )^{3}$$
$5$ $$14\!\cdots\!25$$$$-$$$$29\!\cdots\!00$$$$T +$$$$26\!\cdots\!50$$$$T^{2} - 26145172119140625000 T^{3} + 261577606201171875 T^{4} - 1808601562500000 T^{5} + 12333701562500 T^{6} - 115750500000 T^{7} + 1071421875 T^{8} - 6853800 T^{9} + 43650 T^{10} - 312 T^{11} + T^{12}$$
$7$ $$14\!\cdots\!84$$$$+$$$$98\!\cdots\!40$$$$T +$$$$32\!\cdots\!00$$$$T^{2} +$$$$56\!\cdots\!80$$$$T^{3} +$$$$56\!\cdots\!08$$$$T^{4} + 2603038645501158720 T^{5} + 2577639789737600 T^{6} - 27148735644960 T^{7} + 118648882632 T^{8} - 18658160 T^{9} + 7200 T^{10} - 120 T^{11} + T^{12}$$
$11$ $$( -151973662189123904 + 804032958595744 T + 727368643940 T^{2} - 4738709120 T^{3} - 3517660 T^{4} + 1624 T^{5} + T^{6} )^{2}$$
$13$ $$11\!\cdots\!00$$$$+$$$$82\!\cdots\!00$$$$T +$$$$30\!\cdots\!00$$$$T^{2} +$$$$45\!\cdots\!00$$$$T^{3} +$$$$33\!\cdots\!00$$$$T^{4} -$$$$28\!\cdots\!00$$$$T^{5} +$$$$30\!\cdots\!00$$$$T^{6} + 355167384930000000 T^{7} + 213638052262500 T^{8} - 7228935000 T^{9} + 2205000 T^{10} + 2100 T^{11} + T^{12}$$
$17$ $$26\!\cdots\!84$$$$-$$$$29\!\cdots\!80$$$$T +$$$$16\!\cdots\!00$$$$T^{2} -$$$$44\!\cdots\!60$$$$T^{3} +$$$$66\!\cdots\!08$$$$T^{4} -$$$$42\!\cdots\!40$$$$T^{5} +$$$$10\!\cdots\!00$$$$T^{6} + 39299562620645456320 T^{7} + 7298732058688932 T^{8} - 65671600280 T^{9} + 15345800 T^{10} + 5540 T^{11} + T^{12}$$
$19$ $$12\!\cdots\!00$$$$+$$$$21\!\cdots\!00$$$$T^{2} +$$$$16\!\cdots\!00$$$$T^{4} +$$$$49\!\cdots\!00$$$$T^{6} + 67471168671900000 T^{8} + 428910000 T^{10} + T^{12}$$
$23$ $$61\!\cdots\!84$$$$+$$$$59\!\cdots\!80$$$$T +$$$$28\!\cdots\!00$$$$T^{2} +$$$$33\!\cdots\!40$$$$T^{3} +$$$$19\!\cdots\!08$$$$T^{4} -$$$$39\!\cdots\!60$$$$T^{5} +$$$$20\!\cdots\!00$$$$T^{6} +$$$$21\!\cdots\!20$$$$T^{7} + 113381158104083232 T^{8} - 1852573613120 T^{9} + 284172800 T^{10} + 23840 T^{11} + T^{12}$$
$29$ $$17\!\cdots\!36$$$$+$$$$93\!\cdots\!44$$$$T^{2} +$$$$12\!\cdots\!40$$$$T^{4} +$$$$83\!\cdots\!80$$$$T^{6} + 2113448470828223640 T^{8} + 2376129384 T^{10} + T^{12}$$
$31$ $$( -$$$$20\!\cdots\!04$$$$+$$$$11\!\cdots\!56$$$$T - 17111171277293760 T^{2} - 36316810695680 T^{3} + 8888640 T^{4} + 63576 T^{5} + T^{6} )^{2}$$
$37$ $$33\!\cdots\!64$$$$-$$$$69\!\cdots\!00$$$$T +$$$$72\!\cdots\!00$$$$T^{2} -$$$$33\!\cdots\!00$$$$T^{3} +$$$$10\!\cdots\!48$$$$T^{4} -$$$$31\!\cdots\!00$$$$T^{5} +$$$$15\!\cdots\!00$$$$T^{6} -$$$$66\!\cdots\!00$$$$T^{7} +$$$$18\!\cdots\!12$$$$T^{8} - 3391252433649000 T^{9} + 40016205000 T^{10} - 282900 T^{11} + T^{12}$$
$41$ $$($$$$71\!\cdots\!00$$$$-$$$$82\!\cdots\!00$$$$T - 6491369169769030000 T^{2} + 11240593684000 T^{3} + 7571165000 T^{4} + 160360 T^{5} + T^{6} )^{2}$$
$43$ $$22\!\cdots\!84$$$$-$$$$22\!\cdots\!40$$$$T +$$$$11\!\cdots\!00$$$$T^{2} -$$$$12\!\cdots\!20$$$$T^{3} -$$$$25\!\cdots\!92$$$$T^{4} +$$$$31\!\cdots\!80$$$$T^{5} +$$$$14\!\cdots\!00$$$$T^{6} +$$$$59\!\cdots\!40$$$$T^{7} +$$$$12\!\cdots\!32$$$$T^{8} - 167191585683840 T^{9} + 1976947200 T^{10} + 62880 T^{11} + T^{12}$$
$47$ $$26\!\cdots\!84$$$$-$$$$11\!\cdots\!00$$$$T +$$$$26\!\cdots\!00$$$$T^{2} -$$$$15\!\cdots\!00$$$$T^{3} +$$$$14\!\cdots\!08$$$$T^{4} -$$$$57\!\cdots\!00$$$$T^{5} +$$$$15\!\cdots\!00$$$$T^{6} -$$$$55\!\cdots\!00$$$$T^{7} +$$$$22\!\cdots\!32$$$$T^{8} - 4579127590406400 T^{9} + 72809280000 T^{10} - 381600 T^{11} + T^{12}$$
$53$ $$24\!\cdots\!44$$$$+$$$$44\!\cdots\!00$$$$T +$$$$41\!\cdots\!00$$$$T^{2} +$$$$28\!\cdots\!00$$$$T^{3} +$$$$18\!\cdots\!88$$$$T^{4} +$$$$34\!\cdots\!00$$$$T^{5} +$$$$34\!\cdots\!00$$$$T^{6} +$$$$16\!\cdots\!00$$$$T^{7} +$$$$45\!\cdots\!92$$$$T^{8} + 8849473594897400 T^{9} + 80120045000 T^{10} + 400300 T^{11} + T^{12}$$
$59$ $$49\!\cdots\!16$$$$+$$$$14\!\cdots\!44$$$$T^{2} +$$$$10\!\cdots\!40$$$$T^{4} +$$$$31\!\cdots\!80$$$$T^{6} +$$$$48\!\cdots\!40$$$$T^{8} + 360600374904 T^{10} + T^{12}$$
$61$ $$( -$$$$22\!\cdots\!36$$$$-$$$$22\!\cdots\!92$$$$T -$$$$40\!\cdots\!60$$$$T^{2} + 70206027853047840 T^{3} - 151174420440 T^{4} - 403512 T^{5} + T^{6} )^{2}$$
$67$ $$18\!\cdots\!84$$$$-$$$$20\!\cdots\!80$$$$T +$$$$11\!\cdots\!00$$$$T^{2} -$$$$22\!\cdots\!60$$$$T^{3} +$$$$35\!\cdots\!08$$$$T^{4} -$$$$34\!\cdots\!40$$$$T^{5} +$$$$18\!\cdots\!00$$$$T^{6} -$$$$45\!\cdots\!80$$$$T^{7} +$$$$64\!\cdots\!32$$$$T^{8} - 62863362987601280 T^{9} + 282872332800 T^{10} - 752160 T^{11} + T^{12}$$
$71$ $$($$$$34\!\cdots\!00$$$$-$$$$49\!\cdots\!00$$$$T +$$$$13\!\cdots\!00$$$$T^{2} + 71085323185376000 T^{3} - 309781043200 T^{4} - 101200 T^{5} + T^{6} )^{2}$$
$73$ $$30\!\cdots\!64$$$$+$$$$12\!\cdots\!60$$$$T +$$$$25\!\cdots\!00$$$$T^{2} +$$$$22\!\cdots\!20$$$$T^{3} +$$$$96\!\cdots\!48$$$$T^{4} -$$$$28\!\cdots\!20$$$$T^{5} +$$$$89\!\cdots\!00$$$$T^{6} +$$$$61\!\cdots\!60$$$$T^{7} +$$$$21\!\cdots\!12$$$$T^{8} - 24859429952288440 T^{9} + 51848440200 T^{10} + 322020 T^{11} + T^{12}$$
$79$ $$14\!\cdots\!56$$$$+$$$$99\!\cdots\!56$$$$T^{2} +$$$$11\!\cdots\!40$$$$T^{4} +$$$$61\!\cdots\!20$$$$T^{6} +$$$$15\!\cdots\!40$$$$T^{8} + 1995476079936 T^{10} + T^{12}$$
$83$ $$56\!\cdots\!84$$$$+$$$$11\!\cdots\!80$$$$T +$$$$11\!\cdots\!00$$$$T^{2} +$$$$45\!\cdots\!40$$$$T^{3} +$$$$36\!\cdots\!08$$$$T^{4} -$$$$46\!\cdots\!60$$$$T^{5} +$$$$14\!\cdots\!00$$$$T^{6} -$$$$69\!\cdots\!80$$$$T^{7} +$$$$12\!\cdots\!32$$$$T^{8} - 672256385117377920 T^{9} + 1794678796800 T^{10} - 1894560 T^{11} + T^{12}$$
$89$ $$83\!\cdots\!00$$$$+$$$$18\!\cdots\!00$$$$T^{2} +$$$$89\!\cdots\!00$$$$T^{4} +$$$$13\!\cdots\!00$$$$T^{6} +$$$$36\!\cdots\!00$$$$T^{8} + 3397006520400 T^{10} + T^{12}$$
$97$ $$91\!\cdots\!64$$$$+$$$$22\!\cdots\!00$$$$T +$$$$26\!\cdots\!00$$$$T^{2} +$$$$17\!\cdots\!00$$$$T^{3} +$$$$69\!\cdots\!48$$$$T^{4} +$$$$13\!\cdots\!00$$$$T^{5} +$$$$15\!\cdots\!00$$$$T^{6} +$$$$10\!\cdots\!00$$$$T^{7} +$$$$53\!\cdots\!12$$$$T^{8} + 4354668225994134600 T^{9} + 4998173445000 T^{10} + 3161700 T^{11} + T^{12}$$