# Properties

 Label 245.6.a.l Level $245$ Weight $6$ Character orbit 245.a Self dual yes Analytic conductor $39.294$ Analytic rank $1$ Dimension $10$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$245 = 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 245.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$39.2940358542$$ Analytic rank: $$1$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ Defining polynomial: $$x^{10} - 246 x^{8} - 192 x^{7} + 20336 x^{6} + 25380 x^{5} - 639206 x^{4} - 722920 x^{3} + 7583055 x^{2} + 5935300 x - 22888100$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{3}\cdot 7^{5}$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{2} + ( -6 + \beta_{3} ) q^{3} + ( 18 - \beta_{1} - \beta_{3} + \beta_{4} ) q^{4} -25 q^{5} + ( -15 + 8 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{6} + ( 27 - 16 \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{8} + ( 73 - 2 \beta_{1} - 2 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{2} + ( -6 + \beta_{3} ) q^{3} + ( 18 - \beta_{1} - \beta_{3} + \beta_{4} ) q^{4} -25 q^{5} + ( -15 + 8 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{6} + ( 27 - 16 \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{8} + ( 73 - 2 \beta_{1} - 2 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} - \beta_{7} - \beta_{8} + 2 \beta_{9} ) q^{9} + ( -25 + 25 \beta_{1} ) q^{10} + ( 80 - 7 \beta_{1} - 8 \beta_{2} + \beta_{3} - 3 \beta_{4} + 3 \beta_{7} - \beta_{8} - \beta_{9} ) q^{11} + ( -253 + 8 \beta_{1} - 4 \beta_{2} + 13 \beta_{3} - 13 \beta_{4} + 4 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{12} + ( -47 + 13 \beta_{1} + 4 \beta_{2} - 9 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 3 \beta_{6} + 6 \beta_{7} + \beta_{8} + 3 \beta_{9} ) q^{13} + ( 150 - 25 \beta_{3} ) q^{15} + ( 247 - 13 \beta_{1} - 5 \beta_{2} - 39 \beta_{3} + 8 \beta_{4} - \beta_{5} - 10 \beta_{6} - 4 \beta_{7} + 4 \beta_{8} - 2 \beta_{9} ) q^{16} + ( -76 + 102 \beta_{1} + \beta_{2} + 13 \beta_{3} - 17 \beta_{4} - 8 \beta_{5} - 2 \beta_{6} - 11 \beta_{7} - 5 \beta_{8} + 2 \beta_{9} ) q^{17} + ( 375 - 15 \beta_{1} - 35 \beta_{2} - 27 \beta_{3} - 2 \beta_{4} - 7 \beta_{5} + 2 \beta_{6} + 18 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{18} + ( -727 - 17 \beta_{1} - 29 \beta_{2} + 38 \beta_{3} - 23 \beta_{4} + 7 \beta_{5} - 5 \beta_{6} + \beta_{7} + 6 \beta_{8} - 7 \beta_{9} ) q^{19} + ( -450 + 25 \beta_{1} + 25 \beta_{3} - 25 \beta_{4} ) q^{20} + ( 402 + 46 \beta_{1} + 37 \beta_{2} - 43 \beta_{3} - 14 \beta_{4} - 7 \beta_{5} + 28 \beta_{6} - 16 \beta_{7} - 8 \beta_{8} + 14 \beta_{9} ) q^{22} + ( 391 + 90 \beta_{1} + 25 \beta_{2} - 66 \beta_{3} + 4 \beta_{4} + 25 \beta_{5} - 23 \beta_{6} - 14 \beta_{7} + 5 \beta_{8} - 14 \beta_{9} ) q^{23} + ( -192 + 408 \beta_{1} + 55 \beta_{2} + 3 \beta_{3} - 38 \beta_{4} - 13 \beta_{5} - 5 \beta_{6} - 29 \beta_{7} - 19 \beta_{8} + 8 \beta_{9} ) q^{24} + 625 q^{25} + ( -641 - 11 \beta_{1} - 98 \beta_{2} - 61 \beta_{3} - 19 \beta_{4} - 34 \beta_{5} + 20 \beta_{6} + 38 \beta_{7} + 8 \beta_{8} + 29 \beta_{9} ) q^{26} + ( -1191 + 113 \beta_{1} - 39 \beta_{2} - 27 \beta_{3} + 47 \beta_{4} + 17 \beta_{5} - 19 \beta_{6} - 9 \beta_{7} + 26 \beta_{8} - 17 \beta_{9} ) q^{27} + ( -916 + 160 \beta_{1} + 136 \beta_{2} - 9 \beta_{3} + 15 \beta_{4} + 42 \beta_{5} + 60 \beta_{6} - 3 \beta_{7} + 9 \beta_{8} - 24 \beta_{9} ) q^{29} + ( 375 - 200 \beta_{1} - 100 \beta_{2} - 50 \beta_{3} - 25 \beta_{5} - 25 \beta_{6} ) q^{30} + ( -1341 + 336 \beta_{1} + 103 \beta_{2} + 22 \beta_{3} + 46 \beta_{4} - 15 \beta_{5} - 71 \beta_{6} - 12 \beta_{7} + 3 \beta_{8} - 36 \beta_{9} ) q^{31} + ( 291 - 3 \beta_{1} + 185 \beta_{2} - 169 \beta_{3} + 56 \beta_{4} - \beta_{5} - 38 \beta_{6} + 2 \beta_{7} - 6 \beta_{8} + 14 \beta_{9} ) q^{32} + ( -1045 + 237 \beta_{1} + 102 \beta_{2} + 23 \beta_{3} + 108 \beta_{4} + 45 \beta_{5} + 39 \beta_{6} - 16 \beta_{7} + 27 \beta_{8} - 13 \beta_{9} ) q^{33} + ( -4429 + 534 \beta_{1} - 102 \beta_{2} + 52 \beta_{3} - 84 \beta_{4} + 35 \beta_{5} + 15 \beta_{6} + 38 \beta_{7} - 26 \beta_{8} - 16 \beta_{9} ) q^{34} + ( -1105 - 139 \beta_{1} + 5 \beta_{2} - 113 \beta_{3} + 44 \beta_{4} - 29 \beta_{5} + 110 \beta_{6} + 34 \beta_{7} + 26 \beta_{8} + 22 \beta_{9} ) q^{36} + ( 1671 + 505 \beta_{1} - 274 \beta_{2} - 120 \beta_{3} - 63 \beta_{4} + 15 \beta_{5} + 47 \beta_{6} + 15 \beta_{7} - 38 \beta_{8} + 63 \beta_{9} ) q^{37} + ( -238 + 1489 \beta_{1} + 268 \beta_{2} - 43 \beta_{3} + 71 \beta_{4} + 43 \beta_{5} + 41 \beta_{6} - 32 \beta_{7} - 14 \beta_{8} + 19 \beta_{9} ) q^{38} + ( -2454 + 283 \beta_{1} + 488 \beta_{2} + 107 \beta_{3} - 73 \beta_{4} + 52 \beta_{5} - 80 \beta_{6} - 15 \beta_{7} + 13 \beta_{8} - 15 \beta_{9} ) q^{39} + ( -675 + 400 \beta_{1} + 100 \beta_{2} + 25 \beta_{3} - 25 \beta_{4} + 25 \beta_{5} + 50 \beta_{6} + 25 \beta_{7} - 25 \beta_{8} + 25 \beta_{9} ) q^{40} + ( -3522 + 626 \beta_{1} - 265 \beta_{2} - 98 \beta_{3} + 128 \beta_{4} - 42 \beta_{5} + 58 \beta_{6} + 24 \beta_{7} + 2 \beta_{8} + 62 \beta_{9} ) q^{41} + ( -2888 + 44 \beta_{1} + 110 \beta_{2} + 16 \beta_{3} - 94 \beta_{4} + 22 \beta_{5} + 110 \beta_{6} + 58 \beta_{7} - 28 \beta_{8} - 56 \beta_{9} ) q^{43} + ( -3185 - 256 \beta_{1} - 1025 \beta_{2} + 318 \beta_{3} + 11 \beta_{4} - \beta_{5} + 146 \beta_{7} - 6 \beta_{8} - 12 \beta_{9} ) q^{44} + ( -1825 + 50 \beta_{1} + 50 \beta_{2} + 175 \beta_{3} + 75 \beta_{4} + 50 \beta_{5} + 25 \beta_{7} + 25 \beta_{8} - 50 \beta_{9} ) q^{45} + ( -3931 - 613 \beta_{1} + 769 \beta_{2} + 294 \beta_{3} - 57 \beta_{4} + 21 \beta_{5} - 290 \beta_{6} - 240 \beta_{7} + 14 \beta_{8} - 105 \beta_{9} ) q^{46} + ( -1797 - 245 \beta_{1} + 69 \beta_{2} - 359 \beta_{3} + 175 \beta_{4} - 33 \beta_{5} - 81 \beta_{6} + 15 \beta_{7} - 92 \beta_{8} - 35 \beta_{9} ) q^{47} + ( -10190 + 863 \beta_{1} + 150 \beta_{2} + 145 \beta_{3} - 63 \beta_{4} - 45 \beta_{5} - 11 \beta_{6} + 6 \beta_{7} - 42 \beta_{8} - 7 \beta_{9} ) q^{48} + ( 625 - 625 \beta_{1} ) q^{50} + ( 3090 - 603 \beta_{1} - 1506 \beta_{2} + 121 \beta_{3} + 173 \beta_{4} - 142 \beta_{5} - 150 \beta_{6} - 97 \beta_{7} + 81 \beta_{8} - 21 \beta_{9} ) q^{51} + ( 2327 + 701 \beta_{1} - 1130 \beta_{2} - 612 \beta_{3} - 26 \beta_{4} - 155 \beta_{5} + 208 \beta_{6} + 217 \beta_{7} + 11 \beta_{8} + 151 \beta_{9} ) q^{52} + ( 3601 + 1031 \beta_{1} + 700 \beta_{2} + 354 \beta_{3} - 71 \beta_{4} - 99 \beta_{5} + 113 \beta_{6} + 63 \beta_{7} + 36 \beta_{8} + 161 \beta_{9} ) q^{53} + ( -8087 - 189 \beta_{1} + 116 \beta_{2} - 27 \beta_{3} + 201 \beta_{4} + 8 \beta_{5} - 266 \beta_{6} - 112 \beta_{7} + 110 \beta_{8} - 91 \beta_{9} ) q^{54} + ( -2000 + 175 \beta_{1} + 200 \beta_{2} - 25 \beta_{3} + 75 \beta_{4} - 75 \beta_{7} + 25 \beta_{8} + 25 \beta_{9} ) q^{55} + ( 12744 + 362 \beta_{1} + 848 \beta_{2} - 980 \beta_{3} + 286 \beta_{4} - 222 \beta_{5} + 114 \beta_{6} - 54 \beta_{7} - 116 \beta_{8} + 134 \beta_{9} ) q^{57} + ( -10196 + 10 \beta_{1} - 2063 \beta_{2} + 1253 \beta_{3} - 178 \beta_{4} + 75 \beta_{5} - 112 \beta_{6} + 42 \beta_{7} - 66 \beta_{8} - 102 \beta_{9} ) q^{58} + ( -1756 - 415 \beta_{1} - 132 \beta_{2} - 850 \beta_{3} + 157 \beta_{4} + 88 \beta_{5} + 124 \beta_{6} + 91 \beta_{7} + 99 \beta_{8} + 83 \beta_{9} ) q^{59} + ( 6325 - 200 \beta_{1} + 100 \beta_{2} - 325 \beta_{3} + 325 \beta_{4} - 100 \beta_{5} + 25 \beta_{6} - 25 \beta_{7} + 25 \beta_{8} + 50 \beta_{9} ) q^{60} + ( -6720 - 1790 \beta_{1} - 556 \beta_{2} - 146 \beta_{3} - 68 \beta_{4} - 270 \beta_{5} - 226 \beta_{6} - 324 \beta_{7} - 118 \beta_{8} - 2 \beta_{9} ) q^{61} + ( -17827 + 819 \beta_{1} + 1757 \beta_{2} - 610 \beta_{3} - 69 \beta_{4} + 25 \beta_{5} - 370 \beta_{6} - 512 \beta_{7} + 54 \beta_{8} - 61 \beta_{9} ) q^{62} + ( -5979 - 1575 \beta_{1} + 1685 \beta_{2} + 1181 \beta_{3} - 370 \beta_{4} - 139 \beta_{5} - 170 \beta_{6} - 76 \beta_{7} - 24 \beta_{8} + 12 \beta_{9} ) q^{64} + ( 1175 - 325 \beta_{1} - 100 \beta_{2} + 225 \beta_{3} - 50 \beta_{4} + 75 \beta_{5} - 75 \beta_{6} - 150 \beta_{7} - 25 \beta_{8} - 75 \beta_{9} ) q^{65} + ( -16007 - 2561 \beta_{1} - 1286 \beta_{2} + 1543 \beta_{3} - 35 \beta_{4} + 134 \beta_{5} - 332 \beta_{6} + 50 \beta_{7} + 124 \beta_{8} - 235 \beta_{9} ) q^{66} + ( 8881 - 1072 \beta_{1} + 1165 \beta_{2} + 420 \beta_{3} + 590 \beta_{4} + 279 \beta_{5} + 119 \beta_{6} + 168 \beta_{7} - 31 \beta_{8} - 100 \beta_{9} ) q^{67} + ( -28873 + 3976 \beta_{1} + 1040 \beta_{2} + 633 \beta_{3} - 581 \beta_{4} + 290 \beta_{5} + 533 \beta_{6} + 17 \beta_{7} - 49 \beta_{8} + 76 \beta_{9} ) q^{68} + ( -17373 - 2483 \beta_{1} - 944 \beta_{2} - 672 \beta_{3} + 541 \beta_{4} - 247 \beta_{5} + 401 \beta_{6} + 423 \beta_{7} - 48 \beta_{8} + 19 \beta_{9} ) q^{69} + ( 512 + 580 \beta_{1} + 1052 \beta_{2} + 190 \beta_{3} - 40 \beta_{4} + 372 \beta_{5} + 60 \beta_{6} + 336 \beta_{7} + 276 \beta_{8} - 52 \beta_{9} ) q^{71} + ( -6471 - 527 \beta_{1} - 4175 \beta_{2} + 373 \beta_{3} + 458 \beta_{4} - 75 \beta_{5} + 194 \beta_{6} + 292 \beta_{7} + 120 \beta_{8} + 112 \beta_{9} ) q^{72} + ( -16973 - 2573 \beta_{1} + 1437 \beta_{2} + 832 \beta_{3} - 477 \beta_{4} + 269 \beta_{5} + 45 \beta_{6} + 29 \beta_{7} + 118 \beta_{8} - 43 \beta_{9} ) q^{73} + ( -21675 - 715 \beta_{1} - 43 \beta_{2} + 1586 \beta_{3} - 1285 \beta_{4} - 97 \beta_{5} + 540 \beta_{6} + 320 \beta_{7} - 98 \beta_{8} + 55 \beta_{9} ) q^{74} + ( -3750 + 625 \beta_{3} ) q^{75} + ( -50522 - 2121 \beta_{1} + 610 \beta_{2} + 2443 \beta_{3} - 1065 \beta_{4} - 71 \beta_{5} - 257 \beta_{6} + 10 \beta_{7} - 166 \beta_{8} - 75 \beta_{9} ) q^{76} + ( -15632 + 4984 \beta_{1} + 4839 \beta_{2} + 1335 \beta_{3} - 282 \beta_{4} + 243 \beta_{5} - 764 \beta_{6} - 504 \beta_{7} + 16 \beta_{8} - 78 \beta_{9} ) q^{78} + ( 2802 + 243 \beta_{1} - 1724 \beta_{2} - 449 \beta_{3} - 233 \beta_{4} + 424 \beta_{5} + 340 \beta_{6} - 575 \beta_{7} + 121 \beta_{8} + 65 \beta_{9} ) q^{79} + ( -6175 + 325 \beta_{1} + 125 \beta_{2} + 975 \beta_{3} - 200 \beta_{4} + 25 \beta_{5} + 250 \beta_{6} + 100 \beta_{7} - 100 \beta_{8} + 50 \beta_{9} ) q^{80} + ( -5451 + 310 \beta_{1} + 1596 \beta_{2} - 688 \beta_{3} - 230 \beta_{4} - 182 \beta_{5} - 230 \beta_{6} + 246 \beta_{7} - 320 \beta_{8} - 262 \beta_{9} ) q^{81} + ( -35393 - 786 \beta_{1} - 2497 \beta_{2} + 421 \beta_{3} - 674 \beta_{4} - 256 \beta_{5} + 385 \beta_{6} + 668 \beta_{7} + 200 \beta_{8} + 54 \beta_{9} ) q^{82} + ( -35461 - 2313 \beta_{1} - 1835 \beta_{2} + 316 \beta_{3} + 173 \beta_{4} + 485 \beta_{5} + 161 \beta_{6} + 449 \beta_{7} - 48 \beta_{8} - 63 \beta_{9} ) q^{83} + ( 1900 - 2550 \beta_{1} - 25 \beta_{2} - 325 \beta_{3} + 425 \beta_{4} + 200 \beta_{5} + 50 \beta_{6} + 275 \beta_{7} + 125 \beta_{8} - 50 \beta_{9} ) q^{85} + ( -5302 + 5804 \beta_{1} - 3732 \beta_{2} + 310 \beta_{3} - 554 \beta_{4} - 86 \beta_{5} + 630 \beta_{6} - 92 \beta_{7} - 400 \beta_{8} + 254 \beta_{9} ) q^{86} + ( 797 - 8545 \beta_{1} - 969 \beta_{2} - 2063 \beta_{3} - 345 \beta_{4} + 45 \beta_{5} - 51 \beta_{6} + 515 \beta_{7} - 230 \beta_{8} - 255 \beta_{9} ) q^{87} + ( -11798 + 3485 \beta_{1} + 1071 \beta_{2} - 388 \beta_{3} + 89 \beta_{4} + 108 \beta_{5} + 1046 \beta_{6} + 99 \beta_{7} + 225 \beta_{8} + 121 \beta_{9} ) q^{88} + ( -8465 - 117 \beta_{1} + 3375 \beta_{2} - 336 \beta_{3} - 369 \beta_{4} - 279 \beta_{5} - 607 \beta_{6} - 327 \beta_{7} - 314 \beta_{8} + 509 \beta_{9} ) q^{89} + ( -9375 + 375 \beta_{1} + 875 \beta_{2} + 675 \beta_{3} + 50 \beta_{4} + 175 \beta_{5} - 50 \beta_{6} - 450 \beta_{7} + 50 \beta_{8} - 50 \beta_{9} ) q^{90} + ( 17790 + 3791 \beta_{1} + 7709 \beta_{2} + 1660 \beta_{3} + 1823 \beta_{4} + 242 \beta_{5} - 1610 \beta_{6} - 1081 \beta_{7} - 95 \beta_{8} - 483 \beta_{9} ) q^{92} + ( 19715 + 2147 \beta_{1} - 5046 \beta_{2} - 4108 \beta_{3} - 245 \beta_{4} - 455 \beta_{5} + 89 \beta_{6} - 107 \beta_{7} + 98 \beta_{8} + 165 \beta_{9} ) q^{93} + ( 12693 - 2247 \beta_{1} + 2288 \beta_{2} - 1429 \beta_{3} - 649 \beta_{4} - 378 \beta_{5} - 352 \beta_{6} - 1332 \beta_{7} - 90 \beta_{8} - 141 \beta_{9} ) q^{94} + ( 18175 + 425 \beta_{1} + 725 \beta_{2} - 950 \beta_{3} + 575 \beta_{4} - 175 \beta_{5} + 125 \beta_{6} - 25 \beta_{7} - 150 \beta_{8} + 175 \beta_{9} ) q^{95} + ( -44440 - 235 \beta_{1} - 1046 \beta_{2} + 267 \beta_{3} + 71 \beta_{4} + 495 \beta_{5} + 467 \beta_{6} + 768 \beta_{7} + 416 \beta_{8} - 121 \beta_{9} ) q^{96} + ( -22336 - 5170 \beta_{1} + 1885 \beta_{2} - 3957 \beta_{3} - 295 \beta_{4} + 308 \beta_{5} - 578 \beta_{6} - 685 \beta_{7} + 257 \beta_{8} + 250 \beta_{9} ) q^{97} + ( 12314 - 6926 \beta_{1} + 1716 \beta_{2} - 2586 \beta_{3} - 1664 \beta_{4} - 644 \beta_{5} - 500 \beta_{6} - 232 \beta_{7} - 572 \beta_{8} + 394 \beta_{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 10 q^{2} - 58 q^{3} + 182 q^{4} - 250 q^{5} - 144 q^{6} + 270 q^{8} + 700 q^{9} + O(q^{10})$$ $$10 q + 10 q^{2} - 58 q^{3} + 182 q^{4} - 250 q^{5} - 144 q^{6} + 270 q^{8} + 700 q^{9} - 250 q^{10} + 794 q^{11} - 2560 q^{12} - 474 q^{13} + 1450 q^{15} + 2394 q^{16} - 802 q^{17} + 3702 q^{18} - 7292 q^{19} - 4550 q^{20} + 3948 q^{22} + 3708 q^{23} - 2092 q^{24} + 6250 q^{25} - 6576 q^{26} - 11818 q^{27} - 8866 q^{29} + 3600 q^{30} - 13292 q^{31} + 2590 q^{32} - 9854 q^{33} - 44468 q^{34} - 10690 q^{36} + 16124 q^{37} - 2180 q^{38} - 24982 q^{39} - 6750 q^{40} - 34836 q^{41} - 28604 q^{43} - 31120 q^{44} - 17500 q^{45} - 39732 q^{46} - 18106 q^{47} - 101788 q^{48} + 6250 q^{50} + 31602 q^{51} + 22480 q^{52} + 36440 q^{53} - 80836 q^{54} - 19850 q^{55} + 126988 q^{57} - 100356 q^{58} - 18644 q^{59} + 64000 q^{60} - 68120 q^{61} - 181052 q^{62} - 59358 q^{64} + 11850 q^{65} - 157780 q^{66} + 92328 q^{67} - 288540 q^{68} - 170888 q^{69} + 5044 q^{71} - 61654 q^{72} - 170160 q^{73} - 216584 q^{74} - 36250 q^{75} - 505180 q^{76} - 158008 q^{78} + 26442 q^{79} - 59850 q^{80} - 56314 q^{81} - 353948 q^{82} - 353360 q^{83} + 20050 q^{85} - 52940 q^{86} + 3190 q^{87} - 114916 q^{88} - 90704 q^{89} - 92550 q^{90} + 183520 q^{92} + 188560 q^{93} + 121388 q^{94} + 182300 q^{95} - 442220 q^{96} - 236382 q^{97} + 109024 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 246 x^{8} - 192 x^{7} + 20336 x^{6} + 25380 x^{5} - 639206 x^{4} - 722920 x^{3} + 7583055 x^{2} + 5935300 x - 22888100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-433 \nu^{9} - 1947 \nu^{8} + 98973 \nu^{7} + 554383 \nu^{6} - 6721179 \nu^{5} - 43954077 \nu^{4} + 111936823 \nu^{3} + 837690357 \nu^{2} - 258750600 \nu - 2613859980$$$$)/69843200$$ $$\beta_{3}$$ $$=$$ $$($$$$83137 \nu^{9} + 200027 \nu^{8} - 19567597 \nu^{7} - 57068271 \nu^{6} + 1395513643 \nu^{5} + 4236045917 \nu^{4} - 27571140327 \nu^{3} - 50966745557 \nu^{2} + 113744633960 \nu - 107486394420$$$$)/ 9847891200$$ $$\beta_{4}$$ $$=$$ $$($$$$83137 \nu^{9} + 200027 \nu^{8} - 19567597 \nu^{7} - 57068271 \nu^{6} + 1395513643 \nu^{5} + 4236045917 \nu^{4} - 27571140327 \nu^{3} - 41118854357 \nu^{2} + 103896742760 \nu - 590033063220$$$$)/ 9847891200$$ $$\beta_{5}$$ $$=$$ $$($$$$76219 \nu^{9} + 438897 \nu^{8} - 21012719 \nu^{7} - 98631325 \nu^{6} + 1760844265 \nu^{5} + 6994915287 \nu^{4} - 45551323789 \nu^{3} - 140160241487 \nu^{2} + 252933052440 \nu + 621408182180$$$$)/ 4923945600$$ $$\beta_{6}$$ $$=$$ $$($$$$-95695 \nu^{9} - 431909 \nu^{8} + 23439115 \nu^{7} + 118409649 \nu^{6} - 1626243037 \nu^{5} - 9503152979 \nu^{4} + 23203382625 \nu^{3} + 187758838379 \nu^{2} + 53779123840 \nu - 737664945060$$$$)/ 4923945600$$ $$\beta_{7}$$ $$=$$ $$($$$$-271659 \nu^{9} - 2579273 \nu^{8} + 61777919 \nu^{7} + 579871301 \nu^{6} - 3997103913 \nu^{5} - 38975931263 \nu^{4} + 55560393629 \nu^{3} + 688500755943 \nu^{2} + 31720273720 \nu - 1980608051620$$$$)/ 9847891200$$ $$\beta_{8}$$ $$=$$ $$($$$$91631 \nu^{9} - 419483 \nu^{8} - 20161043 \nu^{7} + 77430063 \nu^{6} + 1507316885 \nu^{5} - 4622163965 \nu^{4} - 44108638233 \nu^{3} + 97163101973 \nu^{2} + 436039731880 \nu - 489826503660$$$$)/ 1969578240$$ $$\beta_{9}$$ $$=$$ $$($$$$25091 \nu^{9} + 50621 \nu^{8} - 5627811 \nu^{7} - 16286853 \nu^{6} + 381410769 \nu^{5} + 1347458171 \nu^{4} - 6897894681 \nu^{3} - 24282359591 \nu^{2} + 16776991880 \nu + 64943739940$$$$)/ 205164400$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} + \beta_{1} + 49$$ $$\nu^{3}$$ $$=$$ $$\beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 80 \beta_{1} + 57$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{9} - 2 \beta_{6} + 3 \beta_{5} + 106 \beta_{4} - 137 \beta_{3} + 11 \beta_{2} + 209 \beta_{1} + 3956$$ $$\nu^{5}$$ $$=$$ $$114 \beta_{9} - 112 \beta_{8} + 116 \beta_{7} + 264 \beta_{6} + 134 \beta_{5} + 336 \beta_{4} - 378 \beta_{3} + 342 \beta_{2} + 7421 \beta_{1} + 10834$$ $$\nu^{6}$$ $$=$$ $$366 \beta_{9} - 76 \beta_{8} - 116 \beta_{6} + 480 \beta_{5} + 10577 \beta_{4} - 14513 \beta_{3} + 2852 \beta_{2} + 30111 \beta_{1} + 369337$$ $$\nu^{7}$$ $$=$$ $$11979 \beta_{9} - 11273 \beta_{8} + 11413 \beta_{7} + 28830 \beta_{6} + 13957 \beta_{5} + 45098 \beta_{4} - 55466 \beta_{3} + 31280 \beta_{2} + 728286 \beta_{1} + 1524905$$ $$\nu^{8}$$ $$=$$ $$50612 \beta_{9} - 17680 \beta_{8} - 1948 \beta_{7} + 11354 \beta_{6} + 64475 \beta_{5} + 1060034 \beta_{4} - 1476853 \beta_{3} + 432195 \beta_{2} + 3821255 \beta_{1} + 36321428$$ $$\nu^{9}$$ $$=$$ $$1265068 \beta_{9} - 1114544 \beta_{8} + 1075408 \beta_{7} + 3012396 \beta_{6} + 1388854 \beta_{5} + 5559868 \beta_{4} - 7296106 \beta_{3} + 3305446 \beta_{2} + 73448747 \beta_{1} + 191858062$$

## Embeddings

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

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

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 10.4745 10.2693 4.72637 4.48290 1.63335 −2.66104 −4.37629 −6.46320 −8.75161 −9.33424
−9.47445 −20.3001 57.7653 −25.0000 192.333 0 −244.112 169.095 236.861
1.2 −9.26930 −1.27429 53.9199 −25.0000 11.8118 0 −203.182 −241.376 231.732
1.3 −3.72637 21.0476 −18.1142 −25.0000 −78.4311 0 186.744 200.001 93.1592
1.4 −3.48290 −0.931256 −19.8694 −25.0000 3.24347 0 180.656 −242.133 87.0725
1.5 −0.633352 −19.5171 −31.5989 −25.0000 12.3612 0 40.2804 137.918 15.8338
1.6 3.66104 −29.0676 −18.5968 −25.0000 −106.418 0 −185.237 601.928 −91.5259
1.7 5.37629 16.5213 −3.09546 −25.0000 88.8236 0 −188.684 29.9545 −134.407
1.8 7.46320 9.93685 23.6994 −25.0000 74.1607 0 −61.9494 −144.259 −186.580
1.9 9.75161 −23.6367 63.0938 −25.0000 −230.496 0 303.214 315.692 −243.790
1.10 10.3342 −10.7786 74.7964 −25.0000 −111.389 0 442.268 −126.821 −258.356
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.6.a.l 10
7.b odd 2 1 245.6.a.m yes 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.6.a.l 10 1.a even 1 1 trivial
245.6.a.m yes 10 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(245))$$:

 $$T_{2}^{10} - \cdots$$ $$T_{3}^{10} + \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-10686592 - 16621440 T + 2127024 T^{2} + 2639600 T^{3} - 230996 T^{4} - 129840 T^{5} + 12314 T^{6} + 2040 T^{7} - 201 T^{8} - 10 T^{9} + T^{10}$$
$3$ $$-12031198596 - 23098421592 T - 11335365576 T^{2} - 285902184 T^{3} + 168931223 T^{4} + 9277842 T^{5} - 571489 T^{6} - 44916 T^{7} + 117 T^{8} + 58 T^{9} + T^{10}$$
$5$ $$( 25 + T )^{10}$$
$7$ $$T^{10}$$
$11$ $$75\!\cdots\!00$$$$+$$$$53\!\cdots\!00$$$$T -$$$$12\!\cdots\!00$$$$T^{2} + 307081920538510000 T^{3} + 6029263259895625 T^{4} - 28430420463450 T^{5} - 42903968825 T^{6} + 385404580 T^{7} - 325049 T^{8} - 794 T^{9} + T^{10}$$
$13$ $$-$$$$10\!\cdots\!48$$$$+$$$$87\!\cdots\!64$$$$T +$$$$13\!\cdots\!00$$$$T^{2} - 2984566971007482312 T^{3} - 40240928938174865 T^{4} + 82398110929386 T^{5} + 448517786815 T^{6} - 589549348 T^{7} - 1549415 T^{8} + 474 T^{9} + T^{10}$$
$17$ $$31\!\cdots\!28$$$$+$$$$90\!\cdots\!92$$$$T +$$$$16\!\cdots\!92$$$$T^{2} -$$$$80\!\cdots\!76$$$$T^{3} - 9683532744285172881 T^{4} + 13887116196422138 T^{5} + 14891600690343 T^{6} - 6772710084 T^{7} - 7290259 T^{8} + 802 T^{9} + T^{10}$$
$19$ $$-$$$$53\!\cdots\!24$$$$+$$$$24\!\cdots\!72$$$$T +$$$$18\!\cdots\!88$$$$T^{2} +$$$$38\!\cdots\!44$$$$T^{3} + 6768632076107608176 T^{4} - 59506244550301392 T^{5} - 59749328775548 T^{6} - 7606711664 T^{7} + 14793984 T^{8} + 7292 T^{9} + T^{10}$$
$23$ $$-$$$$10\!\cdots\!96$$$$-$$$$87\!\cdots\!88$$$$T +$$$$33\!\cdots\!20$$$$T^{2} +$$$$27\!\cdots\!72$$$$T^{3} -$$$$91\!\cdots\!00$$$$T^{4} - 1079181355872719088 T^{5} + 328152338044300 T^{6} + 128454796112 T^{7} - 35742650 T^{8} - 3708 T^{9} + T^{10}$$
$29$ $$20\!\cdots\!00$$$$-$$$$14\!\cdots\!00$$$$T -$$$$14\!\cdots\!00$$$$T^{2} -$$$$34\!\cdots\!00$$$$T^{3} +$$$$78\!\cdots\!25$$$$T^{4} + 38063472197066473650 T^{5} + 998823094777975 T^{6} - 1092360000940 T^{7} - 94954233 T^{8} + 8866 T^{9} + T^{10}$$
$31$ $$52\!\cdots\!48$$$$+$$$$24\!\cdots\!32$$$$T -$$$$22\!\cdots\!60$$$$T^{2} -$$$$98\!\cdots\!84$$$$T^{3} +$$$$10\!\cdots\!40$$$$T^{4} + 67459860181125444528 T^{5} + 610463391386060 T^{6} - 1650292802256 T^{7} - 87166650 T^{8} + 13292 T^{9} + T^{10}$$
$37$ $$-$$$$14\!\cdots\!24$$$$+$$$$18\!\cdots\!96$$$$T +$$$$89\!\cdots\!52$$$$T^{2} -$$$$15\!\cdots\!04$$$$T^{3} +$$$$52\!\cdots\!92$$$$T^{4} -$$$$61\!\cdots\!04$$$$T^{5} - 10102501751260468 T^{6} + 6716080541536 T^{7} - 281018978 T^{8} - 16124 T^{9} + T^{10}$$
$41$ $$20\!\cdots\!16$$$$+$$$$19\!\cdots\!56$$$$T -$$$$28\!\cdots\!72$$$$T^{2} -$$$$21\!\cdots\!08$$$$T^{3} +$$$$90\!\cdots\!56$$$$T^{4} +$$$$88\!\cdots\!44$$$$T^{5} - 76532435095905128 T^{6} - 10351579129472 T^{7} + 27480514 T^{8} + 34836 T^{9} + T^{10}$$
$43$ $$-$$$$31\!\cdots\!00$$$$-$$$$26\!\cdots\!00$$$$T -$$$$70\!\cdots\!00$$$$T^{2} -$$$$34\!\cdots\!00$$$$T^{3} +$$$$12\!\cdots\!00$$$$T^{4} +$$$$16\!\cdots\!00$$$$T^{5} - 37973016134941600 T^{6} - 12939741197440 T^{7} - 257197916 T^{8} + 28604 T^{9} + T^{10}$$
$47$ $$99\!\cdots\!00$$$$+$$$$39\!\cdots\!00$$$$T -$$$$32\!\cdots\!00$$$$T^{2} -$$$$14\!\cdots\!00$$$$T^{3} -$$$$24\!\cdots\!25$$$$T^{4} +$$$$12\!\cdots\!50$$$$T^{5} + 497622361481415175 T^{6} - 27469826505020 T^{7} - 1353838719 T^{8} + 18106 T^{9} + T^{10}$$
$53$ $$-$$$$52\!\cdots\!48$$$$-$$$$14\!\cdots\!80$$$$T +$$$$21\!\cdots\!04$$$$T^{2} +$$$$36\!\cdots\!60$$$$T^{3} -$$$$23\!\cdots\!24$$$$T^{4} -$$$$26\!\cdots\!80$$$$T^{5} + 971455064883012524 T^{6} + 57934173267440 T^{7} - 1725587374 T^{8} - 36440 T^{9} + T^{10}$$
$59$ $$-$$$$26\!\cdots\!64$$$$+$$$$71\!\cdots\!64$$$$T +$$$$60\!\cdots\!64$$$$T^{2} -$$$$11\!\cdots\!68$$$$T^{3} -$$$$65\!\cdots\!68$$$$T^{4} +$$$$39\!\cdots\!96$$$$T^{5} + 2070900034332362856 T^{6} - 47552671613152 T^{7} - 2501472342 T^{8} + 18644 T^{9} + T^{10}$$
$61$ $$-$$$$11\!\cdots\!68$$$$+$$$$17\!\cdots\!00$$$$T +$$$$42\!\cdots\!80$$$$T^{2} -$$$$10\!\cdots\!40$$$$T^{3} -$$$$84\!\cdots\!20$$$$T^{4} +$$$$22\!\cdots\!40$$$$T^{5} + 1876283134488964240 T^{6} - 203261836534880 T^{7} - 2332402240 T^{8} + 68120 T^{9} + T^{10}$$
$67$ $$-$$$$35\!\cdots\!32$$$$-$$$$29\!\cdots\!68$$$$T -$$$$48\!\cdots\!52$$$$T^{2} +$$$$21\!\cdots\!88$$$$T^{3} +$$$$31\!\cdots\!32$$$$T^{4} -$$$$46\!\cdots\!48$$$$T^{5} - 1181396724413154932 T^{6} + 375249336989568 T^{7} - 2307190358 T^{8} - 92328 T^{9} + T^{10}$$
$71$ $$13\!\cdots\!28$$$$-$$$$54\!\cdots\!60$$$$T +$$$$41\!\cdots\!04$$$$T^{2} +$$$$16\!\cdots\!84$$$$T^{3} -$$$$19\!\cdots\!72$$$$T^{4} -$$$$71\!\cdots\!76$$$$T^{5} + 21664033512625829360 T^{6} + 24425163329568 T^{7} - 8170051692 T^{8} - 5044 T^{9} + T^{10}$$
$73$ $$-$$$$37\!\cdots\!32$$$$-$$$$14\!\cdots\!80$$$$T -$$$$70\!\cdots\!20$$$$T^{2} +$$$$21\!\cdots\!40$$$$T^{3} +$$$$13\!\cdots\!20$$$$T^{4} -$$$$19\!\cdots\!40$$$$T^{5} - 24814414659976306460 T^{6} - 326747854419200 T^{7} + 6379859140 T^{8} + 170160 T^{9} + T^{10}$$
$79$ $$56\!\cdots\!12$$$$+$$$$20\!\cdots\!56$$$$T +$$$$66\!\cdots\!04$$$$T^{2} -$$$$98\!\cdots\!60$$$$T^{3} -$$$$43\!\cdots\!51$$$$T^{4} +$$$$14\!\cdots\!42$$$$T^{5} +$$$$16\!\cdots\!87$$$$T^{6} + 181431547342388 T^{7} - 22211916865 T^{8} - 26442 T^{9} + T^{10}$$
$83$ $$-$$$$42\!\cdots\!00$$$$-$$$$10\!\cdots\!60$$$$T +$$$$16\!\cdots\!56$$$$T^{2} +$$$$78\!\cdots\!40$$$$T^{3} -$$$$60\!\cdots\!92$$$$T^{4} -$$$$87\!\cdots\!80$$$$T^{5} -$$$$12\!\cdots\!96$$$$T^{6} + 1193163138278080 T^{7} + 41868858212 T^{8} + 353360 T^{9} + T^{10}$$
$89$ $$-$$$$31\!\cdots\!96$$$$+$$$$73\!\cdots\!84$$$$T +$$$$98\!\cdots\!24$$$$T^{2} -$$$$29\!\cdots\!52$$$$T^{3} -$$$$29\!\cdots\!32$$$$T^{4} +$$$$16\!\cdots\!76$$$$T^{5} +$$$$16\!\cdots\!56$$$$T^{6} - 2651277769037568 T^{7} - 27113027428 T^{8} + 90704 T^{9} + T^{10}$$
$97$ $$30\!\cdots\!88$$$$+$$$$27\!\cdots\!72$$$$T -$$$$27\!\cdots\!88$$$$T^{2} -$$$$50\!\cdots\!56$$$$T^{3} +$$$$27\!\cdots\!79$$$$T^{4} +$$$$15\!\cdots\!18$$$$T^{5} +$$$$22\!\cdots\!43$$$$T^{6} - 11490514518736604 T^{7} - 39117236019 T^{8} + 236382 T^{9} + T^{10}$$