# Properties

 Label 245.6.a.k Level $245$ Weight $6$ Character orbit 245.a Self dual yes Analytic conductor $39.294$ Analytic rank $0$ Dimension $8$ 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: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 3 x^{7} - 225 x^{6} + 537 x^{5} + 15166 x^{4} - 25016 x^{3} - 317696 x^{2} + 463952 x + 1012704$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{5}\cdot 7^{3}$$ Twist minimal: no (minimal twist has level 35) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{3} q^{3} + ( 25 + \beta_{1} + \beta_{2} ) q^{4} -25 q^{5} + ( 6 + 2 \beta_{1} + \beta_{3} + \beta_{7} ) q^{6} + ( 20 + 29 \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} ) q^{8} + ( 151 + 5 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{3} q^{3} + ( 25 + \beta_{1} + \beta_{2} ) q^{4} -25 q^{5} + ( 6 + 2 \beta_{1} + \beta_{3} + \beta_{7} ) q^{6} + ( 20 + 29 \beta_{1} + \beta_{2} + \beta_{6} + \beta_{7} ) q^{8} + ( 151 + 5 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{9} -25 \beta_{1} q^{10} + ( 13 + 5 \beta_{1} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{11} + ( 101 + 6 \beta_{1} + 6 \beta_{2} + 28 \beta_{3} + 2 \beta_{4} + \beta_{5} + 3 \beta_{7} ) q^{12} + ( -255 + 15 \beta_{1} - 8 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{7} ) q^{13} -25 \beta_{3} q^{15} + ( 783 + 45 \beta_{1} + 29 \beta_{2} + 28 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{16} + ( -268 + 94 \beta_{1} + 8 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{17} + ( 177 + 177 \beta_{1} + 27 \beta_{2} + 36 \beta_{3} + 4 \beta_{4} - 3 \beta_{5} - 5 \beta_{6} - 4 \beta_{7} ) q^{18} + ( 241 - 39 \beta_{1} + \beta_{2} + 10 \beta_{3} - 7 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{19} + ( -625 - 25 \beta_{1} - 25 \beta_{2} ) q^{20} + ( 289 + 3 \beta_{1} + 13 \beta_{2} - 54 \beta_{3} + 4 \beta_{4} - \beta_{5} - 9 \beta_{6} - 4 \beta_{7} ) q^{22} + ( 591 + 9 \beta_{1} + 4 \beta_{2} + 32 \beta_{3} - 5 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{23} + ( 98 + 284 \beta_{1} + 32 \beta_{2} + 176 \beta_{3} + 10 \beta_{4} + 9 \beta_{5} + 5 \beta_{6} + 16 \beta_{7} ) q^{24} + 625 q^{25} + ( 1099 - 527 \beta_{1} + 13 \beta_{2} + 130 \beta_{3} + 4 \beta_{4} + \beta_{5} - 9 \beta_{6} - 6 \beta_{7} ) q^{26} + ( -1102 + 248 \beta_{1} - 19 \beta_{2} + 136 \beta_{3} + 14 \beta_{4} + 3 \beta_{5} - 6 \beta_{6} - 18 \beta_{7} ) q^{27} + ( 1294 + 145 \beta_{1} - 27 \beta_{2} - 134 \beta_{3} - 13 \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - 12 \beta_{7} ) q^{29} + ( -150 - 50 \beta_{1} - 25 \beta_{3} - 25 \beta_{7} ) q^{30} + ( -554 - 222 \beta_{1} - 15 \beta_{2} - 16 \beta_{3} - 12 \beta_{4} + 4 \beta_{5} - 3 \beta_{6} - 13 \beta_{7} ) q^{31} + ( 1080 + 1009 \beta_{1} - 39 \beta_{2} + 176 \beta_{3} - 12 \beta_{4} + 10 \beta_{5} + 27 \beta_{6} + 41 \beta_{7} ) q^{32} + ( 462 - 384 \beta_{1} - 22 \beta_{2} + 146 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} + 16 \beta_{6} - 8 \beta_{7} ) q^{33} + ( 5066 + 174 \beta_{1} + 52 \beta_{2} - 194 \beta_{3} - 20 \beta_{4} - 14 \beta_{5} + 20 \beta_{6} - 4 \beta_{7} ) q^{34} + ( 4643 + 1291 \beta_{1} + 47 \beta_{2} - 68 \beta_{3} + 2 \beta_{4} + 3 \beta_{5} + 33 \beta_{6} + 34 \beta_{7} ) q^{36} + ( 763 + 483 \beta_{1} + 14 \beta_{2} + 132 \beta_{3} + 7 \beta_{4} - 7 \beta_{5} - 7 \beta_{6} + 25 \beta_{7} ) q^{37} + ( -2387 + 297 \beta_{1} - 43 \beta_{2} - 168 \beta_{3} - 12 \beta_{4} - 27 \beta_{5} - 7 \beta_{6} - 22 \beta_{7} ) q^{38} + ( -608 + 852 \beta_{1} - 43 \beta_{2} - 110 \beta_{3} - 6 \beta_{4} - 20 \beta_{5} - 19 \beta_{6} - 21 \beta_{7} ) q^{39} + ( -500 - 725 \beta_{1} - 25 \beta_{2} - 25 \beta_{6} - 25 \beta_{7} ) q^{40} + ( 2129 + 422 \beta_{1} + 35 \beta_{2} - 39 \beta_{3} - 12 \beta_{4} + 15 \beta_{5} - 8 \beta_{6} - 56 \beta_{7} ) q^{41} + ( 7676 + 480 \beta_{1} - 34 \beta_{2} + 165 \beta_{3} + 20 \beta_{4} - 14 \beta_{5} - 4 \beta_{6} - 12 \beta_{7} ) q^{43} + ( -787 + 541 \beta_{1} - 219 \beta_{2} - 80 \beta_{3} + 2 \beta_{4} - 9 \beta_{5} + 33 \beta_{6} + 2 \beta_{7} ) q^{44} + ( -3775 - 125 \beta_{1} - 25 \beta_{2} + 75 \beta_{3} + 25 \beta_{4} - 25 \beta_{6} - 25 \beta_{7} ) q^{45} + ( 591 + 713 \beta_{1} + 19 \beta_{2} - 123 \beta_{3} + 24 \beta_{4} + 3 \beta_{5} - 39 \beta_{6} + 21 \beta_{7} ) q^{46} + ( -1551 + 323 \beta_{1} - 77 \beta_{2} - 229 \beta_{3} + 19 \beta_{4} - 6 \beta_{5} + 31 \beta_{6} + 41 \beta_{7} ) q^{47} + ( 12825 + 1458 \beta_{1} + 342 \beta_{2} + 184 \beta_{3} + 14 \beta_{4} + 15 \beta_{5} - 10 \beta_{6} + 187 \beta_{7} ) q^{48} + 625 \beta_{1} q^{50} + ( 4118 - 1282 \beta_{1} - 277 \beta_{2} - 120 \beta_{3} - 20 \beta_{4} + 12 \beta_{5} - 25 \beta_{6} + 145 \beta_{7} ) q^{51} + ( -21241 + 847 \beta_{1} - 513 \beta_{2} + 4 \beta_{3} + 6 \beta_{4} + 29 \beta_{5} + 59 \beta_{6} + 90 \beta_{7} ) q^{52} + ( -5031 + 113 \beta_{1} - 126 \beta_{2} - 43 \beta_{3} + 37 \beta_{4} + 54 \beta_{5} + 28 \beta_{6} - 118 \beta_{7} ) q^{53} + ( 16418 - 1188 \beta_{1} + 130 \beta_{2} - 819 \beta_{3} - 36 \beta_{4} + 26 \beta_{5} + 16 \beta_{6} + 135 \beta_{7} ) q^{54} + ( -325 - 125 \beta_{1} + 25 \beta_{4} - 25 \beta_{5} - 25 \beta_{6} + 25 \beta_{7} ) q^{55} + ( 5200 - 1572 \beta_{1} - 528 \beta_{2} + 975 \beta_{3} - 20 \beta_{4} + 5 \beta_{5} - 33 \beta_{6} - 63 \beta_{7} ) q^{57} + ( 8513 + 222 \beta_{1} - 107 \beta_{2} - 562 \beta_{3} - 16 \beta_{4} - 7 \beta_{5} - 39 \beta_{6} - 216 \beta_{7} ) q^{58} + ( -6658 + 1150 \beta_{1} + 583 \beta_{2} - 494 \beta_{3} - 20 \beta_{4} + 4 \beta_{5} + 35 \beta_{6} - 19 \beta_{7} ) q^{59} + ( -2525 - 150 \beta_{1} - 150 \beta_{2} - 700 \beta_{3} - 50 \beta_{4} - 25 \beta_{5} - 75 \beta_{7} ) q^{60} + ( 1016 - 2015 \beta_{1} - 294 \beta_{2} + 97 \beta_{3} - 19 \beta_{4} - 6 \beta_{5} - 56 \beta_{6} - 42 \beta_{7} ) q^{61} + ( -12070 - 1344 \beta_{1} - 502 \beta_{2} - 458 \beta_{3} - 16 \beta_{4} - 24 \beta_{6} - 82 \beta_{7} ) q^{62} + ( 33749 - 771 \beta_{1} + 821 \beta_{2} + 1244 \beta_{3} + 48 \beta_{4} + 52 \beta_{5} - 88 \beta_{6} + 112 \beta_{7} ) q^{64} + ( 6375 - 375 \beta_{1} + 200 \beta_{2} - 25 \beta_{3} + 25 \beta_{4} - 50 \beta_{7} ) q^{65} + ( -20152 - 442 \beta_{1} + 60 \beta_{2} - 816 \beta_{3} + 24 \beta_{4} - 44 \beta_{5} - 104 \beta_{6} + 108 \beta_{7} ) q^{66} + ( 7110 - 2024 \beta_{1} + 35 \beta_{2} - 130 \beta_{3} - 14 \beta_{4} - 75 \beta_{5} - 18 \beta_{6} + 10 \beta_{7} ) q^{67} + ( 14536 + 3810 \beta_{1} + 346 \beta_{2} - 1340 \beta_{3} + 40 \beta_{4} - 44 \beta_{5} + 6 \beta_{6} - 162 \beta_{7} ) q^{68} + ( 14934 - 997 \beta_{1} + 24 \beta_{2} + 1067 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} + 102 \beta_{6} - 20 \beta_{7} ) q^{69} + ( 23258 - 190 \beta_{1} - 219 \beta_{2} - 778 \beta_{3} - 28 \beta_{4} - 52 \beta_{5} + 33 \beta_{6} + 39 \beta_{7} ) q^{71} + ( 64765 + 1501 \beta_{1} + 1489 \beta_{2} - 184 \beta_{3} + 18 \beta_{4} + 45 \beta_{5} + 35 \beta_{6} + 154 \beta_{7} ) q^{72} + ( -6126 - 1636 \beta_{1} + 854 \beta_{2} - 279 \beta_{3} - 8 \beta_{4} + 59 \beta_{5} + 23 \beta_{6} - 143 \beta_{7} ) q^{73} + ( 27435 + 2033 \beta_{1} + 499 \beta_{2} + 1572 \beta_{3} + 8 \beta_{4} + 25 \beta_{5} + 77 \beta_{6} + 210 \beta_{7} ) q^{74} + 625 \beta_{3} q^{75} + ( 9563 - 2181 \beta_{1} + 23 \beta_{2} - 2112 \beta_{3} + 58 \beta_{4} - 25 \beta_{5} - 3 \beta_{6} - 274 \beta_{7} ) q^{76} + ( 49660 - 1046 \beta_{1} + 296 \beta_{2} - 1194 \beta_{3} - 160 \beta_{4} - 50 \beta_{5} + 106 \beta_{6} - 234 \beta_{7} ) q^{78} + ( 7496 - 4032 \beta_{1} - 427 \beta_{2} + 1500 \beta_{3} + 102 \beta_{4} + 24 \beta_{5} + 33 \beta_{6} - 33 \beta_{7} ) q^{79} + ( -19575 - 1125 \beta_{1} - 725 \beta_{2} - 700 \beta_{3} - 100 \beta_{4} + 50 \beta_{5} + 100 \beta_{6} - 50 \beta_{7} ) q^{80} + ( 18641 - 6220 \beta_{1} + 139 \beta_{2} - 2977 \beta_{3} - 38 \beta_{4} + 35 \beta_{5} + 68 \beta_{6} + 360 \beta_{7} ) q^{81} + ( 23746 + 3709 \beta_{1} - 236 \beta_{2} - 2588 \beta_{3} - 68 \beta_{4} + 16 \beta_{5} + 18 \beta_{6} - 114 \beta_{7} ) q^{82} + ( -8550 + 2308 \beta_{1} - 547 \beta_{2} + 606 \beta_{3} - 78 \beta_{4} + 45 \beta_{5} - 48 \beta_{6} - 72 \beta_{7} ) q^{83} + ( 6700 - 2350 \beta_{1} - 200 \beta_{2} - 125 \beta_{3} + 50 \beta_{4} + 75 \beta_{5} + 25 \beta_{6} + 75 \beta_{7} ) q^{85} + ( 30026 + 7586 \beta_{1} + 604 \beta_{2} - 989 \beta_{3} - 88 \beta_{4} - 36 \beta_{5} + 48 \beta_{6} + 143 \beta_{7} ) q^{86} + ( -50164 - 4154 \beta_{1} - 1076 \beta_{2} + 2292 \beta_{3} + 42 \beta_{4} - 109 \beta_{5} - 167 \beta_{6} - 49 \beta_{7} ) q^{87} + ( 28581 - 8137 \beta_{1} + 1131 \beta_{2} + 496 \beta_{3} - 94 \beta_{4} - 99 \beta_{5} - 67 \beta_{6} - 212 \beta_{7} ) q^{88} + ( 11076 + 1897 \beta_{1} + 720 \beta_{2} + 1236 \beta_{3} + 113 \beta_{4} - 29 \beta_{5} - 103 \beta_{6} - 171 \beta_{7} ) q^{89} + ( -4425 - 4425 \beta_{1} - 675 \beta_{2} - 900 \beta_{3} - 100 \beta_{4} + 75 \beta_{5} + 125 \beta_{6} + 100 \beta_{7} ) q^{90} + ( 20972 + 1447 \beta_{1} - 201 \beta_{2} + 1112 \beta_{3} + 136 \beta_{4} - 18 \beta_{5} + 101 \beta_{6} + 183 \beta_{7} ) q^{92} + ( -4670 - 4320 \beta_{1} - 812 \beta_{2} - 487 \beta_{3} - 56 \beta_{4} - 105 \beta_{5} - 39 \beta_{6} - 257 \beta_{7} ) q^{93} + ( 18939 - 4529 \beta_{1} + 1581 \beta_{2} + 870 \beta_{3} + 120 \beta_{4} - 57 \beta_{5} - 195 \beta_{6} - 210 \beta_{7} ) q^{94} + ( -6025 + 975 \beta_{1} - 25 \beta_{2} - 250 \beta_{3} + 175 \beta_{4} + 75 \beta_{5} - 50 \beta_{6} + 50 \beta_{7} ) q^{95} + ( 65572 + 16604 \beta_{1} + 952 \beta_{2} + 6116 \beta_{3} + 94 \beta_{4} + 3 \beta_{5} + 201 \beta_{6} + 470 \beta_{7} ) q^{96} + ( 22886 + 1692 \beta_{1} - 1002 \beta_{2} + 3476 \beta_{3} - 40 \beta_{4} - 64 \beta_{5} + 118 \beta_{6} + 250 \beta_{7} ) q^{97} + ( 52265 - 7403 \beta_{1} + 102 \beta_{2} + 460 \beta_{3} + 67 \beta_{4} - 53 \beta_{5} + 173 \beta_{6} - 93 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 3 q^{2} + 2 q^{3} + 203 q^{4} - 200 q^{5} + 56 q^{6} + 249 q^{8} + 1218 q^{9} + O(q^{10})$$ $$8 q + 3 q^{2} + 2 q^{3} + 203 q^{4} - 200 q^{5} + 56 q^{6} + 249 q^{8} + 1218 q^{9} - 75 q^{10} + 120 q^{11} + 884 q^{12} - 1994 q^{13} - 50 q^{15} + 6451 q^{16} - 1856 q^{17} + 2013 q^{18} + 1828 q^{19} - 5075 q^{20} + 2199 q^{22} + 4822 q^{23} + 2008 q^{24} + 5000 q^{25} + 7457 q^{26} - 7798 q^{27} + 10502 q^{29} - 1400 q^{30} - 5148 q^{31} + 12061 q^{32} + 2872 q^{33} + 40682 q^{34} + 40949 q^{36} + 7810 q^{37} - 18567 q^{38} - 2572 q^{39} - 6225 q^{40} + 18192 q^{41} + 63190 q^{43} - 4765 q^{44} - 30450 q^{45} + 6567 q^{46} - 11816 q^{47} + 107336 q^{48} + 1875 q^{50} + 28788 q^{51} - 167255 q^{52} - 39902 q^{53} + 126138 q^{54} - 3000 q^{55} + 38748 q^{57} + 67552 q^{58} - 50752 q^{59} - 22100 q^{60} + 2146 q^{61} - 101572 q^{62} + 270039 q^{64} + 49850 q^{65} - 164358 q^{66} + 50498 q^{67} + 125090 q^{68} + 118822 q^{69} + 183976 q^{71} + 522343 q^{72} - 54436 q^{73} + 228885 q^{74} + 1250 q^{75} + 65789 q^{76} + 391806 q^{78} + 51040 q^{79} - 161275 q^{80} + 124612 q^{81} + 195887 q^{82} - 60438 q^{83} + 46400 q^{85} + 260996 q^{86} - 409482 q^{87} + 205001 q^{88} + 96678 q^{89} - 50325 q^{90} + 174679 q^{92} - 51428 q^{93} + 139395 q^{94} - 45700 q^{95} + 587116 q^{96} + 195312 q^{97} + 397244 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} - 225 x^{6} + 537 x^{5} + 15166 x^{4} - 25016 x^{3} - 317696 x^{2} + 463952 x + 1012704$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 57$$ $$\beta_{3}$$ $$=$$ $$($$$$5 \nu^{7} - 52 \nu^{6} - 762 \nu^{5} + 8738 \nu^{4} + 16161 \nu^{3} - 304774 \nu^{2} + 386020 \nu + 915528$$$$)/10464$$ $$\beta_{4}$$ $$=$$ $$($$$$67 \nu^{7} - 457 \nu^{6} - 12151 \nu^{5} + 75255 \nu^{4} + 524526 \nu^{3} - 2509532 \nu^{2} - 3802392 \nu + 9588768$$$$)/20928$$ $$\beta_{5}$$ $$=$$ $$($$$$11 \nu^{7} + 60 \nu^{6} - 2374 \nu^{5} - 12866 \nu^{4} + 142287 \nu^{3} + 736382 \nu^{2} - 1822564 \nu - 6403080$$$$)/10464$$ $$\beta_{6}$$ $$=$$ $$($$$$42 \nu^{7} - 415 \nu^{6} - 6815 \nu^{5} + 68407 \nu^{4} + 206319 \nu^{3} - 2289738 \nu^{2} + 848492 \nu + 6429000$$$$)/10464$$ $$\beta_{7}$$ $$=$$ $$($$$$-42 \nu^{7} + 415 \nu^{6} + 6815 \nu^{5} - 68407 \nu^{4} - 195855 \nu^{3} + 2279274 \nu^{2} - 1811180 \nu - 6041832$$$$)/10464$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 57$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{6} + \beta_{2} + 93 \beta_{1} + 20$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{7} - 4 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} + 28 \beta_{3} + 125 \beta_{2} + 141 \beta_{1} + 5231$$ $$\nu^{5}$$ $$=$$ $$169 \beta_{7} + 155 \beta_{6} + 10 \beta_{5} - 12 \beta_{4} + 176 \beta_{3} + 89 \beta_{2} + 9841 \beta_{1} + 3640$$ $$\nu^{6}$$ $$=$$ $$432 \beta_{7} - 728 \beta_{6} - 268 \beta_{5} + 688 \beta_{4} + 5724 \beta_{3} + 14677 \beta_{2} + 15645 \beta_{1} + 553269$$ $$\nu^{7}$$ $$=$$ $$23521 \beta_{7} + 19809 \beta_{6} + 2232 \beta_{5} - 1664 \beta_{4} + 39512 \beta_{3} + 5477 \beta_{2} + 1099221 \beta_{1} + 393712$$

## 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
 −10.9703 −6.77005 −6.63433 −1.26911 3.11373 4.91557 9.65190 10.9626
−10.9703 4.62268 88.3466 −25.0000 −50.7120 0 −618.137 −221.631 274.256
1.2 −6.77005 23.4913 13.8335 −25.0000 −159.037 0 122.988 308.841 169.251
1.3 −6.63433 −30.5785 12.0144 −25.0000 202.868 0 132.591 692.045 165.858
1.4 −1.26911 −7.00999 −30.3894 −25.0000 8.89643 0 79.1787 −193.860 31.7277
1.5 3.11373 20.6063 −22.3047 −25.0000 64.1623 0 −169.090 181.619 −77.8431
1.6 4.91557 −9.91717 −7.83720 −25.0000 −48.7485 0 −195.822 −144.650 −122.889
1.7 9.65190 −22.8595 61.1592 −25.0000 −220.637 0 281.441 279.555 −241.297
1.8 10.9626 23.6449 88.1776 −25.0000 259.208 0 615.850 316.080 −274.064
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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.k 8
7.b odd 2 1 245.6.a.j 8
7.d odd 6 2 35.6.e.b 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.6.e.b 16 7.d odd 6 2
245.6.a.j 8 7.b odd 2 1
245.6.a.k 8 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{2}^{8} - \cdots$$ $$T_{3}^{8} - \cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1012704 + 463952 T - 317696 T^{2} - 25016 T^{3} + 15166 T^{4} + 537 T^{5} - 225 T^{6} - 3 T^{7} + T^{8}$$
$3$ $$2571136884 - 76563792 T - 109744365 T^{2} - 2145922 T^{3} + 752739 T^{4} + 5436 T^{5} - 1579 T^{6} - 2 T^{7} + T^{8}$$
$5$ $$( 25 + T )^{8}$$
$7$ $$T^{8}$$
$11$ $$12919353559622856000 - 906641634227520 T - 2363688034857584 T^{2} - 968153862560 T^{3} + 96474126076 T^{4} + 42711724 T^{5} - 637969 T^{6} - 120 T^{7} + T^{8}$$
$13$ $$-52824372858032967744 + 4873941523016782784 T + 45920118717960240 T^{2} + 29552766164736 T^{3} - 511451870892 T^{4} - 904624564 T^{5} + 582245 T^{6} + 1994 T^{7} + T^{8}$$
$17$ $$-$$$$17\!\cdots\!56$$$$-$$$$67\!\cdots\!76$$$$T + 8851759980776497280 T^{2} + 18687380036068096 T^{3} + 2101748921984 T^{4} - 13879027872 T^{5} - 5936296 T^{6} + 1856 T^{7} + T^{8}$$
$19$ $$-$$$$14\!\cdots\!32$$$$+$$$$51\!\cdots\!36$$$$T - 22561424176760628416 T^{2} - 55770530826038528 T^{3} + 33354363075864 T^{4} + 18068720336 T^{5} - 10913701 T^{6} - 1828 T^{7} + T^{8}$$
$23$ $$-$$$$73\!\cdots\!19$$$$+$$$$16\!\cdots\!22$$$$T + 12951079092455377092 T^{2} - 27478588011647594 T^{3} - 38705380702662 T^{4} + 60236567362 T^{5} - 9202212 T^{6} - 4822 T^{7} + T^{8}$$
$29$ $$54\!\cdots\!00$$$$+$$$$46\!\cdots\!00$$$$T -$$$$47\!\cdots\!75$$$$T^{2} - 8913703839673994430 T^{3} + 407697784901499 T^{4} + 547950020424 T^{5} - 37479333 T^{6} - 10502 T^{7} + T^{8}$$
$31$ $$-$$$$13\!\cdots\!16$$$$+$$$$31\!\cdots\!20$$$$T -$$$$10\!\cdots\!88$$$$T^{2} - 1466364680147059776 T^{3} + 1026345174305520 T^{4} - 91725816864 T^{5} - 51249220 T^{6} + 5148 T^{7} + T^{8}$$
$37$ $$-$$$$35\!\cdots\!00$$$$-$$$$60\!\cdots\!20$$$$T -$$$$25\!\cdots\!16$$$$T^{2} + 4297580465634570240 T^{3} + 4256939859120896 T^{4} + 257638396960 T^{5} - 140391327 T^{6} - 7810 T^{7} + T^{8}$$
$41$ $$-$$$$44\!\cdots\!75$$$$+$$$$26\!\cdots\!20$$$$T +$$$$27\!\cdots\!52$$$$T^{2} -$$$$27\!\cdots\!76$$$$T^{3} + 15254305419725974 T^{4} + 4497119060848 T^{5} - 250788004 T^{6} - 18192 T^{7} + T^{8}$$
$43$ $$-$$$$19\!\cdots\!12$$$$+$$$$20\!\cdots\!80$$$$T -$$$$74\!\cdots\!77$$$$T^{2} +$$$$10\!\cdots\!90$$$$T^{3} - 1905241260189869 T^{4} - 13829772307980 T^{5} + 1472693757 T^{6} - 63190 T^{7} + T^{8}$$
$47$ $$45\!\cdots\!00$$$$+$$$$19\!\cdots\!00$$$$T -$$$$29\!\cdots\!84$$$$T^{2} +$$$$40\!\cdots\!04$$$$T^{3} + 91580598554929412 T^{4} - 4840624576268 T^{5} - 624527769 T^{6} + 11816 T^{7} + T^{8}$$
$53$ $$18\!\cdots\!48$$$$-$$$$61\!\cdots\!68$$$$T +$$$$16\!\cdots\!24$$$$T^{2} +$$$$61\!\cdots\!52$$$$T^{3} + 951654984164656632 T^{4} - 94869230888056 T^{5} - 2112549535 T^{6} + 39902 T^{7} + T^{8}$$
$59$ $$-$$$$35\!\cdots\!00$$$$-$$$$35\!\cdots\!00$$$$T -$$$$43\!\cdots\!84$$$$T^{2} +$$$$47\!\cdots\!88$$$$T^{3} + 694563837609915328 T^{4} - 112128857363648 T^{5} - 2066763720 T^{6} + 50752 T^{7} + T^{8}$$
$61$ $$-$$$$26\!\cdots\!32$$$$+$$$$11\!\cdots\!32$$$$T +$$$$40\!\cdots\!69$$$$T^{2} -$$$$18\!\cdots\!78$$$$T^{3} + 1100663151905115023 T^{4} + 27052060677700 T^{5} - 2376519969 T^{6} - 2146 T^{7} + T^{8}$$
$67$ $$29\!\cdots\!00$$$$-$$$$73\!\cdots\!00$$$$T +$$$$12\!\cdots\!71$$$$T^{2} -$$$$55\!\cdots\!02$$$$T^{3} - 1169936293570099513 T^{4} + 135775071302764 T^{5} - 1801494719 T^{6} - 50498 T^{7} + T^{8}$$
$71$ $$-$$$$11\!\cdots\!24$$$$-$$$$39\!\cdots\!24$$$$T -$$$$38\!\cdots\!52$$$$T^{2} +$$$$65\!\cdots\!96$$$$T^{3} + 260110765293339584 T^{4} - 293703631622656 T^{5} + 11966413688 T^{6} - 183976 T^{7} + T^{8}$$
$73$ $$-$$$$21\!\cdots\!24$$$$-$$$$72\!\cdots\!64$$$$T -$$$$45\!\cdots\!28$$$$T^{2} +$$$$24\!\cdots\!76$$$$T^{3} + 14831448861501128368 T^{4} - 304976015167488 T^{5} - 7643372172 T^{6} + 54436 T^{7} + T^{8}$$
$79$ $$-$$$$56\!\cdots\!64$$$$-$$$$28\!\cdots\!88$$$$T -$$$$34\!\cdots\!40$$$$T^{2} +$$$$12\!\cdots\!08$$$$T^{3} + 20939360406239602272 T^{4} + 248013417737248 T^{5} - 9501653820 T^{6} - 51040 T^{7} + T^{8}$$
$83$ $$20\!\cdots\!56$$$$+$$$$43\!\cdots\!12$$$$T +$$$$32\!\cdots\!11$$$$T^{2} +$$$$96\!\cdots\!14$$$$T^{3} - 43959940092758889 T^{4} - 512367855896564 T^{5} - 6008215879 T^{6} + 60438 T^{7} + T^{8}$$
$89$ $$-$$$$10\!\cdots\!12$$$$-$$$$43\!\cdots\!60$$$$T +$$$$16\!\cdots\!77$$$$T^{2} +$$$$85\!\cdots\!78$$$$T^{3} - 39133219469201673749 T^{4} + 1347206053507632 T^{5} - 9733082989 T^{6} - 96678 T^{7} + T^{8}$$
$97$ $$10\!\cdots\!16$$$$+$$$$56\!\cdots\!12$$$$T -$$$$33\!\cdots\!80$$$$T^{2} -$$$$27\!\cdots\!08$$$$T^{3} + 94519544348657644768 T^{4} + 4147590852738368 T^{5} - 17233894960 T^{6} - 195312 T^{7} + T^{8}$$